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\section*{Acknowledgements}
We wish to thank Dr. Andrea Geralico and Dr. Damiano Tommasini for the support with the numerical analysis and Prof. Salvatore Capozziello for useful discussions. This work was partially supported by DGAPA-UNAM, Grant No. 113514, and CONACyT, Grant No. 166391.
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BkiUdoU5qsBC8zW3y8KN
| 5 | 1 |
\section{Introduction}
\begin{figure}
\centering
\includegraphics[trim={0.2cm 0cm 0.4cm 0cm},clip,width=0.3\textwidth]{freq_bias.pdf}
\vspace{-2mm}
\caption{Typical popularity distribution of items depicting the long tail.\label{fig:freq_bias}}
\vspace{-4mm}
\end{figure}
The goal of session-based recommendation (SR) models is to recommend top-K items to a user based on the sequence of items clicked so far.
Recently, several effective models for SR based on deep neural networks architectures have been proposed \cite{wu2018session,liu2018stamp,li2017neural,wang2019collaborative}.
These approaches consider SR as a multi-class classification problem where input is a sequence of items clicked in the past in a session and target classes correspond to available items in the catalog.
Many of these approaches use sequential models like recurrent neural networks considering a session as a sequence of item click events \cite{hidasi2015session,jannach2017recurrent,wang2019collaborative}.
On the other hand, approaches like STAMP \cite{liu2018stamp} consider a session to be a set of items, and use attention models while learning to weigh (attend to) items as per their relevance to predict the next item.
Approaches like NARM \cite{li2017neural} and CSRM \cite{wang2019collaborative} use a combination of sequential and attention models.
An important building block in most of these deep learning approaches is their ability to learn representations or embeddings for items and sessions.
Recently, an alternative approach, namely SR-GNN \cite{wu2018session}, has been proposed to model the sessions as graph-structured data using GNNs \cite{li2015gated} rather than as sequences or sets, noting that users tend to make complex \textit{to-and-fro} transitions across items within a session:
for example, consider a session $s = i_1,i_2,i_1,i_3,i_4$ of item clicks by a user. Here, the user clicks on item $i_1$, then clicks on item $i_2$ and then re-clicks on item $i_1$.
This sequence of clicks induces a graph where nodes and edges correspond to items and transitions across items, respectively.
For session $s$ in the above example, the fact that $i_2$ and $i_3$ are neighbors of $i_1$ in the induced session-graph, the representation of $i_1$ can be updated using representations of its neighbors, i.e. $i_2$ and $i_3$, and thus obtain a more context-aware and informative representations.
It is worth noting that this way of capturing neighborhood information has also been found to be effective in neighborhood-based SR methods such as SKNN \cite{jannach2017recurrent} and STAN \cite{garg2019sequence}.
It is well-known that more popular items are presented and interacted-with more often on online platforms. This results in a skewed distribution of items clicked by users \cite{steck2011item,abdollahpouri2017controlling,yang2018unbiased}, as illustrated in Fig. \ref{fig:freq_bias}.
The models trained using the resulting data tend to have \textit{popularity bias}, i.e. they tend to recommend more popular items over rarely clicked items.
We note that SR-GNN (referred to as GNN hereafter) also suffers from popularity bias.
This problem is even more severe in a practical online setting where new items are frequently added to the catalog, and are inherently less popular during initial days.
To mitigate this problem, we study GNN through the lens of an \textit{item and session-graph representation learning mechanism}, where the goal is to obtain a session-graph representation that is \textit{similar} to the representation of the item likely to be clicked next.
We motivate the advantage of restricting the item and session-graph representations to lie on a unit hypersphere both during training and inference, and propose \textbf{NISER}: \textbf{N}ormalized \textbf{I}tem and \textbf{Se}ssion \textbf{R}epresentations model for SR.
We demonstrate the enhanced ability of NISER to deal with popularity bias in comparison to a vanilla GNN model in the offline as well as online settings.
We also extend NISER to incorporate the sequential nature of a session via position embeddings \cite{vaswani2017attention}, thereby leveraging the benefits of both sequence-aware models (like RNNs) and graph-aware models.
\section{Related Work}
Recent results in computer vision literature, e.g. \cite{wang2017normface,zheng2018ring}, indicate the effectiveness of normalizing the final image features during training, and argue in favor of cosine similarity over inner product for learning and comparing feature vectors.
\cite{zheng2018ring} introduces the ring loss for soft feature normalization which eventually learns to constrain the feature vectors on a unit hypersphere. Normalizing words embeddings is also popular in NLP applications, e.g. \cite{peng2015comparative} proposes penalizing the L$_2$ norm of word embeddings for regularization.
However, to the best of our knowledge, the idea of normalizing item and session-graph embeddings or representations has not been explored.
In this work, we study the effect of normalizing the embeddings on popularity bias which has not been established and studied so far.
Several approaches to deal with popularity bias exist in the collaborative filtering settings, e.g. \cite{abdollahpouri2017controlling,steck2011item,yang2018unbiased}.
To deal with popularity bias, \cite{abdollahpouri2017controlling} introduces the notion of flexible regularization in a learning-to-rank algorithm. Similarly \cite{steck2011item,yang2018unbiased} uses the power-law of popularity where the probability of recommending an item is a smooth function of the items' popularity, controlled by an exponent factor. However, to the best of our knowledge, this is the first attempt to study and address popularity bias in DNN-based SR using SR-GNN \cite{wu2018session} as a working example.
Furthermore, SR-GNN does not incorporate the sequential information explicitly to obtain the session-graph representation. We study the effect of incorporating position embeddings \cite{vaswani2017attention} and show that it leads to minor but consistent improvement in recommendation performance.
\section{Problem Definition\label{sec:prob_def}}
Let $\mathcal{S}$ denote all past sessions, and $\mathcal{I}$ denote the set of $m$ items observed in the set $\mathcal{S}$.
Any session $s \in \mathcal{S}$ is a chronologically ordered tuple of item-click events:
$s = (i_{s,1},i_{s,2},\ldots,i_{s,l})$, where each of the $l$ item-click events $i_{s,j}$ ($j=1\ldots l$) corresponds to an item in $\mathcal{I}$, and $j$ denotes the position of the item $i_{s,j}$ in the session $s$.
A session $s$ can be modeled as a graph $\mathcal{G}_s = (\mathcal{V}_s, \mathcal{E}_s)$, where $i_{s,j}\in \mathcal{V}_s$ is a node in the graph.
Further, $(i_{s,j},i_{s,j+1})\in \mathcal{E}_s$ is a directed edge from $i_{s,j}$ to $i_{s,j+1}$.
Given $s$, the goal of SR is to predict the next item $i_{s,l+1}$ by estimating the $m$-dimensional probability vector $\mathbf{\hat{y}}_{s,l+1}$ corresponding to the relevance scores for the $m$ items.
The $K$ items with highest scores constitute the top-K recommendation list.
\section{Learning Item and Session Representations\label{sec:approach}}
Each item is mapped to a $d$-dimensional vector from the trainable embedding look-up table or matrix $\mathbf{I} = [\mathbf{i}_1,\mathbf{i}_2,\ldots,\mathbf{i}_m]^T \in \mathbb{R}^{m\times d}$ such that each row $\mathbf{i}_j\in \mathbb{R}^d$ is the $d$-dimensional representation or embedding\footnote{We use the terms representation and embedding interchangeably.} vector corresponding to item $i_j \in \mathcal{I}$ ($j=1\ldots m$).
Consider any function $f$ (e.g. a neural network as in \cite{liu2018stamp,wu2018session})---parameterized by $\boldsymbol\theta$---that maps the items in a session $s$ to session embedding $\mathbf{s} = f(\mathbf{I}_s;\boldsymbol\theta)$, where\footnote{To ensure same dimensions of $\mathbf{I}_s \in \mathbb{R}^{L\times d}$ across sessions, we can pad with a dummy vector $L-l$ times.} $\mathbf{I}_s = [\mathbf{i}_{s,1},\mathbf{i}_{s,2},\ldots,\mathbf{i}_{s,l}]^T \in \mathbb{R}^{l\times d}$.
Along with $\mathbf{I}_s$ as an input which considers $s$ as a sequence, we also introduce an adjacency matrix $\mathbf{A}_s$ to incorporate the graph structure. We discuss this in more detail later in Section \ref{ssec:niser-gnn}.
The goal is to obtain $\mathbf{s}$ that is close to the embedding $\mathbf{i}_{s,l+1}$ of the target item $i_{k}=i_{s,l+1}$, such that the estimated index/class for the target item is $k = \argmax_j~ \mathbf{i}_j^T\mathbf{s}$ with $j=1\ldots m$.
In a DNN-based model $f$, this is approximated via a differentiable softmax function such that the probability of next item being $i_k$ is given by:
\begin{equation}
\label{eq:softmax1}
p_k(\mathbf{s}) = \hat{\mathbf{y}}_k = \frac{\text{exp}(\mathbf{i}_k^T\mathbf{s})}{\sum_{j=1}^{m}\text{exp}(\mathbf{i}_j^T\mathbf{s})}.
\end{equation}
For this $m$-way classification task, softmax (cross-entropy) loss is used during training for estimating $\boldsymbol \theta$ by minimizing the sum of $\mathcal{L(\hat{\mathbf{y}})} = - \sum_{j=1}^{m}\mathbf{y}_j\text{log}(\hat{\mathbf{y}}_j)$
over all training samples, where $\mathbf{y} \in \{0,1\}^m$ is a 1-hot vector with $\mathbf{y}_k=1$ corresponding to the correct (target) class $k$.
We next introduce the radial property \cite{wang2017normface,zheng2018ring} of softmax loss, and then use it to motivate the need for normalizing the item and session representations in order to reduce popularity bias.
\subsection{Radial Property of Softmax Loss}
\begin{figure*}[h]
\includegraphics[scale=0.5,trim={1cm 4.5cm 2cm 4cm},clip]{nise-15.pdf}
\caption{Illustrative flow diagram for NISER.\label{fig:niser}}
\end{figure*}
It is well-known that optimizing for the softmax loss leads to a radial distribution of features for a target class \cite{wang2017normface,zheng2018ring}:
If $k = \argmax_j~ \mathbf{i}_j^T\mathbf{s}$, then it is easy to show that
\begin{equation}
\frac{\text{exp}(\sigma \mathbf{i}_k^T\mathbf{s})}{\sum_{j\neq k}\text{exp}(\mathbf{i}_j^T\mathbf{s})~+~\text{exp}(\sigma \mathbf{i}_k^T\mathbf{s})} > \frac{\text{exp}(\mathbf{i}_k^T\mathbf{s})}{\sum_{j=1}^{m}\text{exp}(\mathbf{i}_j^T\mathbf{s})}
\end{equation}
for any $\sigma>1$.
This, in turn, implies that softmax loss favors large norm of features for easily classifiable instances.
We omit details for brevity and refer the reader to \cite{wang2017normface,zheng2018ring} for a thorough analysis and proof.
This means that a high value of $\hat{\mathbf{y}}_k$ can be attained by multiplying vector $\mathbf{i}_k$ by a scalar value $\sigma >1$; or simply by ensuring a large norm for the item embedding vector.
\begin{figure}[h]
\centering
\includegraphics[trim={0.25cm 0cm 0.4cm 0cm},clip,width=0.35\textwidth]{pop-avg-norm.pdf}
\vspace{-2mm}
\caption{Recall@20 and L$_2$ norm of learned item embeddings decreases with decreasing popularity in GNN \cite{wu2018session}.\label{fig:pop}}
\vspace{-4mm}
\end{figure}
\subsection{Normalizing the Representations \label{ssec:motiv}}
We note that the radial property has an interesting implication in the context of popularity bias: target items that are easier to predict are likely to have higher L$_2$ norm.
We illustrate this with the help of an example: Items that are popular are likely to be clicked more often, and hence the trained parameters $\boldsymbol{\theta}$ and $\mathbf{I}$ should have values that ensure these items get recommended more often.
The radial property implies that for a given input $s$ and a popular target item $i_k$, a correct classification decision can be obtained as follows: learn the embedding vector $\mathbf{i}_k$ with high $||\mathbf{i}_k||_2$ such that the inner product $\mathbf{i}_k^T\mathbf{s} = ||\mathbf{i}_k||_2||\mathbf{s}||_2cos\alpha$ (where $\alpha$ is the angle between the item and session embeddings) is likely to be high to ensure large value for $\hat{\mathbf{y}}_k$ (even when $\alpha$ is not small enough and $||s||_2$ is not large enough).
When analyzing the item embeddings from a GNN model \cite{wu2018session}, we observe that this is indeed the case: As shown in Fig. \ref{fig:pop}, items with high popularity have high L$_2$ norm while less popular items have significantly lower L$_2$ norm.
Further, performance of GNN (depicted in terms of Recall@20) degrades as the popularity of the target item decreases.
\subsection{NISER}
Based on the above observations, we consider to minimize the influence of embedding norms in the final classification and recommendation decision. We propose optimizing for cosine similarity as a measure of similarity between item and session embeddings instead of the above-stated inner product.
Therefore, during training as well as inference, we normalize the item embeddings as
$\tilde{\mathbf{i}}_k = \frac{\mathbf{i}_{k}}{||\mathbf{i}_{k}||_{_2}}$,
and use them to get $\tilde{\mathbf{I}}_s$.
The session embedding is then obtained as
$\mathbf{s} = f(\tilde{\mathbf{I}}_s;\boldsymbol\theta)$, and is similarly normalized to $\tilde{\mathbf{s}}$ to enforce a unit norm.
The normalized item and session embeddings are then used to obtain the relevance score for next clicked item $i_k$ computed as,
\begin{equation} \label{eq:softmax}
\hat{\mathbf{y}}_k=\frac{\text{exp}(\sigma\tilde{\mathbf{i}}_k^T\tilde{\mathbf{s}})}{\sum_{j=1}^{m}\text{exp}(\sigma \tilde{\mathbf{i}}_j^T\tilde{\mathbf{s}})}.
\end{equation}
Note that the cosine similarity $\tilde{\mathbf{i}}_k^T\tilde{\mathbf{s}}$ is restricted to $[-1,1]$. As shown in \cite{wang2017normface}, this implies that the softmax loss is likely to get saturated at high values for the training set: a scaling factor $\sigma>1$ is useful in practice to allow for better convergence.
\section{Leveraging NISER with GNN\label{ssec:niser-gnn}}
We consider learning the representations of items and session-graphs with GNNs where the session-graph is represented by $\mathcal{G}_s$ as introduced in Section \ref{sec:prob_def}.
Consider two normalized adjacency matrices $\mathbf{A}^{in}_s \in \mathbb{R}^{l,l}$ and $\mathbf{A}^{out}_s \in \mathbb{R}^{l,l}$ corresponding to the incoming and outgoing edges in graph $\mathcal{G}_s$ as illustrated in Fig. \ref{fig:niser}.
GNN takes adjacency matrices $\mathbf{A}^{in}_s$ and $\mathbf{A}^{out}_s$, and the normalized item embeddings $\tilde{\mathbf{I}}_s$ as input, and returns an updated set of embeddings after $\tau$ iterations of message propagation across vertices in the graph using gated recurrent units \cite{li2015gated}: $[\tilde{\mathbf{i}}^\tau_{s,1},\tilde{\mathbf{i}}^\tau_{s,2},\ldots,\tilde{\mathbf{i}}^\tau_{s,l}] = G(\mathbf{A}^{in}_s,\mathbf{A}^{out}_s,\tilde{\mathbf{I}}_s;\boldsymbol{\theta}_g)$, where $\boldsymbol{\theta}_g$ represents the parameters of the GNN function $G$.
For any node in the graph, the current representation of the node and the representations of its neighboring nodes are used to iteratively update the representation of the node $\tau$ times.
More specifically, the representation of node $i_{s,j}$ in the $t$-th message propagation step is updated as follows:
\begin{align}
\label{eq:adj}
\mathcal{\mathbf{a}}_{s,j}^t &= [\mathcal{\mathbf{A}}^{in}_{s,j:} \tilde{\mathbf{I}}^{t-1}_s \mathbf{H}_1, \mathcal{\mathbf{A}}^{out}_{s,j:} \tilde{\mathbf{I}}^{t-1}_s \mathbf{H}_2]^T + \mathbf{b},\\
\mathcal{\mathbf{z}}_{s,j}^t &= \sigma (\mathbf{W}_z \mathcal{\mathbf{a}}_{s,j}^t + \mathbf{U}_z\tilde{\mathbf{i}}_{s,j}^{t-1}),\\
\mathbf{r}_{s,j}^t &= \sigma (\mathbf{W}_r \mathcal{\mathbf{a}}_{s,j}^t + \mathbf{U}_r\tilde{\mathbf{i}}_{s,j}^{t-1}),\\
\hat{\mathbf{i}}_{s,j}^t &= \tanh (\mathbf{W}_o \mathcal{\mathbf{a}}_{s,j}^t + \mathbf{U}_o (\mathbf{r}_{s,j}^t \odot \tilde{\mathbf{i}}_{s,j}^{t-1})),\\
\tilde{\mathbf{i}}_{s,j}^t &= (1-\mathbf{z}_{s,j}^t)\odot \tilde{\mathbf{i}}_{s,j}^{t-1} + \mathbf{z}_{s,j}^t \odot \hat{\mathbf{i}}_{s,j}^t,
\end{align}
\iffalse
\begin{align}
\hat{\mathbf{i}}_{s,j}^t &= \tanh (\mathbf{W}_o \mathcal{\mathbf{a}}_{s,j}^t + \mathbf{U}_o (\mathbf{r}_{s,j}^t \odot \tilde{\mathbf{i}}_{s,j}^{t-1})),\\
\tilde{\mathbf{i}}_{s,j}^t &= (1-\mathbf{z}_{s,j}^t)\odot \tilde{\mathbf{i}}_{s,j}^{t-1} + \mathbf{z}_{s,j}^t \odot \hat{\mathbf{i}}_{s,j}^t,
\end{align}
\fi
where $\mathbf{A}^{in}_{s,j:}$, $\mathbf{A}^{out}_{s,j:} \in \mathbb{R}^{1\times l}$ depicts the $j$-th row of $\mathbf{A}^{in}_{s}$ and $\mathbf{A}^{out}_{s}$ respectively, $\mathbf{H}_1$, $\mathbf{H}_2 \in \mathbb{R}^{d \times d}$, $\mathbf{W}_{(.)}$ and $\mathbf{U}_{(.)}$ are trainable parameters, $\sigma (.)$ is the sigmoid function, and $\odot$ is the element-wise multiplication operator.
To incorporate sequential information of item interactions, we optionally learn \textit{position embeddings} and add them to item embeddings to effectively obtain position-aware item (and subsequently session) embeddings.
The final embeddings for items in a session are computed as $\tilde{\mathbf{i}}^{\tau,p}_{s,j} = \tilde{\mathbf{i}}^\tau_{s,j} + \mathbf{p}_j$, where $\mathbf{p}_j \in \mathbb{R}^d$ is embedding vector for position $j$ obtained via a lookup over the position embeddings matrix $\mathbf{P} = [\mathbf{p}_1,\mathbf{p}_2,\ldots,\mathbf{p}_L]^T \in \mathbb{R}^{L\times d}$, where $L$ denotes the maximum length of any input session such that position $l\leq L$.
The soft-attention weight of the $j$-th item in session $s$ is computed as
$\alpha_j = \mathbf{q}^T\text{sigmoid}(\mathbf{W}_1\tilde{\mathbf{i}}^{\tau,p}_{s,l} + \mathbf{W}_2\tilde{\mathbf{i}}^{\tau,p}_{s,j} + \mathbf{c})$,
where $\mathbf{q}, \mathbf{c}\in \mathbb{R}^d$, $\mathbf{W}_1,\mathbf{W}_2 \in \mathbb{R}^{d\times d}$.
The $\alpha_j$s are further normalized using softmax.
An intermediate session embedding $\mathbf{s}'$ is computed as:
$\mathbf{s}' = \sum_{j=1}^{t}\alpha_j\tilde{\mathbf{i}}^{\tau,p}_{s,j}$.
The session embedding $\mathbf{s}$ is a linear transformation over the concatenation of intermediate session embedding $\mathbf{s}'$ and the embedding of most recent item $\tilde{\mathbf{i}}^{\tau,p}_{s,l}$, s.t. $\mathbf{s} = \mathbf{W}_3[\mathbf{s}';\tilde{\mathbf{i}}^{\tau,p}_{s,l}]
$, where $\mathbf{W}_3\in \mathbb{R}^{d\times2d}$.
The final recommendation scores for the $m$ items are computed as per Eq. \ref{eq:softmax}.
Note that while the session-graph embedding is obtained using item embeddings $\tilde{\mathbf{i}}^{\tau,p}_{s,t}$ that are aware of the session-graph and sequence, the normalized item embeddings $\tilde{\mathbf{i}}_j$ ($j=1\ldots m$) independent of a particular session are used to compute the recommendation scores.
\section{Experimental Evaluation}
\textbf{Dataset Details:}
We evaluate NISER on publicly available benchmark datasets: i) Yoochoose (YC), ii) Diginetica (DN), and iii) RetailRocket (RR).
The YC\footnote{http://2015.recsyschallenge.com/challege.html} dataset is from RecSys Challenge 2015, which contains a stream of user clicks on an e-commerce website within six months.
Given the large number of sessions in YC, the recent 1/4 and 1/64 fractions of the training set are used to form two datasets: YC-1/4 and YC-1/64, respectively, as done in \cite{wu2018session}.
The DN\footnote{http://cikm2016.cs.iupui.edu/cikm-cup} dataset is transactional data from CIKM Cup 2016 challenge.
The RR\footnote{https://www.dropbox.com/sh/dbzmtq4zhzbj5o9/AACldzQWbw-igKjcPTBI6ZPAa?dl=0} dataset is from an e-commerce personalization company retailrocket, which published dataset with six month of user browsing activities.
\\
\textbf{Offline and Online setting:}
We consider two evaluation settings: i. offline and ii. online.
For evaluation in offline setting, we consider static splits of train and test as used in \cite{wu2018session} for YC and DN.
For RR, we consider sessions from last 14 days for testing and remaining 166 days for training.
The statistics of datasets are summarized in Table \ref{tab:datastats}.
For evaluation in online setting, we re-train the models every day for 2 weeks (number of sessions per day for YC is much larger, so we evaluate for 1 week due to computational constraints) by appending the sessions from that day to the previous train set, and report the test results of the trained model on sessions from the subsequent day.\\
\textbf{NISER and its variants:}
We apply our approach over GNN and adapt code\footnote{https://github.com/CRIPAC-DIG/SR-GNN} from \cite{wu2018session} with suitable modification described later.
We found that for sessions with length $l>10$, considering only the most recently clicked 10 items to make recommendations worked consistently better across datasets.
We refer to this variant as GNN+ and use this additional pre-processing step in all our experiments.
\begin{table}[t]
\caption{Statistics of the datasets used for offline experiments. \label{tab:datastats}}
\vspace{-2mm}
\scalebox{0.9}{
\begin{tabular}{|p{0.25\linewidth}|c|c|c|c|}
\hline
\textbf{Statistics} & \textbf{DN} & \textbf{RR} & \textbf{YC-1/64} & \textbf{YC-1/4}\\
\hline
\#train sessions&0.7 M&0.7 M&0.4 M&0.6 M\\
\#test sessions&60,858&60,594&55,898&55,898\\
\#items&43,097&48,759&16,766&29,618\\
Average length&5.12&3.55&6.16&5.17\\
\hline
\end{tabular}}
\vspace{-4mm}
\end{table}
We consider enhancement over GNN+, and proposed following variants of the embedding normalization approach:
\begin{itemize}
\item \textit{Normalized Item Representations (NIR)}: only item embeddings are normalized and scale factor $\sigma$ is not used\footnote{$\tilde{\mathbf{i}}_k^T\mathbf{s}$ is not restricted to [-1,1], in general $||\mathbf{s}||_{_2}\neq1$.},
\item \textit{Normalized Item and Session Representations (NISER)}: both item and session embeddings are normalized,
\item \textit{NISER+}: NISER with position embeddings and dropout applied to input item embeddings.
\end{itemize}
\textbf{Hyperparameter Setup:}
Following \cite{wu2018session},
we use $d=100$ and learning rate of 0.001 with Adam optimizer.
We use 10\% of train set which is closer to test test in time as hold-out validation set for hyperparameter tuning including scale factor $\sigma$.
We found $\sigma = 16.0$ to work best across most models trained on respective hold-out validation set chosen from \{$4.0,9.0,16.0,25.0$\}, and hence, we use the same value across datasets for consistency.
We use dropout probability of 0.1 on dimensions of item embeddings in NISER+ across all models.
Since the validation set is closer to the test set in time, therefore, it is desirable to use it for training the models.
After finding the best epoch
via early stopping based on performance on validation set, we re-train the model for same number of epochs on combined train and validation set.
We train five models for the best hyperparameters with random initialization, and report average and standard deviation of various metrics for all datasets except for YC-1/4 where we train three models (as it is a large dataset and takes around ~15 hours for training one model).
\\
\textbf{Evaluation Metrics:}
We use same evaluation metrics Recall@K and Mean Reciprocal Rank (MRR@K) as in \cite{wu2018session} with $K=20$.
\textbf{Recall@K} represents the proportion of test instances which has desired item in the top-K items.
\textbf{MRR@K} (Mean Reciprocal Rank) is the average of reciprocal ranks of desired item in recommendation list.
Large value of MRR indicates that desired item is in the top of the recommendation list.
For evaluating popularity bias, we consider the following metrics as used in \cite{abdollahpouri2019managing}:
\begin{figure*}[t]
\subfigure[DN]{\includegraphics[trim={0.250cm 0cm 0.4cm 0cm},clip,width=0.22\textwidth]{plot-target-pop-digi-sea.pdf}}
\subfigure[RR]{\includegraphics[trim={0.250cm 0cm 0.4cm 0cm},clip,width=0.21\textwidth]{plot-target-pop-retail-sea.pdf}}
\subfigure[YC-1/64]{\includegraphics[trim={0.250cm 0cm 0.4cm 0cm},clip,width=0.21\textwidth]{plot-target-pop-yc1_64-sea.pdf}}
\subfigure[YC-1/4]{\includegraphics[trim={0.250cm 0cm 0.4cm 0cm},clip,width=0.21\textwidth]{plot-target-pop-yc_4-sea.pdf}}
\vspace{-2mm}
\caption{Offline setting evaluation: Recall@20 and MRR@20 with varying $\phi^*$ indicating larger gains by using NISER+ over GNN+ for less popular items. \label{fig:recall-vs-phi}}
\end{figure*}
\begin{figure*}[t]
\subfigure[DN]{\includegraphics[trim={0.0cm 0cm 0.0cm 0cm},clip,width=0.29\textwidth]{plot-day-r-mrr-dn.pdf}}
\subfigure[RR]{\includegraphics[trim={0.0cm 0cm 0.0cm 0cm},clip,width=0.265\textwidth]{plot-day-r-mrr-rr.pdf}}
\subfigure[YC]{\includegraphics[trim={0.0cm 0cm 0.0cm 0cm},clip,width=0.27\textwidth]{plot-day-r-mrr-yc.pdf}}
\vspace{-2mm}
\caption{Online setting evaluation: Recall@20 and MRR@20 for sessions where target item is one of the less popular newly introduced items from the previous day. $f$ denotes the fraction of such sessions in the training set for $\phi^*=0.01$. Standard deviations over five models are shown in lighter-shaded region around the solid lines. \label{fig:plot-online}}
\end{figure*}
\textbf{Average Recommendation Popularity (ARP)}:
This measure calculates the average popularity of the recommended items in each list given by:
\begin{equation}\label{eq:arp}
ARP=\frac{1}{|S|}\sum_{s \in S} \frac{\sum_{i \in L_s}\phi(i)}{K},
\end{equation}
where $\phi(i)$ is popularity of item $i$, i.e. the number of times item $i$ appears in the training set, $L_s$ is the recommended list of items for session $s$, and $|S|$ is the number of sessions in the test set.
An item $i$ belongs to the set $\Gamma_{\phi^*}$ of long-tail items or less popular items if $\frac{\phi(i)}{max_i\phi(i)} \leq \phi^*$.
We evaluate the performance in terms of Recall@20 and MRR@20 for the sessions with target item in the set $\Gamma_{\phi^*}$ by varying $\phi^*$.
\subsection{Results and Observations}
\begin{table}[t]
\caption{Offline setting evaluation: NISER+ versus GNN+ in terms of Average Recommendation Popularity (ARP). Lower values of ARP indicate lower popularity bias. \label{tab:pop-bias}}
\scalebox{0.82}{
\begin{tabular}{|c|c|c|c|c|}
\hline
\textbf{Method}&\textbf{DN}&\textbf{RR}&\textbf{YC-1/64}&\textbf{YC-1/4}\\
\hline
GNN+ &495.25$\pm$2.52&453.39$\pm$8.97&4128.54$\pm$27.80&17898.10$\pm$126.93\\
NISER+ &\textbf{487.31$\pm$0.30}&\textbf{398.53$\pm$3.09}&\textbf{3972.40$\pm$41.04}&\textbf{16683.52$\pm$120.74}\\
\hline
\end{tabular}}
\vspace{-2mm}
\end{table}
\begin{table}[t]
{\caption{NISER+ versus other benchmark methods in offline setting. Numbers after $\pm$ are standard deviation values over five models. \label{tab:gnn_all_r}}}
\scalebox{0.85}{
\begin{tabular}{|c|c|c|c|c|}
\hline
\textbf{Method}&\textbf{DN}&\textbf{RR}&\textbf{YC-1/64}&\textbf{YC-1/4}\\
\hline
\multicolumn{5}{|c|}{\textbf{Recall@20}}\\
\hline
SKNN \cite{jannach2017recurrent}&48.06&56.42&63.77&62.13\\
STAN \cite{garg2019sequence} &50.97&59.80&69.45&70.07\\
\hline
GRU4REC \cite{hidasi2015session}&29.45&-&60.64&59.53\\
NARM \cite{li2017neural}&49.70&-&68.32&69.73\\
STAMP \cite{liu2018stamp}&45.64&53.94&68.74&70.44\\
GNN \cite{wu2018session}&51.39$\pm$0.38&57.63$\pm$0.15&70.54$\pm$0.14&70.95$\pm$0.04\\
GNN+&51.81$\pm$0.11&58.59$\pm$0.10&70.85$\pm$0.08&71.10$\pm$0.07\\
\hline
NIR&52.40$\pm$0.06&60.67$\pm$0.08&71.12$\pm$0.05&71.32$\pm$0.11\\
NISER&52.63$\pm$0.09&60.85$\pm$0.06&70.86$\pm$0.15&71.69$\pm$0.03\\
NISER+&\textbf{53.39$\pm$0.06}&\textbf{61.41$\pm$0.09}&\textbf{71.27$\pm$0.05}&\textbf{71.80$\pm$0.09}\\
\hline\hline
\multicolumn{5}{|c|}{\textbf{MRR@20}}\\
\hline
SKNN \cite{jannach2017recurrent}&16.95&33.16&25.22&24.82\\
STAN \cite{garg2019sequence}&18.48&35.32&28.74&28.89\\
\hline
GRU4REC \cite{hidasi2015session}&8.33&-&22.89&22.60\\
NARM \cite{li2017neural}&16.17&-&28.63&29.23\\
STAMP \cite{liu2018stamp}&14.32&28.49&29.67&30.00\\
GNN \cite{wu2018session}&17.79$\pm$0.16&32.74$\pm$0.09&30.80$\pm$0.09&31.37$\pm$0.13\\
GNN+&18.03$\pm$0.05&33.29$\pm$0.03&30.84$\pm$0.10&31.51$\pm$0.05\\
\hline
NIR &18.52$\pm$0.06&35.57$\pm$0.05&30.99$\pm$0.10&31.73$\pm$0.11\\
NISER &18.27$\pm$0.10&36.09$\pm$0.03&31.50$\pm$0.11&\textbf{31.80$\pm$0.12}\\
NISER+&\textbf{18.72$\pm$0.06}&\textbf{36.50$\pm$0.05}&\textbf{31.61$\pm$0.02}&31.77$\pm$0.10\\
\hline
\end{tabular}}
\vspace{-2mm}
\end{table}
\begin{table}
\caption{Ablation results for NISER+ indicating that normalization of embeddings (L$_2$ norm) contributes the most to performance improvement. Here PE: Position Embeddings. \label{tab:gnn_plus_ablation_recall}}
\scalebox{0.85}{
\begin{tabular}{|p{0.2\linewidth}|c|c|c|c|}
\hline
\textbf{Method}&\textbf{DN}&\textbf{RR}&\textbf{YC-1/64}&\textbf{YC-1/4}\\
\hline
\multicolumn{5}{|c|}{\textbf{Recall@20}}\\
\hline
NISER+&\textbf{53.39$\pm$0.06}&\textbf{61.41$\pm$0.09}&\textbf{71.27$\pm$0.05}&71.80$\pm$0.09\\
-L$_2$ norm &52.23$\pm$0.10&59.16$\pm$0.10&71.10$\pm$0.09&71.46$\pm$0.19\\
-Dropout&52.81$\pm$0.12&60.99$\pm$0.09&71.07$\pm$0.13&\textbf{71.90$\pm$0.03}\\
-PE&53.11$\pm0.12$&61.22$\pm$0.03&71.13$\pm$0.04&71.70$\pm$0.11\\
\hline\hline
\multicolumn{5}{|c|}{\textbf{MRR@20}}\\
\hline
NISER+&\textbf{18.72$\pm$0.06}&\textbf{36.50$\pm$0.05}&31.61$\pm$0.02&31.77$\pm$0.10\\
-L$_2$ norm&18.11$\pm$0.05&33.78$\pm$0.04&30.90$\pm$0.07&31.49$\pm$0.07\\
-Dropout&18.43$\pm$0.11&35.99$\pm$0.02&31.56$\pm$0.06&\textbf{31.93$\pm$0.17}\\
-PE &18.60$\pm$0.09&36.32$\pm$0.03&\textbf{31.68$\pm$0.05}&31.71$\pm$0.06\\
\hline
\end{tabular}}
\vspace{-2mm}
\end{table}
\textbf{(1)} \textbf{NISER+ reduces popularity bias in GNN+ in :}
Table \ref{tab:pop-bias} shows that ARP for NISER+ is significantly lower than GNN+ indicating that NISER+ is able to recommend less popular items more often than GNN+, thus reducing popularity bias.
Furthermore, from Fig. \ref{fig:recall-vs-phi}, we observe that NISER+ outperforms GNN+ for sessions with less popular items as targets (i.e. when $\phi^*$ is small), with gains 13\%, 8\%, 5\%, and 2\% for DN, RR, YC-1/64, and YC-1/4 respectively for $\phi^*=0.01$ in terms of Recall@20.
Similarly, gains are 28\%, 18\%, 6\%, and 2\% in terms of MRR@20.
Gains for DN and RR are high as compared to YC. This is due to the high value of $max_i\phi(i)$. If instead we consider $\phi^*=0.001$, gains are as high as 26\% and 9\% for YC-1/64 and YC-1/4 respectively in terms of Recall@20. Similarly, gains are as high as 34\% and 19\% in terms of MRR@20.
We also note that NISER+ is at least as good as GNN+ even for the sessions with more popular items as targets (i.e. when $\phi^*$ is large).
\noindent \textbf{(2)} \textbf{NISER+ improves upon GNN+ in online setting}
for newly introduced items in the set of long-tail items $\Gamma_{\phi^*}$. These items have small number of sessions available for training at the end of the day they are launched.
From Fig. \ref{fig:plot-online}, we observe that for the less popular newly introduced items, NISER+ outperforms GNN+ for sessions where these items are target items on the subsequent day.
This proves the ability of NISER+ to recommend new items on very next day, due to its ability to reduce popularity bias.
Furthermore, for DN and RR, we observe that during initial days, when training data is less, GNN+ performs poorly while performance of NISER+ is relatively consistent across days indicating potential regularization effect of NISER on GNN models in less data scenarios.
Also, as days pass by and more data is available for training, performance of GNN+ improves with time but still stays significantly below NISER+.
For YC, as days pass by, the popularity bias becomes more and more severe (as depicted by very small value for $f$, i.e. the fraction of sessions with less popular newly introduced items) such that the performance of both GNN+ and NISER+ degrades with time. However, importantly, NISER+ still performs consistently better than GNN+ on any given day as it better handles the increasing popularity bias.
\noindent \textbf{(3)} \textbf{NISER and NISER+ outperform GNN and GNN+ in offline setting:}
From Table \ref{tab:gnn_all_r}, we observe that NISER+ shows consistent and significant improvement over GNN in terms of Recall and MRR, establishing a \textit{new state-of-the-art} in SR.
We also conduct an \textbf{ablation study} (removing one feature of NISER+ at a time) to understand the effect of each of the following features of NISER+: i. L$_2$ norm of embeddings, ii. including position embeddings, and iii. applying dropout on item embeddings. As shown in Table \ref{tab:gnn_plus_ablation_recall}, we observe that L$_2$ norm is the most important factor across datasets while dropout and position embeddings contribute in varying degrees to the overall performance of NISER+.
\section{Discussion}
In this work, we highlighted that the typical item-frequency distribution with long tail leads to popularity bias in state-of-the-art deep learning models such as GNNs \cite{wu2018session} for session-based recommendation.
We then argued that this is partially related to the `radial' property of the softmax loss that, in our setting, implies that the norm for popular items will likely be larger than the norm of less popular items.
We showed that learning the representations for items and session-graphs by optimizing for cosine similarity instead of inner product can help mitigate this issue to a large extent.
Importantly, this ability to reduce popularity bias is found to be useful in the online setting where the newly introduced items tend to be less popular and are poorly modeled by existing approaches.
We observed significant improvements in overall recommendation performance by normalizing the item and session-graph representations and improve upon the existing state-of-the-art results.
In future, it would be worth exploring NISER to improve other algorithms like STAMP \cite{liu2018stamp} that rely on similarity between embeddings for items, sessions, users, etc.
\bibliographystyle{ACM-Reference-Format}
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train/arxiv
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BkiUaXnxK5YsWTJsNBiq
| 5 | 1 |
\section{Introduction}
The classical theorem of B\'ezout~\cite{Bezout} bounds the number of solutions
to a system of polynomials by the product of their degrees.
While this B\'ezout bound is sharp for generic systems of polynomial equations,
that is no longer the case when the equations possess additional structure.
For example, Kushnirenko~\cite{BKK} showed that if all the polynomials have
the same Newton polytope, then the number of nondegenerate solutions to
such a system is at most the volume of the Newton polytope, suitably normalized.
Bounds for the number of nondegenerate real solutions are governed by
Kushnirenko's ``fewnomial principle'': roughly, few monomials implies few
solutions or restricted topology~\cite{Kush_note}.
This principle was established by Khovanskii in his fundamental work on
fewnomials~\cite{Kh91} in which he showed that a system of $n$ polynomials in $n$ variables
where the polynomials have $1{+}k{+}n$ distinct monomials has fewer than
\begin{equation}\label{Eq:Khovanskii_bound}
2^{\binom{k{+}n}{2}}\, (n{+}1)^{k{+}n}
\end{equation}
nondegenerate positive solutions.
This bound is remarkable as it is independent of the degrees and Newton polytopes of the
polynomials, which control the number of complex solutions.
Few of the complex solutions to a fewnomial system can be real.
Khovanskii's bound was lowered by Bihan and Sottile~\cite{BS_Khovanskii} to
\begin{equation}\label{Eq:BS}
\frac{e^2{+}3}{4}\, 2^{\binom{k}{2}}\, n^k\,.
\end{equation}
For this, they transformed the original polynomial system into an
equivalent Gale dual~\cite{BS_Gale} system of rational functions, whose number of solutions
they bounded.
An essential step for the bound on Gale systems uses Khovanskii's generalization of the
classical Rolle Theorem.
The bound~\eqref{Eq:BS} is smaller than Khovanskii's
bound~\eqref{Eq:Khovanskii_bound} because Khovanskii's bound is a specialization to
polynomials of a bound for more general functions and the proof
of~\eqref{Eq:BS} takes advantage of some special geometry enjoyed by polynomials.
We derive bounds of fewnomial type for polynomial systems with structure
intermediate between that of fewnomials and general polynomials.
These bounds can be dramatically smaller than the fewnomial bound~\eqref{Eq:BS}, and like
that bound do not depend upon the degrees of the polynomials involved.
A collection ${\mathcal{A}}\subset\mathbb{Z}^n$ of exponent vectors is
\demph{$(d,\ell)$-dense} if there are integers $d,\ell$ such that
${\mathcal{A}}$ admits a decomposition of the form
\begin{equation}\label{Eq:strFew}
{\mathcal{A}}\ =\ \psi(d\includegraphics{figures/Simplex.eps}^\ell\cap\mathbb{Z}^\ell)\; \bigcup\; {\mathcal{W}}\,,
\end{equation}
where ${\mathcal{W}}$ consists of $n$ affinely independent vectors, $\psi\colon\mathbb{Z}^\ell\to\mathbb{Z}^n$
is an affine-linear map, and $\includegraphics{figures/Simplex.eps}^\ell$ is the unit simplex in $\mathbb{R}^\ell$.
A Laurent polynomial whose support ${\mathcal{A}}$ is $(d,\ell)$-dense~\eqref{Eq:strFew} is a
\demph{$(d,\ell)$-dense fewnomial}.
We show that a system of $(d,\ell)$-dense fewnomials
in $n$ variables has fewer than
\begin{equation}\label{Eq:structured_bound}
\frac{e^2{+}3}{4}\, 2^{\binom{\ell }{2}}\, n^\ell \cdot d^\ell
\end{equation}
nondegenerate positive solutions.
To compare this to~\eqref{Eq:BS}, observe that $k{+}1\ =\ \binom{d+\ell}{\ell}$,
so the bounds coincide when $d=1$, but otherwise~\eqref{Eq:structured_bound} is dramatically
smaller.
In~\cite{BBS_all}, the methods of~\cite{BS_Khovanskii} were extended to establish the bound
\begin{equation}\label{Eq:BBS_bound}
\frac{e^{\Red{4}}+3}{4}\, 2^{\binom{k}{2}}\, n^k\,,
\end{equation}
for the number of nonzero real solutions to a fewnomial system---not just positive
solutions---when the exponent vectors span a sublattice of $\mathbb{Z}^n$ of odd index.
The same arguments show that if the exponent vectors ${\mathcal{A}}$ span a
sublattice of $\mathbb{Z}^n$ of odd index, then the number of nondegenerate real solutions to a system
of $(d,\ell)$-dense fewnomials is at most
\begin{equation}\label{Eq:structured_real}
\frac{e^{\Red{4}}+3}{4}\, 2^{\binom{\ell }{2}}\, n^\ell \cdot d^\ell\,.
\end{equation}
Khovanskii also gave a bound for the sum \demph{$b_*(X)$} of the Betti
numbers of a smooth hypersurface in the positive orthant $\mathbb{R}^n_>$ defined by a polynomial
with $1{+}k{+}n$ monomial terms~\cite{Kh91} (Corollary 4, p. 91),
\begin{equation}\label{eq:Khovanskii}
b_*(X)\ \leq\ (2n^2{-}n{+}1)^{k{+}n}\, (2n)^{n{-}1}\, 2^{\binom{k{+}n}{2}}\,.
\end{equation}
Bihan and Sottile~\cite{BS_Betti} used the fewnomial bound and stratified Morse theory for
a manifold with corners~\cite{GM88} to lower this to
\begin{equation}\label{eq:Betti}
b_*(X) \ <\ \frac{e^2{+}3}{4}\, 2^{\binom{k}{2}}
\cdot \sum_{i=0}^{n} \tbinom{n}{i}\, i^k\,.
\end{equation}
The same arguments show that when $X$ is defined by a $(d,\ell)$-dense fewnomial, we have
\begin{equation}\label{Eq:structured_Betti}
b_*(X)\ <\ \frac{e^2{+}3}{4}\, 2^{\binom{\ell}{2}} \cdot d^\ell
\cdot \sum_{i=0}^{n} \tbinom{n}{i}\, i^\ell\,.
\end{equation}
An important step in these arguments is a version of Gale duality for
dense fewnomials, which generalizes Gale duality for polynomial
systems as established in~\cite{BS_Gale}.
These bounds~\eqref{Eq:structured_bound},~\eqref{Eq:structured_real},
and~\eqref{Eq:structured_Betti} simultaneously generalize the results
of~\cite{BS_Khovanskii},~\cite{BBS_all},
and~\cite{BS_Betti}, which are the cases when $d=1$.
The case of $\ell=1$ of the bound~\eqref{Eq:BBS_bound} was established
in~\cite{BBS_few}, where a $(d,1)$-dense fewnomial was called a near circuit.
This paper is structured as follows.
We begin in Section~\ref{Sec:one} with definitions and examples of dense fewnomials, give the
precise statements of our main theorems, and study an example when $n=2$.
Section~\ref{S:gale} is devoted to establishing the variant of Gale duality appropriate for
dense fewnomials, and in Section~\ref{S:bounds}, which is the heart of this paper, we establish
the bounds~\eqref{Eq:structured_bound} and~\eqref{Eq:structured_real}.
We develop the necessary tools and give the proof of our bound~\eqref{Eq:structured_Betti}
for the sum of Betti numbers in Section~\ref{Sec:Betti}.
We thank Maurice Rojas, who suggested looking for extentions of the fewnomial bound as a
class project to Rusek and Shakalli.
\section{Dense Fewnomials}\label{Sec:one}
An integer vector $\alpha=(a_1,\dotsc,a_n)\in\mathbb{Z}^n$ is the exponent of a Laurent
monomial,
\[
x^\alpha\ :=\ x_1^{a_1} x_2^{a_2}\dotsb x_n^{a_n}\,.
\]
A polynomial $f$ with \demph{support} ${\mathcal{A}}\subset\mathbb{Z}^n$ is one whose exponent vectors
lie in ${\mathcal{A}}$,
\[
f\ =\ \sum_{\alpha\in{\mathcal{A}}} a_\alpha x^\alpha
\qquad(a_\alpha\in\mathbb{R})\,.
\]
We are interested in systems of $n$ polynomials, each with support ${\mathcal{A}}$.
We obtain novel bounds on the number of real solutions to a system of polynomials when the set
of exponent vectors has structure that is intermediate between fewnomials
and dense polynomials.
Kushnirenko~\cite{BKK} showed that a general system of polynomials, all with support ${\mathcal{A}}$,
will have $n!\operatorname{vol}(\operatorname{conv}({\mathcal{A}}))$ complex solutions, the normalized volume of the convex hull of
the exponent vectors.
While this is also a bound for the number of real solutions, there is another bound which
depends only upon the number of exponents.
Specifically, a \demph{fewnomial system} is one in which the support ${\mathcal{A}}$ consists of
$1{+}k{+}n$ monomials, but is otherwise unstructured, and such a system has the
bound~\eqref{Eq:BS} on its number of positive solutions.
When the exponents ${\mathcal{A}}$ affinely span a sublattice of odd index in $\mathbb{Z}^n$, a fewnomial
system has the bound~\eqref{Eq:BBS_bound} for its number of nondegenerate nonzero real solutions.
A \demph{dense fewnomial} is a polynomial whose support ${\mathcal{A}}$ is intermediate
between fewnomials and general polynomials in the following way.
Let $d,\ell,n$ be positive integers, $\includegraphics{figures/Simplex.eps}^\ell\subset\mathbb{R}^\ell$ be the standard
unit simplex, the convex hull of the original and the $\ell$ unit basis vectors, and
$\psi\colon\mathbb{Z}^\ell\to\mathbb{Z}^n$ be an affine-linear map.
A \demph{$(d,\ell)$-dense fewnomial} is a Laurent polynomial $f$ whose support
${\mathcal{A}}\subset\mathbb{Z}^n$ admits a decomposition
\begin{equation}\label{Eq:near_polyhedral}
{\mathcal{A}}\ =\ \psi(d\includegraphics{figures/Simplex.eps}^\ell\cap\mathbb{Z}^\ell)\;\bigcup\; {\mathcal{W}}\,,
\end{equation}
where ${\mathcal{W}}=\{w_1,\dotsc,w_n\}$ consists of $n$ affinely independent vectors.
Such a set ${\mathcal{A}}$~\eqref{Eq:near_polyhedral} is \demph{$(d,\ell)$-dense}.
We give some examples of $(d,\ell)$-dense sets of exponent vectors.
\begin{enumerate}
\item Any collection ${\mathcal{A}}$ of $1{+}\ell{+}n$ exponent vectors that affinely spans
$\mathbb{R}^n$ is $(1,\ell)$-dense.
To see this, let ${\mathcal{W}}\subset{\mathcal{A}}$ be $n$ affinely independent vectors.
Writing ${\mathcal{A}}-{\mathcal{W}}=\{v_0,v_1,\dotsc,v_\ell\}$, these vectors
are the image of the integer points in $\includegraphics{figures/Simplex.eps}^\ell$ under the affine map $\psi$ that
takes the $i$th unit vector to $v_i$ and the origin to $v_0$.
Thus ordinary fewnomials with $1{+}k{+}n$ monomials are $(1,k)$-dense fewnomials.
\item When $\ell=1$, the exponent vectors ${\mathcal{A}}$ of a dense
fewnomial form a \demph{near circuit} in the terminology of~\cite{BBS_few}.
There, it was shown that if ${\mathcal{A}}$ spans $\mathbb{Z}^n$, then a system with support ${\mathcal{A}}$
has at most $2dn{+}1$ nonzero real solutions.
\item A general $(d,\ell)$-dense set ${\mathcal{A}}$ has the following form,
\begin{equation}\label{Eq:general_form}
{\mathcal{A}}\ :=\
\Bigl\{ v_0\ +\ \sum_{m=1}^\ell \lambda_mv_m \mid
0\leq \lambda_m\,,\ \sum_i\lambda_i\leq d \Bigr\}
\bigcup {\mathcal{W}}\,,
\end{equation}
where ${\mathcal{W}}=\{w_1,\dotsc,w_n\}\subset\mathbb{Z}^n$ is affinely independent and
$v_0,v_1,\dotsc,v_\ell$ are integer vectors.
Below is an example of such a set ${\mathcal{A}}$ in $\mathbb{Z}^2$ ($v_0=(0,0)$ is the open circle).
\begin{equation}\label{Eq:triangle}
\raisebox{-38pt}{\begin{picture}(157,76)(-11,0)
\put(0,0){\includegraphics{figures/triangle.eps}}
\put(-11,1){$v_0$} \put(64,21){$v_1$}
\put(7,32){$v_2$}
\put(18,63){$w_2$} \put(98,1){$w_1$}
\end{picture}}
\end{equation}
For this, $n=\ell=d=2$, ${\mathcal{W}}=\{(9,0),(2,7)\}$ and
$v_1=(7,1)$ and $v_2=(2,3)$.
Here is a $(2,3)$-dense set in $\mathbb{Z}^2$.
\[
\begin{picture}(187,106)(-11,0)
\put(0,0){\includegraphics{figures/tetrahedron.eps}}
\put(-3,32){$v_2$} \put(78,13){$v_1$}
\put(45,53){$v_3$} \put(-11,1){$v_0$}
\put(8,89){$w_2$} \put(138,13){$w_1$}
\end{picture}
\]
\end{enumerate}
\begin{theorem}\label{Th:Structured_bound}
Suppose that ${\mathcal{A}}\subset\mathbb{Z}^n$ is $(d,\ell)$-dense.
Then a system
\[
f_1(x_1,\dotsc,x_n)\ =\
f_2(x_1,\dotsc,x_n)\ =\ \dotsb \ =\
f_n(x_1,\dotsc,x_n)\ =\ 0
\]
of real polynomials with support ${\mathcal{A}}$ has fewer than
\[
\frac{e^2{+}3}{4}\, 2^{\binom{\ell}{2}}\, n^\ell \cdot d^\ell
\]
nondegenerate positive solutions.
If the affine span of ${\mathcal{A}}$ is a sublattice of\/ $\mathbb{Z}^n$ with odd index, then
the number of nondegenerate real solutions is less than
\[
\frac{e^4{+}3}{4}\, 2^{\binom{\ell}{2}}\, n^\ell \cdot d^\ell\,.
\]
\end{theorem}
The bounds in Theorem~\ref{Th:Structured_bound} hold if the support of the system is only a
subset of ${\mathcal{A}}$, for in that case, we may perturb the coefficients of the system to obtain
one whose support is exactly ${\mathcal{A}}$ without decreasing the number of nondegenerate solutions.
When $n=d=\ell=2$, this bound for positive solutions is $83$.
This is already significant for the $(2,2)$-dense set~\eqref{Eq:triangle}.
Since the Kushnirenko bound for~\eqref{Eq:triangle} is 112,
a general system with support~\eqref{Eq:triangle} will have 112 complex solutions, at most 83
of which can be positive.
A generic system of $(d,\ell)$-dense fewnomials (as in~\eqref{Eq:near_polyhedral}),
\begin{equation}\label{Eq:structured_system}
f_1(x)\ =\ f_2(x)\ =\ \dotsb\ =\ f_n(x)\ =\ 0\,,
\end{equation}
will have an invertible matrix of coefficients of the monomials
$\{x^{w_i}\mid i=1,\dotsc,n\}$, and so we may solve~\eqref{Eq:structured_system} for these
monomials to get the equivalent system
\begin{equation}\label{Eq:diagonal}
x^{w_i}\ =\ \sum_{p\in d\includegraphics{figures/Simplex.sm.eps}^\ell\cap\mathbb{Z}^\ell} a_{i,p} x^{\psi(p)}\,,
\qquad\mbox{for}\quad i=1,\dotsc,n,.
\end{equation}
For each $i=1,\dotsc,n$, define the degree $d$ polynomial in variables $y\in\mathbb{R}^\ell$,
\[
\dcol{h_i(y)}\ :=\ \sum_{p\in d\includegraphics{figures/Simplex.sm.eps}^\ell\cap\mathbb{Z}^\ell} a_{i,p} y^{p}\,.
\]
Following the notation in~\eqref{Eq:general_form}, we translate the set ${\mathcal{A}}$ by $-v_0$,
which amounts to multiplying the equations~\eqref{Eq:structured_system}
and~\eqref{Eq:diagonal} by $x^{-v_0}$, and does not change their solutions.
Thus we may assume that $v_0=0$ so that $\psi$ is a linear map, and then let
${\mathcal{V}}:=\{v_1,\dotsc,v_\ell\}\subset\mathbb{Z}^n$ be the images of the standard basis vectors
under $\psi$.
Then we have $x^{w_i}=h_i(x^{v_1},\dotsc,x^{v_\ell})$.
A linear relation among
the vectors in ${\mathcal{V}}$ and ${\mathcal{W}}$,
\begin{equation}\label{Eq:lin_reln}
\sum_{m=1}^\ell b_m v_m\ +\ \sum_{i=1}^n c_i w_i\ =\ 0\,,
\end{equation}
implies the multiplicative relation among the monomials
\[
\prod_{m=1}^\ell (x^{v_m})^{b_m}\ \cdot\ \prod_{i=1}^n (x^{w_i})^{c_i}\ =\ 1\,.
\]
If we use~\eqref{Eq:diagonal} to first substitute $h_i(x^{v_1},\dotsc,x^{v_\ell})$
for $x^{w_i}$ for $i=1,\dotsc,n$ in this expression,
and then substitute $y_m$ for $x^{v_m}$, for $m=1,\dotsc,\ell$, we obtain
\[
\prod_{m=1}^\ell(y_m)^{b_m}\ \cdot\ \prod_{i=1}^n (h_i(y))^{c_i}\ =\ 1\,.
\]
Write $\beta=(b_1,\dotsc,b_\ell)$ for the vector of the coefficients of ${\mathcal{V}}$
in~\eqref{Eq:lin_reln} and $\gamma=(c_1,\dotsc,c_n)$ for the vector of the coefficients of
${\mathcal{W}}$.
Then we may write the left hand side of this last expression compactly as
$y^\beta\cdot h(y)^\gamma$.
Now suppose that $(\beta_j,\gamma_j)\in\mathbb{Z}^\ell\oplus\mathbb{Z}^n$ for $j=1,\dotsc,\ell$ is a basis for
the $\mathbb{Z}$-module of linear relations among the vectors in ${\mathcal{V}}\cup{\mathcal{W}}$.
Then the system
\begin{equation}\label{Eq:str_Gale}
y^{\beta_j}\ \cdot\ h(y)^{\gamma_j}
\ =\ 1\,,\qquad j=1,\dotsc,\ell
\end{equation}
is a \demph{$(d,\ell)$-dense Gale system} dual to the original
system~\eqref{Eq:structured_system} of polynomials.
\begin{theorem}\label{Th:GaleDual}
Let~$\eqref{Eq:structured_system}$ be a system of $(d,\ell)$-dense fewnomials
and~$\eqref{Eq:str_Gale}$ be its corresponding dual $(d,\ell)$-dense Gale system.
Then the number of nondegenerate positive solutions to~$\eqref{Eq:structured_system}$ is equal
to the number of nondegenerate positive solutions to~$\eqref{Eq:str_Gale}$ where
$h_i(y)>0$ for each $i=1,\dotsc,n$.
If the exponents ${\mathcal{A}}$ affinely span a sublattice of\/ $\mathbb{Z}^n$ of odd index and the relations
$(\beta^j,\gamma^j)\in\mathbb{Z}^\ell\oplus\mathbb{Z}^n$ for $j=1,\dotsc,\ell$ span a sub $\mathbb{Z}$-module of odd index
in the module of all linear relations, then the number of nondegenerate real solutions
to~$\eqref{Eq:structured_system}$ is equal to the number of nondegenerate real solutions
to~$\eqref{Eq:str_Gale}$ in $(\mathbb{R}^\times)^\ell$ where no $h_i(y)$ vanishes.
\end{theorem}
This follows from Theorem~\ref{Th:Str_Gale} on Gale duality for $(d,\ell)$-dense fewnomials.
Thus we may prove Theorem~\ref{Th:Structured_bound} by establishing bounds for
$(d,\ell)$-dense Gale systems.\smallskip
Let us consider an example of this duality.
The system of Laurent polynomials,
\begin{equation}\label{Eq:Laurent_system}
\begin{array}{rcl}
\Color{1 0 1 0.4}{f}\ :=\
27t^{-5}\ +\ 31 - 16t^2u - 16t^2u^{-1} -16t^4u^2+ 40t^4-16t^4u^{-2}&=& 0\,,\\
\Color{0 1 1 0.3}{g}\ :=\
12t\hspace{15.5pt}+\ 40 -32t^2u -32t^2u^{-1}
+\ \,5t^4u^2+\ \,6t^4+\ \,5t^4u^{-2}&=& 0\,,\rule{0pt}{14pt}
\end{array}
\end{equation}
has 36 complex solutions with nonzero coordinates,
ten of which are real and eight of which lie in the positive quadrant.
We show the curves $\Color{1 0 1 0.4}{f=0}$ and $\Color{0 1 1 0.3}{g=0}$ defined by the
polynomials~\eqref{Eq:Laurent_system}.
In the picture on the left, the horizontal scale has been exaggerated.
Its shaded region is shown on the right, where now the vertical scale is exaggerated.
\begin{equation}\label{Eq:Laurent_Curves}
%
\raisebox{-60pt}{
\begin{picture}(240,120)
\put(0,0){\includegraphics[height=120pt]{figures/poly_big.eps}}
\put(236,3){$t$} \put(121,110){$u$}
\put(68,88){$\Color{0 1 1 0.3}{g}$} \put(88,88){$\Color{0 1 1 0.3}{g}$}
\put(138,88){$\Color{0 1 1 0.3}{g}$} \put(180,44){$\Color{0 1 1 0.3}{g}$}
\put(1,49){$\Color{1 0 1 0.4}{f}$} \put(4,44){\Color{1 0 1 0.4}{\vector(0,-1){25}}}
\put(228,14){$\Color{1 0 1 0.4}{f}$}
\end{picture}
\qquad
\begin{picture}(141,120)
\put(133,6){$t$} \put(13,110){$u$}
\put(0,0){\includegraphics[height=120pt]{figures/poly_sm.eps}}
\put(60,31){$\Color{0 1 1 0.3}{g}$} \put(117,40){$\Color{0 1 1 0.3}{g}$}
\put(133,104){$\Color{1 0 1 0.4}{f}$}
\end{picture}}
\end{equation}
Here are numerical approximations to the ten real solutions.
\begin{eqnarray*}
&( 0.619, 0.093),\,
( 0.839, 0.326),\,
( 1.003, 0.543),\,
( 1.591, 0.911),\,
(-1.911, 0.864),\,&\\
&( 0.619, 10.71),\,
( 0.839, 3.101),\,
( 1.003, 1.843),\,
( 1.591, 1.097),\,
(-1.911, 1.158)\,.&
\end{eqnarray*}
(The repetition in the $t$-coordinates is explained by the symmetry $u\mapsto u^{-1}$
of~\eqref{Eq:Laurent_system}.)
The system~\eqref{Eq:Laurent_system} is a system of $(2,2)$-dense fewnomials, as we may see
from its support.
\[
\begin{picture}(206,108)
\put(0,0){\includegraphics{figures/Laurent_supp.eps}}
\put(197,65){$t$} \put(89,95){$u$}
\put(129 ,76){$v_2$} \put(131,28){$v_1$}
\put(124.5,46){$w_2$} \put( 14,45){$w_1$}
\end{picture}
\]
If we solve~\eqref{Eq:Laurent_system} for the monomials $t^{-5}$ and $t$, we obtain
\begin{equation}\label{Eq:solved_system}
\begin{array}{rcl}
t^{-5} &=&\frac{1}{27}(-\ 31 + 16t^2u + 16t^2u^{-1}+16t^4u^2- 40t^4+16t^4u^{-2})
\ =:\ h_1(t^2u,t^2u^{-1})\\
t\hspace{11pt}&=&\frac{1}{12}(-\ 40 +32t^2u + 32t^2u^{-1}
-\ \,5t^4u^2-\ \,6t^4-\ \hspace{1.5pt}5t^4u^{-2})
\ =:\ h_2(t^2u,t^2u^{-1}) \,.\rule{0pt}{14pt}
\end{array}
\end{equation}
We convert this into a $(d,\ell)$-dense Gale system dual to~\eqref{Eq:Laurent_system}.
First observe that
\begin{equation}\label{Eq:monoms}
\begin{array}{rcl}
\left(t^2u\right)^{\Red{1}} \left(t^2u^{-1}\right)^{\Red{1}}
\left(t^{-5}\right)^{\Red{1}} \left(t\right)^{\Red{1}} \hspace{6.5pt}
&=&1\\
\left(t^2u\right)^{\Red{2}} \left(t^2u^{-1}\right)^{\Red{2}}
\left(t^{-5}\right)^{\Red{1}} \left(t\right)^{\Red{-3}}
&=&1\,.\rule{0pt}{14pt}
\end{array}
\end{equation}
We use the equations~\eqref{Eq:solved_system} to replace the monomials
$t^{-5}$ and $t$ in~\eqref{Eq:monoms} and then apply the substitutions
$x:=t^2u$ and $y:=t^2u^{-1}$ (so that $x^2=t^4u^2$, $xy=t^4$, and $y^2=t^4u^{-2}$).
Then, after clearing denominators and
rearranging, we have
\begin{equation}\label{Eq:Gale}
\begin{array}{rcl}
x^{\Red{1}} y^{\Red{1}} h_1(x,y)^{\Red{1}} h_2(x,y)^{\Red{1}}
\ -\ 1
&=&0 \\
x^{\Red{2}} y^{\Red{2}} h_1(x,y)^{\Red{1}}
\ -\ h_2(x,y)^{\Red{3}}
&=&0 \,.\rule{0pt}{15pt}
\end{array}
\end{equation}
This system has 36 complex solutions, ten of which are real and eight of which lie in the
shaded region in the picture below where $x,y,h_1(x,y),$ and $h_2(x,y)$ are all positive.
\begin{equation}\label{Eq:Gale_Curves}
\raisebox{-90pt}{
\begin{picture}(275,190)(-70,0)
\put(0,0){\includegraphics{figures/Gale.eps}}
\put(-62.5,65){$h_2(x,y)=0$}
\put(0,68){\vector(1,0){45}}
\put(-62.5,47.6){$h_1(x,y)=0$} \put(141,150){$h_1(x,y)=0$}
\put(155,145){\vector(-1,-1){44}}
\put(0,50.5){\vector(4,-1){67}}
\put(71,3){$x=0$} \put(-30,26){$y=0$}
\put(69,6){\vector(-1,0){29}}
\end{picture}
}
\end{equation}
The numbers of complex, real, and suitably positive solutions to the
two systems~\eqref{Eq:Laurent_system} and~\eqref{Eq:Gale}
is a consequence of Theorem~\ref{Th:Str_Gale} on
structured Gale duality.
Here are numerical approximations to the ten real solutions of~\eqref{Eq:Gale}
\begin{eqnarray*}
&(4.229, 3.154),\,
(4.098, 0.036),\,
(2.777, 2.306),\,
(2.184, 0.227),\,
(1.853, 0.546),\,&
\\
&(3.154, 4.229),\,
(0.036, 4.098),\,
(2.306, 2.777),\,
(0.227, 2.184),\,
(0.546, 1.853)\,.&
\end{eqnarray*}
We remark that there is no relation between the two pairs of curves
in~\eqref{Eq:Laurent_Curves} and~\eqref{Eq:Gale_Curves}.
Gale duality only
asserts a scheme-theoretic equality between the points of intersection of each pair of curves.
\section{Gale Duality for $(d,\ell)$-dense fewnomials}\label{S:gale}
Gale duality~\cite{BS_Gale} asserts that a system of $n$
polynomials in $n$ variables involving a total of $1{+}k{+}n$ distinct monomials is
equivalent to a system of $k$ rational functions of a particular form in the complement of
an arrangement of $k{+}n$ hyperplanes in $\mathbb{R}^k$.
A modification of Gale duality asserts that a system of $(d,\ell)$-dense fewnomials is
equivalent to a system $\ell\; (\leq k)$ rational functions in the complement ${\mathcal{M}}(\mathbb{R})$ of the
coordinate axes of $\mathbb{R}^\ell$ and of $n$ degree $d$ hypersurfaces.
We will call such a system a $(d,\ell)$-dense Gale system.
Write $\dcol{\mathbb{T}}$ for the non-zero complex numbers, $\mathbb{C}^\times$.
\subsection{$(d,\ell)$-dense polynomials}
\label{S:nearpolyhedral}
Suppose that ${\mathcal{A}}=\psi(d\includegraphics{figures/Simplex.eps}^\ell\cap\mathbb{Z}^\ell)\cup{\mathcal{W}}$ is $(d,\ell)$-dense and
that it affinely spans $\mathbb{Z}^n$.
Translating ${\mathcal{A}}$ by $\psi(0)$ if necessary, we may assume that $\psi$ is linear.
Write ${\mathcal{V}}=\{v_1,\dotsc,v_\ell\}\subset\mathbb{Z}^n$ for the images under $\psi$ of the standard
basis vectors of $\mathbb{Z}^\ell$ and list the elements of ${\mathcal{W}}$ as $\{w_1,\dotsc,w_n\}$.
Consider the map
\[
\begin{array}{rclcl}
\varphi&\colon&\mathbb{T}^n&\longrightarrow&\mathbb{T}^\ell\times\mathbb{T}^n\\
&&x&\longmapsto&(x^{v_1},\dotsc,x^{v_\ell},\,x^{w_1},\dotsc,x^{w_n})\,.
\rule{0pt}{14pt}
\end{array}
\]
Write $y=(y_1,\dotsc,y_\ell)$ for the coordinates of the first factor $\mathbb{T}^\ell$ and
$z=(z_1,\dotsc,z_n)$ for the coordinates of the second factor $\mathbb{T}^n$.
A polynomial with support ${\mathcal{A}}$ has the form
\begin{eqnarray}
f&=& \sum_{\substack{\lambda=(\lambda_1,\dotsc,\lambda_\ell)\\|\lambda|=d}}
a_\lambda x^{\lambda_1 v_1}\dotsb x^{\lambda_\ell v_\ell}\ +\
\sum_{i=1}^n a_ix^{w_i}\nonumber\\
&=& \varphi^*\Bigl(\sum_{|\lambda|=d} a_\lambda y_1^{\lambda_1}\dotsb y_\ell^{\lambda_\ell}\ +\
\sum_{i=1}^n a_i z_i\Bigr)
\ =\ \varphi^*(h(y) + \Lambda(z))\,,\label{Eq:pullback}
\end{eqnarray}
the pullback along $\varphi$ of a polynomial $h$ of degree $d$ and a linear form $\Lambda$.
\subsection{Proof of Theorem~\ref{Th:GaleDual}}
Let ${\mathcal{A}}$ and ${\mathcal{W}}$ be as in the previous subsection.
Suppose that $f_1(x)=\dotsb=f_n(x)=0$ is a system of $(d,\ell)$-dense fewnomials with
support ${\mathcal{A}}$.
By~\eqref{Eq:pullback}, there exist polynomials $h_1(y),\dotsc,h_n(y)$ of degree $d$ in
the variables $y=(y_1,\dotsc,y_\ell)$ and linear forms $\Lambda_1(z),\dotsc,\Lambda_n(z)$
in variables $z=(z_1,\dotsc,z_n)$ such that
\[
f_i(x)\ =\ \varphi^*( h_i(y) + \Lambda_i(z))
\qquad i=1,\dotsc,n\,.
\]
Since we wish to enumerate non-degenerate solutions, we may assume that the polynomials
$h_i(y),\Lambda_i(z)$ are generic, for perturbing the coefficients of the $f_i$ can only
increase their number of non-degenerate solutions.
Thus we may assume that $\Lambda_1(z),\dotsc,\Lambda_n(z)$ are linearly independent.
Replacing the polynomials $f_1,\dotsc,f_n$ by appropriate linear combinations, we may
assume that $\Lambda_i(z)=-z_i$ for each $i$.
Then our system becomes
\begin{equation}\label{Eq:graph_pullback}
\varphi^*(h_1(y)-z_1)\ =\
\varphi^*(h_2(y)-z_2)\ =\ \dotsb\ =\
\varphi^*(h_n(y)-z_n)\ =\ 0\,.
\end{equation}
If we define $H\subset\mathbb{C}^\ell\times\mathbb{C}^n$ by the equations
\[
H\ =\ \{(y,z)\in\mathbb{C}^\ell\times\mathbb{C}^n\mid
z_1=h_1(y),\dotsc,z_n=h_n(y)\}\,,
\]
then our system~\eqref{Eq:graph_pullback} has the alternative geometric description
as $\varphi^*(H)$.
Since $\mathbb{Z}{\mathcal{A}}=\mathbb{Z}^n$, $\varphi$ is an isomorphism onto its image, and we deduce the
following lemma.
\begin{lemma}
The system~$\eqref{Eq:graph_pullback}$ is isomorphic to the intersection
$\varphi(\mathbb{T}^n)\cap H$ in $\mathbb{C}^\ell\times\mathbb{C}^n$.
\end{lemma}
This is the first step in Gale duality for $(d,\ell)$-dense fewnomials.
For the second step observe that $H$ is isomorphic to $\mathbb{C}^\ell$, as it
is the graph of the function $\mathbb{C}^\ell\to\mathbb{C}^n$ given by $y\mapsto(h_1(y),\dotsc,h_n(y))$.
Let $\Psi\colon\mathbb{C}^\ell\to H$ be the isomorphism between $\mathbb{C}^\ell$ and this graph.
Then the system~\eqref{Eq:graph_pullback} is equivalent to
\[
\varphi(\mathbb{T}^n)\cap \Psi(\mathbb{C}^\ell)\,.
\]
We determine the equations in $\mathbb{T}^\ell\times\mathbb{T}^n$ that define $\varphi(\mathbb{T}^n)$.
For $\beta=(b_1,\dotsc,b_\ell)\in\mathbb{Z}^\ell$ and $\gamma=(c_1,\dotsc,c_n)\in\mathbb{Z}^n$, let
\[
y^\beta\cdot z^\gamma\ =\ y_1^{\beta_1}\dotsb y_\ell^{\beta_\ell}
\cdot z_1^{c_1}\dotsb z_n^{c_n}\,.
\]
We similarly write $h(y)^\gamma$ for $h_1(y)^{c_1}\dotsb h_n(y)^{c_n}$.
Suppose that ${\mathcal{B}}\subset\mathbb{Z}^{\ell}\oplus\mathbb{Z}^n$
is a basis for the $\mathbb{Z}$-linear relations among the exponent vectors ${\mathcal{V}}\cup{\mathcal{W}}$.
As ${\mathcal{A}}$ spans $\mathbb{R}^n$, so does ${\mathcal{V}}\cup{\mathcal{W}}$, and so ${\mathcal{B}}$ consists of
$\ell$ vectors, $\{(\beta_1,\gamma_1),\dotsc,(\beta_\ell,\gamma_\ell)\}$.
Then the image
$\varphi(\mathbb{T}^{n})\subset \mathbb{T}^\ell\times \mathbb{T}^n$ is
the subtorus defined by
\begin{equation}\label{Eq:sGale}
y^{\beta_j}\cdot z^{\gamma_j}\ =\ 1 \qquad\mbox{for } j=1,\dotsc,\ell\,.
\end{equation}
\begin{proposition}\label{prop:npGale}
The pullback of $\varphi(\mathbb{T}^{n})\cap H$ along the map
$\Psi$ is the system
\begin{equation}\label{Eq:npmf}
y^{\beta_j} \cdot h(y)^{\gamma_j}\ =\ 1
\qquad\mbox{for } j=1,\dotsc,\ell\,.
\end{equation}
This is well-defined in $\mathbb{C}^{\ell}$ in the complement ${\mathcal{M}}(\mathbb{C})$ of the
coordinate planes and the hypersurfaces $h_i(y)=0$ for $i=1,\dotsc,n$.
\end{proposition}
We may now state our main theorem on structured Gale duality.
The \demph{saturation} of a submodule ${\mathcal{B}}\subset\mathbb{Z}^{\ell}\oplus\mathbb{Z}^n$ is the set
$({\mathcal{B}}\otimes_\mathbb{Z}\mathbb{R}) \cap \mathbb{Z}^{\ell}\oplus\mathbb{Z}^n$ of integer points in its linear span.
\begin{theorem}\label{Th:Str_Gale}
Suppose that ${\mathcal{A}},{\mathcal{W}},\psi$, and ${\mathcal{V}}$ are as above and that ${\mathcal{A}}$
spans $\mathbb{Z}^{n}$.
Then the solution set to~\eqref{Eq:graph_pullback} in $\mathbb{T}^{n}$
is scheme-theoretically isomorphic to the solution set of the system of
rational functions~\eqref{Eq:npmf} defined in $\mathbb{T}^\ell$ in the
complement of the hypersurfaces $h_i(y)=0$, for $i=1,\dotsc,n$.
If the coefficients of the polynomials $f_i$ are real, then so are those of $h_i(y)$.
If the span of ${\mathcal{A}}$ has odd index in $\mathbb{Z}^{n}$ and the integer span of the
exponents ${\mathcal{B}}$ has odd index in its saturation, then the analytic subscheme
defined in $(\mathbb{R}^\times)^{n}$ by~\eqref{Eq:graph_pullback} is isomorphic to the analytic
subscheme defined by~\eqref{Eq:npmf} in the complement of the
hypersurfaces $h_i(y)=0$ in $(\mathbb{R}^\times)^\ell$.
If now the exponents ${\mathcal{A}}$ only span a full rank sublattice of
$\mathbb{Z}^{n}$ and the exponents ${\mathcal{B}}$ only span a full rank sublattice of
the module of linear relations among ${\mathcal{V}}\cup{\mathcal{W}}$, then the analytic subscheme of\/
$\mathbb{R}_{>0}^{n}$ defined by~\eqref{Eq:graph_pullback} is isomorphic to the analytic subscheme
defined by~\eqref{Eq:npmf} in the subset of\/ $\mathbb{R}_{>0}^\ell$ defined by $h_i(y)>0$ for
$i=1,\dotsc,n$.
\end{theorem}
\begin{proof}
The first statement concerning complex solutions is immediate from
Proposition~\ref{prop:npGale} and the observation that the system~\eqref{Eq:graph_pullback} is
the pullback of the intersection
$\varphi(\mathbb{T}^n)\cap H$ along the
map $\varphi$, if we know that the map $\varphi$ is injective.
Since ${\mathcal{A}}$ spans $\mathbb{Z}^{n}$, so does ${\mathcal{V}}\cup{\mathcal{W}}$, and the map
$\varphi$ is injective.
As the affine span of ${\mathcal{A}}$ has odd index in $\mathbb{Z}^{n}$, the map
$\varphi$ is injective on $(\mathbb{R}^\times)^{n}$.
As $\mathbb{Z}{\mathcal{B}}$ has odd index in its saturation, the equations
$y^\beta\cdot z^\gamma=1$ for $(\beta,\gamma)\in{\mathcal{B}}$ define the image
$\varphi((\mathbb{R}^\times)^{n})$ in the real torus
$(\mathbb{R}^\times)^\ell\times(\mathbb{R}^\times)^n$.
These facts in turn imply the second statement.
Similarly, the hypotheses of the third statement imply the same facts about the
positive part of the real torus, $\mathbb{R}_{>0}^{n}$.
Observing that the subset of $\mathbb{R}_{>0}^\ell$ defined by $h_i(y)>0$ for $i=1,\dotsc,n$ is the
pullback of $\mathbb{R}_{>0}^\ell\times\mathbb{R}_{>0}^n$ under the map $\Psi$ completes the proof.
\end{proof}
\section{Bounds for $(d,\ell)$-dense Gale systems}\label{S:bounds}
By Theorem~\ref{Th:GaleDual}, Theorem~\ref{Th:Structured_bound} follows from
bounds for $(d,\ell)$-dense Gale systems, which we give and prove below.
Let $d,\ell,n$ be positive integers and $y_1,\dots,y_\ell$ be indeterminates.
Suppose that $h_i(y)$ for $i=1,\dots,n$ are generic degree $d$ polynomials.
Define
\begin{eqnarray*}
\dcol{\Delta}&:=&\{y\in\mathbb{R}^\ell_{>0}\mid h_i(y)>0\ \mbox{for}\ i=1,\dots,n\}\,\qquad\mbox{and}\\
\dcol{{\mathcal{M}}(\mathbb{R})}&:=&\{y\in(\mathbb{R}^\times)^\ell\mid h_i(y)\neq0\ \mbox{for}\ i=1,\dots,n\}\,,
\end{eqnarray*}
and write ${\mathcal{M}}(\mathbb{C})$ for the complexification of ${\mathcal{M}}(\mathbb{R})$.
\begin{theorem}\label{TH:main}
With these definitions, suppose that
$\beta_1,\dotsc,\beta_\ell\in\mathbb{Z}^\ell$ and $\gamma_1,\dotsc,\gamma_\ell\in\mathbb{Z}^n$
are vectors such that
${\mathcal{B}}:=\{(\beta_1,\gamma_1),\dotsc,(\beta_\ell,\gamma_\ell)\}\subset\mathbb{Z}^\ell\oplus\mathbb{Z}^n$
are linearly independent.
Then the number of solutions to
\begin{equation}\label{Eq:Gale_Sy}
y^{\beta_j}\cdot h(y)^{\gamma_{j}}\ =\ 1\,,
\qquad\mbox{for}\quad j\ =\ 1,\dotsc,\ell\,,
\end{equation}
in the positive region $\Delta$ is less than
\[
\frac{e^2{+}3}{4}\, 2^{\binom{\ell}{2}}\, n^\ell \cdot d^\ell\,.
\]
If the integer span of ${\mathcal{B}}$ has odd index in its saturation,
then the number of solutions in ${\mathcal{M}}(\mathbb{R})$ is less
than
\[
\frac{e^4{+}3}{4}\, 2^{\binom{\ell}{2}}\, n^\ell \cdot d^\ell\,.
\]
\end{theorem}
We will deduce these bounds from several lemmata which we now formulate.
Their proofs are given in subsequent subsections.
For a vector $\alpha$, let $\alpha^\pm$ be the coordinatewise maximum of $\pm\alpha$ and $0$ so
that $\alpha^\pm$ is nonnegative, and $\alpha=\alpha^+-\alpha^-$.
Hence $(1,-2)^+=(1,0)$ and $(1,-2)^-=(0,2)$.
Set
\begin{equation}\label{Eq:form_g}
\dcol{g_k(y)}\ :=\ y^{2\beta_k^+}h(y)^{2\gamma_k^+}-y^{2\beta_k^-}h(y)^{2\gamma_k^-}\,.
\end{equation}
Then $g_k(y)=0$ for $y\in{\mathcal{M}}(\mathbb{C})$ if and only if
\begin{equation}\label{Eq:form_prod}
\prod_{i=1}^\ell (y_i^2)^{\beta_{k,i}}\cdot
\prod_{i=1}^n (h_i(y)^2)^{\gamma_{k,i}}\ =\ 1
\end{equation}
Notice that the system
\[
g_1(y)\ =\ g_2(y)\ =\ \dotsb\ =\ g_\ell(y)\ =\ 0\,,
\]
is equivalent in $\Delta$ to the system~\eqref{Eq:Gale_Sy} and in ${\mathcal{M}}(\mathbb{R})$, it
contains the system~\eqref{Eq:Gale_Sy} as a subsystem.
We will bound the number of solutions to this expanded system in $\Delta$ and in ${\mathcal{M}}(\mathbb{R})$ to obtain
our bounds for the system~\eqref{Eq:Gale_Sy} in Theorem~\ref{TH:main}.
We state two important reductions.
\begin{reduction}\label{R:reduction}
It suffices to prove Theorem~$\ref{TH:main}$ under the following additional assumptions.
\begin{enumerate}
\item
For each $j=1,\dotsc,\ell$, the set
$\dcol{\mu_j}\subset{\mathcal{M}}(\mathbb{C})$ defined by the equations
\[
\dcol{y^{2\beta_k}\cdot h(y)^{2\gamma_k}}\ :=\
\prod_{i=1}^\ell (y_i^2)^{\beta_{k,i}}\cdot
\prod_{i=1}^n (h_i(y)^2)^{\gamma_{k,i}}\ =\ 1\qquad
k=1,\dotsc,j\,,
\]
is smooth and has codimension $j$.
This condition holds for all sufficiently generic polynomials $h_i(y)$ of degree $d$.
\item
For each $k=1,\dotsc,\ell$ define
$b_k:= \beta_{k,1}+\dotsb+\beta_{k,\ell} + d(\gamma_{k,1}+\dotsb+\gamma_{k,n})$.
Then every minor of the $\ell\times(1+\ell+n)$ matrix whose $k$th row is
$(-b_k,\beta_k,\gamma_k)$ is nonzero.
\end{enumerate}
\end{reduction}
We establish these reductions in Subsection~\ref{S:reduction}.
Our bounds are based on an induction which comes from
the Khovanskii-Rolle Theorem, or more precisely, the induction is based on a modified form
which was used in~\cite{BS_mixed},
and which ensures that the hypotheses in subsequent lemmata hold.
See~\cite{BS_Khovanskii} and~\cite{BS_mixed} for more discussion.
For $D$ equal to either ${\mathcal{M}}(\mathbb{R})$ or its positive chamber $\Delta$ and $C$ an algebraic curve in
$D$, let \dcol{$\operatorname{ubc}_D(C)$} be the number of noncompact components of $C$.
Write \dcol{$V_D(f_1,\dotsc,f_\ell)$} for the common zeroes in $D$ of functions
$f_1,\dotsc,f_\ell$.
\begin{lemma}[Modified Khovanskii-Rolle Theorem]\label{L:KhRo}
There exist polynomials $G_1,G_2,\dotsc,G_\ell$ where $G_{\ell-j}(y)$ is a generic
polynomial with degree $2^jn\cdot d$ such that the following hold.
\begin{enumerate}
\item The system
\[
g_1(y)\ =\ \dotsb\ =\ g_j(y)\ =\
G_{j+1}(y)\ =\ \dotsb\ =\ G_\ell(y)\ =\ 0\,,
\]
has only nondegenerate solutions in ${\mathcal{M}}(\mathbb{C})$ and the system
%
\begin{equation}\label{Eq:C_j}
g_1(y)\ =\ \dotsb\ =\ g_{j-1}(y)\ =\
G_{j+1}(y)\ =\ \dotsb\ =\ G_\ell(y)\ =\ 0\,,
\end{equation}
($g_j$ is omitted) defines a smooth curve in ${\mathcal{M}}(\mathbb{C})$.
\item Let $C_j\subset{\mathcal{M}}(\mathbb{R})$ be the solutions to~\eqref{Eq:C_j} in ${\mathcal{M}}(\mathbb{R})$, which is a smooth
real algebraic curve.
For $D$ be equal to either of ${\mathcal{M}}(\mathbb{R})$ or $\Delta$, we have
\[
|V_D(g_1,\dotsc,g_j,G_{j+1},\dotsc,G_\ell)|\
\leq\ |V_D(g_1,\dotsc,g_{j-1},G_j,\dotsc,G_\ell)|+ \operatorname{ubc}_D(C_j)\,.
\]
\end{enumerate}
\end{lemma}
This implies the following estimate.
Let $D$ equal to either of ${\mathcal{M}}(\mathbb{R})$ or $\Delta$.
Then we have
\begin{equation}\label{Eq:Fundamental_Estimate}
|V_D(g_1,\dotsc,g_\ell)|\ \leq\ |V_D(G_1,\dotsc,G_\ell)| +
\operatorname{ubc}_D(C_1)+\dotsb+\operatorname{ubc}_D(C_\ell)\,.
\end{equation}
Our next lemma estimates these quantities.
\begin{lemma}\label{L:estimates}
We have
\begin{enumerate}
\item $V_\Delta(G_1,\dotsc,G_\ell)\leq V_{\mathcal{M}}(G_1,\dotsc,G_\ell)\leq
2^{\binom{\ell}{2}}n^\ell\cdot d^\ell$,
\item \rule{0pt}{20pt}${\displaystyle\operatorname{ubc}_\Delta(C_{j})\leq
\tfrac{1}{2} 2^{\binom{\ell-j}{2}} n^{\ell-j}
\tbinom{1+\ell+n}{j}}\cdot d^{\ell}$.
\item \rule{0pt}{20pt}${\displaystyle\operatorname{ubc}_{\mathcal{M}}(C_{j})\leq
\tfrac{1}{2} 2^{\binom{\ell-j}{2}} 2^j n^{\ell-j}
\tbinom{1+\ell+n}{j}}\cdot d^{\ell}$.
\item \rule{0pt}{20pt}$ 2^{\binom{\ell-j}{2}} n^{\ell-j}\tbinom{1+\ell+n}{j}\leq
\frac{1}{2} \frac{2^j}{j!} \cdot 2^{\binom{\ell}{2}}n^\ell$.
\end{enumerate}
\end{lemma}
Statement (1) follows from Lemma~\ref{L:KhRo} by B\'ezout's Theorem, as
$2^{\binom{\ell}{2}}n^\ell d^\ell$ is the product of the degrees of the polynomials
$G_1,\dotsc,G_n$.
Statement (4) is from the proof of Lemma 3.5 of~\cite{BS_Khovanskii}.
We prove the other statements of Lemma~\ref{L:estimates} in Subsection~\ref{S:estimates}.
Lemma~\ref{L:estimates} and the estimate~\eqref{Eq:Fundamental_Estimate} give us the
estimate
\[ |V_\Delta(g_1,\dotsc,g_\ell)|\ \leq\
2^{\binom{\ell}{2}}n^\ell\cdot d^\ell\ +\ \sum_{j=1}^\ell
\frac{1}{4} \frac{2^{j}}{j!}\cdot 2^{\binom{\ell}{2}}n^\ell\cdot d^\ell
\ \leq\ \Bigl(1\ +\ \frac{1}{4} \sum_{j=1}^\ell \frac{2^j}{j!}\Bigr)
\cdot 2^{\binom{\ell}{2}}n^\ell\cdot d^\ell \,.
\]
The sum is a partial sum of the power series for $e^2-1$,
and so we obtain
\[
|V_\Delta(g_1,\dotsc,g_\ell)|\ <\
\frac{e^2+3}{4}2^{\binom{\ell}{2}}n^\ell\cdot d^\ell\,.
\]
The estimation for $|V_{\mathcal{M}}(g_1,\dotsc,g_\ell)|$ is similar.
Using Lemma~\ref{L:estimates}(3) for $\operatorname{ubc}_{\mathcal{M}}(C_j)$, the corresponding sum is now a
partial sum for $e^4-1$, and so we obtain
\[
|V_{\mathcal{M}}(g_1,\dotsc,g_\ell)|\ <\
\frac{e^4+3}{4}2^{\binom{\ell}{2}}n^\ell\cdot d^\ell\,,
\]
which completes the proof of Theorem~\ref{TH:main}.
\subsection{Proof of reductions}\label{S:reduction}
The Reduction~\ref{R:reduction}(1) will follow from Bertini's Theorem that a general linear
section of a smooth quasi-projective variety is smooth and of the expected dimension.
First, define $\mathbb{G}_j\subset\mathbb{T}^\ell\times\mathbb{T}^n$ to be the subtorus defined by the equations
\[
y^{2\beta_k}z^{2\gamma_k}\ =\ 1\qquad\mbox{for}\quad k=1,\dots,j\,.
\]
As in Section~\ref{S:gale}, let $\Psi\colon\mathbb{C}^\ell\to\mathbb{C}^\ell\times\mathbb{C}^n$ be the map
defined by
\[
\Psi\ \colon\ y\ \longmapsto\ (y,\,h_1(y),\dotsc,h_n(y))\,.
\]
Then $\mu_j=\Psi^{-1}(\mathbb{G}_j\cap \Psi(\mathbb{C}^\ell))$.
Since $\Psi$ is an isomorphism onto its image and $\mathbb{G}_j$ has codimension $j$, it suffices to
show that $\mathbb{G}_j\cap\Psi(\mathbb{C}^\ell)$ is transverse.
But this follows because $\Psi(\mathbb{C}^\ell)$ is the pullback of a linear subspace $L$ along
the map
\begin{eqnarray*}
\mathbb{C}^\ell\times\mathbb{C}^n &\longrightarrow& \mathbb{C}^{\binom{\ell+d}{d}}\times\mathbb{C}^n\\
(y,z)&\longmapsto&((y^\lambda\colon |\lambda|=d),\, z)\,.
\end{eqnarray*}
The linear space $L$ is defined by the coefficients of the polynomials
in a system Gale dual to the system~\eqref{Eq:Gale_Sy}.
Choosing $L$ to be generic, we may apply Bertini's Theorem and deduce that $\mu_j$ is smooth and
of codimension $j$.
We also see that this may be accomplished by choosing the polynomials $h_i(y)$
to be sufficiently generic.
For the second reduction, observe that our equations~\eqref{Eq:form_g} and~\eqref{Eq:form_prod}
are equivalent to
\begin{equation}\label{eq:f_k}
f_k(y)\ :=\
\sum_{m=1}^\ell \beta_{k,m}\log|y_m|\ +\
\sum_{i=1}^n \gamma_{k,i}\log|h_i(y)|\ =\ 0\,,\qquad
k=1,\dotsc,\ell\,,
\end{equation}
in ${\mathcal{M}}(\mathbb{R})$.
We may perturb them by changing the coefficients $\beta_{k,m}$ and $\gamma_{k,i}$ without
increasing their numbers of nondegenerate solutions.
Thus we can satisfy Reduction~\ref{R:reduction}(2) with real exponents.
Since the rational numbers are dense in the real numbers, we may satisfy
Reduction~\ref{R:reduction}(2) with rational exponents.
Finally, by clearing denominators, we may assume the exponents are integral.
\subsection{Proof of Lemma \ref{L:KhRo}}\label{S:LhRo}
We will establish Lemma~\ref{L:KhRo} by downward induction on $j$.
The main step is provided by the Khovanskii-Rolle Theorem, which we present in the
simplified form in which we need it.
\begin{theorem}[Khovanskii-Rolle]
Let $f_1,\dots,f_\ell$ be smooth functions defined on a domain $D\subset\mathbb{R}^\ell$
where
\[
f_1(y)\ =\ f_2(y)\ =\ \dotsb\ =\ f_{\ell-1}(y)\ =\ 0\,,
\]
defines a smooth curve $C$ in $D$.
Let
\[
J\ :=\ J(f_1,\dots,f_{\ell})\ :=\
\det\left( \frac{\partial f_i}{\partial y_j} \right)_{i,j=1,\dots,\ell}
\]
%
be the Jacobian determinants of $f_1,\dotsc,f_\ell$.
If $V_D(f_1,\dotsc,f_{\ell-1},J)$ is finite and if $C$ has finitely many components in $D$,
then $V_D(f_1,\dotsc,f_\ell)$ is finite and we have
\begin{equation}\label{Eq:KhRoStep}
|V_D(f_1,\dots,f_{\ell})|\ \leq\ |V_D(f_1,\dots,f_{\ell-1},J)|+ \operatorname{ubc}_D(C)\,.
\end{equation}
\end{theorem}
To see this, note that on $C$, the Jacobian $J$ is proportional to the evaluation of the
differential of $f_\ell$ on a tangent vector to $C$.
Given two consecutive solutions $a,b$ to $f_\ell=0$ along an arc of $C$,
The Jacobian will have different signs at $a$ and at $b$, and therefore will vanish at least
once between $a$ and $b$.
\[
\begin{picture}(282,130)(-40,0)
\put(0,0){\includegraphics[height=120pt]{figures/KhovRolle.eps}}
\put(-40,20){$C=V_D(f_1,\dotsc,f_{\ell-1})$}
\put(-20,65){$f_\ell=0$}
\put(148,120){$V_D(f_1,\dotsc,f_{\ell-1},J)$}
\put(146,122){\vector(-3,-1){30}}
\put(19.2,110.3){$a$} \put(26.2,109.3){\vector(1,-1){30}}
\put(200,85.5){$b$} \put(198.2,83.7){\vector(-1,-1){30}}
\end{picture}
\]
The estimate~\eqref{Eq:KhRoStep} follows as
compact components of $C$ contain as many arcs connecting zeroes of $f_\ell$ as zeroes of
$f_\ell$, while noncompact components contain one arc fewer.
To deduce Lemma~\ref{L:KhRo}, we will iterate the Khovanskii-Rolle Theorem, showing
that the appropriate Jacobians have the claimed degrees and ensuring that its
hypotheses are satisfied.
Observe that Lemma~\ref{L:KhRo}(1) with $j=\ell$ holds by the assumptions we make in
Reduction~\ref{R:reduction}.
We prove Lemma~\ref{L:KhRo} by downward induction on $j=\ell,\dotsc,1$.
Specifically, we will assume that Statement (1) holds of some $j$ and then
construct a polynomial $G_j$ such that (2) holds for $j$ and (1) holds for $j{-}1$.
To construct the polynomials $G_j$, we replace the rational functions
$g_k(y)$ for $k=1,\dotsc,\ell$ in Lemma~\ref{L:KhRo}(2) by the logarithmic functions
$f_k(y)$ for $k=1,\dotsc,\ell$ defined in~\eqref{eq:f_k}.
We may do this, as
if $y\in{\mathcal{M}}(\mathbb{R})$, then $f_k(y)=0$ if and only if $g_k(y)=0$.
First, we need to determine the degrees of the Jacobians.
\begin{lemma}\label{L:Newton}
Let $1\leq j\leq \ell$ and suppose that $G_{j{+}1},\dotsc,G_\ell$ are polynomials, where $G_i$ has
degree $2^{\ell-i}n\cdot d$, but is otherwise general, for each $i=1,\dotsc,\ell$.
Then
\begin{equation}\label{Eq:expression}
\prod_{m=1}^\ell y_m\cdot\prod_{i=1}^n h_i(y)\cdot
J(f_1,\dotsc,f_j,\, G_{j+1},\dotsc,G_\ell)
\end{equation}
is a polynomial with degree $2^{\ell-j}n\cdot d$.
\end{lemma}
We use this to deduce Lemma~\ref{L:KhRo}.
Suppose that we have polynomials $G_{j+1},\dotsc,G_\ell$ where $G_i$ is a generic
polynomial with degree $2^{\ell-i}n\cdot d$, for each $i=j{+}1,\dotsc,\ell$,
and Lemma~\ref{L:KhRo}(1) holds for $j$.
Let $C_j\subset{\mathcal{M}}(\mathbb{R})$ be the smooth real algebraic curve defined by
\[
f_1(y)\ = \ \dotsb\ =\ f_{j-1}(y)\ \;=\;\
G_{j+1}(y)\ =\ \dotsb\ =\ G_\ell(y)\ =\ 0\,.
\]
Let $\overline{G}_j$ be the product of the Jacobian
$J(f_1,\dotsc,f_j,\, G_{j{+}1},\dotsc,G_\ell)$ with the polynomial
\[
\Upsilon(y)\ :=\ \prod_{m=1}^\ell y_m\cdot\prod_{i=1}^n h_i(y)\,.
\]
Then $\overline{G}_j$ is a polynomial with degree $2^{\ell-j}n\cdot d$, by
Lemma~\ref{L:Newton}.
Since $\Upsilon$ does not vanish in ${\mathcal{M}}(\mathbb{R})$, the polynomial $\overline{G}_\ell$
and the Jacobian define the same sets, and we have
\begin{multline}\label{Eq:KhRo_est}
\qquad|V_D(f_1,\dotsc,f_j,\,G_{j+1},\dotsc,G_\ell)|\ \leq\ \\
|V_D(f_1,\dotsc,f_{j-1},\,\overline{G}_j,\,G_{j+1},\dotsc,G_\ell)|\ +\
\operatorname{ubc}_D(C_j)\,,\qquad
\end{multline}
by the Khovanskii-Rolle Theorem.
Note however that we do not know if $\overline{G}_j$ is a generic polynomial
with degree $2^{\ell-j}n\cdot d$, and in particular, we do not know if the
hypotheses of Lemma~\ref{L:KhRo}(1) for $j{-}1$ hold.
These hypotheses may be achieved by perturbing $\overline{G}_j$ to a nearby generic
polynomial $G_j$ with degree $2^{\ell-j}n\cdot d$.
To ensure that this perturbation does not destroy the estimate~\eqref{Eq:KhRo_est},
we only need to guarantee that the signs of $G_j$ and $\overline{G}_j$ are the same at
every point of $V_D(f_1,\dotsc,f_j,\,G_{j+1},\dotsc,G_\ell)$, but this will hold for all
sufficiently small perturbations, as there are only finitely many such points
and $\overline{G}_j$ is nonzero at each.
These conditions are equivalent to every point of
$V_D(f_1,\dotsc,f_j,\,G_{j+1},\dotsc,G_\ell)$ being nondegenerate, which
is ensured by the genericity of $G_{j{+}1},\dotsc,G_\ell$.
This completes the proof of Lemma~\ref{L:KhRo}.
\begin{proof}[Proof of Lemma~$\ref{L:Newton}$]
Write $\partial_m$ for $\partial/\partial y_m$,
and consider the expression~\eqref{Eq:expression}, writing the Jacobian in block form,
with $j$ rows for $f_1,\dotsc,f_j$ and $\ell-j$ rows for $G_{j+1},\dotsc,G_\ell$.
\begin{multline}\label{Eq:block}
\qquad \prod_{m=1}^\ell y_m\cdot\prod_{i=1}^n h_i(y)\cdot
\det\left(\, \begin{matrix}
\bigl(\partial_m f_k(y)\bigr)_{k=1,\dotsc,j}^{m=1,\dotsc,\ell}
\smallskip\\\hline\rule{0pt}{15pt}
\bigl(\partial_m G_k(y)\bigr)_{k=j{+}1,\dotsc,\ell}^{m=1,\dotsc,\ell}
\end{matrix}\,\right)\\
\ =\
\prod_{i=1}^n h_i(y)\cdot
\det\left(\, \begin{matrix}
\bigl(y_m\partial_m f_k(y)\bigr)_{k=1,\dotsc,j}^{m=1,\dotsc,\ell}
\smallskip\\\hline\rule{0pt}{15pt}
\bigl(y_m\partial_m G_k(y)\bigr)_{k=j{+}1,\dotsc,\ell}^{m=1,\dotsc,\ell}
\end{matrix}\,\right)\,.\qquad
\end{multline}
Laplace expansion along the first $j$ rows of the matrix on the right expresses its
determinant as a sum of products of maximal minors of the two blocks.
We will prove Lemma~\ref{L:Newton} by showing that each term in that sum is a polynomial with
degree $2^{\ell-j}n\cdot d$.
First, the lower block
$\bigl(y_m\partial_m G_k(y)\bigr)_{k=j{+}1,\dotsc,\ell}^{m=1,\dotsc,\ell}$
is a matrix of polynomials whose entries in row $k$ are the toric derivatives
$y_m\partial_m G_k(y)$ of $G_k$.
Thus every entry in row $k$ has degree $\deg(G_k)=2^{\ell-k}n\cdot d$, and
therefore each minor has degree
\begin{equation}\label{Eq:lower_degree}
2^{\ell-(j+1)}n\cdot d \ +\ \dotsb\ +\
2n\cdot d \ +\ n\cdot d\ =\
(2^{\ell-j}-1)n\cdot d\,.
\end{equation}
For the upper block, note that $y_m\partial_m f_k(y)$ is
\[
\beta_{k,m}\ +\ \sum_{i=1}^k \gamma_{k,i}y_m\partial_m \log|h_i(y)|\ =\
\beta_{k,m}\ +\ \sum_{i=1}^n\gamma_{k,i} \frac{y_m \partial_m h_i(y)}{h_i(y)}\,.
\]
In particular, the upper block is a product of a $j\times(\ell+n)$ matrix and a
$(\ell+n)\times \ell$ matrix,
\[
\bigl(y_m\partial_m f_k(y)\bigr)_{k=1,\dotsc,j}^{m=1,\dotsc,\ell}\ =\
\bigl(\, \beta_{k,q}\ \mid\ \gamma_{k,i}\,\bigr)\cdot
\left(\begin{matrix} I_\ell\medskip\\
y_m \partial_m h_i(y)/h_i(y)\end{matrix}\right)\ ,
\]
where $I_\ell=(\delta_{q,m})$ is the $\ell\times\ell$ identity matrix.
By the Cauchy-Binet formula, a $j\times j$ minor of $\bigl(y_m\partial_m f_k(y)\bigr)$ is a
sum of products of $j\times j$ minors of the two matrices on the right.
Consider now the product of $\prod_{i=1}^nh_i(y)$ with a term in this sum.
The first matrix $\bigl(\beta_{k,q}\, \mid\, \gamma_{k,i}\bigr)$ contains constants, and
a $j\times j$ minor of the second involves no more than $p:=\min\{n,j\}$ rows from
its lower $n\times\ell$ block $(y_m\partial_mh_i(y)/h_i(y))$.
This minor is a sum of $j!$ terms, each one a product of a constant and $p$ entries of the
matrix of the form $y_m\partial_mh_i(y)/h_i(y)$, for different rows $i$.
Multiplying this term by $\prod_{i=1}^nh_i(y)$ will clear all
denominators and result in a product of $p$ terms of the form
$y_m\partial_mh_i(y)$ and complementary $n{-}p$ terms of the form $h_i(y)$.
Each of these terms has degree $d$, so each term coming from the
expansion of this $j\times j$ minor has degree $n\cdot d$, and therefore
the product of $\prod_{i=1}^nh_i(y)$ by each $j\times j$ minor of the upper block
of~\eqref{Eq:block} will have degree $n\cdot d$.
Together with~\eqref{Eq:lower_degree}, this completes the proof.
\end{proof}
\subsection{Proof of Lemma~\ref{L:estimates}}\label{S:estimates}
We only need to prove Statements (2) and (3) of Lemma~\ref{L:estimates}.
By Reduction~\ref{R:reduction} and Lemma~\ref{L:KhRo}, we may assume that the polynomials
$h_i(y)$ and $G_j(y)$ are generic given their degrees.
The complexification of the real curve $C_j$ is defined in $\mathbb{C}^\ell\supset{\mathcal{M}}(\mathbb{C})$ by
\begin{equation}\label{Eq:curvej}
g_1(y)\ =\ \dotsb\ =\ g_{j-1}(y)\ \ =\ \
G_{j+1}(y)\ =\ \dotsb\ =\ G_\ell(y)\ =\ 0\,,
\end{equation}
and it lies on the codimension $j{-}1$ subvariety $\mu_{j-1}\subset{\mathcal{M}}(\mathbb{C})$ defined by
\[
g_1(y)\ =\ \dotsb\ =\ g_{j-1}(y)\ =\ 0\,.
\]
We bound the number of unbounded components of $C_j$ by first describing the points where
$\mu_{j-1}(\mathbb{C})$ meets the boundary of ${\mathcal{M}}(\mathbb{C})$, then bound the number of real solutions to
\begin{equation}\label{Eq:bigG}
G_{j+1}(y)\ =\ \dotsb\ =\ G_\ell(y)\ =\ 0
\end{equation}
on these boundary points, and lastly by determining the number of components of $C_j$ incident
upon each such real solution.
To accomplish this, consider ${\mathcal{M}}(\mathbb{C})\subset\mathbb{C}^\ell$ as a subset of projective space
$\P^\ell$.
Its boundary $\partial{\mathcal{M}}(\mathbb{C}):=\P^\ell\setminus{\mathcal{M}}(\mathbb{C})$
consists of the finite coordinate planes $y_m=0$ for $m=1,\dotsc,\ell$, the coordinate
plane at inifinty $y_0=0$, and the degree $d$ hypersurfaces $h_i(y)=0$ for $i=1,\dotsc,n$.
Strictly speaking, we must homogenize polynomials $h_i,g_j,G_k$ with respect to the coordinate
$y_0$ at infinity.
When working on an affine patch where $y_m\neq 0$, we de-homogenize them by setting
$y_m=1$.
By our assumption that the polynomials $h_i$ were general, this boundary
$\partial{\mathcal{M}}(\mathbb{C})$ forms a divisor with normal crossings whose components are the coordinate
planes and the hypersurfaces $h_i(y)=0$.
The common zeroes of any $q$ of the polynomials $h_1,\dotsc,h_n$ and $j{-}q$ of the
coordinates $y_0,\dotsc,y_\ell$ is a smooth subvariety of codimension $j$, called a
\demph{codimension-$j$ stratum}.
The union of these $j$-fold intersections of the components of the boundary divisor
is called the \demph{codimension-$j$ skeleton} of $\partial{\mathcal{M}}(\mathbb{C})$.
\begin{lemma}
The closure $\overline{\mu_{j-1}}$ meets $\partial{\mathcal{M}}(\mathbb{C})$ in a union of
codimension-$j$ strata and in the neighborhood of a real point of\/
$\overline{\mu_{j-1}}$ lying in the relative interior of a codimension-$j$ stratum,
$\mu_{j-1}$ has one branch in each of the $2^j$ components of ${\mathcal{M}}(\mathbb{R})$
incident on that point.
\end{lemma}
\begin{proof}
Since $\mu_{j-1}\subset{\mathcal{M}}(\mathbb{C})$ has codimension $j{-}1$ in $\P^\ell$, the
intersection of its closure $\overline{\mu_{j-1}}$ with the
boundary divisor $\partial{\mathcal{M}}(\mathbb{C})$ will have codimension $j$ in $\P^\ell$.
We prove the first part of the lemma by showing that this intersection consists of points
lying within the codimension-$j$ skeleton of $\partial{\mathcal{M}}(\mathbb{C})$, and therefore is a union of
components of the codimension-$j$ skeleton.
Let $Y$ be a point of $\partial{\mathcal{M}}(\mathbb{C})$ that does not lie in the codimension-$j$ skeleton.
We show that $Y\not\in\overline{\mu_{j-1}}$.
Since $\partial{\mathcal{M}}(\mathbb{C})$ is a divisor with normal crossings whose components are defined by
the coordinates $y_i$ and forms $h_i$, at least one, but no more than
$j{-}1$ of the coordinates $y_0,y_1,\dotsc,y_\ell$ and forms
$h_1,\dotsc,h_n$ vanish at $Y$.
Reordering the coordinates and forms if necessary, we may assume that the forms which
vanish at $Y$ are among $h_1,\dotsc,h_q$ and the coordinates are among
$y_{q+1},\dotsc,y_{j-1}$.
Since the assertion about $Y$ is local, we may restrict to the affine patch $U$ where
none of the remaining coordinates or forms vanish.
The equations
\begin{equation}\label{Eq:mujeqs}
y^{2\beta_k} h(y)^{2\gamma_k}\ =\ 1\qquad k=1,\dotsc,j{-}1\,,
\end{equation}
define $\mu_{j-1}\subset{\mathcal{M}}(\mathbb{C})$.
By Reduction~\ref{R:reduction}(2) on the homogenized exponent vectors,
there is an integer linear combination of the first $j{-}1$
rows of the matrix $(-b_k,\beta_k,\gamma_k)$ so that the columns corresponding to
$h_1,\dotsc,h_q$ and $y_{q+1},\dotsc,y_{j-1}$ become diagonal.
These same linear combinations transform the equations~\eqref{Eq:mujeqs} into equations
of the form
\begin{equation}\label{Eq:skeletondiagonal}
\begin{array}{rcl}
h_i(y)^{a_i} &=& y^{\alpha_i}\cdot h(y)^{\delta_i}\qquad i=1,\dotsc,q\,,\\
y_i^{a_i} &=& y^{\alpha_i}\cdot h(y)^{\delta_i}\qquad
i=q{+}1,\dotsc,j{-}1\,,\rule{0pt}{15pt}
\end{array}
\end{equation}
where $a_i> 0$ and the components of $\alpha_i$ in positions $q{+}1,\dotsc,j{-}1$
vanish as do the components of $\delta_i$ in positions $1,\dotsc,q$.
That is, $h_1,\dotsc,h_q$ and $y_{q+1},\dotsc,y_{j-1}$ do not appear on the right of these
expressions.
Since the expressions in~\eqref{Eq:skeletondiagonal} are well-defined functions in $U$,
the regular functions
\[
\begin{array}{rl}
h_i(y)^{a_i}\ -\ y^{\alpha_i}\cdot h(y)^{\delta_i}\quad &\mbox{for}\quad i=1,\dotsc,q\,,
\quad\mbox{and}\\
y_i^{a_i}\ -\ y^{\alpha_i}\cdot h(y)^{\delta_i}\quad&
\mbox{for}\quad i=q{+}1,\dotsc,j{-}1\rule{0pt}{15pt}
\end{array}
\]
vanish on $\mu_{j-1}\cap U$, and hence on $\overline{\mu_{j-1}}\cap U$.
However, these cannot all vanish at $Y$, for none of the functions
$y^{\alpha_i}h^{\delta_i}$ for $i=1,\dotsc,j{-}1$ vanish at $Y$, but at least one of
$h_1^{a_1},\dotsc,h_q^{a_q}$, $y_{q+1}^{a_{q+1}},\dotsc,y_{j-1}^{a_{j-1}}$ vanishes at
$Y$.
To complete the proof, suppose that $Y\in\overline{\mu_{j-1}}(\mathbb{R})\cap\partial{\mathcal{M}}(\mathbb{R})$ is a
real point lying on a
codimension-$j$ stratum of $\partial{\mathcal{M}}(\mathbb{C})$ but not on a stratum of larger codimension.
Reordering the coordinates and functions if necessary and working locally, we may assume that
the polynomials $h_1(y),\dotsc,h_q(y)$ and coordinates $y_{q+1},\dotsc,y_{j}$ vanish at
$Y$.
Thus in the affine neighborhood $U$ of $Y$ where none of the other polynomials or coordinates
vanish, $\mu_{j-1}$ is defined by equations of the form
\begin{equation}\label{Eq:newdiagonal}
\begin{array}{rcl}
h_i(y)^{a_i} &=& y_j^{c_i}\cdot y^{\alpha_i}\cdot h(y)^{\delta_i}\qquad i=1,\dotsc,q\,,\\
y_i^{a_i} &=& y_j^{c_i}\cdot y^{\alpha_i}\cdot h(y)^{\delta_i}\qquad
i=q{+}1,\dotsc,j\,,\rule{0pt}{15pt}
\end{array}
\end{equation}
where $a_i>0$ and none of $h_1,\dotsc,h_q$ and $y_{q+1},\dotsc,y_j$ appear in the expressions
$y^{\alpha_i}\cdot h(y)^{\delta_i}$ for $i=1,\dotsc,j{-}1$.
In fact, we must have $c_i>0$ as $Y\in\overline{\mu_{j-1}}$.
In a neighborhood of $Y$ in $\mathbb{R}\P^\ell$, the complement ${\mathcal{M}}(\mathbb{R})$ has $2^j$ chambers given
by the signs of the functions $h_1(y),\dotsc,h_q(y)$, $y_{q+1},\dotsc,y_j$.
Since the exponents in~\eqref{Eq:newdiagonal} have every component even
(this comes from the evenness of the exponents in~\eqref{Eq:mujeqs}), there is a
component of $\mu_{j-1}$ in each of these chambers.
\end{proof}
We complete the proof of Statements (2) and (3) of Lemma~\ref{L:estimates}.
We estimate the number of unbounded components of the curve $C_j$ by first bounding the
number of points where its closure in $\mathbb{R}\P^\ell$ meets the boundary divisor, and then
bounding the number of components of $C_j$ incident upon each point.
The estimate for the number of points in $\overline{C_j}\cap\partial{\mathcal{M}}(\mathbb{R})$ is simply
the number of points in the codimension-$j$ skeleton where
\begin{equation}\label{eq:Gs}
G_{j+1}(y)\ =\ G_{j+2}(y)\ =\ \dotsb\ =\ G_\ell(y)\ =\ 0\,.
\end{equation}
Consider a stratum where $q$ of the polynomials $h_i$ vanish and $j{-}q$ of the coordinates
$y_m$ vanish.
Since the polynomials $G_i$ and $h_i$ are general given their degrees, the number of
points on this stratum will be at most the product of these degrees, which is
\[
d^q \cdot 2^{\ell-j-1} nd\cdot 2^{\ell-j-2} nd \dotsb 2 nd \cdot nd \ =\
2^{\binom{\ell-j}{2}} \cdot n^{\ell-j} d^{\ell-j} d^q\,.
\]
Since there are $\binom{\ell+1}{j-q}\cdot\binom{n}{q}$ such strata, the number of points
where $C_j$ meets the boundary is at most
\begin{equation}\label{eq:cjbdry}
2^{\binom{\ell-j}{2}} \cdot n^{\ell-j} d^{\ell-j}
\cdot\ \sum_{q=0}^j \binom{\ell+1}{j-q}\binom{n}{q}\cdot d^q\ <\
2^{\binom{\ell-j}{2}} \cdot n^{\ell-j} \binom{1+\ell+n}{j}
\cdot d^\ell\ .
\end{equation}
As the polynomials $G_i$ are general the variety defined by~\eqref{eq:Gs} is transverse to
the codimension-$j$ stratum, so there is at most one branch of $C_j$ in each component of
${\mathcal{M}}(\mathbb{R})$ meeting such a point.
Thus the number~\eqref{eq:cjbdry} bounds the number of ends of components
of $C_j$ in $\Delta$, so it bounds twice the number of unbounded components of $C_j$ in
$\Delta$.
For the bound in ${\mathcal{M}}(\mathbb{R})$, we multiply this by $2^j$, as there are $2^j$ components of
${\mathcal{M}}(\mathbb{R})$ meeting each such point, and each component of ${\mathcal{M}}(\mathbb{R})$ contains at most one branch of
$C_j$.
This completes the proof of the Lemma~\ref{L:estimates}.
\section{Betti Number Bounds}\label{Sec:Betti}
Using the bound~\eqref{Eq:structured_bound} and stratified Morse theory for
a manifold with corners~\cite{GM88}, we prove the following theorem.
\begin{theorem}\label{betti}
Let $X$ be a hypersurface in $\mathbb{R}^n_>$ defined by a $(d,\ell)$-dense fewnomial.
Then
\[
b_*(X)\ <\ \frac{e^2{+}3}{4}\, 2^{\binom{\ell}{2}} \cdot d^\ell
\cdot \sum_{i=0}^{n} \tbinom{n}{i}\, i^\ell\,.
\]
\end{theorem}
Let $f(x_1,\dots,x_n)$ be a real Laurent polynomial with $(d,\ell)$-dense support
${\mathcal{A}} \subset \mathbb{Z}^n$ such that
$X := {\mathcal{V}}(f) \subset \mathbb{R}^n_>$ is a smooth hypersurface.
By a logarithmic change of coordinates, $x_i=e^{z_i}$, we may work with exponential sums in
$\mathbb{R}^n$ instead of polynomials in $\mathbb{R}^n_>$.
Then the $(d,\ell)$-dense fewnomial
$f = \sum_{\alpha \in {\mathcal{A}}} c_{\alpha} x^{\alpha}$ becomes
the exponential sum
\[
\varphi\ :=\ \sum_{\alpha \in {\mathcal{A}}} c_{\alpha} e^{z \cdot \alpha}.
\]
In this way, the bounds~\eqref{Eq:BS} and~\eqref{Eq:structured_bound} for positive
solutions to fewnomial systems hold for real solutions to systems of exponential sums with the
same exponents.
Let $Z := {\mathcal{V}}(\varphi) \subset \mathbb{R}^n$ be the hypersurface defined by $\varphi$, which is
homeomorphic to $X$. We will prove Theorem~\ref{betti} in these logarithmic coordinates
and with real exponents.
\begin{theorem}\label{bettilog}
The sum of the Betti numbers of a hypersurface in $\mathbb{R}^n$ defined by an exponential sum
whose exponent vectors are $(d,\ell)$-dense
${\mathcal{A}} = {\mathcal{W}}\, \cup\, \psi(d\includegraphics{figures/Simplex.eps}^\ell\cap\mathbb{Z}^\ell) \subset \mathbb{R}^n$ is at most
\[
\frac{e^2{+}3}{4}\, 2^{\binom{\ell}{2}} \cdot d^\ell
\cdot \sum_{i=0}^{n} \tbinom{n}{i}\, i^\ell\,.
\]
\end{theorem}
An affine change of coordinates replaces ${\mathcal{W}}$ with another
set of independent vectors and replaces $\psi$ with another affine map but it
does not change the $(d,\ell)$-dense structure.
We may thus assume that the vectors in ${\mathcal{W}}$ are the standard unit basis vectors
in $\mathbb{R}^n$, and so $\varphi$ includes the coordinate exponentials $e^{z_i}$ for
$i = 1,\dots,n$.
Let $M \colon = (M_0,M_1,\dots,M_n)$ be a list of positive numbers and set
\[
\Delta(M) \ :=\ \{z \in \mathbb{R}^n \mid z_i \geq -M_i,\, i = 1,\dots,n \quad \text{ and }
\quad \sum_i z_i \leq M_0\},
\]
which is a nonempty simplex. We will use stratified Morse theory to bound the Betti
numbers of $Y := Z \cap \Delta_M$ when $M$ is general.
\begin{theorem}\label{bettigeneral}
For $M$ general, the sum of the Betti numbers of\/ $Y$ is at most
\[
\frac{e^2{+}3}{4}\, 2^{\binom{\ell}{2}} \cdot d^\ell
\cdot \sum_{i=0}^{n} \tbinom{n}{i}\, i^\ell\,.
\]
\end{theorem}
Theorem~\ref{bettilog} is a consequence of Theorem~\ref{bettigeneral}.
See Theorem $1'$ in~\cite{BS_Betti} for a detailed proof.
\begin{proof}[Proof of Theorem~$\ref{bettigeneral}$]
Given positive numbers $M = (M_0,M_1,\dots,M_n)$, define affine hyperplanes in $\mathbb{R}^n$
by
\[
H_0\ :=\ \{z \mid \sum_i z_i = M_0\} \quad \text{ and } \quad
H_i\ :=\ \{z \mid z_i = -M_i\}, \quad \text{ for } i = 1,\dots,n\,.
\]
For each proper subset $S \subset \{0,\dots,n\}$, define an affine linear subspace
\[
H_S\ :=\ \bigcap_{i \in S} H_i\,.
\]
Since each $M_i > 0$, this has dimension $n-|S|$, and these subspaces are the affine
linear subspaces supporting the faces of the simplex $\Delta_M$.
Choose $M$ generic so that for all $S$ the subspace $H_S$ meets $Z$ transversally.
For each subset $S$, set $Z_S := Z \cap H_S$. This is a smooth hypersurface in $H_S$ and
therefore has dimension $n-|S|-1$. The boundary stratum $Y_S$ of $Y = Z \cap \Delta_M$
lying in the relative interior of the face supported by $H_S$ is an open subset of $Z_S$.
For a nonzero vector $u \in \mathbb{R}^n$, the directional derivative $D_u\varphi$ is
\[
\sum_{\alpha \in {\mathcal{A}}} (u \cdot \alpha)c_{\alpha}e^{z \cdot \alpha}
\]
which is an exponential sum having the same exponents as $\varphi$. Let $L_u$ be the
linear function on $\mathbb{R}^n$ defined by $z \mapsto u \cdot z$.
The critical points of the function $L_u$ restricted to $Z$ are the zeroes of the
system
\[
\varphi(z) = 0 \quad \text{ and } \quad
D_v \varphi(z) = 0 \quad \text{ for } v \in u^{\perp}\,.
\]
When $u$ is general and we choose a basis for $u^\perp$, this becomes a system of
$n$ exponential sums in $n$ variables having the same support as the original polynomial.
Therefore, the whole system has $(d,\ell)$-dense support,
${\mathcal{A}}=\psi(d\includegraphics{figures/Simplex.eps}^\ell\cap\mathbb{Z}^\ell)\cup \{e_1,\dotsc,e_n\}$.
By Theorem~\ref{Th:Structured_bound}, the number of solutions is at most
\[
\frac{e^2{+}3}{4}\, 2^{\binom{\ell}{2}}\, n^\ell \cdot d^\ell\,.
\]
We use this to estimate the number of critical points of the function $L_u$ restricted
to $Z_S$.
The restriction $\varphi_S$ of $\varphi$ to $H_S$ defines $Z_S$ as a hypersurface in $H_S$.
We determine this restriction.
Suppose first that $0\notin S$.
If $i\in S$, then we may use the equation $z_i = -M_i$ to eliminate the variable $z_i$ and
the exponential $e^{z_i}$ from $\varphi$.
The effect of these substitutions for $i\in S$ on the exponents is the projection $\pi$
sending $e_i\mapsto 0$ for $i\in S$.
Then $\pi \circ \psi$ is still affine and so
$\varphi_S$ is still a $(d,\ell)$-dense fewnomial but with $n$ replaced by $n{-}|S|$,
and thus the number of critical points of $L_u|_{H_S}$ on $Z_S$ is bounded by
\[
\frac{e^2{+}3}{4}\, 2^{\binom{\ell}{2}}\, (n-|S|)^\ell \cdot d^\ell\,.
\]
If $0 \in S$, then we could use fewnomial theory to estimate the number of critical
points of $L_u|_{H_S}$ on $Z_S$, but will not need that estimate.
Let $u$ be a general vector in $\mathbb{R}^n$ such that $L_u$ is a Morse function for the
stratified space $Y$. By Proposition $2$ in~\cite{BS_Betti}, the sum of the Betti numbers
of $Y$ is bounded by the number of critical points $p$ of $L_u$ for which $L_u$ achieves
its minimum on the normal slice $N(p)$ at $p$. Since the strata $Y_S$ of $Y$ are open
subsets of the manifolds $Z_S$, this number is bounded above by the number of such
critical points of $L_u$ on the manifolds $Z_S$. Just as in~\cite{BS_Betti}, we can alter
$u$ so that no critical point in any $Z_S$ with $0 \in S$ contributes. Therefore,
the sum of the Betti numbers of $Y$ is bounded above by
\[
\frac{e^2{+}3}{4}\, 2^{\binom{\ell}{2}} \cdot d^\ell \cdot
\sum_{S \subset \{1,\dots,n\}} (n-|S|)^\ell
\ =\
\frac{e^2{+}3}{4}\, 2^{\binom{\ell}{2}} \cdot d^\ell \cdot
\sum_{i=0}^{n} {\binom{n}{i}} (n-i)^\ell\,.\vspace{-30pt}
\]
\end{proof}\vspace{15pt}
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train/arxiv
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BkiUbbM5qsNCPSsgaKqk
| 5 | 1 |
\section{Introduction}
Since the pioneering Bell paper \cite{Bell} the Bell inequalities became the subject of intensive study (for a review see \cite{Liang},\cite{Brunner}). Their importance stems from the fact that their violation at the quantum level provides the evidence that the quantum theory cannot be viewed as a local realistic theory. Another important notion in physics is that of symmetry. On the formal level various symmetries are described in the framework of group theory. Therefore, it appears natural to study Bell inequalities for the systems described by the sets of states classified by the representations of some groups. This idea has been proposed in the interesting papers by G\"uney and Hillery \cite{Guney},\cite{Guney1} and studied in some detail by the present authors \cite{Bolonek,Bolonek1,Bolonek2,Bolonek3}. Within this approach the states of quantum system entering some Bell inequality form the orbit(s) of a particular unitary representation of some group $G$. The orbit is chosen in such a way as to consist of a disjoint sum of subsets, each forming an orthonormal basis in the space of states. Each such basis provides the spectral decomposition of some observable. Grouptheoretical methods allow for simple calculation of quantum bound on the combination of probabilities entering the particular Bell's inequality. Computing classical bound calls for more effort. However, one can use here Fine's theorem \cite{Fine} which considerably reduces the relevant combinatorics.
Recently, Tavakoli and Gisin \cite{Tavakoli} in the very nice paper constructed Bell inequalities which violations are attained with measurements pointing to the vertices of the Platonic and Archimedean solids. Again we are dealing with the systems exhibiting high degree of symmetry; it is the symmetry which makes the Platonic solids so beautiful. In the present paper we generalize the Tavakoli \& Gisin approach using the tools from elementary group theory. Our starting point is the idea to look at the Platonic and Archimedean solids (as well as some other threedimensional objects) as generic and nongeneric orbits of threedimensional representations of some finite groups (basically, the subgroups of relevant symmetry groups). It appears that this point of view brings some advantages. We obtain an extremally simple formula for quantum value of the combination of correlation functions introduced in \cite{Tavakoli} under the assumption that our quantum system is described by the maximally entangled $SO(3)$ singlet state. We find also nice geometric picture, following from Fine's theorem and group theory, of classical states saturating Bell inequality. A number of examples of considerable violation of Bell inequality is presented.
\section{The general scheme}
We consider a Bell experiment with Alice and Bob having a number of possible settings characterized by the sets of directions in space, $\poisson{\vec{v}_i}_{i=1}^{N_A}$ and $\poisson{\vec{w}_j}_{j=1}^{N_B}$, respectively. They measure the spin $-\frac{1}{2}$ projections in the $\vec{v}_i$ (Alice) and $\vec{w}_j$ (Bob) directions; the corresponding observables $A_i$ and $B_j$ acquire the values $\pm 1$ depending on whether the projection is positive or negative.
One expects the Bell inequality to be violated the more the larger is the degree of entanglement of the state of the system. Therefore, we choose this state to be the $SO(3)$ singlet
\begin{equation}
\ket{\phi}\equiv\frac{1}{\sqrt{2}}\naw{\ket{\downarrow\uparrow}-\ket{\uparrow\downarrow}}.\label{1}
\end{equation}
It differs from the state $\ket{\phi^+}\equiv\frac{1}{\sqrt{2}}\naw{\ket{\uparrow\uparrow}+\ket{\downarrow\downarrow}}$ used by Tavakoli and Gisin \cite{Tavakoli}; however it is maximally entangled, $\mathrm{Tr}_A\naw{\ket{\phi}\bra{\phi}}=\frac{1}{2}\mathbbm{1}=\mathrm{Tr}_B\naw{\ket{\phi}\bra{\phi}}$ and slightly more convenient from the theoretical point of view since the relevant correlation functions are rotationally invariant. Moreover, most results obtained for $\ket{\phi^+}$ remain valid here (see below).
Simple calculation yields the well-known formula for the correlation function
\begin{equation}
\av{A_iB_j}_\phi\equiv \bra{\phi}\naw{\vec{v}_i\cdot\vec{\sigma}}\otimes\naw{\vec{w}_j\cdot\vec{\sigma}}\ket{\phi}=-\vec{v}_i\cdot\vec{w_j} \label{2}
\end{equation}
which is rotationally invariant as expected.
The Bell inequality we are considering concerns a linear combination of correlation functions
\begin{equation}
\mathcal{B}=\sum_{i=1}^{N_A}\sum_{j=1}^{N_B}c_{ij}\av{A_iB_j}.\label{3}
\end{equation}
Following \cite{Tavakoli} we adjust the coefficients $c_{ij}$ in such a way as to optimize the quantum bound on $\mathcal{B}$. Therefore, we put
\begin{equation}
c_{ij}=-\vec{v}_i\cdot\vec{w}_j\label{4}
\end{equation}
and eq.~(\ref{3}) takes the form
\begin{equation}
\mathcal{B}=-\sum_{i=1}^{N_A}\sum_{j=1}^{N_B}\naw{\vec{v}_i\cdot\vec{w}_j}\av{A_iB_j}.\label{5}
\end{equation}
In particular, for the singlet state (\ref{1})
\begin{equation}
\mathcal{B}=\sum_{i=1}^{N_A}\sum_{j=1}^{N_B}\naw{\vec{v}_i\cdot\vec{w}_j}^2.\label{6}
\end{equation}
In Ref.~\cite{Tavakoli} the directions $\vec{v}_i$, $\vec{w}_j$ correspond to the vertices of Platonic solids (and Archimedean ones). Therefore, they enjoy a high degree of symmetry. In particular, the most symmetric case corresponds to dual Platonic solids determining Alice and Bob settings. Dual Platonic solids share the same symmetry group. Guided by this example we consider the following general situation. Let $G=\poisson{g_\alpha}_{\alpha=1}^{|G|}$ $(g_1=e)$ be some finite group admitting threedimensional real irreducible representation; some reducible representations can be also considered (see below). We assume that $\poisson{v_i}_{i=1}^{N_A}$, $\poisson{w_j}_{j=1}^{N_B}$ are two arbitrary orbits of $G$, generated by the representation $D(g)$ under consideration. The orbits of $G$ are obtained by acting with all matrices $D(g)$ on some fixed vectors,
\begin{equation}
\vec{v}_\alpha=D(g_\alpha)\vec{v}\label{7}
\end{equation}
\begin{equation}
\vec{w}_\beta=D(g_\beta)\vec{w}.\label{8}
\end{equation}
For generic $\vec{v}$ ($\vec{w}$) we get $|G|$ vectors $\vec{v}_\alpha$ ($\vec{w}_\beta$). However, it can happen that the stability subgroup of $\vec{v}$ ($\vec{w}$), $G_v$ ($G_w$) is nontrivial. Then the vectors $\vec{v}_\alpha$ ($\vec{w}_\beta$) are in one-to-one correspondence with the cosets in $^G/_{G_v}$ ($^G/_{G_w}$), i.e. $\vec{v}_\alpha=\vec{v}_{\alpha'}$ ($\vec{w}_\beta=\vec{w}_{\beta'}$) iff $g_\alpha$, $g_{\alpha'}$ ($g_\beta, g_{\beta'}$) belong to the same coset of $^G/_{G_v}$ ($^G/_{G_w}$). Note that $^{|G|}/_{|G_v|}=N_A$, $^{|G|}/_{|G_w|}=N_B$.
Consider now the following sum
\begin{equation}
\begin{split}
&\sum_{g,g'\in G}\naw{D(g')\vec{v}\cdot D(g)\vec{w}}^2=\sum_{g,g'\in G}\naw{\vec{v}\cdot D(g'^{-1})\cdot D(g)\vec{w}}^2\\
&=\sum_{g,g'\in G}\naw{\vec{v}\cdot D((g'^{-1}\cdot g))\vec{w}}^2=|G|\sum_{g\in G}\naw{\vec{v}\cdot D(g)\vec{w}}^2\\
&=|G|\sum_{a,b,c,d=1}^3\sum_{g\in G}v_a w_b v_c w_d D_{ab}(g) D_{cd}(g)
\end{split} \label{9}
\end{equation}
where we have changed the summation indices, $g'\rightarrow g'$, $g'^{-1}\cdot g\rightarrow g$.
By applying the orthogonality relations
\begin{equation}
\sum_{g\in G}\overline{D_{ab}^{(\mu)}(g)}D_{cd}^{(\nu)}(g)=\frac{|G|}{\dim D^{(\mu)}}\delta_{\mu\nu}\delta_{ac}\delta_{bd}\label{10}
\end{equation}
and the normalization conditions $|\vec{v}|=|\vec{w}|=1$ we find
\begin{equation}
\sum_{g,g'\in G}\naw{D(g')\vec{v}\cdot D(g)\vec{w}}^2=\frac{1}{3}|G|^2.\label{11}
\end{equation}
Now, any term of the sum defining $\mathcal{B}$, eq.~(\ref{6}), enters the left-hand side of (\ref{11}) $|G_{v}|\cdot|G_w|$ times. Therefore, one finds finally the following expression for $\mathcal{B}$ for the maximally entangled state (\ref{1})
\begin{equation}
\mathcal{B}=\frac{1}{3}N_A\cdot N_B. \label{12}
\end{equation}
This is an extremally simple formula in terms of the Alice and Bob numbers of settings. It is important to point out that the only assumption we made is that $G$ generates the orbits of Alice and Bob; $G$ does not have to be the full symetry group of either orbit; the latter is much stronger assumption. For example, the symmetric group $S_4$ is the symmetry group of the regular tetrahedron which is some nongeneric orbit of threedimensional representation of $S_4$; however, for judicious choice of initial vector one can also obtain octahedron which has larger symmetry group $S_4\times S_2$.
Consider now the classical bound $C$ on $\mathcal{B}$. Its determination is slightly more involved. According to Fine's theorem \cite{Fine} it is sufficient to compute $\mathcal{B}$, eq.~(\ref{6}), for all deterministic responses of Alice and Bob and pick the largest one \cite{Tavakoli}, \cite{Bolonek}.
Therefore, we obtain \cite{Tavakoli}
\begin{equation}
\begin{split}
&\mathcal{B}\leq C\equiv \max\limits_{\substack{A_1,\ldots,A_{N_A}\in\{\pm 1\}^{N_A}\\ B_1,\ldots,B_{N_B}\in\{\pm 1\}^{N_B}}}\naw{-\sum_{j=1}^{N_B}B_j \sum_{i=1}^{N_A}A_i\naw{\vec{v}_i\cdot\vec{w}_j}}\\
&=\max\limits_{A_1,\ldots,A_{N_A}\in\{\pm 1\}^{N_A}}\naw{\sum_{j=1}^{N_B}\modu{\sum_{i=1}^{N_A}A_i\naw{\vec{v}_i\cdot\vec{w}_j}}}.\end{split}\label{13}
\end{equation}
The bound (\ref{13}) can be effectively computed provided $N_A$ and $N_B$ are not too large.
In order to get more insight into the structure of classical bound let us rewrite $C$ in the form
\begin{equation}
C=\max\limits_{\substack{A_1,\ldots,A_{N_A}\in\{\pm 1\}^{N_A}\\ B_1,\ldots,B_{N_B}\in\{\pm 1\}^{N_B}}}\naw{\sum_{i=1}^{N_A}A_i\vec{v}_i}\cdot\naw{\sum_{j=1}^{N_B}B_j\vec{w}_j}\label{14}
\end{equation}
where we have redefined $A_i\rightarrow-A_i$. Therefore, $C$ is expressed in terms of scalar products of vectors which are linear combinations, with the coefficients $\pm 1$, of the elements of Alice (Bob) orbit. The set of such vectors may be classified in group-theoretical terms. Let us consider a particular combination (for definiteness, we consider Alice orbit)
\begin{equation}
\sum_{i=1}^{N_A}A_i\vec{v}_i,\quad A_i=\pm 1.\label{15}
\end{equation}
Let $N_A^+$ ($N_A^-$) be the number of $+1$ ($-1$) coefficients entering (\ref{15}), $N_A^++N_A^-=N_A$. The group $G$ acts as some subgroup of permutations of the vectors $\vec{v}_i$. Therefore, by acting with $G$ on the combination (\ref{15}) one obtains the set of vectors of the same form with fixed $N_A^{\pm}$. One concludes that the set of vectors (\ref{15}) with $N_A^{\pm}$ fixed decomposes into disjoint sum of $G$ orbits. Accordingly, the set of all vectors of the above form is also a disjoint sum of such orbits. In order to determine their form it is sufficient to consider the configurations with $N_A^+ \leq N_A^-$; the remaining one are obtained by applying space inversion. Once the numbers $N_A^+\leq N_A^-$ are chosen one picks a particular combination (\ref{15}). In order to find the orbit obtained by acting with the elements of $G$ on this combination one has to determine the stability subgroup. The latter contains those elements of $G$ which permute $\vec{v}_i$'s with the same coefficients. However, in general it is larger; this is because the initial orbit $\poisson{\vec{v}_i}_{i=1}^{N_A}$ may contain the subset of vectors which are linearly dependent with the coefficients $\pm 1$ (this is, for example, the case for the cube which contains four pairs of opposite vectors). The same procedure may be applied to Bob's orbit. Once this is completed one can compute $C$ from eq.~(\ref{14}).
Contrary to the quantum bound the above described procedure does not seem to be more effective than direct evaluation of the bound $C$ from eq. (\ref{13}). However, it provides a nice geometrical picture of classical configurations. What is more it can serve for finding new orbits generated by the same group in order to use them as a starting point in our construction, i.e. as the initial orbits of Alice and Bob.
As a simple example let us take $G=S_4$ and the regular tetrahedron as its orbit. $S_4$ is the symmetry group of Platonic tetrahedron; simply, it permutes its four vertices $\vec{v}_i$, $i=1,2,3,4$. Since any three of them are linearly independent the stability subgroup of any vector (\ref{15}) consists exactly of all permutations exchanging vectors $\vec{v}_i$ entering (\ref{15}) with the same sign. Moreover, since our $G=S_4$ involves all permutations of vertices there is only one orbit corresponding to each $N_A^+$. Now, $N_A^+\leq N_A^-$ takes only three values, $N_A^+=0,\,1,\,2$. For $N_A^+=0$ the sum (\ref{15}) is obviously 0. For $N_A^+=1$ one finds tetrahedron twice as large as the initial one. It is also easy to find the orbit corresponding to $N_A^+=2$. To this end we connect the middle points of all edges obtaining thus regular octahedron and take the octahedron four times as large. The length of the vectors forming the vertices of the latter equals $\frac{4}{\sqrt{3}}$.
The orbits with $N_A^+>N_A^-$ are obtained by inversion. Concluding, the 16 combinations (\ref{15}) form two trivial orbits of $S_4$, two tetrahedrons in dual position and one octahedron.
\begin{figure}[!h]
\centering
\subfloat[two trivial orbits (red point) corresponding to $N_A^+=0,4$; the vertices of black tetrahedron represent Alice's settings]
{\includegraphics[width=0.35\textwidth]{Fig1a.pdf}}
\hspace{1cm}
\subfloat[the orbit (red octahedron) corresponding to $N_A^+=2$]{
\includegraphics[width=0.4\textwidth]{Fig1c.pdf}}
\\
\subfloat[two orbits (red and blue tetrahedrons) corresponding to $N_A^+=1,3$]{
\includegraphics[width=0.55\textwidth]{Fig1b.pdf}}
\caption{The "classical" orbits for tetrahedron.}
\end{figure}
Assume now that Bob's directions are determined by the vertices of regular tetrahedron which provides another orbit of $S_4$ (within the same representation). Note that the tetrahedrons are obtained by acting with the group elements on the vectors possessing stability subgroups of order 6. $S_4$ has four such subgroups, all in one conjugacy class. Taking into account that by inverting the initial vectors one obtains the tetrahedrons in dual position we conclude that there are only two tetrahedrons in dual position which form $S_4$ orbits of order 4. For dual tetrahedrons the sets of vectors (\ref{15}) coincide.
Assuming that both Alice and Bob settings form regular tetrahedrons generated by the same representation of $S_4$ we find that their "classical" configurations coincide. Therefore, we infer from eq.~(\ref{14}) that the classical bound (\ref{14}) is simply the square of the length of the largest vector which gives $\frac{16}{3}$. On the other hand eq.~(\ref{12}) yields also $\frac{16}{3}$ as the quantum bound. Thus there is no violation of Bell's inequality in this case.
\section{Some examples}
Let us consider some examples. In all cases considered below the group $G$ generating the orbit is either $S_4$ or $O_h$. The threedimensional irreducible representations of both groups are explicitly described in Appendix. The sets $\{\vec{v}_i\}_{i=1}^{N_A}$, $\{\vec{w}_j\}_{j=1}^{N_B}$ of Alice and Bob settings are the orbits of $G$ generated by the action of the representation matrices on some carefully selected initial vectors; the latter must be the eigenvectors, corresponding to the eigenvalue 1, of elements belonging to some subgroups of $G$ (stability subgroups). These initial vectors, together with explicit description of the corresponding orbits for all cases considered below, are given in the Appendix. We also give there the example of classical vectors, $\sum_{i=1}^{N_A}A_i\vec{v}_i$ and $\sum_{i=1}^{N_B}B_i\vec{w}_i$, saturating the classical bound (\ref{14}) for the configuration tetrahedron (Alice) - octahedron (Bob), as well as the orbits they generate according to the discussion of the previous section.
The orbits we are considering are:
\begin{itemize}
\item tetrahedron - generated by $S_4$
\item octahedron - generated by $S_4$ or $O_h$
\item cube - generated by $O_h$
\item cuboctahedron - generated by $S_4$ or $O_h$
\item truncated octahedron - generated by $S_4$ or $O_h$.
\end{itemize}
The algorithm described in the previous section can be directly applied to all pairs of the above solids except the pair tetrahedron - cube (see, however, below). The results are presented in Table \ref{t1}.
\begin{table}
\begin{tabular}{|l|c|c|}
\hline
Alice - Bob & Classical bound & Quantum value\\
\hline
cuboctahedron - tetrahedron & 13.0639 & 16\\
cuboctahedron - octahedron & 16.9706 & 24\\
cuboctahedron - cube & 26.1279 & 32\\
cuboctahedron - cuboctahedron & 40 & 48\\
truncated octahedron - tetrahedron & 24.7871 & 32\\
truncated octahedron - octahedron & 42.9325 & 48\\
truncated octahedron - cube & 49.5742 & 64\\
truncated octahedron - cubocthedron & 75.8947 & 96\\
truncated octahedron - truncated octahedron & 160 & 192\\
tetrahedron - octahedron & 6.9282 & 8\\
cube - octahedron & 13.8564 & 16\\
\hline
\end{tabular}
\caption{Classical bounds and quantum values for various combinations of Alice vs. Bob orbits.}\label{t1}
\end{table}
One can compare some of the above results with those obtained by Tavakoli and Gisin \cite{Tavakoli}. For example, for cube - octahedron we find the same value. On the other hand in the tetrahedron - octahedron case the classical bound obtained here is better than the one quoted in \cite{Tavakoli} - 7.82. This is because the relative positions of solids differ: in \cite{Tavakoli} they are chosen to be those given by "PolyhedronData" of Mathematica's software while here they are in position, described in previous section, resulting from group-theoretical considerations.
The "classical" numbers presented in Table \ref{t1} are approximate values. Indeed, from eqs.~(\ref{13}), (\ref{14}) and the forms of $\vec{v}_i$'s and $\vec{w}_j$'s presented in Appendix it follows that they are expressible in terms of square roots of integers. For example, for octahedron - cuboctahedron the ratio quantum - to classical bound is $1.4142\simeq\sqrt{2}$, the same as for CHSH inequality. It is likely that in this case the set of all Alice and Bob settings can be decomposed into the subsets leading to the original CHSH inequality (cf. Sec.~8 of Ref \cite{Tavakoli}). On Fig.~\ref{Fig2} we present some pairs of orbits.
\begin{figure}[!h]
\subfloat[cuboctahedron - tetrahedron]{\includegraphics[width=0.5\textwidth]{Fig2a.pdf}}
\subfloat[cuboctahedron - octahedron]{\includegraphics[width=0.5\textwidth]{Fig2b.pdf}}\\
\subfloat[truncated octahedron - tetrahedron]{\includegraphics[width=0.5\textwidth]{Fig2c.pdf}}
\subfloat[truncated octahedron - octahedron]{\includegraphics[width=0.5\textwidth]{Fig2d.pdf}}
\caption{Examples of Alice (black) and Bob (red) settings.}
\label{Fig2}
\end{figure}
Let us note that even the case tetrahedron - cube can be dealt with grouptheoretical method. The cube obtained by acting with elements of $O_h$ on the initial vector may be viewed as two tetrahedrons in dual position (see Fig. \ref{Fig3}).
Now, cube - cube configuration leads to $\frac{64}{3}$ both for classical and quantum cases. Obviously, for tetrahedron - cube configuration one finds one half of this result, i.e.~$\frac{32}{3}$.
\begin{figure}[!htp]
\centering\includegraphics[scale=0.5]{Fig3}
\caption{Cube as two (red and black) tetrahedrons in dual position.}\label{Fig3}
\end{figure}
Finally, let us come back to the case of two tetrahedrons. We found no violation of Bell inequality because tetrahedrons corresponding to Alice and Bob settings must be dual. The reason for that is that they are nongeneric orbits with stability groups of order 6. This fact strongly reduces the number of possibilities. Therefore, we should look for the group for which the generic orbit has four vertices i.e.~a group of order 4. Now, the regular tetrahedron is generated by cyclic subgroup of $S_4$ of order four, generated by cyclic permutation $(1234)$. So let us take $G=Z_4$ to be cyclic group of order four. Its threedimensional representation can be obtained from that of $S_4$ by subducing it to $Z_4$. In the suitably chosen basis it takes the form (for simplicity we write simply $g$ instead of $D(g)$):
\begin{equation}
\begin{split}
& e=\mathbbm{I},\quad g_1\equiv g=\left(\begin{array}{ccc}
-1 & 0 & 0\\
0 & 0 & 1\\
0 & -1 & 0
\end{array}\right),\quad g_2\equiv g^2=\left(\begin{array}{ccc}
1 & 0 & 0\\
0 & -1 & 0\\
0 & 0 & -1
\end{array}\right), \\
& g_3\equiv g^3=\left(\begin{array}{ccc}
-1 & 0 & 0\\
0 & 0 & -1\\
0 & 1 & 0
\end{array}\right).
\end{split}\label{16}
\end{equation}
This representation is reducible as the sum of one- and twodimensional ones. The generic orbits consist of four vertices. Taking
\begin{equation}
\vec{v}_1=\left(\begin{array}{c}
a\\
b\\
c
\end{array}\right ),\quad a^2+b^2+c^2=1\label{17}
\end{equation}
and denoting $\vec{v}_{k+1}=g_k\vec{v}_1$, $k=1,2,3$, one finds
\begin{equation}
\vec{v}_2=\left(\begin{array}{c}
-a\\
c\\
-b
\end{array}\right ),\quad \vec{v}_3=\left(\begin{array}{c}
a\\
-b\\
-c
\end{array}\right ), \quad \vec{v}_4=\left(\begin{array}{c}
-a\\
-c\\
b
\end{array}\right ).\label{18}
\end{equation}
Note that $\vec{v}_1\cdot\vec{v}_2=\vec{v}_1\cdot\vec{v}_4=\vec{v}_2\cdot\vec{v}_3=\vec{v}_3\cdot\vec{v}_4=-a^2\equiv\cos\psi$, $\vec{v}_1\cdot\vec{v}_3=\vec{v}_2\cdot\vec{v}_4=a^2-b^2-c^2=\cos\varphi$, $(\vec{v}_2-\vec{v}_4)\cdot(\vec{v}_1-\vec{v}_3)=0$. The resulting orbit is depicted on Fig.~\ref{Fig4}.
\begin{figure}
\begin{center}\includegraphics[width=0.6\textwidth]{Fig4a.pdf}\end{center}
\caption{The generic orbit of $Z_4$; $\measuredangle 1O3=\measuredangle 2O4=\varphi$, $\measuredangle 1O2=\measuredangle 1O4=\measuredangle 2O3=\measuredangle 3O4=\psi$.}
\label{Fig4}
\end{figure}
In particular, taking $a^2=b^2=c^2=\frac{1}{3}$ one obtains the regular tetrahedron.
Let us assume that the Alice orbit is the regular tetrahedron (say $a=b=c=\frac{1}{\sqrt{3}}$) while Bob's orbit corresponds to the vector $\vec{w}$ with arbitrary $a$, $b$, $c$ obeying $a^2+b^2+c^2=1$. First we determine the quantum bound. Eq.~(\ref{12}) cannot be applied directly since the representation is reducible. Actually, the orthogonality relations are valid for the representations irreducible in the complex domain. $Z_4$ is abelian so each representation irreducible in the complex domain is onedimensional. In fact $g_1=g$ (and, consequently, all $g_{{\alpha}}$, ${\alpha}=1,2,3$) can be diagonalized using the unitary matrix
\begin{equation}
U=\left(\begin{array}{ccc}
1 & 0 & 0\\
0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\\
0 & \frac{i}{\sqrt{2}} & -\frac{i}{\sqrt{2}}
\end{array}\right )\label{19}
\end{equation}
so that ($\tilde{g}_\alpha\equiv U^+gU$, $\alpha=1,2,3$)
\begin{equation}
\tilde{g}\equiv U^+gU=\left(\begin{array}{ccc}
-1 & 0 & 0\\
0 & i & 0\\
0 & 0 & -i
\end{array}\right ).\label{20}
\end{equation}
Let $e_\mu(\tilde{g}_\alpha)$, $\mu=1,2,3$, be the character corresponding to the first, second and third row of $\tilde{g}_\alpha$, respectively. Then any matrix $\tilde{g}_\alpha$ can be written as
\begin{equation}
\naw{\tilde{g}_\alpha}_{ij}=\sum_{\mu=1}^3e_\mu(\tilde{g}_\alpha)\delta_{i\mu}\delta_{j\mu}. \label{21}
\end{equation}
Using eq.~(\ref{21}) and the orthogonality relations
\begin{equation}
\sum_{g_\alpha\in Z_4}\overline{e_\mu(\tilde{g}_\alpha)}e_\nu(\tilde{g}_\alpha)=4\delta_{\mu\nu}\label{22}
\end{equation}
one finds
\begin{equation}
\sum_{g_\alpha,g_\beta\in Z_4}\naw{g_\alpha\vec{v},g_\beta\vec{w}}^2=4\sum_{g_\alpha\in G}\naw{\vec{v},g_\alpha\vec{w}}^2=4\sum_{g_\alpha\in Z_4}\modu{\naw{\tilde{\vec{v}},\tilde{g}_\alpha\tilde{\vec{w}}}}^2=16\sum_{i=1}^3\modu{\tilde{v}_i}^2\modu{\tilde{w}_i}^2 \label{23}
\end{equation}
where $\tilde{\vec{v}}\equiv U^+\vec{v}$, $\tilde{\vec{w}}\equiv U^+\vec{w}$. Now, $v_i=\frac{1}{\sqrt{3}}$, $i=1,2,3$ and by virtue of eq.~(\ref{19}), $\modu{\tilde{v}_i}=\frac{1}{\sqrt{3}}$. Therefore, eq.~(\ref{23}) yields $\frac{16}{3}$ as the quantum bound. As far as the classical bound is concerned one may either classify all orbits consisting of the vectors of the form (\ref{15}) and use eq.~(\ref{14}) (again quite nice geometrical picture emerges) or use directly eq.~(\ref{13}).
The final result reads
\begin{equation}
C=\frac{1}{\sqrt{3}}\max\naw{16|a|,8(\modu{b}+\modu{c})}.\label{24}
\end{equation}
Minimal value of $C$ is acquainted, for example, for $b=\pm\frac{2}{\sqrt{5}}$, $c=0$; then
\begin{equation}
C=\frac{16}{\sqrt{15}}<\frac{16}{3}\label{25}
\end{equation}
and the Bell inequality is violated.
\section{Summary}
Tavakoli and Gisin presented a very nice picture relating various settings for Alice and Bob, leading to the violation of Bell inequalities, to the geometry of Platonic solids. We have shown here that one can take as a starting point the symmetry groups of Platonic solids or, more precisely, the groups generating these solids as orbits of their threedimensional real representations. This point of view has some advantages. First, it allows the generalization to various groups and generic and nongeneric orbits (including, for example, in unified way the Archimedean solids).
Second, it leads to extremally simple expression for the quantum value of $\mathcal{B}$, given by eq.~(\ref{5}), in terms of the number od Alice's and Bob's settings only (eq.~(\ref{12})). Third, it provides a nice picture of the configurations (eq.~(\ref{15})) entering the formula (\ref{14}) for the classical bound. It should be also noted that the grouptheoretical approach determines some natural relative positions of the solids determining Alice's and Bob's settings.
Finally, let us show that the choice of $\ket{\phi}$ instead of $\ket{\phi^+}$, used in Ref.~\cite{Tavakoli}, does not influence the results. $\ket{\phi^+}$ belongs to the triplet representation of the rotation group; consequently, the relevant correlation function is no longer rotationally invariant
\begin{equation}
\bra{\phi^+}(\vec{v}_i\cdot\vec{\sigma})\otimes(\vec{w}_j\cdot\vec{\sigma})\ket{\phi^+}=\vec{v}_i\cdot\mathcal{I}_y\vec{w}_j\label{26}
\end{equation}
where $\mathcal{I}_y$ is the reflection in $x-z$ plane. Instead of $\mathcal{B}$ defined by eq.~(\ref{5}) we consider the following one
\begin{equation}
\mathcal{B}=\sum_{i=1}^{N_A}\sum_{j=1}^{N_B}\naw{\vec{v}_i\cdot\mathcal{I}_y\vec{w}_j}\av{A_iB_j}.\label{27}
\end{equation}
In particular, for the $\ket{\phi^+}$ state we get
\begin{equation}
\mathcal{B}=\sum_{i=1}^{N_A}\sum_{j=1}^{N_B}\naw{\vec{v}_i\cdot\mathcal{I}_y\vec{w}_j}^2.\label{28}
\end{equation}
Now, repeating the previous reasoning we find
\begin{equation}
\begin{split}
&\sum_{g,g'\in G}\naw{D(g)\vec{v}\cdot\mathcal{I}_y D(g')\vec{w}}^2\\
&=\sum_{a,b,c,d,e,f=1}^3 v_b v_e w_c'w_f'\sum_{g\in G}D(g)_{ab}D(g)_{de}\sum_{g'\in G}\naw{\mathcal{I}_yD(g')\mathcal{I}_y}_{ac}\naw{\mathcal{I}_yD(g')\mathcal{I}_y}_{df}\\
& =\sum_{a,b,c,d,e,f=1}^3 v_bv_ew_c'w_f'\frac{|G|^2}{9}\delta_{ad}\delta_{be}\delta_{ad}\delta_{cf}=\frac{|G|^2}{3}
\end{split}\label{29}
\end{equation}
because $D(g')$ and $\mathcal{I}_yD(g')\mathcal{I}_y$ are equivalent representations and $\vec{w}'=\mathcal{I}_y\vec{w}$.
As far as classical bound is concerned both formulae, eq.~(\ref{5}) and (\ref{27}), give the same results if at least one of the orbits (i.e.~defining Alice's and/or Bob's settings) is invariant under reflection in $x-z$ plane. This is, for example, the case for all pairs of Platonic solids considered in \cite{Tavakoli} which lead to Bell inequality violation.
\begin{appendices}
\section{}
Below we present all solids which form the $G$ - orbits appearing in the text.
\begin{figure}[!h]
\subfloat[the tetrahedron]{\centering\includegraphics[width=0.33\textwidth]{tetrahedron.pdf}}
\subfloat[the octahedron]{\includegraphics[width=0.33\textwidth]{octahedron.pdf}}
\subfloat[the cube]{\includegraphics[width=0.33\textwidth]{cube.pdf}}\\
\centering
\subfloat[the cuboctahedron]{\includegraphics[width=0.4\textwidth]{cuboctahedron.pdf}}
\subfloat[the truncated octahedron]{\includegraphics[width=0.4\textwidth]{truncated_octahedron.pdf}}
\caption{Solids which form the $S_4$-orbits or $O_h$-orbits.}
\end{figure}
\begin{enumerate}
\item The tetrahedron:
\begin{itemize}
\item[] Symmetry group: $S_4$
\item[] Orbit generating group: $S_4$ (or $Z_4$)
\item[] The initial vector: $\vec{v}_1=(1,0,0)$
\item[] The vertices:
\begin{displaymath}\begin{array}{ll}
v_1=(1,0,0), & v_2=\naw{-\frac{1}{3},-\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}}}\\
v_3\naw{-\frac{1}{3},\frac{2\sqrt{2}}{3},0}, & v_4\naw{-\frac{1}{3},-\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}}}. \end{array}\end{displaymath}
\end{itemize}
\item The octahedron:
\begin{itemize}
\item[] Symmetry group: $O_h$
\item[] Orbit generating group: $S_4$ or $O_h$
\item[] The initial vector: $v_1=\naw{\frac{1}{\sqrt{3}},\sqrt{\frac{2}{3}},0}$
\item[] The vertices:
\begin{displaymath}\begin{array}{ll}
v_1=\naw{-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{6}},\frac{1}{\sqrt{2}}}, &
v_2=\naw{\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{6}},-\frac{1}{\sqrt{2}}},\\
v_3=\naw{\frac{1}{\sqrt{3}},\sqrt{\frac{2}{3}},0}, & v_4=\naw{-\frac{1}{\sqrt{3}},-\sqrt{\frac{2}{3}},0},\\
v_5=\naw{-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{6}},-\frac{1}{\sqrt{2}}}, & v_6=\naw{\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{6}},\frac{1}{\sqrt{2}}}. \end{array}\end{displaymath}
\end{itemize}
\newpage
\item The cube:
\begin{itemize}
\item[] Symmetry group: $O_h$
\item[] Orbit generating group: $O_h$
\item[] The initial vector: $v_1=\naw{1,0,0}$
\item[] The vertices:
\begin{displaymath}\begin{array}{ll}
v_1=(1,0,0), & v_2=\naw{-\frac{1}{3},-\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}}},\\
v_3=\naw{-\frac{1}{3},\frac{2\sqrt{2}}{3},0}, & v_4=\naw{-\frac{1}{3},-\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}}},\\
v_5=(-1,0,0), & v_6=\naw{\frac{1}{3},\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}}},\\
v_7=\naw{\frac{1}{3},-\frac{2\sqrt{2}}{3},0}, & v_8=\naw{\frac{1}{3},\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}}}.\end{array}\end{displaymath}
\end{itemize}
\item The cuboctahedron:
\begin{itemize}
\item[] Symmetry group: $O_h$
\item[] Orbit generating group: $S_4$ or $O_h$
\item[] The initial vector: $v_1=\naw{-\sqrt{\frac{2}{3}},\frac{1}{\sqrt{3}},0}$
\item[] The vertices:
\begin{displaymath}\begin{array}{ll}
v_1=\naw{-\sqrt{\frac{2}{3}},\frac{1}{\sqrt{3}},0},& v_2=\naw{0,\frac{\sqrt{3}}{2},\frac{1}{2}},\\
v_3=\naw{0,0,1},& v_4=\naw{-\sqrt{\frac{2}{3}},-\frac{1}{2\sqrt{3}},\frac{1}{2}}, \\
v_5=\naw{\sqrt{\frac{2}{3}},-\frac{1}{\sqrt{3}},0},&
v_6=\naw{0,-\frac{\sqrt{3}}{2},-\frac{1}{2}},\\
v_7=\naw{0,0,-1}, &v_8=\naw{\sqrt{\frac{2}{3}},\frac{1}{2\sqrt{3}},-\frac{1}{2}},\\
v_9=\naw{0,-\frac{\sqrt{3}}{2},\frac{1}{2}}, &
v_{10}=\naw{0,\frac{\sqrt{3}}{2},-\frac{1}{2}},\\
v_{11}=\naw{\sqrt{\frac{2}{3}},\frac{1}{2 \sqrt{3}},\frac{1}{2}}, &
v_{12}=\naw{-\sqrt{\frac{2}{3}},-\frac{1}{2 \sqrt{3}},-\frac{1}{2}}.\end{array}\end{displaymath}
\end{itemize}
\item The truncated octahedron
\begin{itemize}
\item[] Symmetry group: $O_h$
\item[] Orbit generating group: $S_4$ or $O_h$
\item[] The initial vector: $v_1=\naw{\sqrt{\frac{3}{5}},0,\sqrt{\frac{2}{5}}}$
\item[] The vertices:
\begin{displaymath}\begin{array}{ll}
v_1=\naw{\sqrt{\frac{3}{5}},0,\sqrt{\frac{2}{5}}}, & v_2=\naw{\sqrt{\frac{3}{5}},\sqrt{\frac{3}{10}},\frac{1}{\sqrt{10}}}, \\ v_3=\naw{\sqrt{\frac{3}{5}},0,-\sqrt{\frac{2}{5}}}, &
v_4=\naw{\sqrt{\frac{3}{5}},\sqrt{\frac{3}{10}},-\frac{1}{\sqrt{10}}},\\
v_5=\naw{\sqrt{\frac{3}{5}},-\sqrt{\frac{3}{10}},\frac{1}{\sqrt{10}}}, & v_6=\naw{\sqrt{\frac{3}{5}},-\sqrt{\frac{3}{10}},-\frac{1}{\sqrt{10}}},\\
v_7=\naw{-\sqrt{\frac{3}{5}},0,\sqrt{\frac{2}{5}}}, & v_8=\naw{-\sqrt{\frac{3}{5}},\sqrt{\frac{3}{10}},\frac{1}{\sqrt{10}}}, \\
v_9=\naw{-\sqrt{\frac{3}{5}},0,-\sqrt{\frac{2}{5}}}, &
v_{10}=\naw{-\sqrt{\frac{3}{5}},\sqrt{\frac{3}{10}},-\frac{1}{\sqrt{10}}},\\ v_{11}=\naw{-\sqrt{\frac{3}{5}},-\sqrt{\frac{3}{10}},\frac{1}{\sqrt{10}}}, & v_{12}=\naw{-\sqrt{\frac{3}{5}},-\sqrt{\frac{3}{10}},-\frac{1}{\sqrt{10}}},\\
v_{13}=\naw{-\frac{1}{\sqrt{15}},2 \sqrt{\frac{2}{15}},-\sqrt{\frac{2}{5}}},&
v_{14}=\naw{\frac{1}{\sqrt{15}},\sqrt{\frac{5}{6}},-\frac{1}{\sqrt{10}}},\\
v_{15}=\naw{-\frac{1}{\sqrt{15}},2 \sqrt{\frac{2}{15}},\sqrt{\frac{2}{5}}}, & v_{16}=\naw{\frac{1}{\sqrt{15}},\sqrt{\frac{5}{6}},\frac{1}{\sqrt{10}}},\\
v_{17}=\naw{\frac{1}{\sqrt{15}},-2 \sqrt{\frac{2}{15}},\sqrt{\frac{2}{5}}}, & v_{18}=\naw{-\frac{1}{\sqrt{15}},-\sqrt{\frac{5}{6}},\frac{1}{\sqrt{10}}},\\
v_{19}=\naw{\frac{1}{\sqrt{15}},-2 \sqrt{\frac{2}{15}},-\sqrt{\frac{2}{5}}}, & v_{20}=\naw{-\frac{1}{\sqrt{15}},-\sqrt{\frac{5}{6}},-\frac{1}{\sqrt{10}}},\\
v_{21}=\naw{-\frac{1}{\sqrt{15}},\frac{1}{\sqrt{30}},\frac{3}{\sqrt{10}}}, & v_{22}=\naw{\frac{1}{\sqrt{15}},-\frac{1}{\sqrt{30}},\frac{3}{\sqrt{10}}},\\
v_{23}=\naw{\frac{1}{\sqrt{15}},-\frac{1}{\sqrt{30}},-\frac{3}{\sqrt{10}}}, & v_{24}=\naw{-\frac{1}{\sqrt{15}},\frac{1}{\sqrt{30}},-\frac{3}{\sqrt{10}}}.\end{array}\end{displaymath}
\end{itemize}
\end{enumerate}
Below we give an example of "classical" vectors (\ref{15}) saturating the bound (\ref{14}) for the settings: tetrahedron (Alice) and octahedron (Bob). The $G$-orbits they generate are also presented.\\
Alice's vector: $\vec{V}\equiv\sum\limits_{i=1}^{4}A_i\vec{v}_i=\vec{v}_1-\vec{v}_2+\vec{v}_3+\vec{v}_4=\naw{\frac{2}{3},-\frac{4\sqrt{2}}{3},0}$\\
Bob's vector: $\vec{W}\equiv\sum\limits_{j=1}^{6}B_j\vec{w}_j=-\vec{w}_1+\vec{w}_2-\vec{w}_3+\vec{w}_4-\vec{w}_5+\vec{w}_6=\naw{\frac{2}{\sqrt{3}},-4\sqrt{\frac{2}{3}},0}$\\
The $S_4$ orbits generated by Alice's and Bob's vectors are depicted on Fig. \ref{Fig10}.
\begin{figure}[!h]
\subfloat[the initial Alice's settings (magenta) and the $S_4$ orbit generated by the vector $\vec{V}$]{\includegraphics[width=0.5\textwidth]{AppenRys1.pdf}}
\hspace{1 cm}\subfloat[the initial Bob's settings (black) and two $S_4$ orbits generated by the vectors $\vec{W}$ and $-\vec{W}$]{\includegraphics[width=0.5\textwidth]{AppenRys2.pdf}}\\
\subfloat[the $S_4$ orbits generated by $\vec{V}$, $\vec{W}$ and $-\vec{W}$]{\includegraphics[width=0.5\textwidth]{AppenRys3.pdf}}\hspace{1cm}
\subfloat[the initial Alice's and Bob's settings and the $S_4$ orbits generated by $\vec{V}$, $\vec{W}$ and $-\vec{W}$]{\includegraphics[width=0.5\textwidth]{AppenRys4.pdf}}
\caption{The $S_4$ orbits generated by Alice's (magenta and blue) and Bob's (black and red) vectors. Figs.~10c and 10d present the relative position of the Alice and Bob orbits generated by "classical" vectors.}\label{Fig10}\end{figure}
\section{}
In order to describe the threedimensional irreducible representation of $S_4$ group it is sufficient to write out the matrices representing transpositions. They read:
\begin{equation}
\begin{split}
& D\naw{12}=\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & -1
\end{array}\right],\qquad
D\naw{13}=\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & -\frac{1}{2} & -\frac{\sqrt{3}}{2}\\
0 & -\frac{\sqrt{3}}{2} & \frac{1}{2}
\end{array}\right]\\
& D\naw{14}=\left[\begin{array}{ccc}
-\frac{1}{3} & -\frac{\sqrt{2}}{3} & -\frac{\sqrt{6}}{3}\\
-\frac{\sqrt{2}}{3} & \frac{5}{6} & -\frac{\sqrt{3}}{6}\\
-\frac{\sqrt{6}}{3} & -\frac{\sqrt{3}}{6}& \frac{1}{2}
\end{array}\right],\qquad
D\naw{23}=\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & -\frac{1}{2} & \frac{\sqrt{3}}{2}\\
0 & \frac{\sqrt{3}}{2} & \frac{1}{2}
\end{array}\right]
\\
&D\naw{24}=\left[\begin{array}{ccc}
-\frac{1}{3} & -\frac{\sqrt{2}}{3} & \frac{\sqrt{6}}{3}\\
-\frac{\sqrt{2}}{3} & \frac{5}{6} & \frac{\sqrt{3}}{6}\\
\frac{\sqrt{6}}{3} & \frac{\sqrt{3}}{6}& \frac{1}{2}
\end{array}\right],\qquad
D\naw{34}=\left[\begin{array}{ccc}
-\frac{1}{3} & \frac{\sqrt{8}}{3} & 0\\
\frac{\sqrt{8}}{3} & \frac{1}{3} & 0\\
0 & 0 & 1
\end{array}\right].\end{split}\label{30}
\end{equation}
The $O_h$ group it the direct product of $S_4\times S_2$; for this reason the threedimensional irreducible representation of this group can be obtained by multiplying all 24 matrices representing $S_4$ by $\pm 1$.
\end{appendices}
\newpage
|
train/arxiv
|
BkiUaF825V5hcGj03HSJ
| 5 | 1 |
\section{Introduction}
\label{sec:intro}
Research on wave energy converters (WECs) has concentrated traditionally on systems of small floating bodies, like offshore heaving buoys (see \cite{EV76}--\cite{ FA02}). However, the seminal theories on WECs that originated from this first scientific approach to wave energy extraction in the 1970s, do not capture exhaustively the dynamics of the last-generation WECs. The latter are usually large-scale devices designed to be deployed in arrays, some of them in the near-shore environment. For example, while studying the dynamics of an offshore heaving WEC in a channel, Srokosz (1980) \cite{SR80} showed that resonance of the channel sloshing modes is detrimental to the efficiency of power absorption. Conversely, in a recent analysis of a large flap-type WEC in a channel, Renzi \& Dias \cite{RD12} noted that the trapping of transverse modes near the flap increases the efficiency of the converter. Because of the image effect of the channel walls, this fact is also expected to occur in an infinite array of flap-type converters. The aim of this work is to discover the dynamics of a system of last-generation flap-type WECs and to outline its similarities and differences with respect to the systems of the first generation. As a result of this analysis, an optimisation criterion for an array of flap-type WECs is devised, depending on the physical and geometrical parameters of the system.
In Section \ref{sec:model} the behaviour of an array of converters in the open ocean is investigated by taking as a reference the theoretical framework of Renzi and Dias \cite{RD12}. The expressions of the free-surface elevation for the diffracted and radiated wave field in the fluid domain are derived accordingly. Analysis of the wave motion in the far field allows to obtain new formulae for the reflection, transmission and radiation coefficients. Various relations between the hydrodynamic coefficients are then shown in Section \ref{sec:rel}. Some of these relations correspond directly to Srokosz's results \cite{SR80} for floating bodies of symmetric shape in a channel. Some others, on the other hand, incorporate specific properties of the wave field generated by the flap-type converter, not considered before, and point out the peculiarity of such WEC with respect to the converters of the first generation. The analytical model is validated against known theories in the small-gap and in the point-absorber limit. In Section \ref{sec:paran}, a parametric analysis is undertaken for optimising the performance of the system. It is shown that the maximum capture factor is attained at complete trapping of the transverse modes of the array. When complete trapping is not possible, partial trapping can still increase the performance of the system. Finally, in Section \ref{sec:waven} a practical application of an array of large flap-type WECs is devised. Comparison with available data obtained with a finite-element numerical code is very satisfactory (see \ref{sec:appD}).
\section{Mathematical model}
\label{sec:model}
\subsection{Theoretical background}
Consider an in-line array of identical flap-type wave energy converters, each hinged on a bottom foundation of height $c'$ in an ocean of constant depth $h'$, as shown in figure \ref{fig:geom}.
\begin{figure}
\centerline{\includegraphics[width=9cm, trim=3.5cm 0cm 3.5cm 0cm]{geometry.eps}}
\caption{Geometry of the array ($a$) and the reference flap ($b$) in physical variables.}
\label{fig:geom}
\end{figure}
Primes denote dimensional quantities. Monochromatic incident waves of amplitude $A_I'$, period $T'$ and frequency $\omega'=2\pi/T'$ are incoming from the right and set the flaps into motion, which is converted into useful energy by means of generators linked to each device. Since the practical applications of such a system are usually in the nearshore \citep{WF12}, where wave fronts are almost parallel to the shoreline because of refraction, normal incidence is also assumed. Let $w'$ and $b'$ be the width of each flap and the spatial period of the array, respectively. Then the gap between two consecutive flaps is $a'=b'-w'$ (see again figure \ref{fig:geom}). A Cartesian coordinate system is set, with the $x'$ direction orthogonal to the flaps, the $y'$ axis along the flap lineup and the $z'$ axis rising from the undisturbed water level $z'=0$, positive upwards; $t'$ denotes time. Due to periodicity, the origin of the system can be set arbitrarily on any flap, which is therefore identified as the reference flap. The analysis is performed in the framework of a linear inviscid potential-flow theory for small-amplitude oscillations. The velocity potential $\Phi'$ must satisfy the Laplace equation
\begin{equation}
\nabla'^2\Phi'(x',y',z',t') =0
\label{eq:lapl}
\end{equation}
in the fluid domain. The linearised kinematic-dynamic boundary condition on the free surface reads
\begin{equation}
\Phi'_{,t't'}+g \Phi'_{,z'}=0, \quad z'=0,
\label{eq:bcsurf}
\end{equation}
where $g$ is the acceleration due to gravity and subscripts with commas denote differentiation with respect to the relevant variables. Absence of normal flux at the bottom yields
\begin{equation}
\Phi'_{,z'}=0, \quad z'=-h'.
\label{eq:bottom}
\end{equation}
Because of normal incidence of the incoming wave field, periodicity of the problem requires
\begin{equation}
f'(x',y'+mb',z',t')=f'(x',y',z',t'),\quad m=0,\pm 1,\pm 2,\dots,\; y'\in(-b'/2,b'/2),
\label{eq:period}
\end{equation}
where $f'$ indicates any physical quantity associated to the problem and $m$ each of the flaps; $m=0$ denotes the reference flap. Extension to an obliquely-incident wave field can be easily made \citep[see for example][]{PE96}. However, since flap-type WECs are usually designed to operate under normally-incident waves \citep[see][]{WF12}, only normal incidence will be considered here. Because of the periodicity condition (\ref{eq:period}), the solution to the complete problem can be obtained by investigating the wave interaction with the reference flap centred at the origin, with $|y'|<b'/2$. Symmetry of the problem requires also
\begin{equation}
\Phi'_{,y'}=0,\quad y'=\pm b'/2,
\label{eq:simm}
\end{equation}
which can be regarded as a no-flux boundary condition on two imaginary waveguides at $y'=\pm b'/2$ (see again figure \ref{fig:geom}). Let $\theta'(t')$ be the angle of rotation of the flap, positive if anticlockwise; then the kinematic boundary condition on the flap yields
\begin{equation}
\Phi'_{,x'}=-\theta'_{,t'}(t') (z'+h'-c') H(z'+h'-c'), \quad x'=\pm 0, |y'|<w'/2,
\label{eq:plate}
\end{equation}
where the thin-body approximation has been applied \citep{LM01}. The Heaviside step function in (\ref{eq:plate}) assures absence of flux through the bottom foundation. The problem defined above is formally equivalent to that solved by Renzi and Dias \cite{RD12} for a single converter in a channel. Here the main arguments of the theory in \cite{RD12} are retraced and applied to the array configuration. First, the system (\ref{eq:lapl})--(\ref{eq:plate}) is non-dimensionalised as follows \citep[see][eqn. (2.1)]{RD12}
\begin{eqnarray}
&(x,y,z,h,w,a,c)=(x',y',z',h',w',a',c')/b',\; t=\sqrt{g/b'}\,t',\nonumber \\
&\Phi=\left(\sqrt{gb'}A'\right)^{-1}\Phi',\; \theta=(b'/A') \theta',
\label{eq:nondimvar}
\end{eqnarray}
where the wave amplitude scale $A'\ll b'$ because of the hypothesis of small-amplitude oscillations. In expression (\ref{eq:nondimvar}), $a=(1-w)\in(0,1)$ defines the aperture of the array. Then time is factored out by setting
\begin{equation}
\Phi(x,y,z,t)=\Re\left\lbrace \phi(x,y,z) e^{-i\omega t}\right\rbrace,\quad \theta(t)=\Re\left\lbrace \Theta e^{-i\omega t}\right\rbrace,
\label{eq:separ}
\end{equation}
with $\omega=\sqrt{b'/g}\,\omega'$ \citep[see][eqn. (2.11)]{RD12}. The global spatial potential
$$
\phi=\phi^R+\phi^S
$$
is the sum of the radiation potential $\phi^R$ and the scattering potential $\phi^S$. The latter is in turn decomposed into
$$\phi^S=\phi^I+\phi^D,$$ where
\begin{equation}
\phi^I(x,y,z)=-\frac{i A_I}{\omega\cosh kh}\cosh k(z+h) e^{-i kx}
\label{eq:incidentwav}
\end{equation}
is the incident wave potential and $\phi^D$ is the diffraction potential. $\phi^R$ and $\phi^D$ must be both outgoing at large $|x|$. In (\ref{eq:incidentwav}), $A_I=A'_I/A'$ is the non-dimensional amplitude of the incident wave and $k$ is the wavenumber, corresponding to the real solution of the dispersion relationship $\omega^2=k \tanh kh$.
Following the method described in Appendix B of \cite{RD12}, application of the Green integral theorem yields two hypersingular integral equations, in terms of the jump in radiation and scattering potentials across the plate \citep[see][eqn. (B10)]{RD12}. Those equations are solved by expanding the jumps in potential into series of Chebyshev polynomials of even order \citep[for details, see eqn.s (B11)--(B18) of][]{RD12}. Careful treatment of the singularity \citep[see][eqn. (B19)]{RD12} ultimately allows to write the potentials in the reference domain $|y|<1/2$ in a new semi-analytical form \citep[see][eqn.s (B24) and (B25)]{RD12}. The radiation potential is
\begin{equation}
\phi^R(x,y,z)=\sum_{n=0}^{+\infty}\sum_{p=0}^P\sum_{m=-\infty}^{+\infty}\phi^R_{npm}(x,y,z),
\label{eq:radpotsum}
\end{equation}
where
\begin{eqnarray}
\phi^R_{npm}(x,y,z)&=&-\frac{i w V}{8}\,\kappa_n x Z_n(z) \alpha_{(2p)n}\int_{-1}^{1} \left(1-u^2\right)^{1/2} \nonumber\\
&\times& U_{2p}(u)
\frac{H_1^{(1)}\left(\kappa_n\sqrt{x^2+(y-\frac{1}{2} wu-m)^2} \right)}{\sqrt{x^2+(y-\frac{1}{2} wu-m)^2}}\,du,
\label{eq:radpot}
\end{eqnarray}
$H_1^{(1)}$ being the Hankel function of the first kind and first order. In (\ref{eq:radpot}), $V=i\omega\Theta$ is the complex angular velocity of the flap, the subscript $m$ identifies the contribution of each single flap, while the subscript $p$ indicates the order of the Chebyshev expansion, $U_{2p}$ being the Chebyshev polynomial of the second kind and even order $2p$, $p=0,1,...,P\in\mathbb{N}$. The subscript $n$ identifies the contribution of each depth mode
\begin{equation}
Z_n(z)=\frac{\sqrt{2}\cosh \kappa_n (z+h)}{\left(h+\omega^{-2}\sinh^2 \kappa_n h \right)^{1/2}},\quad n=0,1,\dots,
\label{eq:Zn}
\end{equation}
where $\kappa_0=k$, while $\kappa_n=ik_n$ denote the complex solutions of the dispersion relationship
\begin{equation}
\omega^2=-k_n\tan k_nh,\quad n=1,2,\dots
\label{eq:disprel}
\end{equation}
Finally, the $\alpha_{(2p)n}$ are the complex solutions of a system of linear equations ensuring that $\phi^R$ satisfies the kinematic condition on the flap \citep[see][eqn.s (B22) and (B23)]{RD12}. This system is solved numerically with a collocation scheme, therefore the solution (\ref{eq:radpot}) is partly numerical. In summary, $\phi_{npm}$ (\ref{eq:radpot}) indicates the $n$-th depth mode, $p$-th order potential of the wave field radiated by the $m$-th flap ($m$-th array mode), moving at unison with all the other flaps. The diffraction potential is given by
\begin{equation}
\phi^D(x,y,z)=\sum_{p=0}^P\sum_{m=-\infty}^{+\infty}\phi^D_{pm},
\label{eq:difpotsum}
\end{equation}
where
\begin{eqnarray}
\phi^D_{pm}(x,y,z)&=&-\frac{i w A_I}{8}\,kx\, Z_0(z)\beta_{2p}\int_{-1}^{1} \left(1-u^2\right)^{1/2} \nonumber\\
&\times& U_{2p}(u)
\frac{H_1^{(1)}\left(k\sqrt{x^2+(y-\frac{1}{2} wu-m)^2} \right)}{\sqrt{x^2+(y-\frac{1}{2} wu-m)^2}}\,du.
\label{eq:difpot}
\end{eqnarray}
In the latter, the $\beta_{2p}$ are the complex solutions of a system of linear equations, which ensures that $\phi^D$ satisfies the no-flux condition on the flap \citep[see][eqn.s (B22) and (B23)]{RD12}. Again, $\phi^D_{pm}$ indicates the $p$-th order potential diffracted by the $m$-th flap, in the presence of all the other flaps. Note that in $\phi^D$ (\ref{eq:difpot}) only the $0$-th depth mode is present, as required by the solvability of the whole radiation-diffraction problem \citep[see Appendix B.2 of][]{RD12}. Computational aspects involved in the numerical evaluation of (\ref{eq:radpot}) and (\ref{eq:difpot}) are detailed in \S 2.2 of \cite{RD12}.
\subsection{Body motion}
The equation of motion of the reference flap in the frequency domain is that of a damped harmonic oscillator \citep[see eqn. (2.33) of][]{RD12}, namely
\begin{equation}
\left[-\omega^2(I+\mu)+C-i\omega(\nu+\nu_{pto})\right]\Theta=F,
\label{eq:bodymot}
\end{equation}
depending on the moment of inertia of the flap $I=I'/(\rho b'^5)$, on the flap buoyancy torque $C=C'/(\rho g b'^4)$ and on the power take-off (PTO) coefficient $\nu_{pto}=\nu'_{pto}/(\rho b'^4\sqrt{gb'})$, where $\rho$ is the water density. The latter parameters are assumed to be all known. In (\ref{eq:bodymot})
\begin{equation}
\mu=\frac{\pi w}{2\sqrt{2}}\,\Re\left\lbrace \sum_{n=0}^{\infty} \alpha_{0n} \frac{\kappa_n(h-c)\sinh \kappa_n h+\cosh \kappa_n c-\cosh\kappa_n h}{\kappa_n^2\left(h+\omega^{-2}\sinh^2\kappa_n h \right)^{1/2}} \right\rbrace
\label{eq:mu}
\end{equation}
is the added inertia torque \citep[see eqn. (2.34) of][]{RD12}, while
\begin{equation}
\nu=\frac{\pi w}{2\sqrt{2}}\Im\left\lbrace \alpha_{00} \right\rbrace \frac{\omega\left[k(h-c)\sinh k h+\cosh k c-\cosh k h\right]}{k^2\left(h+\omega^{-2}\sinh^2 kh \right)^{1/2}}
\label{eq:nu}
\end{equation}
and
\begin{equation}
F=-\frac{i\pi w A_I}{2\sqrt{2}}\beta_{0}\frac{\omega\left[k(h-c)\sinh kh+\cosh kc-\cosh kh\right]}{k^2\left(h+\omega^{-2}\sinh^2 kh\right)^{1/2}}
\label{eq:F}
\end{equation}
denote, respectively, the radiation damping \citep[see eqn. (2.35) of][]{RD12} and the complex exciting torque \citep[see eqn. (2.36) of][]{RD12}. If the PTO system is designed such that
$$\nu_{pto}=\sqrt{\frac{\left[C-(I+\mu)\omega^2\right]^2}{\omega^2}+\nu^2},$$
which corresponds to the optimum PTO damping \citep[see eqn. (2.40) of][]{RD12},
then the average generated power over a period is
\begin{equation}
P=\frac{1}{4}|F|^2\left[\sqrt{\frac{\left[C-(I+\mu)\omega^2\right]^2}{\omega^2}+\nu^2}+\nu\right]^{-1}.
\label{eq:pow}
\end{equation}
Now, the generated power (\ref{eq:pow}) is maximum under resonant amplification of the body motion, which occurs when
\begin{equation}
\omega=\sqrt{\frac{C}{I+\mu}}.
\label{eq:bodyres}
\end{equation}
By substitution of the latter expression into (\ref{eq:pow}), the optimum power available for extraction from each flap is therefore
\begin{equation}
P_{opt}=\frac{1}{8}\frac{|F|^2}{\nu},
\label{eq:Popt}
\end{equation}
which matches the well-known result of Srokosz \cite{SR80}.
The performance of each element of the array is assessed quantitatively by using two main factors. The amplitude factor
\begin{equation}
A_F=\frac{(h-c)\tan|\Theta|}{A_I}
\label{eq:AF}
\end{equation}
is defined as the ratio between the flap horizontal stroke and the amplitude of the incident waves, $\Theta$ being the solution of the equation of motion (\ref{eq:bodymot}). Finally, the capture factor is defined as the ratio between the power extracted per unit flap width and the power available per unit crest length
\begin{equation}
C_F=\frac{P}{\frac{1}{2} A_I^2 C_g w},
\label{eq:CF}
\end{equation}
where
\begin{equation}
C_g=\frac{\omega}{2k}\left(1+\frac{2kh}{\sinh 2kh}\right)
\end{equation}
is the group velocity of the incident waves. Since $P=P(a)$ and $w=1-a$, the capture factor (\ref{eq:CF}) depends intrinsically on the aperture of the array. A strength point of the method of \cite{RD12} is that knowing the coefficients $\alpha_{0n}$ and $\beta_0$ is sufficient to obtain immediately all the physical quantities describing the performance of the device (eqn.s \ref{eq:mu}--\ref{eq:CF}), without need to evaluate the potentials (\ref{eq:radpotsum}) and (\ref{eq:difpotsum}).
The wave motion at large distance from the array will be now analysed.
\section{The far field}
In this section the behaviour of the wave field is investigated at large distance from the array. First, consider the radiation potential $\phi^R_{npm}$ given by (\ref{eq:radpot}). For $n>0$ the Hankel function in (\ref{eq:radpot}) can be rewritten as
\begin{eqnarray}
H_1^{(1)}\left(\kappa_n|x|\sqrt{1+\left(\frac{y}{x}-\frac{wu}{2x}-\frac{m}{x}\right)^2}\right)\nonumber\\
=-\frac{2}{\pi}K_1\left(\kappa_n|x|\sqrt{1+\left(\frac{y}{x}-\frac{wu}{2x}-\frac{m}{x}\right)^2} \right),
\label{eq:modbes}
\end{eqnarray}
where $K_n$ denotes the modified Bessel function of the second kind and order $n$ \citep[see \S 8.407 of][]{GR07}. Since $K_1(z)\propto e^{-z}$ as $z\rightarrow\infty$ in (\ref{eq:modbes}) and hence in (\ref{eq:radpot}), the argument of $\phi^R_{npm}$ for $n>0$ decays exponentially in the far field, so that at leading order
\begin{equation}
\phi^R_{npm}\sim 0, \quad |x|\rightarrow\infty\,,n>0.
\end{equation}
This happens since the modes $n>0$ physically represent the parasite waves generated by the motion of the flaps. These remain trapped near the device and do not contribute to the wave motion in the far field \cite{ME05}. As a consequence,
\begin{equation}
\phi^R\sim \sum_{p=0}^P\sum_{m=-\infty}^{+\infty}\phi^R_{0pm},\quad |x|\rightarrow\infty.
\label{eq:phirapp}
\end{equation}
Now substituting (\ref{eq:radpot}) into (\ref{eq:phirapp}), using the asymptotic expression (\ref{eq:Sff}) with $(X,Y)=(x,y-wu/2)$ and the integral formulae (\ref{eq:Ip0}), (\ref{eq:Ipq2}) and finally developing some straightforward algebra, yields
\begin{equation}
\phi^R\sim -\frac{iV}{\omega}\frac{\cosh k(z+h)}{\cosh kh}\,\sum_{q=0}^{\bar{q}}\mathcal{A}_q^{\pm}e^{\pm i\gamma_q kx} \cos(2q\pi y),\quad x\rightarrow\pm\infty.
\label{eq:phirff}
\end{equation}
In the latter expression, $\gamma_q=\sqrt{1-(2q\pi/k)^2}$ and $\bar{q}$ is the largest integer for which $\gamma_q$ is real, while
\begin{eqnarray}
\mathcal{A}_0^{\pm}&=&\mp\frac{i\pi}{8}\,w\,\omega\alpha_{00}Z_0(0),\label{eq:A0}\\
\mathcal{A}_q^{\pm}&=&\mp \frac{i}{4}\,w\,\omega Z_0(0)\, \epsilon_q\sum_{p=0}^{P} \alpha_{(2p)0}(-1)^p(2p+1)\frac{J_{2p+1}(q\pi w)}{qw}.\label{eq:Aq}
\end{eqnarray}
In (\ref{eq:Aq}), $\epsilon_q$ is the Jacobi symbol, while $J_{2p+1}$ is the Bessel function of first kind and order $2p+1$. Note that the radiation potential in the far field (\ref{eq:phirff}) is the sum of a progressive long-crested wave (term $q=0$) and several progressive short-crested waves (terms $0<q<\bar{q}$), which correspond to the propagating sloshing modes of the equivalent channel configuration of \cite{RD12}. Expression (\ref{eq:phirff}) is similar in form to (2.24) of \citep{SR80} (accounting for the various differences in the nomenclature), which gives the far-field expression of the radiation potential for a floating body, symmetric with respect to the $x$ axis, in a channel. In (2.24) of \cite{SR80}, however, the coefficients $\mathcal{A}_{q}^{\pm}$ are left in a general form, while here they are determined explicitly for the flap-type converter. The same steps can be repeated to find the far-field expression of the diffraction potential $\phi^D$ (\ref{eq:difpotsum}). By substituting (\ref{eq:difpot}), (\ref{eq:Sff}), (\ref{eq:Ip0}) and (\ref{eq:Ipq2}) in (\ref{eq:difpotsum}) and developing the algebra, the diffraction potential in the far field becomes
\begin{equation}
\phi^D(x,y)\sim \mp\frac{iA_I}{\omega}\frac{\cosh k(z+h)}{\cosh kh}\sum_{q=0}^{\bar{q}}R_q e^{\pm i\gamma_q kx}\cos(2q\pi y),\quad x\rightarrow\pm\infty,
\label{eq:phidff}
\end{equation}
where
\begin{eqnarray}
R_0&=&-\frac{i\pi}{8}\,w\,\omega\beta_0Z_0(0),\label{eq:R0}\\
R_q&=&-\frac{i}{4}\,w\,\omega Z_0(0)\,\epsilon_q\sum_{p=0}^P\beta_{2p}(-1)^p(2p+1)\frac{J_{2p+1}(q\pi w)}{qw}.\label{eq:Rq}
\end{eqnarray}
Equation (\ref{eq:phidff}) is similar to (2.25) of \cite{SR80}, in which, however, the $R_q$ are left in a general form.
Note that the calculation of the coefficients $\mathcal{A}_q^\pm$ and $R_q$ is straightforward once the linear system for $\alpha_{(2p)0}$ and $\beta_{2p}$ is solved \citep[see eqn. B23 of][]{RD12}.
\subsection{The free-surface elevation}
The amplitude of the free surface in the far field is an important parameter in order to assess the impact of the array on the wave climate of the surrounding area. Given the total potential $\Phi(x,y,t)$, the free-surface elevation is
$$\zeta(x,y,t)=-\Phi_{,t}|_{z=0}=\Re\left\lbrace\eta(x,y) e^{-i\omega t}\right\rbrace,$$
where
\begin{equation}
\eta(x,y)=i\omega(\phi^I+\phi^R+\phi^D)
\end{equation}
is the relevant complex spatial component. Substituting (\ref{eq:incidentwav}) for $\phi^I$ and the far-field expressions (\ref{eq:phirff}) and (\ref{eq:phidff}) for $\phi^R$ and $\phi^D$, respectively, yields
\begin{equation}
\eta(x,y)\sim \left\lbrace
\begin{tabular}{l l}
$A_Ie^{-ikx}+\sum_{q=0}^{\bar{q}}(A_I R_q+V\mathcal{A}_q^+)e^{i\gamma_q kx}\cos(2q\pi y)$, & $x\rightarrow+\infty$\\
$\sum_{q=0}^{\bar{q}}(A_I T_q+V\mathcal{A}_q^-)e^{-i\gamma_q kx}\cos(2q\pi y)$,& $x\rightarrow-\infty$
\end{tabular}
\right.,
\label{eq:etaff}
\end{equation}
where
\begin{equation}
T_0=1-R_0, \quad T_q=-R_q.
\label{eq:T0}
\end{equation}
Overall, the free-surface elevation is the sum of a long-crested wave (term $q=0$) and several short-crested waves (terms $0<q<\bar{q}$), namely the propagating transverse modes of the array. Physically, in (\ref{eq:etaff}) the terms $\mathcal{A}_q^{\pm}$ represent the $q$th-mode radiation coefficients, $R_q$ is the $q$th-mode reflection coefficient and finally $T_q$ represents the $q$th-mode transmission coefficient. They enjoy all the general properties of the analogous terms introduced by Srokosz \cite{SR80} for bodies of symmetric shape in a channel. In addition, such coefficients have some specific properties, peculiar to flap-type bodies, which derive from their analytical structure, as shown in detail in \S\ref{sec:rel}. Figure \ref{fig:porevcom} shows the behaviour of $R_0$ and $T_0$ against the nondimensional wavenumber $k$ for a typical configuration where $a=1/2$. The plots in figure \ref{fig:porevcom} compare favourably with those of Williams \& Crull \cite[fig. 3]{WC93} and Porter \& Evans \cite[fig. 2]{PE96}, who studied the scattering of incident waves by an array of thin screens.
\begin{figure}[t]
\begin{center}
\includegraphics[width=8cm, trim= 5cm 9cm 5cm 0cm]{porevcom.eps}
\end{center}
\caption{Magnitude of the fundamental reflection and transmission coefficients, respectively $R_0$ (\ref{eq:R0}) and $T_0$ (\ref{eq:T0}), versus the non-dimensional wavenumber $k$. In this layout the flap width equals the gap size, i.e. $a=w=1/2$.
}
\label{fig:porevcom}
\end{figure}
Note the spiky behaviour of the coefficients, with spikes occurring at the resonant wavenumbers $k=2(\bar{q}+1) \pi$, $\bar{q}=0,1,\dots$ of the transverse short-crested waves, for which the $(\bar{q}+1)$th transverse mode turns from trapped to propagating.
In the following, the coefficients $R_0$, $T_0$ and $\mathcal{A}_0^\pm$ will be shown to enjoy some interesting properties and to be very useful for determining some relations between the hydrodynamic coefficients of the system.
\section{Derivation of relations for an array of flap-type WECs}
\label{sec:rel}
In this section, relations are derived for an array of flap-type WECs, based on the results obtained in the previous section. Some of these relations correspond directly to Srokosz's results \cite{SR80} for floating bodies of symmetric shape, while some others incorporate specific properties of the wave field (\ref{eq:etaff}) generated by the array of flap-type converters. In this sense, such expressions are new and point out the peculiarity of flap-type WECs with respect to the converters of the first generation.
\subsection{Extended Bessho-Newman relation}
First consider $\mathcal{A}_0^+$ and $R_0$. From (\ref{eq:A0}) and (\ref{eq:R0}) it is immediate to get, respectively,
$\arg(\mathcal{A}_0^+)=\arg(\alpha_{00})-\pi/2$ and $\arg(R_0)=\arg(\beta_0)-\pi/2$. Since $\arg(\alpha_{00})=\arg(\beta_0)$ \citep[see Appendix C of][]{RD12}, then the complex coefficients $\mathcal{A}_0^+$ and $R_0$ must have the same argument, say $\delta$, for which
\begin{equation}
\mathcal{A}_0^+=|\mathcal{A}_0^+|e^{i\delta},\quad R_0=|R_0|e^{i\delta},
\label{eq:delta}
\end{equation}
for any wavenumber $k$. The same relation, but only for $k<2\pi$, can be also derived from the Bessho-Newman relation
\begin{equation}
A_0^+-\sum_{q=0}^{\bar{q}}\frac{\gamma_q}{\epsilon_q}\left(A_q^{+\ast}R_q+A_q^{-\ast}T_q \right)=0,
\label{eq:BN}
\end{equation}
where $()^\ast$ denotes the complex conjugate. Expression (\ref{eq:BN}) is obtained by applying Green's integral theorem to $\phi^S$ and $(\phi^R-\phi^{R\ast})$ and corresponds to (3.2) of \citep{SR80}, with small variations due to the difference in the nomenclature. Note that the Bessho-Newman relation (\ref{eq:BN}) is a general form valid for any floating body, symmetric with respect to the $x$ axis, in a channel (or for an infinite array of such bodies). Considering $k<2\pi$, i.e. $\bar{q}=0$, and using the identities $\mathcal{A}_0^-=-\mathcal{A}_0^+$ (see \ref{eq:A0}), with $\mathcal{A}_0^+=|\mathcal{A}_0^+|e^{i\delta}$, and $T_0=1-R_0$ (see \ref{eq:T0}), (\ref{eq:BN}) becomes: $2 R_0-1=e^{2i\delta}$, which implies (\ref{eq:delta}). However, while with the general Bessho-Newman relation (\ref{eq:BN}) it is possible to obtain (\ref{eq:delta}) only in the domain $k<2\pi$, usage of the explicit forms (\ref{eq:A0}) and (\ref{eq:R0}), respectively for $\mathcal{A}_0^+$ and $R_0$, has allowed to extend (\ref{eq:delta}) to any wavenumber. Furthermore, by using (\ref{eq:T0}) and (\ref{eq:delta}), (\ref{eq:BN}) yields
\begin{equation}
\cos\delta=|\mathcal{A}_0^+|^{-1}\sum_{q=0}^{\bar{q}}\frac{\gamma_q}{\epsilon_q}\,\left|\mathcal{A}_q^+R_q\right|,
\label{eq:cosdelta}
\end{equation}
for any $k$. Expression (\ref{eq:cosdelta}) is a particular form of the Bessho-Newman relation, valid for a periodic array of flap-type converters under normally-incident waves. Note that for $k<2\pi$, i.e. $\bar{q}=0$, all the transverse modes are trapped near the array and (\ref{eq:cosdelta}) reduces to
\begin{equation}
\cos\delta=|R_0|,\quad k<2\pi.
\label{eq:cosdeltaR0}
\end{equation}
\subsection{Relation between $F$ and $\mathcal{A}_0^\pm$ (array Haskind relation)}
Consider the complex exciting torque (\ref{eq:F}) and the fundamental radiation coefficient (\ref{eq:A0}). Isolating the term $\alpha_{00}$ from (\ref{eq:A0}) and substituting it into (\ref{eq:F}), yields after some algebra
\begin{equation}
F=\pm2A_I\mathcal{A}_0^{\pm}C_g.
\label{eq:haskind}
\end{equation}
According to (\ref{eq:haskind}), the long-crested component of the radiated wave field is sufficient to obtain the exciting torque acting on each flap, for any value of $k$. Furthermore, since $A_I$ and $C_g$ are real numbers, (\ref{eq:haskind}) requires
\begin{equation}
F=|F|e^{i\delta}.
\label{eq:phaseF}
\end{equation}
Expression (\ref{eq:haskind}) can be transformed into physical variables via (\ref{eq:nondimvar}), thus giving
$$F'=2\rho g b'A_I'\mathcal{A}_0^+{'}C_g'.$$
The latter is similar in form to the well-known two-dimensional Haskind relation \cite{ME05} except for the factor $b'$, which represents the array spacing. Finally, note that (\ref{eq:haskind}) is an extension to intermediate water depth of Srokosz's equation (4.3) in \cite{SR80}.
\subsection{Relation between $F$ and $R_0$}
The relation between the exciting torque and the fundamental reflection coefficient can be easily obtained by isolating $\beta_0$ from (\ref{eq:R0}), substituting it in (\ref{eq:F}) together with (\ref{eq:Zn}) and developing the algebra, so that
\begin{equation}
F=2A_IR_0\frac{\tanh kh}{k}\left(h-c+\frac{\cosh kc-\cosh kh}{k \sinh kh}\right).
\label{eq:relFR0}
\end{equation}
Hence the exciting torque acting on each flap is related to the amplitude of the long-crested component of the reflected wave field, for any value of $k$.
\subsection{Relation between $R_0$ and $\mathcal{A}_0^+$}
By equating (\ref{eq:haskind}) and (\ref{eq:relFR0}) it is immediate to obtain
\begin{equation}
\frac{R_0}{\mathcal{A}_0^+}=\frac{kC_g}{\tanh kh \left(h-c+\frac{\cosh kc-\cosh kh}{k \sinh kh} \right)},
\label{eq:relR0A0p}
\end{equation}
valid for any $k$. Physically, (\ref{eq:relR0A0p}) measures the ratio between the reflective capacity of the system as an array of screens and the radiative capacity of the system as an array of wavemakers, oscillating at unison. In short waves, where the flaps are deemed to be operating \cite{HE10}, it is roughly $k\gg 1$, so that (\ref{eq:relR0A0p}) becomes
\begin{equation}
\frac{R_0}{\mathcal{A}_0^+}\simeq \frac{\sqrt{k}}{2\left(h-c-k^{-1}\right)},
\label{eq:approxex}
\end{equation}
as shown in figure \ref{fig:shortwav}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=7.5cm, trim= 5cm 10cm 5cm 0cm]{shortwav.eps}
\end{center}
\caption{Ratio $R_0/\mathcal{A}_0^+$ (\ref{eq:relR0A0p}) versus non-dimensional wavenumber $k$ (solid line) and approximate expression (\ref{eq:approxex}) for large $k$ (dashed line). Parameters of the system are $b'=91.6\,\mathrm{m}$, $h'=10.9\,\mathrm{m}$, $c'=1.5\,\mathrm{m}$.}
\label{fig:shortwav}
\end{figure}
Since the ratio (\ref{eq:approxex}) is $O(k^{1/2})$, the diffractive phenomena occurring in the system dominate over the radiative ones in short waves. This suggests that the effects of diffraction are not to be neglected if an accurate description of the system dynamics is to be pursued.
\subsection{Relation between $\nu$ and $\mathcal{A}_0^+$}
Consider expression (\ref{eq:nu}), which defines the radiation damping $\nu$ for the reference plate. Isolating $\Im\left\lbrace\alpha_{00}\right\rbrace$ in (\ref{eq:A0}) and substituting it into (\ref{eq:nu}) yields, after some algebra,
\begin{equation}
\nu=2\Re\left\lbrace \mathcal{A}_0^+\right\rbrace \frac{\tanh kh}{k}\left(h-c+\frac{\cosh kc-\cosh kh}{k\sinh kh}\right),
\label{eq:relnua0}
\end{equation}
for any $k$. Incidentally, by isolating $\mathcal{A}_0^+$ in the array Haskind relation (\ref{eq:haskind}) and substituting it into (\ref{eq:relnua0}), the latter becomes
$$\nu=\Re\left\lbrace \frac{F}{A_I}\right\rbrace \frac{\tanh kh}{kC_g}\left(h-c+\frac{\cosh kc-\cosh kh}{k\sinh kh}\right),$$
which corresponds to expression (C3) of Renzi \& Dias \cite{RD12}. Note that (\ref{eq:relFR0}), (\ref{eq:relR0A0p}) and (\ref{eq:relnua0}) allow to obtain the exciting torque and the radiation damping - and consequently the optimum generated power (\ref{eq:Popt}) - directly from the fundamental reflection coefficient $R_0$. This is a peculiar property of the flap-type converter and does not hold in general for converters of different shape.
The above relations have been used to check the numerical calculations in this paper. In order to assess the accuracy of computations for a given equation of the form $\mathrm{l.h.s}=\mathrm{r.h.s.}$, the relative error
\begin{equation}
\epsilon=\frac{|\mathrm{l.h.s.}-\mathrm{r.h.s.}|}{|\mathrm{r.h.s}|}
\label{eq:error}
\end{equation}
is defined. For a typical system configuration (see \ref{sec:appD}), taking 40 array modes, 5 depth modes and 5 terms in the Chebyshev expansion, is sufficient to obtain a relative error of $O (10^{-16})$ in calculating the Haskind relation (\ref{eq:haskind}) and $O(10^{-15})$ in calculating the remaining relations (\ref{eq:relFR0})--(\ref{eq:relnua0}). Hence the method of solution based on the Green's theorem of \cite{RD12} reveals to be fast convergent and very efficient. In the following, the influence of the array aperture on the performance of the system is assessed, based on the relations found in this section.
\section{Performance evaluation}
\label{sec:paran}
Consider the optimum capture factor
\begin{equation}
C_F^{opt}=\frac{1}{4}\frac{|F|^2}{\nu A_I^2 C_g w},
\label{eq:CFopt}
\end{equation}
obtained by substituting the optimum power output (\ref{eq:Popt}) into (\ref{eq:CF}). By replacing $F$ with (\ref{eq:relFR0}), $\nu$ with (\ref{eq:relnua0}), and by performing some algebra, (\ref{eq:CFopt}) can be rewritten as
\begin{equation}
C_F^{opt}=\frac{1}{2}\frac{|R_0|}{(1-a)\cos\delta},
\label{eq:CFopt2}
\end{equation}
where $\delta$ is still the argument of $R_0$. According to (\ref{eq:CFopt2}), the performance of the array depends on the reflection coefficient magnitude and argument, which in turn are functions of the array aperture. Hence the solution of the scattering problem alone is sufficient to assess the performance of the system via (\ref{eq:CFopt2}). This result confirms that diffraction effects are fundamental in wave-power extraction from flap-type WECs. Therefore, the empiric criterion for which ``to absorb waves means to generate waves'' \cite{FA02}, valid for small floating bodies in the absence of diffraction, does not apply here in full. In the following, expression (\ref{eq:CFopt2}) will be validated against known theories in the small-gap and point-absorber limits. Then the maximum capture factor will be assessed.
\subsection{Small-gap limit}
In the limit $a\rightarrow0$, the flaps become joined to each other and the system is two-dimensional. In this case it is $R_0\rightarrow1$, because of complete reflection of the incident wave in the diffraction problem. Then it is straightforward to show that (\ref{eq:CFopt2}) becomes
\begin{equation}
C_F^{opt}\rightarrow\frac{1}{2},
\label{eq:CF2d}
\end{equation}
i.e. the capture factor coincides with the classical hydrodynamic efficiency for 2D devices \cite{ME05}.
\subsection{Point-absorber limit}
Consider now the limit $w=w'/b'\rightarrow 0$, for fixed array spacing $b'$. In this limit, the wavelength $\lambda=2\pi/k$ of the incident wave is much larger than the flap width, $\lambda\gg w$, and the interaction between the flaps is weak. Hence the results of the present theory can be compared to those of Budal \cite{BU77} and Srokosz \cite{SR80} for an infinite array of point absorbers. For such a system, the efficiency is assessed via the absorption length
\begin{equation}
L'=\frac{P}{\frac{1}{2} A_I^2 C_g}\,b'=C_F w',
\label{eq:abslen}
\end{equation}
which is the ratio between the power captured by the single device and that incident per unit wave crest length. For an array of converters the optimum absorption length is
\begin{equation}
L'_{opt}=l'_{opt}\,s,
\label{eq:abslen2}
\end{equation}
where $l'_{opt}$ is the optimum absorption length for an isolated body and $s$ is an interaction factor \cite{BU77, SR80}. When $w\ll \lambda $, each flap can be considered as a three-dimensional axisymmetric body, whose optimum absorption length for given wavelength of the incident wave is
\begin{equation}
l'_{opt}=\xi \frac{\lambda}{2\pi}\,b',
\label{eq:pointabs}
\end{equation}
where $\xi=1$ for heave and $\xi=2$ for surge (see \cite{EV76,HE10}). By substituting (\ref{eq:pointabs}) into (\ref{eq:abslen2}), then the latter into (\ref{eq:abslen}) and employing (\ref{eq:CFopt2}) for the optimum capture factor, the interaction factor becomes
\begin{equation}
s=\frac{k}{2\xi}\frac{|R_0|}{\cos\delta},
\label{eq:qfactor}
\end{equation}
where $\xi=2$, because in the point-absorber approximation the flap moves essentially in surge \cite{HE10}. Figure \ref{fig:qfactor} shows the plot of $s$ versus the non-dimensional wavenumber $k=k'b'$ for a typical configuration in which $w=0.05$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=7cm, trim= 5cm 9cm 5cm 0cm]{qfactor.eps}
\end{center}
\caption{Interaction factor (\ref{eq:qfactor}) versus non-dimensional wavelength $k$ for the point-absorber approximation. Parameters of the system are $b'=91.6\,\mathrm{m}$, $w'=4.58\,\mathrm{m}$, $h'=10.9\,\mathrm{m}$, $c'=1.5\,\mathrm{m}$. The ratio $w=w'/b'=0.05$ justifies the use of the point-absorber approximation. The vertical dash-dotted lines show the resonant wavenumbers $k=2(\bar{q}+1)\pi$; the grey dashed line shows the linear behaviour $s=k/4$ for $k<2\pi$.}
\label{fig:qfactor}
\end{figure}
When $k< 2\pi$ (i.e. $\lambda> 1$) all the transverse modes are trapped and the behaviour of the curve is linear: $s\simeq k'b'/(2\xi)$. This agrees formally with the results shown by Budal \cite{BU77} and Srokosz \cite{SR80} for an infinite array of heaving point-absorbers, where $\xi=1$. When $k>2\pi$, i.e. $\lambda<1$, incomplete trapping of the transverse modes strongly modifies the behaviour of the curve. The interaction factor globally decreases, but spikes occur near the resonant wavenumbers $k=2(\bar{q}+1)\pi$. Note that this dynamics is different from that shown in \cite{SR80} for a system of heaving point absorbers. In the latter, $s$ drops to zero when $k\rightarrow2(\bar{q}+1)\pi$ from the left, so that the trapping of the transverse modes has a detrimental effect on the performance of the system \cite[see fig. 2]{SR80}. Here, instead, resonance produces local maxima of $s$ near $k=2(\bar{q}+1)\pi$ (see again figure \ref{fig:qfactor}) and therefore is beneficial in increasing the optimum efficiency, even away from complete trapping. This happens because the resonance of the transverse modes enhances the horizontal (surge) actions and drops the vertical (heave) loads \cite{CH94}. Therefore surging WECs benefit the most from the resonant mechanisms activating in an array configuration.
\subsection{Maximum capture factor}
Consider again the optimum capture factor, given by (\ref{eq:CFopt2}). Expression (\ref{eq:cosdelta}) shows that $|R_0|/\cos\delta$ has a unit upper limit (\ref{eq:cosdeltaR0}) when all the transverse modes are trapped near the array. This situation is the most favourable for energy extraction and yields the maximum capture factor attainable by an array of oscillating wave energy converters of given aperture $a$. Substitution of (\ref{eq:cosdeltaR0}) into (\ref{eq:CFopt2}) yields
\begin{equation}
C_F^{opt}= C_F^{max}=\frac{1}{2(1-a)},\quad k<2\pi
\label{eq:maxCF}
\end{equation}
for the maximum capture factor. Incidentally, note that (\ref{eq:maxCF}) correspond to Srokosz's \cite{SR80} maximum efficiency $E^{max}=wC_F^{max}$ for a symmetric floating body in a channel. Since $a<1$, the maximum capture factor (\ref{eq:maxCF}) for the array configuration is larger than the well-known limit value of $1/2$, obtained in the small-gap approximation $a\rightarrow 0$ (see expression \ref{eq:CF2d}). Hence the mutual interaction between the flaps, which is responsible for the trapping of energy near the array in the form of short-crested waves, can increase the capture factor of the system \cite{RD12}.
\section{Application to wave energy extraction}
\label{sec:waven}
In applications, the incident wave period $T'$ and wavelength $\lambda'$ are known, together with the flap width $w'$. The array aperture $a$ needs to be optimised so that the capture factor is maximum. Expression (\ref{eq:maxCF}) for the maximum capture factor would suggest to increase the aperture as much as possible, so that $a\rightarrow 1$ and consequently $C_F^{max}\rightarrow \infty$. However, (\ref{eq:maxCF}) is to be regarded as a theoretical upper limit. This is motivated by two reasons. First and most important, as $a\rightarrow 1$, then $w=1-a\rightarrow 0$. Now recall that $w=w'/b'$, being $w'$ the width of the single flap, which in practical applications is large. As a consequence, in order to have $w\rightarrow 0$, it must be $b'\rightarrow \infty$. In this limit, $\lambda=\lambda'/b'\rightarrow 0$, $k\rightarrow \infty$ and expression (\ref{eq:maxCF}) for the maximum capture factor is no longer valid. Physically, by increasing the array aperture $a$, the spatial period $b'$ increases so much, that the array is no longer able to trap all the transverse modes, resulting in more energy leakage. Second, recall that expression (\ref{eq:maxCF}) for the theoretical maximum capture factor is obtained under the assumption (\ref{eq:bodyres}), i.e. that the self-oscillation frequency of each flap is tuned to the frequency of the incoming waves. However, flap-type converters are usually designed to avoid this eventuality. At body resonance, the stroke of the flap would exceed by far the amplitude of the incident wave \cite{HE10}. It is then clear that this condition is undesirable and not compatible with the power take-off mechanism \cite{CR08}. Away from body resonance, the maximum values of $C_F$ attained are smaller than those predicted by (\ref{eq:maxCF}). This is due to the presence of the inertial terms at the denominator of $P$ (\ref{eq:pow}), which in turn reduce $C_F$ (\ref{eq:CF}).
The mathematical model of \S\ref{sec:model} is now applied to determine an optimisation criterion for the array aperture $a$, which maximises the power output of an array of flap-type WECs. The configuration investigated here is that of an infinite array of converters similar to Oyster 800\texttrademark\footnote{Oyster is a trademark of Aquamarine Power Limited.}. Each converter has a width $w'=26\,\mathrm{m}$ and is placed upon a foundation of height $c'=1.5\,\mathrm{m}$ from the bottom of the ocean; water depth is $h'=10.9\,\mathrm{m}$. Monochromatic incident waves of amplitude $A'_I=1\mathrm{m}$, period $T'=7\,\mathrm{s}$ (wavelength $\lambda'= 62\,\mathrm{m}$), representative on average of the wave climate off the west coast of Ireland \cite{R12}, are considered. For these parameters, several different layouts, from compact ($a=0.30$) to sparse ($a=0.95$), are analysed to determine the optimum array aperture $a_{opt}$. In each case the power $P'=\rho A{'}^2b{'}^{3/2}g^{3/2}P$ extracted by a single flap and the relevant capture factor $C_F$ (\ref{eq:CF}) are calculated with the mathematical model of \S\ref{sec:model}. Table \ref{tab:1} shows the selected values of $a$, the corresponding value of the array spacing $b'$, the generated power $P'$ in $\mathrm{kW}$, the capture factor $C_F$ and the theoretical maximum $C_F^{max}$ .
\begin{table}\centering
\begin{tabular}{c c c c c c c}
\hline
$a$ & $0.3$& $0.40$ & $0.50$ & $a_{opt}=0.58$ & $0.70$ & $0.95$ \\ \hline
$b'\,(\mathrm{m})$ & 37 & 43 & 52 & 62 &87 & 520\\\hline
$P'\,(\mathrm{kW})$ & 504 & 564 & 660 & 795 & 574 & 605\\ \hline
$C_F$ & 0.60 & 0.68 & 0.79 & 0.95 & 0.69 & 0.73\\ \hline
$C_F^{max}$ & 0.71 & 0.83 & 1 & 1.19 & (1.67) & (10)\\\hline
\end{tabular}
\caption{Array spatial period $b'$, power output $P'$ , capture factor $C_F$ and maximum theoretical capture factor $C_F^{max}$ for an infinite array of flap-type converters similar to Oyster 800\texttrademark. Different apertures are considered, from compact ($a=0.3$) to sparse ($a=0.95$). Calculations are made with the mathematical model of \S\ref{sec:model}.}
\label{tab:1}
\end{table}
The largest power output and capture factor are attained at the optimum configuration $a=a_{opt}=0.58$, which corresponds to $\lambda'=b'$ (i.e. $k=2\pi$), the trapping wavelength of the first transverse mode \cite{RD12}. For $a<a_{opt}$, $\lambda'>b'$ (i.e. $k<2\pi$) and all the transverse modes are perfectly trapped (see table \ref{tab:1}). However, since $a$ is small, the theoretical maximum (\ref{eq:maxCF}) sets a relatively small upper limit for $C_F$. By increasing $a$, $C_F^{max}$ increases and so does the actual capture factor $C_F$, until it reaches its maximum at $a=a_{opt}$. For $a>a_{opt}$ the theoretical limit $C_F^{max}$ still increases, while the actual capture factor $C_F$ decreases. This happens since in these cases $\lambda'<b'$ (i.e. $k>2\pi$) and complete trapping of the transverse modes is not possible, so that (\ref{eq:maxCF}) does not hold in practice. Energy leakage associated to the propagating transverse waves lowers the power absorption of the array well below the theoretical maximum values. In conclusion, the optimum aperture that maximises the capture factor is the one for which $\lambda'=b'$, i.e.
\begin{equation}
a_{opt}=1-\frac{w'}{\lambda'},
\end{equation}
which can be used as a preliminary design formula.
The theory exposed here reveals to be useful for the optimisation of the efficiency of an infinite array of flap-type wave energy converters in incident monochromatic waves of given period. Of course, in real seas superposition of different wave components must be considered. The power output may thus vary, depending on the coupling between the spectrum energy period and the torque peak period \cite{R12, CL12}. Further analysis is therefore necessary to obtain more accurate estimates of wave power generation in random seas. Finally, due to real sea bottom conditions, converters in array are likely to be deployed in a staggered configuration and in a finite number. Ongoing work is investigating the dynamics of a finite array of staggered converters and will be disclosed in the near future.
\section{Conclusions}
A periodic array of flap-type WECs has been analysed in this work by using the semi-analytical model of Renzi \& Dias \cite{RD12}. Asymptotic analysis in the far field has allowed to obtain new expressions for the radiation, reflection and transmission coefficients. Relations have been determined between the $0$th-mode coefficients and the hydrodynamic parameters of the system. Some of these relations constitute an extension to intermediate water of the previous results obtained by Srokosz \cite{SR80} for an array of floating bodies in deep water, while some others are peculiar to the flap-type converter. The efficiency of the system, evaluated via the capture factor, has been shown to depend on the reflection coefficient magnitude and argument, which in turn are functions of the array aperture. This result shows that diffraction effects are fundamental in wave-power extraction from flap-type WECs. Unlike a line of heaving buoys \cite{SR80}, an array of flap-type WECs can exploit the resonance of transverse modes to attain high capture factor levels, even when complete trapping of the transverse modes does not occur. The maximum capture factor is attained in the regime of complete trapping, for which the amount of energy available for extraction is the largest. Given the wave period and the flap width, the capture factor can be maximised by varying the spacing between the flaps, such that complete trapping of the transverse modes occurs. These results have been obtained under the assumptions that the fluid is inviscid and the flow is irrotational. Viscous effects and turbulent dissipations may reduce the values predicted here, especially near trapping frequencies (see \cite{CH94}). \\
This work was funded by Science Foundation Ireland (SFI) under the research project ``High-end computational modelling for wave energy systems''. Discussions with Prof. D.V. Evans and Dr X. B. Chen have been illuminating. Numerical data provision by Dr G. Bellotti and Mr A. Abdolali is gratefully acknowledged.
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train/arxiv
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BkiUdfPxaKPQoka4R7Xr
| 5 | 1 |
\section{Introduction}
During the last ten years, much have been done about classification of special solutions of isomonodromy equations.
Recall how Painlev\'e and Garnier differential equations arise in the computation of monodromy-preserving
deformations of linear
equations over the Riemann sphere. Consider the general $2^{\text{nd}}$ order linear differential equation
\begin{equation}\label{eq:Gal2ndOrderGarnierReg}
u''+f(x)u'+g(x)u=0,\ \ \ \ (u'=\frac{du}{dx})
\end{equation}
$$\left\{\begin{matrix}
f(x)=&\frac{\theta_{N+1}}{x}+\frac{\theta_{N+2}}{x-1}+\sum_{i=1}^N\frac{\theta_i}{x-t_i}-\sum_{j=1}^N\frac{1}{x-q_j}\\
g(x)=&\frac{c_0}{x}+\frac{c_1}{x-1}-\sum_{i=1}^N\frac{H_i}{x-t_i}+\sum_{j=1}^N\frac{p_j}{x-q_j}
\end{matrix}\right.$$
with $2N+3$ regular-singular points distributed as follows:
\begin{itemize}
\item $N+3$ essential singular points $x=t_1,\ldots,t_N,0,1,\infty$ with exponents $\theta_i$, $i=1,\ldots,N+3$, and
\item $N$ apparent singular points $x=q_1,\ldots,q_N$ (with trivial local monodromy).
\end{itemize}
Coefficients $c_0,c_1$ and $H_i$ can be explicitely determined as rational functions of all other parameters
$t_i$'s, $q_i$'s, $p_i$'s, and $\theta_t$'s after imposing the following constraints
\begin{itemize}
\item the singular point at $x=\infty$ is regular-singular
with exponent $\theta_{N+3}$,
\item the singular points $x=q_1,\ldots,q_N$ are apparent.
\end{itemize}
Then, it follows from the works of Fuchs, Garnier, Okamoto, Kimura that
an analytic deformation
$$t\to (p_1(t),\ldots,p_N(t),q_1(t),\ldots,q_N(t))\ \ \ \text{with}\ \ \ t=(t_1,\ldots,t_N),$$
of equation (\ref{eq:Gal2ndOrderGarnierReg})
is isomonodromic (i.e. with constant monodromy) if,
and only if\footnote{The ``only if'' needs that exponents $\theta_t$'s are not integers.}
all $\theta_t$'s are fixed and other parameters satisfy the Hamiltonian system
\begin{equation}\label{eq:HamiltonianGarnierSystem}
\frac{dq_j}{dt_i}=\frac{\partial H_i}{\partial p_j}\ \ \ \text{and}\ \ \ \frac{dp_j}{dt_i}=-\frac{\partial H_i}{\partial q_j}\ \ \
\forall i,j=1,\ldots,N.
\end{equation}
The system (\ref{eq:HamiltonianGarnierSystem}) reduces to the Painlev\'e VI equation for $N=1$, and to the Garnier system
for $N>1$. It is integrable, in the sense that it admits a local solution for each initial data. These local solutions are
expected to be very transcendental in general, and this has been proved by Umemura in the Painlev\'e case $N=1$:
for any choice of $\theta_i$'s, the general solution cannot be explicitely expressed in terms of solutions of linear
differential equations (of any order), non linear differential equations of order $1$, or algebraic functions.
However, for special choices of $\theta_i$'s, there are Riccati solutions or algebraic solutions.
The first ones, called ``classical'', have been classified by Watanabe; the later ones have been
classified mainly by Boalch, and by Lisovyy and Tykhyy (see \cite{Boalch,LisovyyTykhyy}),
after a long period of works by Hitchin, Dubrovin, Mazzocco, Andreev, Kitaev...
\cite{Hitchin,DM0,MazzoccoPicard,Doran,AK,AK2,Kitaev,Kitaev2,Kitaev3,Kitaev4,BoalchIco,BoalchTetraOcta,Boalch6,BoalchHigherGenus,VK,VK2}. In the Garnier case $N>1$, we expect a similar feature; classical solutions have been classified by
Okamoto and Kimura in \cite{OK}, and by
Mazzocco in \cite{MazzoccoGarnier}, but the classification of algebraic solutions is still open.
Following Cousin, and Heu \cite{Cousin,CousinHeu}, algebraic solutions are finite branch solutions and come from finite orbits
of the Mapping-Class-Group on character varieties, or equivalently representations on the total space of an algebraic deformation of the punctured curve, extending the monodromy representation of (\ref{eq:Gal2ndOrderGarnierReg}).
On the other hand, the result of Corlette and Simpson \cite{CS}
shows that such representations are of three different origins:
\begin{itemize}
\item degenerate representations, i.e. taking values into a finite, dihedral or reducible group,
\item factorization through a representation on a fixed curve,
\item arithmetic quotient of a polydisc.
\end{itemize}
The reader will find a more precise statement in \cite{CS}. Let us just mention in the first case
the works of Girand \cite{Girand} and Komyo \cite{Komyo} for deformations of equation (\ref{eq:Gal2ndOrderGarnierReg})
with dihedral monodromy, and Cousin and Moussard \cite{CousinMoussard} in the reducible case.
Deformations of equation (\ref{eq:Gal2ndOrderGarnierReg}) having a finite group are algebraic
and provide an algebraic Garnier solution in a systematic way, but computations can be very tedious
as it has been in the works of Boalch for the Painlev\'e case $N=1$.
In the second case, solutions are said of ``pull-back type'': the deformation of equation (\ref{eq:Gal2ndOrderGarnierReg})
is given in this case by the pull-back of a fixed differential equation (or instance rigid, hypergeometric, i.e. $N=0$)
by a family of ramified covers $f_t:\mathbb P^1\to\mathbb P^1$ (see section \ref{sec:pullback}). This method
has been used by Doran, Andreev, Kitaev, Vidunas to construct Painlev\'e VI algebraic solutions. For Garnier
systems $N>1$, all pull-back solutions with non degenerate linear monodromy have been classified by the first author
in \cite{Diarra1} (see Proposition \ref{Prop:DiarraPullBack}). However, in the last case, we do not know
how to bound the arithmetic data in order to be able to classify. This is not using this trichotomy that
algebraic solutions of Painlev\'e VI equation were found, but by brute force, which seems out of reach
even in the case $N=2$. Recently, Calligaris and Mazzocco \cite{CM} gave a partial classification
by using confuence of poles in order to exploit the Painlev\'e classification \cite{LisovyyTykhyy}.
So far, we have only considered linear differential equations with regular-singular points,
leading to the Painlev\'e VI equation and Garnier systems. There is a similar approach
for linear differential equations with irregular-singular points leading for instance, in the case $N=1$,
to the other Painlev\'e equations (see section \ref{sec:IsomDef}). They can be deduced from
the regular-singular case by confluence of poles, and whose solutions parametrize isomonodromic and
iso-Stokes deformations of linear differential equations with $4$ poles counted with multiplicity.
In a similar way, we can define irregular Garnier systems and, in the case $N=2$, they are listed
in the papers of Kimura \cite{Kimura} and Kawamuko \cite{Kawamuko}. For general case $N>1$,
such integrable systems also exist, due to the work of Malgrange (see Heu \cite{Heu} in the ramified case),
and a general formula can be found in the work of Krichever \cite{Krichever}.
Again we expect the general solution to be very transcendental, but there are classical and algebraic solutions.
In the Painlev\'e case, a complete classification of these special solutions can be found in \cite{OO};
see section \ref{sec:AlgebraicPainleveI-V} for the list of algebraic solutions for Painlev\'e I to V equations.
For Garnier systems, classical solutions have been investigated by Suzuki in \cite{Suzuki}.
For several formal types, Kawamuko and Suzuki listed rational/algebraic solutions in \cite{KawamukoRat,Suzuki,KawamukoAlg}.
This is all what is known so far about algebraic solutions of irregular Garnier systems.
Our main result is a complete classification, as well as a complete list in the case $N=2$.
Recently, the second author, together with Pereira and Touzet, proved an irregular version of Corlette-Simpson
Theorem in \cite{LPT}. An immediate consequence is that an algebraic solution of an irregular Garnier
system is of one of the two following types
\begin{itemize}
\item {\bf classical}: comes from the deformation of a rank $2$ differential system with diagonal
or dihedral differential Galois group,
\item {\bf pull-back}: comes from the deformation obtained by pull-back of a fixed linear differential equation
by a family ramified covers.
\end{itemize}
(see Corollary \ref{cor:structure}).
The main result of the paper is the classification of solutions of pull-back type. Let us describe more precisely the construction.
We consider a fixed meromorphic linear differential
equation $\mathcal E_0$ on $\mathbb P^1$, which can be a two-by-two system, a second-order scalar equation, or a rank two
vector bundle with a connection. Then, we consider a family of ramified covers $(\phi_t:\mathbb P^1\to \mathbb P^1)_t$
and the family of pull-back $\mathcal E_t:=\phi_t^*\mathcal E_0$. Clearly, the deformation $t\mapsto \mathcal E_t$
is isomonodromic and isoStokes, and this gives rise to a partial solution of a (possibly irregular) Garnier system;
moreover, if the family $(\phi_t)_t$ is algebraic, we get an algebraic partial solution.
Here, partial means that the time variable is a function of $t$ which may not be dominant,
and it won't be for general $\mathcal E_0$ and $(\phi_t)_t$. When the dimension of deformation
has the right dimension, namely $n-3$ where $n$ is the number of poles of $\mathcal E_t$
counted with multiplicity, then we get a complete algebraic solution. Sections \ref{sec:RamifiedCovers},
\ref{sec:Scattering}, \ref{sec:IrregEuler} and \ref{sec:Classifcover} are devoted to the classification
of such solutions. Inspired by the similar classification in the logarithmic case established by the first author in \cite{Diarra1},
we define the irregular analogues of curve, Teichm\"uller and moduli spaces, Euler characteristic and Riemann-Hurwitz
formula. Then we prove that, assuming $\mathcal E_0$ irregular with differential Galois group not reduced to
the diagonal or dihedral group (to avoid classical solutions), $\mathcal E_0$ is of degenerate hypergeometric type
(at most $3$ poles counted with multiplicity) and the cover degree of $\phi_t$ is bounded by $6$.
Finally, the list of solutions is obtained by scattering poles to reduce to the list of \cite{Diarra1}.
In the pure Garnier case $N>1$ (i.e. excluding Painlev\'e equations) we obtain $3$ solutions,
for Garnier systems of rank $N=2$ or $3$ (see Tables \ref{table:irregular} and \ref{table:irregularConfl}).
They all come from pull-back of the degenerate (or ramified) Kummer equation $u''+\frac{2}{3x}u'-\frac{1}{x}u=0$
by coverings of degree $4$ and $6$.
Consequently:
{\it irregular Garnier systems of rank $N>3$ admit only classical algebraic solutions.}
In order to describe our classification result, let us introduce for each singular point of the linear differential
equation the following invariants:
\begin{itemize}
\item the Poincar\'e-Katz irregularity index $\kappa\in\frac{1}{2}\mathbb Z_{\ge0}$ which is such that,
after putting the linear differential equation $u''=g(x)u$ into Sturm-Liouville normal form,
the coefficient $g(x)$ has a pole of order $2\kappa+2$;
\item the exponent $\theta\in\mathbb C$, defined up to a sign, which is the difference of eigenvalues
of the residue for the differential equation in matrix form when $\kappa\in\mathbb Z_{\ge0}$, and
$\theta=0$ in the ramified case $\kappa\in\frac{1}{2}+\mathbb Z_{\ge0}$.
\end{itemize}
Denote by $\lceil\kappa\rceil$ the smallest integer satisfying $\kappa\le k$.
This formal data can be algebraically computed from the differential equation, and is invariant under isomonodromic/isoStokes deformations. In fact, after normalizing the linear differential equation by birational gauge transformation,
in order to minimize the number and order of poles, the differential equation is determined by
\begin{itemize}
\item its irregular curve, i.e. the base curve equipped with local coordinates up to order $\lceil\kappa\rceil$ at each pole
(only the position for simple poles),
\item the monodromy data including Stokes matrices.
\end{itemize}
(see \cite{MRGalois,Krichever,BMM,vdPS}, and sections \ref{sec:IrregCurve} and \ref{sec:MonodStokes} for details).
With this in hand, for each (global) formal data (and fixed genus $g$)
$$\begin{pmatrix}\kappa_1&\cdots&\kappa_n\\
\theta_1&\cdots&\theta_n\end{pmatrix}\ \ \ \text{where}\ \ \
\left\{\begin{matrix}
\kappa_i\in\mathbb Z_{\ge0}&\Rightarrow&\theta_i\in\mathbb C\\
\kappa_i\in\frac{1}{2}+\mathbb Z_{\ge0}&\Rightarrow&\theta_i=0
\end{matrix}\right.$$
we get a quasi-projective moduli space of linear differential equations (see \cite{InabaSaito,Inaba}),
and on this moduli space we get a polynomial foliation whose leaves
correspond to deformations of the differential equation (in fact of the spectral curve)
with constant monodromy data: we call it {\it isomonodromic foliation}. In the genus $g=0$ case, this is known as (degenerate)
Garnier systems: we get a $N$ dimensional foliation on a $3N$-dimensional moduli space,
where $N=\sum_{i=1}^n \lceil\kappa_i\rceil+1$ is called the rank of the Garnier system.
Leaves with algebraic closure correspond to algebraic solutions of the Garnier system.
For $N=1$, we find all Painlev\'e equations (see section \ref{sec:IsomDef}) and algebraic solutions,
in the irregular case ($\kappa_i>0$ for one $i$ at least), are listed in section \ref{sec:AlgebraicPainleveI-V}.
There are biregular isomorphisms between these foliated moduli spaces, due to the fact that
the normalization of a linear differential equation is not unique: after birational gauge transformation,
one can shift $\theta_i$'s by integers (and we can change their sign).
We obviously classify algebraic solutions up to these isomorphisms.
\begin{thm}\label{thm:3nonclassical}
Up to isomorphisms, there are exactly $3$ non classical algebraic solutions, for irregular Garnier systems
of rank $N>1$. The list of corresponding formal data is as follows:
$$\begin{pmatrix}0& 1&1\\ \frac{1}{3} & 0&1\end{pmatrix},\ \ \ \begin{pmatrix}1&2\\ 0 & 1\end{pmatrix}\ \ \ \text{and}\ \ \
\begin{pmatrix}1& 1&1\\ 0&0 & 1\end{pmatrix}.$$
\end{thm}
For the rank $N=2$ case, the list of explicit Garnier systems is provided in \cite{Kimura,Kawamuko}.
The two first algebraic solutions are as follows under Kimura's notations \cite{Kimura}:
\begin{itemize}
\item $H(1,2,2;2)$ with parameters $\varkappa_0=\varkappa_1=0$ and $\varkappa=\frac{2}{9}$ (and $\eta_0=\eta_1=1$):
$q_1$ is implicitely defined by\footnote{In Kimura's paper,
canonical coordinates are denoted $\lambda_i$ and $\mu_i$ instead of variables $q_i$ and $p_i$ respectively,
and Hamiltonians $K_i$ instead of $H_i$}
$$\left(\frac{q_1(q_1+1)}{(q_1-1)(q_1-2)}\right)^3=\left(\frac{t_2}{t_1}\right)^2,$$
and other variables are given by
$$q_2=\frac{q_1+1}{2q_1-1},\ \ \ p_1=-\frac{1}{2}\frac{t_1}{(q_1-1)^2}-\frac{1}{2}\frac{t_2}{q_1^2}-\frac{1}{6}\frac{2q_1-1}{q_1(q_1-1)}$$
$$\text{and}\ \ \ p_2=-\frac{1}{2}\frac{(2q_1-1)^2 t_1}{(q_1-2)^2}-\frac{1}{2}\frac{(2q_1-1)^2 t_2}{(q_1+1)^2}+\frac{1}{2}\frac{2q_1-1}{(q_1-2)(q_1+1)}.$$
\item $H(2,3;2)$ with parameters $\varkappa_0=0$ and $\varkappa_\infty=-\frac{1}{2}$ (and $\eta=1$):
$q_1$ is a solution of
$$\left(\frac{q_1(3q_1+2t_1)}{3}\right)^3=2t_2^2$$
and other variables are given by
$$q_2=-q_1-\frac{2}{3}t_1,\ \ \ p_1=\frac{q_1}{4}+\frac{t_1}{2}-\frac{1}{6q_1}-\frac{t_2}{2q_1^2}$$
$$\text{and}\ \ \ p_2=-\frac{q_1}{4}+\frac{t_1}{3}+\frac{1}{2(3q_1+2t_1)}-\frac{9t_2}{2(3q_1+2t_1)^2}.$$
\end{itemize}
In each case, the solutions $(t_1,t_2)\mapsto(p_1,p_2,q_1,q_2)$ satisfy the Hamiltonian system (\ref{eq:HamiltonianGarnierSystem})
for explicit Hamiltonians $H_i$ given in section \ref{sec:GarnierHamiltonians}.
The second solution coincides with one of the two solutions found by Kawamuko in \cite{KawamukoAlg}.
For the third solution, we are able to compute the algebraic isomonodromic deformation of the pull-back
linear differential equation, but we don't know the explicit form of the Garnier system in that case.
\begin{thm}\label{thm:rank2classical}
Up to isomorphisms, classical algebraic solutions of irregular Garnier systems
of rank $N=2$ occur exactly for the following formal data
\begin{itemize}
\item infinite discrete family:
$$\begin{pmatrix}0&0&0&\frac{1}{2}\\ \frac{1}{2}&\frac{1}{2}&\frac{1}{2}&0\end{pmatrix}$$
\item two-parameter families
$$\begin{pmatrix}0&0&0&1\\ \frac{1}{2}&\frac{1}{2}&\theta_1&\theta_2\end{pmatrix},\ \ \
\begin{pmatrix}0&\frac{1}{2}&0&0\\ \frac{1}{2}&0&\theta_1&\theta_2\end{pmatrix},\ \ \
\begin{pmatrix}0&0&0&1\\ 0&\theta_1&\theta_2&-\theta_1-\theta_2\end{pmatrix}$$
\item one-parameter families
$$\begin{pmatrix}0&0&2\\ \frac{1}{2}&\frac{1}{2}&\theta\end{pmatrix},\ \
\begin{pmatrix}0&\frac{1}{2}&1\\ \frac{1}{2}&0&\theta\end{pmatrix},\ \
\begin{pmatrix}\frac{1}{2}&\frac{1}{2}&0\\ 0&0&\theta\end{pmatrix},\ \
\begin{pmatrix}0&\frac{3}{2}&0\\ \frac{1}{2}&0&\theta\end{pmatrix},\ \
\begin{pmatrix}0&0&2\\ 0&\theta&-\theta\end{pmatrix},\ \
\begin{pmatrix}0&1&1\\ 0&\theta&-\theta\end{pmatrix}$$
\item sporadic solutions
$$\begin{pmatrix}\frac{1}{2}&\frac{3}{2}\\ 0&0\end{pmatrix},\ \ \
\begin{pmatrix}0&\frac{5}{2}\\ \frac{1}{2}&0\end{pmatrix}\ \ \ \text{and}\ \ \
\begin{pmatrix}0&3\\ 0&0\end{pmatrix}.$$
\end{itemize}
In the first case, there are countably many distinct algebraic solutions of unbounded
degree. In any other case, there is exactly one algebraic solution for each formal data.
\end{thm}
Kawamuko already discovered the fourth one-parameter family of solutions in \cite{KawamukoAlg}
and the third sporadic solution in \cite{KawamukoRat}. We provide the first sporadic solution
in section \ref{sec:GarnierHamiltonians}.
\begin{cor}In the rank $N=2$ case, irregular Garnier systems with the following formal data:
$$\begin{pmatrix}\frac{1}{2}&2\\ 0&\theta\end{pmatrix},\ \ \
\begin{pmatrix}1&\frac{3}{2}\\ \theta&0\end{pmatrix},\ \ \
\begin{pmatrix}4\\ \theta\end{pmatrix},\ \ \
\begin{pmatrix}\frac{7}{2}\\ 0\end{pmatrix},$$
have no algebraic solution.
\end{cor}
Sections \ref{sec:DiffEq}, \ref{sec:ConfluentHypergeometric} and \ref{sec:IsomDef} are folklore \cite{MR,MRGalois,Krichever,BMM,vdPS}. The Structure Theorem is presented in section \ref{sec:algsol}.
Sections \ref{sec:RamifiedCovers}, \ref{sec:Scattering}, \ref{sec:IrregEuler} and \ref{sec:Classifcover} are devoted
to the classification of pull-back type solutions; in these sections, the irregular Euler characteristic is introduced and the
irregular Riemann-Hurwitz formula is established. Classical solutions are classified in section \ref{sec:ClassicalN=2}
for the case $N=2$. Finally, explicit Hamiltonians are given in section \ref{sec:GarnierHamiltonians} for the above explicit
algebraic solutions.
\section{Linear differential equations}\label{sec:DiffEq}
In this paper, we consider rank 2 {\bf meromorphic connections} on curves.
This consists in the data of a rank 2 holomorphic vector bundle $E\to C$ on a complete smooth curve $C$,
together with a linear connection $\nabla:E\to E\otimes\Omega^1(D)$ where $D$ is the effective divisor of poles.
Precisely, $\nabla$ is a $\mathbb C$-linear map satisfying the Leibniz
rule $\nabla(f\cdot s)=df\otimes s+f\otimes\nabla(s)$ for any local function $f$ on $C$ and local section $s$ of $E$.
Locally on $C$, in trivializing coordinates for $E$, the connection writes
\begin{equation}\label{eq:generalsystem}
Y\mapsto\nabla(Y)=dY+A\cdot Y\ \ \ \text{with}\ \ \ A=\begin{pmatrix}\alpha&\beta\\ \gamma&\delta\end{pmatrix}
\ \ \ \text{and}\ \ \ Y=\begin{pmatrix}y_1\\ y_2\end{pmatrix}
\end{equation}
where $\alpha,\beta,\gamma,\delta$ are local sections of $\Omega^1(D)$ (meromorphic $1$-forms).
In general, the vector bundle $E$ is not trivial (globally) and we can only give such description in local charts on $C$.
\subsection{Bundle transformations}
There are two kinds of transformations we use to consider on connections. First of all, given a {\bf birational bundle transformation}
$\phi:E'\dashrightarrow E$ over $C$, we can define $\nabla':=\phi^*\nabla$ on $E'$ as the unique connection whose
horizontal sections are preimages by $\phi$ of $\nabla$-horizontal sections at a generic point of $C$.
Locally, $\phi$ is defined by $Y=M\cdot Y'$ with $M$ meromorphic, $\det(M)\not\equiv0$, and $\nabla'$
is defined by
\begin{equation}\label{eq:gauge}
A'=M^{-1}AM+M^{-1}dM.
\end{equation}
Also, given a rank one meromorphic connection $(L,\zeta)$ over $C$, we can consider the {\bf twist}
$(E',\nabla')=(E\otimes L,\nabla\otimes\zeta)$ locally defined by $A'=A+\omega I$ where $\zeta=d+\omega$
and $I$ is the identity matrix.
All these transformations can be equivalently considered locally, in the holomorphic/meromorphic setting.
Note that they can add, simplify or delete singular points of $\nabla$.
We simply call {\bf bundle transformation} the combination of these two kinds of transformations.
They can be used to trivialize the vector bundle, or also to minimize the support and order of poles.
In general, we cannot do this simultaneously, except on $C=\mathbb P^1$ (see Dekker's Theorem).
We will also use local/global biholomorphic/bimeromorphic bundle transformations depending
of the nature of $\zeta$ and $M$.
\subsection{Local formal data}
At the neighborhood of a singular point $x=0$ on $C$,
up to bimeromorphic bundle transformation and change of coordinate $x\to\varphi(x)$,
we are in one of the following models:
\begin{equation}\label{eq:ModelSing}
\begin{matrix}
\Log\hfill\hfill & A=\begin{pmatrix}\frac{\theta}{2}&0\\ 0&-\frac{\theta}{2}\end{pmatrix}\frac{dx}{x}, \hfill\hfill
& \theta\in\mathbb C\setminus\mathbb Z
& \text{logarithmic non resonant}\hfill\begin{pmatrix}0\\ \theta\end{pmatrix}\\
\Logres & A=\begin{pmatrix}\frac{n}{2}&x^n\\ 0&-\frac{n}{2}\end{pmatrix}\frac{dx}{x}, \hfill\hfill
& n\in\mathbb Z_{\ge0}
& \text{logarithmic resonant}\hfill\begin{pmatrix}0\\ n\end{pmatrix}\\
\Irrun\hfill\hfill & A=\begin{pmatrix}\frac{1}{2}&0\\ 0&-\frac{1}{2}\end{pmatrix}\left(\frac{dx}{x^{k+1}}+\theta\frac{dx}{x}\right)+\tilde A,
& \left\{\begin{matrix} k\in\mathbb Z_{>0}\\ \theta\in\mathbb C\end{matrix}\right.
& \text{irregular unramified case}\hfill\begin{pmatrix}k\\ \theta\end{pmatrix}\\
\Irram & A=\begin{pmatrix}0&x\\ 1&0\end{pmatrix}\frac{dx}{x^{k+1}}+\tilde A,\hfill\hfill
& k\in\mathbb Z_{>0}
& \text{irregular ramified case}\begin{pmatrix}k-\frac{1}{2}\\ 0\end{pmatrix}
\end{matrix}
\end{equation}
where $\tilde A$ is holomorphic. The matrix column on the right will be explained later.
The order of pole is minimal up to bimemorphic bundle transformation in all these models,
and it is therefore an invariant.
We call $\theta$ the {\bf exponent}; we set $\theta=n$ in the logarithmic non diagonal case $\Logres$, and
$\theta=0$ in the irregular ramified case $\Irram$.
In fact, only $\cos(2\pi\theta)$ really makes sense up to bimeromorphic bundle transformations, since
$\theta$ can be shifted by integers under birational bundle transformations;
moreover, the variable permutation $y_1\leftrightarrow y_2$ in $Y$ changes the sign of $\theta$.
We now define the Katz {\bf irregularity index} $\kappa\in\frac{1}{2}\mathbb Z_{\ge0}$ by
\begin{itemize}
\item $\kappa=0$ in the logarithmic case $\Log$ and $\Logres$,
\item $\kappa=k$ in the irregular unramified case $\Irrun$,
\item $\kappa=k-\frac{1}{2}$ in the irregular ramified case $\Irram$.
\end{itemize}
The main property of $\theta$ and $\kappa$ is that they are multiplicative under ramified cover:
\begin{prop}\label{Prop:ramificationindex}
If $\varphi(x)=x^n$, $n\in\mathbb Z_{>0}$, then $(\tilde E,\tilde\nabla)=\varphi^*(E,\nabla)$ has,
up to bundle transformation, the following invariants
$$ \tilde \kappa=n\kappa\ \ \ \text{and}\ \ \ \tilde\theta=n\theta.$$
In particular, the class of irregular singular points is characterized by $\kappa\not=0$ and is stable
under ramified covers.
\end{prop}
The proof is straightforward.
\begin{remark}\label{Rem:ramificationindex}In case of model $\Log$ with exponent $\theta=\frac{p}{q}$
rational, the pole becomes apparent after a ramification $\varphi(x)=x^n$ with $n=mq$ a multiple of $q$:
we get $\tilde\theta=mp\in\mathbb Z$ and the pole disappear after bundle transformation.
In a similar way, in case of model $\Irram$, when $n=2m$ is even, we get after bundle transformation
an unramified pole $\Irrun$ with $\tilde\theta=m$, that can be normalized to $\tilde\theta=0$
after an additional bundle transformation.
\end{remark}
In irregular models $\Irrun$-$\Irram$, we can further
kill the holomorphic part $\tilde A$ by formal bundle transformation; however it is divergent in general.
This already shows that $\kappa$ and $\theta$ are the only formal invariant, i.e. that can be
algebraically computed. In the sequel, we denote by $\bar\kappa:=k$ the smallest integer $\ge\kappa$, i.e. $\bar\kappa=\kappa$ or $\kappa+\frac{1}{2}$. There are other invariants, called Stokes matrices (see below),
whose computation is very transcendental.
To resume, each singular point is characterized, up to base change and formal bundle transformation,
by its irregularity $\kappa\in\frac{1}{2}\mathbb Z_{\ge0}$, and $\theta\in\mathbb C$, its exponent.
We will call {\bf local formal data} of a differential
equation the matrix
\begin{equation}\label{eq:LocForData}
\begin{pmatrix}\kappa_1&\cdots&\kappa_n\\
\theta_1&\cdots&\theta_n\end{pmatrix}\ \ \ \text{where}\ \ \
\left\{\begin{matrix}
\kappa_i\in\mathbb Z_{\ge0}&\Rightarrow&\theta_i\in\mathbb C\\
\kappa_i\in-\frac{1}{2}+\mathbb Z_{>0}&\Rightarrow&\theta_i=0
\end{matrix}\right.
\end{equation}
specifying the formal type at each singular point. In (\ref{eq:ModelSing}), the formal type is indicated on the rightside.
\subsection{Normalization}
\begin{prop}\label{prop:normalizedequation}
Any (global) connection $(E,\nabla)$ is equivalent, up to birational bundle tranformation,
to a $\SL$-connection $(E_0,\nabla_0)$ which locally fits with one of the models
$\Log$, $\Logres$, $\Irrun$ or $\Irram$
at any pole along the curve, for a convenient choice of coordinate $x$ and trivialization of $E$.
Such a reduction is not unique, and we can moreover assume, up to additional birational bundle transformation,
that formal data satisfies
$$0\le\Re(\theta_i)\le\frac{1}{2}\ \ \ \text{for}\ i=1,\ldots,n-1\ \ \ \text{and}\ \ \ 0\le\theta_n<1.$$
If $\kappa_i,\theta_i\not\in\frac{1}{2}\mathbb Z$, then this latter reduction is unique (up to permution of poles).
\end{prop}
We call {\bf normalized equation} a connection like $(E_0,\nabla_0)$ in the statement.
\begin{proof}The algorithm is as follows. We alternate birational bundle transformations and twists in order to simplify
the poles (minimize) and then apply a final twist to get the $\SL$-form. Let us firstly dicuss the second step.
Given a connection $(E,\nabla)$, then its trace admits a square root
$$(\det(E),\tr(\nabla))=(L,\zeta)^{\otimes 2}$$
if, and only if, $\deg(E)$ is even. In that case, $(E,\nabla)\otimes (L,\zeta)^{\otimes(-1)}$ is in $\SL$-form.
The first step is done by applying
successive elementary transformations $E'\dashrightarrow E$ at each pole $p$, i.e. of the form
$$Y=M\cdot Y'\ \ \ \text{with}\ \ \ M=\begin{pmatrix}1&0\\ 0&x\end{pmatrix}$$
in convenient local coordinate $x$ and trivializations $Y,Y'$.
In a more intrinsic way, if $l\subset E\vert_p$ denotes the direction spanned by $Y=\begin{pmatrix}1\\ 0\end{pmatrix}$,
then $E'$ is defined as the locally free sheaf whose sections are those sections of $E$ which, in restriction to $E\vert_p$,
belong to the direction $l$. We note that $E$ and $E'$ are canonically isomorphic over the complement $C\setminus p$;
moreover, $\deg(E')=\deg(E)+1$.
Now, if $\nabla$ is a connection on $E$ with a pole at $p$, we say that the elementary transformation is $\nabla$-adapted
if $l$ is an eigendirection of the leading term of the matrix connection; it is equivalent to the fact that
the induced connection $\nabla'$ on $E'$ has a pole of order not greater than $\nabla$ at $p$:
$$
A=\begin{pmatrix}a(x)&b(x)\\ c(x)&d(x)\end{pmatrix}\frac{dx}{x^{k+1}}\ \ \ \Rightarrow\ \ \
A'=M^{-1}AM+M^{-1}dM=\begin{pmatrix}a(x)&xb(x)\\ \frac{c(x)}{x}&d(x)+x^k\end{pmatrix}\frac{dx}{x^{k+1}}.$$
We now proceed to simplify poles by applying adapted elementary transformations.
\begin{itemize}
\item If $A(0)$ is scalar, i.e. of the form $I\frac{dx}{x^{k+1}}$, then it can be killed by a twist, and the order $k$ decreases.
\item If $A(0)$ is semi-simple but not scalar, then we can reduce to model $\Log$, $\Logres$ or $\Irrun$
by biholomorphic bundle equivalence and change of coordinate
(with possibly $\theta\in\mathbb Z$ in $\Log$, in which case the pole can be deleted by birational bundle equivalence)
and passing to the $\SL$-form.
\item If $A(0)$ is not semi-simple, then we apply a $\nabla$-adapted elementary transformation;
if $\nabla$ and $\nabla'$ have same order at $p$, then one can check that we are in models $\Logres$ or
$\Irram$ up to holomorphic bundle equivalence and change of coordinate.
\end{itemize}
Finally, after finitely many elementary steps, we arrive at one of the models, up to biholomorphic bundle transformation.
At the end, if $\deg(E)$ is even, we obtain the $\SL$-form after a twist. If not, we can apply a $\nabla$-adapted
elementary transformation at one of the poles to shift $\deg(E)$ by $+1$, so that it becomes even, and then normalize by a twist.
The first part of the statement is proved.
The lack of unicity comes from
\begin{itemize}
\item the possibility of performing an even number of additional $\nabla$-adapted elementary transformation,
\item the possibility of changing the $\SL$-normalization by twisting with a $2$-torsion holomorphic connection
(this freedom does not occur on $C=\mathbb P^1$).
\end{itemize}
The first operation has the effect to shift exponents $\theta_i\mapsto\theta_i+n_i$, $n_i\in\mathbb Z$, with $\sum\theta_i\in 2\mathbb Z$
(except in case $\Irram$ where $\theta$ is always zero). This does not affect neither the type $\Log$, $\Logres$, $\Irrun$ or $\Irram$
of the pole, nor irregularity $\kappa$, but only $\theta$. Recall also that $\theta_i$'s are defined up to a sign. We promptly
deduce that all $\theta_i$ can be normalized with $0\le\Re(\theta_i)\le\frac{1}{2}$ by birational bundle transformation:
first put $-\frac{1}{2}\le\Re(\theta_i)\le\frac{1}{2}$ by shifting it by integers, and then switch to $-\theta_i$ if necessary.
But since we need $\det(E)$ even for the final $\SL$-normalization, then we possibly need to apply an additional
elementary transformation, and then shift one of the $\theta_i$'s by one.
\end{proof}
\begin{remark}\label{rem:normalizedequation}
If one of the poles has formal data of one of the following types
$$\begin{pmatrix}\kappa\\ \theta\end{pmatrix}=\begin{pmatrix}0\\ \frac{1}{2}\end{pmatrix}\ \ \ \text{or}\ \ \
\begin{pmatrix}\frac{1}{2}\\ 0\end{pmatrix}$$
then one can furthermore assume that
$$0\le\Re(\theta_i)\le\frac{1}{2}\ \ \ \text{for all}\ i=1,\ldots,n.$$
Indeed, it suffices to note that these two types of poles are invariant by some $\nabla$-adapted
elementary transformation. So the last one needed to get $\det(E)$ even can be performed
on this pole.
Also note, in case $C=\mathbb P^1$ and no more than one pole is of the above type, that the normalization
is unique in that case.
\end{remark}
\subsection{Irregular curve}\label{sec:IrregCurve}
To encode the global formal structure of the differential equation, we have to take
into account the conformal type of the curve $C$ (when genus $g>0$), and the position
of singular points.
But it is important to recall here that, to reach the models $\Irrun$-$\Irram$,
we really need a change of coordinate $x:=\varphi(x)$ in general, in order to normalize the principal part
of the differential equation. In fact, it is enough to consider
$\varphi$ polynomial of order $\bar\kappa$ (recall $\bar\kappa\in\mathbb Z_{\ge0}$ is $\kappa$ or $\kappa+\frac{1}{2}$);
in the logarithmic case $\kappa=0$, no change is needed. In other words, for irregular singular points,
there are $\bar\kappa$ other formal invariants when considering only bundle transformations,
and they can be killed by base change. For this reason, the conformal type of the base curve $C$ should
be enriched with the additional data, at each irregular point $t_i$, of a $\bar\kappa_i$-jet of coordinate
$x_i:(C,t_i)\to(\mathbb C,0)$ in which the
equation can be reduced to the models $\Irrun$-$\Irram$. We call {\bf irregular curve}
the data $X:=(C,D,\{x_i\})$. The deformation space of the irregular curve locally identifies with $H^0((\Omega^1)^{\otimes 2}(D))$
and has dimension
\begin{equation}\label{eq:dimirrTeich}
T:=3g-3+\deg(D)=3g-3+\sum_{i=1}^n(1+\bar\kappa_i).
\end{equation}
There is a global deformation space which is a principal bundle over the moduli space $M_{g,n}$
for the punctured curve $(C,\vert D\vert)$, whose fiber is the group product of diffeomorphism jets.
Once we know the irregular curve and local formal data, it remains to add some extra analytic invariants
given by the monodromy representation together with Stokes data.
They can only be computed algebraically from the differential equation, i.e. by means of above invariants,
when $T\le3$. For larger $T$, one can deform the equation with fixed irregular curve and formal data
and these invariants we are going to describe are transcendental functions of the coefficients of the equation.
\subsection{Monodromy and Stokes matrices}\label{sec:MonodStokes}
The {\bf monodromy representation} of the equation is a group morphism
$$\rho_\nabla:\pi_1(C\setminus\vert D\vert,t_0)\to\mathrm{GL}_2(\mathbb C)$$
defined as the monodromy of a local basis of solutions $B_0$.
More precisely, we can cover $C\setminus\vert D\vert$ (where $\vert D\vert$ is the support of $D$, i.e. set of poles)
by open sets $U_i$ over each of which the differential equation admits a basis of solutions $B_i$;
on overlappings $U_i\cap U_{j}$, we get $B_i=M_{i,j}B_{j}$ for a transition matrix $M_{i,j}\in\mathrm{GL}_2(\mathbb C)$.
The collection $(M_{i,j})$ defines an element $\rho_\nabla$ of
$$H^1(C\setminus\vert D\vert,\mathrm{GL}_2(\mathbb C))\simeq\mathrm{Hom}(\pi_1(C\setminus\vert D\vert),\mathrm{GL}_2(\mathbb C))$$
(point of view of {\bf local systems}).
In fact, any loop $\gamma\in\pi_1(C\setminus\vert D\vert,x_0)$ based at $x_0$ can be covered, by compacity,
by a finite number of such open sets $U_i$, where $\gamma$ crosses successively these open sets following
the index order $i=0,1,\ldots,m$. Then, the analytic continuation of the initial basis of solutions $B_0$ writes:
$$\underbrace{B_0}_{\text{on}\ U_0}=\underbrace{M_{0,1}B_1}_{\text{on}\ U_1}=\underbrace{M_{0,1}M_{1,2}B_2}_{\text{on}\ U_2}=\cdots
=\underbrace{M_{0,1}M_{1,2}\cdots M_{m,0}B_0}_{\text{(back to $U_0$)}}=:B_0^\gamma,$$
$$\text{i.e.}\ \ \ B_0^\gamma=M^\gamma B_0\ \text{on}\ U_0,\ \ \ \text{with}\ M^\gamma:=M_{0,1}M_{1,2}\cdots M_{m,0}.$$
We then define the monodromy morphism by setting $\rho_\nabla(\gamma)= M^\gamma$. It depends on the choice
of the basis $B_0$: it is well-defined up to conjugacy by an element $M\in\mathrm{GL}_2(\mathbb C)$:
$$B_0=M\tilde B_0\ \ \ \Rightarrow\ \ \ \tilde M^\gamma=M^{-1}M^\gamma M \ \ \ \Rightarrow\ \ \ \tilde\rho_\nabla=M^{-1}\rho_\nabla M.$$
For instance, in the first logarithmic model $\Log$, one easily calculate
$$B_0(x)=\begin{pmatrix}x^{-\frac{\theta}{2}}&0\\ 0&x^{\frac{\theta}{2}}\end{pmatrix}\ \ \ \text{and}\ \ \ M^\gamma=\begin{pmatrix}e^{i\pi\theta}&0\\ 0&e^{-i\pi\theta}\end{pmatrix}$$
where $\gamma(t)=e^{2i\pi t}$, $t\in[0,1]$.
For irregular singular points, it is more subtle since there are {\bf Stokes matrices}, playing the role of infinitesimal monodromy.
In the unramified case, the model $\Irrun$ with $\tilde A\equiv0$ has the fundamental basis and monodromy
$$B_0(x)=\begin{pmatrix}x^{-\frac{\theta}{2}}e^{\frac{1}{2kx^k}}&0\\ 0&x^{\frac{\theta}{2}}e^{-\frac{1}{2kx^k}}\end{pmatrix}\ \ \ \text{and}\ \ \ M^\gamma=\begin{pmatrix}e^{i\pi\theta}&0\\ 0&e^{-i\pi\theta}\end{pmatrix}$$
We call it the {\bf formal monodromy}. In the general case $\tilde A\not\equiv0$,
equation $\Irrun$ cannot be reduced to the formal normal form $\tilde A\equiv0$
by holomorphic bundle transformation, but only by (generically divergent) formal bundle transformation;
we thus get a formal fundamental basis $\hat B(x)$ with monodromy $M^\gamma$.
The same reduction can also be done holomorphically over each of the following $2\kappa=2k$ sectors
$$V_0:=\left\{x\ ;\ 0<\vert x\vert<r,\ \ \ \vert \arg(x)-\frac{\pi}{2k}\vert<\frac{\pi}{k}-\epsilon\right\}$$
(where $r,\epsilon>0$ are small enough) and
$$V_l:=\left\{x\ ;\ e^{-i\pi\frac{l}{k}}x\in V_0 \right\},\ l=0,\ldots,2k-1.$$
Note that these sectors are covering the punctured disc $\{0<\vert x\vert<r\}$.
Going back to the initial model $\Irrun$, there exists a fundamental matrix $B_l$ over each sector $V_l$,
asymptotic to the formal one $\hat B(x)$.
Then, we have
$B_l=S_lB_{l+1}$ with $S_l\in\mathrm{GL}_2(\mathbb C)$ is unipotent, upper-triangular when $l$ is even,
and lower-triangular for $l$ odd. Therefore, the monodromy around $x=0$ canonically splits as
\begin{equation}\label{eq:StokesDecomp}
M^\gamma=\underbrace{
\begin{pmatrix}e^{i\pi\theta}&0\\ 0&e^{-i\pi\theta}\end{pmatrix}}_{\text{formal monodromy}}
\underbrace{\begin{pmatrix}1&s_1\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ t_1&1\end{pmatrix}\cdots
\begin{pmatrix}1&s_k\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ t_k&1\end{pmatrix}}_{\text{Stokes matrices}}
\end{equation}
and this {\bf Stokes decomposition} is unique up to conjugacy by a diagonal matrix (the choice of $\hat B(x)$),
so that equivalent decompositions write
$$\begin{pmatrix}e^{i\pi\theta}&0\\ 0&e^{-i\pi\theta}\end{pmatrix}
\begin{pmatrix}1&cs_1\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ c^{-1}t_1&1\end{pmatrix}\cdots
\begin{pmatrix}1&cs_k\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ c^{-1}t_k&1\end{pmatrix},\ \ \ \text{for}\ c\in\mathbb C^*.$$
In fact, any two models $\Irrun$ with the same $k$ and $\lambda$
are equivalent by holomorphic bundle transformation if, and only if, their Stokes decomposition coincide up to diagonal conjugacy.
Moreover, any $2k$-uple $(s_1,\ldots,s_k,t_1,\ldots,t_k)$ is realisable as Stokes decomposition of a model $\Irrun$.
In particular, the differential equation is equivalent to the formal model by holomorphic bundle transformation
if, and only if, $s_l=t_l=0$ for $l=1,\ldots,k$.
In the {\bf ramified case}, we have a similar story and Stokes decomposition looks like
\begin{equation}\label{eq:ramStokesDecomp}
M^\gamma=\underbrace{
\begin{pmatrix}0&1\\ -1&0\end{pmatrix}}_{\text{formal monodromy}}
\underbrace{\begin{pmatrix}1&s_1\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ t_1&1\end{pmatrix}\cdots
\begin{pmatrix}1&s_k\\ 0&1\end{pmatrix}}_{\text{Stokes matrices}}
\end{equation}
where $k=\kappa+\frac{1}{2}$: there are $2\kappa$ matrices (like in the unramified case where irregularity is $\kappa=k$).
This decomposition characterizes the analytic equivalence class of the differential
equation, and any such Stokes data can be realized. We refer to \cite{MR,MRGalois,Krichever,BMM,vdPS} for more details.
\subsection{Differential Galois group}
The (differential) Galois group of a normalized differential equation $(C,E,\nabla)$ can be computed
from the monodromy and Stokes data (see \cite{MRGalois}). In the logarithmic case,
the Galois group is just the Zariski closure of the monodromy goup in $\SL(\mathbb C)$.
Recall that algebraic subgroups of $\SL(\mathbb C)$ are in the following list (up conjugacy):
\begin{equation}\label{eq:listGaloisGroup}
\begin{matrix}
\text{infinite} &
\left\{\begin{matrix} C_\infty=\left\{\begin{pmatrix}\lambda&0\\ 0&\lambda^{-1}\end{pmatrix}\ ;\ \lambda\in\mathbb C^*\right\},\hfill\hfill\\
D_\infty=\left\{\begin{pmatrix}\lambda&0\\ 0&\lambda^{-1}\end{pmatrix},\begin{pmatrix}0&\lambda\\ -\lambda^{-1}&0\end{pmatrix},\ ;\ \lambda\in\mathbb C^*\right\}=\langle C_\infty,\begin{pmatrix}0&1\\-1&0\end{pmatrix}\rangle,\\
P_\infty=\left\{\begin{pmatrix}1&\mu\\ 0&1\end{pmatrix}\ ;\ \mu\in\mathbb C\right\},\hfill\hfill\\
T_\infty=\left\{\begin{pmatrix}\lambda&\mu\\ 0&\lambda^{-1}\end{pmatrix}\ ;\ \lambda\in\mathbb C^*,\ \mu\in\mathbb C\right\},\hfill\hfill\\
T_n=\left\{\begin{pmatrix}\lambda&\mu\\ 0&\lambda^{-1}\end{pmatrix}\ ;\ \lambda^n=1,\ \mu\in\mathbb C\right\},\hfill\hfill\\
\SL(\mathbb C)\hfill\hfill\end{matrix}\right. \\
\text{finite} &
\left\{\begin{matrix} C_n=\left\{\begin{pmatrix}\lambda&0\\ 0&\lambda^{-1}\end{pmatrix}\ ;\ \lambda^n=1\right\},\hfill \# n\\
D_n=\left\{\begin{pmatrix}\lambda&0\\ 0&\lambda^{-1}\end{pmatrix},\begin{pmatrix}0&\lambda\\ -\lambda^{-1}&0\end{pmatrix},\ ;\ \lambda^n=1\right\},\ \# 2n\\
\text{tetrahedral}\simeq A_4\ltimes\mathbb Z/2,\hfill \#24\\
\text{octahedral}\simeq S_4\ltimes\mathbb Z/2,\hfill \#48\\
\text{icosahedral}\simeq A_5\ltimes\mathbb Z/2,\hfill\#120
\end{matrix}\right.
\end{matrix}
\end{equation}
In the irregular case, we first define the local Galois group as
\begin{itemize}
\item at an unramified pole, it is generated in the same basis as (\ref{eq:StokesDecomp}) by the {\bf exponential torus}
$$C_\infty=\left\{\begin{pmatrix}\lambda&0\\ 0&\lambda^{-1}\end{pmatrix}\ ;\ \lambda\in\mathbb C^*\right\}$$
together with the Stokes matrices: we obtain $C_\infty$, $T_\infty$ or $\SL(\mathbb C)$
depending if all Stokes matrices are trivial, if one over two is trivial (i.e. all $s_i$'s or all $t_i$'s), or else.
\item at a ramified pole, it is generated in the same basis as (\ref{eq:ramStokesDecomp}) by the exponential torus, the permutation matrix
$$\begin{pmatrix}0&1\\ -1&0\end{pmatrix}$$
and the Stokes matrices: we obtain $D_\infty$ or $\SL(\mathbb C)$
depending if all Stokes matrices are trivial, or not. \end{itemize}
The global Galois group is the Zariski closure in $\SL(\mathbb C)$ of all local Galois groups and the global monodromy group. In particular, irregular differential equations have always $>0$ dimension due to the exponential torus:
we can only have $C_\infty$, $D_\infty$, $T_\infty$ or $\SL(\mathbb C)$.
\subsection{Link with scalar equations}
On $C=\mathbb P^1$, meromorphic connections were historically defined by
higher order {\bf scalar differential equations}, and it is sometimes more convenient to work with them.
Consider the differential equation
\begin{equation}\label{eq:scalar}
u''+f(x)u'+g(x)u=0,
\end{equation}
where $f,g$ rational/meromorphic, and $u'=\frac{du}{dx}$.
Then, thinking of $(u(x_0),u'(x_0))$ as the space of initial conditions at a generic point $x_0$,
it is natural to associate the {\bf companion system}
\begin{equation}\label{eq:companion}
\nabla=d+A\ \ \ \text{with}\ \ \ A=\begin{pmatrix}0&-g\\ 1&-f\end{pmatrix}dx.
\end{equation}
Indeed, identifying the standard basis with $(u(x),u'(x))$, then the connection satisfies
$\nabla\cdot u=u'$ and $\nabla\cdot u'=u''=-fu'-gu$. In the projective coordinate $y=-y_1/y_2$,
it induces the Riccati equation $y'+y^2+fy+g=0$, and we recover the initial scalar equation
by setting $y=u'/u$.
Conversely, given a more general system (\ref{eq:generalsystem}), then we can first apply a twist to set $\alpha=0$
in the matrix $A$, and then use a gauge transformation of the form
\begin{equation}\label{eq:gaugecyclic}
M=\begin{pmatrix}1&G\\0&F\end{pmatrix},\ \ \ F\not\equiv0,
\end{equation}
(and a twist) to reduce the matrix $A$ in the companion form (\ref{eq:companion}): set $Fdx=\gamma$ and $G=0$.
One can further reduce (\ref{eq:companion}), or accordingly (\ref{eq:scalar}), by gauge transformation (\ref{eq:gaugecyclic})
with $F\equiv1$, or equivalently setting $u:=u/\exp(\int G)$; by this way, we can arrive to the unique SL-form
\begin{equation}\label{eqSLscalar}
u''=\frac{s(x)}{2}u,\ \ \ s=f'+\frac{f^2}{2}-g
\end{equation}
also called ``Sturm-Liouville operator''. In fact, $s(x)$ is the {\bf Schwarzian derivative} of the quotient of any two
independant solutions $u_1(x),u_2(x)$ of the initial equation (\ref{eq:scalar})
$$s(x):=\{\varphi,x\}=\left(\frac{\varphi''}{\varphi'}\right)'-\frac{1}{2}\left(\frac{\varphi''}{\varphi'}\right)^2,\ \ \ \varphi=\frac{u_1}{u_2}.$$
The above reduction to scalar equation depends on the choice of coordinate $Y$, or more precisely on the choice
of the so called ``cyclic vector'' $\begin{pmatrix}1\\0\end{pmatrix}$. For a general rank 2 meromorphic connection $(E,\nabla)$,
the reduction to 2nd order scalar equation depend on the choice of a line subbundle $L\subset E$.
It is however important to notice that the irregularity index of an arbitrary connection $\nabla$
(not necessarily of the form $\Log$-$\Logres$-$\Irrun$-$\Irram$)
is directly given by the order of poles of any reduction to scalar equation, namely
$$1+\kappa=\max\{\ord(f),\ord(g)/2\}.$$
\section{Confluent hypergeometric equations}\label{sec:ConfluentHypergeometric}
On $C=\mathbb P^1$, when the polar locus $D$ has degree $3$, the connection can be determined,
up to bundle equivalence, by its local formal data. After base change (i.e. applying
a Moebius transformation in $x$-variable), we can reduce to the following list
of classical scalar equations.
$$
\xymatrix{
{\begin{matrix}
\begin{pmatrix}0&0&0\\ \theta_0&\theta_1&\theta_\infty\end{pmatrix}\\
\text{Hypergeometric}
\end{matrix}}
\ar[r]
&
{\begin{matrix}
\begin{pmatrix}0&1\\ \theta_0&\theta_\infty\end{pmatrix}\\
\text{Kummer}
\end{matrix}}
\ar[r]\ar[rd]
&
{\begin{matrix}
\begin{pmatrix}2\\ \theta_\infty\end{pmatrix}\\
\text{Weber}
\end{matrix}}
\ar[rd]
&\\
&&
{\begin{matrix}
\begin{pmatrix}0&\frac{1}{2}\\ \theta_0&0\end{pmatrix}
\end{matrix}}
\ar[r]
&
{\begin{matrix}
\begin{pmatrix}\frac{3}{2}\\ 0\end{pmatrix}\\
\text{Airy}
\end{matrix}}
}$$
The {\bf Gauss hypergeometric equation}
\begin{equation}\label{eq:Gauss}
u''+\left(\frac{(a+b+1)x-c}{x(x-1)}\right)u'+\frac{ab}{x(x-1)}u=0
\end{equation}
has $3$ simple poles at $x=0,1,\infty$ with respective exponents $\theta_0=c-1$, $\theta_1=a+b-c$ and $\theta_\infty=a-b$.
The {\bf Kummer equation} (also called ``confluent hypergeometric'')
\begin{equation}\label{eq:Kummer}
u''+\left(\frac{c}{x}-1\right)u'-\frac{a}{x}u=0
\end{equation}
has a logarithmic pole at $x=0$ with exponent $\theta_0=c$, and an irregular point at $x=\infty$ having
irregularity index $\kappa=1$ and exponent $\theta_\infty=2a-c$. Its SL-form
$$u''=\left(\frac{1}{4}+\frac{2a-c}{2x}+\frac{(c-1)^2-1}{4x^2}\right)u$$
is also known as Whittaker equation. A particular case is Bessel equation when $\lambda=0$.
The monodromy data can be described as follows:
$$M_0\cdot M_\infty=I\ \ \ \text{with}\ \ \ M_\infty=\begin{pmatrix}e^{i\pi\theta_\infty}&0\\ 0&e^{-i\pi\theta_\infty}\end{pmatrix}
\begin{pmatrix}1&s\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ t&1\end{pmatrix}$$
$$\text{and}\ \ \ \mathrm{trace}(M_0)=2\cos(\pi\theta_0)=\mathrm{trace}(M_\infty)=2\cos(\pi\theta_\infty)+e^{i\pi\theta_\infty}st.$$
In particular, one easily check that are equivalent:
$$\text{the Galois group reducible}\ \Leftrightarrow\ st=0\ \Leftrightarrow\ \mathrm{trace}(M_0)=\mathrm{trace}(M_\infty)$$
$$\ \Leftrightarrow\ \theta_\infty=\pm\theta_0\mod 2\mathbb Z
\ \Leftrightarrow\ a\in\mathbb Z\ \text{or}\ c-a\in\mathbb Z.$$
To determine when the differential equation is totally reducible (i.e. with diagonal monodromy),
we refer to \cite{DLR}. When irreducible, the Galois group of the normalized equation is $\SL(\mathbb C)$.
The {\bf Weber equation}
\begin{equation}\label{eq:Weber}
u''=(x^2-2a)u
\end{equation}
has a single irregular pole at $x=\infty$ with irregularity index $\kappa=2$ and exponent $\theta_\infty=2a-1$.
The monodromy data can be described as follows:
$$M_\infty=\begin{pmatrix}e^{i\pi\theta_\infty}&0\\ 0&e^{-i\pi\theta_\infty}\end{pmatrix}
\begin{pmatrix}1&s_1\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ t_1&1\end{pmatrix}
\begin{pmatrix}1&s_2\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ t_2&1\end{pmatrix}=I.$$
In particular, one easily check that are equivalent:
$$\text{the Galois group reducible}\ \Leftrightarrow\ \theta_\infty\in 2\mathbb Z
\ \Leftrightarrow\ a\in\frac{1}{2} +\mathbb Z.$$
When irreducible, the Galois group of the normalized equation is $\SL(\mathbb C)$.
The {\bf degenerate confluent hypergeometric equation}
\begin{equation}\label{eq:DC}
u''+\frac{c}{x}u'-\frac{1}{x}u=0
\end{equation}
has one logarithmic pole at $x=0$ with exponent $\theta_0=c$ and a ramified irregular point at $x=\infty$
with irregularity index $\kappa=\frac{1}{2}$. Its SL-form is the ``degenerate Whittaker'' equation
\begin{equation}\label{eq:DW}
u''=\left(\frac{1}{x}+\frac{(c-1)^2-1}{4x^2}\right)u
\end{equation}
that will be used in our computations. The monodromy data can be described as follows:
$$M_0\cdot M_\infty=I\ \ \ \text{with}\ \ \ M_\infty=\begin{pmatrix}0&1\\ -1&0\end{pmatrix}
\begin{pmatrix}1&s\\ 0&1\end{pmatrix}=\begin{pmatrix}0&1\\ -1&-s\end{pmatrix}$$
$$\text{and}\ \ \ s+2\cos(\pi\theta_0)=0.$$
The Galois group of the normalized equation is $\SL(\mathbb C)$, except when $s=0$
where the Galois group is dihedral. One easily check that the latter case holds if, and only if $c\in\frac{1}{2}+\mathbb Z$.
Finally, the most degenerate one, the {\bf Airy equation}
\begin{equation}\label{eq:Airy}
u''=xu
\end{equation}
has a single irregular pole at $x=\infty$ with irregularity index $\kappa=\frac{3}{2}$.
The monodromy data can be described as follows:
$$M_\infty=\begin{pmatrix}0&1\\ -1&0\end{pmatrix}
\begin{pmatrix}1&-1\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ 1&1\end{pmatrix}\begin{pmatrix}1&-1\\ 0&1\end{pmatrix}=I$$
and the Galois group is $\SL(\mathbb C)$.
\section{Isomonodromic deformations}\label{sec:IsomDef}
When the degree of the polar divisor $N:=\deg(D)$ satisfies $N>3$, then local formal data fail to determine
the differential equation and we have non trivial deformations, even with constant monodromy
and Stokes data. We call them isomonodromic deformations.
To define monodromy data in family, we need to introduce the {\bf irregular Teichm\"uller} space,
which is the moduli space of $(C,D,\{x_i\},\{\alpha_j,\beta_j,\gamma_i\})$
where we take into account a basis $(\alpha_j,\beta_j)$ for the fundamental group (with some base point $t_0$),
as well as a loop $\gamma_i$ from the base point
to the singular point $t_i$, ending along $x_i\in\mathbb R>0$, for each irregular singular point.
This irregular Teichm\"uller space is described in \cite{Krichever,Heu}. Then we can define monodromy
representation in family, as well as Stokes matrices. This makes sense so talk about deformation with constant
monodromy, that we call isomonodromic deformations.
Precisely, fix a genus $g$ and
a local formal data (\ref{eq:LocForData}); denote by $T$ the corresponding irregular Teichm\"uller dimension
(\ref{eq:dimirrTeich}). Then, we can consider the moduli space of triples $(X,E,\nabla)$ where
\begin{itemize}
\item $X=(C,D,\{x_i\})$ is an irregular curve of genus $g$, i.e. $C$ is a curve of genus $g$,
$$D=\sum_{i=1}^n(1+\bar\kappa_i)[t_i]\ \ \ \text{and}\ \ \ x_i\text{ is a local coordinate at }t_i;$$
\item $E$ is a rank $2$ vector bundle with trivial determinant $\det(E)=\mathcal O_C$;
\item $\nabla:E\to E\otimes\Omega^1_C(D)$ is a meromorphic trace-free connection with polar divisor $D$;
\item in local coordinate $x_i$, the connection is defined by $\Log$-$\Logres$-$\Irrun$-$\Irram$ with local formal data (\ref{eq:LocForData}).
\end{itemize}
We can consider $C$ belonging to the irregular Teichm\"uller space if we want to define the monodromy, or
belonging to the irregular moduli space of curves (or a smooth finite cover) for explicit computations.
The latter one is a quasi-projective variety of dimension $T=3g-3+\deg(D)$, and the former one
is its universal cover, with Mapping-Class-Group acting as covering transformations.
Once the irregular curve $X$ is fixed, the moduli space $\mathcal M(X)$ of $(E,\nabla)$ with above constraints
is a quasi-projective variety of dimension $2T$, provided the local formal data (\ref{eq:LocForData})
is generic enough; in special cases, we have to consider semi-stable connections for some choice of
weights, or for instance irreducible connections, in order to get such a nice moduli space. The total space
$\mathcal M$ of connections with fixed genus $g$ and local formal data (\ref{eq:LocForData}) has therefore
dimension $3T$. A point $(X,E,\nabla)$ on this moduli space is locally determined by an irregular curve
and an irregular monodromy (representation + Stokes matrices).
If we fix the curve $X$ and deform the monodromy, we then get the moduli space $\mathcal M(X)$.
If we now fix the monodromy and deform the curve, then we get the so-called (universal) {\bf isomonodromic deformation}.
There is a $T$-dimensional foliation on the moduli space $\mathcal M$ whose leaves
correspond to maximal isomonodromic/isoStokes deformations. We call it {\bf isomonodromic foliation}.
Isomonodromic leaves are locally parametrized
by the irregular Teichm\"uller space. When considering $X$ living in the moduli space of curves,
the $3T$-dimensional space is algebraic and the isomonodromic foliation is expected to be
a polynomial foliation (i.e. defined by polynomial differential equations). This fact is well-known
when $C=\mathbb P^1$ and $D$ is reduced: isomonodromic differential equations are known as {\bf Garnier systems}
of rank $T$ in that case; the particular case $T=1$ leads to {\bf Painlev\'e equations} (where $\dot{q}$ denotes $\frac{dq}{dt}$):
\begin{equation}\label{eq:PainleveEquations}
\begin{matrix}
P_I\hfill :& \ddot{q}=6q^2+t\hfill\hfill\\
P_{II}(\alpha)\hfill :& \ddot{q}=2q^3+tq+\alpha\hfill\hfill\\
P_{III}(\alpha,\beta,\gamma,\delta)\hfill :& \ddot{q}=\frac{(\dot{q})^2}{q}-\frac{\dot{q}}{t}+\frac{\alpha q^2+\beta}{t}+\gamma q^3+\frac{\delta}{q}\hfill\hfill\\
P_{IV}(\alpha,\beta)\hfill :& \ddot{q}=\frac{(\dot{q})^2}{2q}+\frac{3}{2}q^3+4tq^2+2(t^2-\alpha)q+\frac{\beta}{q}\hfill\hfill\\
P_{V}(\alpha,\beta,\gamma,\delta)\hfill :& \ddot{q}=\left(\frac{1}{2q}+\frac{1}{q-1}\right)(\dot{q})^2-\frac{\dot{q}}{t}+\frac{(q-1)^2}{t^2}\left(\alpha q+\frac{\beta}{q}\right)
+\frac{\gamma q}{t}+\frac{\delta q(q+1)}{q-1}\\
P_{VI}(\alpha,\beta,\gamma,\delta)\hfill :& \ddot{q}=\frac{1}{2}\left(\frac{1}{q}+\frac{1}{q-1}+\frac{1}{q-t}\right)(\dot{q})^2-\left(\frac{1}{t}+\frac{1}{t-1}+\frac{1}{q-t}\right)\dot{q}\hfill\hfill\\
&\hfill+\frac{q(q-1)(q-t)}{t^2(t-1)^2}\left(\alpha+ \frac{\beta t}{q^2}+ \frac{\gamma (t-1)}{(q-1)^2}+ \frac{\delta t(t-1)}{(q-t)^2}\right)
\end{matrix}
\end{equation}
The corresponding isomonodromy equations are given by Table \ref{table:Painleve}.
\begin{table}[h]
\caption{Painlev\'e equations as isomonodromy equations}
\begin{center}
\begin{tabular}{|c|c|}
\hline
Local formal data & Isomonodromy equation \\
\hline
$\begin{pmatrix}0&0&0&0\\ \theta_0&\theta_1&\theta_t&\theta_\infty\end{pmatrix}$ & $P_{VI}\left(\frac{(\theta_\infty-1)^2}{2},-\frac{(\theta_0)^2}{2},\frac{(\theta_1)^2}{2},\frac{1-(\theta_t)^2}{2}\right)$\\
\hline
$\begin{pmatrix}0&1&0\\ \theta_0&\theta_1&\theta_\infty\end{pmatrix}$ & $P_{V}\left(\frac{(\theta_\infty)^2}{2},-\frac{(\theta_0+1)^2}{2},\theta_1,-\frac{1}{2}\right)$\\
\hline
$\begin{pmatrix}0&\frac{1}{2}&0\\ \theta_0&0&\theta_\infty\end{pmatrix}$ & $P_{V}\left(\frac{(\theta_\infty)^2}{2},-\frac{(\theta_0+1)^2}{2},-2,0\right)\sim $\\
&{\small $P_{III}\left(-4(\theta_0-\theta_\infty-1),-4(\theta_0+\theta_\infty),4,-4\right)$}\\
\hline
$\begin{pmatrix}0&2\\ \theta_0&\theta_\infty\end{pmatrix}$ & $P_{IV}\left(\theta_\infty,-2(\theta_0+1)^2\right)$\\
\hline
$\begin{pmatrix}0&\frac{3}{2}\\ \theta_0&0\end{pmatrix}$ & $P_{II}\left(\theta_0-\frac{1}{2}\right)$\\
\hline
$\begin{pmatrix}1&1\\ \theta_0&\theta_\infty\end{pmatrix}$ & $P_{III}\left(4\theta_\infty,-4\theta_0,4,-4\right)$\\
\hline
$\begin{pmatrix}1&\frac{1}{2}\\ \theta_0&0\end{pmatrix}$ & $P_{III}\left(-8,-4\theta_0,0,-4\right)$\\
\hline
$\begin{pmatrix}\frac{1}{2}&\frac{1}{2}\\ 0&0\end{pmatrix}$ & $P_{III}\left(4,-4,0,0\right)$\\
\hline
$\begin{pmatrix}3\\ \theta_\infty\end{pmatrix}$ & $P_{II}\left(\frac{1-\theta_\infty}{2}\right)$\\
\hline
$\begin{pmatrix}\frac{5}{2}\\ 0\end{pmatrix}$ & $P_{I}$\\
\hline
\end{tabular}
\end{center}
\label{table:Painleve}
\end{table}
Let us recall the case of Painlev\'e II equation $P_{II}(\alpha)$. The general linear differential equation
having a single pole at infinity with local formal data
$$\begin{pmatrix}3\\ 1-2\alpha\end{pmatrix}$$
can be normalized in scalar $\SL$-form (i.e. Sturm-Liouville operator)
\begin{equation}\label{eq:LinearP2}
u''=\left(x^4+tx^2+2\alpha x+2H_{II}+\frac{3}{4(x-q)^2}-\frac{p}{x-q}\right)u
\end{equation}
(coefficients of $x^4$ and $x^3$ have been normalized to $1$ and $0$ by an affine transformation in $x$-variable).
Here, $p,q$ are accessory parameters (i.e. initial conditions for the Painlev\'e equation) and
\begin{equation}\label{eq:HamiltonianP2}
H_{II}=\frac{1}{2}\left(p^2-q^4-tq^2-2\alpha q\right)
\end{equation}
is determined so that the pole $x=q$ is apparent. After getting rid of apparent singular point, we can transform equation (\ref{eq:LinearP2}) into the Riccati equation
\begin{equation}\label{eq:RiccatiP2}
y'+(x-q)y^2+(2x^2-2q^2+2p)y+(2p-2q^2-t)x+(1-2\alpha+2pq-2q^3-qt)=0
\end{equation}
or equivalently into the $\SL$-system (setting $y=y_1/y_2$)
\begin{equation}\label{eq:SystemP2}
\begin{pmatrix}y_1\\ y_2\end{pmatrix}'=\begin{pmatrix}-x^2+q^2-p & 2\alpha-1-2pq+2q^3+qt\\ x-q & x^2-q^2+p\end{pmatrix}\begin{pmatrix}y_1\\ y_2\end{pmatrix}.
\end{equation}
One can check, by direct computation, that the singularity at infinity (in variable $z=1/x$) can be
normalized by holomorphic gauge transformation to
\begin{equation}\label{eq:SystemP2infty}
dY+AY=0\ \ \ \text{with}\ \ \
A=\begin{pmatrix}\frac{1}{2}&0\\ 0 & -\frac{1}{2}\end{pmatrix}\left(2\frac{dz}{z^4}+t\frac{dz}{z^2}+\left(1-2\alpha\right)\frac{dz}{z}+\text{holomorphic}\right).
\end{equation}
We retrieve, in the principal part, the fact that the first two coefficients have been normalized by affine transformation in $x$,
the third coefficient stands for the time variable, and the fourth one, for the local formal data $\theta_\infty$.
A deformation $t\mapsto(p(t),q(t))$ of equation (\ref{eq:LinearP2})
is isomonodromic
(in fact, iso-Stokes in that case) if, and only if, it satisfies the Hamiltonian equations
\begin{equation}\label{eq:HamiltonianSystemP2}
\frac{dp}{dt}=-\frac{\partial H_{II}}{\partial q}\ \ \ \text{and}\ \ \ \frac{dq}{dt}=\frac{\partial H_{II}}{\partial p}.
\end{equation}
Equivalently, the corresponding curve in $(t,p,q)$-variables is in the kernel of the $2$-form
\begin{equation}\label{eq:PoissonP2}
\omega=dp\wedge dq+dt\wedge dH_{II}.
\end{equation}
Using the second equation (\ref{eq:HamiltonianSystemP2}), we can express $p$ in terms of $q$ and $\dot{q}$;
substituting in the first equation, we deduce that $q(t)$ is a solution of Painlev\'e II equation.
In (\ref{eq:DegSchemePainleve}) we see the list of formal data of Painlev\'e type, and how they conflue to each other.
\begin{equation}\label{eq:DegSchemePainleve}
{\small
\xymatrix {
{P_{VI}\begin{pmatrix}0&0&0&0\\ \theta_0&\theta_1&\theta_t&\theta_\infty\end{pmatrix}}\ar[dd] &&&\\
&&&\\
{P_V\begin{pmatrix}0&0&1\\ \theta_0&\theta_1&\theta_\infty\end{pmatrix}} \ar[rr] \ar[dd] \ar[dr] && {P_{IV}\begin{pmatrix}0&2\\ \theta_0&\theta_\infty\end{pmatrix}} \ar[dr] \ar[dd] |!{[dl];[dr]}\hole \\
& {P_{III}^{D_6}\begin{pmatrix}0&0&\frac{1}{2}\\ \theta_0&\theta_1&0\end{pmatrix}} \ar[rr] \ar[dd] && {P_{II}\begin{pmatrix}0&\frac{3}{2}\\ \theta_0&0\end{pmatrix}} \ar[dd] \\
{P_{III}^{D_6}\begin{pmatrix}1&1\\ \theta_0&\theta_\infty\end{pmatrix}}\ar@{=}[ur] \ar[rr] |!{[ur];[dr]}\hole \ar[dr] && {P_{II}\begin{pmatrix}3\\ \theta_\infty\end{pmatrix}}\ar@{=}[ur] \ar[rd]& \\
& {P_{III}^{D_7}\begin{pmatrix}1&\frac{1}{2}\\ \theta_0&0\end{pmatrix}}\ar[dr] \ar[rr] && {P_I\begin{pmatrix}\frac{5}{2}\\ 0\end{pmatrix}} \\
&&{P_{III}^{D_8}\begin{pmatrix}\frac{1}{2}&\frac{1}{2}\\ 0&0\end{pmatrix}}
}}
\end{equation}
In the case $T=2$, the degeneration diagram is given in picture (\ref{eq:DegSchemeGarnier2});
the unramified part (and also the case $(7/2)$) of the list is treated by Kimura in \cite{Kimura},
while the ramified part is studied by Kawamuko in \cite{Kawamuko}. We named
$\mathrm{Kim}_1,\ldots,\mathrm{Kim}_8, \mathrm{Kaw}_1,\ldots, \mathrm{Kaw}_8$
these equations following the order of appearance in these two papers.
\begin{equation}\label{eq:DegSchemeGarnier2}
\tiny \xymatrix {
{\mathrm{Kim}_1\left(\begin{matrix}0\\ \theta_0\end{matrix}\begin{matrix}0\\ \theta_1\end{matrix}\begin{matrix}0\\ \theta_2\end{matrix}\begin{matrix}0\\ \theta_3\end{matrix}\begin{matrix}0\\ \theta_\infty\end{matrix}\right)}\ar[dd]&& && \\
\\
{\mathrm{Kim}_2\left(\begin{matrix}0\\ \theta_0\end{matrix}\begin{matrix}0\\ \theta_1\end{matrix}\begin{matrix}0\\ \theta_2\end{matrix}\begin{matrix}1\\ \theta_\infty\end{matrix}\right)}
\ar[rr] \ar[dd] \ar[dr] &&
{\mathrm{Kim}_3\left(\begin{matrix}0\\ \theta_0\end{matrix}\begin{matrix}0\\ \theta_1\end{matrix}\begin{matrix}2\\ \theta_\infty\end{matrix}\right)} \ar[rr]\ar[dr] \ar[dd] |!{[dl];[dr]}\hole &&
{\mathrm{Kim}_5\left(\begin{matrix}0\\ \theta_0\end{matrix}\begin{matrix}3\\ \theta_\infty\end{matrix}\right)} \ar[dr] \ar[dd] |!{[dl];[dr]}\hole\\
& {\mathrm{Kaw}_8\left(\begin{matrix}0\\ \theta_0\end{matrix}\begin{matrix}0\\ \theta_1\end{matrix}\begin{matrix}0\\ \theta_2\end{matrix}\begin{matrix}\frac{1}{2}\\ 0\end{matrix}\right)} \ar[rr] \ar[dd] &&
{\mathrm{Kaw}_5\left(\begin{matrix}0\\ \theta_0\end{matrix}\begin{matrix}0\\ \theta_1\end{matrix}\begin{matrix}\frac{3}{2}\\ 0\end{matrix}\right)} \ar[dd]\ar[rr] &&
{\mathrm{Kaw}_1\left(\begin{matrix}0\\ \theta\end{matrix}\begin{matrix}\frac{5}{2}\\ 0\end{matrix}\right)} \ar[dd] \\
{\mathrm{Kim}_4\left(\begin{matrix}0\\ \theta_0\end{matrix}\begin{matrix}1\\ \theta_1\end{matrix}\begin{matrix}1\\ \theta_\infty\end{matrix}\right)} \ar[rr] |!{[ur];[dr]}\hole \ar[dr] &&
{\mathrm{Kim}_6\left(\begin{matrix}1\\ \theta_0\end{matrix}\begin{matrix}2\\ \theta_\infty\end{matrix}\right)} \ar[rd]\ar[rr] |!{[ur];[dr]}\hole \ar[lddd]|!{[dl];[dr]}\hole |!{[dl];[dd]}\hole&&
{\mathrm{Kim}_7\begin{pmatrix}4\\ \theta\end{pmatrix}}\ar[rd] \\
& {\mathrm{Kaw}_6\left(\begin{matrix}0\\ \theta_0\end{matrix}\begin{matrix}1\\ \theta_1\end{matrix}\begin{matrix}\frac{1}{2}\\ 0\end{matrix}\right)}\ar[dr]\ar[dd] \ar[rr] &&
{\mathrm{Kaw}_2\left(\begin{matrix}1\\ \theta\end{matrix}\begin{matrix}\frac{3}{2}\\ 0\end{matrix}\right)}
\ar[rr]\ar[rd]&&
{\mathrm{Kim}_8\begin{pmatrix}\frac{7}{2}\\ 0\end{pmatrix}} \\
&&
{\mathrm{Kaw}_7\left(\begin{matrix}0\\ \theta\end{matrix}\begin{matrix}\frac{1}{2}\\ 0\end{matrix}\begin{matrix}\frac{1}{2}\\ 0\end{matrix}\right)}\ar[rr]&&
{\mathrm{Kaw}_4\left(\begin{matrix}\frac{1}{2}\\ 0\end{matrix}\begin{matrix}\frac{3}{2}\\ 0\end{matrix}\right)}\\
&
{\mathrm{Kaw}_3\left(\begin{matrix}\frac{1}{2}\\ 0\end{matrix}\begin{matrix}2\\ 0\end{matrix}\right)}\ar[urrr]
}
\end{equation}
It is important to notice that isomonodromy condition is equivalent to the fact that the deformation of equation
is induced by a flat meromorphic connection on the universal irregular curve, i.e. the total space of the family
of irregular curve.
\section{Algebraic solutions of irregular Garnier systems: examples and structure}\label{sec:algsol}
There are several methods to construct algebraic solutions of classical Garnier systems (logarithmic case),
see \cite{AK,Boalch,CM,Diarra1,Doran,DM0,LisovyyTykhyy} and references therein.
In the irregular case, let us describe two methods to produce algebraic isomonodromic deformations.
\subsection{Classical solutions: the Galois group is $C_\infty$ or $D_\infty$}\label{sec:classicalsol}
The rough idea is as follows. Since the differential Galois group of a linear differential equation can be determined
from its coefficients by algebraic operations, it follows that iso-Galois deformations are of algebraic nature:
we have an algebraic stratification of each moduli space
$$\mathcal M_g\begin{pmatrix}\kappa_1&\cdots&\kappa_n\\
\theta_1&\cdots&\theta_n\end{pmatrix}$$
as defined in section \ref{sec:IsomDef} where strata are defined in term of the Galois group. In fact,
this is not exactly true since there might be infinitely many strata corresponding to finite groups in the dihedral case.
However, the locus of each finite group is algebraic and coincides with a finite number of isomonodromic leaves.
It follows that the leaf associated to a finite linear group is algebraic. This has been extensively used in \cite{Hitchin,BoalchIco,BoalchTetraOcta,Boalch6,BoalchHigherGenus,Boalch}
and references therein in the logarithmic case; this does not occur in the irregular case since the Galois group is never finite
in that case.
However, in a similar way, the locus of $C_\infty$ or $D_\infty$ is algebraic and is a finite union of isomonodromy leaves
in many cases when $C=\mathbb P^1$.
Let us first explain the diagonal case $C_\infty$.
The poles can only be of type $\Log$ and $\Irrun$, and in this latter case,
the Galois group $C_\infty$ coincides with all local exponential torii at irregular singular points
and all Stokes matrices are trivial. The two eigendirections of the Galois group correspond to
two $\nabla$-invariant line bundles $L,L^{-1}\subset E$ of the vector bundle for the normalized equation,
and we have $E=L\oplus L^{-1}$.
The connection $\nabla$ restricts as meromorphic connections on $(L,\nabla\vert_L)$ and $(L,\nabla\vert_L)^{\otimes(-1)}$
and, at each pole $t_i$,
the corresponding residues are $\pm\frac{\theta_i}{2}$: they are opposite for $L$ and $L^{-1}$.
Fuchs relation yields:
$$\sum_{i=1}{n}\epsilon_k\theta_k\in\mathbb Z,\ \ \ \epsilon_i=\pm1.$$
There are finitely many such relations for each formal data, and given one relation,
the connection $(L,\nabla\vert_L)$ (and therefore $(E,\nabla)$) can be uniquely determined by Mittag-Leffler's Theorem
from the data of the irregular curve ($C=\mathbb P^1$ + principal parts).
Something similar occur when the Galois group is $D_\infty$. The poles must be of type $\Log$, $\Irrun$ or $\Irram$
and the normal subgroup $C_\infty\subset D_\infty$ must coincide with all local exponential torii at irregular singular points.
In that case, Stokes matrices are trivial and, if $C=\mathbb P^1$, the global monodromy group is generated by matrices $M_i\in D_\infty$
(local monodromy at $t_i$) satisfying
$$M_1\cdots M_n=I.$$
Precisely, for a normalized equation, we have:
$$
\begin{matrix}
M_i\ \text{diagonal}\hfill\hfill & \Rightarrow & \kappa_i\in\mathbb Z_{\ge0}\\
M_i\ \text{anti-diagonal} & \Rightarrow &
\begin{pmatrix}\kappa_i\\ \theta_i\end{pmatrix}=\begin{pmatrix}0\\ \frac{1}{2}\end{pmatrix}\ \text{or}\ \begin{pmatrix}\frac{1}{2}+n\\0\end{pmatrix},\ n\ge0.
\end{matrix}
$$
The number of anti-diagonal matrices among $M_1,\ldots,M_n$ is even.
\begin{remark}\label{rem:2antidiagDinfty}
One can check that, if $C=\mathbb P^1$ and there are only two anti-diagonal matrices among $M_1,\ldots,M_n$,
say $M_{n-1}$ and $M_n$, then again the (algebraic) locus of $D_\infty$ consists in a finite number of isomonodromic leaves.
Indeed, the monodromy representation is determined by the $M_i$'s; for $i=1,\ldots,n-2$, we have
$$M_i=\begin{pmatrix}\lambda_i&0\\ 0&\lambda_i^{-1}\end{pmatrix}\ \ \ \text{with}\ \lambda_i=e^{\pm \sqrt{-1}\pi\theta_i};$$
we can rescale $M_n=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$ by diagonal conjugacy, and $M_{n-1}$ is determined by the relation.
The connection is also determined by the irregular curve in this case (principal parts).
\end{remark}
\subsection{Pull-back algebraic solutions}\label{sec:pullback}
Another way to construct algebraic isomonodromic deformations (see \cite{Doran,AK,Kitaev,AK2,Kitaev2,Kitaev3,Kitaev4,VK,VK2,Diarra1,OO}) is to fix a differential equation
$(C_0,E_0,\nabla_0)$ and consider an algebraic family $\phi_t:C_t\to C_0$ of ramified covers,
where $t\in P$ a projective variety. The pull-back
$(C_t,\phi_t^*E_0,\phi_t^*\nabla_0)$ provides an algebraic isomonodromic deformation. Indeed, it is induced by
the flat connection $(\Phi^*E_0,\Phi^*\nabla_0)$ over the total space $t:\mathcal C=\sqcup_t C_t\to P$ defined by pull-back
via the total ramified cover $\Phi:=(\phi_t,t):\mathcal C\to C_0\times P$.
\subsection{Garnier algebraic solutions and apparent singular points}
So far, we have neglected to consider apparent singular points in isomonodromic deformations
since we were dealing with normalized equations. However, Garnier systems are derived from
isomonodromic deformations of scalar differential equations (\ref{eq:scalar}) with $N$
poles (counted with multiplicity) and $N-3$ apparent singular points (see \cite{Kimura,Kawamuko}).
This can be reinterpreted as the data of a connection $\nabla$ on the bundle
$E=\mathcal O_{\mathbb P^1}\oplus\mathcal O_{\mathbb P^1}(-1)$ with $N$ poles,
and we recover the scalar equation by taking $\mathcal O_{\mathbb P^1}\subset E$ as a cyclic vector.
Equivalently, and closer to our point of view, one can consider a $\SL$-connection $(E,\nabla)$
with $E=\mathcal O_{\mathbb P^1}\oplus\mathcal O_{\mathbb P^1}$,
and the cyclic vector (with the right number of apparent singular points) is choosen to be
the constant line bundle $L\subset E$ fitting with one of the eigendirection over one pole (typically at $\infty$
when normalizing the position of poles on $\mathbb P^1$).
Here, the poles are of type $\Log$, $\Logres$, $\Irrun$ or $\Irram$, but in the case $\Log$, we also
allow $\theta\in\mathbb Z$ as a degenerate case of $\Logres$ when the singular point becomes apparent.
These considerations lead us to the following facts.
\begin{prop}\label{prop:cyclicvector}
Let $(E=\mathcal O_{\mathbb P^1}\oplus\mathcal O_{\mathbb P^1},\nabla)$ be an irregular $\SL$-connection,
and $L\subset E$ be a cyclic vector like above.
The (local) isomonodromic deformation of $(E,\nabla)$ provides a (local) solution of the Garnier system
if the line bundle $L$ is not $\nabla$-invariant. If $\mathrm{Gal}(E,\nabla)=C_\infty$ and there is no apparent singular point,
then $L$ is $\nabla$-invariant and the deformation fails to provide a solution of the corresponding Garnier system.
\end{prop}
\begin{proof} The condition that $L$ is not $\nabla$-invariant is equivalent to the fact that it defines
a cyclic vector, which allow to define the scalar system with $N$ poles and $N-3$ apparent singular points.
This condition is preserved under isomonodromic deformations and the deformation of the scalar system
leads to a Garnier solution. In the case $(E,\nabla)$ is normalized (i.e. without apparent singular points)
and the Galois group is $C_\infty$, then $E=L_0\oplus L_0^{-1}$ with $L_0,L_0^{-1}$ two $\nabla$-invariant line bundles.
Automatically, we have $L_0\simeq\mathcal O_{\mathbb P^1}$ (a constant line bundle), and $L_0$ (resp. $L_0^{-1}$)
coincide with an eigendirection over each pole. It follows that $L_0$ or $L_0^{-1}$ coincides with $L$
everywhere, and is $L$ therefore $\nabla$-invariant.
\end{proof}
In the presence of {\bf apparent singular points}, the condition for a deformation to be isomonodromic
and giving rise to a Garnier solution is more subtle.
Let $(E_0,\nabla_0)$ be a meromorphic $\SL$-connection, which is normalized except
at an apparent singular point $p_0\in C$: it is of type $\Log$
with $\theta=n\in\mathbb Z_{>0}$. Then after a birational bundle transformation $\phi:E_0\dashrightarrow E_0'$,
we can erase the singular point; moreover, we can assume $\phi$ supported by $p_0$, i.e. inducing a biholomorphic
bundle tranformation outside of $p_0$. Generic local sections of $E_0$ at $p_0$ are transformed into sections of $E_0'$,
all tangent at the order $n-1$ to a given $\nabla$-invariant analytic subbundle $L_0\subset E_0'$. Then we have
\begin{prop}\label{prop:singappIsom}Under notations above, given a a deformation of $(E_0,\nabla_0)$
induced by a flat connection $(E,\nabla)$, are equivalent
\begin{itemize}
\item $(E,\nabla)$ is logarithmic near $p$,
\item the deformation $p$ of the singular point $p_0$ remains apparent
and the corresponding deformation of line bundle $L_0\subset E_0'$ is induced by a local analytic $\nabla$-invariant
line bundle $L\subset E'$.
\end{itemize}
In this case we say that the deformation is isomonodromic.
\end{prop}
Consequently, we will encode the data of such an equation $(E_0,\nabla_0)$ by the data of
the normalized equation $(E_0',\nabla_0')$ together with the data of $p_0$ and the local
$\nabla_0'$-invariant analytic line bundle$L_0$, or better by its fiber $l_0:=L_0\vert_{p_0}\subset E_0'\vert_{p_0}$
(initial condition) that is usually called {\bf parabolic data}. This method to deal with
apparent singular points is used in \cite{IIS}.
\begin{remark}
The case $n=0$ also occur in solutions of Garnier systems as degenerate case
of singular points of type $\Logres$.
Garnier solutions can also be interpreted as before as deformation of a normalized equation $(E_0,\nabla_0)$
together with the parabolic data $(p_0\in C,l_0\in E_0\vert_{p_0})$ and there are as many Garnier solutions
as choices of initial condition $l_0$.
\end{remark}
If we have several apparent singular points, the condition above must be imposed for each of them
to get an isomonodromic solution, and therefore giving rise to Garnier solutions.
By the way, we can have Garnier solutions corresponding to equation with Galois group $C_\infty$
in the presence of apparent singular points as we will see for Painlev\'e IV equation
(second line of Table \ref{table:AlgSolPainleve}).
\begin{prop}\label{prop:singappGarniersol}
The isomonodromic deformation of an irregular normalized connection $(E_0,\nabla_0)$
with an apparent singular point $(p_0,l_0)$ (notations above) is algebraic if and only if
\begin{itemize}
\item the deformation of $(E_0,\nabla_0)$ is algebraic, with Galois group $C_\infty$ or $D_\infty$,
\item the local $\nabla_0$-invariant analytic line bundle $L_0$ defined by $l_0$ has algebraic closure,
i.e. corresponds to one of the two $\nabla_0$ invariant line bundles in diagonal case $C_\infty$,
or of the $2$-multivalued line bundle in dihedral case $D_\infty$.
\end{itemize}
\end{prop}
\begin{proof}The global deformation space of the irregular curve including the apparent singular point
may be viewed as a fiber bundle over the deformation space of the strict irregular curve, with fiber $C$
corresponding to the position of the apparent singular point for a fixed normalized equation.
Clearly, the deformation of $(E_0,\nabla_0)$ is algebraic if and only if it is algebraic in each of these two directions.
But to be algebraic along $C$-fibers implies, due to Proposition \ref{prop:singappIsom}, that
the local analytic line bundle $L_0$ have algebraic closure. And this implies that the Galois group
must be $C_\infty$ or $D_\infty$ (the third possibility $\SL$ in the irregular case is excluded here).
\end{proof}
Proposition \ref{prop:singappGarniersol} can immediately be generalized to the case of several
apparent singular points.
\begin{remark}\label{rem:ApparentDihedral}
In the dihedral case $D_\infty$, an algebraic solution of a Garnier system with apparent singular point,
like in Proposition \ref{prop:singappGarniersol}, always arises in family, as limit of algebraic
solutions with linear monodromy $D_\infty$ and no apparent singular points like in section \ref{sec:classicalsol}.
Indeed, from the monodromy side, this can be written as
$$M_1\cdots M_n=I\ \text{with}\ M_{n-2}=\begin{pmatrix}0&-\lambda\\ \lambda^{-1}&0\end{pmatrix},\
M_{n-1}=\begin{pmatrix}0&1\\ -1&0\end{pmatrix},\
\underbrace{M_{n}=\begin{pmatrix}1&0\\ 0&1\end{pmatrix}}_{\text{apparent}}$$
$$\rightsquigarrow
\tilde M_{n-2}=\begin{pmatrix}0&-\lambda_1\\ \lambda_1^{-1}&0\end{pmatrix},\
\tilde M_{n-1}=\begin{pmatrix}0&1\\ -1&0\end{pmatrix},\
\underbrace{\tilde M_{n}=\begin{pmatrix}\lambda_2&0\\ 0&\lambda_2^{-1}\end{pmatrix}}_{\text{non apparent}}$$
where $\lambda_1\lambda_2=\lambda$. For the differential equation, the apparent point is replaced by a non apparent
logarithmic singular point.
\end{remark}
\subsection{Structure theorem}
The key result for our classification is the:
\begin{thm}[\cite{LPT}]Let $(E,\nabla)$ be a flat meromorphic $\mathfrak{sl}_2$-connection
on a projective manifold $X$. Then at least one of the following assertions holds true.
\begin{enumerate}
\item Maybe after passing to a (possibly ramified) two-fold cover $f: X' \to X$,
the connection $(E,\nabla)$ is equivalent to a diagonal connection on the trivial bundle:
$$\nabla=d+\begin{pmatrix}\omega&0\\ 0&-\omega\end{pmatrix}$$
with $\omega$ a rational closed $1$-form on $X$.
\item There exists a rational map $\Phi: X \dashrightarrow C$ to a curve and a meromorphic connection $(E_0,\nabla_0)$ on $C$
such that $(E,\nabla)$ is equivalent to $\Phi^*(E_0,\nabla_0)$ by bundle transformation.
\end{enumerate}
\end{thm}
A direct consequence is the
\begin{cor}\label{cor:structure}
Any algebraic solution of an irregular Garnier system comes from
\begin{enumerate}
\item either the deformation pull-back from a fixed connection $(C_0,E_0,\nabla_0)$
with $\mathrm{SL}_2(\mathbb C)$ Galois group via an algebraic family of ramified covers,
\item or the deformation of a connection having Galois group $D_\infty$,
\item or the deformation of a connection having Galois group $C_\infty$ with apparent
singular point(s) like in Proposition \ref{prop:singappGarniersol}.
\end{enumerate}
\end{cor}
This result has been recently proved in the Painlev\'e case in \cite{OO} by checking {\it a posteriori}
the known list of algebraic solutions.
We expect to have a similar result for isomonodromy equations for curves of genus $g>0$.
However, it is still not known if these equations are polynomial in such a generality, so that the question
of algebraicity of solutions does not make sense so far.
To prove the Corollary, we just notice that any solution (algebraic of not) comes from an isomonodromic deformation,
or equivalently a flat connection over the total space $\mathcal C$ of the deformation curve.
If the solution is algebraic, then $\mathcal C$
is algebraic, as well as the flat connection. We can therefore apply the above Theorem, and deduce the Corollary.
\subsection{Algebraic solutions of Painlev\'e I-V equations}\label{sec:AlgebraicPainleveI-V}
Table \ref{table:AlgSolPainleve} provides the list of algebraic solutions for irregular Painlev\'e equations
up to symmetries.
\begin{table}[h]
\caption{Algebraic solutions of irregular Painlev\'e equations}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Isomonodromy & solution & name & Local formal & Galois & pull-back ? & apparent\\
equation & & & data & group & & pole ? \\
\hline
$P_{V}\left(\frac{\theta^2}{2},-\frac{\theta^2}{2},0,-\frac{1}{2}\right)$ &
$q=-1$ & $P_V$-rat & $\begin{pmatrix}0&1&0\\ \theta-1&0&\theta\end{pmatrix}$ & $\SL$ & yes & no \\
\hline
$P_{V}\left(\frac{\theta^2}{2},-\frac{1}{2},\theta,-\frac{1}{2}\right)$ &
$q=\frac{t}{\theta}+1$ & $P_V$-Lag & $\begin{pmatrix}0&1&0\\ 0&\theta&\theta\end{pmatrix}$ & $C_\infty$ & no & yes\\
\hline
$P_{V}\left(\frac{\theta^2}{2},-\frac{1}{8},-2,0\right)$ &
$q=2\frac{\sqrt{t}}{\theta}+1$ & $P_V$-alg & $\begin{pmatrix}0&\frac{1}{2}&0\\ \frac{1}{2}&0&\theta\end{pmatrix}$ & $D_\infty$ & no & no\\
\hline
$P_{IV}\left(0,-\frac{2}{9}\right)$ & $q=-2t/3$ & $P_{IV}$-rat & $\begin{pmatrix}0&2\\ -\frac{2}{3}&0\end{pmatrix}$
& $\SL$ & yes & no \\
\hline
$P_{IV}\left(0,-2\right)$ & $q=-2t$ & $P_{IV}$-Her & $\begin{pmatrix}0&2\\ 0&0\end{pmatrix}$ & $C_\infty$ & yes & yes \\
\hline
$P_{III}\left(4\theta,-4\theta,4,-4\right)$ & $q=\sqrt{t}$ & $P_{III}^{D_6}$-alg & $\begin{pmatrix}1&1\\ \theta&\theta\end{pmatrix}$ & $\SL$ & yes & no \\
\hline
$P_{III}\left(4,-4,0,0\right)$ & $q=\sqrt{t}$ & $P_{III}^{D_8}$-alg & $\begin{pmatrix}\frac{1}{2}&\frac{1}{2}\\ 0&0\end{pmatrix}$ & $D_\infty$ & yes & no \\
\hline
$P_{III}\left(-8,0,0,-4\right)$ & $q=\left(-\frac{t}{2}\right)^{1/3}$ & $P_{III}^{D_7}$-alg & $\begin{pmatrix}1&\frac{1}{2}\\ 0&0\end{pmatrix}$ & $\SL$ & yes & no \\
\hline
$P_{II}\left(0\right)$ & $q=0$ & $P_{34}$-rat & $\begin{pmatrix}0&\frac{3}{2}\\ \frac{1}{2}&0\end{pmatrix}$
& $D_\infty$ & yes & no \\
&& $P_{II}$-rat & $\begin{pmatrix}3\\ 1\end{pmatrix}$ & $\SL$ & yes & no \\
\hline
\end{tabular}
\end{center}
\label{table:AlgSolPainleve}
\end{table}
The name of solutions follows from \cite{OO}. Here ``rat'' and ``alg'' stand for ``rational'' and ``algebraic''
while ``Lag'' and ``Her'' stand for ``Laguerre'' and ``Hermite'' polynomials; also notation $P_{34}$ refers to Gambier's list.
The $P_V$ equation, when $\delta=0$, is equivalent to $P_{III}$ and solutions $P_{V}$-alg and $P_{III}^{D_6}$-alg
are therefore equivalent, but with different kind of linear local data, and this is why we keep the two solutions in the list.
For the same reason, we might consider $P_{34}$-rat and $P_{II}$-rat as different solutions.
\begin{remark}In table \ref{table:AlgSolPainleve}, apart pull-back solutions listed in tables
\ref{table:irregular} and \ref{table:irregularConfl}, there are two more pull-back solutions
that correspond to dihedral Galois groups.
\end{remark}
\section{Ramified covers and differential equations}\label{sec:RamifiedCovers}
Given a ramified cover $\phi:C\to C_0$ of degree $d$, and given a point $c_0\in C_0$, we can associate
the pull-back divisor $\phi^*[c_0]=m_1[t_1]+\cdots+m_s[t_s]$ with $t_i\in C$ pair-wise distinct for $i=1,\ldots,s$.
This means that $\{t_1,\ldots,t_s\}$ is the fiber of $\phi$, and through convenient local coordinates $x_i$ near $t_i$,
$\phi(x_i)=(x_i)^{m_i}$; we have $m_1+\cdots+m_s=d$, and for generic $t_0$, we have all $m_i=1$.
We can therefore associate to $\phi$ its {\bf passport}
\begin{equation}\label{eq:passport}
\begin{pmatrix}\left[\begin{matrix}m_{1,1}\\ \vdots\\ m_{1,s_1}\end{matrix}\right] & \cdots &
\left[\begin{matrix}m_{\nu,1}\\ \vdots\\ m_{\nu,s_\nu}\end{matrix}\right]\end{pmatrix},\ \ \ \sum_{l=1}^{s_k}m_{k,l}=d\ \text{for}\ k=1,\ldots,\nu
\end{equation}
It is the data, for each critical value $c_1,\ldots,c_\nu\in C_0$ of $\phi$, of the corresponding partition
$m_{k,1}+\cdots+m_{k,s_k}=d$: there are $s_k$ points in the fiber $\phi^{-1}(c_k)$ with multiplicities $m_{k,l}$, $l=1,\ldots,s_k$.
The {\bf Riemann-Hurwitz} formula writes
\begin{equation}\label{eq:RiemHurw}
\underbrace{2-2g(C)}_{\chi(C)}=d(\underbrace{2-2g(C_0)}_{\chi(C_0)})-R\ \ \ \text{where}\ \ \ R=\text{total ramification}:=
\sum_{k=1}^n(d-s_k).
\end{equation}
Given a differential equation $(E_0,\nabla_0)$ with polar divisor $D$ on $C_0$, we can consider the pull-back
$(E,\nabla):=\phi^*(E_0,\nabla_0)$. It is easy to deduce the local formal data of $(E,\nabla)$ from that one
of $(E_0,\nabla_0)$ and the passport. Precisely, let $p\in C$ be a point of multiplicity $m$ for $\phi$ (ramification $r=m-1$);
then the local formal datas of $(E_0,\nabla_0)$ at $\phi(p)$ and $(E,\nabla)$ at $p$ are related by:
$$(\kappa,\theta)=(m\kappa_0,m\theta_0)$$
{\bf except} when $\kappa=0$, $\theta_0\in\mathbb Q\setminus\mathbb Z$ and $m\theta_0\in\mathbb Z$
where the singular point becomes apparent, i.e. can be deleted by bundle transformation; in that latter case,
we indeed delete the pole. We deduce, for the respective order of poles, that
$$\ord_p(\nabla)\le m\cdot \ord_{\phi(p)}(\nabla_0)+1-m$$
with strict inequality in the special case above, and ramified case $\kappa_0\not\in\mathbb Z$ and $m>1$.
On the other hand, when $\phi(p)$ is not a pole for $\nabla_0$, then $p$ is also non singular for $\nabla$.
In the sequel, we still fix the differential equation $(C_0,E_0,\nabla_0)$ and we want to deform the ramified cover
and pull-back equation
$$\phi_t:C_t\to C_0,\ \ \ (E_t,\nabla_t):=\phi_t^*(E_0,\nabla_0).$$
We consider an irreducible algebraic family, parametrized by say $P\ni t$ (irreducible and projective),
and there is a Zariski open subset $U\subset P$
where the passport of $\phi_t$ is locally constant, as well as the number $B$ of its critical values outside
of the (fixed) polar locus of $\nabla_0$. Note that $B$ bounds the dimension of $P$, and maybe switching
to a larger family, i.e. with a larger parameter space $P$, we can assume $B=\dim(P)$.
The deformation $t\mapsto(C_t,E_t,\nabla_t)$, being automatically isomonodromic, locally factors through
the universal isomonodromic deformation (see \cite{Heu}). In general, it defines a projective subvariety contained in the
transcendental isomonodromy leaf $\mathcal L$ of the corresponding isomonodromy foliation $\mathcal F$.
But if $P\to\mathcal L$ is locally dominant, then it is globally dominant and the entire leaf $\mathcal L$ itself is algebraic,
giving rise to an algebraic solution of the corresponding isomonodromy equation.
A necessary condition for this is that the dimension $T$ of $\mathcal L$ is bounded by the dimension $B$ of $P$.
We will call {\bf admissible} the data of a differential equation $(C_0,E_0,\nabla_0)$ and a passport (\ref{eq:passport})
such that $B\ge T$. For the sequel, it is convenient to add in the passport all trivial fibers
$$\underbrace{1+\cdots+1}_{d\ \text{times}}$$
appearing over poles so that we may assume that $\nu=n+B$ in (\ref{eq:passport}) with entries $k=1,\ldots,n$
corresponding to fibers over the poles of $\nabla_0$, and entries $k=n+1,\ldots,\nu$ corresponding to free critical points
that are deformed along the family. Our aim now is to classify those admissible data such that the pull-back differential
equation has irregular Teichm\"uller dimension $T\le B$. We will see in the next section that this inequality gives
very strong constraints.
\section{Scattering ramifications}\label{sec:Scattering}
Suppose we are given a normalized differential equation $(C_0,E_0,\nabla_0)$ with poles $p_1,\ldots,p_n\in C_0$, and local formal data
$(\kappa_i,\theta_i)_{i=1,\ldots,n}$ like (\ref{eq:LocForData}). Suppose we are given
a ramified covering $\phi:C\to C_0$ with passport $(d=m_{k,1}+\cdots+m_{k,s_k})_{k=1,\ldots,\nu}$ over $c_1,\ldots,c_\nu\in C_0$ like (\ref{eq:passport})
where
\begin{itemize}
\item $(m_{k,l})_l$ for $k=1,\cdots,n$ correspond to the multiplicities of $\phi$ along fibers over the poles of $\nabla_0$ ($c_k=p_k$), some of them being possibly unbranched, i.e. $m_{k,l}=1$ for all $l$;
\item $(m_{k,l})_l$ for $k=n+1,\ldots,n+b$ correspond to fibers of $\phi$ over non singular points of $\nabla_0$,
all of which are branching, i.e. $m_{k,l}>1$ for at least one $l$.
\end{itemize}
Denote by $R_k:=d-s_k$ the total ramification number of the fiber over $c_k$. We denote by $(C,E,\nabla)$ the normalized
equation that can be deduced from the pull-back $\phi^*(E_0,\nabla_0)$ by bundle transformation.
Let $N_k$ denote the number of poles counted with multiplicity in the fiber over $c_k$.
The irregular Teichm\"uller dimension of $(C,E,\nabla)$ is given by
$$T=3g-3+N=3g-3+N_1+\cdots+N_n$$
where $g$ is the genus of $C$, given by Riemann-Hurwitz Formula (\ref{eq:RiemHurw}) with $R:=\sum_{k=1}^\nu R_k$.
We now explain how to compute $N_k$ by means of $(\kappa_k,\theta_k)$ and $(m_{k,l})_l$.
\begin{itemize}
\item If $\kappa_k=0$ and $\theta_k\not\in\mathbb Q\setminus\mathbb Z$, then
$$N_k=s_k=d-R_k.$$
\item If $\kappa_k=0$ and $\theta_k$ has order $m>1$ modulo $\mathbb Z$, then
$$N_k=d-R_k-\#\{l=1,\ldots,s_k\ \vert\ m\ \text{divides}\ m_{k,l}\}.$$
\item If $\kappa_k\in\mathbb Z_{>0}$, then
$$N_k=\sum_{l=1}^{s_k}(m_{k,l}+1)=d(\kappa_k+1)-R_k.$$
\item If $\kappa_k\in\mathbb Z_{>0}-\frac{1}{2}$, then
$$N_k=d(\kappa_k+1)-R_k+\#\{l=1,\ldots,s_k\ \vert\ 2\ \text{does not divide}\ m_{k,l}\}.$$
\end{itemize}
We assume $\phi$ admissible, i.e. that $T\le B$.
In this section, we show that we can replace $\phi$ by another ramified cover $\phi'$ with more
critical points but less multiplicity in fibers in such a way that $T-B$ can only decrease. The total ramification
will be unchanged, but will be scattered oustide of the polar locus. This will
allow us to replace the deformation of $\phi$ by the wider deformation of $\phi'$, so that we will
be able to recover $\phi$ (and its deformation) by confluence of critical values. By the way,
the passport of $\phi'$ will be as simple as possible and it will be easy to classify such covers.
The first step consists in scattering the branching points over critical values outside of the polar locus.
We call {\bf simple branching} a fiber of the form
$$\left[\begin{matrix}m_{k,1}\\ m_{k,2}\\ \vdots\\ m_{k,s_k}\end{matrix}\right]=\left[\begin{matrix}2\\ 1\\ \vdots\\ 1\end{matrix}\right]$$
i.e. with $R_k=1$.
\begin{lem}\label{lem:FibreCritique}
Let $n<k\le\nu$, i.e. $c_k$ not a pole of $\nabla$.
We can deform $\phi\rightsquigarrow\phi'$ over a neighborhood of $c_k$ so that
the single fiber of $\phi$ with total ramification $R_k$ is replaced by $R_k$ simple branching fibers for $\phi'$.
We have increased $B$ without changing $R$ or $T$.
\end{lem}
\begin{proof}Fix a disc $\Delta\subset C_0$ in which $c_k$ is the unique critical value of $\phi$.
The monodromy of the ramified covering is given by the permutation that decomposes into
the product of $s_k$ cyclic permutations of orders $m_{k,1},\ldots,m_{k,s_k}$ with disjoint support.
In order to construct $\phi'$, it is enough to define its monodromy in $\Delta$,
namely the data of $R_k$ transpositions whose product is the monodromy of $\phi$.
But it suffices to decompose each cyclic permutation of length $m_{k,l}$ above as the product of
$m_{k,l}-1$ transpositions, which indeed gives
$$\sum_{l=1}^{s_k}(m_{k,l}-1)=d-s_k=R_k$$
transpositions making the job.
\end{proof}
After applying Lemma (\ref{lem:FibreCritique}) to each critical fiber of $\phi$ outside the polar locus,
we can now assume that all fibers $\phi^{-1}(c_k)$ are simple branching for $k>n$. By this way,
we have maximized $B$ without touching at fibers over poles of $\nabla_0$ so far, so $N$ has not changed.
We now discuss how to simplify fibres over poles of $\nabla$ by putting some of their branch points out,
in additional simple branching fibers, without increasing $N-B$; this will however increase $B$.
We do this in successive lemmae discussing on the type of poles.
\begin{lem}\label{lem:FibreNonPeriodic}
Let $0\le k\le n$, i.e. $c_k=p_k$ is a pole of $\nabla$, and assume $\kappa_k=0$
and $\theta_k\not\in\mathbb Q\setminus\mathbb Z$.
We can deform $\phi\rightsquigarrow\phi'$ over a neighborhood of $c_k$ so that
the single fiber of $\phi$ with total ramification $R_k$ is replaced by a non branching fiber
over $p_k$ (i.e. $R_k'=0$), and $R_k$ simple branching fibers nearby.
We have $N_k'=d=N_k+R_k$ and $B'=B+R_k$.
\end{lem}
\begin{proof}We proceed like in the proof of Lemma \ref{lem:FibreCritique} by replacing
the monodromy of $\phi^{-1}(p_k)$ by the product of the identity and $R_k$ transpositions.
Here, the identity stands for the trivial monodromy of the non branching fiber over $p_k$.
\end{proof}
\begin{lem}\label{lem:FibreIrregUnram}
Let $0\le k\le n$, i.e. $c_k=p_k$ is a pole of $\nabla$, and assume $\kappa_k\in\mathbb Z_{>0}$.
We can deform $\phi\rightsquigarrow\phi'$ over a neighborhood of $c_k$ so that
the single fiber of $\phi$ with total ramification $R_k$ is replaced by a non branching fiber
over $p_k$ (i.e. $R_k'=0$), and $R_k$ simple branching fibers nearby.
We have $N_k'=d(\kappa_k+1)=N_k+R_k$ and $B'=B+R_k$.
\end{lem}
The proof is the same as before.
\begin{lem}\label{lem:FibrePeriodic}
Let $0\le k\le n$, i.e. $c_k=p_k$ is a pole of $\nabla$, and assume $\kappa_k=0$
and $\theta_k$ has order $m>1$ modulo $\mathbb Z$.
We can deform $\phi\rightsquigarrow\phi'$ over a neighborhood of $p_k$ so that
the single fiber of $\phi$ with total ramification $R_k$ is replaced by fiber with passport
$$\left[\begin{matrix}m_{k,1}'\\ \vdots\\ m_{k,s_k'}'\end{matrix}\right]=\left[\begin{matrix}m\\ \vdots\\ m\\ 1\\ \vdots\\ 1\end{matrix}\right]$$
and only simple branching fibers nearby.
We have $N_k'\ge N_k$, $B'\ge B$ and $N_k'-B'\le N_k-B$.
\end{lem}
\begin{proof}Here, we cannot just proceed as before. Indeed, if $m$ divides $m_{k,l}$, then there
is not pole on the preimage (or an apparent one that disappears after normalization); however, replacing
by $m_{k,l}$ non branching points would increase $N_k$ by $m_{k,l}$, but increase $B$ only by $m_{k,l}-1$
so that $N-B$ increases by $1$. We thus have to take care of those points with $m$ dividing $m_{k,l}$.
Consider the euclidean division $m_{k,l}=s^0\cdot m+s^1$. Then, we can replace the point with
multiplicity $m_{k,l}$ by
\begin{itemize}
\item $s^0$ points of multiplicity $m$ in the fiber $\phi^{-1}(p_k)$ (contributing to no pole),
\item $s^1$ non branching points in the fiber $\phi^{-1}(p_k)$ (contributing to $s^1$ poles),
\item and $s^0+s^1-1$ additional simple branching fibers around.
\end{itemize}
To realize the deformation of $\phi$, we have to realize the corresponding monodromy representation,
which is easy in this case. Indeed, the concatenation of a cyclic permutation (or a cycle inside a permutation)
runs as follows:
$$\underbrace{(1\ldots \mu)}_{\text{over }p_k}=\underbrace{(1\ldots \mu')(\mu'+1\ldots\mu)}_{\text{over }p_k\text{ after concatenation}}\ \ \ \cdot \underbrace{(1,\mu'+1)}_{\text{simple branching fiber}}$$
where $1<\mu'<\mu$. We just have to repeat this procedure $s_0+s_1-1$ times.
By the way, we get $N_k'=N_k+s^1-1$ (or $N_k'=N_k=0$ if $m$ divides $m_{k,l}$)
and $B'=B+s^0+s^1-1$. We proceed similarly with all $m_{k,l}\not=m,1$ in the fiber.
\end{proof}
\begin{lem}\label{lem:FibreIrregRam}
Let $0\le k\le n$, i.e. $c_k=p_k$ is a pole of $\nabla$, and assume $\kappa_k\in\mathbb Z_{>0}-\frac{1}{2}$.
We can deform $\phi\rightsquigarrow\phi'$ over a neighborhood of $p_k$ so that
the single fiber of $\phi$ with total ramification $R_k$ is replaced by fiber with passport
$$\left[\begin{matrix}m_{k,1}'\\ \vdots\\ m_{k,s_k'}'\end{matrix}\right]=\left[\begin{matrix}2\\ \vdots\\ 2\\ 1\\ \vdots\\ 1\end{matrix}\right]$$
and only simple branching fibers nearby.
We have $N_k'-B'= N_k-B$.
\end{lem}
The proof is similar to that of Lemma \ref{lem:FibrePeriodic}. We call {\bf scattered admissible covering}
(with respect to $(C_0,E_0,\nabla_0)$)
an admissible covering $\phi$ satisfying the conclusion for $\phi'$ in Lemmae \ref{lem:FibreCritique} - \ref{lem:FibreIrregRam}.
We assume from now on that $\phi$ is scattered. We now deform the differential equation $(E_0,\nabla_0)$ on $C_0$
into a logarithmic one $(E_0',\nabla_0')$ without changing the ramified cover, in such a way that $N-B$ does not increase.
This will allow us to conclude with the classification established by the first author \cite{Diarra1} in the logarithmic case.
\begin{lem}\label{lem:IrregPoleLogPole}
Let $p_k$ be an irregular unramified pole of $\nabla$, i.e. $\kappa_k\in\mathbb Z_{>0}$.
We can deform the differential equation $(E_0,\nabla_0)\rightsquigarrow(E_0',\nabla_0')$ over a neighborhood of $p_k$
so that the deformed equation is also normalized with $\kappa_k+1$ simple poles instead
of a single pole of multiplicity $\kappa_k+1$. If $\phi$ is a scattered covering, then $N'=N$ (and $B'=B$).
\end{lem}
\begin{proof}It is similar to the previous proofs: instead of dealing with the monodromy of the covering,
we use the monodromy of the differential equation. We can write the monodromy $M$ of $\nabla_0$ around $p_k$
as the product of $\kappa_k+1$ non trivial linear transformations:
$$M=M_0\cdot M_1\cdots M_{\kappa_k},\ \ \ M_i\in\SL(\mathbb C),\ \ \ M_i\not=\pm I.$$
By standard arguments {\it \`a la Riemann-Hilbert}, we can first realize these matrices as local monodromy
of a differential equation over a disc with $(\kappa_k+1)$ simple poles in normal form, and total monodromy $M$.
Next, by surgery over the disc, we replace the single pole $p_k$ in $(E_0,\nabla_0)$ by this new differential
equation, and get $(E_0',\nabla_0')$ with the desired properties.
\end{proof}
\begin{lem}\label{lem:IrregRamPoleLogPole}
Let $p_k$ be an irregular and ramified pole of $\nabla$, i.e. $\kappa_k\in\mathbb Z_{>0}-\frac{1}{2}$.
We can deform the differential equation $(E_0,\nabla_0)\rightsquigarrow(E_0',\nabla_0')$ over a neighborhood of $p_k$
so that the deformed equation is also normalized with $\bar\kappa_k+1=\kappa_k+\frac{3}{2}$ simple poles instead
of a single pole of multiplicity $\bar\kappa_k+1$, one of which is at the critical point $p_k$ for $\phi$, with $\theta_k'=\frac{1}{2}$. If $\phi$ is a scattered covering, then $N'=N$ (and $B'=B$).
\end{lem}
\begin{proof}The proof is similar, except that we need $M=M_0\cdot M_1\cdots M_{\bar\kappa_k}$ with
$\mathrm{trace}(M_0)=0$.
Then we can realize $M_0$ as the monodromy of a logarithmic differential equation with exponent $\frac{1}{2}$.
\end{proof}
\section{Irregular Euler characteristic}\label{sec:IrregEuler}
Following Poincar\'e, we can associate, to a fuchsian differential equation,
an orbifold structure on the base curve $C_0$ (see also \cite{Diarra1})
and get the notion of orbifold Euler characteristic. Here we define an irregular version of it.
Given $(C_0,E_0,\nabla_0)$ in normal form,
we define the {\bf orbifold order} $\nu_k$ at a logarithmic
singular point as the order of the local monodromy:
\begin{itemize}
\item $\nu_k\in\mathbb Z_{>1}$ is the order of $[\theta_k\ \mod\ \mathbb Z]$ if $\theta_k\in\mathbb Q\setminus\mathbb Z$,
\item $\nu_k=\infty$ if not.
\end{itemize}
We define the {\bf irregular Euler characteristic} of the differential equation as
$$\chi^{\mathrm{irr}}(C_0,E_0,\nabla_0):=2-2g_0-\sum_{k=1}^{n}(1+\kappa_k)+\sum_{\kappa_k=0}\frac{1}{\nu_k}.$$
In the logarithmic case, this notion coincide to the orbifold Euler charateristic $\chi^{\mathrm{irr}}=\chi^{\mathrm{orb}}$.
We note that the two operations of Lemmae \ref{lem:IrregPoleLogPole} and \ref{lem:IrregRamPoleLogPole},
replacing $(E_0,\nabla_0)$ by a logarithmic equation $(E_0',\nabla_0')$, does not change the irregular Euler characteristic.
Moreover, likely as in \cite[Prop. 2.5]{Diarra1}, we have the following characterization:
\begin{prop}\label{prop:chiPositive}
If $\chi^{\mathrm{irr}}(C_0,E_0,\nabla_0)\ge0$, then the Galois group $\mathrm{Gal}(E_0,\nabla_0)$
is virtually abelian. In particular, in the irregular case, $\nabla_0$ has only trivial Stokes and the Galois group
is one-dimensional: it is diagonal, or dihedral.
\end{prop}
\begin{proof}In the logarithmic case, the curve $C_0$ with its orbifold structure is a finite quotient of
the sphere or the torus. The differential equation lifts as a differential equation with trivial or abelian
monodromy group respectively. Since the Galois group is the Zariski closure of the monodromy
group in the logarithmic case, we get that $\mathrm{Gal}(E_0,\nabla_0)$ is virtually abelian.
In the irregular case, $\chi^{\mathrm{irr}}\ge0$ gives us
$$-\chi^{\mathrm{irr}}=2g_0-2+\underbrace{\sum_{\kappa_k=0}(1-\frac{1}{\nu_k})}_{\ge 0}
+\underbrace{\sum_{\kappa_k>0}(1+\kappa_k)}_{\ge 1+\frac{1}{2}}\le0.$$
We promptly see that $g_0=0$, likely as in the logarithmic case, and we have
the following possible formal local data up to bundle transformation
$$\begin{pmatrix}\kappa_1\cdots \kappa_n\\ \theta_1\cdots \theta_n\end{pmatrix}=\begin{pmatrix}1\\ 0\end{pmatrix},\ \ \
\begin{pmatrix}\frac{1}{2}\\ 0\end{pmatrix}\ \ \ \text{or}\ \ \ \begin{pmatrix}0&\frac{1}{2}\\ \frac{1}{2}&0\end{pmatrix}.$$
In the first case, we assume that we have an unramified irregular singular point, and the inequality
for $\chi^{\mathrm{irr}}$ gives no place for any other singular point; moreover, $\kappa=1$.
The monodromy around the unique singular point decomposes as
$$M=
\begin{pmatrix}\lambda&0\\ 0&\lambda^{-1}\end{pmatrix}
\begin{pmatrix}1&s\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ t&1\end{pmatrix}=\begin{pmatrix}\lambda(1+st)&\lambda s\\ \lambda^{-1}t&\lambda^{-1}\end{pmatrix}$$
which must be trivial, implying $\lambda=e^{i\pi\theta}=1$ and $s=t=0$. This means that $\theta\in\mathbb Z$,
or equivalently $\theta=0$ after bundle transformation, and we have trivial Stokes matrices.
The local (and therefore global) Galois group is diagonal like the differential equation.
In the second and third cases, the irregular point is ramified and there is a place for a single logarithmic pole with orbifold
order $\nu=2$. The local monodromy decomposes as
$$M=
\begin{pmatrix}0&1\\ -1&0\end{pmatrix}
\begin{pmatrix}1&s\\ 0&1\end{pmatrix}
=\begin{pmatrix}0&1\\ -1&-s\end{pmatrix}$$
which is never the identity. The second case is therefore impossible. In the third case,
$M$ is also the local monodromy at the logarithmic pole: the trace must be zero, $s=0$, implying trivial Stokes again.
\end{proof}
\begin{prop}\label{prop:bounds} Let $(C_0,E_0,\nabla_0)$ be a normalized differential equation
with local formal data $(\kappa_k,\theta_k)_k$. Let $\phi:C\to C_0$ be a degree $d$ ramified cover
and $(E,\nabla)$ be the pull-back equation. Let $T$ be the dimension of the irregular Teichm\"uller
deformation space for $(C,E,\nabla)$. Let $B$ be the number of critical values of $\phi$ outside
the poles of $\nabla_0$. Then we have
$$T-B\ge g-1-d\cdot\chi^{\mathrm{irr}}(C_0,E_0,\nabla_0)$$
where $g$ is the genus of $C$.
\end{prop}
\begin{proof}Let us decompose
$$\begin{matrix}
N=N_1+\cdots+N_n\hfill\hfill\\
R=R_1+\cdots+R_n+B
\end{matrix}$$
where $N_k$ is the number of poles of $\nabla$ (counted with multiplicity) along the fiber $\phi^{-1}(p_k)$,
and $R_k$ is the total ramification along $\phi^{-1}(p_k)$. Now we have
$$T=3g-3+N=g-1+\underbrace{2g-2}_{=d(2g_0-2)+R}+N$$
(by Riemann-Hurwitz) which gives
$$T-B=g-1+d(2g_0-2)+\sum_{k=1}^n(N_k+R_k).$$
Let us lower bound $N_k+R_k$ in fonction of the type of pole $p_k$ for $\nabla_0$.
We note that, along scattering in Lemmae \ref{lem:FibreCritique} - \ref{lem:FibreIrregRam},
the value of $N_k+R_k$ can only decrease, so that it is enough to estimate a lower bound
for a scattered covering $\phi$.
If $\kappa_k=0$ and $\nu_k\in\mathbb Z_{>0}\cup\{\infty\}$ is the orbifold order,
then we have
$$d=N_k+m\cdot\nu_k\ \ \ \text{and}\ \ \ R_k=m\cdot(\nu_k-1)$$
so that
$$N_k+R_k=N_k+\frac{d-N_k}{\nu_k}(\nu_k-1)
=d\left(1-\frac{1}{\nu_k}\right)+\frac{N_k}{\nu_k}\ge d\left(1-\frac{1}{\nu_k}\right).$$
If $\kappa_k\in\mathbb Z_{>0}$, then we find (after scattering)
$$N_k=d(1+\kappa_k)\ \ \ \text{and}\ \ \ R_k=0.$$
If $\kappa_k\in\mathbb Z_{>0}-\frac{1}{2}$, then we have
$$d=m_0+2m_1,\ \ \ N_k=m_0(1+\kappa_k+\frac{1}{2})+m_1(1+2\kappa_k)\ \ \ \text{and}\ \ \ R_k=m_1$$
so that $m_1=\frac{d-m_0}{2}$, and after substitution, we find
$$N_k+R_k=d\left(1+\kappa_k\right)+\frac{m_0}{2}\ge d\left(1+\kappa_k\right).$$
After summing for $k=1,\ldots,n$, we find the expected lower bound.
\end{proof}
\begin{cor}Under assumptions of Proposition \ref{prop:bounds}, if $\nabla_0$ is irregular with non trivial Stokes matrices
and $T-B\le0$, then we have $\chi^{\mathrm{irr}}<0$, $g_0=g=0$, and
$$d\vert \chi^{\mathrm{irr}}\vert\le 1.$$
\end{cor}
\begin{proof}The inequality $\chi^{\mathrm{irr}}<0$ directly follows from Proposition \ref{prop:chiPositive}
and the fact we are assuming non trivial Stokes matrices. Then Proposition \ref{prop:bounds} gives
$$0\ge T-B\ge g-1+\underbrace{d\vert \chi^{\mathrm{irr}}\vert}_{>0}$$
which implies $g=0$ (and therefore $g_0=0$). Then we deduce the expected inequality.
\end{proof}
\section{Classification of covers}\label{sec:Classifcover}
In this section, we classify pull-back algebraic solutions of irregular Garnier systems
(see section \ref{sec:algsol}). In other words,
we list all differential equations $(C_0,E_0,\nabla_0)$ and ramified coverings $\phi:C\to C_0$
such that, by deforming $\phi\rightsquigarrow\phi_t$, we get a complete isomonodromic deformation $\phi_t^*(E_0,\nabla_0)$.
In fact, we omit classical solutions which will be discussed in section \ref{sec:classicalsol}
and therefore assume $\chi^{\mathrm{irr}}(C_0,E_0,\nabla_0)<0$ (see Proposition \ref{prop:chiPositive}).
Moreover, equations $(E_0,\nabla_0)$ are listed up to bundle transformation;
in particular, we can assume without lack of generality that $(E_0,\nabla_0)$ is a normalized equation.
In the sequel, we use notations of previous sections. In particular,
$T$ is the dimension of the irregular Teichm\"uller space of the irregular curve
given by $\phi^*(C_0,E_0,\nabla_0)$, and $B$ is the dimension of deformation
of $\phi$, obtained by moving the critical values of $\phi$ outside the poles of $\nabla_0$.
To get a complete deformation, we need $T\le B$, and we will assume $T>0$ (otherwise there is no deformation).
\begin{prop}[\cite{Diarra1}]\label{Prop:DiarraPullBack}
Assume $(C_0,E_0,\nabla_0)$ is a (normalized) logarithmic connection
with at least one pole $p_k$ having infinite orbifold order $\nu_k=\infty$, and assume
$\phi:C\to C_0$ is a scattered ramified cover of degree $d\ge 2$. If $T\le B$ and
$\chi^{\mathrm{irr}}(C_0,E_0,\nabla_0)<0$, then $(C_0,E_0,\nabla_0)$ is hypergeometric,
and up to bundle transformation, we are
in the list of table \ref{table:logarithmic}.
\end{prop}
\begin{proof}This proposition is proved in \cite[second table, page 142]{Diarra1}. It can
be proved directly by using the inequality of Lemma \ref{prop:bounds}.
\end{proof}
\begin{table}[h]
\caption{Logarithmic classification}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Local formal data $\theta_k$ & Degree & Covering passport \\
\hline
$( \frac{1}{2}, \frac{1}{3}, \theta)$ & 6 &
$\begin{pmatrix} \left[\begin{matrix}2\\2\\2\end{matrix}\right] & \left[\begin{matrix}3\\3\end{matrix}\right] &\left[\begin{matrix}1\\ \vdots\\1\end{matrix}\right] & 3\end{pmatrix}$ \\
\hline
$( \frac{1}{2}, \frac{1}{3}, \theta)$ & 4 &
$\begin{pmatrix} \left[\begin{matrix}2\\2\end{matrix}\right] & \left[\begin{matrix}3\\1\end{matrix}\right] & \left[\begin{matrix}1\\ \vdots\\1\end{matrix}\right] & 2\end{pmatrix}$ \\
\hline$( \frac{1}{2}, \frac{1}{3}, \theta)$ & 3 &
$\begin{pmatrix} \left[\begin{matrix}2\\1\end{matrix}\right] & \left[\begin{matrix}3\end{matrix}\right] & \left[\begin{matrix}1\\ 1\\1\end{matrix}\right] & 1\end{pmatrix}$ \\
\hline
$( \frac{1}{2}, \frac{1}{4}, \theta)$ & 4 &
$\begin{pmatrix} \left[\begin{matrix}2\\2\end{matrix}\right] & \left[\begin{matrix}4\end{matrix}\right] & \left[\begin{matrix}1\\ \vdots\\1\end{matrix}\right] & 1\end{pmatrix}$ \\
\hline
$( \frac{1}{2}, \theta_1, \theta_\infty)$ & 2 &
$\begin{pmatrix} \left[\begin{matrix}2\end{matrix}\right] & \left[\begin{matrix}1\\1\end{matrix}\right] & \left[\begin{matrix}1\\ 1\end{matrix}\right] & 1\end{pmatrix}$ \\
\hline\end{tabular}
\end{center}
\label{table:logarithmic}
\end{table}
By confluence of the poles of the differential equation, we deduce the following:
\begin{prop}\label{prop:classificationIrreg}
Assume $(C_0,E_0,\nabla_0)$ is a (normalized) differential equation
with at least one irregular pole $p_k$ having non trivial Stokes matrices, and assume
$\phi:C\to C_0$ is a scattered ramified cover of degree $d\ge 2$. If $T\le B$,
then $(C_0,E_0,\nabla_0)$ is a degenerate hypergeometric equation,
and up to bundle transformation, we are
in the list of table \ref{table:irregular}.
\end{prop}
\begin{table}[h]
\caption{Irregular classification with scattered cover}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
Local formal data & Degree & Covering passport & Local formal data & Isomonodromy \\
for $(E_0,\nabla_0)$ &&& for $(E,\nabla)$& equation\\
\hline
$\begin{pmatrix}0& \frac{1}{2}\\ \frac{1}{3} & 0\end{pmatrix}$ & 6 &
$\begin{pmatrix} \left[\begin{matrix}3\\3\end{matrix}\right] & \left[\begin{matrix}2\\2\\2\end{matrix}\right] & 3\end{pmatrix}$ &
$\begin{pmatrix}1& 1&1\\ 0&0 & 1\end{pmatrix}$& $\mathrm{Gar}^3(1,1,1)$ \\
\hline
$\begin{pmatrix}0& \frac{1}{2}\\ \frac{1}{3} & 0\end{pmatrix}$ & 4 &
$\begin{pmatrix} \left[\begin{matrix}3\\1\end{matrix}\right] & \left[\begin{matrix}2\\2\end{matrix}\right] & 2\end{pmatrix}$ &
$\begin{pmatrix}0& 1&1\\ \frac{1}{3} & 0&1\end{pmatrix}$ & $\mathrm{Gar}^2(0,1,1)$\\
\hline
$\begin{pmatrix}0& \frac{1}{2}\\ \frac{1}{3} & 0\end{pmatrix}$ & 3 &
$\begin{pmatrix} \left[\begin{matrix}3\end{matrix}\right] & \left[\begin{matrix}2\\1\end{matrix}\right] & 1\end{pmatrix}$ &
$\begin{pmatrix} \frac{1}{2}&1\\ 0&0\end{pmatrix}$ & $P_{III}^{D_7}$\\
\hline
$\begin{pmatrix}0& \frac{1}{2}\\ \frac{1}{4} & 0\end{pmatrix}$ & 4 &
$\begin{pmatrix} \left[\begin{matrix}4\end{matrix}\right] & \left[\begin{matrix}2\\2\end{matrix}\right] & 1\end{pmatrix}$ &
$\begin{pmatrix}1&1\\ 0&1\end{pmatrix}$&$P_{III}^{D_6}$\\
\hline
$\begin{pmatrix}0& \frac{1}{2}\\ \theta & 0\end{pmatrix}$ & 2 &
$\begin{pmatrix} \left[\begin{matrix}1\\1\end{matrix}\right] & \left[\begin{matrix}2\end{matrix}\right] & 1\end{pmatrix}$ &
$\begin{pmatrix}0&1&0 \\ \theta-1&0 & \theta \end{pmatrix}$& $P_{V}$\\
\hline
$\begin{pmatrix}0& 1\\ \frac{1}{2} & \theta\end{pmatrix}$ & 2 &
$\begin{pmatrix} \left[\begin{matrix}2\end{matrix}\right] & \left[\begin{matrix}1\\1\end{matrix}\right] & 1\end{pmatrix}$ &
$\begin{pmatrix}1& 1\\ \theta-1 & \theta\end{pmatrix}$& $P_{III}^{D_6}$\\
\hline
$\begin{pmatrix}\frac{3}{2}\\ 0\end{pmatrix}$ & 2 &
$\begin{pmatrix} \left[\begin{matrix}2\end{matrix}\right] & 1\end{pmatrix}$ &
$\begin{pmatrix}3\\ 0\end{pmatrix}$& $P_{II}$\\
\hline
\end{tabular}
\end{center}
\label{table:irregular}
\end{table}
\begin{proof}From Lemmae \ref{lem:IrregPoleLogPole} and \ref{lem:IrregRamPoleLogPole} the irregular
poles can be scattered as several logarithmic poles with all of them having exponent $\theta$ with infinite
orbifold order, except one in the ramified case, having exponent $\frac{1}{2}$. We are led to the list of table
\ref{table:logarithmic}. We then deduce the list of table \ref{table:irregular} by confluence of poles with
infinite order $\theta$, and possibly one of them $\frac{1}{2}$. In the first four entries of table \ref{table:logarithmic},
we have no other choice than make the two poles with exponent $\frac{1}{2}$ and $\theta$ confluing into
a ramified irregular pole with $\kappa=\frac{1}{2}$. In the last entry however, we have several possible
confluences, namely $\{\frac{1}{2},\theta_1\}\rightsquigarrow \kappa=\frac{1}{2}$, $\{\theta_1,\theta_2\}\rightsquigarrow \kappa=1$ and $\{\frac{1}{2},\theta_1,\theta_2\}\rightsquigarrow \kappa=\frac{3}{2}$.
\end{proof}
Finally, by confluence of ramification fibers of $\phi$, we complete the list for pull-back solutions:
\begin{prop}\label{prop:classificationIrregConfl}
Assume $(C_0,E_0,\nabla_0)$ is a (normalized) differential equation
with at least one irregular pole $p_k$ having non trivial Stokes matrices, and assume
$\phi:C\to C_0$ is a non scattered ramified cover of degree $d\ge 2$. If $T\le B$,
then $(C_0,E_0,\nabla_0)$ is a degenerate hypergeometric equation,
and up to bundle transformation, we are in the list of table \ref{table:irregularConfl}.
\end{prop}
\begin{table}[h]
\caption{Irregular classification with confluent cover}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
Local formal data & Degree & Covering passport & Local formal data & Isomonodromy \\
for $(E_0,\nabla_0)$ &&& for $(E,\nabla)$& equation\\
\hline
$\begin{pmatrix}0& \frac{1}{2}\\ \frac{1}{3} & 0\end{pmatrix}$ & 6 &
$\begin{pmatrix} \left[\begin{matrix}3\\3\end{matrix}\right] & \left[\begin{matrix}4\\2\end{matrix}\right] & 2\end{pmatrix}$ &
$\begin{pmatrix}1&2\\ 0 & 1\end{pmatrix}$& $\mathrm{Gar}^2(1,2)$ \\
\hline
$\begin{pmatrix}0& \frac{1}{2}\\ \frac{1}{3} & 0\end{pmatrix}$ & 6 &
$\begin{pmatrix} \left[\begin{matrix}3\\3\end{matrix}\right] & \left[\begin{matrix}6\end{matrix}\right] & 1\end{pmatrix}$ &
$\begin{pmatrix}3\\ 1\end{pmatrix}$& $P_{II}$ \\
\hline
$\begin{pmatrix}0& \frac{1}{2}\\ \frac{1}{3} & 0\end{pmatrix}$ & 4 &
$\begin{pmatrix} \left[\begin{matrix}3\\1\end{matrix}\right] & \left[\begin{matrix}4\end{matrix}\right] & 1\end{pmatrix}$ &
$\begin{pmatrix}0& 2\\ \frac{1}{3} & 1\end{pmatrix}$ & $P_{IV}$\\
\hline
\end{tabular}
\end{center}
\label{table:irregularConfl}
\end{table}
\begin{proof}We now inverse the scattering process of Lemmae \ref{lem:FibreCritique} - \ref{lem:FibreIrregRam}.
To do this, we replace simple branching fibers outside the poles of $\nabla_0$ by additional ramifications
over poles. Note that in table \ref{table:irregular}, each entry satisfies $T=B$ so that we cannot add ramifications
over logarithmic poles with finite orbifold order, otherwise $T-B$ becomes $>0$ (see Lemma \ref{lem:FibrePeriodic}).
The only possibility is therefore to add ramifications over irregular poles of $\nabla_0$, or logarithmic poles
with exponent $\theta$ having infinite orbifold order. Only the first two lines give examples with $T>0$.
\end{proof}
\begin{remark}We observe that the algebraic solution of $P_{II}(0)$ can be constructed from
two pull-back constructions (see table \ref{table:irregular} last line and table \ref{table:irregularConfl} line $2$).
This comes from the fact that Airy equation $\begin{pmatrix}\frac{3}{2}\\ 0\end{pmatrix}$ is itself pull-back
from Kummer equation $\begin{pmatrix}0& \frac{1}{2}\\ \frac{1}{3} & 0\end{pmatrix}$ by a $3$-fold ramified cover.
Similarly, the algebraic solution of $P_{III}^{D_6}$ appears twice in table \ref{table:irregular} for $\theta=0$
due to the fact
that Kummer equation $\begin{pmatrix}0& 1\\ \frac{1}{2} & \theta\end{pmatrix}$ is the double cover
of $\begin{pmatrix}0& \frac{1}{2}\\ \frac{1}{4} & 0\end{pmatrix}$ in that case.
\end{remark}
\section{Classification of classical solutions in the case $N=2$.}\label{sec:ClassicalN=2}
In section \ref{sec:Classifcover}, we have given a complete classification of algebraic
solutions of irregular Garnier systems whose linear Galois group is $\SL$. Indeed,
we have classified those solutions of type (1) in Corollary \ref{cor:structure}.
It does not make sense to do the same for solutions of type (2) or (3) in Corollary \ref{cor:structure},
since there are infinitely many, for arbitrary large rank $N$. However, for a given rank,
it makes sense to classify, and we do this in this section for the case $N=2$
which is the first open case after Painlev\'e equations. Recall that we have two solutions
of type (1) in the case $N=2$, one for each formal data:
$$\begin{pmatrix}0&1&1\\ \frac{1}{3}&0&1\end{pmatrix}\ \ \
\text{and}\ \ \ \begin{pmatrix}1&2\\ 0&1\end{pmatrix}$$
\subsection{Without apparent singular point}\label{sec:ClassifDihedral5}
They correspond to the type (2) of Corollary \ref{cor:structure} and have Galois group $D_\infty$.
\begin{equation}
\left\{\begin{matrix}
\begin{pmatrix}0&0&0&\frac{1}{2}\\ \frac{1}{2}&\frac{1}{2}&\frac{1}{2}&0\end{pmatrix}\hfill\hfill
\text{infinite discrete family}\\
\begin{pmatrix}0&0&0&1\\ \frac{1}{2}&\frac{1}{2}&\theta_1&\theta_2\end{pmatrix},\ \ \
\begin{pmatrix}0&\frac{1}{2}&0&0\\ \frac{1}{2}&0&\theta_1&\theta_2\end{pmatrix}\hfill\hfill
\text{two-parameter families}\\
\begin{pmatrix}0&0&2\\ \frac{1}{2}&\frac{1}{2}&\theta\end{pmatrix},\ \
\begin{pmatrix}0&\frac{1}{2}&1\\ \frac{1}{2}&0&\theta\end{pmatrix},\ \
\begin{pmatrix}\frac{1}{2}&\frac{1}{2}&0\\ 0&0&\theta\end{pmatrix},\ \
\begin{pmatrix}0&\frac{3}{2}&0\\ \frac{1}{2}&0&\theta\end{pmatrix}\ \ \
\text{one-parameter families}\\
\begin{pmatrix}\frac{1}{2}&\frac{3}{2}\\ 0&0\end{pmatrix}\ \ \ \text{and}\ \ \
\begin{pmatrix}0&\frac{5}{2}\\ \frac{1}{2}&0\end{pmatrix}\hfill\hfill
\text{sporadic solutions}
\end{matrix}\right.
\end{equation}
These formal data correspond to those for which the dihedral group $D_\infty$ occur
as a Galois group of the linear equation, up to bundle transformation. In order to find this list,
we have to take into account the following constraints:
\begin{itemize}
\item there are $2$ or $4$ poles where the local monodromy (or Galois group) is anti-diagonal,
and the local formal type must be
$$\begin{pmatrix}0\\ \frac{1}{2}\end{pmatrix}
\ \ \ \text{or}\ \ \ \begin{pmatrix}\frac{k}{2}\\ 0\end{pmatrix},\ \ \ \text{with }k\in\mathbb Z_{>0}\text{ odd.}$$
\item other poles are of local formal type
$$\begin{pmatrix}k\\ \theta\end{pmatrix}\ \ \ \text{with }k\in\mathbb Z_{\ge0},\ \theta\in\mathbb C.$$
\end{itemize}
We would like to insist that it is not necessary to consider particular values for $\theta$ as normalized equations
with differential Galois group $D_\infty$ having poles with diagonal local monodromy occur in family where
each exponant $\theta$ can be deformed arbitrarily. This comes from the fact that the monodromy representation
itself can be deformed as well.
The first entry corresponds to the unique case with $4$ poles having local anti-diagonal monodromy.
It is an irregular version of Picard-Painlev\'e equation (see \cite{MazzoccoPicard,LPU}):
there are infinitely many algebraic solutions, in bijection with the orbits of $\mathbb Q\times\mathbb Q$
under the standard action of $\SL(\mathbb Z)$. In fact, if $\tilde C\to C\simeq\mathbb P^1$ denotes the elliptic curve
given by the $2$-fold cover ramifying over the $4$ poles of $(E,\nabla)$, then Picard solutions are related with
torsion points on $\tilde C$ and how they varry when deforming the poles, and the curve $\tilde C$.
Here, the story is the same. Indeed, the locus of $D_\infty$ Galois group in the moduli space is closed algebraic,
and a differential equation in this closed set has trivial Stokes matrices and is characterized by its monodromy
representation.
For all other cases, recall that the differential equation with dihedral Galois group can be determined
by means of exponents $\theta_i$'s, as for its monodromy, once we know the irregular curve. Therefore,
for each $\theta_i$'s, we get exactly one algebraic solution.
\subsection{With apparent singular point}
They correspond to the type (3) of Corollary \ref{cor:structure} and have Galois group $C_\infty$ or $D_\infty$.
However, as noticed in Remark \ref{rem:ApparentDihedral}, algebraic solutions with Galois group $D_\infty$
and apparent singular points always arise as particular cases of more general algebraic solutions with Galois group $D_\infty$
and arbitrary singular points as listed in section \ref{sec:ClassifDihedral5}. It just remains to complete the list
with those algebraic solutions with Galois group $C_\infty$ and (at least one) apparent singular point.
\begin{equation}
\left\{\begin{matrix}
\begin{pmatrix}0&0&0&1\\ 0&\theta_1&\theta_2&\theta_3\end{pmatrix},\ \ \theta_1+\theta_2+\theta_3=0\ \ \
\text{two-parameter family}\\
\begin{pmatrix}0&0&2\\ 0&\theta&-\theta\end{pmatrix},\ \
\begin{pmatrix}0&1&1\\ 0&\theta&-\theta\end{pmatrix}\hfill\hfill
\text{one-parameter families}\\
\begin{pmatrix}0&3\\ 0&0\end{pmatrix}\hfill\hfill
\text{sporadic solution}
\end{matrix}\right.
\end{equation}
For each value of $\theta_i$'s, there is exactly one normalized equation once the irregular curve is fixed,
and therefore exactly one algebraic solution.
Solutions with $2$ or more apparent singular points arise as particular cases of these ones by specifying $\theta_i$'s.
\section{Explicit Hamiltonians for some irregular Garnier systems}\label{sec:GarnierHamiltonians}
Here we provide the linear differential equation and the Hamiltonians for some particular formal types
(the complete list comes from \cite{Kimura,Kawamuko}). We translate our notations with Kimura and Kawamuko's.
\subsection{$\mathrm{Kim}_4(1,2,2)$}
The general linear differential equation with (non apparent) poles $x=0,1,\infty$
and corresponding formal type $\begin{pmatrix}2&2&1\\ \theta_0&\theta_1&\theta_\infty\end{pmatrix}$ can be normalized into the form\footnote{From Kimura's formulae \cite{Kimura}, set $\eta_i=1$,
$\varkappa_i=\theta_i$, $\varkappa=\frac{1}{4}\left[ (\theta_0+\theta_1-1)^2-\theta_\infty^2\right]$
and $K_i=H_i$.}
\begin{equation}\label{eq:Lin(1,2,2)}
L(1,2,2):\ \left\{\begin{matrix}u''+f(x)u'+g(x)u=0\ \ \ \hfill\text{with}\\
f(x)=\frac{t_2}{x^2}+\frac{2-\theta_0}{x}+\frac{t_1}{(x-1)^2}+\frac{2-\theta_1}{x-1}-\sum_{k=1,2}\frac{1}{x-q_k}\hfill\hfill\\
g(x)=\frac{(\theta_0+\theta_1-1)^2-\theta_\infty^2}{4x(x-1)}-\frac{t_1H_1}{x(x-1)^2}+\frac{t_2H_2}{x^2(x-1)}+\sum_{k=1,2}\frac{q_k(q_k-1)p_k}{x(x-1)(x-q_k)}
\end{matrix}\right.
\end{equation}
Singular points $x=q_1,q_2$ are apparent if and only if coefficients $H_1,H_2$ are given by
\begin{equation}\label{eq:Ham(1,2,2)}
H(1,2,2):\ \left\{\begin{matrix}
H_1=\hfill-\frac{q_1^2(q_1-1)^2(q_2-1)}{t_1(q_1-q_2)}
\left(p_1^2-\left(\frac{\theta_0}{q_1}-\frac{t_2}{q_1^2}+\frac{\theta_1-1}{q_1-1}-\frac{t_1}{(q_1-1)^2}\right)p_1+\frac{(\theta_0+\theta_1-1)^2-\theta_\infty^2}{4q_1(q_1-1)}\right)\\
\hfill+\frac{(q_1-1)q_2^2(q_2-1)^2}{t_1(q_1-q_2)}
\left(p_2^2-\left(\frac{\theta_0}{q_2}-\frac{t_2}{q_2^2}+\frac{\theta_1-1}{q_2-1}-\frac{t_1}{(q_2-1)^2}\right)p_1+\frac{(\theta_0+\theta_1-1)^2-\theta_\infty^2}{4q_2(q_2-1)}\right)\\
H_2=\hfill-\frac{q_1^2(q_1-1)^2q_2}{t_2(q_1-q_2)}
\left(p_1^2-\left(\frac{\theta_0-1}{q_1}-\frac{t_2}{q_1^2}+\frac{\theta_1}{q_1-1}-\frac{t_1}{(q_1-1)^2}\right)p_1+\frac{(\theta_0+\theta_1-1)^2-\theta_\infty^2}{4q_1(q_1-1)}\right)\\
\hfill+\frac{q_1q_2^2(q_2-1)^2}{t_2(q_1-q_2)}
\left(p_2^2-\left(\frac{\theta_0-1}{q_2}-\frac{t_2}{q_2^2}+\frac{\theta_1}{q_2-1}-\frac{t_1}{(q_2-1)^2}\right)p_1+\frac{(\theta_0+\theta_1-1)^2-\theta_\infty^2}{4q_2(q_2-1)}\right)
\end{matrix}\right.
\end{equation}
A deformation of (\ref{eq:Lin(1,2,2)}) is isomonodromic if, and only if, parameters satisfies Hamiltonian system
(\ref{eq:HamiltonianGarnierSystem}).
\begin{equation}\label{eq:HamiltonianGarnierSystemK}
\frac{dq_j}{dt_i}=\frac{\partial H_i}{\partial p_j}\ \ \ \text{and}\ \ \ \frac{dp_j}{dt_i}=-\frac{\partial H_i}{\partial q_j}\ \ \
\forall i,j=1,2.
\end{equation}
To construct the pull-back solution (second line of Table \ref{table:irregular}),
we start with the differential equation $\frac{d^2u}{d z^2}+\frac{2}{3z}\frac{du}{d z}-\frac{1}{z}u=0$
and consider its pull-back
by the branch cover
$$z=\phi(x)=\frac{t_2^2(2x-4q_1x+q_1^2+q_1)^3}{16q_1^3(q_1+1)^3x^2(x-1)^2}.$$
Comparing with (\ref{eq:Lin(1,2,2)}), we get the first solution of Theorem \ref{thm:3nonclassical}.
\subsection{$\mathrm{Kim}_6(2,3)$}
The general linear differential equation with (non apparent) poles $x=0,\infty$
and corresponding formal type $\begin{pmatrix}2&3\\ \theta_0&\theta_\infty\end{pmatrix}$ can be normalized into the form\footnote{From Kimura's formulae \cite{Kimura}, set $\eta_0=1$,
$\varkappa_0=\theta_0$, $\varkappa_\infty=\frac{\theta_0+\theta_\infty-1}{4}$
and $K_i=H_i$.}
\begin{equation}\label{eq:Lin(2,3)}
L(2,3):\ \left\{\begin{matrix}u''+f(x)u'+g(x)u=0\ \ \ \hfill\text{with}\\
f(x)=\frac{t_2}{x^2}+\frac{2-\theta_0}{x}-t_1-\frac{x}{2}-\sum_{k=1,2}\frac{1}{x-q_k}\hfill\hfill\\
g(x)=\frac{\theta_0+\theta_\infty-1}{8}-\frac{H_1}{2x}+\frac{t_2H_2}{x^2}+\sum_{k=1,2}\frac{q_k(q_k-1)p_k}{x(x-1)(x-q_k)}
\end{matrix}\right.
\end{equation}
Singular points $x=q_1,q_2$ are apparent if and only if coefficients $H_1,H_2$ are given by
\begin{equation}\label{eq:Ham(2,3)}
H(2,3):\ \left\{\begin{matrix}
H_1=&\hfill\frac{2q_1^2}{q_1-q_2}\left(p_1^2-\left(\frac{\theta_0}{q_1}-\frac{t_2}{q_1^2}+\frac{q_1}{2}+t_1\right)p_1+\frac{\theta_0+\theta_\infty-1}{8}\right)\\
&\hfill-\frac{2q_2^2}{q_1-q_2}\left(p_2^2-\left(\frac{\theta_0}{q_2}-\frac{t_2}{q_2^2}+\frac{q_2}{2}+t_1\right)p_2+\frac{\theta_0+\theta_\infty-1}{8}\right)\\
H_2=&-\frac{q_1^2q_2}{t_2(q_1-q_2)}\left(p_1^2-\left(\frac{\theta_0-1}{q_1}-\frac{t_2}{q_1^2}+\frac{q_1}{2}+t_1\right)p_1+\frac{\theta_0+\theta_\infty-1}{8}\right)\\
&\hfill+\frac{q_1q_2^2}{t_2(q_1-q_2)}\left(p_2^2-\left(\frac{\theta_0-1}{q_2}-\frac{t_2}{q_2^2}+\frac{q_2}{2}+t_1\right)p_2+\frac{\theta_0+\theta_\infty-1}{8}\right)
\end{matrix}\right.
\end{equation}
and isomonodromic deformations are defined by (\ref{eq:HamiltonianGarnierSystemK}).
To construct the pull-back solution (first line of Table \ref{table:irregularConfl}),
we start with the differential equation $\frac{d^2u}{d z^2}+\frac{2}{3z}\frac{du}{d z}-\frac{1}{z}u=0$
and consider its pull-back
by the branch cover
$$z=\phi(x)=\frac{(3x^2+8t_1x+4q_1t_1+6q_1^2)^3}{6912x^2}.$$
Comparing with (\ref{eq:Lin(2,3)}), we get the second solution of Theorem \ref{thm:3nonclassical}.
\subsection{$\mathrm{Kaw}_4(5/2,3/2)$}
The general linear differential equation with (non apparent) poles $x=0,\infty$
and corresponding formal type $\begin{pmatrix}\frac{3}{2}&\frac{1}{2}\\ 0&0\end{pmatrix}$ can be normalized into the form\footnote{From Kawamuko's formulae \cite{Kawamuko}, set $\lambda_i=q_i$, $\mu_i=p_i$
and $h_i^{04}=H_i$.} $u''=g(x)u$ with
\begin{equation}\label{eq:Lin(5/2,3/2)}
g(x)=\frac{t_2^2}{4x^3}+\frac{H_1}{x^2}+\frac{H_2}{x}+\frac{t_1}{2}+\frac{x}{4}+\sum_{k=1,2}\left(\frac{3}{4(x-q_k)^2}-\frac{p_k}{x-q_k}\right)
\end{equation}
Then set
$$u_1=q_1+q_2,\ \ \ u_2=q_1q_2,\ \ \ v_1=\frac{q_1+q_2}{2(q_1-q_2)^2}+\frac{p_1q_1-p_2q_2}{q_1-q_2}$$
$$\text{and}\ \ \ v_2=-\frac{1}{(q_1-q_2)^2}-\frac{p_1-p_2}{q_1-q_2}$$
and
$$\left\{\begin{matrix}
K_1=2u_1v_1^2+4u_2v_1v_2-4v_1-\frac{u_1^2}{2}-t_1u_1+\frac{u_2}{2}+\frac{t_2^2}{2u_2}\\
t_2K_2=-2u_2v_1^2+2u_2^2v_2^2-2u_2v_2+\frac{u_1u_2}{2}+t_1u_2-\frac{t_2^2u_1}{2u_2}
\end{matrix}\right.$$
Deformation of (\ref{eq:Lin(5/2,3/2)}) is isomonodromic if, and only if
$$\frac{du_j}{dt_i}=\frac{\partial K_i}{\partial v_j}\ \ \ \text{and}\ \ \ \frac{dv_j}{dt_i}=-\frac{\partial K_i}{\partial u_j}\ \ \
\forall i,j=1,2.$$
The first classical sporadic solution of Theorem \ref{thm:rank2classical},
with dihedral linear Galois group, can be constructed by
pulling back the differential equation $\frac{d^2u}{d z^2}=\left(\frac{1}{z}-\frac{3}{16x^2}\right)u$ by the ramified covering
$$z=\phi(x)=\frac{(x^2+3t_1x-3t_2)^2}{36 x};$$
after normalizing, and comparing with equation (\ref{eq:Lin(5/2,3/2)}), we get the rational solution
$$(t_1,t_2)\mapsto(u_1,u_2,v_1,v_2):=\left(-t_1,t_2,0,\frac{3}{4t_2}\right).$$
\bibliographystyle{amsplain}
|
train/arxiv
|
BkiUfGk5qsFCiE6Ue_hi
| 5 | 1 |
\section*{Data Analysis Recipes:\\
Products of multivariate Gaussians\\
in Bayesian inferences}
\noindent\textbf{David W. Hogg}\footnote{%
The authors would like to thank
Will Farr (Stony Brook),
Dan Foreman-Mackey (Flatiron),
Rodrigo Luger (Flatiron),
Hans-Walter Rix (MPIA),
and
Sam Roweis (deceased),
for help with all these concepts.
This project was developed
in part at \textsl{AstroHackWeek 2016}, which was
hosted by the Moore--Sloan Data Science Environment.
This research was supported by the National Science Foundation and National Aeronautics and Space Administration.
The source text, example IPython notebooks, and data files used below are
available via \href{https://doi.org/10.5281/zenodo.3855689}{Zenodo archive}
\citep{zenodo}.
}\\
{\footnotesize%
\textsl{Center for Cosmology and Particle Physics, Dept.\ Physics, New York University}\\
\textsl{Max-Planck-Institut f\"ur Astronomie, Heidelberg}\\
\textsl{Flatiron Institute, a division of the Simons Foundation}%
}
\medskip\noindent\textbf{Adrian~M.~Price-Whelan}\\
{\footnotesize%
\textsl{Flatiron Institute, a division of the Simons Foundation}%
}
\medskip\noindent\textbf{Boris Leistedt}\\
{\footnotesize%
\textsl{Department of Physics, Imperial College, London}\\
\textsl{Center for Cosmology and Particle Physics, Dept.\ Physics, New York University}%
}
\paragraph{Abstract:}
A product of two Gaussians---or normal distributions---is another Gaussian.
That's a valuable and useful fact!
Here we use it to derive a refactoring of a common product of
multivariate Gaussians:
The product of a Gaussian likelihood times a Gaussian prior, where some or all
of those parameters enter the likelihood only in the mean and only linearly.
That is, a linear, Gaussian, Bayesian model.
This product of a likelihood times a prior pdf can be refactored into a product of a
marginalized likelihood (or a Bayesian evidence) times a posterior pdf, where
(in this case) both of these are also Gaussian.
The means and variance tensors of the refactored Gaussians are straightforward
to obtain as closed-form expressions;
here we deliver these expressions, with discussion.
The closed-form expressions can be used to speed up and improve the precision
of inferences that contain linear parameters with Gaussian priors.
We connect these methods to inferences that arise frequently in physics
and astronomy.
If all you want is the answer, the question is posed and answered at the
beginning of Section~\ref{sec:problemsolution}.
We show two toy examples, in the form of worked exercises, in
Section~\ref{sec:examples}.
The solutions, discussion, and exercises in this \textsl{Note}\ are aimed at
someone who is already familiar with the basic ideas of
Bayesian inference and probability.
\section{Inferences with linear parameters}
It is common in physics, astronomy, engineering, machine learning, and many other fields that likelihood functions
(probabilities of data given parameters) are chosen to be Gaussian
(or normal\footnote{In this \textsl{Note}, we obey physics and astronomy
conventions and refer to the normal pdf as the Gaussian pdf. We realize that it
is irresponsible to name things after people when those same things also have generic,
descriptive names. On the other hand, ``normal'' isn't the finest name either.
Perhaps Gaussian pdfs should be called ``central'' since they are produced by
the central limit theorem. Anyway, we will continue with the name ``Gaussian''
despite our own reservations. We apologize to our reader.}):
One reason is that a likelihood function is basically a noise model,
and it is often case that the noise is treated as Gaussian.
This assumption for the likelihood function is \emph{accurate} when the noise
model has benefitted from the central limit theorem.
This is true, for example, when the noise is thermal, or when the
noise is shot noise and the numbers (numbers of detected photons or other
particles) are large.
Another reason that the likelihood function is often treated as
Gaussian is that Gaussians are generally \emph{tractable}:
Many computations we like to perform on Gaussians, like integrals and
derivatives and optimizations, have closed-form solutions.
Even when we don't use the closed-form solutions, there are many
contexts in which Gaussians lead to convex optimizations,
providing guarantees to resulting inferences.
It is also common in physics and astronomy that models for data
include parameters such that the expectation value for the data (in,
say, a set of repeated experiments) is linearly proportional to some
subset of the parameters.
This is true, for example, when we fit a histogram of \textsl{Large Hadron
Collider} events affected by the Higgs boson,\footnote{See, for example,
\cite{atlas}, and \cite{cms}, and references therein.}
where the expected number of counts in each
energy bin is proportional to a linear combination of the amplitudes
of various backgrounds and some coupling to the Higgs.
Another linear-parameter context, for example, arises when we fit for the radial-velocity
variation of a star in response to a faint, orbiting companion.\footnote{See,
for an example in our own work, \cite{Price-Whelan:2017, Price-Whelan:2020}.
That project and those papers would have been
impossible without the speed-ups provided by the expressions derived in this
\textsl{Note}. Indeed, the writing of this \textsl{Note}\ was motivated by
the work presented in those papers. (Okay full disclosure: It was motivated in
part by a \emph{mistake} made by DWH in one of those papers!)}
In this problem, the expectation of the
radial-velocity measurements depends linearly on the binary system
velocity and some combination of masses and system inclination (with
respect to the line of sight).
In both of these cases, there are both linear parameters (like the
amplitudes) and nonlinear parameters (like the mass of the Higgs, or
the orbital period of the binary-star system).
In what follows, we will spend our energies on the linear parameters,
though our work on them is in service of learning the nonlinear
parameters too, of course.
In Bayesian inference contexts, the ``models'' to which we are referring are
expressions for likelihood functions and prior pdfs; these are the things
that will be Gaussians here.
Bayes' theorem is often written as a ratio of probability density functions
(pdfs in what follows), but it can also be written as a pdf factorization:\footnote{For
a tutorial on probability factorizations, see \cite{probcalc}.}
\begin{equation}\label{eq:bayes}
p(\vector{y},\vector{\theta}\,|\,\bH) = p(\vector{y}\,|\,\vector{\theta},\bH)\,p(\vector{\theta}\,|\,\bH) = p(\vector{\theta}\,|\,\vector{y},\bH)\,p(\vector{y}\,|\,\bH)
\end{equation}
where
$p(\vector{y},\vector{\theta}\,|\,\bH)$ is the joint probability of data $\vector{y}$ and
parameters $\vector{\theta}$ given your model assumptions and hyper parameters
(symbolized jointly as $\bH$),\footnote{%
In this \textsl{Note}, we typeset different mathematical objects according to their mathematical or transformation properties.
We typeset vectors (which are column vectors) as $\vector{a}, \vector{b}, \vector{\theta}$,
we typeset variance tensors (which in this case are square, non-negative semi-definite matrices) as $\tensor{C}, \tensor{\Lambda}$,
we typeset other matrices (which will in general be non-square) as $\matrix{M}, \matrix{U}$,
and we typeset blobs or unstructured
collections of information as $\bH, \bP$.
Related to this typography is an implicit terminology:
We distinguish variance tensors from matrices.
This distinction is somewhat arbitrary, but the strong constraints on the
variance tensors (non-negative, real eigenvalues) make them special beasts,
with special geometric properties, like that they can be used as metrics
in their respective vector spaces.}
$p(\vector{y}\,|\,\vector{\theta},\bH)$ is the likelihood, or probability of data $\vector{y}$
given parameters (and assumptions),
$p(\vector{\theta}\,|\,\bH)$ is the prior pdf for the parameters $\vector{\theta}$,
$p(\vector{\theta}\,|\,\vector{y},\bH)$ is the posterior pdf for the parameters $\vector{\theta}$
given the data,
and
$p(\vector{y}\,|\,\bH)$ is the pdf for the data, marginalizing out all of the linear
parameters (hereafter, we refer to this as the \textsl{marginalized
likelihood}\footnote{In the case that the problem has no parameters other than
the linear parameters $\vector{\theta}$, this term, $p(\vector{y} \,|\, \bH)$, is sometimes
called the \textsl{Bayesian evidence} or the \textsl{fully marginalized
likelihood}.}).
If the likelihood is Gaussian, and the expectation of the data depends linearly
on the parameters, and if we choose the prior pdf to also be Gaussian, then
all the other pdfs (the joint, the posterior, and the marginalized likelihood)
all become Gaussian too.
The main point of this \textsl{Note}\ is that the means and variances of these
five Gaussians are all related by simple, closed-form expressions, given below.
One consequence of this math is that \emph{if} you have a Gaussian
likelihood function, and \emph{if} you have a subset of parameters that are
linearly related to the expectation of the data, \emph{then} you can obtain both
the posterior pdf $p(\vector{\theta}\,|\,\vector{y},\bH)$ and the marginalized likelihood
$p(\vector{y}\,|\,\bH)$ with closed-form transformations of the means and variances of
the likelihood and prior pdf.
A currently popular data-analysis context in which Gaussian likelihoods are
multiplied by Gaussian priors is \textsl{Gaussian processes} (\acronym{GP}s),
which is a kind of
non-parametric fitting in which a kernel function sets the flexibility of a
data-driven model. A full discussion of \acronym{GP}s is beyond the scope of this
\textsl{Note}, but excellent discussions abound.\footnote{We like the free
book by \cite{Rasmussen:2005}.}
The math below can be applied in many \acronym{GP} contexts.
Indeed, most linear model fits of the kind we describe below can be translated
into the language of \acronym{GP}s, because any noise process that delivers
both a prior pdf and a likelihood with Gaussian form is technically identical
to a (probably non-stationary) \acronym{GP}.
We leave that translation as an exercise to the ambitious
reader.\footnote{If you want a cheat sheet,
we come close to performing this translation in \cite{luger}.}
\section{Marginalization by refactorization}
Imagine that we are doing an inference using data $\vector{y}$ (which is a
$N$-dimensional vector, say).
We are trying to learn linear parameters $\vector{\theta}$ (a $K$-dimensional vector)
and also nonlinear parameters $\bP$ (an arbitrary vector, list, or
blob).\footnote{Here, $\bP$ represents the nonlinear parameters \emph{and}
assumptions or hyper parameters. That is, it contains everything on which
the linear model is conditioned, including not just nonlinear parameters
but also investigator choices. Note our subjectivism here!}
Whether we are Bayesian or frequentist, the inference is based on
a likelihood function, or probability for the data given parameters
\begin{equation}
\mbox{\small likelihood:} ~~ p(\vector{y}\,|\,\vector{\theta},\bP) ~~.
\end{equation}
Now let's imagine that the parameters $\vector{\theta}$ are either nuisance
parameters, or else easily marginalized, so we want to marginalize
them out.
This will leave us with a lower-dimensional marginalized likelihood
function
\begin{equation}
\mbox{\small marginalized likelihood:} ~~ p(\vector{y}\,|\,\bP) ~~.
\end{equation}
That's good, but the marginalization comes at a cost:
We have to become Bayesian, and we have to choose a prior
\begin{equation}
\mbox{\small prior on nuisance parameters:} ~~ p(\vector{\theta}\,|\,\bP) ~~.
\end{equation}
This is the basis for the claim\footnote{A claim that perhaps hasn't been made clearly
yet, but will eventually be by at least one of these authors.} that Bayesian
inference requires a likelihood function, and priors on the nuisance parameters.
It does not require a prior on everything, contrary to some statements
in the literature.\footnote{It is very common for papers or projects with
Bayesian approaches to claim that the
goal of Bayesian inference is to create posterior pdfs. That isn't correct.
Different Bayesian inferences have different objectives. The fundamental
point
of Bayesian inference is that consistently held beliefs obey the rules of
probability. That, in turn, says that if you want
to communicate to \emph{others} things useful to the
updating of \emph{their} beliefs, you want to communicate about your likelihood.
Your posterior pdf isn't all that useful to them!\label{note:lf}}
We have said ``$p(\vector{\theta}\,|\,\bP)$'' because this prior pdf may depend on
the nonlinear parameters $\bP$, but it certainly doesn't have to.
Armed with the likelihood and prior---if you want it---you can construct
the posterior pdf for the linear parameters
\begin{equation}
\mbox{\small posterior for nuisance parameters:} ~~ p(\vector{\theta}\,|\,\vector{y},\bP) ~~.
\end{equation}
To perform a marginalization of the likelihood, we have two choices.
We can either do an integral:
\begin{equation}\label{eq:integral}
p(\vector{y}\,|\,\bP) = \int p(\vector{y}\,|\,\vector{\theta},\bP)\,p(\vector{\theta}\,|\,\bP)\,\mathrm{d}\vector{\theta}
~~,
\end{equation}
where the integral is implicitly over the entire domain of the
linear parameters $\vector{\theta}$ (or the entire support of the prior).
Or we can re-factorize the expression using Bayes' theorem:
\begin{equation}
p(\vector{y}\,|\,\vector{\theta},\bP)\,p(\vector{\theta}\,|\,\bP)
= p(\vector{\theta}\,|\,\vector{y},\bP)\,p(\vector{y}\,|\,\bP)
~~.
\end{equation}
That is, in certain magical circumstances it is possible to do this
re-factorization without explicitly doing any integral.
When this is true, the marginalization is sometimes far easier than
the relevant integral.
The point of this \textsl{Note}\ is that this magical circumstance
arises when the two probability
distributions---the likelihood and the prior---are both Gaussian in
form, and when the model is linear over the parameters we would like to marginalize over.
In detail we will assume
\begin{enumerate}
\item
the likelihood $p(\vector{y}\,|\,\vector{\theta},\bP)$ is a Gaussian in $\vector{y}$,
\item
the prior $p(\vector{\theta}\,|\,\bP)$ is a Gaussian in $\vector{\theta}$,
\item
the mean of the likelihood Gaussian depends linearly on the linear
parameters $\vector{\theta}$, and
\item
the linear parameters $\vector{\theta}$ don't enter the likelihood anywhere
other than in the mean.
\end{enumerate}
In equations, this becomes:
\begin{equation}
p(\vector{y}\,|\,\vector{\theta},\bP) = \mathcal{N}\!\,(\vector{y}\,|\,\matrix{M}\cdot\vector{\theta},\tensor{C})
\end{equation}
\begin{equation}
p(\vector{\theta}\,|\,\bP) = \mathcal{N}\!\,(\vector{\theta}\,|\,\vector{\mu},\tensor{\Lambda})
\end{equation}
\begin{equation}\label{eq:Gaussian}
\mathcal{N}\!\,(\vector{x}\,|\,\vector{m},\tensor{V}) \equiv \frac{1}{||2\pi\,\tensor{V}||^{1/2}}\,\exp\left(-\frac{1}{2}\,[\vector{x}-\vector{m}]^{\!\mathsf{T}\!} \cdot \tensor{V}^{-1} \cdot [\vector{x} - \vector{m}]\right)
~~,
\end{equation}
where $\mathcal{N}\!\,(\vector{x}\,|\,\vector{m},\tensor{V})$ is the multivariate Gaussian
pdf\footnote{Check out what we did with the ``$2\pi$'' in the determinant in
equation~(\ref{eq:Gaussian}): We wrote an expression for the
multivariate Gaussian that never makes any reference to the
dimension $d$ of the $\vector{x}$-space. Most expressions in the literature
have a pesky $d/2$ in them, which is ugly and implies some need for
code or equations to know the dimension explicitly, even though all
the terms (determinant, inner product) are coordinate-free scalar
forms. If you use the expression as we have written it here, you never
have to explicitly access the dimensions.} for a vector $\vector{x}$
given a mean vector $\vector{m}$ and a variance tensor $\tensor{V}$,
$\matrix{M}$ is a $N\times K$ rectangular design matrix (which depends, in
general, on the nonlinear parameters $\bP$),
$\tensor{C}$ is a $N\times N$ covariance matrix of uncertainties for the
data (diagonal if the data dimensions are independent).
That is, the likelihood is a Gaussian with a mean that depends
linearly on the parameters $\vector{\theta}$, and
$\vector{\mu}$ and $\tensor{\Lambda}$ are the $K$-vector mean and $K\times K$ variance tensor
for the Gaussian prior.
\marginpar{\rule{0pt}{1ex}\\ \rule{\marginparwidth}{0.5pt}\\ \rule{0pt}{0.25ex}\\
\includegraphics[width=\marginparwidth, trim=0ex 0ex 0ex 0.5in, clip]{oned.pdf}
\caption{The one-dimensional case: If the prior pdf and the likelihood are both
Gaussian in a single parameter, their product (and hence the posterior pdf) is
also Gaussian, with a narrower width (smaller variance) than
either the prior pdf or the likelihood.\label{fig:oned}}\\
\rule{0pt}{0.25ex}\\ \rule{\marginparwidth}{0.5pt}\\ \rule{0pt}{1ex}}
In this incredibly restrictive---but also surprisingly
common---situation, the re-factored pdfs $p(\vector{\theta}\,|\,\vector{y},\bP)$
(the posterior for the linear parameters, conditioned on
the nonlinear parameters in $\bP$) and $p(\vector{y}\,|\,\bP)$ (the
marginalized likelihood, similarly conditioned) will also both be Gaussian.
We will solve this problem for general multivariate Gaussians in spaces
of different dimensionality (and units) but the one-dimensional case is
illustrated in \figurename~\ref{fig:oned}.
Obtaining the specific form for the general Gaussian product is the object of this
\textsl{Note}.
\section{Products of two Gaussians}\label{sec:problemsolution}
On the internets, there are many documents, slide decks, and videos
that explain products of Gaussians in terms of other Gaussians.\footnote{Two
good examples are \cite{roweis}, and \cite{cookbook}. The closest---that we know
of---to a
discussion with the generality of
what is shown here is perhaps our own previous contribution \cite{luger}.}
The vast majority of these consider either the univariate case (where
the data $\vector{y}$ and the parameter $\vector{\theta}$ are both simple scalars, which
is not useful for our science cases), or the same-dimension case (where the data
$\vector{y}$ and the parameter vector $\vector{\theta}$ are the same length, which never
occurs in our applications).
Here we solve this problem in the general case:\footnote{We solve this general
case, but we are not claiming \emph{priority} in any sense: This mathematics
has been understood for many many decades or even centuries. This \textsl{Note}\ is
a pedagogical contribution, not a research contribution.}
The inputs are multivariate (vectors) and the two Gaussians we are
multiplying live in spaces of different dimensions.
That is, we solve the following problem:
\paragraph{Problem:}
Find $K$-vector $\vector{a}$, $K\times K$ variance tensor $\tensor{A}$, $N$-vector $\vector{b}$,
and $N\times N$ variance tensor $\tensor{B}$ such that
\begin{equation}\label{eq:problem}
\mathcal{N}\!\,(\vector{y}\,|\,\matrix{M}\cdot\vector{\theta},\tensor{C})\,\mathcal{N}\!\,(\vector{\theta}\,|\,\vector{\mu},\tensor{\Lambda})
= \mathcal{N}\!\,(\vector{\theta}\,|\,\vector{a},\tensor{A})\,\mathcal{N}\!\,(\vector{y}\,|\,\vector{b},\tensor{B}) ~~,
\end{equation}
and such that $\vector{a}$, $\tensor{A}$, $\vector{b}$, and $\tensor{B}$ don't depend on $\vector{\theta}$ at all.
Note that
$\vector{y}$ is a $N$-vector,
$\matrix{M}$ is a $N\times K$ matrix,
$\vector{\theta}$ is a $K$-vector,
$\tensor{C}$ is a $N\times N$ non-negative semi-definite variance tensor,
$\vector{\mu}$ is a $K$-vector,
and
$\tensor{\Lambda}$ is a $K\times K$ non-negative semi-definite variance tensor.
\paragraph{Solution:}
\begin{equation}\label{eq:A}
\tensor{A}^{-1} = \tensor{\Lambda}^{-1} + \matrix{M}^{\!\mathsf{T}\!} \cdot \tensor{C}^{-1} \cdot \matrix{M}
\end{equation}
\begin{equation}\label{eq:a}
\vector{a} = \tensor{A} \cdot (\tensor{\Lambda}^{-1} \cdot \vector{\mu} + \matrix{M}^{\!\mathsf{T}\!} \cdot \tensor{C}^{-1} \cdot \vector{y})
\end{equation}
\begin{equation}\label{eq:B}
\tensor{B} = \tensor{C} + \matrix{M} \cdot \tensor{\Lambda} \cdot \matrix{M}^{\!\mathsf{T}\!}
\end{equation}
\begin{equation}\label{eq:b}
\vector{b} = \matrix{M} \cdot \vector{\mu}
~~.
\end{equation}
This is the complete solution to the problem, and constitutes the main point
of this \textsl{Note}.
For completeness, we will give some discussion!
\paragraph{Proof:}
The two sides of equation~(\ref{eq:problem}) are identical if two things
hold.
The first thing is that the determinant products must be equal:
\begin{equation}
||\tensor{C}||\,||\tensor{\Lambda}|| = ||\tensor{A}||\,||\tensor{B}||
~~,
\end{equation}
because the determinants are involved in the normalizations of the
functions.
This equality of determinant products follows straightforwardly from
the matrix determinant lemma\footnote{See, for example, \cite{Wiki:MDL}, and \cite{Harville:2011}.}
\begin{equation}\label{eq:detlemma}
||\tensor{Q} + \matrix{U}\cdot\matrix{V}^{\!\mathsf{T}\!}|| = ||\tensor{I} + \matrix{V}^{\!\mathsf{T}\!}\cdot\tensor{Q}^{-1}\cdot\matrix{U}||\,||\tensor{Q}||
~~,
\end{equation}
where $\matrix{U}$ and $\matrix{V}$ can be rectangular, and $\tensor{I}$ is the correct-sized identity matrix.
This identity implies that
\begin{equation}
||\tensor{A}^{-1}|| = ||\tensor{I} + \matrix{M}^{\!\mathsf{T}\!}\cdot\tensor{C}^{-1}\cdot\matrix{M}\cdot\tensor{\Lambda}||\,||\tensor{\Lambda}^{-1}||
\end{equation}
\begin{equation}
||\tensor{B}|| = ||\tensor{I} + \matrix{M}^{\!\mathsf{T}\!}\cdot\tensor{C}^{-1}\cdot\matrix{M}\cdot\tensor{\Lambda}||\,||\tensor{C}||
~~,
\end{equation}
where we had to apply the identity twice to get the $||\tensor{A}^{-1}||$ expression.
We can ratio these as follows to prove this first thing:
\begin{equation}
||\tensor{A}||\,||\tensor{B}||
= \frac{||\tensor{B}||}{||\tensor{A}^{-1}||}
= \frac{||\tensor{C}||}{||\tensor{\Lambda}^{-1}||}
= ||\tensor{C}||\,||\tensor{\Lambda}||
~~.
\end{equation}
The second thing required for the proof is that the quadratic scalar form
\begin{equation}\label{eq:LHS}
[\vector{y}-\matrix{M}\cdot\vector{\theta}]^{\!\mathsf{T}\!}\cdot\tensor{C}^{-1}\cdot[\vector{y}-\matrix{M}\cdot\vector{\theta}]
+ [\vector{\theta}-\vector{\mu}]^{\!\mathsf{T}\!}\cdot\tensor{\Lambda}^{-1}\cdot[\vector{\theta}-\vector{\mu}]
\end{equation}
must equal the quadratic scalar form
\begin{equation}\label{eq:RHS}
[\vector{\theta}-\vector{a}]^{\!\mathsf{T}\!}\cdot\tensor{A}^{-1}\cdot[\vector{\theta}-\vector{a}]
+ [\vector{y}-\vector{b}]^{\!\mathsf{T}\!}\cdot\tensor{B}^{-1}\cdot[\vector{y}-\vector{b}]
~~,
\end{equation}
because these quadratic scalar forms appear in the exponents in the functions.
This equality follows from straightforward expansion of
all the quadratic forms, plus some use of the matrix inversion lemma\footnote{This
useful lemma is also called the Woodbury matrix identity. See also \cite{wiki:MIL},
and \cite{Harville:2011}.}
\begin{equation}\label{eq:invlemma}
[\tensor{Q} + \matrix{U}\cdot\tensor{S}\cdot\matrix{V}^{\!\mathsf{T}\!}]^{-1} = \tensor{Q}^{-1} - \tensor{Q}^{-1}\cdot\matrix{U}\cdot[\tensor{S}^{-1} + \matrix{V}^{\!\mathsf{T}\!}\cdot\tensor{Q}^{-1}\cdot\matrix{U}]^{-1}\cdot\matrix{V}^{\!\mathsf{T}\!}\cdot\tensor{Q}^{-1}
~~,
\end{equation}
which gives an expression for the inverse $\tensor{B}^{-1}$ of the marginalized
likelihood variance:
\begin{equation}
\tensor{B}^{-1} = \tensor{C}^{-1} - \tensor{C}^{-1}\cdot\matrix{M}\cdot[\tensor{\Lambda}^{-1} + \matrix{M}^{\!\mathsf{T}\!}\cdot\tensor{C}^{-1}\cdot\matrix{M}]^{-1}\cdot\matrix{M}^{\!\mathsf{T}\!}\cdot\tensor{C}^{-1}
~~.
\end{equation}
After that it's just a lot of grinding through matrix expressions.\footnote{%
We leave this grinding to the avid reader.
For guidance, it might help to realize that there are terms that
contain $\vector{\theta}^{\!\mathsf{T}\!}\cdots\vector{\theta}$, $\vector{\theta}^{\!\mathsf{T}\!}\cdots\vector{y}$, $\vector{y}^{\!\mathsf{T}\!}\cdots\vector{y}$,
$\vector{\theta}^{\!\mathsf{T}\!}\cdots\vector{\mu}$, and $\vector{\mu}^{\!\mathsf{T}\!}\cdots\vector{\mu}$.
If you expand out each of these five kinds of terms, each of the five
should lead to an independent-ish equality.}
\paragraph{Solution notes:}
In principle we found this factorization by expanding the quadratic in
(\ref{eq:LHS}) and then completing the square.
Of course we didn't really; we used arguments (which physicists love)
called \emph{detailed balance}:
We required that the terms that look like
$\vector{\theta}^{\!\mathsf{T}\!}\cdot\tensor{Q}\cdot\vector{\theta}$ were equal between the LHS~(\ref{eq:LHS})
and the RHS~(\ref{eq:RHS}), and then all the terms that look like
$\vector{\mu}^{\!\mathsf{T}\!}\cdot\tensor{S}\cdot\vector{\mu}$, and so on.
It turns out you don't have to consider them all to get the right solution.
There is an alternative derivation or proof involving the
\textsl{canonical form} for the multivariate Gaussian. This form is
\begin{equation}
\mathcal{N}\!\,(\vector{x}\,|\,\vector{m},\tensor{V}) = \exp\left(-\frac{1}{2}\vector{x}^T \cdot\tensor{H} \cdot \vector{x} + \vector{\eta}^T \cdot \vector{x} - \frac{1}{2}\,\xi\right)
\end{equation}
\begin{equation}
\tensor{H}\equiv\tensor{V}^{-1} ~~;~~
\vector{\eta}\equiv\tensor{V}^{-1}\cdot\vector{m} ~~;~~
\xi\equiv\ln||2\pi\,\tensor{V}||+\vector{\eta}^T\cdot\tensor{V}\cdot\vector{\eta} ~~.
\end{equation}
In the canonical form, many products and other manipulations become simpler, so
it is worth trying this route if you get stuck when manipulating Gaussian
expressions.
Because the matrix $\matrix{M}$ is not square, it has no inverse. And because this
is a physics problem, $\matrix{M}$ has units (which are the units of
$\mathrm{d}\vector{y}/\mathrm{d}\vector{\theta}$).
It's beautiful in the solution that $\matrix{M}$ and $\matrix{M}^{\!\mathsf{T}\!}$ appear only where the
units make sense.
They make sense because the units of $\tensor{C}^{-1}$ are inverse data-squared (where $\vector{y}$
is the data vector) and the units of $\tensor{\Lambda}$ are parameters-squared and the units
of $\matrix{M}$ are data over parameters.
And they are all different sizes.
If you remember the Bayesian context around equation~(\ref{eq:bayes}) and the
Bayesian discussion thereafter,
the Gaussian $\mathcal{N}\!\,(\vector{\theta}\,|\,\vector{a},\tensor{A})$ is the posterior pdf for the linear
parameters $\vector{\theta}$, and the Gaussian $\mathcal{N}\!\,(\vector{y}\,|\,\vector{b},\tensor{B})$ is the
marginalized likelihood, marginalizing out the linear parameters $\vector{\theta}$.
This marginalization is usually thought of as being an integral, like the
one given in equation~(\ref{eq:integral}).
How are these linear-algebra expressions in any sense ``doing this integral''?
The answer is: That integral is a correlation of two Gaussians,\footnote{Astronomers
like to say that it is the ``convolution'' of two Gaussians, but it is really the
correlation of two Gaussians. The differences between convolution and correlation
are minimal, though, and we aren't sticklers.} and the correlation of two
Gaussians delivers a new Gaussian with a shifted mean that is wider than either
of the original two Gaussians.
This factorization does, indeed, deliver the correct marginalization integral.
Continuing along these lines,
various parts of the solution are highly interpretable in terms of the
objects of Bayesian inference. For example, because the term $p(\vector{\theta}\,|\,
\vector{a},\tensor{A})$ is the conditional posterior pdf\footnote{We say ``conditional'' here
because it is conditioned on nonlinear parameters $\bP$.
The vector $\vector{a}$ and variance tensor
$\tensor{A}$ will depend on the nonlinear parameters $\bP$ through the design matrix
$\matrix{M}$.} for the linear parameters $\vector{\theta}$, the vector $\vector{a}$ is the maximum
\foreign{a posteriori} (or \acronym{MAP}) value for the parameter vector
$\vector{\theta}$.
It is found by inverse-variance-weighted combinations of the data and the prior.
In some projects, posterior pdfs or \acronym{MAP} parameter values are the goal
(although we don't think they often should be\footnote{Although---in subjective
Bayesian inference---the posterior pdf
is the valid statement of your belief, it is not so useful to your colleagues,
who start with different beliefs from yours. See Note\textsuperscript{\ref{note:lf}}.}).
The variance tensor $\tensor{A}$ is the posterior variance in the parameter space.
It is strictly smaller (in eigenvalues or determinant) than either
the prior variance $\tensor{\Lambda}$ or the parameter-space data-noise
variance $[\matrix{M}^{\!\mathsf{T}\!}\cdot\tensor{C}^{-1}\cdot\matrix{M}]^{-1}$.
The vector $\vector{b}$ is the prior-optimal (maximum \foreign{a priori})
value for the data $\vector{y}$.
It is the most probable data vector (prior to seeing any data),
and also the prior expectation for the data,
under the prior pdf.
The variance tensor $\tensor{B}$ is the prior variance expanded out to the
data space, and including the noise variance in the data.
It is strictly larger than both the data noise variance $\tensor{C}$ and the
data-space prior variance $\matrix{M}\cdot\tensor{\Lambda}\cdot\matrix{M}^{\!\mathsf{T}\!}$.
\paragraph{Implementation notes:}
The solution gives an expression for the variance tensor $\tensor{B}$, but
note that when you actually evaluate the pdfs you probably need to
have either the inverse of $\tensor{B}$, or else an operator that computes
the product of the inverse and vectors, as in $\tensor{B}^{-1}\cdot\vector{y}$ and
the same for $\vector{b}$.
To get the inverse of the tensor $\tensor{B}$ stably, \emph{you might want to use
the matrix inversion lemma} (\ref{eq:invlemma}) given above.
This is often useful because you often know or are given the data inverse variance
tensor $\tensor{C}^{-1}$ for the noise, and the prior variance inverse
$\tensor{\Lambda}^{-1}$, and the lemma manipulates these into the answer without
any heavy linear algebra.
The lemma saves you the most time and precision when the parameter size $K$
is much smaller than the data size $N$ (or vice versa); that is, when $\matrix{M}$
is ``very non-square''.
We also give the general advice that one should \emph{avoid taking an explicit
numerical inverse} (unless you know the inverse exactly in closed form, as you
do for, say, diagonal tensors).
In your code, it is typically stabler to use a \code{solve()} function instead
of the \code{inv()} function.
The reason is that the code operation \code{inv(B)} returns the best
possible inverse to machine precision (if you are lucky), but what you
really want instead is the best possible product of that inverse times
a vector.
So, in general, \code{solve(B,y)} will deliver more precise results than
the mathematically equivalent \code{dot(inv(B),y)}.
The expressions in equation~(\ref{eq:problem}) do not require that the variance tensors
$\tensor{C}$, $\tensor{\Lambda}$, $\tensor{A}$, $\tensor{B}$ be positive definite; they only require
that they be non-negative semi-definite.
That means that they can have zero eigenvalues.
As can their inverses $\tensor{C}^{-1}$, $\tensor{\Lambda}^{-1}$, $\tensor{A}^{-1}$, $\tensor{B}^{-1}$.
If either of these might happen in your problem---like if your prior
freezes the parameters to a subspace of the $\vector{\theta}$-space, which
would lead to a zero eigenvalue in $\tensor{\Lambda}$, or if a data point is
unmeasured or missing, which would lead to a zero eigenvalue in
$\tensor{C}^{-1}$---you might have to \emph{think about how you implement the
linear algebra operations to be zero-safe}.\footnote{A completely
zero-safe implementation
is somewhat challenging, but one comment to make is that if, say, $\tensor{A}$
contains a zero eigenvalue, then there is a direction in the parameter
space (the $\vector{\theta}$ space) in which the variance vanishes. This means that
all valid parameter combinations lie on a linear subspace of the full $K$-dimensional
space. All parameter combinations that wander off the subspace get strictly
zero probability or negative infinities in the log.
If your inference is valid, it will probably be the case that the
vectors at which you want to evaluate \emph{always} lie in the non-zero
subspace. It makes sense, then, in this case, to work in a representation in which it is easy
to enforce or ensure that. This usually involves some kind of coordinate transformation
or rotation or projection. Doing this correctly is beyond the scope of this \textsl{Note}.}
\paragraph{Simplification: single multiplicative scaling}
One interesting case is when $K=1$, so the design matrix in fact
reduces to a model vector $\vector{m}$, multiplied by a scalar $\theta$, and
$\mu$ and $\Lambda$ are now scalars as well:
\begin{equation}
\mathcal{N}\!\,(\vector{y}\,|\,\theta\,\vector{m},\tensor{C})\,\mathcal{N}\!\,(\theta\,|\,\mu,\Lambda) = \mathcal{N}\!\,(\theta\,|\, a,A)\,\mathcal{N}\!\,(\vector{y}\,|\,\vector{b},\tensor{B})
~~.
\end{equation}
This can arise if one wants to multiplicatively scale a model to the data. $a$ would then correspond to the maximum a posteriori value of the multiplicative scaling to fit $\vector{y}$ with $\vector{m}$.
In this case, the previous equations are simplified and no longer
involve many matrix operations. It's a nice exercise to simplify the
solution above for this scalar case.
\paragraph{Special case: wide prior}
Another interesting case that often arises in inferences is the use of an improper
(infinitely wide) prior on the parameters $\vector{\theta}$. Rather than
ignoring the prior pdf on the left-hand side of
equation~(\ref{eq:problem}), which is technically incorrect, the correct
posterior pdf can be obtained by taking the limit $\tensor{\Lambda}^{-1} \rightarrow 0$ in the
fiducial results derived above. It is perhaps
worth noting that in the improper-prior case,
the posterior can still be fine, but the marginalized likelihood will make no sense (it will technically vanish).
\paragraph{Generalization: product of many Gaussians}
A case that arises in some applications is that the
likelihood is made of multiple Gaussian terms, each of which is a
different linear combination of the linear parameters $\vector{\theta}$.
That is, there are $J$ data vectors $\vector{y}_j$, each of which has size or length $N_j$,
and each of which has an expectation
set linearly by the parameters $\vector{\theta}$ but through a different design matrix $\matrix{M}_j$.
Provided that the different data vectors $\vector{y}_j$ are independently ``observed'' (that
is, they have independent noise draws with noise variance tensors $\tensor{C}_j$),
the total likelihood is just the product of
the individual-data-vector likelihoods.
An example of this case arises in astronomy, for example, when considering
radial velocity measurements of a star taken with different instruments that may
have systematic offsets between their velocity zero-points.
In principle we could work around this problem---reduce it to the previously solved problem---by forming a large vector $\vector{y}$ which is the concatenation of all the individual data vectors $\vector{y}_j$, and a large design matrix $\matrix{M}$ which is the concatenation of all the individual design matrices $\matrix{M}_j$, and a large total covariance matrix $\tensor{C}$ which is a block diagonal matrix containing the noise variance tensors $\tensor{C}_j$ on the diagonal blocks.
We could then apply the result of the single-data-vector problem above.
However, this can result in significant unnecessary computation, and it is hard to write the answer in a simple form.
Instead we can take advantage of the separability of the likelihoods, and write the following generalized problem statement:
Find $K$-vector $\vector{a}$, $K\times K$ variance tensor $\tensor{A}$, $J$ vectors $\vector{b}_j$ (each
of which is a different length $N_j$),
and $J$ variance tensors $\tensor{B}_j$ (each of which is a different size $N_j\times N_j$)
such that
\begin{equation}\label{eq:genproblem}
\mathcal{N}\!\,(\vector{\theta}\,|\,\vector{\mu},\tensor{\Lambda})\,\prod_{j=1}^J\mathcal{N}\!\,(\vector{y}_j\,|\,\matrix{M}_j\cdot\vector{\theta},\tensor{C}_j)\,
= \mathcal{N}\!\,(\vector{\theta}\,|\,\vector{a},\tensor{A})\,\prod_{j=1}^J\mathcal{N}\!\,(\vector{y}_j\,|\,\vector{b}_j,\tensor{B}_j) ~~,
\end{equation}
and such that $\vector{a}$, $\tensor{A}$, all the $\vector{b}_j$, and all the $\tensor{B}_j$
don't depend on $\vector{\theta}$ at all.
Note that
$\vector{\theta}$ is a $K$-vector,
$\vector{\mu}$ is a $K$-vector,
$\tensor{\Lambda}$ is a $K\times K$ non-negative semi-definite variance tensor,
each $\vector{y}_j$ is an $N_j$-vector,
each $\matrix{M}_j$ is a $N_j\times K$ matrix,
and
each $\tensor{C}_j$ is a $N_j\times N_j$ non-negative semi-definite variance tensor.
One way to solve this problem is to write all Gaussians in their canonical form, then separate the elements that depend on $\vector{\theta}$ and on the individual $\vector{y}_j$.
The result can be written as an iteration over data vectors $\vector{y}_j$:
\begin{equation}
\mbox{\small initialize:}~~
\tensor{A}_0^{-1} = \tensor{\Lambda}^{-1} ~~;~~ \vector{a}_0 = \vector{\mu} ~~;~~ \vector{x}_0 = \tensor{\Lambda}^{-1}\cdot\vector{\mu}
\end{equation}
\begin{align}
\mbox{\small iterate:}~~
\tensor{B}_j &= \tensor{C}_j + \matrix{M}_j \cdot \tensor{A}_{j-1} \cdot \matrix{M}_j^{\!\mathsf{T}\!} \\
\vector{b}_j &= \matrix{M}_j \cdot \vector{a}_{j-1} \\
\tensor{A}_j^{-1} &= \tensor{A}_{j-1}^{-1} + \matrix{M}_j^{\!\mathsf{T}\!} \cdot \tensor{C}_j^{-1} \cdot \matrix{M}_j \\
\vector{x}_j &= \vector{x}_{j-1} + \matrix{M}_j^{\!\mathsf{T}\!} \cdot \tensor{C}_j^{-1} \cdot \vector{y}_j \\
\vector{a}_j &= \tensor{A}_j \cdot \vector{x}_j
\end{align}
\begin{align}
\mbox{\small finish:}~~
\tensor{A} &= \tensor{A}_J \\
\vector{a} &= \vector{a}_J
~~.
\end{align}
The solution is an iteration because you can think of adding each new data
set $\vector{y}_j$ sequentially, with the prior for set $j$ being the posterior from
set $j-1$.
The way this solution is written is unpleasant, because the specific values you get for the
vectors $\vector{b}_j$ and tensors $\tensor{B}_j$ depend on the order in which you insert the data.
But---and very importantly for the rules of Bayesian inference---the posterior
mean $\vector{a}$ and variance $\tensor{A}$ \emph{do not depend} on the order!\footnote{It
is literally part of the fundamental justification of Bayesian inference
that the knowledge you eventually have (your final beliefs) does not depend
on the order in which you observed the data. This is one of the axioms or inputs
to the theorems that underlie the consistency of Bayesian reasoning.
There is an illuminating discussion of all this in Chapter~1 of \cite{jaynes}.}
\section{Worked Examples}\label{sec:examples}
When working with a probabilistic model that meets the strong requirements
imposed above (Gaussians everywhere; expectations linear in parameters),
the identities described in this
\textsl{Note}\ have practical uses: (1) To simplify the posterior pdf of your
model (which makes generating samples or computing integrals far simpler), and
(2) to reduce the dimensionality of your model (by enabling closed-form
marginalizations over linear parameters).
Reducing the dimensionality of your parameter-space will in general improve
convergence of Markov Chain Monte Carlo\footnote{We have also written a tutorial
on \acronym{MCMC} in this series \citep{Hogg:2018}.} (\acronym{MCMC}) sampling
methods, or enable alternate sampling methods (for example, rejection sampling)
that may be intractable when the parameter dimensionality is large: These two
benefits also typically make inference procedures (like sampling) \emph{far}
faster.
Here, we demonstrate the utility of the identities shown above with two
worked exercises.
\paragraph{Exercise 1: A fully linear model:} We observe
a set of data $(x_i, y_i)$ (indexed by $i$) with
known, Gaussian uncertainties in $y$, $\sigma_y$, and no uncertainty in $x$.
The parametric model we will use for these data is a quadratic polynomial,
\begin{equation}
f(x \,;\, \alpha, \beta, \gamma) = \alpha\,x^2 + \beta\,x + \gamma
\end{equation}
and we assume we have Gaussian prior pdfs on all of the $K=3$ linear parameters
$(\alpha, \beta, \gamma)$,
\begin{align}
p(\alpha) &= \mathcal{N}\!\,(\alpha \,|\, \mu_\alpha, \sigma_\alpha)\\
p(\beta) &= \mathcal{N}\!\,(\beta \,|\, \mu_\beta, \sigma_\beta)\\
p(\gamma) &= \mathcal{N}\!\,(\gamma \,|\, \mu_\gamma, \sigma_\gamma)
~~.
\end{align}
While this example may seem overly simple or contrived, quadratic models are
occasionally useful in astronomy and physics, for example, when centroiding a
peak,\footnote{Fitting second-order polynomials has been shown to be great for
centroiding peaks in astronomy contexts. See, for example, \cite{vakili}, and
\cite{teague}.}
and polynomial models are often used to capture smooth trends in
data.
\marginpar{\rule{0pt}{1ex}\\ \rule{\marginparwidth}{0.5pt}\\ \rule{0pt}{0.25ex}\\
\parbox{\marginparwidth}{
~\hfill\begin{tabular}{c|c|c}
$x$ & $y$ & $\sigma_y$ \\
\hline
$-0.6$ & 12.2 & 0.8 \\
2.0 & 4.1 & 3.2 \\
2.7 & 0.9 & 3.3 \\
3.6 & $-15.0$ & 3.9 \\
\end{tabular}\hfill~}
\includegraphics[width=\marginparwidth]{exercise1.pdf}
\caption{\textsl{Top:} Data generated with ``true'' parameters
$(\alpha, \beta, \gamma) = (3.21, 2.44, 14.82)$.
\textsl{Bottom:} The solution to Exercise 1. The data points (black markers)
show the data from the table. The \acronym{MAP} parameter values were
found using equation~(\ref{eq:a}).
All data files and solution notebooks are available via
\href{https://doi.org/10.5281/zenodo.3855689}{Zenodo} \citep{zenodo}.\label{fig:ex1}}\\
\rule{0pt}{0.25ex}\\ \rule{\marginparwidth}{0.5pt}\\ \rule{0pt}{1ex}}
The data table shown in \figurename~\ref{fig:ex1} contains $N=4$ data points,
$(x_i, y_i, \sigma_{y_i})$, generated using this quadratic model.
Assuming values for the prior means, $\vector{\mu}$, and prior variance tensor,
$\tensor{\Lambda}$,
\begin{align}
\vector{\mu}^{\!\mathsf{T}\!} &= (\mu_\alpha, \mu_\beta, \mu_\gamma) = (1, 3, 9)\\
\tensor{\Lambda} &=
\begin{pmatrix}
5^2 & 0 & 0\\
0 & 2^2 & 0\\
0 & 0 & 8^2
\end{pmatrix}
\end{align}
compute the \acronym{MAP} parameter values $\vector{a}^{\!\mathsf{T}\!} = (\alpha_{\rm MAP}, \beta_{\rm MAP},
\gamma_{\rm MAP})$.
Plot the data (with error bars) and over-plot the model evaluated at the
\acronym{MAP} parameter values.
Generate 4096 posterior samples of the linear parameters.
Over-plot a shaded region showing the 68~percent credible region for the model,
estimated using these samples.
\paragraph{Solution:} Given the assumptions and prior parameter values
above, the design matrix, $\matrix{M}$, is
\begin{align}
\matrix{M} &= \begin{pmatrix}
0.36 & -0.6 & 1\\
4.0 & 2.0 & 1\\
7.29 & 2.7 & 1\\
12.96 & 3.6 & 1
\end{pmatrix} ~~.
\end{align}
By plugging in to equation~(\ref{eq:a}), we find \acronym{MAP} parameter values for the
linear parameters
\begin{equation}
\vector{a}^{\!\mathsf{T}\!} =
(\alpha_{\rm MAP}, \beta_{\rm MAP}, \gamma_{\rm MAP}) =
(3.61, 1.98, 14.26) ~~.
\end{equation}
\figurename~\ref{fig:ex1} shows the data (black points), the model computed with
the \acronym{MAP} parameter values (blue line), and the 68-percent credible region (shaded
blue region) estimated using posterior samples generated from
$\mathcal{N}\!\,(\vector{\theta}\,|\,\vector{a},\tensor{A})$.
The companion \href{https://doi.org/10.5281/zenodo.3855689}{\texttt{IPython}
notebook} (\texttt{Exercise1.ipynb}) contains the full solution.
\paragraph{Exercise 2: A model with a nonlinear parameter:} We
observe a set of data $(x_i, y_i)$ (indexed by $i$) with
known, Gaussian uncertainties in $y$, $\sigma_{y_i}$, and no uncertainty in $x$.
The parametric model we will use for these data is a generalized sinusoid with a
constant offset,
\begin{equation}
f(x \,;\, \alpha, \beta, \gamma, \omega) =
\alpha\,\cos(\omega \, x) + \beta\,\sin(\omega \, x) + \gamma \label{eq:ex2model}
\end{equation}
and we again assume we have Gaussian prior pdfs on all of the \emph{linear}
parameters $(\alpha, \beta, \gamma)$.
Models like this (a periodic model with both linear and nonlinear parameters)
are common in astronomy, especially in the context of asteroseismology, light
curve analysis, and radial velocity variations from massive companions (binary
star systems or exoplanets).
For this setup, we can no longer analytically express the posterior pdf because
of the nonlinear parameter $\omega$, but we can compute the marginal likelihood
(marginalizing over the linear parameters) conditioned on the frequency
$\omega$.
\marginpar{\rule{0pt}{1ex}\\ \rule{\marginparwidth}{0.5pt}\\ \rule{0pt}{0.25ex}\\
\parbox{\marginparwidth}{
~\hfill\begin{tabular}{c|c|c}
$x$ & $y$ & $\sigma_y$ \\
\hline
$-1.2$ & 11.2 & 0.2 \\
1.3 & 16.1 & 0.2 \\
3.1 & 10.2 & 0.3 \\
4.1 & 13.5 & 0.3
\end{tabular}\hfill~}
\includegraphics[width=\marginparwidth]{exercise2a.pdf}
\includegraphics[width=\marginparwidth]{exercise2b.pdf}
\caption{\textsl{Top:} Data generated with ``true'' parameters
$(\alpha, \beta, \gamma, \omega) = (3.21, 2.44, 13.6, 1.27)$.
\textsl{Middle and Bottom:} The solution to Exercise 2.
All data files and solution notebooks are available via
\href{https://doi.org/10.5281/zenodo.3855689}{Zenodo} \citep{zenodo}.\label{fig:ex2}}\\
\rule{0pt}{0.25ex}\\ \rule{\marginparwidth}{0.5pt}\\ \rule{0pt}{1ex}}
The table shown in \figurename~\ref{fig:ex2} contains $N=4$ data points,
$(x_i, y_i, \sigma_{y_i})$, generated with this sinusoid model.
Assuming values for the prior means, $\vector{\mu}$, and prior variance tensor,
$\tensor{\Lambda}$,
\begin{align}
\vector{\mu}^{\!\mathsf{T}\!} &= (\mu_\alpha, \mu_\beta, \mu_\gamma) = (0, 0, 0)\\
\tensor{\Lambda} &=
\begin{pmatrix}
5^2 & 0 & 0\\
0 & 5^2 & 0\\
0 & 0 & 10^2
\end{pmatrix}
\end{align}
write a function to compute the vectors and matrices we need for the linear
parameters (the design matrix and components
$\vector{a}, \tensor{A}, \vector{b}, \tensor{B}$
of the factorization) at a given value of the frequency $\omega$.
Assuming a prior on $\omega$ that is uniform in $\ln\omega$ over the domain
$(0.1, 100)$,
\begin{equation}
p(\omega) \propto \frac{1}{\omega}
\end{equation}
evaluate the log-marginal likelihood $\ln p(\vector{y}\,|\,\omega)$ and add to the
log-frequency prior $\ln p(\omega)$ over a grid of 16,384 frequencies $\omega$
between (0.1, 100).
Plot both the marginal likelihood (not log!) $p(\vector{y}\,|\,\omega)$ and the
posterior pdf $p(\vector{y}\,|\,\omega)\,p(\omega)$ as a function of this frequency
grid.
Generate 512 posterior samples\footnote{The
posterior pdf over $\omega$ will be extremely multimodal. Don't fire up
standard \acronym{MCMC}! Try using rejection sampling instead: Generate a
dense prior sampling in the nonlinear parameter $\omega$, evaluate
the marginalized likelihood at each sample in $\omega$, and use this
to reject prior samples.} in the full set of parameters
$(\alpha,\beta,\gamma,\omega)$.
Make a scatter plot showing a 2D projection of these samples in
$(\alpha, \ln \omega)$.
Plot the data, and over-plot 64 models (equation~\ref{eq:ex2model}) computed
using a fair subset of these posterior samples.
The companion \href{https://doi.org/10.5281/zenodo.3855689}{\texttt{IPython}
notebook} (\texttt{Exercise2.ipynb}) contains the full solution.
\clearpage\raggedright
|
train/arxiv
|
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| 5 | 1 |
\section{Introduction}\label{sec:Introduction}
The pioneering works of Feynman \cite{feynman1986quantum, feynman1982simulating}, Bennett and Brassard \cite{BB84}, Deutsch \cite{deutsch1985quantum}, Shor \cite{shor1994algorithms}, and others have strongly established that using quantum resources in computation, communication, and metrology, one can obtain certain advantages usually referred to as \textit{quantum advantage}. Subsequent theoretical and experimental works including Google's recent experimental demonstration of quantum advantage \cite{arute2019quantum} have further established this fact (\cite{nagali2012experimental, kumar2019experimental, centrone2021experimental, bravyi2020quantum, novo2021quantum, maslov2021quantum} are a few recent examples). A common feature of all these works is that quantum advantage requires nonclassical states, or in other words, the quantum states having no classical analog.
Nonclassical states may be generated by matter-field interactions. The Jaynes-Cummings model \cite{shore1993jaynes}, introduced in 1963 \cite{jaynes1963comparison}, describes one such simple interaction between a monochromatic field and a two-level atom.
This simple system, in the original form, as well as its various generalizations (see \cite{larsonspecial} in this context), is still relevant in several aspects of quantum information and QED. For instance, it plays a significant role in the generation of engineered quantum states \cite{dell2006multiphoton}, and the study of non-Markovian evolution \cite{bellomo2007non}. In fact, it is a convenient toy model---which has recently been generalized from different perspectives \cite{ermann2020jaynes, ghoshal2020population, villas2019multiphoton, huang2020ultrastrong}---often employed as a testbed to study various quantum effects, such as generation of Schr\"odinger-cat states \cite{buzek1992schrodinger}, entanglement protection \cite{fasihi2019entanglement}, catalysis \cite{Messinger_2020}, multiphoton blockade \cite{zou2020multiphoton}, fractional revivals \cite{averbukh1992fractional}, quantum state engineering \cite{gea1990collapse}, strong squeezing \cite{kuklinski1988squeezing}, entanglement generation \cite{phoenix1991entangled}, state discrimination \cite{namkung2019almost}.
In the usual Jaynes-Cummings model there are only two modes (atom mode and field mode), so relevant nonclassicality may appear either as two types of single-mode nonclassicalities involving each of these two modes, or as two-mode nonclassicality reflected as atom-field correlation. Atom-field correlations in the Jaynes-Cummings model have already been studied using negativity \cite{akhtarshenas2007negativity}. Further, using the Wigner-Yanase skew information-based quantifier of nonclassicality described in \cite{luo2019quantifying}, single-mode nonclassicalities present in Jaynes-Cummings model have been studied \cite{fu2021dynamics, dai2020atomic}. Specifically, field nonclassicality was studied in \cite{fu2021dynamics}, and atomic nonclassicality in \cite{dai2020atomic}. Additionally, entanglement dynamics and the field nonclassicality in the double Jaynes-Cummings model are reported in \cite{ghorbani2017wigner}. Specifically, the correlations between the two atoms in the two-mode nonlinear Jaynes-Cummings model were quantified using von Neumann entropy, negativity, and concurrence, while two-mode field nonclassicality was witnessed by the Wigner quasiprobability distribution. The Wigner function neither quantifies the nonclassicality, nor captures all the nonclassicality of states; for example, it fails in the case of squeezed states \cite{agarwal2012quantum}.
Thus, discrete efforts to study nonclassical features associated with the Jaynes-Cummings model have been made, but no attempt has yet been made either to look at the interplay between all three different aspects of nonclassicality which may be present in the Jaynes-Cummings model or to quantify the atom-field correlation, atomic nonclassicality, and field nonclassicality on the same footing.
Here, we aim to do so and study atom-field correlations, atomic nonclassicality, and field nonclassicality using the same measure in the resonant Jaynes-Cummings model. Specifically, we use negativity potential (entanglement potential \cite{asboth2005computable}) to quantify the local nonclassicalities of the atom and field and negativity to measure atom-field correlation. Our main motivation is to understand the dynamics and interplay between single-mode and two-mode nonclassicalities observed in the Jaynes-Cummings model and also to verify whether the dynamics conserve total nonclassicality (\cite{ge2015conservation} and references therein). We consider distinct initial conditions which include four cases: (A) atom in the excited state and field vacuum, and atom in ground state with field in (B) Fock, (C) thermal, and (D) coherent state. The significant feature of Cases B-D is that Fock state are known to be most nonclassical state, while both mixture and superposition of Fock states cannot generate entanglement at a beam splitter. Further, coherent state has non-zero coherence in the Fock basis, unlike Fock and thermal state.
The rest of the paper is organized as follows. In Section \ref{sec:nonclassicality}, we discuss nonclassicality measures in general with a specific focus on the measures used in the present work. In Section \ref{sec:dynamics}, we analyze the dynamics of nonclassicalities in detail for various initial conditions with specific attention to the question: Is total nonclassicality in the Jaynes-Cummings model conserved? Finally, the paper is concluded in Section \ref{sec:conclusion}.
\section{Nonclassicality Measures}\label{sec:nonclassicality}
A nonclassical state cannot be represented as a statistical mixture of coherent states, and thus has a non-positive Glauber-Sudarshan $P$ function. This gives us the necessary and sufficient criterion of nonclassicality. However, this is not easy to compute for most of the states and does not quantify the amount of nonclassicality. Inspired by this, several measures of nonclassicality were proposed over the years, such as nonclassical distance (the distance from the nearest classical states) \cite{hillery1987nonclassical}, nonclassical depth (the amount of noise required to destroy the nonclassicality) \cite{lee1991measure}, nonclassical volume (volume of the negative part of the Wigner function) \cite{kenfack2004negativity}. However, these measures have some inherent limitations \cite{miranowicz2015statistical}. Nonclassical distance requires minimization over all possible classical states, nonclassical depth is unity for all pure non-Gaussian states \cite{lutkenhaus1995nonclassical}, and nonclassical volume fails for some squeezed states as Wigner function is non-negative. Further, entanglement potential \cite{asboth2005computable} was proposed by Asboth to quantify the {amount} of single-mode nonclassicality. It is the amount of entanglement at the output of an auxiliary beam splitter mixing the single-mode nonclassicality with any classical state. This allows us to use the same entanglement measure to quantify local atom and field nonclassicality (as entanglement potential) as well as atom-field correlations.
Specifically, we use negativity---introduced in \cite{zyczkowski1998volume} and shown to be LOCC monotone in \cite{vidal2002computable}---to quantify atom-field correlations (as well as entanglement potential). Negativity in a bipartite entangled state $\rho$ is the sum of the absolute values of the negative eigenvalues ($\lambda_k$) of the partial transposed density matrix $\rho^{\Gamma}$:
\begin{equation}
N(\rho) = \sum_{k: \lambda_k < 0} \left| \lambda_k \right| = \sum_{k} \frac{1}{2}\left(\left| \lambda_k\right| - \lambda_k\right).
\end{equation}
This measure of entanglement is based on the partial transpose condition introduced by Peres \cite{peres1996separability}, which states that a bipartite density matrix is separable (classical) if its partial transpose has only non-negative eigenvalues. This is a necessary condition only for qubit-qubit and qubit-qutrit systems \cite{horodecki2001separability}. We study negativity here because it is easy to compute. As most entanglement measures involve an extremization, usually over complicated regions of the Hilbert space \cite{horodecki2009quantum}, computing the time evolution of such measures is difficult. In addition, negativity has a physical interpretation as the approximate number of entangled degrees of freedom in a bipartite system \cite{eltschka2013negativity}.
We use negativity potential as used in \cite{miranowicz2015statistical} to quantify nonclassicality in a single qubit. An analogous theoretical construction for the atomic states that defines a negativity potential for the atom \cite{naikoo2019interplay} is used here to quantify atomic nonclassicality. For brevity, we omit 'potential' in what follows and call them field negativity and atom negativity.
The choice of negativity as a measure of entanglement is by no means unique, we could have chosen concurrence or logarithmic negativity instead (\cite{horodecki2009quantum} reviews these in detail).
Further, Asboth's idea of entanglement potential is critically analyzed as it is shown that there exists some residual nonclassicality in the single-mode states at the output of the beam splitter which is not captured by entanglement \cite{ge2015conservation, arkhipov2016nonclassicality}. We also verify this in our context by using additional beam splitter layers to \emph{deplete} all the residual nonclassicality.
\section{Dynamics of nonclassiality in the resonant Jaynes-Cummings model\label{sec:dynamics}}
To study the interplay between different types of nonclassicality present in Jaynes-Cummings model, we first need to briefly describe the model.
In a conventional Jaynes-Cummings model, a two-level atom interacts with a single-mode quantized electromagnetic field. In what follows, we denote the field (atom) mode with a subscript $f$ ($a$) and the excited (ground) state of the atom by $\ket{1_a}$ ($\ket{0_a}$). Under this notation, resonant Jaynes-Cummings Hamiltonian can be expressed as \cite{jaynes1963comparison}
\begin{equation}
H = \hbar \omega N_f + \hbar\omega \sigma_3 + \hbar\lambda \left( \sigma_+ a_f + \sigma_- a_f^{\dagger}\right),
\label{eq:H}\end{equation}
where $N_f = a_f^{\dagger}a_f$ is the number operator of the electromagnetic field-mode with frequency $\omega$, $\sigma_3 = \left(\ket{1_a}\bra{1_a} - \ket{0_a}\bra{0_a}\right)$, $\sigma_+ = \ket{1_a}\bra{0_a}$, $\sigma_- = \ket{0_a}\bra{1_a}$, and $\lambda$ is a coupling constant. The first two terms in (\ref{eq:H}) are the free Hamiltonians of a single-mode electromagnetic field and a two-level atom. Due to the interaction between atom and field, the atom may drop (jump) from the excited (ground) to ground (excited) state by emitting (absorbing) a photon. Neglecting the effect of ambient environment, we can obtain the unitary dynamics of an arbitrary initial atom-field state in the Heisenberg picture.
Interestingly, different initial states will lead to different dynamics of nonclassicality. Here, we discuss the dynamics of nonclassicality for a set of initial states. Specifically, we will provide the analytic expressions of both local nonclassicality and correlations for two initial states, which will be followed by some numerical results for two relatively complex initial states. To begin with we will discuss a simple case where the composite atom-field initial state is $\ket{0_f}\ket{1_a} $, implying that the filed is in vacuum and the atom is in excited state.
\subsection{Case A: Initial Field in Vacuum and Atom in Excited State }\label{sec:A Simple Case}
Under Jaynes-Cummings Hamiltonian (\ref{eq:H}), initial state $\ket{0_f}\ket{1_a}$ evolves as (see \cite{gerry2005introductory} for instance)
\begin{equation}\label{01dynamics}
\ket{0_f}\ket{1_a} \to \cos(\lambda t) \ket{0_f}\ket{1_a} - i \sin(\lambda t) \ket{1_f}\ket{0_a}.
\end{equation}
Thus, time evolution of the corresponding density matrix is
\begin{equation}\label{01dmatrix}
\begin{split}
\rho(T) =& \cos^2(T) \ket{0_f,1_a}\bra{0_f,1_a}+ \sin^2(T) \ket{1_f,0_a}\bra{1_f,0_a} \\
& -\left(i \sin(T)\cos(T) \ket{1_f,0_a}\bra{0_f,1_a} + \text{H.c.}\right),
\end{split}
\end{equation}
where $\text{H.c.}$ stands for Hermitian conjugate, and $T = \lambda t$ is a dimensionless time parameter.
Using this we can write the dynamics of the reduced density matrices for the field and atom as
\begin{eqnarray}
\rho_f(T) = \text{Tr}_{a} \left(\rho(T) \right)= \cos^2(T) \ket{0_f}\bra{0_f} + \sin^2(T) \ket{1_f} \bra{1_f}, \label{01fdmatrix}\\
\rho_a(T) = \text{Tr}_{f} \left(\rho(T) \right)= \sin^2(T) \ket{0_a}\bra{0_a} + \cos^2(T) \ket{1_a}\bra{1_a}, \label{01admatrix}
\end{eqnarray}
by partial tracing the atom and field modes, respectively.
Let us now compute the analytic expressions of negativity to quantify the nonclassicalities present in the composite state $\rho(T)$ and the reduced single-mode states $\rho_{f}(T)$ and $\rho_{a}(a)$.
\subsubsection{Atom-Field Correlations}
We can obtain the partial transposed density matrix from (\ref{01dmatrix}) as
\begin{equation}\label{01dmatrixpt}
\begin{split}
\rho^{\Gamma}(T) =& \cos^2(T) \ket{0_f,1_a}\bra{0_f,1_a}+ \sin^2(T) \ket{1_f,0_a}\bra{1_f,0_a} \\
& + \left(i \sin(T)\cos(T) \ket{0_f,0_a}\bra{1_f,1_a} + \text{H.c.}\right).
\end{split}
\end{equation}
As we know that partial transpose with respect to the atom or field gives the same eigenvalues and hence the same negativity.
The eigenvalues of the partial transposed density matrix are
\begin{equation}\label{eigc}
\lambda^{(c)}_1 = \cos^2(T), \quad \lambda^{(c)}_2 = \sin^2(T),\quad \lambda^{(c)}_{3} = - \lambda^{(c)}_4 = \frac{1}{2}|\sin(2T)|.
\end{equation}
As only $\lambda_4^{(c)}$ is negative, a closed form analytic expression for the atom-field negativity can be obtained as
\begin{equation}\label{Nc}
N_c(T) = |\lambda^{(c)}_4| = \frac{1}{2}|\sin(2T)|.
\end{equation}
If $N_c(T)=0$, the state is separable.
\subsubsection{Field Negativity}
The reduced density matrix of the field is given in Eq. (\ref{01fdmatrix}). If we inject it through one port of a symmetric beam splitter and an auxiliary vacuum $\ket{0_{f_0}}$ through the other port, the output would be
\begin{equation}\label{rfbs}
\rho_{f_{BS}}(T) = U_{BS} \left(\rho_f(T) \otimes \ket{0_{f_0}}\bra{0_{f_0}}\right) U_{BS}^{\dagger},
\end{equation}
where $U_{BS}$ is the unitary action of the beam splitter \cite{prasad1987quantum,thapliyal2019optical} and ${f_0}$ stands for the auxiliary bosonic field.
We can compute $\rho_{f_{BS}}(T)$ as
\begin{equation}\label{fbs}
\begin{split}
\rho_{f_{BS}}(T) = & \cos^2(T) \ket{0_f,0_{f_0}}\bra{0_f,0_{f_0}} + \frac{1}{2}\sin^2(T) \left(\ket{1_f,0_{f_0}}\bra{1_f,0_{f_0}} + \ket{0_f,1_{f_0}}\bra{0_f,1_{f_0}}\right) \\
& -\frac{1}{2}\left({i}\sin^2(T)\ket{0_f,1_{f_0}}\bra{1_f,0_{f_0}} + \text{H.c.}\right)
\end{split}
\end{equation}
{using the properties of beam splitter operation in Schr{\"o}dinger picture}
\begin{equation}
\begin{split}
U_{BS} \ket{1_f}\ket{0_{f_0}} =& \frac{1}{\sqrt{2}}\left(\ket{1_f,0_{f_0}} - i \ket{0_f,1_{f_0}}\right).
\end{split}
\end{equation}
Also, the beam splitter operation does not change the vacuum input states $U_{BS}\ket{0_f}\ket{0_{f_0}} = \ket{0_f}\ket{0_{f_0}}$.
Partial transpose of density matrix (\ref{fbs}) with respect to modes $b$ yields
\begin{equation}\label{fbspt}
\begin{split}
\rho_{f_{BS}}^{\Gamma}(T) &= \cos^2(T) \ket{0_f,0_{f_0}}\bra{0_f,0_{f_0}} + \frac{1}{2}\sin^2(T) \left(\ket{1_f,0_{f_0}}\bra{1_f,0_{f_0}} + \ket{0_f,1_{f_0}}\bra{0_f,1_{f_0}}\right) \\
&- \frac{1}{2}\left( {i}\sin^2(T)\ket{0_f,0_{f_0}}\bra{1_f,1_{f_0}} + \text{H.c.}\right).
\end{split}
\end{equation}
We further compute its eigenvalues as
\begin{equation}\label{eigf}
\begin{split}
\lambda^{(f)}_1 = \lambda^{(f)}_2 = \frac{\sin ^2(T)}{2},\quad
\lambda^{(f)}_j = \frac{\cos^2(T)}{2} -(-1)^j\chi(T),
\end{split}
\end{equation}
where
$\chi(T) = \frac{1}{4}\sqrt{3+\cos(4T)}$ and $j = 3,4$. Thus, the field negativity is obtained as
\begin{equation}\label{Nf}
\begin{split}
N_f(T) = \left|\lambda^{(f)}_4\right|=-\frac{1}{4} \left(1+\cos (2 T)-\sqrt{3+\cos (4 T)}\right),
\end{split}
\end{equation}
as only $\lambda^{(f)}_4$ is negative.
\subsubsection{Atom Negativity}
To obtain atom negativity, we use a beam splitter type operation
\begin{equation}
\begin{split}
U_{BS} \ket{1_a}\ket{0_{a_0}} =& \frac{1}{\sqrt{2}}\left(\ket{1_a,0_{a_0}} - i \ket{0_a,1_{a_0}}\right)
\end{split}
\end{equation}
with an auxiliary mode ${a_0}$, in the same way as used for the field mode. The beam splitter operation in context of mapping the nonclassicality of a quantum state to the entanglement of the output \cite{vogel2014unified} is often discussed for single qubit \cite{miranowicz2015statistical} and spin states \cite{markham2003classicality}, too.
The atomic state after the action of the beam splitter, obtained in the same way as Eq. (\ref{fbs}) with the help of an auxiliary mode ${a_0}$, is
\begin{equation}\label{abs}
\rho_{a_{BS}}(T) = \rho_{f_{BS}}\left(T\pm\frac{\pi}{2}\right),
\end{equation}
where the modes ${f_0}$ and $f$ are substituted by ${a_0}$ and $a$, respectively.
Similarly, the partial transpose of density matrix (\ref{abs}) with respect to the auxiliary mode $c$ is obtained and its eigenvalues are
\begin{equation}\label{eiga}
\lambda^{(a)}_1 = \lambda^{(a)}_2 = \frac{\cos ^2(T)}{2}, \quad \lambda^{(a)}_{i} = \frac{\sin^2(T)}{2} -(-1)^j\chi(T), \quad \text{$j=3,4$.}
\end{equation}
This allows us to quantify the atom negativity in terms of entanglement potential as
\begin{equation}\label{Na}
N_a(T) = \left|\lambda_4^{(a)}\right|=N_f\left(T\pm\frac{\pi}{2}\right).
\end{equation}
\begin{figure}
\includegraphics[scale=0.75]{fig1.pdf}
\caption{(Color online) Atom-field correlations $N_{c}$, field negativity $N_{f}$, and atom negativity $N_{a}$, and total nonclassicality $N_{\rm tot}$ are shown as a function of the dimensionless time parameter $T=\lambda t$ for the field initially in vacuum and atom in the excited state.}
\label{fig1}
\end{figure}
The analytic expressions describing the dynamics of the amount of atom-field correlation, and single-mode nonclassicalities present in the system (cf. Eqs. (\ref{Nc}), (\ref{Nf}), and (\ref{Na})) reveals that the periodic evolution of both the field and atom nonclassicality has a period $\pi$, while correlations have one-half the time period of that of local nonclassicalities. Dynamics of atom-field correlation, field nonclassicality, and atom nonclassicality quantified via corresponding negativity are plotted against $T$ in Fig. \ref{fig1}. It is observed that an exchange of nonclassicality between the field and atom is mediated through the correlations. Interestingly, when the correlations are zero, the atom and field negativities are alternatively maximum, which corresponds to the scenario when the nonclassicality is completely localized in the subsystems. Further, it is observed that even when the correlations are maximum the atom and field negativity potentials are non-zero. The origin of this interesting feature will be discussed later in Subsection \ref{sec:where}.
To to see whether the total nonclassicality remains constant when the chosen initial state evolves under the Jaynes-Cummings Hamiltonian \ref{eq:H}, the total nonclassicality ($N_{\rm tot}(T)=N_c(T)+N_f(T)+N_a(T)$) is plotted in Fig. \ref{fig1}. Jaynes-Cummings dynamics is found to generate an additional nonclassicality as the total nonclassicality is found to be bounded from below by the initial amount of nonclassicality in the system, i.e., $N_{\rm tot}(T)\geq N_{a}(T=0)$.
\subsubsection{Residual Nonclassicalities}
To quantify the amount of nonclassicality in $\rho_j(T):\,j\in\{a,f\}$, we mapped it using a linear optical tool, a beam splitter as it cannot enhance the amount of nonclassicality ($\mathcal{N}$) \cite{vogel2014unified}, to the amount of entanglement in $\rho_{j_{BS}}(T)$. However, a more general systematic study should be the amount of nonclassicality $\mathcal{N}^{\rm in}$ in the input of the beam splitter should be equal to the amount of nonclassicality $\mathcal{N}^{\rm out}$ in the output of the beam splitter. We know that $\mathcal{N}^{\rm in}_j(T)=\mathcal{N} \left(\rho_j(T) \right)$ as the auxiliary mode ${j_0}$ is initially in vacuum ($\mathcal{N} \left(\rho_{j_0}(T) \right)=0$ and not correlated with $\rho_j(T)$. However, $\mathcal{N}^{\rm out}_j(T)= \mathcal{E} \left(\rho_{j_{BS}}(T) \right)+\mathcal{N} \left(\rho_{j^{\prime}}(T) \right)+\mathcal{N} \left(\rho_{j_0^{\prime}}(T) \right)$, where $\rho_{j^{\prime}}(T)=\text{Tr}_{j_0} \left(\rho_{j_{BS}}(T) \right)$ and $\rho_{j_0^{\prime}}(T)=\text{Tr}_{j} \left(\rho_{j_{BS}}(T) \right)$ are the reduced single-mode states at the output of the beam splitter, and $\mathcal{E} \left(\rho_{j_{BS}}(T) \right) $ is the amount of entanglement in $\rho_{j_{BS}}(T)$ which is already calculated as $\mathcal{E} \left(\rho_{j_{BS}}(T) \right)= N_j(T)$. {In brief, we obtain the reduced states after the beam splitter as $\left\{\rho_j(T),\rho_{j_0}(T) \right\} \xrightarrow{U_{BS}} \left\{\rho_{j_{BS}}(T) \right\}\xrightarrow{{\mathrm{Partial\, trace}}} \left\{\rho_{j^{\prime}}(T),\rho_{j_0^{\prime}}(T) \right\}$.}
We calculate the local nonclassicalities in the reduced single mode outputs of the beam splitter applied on the field mode $\mathcal{N} \left(\rho_{f^{\prime}}(T) \right)$ and
$\mathcal{N} \left(\rho_{f_0^{\prime}}(T) \right)$ in terms of entanglement potential. Here, from Eq. (\ref{fbs}) we have
\begin{eqnarray}
\rho_{f^{\prime}}(T) = \left(\cos^2(T) + \frac{1}{2}\sin^2(T)\right) \ket{0_f}\bra{0_f} + \frac{1}{2}\sin^2(T)\ket{1_f}\bra{1_f}, \label{ff}\nonumber
\\
\rho_{f_0^{\prime}}(T) = \left(\cos^2(T) + \frac{1}{2}\sin^2(T)\right)\ket{0_{f_0}}\bra{0_{f_0}} + \frac{1}{2}\sin^2(T)\ket{1_{f_0}}\bra{1_{f_0}}. \label{fb}
\end{eqnarray}
This yields $ \rho_{f^{\prime}_{BS}}(T)$ the state after the beam splitter operation on $\rho_{f^{\prime}}(T)$, obtained in the same way as Eq. (\ref{fbs}) with the help of an auxiliary mode ${f_{10}}$, as \begin{equation}\label{ffbs}
\begin{split}
\rho_{f^{\prime}_{BS}}(T) & = \left(\cos^2(T) + \frac{1}{2}\sin^2(T)\right)\ket{0_f,0_{f_{10}}}\bra{0_f,0_{f_{10}}}
+ \frac{1}{4}\sin^2(T) \left(\ket{1_f,0_{f_{10}}}\bra{1_f,0_{f_{10}}} + \ket{0_f,1_{f_{10}}}\bra{0_f,1_{f_{10}}}\right) \\
& -\frac{1}{4}\left({i}\sin^2(T)\ket{0_f,1_{f_{10}}}\bra{1_f,0_{f_{10}}} + \text{H.c.}\right).
\end{split}
\end{equation}
It is worth noting here that $\rho_{f_0^{\prime}}(T)$ and $\rho_{f^{\prime}}(T)$ have the same form in Eq. (\ref{fb}). Consequently, the amount of nonclassicality will be the same.
We compute the eigenvalues of the partial transpose of density matrices $\rho_{f_0^{\prime}}(T)$ and $\rho_{f^{\prime}}(T)$ as
\begin{equation}\label{eigff}
\begin{split}
\lambda^{(f_1)}_1 = \lambda^{(f_1)}_2 = \frac{\sin ^2(T)}{4},\quad
\lambda^{(f_1)}_j =\frac{1}{8} \left(3+\cos (2 T) -(-1)^j\xi(T)\right),
\end{split}
\end{equation}
where $\xi(T) = \sqrt{11+4 \cos (2 T)+\cos (4 T)}$ and $j = 3,4$. This gives us the nonclassicality in the states $\rho_{f_0^{\prime}}(T)$ and $\rho_{f^{\prime}}(T)$ as
\begin{equation}\label{Nff}
\begin{split}
N_{f_1}(T) = \left|\lambda^{({f_1})}_4\right|=-\frac{1}{8} \left(3+\cos (2 T)-\xi(T)\right),
\end{split}
\end{equation}
as only $\lambda^{(f_1)}_4$ is negative.
\begin{figure}
\begin{centering}
\subfloat[]{\begin{centering}
\includegraphics[scale=0.4]{bslayers_v3.pdf}
\par\end{centering}
}\subfloat[]{\begin{centering}
\includegraphics[scale=0.75]{fig2b.pdf}
\par\end{centering}
}
\par\end{centering}
\caption{(Color online) (a) Additional layers of beam splitter used to quantify the nonclassicality and to deplete the residual nonclassicality. Here, $\rho_0$ denotes the density matrix of single mode vacuum and $j\in\{a,f\}$. (b) Total amount of nonclassicality $N_{\rm tot}^{(l)}$ quantified after $l$ layers of beam splitters. The thin (black) line corresponds to $N_{\rm tot}^{(\infty)}$ in (b). }
\label{fig3}
\end{figure}
Interestingly, $N_{f_1}(T)\neq0$ for an arbitrary value of $T$ leads us to conclude that $\mathcal{N}^{\rm out}_f(T)\geq N_f(T)$ as we have obtained $\mathcal{N} \left(\rho_{j^{\prime}}(T) \right)=\mathcal{N} \left(\rho_{j_0^{\prime}}(T) \right)=N_{f_1}(T)$. We call this nonclassicality in the reduced single mode output states as residual nonclassicality \cite{ge2015conservation}. Adopting the similar mechanism for the reduced single mode outputs of the beam splitter applied on the atomic state we obtain $\mathcal{N} \left(\rho_{a^{\prime}}(T) \right)$, which is the same as $\mathcal{N} \left(\rho_{a_0^{\prime}}(T) \right)$, in terms of entanglement potential which gives us
\begin{equation}\label{Naa}
N_{a_1}(T) =N_{f_1}\left(T\pm\frac{\pi}{2}\right).
\end{equation}
However, the result so far allow us to conclude that $\mathcal{N}^{\rm out}_j(T)\geq N_j(T)+2 N_{j_1}(T)$ as we cannot discard the residual nonclassicality in the reduced single mode states of $\rho_{j^{\prime}_{BS}}(T)$ and the rest of the modes. For this reason, we can add a new layer of beam splitters as shown in Fig. \ref{fig3} (a) until the local nonclassicality in the outputs of the beam splitters seizes.
Further, the total nonclassicality after two layers of beam splitters can be written as
\begin{equation}\label{N2}
N_{\rm tot}^{(2)}(T) = N_c (T)+ N_f (T)+ N_a(T) + 2 N_{f_1}(T) + 2 N_{a_1}(T).
\end{equation}
The variation of $N_{\rm tot}^{(2)}(T)$ in Fig. \ref{fig3} (b) clearly shows that $N_{\rm tot}^{(2)}(T)>N_{\rm tot}(T)\,\forall T$. This motivated us to obtain the total nonclassicality $N_{\rm tot}^{(l)}(T)$ obtained as sum of nonclassicality after $l$ layers of beam splitters. We obtained the analytic expressions for $N_{\rm tot}^{(l)}(T)$ upto $l=4$ and reported the variation in Fig. \ref{fig3} (b). We can observe that $N_{\rm tot}^{(l)}(T)> N_{\rm tot}^{(l-1)}(T)$ and the change, i.e., $N_{\rm tot}^{(l)}(T)-N_{\rm tot}^{(l-1)}(T)$, at the higher values of $l$ is less prominent as compared to that for the smaller number of s layers of beam splitter used.
We have already mentioned that due to symmetry of the system we observed that the amount of residual nonclassicality $N_{a_{l}}(T)\, \left(N_{f_{l}}(T) \right)$ in all the atomic (field) modes after $l+1$ layers of beam splitters is the same. Further, the amount of nonclassicality is also observed to follow $N_{a_{l}}(T)=N_{f_{l}}\left(T\pm\frac{\pi}{2} \right)$. We also observed that nonclassicality depletes approximately by a factor of one-fifth, i.e., $N_{j_{l}}(T)\approx5N_{j_{l-1}}(T)$.
Using this we can obtain the total nonclassicality (after an infinitely large number of beam splitter layers) approximately using the sum of geometric series as
\begin{equation}\label{Ninf}
N_{\rm tot}^{(\infty)}(T) \approx N_c (T)+ \sum\limits_{n=0}^{\infty} \left(\frac{2}{5}\right)^n \left(N_f (T)+ N_a(T)\right) = N_c (T)+ \frac{5}{3}\left(N_f (T)+ N_a(T)\right).
\end{equation}
\subsubsection{Origin of extra nonclassicality}\label{sec:where}
We have observed in Figs. \ref{fig1} and \ref{fig3} that $N_{\rm tot}^{(l)} (T)\geq N_{a}(T=0)\forall\, T,\, l$. Specifically, when the correlations are maximum $N_{c} \left(T=\frac{ (2 n+1)\pi}{4}\right)= N_{a}(T=0)\,:n\in \mathbb{Z}$, the atom and field negativity potentials were observed to be non-zero, i.e., $N_{j} \left(T=\frac{ (2 n+1)\pi}{4}\right)\neq 0:\,j\in\{a,f\}$.
Interestingly, upon closer examination of the reduced states in Eqs. (\ref{01fdmatrix}) and (\ref{01admatrix}), we can observe that $\rho_f\left(T=\frac{ (2 n+1)\pi}{4}\right)=\rho_a\left(T=\frac{ (2 n+1)\pi}{4}\right)$ are maximally mixed states. The nonclassicality present in the maximally mixed states, obtained here as a statistical mixture of a classical (vacuum) and a nonclassical (single photon Fock) states with equal probability, can be attributed to the presence of highly nonclassical Fock state.
The observed behavior is consistent with the earlier findings of some of the present authors \cite{miranowicz2015statistical} that statistical mixtures of vacuum and single photon states can be highly nonclassical. Thus, as long as $T\neq \frac{n\pi}{2}$, this contribution increases the total nonclassicality in the system.
In this subsection, we have extensively discussed the dynamics of nonclassicalities in the resonant Jaynes-Cummings model considering that the initial state is $ \ket{0_f}\ket{1_a} $. In the following subsections, we will concisely report how the dynamics of nonclassicality changes if we change the initial state.
\subsection{Case B: Initial Field in Fock State and Atom in Ground State}
In Case A, we have studied the case where initially the field was classical (vacuum) and the atom was excited. Here, we assume the field in the (nonclassical) Fock state $\ket{2_f}$ and the atom in the ground state initially. Under the Jaynes-Cummings dynamics, this state evolves as (see \cite{gerry2005introductory} for reference)
\begin{equation}\label{20dynamics}
\ket{2_f}\ket{0_a} \to \cos(T\sqrt{3})\ket{2_f}\ket{0_a} - i \sin(T\sqrt{3}) \ket{1_f}\ket{1_a}.
\end{equation}
Computation of negativity is performed using similar strategy as was adopted for Case A. Specifically, we obtained the atom-field correlation and atom negativity as
\begin{equation}\label{Nf-CB}
\begin{split}
N_c(T) &=\frac{1}{2} \left| \sin \left(2 \sqrt{3} T\right)\right|,\\
N_a(T) &= -\frac{1}{4} \left(1+\cos \left(2 \sqrt{3} T\right)-\sqrt{3+\cos \left(4 \sqrt{3} T\right)}\right),
\end{split}
\end{equation}
respectively. The analytic expression of the field negativity is not included here as the expression is cumbersome and it's difficult to deduce any physical insight from that. We can clearly observe from Eq. (\ref{Nf-CB}) that both the negativity expressions have periodic behavior where $N_c(T)$ and $N_a(T)$ have period $\frac{\pi}{2\sqrt{3}}$ and $\frac{\pi}{\sqrt{3}}$, respectively.
The obtained results are plotted in Fig. \ref{fig7} (a) which illustrates the evolution of the atom-field correlation, the field negativity, and the atom negativity in this case. Initially, all the nonclassicality resides in the field, in contrast to Case A. Further, we can observe that the total initial nonclassicality (which is nonclassicality in the field) in Case B is higher than that observed in Case A (which is nonclassicality in the atom). This is due to the fact that nonclassicality of Fock state $\ket{n}$ is known to increase with $n$. Interestingly, negativity of field has the same period as that of atom negativity, which is consistent with Case A.
When the correlations are zero (for $T=\frac{m\pi}{2\sqrt{3}}$), two possibilities are observed: (i) the atom negativity is zero and the field negativity is maximum (for even values of $m$); corresponding to $\ket{2_f}\ket{0_a}$, and (ii) atom negativity is maximum and the field negativity is $0.5$ (for odd values of $m$); corresponding to $\ket{1_f}\ket{1_a}$. We again observe that nonclassicality is exchanged between the field and atom through the correlations. Also notice that $N_f\left(T=\frac{\left(2k+1\right)\pi}{2\sqrt{3}}\right)=0.5\neq0\,\forall\,k\in\mathbb{Z}$, which corresponds to the case when field negativity is maximum in Case A.
When the atom-field correlations are maximum (for $T=\frac{\left(2k+1\right)\pi}{4\sqrt{3}}\,\forall\,k\in\mathbb{Z}$), the field negativity and the atom negativity are non-zero. Specifically, in this case, the atom is in maximally mixed states of ground and excited state; whereas the field is in the statistical mixture of two nonclassical (single and two photon Fock) states with equal probability. Therefore, the nonclassicalicality in the field in this particular case is less than that observed for pure Fock states $\ket{n=1}$ and $\ket{n=2}$, which are $1/2$ and $(1+2\sqrt{2})/4$, respectively.
\begin{figure}
\begin{centering}
\subfloat[]{\begin{centering}
\includegraphics[scale=0.6]{fig3a.pdf}
\par\end{centering}
}\subfloat[]{\begin{centering}
\includegraphics[scale=0.6]{fig3b.pdf}
\par\end{centering}
}
\par\end{centering}
\caption{(Color online) (a) Atom-field correlations, field negativity potential, and atom negativity potential as functions of $T$ for the initial state $\ket{2_f}\ket{0_a}$. (b) Total nonclassicality for the initial state $\ket{2_f}\ket{0_a}$ up to four layers of beam splitters.}
\label{fig7}
\end{figure}
We also computed the total nonclassicality $N_{\rm tot}^{(l)}(T)$ including the residual nonclassicality depleted using $l\leq 4$ layers of beam splitters. Variation of $N_{\rm tot}^{(l)}(T)$ is shown in Fig. \ref{fig7} (b). The observation here is consistent with that of Case A, i.e., total nonclassicality in the Jaynes-Cummings dynamics is bounded below by the amount of initial nonclassicality in the atom-field system. Further, $N_{\rm tot}^{(\infty)}$ obtained in Case A is not valid in this case due to larger dimension of the truncated Fock space corresponding to the field mode.
We have considered so far both atom and field in the pure states. In what follows, we consider a case with field initially in the mixture of Fock states and other that with the superposition of Fock states. For the sake of comparative study, we assume both the states with the same average photon number.
\subsection{Case C: Initial Field in Thermal State and Atom in Excited State}
Once again we assume all the initial nonclassicality in the atom and consider the field to be in the thermal state initially, which is defined as
\begin{equation}\label{rth}
\rho_{\rm th} = \sum_{n=0}^{\infty} p_ n \ket{n_f}\bra{n_f}=
\sum_{n=0}^{\infty} \frac{\langle n \rangle^n}{\left(\langle n \rangle + 1\right)^{n+1}}\ket{n_f}\bra{n_f},
\end{equation}
where $\langle n \rangle$ is the average photon number of the thermal state.
We can consider a small value of average photon number to truncate the thermal state to a form as a statistical mixture of first few Fock states with the weight defined by $p_ n$ in Eq. (\ref{rth}). For instance, for $\langle n \rangle = 0.01$, $p_ 0=\frac{100}{101}$ and $p_ 1=\frac{100}{101^2}\approx1-p_0$ with $p_ j=0\,\forall\, j>1$, which allows us to truncate the thermal state at $n=2$.
Thus, combined initial state of the atom and field is $\rho_{\rm th}\otimes \ket{1_a}\bra{1_a}$.
We truncate the series at the second term to ensure that the field remains effectively in the three dimensional space ($\ket{0_f}, \ket{1_f}, \ket{2_f}$) under evolution which allows us to perform our study in qubit-qutrit system, where the partial transpose condition is both necessary and sufficient to detect entanglement \cite{horodecki2001separability}. Specifically, when the atom returns to ground state, it transfers the excitation to the field, which is now obtained in photon added thermal state \cite{agarwal1992nonclassical,zavatta2007experimental}. This is one of the significant methods of adding photons in arbitrary quantum states of field to obtain a desired state (see \cite{malpani2021can} and references therein for a detailed discussion).
\begin{figure}
\begin{centering}
\subfloat[]{\begin{centering}
\includegraphics[scale=0.6]{fig4a.pdf}
\par\end{centering}
}\subfloat[]{\begin{centering}
\includegraphics[scale=0.6]{fig4b.pdf}
\par\end{centering}
}
\par\end{centering}
\caption{(Color online) (a) Atom-field correlations, field negativity, and atom negativity as functions of $T$ for field initially in thermal state with average photon number $\langle n \rangle = 0.1$ and atom in the excited state. (b) Total nonclassicality $N_{\rm tot}^{(l)}(T)$ up to $l=4$ layers of beam splitters in this case.}
\label{fig9}
\end{figure}
We can observe that the variation in Fig. \ref{fig9} (a) is similar to Fig. \ref{fig1},
as $p_1\ll p_0$. It can be easily verified from the reduced atom and field density matrices which are computed as
\begin{eqnarray}
\rho_{a}(T) = \left(p_0 \sin^2(T) +p_1 \sin^2(\sqrt{2}T)\right) \ket{0_a}\bra{0_a} + \left(p_0 \cos^2(T) +p_1 \cos^2(\sqrt{2}T)\right) \ket{1_a}\bra{1_a},\nonumber
\\
\rho_{f}(T) =p_0 \cos^2(T)\ket{0_f}\bra{0_f}+ \left(p_0 \sin^2(T) +p_1 \cos^2(\sqrt{2}T)\right) \ket{1_f}\bra{1_f} + p_1 \sin^2(\sqrt{2}T)\ket{2_f}\bra{2_f}. \label{fb-ther}
\end{eqnarray}
For relatively higher values of average photon number (corresponding to higher values of $p_1$ but $p_2\approx 0$), we can observe that unlike in Case A the curves were not periodic due to contribution from both the cases but the qualitative behaviour remained identical. Specifically, the atom and field exchange nonclassicality through the correlations. When the correlations are maximum, the other two remain equal and non-zero.
We also computed the total nonclassicality $N_{\rm tot}^{(l)}(T)$ up to $l=4$ layers of beam splitters in this case, too. We have shown the variation of total nonclassicality in Fig. \ref{fig9} (b), where $N_{\rm tot}^{(\infty)}$ is also illustrated.
\sloppy Interestingly, when the entanglement is maximum, the atomic state (from Eq. (\ref{fb-ther})) can be written as ~$p_0 \left(\ket{0_a}\bra{0_a}+\ket{1_a}\bra{1_a}\right) +p_1 \left(\sin^2(\frac{\pi}{2\sqrt{2}})\ket{0_a}\bra{0_a} +\cos^2(\frac{\pi}{2\sqrt{2}})\ket{1_a}\bra{1_a}\right).$ Therefore, the observed behavior, i.e., the total nonclassicality is not conserved in this case, can be inferred in the same way as in Case A in the limits of $p_1\ll p_0$. Further, we can observe in Eq. (\ref{fb-ther}) that when field nonclassicality is maximized a hole is burned in the photon number distribution at vacuum, which is a characteristic feature of photon added thermal state.
\subsection{Case D: Initial Field in Coherent State and Atom in Excited State}
Yet another classical state with the distinct feature that it is the only pure classical state is a coherent state, which is defined as
\begin{equation}\label{coherent}
\ket{\alpha} =\sum_{n=0}^{\infty} c_n \ket{n}= e^{-\frac{1}{2}|\alpha|^2}\sum_{n=0}^{\infty} \frac{\alpha^n}{\sqrt{n!}} \ket{n}.
\end{equation}
Similar to the thermal state case, we choose a small value for $\alpha$ such that $\langle n \rangle =\left|\alpha\right|^2= 0.01$ so that we may truncate the infinite series at $n=2$. Specifically, we chose $\alpha = 0.1$ and so the initial field has $c_0=e^{-\frac{1}{200}}$ and $c_1=\frac{c_0}{10}$ with $c_2\approx 0$
in Eq. (\ref{coherent}).
When coupled with the excited atom, the field spans an effective three-dimensional space and the partial transpose condition becomes necessary and sufficient.
Dynamics of quantified nonclassicality in this case is obtained numerically and the same is illustrated in Fig. \ref{fig11}. In Fig. \ref{fig11}, we again observe the same dynamics of nonclassiality as we have been observing for the other initial states. When the atom-field correlations are minimum, the field and the atom nonclassicalities are alternatively maximum. When the atom-field correlations are maximum, the field and atom negativity potentials are equal and non-zero. Nonclassicality is exchanged between the field and atom via correlations. However, Cases A-C correspond to cases where the reduced atomic and field states are always mixed state, i.e., all the quantum coherence present in the state is in the bipartite entangled state only. In contrast, in the present case, the initial field has non-zero coherence which is transferred to atom as well. This fact can be easily observed through the analytic expressions of the reduced atom and field density matrices which are computed as
\begin{equation}
\begin{split}
\rho_a(T) = & \left(c_0^2 \sin^2(T) + c_1^2 \sin^2(\sqrt{2}T)\right)\ket{0_a}\bra{0_a} + \left( i c_0 c_1 \cos(\sqrt{2}T)\sin(T)\ket{1_a}\bra{0_a} + \text{H.c.}\right) \\ & + \left(c_0^2\cos^2(T) + c_1^2 \cos^2(\sqrt{2}T)\right)\ket{1_a}\bra{1_a}, \\
\rho_f(T) & = c_0^2\cos^2(T)\ket{0_f}\bra{0_f} + \left(c_0c_1\cos(T)\cos(\sqrt{2}T)\ket{0_f}\bra{1_f} + \text{H.c}\right) + \left(c_0^2\sin^2(T) + c_1^2\cos^2(\sqrt{2}T)\right)\ket{1_f}\bra{1_f} \\ & + \left(c_0c_1 \sin(T)\sin(\sqrt{2}T)\ket{1_f}\bra{2_f} + \text{H.c}\right) + c_1^2\sin^2(\sqrt{2}T)\ket{2_f}\bra{2_f}.
\end{split} \label{eq:reduced-D}
\end{equation}
It can be observed from the above expressions of the reduced atom and field states that in the absence of the off-diagonal terms Eq. (\ref{eq:reduced-D}) reduces to Eq. (\ref{fb-ther}) as $p_j\equiv c_j^2$. The distinct features observed in Fig. \ref{fig11} may be attributed to the quantum coherence in the subsystems. For instance, the reduced atomic state has non-zero quantum coherence in the Fock basis when the atom-field correlation is maximum, i.e., at $T=\pi/4$. Further, analogous to the photon added thermal state, the maximum nonclassicality in the field is obtained when the field is in photon added coherent state. The maximum and minimum of atom and field negativities as well as correlations can be observed at the similar conditions as in Case A-C because at lower average photon number vacuum term plays more significant role than any other non-zero photon term.
The total nonclassicality after one layer of beam splitters is illustrated in Fig. \ref{fig11}. Due to the computational difficulty of the problem we were unable to proceed beyond one beam splitter layer. We observed the Jaynes-Cummings dynamics failing to conserve the amount of nonclassicality in this case due to truncation of coherent state.
\begin{figure}
\includegraphics[scale=0.6]{fig5.pdf}
\caption{(Color online) Atom-field correlations, field negativity, atom negativity, and total nonclassicality for the field initially in coherent state $\ket{\alpha = 0.1}$ and atom in the excited state.}
\label{fig11}
\end{figure}
For all the different initial states studied here, we have observed residual nonclassicalities and the fact that nonclassicality is not conserved (total nonclassicality is not constant) under Jaynes-Cummings dynamics.
\section{conclusion\label{sec:conclusion}}
Nonclassical features associated with a resonant Jaynes-Cummings model (e.g., atom-field correlations, atomic nonclassicality, and field nonclassicality) have been studied for a set of initial states to reveal the interplay between these three different aspects of nonclassicality. The analysis is performed using negativity potential (for single-mode nonclassicality) and negativity (for atom-field correlation), where atom and field negativity are computed using Asboth's criterion which tells us that if a single-mode nonclassical state is mixed with vacuum at a beam splitter then the output state becomes entangled, and the amount of entanglement present in the output state can be used as a quantitative measure of the input single-mode nonclassicality. An analysis of Asboth's criterion performed here has revealed that there exists some residual nonclassicality in the single-mode states at the output of the beam splitter which is not captured by entanglement. We have verified this using additional layers of beam splitters and tried to deplete all the residual nonclassicality. The investigation has revealed that almost all the residual nonclassicality is captured with three layers of beam splitters. Further, the origin of residual nonclassicality has also been investigated and the same is attributed to the Jaynes-Cummings dynamics.
The analysis of dynamics of nonclassicality in the Jaynes-Cummings model has also revealed that nonclassicality is exchanged between the field and atom through atom-field correlations. {Further, it is observed that the reduced states of the atom and field are found to be a statistical mixture of vacuum and Fock states when the atom-field correlations are maximum and the initial states of atom and field have no quantum coherence. However non-zero quantum coherence (in the Fock basis) in the initial atom-field state is transferred to both reduced atom and field states when the atom-field correlation are maximum. In both these situations, when the two-mode nonclassicality (correlation) has a maximum value, the single-mode nonclassicalities do not vanish. Further, it is found that when the correlations vanish, the atom and field negativities are alternatively maximum, implying that the nonclassicality is completely localized in the subsystems in those situations.} These observations motivated us to address the question: Is total nonclassicality conserved under Jaynes-Cummings dynamics? The study has revealed that it is not strictly conserved, but it can be viewed as quasi-conserved. Specifically, we have observed that Jaynes-Cummings dynamics generates an additional nonclassicality and the total nonclassicality is observed to be bounded below by the initial amount of nonclassicality present in the system. In short, the study has revealed many new features associated with the dynamics of nonclassicality present in the Jaynes-Cummings model, and it is hoped that others interested in nonclassicality dynamics will find the approach developed and used here useful for their studies involving other models of matter-field interaction.
\section*{Acknowledgement}
SA thanks José Luis Gómez-Muñoz and Francisco Delgado for their "Quantum" package for \textit{Mathematica}. It helped in the computations. AP acknowledges the support from the QUEST scheme of Interdisciplinary Cyber Physical Systems (ICPS) program of the Department of Science and Technology (DST), India (Grant No.:
DST/ICPS/QuST/Theme-1/2019/14 (Q80)). KT acknowledges support from
ERDF/ESF project `Nanotechnologies for Future' (CZ.02.1.01/0.0/0.0/16 019/0000754).
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train/arxiv
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BkiUfp7xaKgQZUmnaXGu
| 5 | 1 |
\section{INTRODUCTION}
Magnetically induced ferroelectricity or electric control of magnetic ordering has been intensively %
investigated since a colossal magneto-electric (ME) effect was found in TbMnO$_3$.\cite{Kimura_nature} %
Recent experimental studies have discovered a variety of ferroelectric magnetic compounds, %
which might be suitable for use in advanced ME devices. %
These materials are often termed ME multiferroics. %
In order to develop practical device applications, it is essential to determine the microscopic mechanism %
of spin-polarization coupling in these systems. %
So far, there have been two different microscopic models %
for magnetically induced ferroelectricity. %
One is the spin-current model,\cite{Katsura_PRL_2005} %
which predicts that two noncollinearly aligned neighboring spins $\mbox{\boldmath $S$}_i$ and $\mbox{\boldmath $S$}_{i+1}$
generate a local electric dipole moment $\mbox{\boldmath $p$}$ given by %
$\mbox{\boldmath $p$}\propto\mbox{\boldmath $e$}_{i,i+1}\times (\mbox{\boldmath $S$}_i \times \mbox{\boldmath $S$}_{i+1})$, %
where $\mbox{\boldmath $e$}_{i,i+1}$ is the unit vector connecting the two spins. %
This formula predicts macroscopic uniform electric polarization in a magnetic structure with cycloidal spin-components, %
and shows excellent agreement with the experimentally determined relationships between magnetic structures and electric polarization %
in various transition metal oxides, such as TbMnO$_3$,\cite{Kenzelmann_PRL_Spiral} %
Tb$_{1-x}$Dy$_x$MnO$_3$,\cite{Arima_PRL_Spiral} %
Ni$_3$V$_2$O$_8$,\cite{PRL_Ni3V2O8} %
MnWO$_4$,\cite{PRL_MnWO4} %
and CoCr$_2$O$_4$.\cite{Yamasaki_PRL_conical} %
The other microscopic model is the magnetostriction model, which predicts ferroelectricity in collinear-commensurate magnetic structures. %
In this case, the local electric dipole moment is given by %
$C(\mbox{\boldmath $r$})(\mbox{\boldmath $S$}_i\cdot \mbox{\boldmath $S$}_{i+1})$, where $C(\mbox{\boldmath $r$})$ is a constant dependent on %
the local crystal structure and the exchange interactions.
The spontaneous electric polarization observed in the collinear-commensurate magnetic orderings in orthorhombic %
HoMnO$_3$\cite{HoMnO3} %
may be attributed to this model. %
However, there are several ME multiferroic materials whose ferroelectricity cannot be explained by either of these models,\cite{RFMO} %
for example, delafossite multiferroic CuFeO$_2$.\cite{Kimura_CuFeO2,SpinNoncollinearlity,CFAO_Helicity} %
Hence, identification of microscopic spin-polarization coupling in these systems paves the way for %
a new design of multiferroic materials. %
The crystal structure of CuFeO$_2$ is shown in Figs. \ref{structure_phasediagram}(a)-(c). %
Owing to the geometrical frustration in the triangular lattice planes of the magnetic Fe$^{3+}$ ions, %
CuFeO$_2$ exhibits various magnetically ordered phases.\cite{Mitsuda_1991,Mitsuda_2000,Petrenko_2000} %
The ground state of CuFeO$_2$ is a collinear-commensurate 4-sublattice (4SL) antiferromagnetic state. %
A spontaneous electric polarization %
emerging in the direction perpendicular to the hexagonal $c$ axis was discovered in the first field-induced phase.\cite{Kimura_CuFeO2} %
Subsequent studies\cite{Kanetsuki_JPCM,Seki_PRB_2007} %
revealed that the ferroelectric phase is stabilized even under %
zero field by substituting a small amount of nonmagnetic Al$^{3+}$ ions for magnetic Fe$^{3+}$ ions, as seen in Fig. \ref{x-T-H_PhaseDiagram}. %
Recent neutron diffraction measurements in applied field revealed that %
the ferroelectric phase magnetic structure is an antiferromagnetically %
stacked proper helical magnetic structure.\cite{SpinNoncollinearlity} %
The wave vector is incommensurate. %
The helical axis is parallel to the hexagonal [110] direction, as shown in Fig. \ref{structure_phasediagram}(d). %
Hereafter, we refer to the ferroelectric phase as the ferroelectric incommensurate (FEIC) phase. %
Since neither the spin-current model nor the magnetostriction model is able to explain the ferroelectricity in proper helical magnetic %
ordering, we anticipate that another type of spin-polarization coupling is realized in CuFe$_{1-x}$Al$_x$O$_2$. %
\begin{figure}[t]
\begin{center}
\includegraphics[keepaspectratio,clip,width=7.8cm]{Fig1.eps}
\caption{(Color online) (a) Crystal structure of CuFeO$_2$ with the hexagonal basis ($\mbox{\boldmath $a$}_h ,\mbox{\boldmath $b$}_h, \mbox{\boldmath $c$}_h$). (b)-(c) Relationship between the hexagonal basis %
and the monoclinic basis ($\mbox{\boldmath $a$} ,\mbox{\boldmath $b$}, \mbox{\boldmath $c$}$). %
(d) Schematic drawing of the proper helical magnetic structure in CuFe$_{1-x}$Al$_x$O$_2$. }
\label{structure_phasediagram}
\end{center}
\end{figure}
Recently, Arima presented a theoretical consideration on the ferroelectricity in CuFe$_{1-x}$Al$_x$O$_2$ %
suggesting that the variation in the metal-ligand hybridization with spin-orbit coupling is %
relevant to the ferroelectricity in this system. %
Applying a microscopic theory derived by Jia \textit{et al.}\cite{Jia1,Jia2} to a cluster model with a proper helical spin arrangement, %
Arima has predicted the following intrinsic features of the ferroelectricity in this system: %
Feature (i): the direction of the uniform polarization should be parallel to the helical axis of the proper helical magnetic ordering. %
Feature (ii): the spin helicity (i.e., right- or left-handed helical arrangement of spins) should correspond to the polarity of %
the uniform polarization. %
Feature (iii): There must be spatial modulations with wave numbers of $2q$ and $4q$ (where $q$ is the magnetic modulation wave number) %
in the helical-axis components of the local electric polarization vectors. %
The results of recent polarized neutron diffraction measurements on CuFe$_{1-x}$Al$_x$O$_2$ ($x=0.02$) under an applied %
electric field show excellent agreement with features (i) and (ii).\cite{CFAO_Helicity} %
However, the existence of spatial modulations of the local polarization (feature (iii)) has not yet been confirmed. %
Spatial modulations of the local polarization must result in lattice modulations, which can be observed by X-ray diffraction %
measurements. %
The existence of incommensurate lattice modulations has been reported in the field-induced FEIC phase of undiluted CuFeO$_2$ by %
Terada \textit{et al.}\cite{Terada_LatticeModulation} and Ye \textit{et al.}\cite{Ye_CuFeO2} %
However, application of a magnetic field readily induces slight distortion of the magnetic structure %
as well as the uniform magnetization components. %
Actually, higher harmonic magnetic reflections have been detected in the field-induced FEIC phase of CuFeO$_2$.\cite{Mitsuda_2000} %
As discussed by Terada \textit{et al.} in Ref.\onlinecite{Terada_LatticeModulation}, this situation can easily cause additional lattice modulations through magnetostriction. %
In order to elucidate the existence of the lattice modulations predicted by the metal-ligand hybridization model (Feature (iii)), %
it is essential to remove the contribution of these field-induced lattice modulations. %
In the present study, %
we have performed synchrotron radiation X-ray measurements using a CuFe$_{1-x}$Al$_x$O$_2$ ($x=0.0155$) %
sample, which exhibits a FEIC phase under zero field. %
The present measurements reveal %
the existence of $2q$-lattice modulations in the zero-field FEIC phase. %
We have also investigated the magnetic-field dependence of the $2q$ lattice modulations by %
X-ray and neutron diffraction measurements under a field applied along the helical axis. %
In order to identify the origin of the observed $2q$-lattice modulation, %
we calculate the Fourier spectrum of the spatial modulations in the helical-axis component of the %
local electric polarization using %
the microscopic model presented by Arima.\cite{Arima_Symmetry} %
This calculation reveals that the lattice modulation does not originate from a conventional magnetostriction %
but rather from the variation in the metal-ligand hybridization between the magnetic Fe$^{3+}$ ions and the ligand O$^{2-}$ ions. %
As a result, we conclude that the microscopic mechanism of the ferroelectricity in CuFe$_{1-x}$Al$_x$O$_2$ is %
the variation in the metal-ligand hybridization with spin-orbit coupling. %
\begin{figure}[t]
\begin{center}
\includegraphics[keepaspectratio,clip,width=7.8cm]{Fig2.eps}
\caption{(Color online) Schematic drawing of $x$-$H$-$T$ phase diagram of CuFe$_{1-x}$Al$_x$O$_2$.}
\label{x-T-H_PhaseDiagram}
\end{center}
\end{figure}
\begin{figure*}[t]
\begin{center}
\includegraphics[keepaspectratio,clip,width=17cm]{Fig3.eps}
\caption{The X-ray diffraction intensity maps obtained by the $(H,K,0)_h$ reciprocal lattice scans around the reciprocal lattice position of $(2,2,0)_h$ at (a) $T=5.5$ K, (b) $T=10.5$ K, %
(c) $T=11.0$ K and (d) $T=19.2$ K. These intensity maps were measured in a warming process. %
(e) Temperature variations of the lattice constants $a$ and $b$ in the monoclinic notation, which were measured on heating. %
(f) Temperature variations of the intensities of $(0,3,2)$ and $(0,2+2q^{\rm PD},2)$ reflections. The inset shows the diffraction %
profiles of the $(0,2+2q^{\rm PD},2)$ reflections at typical temperatures. The data of (0, 2 + 2qPD, 2) are %
magnified by a factor of 20. }
\label{BL46XU_results}
\end{center}
\end{figure*}
\section{EXPERIMENTAL DETAILS}
A single crystal of CuFe$_{1-x}$Al$_x$O$_2$ with $x=0.0155$ was %
prepared by the floating zone technique \cite{Zhao_FZ}, and was cut into a disk shape. %
The Al-concentration was determined by chemical analysis. %
We performed X-ray diffraction measurements in zero field at the beamline BL46XU in SPring-8. %
The energies of the incident X-ray beams were tuned to 12 keV. %
The sample was mounted in a closed-cycle $^4$He-refrigerator.
The data obtained in this measurement are presented in sec. III. A.
We have also performed X-ray diffraction measurements in zero field and in fields applied along the hexagonal $[110]$ %
(monoclinic $b$ axis) at the beamline BL22XU in SPring-8. %
The sample was mounted in a horizontal-field cryomagnet whose maximum field is 6 T; %
the hexagonal $(hhl)$ plane was used as the scattering plane. %
The data obtained in these measurements are presented in sec. III, B and C. %
Neutron diffraction measurements in a field applied along the hexagonal $[110]$ axis were performed at the two-axis neutron diffractometer E4 %
installed at the Berlin Neutron Scattering Center in Hahn-Meitner Institute.%
The hexagonal ($hhl$) plane was selected as the scattering plane. %
Incident neutrons with wave numbers of 2.44 \AA were obtained by a pyrolytic graphite (002) monochromator. %
An external magnetic field directed along the hexagonal $[110]$ direction was provided by the horizontal field cryomagnet, %
HM-2, whose maximum field is 4 T. %
As described later, CuFe$_{1-x}$Al$_x$O$_2$ with $x=0.0155$ exhibits a symmetry-lowering structural transition from a rhombohedral structure %
to a structure with lower symmetry (probably monoclinic symmetry, but possibly even lower symmetry). Taking into %
account the monoclinic lattice distortions found in the magnetically %
ordered phases of undiluted CuFeO$_2$,\cite{Terada_14.5T,Terada_CuFeO2_Xray,Ye_CuFeO2} %
it is reasonable to employ a monoclinic basis in addition to the conventional hexagonal basis. %
The definitions of these bases are shown in Figs. \ref{structure_phasediagram}(a)-(c). %
We mainly employed the monoclinic notation. %
To distinguish between the two bases, the subscript 'h' has been added to the hexagonal %
notation when referring to modulation wave numbers and reciprocal lattice indices. %
\section{EXPERIMENTAL RESULTS}
\subsection{Characterization of the crystal structures in the magnetically ordered phases}
Before discussing the lattice modulations in the ferroelectric phase, we present the symmetry-lowering structural transition %
corresponding to the magnetic phase transitions in CuFe$_{1-x}$Al$_x$O$_2$ with $x=0.0155$. %
At first, we should review the magnetic phase transitions in the $x=0.0155$ sample.\cite{Terada_x_T} %
As shown in Fig. \ref{x-T-H_PhaseDiagram}, this system exhibits three magnetically ordered phases in zero field.
For all the phases, the magnetic modulation wave vectors are described by $(0,q,\frac{1}{2}) [=(q_h,q_h,\frac{3}{2})_h]$. %
The highest temperature phase and the intermediate phase are the oblique partially disordered (OPD) phase %
and the partially disordered (PD) phase, respectively. %
Both of them have collinear sinusoidal magnetic structures with incommensurate wave numbers. %
The wave number of the OPD phase, $q^{\rm OPD}\sim0.390$, is independent of temperature, while %
that of the PD phase, $q^{\rm PD}$ varies with temperature ($0.404<q^{\rm PD}<0.430$). %
The lowest temperature phase is the FEIC phase, whose modulation wave number is $q^{\rm FEIC}=0.414$.
Figures \ref{BL46XU_results}(a)-(d) show the X-ray diffraction intensity maps around the reciprocal lattice position %
of $(2,2,0)_h$ in the paramagnetic (PM) phase and the magnetically ordered phases. %
As seen in Fig. \ref{BL46XU_results}(d), a single peak assigned as $(2,2,0)_h$ is observed %
at $T=19.2$ K in the PM phase. %
In the PD and FEIC phases, this fundamental peak splits into several peaks, %
as shown in Figs. \ref{BL46XU_results}(a)-(b). %
This suggests that the threefold rotational symmetry along the $c_h$ axis vanishes, %
resulting in a monoclinic (or even lower symmetry) lattice distortion in the PD phase and the FEIC phase. %
The monoclinic lattice distortions are also %
observed in the undiluted ($x=0.00$) system,\cite{Terada_14.5T,Terada_CuFeO2_Xray} %
in which the PD and FEIC phases show up as the thermally induced phase and the first field-induced phase, %
respectively. %
As discussed in the previous studies,\cite{Terada_14.5T,Terada_CuFeO2_Xray,Ye_CuFeO2} these structural transitions are %
due to the bond order induced by the magnetostriction, which leads to lower-symmetry magnetic orderings and lifts the %
macroscopic degeneracy of the magnetic states. %
On the other hand, splitting of the $(2,2,0)_h$ peak was not observed in the OPD phase, as shown in Fig. \ref{BL46XU_results}(c), %
even though the magnetic structure of the OPD phase does not have threefold rotational symmetry along the $c_h$ axis.\cite{Terada_FONDER} %
This suggests that the coherent bond order is not essential to the OPD magnetic ordering. %
Based on the fact that the OPD phase is never observed without nonmagnetic substitution, it is reasonable to propose that the OPD %
magnetic ordering is stabilized by local symmetry breaking due to the site-random magnetic vacancies. %
Using the monoclinic basis, %
we can identify the splitting peaks in the FEIC and PD phases, as shown in Figs. \ref{BL46XU_results}(a) and \ref{BL46XU_results}(b), %
and we can also estimate the temperature variation of the lattice constants $a$ and $b$ in the $x=0.0155$ sample, %
as shown in Fig. \ref{BL46XU_results}(e). %
The differences between the lattice constants in the FEIC (or PD) phase and those in the PM phase in this system are %
comparable with those in the undiluted system.\cite{Terada_CuFeO2_Xray,Terada_14.5T,Ye_CuFeO2} %
These results suggest that the crystal structures of the FEIC and PD phases in the $x=0.0155$ sample %
are almost the same as those in the undiluted system. %
In the present measurements, %
we observed incommensurate superlattice reflection at the reciprocal lattice point of $(0,2.86,2)$ in the PD phase.
This reflection can be assigned as $\mbox{\boldmath $\tau$}_{even}+(0,2q^{\rm PD},0)$, %
where $\mbox{\boldmath $\tau$}_{even}$ is the reciprocal lattice point of $(H,K,L)$ with the condition `$H+K=2n$' ($n$ is an integer). %
We also observed commensurate superlattice reflection at the reciprocal lattice points of $(0,3,2)$ in the FEIC phase.
The temperature variations of the intensities of these reflections are shown in Fig. \ref{BL46XU_results}(f). %
The incommensurate superlattice reflection in the PD phase and the commensurate superlattice reflection in the FEIC phase are also observed %
in the undiluted system,\cite{Terada_14.5T,Ye_CuFeO2,Terada_LatticeModulation} %
and consistently explained by the magnetostriction model proposed by Terada \textit{et al.}\cite{Terada_LatticeModulation} %
\begin{figure*}[t]
\begin{center}
\includegraphics[keepaspectratio,clip,width=17cm]{Fig4.eps}
\caption{(Color online) (a) Illustration of the reciprocal lattice $(0KL)$ zone. %
[(b)-(c)] Diffraction profiles of the (b) $(0K0)$ and (c) $(0K\frac{1}{2})$ reciprocal lattice scans in the FEIC phase. %
Contamination peaks from the Cu sample holder are masked by the shaded areas. %
The inset of (b) shows the magnification around the reciprocal lattice position of $(0,3-2q^{\rm FEIC},0)$. %
[(d)-(e)] The diffraction profiles of (d) $(0,3+2q^{\rm FEIC},0)$ and (e) $(0,5-2q^{\rm FEIC},0)$ reflections. %
(f) Temperature dependences of the integrated intensity of the $(0,3-2q^{\rm FEIC},0)$ reflection and %
the neutron diffraction intensity of the magnetic Bragg reflection corresponding to the FEIC magnetic ordering %
(taken from Ref. \onlinecite{Terada_x_T}), under zero field. }%
\label{BL22XU_LongProfile}
\end{center}
\end{figure*}
\subsection{Lattice modulations in the FEIC phase}
We now focus on the incommensurate lattice modulations in the FEIC phase.
Figures \ref{BL22XU_LongProfile}(b) and \ref{BL22XU_LongProfile}(c) show the X-ray diffraction profiles of the %
$(0,K,0)$ and $(0,K,\frac{1}{2})$ reciprocal lattice scans at $T=4.5$ K in the FEIC phase under zero field. %
As shown in Fig. 4(b) and its inset, a satellite reflection is found at the reciprocal lattice position of (0, 2.172, 0). %
The intensity of this reflection is smaller than that of the $(0,4,0)$ fundamental reflection by a factor $10^{-7}\sim 10^{-8}$. %
This reflection is successfully identified as being $(0,3-2q^{\rm FEIC},0)$.
It should be noted that the experimental resolution in the present measurement can clearly distinguish the difference %
between the $q^{\rm FEIC}$ and $q^{\rm PD}$ phases at $T=4.5$ K $(\sim0.43)$, %
although the high-temperature PD phase coexists with the FEIC phase under zero field cooling in the $x=0.0155$ sample.\cite{Terada_x_T,CFAO_AnisoPD} %
We confirmed the repetition of this reflection in reciprocal lattice space, as shown in the Figs. \ref{BL22XU_LongProfile}(d) and \ref{BL22XU_LongProfile}(e). %
Hereafter, we refer to these reflections as $\mbox{\boldmath $\tau$}_{odd}\pm(0,2q^{\rm FEIC},0)$ reflections, %
where $\mbox{\boldmath $\tau$}_{odd}$ is the reciprocal lattice point of $(H,K,L)$ with the condition `$H+K=2n+1$'. %
The $\mbox{\boldmath $\tau$}_{odd}\pm(0,2q^{\rm FEIC},0)$ reflections are observed only in the FEIC phase. %
However, as shown in Fig. \ref{BL22XU_LongProfile}(f), the temperature variation of the integrated intensity of the $\mbox{\boldmath $\tau$}_{odd}\pm(0,2q^{\rm FEIC},0)$ reflection is not %
proportional to that of the magnetic order parameter; %
specifically, the intensity of the $(0,3-2q^{\rm FEIC},0)$ reflection obviously increases around $T=6.5$ K with increasing temperature, %
and rapidly decreases at the transition temperature from the FEIC phase to the PD phase, $T=8$ K, %
while the FEIC magnetic order parameter (neutron diffraction intensity) monotonically decreases with increasing temperature, as seen in Fig. \ref{BL22XU_LongProfile}(f). %
No significant reflections were detected in the $(0,K,\frac{1}{2})$ reciprocal lattice scan, as shown in Fig. \ref{BL22XU_LongProfile}(c). %
The satellite reflections assigned as $(0,2n\pm q^{\rm FEIC},\frac{1}{2})$ %
and $(0,(2n+1)\pm q^{\rm FEIC},\frac{1}{2})$ were observed in the field-induced FEIC phase of undiluted CuFeO$_2$.\cite{Terada_LatticeModulation} %
These satellite reflections indicate the existence of a lattice modulation with a wave number $1q^{\rm FEIC}$ %
in the field-induced FEIC phase. %
However, these reflections were not observed in the zero-field FEIC phase of CuFe$_{1-x}$Al$_x$O$_2$ with $x=0.0155$. %
This is consistent with the calculation of Terada \textit{et al.},\cite{Terada_LatticeModulation} which shows %
that the combination of the proper helical spin components and uniform magnetization components along the $c_h$ axis %
induces $1q$-lattice modulation through magnetostriction, %
and suggests that the $1q$-lattice modulation is not essential to the ferroelectricity in this system. %
The present results reveal that the $2q$-lattice modulation corresponding to the satellite reflections at %
$\mbox{\boldmath $\tau$}_{odd}\pm(0,2q^{\rm FEIC},0)$ exists in the zero-field FEIC phase. %
The microscopic origin of the $2q$-lattice modulation is discussed in sec. IV. %
\subsection{$H_{\parallel b}$ dependence of the $2q$-lattice modulation}
We also surveyed the magnetic field dependence of the $\mbox{\boldmath $\tau$}_{odd}\pm(0,2q^{\rm FEIC},0)$ reflections by %
the X-ray diffraction measurements under an applied field. %
As mentioned above, application of a magnetic field along the $c_h$ axis readily leads to %
additional lattice modulations through magnetostriction. %
We thus applied a magnetic field along the helical axis of the proper helical magnetic structure, i.e. along the $b$ axis. %
When a magnetic field is applied along this direction, the proper helical magnetic structure %
deforms into a conical magnetic structure, %
as shown in Fig. \ref{H110_dependence}(d). %
In this configuration, the magnetic field ($H_{\parallel b}$) does not modify the amplitude of the magnetic moment at each Fe$^{3+}$ site. %
Hence, the magnetostriction cannot induce additional lattice modulations. %
\begin{figure}[t]
\begin{center}
\includegraphics[keepaspectratio,clip,width=7.8cm]{Fig5.eps}
\caption{(Color online) (a)The $H_{\parallel b}$-dependence of the integrated intensity of $(0,3-2q^{\rm FEIC},0)$ reflection observed in the X-ray diffraction measurements. %
(b) The $H_{\parallel b}$-dependence of $(\frac{1}{2}-q_h^{\rm FEIC},\frac{1}{2}-q_h^{\rm FEIC},\frac{3}{2})_h$ magnetic Bragg reflection observed in the neutron diffraction measurements at $T=4.5$ K. %
(d) The field dependence of the integrated intensity of $(0,3-2q^{\rm FEIC},0)$ reflection normalized by the data of the neutron %
diffraction measurements. %
(d) Schematic drawings of a conical structure with uniform magnetization %
component along the helical axis. %
[(e)-(f)] Schematic drawings of the fractions of three magnetic domains (e) under zero field, and (f) under applied field along %
the $b$ axis ($[110]_h$ axis). The directions of the filled arrows denote the $[001]_h$ projections of the propagation wave vectors. %
The sizes of arrows qualitatively show the fractions of each domain. }
\label{H110_dependence}
\end{center}
\end{figure}
Figure \ref{H110_dependence}(a) shows the integrated intensity of the $(0,3-2q^{\rm FEIC},0)$ reflection at $T=4.5$ K as a function of %
the magnetic field along the $b$ axis ($H_{\parallel b}$). %
The intensity of the reflection increases with increasing magnetic field. %
This result leads to the following two possibilities: %
that the amplitude of the $2q$-lattice modulation itself increases with increasing magnetic field, or %
the fractions of the three magnetic domains, which reflect the threefold rotational symmetry of the original trigonal crystal structure, %
change with magnetic field intensity. %
We have also performed $(0,K,0)$ and $(0,K,\frac{1}{2})$ reciprocal lattice scans at $T=4.5$ K under $H_{\parallel b}=6$ T. %
In both these scans, no additional (field-induced) reflections are detected. %
\subsection{Neutron diffraction measurements under applied field along the $b$ axis}%
In order to elucidate the origin of the $H_{\parallel b}$-variation of the intensity of the $(0,3-2q^{\rm FEIC},0)$ reflection, %
we performed neutron diffraction measurements under an applied field along the $b$ axis.
We found that the magnetic Bragg reflections corresponding to the FEIC %
magnetic ordering increase with increasing magnetic field.\cite{Comment1} %
Figure \ref{H110_dependence}(b) shows the field dependence of the intensity of the $(\frac{1}{2}-q_h^{\rm FEIC},\frac{1}{2}-q_h^{\rm FEIC},\frac{3}{2})_h$ magnetic %
Bragg reflection, where $q_h^{\rm FEIC}[=\frac{1}{2}q^{\rm FEIC}]=0.207$. %
Since the change in the magnetic structure factor is considered to be negligible for magnetic fields of $0<H_{\parallel b}<4$ T,\cite{CFAO_AnisoPD} %
this enhancement of the magnetic Bragg intensity indicates that the fraction of the FEIC magnetic domain increases with the propagation %
wave vector $(q_h^{\rm FEIC},q_h^{\rm FEIC},\frac{3}{2})_h$.
This result supports the latter scenario for the enhancement of the intensity of the $(0,3-2q^{\rm FEIC},0)$ reflection.
A magnetic field along the $b$ axis ([110]$_h$ axis) should favor the magnetic domain in which a more uniform magnetization component %
is induced along the direction of the magnetic field. %
As a result, the fraction of the magnetic domains with a wave vector $(q_h,q_h,\frac{3}{2})_h$ is enhanced %
by the magnetic field along the $[110]_h$ axis, and the fractions of the other magnetic domains with the wave vectors $(-2q_h,q_h,\frac{3}{2})_h$ and $(q_h,-2q_h,\frac{3}{2})_h$ %
should be reduced, as shown in Figs. \ref{H110_dependence}(e) and \ref{H110_dependence}(f). %
The details of the magnetic domain distributions and the magnetic phase transitions in CuFe$_{1-x}$Al$_x$O$_2$ ($x=0.0155$) %
under applied field along the $[110]_h$, $[1\bar{1}0]_h$ and $[001]_h$ directions will be presented in another paper.\cite{CFAO_AnisoPD} %
Figure \ref{H110_dependence}(c) shows the intensity of the $(0,3-2q^{\rm FEIC},0)$ reflection normalized to the intensity of the magnetic Bragg reflection. %
In the magnetic field region of $H_{\parallel b}\le4$ T, the amplitude of the $2q$-lattice modulation does not exhibit remarkable change, %
and it decreases slightly above $H_{\parallel b}=3$ T. %
\begin{figure}[b]
\begin{center}
\includegraphics[keepaspectratio,clip,width=7.8cm]{Fig6.eps}
\caption{(Color online) (a) A Fe$_4$O$_2$ cluster used for the calculation of the local electric dipole moment in Ref.\onlinecite{Arima_Symmetry}. %
(b) The magnetic structure in the FEIC phase with the $a\times b\times 2c$ cell. %
(c) The phase differences in the magnetic modulations ($\Delta \Phi_{i,j}^{mag}$) between the Fe$^{3+}$ sites. }
\label{unit_cell}
\end{center}
\end{figure}
\section{DISCUSSIONS}
\subsection{Review of Arima's cluster model}
In this section, we discuss whether the microscopic origin of the $2q$-lattice modulation observed in the FEIC phase %
is the variation in the $d$-$p$ hybridization or not. %
At first, we start from the Fe$_4$O$_2$ cluster model used in Ref. \onlinecite{Arima_Symmetry} (see Fig. \ref{unit_cell}(a)). %
Applying the microscopic theory by Jia \textit{et al.}\cite{Jia1,Jia2} to the Fe-O covalent bonding, %
Arima pointed out that the $d$-$p$ hybridization between a magnetic Fe$^{3+}$ ion and a ligand O$^{2-}$ ion is %
slightly modified depending on the direction of the magnetic moment at the Fe$^{3+}$ site.\cite{Arima_Symmetry}
As a result, the charge transfer from the magnetic Fe$_i$ ($i=1,2,3,4$) site to the neighboring O$_j$ ($j=1,2$) site %
can be described by %
\begin{eqnarray}
C_0 + \Delta C\cos2\Theta_{i,j},
\label{2C_eq}
\end{eqnarray}
where $C_0$ is the charge transfer in the PM phase, and $\Delta C$ %
is the constant dependant on the difference in the spin-orbit interaction between Fe$_i$ and O$_j$, and %
the magnitude of the ordered magnetic moment at the Fe site, and so on. %
$\Theta_{i,j}$ is the angle between the direction of the Fe$_i$-O$_j$ bond and the direction of the magnetic moment at the Fe$_i$ site. %
The macroscopic uniform electric polarization is not induced by this term only. %
Arima argued that the many-body effect among the three covalent Fe-O bonds which are concerned with an O$^{2-}$ ion, should be %
taken into account in Eq. (\ref{2C_eq}). %
As shown in Fig. \ref{unit_cell}(a), an O$^{2-}$ ion in this cluster (or CuFeO$_2$) is surrounded by three neighboring Fe$^{3+}$ ions. %
Hence, the charge transfer between Fe$_i$ and O$_j$ should be slightly affected by the amount of charge transfer in the other two Fe-O bonds %
which are concerned with the O$_j$ site. %
For example, the covalency between O$_1$ and Fe$_1$ should be reduced when Fe$_2$-O$_1$ and %
Fe$_4$-O$_1$ bonds are more covalent. %
Taking account of this many-body effect, the charge transfer from Fe$_1$ to O$_1$ can be described as follows:\cite{Comment2} %
\begin{eqnarray}
C_0[1+C'\cos2\Theta_{1,1}][1-\alpha C'(\cos2\Theta_{2,1}+\cos2\Theta_{4,1})],
\label{Eq_CT}
\end{eqnarray}
where $C'=\frac{\Delta C}{C_0}(\ll1)$. $\alpha$ is the parameter representing the efficiency of the many-body effect, and is assumed to be small %
$\alpha \ll 1$. %
Here, we apply to the cluster, a proper helical magnetic structure with a modulation wave number $q$, whose helical axis is parallel to the Fe$_1$-Fe$_4$ bond direction. %
The helical-axis component of the induced electric dipole moment at the O$_1$ site, $p_b$, %
should be proportional to the imbalance between the charge transfer from Fe$_1$ to O$_1$ and that from Fe$_4$ to O$_1$, %
\begin{eqnarray}
p_b&\propto& (1+\alpha)(\cos2\Theta_{1,1}-\cos2\Theta_{4,1})\nonumber\\
&&-\alpha C'\cos2\Theta_{2,1}(\cos2\Theta_{1,1}-\cos2\Theta_{4,1}).
\label{Eq_Pb}
\end{eqnarray}
In the case of the proper helical magnetic structure, the value of $\cos2\Theta_{i,j}$ oscillates with a period of half the magnetic modulation. %
As a result, the first term of the Eq. (\ref{Eq_Pb}) gives the nonuniform polarization oscillating with a wave number of $2q$ ($2q$-modulation), %
and the second term of Eq. (\ref{Eq_Pb}) gives the uniform polarization and the nonuniform polarization oscillating with a wave number of $4q$ %
($4q$-modulation). %
It should be noted that one can derive Eq. (6) in Ref. \onlinecite{Arima_Symmetry} from Eq.(\ref{Eq_Pb}) in this paper. %
The above formula of the local electric polarization suggests that the amplitude of the $4q$-modulation is much smaller than that of the $2q$-modulation. %
In the limit of $\alpha = 0$, the uniform polarization and $4q$-modulation vanish. %
It is worth mentioning here that the same amount of electric dipole moment is induced at the O$_2$ site, %
because of the symmetry of the proper helical magnetic structure.
\begin{figure}[b]
\begin{center}
\includegraphics[keepaspectratio,clip,width=7.8cm]{Fig7.eps}
\caption{(Color online) [(a)-(b)] $\delta_{mag}$ dependence of the difference between the phases of (a) $2q$- and (b) $4q$-modulations. %
[(c)-(d)] The phase differences of the (c) $2q$- and (d) $4q$-components in the spatial modulations in the local electric polarization. }
\label{Fourier_calc}
\end{center}
\end{figure}
\subsection{Calculation results in zero field}
We apply Eq. (\ref{Eq_Pb}) to the magnetic structure in the FEIC phase, which has been determined by a previous magnetic structure analysis.\cite{SpinNoncollinearlity} %
For simplicity, hereafter, we refer to the wave number and the wave vector in the FEIC phase as $q$ and $\mbox{\boldmath $q$}[=(0,q,\frac{1}{2})]$, %
respectively. %
In order to define the spin arrangement in the FEIC phase, we employ $a\times b\times 2c$ cell, as shown in Fig. \ref{unit_cell}(b). %
This cell contains four Fe$^{3+}$ sites, and the fractional coordinates of these sites are:
\begin{eqnarray}
\mbox{\boldmath $d$}_1 &=& (0,0,0)\nonumber\\
\mbox{\boldmath $d$}_2 &=& (0,1/2,0)\nonumber\\
\mbox{\boldmath $d$}_3 &=& (0,0,1)\nonumber\\
\mbox{\boldmath $d$}_4 &=& (0,1/2,1).
\end{eqnarray}
From the results of a previous magnetic structure analysis, the spin components at $\mbox{\boldmath $d$}_i$ site are described, %
using the Cartesian coordinates shown in Fig. \ref{unit_cell}(c), as follows:
\begin{eqnarray}
S_i^x &=& \mu_x \cos(2\pi\mbox{\boldmath $q$}\cdot(\mbox{\boldmath $l$}+\mbox{\boldmath $d$}_i) - \phi_i)\nonumber\\
S_i^y &=& 0 \nonumber\\
S_i^z &=& \mu_z \sin(2\pi\mbox{\boldmath $q$}\cdot(\mbox{\boldmath $l$}+\mbox{\boldmath $d$}_i) - \phi_i)
\label{S_component}
\end{eqnarray}
$\mu_x$ and $\mu_z$ are the magnetic moments along the $x$ and $z$ axes, respectively. %
Since no significant ellipticity has been detected in the magnetic structure analysis for zero field (and for relatively low fields, $H_{\parallel[001]_h}<4$T) on %
the $x=0.0155$ sample,\cite{SpinNoncollinearlity} we assume at this stage that the magnetic structure has no ellipticity, i.e., $\mu_x = \mu_z=\mu$. %
$\phi_i$ is the relative phase shift at the $\mbox{\boldmath $d$}_i$ site; specifically,
\begin{eqnarray}
|\phi_1-\phi_2|&=&|\phi_3-\phi_4|=\delta_{mag}\nonumber\\
|\phi_1-\phi_3|&=&|\phi_2-\phi_4|=0
\end{eqnarray}%
Although $\delta_{mag}$ was determined to be $\sim76^{\circ} (\sim \pi q)$ by magnetic structure analysis,\cite{SpinNoncollinearlity} %
we treat $\delta_{mag}$ as a parameter in the following calculation.
Here, we define the phase difference of the magnetic modulations between the $\mbox{\boldmath $d$}_i$ site and the $\mbox{\boldmath $d$}_j$ site as $\Delta\Phi_{i,j}^{mag}$. %
Using this notation, the relationship among the phases of the magnetic modulations on the four Fe$^{3+}$ sites is %
summarized in Fig. \ref{unit_cell}(c). %
\begin{figure}[t]
\begin{center}
\includegraphics[keepaspectratio,clip,width=6.9cm]{Fig8.eps}
\caption{(Color online) Fourier spectrum mappings of the spatial modulations in the local polarization, which are calculated using %
(a) the $d$-$p$ hybridization model with $\delta_{mag}=0,\pi$, %
(b) that with $\delta_{mag}=76^{\circ}$, and %
(c) the magnetostriction model with finite ellipticity. %
The size of a symbol, except for $\mbox{\boldmath $\tau$}_{even}$ and $\mbox{\boldmath $\tau$}_{odd}$, qualitatively shows the intensity of the spectrum $|P(\mbox{\boldmath $Q$})|^2$, %
which is normalized by the intensity of the most strongest peak in the spectrum.
The black arrows denote the position of the satellite reflections observed in the zero-field FEIC phase. }
\label{Fourier_map}
\end{center}
\end{figure}
We numerically calculated the spatial modulations of $p_b$ at each oxygen site using Eq. (\ref{Eq_Pb}). %
The system size is set to be $a\times Nb\times2c$. %
Since the magnetic propagation wave number along the $b$ axis is incommensurate, $N$ is set to be a large number (typically, $N\sim100$). %
We also calculated Fourier spectra for their nonuniform components. %
Here, we define the phase difference in the $nq$-modulation ($n=2,4$) between the oxygen site neighboring $\mbox{\boldmath $d$}_i$ site and that neighboring $\mbox{\boldmath $d$}_j$ site, %
as $\Delta\Phi_{i,j}^{nq}$, in the same manner as that of the magnetic modulations. %
The results of the calculations for $\Delta\Phi_{i,j}^{2q}$ and $\Delta\Phi_{i,j}^{4q}$ are shown in Figs. \ref{Fourier_calc}(a) and \ref{Fourier_calc}(d). %
For the $2q$-modulations, there is a finite phase shift, $2\delta_{mag}$, between the oxygen %
site neighboring the $\mbox{\boldmath $d$}_1$ ($\mbox{\boldmath $d$}_3$) %
site and that neighboring the $\mbox{\boldmath $d$}_2$ ($\mbox{\boldmath $d$}_4$) site. %
For the $4q$-modulation, there are no phase shifts along the $a$ axis. %
For both of the $2q$- and $4q$-modulations, there are also no phase shifts along the $c$ axis. %
These results are summarized in Figs. \ref{Fourier_calc}(c) and \ref{Fourier_calc}(d). %
The intensities of the calculated Fourier spectra for the spatial modulations of $p_b$, $|P(\mbox{\boldmath $Q$})|^2$, %
where $\mbox{\boldmath $Q$}$ is a vector in the reciprocal lattice space, are mapped onto the reciprocal lattice space, %
as shown in Figs. \ref{Fourier_map}(a) and \ref{Fourier_map}(b). %
Since the $d$-$p$ hybridization model predicts that the local electric polarization arises from the local imbalance of the Fe-O bond-covalency, %
it is reasonable to consider that the spatial modulations of the local electric polarization result in spatial modulations of %
the local atomic displacements of the oxygen ions. %
We thus assume that the local displacement of the oxygen ion is proportional to the magnitude of the local polarization vector. %
In this case, $|P(\mbox{\boldmath $Q$})|$ is proportional to the structure factor in the X-ray diffraction measurements. %
In the case of $\delta_{mag}=0,\pi$, satellite peaks corresponding to the $2q$- and $4q$-modulations %
appear at $\mbox{\boldmath $\tau$}_{even}\pm(0,2q,0)$ and $\tau_{even}\pm(0,4q,0)$, respectively.
In the case of $\delta_{mag}\ne0,\pi$, satellite peaks corresponding to the $2q$-modulation appear,
in addition to the above peaks, at $\mbox{\boldmath $\tau$}_{odd}\pm(0,2q,0)$.
By substituting $\delta_{mag}=76^{\circ}$, the ratio of the intensities of the spectra, $|P(\mbox{\boldmath $\tau$}_{even}\pm(0,2q,0))|^2$ %
and $|P(\mbox{\boldmath $\tau$}_{odd}\pm(0,2q,0))|^2$ is calculated to be %
\begin{eqnarray}
\frac{|P(\mbox{\boldmath $\tau$}_{even}\pm(0,2q,0))|^2}{|P(\mbox{\boldmath $\tau$}_{odd}\pm(0,2q,0))|^2}=0.062.
\end{eqnarray}
This suggests that in X-ray measurements, the satellite reflections corresponding to the $2q$-modulation should be mainly observed at %
the reciprocal lattice position of $\mbox{\boldmath $\tau$}_{odd}\pm(0,2q,0)$. %
This shows good agreement with the results of the present measurements. %
Although reflections at $\mbox{\boldmath $\tau$}_{even}\pm(0,2q,0)$ and $\mbox{\boldmath $\tau$}_{even}\pm(0,4q,0)$ were not observed in the present measurements, %
this can be ascribed to the S/N of the present measurements, because the intensities of those reflections are %
estimated to be much smaller than that of $\mbox{\boldmath $\tau$}_{odd}\pm(0,2q,0)$ reflections. %
It should be noted that this result does not change in the parameter region of $0<\alpha<1$, %
although we assumed $\alpha = 0.1$ in the above calculation. %
This is because the value of $\alpha$ affects only the ratio between the amplitude of the $2q$-modulation and that of the $4q$-modulation (and uniform polarization); %
specifically, $|P(\mbox{\boldmath $\tau$}_{even}\pm(0,4q,0))|^2/|P(\mbox{\boldmath $\tau$}_{odd}\pm(0,2q,0))|^2$ %
is in the order of $\sim\alpha^2 C'^2 (\ll1)$
It is worth mentioning here that neither the magnetostriction model nor the spin-current model explains %
the satellite reflections at $\mbox{\boldmath $\tau$}_{odd}\pm(0,2q,0)$, even if the magnetic structure has finite ellipticity. %
The spin-current term, $\mbox{\boldmath $e$}_{i,i+1}\times (\mbox{\boldmath $S$}_i \times \mbox{\boldmath $S$}_{i+1})$, %
does not produce the helical-axis component of the local polarization vector, in the proper (or elliptic) helical magnetic ordering in this system. %
Therefore, the lattice modulations induced by this term do not contribute to the satellite reflection in the reciprocal lattice position of $(0,K,0)$. %
The magnetostriction term, ($\mbox{\boldmath $S$}_i\cdot \mbox{\boldmath $S$}_{i+1}$), produces the spatial modulation with a wave number of $2q$ %
in the helical-axis components of the local polarization vectors, when the magnetic structure has finite ellipticity.\cite{Terada_LatticeModulation} %
However, as discussed by Terada \textit{et al.},\cite{Terada_LatticeModulation} %
the satellite reflections corresponding to the magnetostriction-induced $2q$-modulation are observed only at %
the reciprocal lattice position of $\mbox{\boldmath $\tau$}_{even}\pm(0,2q,0)$, regardless of the value of $\delta_{mag}$ (see Fig. \ref{Fourier_map}(c)). %
Therefore, we conclude that the origin of the $2q$-lattice modulation observed in the zero-field FEIC phase is the %
variation in the metal-ligand hybridization with spin-orbit coupling, which corresponds to the feature (iii) mentioned in the introduction. %
We should mention the non-monotonic temperature variation of the intensity of the $\mbox{\boldmath $\tau$}_{odd}\pm(0,2q,0)$ reflection observed in the present measurement (see Fig. \ref{BL22XU_LongProfile}(f)). %
The $d$-$p$ hybridization between the Fe$^{3+}$ ions and the O$^{2-}$ ions must depend on the length of the %
Fe-O bonds as well as the angle between the Fe-O bond direction and the magnetic moment at the Fe site. %
We therefore anticipate that the temperature variations of the lattice constants and the fractional coordinates of the %
oxygen sites affect the temperature variation of the intensity of the $\mbox{\boldmath $\tau$}_{odd}\pm(0,2q,0)$ reflection. %
The precise determination of the structural parameters by high-resolution X-ray %
and/or neutron diffraction measurements are desirable for further clarification. %
\begin{figure}[t]
\begin{center}
\includegraphics[clip,keepaspectratio,width=7.5cm]{Fig9.eps}
\caption{(Color online) (a) $\theta$ dependence of the intensities of the Fourier spectra for the spatial modulations in the local polarization. %
To enhance the visibility, the data of $3q$ and $4q$ are magnified by a factor of 200.
(b) Fourier spectrum mapping of the spatial modulations in the local polarization, which are calculated with %
the $d$-$p$ hybridization model with $\delta_{mag}=76^{\circ}$, $\alpha=0.1$ and $\theta=10^{\circ}$. %
The size of the symbols, except for $\mbox{\boldmath $\tau$}_{even}$ and $\mbox{\boldmath $\tau$}_{odd}$, corresponds to the intensity of the spectrum $|P(\mbox{\boldmath $Q$})|^2$, %
which are normalized using the intensity of the most intense peak in the spectrum.
The black arrows denote the positions of the satellite reflections observed in the present measurements. }
\label{H110_calc}
\end{center}
\end{figure}
\subsection{Calculation results for an applied field }
When a magnetic field is applied along the $b$ axis, the magnetic structure is expected to be a conical structure. %
We also numerically calculated the Fourier spectra for the nonuniform polarizations induced by the conical magnetic structure. %
Figures \ref{H110_calc}(a) and \ref{H110_calc}(b) show the calculation results with the parameters of $\alpha =0.1$ and $\delta_{mag}=76^{\circ}$. %
$\theta$ is the angle defined in the Fig. \ref{H110_dependence}(a). %
With increasing $\theta$, the intensities of the Fourier spectra for 2$q$- and 4$q$-modulations decrease. %
In the region of $0<\theta<\frac{\pi}{2}$, $1q$- and $3q$-modulations emerge from %
the first and second terms of Eq. (\ref{Eq_Pb}), respectively. %
In reciprocal lattice space, the satellite peaks corresponding to the $1q$-modulations appear at %
$\mbox{\boldmath $\tau$}_{even}\pm(0,q,\frac{1}{2})$ and $\mbox{\boldmath $\tau$}_{odd}\pm(0,q,\frac{1}{2})$, and the satellite peaks corresponding to the $3q$-modulations appear at %
$\mbox{\boldmath $\tau$}_{even}\pm(0,3q,\frac{1}{2})$ and $\mbox{\boldmath $\tau$}_{odd}\pm(0,3q,\frac{1}{2})$, as shown in Fig. \ref{H110_calc}(b). %
According to the results of recent magnetization measurements,\cite{PulsMag_CFAO} the canting angle $\theta$ at $H_{\parallel b}=4$T %
is roughly estimated to be $5^{\circ}\sim10^{\circ}$. %
In the region of $\theta<10^{\circ}$, %
$|P(\mbox{\boldmath $\tau$}_{odd}\pm(0,2q,0))|^2$ decreases slightly with increasing $\theta$, and the intensities of the other Fourier spectra are quite small %
compared with $|P(\mbox{\boldmath $\tau$}_{odd}\pm(0,2q,0))|^2$, as shown in Fig. \ref{H110_calc}(a). %
Hence, it is expected that the intensity of the $\mbox{\boldmath $\tau$}_{odd}\pm (0,2q,0)$ satellite reflection %
decreases slightly with increasing applied magnetic field along the $b$ axis. %
This is consistent with the results of the present measurements. %
Although we could not survey the region of $\theta>10^{\circ}$, where remarkable changes in the amplitudes of the lattice modulations %
are expected, %
these results imply that the $d$-$p$ hybridization model still works for a finite magnetic field along the $b$ axis. %
\section{SUMMARY}
In summary, we have performed synchrotron radiation X-ray and neutron diffraction measurements under zero field and %
applied field, on the delafossite multiferroic CuFe$_{1-x}$Al$_x$O$_2$ %
with $x=0.0155$, in which the ferroelectric phase shows up under zero field. %
We found that the threefold rotational symmetry along $c_h$ axis vanishes in the PD and FEIC phases. %
Although the present result does not determine the symmetry of the crystal structure in the PD and FEIC phases, %
it is reasonable to propose that a monoclinic lattice distortion occurs in the FEIC and PD phases in CuFe$_{1-x}$Al$_x$O$_2$ with $x=0.0155$,
because the monoclinic lattice distortion is observed in the PD and FEIC phases in undiluted CuFeO$_2$.\cite{Terada_CuFeO2_Xray,Terada_14.5T,Ye_CuFeO2} %
On the other hand, monoclinic lattice distortion was not observed in the OPD phase, which never shows up %
without nonmagnetic substitution. %
This implies that the OPD magnetic ordering is stabilized not by the coherent bond order due to magnetostriction, %
but by the local symmetry breaking due to site-random magnetic vacancies. %
In the FEIC phase, we found satellite reflections identified as $\mbox{\boldmath $\tau$}_{odd}\pm (0,2q^{\rm FEIC},0)$ under zero field. %
This indicates that the FEIC phase is essentially accompanied by $2q$-lattice modulations. %
We have calculated the Fourier spectra of the spatial modulations in the local electric polarization using the experimentally determined magnetic structure in the FEIC phase and %
the microscopic model presented by Arima.\cite{Arima_Symmetry} %
Comparing the experimental results with the calculated Fourier spectrum revealed that the origin of the $2q$-lattice modulation is not %
conventional magnetostriction but the variation in the $d$-$p$ hybridization between the magnetic Fe$^{3+}$ ions and the ligand O$^{2-}$ ions. %
Combining the present results with the results of a previous polarized neutron diffraction study,\cite{CFAO_Helicity} %
we conclude that the microscopic origin of the ferroelectricity in this system is the variation in the $d$-$p$ hybridization %
with spin-orbit coupling. %
\section*{Acknowledgment}
The authors are grateful to T. Arima and S. Onoda for helpful discussions about the mechanism of the microscopic spin-polarization coupling,
and are also grateful to N. Nagaosa for suggesting the importance of the lattice modulations in a new class of magnetoelectric multiferroics.
The neutron diffraction measurement at BENSC was carried out along the proposal PHY-01-2285-DT. %
The synchrotron radiation X-ray diffraction measurements at SPring-8 were performed along the proposal %
2007B3773 (BL22XU) with the approval of the nano-technology network project, and the proposal 2006B1348 (BL46XU). %
This work was supported by a Grant-in-Aid for Scientific Research (C), No. 19540377, from JSPS, Japan. %
The images of the crystal and magnetic structures in this paper were depicted using the software VESTA\cite{VESTA} %
developed by K. Monma.%
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\section{Introduction}
\label{intro}
Most of the known exoplanets orbit late-F, G, or early-K dwarfs,
with masses ranging from 0.7 to 1.2\,M$_{\odot}$. There are nearly
2,000 such stars within 50\,pc brighter than V=8. In contrast
there are just a handful of M dwarfs brighter than V=8.
Although M dwarfs make up 70\% of nearby stars, their faintness in
the optical makes them difficult targets
for which to obtain precision Doppler velocities. As a result
they make up as little as 5\% of the current planet search targets, and
only 11 exoplanets
have been found to date, orbiting a total of 7 M dwarfs.
Just over half
of these M dwarf planets are less massive than Neptune, leading to
speculation that M dwarfs typically host either fewer planets than G dwarfs
\citep{johnson07}, or lower mass planets as a result of their
smaller proto-planetary disks \citep{laughlin04,il05,kk08}.
The primary parameters in planet formation theory are the
mass of the central star, the mass of the protoplanetary
disk, and the metallicity of the system. While it is now
well established for late-F, G, and K dwarfs that metal-rich stars are
enhanced in planets relative to metal-poor stars \citep{g1,g2,g4,g5,g3,sb,s1,s04,g6,re,fv,bond},
it has been harder to establish
the importance of stellar mass on planet formation since most
of the stars under survey lie in the relatively narrow
mass range encompassed by late-F, G and K dwarfs.
The searches which {\em are} being done for planets orbiting M dwarfs,
will ultimately provide the data needed to see if
the metallicity-planet relation extends down to the M dwarf
regime, and whether the mass distribution of exoplanets formed around
M dwarfs is similar to, or different than, that for more massive host stars.
M dwarf Doppler surveys, therefore, have the power to address some of the most important
questions in exoplanetary science, as they extend the
mass range of potential exoplanet host stars down to 0.3\,M$_{\odot}$.
We report here a new extrasolar planet in a long period orbit with
eccentricity consistent with zero, discovered by the Anglo-Australian
Planet Search (AAPS). The AAPS program is
described in Section 2. The characteristics of the host star
and our Doppler measurements are presented in Section 3.
A discussion follows.
\section{The Anglo-Australian Planet Search}
The AAPS began in 1998 January, and is
currently surveying 250 stars. Thirty exoplanets with
$m \sin i$\ ranging from 0.17 to 10\,M$_{\rm JUP}$\ have first been discovered
by the AAPS \citep{AAPSI,AAPSIII,AAPSVII,AAPSXI,AAPSXIII,AAPSII,AAPSV,
AAPSIV,AAPSVI,AAPSVIII,AAPSXII,AAPSIX,AAPSX,AAPSXIV}.
Our precision Doppler measurements are made with the UCLES echelle
spectrometer \citep{Diego90} on the 3.9m Anglo-Australian Telescope (AAT).
An iodine absorption cell provides wavelength calibration from
5000 to 6200 \AA. The spectrometer point-spread function and wavelength calibration
are derived from the iodine absoption lines embedded on every
spectrum by the cell \citep{Valenti95,Butler96}.
Our observing and analysis system has demonstrated long term
precision of 3\,\mbox{m\,s$^{-1}$}\ for
late-F, G, and early-K dwarfs brighter than V=7.5 \citep{AAPSXI,AAPSII}.
\section{GJ 832}
\label{obs}
At 4.93\,pc, GJ\,832 (LHS\,3865, HD\,204961, HIP\,106440) is amongst the nearest
stars in the sky \citep{Perryman97}. It is an M1.5 dwarf
with an optical absolute magnitude and colors of M$_{\rm V}=10.19$, $V=8.66$,
and $B-V=1.52$, and an infrared absolute magnitude and colors of M$_{\rm K}=6.03$, $K=4.50$,
and $V-K=4.16$.
Both Hipparcos and ground-based photometry \citep{Koen02}
find GJ 832 to be photometrically stable at the several milli-magnitude
level.
\citet{Gautier07} have combined Nstars\footnote{http://nstars.nau.edu}
visible photometry with Spitzer far-infrared photometry, to
estimate an ``infrared flux method'' effective temperature of 3657\,K for GJ\,832. The Spitzer
observations reveal no evidence of mid- or far-infrared excess.
The radius of GJ\,832 is estimated to be 0.48\,R$_{\odot}$\ \citep{Pasinetti01}.
Although accurate metallicities for M dwarfs are problematic,
GJ\,832 is likely to be rather metal poor. Matching synthetic spectra to high-resolution
spectra of the FeH band near 9900\AA,
Schiavon et al. (1997) estimate a metallicity for GJ\,832
of [Fe/H] $=$ $-0.7$, and a surface gravity of $\log g = 4.7$.
The photometric metallicity calibration of \citet{Bonfils05a}
gives an estimated metallicity of [Fe/H] = $-0.31\pm0.2$.
Due to its late spectral type, GJ\,832 has not (to date) been subject to detailed
spectroscopic analysis, and so to estimate its mass we must rely on either
theoretical isochrones, or empirical mass-luminosity calibration.
The latter indicate a mass for GJ\,832 of 0.45$\pm$0.05\,M$_{\odot}$\ (with the mass uncertainty
being largely due to the scatter about the mass-luminosity calibration relationship
of \citet{delfosse98}). The Padova theoretical isochrones \citep{marigo08} predict
M$_{\rm K}$ ranging from 5.97 (at $10^9$\,yr) to 6.03 (at $10^{10}$\,yr) for a 0.45\,M$_{\odot}$\
dwarf with [Fe/H] = $-0.3$ which is consistent with the observed luminosity of GJ\,832.
At [Fe/H] = $-0.7$ they predict M$_{\rm K}$ in the range 5.92 (at $10^9$\,yr) to
5.86 (at $10^{10}$\,yr). Given the difficulty in determining metallicities for M dwarfs,
we therefore derive a mass estimate for the primary of 0.45$\pm$0.05\,M$_{\odot}$.
GJ\,832 is chromospherically quiescent. Based on high
resolution spectroscopy of the CaII H\&K lines, \citet{CaHKI}
report $log R^\prime_{\rm HK} = -5.10$.
This would suggest a jitter of 3.9\,\mbox{m\,s$^{-1}$}\ using the
$B-V$, M$_{\rm V}$ \& T$_{\rm eff}$ in the most recent stellar ``jitter''
calibration of J.Wright (priv.comm).
\citet{Bonfils05b}
estimate the stellar jitter of GJ\,832 to be less than 2\,\mbox{m\,s$^{-1}$}.
GJ\,832 is among the fainter stars on the AAT program. The
signal-to-noise of these observations range from 46 to 150 per spectral pixel, with
a median of 98, which is lower than typical for AAPS targets. Four
late dwarfs from the long term AAT program are shown in
Figure \ref{stable_stars}. These stars are shown in order of descending B$-$V.
GJ 887 is an especially close match to GJ\,832 in B-V colour
and V magnitude. Based on this we estimate the combined velocity uncertainty due
to photon statistics, jitter, unknown planets,
and systematic errors is 5\,\mbox{m\,s$^{-1}$}\ for late-K and M dwarfs
in the AAPS. This is comparable to that estimated for late-K and M
dwarfs in the Keck program, as shown in Figures 2-4 of \citet{Butler08}.
A total of 32 precision Doppler measurements of GJ\,832
spanning 9.6 years are listed in Table \ref{velgj832} and shown in
Figure \ref{GJ832_Velocities} (upper panel). The root-mean-square (RMS) scatter
of the residuals about the mean velocity of this data set is 11.6\,\mbox{m\,s$^{-1}$}.Using the
2-Dimensional Keplerian Lomb-Scargle (2DKLS) peridogram of \citet{AAPSXIV} to
identify an initial period and eccentricity, the subsequent
best-fit Keplerian to all 32 epochs of data reduces this to an RMS of 5.5\,\mbox{m\,s$^{-1}$},
and gives a reduced $\chi_{\nu}^2$ of 1.54 (see Table \ref{orbit} -- a stellar jitter
of 3.9\,\mbox{m\,s$^{-1}$}\ was used, together with the internal velocity measurement uncertainty
for each epoch in Table \ref{velgj832}, to determine reduced $\chi_{\nu}^2$).
These fit parameters strongly suggest the presence of an exoplanet with minimum
mass $m \sin i$\ of 0.64\,M$_{\odot}$, period 9.4$\pm$0.4\,yr, eccentricity 0.12$\pm$0.11 (which we consider to be consistent with
zero eccentricity, particularly when the bias against measuring zero eccentricities
demonstrated by \citet{otoole08} is taken into account) and semi-major axis 3.4$\pm$0.4\,AU.
We have determined the False Alarm Probability (FAP, i.e. the probability
that we have falsely identified an exoplanet that is not present) for this orbit
determination using the Monte Carlo ``scrambled velocities'' approach described by
\cite{marcy05}. This method tests the hypothesis that no planet is present and
the Keplerian fit could have been obtained from mere noise, by generating randomly scrambled data sets in which
the order of velocities are changed but the times remain the same. These are
then subjected to the same analysis as our actual data set (i.e. identfying the
strongest peak in the 2DKLS followed by a full Keplerian fit). In this case
2002 random trials were carried out and only one of these yielded a
$\chi_{\nu}^2$ less than the value of 1.54 obtained with the original data. The
histogram of $\chi_{\nu}^2$ is shown in Figure \ref{FAP}. These results
imply a FAP of 0.05\% for the GJ 832 planet detection.
\section{Discussion}
GJ\,832 at a distance of 4.93pc is one of the nearest known exoplanetary
systems. The combination of the small distance and relatively long period
gives a large angular distance from the star
of 0.69 arc seconds for an edge-on circular orbit. This is exceeded only by
$\epsilon$ Eri among radial velocity detected exoplanets, and only six other
systems exceed 0.2 arc sec. GJ\,832b is therefore a potentially interesting
target for direct detection, although the high constrast with the star
\citep[likely to be $<10^{-8}$;][]{Burrows04} still makes this an extremely
challenging observation.
GJ\,832 is an excellent candidate for astrometric orbit determination. The
astrometric orbit semimajor axis is $\alpha \sin i = 0.95$ mas, which is
comparable to that of $\epsilon$ Eri for which an astrometric orbit was
determined by \cite{benedict06} and larger than that of GJ 876
which also has an astrometric orbit determination \citep{benedict02}. The
astrometric orbit would enable the inclination to be determined, removing the
current $\sin i$ uncertainty on the mass.
Seven M dwarfs (including GJ\,832) are currently known to host as many as
11 exoplanets, and
these are listed in Table \ref{mdwarfs} (see table notes for
references).
As noted earlier, determining the metallicities
of M dwarfs is notoriously difficult -- published metallicity estimates are
available for several of the known exoplanet host M dwarfs, and these
are listed in the Table. In addition, we have also derived for all seven
M-dwarfs a photometric
metallicity estimate, using the technique of \citet{Bonfils05a},
which has the advantage of being uniform over all these M dwarfs.
On average the \citet{Schiavon97} metallicities appear to be systematically
0.3-0.4\,dex lower than those derived from the Bonfils et al. calibration.
The \citet{bean06} metallicities are similarly on the metal-poor side
of the Bonfils et al. results, though not by as much ($\approx$ 0.2\,dex).
In general, the metallicity trends are similar across all three calibrations,
and it is clear there is a metallicity spread across the observed M dwarf exoplanet
hosts.
Based on these metallicity estimates, it would appear that four of
the current M dwarf exoplanet host stars are somewhat metal-poor, two have about
solar metallicity, and one is slightly metal rich.
Given the well known correlation between stellar
metallicity and observed exoplanet frequency for F, G, and K dwarf host stars,
this metallicity distribution for M dwarf host stars
is quite unexpected. Whhile the numbers of systems are small there is no obvious
difference in metallicity between the stars hosting Jupiter-mass planets and
those hosting Neptune-mass planets.
The correlation between high stellar metallicity and
planets for late-F, G, and early-K dwarfs points
toward the core accretion model for planet formation.
But there does not appear to be strong evidence to date that
M dwarf planet formation is strongly correlated with
high metallicity. This is puzzling, particularly in view of the fact that
M dwarfs probably have lower mass protoplanetary disks, and therefore would
need even higher metallicity than a F, G or K dwarf to provide enough solid
material (silicates and ices) to build a planetary core. Obviously it must be
kept in mind that measuring metallicities for
M dwarfs is problematic, and that even the \citet{Bonfils05a}
calibration (though empirically based and moderately robust) is only
good to $\pm$0.2\,dex. Nonetheless it is interesting to
consider possible means by which M dwarf exoplanets could be formed in such a
manner as to {\em not} display the strong metallicity correlation seen in
FGK dwarfs. One initially attractive explaneation is that since M dwarfs are essentially
immortal on a Hubble time scale, the vast majority of nearby M dwarfs could be
old metal poor stars. Unfortunately such an explanation
would appear unlikely. The study of M dwarf
kinematics has an extensive and venerable history \citep[e.g.][]{wielen77, wu95, rhg95}
which has contributed to the creation of extensive and sophisticated models of the
stellar populations present in the Solar Neighbourhood \citep[e.g. the Besancon models
of][]{robin2003}.
More recently, the availability of huge numbers of M dwarf spectra from the
SDSS survey have enabled sophisticated tests of the kinematics of the Besancon models
by \citet{bochanski07}, and have substantially born out the
Besancon model predictions for M dwarfs. Those models indicate that
dominant solar neighbourhood M dwarfs will be thin disc members with ages almost
uniformly spread between 0.1 and 10\,Gyr. Thick disc M dwarfs (which would indeed
be expected to have systematically lower metallicities)
will be present at much lower densities \citep[around a factor of one twentieth or
less;][]{robin2003}, and the probability that they would make up four of the
seven M dwarf exoplanet hosts would seem to be negligibly small.
An alternative explanation could be that M dwarf planets might form primarily via
the disk instability mechanism \citep[see e.g.][and references therein]{boss08},
rather than via core accretion,
which would make their formation probability more or less independent of metallicity.
Six of the eleven exoplanets known to orbit M dwarfs have
minimum masses less than 0.1\,M$_{\rm JUP}$. In contrast,
only nine planets with $m \sin i$ $<$ 0.1\,M$_{\rm JUP}$\ have been
found among the 216 Doppler velocity planets with
B-V $<$ 1.2 in the ``Catalog of Nearby Exoplanets'' \citep{butler06b}.
With a minimum mass of
0.64$\pm$0.06\,M$_{\rm JUP}$, GJ\,832b is the fifth jovian mass planet
found orbiting an M dwarf. The most massive M dwarf
planet yet found is 1.93\,M$_{\rm JUP}$. Since massive planets
are by far the easiest ones to find, planets of more than 2\,M$_{\rm JUP}$\
orbiting within 3\,AU of M dwarfs must be rare,
occuring less than around once per 300 M dwarfs.
\acknowledgements
We acknowledge support by NSF grant AST-9988087, NASA grant
NAG5-12182, PPARC grant PP/C000552/1, ARC Grant DP0774000
and travel support from the Carnegie Institution
of Washington (to RPB) and from the Anglo-Australian Observatory (to CGT, BDC
and JB).
We thank the Anglo-Australian time assignment
committees for allocations of AAT time. We are grateful
for the extraordinary support we have received from the
AAT technical staff -- E. Penny, R. Paterson, D. Stafford,
F. Freeman, S. Lee, J. Pogson, S. James, J. Stevenson, K. Fiegert and G. Schaffer.
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\subsection{Discovery and confirmation}
We searched a sample of objects selected as being potentially under-luminous relative to the main sequence using \emph{Gaia} Data Release 2 (DR2) astrometry. The selection is described by
\begin{equation}
\label{eq:selection}
G_\mathrm{mean}+5\times\left[\log(\sigma+\sigma_\mathrm{err})-2\right]>5\times (BP-RP)+8~,
\end{equation} where $G_\mathrm{mean}$ is the mean \emph {Gaia} G apparent magnitude and $(BP-RP)$ is the colour, both from Gaia DR2. This selection yielded a total of 22,912,186 candidate objects, which we then cross-matched with data from the Zwicky Transient Facility\cite{Bellm2019,Graham2019,Dekany2020,Masci2019}. We searched for periodic signals in all objects which had more than 50 combined epochs in their $g$ and $r$ light curves using the conditional entropy algorithm\cite{Graham2013} and selected candidates whose minimum conditional entropy fell more than 20 standard deviations below the mean entropy of the power spectrum. We searched frequencies ranging from $720>f>\frac{2}{\rm baseline}\rm\,cycle\,day^{-1}$, where the baseline is the time elapsed between the first and last epochs in each light curve in days. This selection yielded 25,025 high significance candidate objects, whose phase-folded light curves were visually inspected (investigation of many of these candidates is ongoing). ZTF J1406+1222 immediately captured our attention due to the stark difference in modulation amplitude between its ZTF $g$, ZTF $r$ and ZTF $i$ light curves. In particular, it exhibited a larger amplitude at blue wavelengths, as seen in Extended Data Figure 1. This is unexpected for white dwarf binaries with irradiated companions at such short orbital periods, as they host cool late type stars, which preferentially reprocess radiation at longer wavelengths. We thus immediately suspected that ZTF J1406+1222 could be a black widow in which the companion was so heavily irradiated that its inner face had heated sufficiently to preferentially reprocess radiation in bluer wavelengths. However, we later discovered that this difference in amplitude as a function of wavelength is primarily due to the presence of the tertiary sdK companion which was diluting the apparent amplitude at redder wavelengths.
\begin{figure}
\includegraphics[width=6.5in]{Figures/Lightcurves-eps-converted-to.pdf}
\linespread{1.3}\selectfont{}
\renewcommand{\figurename}{Extended Data Figure}
\setcounter{figure}{0}
\caption{\textbf{CHIMERA and ZTF lightcurve of ZTF J1406+1222} \textbf{a)} The CHIMERA light curve of ZTF J1406+1222. Like the HiPERCAM light curve illustrated in Figure \ref{fig:LC}, the variability is much more pronounced at shorter wavelengths. This light curve was used to confirm the signal observed in the ZTF light curve, prompting further follow-up of the target. \textbf{b)} The ZTF light curve of ZTF J1406+1222. The object appears more variable in the shorter wavelength passbands, but is brighter overall at longer wavelengths, primarily due to the contribution of the sdK tertiary component, and thus the true fractional variability is underestimated in redder passbands. All the error bars illustrated represent $1\sigma$ flux uncertainties.}
\label{fig:LC2}
\end{figure}
Using the high-speed imager CHIMERA\cite{Harding2016} on the 200-inch Hale telescope at Palomar Observatory on the nights of July 15 and 21 2020, we obtained the $g^{\prime}$, $r^{\prime}$, and $i^{\prime}$ light curves illustrated in Extended Data Figure 1, confirming the 62-minute period modulation of over a factor of 5 in $g^{\prime}$ and smaller amplitudes in $r^{\prime}$ and $i^{\prime}$. We reduced these data using a custom pipeline which performed aperture photometry on the source and a nearby reference star in each frame using a dynamic full-width-half-max.
We also used the Wafer Scale Imager at Prime (WASP) to obtain a 1200-s an H$\alpha$ filter ($\lambda_{0}=6570\,\mathrm{\AA},\Delta \lambda=100\,\mathrm{\AA}$) and an additional 1200-s exposure in an off-band filter ($\lambda_{0}=6651\,\mathrm{\AA},\Delta \lambda=107\,\mathrm{\AA}$) to check for the presence of a nebula around the object. We bias subtracted and flat fielded these observations with a custom pipeline. We then subtracted the normalized off-band exposure from the normalized H$\alpha$ exposure and found no discernible extended emission in the resulting image.
Data using the three channel imager ULTRACAM\cite{Dhillon2007} on the 3.5-m New Technology Telescope at LaSilla, taken in $u$, $g$ and $i$ filters, were used to confirm that the amplitude of the light curve modulation increased substantially in the $u$ filter relative to what we had observed previously in redder filters.
After confirming the extreme amplitude of modulation in the $u$ filter, we obtained two hours of data using the quintuple beam high speed photometer HiPERCAM\cite{Dhillon2021} on the 10.4-m Gran Telescopio Canarias and used these data for our light curve modelling.
\subsection{HiPERCAM light curve analysis}
Modelling the HiPERCAM light curve of ZTF J1406+1222 is challenging for three reasons: 1. We lack kinematic constraints and thus the binary mass ratio and the scale of the semi-major axis are not constrained; 2. the two modelling codes we used, LCURVE\cite{Copperwheat2010} and ICARUS\cite{Breton2012}, lack the physics needed to fully describe the behaviour we see in the light curve, in particular, contributions from the wind near minimum flux and the phase shift seen at maximum; and 3. the cool subdwarf companion contributes some amount of light in each band and its assumed spectral energy distribution greatly impacts the amplitude of the modulation, especially in the redder filters. We introduce a correction for this contribution in our ICARUS model.
Despite these limitations, we can still make useful constraints on the nature of the system using light curve models by investigating simple limiting cases. As discussed in the section on distance estimates, we know the distance to the system must be $D<2$\,kpc based on its large proper motion, we are able to precisely measure its apparent magnitude at peak and minimum at wavelengths ranging from the ultraviolet to near infrared, we know that there is little extinction along the line of sight, and thus we can estimate the peak luminosity of the system as a function of wavelength and also get an upper limit on the minimum luminosity of the irradiated object (the wind contributes near minimum flux, so the true depth of the minimum is unknown). We also know the orbital period of the system with a high degree of precision and can therefore assume a total system mass and estimate the scale of its semi-major axis.
We developed a model based on the HiPERCAM $u_{\rm s}$-band light curve due to the minimal contamination from the sdK companion at these wavelengths and high signal-to-noise ratio. We used LCURVE to model the light curve and performed nested sampling using the PyMultiNest \cite{Buchner2014,Feroz2009} package. For this model, we ignored the third light contribution due to the subdwarf, which is small in the $u_{\rm s}$ band.
The purpose of our LCURVE model was to explore whether a model using a neutron star and/or a white dwarf irradiator could acceptably fit the data. Due to the clear contribution of the wind near flux minimum, we adjusted the weights of all but the most restrictive point near the minimum to 0 (even this point likely has some wind contribution). We fixed the orbital period at $P=3720.09213$\,s ,based on the ZTF light curve, and the mass ratio at $q = 0.0357$ (assuming a mass of $0.05\,\mathrm{M}_\odot$ for the irradiated object and $1.4\,\mathrm{M}_\odot$ for the compact object). We adopted limb-darkening coefficients for a model atmosphere of a $10\,000\,$K main sequence star (we use this value to reflect the temperature of the irradiated surface) with a surface gravity $\log(g)=5.5$\,dex (cgs units). The mass ratio was fixed because we found that when allowed to vary freely, the solution strongly favoured irradiated objects of arbitrarily high mass in order to increase the radius of the irradiated object without filling its Roche lobe. We then directly fit the $u_{\rm s}$-band flux level of the light curve by introducing the distance as a parameter in the model (extinction is negligible along this line of sight, with a reddening of $\rm E(g-r)<0.02$ within a distance of $6\,\rm kpc$\cite{Green2019}). We constructed two models using the method described here, with the radius of the compact object responsible for emitting the irradiating flux fixed to 14\,km or 2000\,km, respectively. We found that the latter model was not even remotely able to fit the data, as seen by the fit presented in Figure~\ref{fig:Model}. As a check, we allowed the distance to vary within the 2000~km model and observed that LCURVE was able to converge to an acceptable solution for an object approximately 6.6\,kpc away by increasing the nightside temperature of the irradiated object to significantly outshine the extremely hot 2000~km irradiator even at flux minimum. The model is not able to achieve this at smaller distances, as making the irradiated object's night side this bright increases the overall luminosity of the system too much to be consistent with the observed apparent magnitude. Increasing the assumed mass of the neutron star in the model above $1.4\,\mathrm{M}_\odot$ results in a larger binary semi-major axis, and increases the total required heating flux radiated by the neutron star.
These simplified LCURVE models were meant to test the possibility of a white dwarf compact object. However, LCURVE treats the objects in the system as simple black bodies and neglects atmospheric corrections such as the attenuation in flux seen in the $u_{\rm s}$ band relative to a simple blackbody due to the Balmer ionization. Therefore, to construct the most realistic model possible now that we had ruled out the possibility of a white dwarf solution, we turned to the ICARUS light curve modelling code.
Before constructing the ICARUS model, we corrected for the contribution of the sdK component. The flux-weighted
position vector in the HiPERCAM images of the combined black widow plus sdK star can be written as
\begin{equation}
\mathbf{r} = \frac{f_{\rm bw} \mathbf{r}_{\rm bw} + f_{\rm sd} \mathbf{r}_{\rm sd}}{f_{\rm bw} + f_{\rm sd}}~.
\end{equation}
The flux of the sdK, $f_{\rm sd}$, is constant, and although the positions of the two stars move due to telescope guiding errors, there should be a fixed offset $\mathbf{s}$ between them, i.e.
\begin{equation}
\mathbf{r}_{\rm sd} = \mathbf{r}_{\rm bw} + \mathbf{s}~.
\end{equation}
We cannot measure the individual fluxes because of the blending, and instead measure the total
\begin{equation}
f = f_{\rm sd} + f_{\rm bw}~.
\end{equation}
Using these relations to substitute for $f_{\rm bw}$ and $\mathbf{r}_{\rm sd}$, the position of the centroid can be written as
\begin{equation}
\mathbf{r} = \mathbf{r}_{\rm bw} + \frac{f_{\rm sd}}{f}\, \mathbf{s}~.
\end{equation}
The position vector of the black widow in this expression is still subject to guiding errors, so we reference the position with respect to the comparison stars to remove the variability:
\begin{equation}
\Delta\mathbf{r} = \Delta \mathbf{r}_{\rm bw} + (f_{\rm sd}\,\mathbf{s}) (1/f)~,
\end{equation}
where now $\Delta\mathbf{r}_{\rm bw}$ is now a constant difference relative to the comparison star.
This equation shows that the centroid position of the blended image varies linearly with the inverse of the flux, $1/f$. If the measurements are made along a line parallel to $\mathbf{s}$, then a plot of position versus $1/f$ has gradient $f_{\rm sd}\, \mathbf{s}$. We know $\mathbf{s} = 0.5555$\,arcsec from Gaia, hence the contribution of the sdK to the flux, $f_{\rm sd}$, results. This has to be applied to the data for each filter, and the resultant corrected lightcurves are plotted in Extended Data Figure 2.
In ICARUS, we fixed the mass ratio $q=0.0357$ (again assuming a mass of $0.05\,\mathrm{M}_\odot$ for the irradiated object and $1.4\,\mathrm{M}_\odot$ for the compact object), the orbital period $P=3720.09213$\,s and the distance $d=1140$\,pc. ICARUS is designed to fit multi-band light curves simultaneously. However, because of the significant wavelength dependent phase shifts in the data, ICARUS was unable to converge to an acceptable solution by simply fitting the light curves without correcting for these shifts. We arbitrarily shifted the light curves to have all other maxima at the $u_{\rm s}$-band peak flux phase and performed a series of ICARUS fits. In order to account for the wind contribution, we experimented with not including data near the minimum flux in the fit. We found that because of the extraordinarily high signal-to-noise ratio of the HiPERCAM data, Markov-chain Monte Carlo (MCMC) fits of models which accounted for only data between phases $0.4-0.6$ (where $0.5$ is the flux maximum), vs $0.3-0.7$, vs all of the data converged to radically different solutions, with orbital inclinations in the $35-90$\,deg range. In fact, without any constrains, there were even good fits achieved for inclinations as low as $14$ degrees, but these solutions were unphysical, as they required a mass ratio in which the irradiated object was far more massive than the accretor, and that the system be at a great distance, as the solutions were trying to force the radius of the irradiated object to be as large as possible. Because the ICARUS model does not account for physics such as flux from the intrabinary shock (which may have an asymmetric offset from the line between the pulsar and the companion), the wind being driven off of the companion or the possibility of atmospheric transport on the surface of the irradiated object, rather than derive parameters from these models, we choose not to report these parameter estimates. A future detection of the pulsar, which would provide a precisely measured time of superior conjunction and an estimate of its radial velocity, may provide enough additional constraints for a more sophisticated modelling of the system and avoid simply over-fitting the data. Our ICARUS and LCURVE models both generally preferred solutions in which the irradiated object was underfilling its Roche lobe (with a $<0.5$ filling factor); however, the inferred filling factor was sensitive to which portion of the data was modelled, and thus is poorly constrained. An example ICARUS model fit is illustrated in Extended Data Figure 3.
However, in addition to unambiguously ruling out the possibility of a white dwarf irradiator, the light curve also allows us to estimate the amount of radiation the neutron star is emitting, $\dot{E}$. This is because the peak temperature of the irradiated companion, $T_{\rm day}$, can be estimated from the apparent magnitudes at peak flux, and this temperature is directly related to the amount of incident radiation per unit area by the follow expression:
\begin{equation}
\label{ddot}
L_\mathrm{H}\sim4\pi a^{2}\sigma(T_{\mathrm{day}}^4-T_{\mathrm{night}}^4)~,
\end{equation}where $L_\mathrm{H}$ is the heating luminosity, $\sigma$ is the Stefan-Boltzmann constant, $T_{\rm night}$ is the nightside temperature of the companion and $a$ is the semi-major axis of the binary system. The heating luminosity and the spin down luminosity are related by an efficiency factor, $L_\mathrm{H}=\eta \dot{E}$. Although we do not know the total mass of the system, $M_\mathrm{T}$, and therefore must assume one to estimate the semi-major axis, $a$ only varies as $M_\mathrm{T}^{1/3}$ and is much more sensitive to the orbital period, which we know quite precisely. We consider a range of semi-major axis values, assuming $1.45<M_\mathrm{T}<2.05\,\mathrm{M}_\odot$. We estimate $T_{\rm day}$ by fitting the spectral energy distribution (SED) at the orbital phase of maximum flux in the Swift UVOT light curve, and our best fit model is illustrated in Extended Data Figure 4. This gives a warmer estimate than if we simply take the peak flux for each filter, since redder filters peak earlier than blue filters. The companion, given the short orbital period, must be degenerate or near degenerate, and its atmosphere is clearly dominated by hydrogen lines. Thus, in order to fit the SED of the dayside of the object, we used hydrogen rich low-mass white dwarf atmospheric models\cite{Gianninas2014,Tremblay2015}, which yielded an estimated temperature of $T_\mathrm{day}=10462\pm150$\,K. To obtain the most conservative estimate of $T_\mathrm{night}$, we constructed an ICARUS model in which we fix $T_\mathrm{day}$ to $10462\pm150\,\mathrm{K}$ and fix the inclination to $90\,\mathrm{deg}$. This requires the light curve model to pick the largest possible value of $T_\mathrm{night}$ in order to minimize the amplitude of variability to match the data. In this limiting case, find $T_\mathrm{night}\approx6500\,\mathrm{K}$. Using these boundaries and assuming a semi-major axis corresponding to a total system mass of $1.45<M_\mathrm{T}<2.05\,\mathrm{M}_\odot$, we conclude that $L_{\mathrm{H}}$ is approximately in the range of $(1.13 <L_{\mathrm{H}}< 1.79)\times10^{34}\,\mathrm{erg}\,\mathrm{s}^{-1}$, a value typical of spider binaries.
\begin{figure}
\includegraphics[width=6.5in]{Figures/Mags-eps-converted-to.pdf}
\linespread{1.3}\selectfont{}
\renewcommand{\figurename}{Extended Data Figure}
\caption{\textbf{Swift UVOT and corrected HiPERCAM lightcurve of ZTF J1406+1222} The HiPERCAM and Swift UVOT light curves on a magnitude scale after correcting for the contribution of the sdK star. Only $>1\sigma$ detections have been shown for the Swift data, as the object magnitude lies below the detection threshold for over half of the orbit. The prominent phase shifts in the peak flux indicate that the object transitions from a cooler to a hotter surface over the course of the flux maximum. Our LCURVE and ICARUS models were not able to account for this physical effect, so we arbitrarily shifted the other light curves to match the $u_{\rm s}$-band maximum flux for the purpose of constructing simple models. In order to estimate the temperature of the day side of the irradiated object, we fitted the spectral energy distribution from the apparent magnitudes at orbital phase 0.55, the peak of the Swift UVOT UVW2 light curve. One orbital cycle has been repeated for display purposes. All the error bars illustrated represent $1\sigma$ uncertainties in apparent magnitude.}
\label{fig:Mags}
\end{figure}
\begin{figure}
\includegraphics[width=6.5in]{Figures/ICARUS_v2-eps-converted-to.pdf}
\linespread{1.3}\selectfont{}
\renewcommand{\figurename}{Extended Data Figure}
\caption{\textbf{ICARUS model fit to lightcurve} An example ICARUS toy model fit to data between orbital phases 0.25 and 0.75 (ignoring data outside these phases), with the temperature of the day side of the companion fixed to $10462\,\mathrm{K}$, the distance fixed to $1140\,\mathrm{pc}$, a fitted inclination of $i=\approx66\,\mathrm{deg}$ and an irradiated object with a radius of just $0.029\,\mathrm{R}_\odot$. The data from left to right illustrate the HiPERCAM $u_{\rm s}$, $g_{\rm s}$, $r_{\rm s}$, $i_{\rm s}$, and $z_{\rm s}$ filters, with the dashed black lines illustrating the best fit model in that filter. The light curves have been artificially shifted to line up with the $u_{\rm s}$-band maximum light, as the ICARUS model is unable to capture the strong colour-dependent phase shifts seen in the data. We did not perform an MCMC over these models, as we found that there were acceptable fits for a wide range of inclinations and other parameters given the limitations of using a simple direct heating model. Because we chose to fit the flux at peak, but not at minimum (where the wind contributes significantly), most models fit the peak flux well, but the excess flux at minimum increases gradually to the redder passbands, suggesting that there may be a colour dependence associated with the flux contribution from the wind, though this is difficult to disentangle from the flux contribution of the sdK at minimum light. }
\label{fig:ICARUS}
\end{figure}
\begin{figure}
\includegraphics[width=6.5in]{Figures/Peak_fit-eps-converted-to.pdf}
\linespread{1.3}\selectfont{}
\renewcommand{\figurename}{Extended Data Figure}
\caption{\textbf{Model fit to peak of ZTF J1406+1222's spectral energy distribution} The spectral energy distribution at the orbital phase of the Swift UVOT maximum flux. The red squares illustrate the filter averaged model, which convolves the model spectrum with the respective HiPERCAM and Swift filters. The solid line is the best-fit white dwarf model spectrum. We used a grid of white dwarf model atmospheres because they naturally cover the appropriate surface gravities and temperatures needed to model the irradiated face of the companion, which could be either a brown dwarf or a white dwarf. The Swift detection strongly constrains its dayside temperature, giving a best-fit value of $T_{\mathrm{day}} = 10462\pm150\,\mathrm{K}$. All the error bars illustrated represent $1\sigma$ uncertainties in apparent magnitude.}
\label{fig:Peak_fit}
\end{figure}
\subsection{Spectroscopic analysis}
The spectroscopic data were obtained with LRIS using the 600/4000 grism with 2x2 binning on the blue arm and the 400/8500 grating with 2x1 binning on the red arm, on the nights of July 18 2020 UTC and Jan 10 2021 UTC. One the night of June 5 2021, we used the 600/3400 grism with 2x2 binning on the blue arm and the 600/7500 grating with 2x2 binning on the red arm. On this last night, we adjusted the position angle of the slit to $-$79.42\,degrees East of North in order to capture both sources and were also able to take advantage of a newly upgraded detector on the red arm. Thus, we base much of our spectroscopic analysis on this data set, which is superior to the other nights, though our radial velocity measurements made use of all three nights of data. We reduced all the spectroscopic data using the LPIPE pipeline\cite{Perley2019}.
\subsection{Black widow radial velocity analysis}
In order to measure radial velocity variations, we used a custom code which simultaneously fit six Lorentzian profiles plus a quadratic polynomial to the full spectrum, with the Lorentzians fitting the hydrogen lines at 4861.35, 4340.46, 4101.74, 3970.07, 3889.05 and 3835.40~\AA.
We attempted to fit both the hydrogen absorption and emission lines in order to measure radial velocity variations on the orbital period. However, the hydrogen absorption lines are broad and only visible over a short orbital phase interval, preventing us from extracting reliable radial velocity shifts from them. This is not surprising given that these absorption lines are blended with variable emission from the ablation wind, as well as underlying features of the sdK spectrum.
However, we detected appreciable radial velocity variations in the much narrower Balmer emission lines. As discussed in the main text, we found that these features are blueshifted after superior conjunction of the black widow, and we interpret this as a signature that they originate from a cometary tail spiraling around the irradiated companion, or potentially from the intrabinary shock in the system. Figure~\ref{fig:Trail} presents the best sine fit to the emission-line radial velocity curve, overplotted with the phase-folded trailed spectra.
\subsection{Cool subdwarf atmospheric analysis}
We identified the sdK nature of the hierarchical companion due to the presence of strong CaH absorption lines, a signature of these low metallicity late type stars\cite{Allard1995}. Typically, late type main-sequence stars are characterized in terms of line indices, which are measures of the strength of an absorption line relative to the continuum, the most important of which are the CaH and TiO indices\cite{Reid1995}. Analyzing the spectral indices of ZTF J1406+1222 presents a unique challenge because it is variable, with a non-negligible contribution from the inner binary at the wavelengths of these indices. In order to compensate for the variability, we compute the indices using a coadd of spectra taken within the faintest 20 percent of the inner binary orbit in order to minimize its contribution. This analysis yielded the spectral indices and derived parameters presented in Extended Data Table~\ref{tab:sdK}. We use the empirical relation\cite{Woolf2009} to determine the metallicity of the sdK star based on the indices found. One way to compute its temperature is using the CaH2 index\cite{Woolf2009}, where the effective temperature is given by $T_\mathrm{eff}=2696+1618\times CaH2$, which gives $T_\mathrm{eff}=4020\pm70 \rm \, K$. The uncertainty has been estimated by computing the scatter of the indices of the individual phase resolved spectra. Additionally, by fitting the BT-NextGen (GNS93) atmospheric models for low mass stars\cite{Allard2012} to the region around the TiO+CaH band (between $6250\AA$ and $7250\AA$) in the spectra taken during the faintest portion of the orbit, as illustrated in the inset of Extended Data Figure 5, we also obtain estimates of $T_\mathrm{eff}=3800\pm30 \rm \, K$ and $\rm [Fe/H]=-1.16\pm0.084~dex$, though the uncertainties, derived from the covariance matrix of the fit, are sensitive to the slope of the continuum and dilution due to the black widow component, and thus are underestimated. Additionally, we use the Ca II $8542\,\rm \AA$ line originating from the sdK to estimate the systematic velocity of the system by fitting for velocities in individual spectra, and taking a weighted average of these measurements. We measure a systemic velocity of $\gamma=67\pm30\,\rm km\, s^{-1}$, where the uncertainty is dominated by the systematic uncertainty of the absolute wavelength calibration which we estimate by taking a simple barycentric corrected coadd of the individual spectra, and measuring the velocity of the Ca II $8542\,\rm \AA$ line in this coadd and comparing with the value yielded by the weighted average of fitting the individual epochs. We also verified that the scatter in the velocities estimated using the individual epochs does not track the 62-minute orbital period of the black widow component.
\begin{figure}
\includegraphics[width=6.5in]{Figures/Red_Spectrum_v2-eps-converted-to.pdf}
\linespread{1.3}\selectfont{}
\renewcommand{\figurename}{Extended Data Figure}
\caption{\textbf{Red LRIS spectrum of ZTF J1406+1222} The red LRIS spectrum of ZTF J1406+1222 illustrating the significant contribution from the cool sdK component. The broad feature at $6700-7000\, \rm \AA$ is a combination of titanium oxide (TiO) absorption bands commonly seen in late type stars and strong calcium hydride (CaH) bands, which are more intense in low metallicity sdM/sdK stars. We classify the object as a low metallicity star by measuring the ratio of the TiO to CaH bands in a coadd of spectra at close to the minimum flux of the black widow component to avoid significant contamination of the continuum. Additionally, we also estimated the metallicity by performing a fit of atmospheric models to the region of the spectrum around the TiO+CaH band, as illustrated in the inset.}
\label{fig:Red_spectrum}
\end{figure}
\subsection{Distance Estimate}
We use three approaches to constrain the distance to the system: 1. using the parallax as reported in \emph{Gaia} eDR3\cite{Gaia2021}, 2. using the absolute magnitude of the sdK companion in the $K$ band, and 3. putting an upper limit based on the large proper motion of the system. We discuss all three approaches here.
As of Gaia eDR3, only the sdK component contains a full astrometric solution, and its parallax is measured to be $\varpi=2.1118\pm1.5882~ \rm mas$. Estimating a distance based on this is challenging, as the parallax measurement is hardly larger than its uncertainty. The best distance estimate based on this astrometric solution according to the Bailer-Jones et al. catalog\cite{Bailer-Jones2021} is $1290^{+2233}_{-724}\rm \, pc$. This value is poorly constrained, so we did not use it in our analysis, but instead used the properties of the sdK companion.
In order to estimate the distance to the system using the luminosity of the sdK, we interpolated over evolutionary model grids\cite{Baraffe1997} using the estimated metallicity $\rm [Fe/H]=-1.173~\rm dex$ and the observed colour $(V-K)=3.34\pm0.09$ as input parameters. Because our highest signal-to-noise ratio light curve was obtained in $g$, we empirically correct it to the $V$ band using the sample of objects presented in \cite{Jao2017}, which contains $g$ and $V$ magnitudes for a large sample of objects. We find that over this entire sample of objects, the $g-V$ correction factor is $0.738\pm0.072$. We use the minimum brightness of our HiPERCAM $g$ light curve of $21.289\pm0.0127 \rm \, mag_{AB}$ to estimate an apparent magnitude of the sdK of $V=20.55\pm0.09$. Technically, the black widow still contributes here, but if that were the case it would cause the sdK to appear warmer than expected and cause the distance to be slightly overestimated, which would drive lightcurve models even further in favor of a neutron star solution. The model grids also contain estimates of the effective temperature, which we found to be $T_\mathrm{eff}=3870 \pm 70~ \rm K$, consistent with the estimate made using the CaH2 index as well as with the spectroscopic model. The distance we calculated using this analysis and the observed UKIDSS $K$-band apparent magnitude of $K=17.182\pm0.054$ is $1140\pm200\,\rm pc$, with an approximate absolute $K$-band magnitude of $M_K=6.6\pm0.3$. This does not account for the uncertainty in the metallicity, as the grids used do not extend to metallicities above $\rm [Fe/H]=-1.0~dex$. Rather than extrapolate these grids, we made a more robust but broader estimate of the distance by investigating the measured absolute magnitudes of a nearby sample of cool subdwarf stars with colours similar to our observed colour of $(V-K)=3.34\pm0.09$. Based on the recent sample presented in \cite{Jao2017}, we find that the $K$-band absolute magnitude of the companion should fall between 6 and 8, in good agreement with the $M_K=6.6\pm0.3$ estimated using evolutionary models. The absolute magnitude range of $6<M_K<8$ corresponds to a distance interval of $680<d<1720 \rm \, pc$. Based on its absolute magnitude, spectral type, and metallicity, we estimate that the sdK has a mass of approximately $0.18\,M_{\rm \odot}$ to $0.3\,M_{\rm \odot}$\cite{Jao2016}.
Finally, in order to place a very conservative absolute upper bound on the distance using the proper motion, we assume that the cool companion should have a tangential velocity less than $1200 \rm ~ km \, s^{-1}$, as, if it exceeded this value, it would be the fastest hyper-velocity pulsar known\cite{Chatterjee2005}. This seems implausible, especially considering that it is a wide triple system. Given the $74.486\pm1.769~\rm mas\,yr^{-1}$ proper motion, this corresponds to a distance upper limit of approximately $3400\rm \, pc$. Additionally, an analysis of the object's kinematics revealed that it would be on a trajectory to escape the Galaxy if it were at a distance of $>2000\rm \, pc$, so the latter distance is a more realistic upper bound.
\begin{table}
\centering
\caption{\textbf{Spectroscopic properties of the subdwarf K star}. The values presented here are shown in two columns, one considering indices of spectra taken within the faintest 30 percent of the light curve, centred around minimum brightness (orbital phases $\phi=0.8$ to $\phi=0.1$), and the other from a spectroscopic model fit to the coadd of the spectra taken around the faintest phase. The first four rows show the spectroscopic indices\cite{Reid1995} commonly used to characterize low metallicity, cool subdwarf stars \cite{Allard1995}. $\rm [Fe/H]$ is an estimate of the metallicity of the object based on the $\zeta_\mathrm{TiO/CaH}$ index computed according to the formalism described in \cite{Woolf2009}. The final row gives the metallicity classification of the object.}
\medskip
\begin{tabular}{lcc}
\hline
$\rm{Index/Parameter}$ &$\rm Value~(\phi=0)$ &$\rm Value~(Spectroscopic)$ \\
\hline
$ \rm{TiO5}$ & $0.953\pm0.044$ & \\
$\rm{CaH2}$ & $0.820\pm0.041$ & \\
$ \rm{CaH3}$ & $0.90\pm0.045$ & \\
$ \rm{CaH2+CaH3}$ & $1.718\pm0.084$ & \\
$ \zeta_\mathrm{TiO/CaH}$ & $0.45\pm0.50$ & \\
$ \rm{[Fe/H]}$ & $-0.94\pm0.82$ & $-1.16\pm0.08$ \\
$ \rm{T_{\mathrm{eff}}}$ & $4020\pm70\,\rm K$ & $3800\pm30\,\rm K$ \\
$ \rm{Classification}$ & $\rm esdK$ & $\rm esdM$ \\
\hline
\end{tabular}
\label{tab:sdK}
\end{table}
\subsection{Establishing the hierarchical triple nature of the system}
Archival SDSS\cite{York2000} and Pan-STARRs1\cite{Chambers2016} images (Extended Data Figures 6 and 7) revealed the presence of what appears to be two distinct sources separated by a fraction of an arcsec, even though \emph{Gaia} DR2 only detected a single source at this position. With the release of \emph{Gaia} eDR3, a second source was identified by \emph{Gaia} $0.5555\pm0.0045$ arcsec away from the previously detected source. However, no further astrometric information is available for this source. The \emph{Gaia} source with a full astrometric solution exhibits a large proper motion of $74.486\pm1.769~\rm mas\,yr^{-1}$, which corresponds to the sdK star seen in the spectrum. By monitoring the position of the centroid of the point spread function (PSF) of the source in the HiPERCAM images, we were able to demonstrate that these two sources must be co-moving, and thus members of a hierarchical triple system. In Extended Data Figure 8, we illustrate that the position angle between the two sources has not changed significantly since the \emph{Gaia} measurement in 2016, whereas if one assumed that the variable black widow component was a distant background source with low proper motion and only the sdK exhibited this large proper motion, this position angle would have changed substantially in five years. Thus, we conclude that the sdK is a common proper motion companion to the 62-minute binary, making it the outer companion in a hierarchical triple system.
\begin{figure}
\centering
\begin{minipage}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{Figures/SDSS-eps-converted-to.pdf}
\linespread{1.3}\selectfont{}
\renewcommand{\figurename}{Extended Data Figure}
\setcounter{figure}{5}
\caption{\textbf{SDSS colour image of ZTF J1406+1222} An SDSS DR9 colour image cutout of the ZTF J1406+1222, revealing an asymmetric colour across the PSF with the sdK on the left. The two red boxes indicate the J2016.0 positions of the two \emph{Gaia} eDR3 sources. This image has been centred on the same coordinates as the Pan-STARRS1 cutout shown in Figure~\ref{fig:PS}.}
\label{fig:SDSS}
\end{minipage}
\hfill
\begin{minipage}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{Figures/PS-eps-converted-to.pdf}
\linespread{1.3}\selectfont{}
\renewcommand{\figurename}{Extended Data Figure}
\caption{\textbf{Pan-STARRS1 colour image of ZTF J1406+1222} A Pan-STARRS1 colour image cutout at the same centroid as the SDSS image in Figure~\ref{fig:SDSS}. The Pan-STARRS1 PSF exhibits the same colour asymmetry as seen in the SDSS image, and the centroid is closer to the Gaia source position than the SDSS image due to the $74.5 \rm\, mas\,yr^{-1}$ proper motion of the system. }
\label{fig:PS}
\end{minipage}
\end{figure}
\begin{figure}
\includegraphics[width=6.5in]{Figures/Position_3-eps-converted-to.pdf}
\linespread{1.0}\selectfont{}
\renewcommand{\figurename}{Extended Data Figure}
\caption{\textbf{Astrometric characterization of ZTF J1406+1222} \textbf{a)} The measured \emph{Gaia} eDR3 J2016 position of the sdK is shown by the blue circles and the measured position of the variable black widow component is illustrated by the red diamonds. The magenta stars represent the projected J2021.5 position of the sdK, given the proper motion in its \emph{Gaia} astrometric solution. The solid black line shows the position angle between the two \emph{Gaia} source positions. The pink star indicates the position of the source in an SDSS u image obtained in J2003.4, and the black star indicates the source position in the SDSS z image at this epoch. Because the BW component dominates in u band, and the sdK in z band, these positions should approximately reflect the positions of the two components in J2003.4, and the dashed arrows indicate that these sources are co-moving, having both translated their positions significantly since that epoch (by about an arcsecond). \textbf{b)} The black points indicate the measured centroid of the point spread function of the variable source with respect to a reference star on the HiPERCAM $r$-band images. This centroid moves back and forth between the two sources of luminosity as the black widow component brightens and fades, and gives a precise estimate of the position angle between the two sources when the data were obtained, at epoch J2021.5. The solid blue line is a linear fit to the data used to derive a slope to measure this position angle, whose value is shown in the legend. The slope is consistent with the J2016 position angle and clearly inconsistent with only the sdK having moved since epoch J2016.0, demonstrating that the two sources must indeed be co-moving, and thus part of a hierarchical triple system.}
\label{fig:Position}
\end{figure}
We estimate the separation of the sdK and the BW by using the observed angular separation of $0.5555\pm0.0045$~arcsec between the \emph{Gaia} positions, and the estimated distance of $1140\,\rm pc$, yielding a separation of approximately $600$ astronomical units. Using the estimated sdK mass of $0.25\rm\,M_{\odot}$, and taking the inner black widow binary as having a mass of approximately $1.45\rm\,M_{\odot}$, we estimate that this would correspond to an orbital period of $11000$ years in a circular orbit (around $10500\pm500\rm\, years$ if one instead allows the mass of the inner BW to vary between $1.4\rm\,M_{\odot}$ and $2.0\rm\,M_{\odot}$ rather than fixing it to $1.45\rm\,M_{\odot}$.) It is quite possible that the orbit of the sdK is far from circular, which means we can only estimate a lower limit to the orbital period, by considering that the current separation represents the maximum separation of a highly elliptical orbit.
\subsection{Swift XRT and NuSTAR analysis}
We obtained five separate Swift XRT observations of the object (observation IDs 00013598001, 00013598002, 00013598003, 00013598004, 00013598005) for a total of $12\, \rm ks$ exposure time. We used \texttt{XIMAGE} to coadd all the XRT exposures and compute a 3-$\sigma$ upper limit on the count rate within an 18 arcsec aperture using the \texttt{uplimit} routine of \texttt{XIMAGE}. The aperture only contained 1 count, and the 3-$\sigma$ upper limit on the count rate was $6.648\times10^{-4}\, \rm count~s^{-1}$. We used \texttt{WebPIMMS} to convert this into a 3-$\sigma$ upper limit on the unabsorbed flux in the $0.2-10~\rm keV$ band, which is $2.6\times 10^{-14}\,\rm erg\,cm^{-2}\, s^{-1}$, assuming a column density of $n_{\mathrm H}=2\times 10^{20}$ along the line of sight. Taking our estimated distance of $1140\,\rm pc$, this translates into a 3-$\sigma$ upper limit on the source luminosity of $4.04\times10^{30}\,\rm erg\,s^{-1}$ at $0.2-10\,\rm keV$. We assumed a power law spectrum with a photon index of $2$ when estimating the source flux.
We also obtained a NuSTAR observation (ID 90601325002) with $28\,\rm ks$ on source time with module A and $25\,\rm ks$ with module B. The source was not detected, with 191 counts in a $50$ arcsec aperture in module A, with an average background of 177 counts determined using an annulus centred on the source with an inner and outer radius of $100$ and $200$ arcsec, respectively. The $50$ arcsec aperture in module B produced 192 counts and the background amounts to an average of $205$ in an equivalent area. Using \texttt{XIMAGE}, we summed the two exposures and supplied the summed background rate to the \texttt{uplimit} routine, which yielded a 3-$\sigma$ upper limit of $0.0013899\,\rm count\,s^{-1}$. Using \texttt{WebPIMMS}, we estimate this corresponds to an unabsorbed flux of $1.178 \times 10^{-13}\,\rm erg\,cm^{-2}\, s^{-1}$, translating to a luminosity upper limit of $1.83 \times 10^{31}\,\rm erg\,s^{-1}$ at $0.2-10\,keV$ assuming a distance of $1140\,\rm pc$, and a column density of $n_{\mathrm H}=2\times 10^{20}$ along the line of sight. Like with the Swift observation, we assumed a power law spectrum with a photon index of $2$.
There is a relation between the pulsar spin down luminosity, $\dot{E}$, and the heating luminosity inferred from the irradiated companion, $L_\mathrm{H}$, and the X-ray luminosity in the system\cite{Lee2018}. In Extended Data Figure 9, we illustrate the deeper Swift upper limit and our estimates on the $\dot{E}$ based on light curve modelling and assuming a pulsar heating efficiency factor $0.1<\eta<1$. The black diamonds indicate the measured X-ray luminosity of other black widow systems, and the red triangles are upper limits for other black widows which have not yet been detected in X-rays. Our Swift observation would have failed to recover several known systems at the estimated distance of $1140\,\rm pc$ and similar $\dot{E}$, and thus future deeper X-ray observations will be informative in further constraining ZTF J1406+1222's X-ray luminosity. Unlike other black widow systems, we have not directly detected the pulsar, and thus have no direct measurement of $\dot{E}$, but rather only an estimate of $L_\mathrm{H}$. Typical $L_\mathrm{H}$ values in spider binaries are on the order of $\eta=0.15$, indicating that only a fraction of the spin down luminosity of the neutron star contributes to heating the companion. However, for illustrative purposes, we choose an upper limit of $\eta=1$ because ZTF J1406+1222, with its extremely short orbital period and cometary tail most resembles PSR J1311-3430, an object consistent with $\eta>1$. Light curve models are unable to fully capture the complicated heating physics in such systems, for example, an intrabinary shock which wraps around the companion and subtends a larger cross section of the neutron star spin down flux than possible with only the companion.
\begin{figure}
\includegraphics[width=6.5in]{Figures/x_ray3-eps-converted-to.pdf}
\linespread{1.0}\selectfont{}
\renewcommand{\figurename}{Extended Data Figure}
\caption{\textbf{X-ray luminosity constraint of ZTF J1406+1222} The $0.2-10\,\rm keV$ X-ray luminosity, $L_\mathrm{X}$ vs $\dot{E}$ for known black widow systems\cite{Lee2018,Kong2014} (detections shown as black circles and upper limits as red triangles). The red dashed line indicates the region which we have constrained ZTF J1406+1222 to occupy based on the X-ray luminosity upper limit derived from our Swift observation and the estimate of $\dot{E}$ based on the peak temperature of the irradiated face of the neutron star's companion. Several known millisecond pulsars with a similar $\dot{E}$ are currently below our X-ray flux upper limit, and thus deeper observations may yield an X-ray detection of ZTF J1406+1222.}
\label{fig:xray}
\end{figure}
\subsection{Swift UVOT analysis}
We constructed a UVW2-band light curve of the system from the Swift observations 00013598003, 00013598004 and 00013598005. We used the \texttt{uvotlc} tool to bin event mode data into $10\,\rm s$ snapshots and constructed a binned light curve by computing a weighted sum of the measured fluxes in these snapshots according to orbital phase. These measurements are presented in Extended Data Figure 2, where we have omitted $<1$-$\sigma$ detections (many of which were consistent with negative fluxes and would not have translated to a magnitude scale). One snapshot, 00013598005, was centred around the time of minimum light with an exposure duration of $1736\,\rm s$, covering the full fainter half of the orbit. The source was not detected above background in this deep exposure, and we used XIMAGE and a $5$ arcsec aperture to compute a 3-$\sigma$ upper limit on the apparent magnitude of the object during the fainter half of the orbit, which we found to be $>22.9\,m_\mathrm{AB}$ in the UVW2 band.
\subsection{Fermi Analysis}
The source is not detected by the Fermi Gamma-ray Telescope. We computed an upper limit to the gamma-ray luminosity using a prescription similar to that used for Swift~ J1644+5734\cite{Burrows2011}. We considered a 15-degree region around the source and the $100-10000\,\rm MeV$ energy range, and perform a binned likelihood analysis of Fermi LAT data between 2008-08-04 and 2021-08-27. We find a 3-$\sigma$ upper limit on the photon rate of $9.64\times10^{-10}\rm \, photon\,cm^{-2}\,s^{-2}$. We assumed a power law with photon index 2, which is typical of spider binaries at gamma-ray energies\cite{Hui2019}. This corresponds to a $3\sigma$ uppper limit on the gamma-ray luminosity of $L_{\gamma}<1.45\times10^{32}\,\rm erg\,s^{-1}$ for an assumed distance of $1140\,\rm pc$; this means the source is fainter than most known black widow binaries in the Fermi 4FGL catalog exhibit\cite{Hui2019}. The system is at a similar distance to PSR J0636+5128, a 95.8-minute orbital period black widow which is also not in the Fermi LAT 8 yr catalog\cite{Draghis2018}. As discussed in the work on PSR J0636+5128\cite{Draghis2018}, this may be due to gamma-ray luminosity being preferentially beamed towards the spin equator in the system, our our viewing the system at high inclination. Our heating models of the lightcurve are able achieve good fits with physical solutions for inclinations as low as 35 degrees, and thus it is possible that for if the spin of the pulsar is aligned with the binary orbit, we intercept very little gamma-ray luminosity. Because the ZTF J1406+1222 was selected optically, its selection was not biased towards gamma-ray bright sources like those selected using Fermi.
\subsection{Galactic kinematic analysis}
The high proper motion and low metallicity sdK companion of ZTF J1406+1222 are suggestive that the system is an object in the Galactic halo. To confirm this, we used the \texttt{galpy}\cite{Bovy2015} package to compute its trajectory around the Milky Way over 6 Gyr, using the McMillan2017 potential\cite{McMillan2017}, and it's Gaia astrometric solution and the radial velocity of the sdK measured from the LRIS spectra. The resulting trajectories, shown in Extended Data Figures 10 and 11, are clearly consistent with a halo trajectory in which the object reaches scale heights of $>10\,\rm kpc$ out of the Galactic disk, as seen in Figure~\ref{fig:Orbit1}, and $>10\,\rm kpc$ radially from the Galactic centre, as seen in Extended Data Figure 11. This analysis revealed that the object would escape the Galaxy when placed at a distance of $>2000\,\rm pc$. Significantly, the solution corresponding to ZTF J1406+1222 being 940~pc away from Earth passes within just 50 parsecs of the Galactic center, as illustrated in Extended Data Figure 11. This suggests that the system may have originated from a disrupted globular cluster.
\begin{figure}
\includegraphics[width=6.5in]{Figures/plot1-eps-converted-to.pdf}
\linespread{1.3}\selectfont{}
\renewcommand{\figurename}{Extended Data Figure}
\caption{\textbf{Radial vs scale height orbital solution of ZTF J1406+1222 around the Galaxy} A plot illustrating the Milky Way orbit of ZTF J1406+1222 over the course of 10 Gyr. The colours (green, orange and blue) indicate three different distances corresponding to our best distance estimate of $1140\pm200\,\rm pc$ and the 1-$\sigma$ upper and lower bounds of this distance estimate. In all cases, the object reaches a scale height ($z$) of more than $10 \, \rm kpc$ above the Galactic disk and travels a great distance ($R$) away from the Galactic centre in the radial direction, clearly indicating that it is a halo object. Notably, the green line, illustrating the orbital solution if ZTF J1406+1222 is at $\approx 940\pm200\,\rm pc$ passes within 50 parsecs of the Galactic center.}
\label{fig:Orbit1}
\end{figure}
\begin{figure}
\includegraphics[width=6.5in]{Figures/plot2-eps-converted-to.pdf}
\linespread{1.3}\selectfont{}
\renewcommand{\figurename}{Extended Data Figure}
\caption{\textbf{Cross section of ZTF J1406+1222's orbital solution in the Galaxy} Cross section of the Milky Way orbit of ZTF J1406+1222 over the course of 10 Gyr. The colours (green, orange and blue) indicate three different distances corresponding to our best distance estimate of $1140\pm200\,\rm pc$ and the 1-$\sigma$ upper and lower bounds of this distance estimate. }
\label{fig:Orbit2}
\end{figure}
\subsection{Searches for Radio Pulsations}
Black widow pulsars are capable of producing detectable radio pulsations (e.g. see~\cite{Fruchter+1988, Manchester+1991, Deich+1993, Ransom+2005, Ray+2013}), provided the radio emission is beamed toward Earth and it is not scattered or absorbed. These effects can be mitigated by performing observations at higher radio frequencies. To search for radio pulsations from ZTF~J1406+1222, we performed three radio observations with the NASA Deep Space Network~(DSN; see~\cite{Pearlman+2019a}) radio telescopes, which were carried out with the DSS-13 (34\,m diameter) and DSS-14 (70\,m diameter) antennae. The DSS-13 observation (epoch~1) was carried out for 56 minutes starting at 2020~July~02 21:57:00~UTC, at a center frequency of 2.26\,GHz ($S$-band) with a recording bandwidth of 110.625\,MHz. Two additional observations were performed with DSS-14 during separate epochs, with one starting at (epoch~2) 2020~July~13 22:02:24~UTC for a duration of 183~minutes at a center frequency of 1.54\,GHz ($L$-band) with a recording bandwidth of 320.625\,MHz, and another starting at (epoch~3) 2021~July~31 05:35:00~UTC for a duration of 75~minutes at a center frequency of 2.24\,GHz ($S$-band) with a recording bandwidth of 115.625\,MHz. During each observation, power spectral measurements were recorded across the band at high time resolution in a digital polyphase filterbank with a sampling time of 102.4\,$\mu$s and a frequency resolution of 0.625\,MHz.
The data processing procedures are similar to those described in earlier studies of pulsars and magnetars with the DSN~(e.g. see~\cite{Majid+2017, Pearlman+2018a, Pearlman+2019a, Pearlman+2020b}). We reduced the radio data by first identifying and masking data corrupted by radio frequency interference~(RFI) using the \texttt{rfifind} tool available in the \texttt{PRESTO} pulsar search package\cite{Ransom2001}. The data were then bandpass-corrected, and low frequency fluctuations in the baseline were removed by subtracting the moving average from each data point in each frequency channel, which was calculated using 100~ms of data around each time sample. Next, the sample times were corrected to the solar system barycenter~(SSB) using the \texttt{prepdata} tool available in \texttt{PRESTO} and JPL's DE405 ephemeris.
The maximum predicted DM contribution along the line-of-sight is 23.4/21.9\,pc\,cm$^{\text{--3}}$ according to the NE2001/YMW16 electron density models\cite{Cordes+2002, Yao+2017}, and there is likely an additional, unknown contribution to the DM from the dense, ionized plasma wind produced by evaporation of the companion. We chose to incoherently dedisperse the $L$-band and $S$-band data using dispersion measure~(DM) trials between 0 and 5000\,pc\,cm$^{\text{--3}}$. For each observation, the DM trial spacing was chosen to minimize the total dispersive smearing (e.g. due to dispersive smearing within each frequency channel, dispersive smearing across all frequency channels from the DM step-size, and the sampling time). When determining the dedispersion scheme, we ignored the effects of scattering, which would have resulted in larger DM step-sizes.
We searched for bright, astrophysical single pulses using a Fourier domain matched-filtering algorithm, where each dedispersed time series was convolved with boxcar functions with logarithmically spaced widths between 102.4\,$\mu$s and 30.72\,ms. Candidates identified from each DM trial with a signal-to-noise ratio~(S/N)\,$\geq$\,6 were saved and classified using the \texttt{FETCH} (Fast Extragalactic Transient Candidate Hunter) software package\cite{Agarwal+2020}, which uses a deep-learning convolutional neural network to identify astrophysical radio pulses. No astrophysical single pulses were detected during any of our radio observations. We place the following 6$\sigma$ upper limits on the peak flux density of single pulses from ZTF~J1406+1222, assuming a fiducial width of 1\,ms: (epoch~1) $<$\,1.88\,Jy at $S$-band, (epoch~2) $<$\,0.23\,Jy at $L$-band, and (epoch~3) $<$\,0.41\,Jy at $S$-band.
We also carried out a search for radio pulsations using \texttt{accelsearch}, a Fourier Domain Acceleration Search~(FDAS) pipeline available in \texttt{PRESTO}, which employs a matched-filtering algorithm to correct for Doppler smearing. The search was carried out by summing 16 harmonics and using the \texttt{-zmax 1200} option, which defines the maximum number of Fourier bins that the highest harmonic can linearly drift in the power spectrum (e.g. due to orbital motion). We carried out pulsation searches at $L$-band and $S$-band by independently searching overlapping data segments spanning 10\% of the orbital period (6.2\,min). Searching shorter data segments, spanning 10\% of the orbit, aids in mitigating the sensitivity loss due to orbital motion since the pulsar spin frequency experiences an approximately linear drift in the regime where $T_{\text{obs}}$\,$\lesssim$\,$P_{\text{orb}}$/10~\cite{Ransom+2003}. We also carried out searches for pulsations with periods between 1\,ms and 100\,s in each dedispersed time series segment using a GPU-accelerated Fast Folding Algorithm~(FFA). We folded the dedispersed data modulo each of the period candidates identified by the two algorithms, but we found no statistically significant signals with S/N\,$\geq$\,6 in any individual data segment. We place the following 6$\sigma$ upper limits on the radio flux density of ZTF~J1406+1222, assuming a duty cycle of 10\% and an integration time of 6.2\,min: (epoch~1) $<$\,0.7\,mJy at $S$-band, (epoch~2) $<$\,0.1\,mJy at $L$-band, and (epoch~3) $<$\,0.2\,mJy at $S$-band. These sensitivity limits are comparable to the typical luminosities of known pulsars at 1~kpc in the ATNF pulsar catalog\cite{Manchester2005}, and many pulsars in the catalog at this distance are below this luminosity threshold.
There are several possible explanations for the lack of radio pulsations observed from this system. It is possible that the black widow pulsar is either radio-quiet, producing radio pulsations that are below the above-mentioned detection thresholds, or its radio emission is not beamed toward Earth. Alternatively, if the pulsar is an aligned rotator, then it is unsurprising that pulsations were not be detected. At a distance of 1.14\,kpc, the predicted diffractive interstellar scintillation~(DISS) bandwidth is $\sim$8\,MHz and the predicted DISS timescale is $\sim$13~minutes at 1\,GHz (assuming a transverse velocity of 100\,km\,s$^{\text{--1}}$), according to NE2001\cite{Cordes+2002}. At $S$-band, the DISS bandwidth and DISS timescales are $\sim$177\,MHz and $\sim$30~minutes, respectively, assuming a scaling of $\Delta\nu_{\text{DISS}}$\,$\propto$\,$\nu^{\text{4}}$ and $t_{\text{DISS}}$\,$\propto$\,$\nu^{\text{1.2}}$. Since the scintillation bandwidth at $S$-band is larger than our observing bandwidth, it is possible that scintillation-induced modulation could have reduced the apparent flux density of the radio pulses to a level below our detection threshold. Since ZTF~J1406+1222 is a compact system, with an estimated semi-major axis of 0.59\,$R_{\odot}$ for a 1.4\,$M_{\odot}$ neutron star in a circular orbit with a 0.05\,$M_{\odot}$ companion, extreme scattering and eclipses due to the presence of gas flowing out from the irradiated companion is also likely a contributing factor to the detectability of the pulsar's radio emission.
\subsection{Formation Questions}
The wide tertiary companion and short inner binary orbital period of ZTF J1406+1222 are highly unusual amongst neutron star systems, prompting the question of whether these two peculiarities are related, and challenging formation models. The orbital period of 62 minutes is the shortest known for a black widow system, the previous record holder being PSR J1653.6-0159 with an orbital period of 75~minutes\cite{Romani2014}. Similar to cataclysmic variables, formation models for black widow and redback systems (e.g., \cite{Chen2013,Benvenuto2015,Ginzburg2020}) predict that hydrogen rich donors reach a minimum orbital period at $P_{orb} \approx 80 \, {\rm min}$, slightly longer than that of ZTF J1406+1222. Shorter minimum orbital periods are possible if the donor star is somewhat evolved\cite{Podsiadlowski2003,Benvenuto2014} (i.e., it has a helium-rich composition). Hence, the ablated component of ZTF J1406+1222 may have formed from a helium-rich star that began mass transfer near the end of its progenitor's main sequence evolution.
There is no obvious reason to believe the widely separated tertiary allowed the system to reach unusually short orbital periods. The dynamics of the inner binary would naively be decoupled from the influence of the outer tertiary once general relativistic precession occurs on a timescale shorter than the precession induced by the outer body, which will happen at orbital periods far greater than one hour. Similarly, any formation models involving tidal effects or mass transfer from the third body would probably not occur due to its very wide separation. The same applies to formation models invoking a three-body common envelope event\cite{Sabach2015}.
In fact, the wide tertiary companion in ZTF J1406+1222 poses serious problems for most formation models of the system. It is difficult to understand how the widely separated companion, with an orbital velocity of $\sim \! 1 \, {\rm km~s^{-1}}$, was not unbound due to a kick imparted to the inner binary upon the formation of the neutron star. In formation models of the pulsar triple system PSR J0337+1715\cite{Tauris2014}, the outermost orbit has a period of $P \! \approx \! 17 \, {\rm d}$ (orbital velocity of $\sim \! 100$ km/s) at the time of the supernova, so the system would have more easily stayed bound. Appealing to extremely small supernova kicks\cite{Igoshev2019} is not sufficient for ZTF J1406+1222 to have remained bound, as explained below.
In standard binary formation channels of spider systems and low-mass X-ray binaries, the binary evolution begins with a massive ($M_1 \sim 10-20 \, \rm M_\odot$) star at moderate orbital separation ($a \sim 1 \, {\rm AU}$) with a low-mass ($M_2 \sim 1 \, \rm M_\odot$) companion star. A common envelope phase occurs after the primary expands into a red supergiant, after which the binary is composed of the helium core ($M_1 \sim 3-6~\rm M_\odot$) of the primary in a $\sim \! 1 \, {\rm day}$ orbital period with the low-mass companion. The helium star undergoes a core-collapse explosion to form a neutron star, ejecting its remaining helium/carbon/oxygen envelope, and possibly being kicked in the process.
In the absence of significant envelope stripping via case-BB mass transfer after core He-burning\cite{Tauris2015} (as expected for higher mass helium stars), a few solar masses of material is expected to be ejected during the explosion. This is comparable to the remaining mass of the system, and thus the inner binary could become unbound due to the supernova mass loss, even in the absence of a kick. The instantaneous mass loss will also kick the centre of mass of the inner binary by a velocity $\Delta v = v_{\rm He} (M_{\rm ej}/M_{\rm final})$, where $v_{\rm He} = \sqrt{G(M_1+M_2)/a}[M_2/(M_1+M_2)]$ is the helium star orbital velocity before the explosion, $M_{\rm ej}$ is the ejecta mass and $M_{\rm final} = M_1 + M_2 - M_{\rm ej}$ is the post-explosion mass of the system. For a typical pre-explosion configuration of $M_1 \sim 4 \, \rm M_\odot$, $M_2 \sim 1 \, \rm M_\odot$, $M_{\rm ej} \sim 2.5 \, \rm M_\odot$ and $a \sim 7 \, \rm R_\odot$, we thus expect a kick of $\Delta v \sim v_{\rm He} \sim 70 \, {\rm km~s^{-1}}$. This is far larger than the orbital velocity of the low-mass tertiary object in ZTF J1406+1222, meaning that the third body should become unbound for any choice of parameters near our fiducial values.
There are a few ways in which the net kick could be lower than these estimates. If the companion mass $M_2$ at the time of core-collapse is much smaller, e.g. $M_2 \sim 0.1 \, \rm M_\odot$, this would lower the kick velocity by a factor of $\sim \!10$ to several km~s$^{-1}$, but this would likely still unbind the tertiary. If the ejecta mass was much smaller, e.g. $M_{\rm ej} \sim 0.25 \, \rm M_\odot$ as would be expected for a low-mass helium star that lost most of its envelope via case-BB mass transfer\cite{Tauris2015}, this would lower the kick by a factor of $\sim \! 10$, but once again the tertiary would likely become unbound. A combination of these two mechanisms could potentially operate, but one would expect the companion to accrete some of the transferred mass such that its pre-explosion mass increases well above $0.1 \, \rm M_\odot$. The mass loss kick could be diminished if the pre-explosion orbital period is longer than one day, but it can only be a few times larger in order to ensure that subsequent magnetic braking can shrink the system to the observed short period, which is not enough to greatly decrease the kick velocity. Finally, it is possible that a supernova kick to the neutron star could cancel out the mass loss kick so that the overall kick to the inner binary is very small. But given the typical kick scale of a few hundred kilometers per second, great fine tuning would be required to reduce the binary motion to less than a few km~s$^{-1}$.
Since none of the solutions above are particularly compelling, other formation mechanisms should be considered. One possibility is that the inner binary of ZTF J1406+1222 had a much larger orbital separation than $a \sim 7 \, \rm R_\odot$, e.g., if it never went through a common envelope phase. This would entail a much smaller kick due to mass loss upon core-collapse, perhaps allowing the tertiary to remain bound. Given the right orbital inclination of the tertiary, the inner binary could be hardened by high-eccentricity migration\cite{Naoz2016} (e.g. Kozai oscillations). Upon tidal orbital circularization to short periods, magnetic braking and gravitational wave emission would then bring the system to the short orbital period observed. However, this mechanism would still require an exceptionally small supernova kick (less than a few km~s$^{-1}$) to the neutron star in order for the tertiary to remain bound, which may be unlikely.
Another possibility is that this system was dynamically assembled and ejected from a globular cluster. In order to dynamically eject a bound triple system, a four-body interaction would be required. For instance, a binary-binary interaction could have occurred, ejecting one star and forming a bound triple system, while also imparting enough momentum to the triple system to eject it from the cluster. However, that scenario may still require some fine tuning to retain a widely separated third body whose orbital velocity is much less than a typical globular cluster escape velocity. A remarkably similar system is PSR J1024-0719\cite{Kaplan2016}, which also has a low-mass low-metallicity widely separated companion star, and was suggested to have been ejected from a globular cluster.
If ZTF J1406+1222 formed in the Galactic field, perhaps the most appealing solution is that the neutron star in ZTF J1406+1222 was formed by AIC rather than a core-collapse explosion. In this scenario, the neutron star was formed from a white dwarf that grew to larger than the Chandrasekhar mass $M_{\rm Ch} \sim 1.4 \, \rm M_\odot$ after accreting from the binary companion. In this case, the decrease in gravitational mass of the neutron star upon AIC is $\sim \! 0.1 \, \rm M_\odot$, resulting in a kick to the centre of motion of the binary of a few ${\rm km~s^{-1}}$ (or less for $M_2 < 1 \, \rm M_\odot$), small enough that the tertiary is more likely to remain bound. The neutron star could become a millisecond pulsar immediately (depending on the spin rate of the accreting white dwarf) or after further mass accretion from the binary companion, creating the spider system that we observe today. More sophisticated modelling of the possibilities outlined above should be performed in order to understand the formation of ZTF J1406+1222.
\end{methods}
\section{Data Availability}
Reduced HiPERCAM photometric data and LRIS spectroscopic data are availible at \\ https://github.com/kburdge/ZTFJ1406-1222. The X-ray observations already in the public domain, and their observation IDs have been supplied in the text. The ZTF data is also in the public domain. The proprietary period for the spectroscopic data will expire at the start of 2022, at which point the raw spectroscopic images will also be accessible via the Keck observatory archive.
\section{Code Availability}
Upon request, the first author will provide code (primarily in python) used to analyze the observations, and any data used to generate figures (MATLAB was used to generate most of the figures).
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\section{Introduction} \label{sec:intro}
The fluctuations of virtual charged particles in the vacuum of quantum electrodynamics (QED) give rise to nonlinear, effective couplings between electromagnetic fields.
While this has been realized theoretically already in the early days of QED \cite{Heisenberg:1935qt,Weisskopf}, the pure electromagnetic nonlinearity of the quantum vacuum still awaits its direct experimental verification on macroscopic scales.
The advent and planning of high-intensity laser facilities of the petawatt class has triggered a huge interest in ideas and proposals to probe quantum vacuum nonlinearities in realistic all-optical experimental set-ups;
for recent reviews, see \cite{Dittrich:2000zu,Marklund:2008gj,Dunne:2008kc,Heinzl:2008an,DiPiazza:2011tq}.
Typical examples are proposals intended to verify vacuum birefringence \cite{Toll:1952,Baier,BialynickaBirula:1970vy,Adler:1971wn} that can be searched for using macroscopic magnetic fields
\cite{Cantatore:2008zz,Berceau:2011zz} or with the aid of high-intensity lasers \cite{Heinzl:2006xc}, see also \cite{Dinu:2013gaa}.
Alternative concepts suggest the use of time-varying fields and high-precision interferometry \cite{Zavattini:2008cr,Dobrich:2009kd,Grote:2014hja}.
Other commonly studied nonlinear vacuum effects are direct light-by-light scattering \cite{Euler:1935zz,Karplus:1950zz}, photon splitting \cite{Adler:1971wn}, and spontaneous vacuum decay in terms
of Schwinger pair-production in electric fields \cite{Sauter:1931zz,Heisenberg:1935qt,Schwinger:1951nm}.
Further optical signatures of quantum vacuum nonlinearities are those based on interference effects \cite{King:2013am,Tommasini:2010fb,Hatsagortsyan:2011}, photon-photon scattering in the form of laser-pulse collisions \cite{King:2012aw,Lundin:2006wu}, quantum reflection \cite{Gies:2013yxa}, photon merging \cite{Gies:2014jia},
and harmonic generation from laser-driven vacuum \cite{DiPiazza:2005jc,Fedotov:2006ii}. Related effects have also been discussed in the context of searching for minicharged particles \cite{Villalba-Chavez:2013txu}.
In this paper we study the phenomenon of stimulated photon emission from the vacuum in the presence of a strong space-time dependent electromagnetic field (cf. also \cite{Galtsov:1971xm}).
Focusing on a comparably simple electromagnetic field profile, which should nevertheless capture the essential features of the electromagnetic fields generated in the focal spots of real high-intensity laser beams,
we provide estimates for the numbers of emitted photons attainable with present and near future high-intensity laser facilities.
The experimental set-up we have in mind is as follows: A high-intensity laser pulse is split equally into two pulses, which are separated and directed in a counter-propagation geometry.
Both pulses are focused such that they evolve along the well-defined envelope of a Gaussian beam and their foci overlap. This results in a macroscopic strong-field region about the beam waist (cf. also Fig.~\ref{fig:cartoon}, below).
The superposition of the two counter-propagating electromagnetic waves results in a standing electromagnetic wave which -- in contrast to a single plane wave -- is characterized by at least one nonzero electromagnetic field invariant.
The idea is to look for induced photons emitted from the strong-field region and to be detected in the field free region. These photons can be considered as emitted from the vacuum subjected to the space-time dependent macroscopic laser field -- whose microscopic composition in terms of laser photons is not resolved --
enabling and stimulating the emission process.
Of course, this scenario can alternatively be interpreted in terms of microscopic laser photon scattering and deflection in the collision of two laser pulses.
From this perspective, the emitted photons correspond to the outgoing photons carrying the imprint of the collision process, i.e., outgoing photons whose properties (in particular their polarization characteristics and propagation directions) differ from the incident laser photons brought into collision.
However, we believe that viewing laser pulse collision processes in terms of a stimulated emission process, i.e., viewing the laser pulses as macroscopic fields, rather than in terms of the constituting laser photons, allows for a particularly intuitive and elegant theoretical treatment.
In this framework it is easy to vary detector sizes and ask for the number of photons carrying the signature of vacuum nonlinearity to be registered in any given solid angle interval, which is not so straightforward in other approaches. In addition, and in contrast to previous studies, e.g., \cite{King:2012aw,Lundin:2006wu}, we can straightforwardly study the polarization properties of the outgoing photons.
Moreover, and from a conceptual level even more important, our approach will also allow us to study photon emission from the vacuum subjected to macroscopic field configurations which are hard to describe as a collection of photons,
like, e.g., rotating inhomogeneous magnetic fields.
Our paper is organized as follows: In Sec.~\ref{sec:calculation} we outline the derivation of the stimulated photon emission rate, and provide explicit analytical results for a particular electromagnetic field configuration,
mimicking the superposition of two counter-propagating laser pulses with the same characteristics. In the diffraction limit these expressions are of a particularly simple form.
Most strikingly, the directional emission characteristics of the induced photons becomes independent of the laser parameters and is described by a generic function.
The number of photons emitted in a specific spherical angle is obtained straightforwardly upon integration of the directional emission characteristics and multiplication with an overall factor determined by the parameters of the used lasers.
Hence, we can easily provide estimates of the number of emitted photons for any desired laser parameters. Section~\ref{seq:Ex+Res} is devoted to the discussion of some explicit results.
We end with conclusions and an outlook in Sec.~\ref{seq:Con+Out}.
\section{Calculation} \label{sec:calculation}
Starting point of our calculation is the one-loop effective Lagrangian in constant external electromagnetic fields (``Heisenberg-Euler effective Lagrangian'') \cite{Heisenberg:1935qt}.
It can be compactly represented as \cite{Schwinger:1951nm} (cf. also \cite{Dittrich:2000zu,Jentschura:2001qr}),
\begin{equation}
{\cal L}=\frac{e^2}{8\pi^2}\int_{-i\eta}^{\infty-i\eta}\frac{{\rm d}s}{s}\,{\rm e}^{-i(m^2-i\epsilon)s}\left[ab\coth(eas)\cot(ebs)-\frac{a^2-b^2}{3}
-\frac{1}{(es)^2}\right], \label{eq:effL}
\end{equation}
with $\{\epsilon,\eta\}\to0^+$, elementary charge $e$ and electron mass $m$.
The secular invariants
\begin{equation}
a=(\sqrt{{\cal F}^2+{\cal G}^2}-{\cal F})^{1/2}\,, \quad b=(\sqrt{{\cal F}^2+{\cal G}^2}+{\cal F})^{1/2}\,,
\end{equation}
are made up of the gauge and Lorentz invariants of the electromagnetic field,
\begin{equation}
{\cal F}=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}=\frac{1}{2}(\vec{B}^2-\vec{E}^2)\,, \quad {\cal G}=\frac{1}{4}F_{\mu\nu}{}^*F^{\mu\nu}=-\vec{E}\cdot\vec{B}\,. \label{eq:invs}
\end{equation}
Here $^*F^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$ denotes the dual field strength tensor, and $\epsilon^{\mu\nu\alpha\beta}$ is the totally antisymmetric tensor; $\epsilon^{0123}=1$.
Our metric convention is $g_{\mu \nu}=\mathrm{diag}(-1,+1,+1,+1)$, and we use $c=\hbar=1$.
For completeness note that $ab=\sqrt{{\cal G}^2}$, $a^2-b^2=-2{\cal F}$ and $a^2+b^2=2\sqrt{{\cal F}^2+{\cal G}^2}$.
Strictly speaking, the Heisenberg-Euler Lagrangian~\eqref{eq:effL} describes the effective nonlinear interactions between constant electromagnetic fields mediated by electron-positron fluctuations in the vacuum.
The typical spatial (temporal) extents to be probed by these fluctuations are of the order of the Compton wavelength (time) $\lambda_c=\tau_c=1/m$, with $\lambda_c=3.86\cdot10^{-13}{\rm m}$ and $\tau_c=1.29\cdot10^{-21}{\rm s}$.
Hence, \Eqref{eq:effL} can also be adopted for inhomogeneous electromagnetic fields whose typical spatial (temporal) variation is on scales much larger than the Compton wavelength (time),
i.e., for {\it soft} electromagnetic fields that may locally be approximated by a constant.
Many electromagnetic fields available in the laboratory are compatible with this requirement.
Within the above restrictions, \Eqref{eq:effL} can serve as a starting point to study the effective interaction between dynamical photons and inhomogeneous background electromagnetic fields.
For this purpose it is convenient to decompose the electromagnetic field strength tensor $F^{\mu\nu}$ introduced above as $F^{\mu\nu}\to F^{\mu\nu}(x)+f^{\mu\nu}(x)$ into the field strength tensor of the background field $F^{\mu\nu}(x)$ and the photon field strength tensor $f^{\mu\nu}(x)$ \cite{BialynickaBirula:1970vy}.
To linear order in $f\equiv f^{\mu\nu}$, the Lagrangian can then be compactly written as
\begin{equation}
{\cal L}=f^{\mu\nu}(x)\frac{\partial{\cal L}}{\partial F^{\mu\nu}}(x)+{\cal O}(f^2)\,. \label{eq:Seff}
\end{equation}
Here we neglected higher-order terms with two or more photons.
Equation~\eqref{eq:effL} is straightforwardly differentiated with respect to $F^{\mu\nu}$, yielding
\begin{equation}
\frac{\partial{\cal L}}{\partial F^{\mu\nu}}
=\frac{1}{2}\frac{1}{a^2+b^2}\left[\left(b\frac{\partial{\cal L}}{\partial b}-a\frac{\partial{\cal L}}{\partial a}\right)F_{\mu\nu}
+{\cal G}\left(\frac{1}{b}\frac{\partial{\cal L}}{\partial b}+\frac{1}{a}\frac{\partial{\cal L}}{\partial a}\right){}^*F_{\mu\nu}\right].
\end{equation}
In particular at leading order in a double expansion of the integrand in \Eqref{eq:effL} in terms of $a$ and $b$
the propertime integral can be performed easily, resulting in
\begin{equation}
{\cal L}=\frac{e^2}{8\pi^2}\frac{1}{45}\frac{e^2}{m^4}\Bigl[(a^2+b^2)^2+3(ab)^2+ {\cal O}(\varepsilon^6)\Bigr], \label{eq:effLpert}
\end{equation}
with ${\cal O}(a)\sim{\cal O}(b)\sim{\cal O}(\varepsilon)$;
cf. also \cite{Dunne:2004nc} providing the weak field expansion coefficients of the Heisenberg-Euler effective Lagrangian explicitly to all orders.
From \Eqref{eq:effLpert} we obtain the compact expression
\begin{equation}
\frac{\partial{\cal L}}{\partial F^{\mu\nu}}
=\frac{e^2}{8\pi^2}\frac{1}{45}\frac{e^2}{m^4}\bigl[4{\cal F}F_{\mu\nu}
+7{\cal G}{}^*F_{\mu\nu} \bigr] +{\cal O}(\varepsilon^5)\,, \label{eq:delLdelF}
\end{equation}
where we counted $F^{\mu\nu}$ and ${}^*F^{\mu\nu}$ as ${\cal O}(\varepsilon)$.
In our explicit calculations to be performed subsequently for an all-optical laser experiment, we will always limit ourselves to the leading order terms given explicitly in \Eqref{eq:delLdelF}.
As the field strengths attainable in present and near future high-intensity laser facilities are small in comparison to the {\it critical field strength} $E_{\rm cr}\equiv\frac{m^2}{e}$ \cite{Heisenberg:1935qt},
i.e., $\{\frac{eE}{m^2},\frac{eB}{m^2}\}\ll1$, this approximation is well justified.
The amplitude for emission of a single photon with momentum $\vec{k}$ from the vacuum subjected to the background electromagnetic field $F^{\mu\nu}(x)$ is given by
\begin{equation}
{\cal S}_{(p)}(\vec{k})\equiv\langle\gamma_{p}(\vec{k})|\int{\rm d}^4x\, f^{\mu\nu}(x)\frac{\partial{\cal L}}{\partial F^{\mu\nu}}(x)|0\rangle\,, \label{eq:S}
\end{equation}
with the single photon state denoted by $|\gamma_{p}(\vec{k})\rangle\equiv a^\dag_{\vec{k},p}|0\rangle$ (cf. Fig.~\ref{fig:Feyndiag}). Here $p$ denotes the polarization of the emitted photons.
\begin{figure}
\center
\includegraphics[width=0.3\textwidth]{Fig0}
\caption{Leading order contribution to the stimulated photon emission process in the weak field limit.
The process is cubic in the background field represented by the wiggly lines ending at crosses; cf. Eqs.~\eqref{eq:delLdelF} and \eqref{eq:S}. It results in outgoing photons of wave vector $\vec{k}$ and polarization $p$.}
\label{fig:Feyndiag}
\end{figure}
Representing the photon field in Lorentz gauge as
\begin{equation}
a^{\mu}(x)=\sum_{p}\int\frac{{\rm d}^3{k}}{(2\pi)^3}\frac{1}{\sqrt{2k^0}}\,\Bigl(\epsilon^{\mu}_{(p)}(k)\,{\rm e}^{-ikx}\,a_{\vec{k},p}+\epsilon^{*\mu}_{(p)}(k)\,{\rm e}^{ikx}\,a^\dag_{\vec{k},p}\Bigl) ,
\end{equation}
where $k^0\equiv|\vec{k}|$, and the sum is over the two physical (transversal) photon polarizations,
we obtain
\begin{equation}
{\cal S}_{(p)}(\vec{k})
=\frac{i}{\sqrt{2k^0}} \, \hat f^{\mu\nu}_{(p)}(k)\, \frac{\partial{\cal L}}{\partial F^{\mu\nu}}(-k)\,, \label{eq:Sp}
\end{equation}
where $\frac{\partial{\cal L}}{\partial F^{\mu\nu}}(k)=\int{\rm d}^4 x\,{\rm e}^{-ikx}\frac{\partial{\cal L}}{\partial F^{\mu\nu}}(x)$, and we made use of the shorthand notation $\hat f^{\mu\nu}_{(p)}(k)=k^\mu\epsilon^{*\nu}_{(p)}(k)- k^\nu\epsilon^{*\mu}_{(p)}(k)$.
In the vicinity of its beam waist the electromagnetic field of a Gaussian laser beam, corresponding to a fundamental transverse electromagnetic ${\rm TEM}_{00}$ mode, polarized along $\vec{e}_{\rm x}$ and propagating along $\pm\vec{e}_{\rm z}$ can be approximately modeled by the following field configuration
\begin{gather}
\vec{E}_\pm(x)=\pm{\cal E}\,\vec{e}_{\rm x}\,{\rm e}^{-\frac{4({\rm x}^2+{\rm y}^2)}{w^2}}{\rm e}^{-\frac{{\rm z}^2}{\zeta^2}}\cos\bigl(\Omega(t\mp {\rm z})\bigr), \nonumber\\
\vec{B}_\pm(x)={\cal E}\,\vec{e}_{\rm y}\,{\rm e}^{-\frac{4({\rm x}^2+{\rm y}^2)}{w^2}}{\rm e}^{-\frac{{\rm z}^2}{\zeta^2}}\cos\bigl(\Omega(t\mp {\rm z})\bigr), \label{eq:beamfields}
\end{gather}
i.e., orthogonal electric and magnetic fields, which -- for given space-time coordinates -- are of the same magnitude, and become maximum for ${\rm x}={\rm y}={\rm z}=0$ (and $t=0$).
Here, we have chosen the orientation of the electric and magnetic fields in such a way that the magnetic field vector points in the same direction ($\vec{e}_{\rm y}$) for both propagation directions $\pm\vec{e}_{\rm z}$;
${\cal E}>0$ denotes the electric/magnetic field amplitude.
The transversal field profile in \Eqref{eq:beamfields} is a Gaussian characterized by its full width $w$ at ${\rm e}^{-1}$ of its maximum.
In longitudinal direction the fields feature a plane-wave type modulation of frequency $\Omega>0$; wavelength $\lambda=\frac{2\pi}{\Omega}$.
Without loss of generality, the beam waist is assumed to be located at $z=0$, such that the Gaussian envelope $\sim{\rm exp}(-{\rm z}^2/\zeta^2)$ can be seen as mimicking the decrease of the field over the Rayleigh range $z_{\rm R}$ which is of the order of $\zeta$.
Note that for real Gaussian beams (for ${\rm x}={\rm y}=0$) the field decrease over the Rayleigh range is described by a Lorentzian profile, which is harder to tackle analytically and thus, would result in less transparent and handy expressions for the vacuum emission probability.
We argue that for our purposes the Gaussian profile captures all relevant features, and -- when providing experimental estimates below -- will actually identify $\zeta=z_{\rm R}$.
Moreover, we neglect diffraction spreading and wavefront curvature effects about the beam waist, arguing that within the Rayleigh range they amount to subleading corrections. Outside the Rayleigh range the fields~\eqref{eq:beamfields} rapidly drop to zero.
High-intensity lasers deliver multicycle pulses of finite duration $\tau$. The pulse duration is also not accounted for explicitly here. Given that $\tau\gg\lambda$, which is typically fulfilled for near infrared high-intensity lasers
(cf. also Tab.~\ref{tab:lasers}, below) whose pulse duration is $\gtrsim$ tens of femtoseconds and wavelength of the order of $1000$ nanometers, and $\frac{\tau[{\rm fs}]}{\lambda[{\rm nm}]}\approx 300$,
this is justified.
The time scale $\tau$ enters our calculation only as a measure of the interaction time (cf. below).
Let us emphasize that both invariants~\eqref{eq:invs} vanish for a single Gaussian laser beam, modeled by one of the field configurations labeled by $\pm$ in \Eqref{eq:beamfields}.
However, nonzero invariants are attainable by superimposing multiple, e.g., two, Gaussian beams.
Note that macroscopic, non-vanishing invariants could also be realized by a single laser beam if higher laser modes are utilized.
However, in this case the focus area is increased in comparison to the ${\rm TEM}_{00}$ mode and correspondingly the available laser intensity diminished.
Another option is to consider a single Gaussian beam in the limit of a substantial beam divergence $\theta\to\frac{\pi}{2}$ \cite{Monden:2011}, such that wavefront curvature effects become dominant and cannot be neglected; cf. also \cite{Paredes:2014oxa}. Of course, under theses circumstances the laser beam does no longer correspond to a slight modification of a plane-wave like electromagnetic field configuration and both invariants~\eqref{eq:invs} can assume nonzero values, facilitating stimulated photon emission from the vacuum.
At least one invariant can be rendered nonzero by superimposing the two counter-propagating laser beams introduced in \Eqref{eq:beamfields} above.
The resulting electric and magnetic fields amount to standing waves and read
\begin{gather}
\vec{E}(x)=\vec{E}_+(x)+\vec{E}_-(x)=2{\cal E}\,\vec{e}_{\rm x}\,{\rm e}^{-\frac{4({\rm x}^2+{\rm y}^2)}{w^2}}{\rm e}^{-\frac{{\rm z}^2}{\zeta^2}}\sin(\Omega t)\sin(\Omega {\rm z})\,, \nonumber\\
\vec{B}(x)=\vec{B}_+(x)+\vec{B}_-(x)=2{\cal E}\,\vec{e}_{\rm y}\,{\rm e}^{-\frac{4({\rm x}^2+{\rm y}^2)}{w^2}}{\rm e}^{-\frac{{\rm z}^2}{\zeta^2}}\cos(\Omega t)\cos(\Omega {\rm z})\,. \label{eq:resfields}
\end{gather}
Figure~\ref{fig:cartoon} is a cartoon of the experimental situation we have in mind.
\begin{figure}
\center
\includegraphics[width=0.2\textwidth]{Fig1}
\caption{Two counter-propagating Gaussian laser beams with wave vectors $\vec{k}_{\pm}=\pm\vec{e}_{\rm z}$ (electric and magnetic field vectors $\vec{E}_\pm$ and $\vec{B}_\pm$)
are superimposed to form a standing electromagnetic wave about the beam focus with nonvanishing field invariant $\cal F$.
The idea is to look for induced photons emitted from this strong-field region, to be detected in the field free region.
For later reference, we also depict the beam divergence $\theta$.}
\label{fig:cartoon}
\end{figure}
For the particular electromagnetic field configuration~\eqref{eq:resfields}, the invariants~\eqref{eq:invs} are
\begin{equation}
{\cal F}(x)={\cal E}^2\,{\rm e}^{-\frac{8({\rm x}^2+{\rm y}^2)}{w^2}}{\rm e}^{-\frac{2{\rm z}^2}{\zeta^2}}\bigl[\cos(2\Omega t)+\cos(2\Omega z)\bigr]\,,\quad {\cal G}=0\,,
\end{equation}
and all components of the field strength tensor $F_{\mu\nu}$ apart from
\begin{equation}
F_{10}(x)=-F_{01}(x)=\vec{e}_{\rm x}\cdot\vec{E}(x)\,,\quad F_{31}(x)=-F_{13}(x)=\vec{e}_{\rm y}\cdot\vec{B}(x)\,,
\end{equation}
vanish.
Thus, the emission amplitude~\eqref{eq:Sp} can be expressed concisely as
\begin{equation}
{\cal S}_{(p)}(\vec{k})
=\frac{2i}{\sqrt{2k^0}}\biggl(\hat f^{10}_{(p)}(k)\,\frac{\partial{\cal L}}{\partial F^{10}}(-k)
+\hat f^{31}_{(p)}(k)\,\frac{\partial{\cal L}}{\partial F^{31}}(-k)\biggr)\,, \label{eq:Sp1031}
\end{equation}
with
\begin{equation}
\frac{\partial{\cal L}}{\partial F^{\mu\nu}}(-k)
=\frac{e^2}{\pi^2}\frac{1}{90}\frac{e^2}{m^4}\int{\rm d}^4x\,{\rm e}^{ikx} {\cal F}(x)F_{\mu\nu}(x) + {\cal O}(\varepsilon^5) \,. \label{eq:delLdelFexpl}
\end{equation}
The Fourier integrals in \Eqref{eq:delLdelFexpl} can be performed straightforwardly.
The integration over time yields $\delta$ functions and the spatial integrations are of Gaussian type.
As the resulting expressions are not very elucidating we do not reproduce them here.
For the following discussion it is convenient to switch to spherical momentum coordinates $\vec{k}={\rm k}\hat{\vec{k}}$, where ${\rm k}=\sqrt{k_{\rm x}^2+k_{\rm y}^2+k_{\rm z}^2}$ and $\hat{\vec{k}}=(\cos\varphi\sin\vartheta,\sin\varphi\sin\vartheta,\cos\vartheta)$,
with $\varphi\in[0\ldots2\pi)$ and $\vartheta\in[0\ldots\pi]$.
The orthogonal vectors to $\vec{k}$ can then be parameterized by a single angle $\beta\in[0\ldots2\pi)$,
\begin{equation}
\vec{e}_{\perp,\beta}=
\left(\begin{array}{c}
\cos\varphi\cos\vartheta\cos\beta-\sin\varphi\sin\beta \\
\sin\varphi\cos\vartheta\cos\beta+\cos\varphi\sin\beta \\
-\sin\vartheta\cos\beta
\end{array}\right) . \label{eq:e_perpbeta}
\end{equation}
These vectors live in the tangent space of the unit sphere.
Correspondingly, the two transverse polarization modes of photons with wave vector $\vec{k}$ can be spanned by two orthonormalized four-vectors $\epsilon^\mu_{(p)}(\vec{k})$, with $p\in\{1,2\}$,
\begin{equation}
\epsilon^\mu_{(1)}(\vec{k})=(0,\vec{e}_{\perp,\beta}) \quad\textrm{and}\quad \epsilon^\mu_{(2)}(\vec{k})=\epsilon^\mu_{(1)}(\vec{k})\big|_{\beta\to\beta+\frac{\pi}{2}}\,, \label{eq:epsilons}
\end{equation}
representing linear polarization states in the specific basis characterized by a particular choice of $\beta$. In this work we exclusively focus on linear polarization modes.
Polarizations other than linear can be obtained through linear combinations of the vectors~\eqref{eq:epsilons}.
Resorting to these definitions, the $10$ and $31$ entries of $\hat f^{\mu\nu}_{(p)}(k)$ entering \Eqref{eq:Sp1031} read
\begin{gather}
\hat f^{10}_{(1)}(k)
= {\rm k} \bigl(\sin\varphi\sin\beta-\cos\varphi \,u \cos\beta\bigr),
\quad \hat f^{10}_{(2)}(k)=\hat f^{10}_{(1)}(k)\big|_{\beta\to\beta+\frac{\pi}{2}} , \\
\hat f^{31}_{(1)}(k)
= {\rm k}\bigl( \cos\varphi\cos\beta-\sin\varphi \,u \sin\beta \bigr),
\quad \hat f^{31}_{(2)}(k)=\hat f^{31}_{(1)}(k)\big|_{\beta\to\beta+\frac{\pi}{2}} ,
\end{gather}
where we made use of the shorthand notation $u\equiv\cos\vartheta$.
According to Fermi's golden rule, the number of induced photons with polarization $p\in\{1,2\}$ and momentum ${\rm k}$ in the interval ${\rm dk}$ emitted in the solid angle interval ${\rm d}u {\rm d}\varphi$
is obtained from the modulus squared of \Eqref{eq:Sp} as $\frac{1}{(2\pi)^3}\bigl|{\rm k}\,{\cal S}_{(p)}(\vec{k})\bigr|^2 {\rm dk}\,{\rm d}u\,{\rm d}\varphi$.
A straightforward but somewhat tedious calculation yields the following expressions for the modulus squared of ${\rm k}\,{\cal S}_{(p)}(\vec{k})$,
\begin{multline}
\bigl|{\rm k}\,{\cal S}_{(1)}(u,\varphi)\bigr|^2
=\frac{\pi^2\,\alpha}{218700}\left(\frac{e{\cal E}}{m^2}\right)^6 \left(\frac{w}{2}m\right)^4\,{\rm k}^3\,\zeta^2\,{\rm e}^{-\frac{\zeta^2\Omega^2+(\frac{w}{2})^2{\rm k}^2}{6}}\,{\rm e}^{-\frac{[\zeta^2-(\frac{w}{2})^2]{\rm k}^2}{6}u^2} \\
\times\sum_{n=\pm1}\sum_{l=\pm1}\biggl\{\biggl[
\sum_{j=0,2} c_j\,{\rm e}^{+\tfrac{lj}{3}\zeta^2{\rm k}\Omega u}(1-u^2)\bigl[\cos(2\beta) + \cos(2\varphi)\bigr] \\
+\sum_{j=1,3}c_j\,{\rm e}^{+\tfrac{lj}{3}\zeta^2{\rm k}\Omega u}(1-unl)^2[1+\cos(2\varphi-2nl\beta)] \\
+{\rm e}^{+\tfrac{l}{3}\zeta^2{\rm k}\Omega u}\,8nl\Bigl[2u[1+\cos(2\varphi)\cos(2\beta)]-(1+u^2) \sin(2\varphi)\sin(2\beta)\Bigr]\biggr]\delta^2({\rm k}-n\Omega) \\
+\biggl[(1-u^2)\bigl[\cos(2\beta) + \cos(2\varphi)\bigr]\\
+{\rm e}^{+\tfrac{l}{3}\zeta^2{\rm k}\Omega u}(1-unl)^2[1+\cos(2\varphi-2nl\beta)] \biggr]\,\delta^2({\rm k}-3n\Omega)
\biggr\}, \label{eq:S^2}
\end{multline}
and $\bigl|{\rm k}\,{\cal S}_{(2)}(u,\varphi)\bigr|^2=\bigl|{\rm k}\,{\cal S}_{(1)}(u,\varphi)\bigr|^2\big|_{\beta\to\beta+\frac{\pi}{2}}$,
with $\alpha=\frac{e^2}{4\pi}$ and coefficients
\begin{equation}
c_0=4+{\rm e}^{-\frac{4}{3}\zeta^2\Omega^2}, \quad c_1=4+4{\rm e}^{-\frac{2}{3}\zeta^2\Omega^2}\,, \quad c_2= 4{\rm e}^{-\frac{2}{3}\zeta^2\Omega^2}, \quad c_3={\rm e}^{-\frac{4}{3}\zeta^2\Omega^2}.
\end{equation}
Evidently, only photons with the two distinct frequencies $\omega\in\{\Omega,3\Omega\}$ are induced.
This is in agreement with elementary physical reasoning: In a Feynman diagrammatic expansion of the effective Lagrangian~\eqref{eq:effL}, the leading terms~\eqref{eq:effLpert} taken into account by us
actually amount to an effective four-photon interaction. Our electromagnetic background field configuration~\eqref{eq:resfields} modeling the counter-propagating laser beams is characterized by a single frequency scale $\Omega$.
Each coupling to the background field configuration can be seen as coupling to a laser photon of frequency $\Omega$. The stimulated emission process involves three laser photons.
Three laser photons can either give rise to a emitted photon of frequency $\Omega$ (two laser photons are scattered into one laser photon and one photon to be emitted) or merge to form a $3\Omega$ photon.
Hence, upon performing the integration over all possible values of ${\rm k}\in[0\ldots\infty)$ it is convenient to decompose the total number density $\rho_{(p)}(u,\varphi) \equiv \frac{1}{(2\pi)^3}\int_0^\infty{\rm dk}\,|{\rm k}\,S_{(p)}(u,\varphi)|^2$ of induced photons polarized in mode $p$ and emitted in $(u,\varphi)$ direction as
\begin{equation}
\rho_{(p)}(u,\varphi)=\rho^\Omega_{(p)}(u,\varphi)+\rho^{3\Omega}_{(p)}(u,\varphi),
\end{equation}
where $\rho^{\,\omega}_{(p)}(u,\varphi)$ refers to the number density of induced frequency $\omega$ photons.
These quantities are obtained straightforwardly from \Eqref{eq:S^2}, employing that $\delta^2({\rm k}-\omega)=\frac{\tau}{2\pi}\,\delta({\rm k}-\omega)$, with $\tau$ denoting the time scale of the interaction.
Aiming at the number of photons per laser shot originating from the stimulated emission process, we identify this time scale with the laser pulse duration.
For the $p=1$ polarization mode they read
\begin{equation}
\rho_{(1)}^{\omega}(u,\varphi)=\left(\frac{e{\cal E}}{m^2}\right)^6 \left(\frac{w}{2}m\right)^4\frac{\Omega \tau}{2\pi}\, h_{(1)}^{\omega}(u,\varphi)\,, \label{eq:rho1}
\end{equation}
where
\begin{multline}
h_{(1)}^{\Omega}(u,\varphi)=\frac{\alpha}{1749600\pi}(\Omega\zeta)^2\,{\rm e}^{-\frac{[\zeta^2+(\frac{w}{2})^2]\Omega^2}{6}}\,{\rm e}^{-\frac{[\zeta^2-(\frac{w}{2})^2]\Omega^2}{6}u^2} \\
\times\sum_{l=\pm1}\biggl\{
\sum_{j=0,2} c_j\,{\rm e}^{+\tfrac{lj}{3}\zeta^2\Omega^2u}(1-u^2)\bigl[\cos(2\beta) + \cos(2\varphi)\bigr] \\
+\sum_{j=1,3}c_j\,{\rm e}^{+\tfrac{lj}{3}\zeta^2\Omega^2u}(1-ul)^2[1+\cos(2\varphi-2l\beta)] \\
+{\rm e}^{+\tfrac{l}{3}\zeta^2\Omega^2u}\,8l\Bigl[2u[1+\cos(2\varphi)\cos(2\beta)]-(1+u^2) \sin(2\varphi)\sin(2\beta)\Bigr]\biggr\}, \label{eq:rhop1}
\end{multline}
and
\begin{multline}
h_{(1)}^{3\Omega}(u,\varphi)=\frac{\alpha}{874800\pi}\,3(3\Omega\zeta)^2\,{\rm e}^{-\frac{\zeta^2\Omega^2+(\frac{w}{2})^2(3\Omega)^2}{6}}\,{\rm e}^{-\frac{[\zeta^2-(\frac{w}{2})^2](3\Omega)^2}{6}u^2} \\
\times
\biggl\{(1-u^2)\bigl[\cos(2\beta) + \cos(2\varphi)\bigr]\\
+\frac{1}{2}\sum_{l=\pm1}{\rm e}^{+l\zeta^2\Omega^2u}(1-ul)^2[1+\cos(2\varphi-2l\beta)] \biggr\}. \label{eq:rhop2}
\end{multline}
In \Eqref{eq:rho1} we have pulled out an overall factor, such that, apart from $u$ and $\varphi$, the functions $h_{(1)}^{\omega}(u,\varphi)$ only depend on the dimensionless combinations $\xi^2\Omega^2$ and $(\frac{w}{2})^2\Omega^2$.
The results for $p=2$ again follow by shifting the angle $\beta\to\beta+\frac{\pi}{2}$, i.e., $\rho_{(2)}^{\omega}=\rho_{(1)}^{\omega}\big|_{\beta\to\beta+\frac{\pi}{2}}$.
As $\cos(\chi\pm\pi)=-\cos\chi$ and $\sin(\chi\pm\pi)=-\sin\chi$, the {\it total number densities} $\rho^{\,\omega}(u,\varphi)=\sum_{p=1}^2\rho^{\,\omega}_{(p)}(u,\varphi)$
of photons of frequency $\omega$ obtained in a polarization insensitive measurement obviously become independent of $\beta$,
i.e., independent of the specific polarization basis used, as they should:
The resulting expressions are effectively obtained by multiplying Eqs.~\eqref{eq:rhop1} and \eqref{eq:rhop2} with a factor of two and setting all trigonometric functions involving $\beta$ in their arguments to zero.
They read
\begin{equation}
\rho^{\omega}(u,\varphi)=\left(\frac{e{\cal E}}{m^2}\right)^6 \left(\frac{w}{2}m\right)^4\frac{\Omega \tau}{2\pi}\, h^{\omega}(u,\varphi)\,, \label{eq:rhosum}
\end{equation}
with
\begin{multline}
h^{\Omega}(u,\varphi)=\frac{\alpha}{874800\pi}(\Omega\zeta)^2\,{\rm e}^{-\frac{[\zeta^2+(\frac{w}{2})^2]\Omega^2}{6}}\,{\rm e}^{-\frac{[\zeta^2-(\frac{w}{2})^2]\Omega^2}{6}u^2} \\
\times\sum_{l=\pm1}\biggl\{
\sum_{j=0,2} c_j\,{\rm e}^{+\tfrac{lj}{3}\zeta^2\Omega^2u}(1-u^2)\cos(2\varphi)
+\sum_{j=1,3}c_j\,{\rm e}^{+\tfrac{lj}{3}\zeta^2\Omega^2u}(1-ul)^2 \\
+16\,{\rm e}^{+\tfrac{l}{3}\zeta^2\Omega^2u}\,l u\biggr\}, \label{eq:rhosum1}
\end{multline}
and
\begin{multline}
h^{3\Omega}(u,\varphi)=\frac{\alpha}{874800\pi}\,3(3\Omega\zeta)^2\,{\rm e}^{-\frac{\zeta^2\Omega^2+(\frac{w}{2})^2(3\Omega)^2}{6}}\,{\rm e}^{-\frac{[\zeta^2-(\frac{w}{2})^2](3\Omega)^2}{6}u^2} \\
\times
\biggl\{2(1-u^2)\cos(2\varphi)+\sum_{l=\pm1}{\rm e}^{+l\zeta^2\Omega^2u}(1-ul)^2 \biggr\}. \label{eq:rhosum2}
\end{multline}
The number of photons emitted in a given solid angle interval characterized by $u_2\leq u\leq u_1$ and $\varphi_1\leq\varphi\leq\varphi_2$ is obtained by integration of \Eqref{eq:rho1} or \eqref{eq:rhosum}, respectively.
Note that $\int_{\vartheta_1}^{\vartheta_2}{\rm d}\vartheta\,\sin\vartheta=\int_{u_2}^{u_1}{\rm d}u$, with $u_i=\cos\vartheta_i$ and $i\in\{1,2\}$.
Hence, the total number of frequency $\omega$ photons originating from the stimulated emission process emitted in this solid angle interval ($\Delta u=u_1-u_2$, $\Delta\varphi=\varphi_2-\varphi_1$) is given by
\begin{equation}
N^{\omega}(\Delta u,\Delta\varphi)=\left(\frac{e{\cal E}}{m^2}\right)^6 \left(\frac{w}{2}m\right)^4\frac{\Omega \tau}{2\pi}\,\int_{u_2}^{u_1}{\rm d}u\int_{\varphi_1}^{\varphi_2}{\rm d}\varphi\, h^{\omega}(u,\varphi)\,, \label{eq:Nsum}
\end{equation}
with $\omega\in\{\Omega,3\Omega\}$.
Obviously, the $\varphi$ integration in \Eqref{eq:Nsum} is trivial. Also the $u$ integration can easily be performed analytically and the result be written in terms of exponential and error functions.
As these results are rather lengthy and do not allow for any additional insights we do not represent them here.
Analogously, the number of emitted photons polarized in mode $p=1$ is obtained by
\begin{equation}
N_{(p)}^{\omega}(\Delta u,\Delta\varphi)=\left(\frac{e{\cal E}}{m^2}\right)^6 \left(\frac{w}{2}m\right)^4\frac{\Omega \tau}{2\pi}\,\int_{u_2}^{u_1}{\rm d}u\int_{\varphi_1}^{\varphi_2}{\rm d}\varphi\, h_{(p)}^{\omega}(u,\varphi)\,. \label{eq:Np}
\end{equation}
As before, the result for the $p=2$ mode follows upon substitution of $\beta\to\beta+\frac{\pi}{2}$.
If the angle parameter $\beta$ is chosen independent of the values of $\varphi$ and $\vartheta$ both integrations can again be performed analytically as for \Eqref{eq:Nsum}.
However, note that the integrations over the solid angle interval can be significantly complicated if $\beta=\beta(\vartheta,\varphi)$ as is, e.g., necessary if we are interested in all photons polarized perpendicular to $\vec{e}_{\rm x}$; cf. Sec.~\ref{seq:Ex+Res} below.
To maximize the effect of stimulated photon emission, the laser field strength $\cal E$ is preferably rendered as large as possible.
For given laser parameters, $\cal E$ can be maximized by focusing the laser beam down to the diffraction limit, which will be assumed to be the case when providing experimental estimates for the effect below.
The beam diameter of a Gaussian beam of wavelength $\lambda$ focused down to the diffraction limit is given by $w=2\lambda f^\#$ and its Rayleigh range by $z_{\rm R}=\pi\lambda(f^\#)^2$,
with $f^\#$, the so-called $f$-number, defined as the ratio of the focal length and the diameter of the focusing aperture \cite{Siegman}; $f$-numbers as low as $f^\#=1$ can be realized experimentally.
Recall that in our approximation the length scale $\zeta$ mimics the Rayleigh range $z_{\rm R}$. Correspondingly, aiming at experimental estimates, we identify $\zeta=z_{\rm R}$.
Hence, and perhaps most strikingly, in the diffraction limit the combinations $\zeta^2\Omega^2=(2\pi)^2\pi^2(f^\#)^4$, $(\frac{w}{2})^2\Omega^2=(2\pi)^2(f^\#)^2$
become generic numbers.
In turn, the functions $h_{(p)}^\omega(u,\varphi)$ and $h^\omega(u,\varphi)$ defined in Eqs.~\eqref{eq:rho1}-\eqref{eq:rhosum2} become independent of any explicit laser parameters apart from $f^\#$.
The entire dependence on the laser parameters in Eqs.~\eqref{eq:rho1}, \eqref{eq:rhosum}, \eqref{eq:Nsum} and \eqref{eq:Np} is encoded in the overall prefactor
\begin{equation}
\left(\frac{e{\cal E}}{m^2}\right)^6 \left(\frac{w}{2}m\right)^4\frac{\Omega\tau}{2\pi} \ \ \xrightarrow{\text{diffraction\ limit}} \ \
(f^\#)^4\left(\frac{e{\cal E}}{m^2}\right)^6 \left(\frac{\lambda}{\lambda_c}\right)^3\frac{\tau}{\tau_c}=(f^\#)^4\,2^3\left(\frac{I}{I_{\rm cr}}\frac{\lambda}{\lambda_c}\right)^3\frac{\tau}{\tau_c}\,,
\end{equation}
where $I=\frac{1}{2}{\cal E}^2$ denotes the mean intensity per laser beam and $I_{\rm cr}\equiv(\frac{m^2}{e})^2=4.68\cdot10^{29}\frac{\rm W}{{\rm cm}^2}$ is the {\it critical intensity}.
Moreover, $\lambda_c$ and $\tau_c$ are the Compton wavelength and time introduced above.
\section{Results and Discussion} \label{seq:Ex+Res}
Here, we aim at providing some rough estimates of the number of photons resulting from the stimulated photon emission process.
To this end we assume the original multicycle laser pulse characterized by its wavelength $\lambda$, pulse energy $W$ and pulse duration $\tau$
to be split into two counter-propagating pulses of energy $W/2$ to be focused down to the diffraction limit with $f^\#=1$,
and give the numbers of emitted photons per shot. The experimental scenario is sketched in Fig.~\ref{fig:cartoon}.
The counter-propagating laser pulses are superimposed to form a standing electromagnetic wave within their overlapping foci; cf. \Eqref{eq:resfields} above.
Assuming Gaussian beams, the effective focus area is conventionally defined to contain $86\%$ of the beam energy ($1/e^2$ criterion for the intensity).
Correspondingly, the mean intensity for each beam is estimated as
\begin{equation}
I\approx \frac{0.86\,(W/2)}{\tau\,\sigma}\,,
\label{eq:Ipump}
\end{equation}
with focus area $\sigma\approx\pi\lambda^2$.
For completeness, also note that the divergence $\theta$ of a Gaussian beam in the considered limit is given by $\theta=\frac{1}{\pi}$ \cite{Siegman} (cf. Fig.~\ref{fig:cartoon}).
Therewith, all physical parameters in Eqs.~\eqref{eq:Nsum} and \eqref{eq:Np} are specified and the number of emitted photons can be evaluated.
As $W$ is conventionally given in units of joules, $\tau$ in femtoseconds and $\lambda$ in nanometers, it is helpful to note that
\begin{equation}
\left(\frac{I}{I_{\rm cr}}\frac{\lambda}{\lambda_c}\right)^3\frac{\tau}{\tau_c}\approx 3.40\cdot10^{11}\left(\frac{W[J]}{\tau[{\rm fs}]\,\lambda[{\rm nm}]}\right)^3 \tau[{\rm fs}]\,. \label{eq:J-fs-nm}
\end{equation}
Before providing some explicit estimates of the numbers of photons resulting from the stimulated emission process attainable with present and near future high-intensity laser facilities,
we focus on the directional emission characteristics encoded in the functions $h_{(p)}^\omega(u,\varphi)$.
Let us emphasize again that -- in the diffraction limit, and particularly for $f^\#=1$ -- these characteristics are independent of the actual laser parameters, and thus, are the same for all lasers.
For a polarization insensitive measurement of the emitted photons the relevant directional emission characteristics as a function $\varphi,\vartheta$ are described by $h^\omega(\cos\vartheta,\varphi)\sin\vartheta$; recall that $|\frac{{\rm d}u}{{\rm d}\vartheta}|=\sin\vartheta$.
We depict them in Fig.~\ref{fig:Ntotal}.
\begin{figure}[h]
\center
\subfigure{\includegraphics[width=0.4\textwidth]{Fig2a}}\hspace*{0.2cm}
\subfigure{\includegraphics[width=0.485\textwidth]{Fig2b}}
\caption{{\bf Left:} Directional emission characteristics $h^\omega(\cos\vartheta,\varphi)\sin\vartheta$ for a polarization insensitive measurement of the photons emitted from the laser focus in arbitrary units.
The result exhibits a superficial rotational symmetry about the beam axis $\rm z$ (cf. Fig.~\ref{fig:cartoon}) and a mirror symmetry with respect to the $\rm x$-$\rm y$ plane.
Deviations from the rotational symmetry $\sim\cos(2\varphi)$ [cf. \Eqref{eq:rhosum1}] are extremely tiny and indiscernible here.
The electric field and thus, the polarization vector of the electromagnetic field configuration~\eqref{eq:resfields} in the laser focus oscillates along $\vec{e}_{\rm x}$ (green arrows).
{\bf Right:} Cut through the three dimensional emission characteristics (left) in the $\rm x$-$\rm z$ plane.
For comparison, the divergence $\theta=\frac{1}{\pi}$ of a Gaussian beam focused down to the diffraction limit is depicted in gray, i.e., a Gaussian beam encompasses photons propagating in all gray shaded directions.
While most photons are emitted into these directions, a certain fraction is emitted into directions outside the laser beam.
Practically no photons are emitted under angles $>\frac{\pi}{6}=30^\circ$ about the beam axis.}
\label{fig:Ntotal}
\end{figure}
The total number $N^\omega_{\rm tot}$ of emitted photons of frequency $\omega\in\{\Omega,3\Omega\}$ is obtained straightforwardly from \Eqref{eq:Nsum} with $u_2=-1$, $u_1=1$, $\varphi_1=0$ and $\varphi_2=2\pi$.
This results in
\begin{equation}
\left\{\begin{array}{c}
\! N^\Omega_{\rm tot}\\
\! N^{3\Omega}_{\rm tot}
\end{array}\right\}
\approx
\left\{\begin{array}{c}
2.94\cdot10^{7} \\
8.48\cdot10^{-16}
\end{array}\right\}\left(\frac{W[J]}{\lambda[{\rm nm}]}\right)^3\!\left(\frac{1}{\tau[{\rm fs}]}\right)^2 . \label{eq:res1}
\end{equation}
As the $3\Omega$ signal is severely suppressed, we do not study it any further in the remainder of this paper.
It is instructive to also provide the total number of photons of frequency $\Omega$ emitted into directions outside the laser beam, to be denoted by $N^\Omega_{{\rm tot},>\theta}$ (cf. Fig.~\ref{fig:Ntotal})
and given by
\begin{equation}
N^\Omega_{{\rm tot},>\theta}\approx 9.84\cdot10^{6}\left(\frac{W[J]}{\lambda[{\rm nm}]}\right)^3\!\left(\frac{1}{\tau[{\rm fs}]}\right)^2. \label{eq:res2}
\end{equation}
As the laser field is polarized along $\vec{e}_{\rm x}$, it is particularly interesting to ask for the number of emitted photons with perpendicular polarization, fulfilling
$\vec{e}_{\perp,\beta}\cdot\vec{e}_{\rm x}=0$ $\leftrightarrow$ $\beta=\arctan(\cot\varphi\cos\vartheta)$; cf. Eqs.~\eqref{eq:e_perpbeta} and \eqref{eq:epsilons} above.
Hence, to project out the emitted photons with polarization vector in the $\rm y$-$\rm z$ plane, the angle parameter $\beta$ has to be adjusted as a function of the emission direction parameterized by the angles $\varphi$ and $\vartheta$.
With regard to Eqs.~\eqref{eq:rhop1} and \eqref{eq:rhop2} it is helpful to note that $\sin(2\arctan\chi)=\frac{2\chi}{1+\chi^2}$,
while $\cos(2\varphi-2l\arctan\chi)=\frac{1-\chi^2}{1+\chi^2}\cos(2\varphi)+2l\frac{\chi}{1+\chi^2}\sin(2\varphi)$ for $l=\pm1$.
Defining $h_{\perp}^\omega(u,\varphi)\equiv h_{(1)}^\omega(u,\varphi)\big|_{\beta=\arctan(\cot\varphi\cos\vartheta)}$,
the directional emission characteristics for photons with polarization vector perpendicular to the polarization direction of the laser $\vec{e}_{\rm x}$ are described by
$h_{\perp}^\omega(\cos\vartheta,\varphi)\sin\vartheta$.
We depict them in Fig.~\ref{fig:Nperpx}.
\begin{figure}[h]
\center
\subfigure{\includegraphics[width=0.465\textwidth]{Fig3a}}\hspace*{-1.2cm}
\subfigure{\includegraphics[width=0.515\textwidth]{Fig3b}}
\caption{{\bf Left:} Directional emission characteristics $h_{\perp}^\omega(\cos\vartheta,\varphi)\sin\vartheta$ for the emitted photons polarized perpendicular to the laser field (polarized in $\rm x$ direction) in arbitrary units; cf. also Fig.~\ref{fig:Ntotal}.
For completeness, we note that the maximum value in this plot amounts to one thousandth of the maximum value in Fig.~\ref{fig:Ntotal},
i.e., the emission signal for perpendicular polarized photons is significantly smaller than the result obtained when including all polarizations.
The three dimensional emission characteristics is symmetric with respect to the coordinate planes.
{\bf Right:} Cut through the three dimensional figure (left) along the $d_{\rm xy}$-$\rm z$ plane, where $d_{\rm xy}$ is the diagonal in the $\rm x$-$\rm y$ plane above which the signal becomes maximum.
}
\label{fig:Nperpx}
\end{figure}
The number of frequency $\Omega$ photons polarized perpendicular to $\vec{e}_x$ and emitted in the solid angle interval parameterized by $u_1\leq u\leq u_2$ and $\varphi_1\leq\varphi\leq\varphi_2$ is obtained by integration of $h_{\perp}^\omega(u,\varphi)$ (cf. Sec.~\ref{sec:calculation} above).
Integrating over the full solid angle results in $N^\Omega_{\perp}$, and just integrating over all directions outside the laser beam in $N^\Omega_{\perp,>\theta}$.
Our explicit results are
\begin{equation}
\left\{\begin{array}{c}
\! N^\Omega_{\perp}\\
\! N^{\Omega}_{\perp,>\theta}
\end{array}\right\}
\approx
\left\{\begin{array}{c}
1.35\cdot10^{4} \\
1.18\cdot10^{4}
\end{array}\right\}\left(\frac{W[J]}{\lambda[{\rm nm}]}\right)^3\!\left(\frac{1}{\tau[{\rm fs}]}\right)^2 . \label{eq:Nperp}
\end{equation}
Note that these numbers are about a factor of $10^{-3}$ smaller than those for all polarizations given in the first line of \Eqref{eq:res1} and \Eqref{eq:res2}; cf. also Figs.~\ref{fig:Ntotal} and \ref{fig:Nperpx}.
In Tab.~\ref{tab:lasers} we list some explicit estimates for the numbers~\eqref{eq:res1}-\eqref{eq:Nperp} of photons of frequency $\Omega$ originating from the stimulated emission process for various present and near future high-intensity laser facilities.
\begin{table}[h]
\begin{tabular}{ | c || c | c | c || c | c | c | c |}
\hline
Laser & \ $W$[J]\ \ & \ $\tau$[fs]\ \ & \ $\lambda$[nm]\ \ & $N^\Omega_{\rm tot}$ & $N^\Omega_{{\rm tot},>\theta}$ & $N^\Omega_\perp$ & $N^\Omega_{\perp,>\theta}$ \\
\hline
\hline
POLARIS & $150$ & $150$ & $1030$ & $4.04$ & $1.35$ & \ $1.86\cdot10^{-3}$\ \ & \ $1.62\cdot10^{-3}$ \ \ \\
\hline
Vulcan & $500$ & $500$ & $1054$ & $12.6$ & $4.22$ & $5.77\cdot10^{-3}$ & $5.04\cdot10^{-3}$ \\
\hline
Omega EP & $2000$ & $10000$ & $1054$ & $2.01$ & $6.75\cdot10^{-1}$ & $9.20\cdot10^{-4}$ & $8.08\cdot10^{-4}$ \\
\hline
ELI Prague & $1500$ & $150$ & $1054$ & \ $3.77\cdot10^3$\ \ & \ $1.26\cdot10^3$\ \ & $1.73$ & $1.51$ \\
\hline
ELI-NP & $2\,\times\,250$ & $25$ & $800$ & $1.15\cdot10^4$ & $3.86\cdot10^3$ & $5.28$ & $4.62$ \\
\hline
XCELS & \ $12\,\times\,400$\ \ & $25$ & $910$ & $6.90\cdot10^6$ & $2.32\cdot10^6$ & $3.17\cdot10^3$ & $2.78\cdot10^3$ \\
\hline
\end{tabular}
\caption{Numbers of induced photons resulting from the stimulated vacuum emission process for various present and near future high-intensity laser systems, characterized by their pulse energy $W$, pulse duration $\tau$ and wavelength $\lambda$.
Apart from the total numbers of frequency $\Omega$ photons emitted in all directions, $N^\Omega_{\rm tot}$, and in all directions outside the laser beam, $N^\Omega_{{\rm tot},>\theta}$,
we provide the numbers of emitted photons polarized perpendicular to the initial laser field, $N^\Omega_\perp$ and $N^\Omega_{\perp,>\theta}$.}
\label{tab:lasers}
\end{table}
However, let us emphasize that only those photons emitted in the $\rm y$-$\rm z$ plane ($\vartheta=\frac{\pi}{2}$) can be polarized in the same direction as the original laser beam.
Only here, the polarization vectors which live in the tangent space of the unit sphere [cf. \Eqref{eq:e_perpbeta}] can point in the $\vec{e}_{\rm x}$ direction.
With regard to the total number of emitted photons, these photons amount to a negligible fraction:
This becomes particularly obvious when looking at the directional emission characteristics for the total number of photons depicted in Fig.~\ref{fig:Ntotal} (left).
All photons that might have their polarization vector in the same direction as the original laser field lie on the intersection of the $\rm y$-$\rm z$ plane with the three dimensional emission characteristics.
Clearly, their contribution to the integral~\eqref{eq:Nsum} yielding the number of emitted photon number in three dimensions is negligible as it is to be multiplied with ${\rm d}\varphi\to0$ when performing the integration over any solid angle interval.
In all other emission directions ($\vartheta\neq\frac{\pi}{2}$) the induced photons originating from the stimulated emission process are polarized differently than the laser,
implying that basically all emitted photons are polarized differently than the laser beam triggering the effect.
In turn, this could be used to distinguish the signal (emitted photons) from the laser photons of the same frequency.
For example, equipping a photon detector with a polarizer blocking the polarization of the laser beam along $\vec{e}_{\rm x}$ still a significant fraction of the total numbers of photons emitted in directions outside the laser beam, $N^\Omega_{{\rm tot},>}$, should be detectable: All photons with a nonvanishing polarization component perpendicular to $\vec{e}_{\rm x}$ will actually contribute to the signal.
With respect to such measurement, our result~\eqref{eq:Nperp} for the truly perpendicular polarized emission signal $N^\Omega_{\perp,>}$ (cf. also Fig.~\ref{fig:Nperpx} and Tab.~\ref{tab:lasers}) just amounts to the absolute minimum number of emitted photons to be detected.
\section{Conclusions and Outlook} \label{seq:Con+Out}
In this paper we have studied and interpreted a specific laser pulse collision process in terms of stimulated single photon emission from the vacuum in strong space-time dependent electromagnetic fields.
More specifically, we have focused on a particular field configuration mimicking the electromagnetic field in the focal spot of two counter-propagating, linearly polarized high-intensity laser beams with their polarization vectors pointing in the same direction.
It would be interesting to extend our study to other electromagnetic field configurations attainable in the overlapping foci of two high-intensity laser pulses,
e.g., to deviate from the counter-propagation geometry by letting the beams collide under an relative angle and to study other laser polarizations.
Moreover, the electromagnetic field profiles to mimic the laser beams should eventually be improved to account for more features of real, experimentally attainable pulses.
In particular the Gaussian profile mimicking the finite Rayleigh length in the present study should be replaced by a Lorentzian profile.
Besides, in a latter step of this program also a dedicated detection set-up should be worked out and the precise numbers of the detectable photons originating from the stimulated emission process should be specified.
Let us emphasize again that in the present study we rather intended to underpin our viewpoint of interpreting the vacuum subjected to macroscopic strong electromagnetic (laser) fields as source term for outgoing photons. To this end, we present first estimates of the photon numbers attainable from the effect of stimulated photon emission in an all-optical experimental set-up within this framework.
Finally, and perhaps most importantly, the approach adopted by us can be straightforwardly extended to processes involving $n\geq2$ external photons, with $n\in\mathbb{N}$,
attainable by expanding the effective Lagrangian~\eqref{eq:effL} with $F^{\mu\nu}(x)\to F^{\mu\nu}(x)+f^{\mu\nu}(x)$ to ${\cal O}(f^n)$;
cf. our discussion in the context of \Eqref{eq:Seff} above, and also \cite{Karbstein:2015cpa}.
\section*{Acknowledgments}
The authors thank H.~Gies for many stimulating discussions.
FK is grateful to Matt~Zepf for various helpful and enlightening discussions.
FK acknowledges support by the DFG (SFB-TR18).
RS acknowledges support by the Ministry of Education and Science of the Republic of Kazakhstan.
|
train/arxiv
|
BkiUdh85qoYDgeESlhml
| 5 | 1 |
\section*{Summary}
Malignant mesothelioma is classified into three histological subtypes, Epithelioid, Sarcomatoid, and Biphasic according to the relative proportions of epithelioid and sarcomatoid tumor cells present. Biphasic tumors display significant populations of both cell types. Current guidelines recommend that the sarcomatoid component of each mesothelioma is quantified, specifically, as a higher percentage of sarcomatoid pattern in biphasic mesothelioma shows poorer prognosis. It is therefore important to accurately diagnose morphological subtypes. However, this subtyping is subjective and limited by current diagnostic guidelines. This can affect the reproducibility even between expert thoracic pathologists when characterising the continuum of relative proportions of epithelioid and sarcomatoid components using a three class system. In this work, we develop a novel dual-task Graph Neural Network (GNN) architecture with ranking loss to learn a model capable of scoring regions of tissue down to cellular resolution. This allows quantitative profiling of a tumor sample according to the aggregate sarcomatoid association score of all the cells in the sample. The proposed approach uses only core-level labels and frames the prediction task as a dual multiple instance learning (MIL) problem. Tissue is represented by a cell graph with both cell-level morphological and regional features. We introduce a dataset of 234 mesothelioma tissue micro-array cores, and use an external multi-centric test set from Mesobank, on which we demonstrate the predictive performance of our model. We validate our model predictions through an analysis of the typical morphological features of cells according to their predicted score, finding that some of the morphological differences identified by our model match known differences used by pathologists. We further show that the model score is predictive of patient survival with a hazard ratio of 2.30. The ability to quantitatively score tissue on how much sarcomatoid component is present, and overlay cell-level instance scores, will assist pathologists in assessing a sample more quickly and consistently by defining the sarcomatoid component, particularly in biphasic mesothelioma. This ultimately will improve treatment decisions for patients with mesothelioma. The code for the proposed approach, along with the dataset, is available at: https://github.com/measty/MesoGraph.
Our contributions include releasing a new dataset, introduction of the MesoGraph architecture providing accurate subtype predictions, and insightful analysis of morphological differences between sarcomatoid and epithelioid components.
\keywords{Graph Neural Networks \and Multiple Instance Learning \and Mesothelioma \and Cancer subtyping \and Digital Pathology}
\section{Introduction} \label{intro}
Malignant mesothelioma (MM) is an aggressive cancer of malignant mesothelial cells of the pleural lining, primarily associated with asbestos exposure \cite{WAGNER1960a}. It has a poor prognosis with less than 10\% five year survival rates due to late diagnosis (\cite{hjerpe2018cyto} ; \cite{mesostats}). It has a long latency period from initial exposure to eventual carcinogenesis, and is difficult to diagnose due to its nonspecific clinical manifestations. MM is classified into 3 subtypes \cite{Ai2014}, epithelioid, biphasic and sarcomatoid mesothelioma (EM, BM and SM respectively), with BM characterised by a mix of epithelioid and sarcomatoid components.
The Histological subtype of malignant mesothelioma is essential for prognosis and clincial decisions on treatment pathways for patients \cite{Meyerhoff2015ImpactOM}. Stratification of a given sample into a particular subtype informs treatment and can help gain a more in-depth understanding of disease pathology and outcome. The benefit of surgical treatment has prognostic implications for epithelioid mesothelioma with median survival of 19 months but less so for SM and BM with respective median survival of 4 months and 12 months after surgical treatment \cite{Mansfield2014SystematicRO}.
Epithelioid mesothelioma is characterised by malignant cells that are cytologically round with varying grading of atypia. Sarcomatoid mesothelioma cells are generally recognised as malignant elongated spindle cells \cite{whotumors} and are associated with worse prognosis in comparison to EM. Sarcomatoid mesothelial cells may also include transitional features that are intermediate between epithelioid and sarcomatoid. Although transitional cells are now classified under sarcomatoid mesothelioma, their presence is associated with worse prognosis \cite{courtiol2019meso}.
While the distinction of these three histological subtypes of MM is crucial to patient treatment, management and prognosis, it is challenging to differentiate EM, SM and BM through visual analysis. Currently, there are no clear guidelines on how to perform this stratification in an objective and reproducible manner \cite{dacic2020interobserver}. Furthermore, even though mesotheliomas are divided into these three broad categories, in reality there is a continuous spectrum from EM to SM dependent upon the relative proportion of epithelioid and sarcomatoid cells in a given sample. As a consequence, existing approaches are unable to objectively quantify where on this spectrum a given sample falls based on profiling of cellular morphological patterns in it.
A number of deep learning methods for analyzing mesothelioma images have been developed recently. For example, SpindleMesoNET~\cite{Naso2021} can separate malignant SM from benign spindle cell mesothelial proliferations. A recent approach for survival prediction of MM patients called MesoNet~\cite{courtiol2019meso} uses a Mulitple Instance Learning (MIL) solver originally developed for computer vision applications~\cite{courtiol2018weak} and classification of lymph node metastases \cite{weldon2016}. However, automated subtyping of malignant mesothelioma from hematoxylin and eosin (H\&E) stained tissue sections remains an open problem.
\begin{figure*}[ht]
\centering
\centerline{\includegraphics[width=15.5cm]{Figs/graphical_abstract.png}}
\caption{High level overview/graphical abstract of our approach to mesothelioma subtyping. Pathologist labelled TMA cores are used to learn a graph nueral network model on cell-level features. We show the model can identify sarcomatioid areas, and its predicted scores can be used to stratify patient survival.}
\label{graphical_abs}
\end{figure*}
One challenge in the characterization of mesothelioma is that pathologist-assigned ground-truth labels of mesothelioma subtypes are typically available only at the case level as it is very difficult for pathologists to associate tumor micro-environment or cellular morphometric patterns with image or case level labels in an objective manner. Moreover, it can be very time-consuming to obtain detailed cellular or regional annotations, and those annotations may not be very reliable due to significant inter- and intra-observer variation.
The aim of this study was to develop a GNN approach to predict subtypes of malignant mesothelioma in a MIL setting. This was achieved considering Tissue Micro-array (TMA) cores as bags and individual cells as instances.
On these we have built a weakly-supervised machine learning model to characterize mesothelioma subtypes using only case level labels in its training. The proposed approach can generate a quantitative assessment of where the sample stands in terms of the aforementioned epithelioid to sarcomatoid spectrum, enabling pathologists to perform a more in-depth characterization of tumor samples. A high level overview of the approach presented in the paper can be found in Fig. \ref{graphical_abs}.
\section{Results and Discussion}
We have developed a custom Graph Neural Network based pipeline called MesoGraph that can predict mesothelioma subtypes using H\&E stained tissue images. MesoGraph uses pathologist assigned case-level labels without any cellular or regional annotations in its training. The proposed approach models each cell in a given sample as a node in the graph, which is connected to neighbouring cells. Each node is associated with various features, which can be broadly classified into four types: 1) nuclear morphometric features, 2) stain intensity features of nuclear and cytoplasmic components of the cell, 3) cellular counts in the neighbourhood of node and 4) deep neural network and haralick based texture features. For a given test sample, it generates two probability scores (collectively called MesoScore) representing the probabilities of the sample being epithelioid or sarcomatoid. As a biphasic mesothelioma tumor is composed of both epithelioid and sarcomatoid components, the two outputs in MesoScore allow precise quantification of the two components in the sample. In addition to predicting mesothelioma subtype, MesoGraph also generates cell-level quantitative scores representing the association of each cell with the mesothelial subtype of the given sample. MesoGraph has been trained and independently validated on two datasets: St. George's Hospital (SGH, n = 234) and the multi-centric MesoBank collection (MB, n = 258).
In this section, we present the results of the proposed method in terms of its predictive performance in comparison to existing approaches, as well as its ability to identify histological features and morphological characteristics of cells associated with different types of malignant mesothelioma. We also demonstrate the ability of the proposed approach to stratify mesothelioma patients based on their expected survival.
\subsection{Predictive Performance} \label{performance_results}
Test results from the MesoGraph pipeline for both MesoBank and SGH datasets are shown in Figure \ref{model_auc} and Table~\ref{restable}. Here, the Receiver Operating Characteristic (ROC) curve is obtained by considering both sarcomatoid and biphasic samples as the positive class, whereas the epithelioid samples are associated with the negative label. As can be seen from these results, the proposed approach offers high predictive quality over both cross-validation and independent testing in comparison to other existing approaches. In the table, PINS refers to the Positive Instance Sampling Patch-based MIL approach as detailed in \cite{eastwood2022pins} whereas CLAM is the Clustering-constrained Attention Multiple Instance Learning method described in \cite{lu2020data}, a deep-learning-based weakly-supervised method that uses attention in combination with clustering-based constraints to identify the most predictive areas of the image. Max-MIL and naive-MIL are simple patch based baseline multiple instance learning (MIL) methods detailed further in the methods Section \ref{evaluation}.
As can be seen from Table \ref{restable}, the max-based MIL strategy performs poorly. This is likely due to the relatively small size of the training dataset, as learning only on the maximally scoring instance per bag exacerbates this.
Naive MIL performs surprisingly well. This may be due to the relatively high proportion of positive instances that are expected to be present in many of the positive bags (for example a sarcomatoid core should contain mostly positive instances).
This makes the implicit assumption this model makes, namely that all instances share the label of the bag, less wrong for this dataset compared to other MIL tasks.
PINS and CLAM, as two patch based methods with a mechanism for focussing on the most relevant region of an image, perform similarly with solid performance. However, as patch based methods, the spatial resolution of the prediction maps they can provide is lower than that of our cell graph based model.
Our model outperforms other models tested, achieving an internal cross-validation performance of $0.90 \pm 0.01$. It performs particularly well in terms of its average precision (AP) of $0.86\pm0.02$, indicating that its performance on the positive class (which is the minority class) is very good. While performance drops slightly on the external validation set, an AUROC of $0.86$ and AP of $0.8$ as seen in Fig \ref{model_auc}, panels C and D shows these results generalize well. We attribute the performance improvements achieved by our model firstly to the cell graph representation, with cells and their morhpological features as instances, which is far more natural than an arbitrary division into patch instances. Secondly, our formulation of the learning as a dual task problem with a ranking loss. This acknowledges both the ordering we know exists between EM, BM and SM cores in terms of how much sarcomatoid component is present, and also the possibility that some regions of tissue may not be strongly associated with either a sarcomatoid or epithelioid core label.
\begin{figure*}[ht]
\centering
\centerline{\includegraphics[width=14cm]{Figs/results_auc.png}}
\caption{A) AUROC for each model. B) Model average precision. C) AUROC on cross-cohort evaluation. D) Average precision on cross-cohort evaluation. Our model outperforms other models tested, and from C and D we see good generalization to external validation data.}
\label{model_auc}
\end{figure*}
\begin{table*}[t]
\begin{center}
\begin{tabularx}{0.85 \textwidth}{|>{\raggedright\arraybackslash}X||>{\raggedright\arraybackslash}X|>{\raggedright\arraybackslash}X|>{\raggedright\arraybackslash}X|>{\raggedright\arraybackslash}X|}
\hline
Metric&AUC-ROC&Avg. Precision&Sensitivity&Specificity\\
\hline\hline
max-MIL&$0.70\pm0.01$&$0.54\pm0.12$&$0.54\pm0.07$&$0.73\pm0.09$\\
naive-MIL&$0.84\pm0.05$&$0.72\pm0.11$&$0.72\pm0.08$&$0.71\pm0.1$\\
PINS \cite{eastwood2022pins}&$0.85\pm0.05$&$0.80\pm0.07$&$0.82\pm0.1$&$0.71\pm0.13$\\
CLAM \cite{lu2020data}&$0.85\pm0.07$&$0.74\pm0.11$&$0.75\pm0.11$&$0.77\pm0.02$\\
\small{MesoGraph(Ours)}&$0.90\pm 0.007$&$0.86\pm0.02 $&$ 0.88\pm0.015$&$0.72\pm0.01 $\\
\hline
\end{tabularx}
\caption{Summary of results (mean$\pm$stdev) of models evaluated on primary dataset (SGH). } \label{restable}
\end{center}
\end{table*}
\renewcommand{\arraystretch}{1}
\subsection{Visualization of Model Output}
The output of our model can be visualised in a zoomable, interactive Graphical User Interface (GUI) we have developed. A demo of results from our model can be found at (https://mesograph.dcs.warwick.ac.uk). Examples of the cell level scoring output by the model are shown in Fig. \ref{cores_vis}. For each cell in a given sample, the proposed model generates two prediction scores signifying the probability of the cell being associated with sarcomatoid or epithelioid labels. These scores can be combined and visualized in a colourmap showing cells that are associated with epithelioid (blue) and sarcomatoid (red) subtypes as well as a histogram (called MesoGram) showing the relative distributions of epithelioid and sarcomatoid components.
\begin{figure*}[ht!]
\centering
\centerline{\includegraphics[width=14cm]{Figs/paper_core_fig_small.png}}
\caption{Model visualisation. Each core is shown together with a zoomed in view showing the differences in morphology between regions, and a plot showing the distribution of node scores in that core. A. Epithelioid core. We see a predominantly low-scoring core with only a few small regions displaying slightly more sarcomatoid features. Conversely, in B, a Sarcomatoid core, nodes are predominantly high scoring. C and D. Biphasic cores. In each core we see a bimodal distribution of scores, particularly pronounced in core D. The zoomed in regions show a distinct difference in morphology between high and low scoring regions, with rounder cells seen in lower scoring regions, and a more elongated morphology and less structured cell organisation in higher scoring regions.}
\label{cores_vis}
\end{figure*}
From the zoomed-in masks, we can see the model can distinguish between regions with typical rounded morphology of the Epithelioid subtype and the more elongated morphology displayed in Sarcomatoid regions. The MesoGram plots of most samples tend to be bimodal to some extent, with epithelioid and sarcomatoid cores more heavily skewed toward low and high scores, respectively. This continuum of distribution between sarcomatoid and epithelioid is demonstrated further in Fig. \ref{cores_overview}, where thumbnails of model output on all cores are shown, grouped by subtype and ordered within each subtype by model score. This ability to give a more precise, fine-grained characterisation of a tumor beyond the current three poorly defined and subjective subtypes is a strength of our approach.
\begin{figure*}[ht!]
\centering
\centerline{\includegraphics[width=14cm]{Figs/core_overview.png}}
\caption{Overview of model predictions by subtype. Images of model predictions ordered within each subtype by the predicted predominance of the sarcomatoid component, illustrating the underlying continuous biological expression of tissue on the E to S spectrum.}
\label{cores_overview}
\end{figure*}
\subsection{Explainability of model predictions}
To gain an understanding of the predictions generated by the proposed approach, we have applied GNNExplainer \cite{ying2019gnnexplainer} to the trained graph neural network model. This allows us to understand what node level features are contributing to the prediction of a given sample for each subtype. The top 10 features identified in this analysis are shown in Fig. \ref{feat_imp}.
The most important feature overall is the circularity, which confirms the expected distinction between the rounder morphology of the epithelioid subtype, compared to the more spindle-shaped sarcomatoid morphology. There are also a number of features describing the intensity and texture in the Eosin channel around the nucleus. Looking at the feature importances on specific subtypes, the resnet features are most useful on epithelioid cores. Circularity is specifically important in Epithelioid and sarcomatoid subtypes, as they tend to be composed of more homogeneous populations and therefore are expected to contain mostly either rounded or more elongated cells. In both Biphasic and Sarcomatoid cores, nearby detection counts seem to be an important feature. This may reflect a tendency for non-epithelioid tumors to display a slightly more spread out and disorganised cell distribution. We also notice that the importance of most of the top features has far more spread when considering Epithelioid cores, indicating a wide variety of features can contribute to an epithelioid score, with few features being universally important across all epithelioid cores.
\begin{figure*}[ht!]
\centering
\centerline{\includegraphics[width=14cm]{Figs/box_charts_gnn.png}}
\caption{Top 10 features identified by GNNExplainer, considering A) all cores, and in B-D, importances on cores grouped by subtype. Results shown as a standard box \& whisker plot, with the box showing the 25th, 50th and 75th percentile of a features importance scores over cores. Whiskers show min and max values, limited at box $\pm$ 1.5 $\times$ inter-quartile range. The top feature is circularity, a knows differentiating characteristic between mesothelial subtypes, providing validation for our model.}
\label{feat_imp}
\end{figure*}
To determine the separation between classes based on top scoring features, in the lower half of Fig. \ref{morph_fig} (C and D) we plotted the prediction of each core against the assigned label by a pathologist. While Epithelioid cores and Sarcomatoid cores are mostly well-separated, we observe there can be overlap between some of the cases in terms of morphology. We also observe that Biphasic is not very distinct from Sarcomatoid cores.
\subsection{Characterization of Cellular Morphologies} \label{pca-umap}
Pathologists assess cell morphology when diagnosing and scoring mesothelioma tumours. Therefore, we sought to investigate differences in nuclear morphology between mesothelioma tumours with different diagnoses. We focused on key features assessed qualitatively by pathologists including nuclear area, elongation (width and length) and nucleus shape regularity as measured by both circularity (how close it is to a circular shape), and solidity (which reflects overall concavity of a shape). Interestingly, in Figure \ref{morph_fig}, we found sarcomatoid tumours tend to have larger nuclei on average compared to epithelioid tumours. As expected these nuclei are more elongated and have less circular shapes. For almost all features, measures of nucleus shape in biphasic tumours fall in between epithelioid and sarcomatoid tumours. These results already confirm that our image analysis pipeline reflects inherent differences between these types even when only considering the average measures of each tumour which is consistent with pathological features. This motivates the development of more sophisticated AI methods to detect these differences.
\begin{figure}[ht!]
\centerline{\includegraphics[width=15.8cm]{Figs/morph_fig.png}}
\caption{Upper panel: Examples of cells scored most and least highly by the model, plotted as a 2-D UMAP reduction
of principal components calculated on both high and low scoring cells. For each cluster, the mean of the cells is
displayed, together with individual example cells.
The clusters A-E, on the left side of the panel are predicted as Sarcomatoid and demonstrate a more spindle like morphology, grouped together in size and relative spindle cell characteristics in each cluster as shown by example cells on the right side of the panel. Similarly, the Non-sarcomatoid predicted cells also show clustering into 5 groups, F-J.
Lower panel: Morphological heterogeneity of mesothelioma tumours independent of model prediction.
A) Distribution of average morphology across tumour types. All measurements are normalised to the data average
and standard deviation. B) Heterogeneity of cell morphology across different mesothelioma tumour types based on
standard deviation (SD) of z-scored single cell data. C-D) Morphological heterogeneity based on ground truth labels
(C) and predicted labels (D)}
\label{morph_fig}
\end{figure}
Next, we investigated the extent of variability in morphological measurements across different tumour types. We measured the standard deviation of cell features for each single tumour core as a proxy of heterogeneity. We found that Sarcomatoid tumours exhibit higher morphological heterogeneity in all nuclear features. These analyses motivate the investigation of single cell phenotypes to identify the most relevant subpopulations.
We can gain further insight into the differences in morphology that the model is associating with each subtype by looking at the principal components of cells assigned the highest and lowest scores (i.e most and least likely to be associated with a sarcomatoid core, respectively).
Comparing the first principal component for each subtype in supplementary Fig. 2, we can see that the model has learnt to assign a higher score to cells with a more elongated morphology. This reflects a known distinguishing feature of the sarcomatoid morphology, validating our model scoring. This is further illustrated in the scatter plot in the upper half of Fig. \ref{morph_fig}, where we show a supervised UMAP \cite{McInnes2018} of the principal components of high and low scoring cells. The supervisory signal is provided by the output of our model, as a binary label of whether it is in the top or bottom 10\% of cells by score. Each point in the map represents a cell, colored red if it is in the top 10\% of model scores, or blue if in the bottom 10\%. UMAP attempts to learn an embedding in which examples with similar features are closer together. Thus, by looking at groups of cells in this map we can understand how high scoring and low scoring cell populations look. From Fig \ref{morph_fig}, sarcomatoid groups B and E we can see elongated cells are scored highly sarcomatoid, as are groups C and D showing large, irregular cells. Cells scored in the bottom 10\% tend to be much smaller, as can be seen in groups F, H, I and J. They also are rounder and more regular in their shape. As can be seen comparing epithelioid group G to sarcomatoid group C, while large cells may also be scored as epithelioid, they have a round shape with less texture to the staining.
\subsection{Survival Analysis using MesoScores}
Survival analysis using Kaplan Meier plots are shown in Figure \ref{surv}. Patients were divided into two groups based on model score. The median survival time of patients predicted more sarcomatoid was significantly shorter compared to non-sarcomatoid patients (190 vs 402 days, $p < 0.002$) for both. This difference in survival can be observed in the Kaplan Meier plot (a), where the predicted non-sarcomatoid curve in orange is less steep than the blue predicted sarcomatoid curve. In a Cox-proportional hazard model adjusted for sex and age at diagnosis, the hazard ratio for sarcomatoid cases was 2.43 (95\% CI 1.44 – 4.12, $p<0.005$), indicating that patients with sarcomatoid morphology were 2.43 times more likely to have died at a specific time point than non-sarcomatoid cases. In comparison, the HR for both gender and age were both much smaller, at $<1.1$. Very similar findings were obtained with censoring at 3 years (see supplementary material, Fig. 1).
\begin{figure*}[ht]
\centering
\centerline{\includegraphics[width=15cm]{Figs/surv_figA.png}}
\caption{a) Kaplan-Meier curves for all data. For data right censored at 3 years see supplementary material Fig 1. c) log Hazard Ratio of high Meso-score compared to demographic factors.}
\label{surv}
\end{figure*}
\subsection{General Discussion and Future Work}
We have developed a model capable of learning a cell-level indication of sarcomatoid and epithelioid regions of a TMA core tissue sample, which enables quantitative characterisation of a core according to the relative proportions of S and E components present. In summary:
\begin{enumerate}
\item
We have developed a Graph Neural Network (GNN) model (called MesoGraph) that can predict the mesothelioma subtype of the given patient sample with high accuracy (AUC-ROC $>$ 0.85) over independent multi-centric validation using only H\&E stained images of tumor samples.
\item
For a given sample, the proposed approach can generate a quantitative assessment (called MesoScore) of where the sample stands in terms of the epithelioid to sarcomatoid spectrum.
\item
Model predictions can be mapped onto individual cells in a given sample to generate histograms (called MesoGrams) showing the relative densities of epithelioid and sarcomatoid components within the tumor.
\item
MesoGraph generated scores can be used as a prognostic marker for predicting disease specific survival.
\item
We show that the weakly-supervised model is able to characterize known morphological patterns of cells associated with epithelioid and sarcomatoid mesothelioma.
\item
The code and the dataset used in this study is made publicly available for further development at https://github.com/measty/meso\textunderscore graph.
\end{enumerate}
\noindent The developed approach could help pathologists to subtype a core more accurately, consistently and efficiently, and paves the way to move beyond the three type system of characterising a tumor, towards a more fine-grained characterisation that matches the underlying continuous biological expression of mesothelial tumour cells on a spectrum between epithelioid and sarcomatoid morphology.
Most MIL based methods introduced in the literature have been patch based. One such MIL approach was introduced in \cite{dsmil2021}. Here, a dual stream approach was used where the final bag score is the mean of max instance pooling and an attention based weighted average of instances attended to by the max instance.
In another approach \cite{clinical2019} large scale datasets are used to train a MIL model for tumor detection, backpropagating only the top K instances per bag.
The CLAM algorithm \cite{lu2020data} is a further patch-based MIL method with attention that has been applied to a variety of computational pathology tasks.
As a final example of patch-based approaches, in the IDaRS algorithm proposed in \cite{idars2021} to detect key mutations on colorectal cancer, learning occurs on patches drawn using a ranking-based sampling scheme. While these MIL approaches have been developed for patch-level instances, we develop our method treating each cell as an instance, allowing us to investigate the differences in cell morphology between subtypes. This also removes the limitation on spatial resolution of predictions imposed by a patch-based approach.
Graph Neural Networks (GNNs) have also been applied in this domain. In \cite{wang2019weakly}, a GNN is used on prostate cancer TMA cores with self-supervised and morphological features for the task of classifying examples as high or low risk according to the Gleason score. GNNs are applied to whole slide images in \cite{lu2021slidegraph} by spatially clustering cells to form agglomerate nodes from which to build a slide-level graph to predict HER2 status in breast cancer. Our approach uses a dual-task formulation with ranking loss, on a cell graph, to allow better identification of regions associated with the two components that may be present in a mesothelioma core.
The results for this study show a potential for clinical implications when applied to routine diagnosis of MM. As shown in our work, there is a gradient between epithelioid and sarcomatoid MM and the various cell populations are identifiable and can be quantified using our approach. Improving identification of mesothelioma subtypes, is an essential part of diagnosis for MM. The behavior of biphasic MM is dependant on the ratio of epithelioid and sarcomatoid cells and may also be extended to other biphasic tumours.
One limitation of our method is that, while we have taken care to validate our model by looking at the features that influence its predictions, and the typical morphology of cells found in epithelioid and sarcomatoid regions, we still have some issues with the interpretability of the model outputs. Not all of the features our model learns on have an obvious histomorphological counterpart. For example, if we see from a feature importance analysis that a particular resnet feature is important, it is not clear how that translates into a histomorphological biomarker that a pathologist can look for in a tissue sample. Haralick texture features are a little better as they are constructed to capture specific, well defined properties of textures, but they are still difficult to interpret in comparison to morphological features.
Future work could involve the incorporation of cell classifications, via a segmentation method such as Hovernet \cite{graham2019hover} capable of simultaneous cell segmentation and classification. This would provide a further informative feature that may help identify cell-type specific patterns such as an association of tumor-infiltrating lymphocytes to a specific subtype. Such features could also help move away from difficult to interpret features such as resnet features, without sacrificing performance. A more extensive evaluation considering a larger dataset and including pathologist concordance studies to identify whether pathologists using such a tool would make more consistent and more accurate subtyping could also be considered.
In conclusion, we provide a novel method for more precisely characterising epithelioid and sarcomatoid cell subtypes in a quantifiable and reproducible way. Given the importance of sarcomatoid subtypes for prognosis and deciding on treatment pathway, our method may potentially offer clinical implications for patient care. Improved subtyping of MM allows for gains in both the efficiency and reliability of assessment of mesothelioma tumour cell classification by a reporting pathologist. The method we present and future work using our approach to further define epithelioid and sarcomatoid spectrum of MM, may ultimately form a basis for improving treatment and prognosis for the patient.
\subsection*{Acknowledgements}
\noindent This project was funded by CRUK-STFC Early Detection Innovation Award.
\noindent FM and ME also acknowledge funding support from EPSRC EP/W02909X/1.
\noindent We are grateful to The London Asbestos Support Awareness Group: https://www.lasag.org.uk/, National Mesothelioma Virtual Bank: http://www.mesotissue.org/ and MesoBank UK: http://www.mesobank.com/for their support.
\subsection*{Author Contributions and Funding}
\noindent ME: Conceptualization, Machine Learning Model Design and Implementation, Analysis, Writeup.
\noindent FM: Conceptualization, Lead for Machine Learning Model and Experiment Design, Analysis, Paper Write-up, Funding acquisition, Project Supervision and Management.
\noindent JO: Conceptualization, statistical analysis, paper writing up and revisions
\noindent JLR: Conceptualization, Clinical Lead, Paper Write-up, Funding acquisition, Project Supervision and Management.
\noindent HS: Conceptualization and study design, result analysis, paper writeup and revisions, Funding acquisition.
\noindent STM and XG: Conceptualization, paper writeup and revisions.
\noindent EK: Conceptualization, Paper structuring and revisions.
\noindent AMF, DJ, WC, MF, SP: Data Provision.
\subsection*{Declaration of Interests}
The authors declare no competing interests.
\bibliographystyle{splncs04}
\small{
\section*{Summary}
Malignant mesothelioma is classified into three histological subtypes, Epithelioid, Sarcomatoid, and Biphasic according to the relative proportions of epithelioid and sarcomatoid tumor cells present. Biphasic tumors display significant populations of both cell types. Current guidelines recommend that the sarcomatoid component of each mesothelioma is quantified, specifically, as a higher percentage of sarcomatoid pattern in biphasic mesothelioma shows poorer prognosis. It is therefore important to accurately diagnose morphological subtypes. However, this subtyping is subjective and limited by current diagnostic guidelines. This can affect the reproducibility even between expert thoracic pathologists when characterising the continuum of relative proportions of epithelioid and sarcomatoid components using a three class system. In this work, we develop a novel dual-task Graph Neural Network (GNN) architecture with ranking loss to learn a model capable of scoring regions of tissue down to cellular resolution. This allows quantitative profiling of a tumor sample according to the aggregate sarcomatoid association score of all the cells in the sample. The proposed approach uses only core-level labels and frames the prediction task as a dual multiple instance learning (MIL) problem. Tissue is represented by a cell graph with both cell-level morphological and regional features. We introduce a dataset of 234 mesothelioma tissue micro-array cores, and use an external multi-centric test set from Mesobank, on which we demonstrate the predictive performance of our model. We validate our model predictions through an analysis of the typical morphological features of cells according to their predicted score, finding that some of the morphological differences identified by our model match known differences used by pathologists. We further show that the model score is predictive of patient survival with a hazard ratio of 2.30. The ability to quantitatively score tissue on how much sarcomatoid component is present, and overlay cell-level instance scores, will assist pathologists in assessing a sample more quickly and consistently by defining the sarcomatoid component, particularly in biphasic mesothelioma. This ultimately will improve treatment decisions for patients with mesothelioma. The code for the proposed approach, along with the dataset, is available at: https://github.com/measty/MesoGraph.
Our contributions include releasing a new dataset, introduction of the MesoGraph architecture providing accurate subtype predictions, and insightful analysis of morphological differences between sarcomatoid and epithelioid components.
\keywords{Graph Neural Networks \and Multiple Instance Learning \and Mesothelioma \and Cancer subtyping \and Digital Pathology}
\section{Introduction} \label{intro}
Malignant mesothelioma (MM) is an aggressive cancer of malignant mesothelial cells of the pleural lining, primarily associated with asbestos exposure \cite{WAGNER1960a}. It has a poor prognosis with less than 10\% five year survival rates due to late diagnosis (\cite{hjerpe2018cyto} ; \cite{mesostats}). It has a long latency period from initial exposure to eventual carcinogenesis, and is difficult to diagnose due to its nonspecific clinical manifestations. MM is classified into 3 subtypes \cite{Ai2014}, epithelioid, biphasic and sarcomatoid mesothelioma (EM, BM and SM respectively), with BM characterised by a mix of epithelioid and sarcomatoid components.
The Histological subtype of malignant mesothelioma is essential for prognosis and clincial decisions on treatment pathways for patients \cite{Meyerhoff2015ImpactOM}. Stratification of a given sample into a particular subtype informs treatment and can help gain a more in-depth understanding of disease pathology and outcome. The benefit of surgical treatment has prognostic implications for epithelioid mesothelioma with median survival of 19 months but less so for SM and BM with respective median survival of 4 months and 12 months after surgical treatment \cite{Mansfield2014SystematicRO}.
Epithelioid mesothelioma is characterised by malignant cells that are cytologically round with varying grading of atypia. Sarcomatoid mesothelioma cells are generally recognised as malignant elongated spindle cells \cite{whotumors} and are associated with worse prognosis in comparison to EM. Sarcomatoid mesothelial cells may also include transitional features that are intermediate between epithelioid and sarcomatoid. Although transitional cells are now classified under sarcomatoid mesothelioma, their presence is associated with worse prognosis \cite{courtiol2019meso}.
While the distinction of these three histological subtypes of MM is crucial to patient treatment, management and prognosis, it is challenging to differentiate EM, SM and BM through visual analysis. Currently, there are no clear guidelines on how to perform this stratification in an objective and reproducible manner \cite{dacic2020interobserver}. Furthermore, even though mesotheliomas are divided into these three broad categories, in reality there is a continuous spectrum from EM to SM dependent upon the relative proportion of epithelioid and sarcomatoid cells in a given sample. As a consequence, existing approaches are unable to objectively quantify where on this spectrum a given sample falls based on profiling of cellular morphological patterns in it.
A number of deep learning methods for analyzing mesothelioma images have been developed recently. For example, SpindleMesoNET~\cite{Naso2021} can separate malignant SM from benign spindle cell mesothelial proliferations. A recent approach for survival prediction of MM patients called MesoNet~\cite{courtiol2019meso} uses a Mulitple Instance Learning (MIL) solver originally developed for computer vision applications~\cite{courtiol2018weak} and classification of lymph node metastases \cite{weldon2016}. However, automated subtyping of malignant mesothelioma from hematoxylin and eosin (H\&E) stained tissue sections remains an open problem.
\begin{figure*}[ht]
\centering
\centerline{\includegraphics[width=15.5cm]{Figs/graphical_abstract.png}}
\caption{High level overview/graphical abstract of our approach to mesothelioma subtyping. Pathologist labelled TMA cores are used to learn a graph nueral network model on cell-level features. We show the model can identify sarcomatioid areas, and its predicted scores can be used to stratify patient survival.}
\label{graphical_abs}
\end{figure*}
One challenge in the characterization of mesothelioma is that pathologist-assigned ground-truth labels of mesothelioma subtypes are typically available only at the case level as it is very difficult for pathologists to associate tumor micro-environment or cellular morphometric patterns with image or case level labels in an objective manner. Moreover, it can be very time-consuming to obtain detailed cellular or regional annotations, and those annotations may not be very reliable due to significant inter- and intra-observer variation.
The aim of this study was to develop a GNN approach to predict subtypes of malignant mesothelioma in a MIL setting. This was achieved considering Tissue Micro-array (TMA) cores as bags and individual cells as instances.
On these we have built a weakly-supervised machine learning model to characterize mesothelioma subtypes using only case level labels in its training. The proposed approach can generate a quantitative assessment of where the sample stands in terms of the aforementioned epithelioid to sarcomatoid spectrum, enabling pathologists to perform a more in-depth characterization of tumor samples. A high level overview of the approach presented in the paper can be found in Fig. \ref{graphical_abs}.
\section{Results and Discussion}
We have developed a custom Graph Neural Network based pipeline called MesoGraph that can predict mesothelioma subtypes using H\&E stained tissue images. MesoGraph uses pathologist assigned case-level labels without any cellular or regional annotations in its training. The proposed approach models each cell in a given sample as a node in the graph, which is connected to neighbouring cells. Each node is associated with various features, which can be broadly classified into four types: 1) nuclear morphometric features, 2) stain intensity features of nuclear and cytoplasmic components of the cell, 3) cellular counts in the neighbourhood of node and 4) deep neural network and haralick based texture features. For a given test sample, it generates two probability scores (collectively called MesoScore) representing the probabilities of the sample being epithelioid or sarcomatoid. As a biphasic mesothelioma tumor is composed of both epithelioid and sarcomatoid components, the two outputs in MesoScore allow precise quantification of the two components in the sample. In addition to predicting mesothelioma subtype, MesoGraph also generates cell-level quantitative scores representing the association of each cell with the mesothelial subtype of the given sample. MesoGraph has been trained and independently validated on two datasets: St. George's Hospital (SGH, n = 234) and the multi-centric MesoBank collection (MB, n = 258).
In this section, we present the results of the proposed method in terms of its predictive performance in comparison to existing approaches, as well as its ability to identify histological features and morphological characteristics of cells associated with different types of malignant mesothelioma. We also demonstrate the ability of the proposed approach to stratify mesothelioma patients based on their expected survival.
\subsection{Predictive Performance} \label{performance_results}
Test results from the MesoGraph pipeline for both MesoBank and SGH datasets are shown in Figure \ref{model_auc} and Table~\ref{restable}. Here, the Receiver Operating Characteristic (ROC) curve is obtained by considering both sarcomatoid and biphasic samples as the positive class, whereas the epithelioid samples are associated with the negative label. As can be seen from these results, the proposed approach offers high predictive quality over both cross-validation and independent testing in comparison to other existing approaches. In the table, PINS refers to the Positive Instance Sampling Patch-based MIL approach as detailed in \cite{eastwood2022pins} whereas CLAM is the Clustering-constrained Attention Multiple Instance Learning method described in \cite{lu2020data}, a deep-learning-based weakly-supervised method that uses attention in combination with clustering-based constraints to identify the most predictive areas of the image. Max-MIL and naive-MIL are simple patch based baseline multiple instance learning (MIL) methods detailed further in the methods Section \ref{evaluation}.
As can be seen from Table \ref{restable}, the max-based MIL strategy performs poorly. This is likely due to the relatively small size of the training dataset, as learning only on the maximally scoring instance per bag exacerbates this.
Naive MIL performs surprisingly well. This may be due to the relatively high proportion of positive instances that are expected to be present in many of the positive bags (for example a sarcomatoid core should contain mostly positive instances).
This makes the implicit assumption this model makes, namely that all instances share the label of the bag, less wrong for this dataset compared to other MIL tasks.
PINS and CLAM, as two patch based methods with a mechanism for focussing on the most relevant region of an image, perform similarly with solid performance. However, as patch based methods, the spatial resolution of the prediction maps they can provide is lower than that of our cell graph based model.
Our model outperforms other models tested, achieving an internal cross-validation performance of $0.90 \pm 0.01$. It performs particularly well in terms of its average precision (AP) of $0.86\pm0.02$, indicating that its performance on the positive class (which is the minority class) is very good. While performance drops slightly on the external validation set, an AUROC of $0.86$ and AP of $0.8$ as seen in Fig \ref{model_auc}, panels C and D shows these results generalize well. We attribute the performance improvements achieved by our model firstly to the cell graph representation, with cells and their morhpological features as instances, which is far more natural than an arbitrary division into patch instances. Secondly, our formulation of the learning as a dual task problem with a ranking loss. This acknowledges both the ordering we know exists between EM, BM and SM cores in terms of how much sarcomatoid component is present, and also the possibility that some regions of tissue may not be strongly associated with either a sarcomatoid or epithelioid core label.
\begin{figure*}[ht]
\centering
\centerline{\includegraphics[width=14cm]{Figs/results_auc.png}}
\caption{A) AUROC for each model. B) Model average precision. C) AUROC on cross-cohort evaluation. D) Average precision on cross-cohort evaluation. Our model outperforms other models tested, and from C and D we see good generalization to external validation data.}
\label{model_auc}
\end{figure*}
\begin{table*}[t]
\begin{center}
\begin{tabularx}{0.85 \textwidth}{|>{\raggedright\arraybackslash}X||>{\raggedright\arraybackslash}X|>{\raggedright\arraybackslash}X|>{\raggedright\arraybackslash}X|>{\raggedright\arraybackslash}X|}
\hline
Metric&AUC-ROC&Avg. Precision&Sensitivity&Specificity\\
\hline\hline
max-MIL&$0.70\pm0.01$&$0.54\pm0.12$&$0.54\pm0.07$&$0.73\pm0.09$\\
naive-MIL&$0.84\pm0.05$&$0.72\pm0.11$&$0.72\pm0.08$&$0.71\pm0.1$\\
PINS \cite{eastwood2022pins}&$0.85\pm0.05$&$0.80\pm0.07$&$0.82\pm0.1$&$0.71\pm0.13$\\
CLAM \cite{lu2020data}&$0.85\pm0.07$&$0.74\pm0.11$&$0.75\pm0.11$&$0.77\pm0.02$\\
\small{MesoGraph(Ours)}&$0.90\pm 0.007$&$0.86\pm0.02 $&$ 0.88\pm0.015$&$0.72\pm0.01 $\\
\hline
\end{tabularx}
\caption{Summary of results (mean$\pm$stdev) of models evaluated on primary dataset (SGH). } \label{restable}
\end{center}
\end{table*}
\renewcommand{\arraystretch}{1}
\subsection{Visualization of Model Output}
The output of our model can be visualised in a zoomable, interactive Graphical User Interface (GUI) we have developed. A demo of results from our model can be found at (https://mesograph.dcs.warwick.ac.uk). Examples of the cell level scoring output by the model are shown in Fig. \ref{cores_vis}. For each cell in a given sample, the proposed model generates two prediction scores signifying the probability of the cell being associated with sarcomatoid or epithelioid labels. These scores can be combined and visualized in a colourmap showing cells that are associated with epithelioid (blue) and sarcomatoid (red) subtypes as well as a histogram (called MesoGram) showing the relative distributions of epithelioid and sarcomatoid components.
\begin{figure*}[ht!]
\centering
\centerline{\includegraphics[width=14cm]{Figs/paper_core_fig_small.png}}
\caption{Model visualisation. Each core is shown together with a zoomed in view showing the differences in morphology between regions, and a plot showing the distribution of node scores in that core. A. Epithelioid core. We see a predominantly low-scoring core with only a few small regions displaying slightly more sarcomatoid features. Conversely, in B, a Sarcomatoid core, nodes are predominantly high scoring. C and D. Biphasic cores. In each core we see a bimodal distribution of scores, particularly pronounced in core D. The zoomed in regions show a distinct difference in morphology between high and low scoring regions, with rounder cells seen in lower scoring regions, and a more elongated morphology and less structured cell organisation in higher scoring regions.}
\label{cores_vis}
\end{figure*}
From the zoomed-in masks, we can see the model can distinguish between regions with typical rounded morphology of the Epithelioid subtype and the more elongated morphology displayed in Sarcomatoid regions. The MesoGram plots of most samples tend to be bimodal to some extent, with epithelioid and sarcomatoid cores more heavily skewed toward low and high scores, respectively. This continuum of distribution between sarcomatoid and epithelioid is demonstrated further in Fig. \ref{cores_overview}, where thumbnails of model output on all cores are shown, grouped by subtype and ordered within each subtype by model score. This ability to give a more precise, fine-grained characterisation of a tumor beyond the current three poorly defined and subjective subtypes is a strength of our approach.
\begin{figure*}[ht!]
\centering
\centerline{\includegraphics[width=14cm]{Figs/core_overview.png}}
\caption{Overview of model predictions by subtype. Images of model predictions ordered within each subtype by the predicted predominance of the sarcomatoid component, illustrating the underlying continuous biological expression of tissue on the E to S spectrum.}
\label{cores_overview}
\end{figure*}
\subsection{Explainability of model predictions}
To gain an understanding of the predictions generated by the proposed approach, we have applied GNNExplainer \cite{ying2019gnnexplainer} to the trained graph neural network model. This allows us to understand what node level features are contributing to the prediction of a given sample for each subtype. The top 10 features identified in this analysis are shown in Fig. \ref{feat_imp}.
The most important feature overall is the circularity, which confirms the expected distinction between the rounder morphology of the epithelioid subtype, compared to the more spindle-shaped sarcomatoid morphology. There are also a number of features describing the intensity and texture in the Eosin channel around the nucleus. Looking at the feature importances on specific subtypes, the resnet features are most useful on epithelioid cores. Circularity is specifically important in Epithelioid and sarcomatoid subtypes, as they tend to be composed of more homogeneous populations and therefore are expected to contain mostly either rounded or more elongated cells. In both Biphasic and Sarcomatoid cores, nearby detection counts seem to be an important feature. This may reflect a tendency for non-epithelioid tumors to display a slightly more spread out and disorganised cell distribution. We also notice that the importance of most of the top features has far more spread when considering Epithelioid cores, indicating a wide variety of features can contribute to an epithelioid score, with few features being universally important across all epithelioid cores.
\begin{figure*}[ht!]
\centering
\centerline{\includegraphics[width=14cm]{Figs/box_charts_gnn.png}}
\caption{Top 10 features identified by GNNExplainer, considering A) all cores, and in B-D, importances on cores grouped by subtype. Results shown as a standard box \& whisker plot, with the box showing the 25th, 50th and 75th percentile of a features importance scores over cores. Whiskers show min and max values, limited at box $\pm$ 1.5 $\times$ inter-quartile range. The top feature is circularity, a knows differentiating characteristic between mesothelial subtypes, providing validation for our model.}
\label{feat_imp}
\end{figure*}
To determine the separation between classes based on top scoring features, in the lower half of Fig. \ref{morph_fig} (C and D) we plotted the prediction of each core against the assigned label by a pathologist. While Epithelioid cores and Sarcomatoid cores are mostly well-separated, we observe there can be overlap between some of the cases in terms of morphology. We also observe that Biphasic is not very distinct from Sarcomatoid cores.
\subsection{Characterization of Cellular Morphologies} \label{pca-umap}
Pathologists assess cell morphology when diagnosing and scoring mesothelioma tumours. Therefore, we sought to investigate differences in nuclear morphology between mesothelioma tumours with different diagnoses. We focused on key features assessed qualitatively by pathologists including nuclear area, elongation (width and length) and nucleus shape regularity as measured by both circularity (how close it is to a circular shape), and solidity (which reflects overall concavity of a shape). Interestingly, in Figure \ref{morph_fig}, we found sarcomatoid tumours tend to have larger nuclei on average compared to epithelioid tumours. As expected these nuclei are more elongated and have less circular shapes. For almost all features, measures of nucleus shape in biphasic tumours fall in between epithelioid and sarcomatoid tumours. These results already confirm that our image analysis pipeline reflects inherent differences between these types even when only considering the average measures of each tumour which is consistent with pathological features. This motivates the development of more sophisticated AI methods to detect these differences.
\begin{figure}[ht!]
\centerline{\includegraphics[width=15.8cm]{Figs/morph_fig.png}}
\caption{Upper panel: Examples of cells scored most and least highly by the model, plotted as a 2-D UMAP reduction
of principal components calculated on both high and low scoring cells. For each cluster, the mean of the cells is
displayed, together with individual example cells.
The clusters A-E, on the left side of the panel are predicted as Sarcomatoid and demonstrate a more spindle like morphology, grouped together in size and relative spindle cell characteristics in each cluster as shown by example cells on the right side of the panel. Similarly, the Non-sarcomatoid predicted cells also show clustering into 5 groups, F-J.
Lower panel: Morphological heterogeneity of mesothelioma tumours independent of model prediction.
A) Distribution of average morphology across tumour types. All measurements are normalised to the data average
and standard deviation. B) Heterogeneity of cell morphology across different mesothelioma tumour types based on
standard deviation (SD) of z-scored single cell data. C-D) Morphological heterogeneity based on ground truth labels
(C) and predicted labels (D)}
\label{morph_fig}
\end{figure}
Next, we investigated the extent of variability in morphological measurements across different tumour types. We measured the standard deviation of cell features for each single tumour core as a proxy of heterogeneity. We found that Sarcomatoid tumours exhibit higher morphological heterogeneity in all nuclear features. These analyses motivate the investigation of single cell phenotypes to identify the most relevant subpopulations.
We can gain further insight into the differences in morphology that the model is associating with each subtype by looking at the principal components of cells assigned the highest and lowest scores (i.e most and least likely to be associated with a sarcomatoid core, respectively).
Comparing the first principal component for each subtype in supplementary Fig. 2, we can see that the model has learnt to assign a higher score to cells with a more elongated morphology. This reflects a known distinguishing feature of the sarcomatoid morphology, validating our model scoring. This is further illustrated in the scatter plot in the upper half of Fig. \ref{morph_fig}, where we show a supervised UMAP \cite{McInnes2018} of the principal components of high and low scoring cells. The supervisory signal is provided by the output of our model, as a binary label of whether it is in the top or bottom 10\% of cells by score. Each point in the map represents a cell, colored red if it is in the top 10\% of model scores, or blue if in the bottom 10\%. UMAP attempts to learn an embedding in which examples with similar features are closer together. Thus, by looking at groups of cells in this map we can understand how high scoring and low scoring cell populations look. From Fig \ref{morph_fig}, sarcomatoid groups B and E we can see elongated cells are scored highly sarcomatoid, as are groups C and D showing large, irregular cells. Cells scored in the bottom 10\% tend to be much smaller, as can be seen in groups F, H, I and J. They also are rounder and more regular in their shape. As can be seen comparing epithelioid group G to sarcomatoid group C, while large cells may also be scored as epithelioid, they have a round shape with less texture to the staining.
\subsection{Survival Analysis using MesoScores}
Survival analysis using Kaplan Meier plots are shown in Figure \ref{surv}. Patients were divided into two groups based on model score. The median survival time of patients predicted more sarcomatoid was significantly shorter compared to non-sarcomatoid patients (190 vs 402 days, $p < 0.002$) for both. This difference in survival can be observed in the Kaplan Meier plot (a), where the predicted non-sarcomatoid curve in orange is less steep than the blue predicted sarcomatoid curve. In a Cox-proportional hazard model adjusted for sex and age at diagnosis, the hazard ratio for sarcomatoid cases was 2.43 (95\% CI 1.44 – 4.12, $p<0.005$), indicating that patients with sarcomatoid morphology were 2.43 times more likely to have died at a specific time point than non-sarcomatoid cases. In comparison, the HR for both gender and age were both much smaller, at $<1.1$. Very similar findings were obtained with censoring at 3 years (see supplementary material, Fig. 1).
\begin{figure*}[ht]
\centering
\centerline{\includegraphics[width=15cm]{Figs/surv_figA.png}}
\caption{a) Kaplan-Meier curves for all data. For data right censored at 3 years see supplementary material Fig 1. c) log Hazard Ratio of high Meso-score compared to demographic factors.}
\label{surv}
\end{figure*}
\subsection{General Discussion and Future Work}
We have developed a model capable of learning a cell-level indication of sarcomatoid and epithelioid regions of a TMA core tissue sample, which enables quantitative characterisation of a core according to the relative proportions of S and E components present. In summary:
\begin{enumerate}
\item
We have developed a Graph Neural Network (GNN) model (called MesoGraph) that can predict the mesothelioma subtype of the given patient sample with high accuracy (AUC-ROC $>$ 0.85) over independent multi-centric validation using only H\&E stained images of tumor samples.
\item
For a given sample, the proposed approach can generate a quantitative assessment (called MesoScore) of where the sample stands in terms of the epithelioid to sarcomatoid spectrum.
\item
Model predictions can be mapped onto individual cells in a given sample to generate histograms (called MesoGrams) showing the relative densities of epithelioid and sarcomatoid components within the tumor.
\item
MesoGraph generated scores can be used as a prognostic marker for predicting disease specific survival.
\item
We show that the weakly-supervised model is able to characterize known morphological patterns of cells associated with epithelioid and sarcomatoid mesothelioma.
\item
The code and the dataset used in this study is made publicly available for further development at https://github.com/measty/meso\textunderscore graph.
\end{enumerate}
\noindent The developed approach could help pathologists to subtype a core more accurately, consistently and efficiently, and paves the way to move beyond the three type system of characterising a tumor, towards a more fine-grained characterisation that matches the underlying continuous biological expression of mesothelial tumour cells on a spectrum between epithelioid and sarcomatoid morphology.
Most MIL based methods introduced in the literature have been patch based. One such MIL approach was introduced in \cite{dsmil2021}. Here, a dual stream approach was used where the final bag score is the mean of max instance pooling and an attention based weighted average of instances attended to by the max instance.
In another approach \cite{clinical2019} large scale datasets are used to train a MIL model for tumor detection, backpropagating only the top K instances per bag.
The CLAM algorithm \cite{lu2020data} is a further patch-based MIL method with attention that has been applied to a variety of computational pathology tasks.
As a final example of patch-based approaches, in the IDaRS algorithm proposed in \cite{idars2021} to detect key mutations on colorectal cancer, learning occurs on patches drawn using a ranking-based sampling scheme. While these MIL approaches have been developed for patch-level instances, we develop our method treating each cell as an instance, allowing us to investigate the differences in cell morphology between subtypes. This also removes the limitation on spatial resolution of predictions imposed by a patch-based approach.
Graph Neural Networks (GNNs) have also been applied in this domain. In \cite{wang2019weakly}, a GNN is used on prostate cancer TMA cores with self-supervised and morphological features for the task of classifying examples as high or low risk according to the Gleason score. GNNs are applied to whole slide images in \cite{lu2021slidegraph} by spatially clustering cells to form agglomerate nodes from which to build a slide-level graph to predict HER2 status in breast cancer. Our approach uses a dual-task formulation with ranking loss, on a cell graph, to allow better identification of regions associated with the two components that may be present in a mesothelioma core.
The results for this study show a potential for clinical implications when applied to routine diagnosis of MM. As shown in our work, there is a gradient between epithelioid and sarcomatoid MM and the various cell populations are identifiable and can be quantified using our approach. Improving identification of mesothelioma subtypes, is an essential part of diagnosis for MM. The behavior of biphasic MM is dependant on the ratio of epithelioid and sarcomatoid cells and may also be extended to other biphasic tumours.
One limitation of our method is that, while we have taken care to validate our model by looking at the features that influence its predictions, and the typical morphology of cells found in epithelioid and sarcomatoid regions, we still have some issues with the interpretability of the model outputs. Not all of the features our model learns on have an obvious histomorphological counterpart. For example, if we see from a feature importance analysis that a particular resnet feature is important, it is not clear how that translates into a histomorphological biomarker that a pathologist can look for in a tissue sample. Haralick texture features are a little better as they are constructed to capture specific, well defined properties of textures, but they are still difficult to interpret in comparison to morphological features.
Future work could involve the incorporation of cell classifications, via a segmentation method such as Hovernet \cite{graham2019hover} capable of simultaneous cell segmentation and classification. This would provide a further informative feature that may help identify cell-type specific patterns such as an association of tumor-infiltrating lymphocytes to a specific subtype. Such features could also help move away from difficult to interpret features such as resnet features, without sacrificing performance. A more extensive evaluation considering a larger dataset and including pathologist concordance studies to identify whether pathologists using such a tool would make more consistent and more accurate subtyping could also be considered.
In conclusion, we provide a novel method for more precisely characterising epithelioid and sarcomatoid cell subtypes in a quantifiable and reproducible way. Given the importance of sarcomatoid subtypes for prognosis and deciding on treatment pathway, our method may potentially offer clinical implications for patient care. Improved subtyping of MM allows for gains in both the efficiency and reliability of assessment of mesothelioma tumour cell classification by a reporting pathologist. The method we present and future work using our approach to further define epithelioid and sarcomatoid spectrum of MM, may ultimately form a basis for improving treatment and prognosis for the patient.
\subsection*{Acknowledgements}
\noindent This project was funded by CRUK-STFC Early Detection Innovation Award.
\noindent FM and ME also acknowledge funding support from EPSRC EP/W02909X/1.
\noindent We are grateful to The London Asbestos Support Awareness Group: https://www.lasag.org.uk/, National Mesothelioma Virtual Bank: http://www.mesotissue.org/ and MesoBank UK: http://www.mesobank.com/for their support.
\subsection*{Author Contributions and Funding}
\noindent ME: Conceptualization, Machine Learning Model Design and Implementation, Analysis, Writeup.
\noindent FM: Conceptualization, Lead for Machine Learning Model and Experiment Design, Analysis, Paper Write-up, Funding acquisition, Project Supervision and Management.
\noindent JO: Conceptualization, statistical analysis, paper writing up and revisions
\noindent JLR: Conceptualization, Clinical Lead, Paper Write-up, Funding acquisition, Project Supervision and Management.
\noindent HS: Conceptualization and study design, result analysis, paper writeup and revisions, Funding acquisition.
\noindent STM and XG: Conceptualization, paper writeup and revisions.
\noindent EK: Conceptualization, Paper structuring and revisions.
\noindent AMF, DJ, WC, MF, SP: Data Provision.
\subsection*{Declaration of Interests}
The authors declare no competing interests.
\bibliographystyle{splncs04}
\small{
|
train/arxiv
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BkiUdmE4eIOjR-dm5eBC
| 5 | 1 |
\section{Implementation in Mathematica}
\label{sec:appendix_mathematica}
The following short Mathematica script takes as input a graph $G$ (in the example below, this is the edge-graph of the dodecahedron), and an index $k$ of an eigenvalue.
It then compute the $v_i$ (or \texttt{vert} in the code), \shortStyle{i.e.,}\ the vertex-coordinates of the $\theta_k$-eigenpolytope.
If the dimension turns out to be appropriate, the spectral embedding of the graph, as well as the eigenpolytope are plotted.
\vspace{0.5em}
\begin{lstlisting}
(* Input:
* the graph G, and
* the index k of an eigenvalue (k = 1 being the largest eigenvalue).
*)
G = GraphData["DodecahedralGraph"];
k = 2;
(* Computation of vertex coordinates 'vert' *)
n = VertexCount[G];
A = AdjacencyMatrix[G];
eval = Tally[Sort@Eigenvalues[A//N], Round[#1-#2,0.00001]==0 &];
d = eval[[-k,2]]; (* dimension of the eigenpolytope *)
vert = Transpose@Orthogonalize@
NullSpace[eval[[-k,1]] * IdentityMatrix[n] - A];
(* Output:
* the graph G,
* its eigenvalues with multiplicities,
* the spectral embedding, and
* its convex hull (the eigenpolytope).
*)
G
Grid[Join[{{$\theta$,"mult"}}, eval], Frame$\to$All]
Which[
d<2 , Print["Dimension too low, no plot generated."],
d==2, GraphPlot[G, VertexCoordinates$\to$vert],
d==3, GraphPlot3D[G, VertexCoordinates$\to$vert,
d>3 , Print["Dimension too high, 3-dimensional projection is plotted."];
GraphPlot3D[G, VertexCoordinates$\to$vert[[;;,1;;3]] ]
]
If[d==2 || d==3,
Region`Mesh`MergeCells[ConvexHullMesh[vert]]
]
\end{lstlisting}
\section{Motivation}
\label{sec:motivation}
We dive into three curious topics in the interplay between graphs and polytopes, all of which will be addressed in this paper.
\subsection{Reconstructing polytopes from the edge-graph}
\label{sec:reconstructing_polytopes}
Since Steinitz \cite{...} it is known that the edge-graphs of polyhedra are exactly the 3-connected planar graphs, and that any such graph belongs to a unique combinatorial type of polyhedra.
On the other hand, the study of higher-dimensional polytopes taught us that the edge-graph carries very little information about its polytope if $d\ge 4$.
For example, the edge-graph of any simplex is the complete graph.
But so is the edge-graph of every \emph{neighborly polytope}, and so, given only the edge-graph, we will never know from which polytope it was obtained initially.
The same example shows that in many cases not even the dimension of the polytope can be reconstructed.
As a light in the dark, we have the result of Blind and Mani \cite{...} (with its modern proof from Kalai \cite{kalai1988simple}), which states that the combinatorial type of a \emph{simple polytope} can be reconstructed from its edge-graph.
One can then ask what other classes of polytopes allow such a unique reconstruction.
For example, it might be that every \enquote{sufficiently symmetric} polytope is reconstructable in this sense.
\begin{question}\label{q:symmetrc_polytopes_with_same_edge_graph}
Can there be two distinct sufficiently symmetric polytopes with the same edge-graph?
\end{question}
One might ask \enquote{what counts as sufficient symmetric?}, and this paper shall answer this question partially.
For example, vertex-transitivity is not sufficient:
every~even-dimen\-sional \emph{cyclic polytope} $C(n,2d)$ is neighborly and can be realized in a vertex-transitive way: set the coordinates of the $i$-th vertex to be
$$\big(\sin(t_i),\cos(t_i);\sin(2t_i),\cos(2t_i);...;\sin(dt_i),\cos(dt_i)\big)\in\RR^{2d}$$
with $t_i=2\pi i/n$. The edge-graph is $K_n$, independent of $d$.
The next candidate in line, \emph{edge-transitivity}, seems more promising.
And while a complete classification of edge-transitive polytopes might be possible (though, none is currently known), a more satisfying outcome would be a result independent of any such classification.
\subsection{Realizing symmetries of the edge-graph}
\label{sec:realizing_symmetries}
Polytopes are at most as symmetric as there edge-graph.
Any symmetry $T\in\Aut(P)\subset\Ortho(\RR^d)$ of a polytope $P\subset\RR^d$ induces a symmetry $\phi\in\Aut(G_P)\subseteq\Sym(n)$ of the edge-graph.
And thus $\Aut(P)\le\Aut(G_P)$ as abstract groups.
For general polytopes, it cannot be expected that the polytope shows many, or just any symmetries of the edge-graph.
However, one can always try to find a sufficiently symmetric realization of the polytope, so that a maximum of combinatorial symmetries of the edge-graph is realized as geometric symmetries of the polytope.
One can even hope to realize \emph{all} the symmetries of the edge-graph.
In the case of polyhedra, the circle packing version of the Steinitz theorem provides a way to realize all the symmetries of the edge-graph.
Again, higher dimensions tell a different story: there exist 4-dimensional polytopes with edge-graphs that can never be realized as polytopes the edge-graphes of which cannot be realized with all their symmeteries (consider \shortStyle{e.g.}\ the dual of the polytope constructed in \cite{bokowski1984combinatorial}).
Again, it seems that assuming sufficient symmetry of the polytope ensures that the polytope has already \emph{all} the symmetries of its edge-graph.
We can actually proof this in the case of so called \emph{distance-transitive} symmetry, and we hope to find the same in the case of only edge-transitive polytopes.
\begin{question}
Does every sufficiently symmetric polytope realize all the symmetries of its edge-graph?
\end{question}
\subsection{Rigidity and perfect polytopes}
\label{sec:rigid_perfect}
Every polytope can be continuously deformed while preserving its combinatorial type.
In the case of highly symmetric polytopes, one might be additionally interesting in keeping the symmetry properties during the deformation.
A polytope that cannot be deformed without loosing any of its symmetries is be called \emph{rigid} or, occasionally, a \emph{perfect polytope}.
All regular polytopes are examples of such rigid polytopes, and there exist many more.
While quite some examples are known, no general construction procedures has been described.
Again, one might hope to show that all sufficiently symmetric polytopes are rigid, in particular, the edge-transitive polytopes.
\begin{question}
Is every sufficiently symmetric polytope rigid?
\end{question}
\subsection{A motivating example}
\label{sec:motivating_example}
Consider your favorite highly symmetric polytope $P\subset\RR^d$.
For demonstration, we choose $P$ to be the \emph{regular dodecahedron}\footnote{Any other platonic solid (or regular polytope) might do the trick. Ideally, your polytope is transitive on vertices and edges.}
Here are some observations:
\begin{myenumerate}
\item $P$ is really symmetric (it is a regular polyhedron). All its symmetries must also be reflected in the edge-graph $G_P$, since any automorphism of $P$ restricts to an automorphism of $G_P$.
However, it is far from evident whether any particular symmetry of $G_P$ is reflected in $P$ as well?
In the case of the dodecahedron, every combinatorial symmetry of the edge-graph is indeed a geomery symmetry of $P$.
\item The study of higher dimensional polytopes teaches that the edge-graph of a polytope carries usually very little information about the polytope itself.
For example, the edge-graph of a simplex is a complete graph.
But so are the edge-graph of all \emph{neighborly polytopes}.
It was shown in \cite{kalai1988simple} that this problem of reconstruction does not happen for \emph{simple polytopes}, and it is our goal to show that this is also not to be expected for sufficiently symmetric polytope.
\item A polytope is called \emph{perfect} if its combinatorial type has a unique geometric realization of highest symmetry.
Many highly symmetric polytopes one immediately thinks about have this property. But many other do not.
We show that being edge-transitive seems to be the right assumption to ensure being perfect.
\end{myenumerate}
\section{Eigenpolytopes and spectral polytopes}
One way to connect spectral graph theory to polytope theory is via the construction of the so-called \emph{eigenpolytopes}.
In this section, we will survey this construction and its application primarily to the study of highly symmetric graphs.
From this section on, let $P\subset\RR^d$ denote a $d$-dimensio\-nal convex polytope with $d\ge 2$.
If not stated otherwise, we shall~assume that $P$ is \emph{full-dimensional}, that is, that $P$ is not contained in any proper subspace of $\RR^d$.
Let $\F_\delta(P)$ denote the set of \emph{$\delta$-dimensional faces} of $P$.
Fix some bijection $v\: V\to \F_0(P)$ between $V=\{1,...,n\}$ and the vertices of $P$.
Let $G_P=(V,E)$ be the graph with $ij\in E$ if and only if $\conv\{v_i,v_j\}$ is an edge of $P$.
$G_P$ is called \emph{edge-graph} of $P$, and $v$ the \emph{skeleton} of $P$.
Note that we can consider $v$ as a realization of $G_P$.
\subsection{Eigenpolytopes}
\begin{definition}
Given a graph $G$ and an eigenvalue $\theta\in \Spec(G)$ of multiplicity $d$. If $v$ is the $\theta$-realization of $G$, then
%
$$P_G(\theta):=\conv\{v_i\mid i\in V\}\subset\RR^d$$
%
is called \emph{$\theta$-eigenpolytope} of $G$, or just \emph{eigenpolytope} if $\theta$ is not specified.
\end{definition}
The notion of the eigenpolytope goes back to Godsil \cite{godsil1978graphs}, and can~be under\-stood~as a geometric way to study the combinatorial, algebraic~and~spectral properties of a graph.
Several authors investigated the eigenpolytopes of famous graphs or graph families.
For example, Powers \cite{powers1986petersen} gave a description for the face structure of~the~different eigenpolytopes of the Petersen graph, one of which will reappear as distance-transitive polytope in \cref{sec:distance_transitive_polytopes}.
Mohri \cite{mohri1997theta_1} analysed the geometry of the, what he called, \emph{Hamming polytopes}, the $\theta_2$-eigenpolytopes of the Hamming graphs.
As it turns out, but was not realized by himself, these polytopes are merely the cartesian products of regular simplices.
These, as well, are distance-transitive~\mbox{polytopes}. Powers \cite{powers1988eigenvectors} took a first look at the eigenpolytopes of the \emph{distance-regular} graphs.
Those especially will be discussed in greater detail below.
Finally, Padrol and Pfeifle \cite{padrol2010graph} investigated graph operations and their effect on the associated eigenpolytopes.
Before we proceed, we recall the relevant terminology for polytopes.
\subsection{Spectral polytopes}
Most graphs are not edge-graphs of polytopes (so-called \emph{polytopal} graphs), but eigenpolytope provide an operation that to every graph (and an eigenvalue thereof) assignes such a polytopal graph (the edge-graph of the eigenpolytope).
This operation is most interesting in the case of vertex-transitive graphs.
If $G$ is a vertex-transitive graph on $n$ vertices, every vertex of $G$ becomes a vertex of $P_G(\theta)$ (though maybe several vertices are assgined to the same point).
It seems interesting to ask for when this operation is a fixed point, that is, when is the edge-graph of the eigenpolytope isomorphic to the original graph.
We are specifically interested in one particular phenomenon associated with eigenpolytopes, that can best be observed in our running example.
\begin{definition}
\quad
\begin{myenumerate}
\item A polytope $P$ is called \emph{balanced} resp.\ \emph{spectral}, if this is the case for its skeleton.
\item A graph is called \emph{spectral} if it is the edge-graph of a spectral polytope.
\end{myenumerate}
\end{definition}
For a polytope, being spectral seems to be a quite exceptional property.
For example, being spectral implies that the polytope realizes all the symmetries of its edge-graph.
So, no neighborly polytope except for the simplex can be spectral.
In contrast, all (centered) neighborly polytopes are balanced, and many neighborly polytopes are eigenpolytopes (\shortStyle{e.g.}\ one of the Petersen-polytopes is neighborly).
If a graph is spectral to eigenvalue $\theta$, then it is isomorphic to the edge-graph of its $\theta$-eigenpolytop.
However, the converse is not true: the $\theta_3$-realization of the pentagon-graph is a pentagram, whose convex hull is a pentagon. But the pentagon-graph is not $\theta_3$-spectral.
\begin{center}
\includegraphics[width=0.7\textwidth]{img/convex_hull_5_gon}
\end{center}
\begin{example}
Every neighborly polytope has eigenvalue $-1$ (if centered appropriately), but none of these is spectral, except for the simplices.
\end{example}
\begin{example}
It is an easy exercise to show that the $d$-cubes and $d$-crosspolytopes are spectral with eigenvalue $\theta_2$.
In fact, it was shows by Licata and Power \cite{licata1986surprising} that also the regular polygons, all simplices, the regular dodecahedron and the regular icosahedron are spectral in this sense.
The only regular polytopes which they were not able to prove spectral were the 24-cell, 120-cell and 600-cell.
However, they conducted numerical experiments hinting in that direction.
\end{example}
\msays{The $\theta_3$-realization of the cube-graph is a polytope (the tetrahedron) and its edges correspond to edges of the graph, but it is not faithful.}
In a slightly more general setting, Izmestiev \cite{izmestiev2010colin} found that every polytope is~spectral, if we are allowed to add appropriate edge-weights to the edge-graph.
This~results in an adjacency matrix whose entries can be any non-negative real number instead of just zero and one.
This however might break the symmetry in a combinatorially highly symmetric graph and is therefore not our preferred perspective.
First results on this topic were obtained by Licata and Powers \cite{licata1986surprising}, who indentified several polytopes as being spectral.
Among these, all regular polytopes, excluding the exceptional 4-dimensional regular polytopes: 24-cell, 120-cell and 600-cell.
Still kind of \enquote{unexplained} is their finding that the \emph{rhombic dodecahedron} is spectral as well.
Since the edge graph of this polyhedron is not regular, this observation hinges critically on the fact that we consider eigenpolytopes constructed from the adjacency matrix, rather than, say, the Laplace matrix.
Godsil \cite{godsil1998eigenpolytopes} classified all spectral graphs among the distance-regular graphs (for a definition, see \cref{sec:distance_transitive_realizations}).
He found that these are the following:
\begin{theorem}[Godsil, \cite{...}]
\label{res:spectral_distance_regular_graphs}
Let $G$ be a distance-regular graph.
If $G$ is $\theta_2$-spectral,~then $G$ is one of the following:
\begin{enumerate}[label=$(\text{\roman*}\,)$]
\item a cycle graph $C_n,n\ge 3$,
\item the edge-graph of the dodecahedron,
\item the edge-graph of the icosahedron,
\item the complement of a disjoint union of edges,
\item a Johnson graph $J(n,k)$,
\item a Hamming graph $H(d,q)$,
\item a halved $n$-cube $\nicefrac12 Q_n$,
\item the Schläfli graph, or
\item the Gosset graph.
\end{enumerate}
\end{theorem}
One remarkable fact about this list, which seems to have gone unnoticed, is that all these graphs are actually \emph{distance-transitive}.
In general there was not found a single polytope (or graph) that is $\theta_i$-spectral for any $i\not=2$ (assuming that the graph has at least one edge).
It is not implausible to conjecture that skeleta of polytopes can have, if at all, only eigenvalue $\theta_2$.
This comes from the study of so-called \emph{nodal domains}.
There is a remarkable construction due to Izmestiev \cite{izmestiev2010colin}.
\begin{theorem}[Izmestiev \cite{izmestiev2010colin}, Theorem 2.4]\label{res:izmestiev}
Let $P\subset\RR^d$ be a polytope, $G_P$ its~edge-graph and $v\:V\to\RR^d$ its skeleton. For $c=(c_1,...,c_n)\in\RR^n$ define
$$P^\circ(c):=\{x\in\RR^d\mid\<x,v_i\>\le c_i\text{ for all $i\in V$}\}.$$
Note that $P^\circ=P^\circ(1,...,1)$ is the polar dual of $P$.
The matrix $X\in\RR^{n\x n}$ with~compo\-nents
$$X_{ij}:=-\frac{\partial^2 \vol(P^\circ(c))}{\partial c_i\partial c_j}\Big|_{c=(1,...,1)}$$
has the following properties:
\begin{myenumerate}
%
\item $X_{ij}< 0$ whenever $ij\in E(G_P)$,
\item $X_{ij}=0$ whenever $ij\not\in E(G_P)$,
\item $XM=0$, where $M$ is the arrangement matrix of $v$,
\item $X$ has a unique negative eigenvalue, and this eigenvalue is simple,
\item $\dim\ker X=d$.
\end{myenumerate}
\end{theorem}
One can view the matrix $X$ (or better, $-X$) as kind of an adjacency matrix of~a vertex- and edge-weighted version of $G_P$.
Part $(iii)$ states that $v$ is balanced~with eigenvalue zero.
Since $\rank M=d$, part $(v)$ states that the arrangement space~$U:=$ $\Span M$ is already the whole eigenspace.
And part $(iv)$ states that zero is the second largest eigenvalue of $-X$.
\begin{theorem}\label{res:implies_spectral}
Let $P\subset\RR^d$ be a polytope, and $X\in\RR^{n\x n}$ the associated matrix~defined as in \cref{res:izmestiev}. If we have that
\begin{myenumerate}
\item $X_{ii}$ is independent of $i\in V(G_P)$, and
\item $X_{ij}$ is independent of $ij\in E(G_P)$,
\end{myenumerate}
then $P$ is spectral with eigenvalue $\theta_2$.
\begin{proof}
By assumption there are $\alpha,\beta\in\RR$, $\beta>0$, so that $X_{ii}=\alpha$ for all vertices~$i\in$ $V(G_P)$, and $X_{ij}=-\beta$ for all edges $ij\in E(G_P)$.
We then find
$$X=\alpha \Id - \beta A\quad\implies\quad (*)\;A=\frac\alpha\beta \Id-\frac1\beta X,$$
where $A$ is the adjacency matrix of $G_P$.
By \cref{res:izmestiev} $(iv)$ and $(v)$, the matrix $X$ has second smallest eigenvalue zero of multiplicity $d$.
By \cref{res:izmestiev} $(iii)$, the columns of $M$ are the corresponding eigenvectors.
Since $\rank M=d$ we find that these are all the eigenvectors, in particular, that the arrangement space $U:=\Span M$ is the 0-eigenspace of $X$.
By $(*)$, $A$ (that is, $G_P$) has then second \emph{largest} eigenvalue $\theta_2=\alpha/\beta$ of multiplicity $d$ with eigenspace $U$.
The skeleton of $P$ is then the $\theta_2$-realization of $G_P$ and $P$ is spectral.
\end{proof}
\end{theorem}
\begin{corollary}
If $P\subset\RR^d$ is simultaneously vertex- and edge-transitive, then $P$ is spectral with eigenvalue $\theta_2$.
\end{corollary}
It seems tempting to conjecture that the spectral polytopes are \emph{exactly} the polytopes that satisfy the conditions $(i)$ and $(ii)$ in \cref{res:implies_spectral}.
This would give an interesting geometric characterization of eigenpolytopes.
Izmestiev proved the following:
for $ij\in E$ let $\sigma_i,\sigma_j\in\F_{d-1}(P^\circ)$ be the dual facets to the vertices $v_i$ and $v_j$ in $P$. Then holds
$$X_{ij}= \frac{\vol_{d-2}(\sigma_i\cap\sigma_j)}{\|v_i\|\|v_j\|\sin\angle(v_i,v_j)}.$$
There seems to be no equally elementary formula for $X_{ii},i\in V$, but its value can be derived from $XM=0$, which implies
$$X_{ii} v_i = -\sum_{\mathclap{j\in N(i)}} M_{ij} v_j.$$
It seems out of reach to prove this fact.
There are many balanced polytopes that are not spectral (\shortStyle{e.g.}\ all appropriately centered neighborly polytopes that are not simplices).
Further, there are polytope for which being spectral appears completely exceptional, \shortStyle{e.g.}\ the \emph{rhombic dodecahedron} (already mentioned by Licata and Powers \cite{licata1986surprising}).
\subsection{Applications}
Spectral polytopes turn out to be exactly the kind of polytopes we need to address the problems listed in \cref{sec:motivation}.
For example, for fixed $i\in\NN$ consider the set of all $\theta_i$-spectral polytopes.
Given the edge-graph $G_P$ of some polytope in this set, this uniquely determines the polytope, simply since it can be reconstruced via the $\theta_i$-realizaion of $G_P$.
In other~words, the $\theta_i$-spectral polytopes are uniquely determined by their edge-graph.
Also, each $\theta_i$-spectral polytope realizes all the symmetries of its edge-graph by \cref{res:spectral_implies_symmetric}.
The question then is how large the class of these polytopes is, and whether their members are easily recognized.
For example, has every family of vertex-transitive polytopes with the same edge graph a spectral representative?
If so, this would imply that every vertex-transitive edge-graph can be realized as a polytope with all symmetries.
\section{Edge-transitive polytopes}
\subsection{Arc-transitive polytopes}
\label{sec:arc_transitive_polytopes}
Our final goal would be to establish that every arc-transitive polytope is spectral, or more precisely, $\theta_2$-spectral.
To anticipate~
the outcome, we were not able to prove this.
However, we also observe that this statement is true for \emph{all} known arc-transitive polytopes.
In the last section we have learned that the skeleton of an arc-transitive polytope is of full local dimension, hence irreducible, spherical, rigid and has an eigenvalue.
\begin{definition}
Given a polytope $P\subset\RR^d$.
\begin{myenumerate}
\item $P$ is called \emph{rigid} if it cannot be continuously deformed without changing its combinatorial type or symmetry group.
\item $P$ is called \emph{perfect} if any essentially distinct but combinatorially equivalent polytope $Q\subset\RR^d$ has a smaller symmetry group than $P$, that is $\Aut(Q)\subset\Aut(P)$.
\end{myenumerate}
\end{definition}
\enquote{Essentially distinct} in $(ii)$ means that $P$ and $Q$ are not just reoriented or~rescaled versions of one another.
If a polytope is perfect, then it is rigid.
It seems to be open whether the converse holds in general.
In theory it could happen that a combinatorial type might have two \enquote{most symmetric} realizations as a polytope, and those cannot be continuously deformed into each other.
\begin{theorem}
An arc-transitive polytope $P\subset\RR^d$ is $\theta_2$-spectral.
\begin{proof}
The skeleton of $P$ is an arc-transitive realization of $G_P$ of full local dimension, hence has an eigenvalue $\theta\in\Spec(G)$ according to \cref{res:...}.
The tricky part is to show that $\theta=\theta_2$.
For this, we use the following fact:
Let $v$ be the skeleton of $P$ and define
$$P^\circ(\mathbf c) = P^\circ(c_1,...,c_n):=\{x\in\RR^d\mid \<x,v_i\>\le c_i\text{ for all $i\in V$}\}.$$
Finally, consider the matrix
$$M_{ij}:=-\frac{\partial^2\mathrm{vol}(P(\mathbf c))}{\partial c_i \partial c_j}\Big|_{\mathbf c=(1,...,1)}.$$
Since $P$ is edge-transitive, $M_{ij}$ contains the same value for all $ij\in E$.
Since $P$ is vertex-transitive, $M_{ii}$ does not depend in $i\in V$.
Finally, it is proven in \cite{...} that $M_{ij}=0$ whenever $ij\not\in E$, and that $M$ has a single negative eigenvalue which is simple.
Apparently, there are $\alpha,\beta\in\RR$ so that $M=\alpha A+\beta\Id$, where $A$ is the adjacency matrix of $M$.
\end{proof}
\end{theorem}
\begin{theorem}
If $P\subset\RR^d$ is an arc-transitive polytope, then
\begin{myenumerate}
\item $P$ is rigid,
\item $\Aut(P)$ is irreducible,
\item any projection $\pi_U(P)$ onto a subspace $U\subseteq\RR^d$ is either not arc-transitive~or has a different edge-graph as $P$.
\end{myenumerate}
Furthermore, there is an eigenvalue $\theta\in\Spec(G_P)$ of the edge-graph $G_P$ resp.\ its~associated Laplacian eigenvalue $\lambda=\theta-\deg(G_P)$, so that the following holds:
\begin{myenumerate}
\setcounter{enumi}{3}
\item if $P$ has edge length $\ell$ and circumradius $r$, then
%
$$\frac\ell r=\sqrt{\frac{2\lambda}{\deg(G_P)}}.$$
\item if the polar polytope $P^\circ$ has dihedral angle $\alpha$, then
%
$$\cos\alpha=-\frac{\theta}{\deg(G_P)}.$$
\end{myenumerate}
\end{theorem}
\begin{conjecture}\label{conj:arc_transitve}
Each arc-transitive polytope $P\subset\RR^d$ is $\theta_2$-spectral, and satisfies
\begin{myenumerate}
\item $P$ is perfect,
\item $P$ realizes all the symmetries of its edge-graph,
\item $P$ is the only arc-transitive polytope (up to reorientation and rescaling) with edge-graph $G_P$.
\end{myenumerate}
\end{conjecture}
Probably in, in all these result, the eigenvalue $\theta$ resp.\ $\lambda$ can be replaced with $\theta_2$ resp.\ $\lambda_2$, but we were not able to prove that.
However, it holds in any single case of arc-transitive polytope that we have checked.
\begin{example}
There are many ways to compute the circumradius of an $n$-simplex with unit edge-length, and here is another one:
The complete graph $K_n$ is arc-transitive and has spectrum $\{(-1)^{n-1},n^1\}$. The second largest eigenvalue $\theta_2=-1$ gives rise to an $(n-1)$-dimensional polytope with edge-graph $K_n$, that is, the $(n-1)$-dimensional simplex.
Assuming edge length $\ell=1$, we can compute its circumradius:
$$r=\sqrt{\frac{\deg(G)}{\lambda_2}} = \sqrt{\frac{\deg(G)}{\deg(G)-\theta_2}} = \sqrt{\frac{n-1}{n-1-(-1)}} = \sqrt{1-\frac1n}.$$
\end{example}
\begin{example}
Compute the dihedral angle of the rhombic dodecahedron and the rhombic triacontahedron.
\end{example}
But this technique can also be used the other way around, namely, concluding from the knowledge of the metric properties of a polytope and the eigenvalues of its edge-graph to the fact that it is spectral.
\begin{example}
We show that all regular polytopes are in fact $\theta_2$-spectral, as was already conjectured by Licata and Powers \cite{licata1986surprising}.
The essential facts we are using are some tabulated metric and spectral properties of these polytopes, \shortStyle{e.g.}\ found in \cite{buekenhout1998number}.
\begin{itemize}
\item
Remarkable, the 24-cell has edge-length $\ell$ identical to circumradius $r$.
This determines the Laplacian eigenvalue of its skeleton:
$$1=\frac\ell r=\sqrt{\frac{2\lambda}{\deg(G)}} \quad\implies\quad \lambda=\frac12\deg(G) = 4.$$
One checks that $4$ is exactly the second largest Laplacian eigenvalue of its edge-graph, and that this eigenvalue has multiplicity four.
\item
600-cell: $3(1+\sqrt{5})$
\item
120-cell: $2\phi-1$
\end{itemize}
\begin{center}
\begin{tabular}{c|rrrr}
polytope & $\ell$ & $r$ & $\lambda_2$ & $\deg(G_P)$ \\
\hline
$\overset{\phantom.}24$-cell & $1$ & $1$ & $4$ & $8$ \\
$120$-cell & $3-\sqrt 5$ & $2\sqrt 2$ & $2\phi-1$ & $4$ \\
$600$-cell & $2/\phi$ & $2$ & $3(1+\sqrt 5)$ & $12$
\end{tabular}
\end{center}
\end{example}
\begin{theorem}
If a polytope has a connected, full-dimensional and spanning edge-orbit, then it is perfect.
\end{theorem}
\newpage
\section{Edge- and vertex-transitive polytopes}
The goal of this section is to show that every polytope $P\subset\RR^d$ that is simultane\-ously vertex- and edge-transitive is an eigenpolytope to the second smallest eigenvalue of its edge-graph.
Note that being only edge-transitive is just slightly weaker than being vertex- and edge-transitive, as it was shown in \cite{winter2020polytopes} that edge-transitivity of convex polytopes in dimension $d\ge 4$ implies vertex-transitivity anyway.
\begin{theorem}\label{res:vertex_edge_transitive_is_theta2_realization}
If $P\subset\RR^d$ is vertex-transitive and edge-transitive, then it is the~$\theta_2$-eigenpolytopes of its edge-graph.
\begin{proof}
Let $X\in\RR^{n\x n}$ be the matrix defined in \cref{res:izmestiev}.
Since $P$ is transitive on vertices, there is a number $\alpha\in\RR$ with $X_{ii}=\alpha$ for all $i\in V$.
Since $P$ is edge-transitive, there is a number $\beta<0$ so that $X_{ij}=\beta$ for all $ij\in E(G_P)$.
Since we also have $X_{ij}=0$ for all $ij\not\in E(G_P)$, we find that $X$ must be of the form
$$X= \alpha \Id - \beta A \quad\implies\quad A=\alpha\Id-\frac1\beta X,$$
where $A$ is the adjacency matrix of $G_P$.
$X$ has a single negative eigenvalue, which is simple, and $\dim\ker X=d$.
In other words, its second smallest eigenvalue is zero and has multiplicity $d$.
Consequently, $A$ has second \emph{largest} eigenvalue $\theta_2=\alpha$ of multiplicity $d$ to the same eigenvectors as $X$.
Let $v$ be the skeleton of $P$ with arranegment matrix $M$ and arrangement space $U$.
Since $XM=0$ and $\rank M=d$, we find that the $U=\Span M$ must be the $0$-eigenspace of $X$, hence the $\theta_2$-eigenspace of $A$.
Thus, $v$ is the $\theta_2$-realization of $G_P$ and $P$ is spectral.
\end{proof}
\end{theorem}
The argument of \cref{res:vertex_edge_transitive_is_theta2_realization} cannot be easily generalized to any weaker form of symmetry.
For example, one would like to proof that every spectral polytope has eigenvalue $\theta_2$, or more generally, every balanced polytope has eigenvalue $\theta_2$.
\begin{corollary}
If $P\subset\RR^d$ is vertex- and edge-transitive, then holds:
\begin{myenumerate}
\item $\Aut(P)$ is irreducible,
\item $P$ is the unique simultaneously vertex- and edge-transitive polytope for which the edge-graph is isomorphic to $G_P$,
\item $P$ realizes all symmetries of its edge-graph,
\item $P$ is a perfect (rigid) polytope,
\item $P$ is the unique most symmetric realization of its combinatorial type,
\item if $P$ has edge-length $\ell$ and circumradius $r$, then
%
$$\Big[\frac \ell r \Big]^2 = \frac{2\lambda_2(G_P)}{\deg(G_P)} = 2\Big(1-\frac{\theta_2(G_P)}{\deg(G_P)}\Big).$$
%
\item if the polar dual $P^\circ$ has dihedral angle $\alpha$, then
%
$$\cos(\alpha)=-\frac{\theta_2(G_P)}{\deg(G_P)}.$$
\end{myenumerate}
\end{corollary}
\section{Distance-transitive polytopes}
\subsection{Distance-transitive polytopes}
\label{sec:distance_transitive_polytopes}
In general, there is no apparent reason for why a spectral polytope must be $\theta_2$-spectral.
In the case of distance-transitive polytopes there can be found an easy argument involving the cosine sequence of its skeleton.
The argument is based on the number of sign changes in $u_\delta$ and the so called \emph{two-piece property} for embedded graphs.
\begin{definition}
A realization $v$ is said to have the \emph{two-piece property} (or TPP),~if every hyperplane cuts it into at most two pieces.
More formally, given any vector $x\in \RR^d$ and scalar $c\in\RR$, the induced subgraphs
$$
G_+:=G[\,i\in V\mid \<x,v_i\>\ge c\,]
\quad\text{and}\quad
G_-:=G[\,i\in V\mid \<x,v_i\>\le c\,]
$$
must be connected (if non-empty).
\end{definition}
The image below shows two symmetric realizations of the cycle graph $C_5$, one~of which has the TPP (left), and one of which has not (right).
We are going to use the following fact:
\begin{theorem}[\cite{...}, Theorem xx]
The skeleton of a polytope has the TPP.
\begin{proof}[Sketch of proof.]
Let $v$ be the skeleton of a polytope, $x\in\RR^d$ and $c\in \RR$.
Let $i_+\in V$ be the vertex which maximizes the functional $\<v_\cdot,x\>$.
By the simplex algorithm, every vertex on the \enquote{positive} side of the hyperplane $\<\cdot,x\>=c$ can reach $v_{i_+}$ by a path in the edge-graph that is non-decreasing in $\<v_\cdot,x\>$.
Hence the positive side is connected if non-empty.
Equivalently for the negative side.
\end{proof}
\end{theorem}
\begin{theorem}
A distance-transitive polytope $P$ is $\theta_2$-spectral.
\begin{proof}
Let $v$ be the full-dimensional and distance-transitive realization of $P$.
Since it is also arc-transitive, it has an eigenvalue (by \cref{cref:...}).
By \cref{res:...} it must then be the $\theta_k$-realization of $G_P$.
Now consider cutting $v$ by the hyperplane $v_i^\bot$ for some $i\in V$.
The number of components left after the cut is exactly one more than the number of sign changes of the cosine sequence $u_\delta$.
From \cref{res:...} we know that there are exactly $k-1$ sign changes for the $\theta_k$-realization, hence $k$ pieces after the cut.
Since $v$ is the skeleton of a polytope, we have the TPP, and $k\le 2$ pieces.
Since $v$ cannot be the trivial $\theta_1$-realization of $G_P$, it must be the $\theta_2$-realization, and $P$ the $\theta_2$-eigenpolytope.
\end{proof}
\end{theorem}
\begin{corollary}
Let $P$ be a distance-transitive polytope.
\begin{myenumerate}
\item Among distance-transitive polytopes, $P$ is uniquely determined by its edge-graph.
\item $P$ realizes all the symmetries of its edge-graph.
\item $P$ is the unique most symmetric realization of its combinatorial type.
\end{myenumerate}
\end{corollary}
Godsil's classification of the $\theta_2$-spectral distance-regular graphs (see \cref{res:spectral_distance_regular_graphs}; all of which~turned out to be distance-transitive) allows a complete classification of distance-tran\-sitive polytopes.
\begin{theorem}\label{res:distance_transitive_classification}
If $P\subset\RR^d$ is a distance-transitive polytope, then it is one of the following:
\begin{enumerate}[label=$(\text{\roman*}\,)$]
\item a regular polygon $(d=2)$,
\item the regular dodecahedron $(d=3)$,
\item the regular icosahedron $(d=3)$,
\item a cross-polytopes, that is, $\conv\{\pm e_1,...,\pm e_d\}$ where $\{e_1,...,e_d\}\subset\RR^d$ is the standard basis of $\RR^d$,
\item a hyper-simplex $\Delta(d,n)$, that is, the convex hull of all vectors $v\in\{0,1\}^{d+1}$ with exactly $k$ 1-s,
\item a cartesian power of a regular simplex (this includes hypercubes),
\item a demi-cube, that is, the convex hull of all vectors $v\in\{-1,1\}^d$ with~an~even number of 1-s,
\item the $2_{21}$-polytope, also called Gosset-polytope $(d=6)$,
\item the $3_{21}$-polytope, also called Schläfli-polytope $(d=7)$.
\end{enumerate}
The order of this list agrees with the list of graphs in \cref{res:spectral_distance_regular_graphs}.
\end{theorem}
Note that the list in \cref{res:distance_transitive_classification} contains many polytopes that are not regular, and contains all regular polytopes except for the 24-cell, 120-cell and 600-cell.
The distance-transitive polytopes thus form a distinct class of remarkably symmetric polytopes which is not immediately related to the class of regular polytopes.
Noteworthy however, is that all the distance-transitive polytopes are \emph{Wythoffian poly\-topes}, that is, they are orbit polytopes of finite reflection groups.
\Cref{fig:distance_transitive_Coxeter} shows the Coxeter-Dynkin diagrams of these polytopes.
\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{img/distance_transitive_Coxeter}
\caption{Coxeter-Dynkin diagrams of distance-transitive polytopes.}
\label{fig:distance_transitive_Coxeter}
\end{figure}
\section{Conclusion and open questions}
\label{sec:future}
In this paper we have studied \emph{eigenpolytopes} and \emph{spectral polytopes}.
The former are polytopes constructed from a graph and one of its eigenvalues.
A polytope is spectral if it is the eigenpolytopes of its edge-graph.
These are of interest because spectral graph theory then ensures a strong interplay between the combinatorial properties of the edge-graph and the geometric properties of the polytope.
The study of eigenpolytopes and spectral polytopes has left us with many open questions.
Most notably, how to detect spectral polytopes purely from their geome\-try.
We introduced a tool (\cref{res:implies_spectral}), which was sufficient to proof that (most) edge-transitive polytopes are spectral.
We do not know how much more general it can be applied.
\begin{question}
\label{q:characterization}
Does \cref{res:implies_spectral} already characterize $\theta_2$-spectral polytopes (or even spectral polytopes in general)?
\end{question}
If the answer is affirmative, this would provide a geometric characterization of polytopes that are otherwise defined purely in terms of spectral graph theory.
The result of Izmestiev suggests that polytopes with sufficiently regular geometry are $\theta_2$-spectral: the entry of the matrix $X$ in \cref{res:izmestiev} at index $ij\in E$ can be~expressed as
$$X_{ij}=\frac{\vol(\sigma_i\cap\sigma_j)}{\|v_i\|\|v_j\|\sin\angle(v_i,v_j)},$$
where $\sigma_i$ and $\sigma_j$ are the facets of the polar dual $P^\circ$ that correspond to the vertices $v_i,v_j\in\F_0(P)$.
Because of this formula, it might be actually easier to classify the polar duals of $\theta_2$-spectral polytopes.
An affirmative answer to \cref{q:characterization} would also mean a negative answer to the following:
\begin{question}
\label{q:not_theta_2}
Is there a $\theta_k$-spectral polytope/graph for some $k\not=2$?
\end{question}
The answer is known to be negative for edge-transitive polytopes/graphs (see \cref{res:edge_transitive_spectral_graph}), but unknown in general.
The second-largest eigenvalue $\theta_2$ is special for other reasons too.
Even if a graph is not $\theta_2$-spectral, it seems to still imprint its adjacency information onto the edge-graph of its $\theta_2$-eigenpolytope.
\begin{question}
\label{q:realizing_edges}
Given an edge $ij\in E$ of $G$, if $v_i$ and $v_j$ (as defined in \cref{def:eigenpolytope}) are distinct vertices of the $\theta_2$-eigenpolytope $P_G(\theta_2)$, is then also $\conv\{v_i,v_j\}$ an edge of $P_G(\theta_2)$?
\end{question}
This was proven for distance-regular graphs in \cite{godsil1998eigenpolytopes}, and is not necessarily true for eigenvalues other than $\theta_2$.
All known spectral polytopes are exceptionally symmetric.
It is unclear whether this is true in general.
\begin{question}
\label{q:trivial_symmetry}
Are there spectral polytopes with trivial symmetry group?
\end{question}
An example for \cref{q:trivial_symmetry} must be asymmetric, yet with a reasonably large eigenspaces.
Such graphs exist among the distance-regular graphs, but all spectral distance-regular graphs were determined in \cite{godsil1998eigenpolytopes} (see also \cref{res:spectral_distance_regular_graphs}) and turned out to be distance-transitive, \shortStyle{i.e.,}\ highly symmetric.
A clear connection between being spectral and being symmetric is missing.
To emphasize our ignorance, we ask the following:
\begin{question}
\label{q:spectral_non_vertex_transitive}
Can we find more spectral polytopes that are \emph{not vertex-transitive}?
What characterizes them?
\end{question}
The single known spectral polytope that is \emph{not} vertex-transitive is the \emph{rhombic dodecahedron} (see \cref{fig:edge_transitive}).
The fact that it is spectral appears purely accidental, as there seems to be no reason for it to be spectral, except that we can explicitly check that it is. For comparison, the highly related \emph{rhombic triacontahedron} is not spectral.
On the other hand, vertex-transitive spectral polytopes might be quite common.
\begin{question}
\label{q:specific_instance}
Let $P\subset\RR^d$ be a polytope with the following properties:
\begin{myenumerate}
\item $P$ is vertex-transitive,
\item $P$ realizes all the symmetries of its edge-graph, and
\item $\Aut(P)$ is irreducible.
\end{myenumerate}
Is $P$ (combinatorially equivalent to) a spectral polytope?
\end{question}
No condition in \cref{q:specific_instance} can be dropped.
If we drop vertex-transitivity,~we could take some polytope whose edge-graph has trivial symmetry and only small eigenspaces. Dropping $(ii)$ leaves vertex-transitive neighborly polytopes, for which we know that these are mostly not spectral (except for the simplex).
Dropping $(iii)$ leaves us with the prisms and anti-prisms, the eigenspaces of their edge-graphs are rarely of dimension greater than two.
Finally, we wonder whether these spectral techniques can be any help in classifying the edge-transitive polytopes.
\begin{question}
Can we classify the edge-transitive graphs that are spectral, and~by this, the edge-transitive polytopes?
\end{question}
\begin{question}
Can the existence of half-transitive polytopes be excluded by using spectral graph theory (see \cref{sec:arc_transitive})?
\end{question}
\iffalse
\hrulefill
We have seen that in some situations computing the eigenpolytope of the edge-graph of a polytope results in the same polytope again.
Polytopes/graphs for which this happens are called \emph{spectral}.
We have introduced a tool for proving that certain geometrically well-behaved polytopes are necessarily $\theta_2$-spectral.
It is open whether spectral polytopes exist to other eigenvalues than $\theta_2$:
\begin{question}
\label{q:not_theta_2}
Is there a $\theta_k$-spectral polytope/graph for some $k\not=2$?
\end{question}
We have seen that the answer is negative for the distance-transitive (actually distance-regular) graphs (see \cref{res:distance_regular_theta_k}).
Our hope is that the proof can~be~extended to a larger class of graphs as \shortStyle{e.g.}\ 1-walk-regular or arc-transitive graphs.
Even if a graph is not $\theta_2$-spectral, it seems to still imprint its adjacency information on the edge-graph of its $\theta_2$-eigenpolytope:
\begin{question}
\label{q:realizing_edges}
Let $G$ be a graph with $\theta_2$-eigenpolytope $P=P_G(\theta_2)$.
For each~$i\in$ $V$ let $v_i\in\RR^d$ be as defined in \cref{def:eigenpolytope}.
Now, if $ij\in E$ and $v_i\not= v_j$ are distinct vertices of $P$, is then also $\conv\{v_i,v_j\}$ an edge of $P$?
\end{question}
The answer is affirmative for distance-regular graphs as shown by Godsil in \cite{godsil1998eigenpolytopes}.
All known spectral polytopes are highly symmetric.
We therefore ask the following:
\begin{question}
\label{q:trivial_symmetry}
Are there spectral polytopes with trivial symmetry group?
\end{question}
To construct an example, one has to find a graph with no non-trivial symmetries, but whose eigenspaces are still reasonably large.
Such graphs can be found among the distance-regular graphs.
However, all spectral graphs among these were already determined by Godsil and all of them turned out to be distance-transitive.
A clear connection between being spectral and being symmetric is missing.
To emphasize our ignorance, we have the following:
\begin{question}
Can we find more spectral polytopes that are \emph{not vertex-transitive}?
What characterizes them?
\end{question}
Only one such polytope is known: the \emph{rhombic dodecahedron}.
We know that it spectral by explicit computation, but a systematic proof of this fact (that admits a generalization) is missing.
In general, how else can one characterize spectral polytopes?
The result of~Izmes\-tiev suggests that polytopes with sufficiently regular geometry are $\theta_2$-spectral.
In fact, the value of the matrix $X$ in \cref{res:izmestiev} at an index $ij\in E$ can be given~explicitly
$$X_{ij}=\frac{\vol(\sigma_i\cap\sigma_j)}{\|v_i\|\|v_j\|\sin\angle(v_i,v_j)},$$
where $\sigma_i$ and $\sigma_j$ are the facets of the polar dual $P^\circ$ that correspond to the vertices $v_i,v_j\in\F_0(P)$.
Because of this, it might be actually easier to classify the polar duals of $\theta_2$-spectral polytopes.
\begin{question}
\label{q:characterization}
Does \cref{res:implies_spectral} already characterize $\theta_2$-spectral polytopes (or even spectral polytopes in general)?
\end{question}
There are very specific classes of polytopes that might have spectral of their combiantorial type:
\begin{question}
\label{q:specific_instance}
Let $P\subset\RR^d$ be a polytope with the following properties:
\begin{myenumerate}
\item $P$ is vertex-transitive,
\item $P$ realizes all the symmetries of its edge-graph, and
\item $\Aut(P)$ is irreducible.
\end{myenumerate}
Is $P$ (combinatorially equivalent to) a spectral polytope? Or at least an eigenpolytope?
\end{question}
We know that $P$ in \cref{q:specific_instance} is not necessarily spectral if we drop any of the conditions.
If we drop vertex-transitivity we could take some polytope with trivial symmetry. Dropping $(ii)$ gives the neighborly polytopes, for which we know that these are only balanced, but mostly not spectral (except for the simplex).
Dropping $(iii)$ gives us the prisms and anti-prisms which mostly do not have eigenspaces of dimension three or higher.
\cref{q:specific_instance} has equally interesting consequences if it is only true for the eigenpolytope case.
For example, several polytopes with relevance in integer optimization seem to be eigenpolytopes, \shortStyle{e.g.}\ the Birkhoff polytope or the traveling salesman polytopes.
This would allow a formulation of the respective optimization problem in the language of spectral graph theory.
Also, this gives a way to study the symmetry groups of these polytopes (which are oftem much larger than expected from their definition).
Finally, we wonder whether these spectral techniques can be any help in classifying edge-transitive polytopes.
\begin{question}
Can we classify the edge-transitive polytopes, or equivalently, the simultaneously vertex- and edge-transitive graphs that are spectral?
\end{question}
\begin{question}
Can spectral graph theory help to show that no half-transitive polyopes exist?
\end{question}
\fi
\section{Introduction}
\label{sec:introduction}
\emph{Eigenpolytopes} are a construction in the intersection of combinatorics and geometry, using techniques from spectral graph theory.
Eigenpolytopes provide a way~to associate several polytopes to a finite simple graph, one for each eigenvalues of~its adjacency matrix.
A formal definition can be found in \cref{sec:def_eigenpolytope}.
\vspace{0.5em}
\begin{center}
\includegraphics[width=0.7\textwidth]{img/cube_intro_pic}
\end{center}
\vspace{0.5em}
Eigenpolytopes can be applied from two directions:
for the first, one starts~from a given graph, computes its eigenpolytopes, and tries to deduce, from the geometry and combinatorics of these polytopes, something about the original~graph.
For the other direction, one starts with a polytope, asks whether it is an eigenpolytope, and if so, for which graphs, which eigenvalues, and how these relate to the original polytope.
Eigenpolytopes have several interesting geometric and algebraic properties, and establishing that a family of polytopes consists of eigenpolytopes opens up their study to the techniques of spectral graph theory.
For some graphs the connection to their eigenpolytopes is especially strong: it can happen that a graph is the edge-graph of one of its eigenpolytopes, or equivalently, that a polytope is an eigenpolytope of its edge-graph.
Such graphs/polytopes are quite special and we shall call them \emph{spectral}.
For example, all regular polytopes are spectral, but there are many others.
Their properties are not well-understood.
We survey the literature of eigenpolytope and spectral polytopes.
We establish a technique with which to prove that certain polytopes are spectral polytopes and we apply it to \emph{edge-transitive} polytopes. That are polytopes for which the Euclidean symmetry group $\Aut(P)\subset\Ortho(\RR^d)$ acts transitively on the set of edge $\F_1(P)$.
As we shall explain, this characterization suffices to proves that an edge-transitive polytope is uniquely determined by its edge-graph, and also realizes all its combinatorial symmetries.
A complete classification of edge-transitive polytopes is not known as of yet.
However, using results on eigenpolytopes, we are able to give a complete classification of a sub-class of the edge-transitive polytopes, namely, the \emph{distance-transitive polytopes}.
\subsection{Outline of the paper}
\cref{sec:eigenpolytopes} starts with a motivating example for directing the reader towards the definition of the \emph{eigenpolytope} as well as the phenomenon of \emph{spectral} graphs and polytopes.
We include a literature overview for~\mbox{eigenpolytopes} and spectral polytopes.
In \cref{sec:balanced_spectral} we give a first rigorous definition for the notion \enquote{spectral polytope} via \emph{balanced polytopes}. The latter is a notion related to the rigidity theory.
In \cref{sec:izmestiev} we introduce the, as of yet, most powerful tool for proving that~certain polytopes are spectral.
In the final section, \cref{sec:edge_transitive}, we apply this result to \emph{edge-transitive} polytopes.
It is a simple corollary of the previous section that these are $\theta_2$-spectral.
We explore the implications of this finding: edge-transitive polytopes (in dimension $d\ge 4$) are uniquely determined by the edge-graph and realize all of its symmetries.~%
We~discuss sub-classes, such as the arc-, half- and distance-transitive polytopes.
We close with a complete classification of the latter (based on a result of Godsil).
\section{Eigenpolytopes and spectral polytopes}
\label{sec:eigenpolytopes}
\subsection{A motivating example}
\label{sec:example}
Let $G=(V,E)$ be the edge-graph of the cube, with vertex set $V=\{1,...,8\}$,~num\-bers assigned to the vertices as in the figure below.
\begin{figure}[h!]
\centering
\includegraphics[width=0.48\textwidth]{img/cube}
\end{figure}
\noindent
The spectrum of that graph (\shortStyle{i.e.,}\ of its adjacency matrix) is $\{(-3)^1,(-1)^3,1^3,3^1\}$.
Most often, one denotes the largest eigenvalue by $\theta_1$, the second-largest by $\theta_2$, and so on.
In spectral graph theory, there exists the general rule of thumb that the most exciting eigenvalue of a graph is not its largest, but its \emph{second-largest} eigenvalue $\theta_2$ (which is related to the \emph{algebraic connectivity} of $G$).
For the edge-graph of the cube, we have $\theta_2=1$,~of multiplicity \emph{three}.
And here are three linearly independent eigenvectors to $\theta_2$:
\vspace{0.5em}
\begin{center}
\raisebox{-3.3em}{\includegraphics[width=0.23\textwidth]{img/cube_eigenvector}}
\qquad
$
u_1 = \begin{pmatrix}
\phantom+1\\ \phantom+1 \\ \phantom+1 \\ \phantom+1 \\ -1 \\ -1 \\ -1 \\ -1
\end{pmatrix},\quad
u_2 = \begin{pmatrix}
\phantom+1\\ \phantom+1 \\ -1 \\ -1 \\ \phantom+1 \\ \phantom+1 \\ -1 \\ -1
\end{pmatrix},\quad
u_3 = \begin{pmatrix}
\phantom+1\\ -1 \\ \phantom+1 \\ -1 \\ \phantom+1 \\ -1 \\ \phantom+1 \\ -1
\end{pmatrix}.
$
\end{center}
\vspace{0.7em}
\noindent
We can write these more compactly in a single matrix $\Phi\in\RR^{8\x 3}$:
\vspace{0.4em}
\begin{center}
$\Phi= \;\begin{blockarray}{(lll)r
\phantom+1 & \phantom+1 & \phantom+1\;\; &\text{\quad \footnotesize $\leftarrow v_1$}\\
\phantom+1 & \phantom+1 & -1 & \text{\quad \footnotesize $\leftarrow v_2$} \\
\phantom+1 & -1 & \phantom+1 & \text{\quad \footnotesize $\leftarrow v_3$} \\
\phantom+1 & -1 & -1 & \text{\quad \footnotesize $\leftarrow v_4$} \\
-1 & \phantom+1 & \phantom+1 & \text{\quad \footnotesize $\leftarrow v_5$} \\
-1 & \phantom+1 & -1 & \text{\quad \footnotesize $\leftarrow v_6$} \\
-1 & -1 & \phantom+1 & \text{\quad \footnotesize $\leftarrow v_7$} \\
-1 & -1 & -1 & \text{\quad \footnotesize $\leftarrow v_8$} \\
\end{blockarray}.$
\qquad
\raisebox{-4.2em}{\includegraphics[width=0.4\textwidth]{img/cube_coordinates}}
\end{center}
We now take a look at the rows of that matrix, of which it has exactly eight.~These rows are naturally assigned to the vertices of $G$ (assign $i\in V$ to the $i$-th row of~$\Phi$), and each row can be interpreted as~a~vec\-tor in $\RR^3$.
If we place each vertex $i\in V$ at the position $v_i\in\RR^3$ given by the $i$-th row of $\Phi$, we find that this embedds the graph $G$ \emph{exactly} as the skeleton of a cube (see the figure above).
In other words: if we compute the convex hull of the $v_i$, we get back the polyhedron from which we have started.
What a coincidence, isn't it?
This example was specifically chosen for its nice numbers, but in~fact,~the same works out as well for many other polytopes, inclu\-ding all the regular polytopes~in all dimension.
One probably learns to appreciate this magic when suddenly in~need for the vertex coordinates of some not so nice polytope, say, the regular dodecahedron or 120-cell.
With this technique in the toolbox, these coordinates are just one eigenvector-computation away (we included a short Mathematica script in \cref{sec:appendix_mathematica}).
Note also, that we never specified~the~dim\-en\-sion of~the~embed\-ding, but it just so happened, that the second-largest eigenvalue has the right multiplicity.
This phenomenon definitely deserves an explanation.
\subsubsection*{On the choice of eigenvectors}
\label{sec:choice_of_eigenvectors}
One might object that the chosen eigenvectors $u_1, u_2$ and $u_3$ look suspiciously cherry-picked, and we may not get such a nice result if we would have chosen just any eigenvectors.
And this is true.
For an appropriate choice of these vectors, we can, instead of a cube, get a cuboid, or a parallelepiped.
In fact, we can obtain any \emph{linear} transformations of the cube.
\emph{But}, we can also get \emph{only} linear transformations, and nothing else.
The reason is the following well~know fact from linear algebra:
\begin{theorem}
\label{res:same_column_span}
Two matrices $\Phi,\Psi\in\RR^{n\x d}$ have the same column span, \shortStyle{i.e.,}~$\Span \Phi=\Span \Psi$, if and only if their rows are related by an invertible linear transformation, \shortStyle{i.e.,}\ $\Phi=\Psi T$ for some $T\in\GL(\RR^d)$.
\end{theorem}
\noindent
In our case, the column span is the $\theta_2$-eigenspace, and the rows are the coordinates of the $v_i$.
We say that any two polytopes constructed in this way are \emph{linearly equivalent}.
The only notable property of the chosen basis in the example is, that the vectors $u_1, u_2$ and $u_3$ are orthogonal and of the same length.
Any other choice of such a basis of~the~\mbox{$\theta_2$-eigen}\-space (\shortStyle{e.g.}\ an orthonormal basis) would also have given a cube, but reoriented, rescaled and probably with less nice coordinates.
For details on how this choice relates to the orientation, see \shortStyle{e.g.}\ \cite[Theorem 3.2]{winter2019geometry}.
\subsection{Eigenpolytopes}
\label{sec:def_eigenpolytope}
We compile our example into a definition.
\begin{definition}\label{def:eigenpolytope}
Start with a graph $G=(V,E)$, an eigenvalue $\theta\in\Spec(G)$ thereof, as well as an orthonormal basis $\{u_1,...,u_d\}\subset\RR^n$ of the $\theta$-eigenspace.
We define the \emph{eigenpolytope matrix} $\Phi\in\RR^{n\x d}$ as the matrix in which the $u_i$ are the columns:\phantom{mm}
\begin{equation}
\label{eq:eigenpolytope_matrix}
\Phi :=\begin{pmatrix}
\mid & & \mid \\
u_1 & \!\!\cdots\!\!\! & u_d \\
\mid & & \mid
\end{pmatrix}=
\begin{pmatrix}
\;\rule[.5ex]{2.5ex}{0.4pt}\!\!\!\! & v_1^{\Tsymb} & \!\!\!\!\rule[.5ex]{2.5ex}{0.4pt}\;\; \\
& \vdots & \\[0.4ex]
\;\rule[.5ex]{2.5ex}{0.4pt}\!\!\!\! & v_n^{\Tsymb} & \!\!\!\!\rule[.5ex]{2.5ex}{0.4pt}\;\;
\end{pmatrix}.
\end{equation}
Let $v_i\in\RR^d$ denote the $i$-th row of $\Phi$.
The polytope
$$P_G(\theta):=\conv\{v_i\mid i\in V\}\subset\RR^d$$
is called \emph{$\theta$-eigenpolytope} (or just \emph{eigenpolytope}) of $G$.
\end{definition}
For later use we define the \emph{eigenpolytope map}
\begin{equation}
\label{eq:eigenpolytope_map}
\phi:V\ni i\mapsto v_i\in\RR^d
\end{equation}
that to each vertex $i\in V$ assignes the $i$-th row of the eigenpolytope matrix.
Note that the basis $\{u_1,...,u_d\}\subset\Eig_G(\theta)$ in \cref{def:eigenpolytope} is explicitly chosen~to be an \emph{orthonormal basis}.
This is not strictly necessarily, but this choice is convenient from a geometric point of view:
a different choice for this basis gives the same~poly\-tope, but with a different orientation rather than, say, transformed~by~a~general linear transformation.
This preserves metric properties and is closer to how polytopes are usually consider up to rigid motions.
We can also reasonably speak of \emph{the} $\theta$-eigenpolytope, as any two differ only by orientation.
With this terminology in place, our observation in the example of \cref{sec:example}~can be summarized as \enquote{the cube is the $\theta_2$-eigenpolytope of its edge-graph}, or alternatively as \enquote{the cube-graph is the edge-graph of its $\theta_2$-eigenpolytope}.
Here is~a~depic\-tion of all the eigenpolytopes of the cube-graph, one for each eigenvalue:
\vspace{0.5em}
\begin{center}
\includegraphics[width=0.7\textwidth]{img/cube_eigenpolytopes}
\end{center}
\noindent
We observe that the phenomenon from \cref{sec:example} only happens for $\theta_2$.
In general, the $\theta_1$-eigenpolytope of a regular graph will always be a single point (which is, why we rarely care about the largest eigenvalue).
Also, whenever a graph is bipartite, the eigenpolytope to the smallest eigenvalue is 1-dimensional, hence a line segment.
We are now free to compute the eigenpolytopes of all kinds of graphs, \mbox{including} graphs which are not the edge-graph of any polytope (so-called \emph{non-polytopal} graphs).
It is then little surprising that no edge-graph of any of its eigenpolytope gives the original graph again.
But even if we start from a polytopal graph, one is not guaranteed to find an eigen\-polytope that has the initial graph as its edge-graph (\shortStyle{e.g.}\ the edge-graph of the triangular prism has no eigenvalue of multiplicity three, hence no eigenpolytope of dimension three, see also \cref{ex:prism}).
Equivalently, if one starts with a polytope, it~is~not guaranteed that this polytope is the eigenpolytope of its edge-graph (or even combinatorially equivalent to it).
\begin{example}
\label{ex:neighborly_1}
A \emph{neighorly polytope} is a polytope whose edge-graph is the complete graph $K_n$.
The spectrum of $K_n$ is $\{(-1)^{n-1},(n-1)^1\}$.
One checks that~the~eigenpolytopes are a single point (for $\theta_1=n-1$) and the regular simplex of dimension $n-1$ (for $\theta_2=-1$).
Consequently, no neighborly polytope other than a simplex is combinatorially equivalent to an eigenpolytope of its edge-graph.
\end{example}
That a graph and its eigenpolytope translate into each other as well as in the case of the cube in \cref{sec:example} is a very special phenomenon, to which we shall give a name:
a polytope (or graph) for which this happens, will be called \emph{spectral}\footnote{There was at least one previous attempt to give a name to this phenomenon, namely, in \cite{licata1986surprising}, where it was called \emph{self-reproducing}.}.
We cannot formalize this definition right away, as there is some subtlety we have to discuss first (we give a formal definition in \cref{sec:balanced_spectral}, see \cref{def:spectral}).
\begin{example}
\label{ex:pentagon}
The image below shows two spectral realizations of the 5-cycle $C_5$\footnote{Spectral realizations are essentially defined like eigenpolytopes, assinging coordinates $v_i\in\RR^d$ to each vertex $i\in V$ (as in \cref{def:eigenpolytope}), but without taking the convex hull. Instead, one draws the edges between adjacent vertices.}.
\begin{center}
\includegraphics[width=0.4\textwidth]{img/C5}
\end{center}
The left image~shows the realization to the second-largest eigenvalue $\theta_2$, the right image shows the realization to the smallest eigenvalue $\theta_3$.
In both cases, the convex hull (the actual eigenpolytope) is a regular pentagon, whose edge-graph is $C_5$ again.
But we see that only in the case of $\theta_2$ the edges of the graphs get properly mapped into the edges of the pentagon.
While it is true that the 5-cycle $C_5$ is the edge-graph of its $\theta_3$-eigenpolytope, the adjacency informations gets scrambled in the process:
while, say, vertex 1 and 2 are adjacent in $C_5$, their images $v_1$ and $v_2$ do not form an edge in the $\theta_3$-eigenpolytope.
We do not want to call this \enquote{spectral}, as the adjacency information is not preserved.
The same can happen in higher dimensions too, \shortStyle{e.g.}\ with $G$ being the edge-graph of the dodecahedron:
\begin{center}
\includegraphics[width=0.55\textwidth]{img/dodecahedron_eigenpolytopes_2}
\end{center}
\end{example}
\begin{observation}
\label{res:theta_2_observations}
From studying many examples, there are two interesting observations to be made, both concern $\theta_2$, none of which is rigorously proven:
\begin{myenumerate}
\item
It appears as if only $\theta_2$ can give rise to spectral polytopes/graphs.
At~least, all known examples are $\theta_2$-spectral (see also \cref{q:not_theta_2}). Some considerations on nodal domains make this plausible, but no proof is known in the general case (a proof is known in certain special cases, see \cref{res:edge_transitive_spectral_graph}).
\item
If $i\in V$ is a vertex of $G$, then $v_i$ is not necessarily a vertex of every~eigenpolytope ($v_i$ might end up in the interior of $P_G(\theta)$ or one of its faces).
And even if $v_i,v_j\in\F_0(P_G(\theta))$ are distinct vertices and $ij\in E$ is an edge of $G$, it is still not necessarily true that $\conv\{v_i,v_j\}$ is also an edge of the~eigen\-polytope (as seen in \cref{ex:pentagon}).
However, this seems to be no concern in the case $\theta_2$.
It appears as if all edges of $G$ become edges of the $\theta_2$-eigenpolytope, even if $G$ is not spectral (under mild assumptions on the end vertices of the edge).
In other words, the adjacency information of $G$ gets imprinted on the edge-graph of the $\theta_2$-eigenpolytope, whether $G$ is spectral or not.
This is known to be true only in the case of distance-regular graphs \cite[Theorem 3.3 (b)]{godsil1998eigenpolytopes}, but unproven~in general (see also \cref{q:realizing_edges})
\end{myenumerate}
\end{observation}
\subsection{Litarture}
Eigenpolytope were first introduced by Godsil \cite{godsil1978graphs} in 1978.
Godsil proved the existence of a group homomorphism $\Aut(G)\to\Aut(P_G(\theta))$, \shortStyle{i.e.,}\, any combinatorial symmetry of the graph translates into a Euclidean symmetry of the polytope.
From that, he deduces results about the combinatorial symmetry group of the original graph.
We say more about the group homomorphism: for every $\theta\in\Spec(G)$ we have
\begin{theorem}[\!\cite{godsil1978graphs}, Theorem 2.2]
\label{res:realizing_symmetries}
If $\sigma\in\Aut(G)\subseteq\Sym(n)$ is a symmetry of $G$, and $\Pi_\sigma\in\Perm(\RR^n)$ is the associated permutation matrix, then
$$T_\sigma:=\Phi^{\Tsymb} \Pi_\sigma \Phi \;\in\; \Ortho(\RR^d),\qquad (\text{$\Phi$ is the eigenpolytope matrix})$$
is a Euclidean symmetry of the eigenpolytope $P_G(\theta)$ that also permutes the $v_i$ as~prescribed by $\sigma$, \shortStyle{i.e.,}\ $T_\sigma\circ \phi = \phi\circ\sigma$, or $T_\sigma v_i = v_{\sigma(i)}$ for all $i\in V$.
\end{theorem}
This result is also proven (more generally for spectral graph realizations) in \cite[Corollary 2.9]{winter2020symmetric}.
\cref{res:realizing_symmetries} explicitly uses that eigenpolytopes are defined using an \emph{orthonormal} bases rather than any basis of the eigenspace, to conclude that the symmetries $T_\sigma$ are \emph{orthogonal} matrices.
Also, the statement of \cref{res:realizing_symmetries} is not too satisfying in general, as it can happen that non-trivial~symmetries of $G$ are mapped to the identity transformation.
We not necessarily have $\Aut(G)\cong\Aut(P_G(\theta))$.
Several authors construct the eigenpolytopes of certain famous graphs or graph families.
Powers \cite{powers1986petersen} computed the eigenpolytopes of the \emph{Petersen graph}, which he termed the \emph{Petersen polytopes} (one of which will appear as a distance-transitive polytope in \cref{sec:distance_transitive}).
The same author also investigates eigenpolytopes of general distance-regular graphs in \cite{powers1988eigenvectors}.
In \cite{mohri1997theta_1}, Mohri described the face structure of the \emph{Hamming polytopes}, the $\theta_2$-eigenpolytopes of the Hamming graphs.
Seemingly unknown to the author, these polytopes can also by described as the cartesian powers of regular simplices (also distance transitive, see \cref{sec:distance_transitive}).
There exists a wonderful enumeration of the eigenpolytopes (actually, spectral realizations) of the edge-graphs of all uniform polyhedra in \cite{blueSpectral}. Sadly, this write-up was never published formally.
This provides empirical evidence that every uniform polyhedron
has a spectral realization.
The same question might then be asked for uniform polytopes in higher dimensions.
Rooney \cite{rooney2014spectral} used the combinatorial structure of the eigenpolytope (the size of their facets) to deduce statements about the size of cocliques in a graph.
In \cite{padrol2010graph}, the authors investigates how common graph operations translate to operations on their eigenpolytopes.
Particular attention was given to the eigenpolytopes of distance-regular graphs \cite{powers1988eigenvectors,godsil1998eigenpolytopes,godsil1995euclidean}.
It was shown that in a $\theta_2$-eigenpolytope of a distance-regular graph~$G$,~every edge of $G$ corresponds to an edge of the eigenpolytope \cite{godsil1998eigenpolytopes}.
Consequently, $G$~is a spanning subgraph of the edge-graph of the eigenpolytope.
It remains open if the same holds for less regular graphs, \shortStyle{e.g.}\ 1-walk regular graphs or arc-transitive graphs (see also \cref{q:realizing_edges}).
The observation that some polytopes are the eigenpolytopes
of their edge-graph (\shortStyle{i.e.,}\ they are \emph{spectral} in our terminology) was made repeatedly, \shortStyle{e.g.}\ in \cite{godsil1995euclidean} and \cite{licata1986surprising}.
In the latter, this was shown for all regular polytopes, excluding the exceptional 4-dimensional polytopes, the 24-cell, 120-cell and 600-cell.
This gap was filled in \cite{winter2020symmetric} via general considerations concerning spectral realizations of arc-transitive graphs.
In sum, all regular polytopes are known to be $\theta_2$-spectral.
The next major result for spectral polytopes was obtained by Godsil in \cite{godsil1998eigenpolytopes}, where he was able to classify all $\theta_2$-spectral distance-regular graphs (see also \cref{sec:distance_transitive}):
\begin{theorem}[\!\cite{godsil1998eigenpolytopes}, Theorem 4.3]
\label{res:spectral_distance_regular_graphs}
Let $G$ be distance-regular.
If $G$ is $\theta_2$-spectral,~then $G$ is one of the following:
\begin{enumerate}[label=$(\text{\roman*}\,)$]
\item a cycle graph $C_n,n\ge 3$,
\item the edge-graph of the dodecahedron,
\item the edge-graph of the icosahedron,
\item the complement of a disjoint union of edges,
\item a Johnson graph $J(n,k)$,
\item a Hamming graph $H(d,q)$,
\item a halved $n$-cube $\nicefrac12 Q_n$,
\item the Schläfli graph, or
\item the Gosset graph.
\end{enumerate}
\end{theorem}
A second look at this list reveals a remarkable \enquote{coincidence}: while the generic distance-regular graph has few or no symmetries, all the graphs in this list are highly symmetric, in fact, \emph{distance-transitive} (a definition will be given in \cref{sec:distance_transitive}).
It is a widely open question whether being spectral is a property solely reserved for highly symmetric graphs and polytopes (see also \cref{q:trivial_symmetry}).
There is only a single known spectral polytope that is not vertex-transitive (see also \cref{rem:edge_not_vertex} and \cref{q:spectral_non_vertex_transitive}).
\section{Balanced and spectral polytopes}
\label{sec:balanced_spectral}
In this section we give a second approach to \emph{spectral polytopes} that circumvents the mentioned subtleties.
For the rest of the paper, let $P\subset\RR^d$ denote a full-dimensional polytope in~dimen\-sion $d\ge 2$ with vertices $v_1,...,v_n\in\F_0(P)$.
We disinguish the \emph{skeleton} of $P$, which is the graph with vertex set $\F_0(P)$ and edge set $\F_1(P)$, from the~\mbox{\emph{edge-graph}}~$G_P=(V,E)$ of $P$, which is isomor\-phic to the skeleton, but has vertex set $V=\{1,...,n\}$. The isomorphism will be denoted
\begin{equation}
\label{eq:vertex_map}
\psi:V\ni i\mapsto v_i\in\F_0(P),
\end{equation}
and we call it the \emph{skeleton map}.
\subsection{Balanced polytopes}
\begin{definition}
The polytope $P$ is called \emph{$\theta$-balanced} (or just \emph{balanced}) for some~real number $\theta\in\RR$, if
\begin{equation}
\label{eq:balanced}
\sum_{\mathclap{j\in N(i)}} v_j = \theta v_i,\quad\text{for all $i\in V$},
\end{equation}
where $N(i):=\{j\in V\mid ij\in E\}$ denotes the \emph{neighborhood} of a vertex $i\in V$.
\end{definition}
One way to interpret the balancing condition \eqref{eq:balanced} is as a kind of self-stress~con\-dition on the skeleton of $P$ (the term \enquote{balanced} is motivated from this).
For each edge $ij\in E$, the vector $v_j-v_i$ is parallel to the edge $\conv\{v_i,v_j\}$.
If $P$ is $\theta$-balanced, at each vertex $i\in V$ we have the equation
$$\sum_{\mathclap{j\in N(i)}} (v_j-v_i) = \sum_{\mathclap{j\in N(i)}} v_j - \deg(i) v_i = \big(\theta-\deg(i)\big)v_i.$$
This equation can be interpreted as two forces that cancel each other out: on the left, a contracting force along each edge (proportion only to the length of that edge), and on the right, a force repelling each vertex away from the origin (proportional to the distance of that vertex from the origin, and proportional to $\theta-\deg(i)$).
A second interpretation of \eqref{eq:balanced} is via spectral graph theory.
Define the matrix
\begin{equation}
\label{eq:arrangement_matrix}
\Psi :=
\begin{pmatrix}
\;\rule[.5ex]{2.5ex}{0.4pt}\!\!\!\! & v_1^{\Tsymb} & \!\!\!\!\rule[.5ex]{2.5ex}{0.4pt}\;\; \\
& \vdots & \\[0.4ex]
\;\rule[.5ex]{2.5ex}{0.4pt}\!\!\!\! & v_n^{\Tsymb} & \!\!\!\!\rule[.5ex]{2.5ex}{0.4pt}\;\;
\end{pmatrix}
\end{equation}
in which the $v_i$ are the rows.
This matrix will be called the \emph{arrangement matrix} of $P$.
Note that the skeleton map $\psi$ assignes $i\in V$ to the $i$-th row of $\Psi$.
Since we use that $P\subset\RR^d$ is full-dimensional,~we have $\rank \Psi=d$.
\begin{observation}\label{res:eigenvalue}
Suppose that $P$ is $\theta$-balanced.
The defining equation \eqref{eq:balanced} can be equivalently written as the matrix equation $A\Psi=\theta \Psi$.
In this form,~it~is~apparent that $\theta$ is an eigenvalue of the adjacency matrix $A$, and the columns of $\Psi$ are $\theta$-eigenvectors, or $\Span\Psi\subseteq\Eig_{G_P}(\theta)$.
\end{observation}
We have seen that for a balanced polytope, the columns of $\Psi$ must be eigenvectors.
But they are not necessarily a complete set of~$\theta$-eigen\-vectors, \shortStyle{i.e.,}\ they not necessarily span the whole eigenspace.
\begin{example}
\label{ex:neighborly}
Every centered neighborly polytope $P$ is balanced, but except if it is a simplex, it is not spectral (the latter was shown in \cref{ex:neighborly_1}).
Centered~means that
$$\sum_{i\in V} v_i = 0.$$
Since $P$ is neighborly, we have $G_P=K_n$ and $N(i)=V\setminus\{i\}$ for all $i\in V$. Therefore
$$\sum_{\mathclap{j\in N(i)}} v_j = \sum_{\mathclap{j\in V}} v_j - v_i = -v_i,\quad\text{for all $i\in V$}.$$
And indeed, $K_n$ has spectrum $\{(-1)^{n-1},(n-1)^1\}$.
So $P$ is $(-1)$-balanced.
\end{example}
The last example shows that every neighborly polytopes can be made balanced by merely translating it.
More generally, many polytopes have a realization (of~their combinatorial type) that is balanced.
But other polytopes do not:
\begin{example}
\label{ex:prism}
Let $P\subset\RR^3$ be a triangular prism.
The spectrum of the edge-graph of $P$ is $\{(-2)^2,0^2,1^1,3^1\}$.
Note that there~is~no eigenvalue of multiplicity greater than~two.
In particular, we cannot choose three linearly independent eigenvectors to a common eigenvalue.
But if $P$ were balanced, then \cref{res:eigenvalue} tells us that the columns of the arrangement matrix $\Psi$ would be three eigenvectors to the same eigenvalue (linearly independent, since $\rank \Psi=3$), which is not possible.
And so, no realization of $P$ can be balanced.
\end{example}
\subsection{Spectral graphs and polytopes}
In the extreme case, when the columns of $\Psi$ span the whole eigenspace, we can finally give a compact definition of what we want~to~consider as \emph{spectral}:
\begin{definition}
\label{def:spectral}\quad
\begin{myenumerate}
\item A polytope $P$ is called \emph{$\theta$-spectral} (or just \emph{spectral}), if its arrangement matrix $\Psi$ satisfies $\Span \Psi=\Eig_{G_P}(\theta)$.
\item A graph is said to be \emph{$\theta$-spectral} (or just \emph{spectral}) if it is (isomorphic to) the edge-graph of~a $\theta$-spectral polytope.
\end{myenumerate}
\end{definition}
This definition is now perfectly compatible with our initial motivation for the~term \enquote{spectral} in \cref{sec:def_eigenpolytope}.
\begin{lemma}\quad
\label{res:naive}
\begin{myenumerate}
\item If a polytope $P$ is $\theta$-spectral, then $P$ is linearly equivalent to the $\theta$-eigenpoly\-tope of its edge-graph (see also \cref{res:naive_polytope}).
\item If a graph $G$ is $\theta$-spectral, then $G$ is (isomorphic to) the edge-graph of its~$\theta$-eigen\-polytope (see also \cref{res:naive_graph}).
\end{myenumerate}
\end{lemma}
In both cases, the converse is \emph{not} true.
This is intentional, to avoid the problems mentioned in \cref{ex:pentagon}.
Both statement will be proven below by formulating a more technical condition that is then actually equivalent to being spectral.
\begin{proposition}
\label{res:naive_polytope}
A polytope $P$ is $\theta$-spectral if and only if it is linearly equivalent to the $\theta$-eigenpolytope of its edge-graph via some linear map $T\in\GL(\RR^d)$ for which the following diagram commutes:
\begin{equation}
\label{eq:diagram_polytope}
\begin{tikzcd}
P \arrow[r, "T"] & P_{G_P}(\theta) \\
& G_P \arrow[lu, "\psi"] \arrow[u, "\phi"']
\end{tikzcd}
\end{equation}
where $\phi$ and $\psi$ denote the eigenpolytope map and skeleton map respectively.
\end{proposition}
\begin{proof
By definition, the $\theta$-eigenpolytope of $G_P$ satisfies $\Span \Phi=\Eig_{G_P}(\theta)$, where $\Phi$ is the corresponding eigenpolytope matrix.
Now, by definition, $P$ is $\theta$-spectral if and only if $\Span \Psi = \Eig_{G_P}(\theta)$, where $\Psi$ is its arrangement matrix.
But by \cref{res:same_column_span}, $\Phi$ and $\Psi$ have the same span if and only of their rows are related by some invertible linear map $T\in\GL(\RR^d)$, that is, $\Psi T=\Phi$, or $T\circ \psi=\phi$. The latter expresses exactly that \eqref{eq:diagram_polytope} commutes.
\end{proof}
This also proves \cref{res:naive} $(i)$.
\begin{proposition}
\label{res:naive_graph}
A graph $G$ is $\theta$-spectral if and only if the eigenpolytope map~$\phi\:$ $V(G)\to \RR^d$ provides an isomorphism between $G$ and the skeleton of its $\theta$-eigenpoly\-tope $P_G(\theta)$.
\begin{proof}
Suppose first that $G$ is $\theta$-spectral.
Then there is a $\theta$-spectral polytope $Q$~with edge-graph~$G_Q=G$ and skeleton map $\psi\:V(G_Q)\to\F_0(Q)$.
By \cref{res:naive} $(i)$, $Q$ is linearly equivalent to $P_G(\theta)$ via some linear map $T\in\GL(\RR^d)$.
By \cref{res:naive_polytope}, the eigenpolytope map satisfies $\phi=T\circ \psi$.
Since $T$ induces an isomorphism between the skeleta of $Q$ and $P_G(\theta)$, and $\psi$ is an isomorphism between $G$ and the skeleton of $Q$, we find that $\phi$ must be an isomorphism between $G$ and the skeleton of $P_G(\theta)$.
This shows one direction.
For the converse, suppose that $\phi$ is an isomorphism.
Set $P:=P_G(\theta)$ and let~$G_P$ be its edge-graph with skeleton map $\psi\:V(G_P)\to \F_0(P)$.
Then $\sigma:=\psi^{-1}\circ\phi$ is a graph isomorphism between $G$ and $G_P$.
So, since $G\cong G_P$, each eigenpolytope of $G$ is also an eigenpolytope of $G_P$.
We can therefore choose $P_{G_P}(\theta)=P_G(\theta)$, with corresponding eigenpolytope map $\phi':=\sigma^{-1}\circ\phi$.
In sum, the outer square in the following diagram commutes:
\begin{center}
\begin{tikzcd}
G \arrow[r, "\sigma"] \arrow[d, "\phi"'] & G_P \arrow[d, "\phi'"] \arrow[ld, "\psi"'] \\
\mathllap{P:=\,}P_G(\theta) \arrow[r, "\Id"'] & P_{G_P}(\theta)
\end{tikzcd}
\end{center}
Also, by construction of $\sigma$, the upper triangle commutes.
In conclusion, the lower triangle must commute as well, which is exactly \eqref{eq:diagram_polytope} with $T=\Id$. This proves that $P$~is $\theta$-spectral via \cref{res:naive_polytope}.
Since $G$ is isomorphic to $G_P$, $G$ is $\theta$-spectral.
\end{proof}
\end{proposition}
\noindent
This also proves \cref{res:naive} $(ii)$.
\iffalse
\begin{proposition}
\label{res:naive_graph}
A graph $G$ is $\theta$-spectral if and only if it is isomorphic to the edge-graph~of its $\theta$-eigenpolytope $P:=P_G(\theta)$ via some isomorphism $\sigma\:V(G)\to V(G_P)$ for which the following diagram commutes:
\begin{equation}
\label{eq:diagram}
\begin{tikzcd}
G \arrow[rd, "\phi"'] \arrow[r, "\sigma"] & G_P \arrow[d, "\psi"] \\
& P_G(\theta)\mathrlap{\,:=P }
\end{tikzcd}
\end{equation}
\begin{proof}
First, suppose that $G$ is $\theta$-spectral.
By definition, there exists a $\theta$-spectral polytope $Q$ with edge-graph $G_Q=G$, and let $\psi_Q\:V(G_Q)\to\F_0(Q)$ be the map \eqref{eq:vertex_map} (we use subscripts since there will be further such maps).
Since $Q$ is $\theta$-spectral, \cref{res:naive} $(i)$ states that $Q$ and $P_{G_Q}(\theta)=P_G(\theta)= P$ are linearly equivalent via some map $T\in\GL(\RR^d)$.
Let $\psi_P\:V(G_P)\to\F_0(P)$ be~the corresponding map \eqref{eq:vertex_map} for $P$.
We can then define the isomorphism as $\sigma:=\psi_Q\circ T\circ \psi_P^{-1}$ (all involved maps are invertible and preserve adjacency).
The following diagram then commutes:
\begin{center}
\begin{tikzcd}
\mathllap{G=\,}G_Q \arrow[r, "\sigma"] \arrow[d, "\psi_Q"'] & G_P \arrow[d, "\psi_P"] \\
Q \arrow[r, "T"'] & P\mathrlap{\,=P_G(\theta)}
\end{tikzcd}
\end{center}
We can now insert the diagonal arrow $\phi\: V(G)\to\RR^d\supseteq P_G(\theta)$ given by \eqref{eq:eigenpolytope_map}.
\begin{center}
\begin{tikzcd}
\mathllap{G=\,}G_Q \arrow[r, "\sigma"] \arrow[d, "\psi_Q"'] \arrow[rd, "\phi"] & G_P \arrow[d, "\psi_P"] \\
Q \arrow[r, "T"'] & P\mathrlap{\,:= P_G(\theta)}
\end{tikzcd}
\end{center}
Since $Q$ is spectral, the the lower triangle commutes by \cref{res:naive_polytope}.
Since also the outer square commutes, we find that the upper triangle (which is exactly \eqref{eq:diagram}) commutes as well, and we are done.
For the other direction, suppose that there is an isomorphism $\sigma:V(G)\to V(G_P)$ so that \eqref{eq:diagram} commutes.
Since $G$ and $G_P$ are isomorphic, $P=P_G(\theta)$ can be consider as a $\theta$-eigenpolytope of $G_P$ as well, thus $P=P_{G_P}(\theta)$.
If $\phi$ is the map \eqref{eq:eigenpolytope_map} for the eigenpolytope of $G$, then because $G_P$ uses the same eigenpolytope, $\phi\circ\sigma^{-1}$ is the respective map \eqref{eq:eigenpolytope_map} for the eigenpolytope of $G_P$.
The following diagram then commutes:
\begin{center}
\begin{tikzcd}
G \arrow[rd, "\phi"'] \arrow[r, "\sigma"] & G_P \arrow[rd, "\phi\circ\sigma^{-1}"] & \\
& \mathllap{P=:\,}P_G(\theta) \arrow[r, "\Id"'] & P_{G_P}(\theta)
\end{tikzcd}
\end{center}
We can now insert the vertical arrow $\psi\: V(G_P)\to \F_0(P)$ given by \eqref{eq:vertex_map}.
\begin{center}
\begin{tikzcd}
G \arrow[rd, "\phi"'] \arrow[r, "\sigma"] & G_P \arrow[rd, "\phi\circ\sigma^{-1}"] \arrow[d, "\psi"] & \\
& \mathllap{P=:\,}P_G(\theta) \arrow[r, "\Id"'] & P_{G_P}(\theta)
\end{tikzcd}
\end{center}
The left triangle is exactly \eqref{eq:diagram}, hence commutes by assumption.
The outer parallelogram commutes by construction, and so we find that the right triangle commutes as well.
This triangle is exactly \eqref{eq:diagram_polytope} from \cref{res:naive_polytope} with $T=\Id$.
This shows that $P
$ is $\theta$-spectral, and since $G\cong G_P$ is the edge-graph of $P$, we found that $G$ is $\theta$-spectral, and we are done.
\end{proof}
\end{proposition}
\fi
\iffalse
\begin{lemma}
If $P$ is linearly equivalent to the $P_{G_P}(\theta)$, and this linear map $T\in\GL(\RR^d)$ satisfies $\Phi=\Psi T$, then $P$ is $\theta$-spectral.
\end{lemma}
\begin{lemma}
If $G$ is isomorphic to the edge-graph of $P_G(\theta)=:Q$, and the isomorphism $\sigma\:V(G)\to V(G_Q)$ saitsies $\Phi=\Pi_\sigma \Psi$, then $G$ is $\theta$-spectral.
\end{lemma}
\begin{lemma}
\label{res:naive_polytope}
If $P$ is $\theta$-spectral, then $P$ is linearly equivalent to the $\theta$-eigenpolytope of its edge-graph. The converse is not true.
However, $P$ is $\theta$-spectral if and only if $P$ is linearly equivalent to $P_{G_P}(\theta)$ via~some map $T\in\GL(\RR^d)$, and $T$ additionally satisfies $T \phi(i) = \psi(i)$ for all $i\in V$, where~$\phi\:$ $V(G_P)\to\RR^d$ and $\psi\:V(G_P)\to\F_0(P)$ are the maps defined in \eqref{eq:eigenpolytope_map} resp.\ \eqref{eq:vertex_map}.
\begin{proof}
By definition, the $\theta$-eigenpolytope of $G_P$ satisfies $\Span \Phi=\Eig_{G_P}(\theta)$, where $\Phi$ is as defined in \eqref{eq:eigenpolytope_matrix}.
Now, by definition, $P$ is $\theta$-spectral if and only if $\Span \Psi = \Eig_{G_P}(\theta)$, where $\Psi$ is its arrangement matrix \eqref{eq:arrangement_matrix}.
But by \cref{res:same_column_span}, $\Phi$ and $\Psi$ can have the same span if and only of their rows are related by some invertible linear map $T\in\GL(\RR^d)$, that is, $T\phi(i)=\psi(i)$ for all $i\in V$.
A counterexample for the initial converse was given in \cref{ex:pentagon}.
\end{proof}
\end{lemma}
\begin{corollary}
If $P$ is $\theta$-spectral, then so is $G_P$ and $P_{G_P}(\theta)$.
\end{corollary}
The respective statement for graphs is the following:
\begin{lemma}
\label{res:naive_graph}
If $G$ is $\theta$-spectral, then $G$ is isomorphic to the edge-graph of its $\theta$-eigenpolytope. The converse is not true.
However, $G$ is $\theta$-spectral if and only if it is isomorphic to the edge-graph of its $\theta$-eigenpolytope, and the isomorphism is given by $\phi\:V\to\RR^d$~as~de\-fined in \eqref{eq:eigenpolytope_map}.
\begin{proof}
Let $Q:=P_G(\theta)$ be the $\theta$-eigenpolytope of $G$.
Then we try to prove $G\cong G_Q$.
If $G$ is $\theta$-spectral, then there is a $\theta$-spectral polytope $P$ with edge-graph $G_P\cong G$.
By this, $P_{G_P}(\theta)\cong P_G(\theta)=Q$.
By \cref{res:naive_polytope}, $P$ is linearly~equivalent to~$P_{G_P}(\theta)$, hence linearly equivalent to $Q$.
In particular, $G_P\cong G_Q$, and so we found $G\cong G_P\cong G_Q$.
A counterexample for the initial converse was given in \cref{ex:pentagon}.
\end{proof}
\end{lemma}
\fi
\iffalse
\begin{corollary}
\quad
\begin{myenumerate}
\item If a polytope $P$ is $\theta$-spectral, then it is the $\theta$-eigenpolytope of its edge-graph (up to some invertible linear transformation).
\item If a graph $G$ is $\theta$-spectral, then it is (isomorphic to) the edge-graph of its $\theta$-eigenpolytope.
\item If a graph $G$ is $\theta$-spectral, then its $\theta$-eigenpolytope is $\theta$-spectral.
\end{myenumerate}
\end{corollary}
\begin{corollary}
\label{res:naive_definition}
If $P$ is $\theta$-spectral, then it is the $\theta$-eigenpolytope of its edge-graph (up to invertible linear transformation)
\begin{proof}
If $P$ is $\theta$-spectral with arrangement matrix $M'$, then by \mbox{\cref{def:spectral} $(i)$} we have $\Span M'=\Eig_{G_P}(\theta)$.
Let further $M$ be the matrix \eqref{eq:eigenpolytope_matrix} used in \cref{def:eigenpolytope} to define the $\theta$-eigenpolytope of $G_P$.
By definition we have $\Span M=\Eig_{G_P}(\theta)$.
By \cref{res:same_column_span}, since $M$ and $M'$ have the same column span, their rows are related by an invertible linear transformation.
These rows are the vertices of $P$ and $P_{G_P}(\theta)$ respectively, finishing the proof.
\end{proof}
\end{corollary}
The converse of \cref{res:naive_definition} is not true.
This is intentional, since we do not want confiurations as in \cref{ex:pentagon} to be called \enquote{spectral}.
\begin{corollary}
\label{res:spectral_graph_naive}
If a graph $G$ is $\theta$-spectral, then $\phi\:V\ni i\mapsto v_i\in\RR^d$ (as \mbox{introduced} in \cref{def:eigenpolytope}) is an isomorphism between $G$ and the edge-graph of $P_G(\theta)$.
That includes
\begin{myenumerate}
\item for all $i\in V$ holds $v_i\in\F_0(P_G(\theta))$.
\item for distinct $i,j\in V$ holds $v_i\not= v_j$.
\item $ij\in E$ if and only if $\conv\{v_i,v_j\}\in\F_1(P_G(\theta))$.
\end{myenumerate}
\begin{proof}
Since $G$ is $\theta$-spectral, it is (isomorphic to) the edge-graph of some $\theta$-spectral polytope $P$.
Recall that $n$ denotes the number of vertices of $G$.
Since $P_G(\theta)$ is the convex hull of $v_1,...,v_n$, $P_G(\theta)$ has at most $n$ vertices, and $\phi$ is surjective.
Since $P_G(\theta)$ is combinatorially equivalent to $P$ (by \cref{res:naive_definition}), $P_G(\theta)$ has exactly $n$ vertices, and $\phi$ must also be injective.
This also shows $(i)$ and $(ii)$.
\end{proof}
\end{corollary}
Note that not every graph that is isomorphic to the edge-graph of its $\theta$-eigenpoly\-tope is also $\theta$-spectral (again, consider \cref{ex:pentagon}).
However, if the isomorphism is the one from \cref{res:spectral_graph_naive}, then this conclusion is valid.
\iffalse
\begin{lemma}
If $P\subset\RR^d$ is a (full-dimensional) polytope, then the following are~equivalent:
\begin{myenumerate}
\item $P$ is $\theta$-spectral,
\item $P$ is $\theta$-balanced and the eigenvalue $\theta$ has multiplicity $d$.
\end{myenumerate}
If $G$ is a graph, then the following are equivalent:
\begin{myenumerate}
\setcounter{enumi}{2}
\item $G$ is $\theta$-spectral,
\item the $\theta$-eigenpolytope of $G$ is $\theta$-spectral with edge-graph $G$,
\item $G$ is isomorphic to the edge-graph of its $\theta$-eigenpolytope, with isomorphism $V\ni i\mapsto v_i\in\RR^d$ (as constructed in \cref{def:eigenpolytope}).
\end{myenumerate}
\begin{proof}
By \cref{res:eigenvalue}, the balancing equation \eqref{eq:balanced} is equivalent to $\Span M\subseteq\Eig_G(\theta)$.
Recall that the rank~of~$M$~is~$d$.
Now, $\theta$ having multiplicity $d$ (\shortStyle{i.e.,}\ $(ii)$) is equivalent to $\dim\Eig_G(\theta)=d=\rank M$. This is then equivalent to $M=\Eig_G(\theta)$ which is $(i)$.
If $G$ is $\theta$-spectral then it is (isomorphic to) the edge-graph of \emph{some} $\theta$-spectral polytope $P$.
But \cref{res:naive_definition} states that $P$ is (up to linear transformation) the $\theta$-eigenpolytope of its edge-graph, or, since isomorphic, the $\theta$-eigenpolytope of $G$.
\end{proof}
\end{lemma}
\fi
\fi
It is also possible to give a definition of spectral graphs purely in terms of graph theory, without any explicit reference to polytopes:
\begin{lemma}
\label{res:spectral_2}
A graph $G$ is $\theta$-spectral if and only if it satisfies both of the following:
\begin{myenumerate}
\item for each vertex $i\in V$ exists a $\theta$-eigenvector $u=(u_1,...,u_n)\in\Eig_G(\theta)$~whose single largest component is $u_i$, or equivalently,
$$\Argmax_{k\in V} u_k = \{i\}.$$
\item any two vertices $i,j\in V$ form an edge $ij\in E$ in $G$ if and only~if there is a $\theta$-eigenvector $u=(u_1,...,u_n)\in\Eig_G(\theta)$ whose only two largest components are $u_i$ and $u_j$, or equivalently,
$$\Argmax_{k\in V} u_k = \{i,j\}.$$
\end{myenumerate}
\end{lemma}
This characterization of spectral graphs can be interpreted as follows: a spectral graph can be reconstructed from knowing a single eigenspace, rather than, say, all eigenspaces and their associated eigenvalues.
\begin{proof}[Proof of \cref{res:spectral_2}]
Let $P_G(\theta)\subset\RR^d$ be the $\theta$-eigenpolytope of $G$ with eigenpolytope matrix $\Phi$ and eigenpolytope map $\phi\:V\ni i\mapsto v_i\in\RR^d$.
Since $\Span\Phi=\Eig_G(\theta)$, the eigenvectors $u=(u_1,...,u_n)\in\Eig_G(\theta)$ are exactly the vectors that can be written as $u=\Phi x$ for some $x\in\RR^d$.
If then $e_k\in\RR^n$ denotes the $k$-th standard basis vector, we have
$$u_k = \<u,e_k\> = \<\Phi x, e_k\> = \<x,\Phi^{\Tsymb}\! e_k\> = \<x, v_k\>.$$
Therefore, there is a $\theta$-eigenvector $u=(u_1,...,u_n)\in\Eig_G(\theta)$ with
$\Argmax_{k\in V} u_k = \{i_1,...,i_m\}$
if and only if there is a vector $x\in\RR^d$ with
$$\Argmax_{k\in V} \<x,v_k\> = \{i_1,...,i_m\}.$$
But this last line is exactly what it means for $\conv\{v_{i_1},...,v_{i_m}\}$ to be a face of $P_G(\theta)$ $=\conv\{v_1,...,v_n\}$ (and $x$ is a normal vector of that face).
In this light, we can interpret $(i)$ as stating that $v_1,...,v_n$ form $n$ distinct vertices of $P_G(\theta)$, and $(ii)$ as stating that $\conv\{v_i,v_j\}$ is an edge of $P_G(\theta)$ if and only if $ij\in E$.
And this means exactly that $\phi$ is a graph isomorphism between $G$ and the skeleton of $P_G(\theta)$.
By \cref{res:naive_graph}, this is equivalent to $G$ being $\theta$-spectral.
\end{proof}
\iffalse
If the graph satisfies $(i)$ and $(ii)$, then the map $\sigma:=\psi^{-1}\circ\phi:V(G)\to V(G_P)$ is bijective because of $(i)$, and a graph isomorphism because of $(ii)$.
This isomorphism makes \eqref{eq:diagram} commute by definition, and so $G$ is $\theta$-spectral by \cref{res:naive_graph}.
Conversely, if $G$ is $\theta$-spectral, then it is isomorphic to the edge-graph $G_P$ of its $\theta$-eigenpolytope via some isomorphism $\sigma\: V(G)\to V(G_P)$ that makes \eqref{eq:diagram} commute.
We first note that that implies that $P_G(\theta)$ has exactly the vertices $v_1,...,v_n$, which means that $G$ satisfies $(i)$.
Since $\sigma$ and $\psi$ are graph isomorphisms, so is $\psi\circ\sigma$, and since \eqref{eq:diagram} commutes, also $\phi$.
This means $ij\in E$ if and only if $\conv\{\phi(i),\phi(k)\}=\conv\{v_i,v_j\}\in P_G(\theta)$.
This is equivalent to $(ii)$, and we are done.
Suppose that $G$ is $\theta$-spectral, then
We show first that there is a $\theta$-eigenvectors $u=(u_1,...,u_n)\in\Eig_G(\theta)$ with
$$\Argmax_{k\in V} u_k=\{i_1,...,i_m\}$$
if and only if $\{v_{i_1},...,v_{i_m}\}$ (as defined in \cref{def:eigenpolytope}) is the vertex set of a face~of the $\theta$-eigenpolytope of $G$.
Recall that a set $\{v_{i_1},...,v_{i_m}\}\subseteq \F_0(P)$ of vertices is the vertex set of a face of $P$ if and only if there is a vector $x\in\RR^d$ (a normal vector) with
$$\Argmax_{k\in V} \<x,v_k\> = \{i_1,...,i_m\}.$$
Let $M$ be the arrangement matrix of $P$, set $u=(u_1,...,u_n):=Mx$, and let $e_k\in\RR^n$ be the $k$-th standard basis vector.
Then
$$\<x,v_k\> = \<x, M^{\Tsymb}\! e_k\> = \<M x, e_k\> =\<u,e_k\> = u_k.$$
By $u=Mx$, the $x\in\RR^d$ are in one-to-one correspondence with the $u\in\Span M=\Eig_G(\theta)$, which are exactly the $\theta$-eigenvectors of $G$.
So $\{v_{i_1},...,v_{i_m}\}$ forms a face of $P$ if and only if
$$(*)\quad \Argmax_{k\in V} u_k = \Argmax_{k\in V} \<u,v_k\> = \{i_1,...,i_m\},$$
Condition $(i)$ and $(ii)$ are now the version of $(*)$ for vertices and edges of $P$, which correspond to vertices and edges of $G$ via the isomorphism $\phi$.
\fi
In practice, to reconstruct a spectral graph from an eigenspace, the steps could be the following: given a subspace $U\subseteq\RR^n$ (the claimed eigenspace), then
\begin{myenumerate}
\item choose any basis $u_1,...,u_d\in\RR^n$ of $U$,
\item build the matrix $\Phi=(u_1,...,u_d)\in\RR^{n\x d}$ in which the $u_i$ are the columns,
\item define $v_i$ as the $i$-th \emph{row} of $\Phi$,
\item define $P:=\conv\{v_1,...,v_n\}\subset\RR^d$ as the convex hull of the $v_i$,
\item the reconstructed graph $G=G_P$ is then the edge-graph of $P$.
\end{myenumerate}
\subsection{Properties of spectral polytopes}
We discuss two properties of spectral polytopes that make them especially interesting in polytope theory.
\subsubsection*{Reconstruction from the edge-graph}
\label{sec:spectral_reconstruction}
The edge-graph of a general polytope carries little information about that polytope \shortStyle{i.e.,}\ given only its edge-graph, we can often not reconstruct the polytope from this (up to combinatorial equivalence).
Often, one cannot even deduce the dimension of the polytope from its edge-graph.
Reconstruction might be possible in certain special cases, as \shortStyle{e.g.}\ for 3-dimensional polyhedra, simple polytopes or zonotopes.
The spectral polytopes provide another such class.
\begin{theorem}\label{res:reconstruction}
A $\theta_k$-spectral polytope is uniquely~determined by its edge-graph up to invertible linear transformations.
\end{theorem}
The proof is simple:
every $\theta_k$-spectral polytope is linearly equivalent to the~$\theta_k$-eigenpolytope of its edge-graph (by \cref{res:naive} $(i)$).
Our definition of the \mbox{$\theta_k$-eigen}\-polytope already suggests an explicit procedure to construct it (a script for this~is included in \cref{sec:appendix_mathematica}).
This property of spectral polytopes appears more exciting when applied to graph classes that are not obviously spectral (see \cref{sec:edge_transitive}).
\subsubsection*{Realizing symmetries of the edge-graph}
\label{sec:spectral_symmetries}
Every Euclidean symmetry of a polytope induces a combinatorial symmetry on its edge-graph.
The converse is far from true.
Think, for example, about a rectangle that is not a square.
Even worse, it can happen that a polytope does not even have a realization that realizes all the symmetries of its edge-graph (\shortStyle{e.g.}\ the polytope constructed in \cite{bokowski1984combinatorial}).
We have previously discussed (in \cref{res:realizing_symmetries}) the existence of a homomorphism $\Aut(G)\to\Aut(P_G(\theta))$ between the symmetries of a graph $G$ and the symmetries of its eigenpolytopes.
There are two caveats:
\begin{myenumerate}
\item this is not necessarily an isomorphism, and
\item it says nothing about the symmetries of the edge-graph of $P_G(\theta)$, as this one needs not to be isomorphic to $G$
\end{myenumerate}
Still, it suffices to makes statement of the following form:
if $G$ is vertex-transitive, then so are all its eigenpolytopes.
This might not work with other transitivities, as for example edge-transitivity.
This is no concern for spectral graphs/polytopes:
\begin{theorem} \quad
\label{res:symmetries}
\begin{myenumerate}
\item If $G$ is $\theta$-spectral, then $P_G(\theta)$ realizes all its symmetries, which includes
$$\Aut(G)\cong\Aut(P_G(\theta))$$
%
via the map $\sigma\mapsto T_\sigma$ given in \cref{res:realizing_symmetries}, as wells as that $T_\sigma$ permutes the vertices and edges of $P_G(\theta)$ exactly as $\sigma$ permutes the vertices and edges of the graph $G$.
\item If $P$ is $\theta$-spectral, then $P$ has a realization that realizes all the symmetries~of its edge-graph, namely, the $\theta$-eigenpolytope of its edge-graph.
\end{myenumerate}
%
\end{theorem}
This is mostly straight forward, with large parts already addressed in \mbox{\cref{res:realizing_symmetries}}.
The major difference is that for spectral graphs $G$ the eigenpolytope has exactly the distinct vertices $v_1,...,v_n\in\RR^d$.
The statement from \cref{res:realizing_symmetries} that $T_\sigma$~per\-mutes the $v_i$ as prescribed by $\sigma$, then becomes, that $T_\sigma$ permutes the \emph{vertices} as prescribed by $\sigma$, and hence also the edges.
Also, since the $v_i$ are distinct, no non-trivial symmetry $\sigma$ can result in trivial $T_\sigma$, making $\sigma\mapsto T_\sigma$ into a group \emph{isomorphism}.
For part $(ii)$ merely recall that the eigenpolytope $P_{G_P}(\theta)$ is indeed a realization of $P$ by \cref{res:naive} $(i)$.
The major consequence of this is, that for spectral graphs/polytopes also more complicates types of symmetries translate between a polytope and its graph, as \shortStyle{e.g.}\ edge-transitivity (see also \cref{sec:edge_transitive}).
\iffalse
But, we have to be careful: we cannot expect a general statement about symmetries of spectral polytopes, as \shortStyle{e.g.}\ a rectangle and a square are both spectral (since they are linear transformations of each other), but one is apparently more symmetric, if by symmetry we understand a rigid motion.
There are two solutions: either, generalizing symmetries to general invertible linear transformations, or, restricting the polytopes about which we talk.
Both perspectives are equivalent, and we decided to choose the second.
Let us call a polytope \emph{normalized} if $M^{\Tsymb}\! M=\Id$ for its arrangement matrix $M$, that is, the columns of $M$ form a orthonormal basis of $\Span M$.
By \cref{res:same_column_span}, every polytope is just one invertible linear transformation away from being normalized.
Also, the way in which we defined eigenpolytopes in \cref{def:eigenpolytope} ensures that they are normalized.
We have the following:
\begin{theorem}
\label{res:realizing_symmetries}
A normalized spectral polytope realizes all the combinatorial symmetries of its edge-graph as Euclidean symmetries.
More precisely, we have the following: if $\sigma\in\Aut(G_P)\subseteq\Sym(V)$ is a combinatorial symmetry of the edge-graph, and $\Pi_\sigma\in\Perm(\RR^n)$ is the associated permutation matrix, then
$$T_\sigma:=M^{\Tsymb} \Pi_\sigma M \;\in\; \Ortho(\RR^d)$$
is a Euclidean symmetry of $P$ that permutes its vertices exactly as $\sigma$ permutes the vertices of $G_P$ (that is $T_\sigma v_i = v_{\sigma(i)}$).
\end{theorem}
We should mention that this result is more generally true for all eigenpolytopes of a graph and was already proven in \cite[Theorem 2.2]{godsil1978graphs}.
Another proof in the context of spectral graph realizations can be found \cite[Corollary 2.9]{winter2020symmetric}.
As a general consequence we have that if $G$ is vertex-transitive, then so are all its eigenpolytopes.
This does not extend to other kinds of symmetries, \shortStyle{e.g.}\ edge-transitivity.
However, if $G$ is $\theta$-spectral, then the symmetries of $G$ and its~$\theta$-eigenpolytope translate into each other one-to-one.
For example, $G$ is edge-transitive if and only if $P_G(\theta)$ is.
We make use of this in later sections.
\fi
\section{The Theorem of Izmestiev}
\label{sec:izmestiev}
We introduce our, as of yet, most powerful tool for proving that certain polytopes are $\theta_2$-spectral.
For this, we make use of a more general theorem by Izmestiev \cite{izmestiev2010colin}, first proven in the context of the Colin de Verdière graph invariant.
The proof of this theorem requires techniques from convex geometry, most notably, mixed volumes, which we not address here.
We need to introduce some terminology.
As before, let $P\subset\RR^d$ denote a full-dimensional polytope of dimension $d\ge 2$, with~edge-graph $G_P=(V,E),V=\{1,...,n\}$ and vertices $v_i\in \F_0(P),i\in V$.
Recall, that the \emph{polar dual} of $P$ is the polytope
$$P^\circ:=\{x\in\RR^d\mid \<x,v_i\>\le 1\text{ for all $i\in V$}\}.$$
We can replace the $1$-s in this definition by variables $c=(c_1,...,c_n)$, to obtai
$$P^\circ(c):=\{x\in\RR^d\mid\<x,v_i\>\le c_i\text{ for all $i\in V$}\}.$$
The usual polar dual is then $P^\circ=P^\circ(1,...,1)$.
\begin{figure}[h!]
\includegraphics[width=0.85\textwidth]{img/P_dual_2}
\caption{Visualization of $P^\circ(c)$ for different values of $c\in\RR^n$.}
\end{figure}
In the following, $\vol(\free)$ denotes the volume of convex sets in $\RR^d$ (\shortStyle{w.r.t.}\ the usual Lebesgue measure).
Note that the function $\vol(P^\circ(c))$ is differentiable in $c$, and so we can compute partial derivatives \shortStyle{w.r.t.}\ the components of $c$.
\begin{theorem}[Izmestiev \cite{izmestiev2010colin}, Theorem 2.4]\label{res:izmestiev}
Define a matrix $X\in\RR^{n\x n}$ with~compo\-nents
$$X_{ij}:=-\frac{\partial^2 \vol(P^\circ(c))}{\partial c_i\partial c_j}\Big|_{c=(1,...,1)}.$$
The matrix $X$ has the following properties:
\begin{myenumerate}
%
\item $X_{ij}< 0$ whenever $ij\in E(G_P)$,
\item $X_{ij}=0$ whenever $ij\not\in E(G_P)$,
\item $X\Psi=0$ (where $\Psi$ is the arrangement matrix of $P$),
\item $X$ has a unique negative eigenvalue, and this eigenvalue is simple,
\item $\dim\ker X=d$.
\end{myenumerate}
\end{theorem}
One can view the matrix $X$ as some kind of adjacency matrix of~a vertex- and edge-weighted version of $G_P$.
Part $(iii)$ states that $v$ satisfies a weighted form of the balancing condition \eqref{eq:balanced} with eigenvalue zero.
Since $\rank \Psi=d$, part $(v)$ states that $\Span \Psi$ is already the whole 0-eigenspace.
And part $(iv)$ states that zero is the second smallest eigenvalue of $X$.
\begin{theorem}\label{res:implies_spectral}
Let $X\in\RR^{n\x n}$ be the matrix defined in \cref{res:izmestiev}.
If we have
\begin{myenumerate}
\item $X_{ii}$ is independent of $i\in V(G_P)$, and
\item $X_{ij}$ is independent of $ij\in E(G_P)$,
\end{myenumerate}
then $P$ is $\theta_2$-spectral.
\begin{proof}
By assumption there are $\alpha,\beta\in\RR$, $\beta>0$, so that $X_{ii}=\alpha$ for all vertices~$i\in$ $V(G_P)$, and $X_{ij}=\beta<0$ for all edges $ij\in E(G_P)$ (we have $\beta<0$ by \cref{res:izmestiev} $(i)$).
We can write this as
$$X=\alpha \Id + \beta A\quad\implies\quad (*)\;A=\frac\alpha\beta \Id+\frac1\beta X,$$
where $A$ is the adjacency matrix of $G_P$.
By \cref{res:izmestiev} $(iv)$ and $(v)$, the matrix $X$ has second smallest eigenvalue zero of multiplicity $d$.
By \cref{res:izmestiev} $(iii)$, the columns of $M$ are the corresponding eigenvectors.
Since $\rank \Psi=d$ we find that these are all the eigenvectors and $\Span \Psi$ is the 0-eigenspace of $X$.
By $(*)$ the eigenvalues of $A$ are the eigenvalues of $X$, but scaled by $1/\beta$ and shifted by $\alpha/\beta$. Since $1/\beta <0$, the second-\emph{smallest} eigenvalue of $X$ gets mapped onto the second-\emph{largest} eigenvalue of $A$.
Therefore, $A$ (and also $G_P$) has second-largest eigenvalue $\theta_2=\alpha/\beta$ of multiplicity $d$, and $\Span \Psi$ is the corresponding eigenspace.
By definition, $P$ is then the $\theta_2$-eigenpolytope of $G_P$ and is therefore~$\theta_2$-spectral.
\end{proof}
\end{theorem}
It is unclear whether \cref{res:implies_spectral} already characterizes $\theta_2$-spectral polytopes, or even spectral polytopes in general (see also \cref{q:characterization}).
\section{Edge-transitive polytopes}
\label{sec:edge_transitive}
We apply \cref{res:implies_spectral} to edge-transitive polytopes, that is, to polytopes for~which the Euclidean symmetry group $\Aut(P)\subset\Ortho(\RR^d)$ acts transitively on the edge set $\F_1(P)$.
No classification of edge-transitive polytopes is known.
Some \mbox{edge-transitive} polytopes are listed in \cref{sec:classification}.
Despite the name of this section, we are actually going to address polytopes that are simultaneously vertex- and edge-transitive.
This is not a huge deviation from the title: as shown in \cite{winter2020polytopes}, edge-transitive polytopes in dimension $d\ge 4$ are always also vertex-transitive, and the exceptions in lower dimensions are few (a continuous family of $2n$-gons for each $n\ge 2$, and two exceptional polyhedra).
\cref{res:implies_spectral} can be directly applied to simultaneously vertex- and edge-transitive polytopes, and so we have
\begin{corollary}
\label{res:vertex_edge_transitive_cor}
A simultaneously vertex- and edge-transitive polytope is $\theta_2$-spectral.
\end{corollary}
We collect all the notable consequences in the following theorem:
\begin{theorem}\label{res:edge_vertex_transitive}
If $P\subset\RR^d$ is simultaneously vertex- and edge-transitive, then
\begin{myenumerate}
\item $\Aut(P)\subset\RR^d$ is irreducible as a matrix group.
\item $P$ is uniquely determined by its edge-graph up to scale and orientation.\footnote{This shows that $P$ is \emph{perfect}, \shortStyle{i.e.,}\ is the unique maximally symmetric realization of its combinatorial type. See \cite{gevay2002perfect} for an introduction to perfect polytopes.}
\item $P$ realizes all the symmetries of its edge-graph.
\item if $P$ has edge length $\ell$ and circumradius $r$, then
%
\begin{equation}
\label{eq:circumradius}
\frac{\ell}r = \sqrt{\frac{2\lambda_2}{\deg(G_P)}} = \sqrt{2\Big(1-\frac{\theta_2}{\deg(G_P)}\Big)},
\end{equation}
%
where $\deg(G_P)$ is the vertex degree of $G_P$, and $\lambda_2=\deg(G_P)-\theta_2$ denotes its second smallest Laplacian eigenvalue.
\item if $\alpha$ is the dihedral angle of the polar dual $P^\circ$, then
%
\begin{equation}
\label{eq:dihedral_angle}
\cos(\alpha)=-\frac{\theta_2}{\deg(G_P)}.
\end{equation}
\end{myenumerate}
\begin{proof}
The complete proof of $(i)$ and $(ii)$ has to be postponed until \cref{sec:rigidity} (see \cref{res:edge_transitive_rigid}).
Concerning $(ii)$, from \cref{res:vertex_edge_transitive_cor} and \cref{res:reconstruction} already follows that $P$ is determined by its edge-graph up to \emph{invertible linear transformations}, but not necessarily only up to scale and orientation.
Part $(iii)$ follows from \cref{res:symmetries}.
Part $(iv)$ and $(v)$ were proven (in a more general setting) in \cite[Proposition 4.3]{winter2020symmetric}.
This applies literally to $(iv)$.
For $(v)$, note the following: if $\sigma_i\in\F_{d-1}(P^\circ)$ is the facet of the polar dual $P^\circ$ that corresponds to the vertex $v_i\in\F_1(P)$, then the dihedral angle between $\sigma_i$ and $\sigma_j$ is $\pi-\angle(v_i,v_j)$.
The latter expression was proven in \cite{winter2020symmetric} to agree with \eqref{eq:dihedral_angle}.
\end{proof}
\end{theorem}
It is worth emphasizing that large parts of \cref{res:edge_vertex_transitive} do not apply to polytopes of a weaker symmetry, as \shortStyle{e.g.}\ vertex-transitive polytopes.
Prisms are counterexamples to both $(i)$ and $(ii)$.
There are vertex-transitive neighborly polytopes (other than simplices) and they are counterexamples to $(ii)$ and $(iii)$.
\iffalse
A simultaneously vertex- and edge-transitive polytope can neither deformed nor projected to keep its edge-graph and symmetry.
The fact that $P$ is perfect can already be proven
from classical rigidity results and thus can be formulated for a much larger class of polytopes:
\begin{theorem}[Alexandrov, \cite{...}]
\label{res:alexandrov_rigidity}
A polyope is uniquely determined (up to orientation) by its combinatorial type and the shape of its facets.
\end{theorem}
\begin{theorem}
An inscribed polytope with all edges of length $\ell$ is uniquely determined (up to orientation) by its combinatorial type.
\begin{proof}
The proof is by induction.
A 2-dimensional inscribed polytope with all edges of length $\ell$ ic clearly a regular polygon, hence of unique shape.
Now consider $P$ is a $d$-dimensional inscribed polytope with all edges of length $\ell$.
Then also each facet of $P$ is inscribed with this edge-length.
Also, the combinatorial type of each facet is determined by the combinatorial type of $P$.
By induction assumption, the shape of each facet is hence uniquely determined.
By \cref{res:alexandrov_rigidity} this uniquely determines the shape of $P$.
\end{proof}
\end{theorem}
\begin{corollary}
A simultaneously vertex- and edge-transitive polytope is perfect.
\end{corollary}
\fi
\begin{remark}
\label{rem:edge_not_vertex}
There are two edge-transitive polyhedra that are not vertex-transitive: the \emph{rhombic dodecahedron} and the \emph{rhombic triacontahedron} (see also \cref{fig:edge_transitive}).~Only the former is $\theta_2$-spectral, and the latter is not spectral for any eigenvalue (this was already mentioned in \cite{licata1986surprising}).
Since the rhombic dodecahedron is not vertex-transitive, nothing of this follows from \cref{res:vertex_edge_transitive_cor}.
However, this polytope satisfies the conditions of \cref{res:implies_spectral}, which seems purely accidental.
It is the only known spectral polytope that is not vertex-transitive.
\end{remark}
\subsection{Rigidity and irreducibility}
\label{sec:rigidity}
The goal of this section is to prove the missing part of \cref{res:edge_vertex_transitive}:
\begin{theorem}
\label{res:edge_transitive_rigid}
If $P\subset\RR^d$ is simultaneously vertex- and edge-transitive, then
\begin{myenumerate}
\item $\Aut(P)\subset\Ortho(\RR^d)$ is irreducible as a matrix group, and
\item $P$ is determined by its edge-graph up to scale and orientation.
\end{myenumerate}
\end{theorem}
To prove \cref{res:edge_transitive_rigid}, we make use of \emph{Cauchy's rigidity theorem} for polyhedra (with its beautiful proof listed in \cite[Section 12]{aigner2010proofs}).
It states that every~polyhedron is uniquely determined by its combinatorial type and the shape of its faces.
This was generalized by Alexandrov to general dimensions $d\ge 3$ (proven \shortStyle{e.g.}\ in \cite[Theorem 27.2]{pak2010lectures}):
\begin{theorem}[Alexandrov]
\label{res:alexandrov}
Let $P_1,P_2\subset\RR^d, d\ge 3$ be two polytopes, so that
\begin{myenumerate}
\item $P_1$ and $P_2$ are combinatorially equivalent via a face lattice~isomorphism~$\phi:\F(P_1)\to\F(P_2)$, and
\item each facet $\sigma\in\F_{d-1}(P_1)$ is congruent to the facet $\phi(\sigma)\in\F_{d-1}(P_2)$.
\end{myenumerate}
Then $P_1$ and $P_2$ are congruent, \shortStyle{i.e.,}\ are the same up to orientation.
\end{theorem}
\begin{proposition}
\label{res:regular_rigid}
Let $P_1,P_2\subset\RR^d$ be two combinatorially equivalent polytopes,~each of which has
\begin{myenumerate}
\item all vertices on a common sphere (\shortStyle{i.e.,}\ is inscribed), and
\item all edges of the same length $\ell_i$.
\end{myenumerate}
Then $P_1$ and $P_2$ are the same up to scale and orientation.
\begin{proof}[Proof.\!\!]\footnote{This proof was proposed by the user \emph{Fedor Petrov} on MathOverflow \cite{petrovMO}.}
W.l.o.g.\ assume that $P_1$ and $P_2$ have the same circumradius, otherwise~re\-scale $P_2$.
It then suffices to show that $P_1$ and $P_2$ are the same up to orientation.
We proceed with induction by the dimension $d$.
The induction base is given by $d=2$, which is trivial, since any two inscribed polygons with constant edge length are regular and thus completely determined (up to scale and orientation) by their number of vertices.
Suppose now that $P_1$ and $P_2$ are combinatorially equivalent polytopes of dimension $d\ge 3$ that satisfy $(i)$ and $(ii)$.
Let $\phi$ be the face lattice isomorphism between them.
Let $\sigma\in\F_{d-1}(P_1)$ be a facet of $P_1$, and $\phi(\sigma)$ the corresponding facet in $P_2$.
In particular, $\sigma$ and $\phi(\sigma)$ are combinatorially equivalent.
Furthermore, both $\sigma$ and $\phi(\sigma)$ are of dimension $d-1$ and satisfy $(i)$ and $(ii)$. This is obvious for $(ii)$, and for $(i)$ recall that facets of inscribed polytopes are also inscribed.
By induction hypothesis, $\sigma$ and $\phi(\sigma)$ are then congruent.
Since this holds for all facets $\sigma\in\F_{d-1}(P_1)$, \cref{res:alexandrov} tells us that $P_1$ and $P_2$ are congruent, that is, the same up to orientation.
\end{proof}
\end{proposition}
We can now prove the main theorem of this section:
\begin{proof}[Proof of \cref{res:edge_transitive_rigid}]
By \cref{res:edge_vertex_transitive} the combinatorial type of $P$ is determined by its edge-graph.
By vertex-transitivity, all vertices are on a sphere.
By edge-transitivity, all edges are of the same length.
We can then apply \cref{res:regular_rigid} to obtain that $P$ is unique up to scale and orientation.
This proves $(ii)$.
Suppose now, that $\Aut(P)$ is not irreducible, but that $\RR^d$ decomposes as $\RR^d=W_1\oplus W_2$ into non-trivial orthogonal $\Aut(P)$-invariant subspaces.
Let $T_\alpha\in\GL(\RR^d)$ be the linear map that acts as identity on $W_1$, but as $\alpha\Id$ on $W_2$ for some $\alpha >1$.
Then $T_\alpha P$ is a non-orthogonal linear transformation of $P$ (in particular, combinatorially equivalent), on which $\Aut(P)$ still acts vertex- and edge-transitively.
By $(ii)$, this cannot be. Hence $\Aut(P)$ must be irreducible, which proves $(i)$.
\end{proof}
\subsection{A word on classification}
\label{sec:classification}
Despite the simple appearance of the definition of an edge-transitive polytope,
no classification was obtained so far.
There exists a classification of the 3-dimension edge-transitive polyhedra: besides the Platonic solids, these are the ones shown in \cref{fig:edge_transitive} (nine in total).
\begin{figure}[h!]
\includegraphics[width=0.75\textwidth]{img/edge_transitive_polyhedra}
\caption{From left to right, these are: the cuboctahedron, the icosido\-decahedron, the rhombic dodecahedron, and the rhombic triacontahedron.}
\label{fig:edge_transitive}
\end{figure}
There are many known edge-transitive polytopes in dimension $d\ge 4$ (so we~are not talking about a class as restricted as the regular polytopes).
There are 15 known edge-transitive 4-polytopes (and an infinite family of duoprisms\footnote{The $(n,m)$-duoprism is the cartesian product of a regular $n$-gon and a regular $m$-gon. Those are edge-transitive if and only of $n=m$. Technically, the 4-cube is the $(4,4)$-duoprism but is usually not counted as such, because of its exceptionally large symmetry group.}), but already here, no classification is known.
It is known that the number of irreducible\footnote{Being not the cartesian product of lower dimensional edge-transitive polytopes.} edge-transitive polytopes grows at least linearly with the number of dimensions.
For example, there are $\lfloor d/2\rfloor$ \emph{hyper-simplices} in dimension $d$.
These are edge-transitive (even distance-transitive, see \cref{sec:distance_transitive}).
It is the hope of the author, that the classification of the edge-transitive polytopes can be obtained using their spectral properties.
Their classification can now be stated purely as a problem in spectral graph theory:
the classification of the edge-transitive polytopes (in dimension $d\ge 4$) is equivalent to the classification of $\theta_2$-spectral edge-transitive graphs, and since \cref{res:spectral_2}, we have a completely graph theoretic characterization of spectral graphs.
\begin{theorem}
\label{res:edge_transitive_spectral_graph}
Let $G$ be an edge-transitive graph. If $G$ is $\theta_k$-spectral, then
\begin{myenumerate}
\item $k=2$, and
\item if $G$ is \ul{not} vertex-transitive, then $G$ is the edge-graph of the rhombic dodecahedron (see \cref{fig:edge_transitive}).
\end{myenumerate}
\begin{proof}
We first prove $(ii)$.
As shown in \cite{winter2020polytopes} all edge-transitive polytopes in dimen\-sion $d\ge 4$ are vertex-transitive.
If $G$ is edge-transitive, not vertex-transitive and $\theta_k$-spectral, then its $\theta_k$-eigenpolytope is also edge-transitive but not vertex-transitive, hence of dimension $d\le 3$.
One checks that the 2-dimensional spectral polytopes are regular polygons, hence vertex-transitive.
The remaining polytopes are polyhedra, and we mentioned in \cref{rem:edge_not_vertex} that among these, only the rhombic dodecahedron is spectral, in fact $\theta_2$-spectral.
This proves $(ii)$.
Equivalently, if $G$ is vertex- and edge-transitive, then so is its eigenpolytope.
By \cref{res:vertex_edge_transitive_cor} this is a $\theta_2$-eigenpolytope.
Together with part $(ii)$, we find $k=2$~in~all cases, which proves $(i)$.
\end{proof}
\end{theorem}
\subsection{Arc- and half-transitive polytopes}
\label{sec:arc_transitive}
In a graph or polytope, an \emph{arc} is~an incident vertex-edge-pair.
A graph or polytope is called \emph{arc-transitive} if its symmetry group acts transitively on the arcs.
Being arc-transitive implies both, being vertex-transitive, and being edge-transitive.
In addition to that, in an arc-transitive graph, every edge can be mapped, not only onto every other edge, but also onto itself with flipped orientation.
There exist graphs that are simul\-taneously vertex- and edge-transitive, but not arc-transitive.
Those are called \emph{half-transitive} graphs, and are comparatively rare.
The smallest one has $27$ vertices and is known as the \emph{Holt graph} (see \cite{bouwer1970vertex,holt1981graph}).
For polytopes on the other hand, it is unknown whether there eixsts a distinction being arc-transitive and being simultaneously vertex- and edge-transitive.
No \emph{half-transitive polytope} is known.
Because of \cref{res:edge_vertex_transitive} $(i)$, we know that the edge-graph of a half-transitive polytope must itself be half-transitive.
Since such graphs are rare, the existence of half-transitive polytopes seems unlikely.
\begin{example}
The Holt graph is not the edge-graph of a half-transitive polytope: the Holt graph is of degree four, and its second-largest eigenvalue is of multiplicity six, giving rise to a 6-dimensional $\theta_2$-eigenpolytope.
But a 6-dimensional polytope must have an edge-graph of degree at least six, and so the Holt graph is not spectral.
\end{example}
The lack of examples of half-transitive polytopes means that all known edge-transitive polytopes in dimension $d\ge 4$ are in fact arc-transitive.
Likewise, a~classification of arc-transitive polytopes is not known.
\iffalse
\subsection{Half-transitive polytopes}
\label{sec:half_transitive}
Arc-transitivity implies vertex- and edge-transi\-ti\-vity, but for graphs, the converse is not true.
A graph (and we shall adopt this terminology for polytopes) is called \emph{half-transitive}, if it is both vertex- and edge-transitive, but not arc-transitive.
In a half-transitive graph, each edge can be mapped onto any other edge, but not onto itself with flipped orientation.
The existence of half-transitive graphs was proven first by Holt \cite{holt1981graph}.
Such graphs are relatively rare. The smallest half-transitive graph is now known as the Holt graph and has 27 vertices.
It is not known whether there are any \emph{half-transitive polytopes}.
However, by \cref{res:...} we can now conclude that they are at least as rare as the half-transitive graphs.
\begin{corollary}
The edge-graph of half-transitive polytope must be half-transitive.
\end{corollary}
It is known that the Holt graph is not the edge-graph of a half-transitive polytope.
\fi
\subsection{Distance-transitive polytopes}
\label{sec:distance_transitive}
Our previous results about edge-transitive polytopes already allow for a complete classification of a particular subclass, namely, the \emph{distance-transitive polytopes}, thereby also providing a list of examples of edge-transitive polytopes in higher dimensions.
The distance-transitive symmetry is usually only considered for graphs, and the distance-transitive graphs form a subclass of the distance-regular graphs.
The usual reference for these is the classic monograph by Brouwer, Cohen and Neumaier \cite{brouwer1989distance}.
For any two vertices $i,j\in V$ of a graph $G$, let $\dist(i,j)$ denote the graph-theoretic \emph{distance} between those vertices, that is, the length of the shortest path connecting them.
The \emph{diameter} $\diam(G)$ of $G$ is the largest distance between any two vertices in $G$.
\begin{definition}
A graph is called \emph{distance-transitive} if $\Aut(G)$ acts transitively on each of the sets
$$D_{\delta}:=\{(i,j)\in V\times V \mid \dist(i,j)=\delta\},\quad\text{for all $\delta\in\{0,...,\diam(G)\}$}.$$
Analogously, a polytope $P\subset\RR^d$ is said to be \emph{distance-transitive}, if its Euclidean symmetry group $\Aut(P)$ acts transitively on each of the sets
$$D_{\delta}:=\{(v_i,v_j)\in \F_0(P)\times \F_0(P) \mid \dist(i,j)=\delta\},\quad\text{for all $\delta\in\{0,...,\diam(G_P)\}$}.$$
Note that the distance between the vertices is still measured along the edge-graph rather than via the Euclidean distance.
\end{definition}
Being arc-transitive is equivalent to being transitive on the set $D_1$.
Hence, distance-transitivity implies arc-transitivity, thus edge-transitivity.
By our considerations in the previous sections, we know that the \mbox{classification} of distance-transitive polytopes is equivalent~to the classification of the $\theta_2$-spectral distance-transitive graphs.
Those where classified by Godsil (see \cref{res:spectral_distance_regular_graphs}).
In the following theorem we translated each such $\theta_2$-spectral distance-transitive graph into its respective eigenpolytope.
This gives a complete classification of the distance-transitive polytopes.
\begin{theorem}\label{res:distance_transitive_classification}
If $P\subset\RR^d$ is distance-transitive, then it is one of the following:
\begin{enumerate}[label=$(\text{\roman*}\,)$]
\item a regular polygon $(d=2)$,
\item the regular dodecahedron $(d=3)$,
\item the regular icosahedron $(d=3)$,
\item a cross-polytopes, that is, $\conv\{\pm e_1,...,\pm e_d\}$ where $\{e_1,...,e_d\}\subset\RR^d$ is the standard basis of $\RR^d$,
\item a hyper-simplex $\Delta(d,k)$, that is, the convex hull of all vectors $v\in\{0,1\}^{d+1}$ with exactly $k$ 1-entries,
\item a cartesian power of a regular simplex (also known as the Hamming polytopes; this includes regular simplices and hypercubes),
\item a demi-cube, that is, the convex hull of all vectors $v\in\{-1,1\}^d$ with~an~even number of 1-entries,
\item the $2_{21}$-polytope, also called Gosset-polytope $(d=6)$,
\item the $3_{21}$-polytope, also called Schläfli-polytope $(d=7)$.
\end{enumerate}
The ordering of the polytopes in this list agrees with the ordering of graphs in the list in \cref{res:spectral_distance_regular_graphs}.
The latter two polytopes where first constructed by Gosset in \cite{gosset1900regular}.
\end{theorem}
We observe that the list in \cref{res:distance_transitive_classification} contains many polytopes that are not regular, and contains all regular polytopes excluding the 4-dimensional exceptions, the 24-cell, 120-cell and 600-cell.
The distance-transitive polytopes thus form a distinct class of remarkably symmetric polytopes which is not immediately related to the class of regular polytopes.
Another noteworthy observation is that all the distance-transitive polytopes are \emph{Wythoffian poly\-topes}, that is, they are orbit polytopes of finite reflection groups.
\Cref{fig:distance_transitive_Coxeter} shows the Coxeter-Dynkin diagrams of these polytopes.
\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{img/distance_transitive_Coxeter}
\caption{Coxeter-Dynkin diagrams of distance-transitive polytopes.}
\label{fig:distance_transitive_Coxeter}
\end{figure}
\iffalse
Finally, we can show that \cref{res:distance_transitive_classification} is the final word on distance-transitive (actually distance-regular) graphs that are spectral: we can show that such a graph cannot be $\theta_k$-spectral for any $k\not=2$.
\begin{theorem}
\label{res:distance_regular_theta_k}
If $G$ is distance-transitive (actually distance-regular) and $\theta_k$-spectral, then $k=2$.
\end{theorem}
The proof uses some special properties of distance-transitive (actually distance-regular graphs) that we are citing in place.
In general, we recommend the chapters 11 and 13 of \cite{godsil1993algebraic}, the latter of which contains all the tools we need.
\begin{proof}[Proof of \cref{res:distance_regular_theta_k}]
The $\theta_1$-eigenpolytope of any graphs is a single point.
Since we generally assumed that our polytopes ar non-trivial, \shortStyle{i.e.,}\ of dimension $d\ge 2$, we from now on assume $k\ge 2$.
Let~$v_i\in$ $\RR^d$ be the coordinates assigned to the vertices $i\in V$ of $G$ as given~in~\cref{def:eigenpolytope}.
We know that the arrangement of the points $v_i$ in $\RR^d$ is as symmetric as the underlying graph, which in the case of distance-transitive graphs means that the value of $\<v_i,v_j\>$ depends only on the distance $\dist(i,j)$ (more generally, the same holds for distance-regular graphs).
We can then define the so-called \emph{cosine sequence} $u_\delta:=\<v_i,v_j\>$ whenever $\dist(i,j)$ $=\delta$.
A well-known result for distance-transitive (actually distance-regular) graphs is, that the sequence $u_0,...,u_{\diam(G)}$ has \emph{exactly} $k-1$ sign changes (see \cite{godsil1993algebraic}, Chapter 13, Lemma 2.1).
It remains to show, that if the $v_i$ embedd $G$ as the skeleton of a polytope, then there can be at most one signs change, hence $k=2$.
\begin{center}
\includegraphics[width=0.4\textwidth]{img/5_gon_cut}
\end{center}
This property is occasionally known as the \emph{tightness} of the skeleton of a polytope.
It says that the skeleton cannot be cut into more than two pieces by any hyperplane.
We give a formal proof below.
Suppose that there are at least two signs changes in $u_\delta$. This means, there exist $\delta>\delta'>0$ with $u_0u_{\delta'}<0$ and $u_{\delta'}u_{\delta}<0$, or equivalently $u_0 > u_\delta > 0 > u_{\delta'}$.
Suppose that the $v_i$ embedd $G$ as the skeleton of a polytope $P$.
Consider the linear optimization problem maximizing the linear functional $\<v_i,\free\>$ over $P$, which clearly attains its maximum in $v_i$.
Let $j\in V$ be a vertex with $\dist(i,j)=\delta$.
Applying the simplex algorithm yields a monotone path in the skeleton from $v_j$ to $v_i$.
But any such path must pass though a vertex $j'\in V$ with $\dist(i,j')=\delta'<\delta$. But we know that $\<v_i,v_{j'}\> = u_{\delta'} < 0 < u_{\delta}$.
So the path cannot have been monotone.
This is a contradiction, showing that there cannot have been two sign changes.
\end{proof}
Whether a similar argument is possible for larger classes of graphs is unknown.
\fi
\iffalse
\hrulefill
Let $G=(V,E)$ be a (simple, undirected) graph on the vertex set $V=\{1,...,n\}$.
From any eigenvalue $\theta\in\Spec(G)$ of $G$ (\shortStyle{i.e.,}\ of its adjacency matrix $A$) of multiplicity $d$, we can construct a so-called \emph{spectral realization} (or \emph{$\theta$-realization}) $v\: V\to\RR^d$ of $G$ as follows.
Let $\{u_1,...,u_d\}\subset\RR^n$ be an orthonormal basis of the $\theta$-eigenspace of $G$ and define the \emph{arrangement matrix} $M:=(u_1,...,u_d)\in\RR^{n\x d}$ in which the $u_i$ are the columns.
This matrix has one row per vertex, and so we can obtain a realization $v$ by assigning $i\in V$ to the $i$-th row of $M$:
$$M :=
\begin{pmatrix}
\;\rule[.5ex]{2.5ex}{0.4pt}\!\!\!\! & v_1^{\Tsymb} & \!\!\!\!\rule[.5ex]{2.5ex}{0.4pt}\;\; \\
& \vdots & \\[0.4ex]
\;\rule[.5ex]{2.5ex}{0.4pt}\!\!\!\! & v_n^{\Tsymb} & \!\!\!\!\rule[.5ex]{2.5ex}{0.4pt}\;\;
\end{pmatrix}\mathrlap{\in\RR^{n\times d}.}$$
The realization $V\ni i\mapsto v_i$ is called $\theta$-realization of $G$. The $\theta$-realization is unique up to re-orientation (depending only on the choice of the basis $\{u_1,...,u_d\}$).
\begin{definition}
The \emph{$\theta$-eigenpolytope} $P_G(\theta)$ of a graph $G$ is the convex hull of the $\theta$-realization $v$ of $G$:
$$P_G(\theta):=\conv\{v_i\mid i\in V\}.$$
\end{definition}
\fi
\iffalse
\section{The setting}
\label{sec:setting}
Throughout the paper, $P\subset\RR^d$ denotes a $d$-dimensional polytope of full dimen\-sion, that is, $P$ is the convex hull of finitely many points $v_1,...,v_n\in\RR^d$ (its \emph{vertices}), and $P$ is not contained in a proper affine subspace of $\RR^d$.
In particular, we assume the vertice to be labeled with the numbers $1$ to $n$.
By $\F_\delta(P)$, we denote the $\delta$-dimensional faces of $P$, in particular, its vertex set $\F_0(P)$, and its edge set $\F_1(P)$.
By $G_P=(V,E)$ we denote the \emph{edge-graph} of $P$, with vertex set $V=\{1,...,n\}$, so that $i\in V$ corresponds to the vertex $v_i\in \F_0(P)$, and each $ij\in E$ correspond to the edge $\conv\{v_i,v_j\}\in\F_1(P)$.
The term vertex (resp.\ edge) will be used for the elements of $V$ and $\F_0(P)$ (resp.\ $E$ and $\F_1(P)$) likewise, but no confusion is to be expected as these are in a one-to-one correspondence.
Finally, there are two matrices of major importance: for every polytope $P\subset\RR^d$ with labeled vertices $v_1,...,v_n$, the matrix
$$M :=
\begin{pmatrix}
\;\rule[.5ex]{2.5ex}{0.4pt}\!\!\!\! & v_1^{\Tsymb} & \!\!\!\!\rule[.5ex]{2.5ex}{0.4pt}\;\; \\
& \vdots & \\[0.4ex]
\;\rule[.5ex]{2.5ex}{0.4pt}\!\!\!\! & v_n^{\Tsymb} & \!\!\!\!\rule[.5ex]{2.5ex}{0.4pt}\;\;
\end{pmatrix}\mathrlap{\in\RR^{n\times d}}$$
is called \emph{arrangement matrix} of $P$ (the vertices $v_i$ are the rows of $M$).
The matrix $A\in\{0,1\}^{n\x n}$ with
$$A_{ij}=\begin{cases} 1 & \text{if $ij\in E$}\\0&\text{if $ij\not\in E$}\end{cases}$$
is the usual \emph{adjacency matrix} of $G_P$.
\section{Two open questions}
Most of this book talks about the relation between polytopes and their edge-graphs.
It is well-known, that not every graphs is the edge-graph of polytope.
Since Steinitz, we know that the edge-graphs of polyhedra (3-dimensional polytopes) are exactly the 3-connected planar graphs, and each such graph determines a single combinatorial type of polytopes.
In contrast, no characterization of polytopes in dimension $d\ge 4$ is known, and even worse, if a graph is an edge-graph, then in general, it tells us little about the polytope.
That means, there can be many combinatorially distinct polytopes with the same edge-graph, even of the same dimension.
As a light in the dark, we have the result of Blind \& Mani (later refined by Kalai), that every \emph{simple} polytope (that is, a $d$-dimensional polytope whose edge-graph is of degree $d$) is uniquely determined by its edge-graph.
Similar results are known for other classes of polytopes, \shortStyle{e.g.}\ zonotopes.
In this paper, we add another class of polyopes to that list: \emph{spectral polytopes}.
Given the definition of spectral polytope, this is not too surprising (it sounds like an algorithm to compute the polytope from its edge-graph), but the interesting bit is in the graphs that turn out to be spectral and which would not have been expected to be uniquely determined otherwise.
For example, consider highly symmetric polytopes.
Each regular polytopes is uniquely determined (among regular polytopes) by itd edge-graph. In other words, there are no two regular polytopes with the same edge-graph.
On the other hand, there are many distinct vertex-transitive polytopes with the same edge-graph (\shortStyle{e.g.}\ any two vertex-transitive $n$-vertex neighborly polytopes).
So regularity is too strong to be interesting, but vertex-transitivity is too weak.
In this paper, we show that edge-transitivity also already gives unique reconstruction from the edge-graph, since they are spectral.
\fi
\iffalse
\section{Balanced and spectral polytopes}
\label{sec:balanced}
We now come back to the curious observations, that sometimes, a polytope is the eigenpolytope of its edge-graph.
A polytope with this property will be called \emph{spectral}.
However, for this section, we decided to go a different path, and introduce spectral polytopes from a different perspective: via balanced polytopes.
The overall goal of this section is to investigate how to can find out whether a given polytope already is spectral.
The precursor to spectral polytopes are the balanced polytopes that we introduce now.
\begin{definition}
$P\subset\RR^d$ is called \emph{$\theta$-balanced} (or just \emph{balanced})~for~a~num\-ber $\theta\in\RR$ if
\begin{equation}\label{eq:balanced}
\sum_{\mathclap{j\in N(i)}} v_j=\theta v_i,\quad \text{for all $i\in V$}.
\end{equation}
The value $\theta$ is actually an eigenvalue of $G$ (see \cref{res:eigenvalue} below).
If the~multi\-plicity of this eigenvalue $\theta$ is exactly $d$, then $P$ is called \emph{$\theta$-spectral} (or just \emph{spectral}).
\end{definition}
\begin{example}
All neighborly polytopes are balanced (if the barycenter of its vertices is the origin).
We can see this as follows:
$$\sum_{\mathclap{j\in N(i)}} v_j = -v_i + \underbrace{\sum_{\mathclap{j\in V}} v_j}_{=0} = -v_i.$$
Note that the spectrum of $K_n$ is $\{(-1)^{n-1},(n-1)^1\}$, and so $-1$ is indeed the second largest eigenvalue.
\end{example}
\begin{example}
All regular polytope are $\theta_2$-spectral.
This was first shown by Licata and Powers \cite{licata1986surprising} for all regular polytopes excluding the 4-dimensional exceptions (the 24-cell, 120-cell and 600-cell).
For the remaining polytopes this was proven in \cite{winter...} (Theorem ??).
Alternatively, the same follows from \cref{res:edge_vertex_transitive}.
Other polytopes that were recognized as spectral by Licata and Powers were the
\emph{rhombic dodecahedron}, ... \msays{TODO}
\end{example}
\begin{example}
Probably all uniform polytopes have a realization that is spectral.
\end{example}
Note that all these polytopes are spectral to the \emph{second-largest eigenvalue $\theta_2$}.
This is plausible for different reasons, but no proof of this fact is currently known.
\begin{observation}
Nodal domains ...
\end{observation}
Being balanced can be interpreted in at least two different way.
One of which is in terms of rigidity theory (and applies more generally to graph realizations rather than just to skeleta of polytopes).
The other is purely in terms of spectral graph theory.
\begin{remark}
Being $\theta$-balanced can also be interpreted in the context of rigidity theory.
For an edge $ij\in E$, the vector $v_j-v_i$ is pointing from the vertex $v_i$ along the edge $ij$ and can be interpreted as a force pulling $v_i$ along this edge.
At each vertex, the sum of these forces behaves as follows:
\begin{equation}\label{eq:self_stress}
\sum_{\mathclap{j\in N(i)}} (v_j-v_i) = \sum_{\mathclap{j\in N(i)}} v_j - \deg(G)v_i =\big(\theta-\deg(G)\big) v_i.
\end{equation}
In the language of rigidity theory: if we put the ...
Equation \eqref{eq:self_stress} can also be written as $LM=\theta M$, where $L$ is the Laplacian of $G$.
\end{remark}
\begin{remark}
Let us call a graph \emph{$\theta$-spectral} if it is the edge-graph of a $\theta$-spectral polytope.
We can characterize such graphs purely in the language of spectral graph theory.
A graph is $\theta$-spectral if $i,j\in V$ are adjacent in $G$ if and only if there exists a $\theta$-eigenvector $u\in\RR^n$ that is maximized exactly on $i$ and $j$:
$$\Argmax_{k\in V} u_k = \{i,j\}.$$
\end{remark}
Now the central observation concerning balanced polytopes, paving the way to spectral polytopes, is the following:
\begin{observation}\label{res:eigenvalue}
The defining equation \eqref{eq:balanced} can be written as $AM=\theta M$, where $A\in\RR^{n\x n}$ (where the matrices $A$ and $M$ are defined as in \cref{sec:setting}).
In this form, it is apparent that $\theta$ is an eigenvalue of $A$, and the columns of $M$ are the corresponding eigenvectors.
\end{observation}
Since $P$ is assumed to have full dimension, the matrix $M$ has rank $d$, \shortStyle{i.e.,}\ its columns are linearly independent eigenvectors.
But they not necessarily span the whole eigenspace.
If they do, this is called a spectral polytope:
\begin{definition}
A $\theta$-balanced polytope $P\subset\RR^d$ is called \emph{$\theta$-spectral} (or just \emph{spectral}), if any of the following equivalent conditions is satisfied:
\begin{myenumerate}
\item $\theta\in\Spec(G_P)$ has multiplicity $d$ (the dimension of $P$),
\item the columns of $M$ span the $\theta$-eigenspace of $G_P$.
\end{myenumerate}
\end{definition}
\subsection{Applications}
There are at least two interesting facts about spectral polytopes.
\begin{corollary}
A spectral polytope realizes all the symmetries of its edge-graph.
\end{corollary}
\begin{corollary}
A $\theta_2$-spectral polytope is uniquely determined by its edge-graph (among $\theta_2$-spectral polytopes).
\end{corollary}
Of course, $\theta_2$ can be replaced by any $\theta_i$.
\fi
\section{Implementation in Mathematica}
\label{sec:appendix_mathematica}
The following short Mathematica script takes as input a graph $G$ (in the example below, this is the edge-graph of the dodecahedron), and an index $k$ of an eigenvalue.
It then compute the $v_i$ (or \texttt{vert} in the code), \shortStyle{i.e.,}\ the vertex-coordinates of the $\theta_k$-eigenpolytope.
If the dimension turns out to be appropriate, the spectral embedding of the graph, as well as the eigenpolytope are plotted.
\vspace{0.5em}
\begin{lstlisting}
(* Input:
* the graph G, and
* the index k of an eigenvalue (k = 1 being the largest eigenvalue).
*)
G = GraphData["DodecahedralGraph"];
k = 2;
(* Computation of vertex coordinates 'vert' *)
n = VertexCount[G];
A = AdjacencyMatrix[G];
eval = Tally[Sort@Eigenvalues[A//N], Round[#1-#2,0.00001]==0 &];
d = eval[[-k,2]]; (* dimension of the eigenpolytope *)
vert = Transpose@Orthogonalize@
NullSpace[eval[[-k,1]] * IdentityMatrix[n] - A];
(* Output:
* the graph G,
* its eigenvalues with multiplicities,
* the spectral embedding, and
* its convex hull (the eigenpolytope).
*)
G
Grid[Join[{{$\theta$,"mult"}}, eval], Frame$\to$All]
Which[
d<2 , Print["Dimension too low, no plot generated."],
d==2, GraphPlot[G, VertexCoordinates$\to$vert],
d==3, GraphPlot3D[G, VertexCoordinates$\to$vert,
d>3 , Print["Dimension too high, 3-dimensional projection is plotted."];
GraphPlot3D[G, VertexCoordinates$\to$vert[[;;,1;;3]] ]
]
If[d==2 || d==3,
Region`Mesh`MergeCells[ConvexHullMesh[vert]]
]
\end{lstlisting}
\section{Motivation}
\label{sec:motivation}
We dive into three curious topics in the interplay between graphs and polytopes, all of which will be addressed in this paper.
\subsection{Reconstructing polytopes from the edge-graph}
\label{sec:reconstructing_polytopes}
Since Steinitz \cite{...} it is known that the edge-graphs of polyhedra are exactly the 3-connected planar graphs, and that any such graph belongs to a unique combinatorial type of polyhedra.
On the other hand, the study of higher-dimensional polytopes taught us that the edge-graph carries very little information about its polytope if $d\ge 4$.
For example, the edge-graph of any simplex is the complete graph.
But so is the edge-graph of every \emph{neighborly polytope}, and so, given only the edge-graph, we will never know from which polytope it was obtained initially.
The same example shows that in many cases not even the dimension of the polytope can be reconstructed.
As a light in the dark, we have the result of Blind and Mani \cite{...} (with its modern proof from Kalai \cite{kalai1988simple}), which states that the combinatorial type of a \emph{simple polytope} can be reconstructed from its edge-graph.
One can then ask what other classes of polytopes allow such a unique reconstruction.
For example, it might be that every \enquote{sufficiently symmetric} polytope is reconstructable in this sense.
\begin{question}\label{q:symmetrc_polytopes_with_same_edge_graph}
Can there be two distinct sufficiently symmetric polytopes with the same edge-graph?
\end{question}
One might ask \enquote{what counts as sufficient symmetric?}, and this paper shall answer this question partially.
For example, vertex-transitivity is not sufficient:
every~even-dimen\-sional \emph{cyclic polytope} $C(n,2d)$ is neighborly and can be realized in a vertex-transitive way: set the coordinates of the $i$-th vertex to be
$$\big(\sin(t_i),\cos(t_i);\sin(2t_i),\cos(2t_i);...;\sin(dt_i),\cos(dt_i)\big)\in\RR^{2d}$$
with $t_i=2\pi i/n$. The edge-graph is $K_n$, independent of $d$.
The next candidate in line, \emph{edge-transitivity}, seems more promising.
And while a complete classification of edge-transitive polytopes might be possible (though, none is currently known), a more satisfying outcome would be a result independent of any such classification.
\subsection{Realizing symmetries of the edge-graph}
\label{sec:realizing_symmetries}
Polytopes are at most as symmetric as there edge-graph.
Any symmetry $T\in\Aut(P)\subset\Ortho(\RR^d)$ of a polytope $P\subset\RR^d$ induces a symmetry $\phi\in\Aut(G_P)\subseteq\Sym(n)$ of the edge-graph.
And thus $\Aut(P)\le\Aut(G_P)$ as abstract groups.
For general polytopes, it cannot be expected that the polytope shows many, or just any symmetries of the edge-graph.
However, one can always try to find a sufficiently symmetric realization of the polytope, so that a maximum of combinatorial symmetries of the edge-graph is realized as geometric symmetries of the polytope.
One can even hope to realize \emph{all} the symmetries of the edge-graph.
In the case of polyhedra, the circle packing version of the Steinitz theorem provides a way to realize all the symmetries of the edge-graph.
Again, higher dimensions tell a different story: there exist 4-dimensional polytopes with edge-graphs that can never be realized as polytopes the edge-graphes of which cannot be realized with all their symmeteries (consider \shortStyle{e.g.}\ the dual of the polytope constructed in \cite{bokowski1984combinatorial}).
Again, it seems that assuming sufficient symmetry of the polytope ensures that the polytope has already \emph{all} the symmetries of its edge-graph.
We can actually proof this in the case of so called \emph{distance-transitive} symmetry, and we hope to find the same in the case of only edge-transitive polytopes.
\begin{question}
Does every sufficiently symmetric polytope realize all the symmetries of its edge-graph?
\end{question}
\subsection{Rigidity and perfect polytopes}
\label{sec:rigid_perfect}
Every polytope can be continuously deformed while preserving its combinatorial type.
In the case of highly symmetric polytopes, one might be additionally interesting in keeping the symmetry properties during the deformation.
A polytope that cannot be deformed without loosing any of its symmetries is be called \emph{rigid} or, occasionally, a \emph{perfect polytope}.
All regular polytopes are examples of such rigid polytopes, and there exist many more.
While quite some examples are known, no general construction procedures has been described.
Again, one might hope to show that all sufficiently symmetric polytopes are rigid, in particular, the edge-transitive polytopes.
\begin{question}
Is every sufficiently symmetric polytope rigid?
\end{question}
\subsection{A motivating example}
\label{sec:motivating_example}
Consider your favorite highly symmetric polytope $P\subset\RR^d$.
For demonstration, we choose $P$ to be the \emph{regular dodecahedron}\footnote{Any other platonic solid (or regular polytope) might do the trick. Ideally, your polytope is transitive on vertices and edges.}
Here are some observations:
\begin{myenumerate}
\item $P$ is really symmetric (it is a regular polyhedron). All its symmetries must also be reflected in the edge-graph $G_P$, since any automorphism of $P$ restricts to an automorphism of $G_P$.
However, it is far from evident whether any particular symmetry of $G_P$ is reflected in $P$ as well?
In the case of the dodecahedron, every combinatorial symmetry of the edge-graph is indeed a geomery symmetry of $P$.
\item The study of higher dimensional polytopes teaches that the edge-graph of a polytope carries usually very little information about the polytope itself.
For example, the edge-graph of a simplex is a complete graph.
But so are the edge-graph of all \emph{neighborly polytopes}.
It was shown in \cite{kalai1988simple} that this problem of reconstruction does not happen for \emph{simple polytopes}, and it is our goal to show that this is also not to be expected for sufficiently symmetric polytope.
\item A polytope is called \emph{perfect} if its combinatorial type has a unique geometric realization of highest symmetry.
Many highly symmetric polytopes one immediately thinks about have this property. But many other do not.
We show that being edge-transitive seems to be the right assumption to ensure being perfect.
\end{myenumerate}
\section{Eigenpolytopes and spectral polytopes}
One way to connect spectral graph theory to polytope theory is via the construction of the so-called \emph{eigenpolytopes}.
In this section, we will survey this construction and its application primarily to the study of highly symmetric graphs.
From this section on, let $P\subset\RR^d$ denote a $d$-dimensio\-nal convex polytope with $d\ge 2$.
If not stated otherwise, we shall~assume that $P$ is \emph{full-dimensional}, that is, that $P$ is not contained in any proper subspace of $\RR^d$.
Let $\F_\delta(P)$ denote the set of \emph{$\delta$-dimensional faces} of $P$.
Fix some bijection $v\: V\to \F_0(P)$ between $V=\{1,...,n\}$ and the vertices of $P$.
Let $G_P=(V,E)$ be the graph with $ij\in E$ if and only if $\conv\{v_i,v_j\}$ is an edge of $P$.
$G_P$ is called \emph{edge-graph} of $P$, and $v$ the \emph{skeleton} of $P$.
Note that we can consider $v$ as a realization of $G_P$.
\subsection{Eigenpolytopes}
\begin{definition}
Given a graph $G$ and an eigenvalue $\theta\in \Spec(G)$ of multiplicity $d$. If $v$ is the $\theta$-realization of $G$, then
%
$$P_G(\theta):=\conv\{v_i\mid i\in V\}\subset\RR^d$$
%
is called \emph{$\theta$-eigenpolytope} of $G$, or just \emph{eigenpolytope} if $\theta$ is not specified.
\end{definition}
The notion of the eigenpolytope goes back to Godsil \cite{godsil1978graphs}, and can~be under\-stood~as a geometric way to study the combinatorial, algebraic~and~spectral properties of a graph.
Several authors investigated the eigenpolytopes of famous graphs or graph families.
For example, Powers \cite{powers1986petersen} gave a description for the face structure of~the~different eigenpolytopes of the Petersen graph, one of which will reappear as distance-transitive polytope in \cref{sec:distance_transitive_polytopes}.
Mohri \cite{mohri1997theta_1} analysed the geometry of the, what he called, \emph{Hamming polytopes}, the $\theta_2$-eigenpolytopes of the Hamming graphs.
As it turns out, but was not realized by himself, these polytopes are merely the cartesian products of regular simplices.
These, as well, are distance-transitive~\mbox{polytopes}. Powers \cite{powers1988eigenvectors} took a first look at the eigenpolytopes of the \emph{distance-regular} graphs.
Those especially will be discussed in greater detail below.
Finally, Padrol and Pfeifle \cite{padrol2010graph} investigated graph operations and their effect on the associated eigenpolytopes.
Before we proceed, we recall the relevant terminology for polytopes.
\subsection{Spectral polytopes}
Most graphs are not edge-graphs of polytopes (so-called \emph{polytopal} graphs), but eigenpolytope provide an operation that to every graph (and an eigenvalue thereof) assignes such a polytopal graph (the edge-graph of the eigenpolytope).
This operation is most interesting in the case of vertex-transitive graphs.
If $G$ is a vertex-transitive graph on $n$ vertices, every vertex of $G$ becomes a vertex of $P_G(\theta)$ (though maybe several vertices are assgined to the same point).
It seems interesting to ask for when this operation is a fixed point, that is, when is the edge-graph of the eigenpolytope isomorphic to the original graph.
We are specifically interested in one particular phenomenon associated with eigenpolytopes, that can best be observed in our running example.
\begin{definition}
\quad
\begin{myenumerate}
\item A polytope $P$ is called \emph{balanced} resp.\ \emph{spectral}, if this is the case for its skeleton.
\item A graph is called \emph{spectral} if it is the edge-graph of a spectral polytope.
\end{myenumerate}
\end{definition}
For a polytope, being spectral seems to be a quite exceptional property.
For example, being spectral implies that the polytope realizes all the symmetries of its edge-graph.
So, no neighborly polytope except for the simplex can be spectral.
In contrast, all (centered) neighborly polytopes are balanced, and many neighborly polytopes are eigenpolytopes (\shortStyle{e.g.}\ one of the Petersen-polytopes is neighborly).
If a graph is spectral to eigenvalue $\theta$, then it is isomorphic to the edge-graph of its $\theta$-eigenpolytop.
However, the converse is not true: the $\theta_3$-realization of the pentagon-graph is a pentagram, whose convex hull is a pentagon. But the pentagon-graph is not $\theta_3$-spectral.
\begin{center}
\includegraphics[width=0.7\textwidth]{img/convex_hull_5_gon}
\end{center}
\begin{example}
Every neighborly polytope has eigenvalue $-1$ (if centered appropriately), but none of these is spectral, except for the simplices.
\end{example}
\begin{example}
It is an easy exercise to show that the $d$-cubes and $d$-crosspolytopes are spectral with eigenvalue $\theta_2$.
In fact, it was shows by Licata and Power \cite{licata1986surprising} that also the regular polygons, all simplices, the regular dodecahedron and the regular icosahedron are spectral in this sense.
The only regular polytopes which they were not able to prove spectral were the 24-cell, 120-cell and 600-cell.
However, they conducted numerical experiments hinting in that direction.
\end{example}
\msays{The $\theta_3$-realization of the cube-graph is a polytope (the tetrahedron) and its edges correspond to edges of the graph, but it is not faithful.}
In a slightly more general setting, Izmestiev \cite{izmestiev2010colin} found that every polytope is~spectral, if we are allowed to add appropriate edge-weights to the edge-graph.
This~results in an adjacency matrix whose entries can be any non-negative real number instead of just zero and one.
This however might break the symmetry in a combinatorially highly symmetric graph and is therefore not our preferred perspective.
First results on this topic were obtained by Licata and Powers \cite{licata1986surprising}, who indentified several polytopes as being spectral.
Among these, all regular polytopes, excluding the exceptional 4-dimensional regular polytopes: 24-cell, 120-cell and 600-cell.
Still kind of \enquote{unexplained} is their finding that the \emph{rhombic dodecahedron} is spectral as well.
Since the edge graph of this polyhedron is not regular, this observation hinges critically on the fact that we consider eigenpolytopes constructed from the adjacency matrix, rather than, say, the Laplace matrix.
Godsil \cite{godsil1998eigenpolytopes} classified all spectral graphs among the distance-regular graphs (for a definition, see \cref{sec:distance_transitive_realizations}).
He found that these are the following:
\begin{theorem}[Godsil, \cite{...}]
\label{res:spectral_distance_regular_graphs}
Let $G$ be a distance-regular graph.
If $G$ is $\theta_2$-spectral,~then $G$ is one of the following:
\begin{enumerate}[label=$(\text{\roman*}\,)$]
\item a cycle graph $C_n,n\ge 3$,
\item the edge-graph of the dodecahedron,
\item the edge-graph of the icosahedron,
\item the complement of a disjoint union of edges,
\item a Johnson graph $J(n,k)$,
\item a Hamming graph $H(d,q)$,
\item a halved $n$-cube $\nicefrac12 Q_n$,
\item the Schläfli graph, or
\item the Gosset graph.
\end{enumerate}
\end{theorem}
One remarkable fact about this list, which seems to have gone unnoticed, is that all these graphs are actually \emph{distance-transitive}.
In general there was not found a single polytope (or graph) that is $\theta_i$-spectral for any $i\not=2$ (assuming that the graph has at least one edge).
It is not implausible to conjecture that skeleta of polytopes can have, if at all, only eigenvalue $\theta_2$.
This comes from the study of so-called \emph{nodal domains}.
There is a remarkable construction due to Izmestiev \cite{izmestiev2010colin}.
\begin{theorem}[Izmestiev \cite{izmestiev2010colin}, Theorem 2.4]\label{res:izmestiev}
Let $P\subset\RR^d$ be a polytope, $G_P$ its~edge-graph and $v\:V\to\RR^d$ its skeleton. For $c=(c_1,...,c_n)\in\RR^n$ define
$$P^\circ(c):=\{x\in\RR^d\mid\<x,v_i\>\le c_i\text{ for all $i\in V$}\}.$$
Note that $P^\circ=P^\circ(1,...,1)$ is the polar dual of $P$.
The matrix $X\in\RR^{n\x n}$ with~compo\-nents
$$X_{ij}:=-\frac{\partial^2 \vol(P^\circ(c))}{\partial c_i\partial c_j}\Big|_{c=(1,...,1)}$$
has the following properties:
\begin{myenumerate}
%
\item $X_{ij}< 0$ whenever $ij\in E(G_P)$,
\item $X_{ij}=0$ whenever $ij\not\in E(G_P)$,
\item $XM=0$, where $M$ is the arrangement matrix of $v$,
\item $X$ has a unique negative eigenvalue, and this eigenvalue is simple,
\item $\dim\ker X=d$.
\end{myenumerate}
\end{theorem}
One can view the matrix $X$ (or better, $-X$) as kind of an adjacency matrix of~a vertex- and edge-weighted version of $G_P$.
Part $(iii)$ states that $v$ is balanced~with eigenvalue zero.
Since $\rank M=d$, part $(v)$ states that the arrangement space~$U:=$ $\Span M$ is already the whole eigenspace.
And part $(iv)$ states that zero is the second largest eigenvalue of $-X$.
\begin{theorem}\label{res:implies_spectral}
Let $P\subset\RR^d$ be a polytope, and $X\in\RR^{n\x n}$ the associated matrix~defined as in \cref{res:izmestiev}. If we have that
\begin{myenumerate}
\item $X_{ii}$ is independent of $i\in V(G_P)$, and
\item $X_{ij}$ is independent of $ij\in E(G_P)$,
\end{myenumerate}
then $P$ is spectral with eigenvalue $\theta_2$.
\begin{proof}
By assumption there are $\alpha,\beta\in\RR$, $\beta>0$, so that $X_{ii}=\alpha$ for all vertices~$i\in$ $V(G_P)$, and $X_{ij}=-\beta$ for all edges $ij\in E(G_P)$.
We then find
$$X=\alpha \Id - \beta A\quad\implies\quad (*)\;A=\frac\alpha\beta \Id-\frac1\beta X,$$
where $A$ is the adjacency matrix of $G_P$.
By \cref{res:izmestiev} $(iv)$ and $(v)$, the matrix $X$ has second smallest eigenvalue zero of multiplicity $d$.
By \cref{res:izmestiev} $(iii)$, the columns of $M$ are the corresponding eigenvectors.
Since $\rank M=d$ we find that these are all the eigenvectors, in particular, that the arrangement space $U:=\Span M$ is the 0-eigenspace of $X$.
By $(*)$, $A$ (that is, $G_P$) has then second \emph{largest} eigenvalue $\theta_2=\alpha/\beta$ of multiplicity $d$ with eigenspace $U$.
The skeleton of $P$ is then the $\theta_2$-realization of $G_P$ and $P$ is spectral.
\end{proof}
\end{theorem}
\begin{corollary}
If $P\subset\RR^d$ is simultaneously vertex- and edge-transitive, then $P$ is spectral with eigenvalue $\theta_2$.
\end{corollary}
It seems tempting to conjecture that the spectral polytopes are \emph{exactly} the polytopes that satisfy the conditions $(i)$ and $(ii)$ in \cref{res:implies_spectral}.
This would give an interesting geometric characterization of eigenpolytopes.
Izmestiev proved the following:
for $ij\in E$ let $\sigma_i,\sigma_j\in\F_{d-1}(P^\circ)$ be the dual facets to the vertices $v_i$ and $v_j$ in $P$. Then holds
$$X_{ij}= \frac{\vol_{d-2}(\sigma_i\cap\sigma_j)}{\|v_i\|\|v_j\|\sin\angle(v_i,v_j)}.$$
There seems to be no equally elementary formula for $X_{ii},i\in V$, but its value can be derived from $XM=0$, which implies
$$X_{ii} v_i = -\sum_{\mathclap{j\in N(i)}} M_{ij} v_j.$$
It seems out of reach to prove this fact.
There are many balanced polytopes that are not spectral (\shortStyle{e.g.}\ all appropriately centered neighborly polytopes that are not simplices).
Further, there are polytope for which being spectral appears completely exceptional, \shortStyle{e.g.}\ the \emph{rhombic dodecahedron} (already mentioned by Licata and Powers \cite{licata1986surprising}).
\subsection{Applications}
Spectral polytopes turn out to be exactly the kind of polytopes we need to address the problems listed in \cref{sec:motivation}.
For example, for fixed $i\in\NN$ consider the set of all $\theta_i$-spectral polytopes.
Given the edge-graph $G_P$ of some polytope in this set, this uniquely determines the polytope, simply since it can be reconstruced via the $\theta_i$-realizaion of $G_P$.
In other~words, the $\theta_i$-spectral polytopes are uniquely determined by their edge-graph.
Also, each $\theta_i$-spectral polytope realizes all the symmetries of its edge-graph by \cref{res:spectral_implies_symmetric}.
The question then is how large the class of these polytopes is, and whether their members are easily recognized.
For example, has every family of vertex-transitive polytopes with the same edge graph a spectral representative?
If so, this would imply that every vertex-transitive edge-graph can be realized as a polytope with all symmetries.
\section{Edge-transitive polytopes}
\subsection{Arc-transitive polytopes}
\label{sec:arc_transitive_polytopes}
Our final goal would be to establish that every arc-transitive polytope is spectral, or more precisely, $\theta_2$-spectral.
To anticipate~
the outcome, we were not able to prove this.
However, we also observe that this statement is true for \emph{all} known arc-transitive polytopes.
In the last section we have learned that the skeleton of an arc-transitive polytope is of full local dimension, hence irreducible, spherical, rigid and has an eigenvalue.
\begin{definition}
Given a polytope $P\subset\RR^d$.
\begin{myenumerate}
\item $P$ is called \emph{rigid} if it cannot be continuously deformed without changing its combinatorial type or symmetry group.
\item $P$ is called \emph{perfect} if any essentially distinct but combinatorially equivalent polytope $Q\subset\RR^d$ has a smaller symmetry group than $P$, that is $\Aut(Q)\subset\Aut(P)$.
\end{myenumerate}
\end{definition}
\enquote{Essentially distinct} in $(ii)$ means that $P$ and $Q$ are not just reoriented or~rescaled versions of one another.
If a polytope is perfect, then it is rigid.
It seems to be open whether the converse holds in general.
In theory it could happen that a combinatorial type might have two \enquote{most symmetric} realizations as a polytope, and those cannot be continuously deformed into each other.
\begin{theorem}
An arc-transitive polytope $P\subset\RR^d$ is $\theta_2$-spectral.
\begin{proof}
The skeleton of $P$ is an arc-transitive realization of $G_P$ of full local dimension, hence has an eigenvalue $\theta\in\Spec(G)$ according to \cref{res:...}.
The tricky part is to show that $\theta=\theta_2$.
For this, we use the following fact:
Let $v$ be the skeleton of $P$ and define
$$P^\circ(\mathbf c) = P^\circ(c_1,...,c_n):=\{x\in\RR^d\mid \<x,v_i\>\le c_i\text{ for all $i\in V$}\}.$$
Finally, consider the matrix
$$M_{ij}:=-\frac{\partial^2\mathrm{vol}(P(\mathbf c))}{\partial c_i \partial c_j}\Big|_{\mathbf c=(1,...,1)}.$$
Since $P$ is edge-transitive, $M_{ij}$ contains the same value for all $ij\in E$.
Since $P$ is vertex-transitive, $M_{ii}$ does not depend in $i\in V$.
Finally, it is proven in \cite{...} that $M_{ij}=0$ whenever $ij\not\in E$, and that $M$ has a single negative eigenvalue which is simple.
Apparently, there are $\alpha,\beta\in\RR$ so that $M=\alpha A+\beta\Id$, where $A$ is the adjacency matrix of $M$.
\end{proof}
\end{theorem}
\begin{theorem}
If $P\subset\RR^d$ is an arc-transitive polytope, then
\begin{myenumerate}
\item $P$ is rigid,
\item $\Aut(P)$ is irreducible,
\item any projection $\pi_U(P)$ onto a subspace $U\subseteq\RR^d$ is either not arc-transitive~or has a different edge-graph as $P$.
\end{myenumerate}
Furthermore, there is an eigenvalue $\theta\in\Spec(G_P)$ of the edge-graph $G_P$ resp.\ its~associated Laplacian eigenvalue $\lambda=\theta-\deg(G_P)$, so that the following holds:
\begin{myenumerate}
\setcounter{enumi}{3}
\item if $P$ has edge length $\ell$ and circumradius $r$, then
%
$$\frac\ell r=\sqrt{\frac{2\lambda}{\deg(G_P)}}.$$
\item if the polar polytope $P^\circ$ has dihedral angle $\alpha$, then
%
$$\cos\alpha=-\frac{\theta}{\deg(G_P)}.$$
\end{myenumerate}
\end{theorem}
\begin{conjecture}\label{conj:arc_transitve}
Each arc-transitive polytope $P\subset\RR^d$ is $\theta_2$-spectral, and satisfies
\begin{myenumerate}
\item $P$ is perfect,
\item $P$ realizes all the symmetries of its edge-graph,
\item $P$ is the only arc-transitive polytope (up to reorientation and rescaling) with edge-graph $G_P$.
\end{myenumerate}
\end{conjecture}
Probably in, in all these result, the eigenvalue $\theta$ resp.\ $\lambda$ can be replaced with $\theta_2$ resp.\ $\lambda_2$, but we were not able to prove that.
However, it holds in any single case of arc-transitive polytope that we have checked.
\begin{example}
There are many ways to compute the circumradius of an $n$-simplex with unit edge-length, and here is another one:
The complete graph $K_n$ is arc-transitive and has spectrum $\{(-1)^{n-1},n^1\}$. The second largest eigenvalue $\theta_2=-1$ gives rise to an $(n-1)$-dimensional polytope with edge-graph $K_n$, that is, the $(n-1)$-dimensional simplex.
Assuming edge length $\ell=1$, we can compute its circumradius:
$$r=\sqrt{\frac{\deg(G)}{\lambda_2}} = \sqrt{\frac{\deg(G)}{\deg(G)-\theta_2}} = \sqrt{\frac{n-1}{n-1-(-1)}} = \sqrt{1-\frac1n}.$$
\end{example}
\begin{example}
Compute the dihedral angle of the rhombic dodecahedron and the rhombic triacontahedron.
\end{example}
But this technique can also be used the other way around, namely, concluding from the knowledge of the metric properties of a polytope and the eigenvalues of its edge-graph to the fact that it is spectral.
\begin{example}
We show that all regular polytopes are in fact $\theta_2$-spectral, as was already conjectured by Licata and Powers \cite{licata1986surprising}.
The essential facts we are using are some tabulated metric and spectral properties of these polytopes, \shortStyle{e.g.}\ found in \cite{buekenhout1998number}.
\begin{itemize}
\item
Remarkable, the 24-cell has edge-length $\ell$ identical to circumradius $r$.
This determines the Laplacian eigenvalue of its skeleton:
$$1=\frac\ell r=\sqrt{\frac{2\lambda}{\deg(G)}} \quad\implies\quad \lambda=\frac12\deg(G) = 4.$$
One checks that $4$ is exactly the second largest Laplacian eigenvalue of its edge-graph, and that this eigenvalue has multiplicity four.
\item
600-cell: $3(1+\sqrt{5})$
\item
120-cell: $2\phi-1$
\end{itemize}
\begin{center}
\begin{tabular}{c|rrrr}
polytope & $\ell$ & $r$ & $\lambda_2$ & $\deg(G_P)$ \\
\hline
$\overset{\phantom.}24$-cell & $1$ & $1$ & $4$ & $8$ \\
$120$-cell & $3-\sqrt 5$ & $2\sqrt 2$ & $2\phi-1$ & $4$ \\
$600$-cell & $2/\phi$ & $2$ & $3(1+\sqrt 5)$ & $12$
\end{tabular}
\end{center}
\end{example}
\begin{theorem}
If a polytope has a connected, full-dimensional and spanning edge-orbit, then it is perfect.
\end{theorem}
\newpage
\section{Edge- and vertex-transitive polytopes}
The goal of this section is to show that every polytope $P\subset\RR^d$ that is simultane\-ously vertex- and edge-transitive is an eigenpolytope to the second smallest eigenvalue of its edge-graph.
Note that being only edge-transitive is just slightly weaker than being vertex- and edge-transitive, as it was shown in \cite{winter2020polytopes} that edge-transitivity of convex polytopes in dimension $d\ge 4$ implies vertex-transitivity anyway.
\begin{theorem}\label{res:vertex_edge_transitive_is_theta2_realization}
If $P\subset\RR^d$ is vertex-transitive and edge-transitive, then it is the~$\theta_2$-eigenpolytopes of its edge-graph.
\begin{proof}
Let $X\in\RR^{n\x n}$ be the matrix defined in \cref{res:izmestiev}.
Since $P$ is transitive on vertices, there is a number $\alpha\in\RR$ with $X_{ii}=\alpha$ for all $i\in V$.
Since $P$ is edge-transitive, there is a number $\beta<0$ so that $X_{ij}=\beta$ for all $ij\in E(G_P)$.
Since we also have $X_{ij}=0$ for all $ij\not\in E(G_P)$, we find that $X$ must be of the form
$$X= \alpha \Id - \beta A \quad\implies\quad A=\alpha\Id-\frac1\beta X,$$
where $A$ is the adjacency matrix of $G_P$.
$X$ has a single negative eigenvalue, which is simple, and $\dim\ker X=d$.
In other words, its second smallest eigenvalue is zero and has multiplicity $d$.
Consequently, $A$ has second \emph{largest} eigenvalue $\theta_2=\alpha$ of multiplicity $d$ to the same eigenvectors as $X$.
Let $v$ be the skeleton of $P$ with arranegment matrix $M$ and arrangement space $U$.
Since $XM=0$ and $\rank M=d$, we find that the $U=\Span M$ must be the $0$-eigenspace of $X$, hence the $\theta_2$-eigenspace of $A$.
Thus, $v$ is the $\theta_2$-realization of $G_P$ and $P$ is spectral.
\end{proof}
\end{theorem}
The argument of \cref{res:vertex_edge_transitive_is_theta2_realization} cannot be easily generalized to any weaker form of symmetry.
For example, one would like to proof that every spectral polytope has eigenvalue $\theta_2$, or more generally, every balanced polytope has eigenvalue $\theta_2$.
\begin{corollary}
If $P\subset\RR^d$ is vertex- and edge-transitive, then holds:
\begin{myenumerate}
\item $\Aut(P)$ is irreducible,
\item $P$ is the unique simultaneously vertex- and edge-transitive polytope for which the edge-graph is isomorphic to $G_P$,
\item $P$ realizes all symmetries of its edge-graph,
\item $P$ is a perfect (rigid) polytope,
\item $P$ is the unique most symmetric realization of its combinatorial type,
\item if $P$ has edge-length $\ell$ and circumradius $r$, then
%
$$\Big[\frac \ell r \Big]^2 = \frac{2\lambda_2(G_P)}{\deg(G_P)} = 2\Big(1-\frac{\theta_2(G_P)}{\deg(G_P)}\Big).$$
%
\item if the polar dual $P^\circ$ has dihedral angle $\alpha$, then
%
$$\cos(\alpha)=-\frac{\theta_2(G_P)}{\deg(G_P)}.$$
\end{myenumerate}
\end{corollary}
\section{Distance-transitive polytopes}
\subsection{Distance-transitive polytopes}
\label{sec:distance_transitive_polytopes}
In general, there is no apparent reason for why a spectral polytope must be $\theta_2$-spectral.
In the case of distance-transitive polytopes there can be found an easy argument involving the cosine sequence of its skeleton.
The argument is based on the number of sign changes in $u_\delta$ and the so called \emph{two-piece property} for embedded graphs.
\begin{definition}
A realization $v$ is said to have the \emph{two-piece property} (or TPP),~if every hyperplane cuts it into at most two pieces.
More formally, given any vector $x\in \RR^d$ and scalar $c\in\RR$, the induced subgraphs
$$
G_+:=G[\,i\in V\mid \<x,v_i\>\ge c\,]
\quad\text{and}\quad
G_-:=G[\,i\in V\mid \<x,v_i\>\le c\,]
$$
must be connected (if non-empty).
\end{definition}
The image below shows two symmetric realizations of the cycle graph $C_5$, one~of which has the TPP (left), and one of which has not (right).
We are going to use the following fact:
\begin{theorem}[\cite{...}, Theorem xx]
The skeleton of a polytope has the TPP.
\begin{proof}[Sketch of proof.]
Let $v$ be the skeleton of a polytope, $x\in\RR^d$ and $c\in \RR$.
Let $i_+\in V$ be the vertex which maximizes the functional $\<v_\cdot,x\>$.
By the simplex algorithm, every vertex on the \enquote{positive} side of the hyperplane $\<\cdot,x\>=c$ can reach $v_{i_+}$ by a path in the edge-graph that is non-decreasing in $\<v_\cdot,x\>$.
Hence the positive side is connected if non-empty.
Equivalently for the negative side.
\end{proof}
\end{theorem}
\begin{theorem}
A distance-transitive polytope $P$ is $\theta_2$-spectral.
\begin{proof}
Let $v$ be the full-dimensional and distance-transitive realization of $P$.
Since it is also arc-transitive, it has an eigenvalue (by \cref{cref:...}).
By \cref{res:...} it must then be the $\theta_k$-realization of $G_P$.
Now consider cutting $v$ by the hyperplane $v_i^\bot$ for some $i\in V$.
The number of components left after the cut is exactly one more than the number of sign changes of the cosine sequence $u_\delta$.
From \cref{res:...} we know that there are exactly $k-1$ sign changes for the $\theta_k$-realization, hence $k$ pieces after the cut.
Since $v$ is the skeleton of a polytope, we have the TPP, and $k\le 2$ pieces.
Since $v$ cannot be the trivial $\theta_1$-realization of $G_P$, it must be the $\theta_2$-realization, and $P$ the $\theta_2$-eigenpolytope.
\end{proof}
\end{theorem}
\begin{corollary}
Let $P$ be a distance-transitive polytope.
\begin{myenumerate}
\item Among distance-transitive polytopes, $P$ is uniquely determined by its edge-graph.
\item $P$ realizes all the symmetries of its edge-graph.
\item $P$ is the unique most symmetric realization of its combinatorial type.
\end{myenumerate}
\end{corollary}
Godsil's classification of the $\theta_2$-spectral distance-regular graphs (see \cref{res:spectral_distance_regular_graphs}; all of which~turned out to be distance-transitive) allows a complete classification of distance-tran\-sitive polytopes.
\begin{theorem}\label{res:distance_transitive_classification}
If $P\subset\RR^d$ is a distance-transitive polytope, then it is one of the following:
\begin{enumerate}[label=$(\text{\roman*}\,)$]
\item a regular polygon $(d=2)$,
\item the regular dodecahedron $(d=3)$,
\item the regular icosahedron $(d=3)$,
\item a cross-polytopes, that is, $\conv\{\pm e_1,...,\pm e_d\}$ where $\{e_1,...,e_d\}\subset\RR^d$ is the standard basis of $\RR^d$,
\item a hyper-simplex $\Delta(d,n)$, that is, the convex hull of all vectors $v\in\{0,1\}^{d+1}$ with exactly $k$ 1-s,
\item a cartesian power of a regular simplex (this includes hypercubes),
\item a demi-cube, that is, the convex hull of all vectors $v\in\{-1,1\}^d$ with~an~even number of 1-s,
\item the $2_{21}$-polytope, also called Gosset-polytope $(d=6)$,
\item the $3_{21}$-polytope, also called Schläfli-polytope $(d=7)$.
\end{enumerate}
The order of this list agrees with the list of graphs in \cref{res:spectral_distance_regular_graphs}.
\end{theorem}
Note that the list in \cref{res:distance_transitive_classification} contains many polytopes that are not regular, and contains all regular polytopes except for the 24-cell, 120-cell and 600-cell.
The distance-transitive polytopes thus form a distinct class of remarkably symmetric polytopes which is not immediately related to the class of regular polytopes.
Noteworthy however, is that all the distance-transitive polytopes are \emph{Wythoffian poly\-topes}, that is, they are orbit polytopes of finite reflection groups.
\Cref{fig:distance_transitive_Coxeter} shows the Coxeter-Dynkin diagrams of these polytopes.
\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{img/distance_transitive_Coxeter}
\caption{Coxeter-Dynkin diagrams of distance-transitive polytopes.}
\label{fig:distance_transitive_Coxeter}
\end{figure}
\section{Conclusion and open questions}
\label{sec:future}
In this paper we have studied \emph{eigenpolytopes} and \emph{spectral polytopes}.
The former are polytopes constructed from a graph and one of its eigenvalues.
A polytope is spectral if it is the eigenpolytopes of its edge-graph.
These are of interest because spectral graph theory then ensures a strong interplay between the combinatorial properties of the edge-graph and the geometric properties of the polytope.
The study of eigenpolytopes and spectral polytopes has left us with many open questions.
Most notably, how to detect spectral polytopes purely from their geome\-try.
We introduced a tool (\cref{res:implies_spectral}), which was sufficient to proof that (most) edge-transitive polytopes are spectral.
We do not know how much more general it can be applied.
\begin{question}
\label{q:characterization}
Does \cref{res:implies_spectral} already characterize $\theta_2$-spectral polytopes (or even spectral polytopes in general)?
\end{question}
If the answer is affirmative, this would provide a geometric characterization of polytopes that are otherwise defined purely in terms of spectral graph theory.
The result of Izmestiev suggests that polytopes with sufficiently regular geometry are $\theta_2$-spectral: the entry of the matrix $X$ in \cref{res:izmestiev} at index $ij\in E$ can be~expressed as
$$X_{ij}=\frac{\vol(\sigma_i\cap\sigma_j)}{\|v_i\|\|v_j\|\sin\angle(v_i,v_j)},$$
where $\sigma_i$ and $\sigma_j$ are the facets of the polar dual $P^\circ$ that correspond to the vertices $v_i,v_j\in\F_0(P)$.
Because of this formula, it might be actually easier to classify the polar duals of $\theta_2$-spectral polytopes.
An affirmative answer to \cref{q:characterization} would also mean a negative answer to the following:
\begin{question}
\label{q:not_theta_2}
Is there a $\theta_k$-spectral polytope/graph for some $k\not=2$?
\end{question}
The answer is known to be negative for edge-transitive polytopes/graphs (see \cref{res:edge_transitive_spectral_graph}), but unknown in general.
The second-largest eigenvalue $\theta_2$ is special for other reasons too.
Even if a graph is not $\theta_2$-spectral, it seems to still imprint its adjacency information onto the edge-graph of its $\theta_2$-eigenpolytope.
\begin{question}
\label{q:realizing_edges}
Given an edge $ij\in E$ of $G$, if $v_i$ and $v_j$ (as defined in \cref{def:eigenpolytope}) are distinct vertices of the $\theta_2$-eigenpolytope $P_G(\theta_2)$, is then also $\conv\{v_i,v_j\}$ an edge of $P_G(\theta_2)$?
\end{question}
This was proven for distance-regular graphs in \cite{godsil1998eigenpolytopes}, and is not necessarily true for eigenvalues other than $\theta_2$.
All known spectral polytopes are exceptionally symmetric.
It is unclear whether this is true in general.
\begin{question}
\label{q:trivial_symmetry}
Are there spectral polytopes with trivial symmetry group?
\end{question}
An example for \cref{q:trivial_symmetry} must be asymmetric, yet with a reasonably large eigenspaces.
Such graphs exist among the distance-regular graphs, but all spectral distance-regular graphs were determined in \cite{godsil1998eigenpolytopes} (see also \cref{res:spectral_distance_regular_graphs}) and turned out to be distance-transitive, \shortStyle{i.e.,}\ highly symmetric.
A clear connection between being spectral and being symmetric is missing.
To emphasize our ignorance, we ask the following:
\begin{question}
\label{q:spectral_non_vertex_transitive}
Can we find more spectral polytopes that are \emph{not vertex-transitive}?
What characterizes them?
\end{question}
The single known spectral polytope that is \emph{not} vertex-transitive is the \emph{rhombic dodecahedron} (see \cref{fig:edge_transitive}).
The fact that it is spectral appears purely accidental, as there seems to be no reason for it to be spectral, except that we can explicitly check that it is. For comparison, the highly related \emph{rhombic triacontahedron} is not spectral.
On the other hand, vertex-transitive spectral polytopes might be quite common.
\begin{question}
\label{q:specific_instance}
Let $P\subset\RR^d$ be a polytope with the following properties:
\begin{myenumerate}
\item $P$ is vertex-transitive,
\item $P$ realizes all the symmetries of its edge-graph, and
\item $\Aut(P)$ is irreducible.
\end{myenumerate}
Is $P$ (combinatorially equivalent to) a spectral polytope?
\end{question}
No condition in \cref{q:specific_instance} can be dropped.
If we drop vertex-transitivity,~we could take some polytope whose edge-graph has trivial symmetry and only small eigenspaces. Dropping $(ii)$ leaves vertex-transitive neighborly polytopes, for which we know that these are mostly not spectral (except for the simplex).
Dropping $(iii)$ leaves us with the prisms and anti-prisms, the eigenspaces of their edge-graphs are rarely of dimension greater than two.
Finally, we wonder whether these spectral techniques can be any help in classifying the edge-transitive polytopes.
\begin{question}
Can we classify the edge-transitive graphs that are spectral, and~by this, the edge-transitive polytopes?
\end{question}
\begin{question}
Can the existence of half-transitive polytopes be excluded by using spectral graph theory (see \cref{sec:arc_transitive})?
\end{question}
\iffalse
\hrulefill
We have seen that in some situations computing the eigenpolytope of the edge-graph of a polytope results in the same polytope again.
Polytopes/graphs for which this happens are called \emph{spectral}.
We have introduced a tool for proving that certain geometrically well-behaved polytopes are necessarily $\theta_2$-spectral.
It is open whether spectral polytopes exist to other eigenvalues than $\theta_2$:
\begin{question}
\label{q:not_theta_2}
Is there a $\theta_k$-spectral polytope/graph for some $k\not=2$?
\end{question}
We have seen that the answer is negative for the distance-transitive (actually distance-regular) graphs (see \cref{res:distance_regular_theta_k}).
Our hope is that the proof can~be~extended to a larger class of graphs as \shortStyle{e.g.}\ 1-walk-regular or arc-transitive graphs.
Even if a graph is not $\theta_2$-spectral, it seems to still imprint its adjacency information on the edge-graph of its $\theta_2$-eigenpolytope:
\begin{question}
\label{q:realizing_edges}
Let $G$ be a graph with $\theta_2$-eigenpolytope $P=P_G(\theta_2)$.
For each~$i\in$ $V$ let $v_i\in\RR^d$ be as defined in \cref{def:eigenpolytope}.
Now, if $ij\in E$ and $v_i\not= v_j$ are distinct vertices of $P$, is then also $\conv\{v_i,v_j\}$ an edge of $P$?
\end{question}
The answer is affirmative for distance-regular graphs as shown by Godsil in \cite{godsil1998eigenpolytopes}.
All known spectral polytopes are highly symmetric.
We therefore ask the following:
\begin{question}
\label{q:trivial_symmetry}
Are there spectral polytopes with trivial symmetry group?
\end{question}
To construct an example, one has to find a graph with no non-trivial symmetries, but whose eigenspaces are still reasonably large.
Such graphs can be found among the distance-regular graphs.
However, all spectral graphs among these were already determined by Godsil and all of them turned out to be distance-transitive.
A clear connection between being spectral and being symmetric is missing.
To emphasize our ignorance, we have the following:
\begin{question}
Can we find more spectral polytopes that are \emph{not vertex-transitive}?
What characterizes them?
\end{question}
Only one such polytope is known: the \emph{rhombic dodecahedron}.
We know that it spectral by explicit computation, but a systematic proof of this fact (that admits a generalization) is missing.
In general, how else can one characterize spectral polytopes?
The result of~Izmes\-tiev suggests that polytopes with sufficiently regular geometry are $\theta_2$-spectral.
In fact, the value of the matrix $X$ in \cref{res:izmestiev} at an index $ij\in E$ can be given~explicitly
$$X_{ij}=\frac{\vol(\sigma_i\cap\sigma_j)}{\|v_i\|\|v_j\|\sin\angle(v_i,v_j)},$$
where $\sigma_i$ and $\sigma_j$ are the facets of the polar dual $P^\circ$ that correspond to the vertices $v_i,v_j\in\F_0(P)$.
Because of this, it might be actually easier to classify the polar duals of $\theta_2$-spectral polytopes.
\begin{question}
\label{q:characterization}
Does \cref{res:implies_spectral} already characterize $\theta_2$-spectral polytopes (or even spectral polytopes in general)?
\end{question}
There are very specific classes of polytopes that might have spectral of their combiantorial type:
\begin{question}
\label{q:specific_instance}
Let $P\subset\RR^d$ be a polytope with the following properties:
\begin{myenumerate}
\item $P$ is vertex-transitive,
\item $P$ realizes all the symmetries of its edge-graph, and
\item $\Aut(P)$ is irreducible.
\end{myenumerate}
Is $P$ (combinatorially equivalent to) a spectral polytope? Or at least an eigenpolytope?
\end{question}
We know that $P$ in \cref{q:specific_instance} is not necessarily spectral if we drop any of the conditions.
If we drop vertex-transitivity we could take some polytope with trivial symmetry. Dropping $(ii)$ gives the neighborly polytopes, for which we know that these are only balanced, but mostly not spectral (except for the simplex).
Dropping $(iii)$ gives us the prisms and anti-prisms which mostly do not have eigenspaces of dimension three or higher.
\cref{q:specific_instance} has equally interesting consequences if it is only true for the eigenpolytope case.
For example, several polytopes with relevance in integer optimization seem to be eigenpolytopes, \shortStyle{e.g.}\ the Birkhoff polytope or the traveling salesman polytopes.
This would allow a formulation of the respective optimization problem in the language of spectral graph theory.
Also, this gives a way to study the symmetry groups of these polytopes (which are oftem much larger than expected from their definition).
Finally, we wonder whether these spectral techniques can be any help in classifying edge-transitive polytopes.
\begin{question}
Can we classify the edge-transitive polytopes, or equivalently, the simultaneously vertex- and edge-transitive graphs that are spectral?
\end{question}
\begin{question}
Can spectral graph theory help to show that no half-transitive polyopes exist?
\end{question}
\fi
\section{Introduction}
\label{sec:introduction}
\emph{Eigenpolytopes} are a construction in the intersection of combinatorics and geometry, using techniques from spectral graph theory.
Eigenpolytopes provide a way~to associate several polytopes to a finite simple graph, one for each eigenvalues of~its adjacency matrix.
A formal definition can be found in \cref{sec:def_eigenpolytope}.
\vspace{0.5em}
\begin{center}
\includegraphics[width=0.7\textwidth]{img/cube_intro_pic}
\end{center}
\vspace{0.5em}
Eigenpolytopes can be applied from two directions:
for the first, one starts~from a given graph, computes its eigenpolytopes, and tries to deduce, from the geometry and combinatorics of these polytopes, something about the original~graph.
For the other direction, one starts with a polytope, asks whether it is an eigenpolytope, and if so, for which graphs, which eigenvalues, and how these relate to the original polytope.
Eigenpolytopes have several interesting geometric and algebraic properties, and establishing that a family of polytopes consists of eigenpolytopes opens up their study to the techniques of spectral graph theory.
For some graphs the connection to their eigenpolytopes is especially strong: it can happen that a graph is the edge-graph of one of its eigenpolytopes, or equivalently, that a polytope is an eigenpolytope of its edge-graph.
Such graphs/polytopes are quite special and we shall call them \emph{spectral}.
For example, all regular polytopes are spectral, but there are many others.
Their properties are not well-understood.
We survey the literature of eigenpolytope and spectral polytopes.
We establish a technique with which to prove that certain polytopes are spectral polytopes and we apply it to \emph{edge-transitive} polytopes. That are polytopes for which the Euclidean symmetry group $\Aut(P)\subset\Ortho(\RR^d)$ acts transitively on the set of edge $\F_1(P)$.
As we shall explain, this characterization suffices to proves that an edge-transitive polytope is uniquely determined by its edge-graph, and also realizes all its combinatorial symmetries.
A complete classification of edge-transitive polytopes is not known as of yet.
However, using results on eigenpolytopes, we are able to give a complete classification of a sub-class of the edge-transitive polytopes, namely, the \emph{distance-transitive polytopes}.
\subsection{Outline of the paper}
\cref{sec:eigenpolytopes} starts with a motivating example for directing the reader towards the definition of the \emph{eigenpolytope} as well as the phenomenon of \emph{spectral} graphs and polytopes.
We include a literature overview for~\mbox{eigenpolytopes} and spectral polytopes.
In \cref{sec:balanced_spectral} we give a first rigorous definition for the notion \enquote{spectral polytope} via \emph{balanced polytopes}. The latter is a notion related to the rigidity theory.
In \cref{sec:izmestiev} we introduce the, as of yet, most powerful tool for proving that~certain polytopes are spectral.
In the final section, \cref{sec:edge_transitive}, we apply this result to \emph{edge-transitive} polytopes.
It is a simple corollary of the previous section that these are $\theta_2$-spectral.
We explore the implications of this finding: edge-transitive polytopes (in dimension $d\ge 4$) are uniquely determined by the edge-graph and realize all of its symmetries.~%
We~discuss sub-classes, such as the arc-, half- and distance-transitive polytopes.
We close with a complete classification of the latter (based on a result of Godsil).
\section{Eigenpolytopes and spectral polytopes}
\label{sec:eigenpolytopes}
\subsection{A motivating example}
\label{sec:example}
Let $G=(V,E)$ be the edge-graph of the cube, with vertex set $V=\{1,...,8\}$,~num\-bers assigned to the vertices as in the figure below.
\begin{figure}[h!]
\centering
\includegraphics[width=0.48\textwidth]{img/cube}
\end{figure}
\noindent
The spectrum of that graph (\shortStyle{i.e.,}\ of its adjacency matrix) is $\{(-3)^1,(-1)^3,1^3,3^1\}$.
Most often, one denotes the largest eigenvalue by $\theta_1$, the second-largest by $\theta_2$, and so on.
In spectral graph theory, there exists the general rule of thumb that the most exciting eigenvalue of a graph is not its largest, but its \emph{second-largest} eigenvalue $\theta_2$ (which is related to the \emph{algebraic connectivity} of $G$).
For the edge-graph of the cube, we have $\theta_2=1$,~of multiplicity \emph{three}.
And here are three linearly independent eigenvectors to $\theta_2$:
\vspace{0.5em}
\begin{center}
\raisebox{-3.3em}{\includegraphics[width=0.23\textwidth]{img/cube_eigenvector}}
\qquad
$
u_1 = \begin{pmatrix}
\phantom+1\\ \phantom+1 \\ \phantom+1 \\ \phantom+1 \\ -1 \\ -1 \\ -1 \\ -1
\end{pmatrix},\quad
u_2 = \begin{pmatrix}
\phantom+1\\ \phantom+1 \\ -1 \\ -1 \\ \phantom+1 \\ \phantom+1 \\ -1 \\ -1
\end{pmatrix},\quad
u_3 = \begin{pmatrix}
\phantom+1\\ -1 \\ \phantom+1 \\ -1 \\ \phantom+1 \\ -1 \\ \phantom+1 \\ -1
\end{pmatrix}.
$
\end{center}
\vspace{0.7em}
\noindent
We can write these more compactly in a single matrix $\Phi\in\RR^{8\x 3}$:
\vspace{0.4em}
\begin{center}
$\Phi= \;\begin{blockarray}{(lll)r
\phantom+1 & \phantom+1 & \phantom+1\;\; &\text{\quad \footnotesize $\leftarrow v_1$}\\
\phantom+1 & \phantom+1 & -1 & \text{\quad \footnotesize $\leftarrow v_2$} \\
\phantom+1 & -1 & \phantom+1 & \text{\quad \footnotesize $\leftarrow v_3$} \\
\phantom+1 & -1 & -1 & \text{\quad \footnotesize $\leftarrow v_4$} \\
-1 & \phantom+1 & \phantom+1 & \text{\quad \footnotesize $\leftarrow v_5$} \\
-1 & \phantom+1 & -1 & \text{\quad \footnotesize $\leftarrow v_6$} \\
-1 & -1 & \phantom+1 & \text{\quad \footnotesize $\leftarrow v_7$} \\
-1 & -1 & -1 & \text{\quad \footnotesize $\leftarrow v_8$} \\
\end{blockarray}.$
\qquad
\raisebox{-4.2em}{\includegraphics[width=0.4\textwidth]{img/cube_coordinates}}
\end{center}
We now take a look at the rows of that matrix, of which it has exactly eight.~These rows are naturally assigned to the vertices of $G$ (assign $i\in V$ to the $i$-th row of~$\Phi$), and each row can be interpreted as~a~vec\-tor in $\RR^3$.
If we place each vertex $i\in V$ at the position $v_i\in\RR^3$ given by the $i$-th row of $\Phi$, we find that this embedds the graph $G$ \emph{exactly} as the skeleton of a cube (see the figure above).
In other words: if we compute the convex hull of the $v_i$, we get back the polyhedron from which we have started.
What a coincidence, isn't it?
This example was specifically chosen for its nice numbers, but in~fact,~the same works out as well for many other polytopes, inclu\-ding all the regular polytopes~in all dimension.
One probably learns to appreciate this magic when suddenly in~need for the vertex coordinates of some not so nice polytope, say, the regular dodecahedron or 120-cell.
With this technique in the toolbox, these coordinates are just one eigenvector-computation away (we included a short Mathematica script in \cref{sec:appendix_mathematica}).
Note also, that we never specified~the~dim\-en\-sion of~the~embed\-ding, but it just so happened, that the second-largest eigenvalue has the right multiplicity.
This phenomenon definitely deserves an explanation.
\subsubsection*{On the choice of eigenvectors}
\label{sec:choice_of_eigenvectors}
One might object that the chosen eigenvectors $u_1, u_2$ and $u_3$ look suspiciously cherry-picked, and we may not get such a nice result if we would have chosen just any eigenvectors.
And this is true.
For an appropriate choice of these vectors, we can, instead of a cube, get a cuboid, or a parallelepiped.
In fact, we can obtain any \emph{linear} transformations of the cube.
\emph{But}, we can also get \emph{only} linear transformations, and nothing else.
The reason is the following well~know fact from linear algebra:
\begin{theorem}
\label{res:same_column_span}
Two matrices $\Phi,\Psi\in\RR^{n\x d}$ have the same column span, \shortStyle{i.e.,}~$\Span \Phi=\Span \Psi$, if and only if their rows are related by an invertible linear transformation, \shortStyle{i.e.,}\ $\Phi=\Psi T$ for some $T\in\GL(\RR^d)$.
\end{theorem}
\noindent
In our case, the column span is the $\theta_2$-eigenspace, and the rows are the coordinates of the $v_i$.
We say that any two polytopes constructed in this way are \emph{linearly equivalent}.
The only notable property of the chosen basis in the example is, that the vectors $u_1, u_2$ and $u_3$ are orthogonal and of the same length.
Any other choice of such a basis of~the~\mbox{$\theta_2$-eigen}\-space (\shortStyle{e.g.}\ an orthonormal basis) would also have given a cube, but reoriented, rescaled and probably with less nice coordinates.
For details on how this choice relates to the orientation, see \shortStyle{e.g.}\ \cite[Theorem 3.2]{winter2019geometry}.
\subsection{Eigenpolytopes}
\label{sec:def_eigenpolytope}
We compile our example into a definition.
\begin{definition}\label{def:eigenpolytope}
Start with a graph $G=(V,E)$, an eigenvalue $\theta\in\Spec(G)$ thereof, as well as an orthonormal basis $\{u_1,...,u_d\}\subset\RR^n$ of the $\theta$-eigenspace.
We define the \emph{eigenpolytope matrix} $\Phi\in\RR^{n\x d}$ as the matrix in which the $u_i$ are the columns:\phantom{mm}
\begin{equation}
\label{eq:eigenpolytope_matrix}
\Phi :=\begin{pmatrix}
\mid & & \mid \\
u_1 & \!\!\cdots\!\!\! & u_d \\
\mid & & \mid
\end{pmatrix}=
\begin{pmatrix}
\;\rule[.5ex]{2.5ex}{0.4pt}\!\!\!\! & v_1^{\Tsymb} & \!\!\!\!\rule[.5ex]{2.5ex}{0.4pt}\;\; \\
& \vdots & \\[0.4ex]
\;\rule[.5ex]{2.5ex}{0.4pt}\!\!\!\! & v_n^{\Tsymb} & \!\!\!\!\rule[.5ex]{2.5ex}{0.4pt}\;\;
\end{pmatrix}.
\end{equation}
Let $v_i\in\RR^d$ denote the $i$-th row of $\Phi$.
The polytope
$$P_G(\theta):=\conv\{v_i\mid i\in V\}\subset\RR^d$$
is called \emph{$\theta$-eigenpolytope} (or just \emph{eigenpolytope}) of $G$.
\end{definition}
For later use we define the \emph{eigenpolytope map}
\begin{equation}
\label{eq:eigenpolytope_map}
\phi:V\ni i\mapsto v_i\in\RR^d
\end{equation}
that to each vertex $i\in V$ assignes the $i$-th row of the eigenpolytope matrix.
Note that the basis $\{u_1,...,u_d\}\subset\Eig_G(\theta)$ in \cref{def:eigenpolytope} is explicitly chosen~to be an \emph{orthonormal basis}.
This is not strictly necessarily, but this choice is convenient from a geometric point of view:
a different choice for this basis gives the same~poly\-tope, but with a different orientation rather than, say, transformed~by~a~general linear transformation.
This preserves metric properties and is closer to how polytopes are usually consider up to rigid motions.
We can also reasonably speak of \emph{the} $\theta$-eigenpolytope, as any two differ only by orientation.
With this terminology in place, our observation in the example of \cref{sec:example}~can be summarized as \enquote{the cube is the $\theta_2$-eigenpolytope of its edge-graph}, or alternatively as \enquote{the cube-graph is the edge-graph of its $\theta_2$-eigenpolytope}.
Here is~a~depic\-tion of all the eigenpolytopes of the cube-graph, one for each eigenvalue:
\vspace{0.5em}
\begin{center}
\includegraphics[width=0.7\textwidth]{img/cube_eigenpolytopes}
\end{center}
\noindent
We observe that the phenomenon from \cref{sec:example} only happens for $\theta_2$.
In general, the $\theta_1$-eigenpolytope of a regular graph will always be a single point (which is, why we rarely care about the largest eigenvalue).
Also, whenever a graph is bipartite, the eigenpolytope to the smallest eigenvalue is 1-dimensional, hence a line segment.
We are now free to compute the eigenpolytopes of all kinds of graphs, \mbox{including} graphs which are not the edge-graph of any polytope (so-called \emph{non-polytopal} graphs).
It is then little surprising that no edge-graph of any of its eigenpolytope gives the original graph again.
But even if we start from a polytopal graph, one is not guaranteed to find an eigen\-polytope that has the initial graph as its edge-graph (\shortStyle{e.g.}\ the edge-graph of the triangular prism has no eigenvalue of multiplicity three, hence no eigenpolytope of dimension three, see also \cref{ex:prism}).
Equivalently, if one starts with a polytope, it~is~not guaranteed that this polytope is the eigenpolytope of its edge-graph (or even combinatorially equivalent to it).
\begin{example}
\label{ex:neighborly_1}
A \emph{neighorly polytope} is a polytope whose edge-graph is the complete graph $K_n$.
The spectrum of $K_n$ is $\{(-1)^{n-1},(n-1)^1\}$.
One checks that~the~eigenpolytopes are a single point (for $\theta_1=n-1$) and the regular simplex of dimension $n-1$ (for $\theta_2=-1$).
Consequently, no neighborly polytope other than a simplex is combinatorially equivalent to an eigenpolytope of its edge-graph.
\end{example}
That a graph and its eigenpolytope translate into each other as well as in the case of the cube in \cref{sec:example} is a very special phenomenon, to which we shall give a name:
a polytope (or graph) for which this happens, will be called \emph{spectral}\footnote{There was at least one previous attempt to give a name to this phenomenon, namely, in \cite{licata1986surprising}, where it was called \emph{self-reproducing}.}.
We cannot formalize this definition right away, as there is some subtlety we have to discuss first (we give a formal definition in \cref{sec:balanced_spectral}, see \cref{def:spectral}).
\begin{example}
\label{ex:pentagon}
The image below shows two spectral realizations of the 5-cycle $C_5$\footnote{Spectral realizations are essentially defined like eigenpolytopes, assinging coordinates $v_i\in\RR^d$ to each vertex $i\in V$ (as in \cref{def:eigenpolytope}), but without taking the convex hull. Instead, one draws the edges between adjacent vertices.}.
\begin{center}
\includegraphics[width=0.4\textwidth]{img/C5}
\end{center}
The left image~shows the realization to the second-largest eigenvalue $\theta_2$, the right image shows the realization to the smallest eigenvalue $\theta_3$.
In both cases, the convex hull (the actual eigenpolytope) is a regular pentagon, whose edge-graph is $C_5$ again.
But we see that only in the case of $\theta_2$ the edges of the graphs get properly mapped into the edges of the pentagon.
While it is true that the 5-cycle $C_5$ is the edge-graph of its $\theta_3$-eigenpolytope, the adjacency informations gets scrambled in the process:
while, say, vertex 1 and 2 are adjacent in $C_5$, their images $v_1$ and $v_2$ do not form an edge in the $\theta_3$-eigenpolytope.
We do not want to call this \enquote{spectral}, as the adjacency information is not preserved.
The same can happen in higher dimensions too, \shortStyle{e.g.}\ with $G$ being the edge-graph of the dodecahedron:
\begin{center}
\includegraphics[width=0.55\textwidth]{img/dodecahedron_eigenpolytopes_2}
\end{center}
\end{example}
\begin{observation}
\label{res:theta_2_observations}
From studying many examples, there are two interesting observations to be made, both concern $\theta_2$, none of which is rigorously proven:
\begin{myenumerate}
\item
It appears as if only $\theta_2$ can give rise to spectral polytopes/graphs.
At~least, all known examples are $\theta_2$-spectral (see also \cref{q:not_theta_2}). Some considerations on nodal domains make this plausible, but no proof is known in the general case (a proof is known in certain special cases, see \cref{res:edge_transitive_spectral_graph}).
\item
If $i\in V$ is a vertex of $G$, then $v_i$ is not necessarily a vertex of every~eigenpolytope ($v_i$ might end up in the interior of $P_G(\theta)$ or one of its faces).
And even if $v_i,v_j\in\F_0(P_G(\theta))$ are distinct vertices and $ij\in E$ is an edge of $G$, it is still not necessarily true that $\conv\{v_i,v_j\}$ is also an edge of the~eigen\-polytope (as seen in \cref{ex:pentagon}).
However, this seems to be no concern in the case $\theta_2$.
It appears as if all edges of $G$ become edges of the $\theta_2$-eigenpolytope, even if $G$ is not spectral (under mild assumptions on the end vertices of the edge).
In other words, the adjacency information of $G$ gets imprinted on the edge-graph of the $\theta_2$-eigenpolytope, whether $G$ is spectral or not.
This is known to be true only in the case of distance-regular graphs \cite[Theorem 3.3 (b)]{godsil1998eigenpolytopes}, but unproven~in general (see also \cref{q:realizing_edges})
\end{myenumerate}
\end{observation}
\subsection{Litarture}
Eigenpolytope were first introduced by Godsil \cite{godsil1978graphs} in 1978.
Godsil proved the existence of a group homomorphism $\Aut(G)\to\Aut(P_G(\theta))$, \shortStyle{i.e.,}\, any combinatorial symmetry of the graph translates into a Euclidean symmetry of the polytope.
From that, he deduces results about the combinatorial symmetry group of the original graph.
We say more about the group homomorphism: for every $\theta\in\Spec(G)$ we have
\begin{theorem}[\!\cite{godsil1978graphs}, Theorem 2.2]
\label{res:realizing_symmetries}
If $\sigma\in\Aut(G)\subseteq\Sym(n)$ is a symmetry of $G$, and $\Pi_\sigma\in\Perm(\RR^n)$ is the associated permutation matrix, then
$$T_\sigma:=\Phi^{\Tsymb} \Pi_\sigma \Phi \;\in\; \Ortho(\RR^d),\qquad (\text{$\Phi$ is the eigenpolytope matrix})$$
is a Euclidean symmetry of the eigenpolytope $P_G(\theta)$ that also permutes the $v_i$ as~prescribed by $\sigma$, \shortStyle{i.e.,}\ $T_\sigma\circ \phi = \phi\circ\sigma$, or $T_\sigma v_i = v_{\sigma(i)}$ for all $i\in V$.
\end{theorem}
This result is also proven (more generally for spectral graph realizations) in \cite[Corollary 2.9]{winter2020symmetric}.
\cref{res:realizing_symmetries} explicitly uses that eigenpolytopes are defined using an \emph{orthonormal} bases rather than any basis of the eigenspace, to conclude that the symmetries $T_\sigma$ are \emph{orthogonal} matrices.
Also, the statement of \cref{res:realizing_symmetries} is not too satisfying in general, as it can happen that non-trivial~symmetries of $G$ are mapped to the identity transformation.
We not necessarily have $\Aut(G)\cong\Aut(P_G(\theta))$.
Several authors construct the eigenpolytopes of certain famous graphs or graph families.
Powers \cite{powers1986petersen} computed the eigenpolytopes of the \emph{Petersen graph}, which he termed the \emph{Petersen polytopes} (one of which will appear as a distance-transitive polytope in \cref{sec:distance_transitive}).
The same author also investigates eigenpolytopes of general distance-regular graphs in \cite{powers1988eigenvectors}.
In \cite{mohri1997theta_1}, Mohri described the face structure of the \emph{Hamming polytopes}, the $\theta_2$-eigenpolytopes of the Hamming graphs.
Seemingly unknown to the author, these polytopes can also by described as the cartesian powers of regular simplices (also distance transitive, see \cref{sec:distance_transitive}).
There exists a wonderful enumeration of the eigenpolytopes (actually, spectral realizations) of the edge-graphs of all uniform polyhedra in \cite{blueSpectral}. Sadly, this write-up was never published formally.
This provides empirical evidence that every uniform polyhedron
has a spectral realization.
The same question might then be asked for uniform polytopes in higher dimensions.
Rooney \cite{rooney2014spectral} used the combinatorial structure of the eigenpolytope (the size of their facets) to deduce statements about the size of cocliques in a graph.
In \cite{padrol2010graph}, the authors investigates how common graph operations translate to operations on their eigenpolytopes.
Particular attention was given to the eigenpolytopes of distance-regular graphs \cite{powers1988eigenvectors,godsil1998eigenpolytopes,godsil1995euclidean}.
It was shown that in a $\theta_2$-eigenpolytope of a distance-regular graph~$G$,~every edge of $G$ corresponds to an edge of the eigenpolytope \cite{godsil1998eigenpolytopes}.
Consequently, $G$~is a spanning subgraph of the edge-graph of the eigenpolytope.
It remains open if the same holds for less regular graphs, \shortStyle{e.g.}\ 1-walk regular graphs or arc-transitive graphs (see also \cref{q:realizing_edges}).
The observation that some polytopes are the eigenpolytopes
of their edge-graph (\shortStyle{i.e.,}\ they are \emph{spectral} in our terminology) was made repeatedly, \shortStyle{e.g.}\ in \cite{godsil1995euclidean} and \cite{licata1986surprising}.
In the latter, this was shown for all regular polytopes, excluding the exceptional 4-dimensional polytopes, the 24-cell, 120-cell and 600-cell.
This gap was filled in \cite{winter2020symmetric} via general considerations concerning spectral realizations of arc-transitive graphs.
In sum, all regular polytopes are known to be $\theta_2$-spectral.
The next major result for spectral polytopes was obtained by Godsil in \cite{godsil1998eigenpolytopes}, where he was able to classify all $\theta_2$-spectral distance-regular graphs (see also \cref{sec:distance_transitive}):
\begin{theorem}[\!\cite{godsil1998eigenpolytopes}, Theorem 4.3]
\label{res:spectral_distance_regular_graphs}
Let $G$ be distance-regular.
If $G$ is $\theta_2$-spectral,~then $G$ is one of the following:
\begin{enumerate}[label=$(\text{\roman*}\,)$]
\item a cycle graph $C_n,n\ge 3$,
\item the edge-graph of the dodecahedron,
\item the edge-graph of the icosahedron,
\item the complement of a disjoint union of edges,
\item a Johnson graph $J(n,k)$,
\item a Hamming graph $H(d,q)$,
\item a halved $n$-cube $\nicefrac12 Q_n$,
\item the Schläfli graph, or
\item the Gosset graph.
\end{enumerate}
\end{theorem}
A second look at this list reveals a remarkable \enquote{coincidence}: while the generic distance-regular graph has few or no symmetries, all the graphs in this list are highly symmetric, in fact, \emph{distance-transitive} (a definition will be given in \cref{sec:distance_transitive}).
It is a widely open question whether being spectral is a property solely reserved for highly symmetric graphs and polytopes (see also \cref{q:trivial_symmetry}).
There is only a single known spectral polytope that is not vertex-transitive (see also \cref{rem:edge_not_vertex} and \cref{q:spectral_non_vertex_transitive}).
\section{Balanced and spectral polytopes}
\label{sec:balanced_spectral}
In this section we give a second approach to \emph{spectral polytopes} that circumvents the mentioned subtleties.
For the rest of the paper, let $P\subset\RR^d$ denote a full-dimensional polytope in~dimen\-sion $d\ge 2$ with vertices $v_1,...,v_n\in\F_0(P)$.
We disinguish the \emph{skeleton} of $P$, which is the graph with vertex set $\F_0(P)$ and edge set $\F_1(P)$, from the~\mbox{\emph{edge-graph}}~$G_P=(V,E)$ of $P$, which is isomor\-phic to the skeleton, but has vertex set $V=\{1,...,n\}$. The isomorphism will be denoted
\begin{equation}
\label{eq:vertex_map}
\psi:V\ni i\mapsto v_i\in\F_0(P),
\end{equation}
and we call it the \emph{skeleton map}.
\subsection{Balanced polytopes}
\begin{definition}
The polytope $P$ is called \emph{$\theta$-balanced} (or just \emph{balanced}) for some~real number $\theta\in\RR$, if
\begin{equation}
\label{eq:balanced}
\sum_{\mathclap{j\in N(i)}} v_j = \theta v_i,\quad\text{for all $i\in V$},
\end{equation}
where $N(i):=\{j\in V\mid ij\in E\}$ denotes the \emph{neighborhood} of a vertex $i\in V$.
\end{definition}
One way to interpret the balancing condition \eqref{eq:balanced} is as a kind of self-stress~con\-dition on the skeleton of $P$ (the term \enquote{balanced} is motivated from this).
For each edge $ij\in E$, the vector $v_j-v_i$ is parallel to the edge $\conv\{v_i,v_j\}$.
If $P$ is $\theta$-balanced, at each vertex $i\in V$ we have the equation
$$\sum_{\mathclap{j\in N(i)}} (v_j-v_i) = \sum_{\mathclap{j\in N(i)}} v_j - \deg(i) v_i = \big(\theta-\deg(i)\big)v_i.$$
This equation can be interpreted as two forces that cancel each other out: on the left, a contracting force along each edge (proportion only to the length of that edge), and on the right, a force repelling each vertex away from the origin (proportional to the distance of that vertex from the origin, and proportional to $\theta-\deg(i)$).
A second interpretation of \eqref{eq:balanced} is via spectral graph theory.
Define the matrix
\begin{equation}
\label{eq:arrangement_matrix}
\Psi :=
\begin{pmatrix}
\;\rule[.5ex]{2.5ex}{0.4pt}\!\!\!\! & v_1^{\Tsymb} & \!\!\!\!\rule[.5ex]{2.5ex}{0.4pt}\;\; \\
& \vdots & \\[0.4ex]
\;\rule[.5ex]{2.5ex}{0.4pt}\!\!\!\! & v_n^{\Tsymb} & \!\!\!\!\rule[.5ex]{2.5ex}{0.4pt}\;\;
\end{pmatrix}
\end{equation}
in which the $v_i$ are the rows.
This matrix will be called the \emph{arrangement matrix} of $P$.
Note that the skeleton map $\psi$ assignes $i\in V$ to the $i$-th row of $\Psi$.
Since we use that $P\subset\RR^d$ is full-dimensional,~we have $\rank \Psi=d$.
\begin{observation}\label{res:eigenvalue}
Suppose that $P$ is $\theta$-balanced.
The defining equation \eqref{eq:balanced} can be equivalently written as the matrix equation $A\Psi=\theta \Psi$.
In this form,~it~is~apparent that $\theta$ is an eigenvalue of the adjacency matrix $A$, and the columns of $\Psi$ are $\theta$-eigenvectors, or $\Span\Psi\subseteq\Eig_{G_P}(\theta)$.
\end{observation}
We have seen that for a balanced polytope, the columns of $\Psi$ must be eigenvectors.
But they are not necessarily a complete set of~$\theta$-eigen\-vectors, \shortStyle{i.e.,}\ they not necessarily span the whole eigenspace.
\begin{example}
\label{ex:neighborly}
Every centered neighborly polytope $P$ is balanced, but except if it is a simplex, it is not spectral (the latter was shown in \cref{ex:neighborly_1}).
Centered~means that
$$\sum_{i\in V} v_i = 0.$$
Since $P$ is neighborly, we have $G_P=K_n$ and $N(i)=V\setminus\{i\}$ for all $i\in V$. Therefore
$$\sum_{\mathclap{j\in N(i)}} v_j = \sum_{\mathclap{j\in V}} v_j - v_i = -v_i,\quad\text{for all $i\in V$}.$$
And indeed, $K_n$ has spectrum $\{(-1)^{n-1},(n-1)^1\}$.
So $P$ is $(-1)$-balanced.
\end{example}
The last example shows that every neighborly polytopes can be made balanced by merely translating it.
More generally, many polytopes have a realization (of~their combinatorial type) that is balanced.
But other polytopes do not:
\begin{example}
\label{ex:prism}
Let $P\subset\RR^3$ be a triangular prism.
The spectrum of the edge-graph of $P$ is $\{(-2)^2,0^2,1^1,3^1\}$.
Note that there~is~no eigenvalue of multiplicity greater than~two.
In particular, we cannot choose three linearly independent eigenvectors to a common eigenvalue.
But if $P$ were balanced, then \cref{res:eigenvalue} tells us that the columns of the arrangement matrix $\Psi$ would be three eigenvectors to the same eigenvalue (linearly independent, since $\rank \Psi=3$), which is not possible.
And so, no realization of $P$ can be balanced.
\end{example}
\subsection{Spectral graphs and polytopes}
In the extreme case, when the columns of $\Psi$ span the whole eigenspace, we can finally give a compact definition of what we want~to~consider as \emph{spectral}:
\begin{definition}
\label{def:spectral}\quad
\begin{myenumerate}
\item A polytope $P$ is called \emph{$\theta$-spectral} (or just \emph{spectral}), if its arrangement matrix $\Psi$ satisfies $\Span \Psi=\Eig_{G_P}(\theta)$.
\item A graph is said to be \emph{$\theta$-spectral} (or just \emph{spectral}) if it is (isomorphic to) the edge-graph of~a $\theta$-spectral polytope.
\end{myenumerate}
\end{definition}
This definition is now perfectly compatible with our initial motivation for the~term \enquote{spectral} in \cref{sec:def_eigenpolytope}.
\begin{lemma}\quad
\label{res:naive}
\begin{myenumerate}
\item If a polytope $P$ is $\theta$-spectral, then $P$ is linearly equivalent to the $\theta$-eigenpoly\-tope of its edge-graph (see also \cref{res:naive_polytope}).
\item If a graph $G$ is $\theta$-spectral, then $G$ is (isomorphic to) the edge-graph of its~$\theta$-eigen\-polytope (see also \cref{res:naive_graph}).
\end{myenumerate}
\end{lemma}
In both cases, the converse is \emph{not} true.
This is intentional, to avoid the problems mentioned in \cref{ex:pentagon}.
Both statement will be proven below by formulating a more technical condition that is then actually equivalent to being spectral.
\begin{proposition}
\label{res:naive_polytope}
A polytope $P$ is $\theta$-spectral if and only if it is linearly equivalent to the $\theta$-eigenpolytope of its edge-graph via some linear map $T\in\GL(\RR^d)$ for which the following diagram commutes:
\begin{equation}
\label{eq:diagram_polytope}
\begin{tikzcd}
P \arrow[r, "T"] & P_{G_P}(\theta) \\
& G_P \arrow[lu, "\psi"] \arrow[u, "\phi"']
\end{tikzcd}
\end{equation}
where $\phi$ and $\psi$ denote the eigenpolytope map and skeleton map respectively.
\end{proposition}
\begin{proof
By definition, the $\theta$-eigenpolytope of $G_P$ satisfies $\Span \Phi=\Eig_{G_P}(\theta)$, where $\Phi$ is the corresponding eigenpolytope matrix.
Now, by definition, $P$ is $\theta$-spectral if and only if $\Span \Psi = \Eig_{G_P}(\theta)$, where $\Psi$ is its arrangement matrix.
But by \cref{res:same_column_span}, $\Phi$ and $\Psi$ have the same span if and only of their rows are related by some invertible linear map $T\in\GL(\RR^d)$, that is, $\Psi T=\Phi$, or $T\circ \psi=\phi$. The latter expresses exactly that \eqref{eq:diagram_polytope} commutes.
\end{proof}
This also proves \cref{res:naive} $(i)$.
\begin{proposition}
\label{res:naive_graph}
A graph $G$ is $\theta$-spectral if and only if the eigenpolytope map~$\phi\:$ $V(G)\to \RR^d$ provides an isomorphism between $G$ and the skeleton of its $\theta$-eigenpoly\-tope $P_G(\theta)$.
\begin{proof}
Suppose first that $G$ is $\theta$-spectral.
Then there is a $\theta$-spectral polytope $Q$~with edge-graph~$G_Q=G$ and skeleton map $\psi\:V(G_Q)\to\F_0(Q)$.
By \cref{res:naive} $(i)$, $Q$ is linearly equivalent to $P_G(\theta)$ via some linear map $T\in\GL(\RR^d)$.
By \cref{res:naive_polytope}, the eigenpolytope map satisfies $\phi=T\circ \psi$.
Since $T$ induces an isomorphism between the skeleta of $Q$ and $P_G(\theta)$, and $\psi$ is an isomorphism between $G$ and the skeleton of $Q$, we find that $\phi$ must be an isomorphism between $G$ and the skeleton of $P_G(\theta)$.
This shows one direction.
For the converse, suppose that $\phi$ is an isomorphism.
Set $P:=P_G(\theta)$ and let~$G_P$ be its edge-graph with skeleton map $\psi\:V(G_P)\to \F_0(P)$.
Then $\sigma:=\psi^{-1}\circ\phi$ is a graph isomorphism between $G$ and $G_P$.
So, since $G\cong G_P$, each eigenpolytope of $G$ is also an eigenpolytope of $G_P$.
We can therefore choose $P_{G_P}(\theta)=P_G(\theta)$, with corresponding eigenpolytope map $\phi':=\sigma^{-1}\circ\phi$.
In sum, the outer square in the following diagram commutes:
\begin{center}
\begin{tikzcd}
G \arrow[r, "\sigma"] \arrow[d, "\phi"'] & G_P \arrow[d, "\phi'"] \arrow[ld, "\psi"'] \\
\mathllap{P:=\,}P_G(\theta) \arrow[r, "\Id"'] & P_{G_P}(\theta)
\end{tikzcd}
\end{center}
Also, by construction of $\sigma$, the upper triangle commutes.
In conclusion, the lower triangle must commute as well, which is exactly \eqref{eq:diagram_polytope} with $T=\Id$. This proves that $P$~is $\theta$-spectral via \cref{res:naive_polytope}.
Since $G$ is isomorphic to $G_P$, $G$ is $\theta$-spectral.
\end{proof}
\end{proposition}
\noindent
This also proves \cref{res:naive} $(ii)$.
\iffalse
\begin{proposition}
\label{res:naive_graph}
A graph $G$ is $\theta$-spectral if and only if it is isomorphic to the edge-graph~of its $\theta$-eigenpolytope $P:=P_G(\theta)$ via some isomorphism $\sigma\:V(G)\to V(G_P)$ for which the following diagram commutes:
\begin{equation}
\label{eq:diagram}
\begin{tikzcd}
G \arrow[rd, "\phi"'] \arrow[r, "\sigma"] & G_P \arrow[d, "\psi"] \\
& P_G(\theta)\mathrlap{\,:=P }
\end{tikzcd}
\end{equation}
\begin{proof}
First, suppose that $G$ is $\theta$-spectral.
By definition, there exists a $\theta$-spectral polytope $Q$ with edge-graph $G_Q=G$, and let $\psi_Q\:V(G_Q)\to\F_0(Q)$ be the map \eqref{eq:vertex_map} (we use subscripts since there will be further such maps).
Since $Q$ is $\theta$-spectral, \cref{res:naive} $(i)$ states that $Q$ and $P_{G_Q}(\theta)=P_G(\theta)= P$ are linearly equivalent via some map $T\in\GL(\RR^d)$.
Let $\psi_P\:V(G_P)\to\F_0(P)$ be~the corresponding map \eqref{eq:vertex_map} for $P$.
We can then define the isomorphism as $\sigma:=\psi_Q\circ T\circ \psi_P^{-1}$ (all involved maps are invertible and preserve adjacency).
The following diagram then commutes:
\begin{center}
\begin{tikzcd}
\mathllap{G=\,}G_Q \arrow[r, "\sigma"] \arrow[d, "\psi_Q"'] & G_P \arrow[d, "\psi_P"] \\
Q \arrow[r, "T"'] & P\mathrlap{\,=P_G(\theta)}
\end{tikzcd}
\end{center}
We can now insert the diagonal arrow $\phi\: V(G)\to\RR^d\supseteq P_G(\theta)$ given by \eqref{eq:eigenpolytope_map}.
\begin{center}
\begin{tikzcd}
\mathllap{G=\,}G_Q \arrow[r, "\sigma"] \arrow[d, "\psi_Q"'] \arrow[rd, "\phi"] & G_P \arrow[d, "\psi_P"] \\
Q \arrow[r, "T"'] & P\mathrlap{\,:= P_G(\theta)}
\end{tikzcd}
\end{center}
Since $Q$ is spectral, the the lower triangle commutes by \cref{res:naive_polytope}.
Since also the outer square commutes, we find that the upper triangle (which is exactly \eqref{eq:diagram}) commutes as well, and we are done.
For the other direction, suppose that there is an isomorphism $\sigma:V(G)\to V(G_P)$ so that \eqref{eq:diagram} commutes.
Since $G$ and $G_P$ are isomorphic, $P=P_G(\theta)$ can be consider as a $\theta$-eigenpolytope of $G_P$ as well, thus $P=P_{G_P}(\theta)$.
If $\phi$ is the map \eqref{eq:eigenpolytope_map} for the eigenpolytope of $G$, then because $G_P$ uses the same eigenpolytope, $\phi\circ\sigma^{-1}$ is the respective map \eqref{eq:eigenpolytope_map} for the eigenpolytope of $G_P$.
The following diagram then commutes:
\begin{center}
\begin{tikzcd}
G \arrow[rd, "\phi"'] \arrow[r, "\sigma"] & G_P \arrow[rd, "\phi\circ\sigma^{-1}"] & \\
& \mathllap{P=:\,}P_G(\theta) \arrow[r, "\Id"'] & P_{G_P}(\theta)
\end{tikzcd}
\end{center}
We can now insert the vertical arrow $\psi\: V(G_P)\to \F_0(P)$ given by \eqref{eq:vertex_map}.
\begin{center}
\begin{tikzcd}
G \arrow[rd, "\phi"'] \arrow[r, "\sigma"] & G_P \arrow[rd, "\phi\circ\sigma^{-1}"] \arrow[d, "\psi"] & \\
& \mathllap{P=:\,}P_G(\theta) \arrow[r, "\Id"'] & P_{G_P}(\theta)
\end{tikzcd}
\end{center}
The left triangle is exactly \eqref{eq:diagram}, hence commutes by assumption.
The outer parallelogram commutes by construction, and so we find that the right triangle commutes as well.
This triangle is exactly \eqref{eq:diagram_polytope} from \cref{res:naive_polytope} with $T=\Id$.
This shows that $P
$ is $\theta$-spectral, and since $G\cong G_P$ is the edge-graph of $P$, we found that $G$ is $\theta$-spectral, and we are done.
\end{proof}
\end{proposition}
\fi
\iffalse
\begin{lemma}
If $P$ is linearly equivalent to the $P_{G_P}(\theta)$, and this linear map $T\in\GL(\RR^d)$ satisfies $\Phi=\Psi T$, then $P$ is $\theta$-spectral.
\end{lemma}
\begin{lemma}
If $G$ is isomorphic to the edge-graph of $P_G(\theta)=:Q$, and the isomorphism $\sigma\:V(G)\to V(G_Q)$ saitsies $\Phi=\Pi_\sigma \Psi$, then $G$ is $\theta$-spectral.
\end{lemma}
\begin{lemma}
\label{res:naive_polytope}
If $P$ is $\theta$-spectral, then $P$ is linearly equivalent to the $\theta$-eigenpolytope of its edge-graph. The converse is not true.
However, $P$ is $\theta$-spectral if and only if $P$ is linearly equivalent to $P_{G_P}(\theta)$ via~some map $T\in\GL(\RR^d)$, and $T$ additionally satisfies $T \phi(i) = \psi(i)$ for all $i\in V$, where~$\phi\:$ $V(G_P)\to\RR^d$ and $\psi\:V(G_P)\to\F_0(P)$ are the maps defined in \eqref{eq:eigenpolytope_map} resp.\ \eqref{eq:vertex_map}.
\begin{proof}
By definition, the $\theta$-eigenpolytope of $G_P$ satisfies $\Span \Phi=\Eig_{G_P}(\theta)$, where $\Phi$ is as defined in \eqref{eq:eigenpolytope_matrix}.
Now, by definition, $P$ is $\theta$-spectral if and only if $\Span \Psi = \Eig_{G_P}(\theta)$, where $\Psi$ is its arrangement matrix \eqref{eq:arrangement_matrix}.
But by \cref{res:same_column_span}, $\Phi$ and $\Psi$ can have the same span if and only of their rows are related by some invertible linear map $T\in\GL(\RR^d)$, that is, $T\phi(i)=\psi(i)$ for all $i\in V$.
A counterexample for the initial converse was given in \cref{ex:pentagon}.
\end{proof}
\end{lemma}
\begin{corollary}
If $P$ is $\theta$-spectral, then so is $G_P$ and $P_{G_P}(\theta)$.
\end{corollary}
The respective statement for graphs is the following:
\begin{lemma}
\label{res:naive_graph}
If $G$ is $\theta$-spectral, then $G$ is isomorphic to the edge-graph of its $\theta$-eigenpolytope. The converse is not true.
However, $G$ is $\theta$-spectral if and only if it is isomorphic to the edge-graph of its $\theta$-eigenpolytope, and the isomorphism is given by $\phi\:V\to\RR^d$~as~de\-fined in \eqref{eq:eigenpolytope_map}.
\begin{proof}
Let $Q:=P_G(\theta)$ be the $\theta$-eigenpolytope of $G$.
Then we try to prove $G\cong G_Q$.
If $G$ is $\theta$-spectral, then there is a $\theta$-spectral polytope $P$ with edge-graph $G_P\cong G$.
By this, $P_{G_P}(\theta)\cong P_G(\theta)=Q$.
By \cref{res:naive_polytope}, $P$ is linearly~equivalent to~$P_{G_P}(\theta)$, hence linearly equivalent to $Q$.
In particular, $G_P\cong G_Q$, and so we found $G\cong G_P\cong G_Q$.
A counterexample for the initial converse was given in \cref{ex:pentagon}.
\end{proof}
\end{lemma}
\fi
\iffalse
\begin{corollary}
\quad
\begin{myenumerate}
\item If a polytope $P$ is $\theta$-spectral, then it is the $\theta$-eigenpolytope of its edge-graph (up to some invertible linear transformation).
\item If a graph $G$ is $\theta$-spectral, then it is (isomorphic to) the edge-graph of its $\theta$-eigenpolytope.
\item If a graph $G$ is $\theta$-spectral, then its $\theta$-eigenpolytope is $\theta$-spectral.
\end{myenumerate}
\end{corollary}
\begin{corollary}
\label{res:naive_definition}
If $P$ is $\theta$-spectral, then it is the $\theta$-eigenpolytope of its edge-graph (up to invertible linear transformation)
\begin{proof}
If $P$ is $\theta$-spectral with arrangement matrix $M'$, then by \mbox{\cref{def:spectral} $(i)$} we have $\Span M'=\Eig_{G_P}(\theta)$.
Let further $M$ be the matrix \eqref{eq:eigenpolytope_matrix} used in \cref{def:eigenpolytope} to define the $\theta$-eigenpolytope of $G_P$.
By definition we have $\Span M=\Eig_{G_P}(\theta)$.
By \cref{res:same_column_span}, since $M$ and $M'$ have the same column span, their rows are related by an invertible linear transformation.
These rows are the vertices of $P$ and $P_{G_P}(\theta)$ respectively, finishing the proof.
\end{proof}
\end{corollary}
The converse of \cref{res:naive_definition} is not true.
This is intentional, since we do not want confiurations as in \cref{ex:pentagon} to be called \enquote{spectral}.
\begin{corollary}
\label{res:spectral_graph_naive}
If a graph $G$ is $\theta$-spectral, then $\phi\:V\ni i\mapsto v_i\in\RR^d$ (as \mbox{introduced} in \cref{def:eigenpolytope}) is an isomorphism between $G$ and the edge-graph of $P_G(\theta)$.
That includes
\begin{myenumerate}
\item for all $i\in V$ holds $v_i\in\F_0(P_G(\theta))$.
\item for distinct $i,j\in V$ holds $v_i\not= v_j$.
\item $ij\in E$ if and only if $\conv\{v_i,v_j\}\in\F_1(P_G(\theta))$.
\end{myenumerate}
\begin{proof}
Since $G$ is $\theta$-spectral, it is (isomorphic to) the edge-graph of some $\theta$-spectral polytope $P$.
Recall that $n$ denotes the number of vertices of $G$.
Since $P_G(\theta)$ is the convex hull of $v_1,...,v_n$, $P_G(\theta)$ has at most $n$ vertices, and $\phi$ is surjective.
Since $P_G(\theta)$ is combinatorially equivalent to $P$ (by \cref{res:naive_definition}), $P_G(\theta)$ has exactly $n$ vertices, and $\phi$ must also be injective.
This also shows $(i)$ and $(ii)$.
\end{proof}
\end{corollary}
Note that not every graph that is isomorphic to the edge-graph of its $\theta$-eigenpoly\-tope is also $\theta$-spectral (again, consider \cref{ex:pentagon}).
However, if the isomorphism is the one from \cref{res:spectral_graph_naive}, then this conclusion is valid.
\iffalse
\begin{lemma}
If $P\subset\RR^d$ is a (full-dimensional) polytope, then the following are~equivalent:
\begin{myenumerate}
\item $P$ is $\theta$-spectral,
\item $P$ is $\theta$-balanced and the eigenvalue $\theta$ has multiplicity $d$.
\end{myenumerate}
If $G$ is a graph, then the following are equivalent:
\begin{myenumerate}
\setcounter{enumi}{2}
\item $G$ is $\theta$-spectral,
\item the $\theta$-eigenpolytope of $G$ is $\theta$-spectral with edge-graph $G$,
\item $G$ is isomorphic to the edge-graph of its $\theta$-eigenpolytope, with isomorphism $V\ni i\mapsto v_i\in\RR^d$ (as constructed in \cref{def:eigenpolytope}).
\end{myenumerate}
\begin{proof}
By \cref{res:eigenvalue}, the balancing equation \eqref{eq:balanced} is equivalent to $\Span M\subseteq\Eig_G(\theta)$.
Recall that the rank~of~$M$~is~$d$.
Now, $\theta$ having multiplicity $d$ (\shortStyle{i.e.,}\ $(ii)$) is equivalent to $\dim\Eig_G(\theta)=d=\rank M$. This is then equivalent to $M=\Eig_G(\theta)$ which is $(i)$.
If $G$ is $\theta$-spectral then it is (isomorphic to) the edge-graph of \emph{some} $\theta$-spectral polytope $P$.
But \cref{res:naive_definition} states that $P$ is (up to linear transformation) the $\theta$-eigenpolytope of its edge-graph, or, since isomorphic, the $\theta$-eigenpolytope of $G$.
\end{proof}
\end{lemma}
\fi
\fi
It is also possible to give a definition of spectral graphs purely in terms of graph theory, without any explicit reference to polytopes:
\begin{lemma}
\label{res:spectral_2}
A graph $G$ is $\theta$-spectral if and only if it satisfies both of the following:
\begin{myenumerate}
\item for each vertex $i\in V$ exists a $\theta$-eigenvector $u=(u_1,...,u_n)\in\Eig_G(\theta)$~whose single largest component is $u_i$, or equivalently,
$$\Argmax_{k\in V} u_k = \{i\}.$$
\item any two vertices $i,j\in V$ form an edge $ij\in E$ in $G$ if and only~if there is a $\theta$-eigenvector $u=(u_1,...,u_n)\in\Eig_G(\theta)$ whose only two largest components are $u_i$ and $u_j$, or equivalently,
$$\Argmax_{k\in V} u_k = \{i,j\}.$$
\end{myenumerate}
\end{lemma}
This characterization of spectral graphs can be interpreted as follows: a spectral graph can be reconstructed from knowing a single eigenspace, rather than, say, all eigenspaces and their associated eigenvalues.
\begin{proof}[Proof of \cref{res:spectral_2}]
Let $P_G(\theta)\subset\RR^d$ be the $\theta$-eigenpolytope of $G$ with eigenpolytope matrix $\Phi$ and eigenpolytope map $\phi\:V\ni i\mapsto v_i\in\RR^d$.
Since $\Span\Phi=\Eig_G(\theta)$, the eigenvectors $u=(u_1,...,u_n)\in\Eig_G(\theta)$ are exactly the vectors that can be written as $u=\Phi x$ for some $x\in\RR^d$.
If then $e_k\in\RR^n$ denotes the $k$-th standard basis vector, we have
$$u_k = \<u,e_k\> = \<\Phi x, e_k\> = \<x,\Phi^{\Tsymb}\! e_k\> = \<x, v_k\>.$$
Therefore, there is a $\theta$-eigenvector $u=(u_1,...,u_n)\in\Eig_G(\theta)$ with
$\Argmax_{k\in V} u_k = \{i_1,...,i_m\}$
if and only if there is a vector $x\in\RR^d$ with
$$\Argmax_{k\in V} \<x,v_k\> = \{i_1,...,i_m\}.$$
But this last line is exactly what it means for $\conv\{v_{i_1},...,v_{i_m}\}$ to be a face of $P_G(\theta)$ $=\conv\{v_1,...,v_n\}$ (and $x$ is a normal vector of that face).
In this light, we can interpret $(i)$ as stating that $v_1,...,v_n$ form $n$ distinct vertices of $P_G(\theta)$, and $(ii)$ as stating that $\conv\{v_i,v_j\}$ is an edge of $P_G(\theta)$ if and only if $ij\in E$.
And this means exactly that $\phi$ is a graph isomorphism between $G$ and the skeleton of $P_G(\theta)$.
By \cref{res:naive_graph}, this is equivalent to $G$ being $\theta$-spectral.
\end{proof}
\iffalse
If the graph satisfies $(i)$ and $(ii)$, then the map $\sigma:=\psi^{-1}\circ\phi:V(G)\to V(G_P)$ is bijective because of $(i)$, and a graph isomorphism because of $(ii)$.
This isomorphism makes \eqref{eq:diagram} commute by definition, and so $G$ is $\theta$-spectral by \cref{res:naive_graph}.
Conversely, if $G$ is $\theta$-spectral, then it is isomorphic to the edge-graph $G_P$ of its $\theta$-eigenpolytope via some isomorphism $\sigma\: V(G)\to V(G_P)$ that makes \eqref{eq:diagram} commute.
We first note that that implies that $P_G(\theta)$ has exactly the vertices $v_1,...,v_n$, which means that $G$ satisfies $(i)$.
Since $\sigma$ and $\psi$ are graph isomorphisms, so is $\psi\circ\sigma$, and since \eqref{eq:diagram} commutes, also $\phi$.
This means $ij\in E$ if and only if $\conv\{\phi(i),\phi(k)\}=\conv\{v_i,v_j\}\in P_G(\theta)$.
This is equivalent to $(ii)$, and we are done.
Suppose that $G$ is $\theta$-spectral, then
We show first that there is a $\theta$-eigenvectors $u=(u_1,...,u_n)\in\Eig_G(\theta)$ with
$$\Argmax_{k\in V} u_k=\{i_1,...,i_m\}$$
if and only if $\{v_{i_1},...,v_{i_m}\}$ (as defined in \cref{def:eigenpolytope}) is the vertex set of a face~of the $\theta$-eigenpolytope of $G$.
Recall that a set $\{v_{i_1},...,v_{i_m}\}\subseteq \F_0(P)$ of vertices is the vertex set of a face of $P$ if and only if there is a vector $x\in\RR^d$ (a normal vector) with
$$\Argmax_{k\in V} \<x,v_k\> = \{i_1,...,i_m\}.$$
Let $M$ be the arrangement matrix of $P$, set $u=(u_1,...,u_n):=Mx$, and let $e_k\in\RR^n$ be the $k$-th standard basis vector.
Then
$$\<x,v_k\> = \<x, M^{\Tsymb}\! e_k\> = \<M x, e_k\> =\<u,e_k\> = u_k.$$
By $u=Mx$, the $x\in\RR^d$ are in one-to-one correspondence with the $u\in\Span M=\Eig_G(\theta)$, which are exactly the $\theta$-eigenvectors of $G$.
So $\{v_{i_1},...,v_{i_m}\}$ forms a face of $P$ if and only if
$$(*)\quad \Argmax_{k\in V} u_k = \Argmax_{k\in V} \<u,v_k\> = \{i_1,...,i_m\},$$
Condition $(i)$ and $(ii)$ are now the version of $(*)$ for vertices and edges of $P$, which correspond to vertices and edges of $G$ via the isomorphism $\phi$.
\fi
In practice, to reconstruct a spectral graph from an eigenspace, the steps could be the following: given a subspace $U\subseteq\RR^n$ (the claimed eigenspace), then
\begin{myenumerate}
\item choose any basis $u_1,...,u_d\in\RR^n$ of $U$,
\item build the matrix $\Phi=(u_1,...,u_d)\in\RR^{n\x d}$ in which the $u_i$ are the columns,
\item define $v_i$ as the $i$-th \emph{row} of $\Phi$,
\item define $P:=\conv\{v_1,...,v_n\}\subset\RR^d$ as the convex hull of the $v_i$,
\item the reconstructed graph $G=G_P$ is then the edge-graph of $P$.
\end{myenumerate}
\subsection{Properties of spectral polytopes}
We discuss two properties of spectral polytopes that make them especially interesting in polytope theory.
\subsubsection*{Reconstruction from the edge-graph}
\label{sec:spectral_reconstruction}
The edge-graph of a general polytope carries little information about that polytope \shortStyle{i.e.,}\ given only its edge-graph, we can often not reconstruct the polytope from this (up to combinatorial equivalence).
Often, one cannot even deduce the dimension of the polytope from its edge-graph.
Reconstruction might be possible in certain special cases, as \shortStyle{e.g.}\ for 3-dimensional polyhedra, simple polytopes or zonotopes.
The spectral polytopes provide another such class.
\begin{theorem}\label{res:reconstruction}
A $\theta_k$-spectral polytope is uniquely~determined by its edge-graph up to invertible linear transformations.
\end{theorem}
The proof is simple:
every $\theta_k$-spectral polytope is linearly equivalent to the~$\theta_k$-eigenpolytope of its edge-graph (by \cref{res:naive} $(i)$).
Our definition of the \mbox{$\theta_k$-eigen}\-polytope already suggests an explicit procedure to construct it (a script for this~is included in \cref{sec:appendix_mathematica}).
This property of spectral polytopes appears more exciting when applied to graph classes that are not obviously spectral (see \cref{sec:edge_transitive}).
\subsubsection*{Realizing symmetries of the edge-graph}
\label{sec:spectral_symmetries}
Every Euclidean symmetry of a polytope induces a combinatorial symmetry on its edge-graph.
The converse is far from true.
Think, for example, about a rectangle that is not a square.
Even worse, it can happen that a polytope does not even have a realization that realizes all the symmetries of its edge-graph (\shortStyle{e.g.}\ the polytope constructed in \cite{bokowski1984combinatorial}).
We have previously discussed (in \cref{res:realizing_symmetries}) the existence of a homomorphism $\Aut(G)\to\Aut(P_G(\theta))$ between the symmetries of a graph $G$ and the symmetries of its eigenpolytopes.
There are two caveats:
\begin{myenumerate}
\item this is not necessarily an isomorphism, and
\item it says nothing about the symmetries of the edge-graph of $P_G(\theta)$, as this one needs not to be isomorphic to $G$
\end{myenumerate}
Still, it suffices to makes statement of the following form:
if $G$ is vertex-transitive, then so are all its eigenpolytopes.
This might not work with other transitivities, as for example edge-transitivity.
This is no concern for spectral graphs/polytopes:
\begin{theorem} \quad
\label{res:symmetries}
\begin{myenumerate}
\item If $G$ is $\theta$-spectral, then $P_G(\theta)$ realizes all its symmetries, which includes
$$\Aut(G)\cong\Aut(P_G(\theta))$$
%
via the map $\sigma\mapsto T_\sigma$ given in \cref{res:realizing_symmetries}, as wells as that $T_\sigma$ permutes the vertices and edges of $P_G(\theta)$ exactly as $\sigma$ permutes the vertices and edges of the graph $G$.
\item If $P$ is $\theta$-spectral, then $P$ has a realization that realizes all the symmetries~of its edge-graph, namely, the $\theta$-eigenpolytope of its edge-graph.
\end{myenumerate}
%
\end{theorem}
This is mostly straight forward, with large parts already addressed in \mbox{\cref{res:realizing_symmetries}}.
The major difference is that for spectral graphs $G$ the eigenpolytope has exactly the distinct vertices $v_1,...,v_n\in\RR^d$.
The statement from \cref{res:realizing_symmetries} that $T_\sigma$~per\-mutes the $v_i$ as prescribed by $\sigma$, then becomes, that $T_\sigma$ permutes the \emph{vertices} as prescribed by $\sigma$, and hence also the edges.
Also, since the $v_i$ are distinct, no non-trivial symmetry $\sigma$ can result in trivial $T_\sigma$, making $\sigma\mapsto T_\sigma$ into a group \emph{isomorphism}.
For part $(ii)$ merely recall that the eigenpolytope $P_{G_P}(\theta)$ is indeed a realization of $P$ by \cref{res:naive} $(i)$.
The major consequence of this is, that for spectral graphs/polytopes also more complicates types of symmetries translate between a polytope and its graph, as \shortStyle{e.g.}\ edge-transitivity (see also \cref{sec:edge_transitive}).
\iffalse
But, we have to be careful: we cannot expect a general statement about symmetries of spectral polytopes, as \shortStyle{e.g.}\ a rectangle and a square are both spectral (since they are linear transformations of each other), but one is apparently more symmetric, if by symmetry we understand a rigid motion.
There are two solutions: either, generalizing symmetries to general invertible linear transformations, or, restricting the polytopes about which we talk.
Both perspectives are equivalent, and we decided to choose the second.
Let us call a polytope \emph{normalized} if $M^{\Tsymb}\! M=\Id$ for its arrangement matrix $M$, that is, the columns of $M$ form a orthonormal basis of $\Span M$.
By \cref{res:same_column_span}, every polytope is just one invertible linear transformation away from being normalized.
Also, the way in which we defined eigenpolytopes in \cref{def:eigenpolytope} ensures that they are normalized.
We have the following:
\begin{theorem}
\label{res:realizing_symmetries}
A normalized spectral polytope realizes all the combinatorial symmetries of its edge-graph as Euclidean symmetries.
More precisely, we have the following: if $\sigma\in\Aut(G_P)\subseteq\Sym(V)$ is a combinatorial symmetry of the edge-graph, and $\Pi_\sigma\in\Perm(\RR^n)$ is the associated permutation matrix, then
$$T_\sigma:=M^{\Tsymb} \Pi_\sigma M \;\in\; \Ortho(\RR^d)$$
is a Euclidean symmetry of $P$ that permutes its vertices exactly as $\sigma$ permutes the vertices of $G_P$ (that is $T_\sigma v_i = v_{\sigma(i)}$).
\end{theorem}
We should mention that this result is more generally true for all eigenpolytopes of a graph and was already proven in \cite[Theorem 2.2]{godsil1978graphs}.
Another proof in the context of spectral graph realizations can be found \cite[Corollary 2.9]{winter2020symmetric}.
As a general consequence we have that if $G$ is vertex-transitive, then so are all its eigenpolytopes.
This does not extend to other kinds of symmetries, \shortStyle{e.g.}\ edge-transitivity.
However, if $G$ is $\theta$-spectral, then the symmetries of $G$ and its~$\theta$-eigenpolytope translate into each other one-to-one.
For example, $G$ is edge-transitive if and only if $P_G(\theta)$ is.
We make use of this in later sections.
\fi
\section{The Theorem of Izmestiev}
\label{sec:izmestiev}
We introduce our, as of yet, most powerful tool for proving that certain polytopes are $\theta_2$-spectral.
For this, we make use of a more general theorem by Izmestiev \cite{izmestiev2010colin}, first proven in the context of the Colin de Verdière graph invariant.
The proof of this theorem requires techniques from convex geometry, most notably, mixed volumes, which we not address here.
We need to introduce some terminology.
As before, let $P\subset\RR^d$ denote a full-dimensional polytope of dimension $d\ge 2$, with~edge-graph $G_P=(V,E),V=\{1,...,n\}$ and vertices $v_i\in \F_0(P),i\in V$.
Recall, that the \emph{polar dual} of $P$ is the polytope
$$P^\circ:=\{x\in\RR^d\mid \<x,v_i\>\le 1\text{ for all $i\in V$}\}.$$
We can replace the $1$-s in this definition by variables $c=(c_1,...,c_n)$, to obtai
$$P^\circ(c):=\{x\in\RR^d\mid\<x,v_i\>\le c_i\text{ for all $i\in V$}\}.$$
The usual polar dual is then $P^\circ=P^\circ(1,...,1)$.
\begin{figure}[h!]
\includegraphics[width=0.85\textwidth]{img/P_dual_2}
\caption{Visualization of $P^\circ(c)$ for different values of $c\in\RR^n$.}
\end{figure}
In the following, $\vol(\free)$ denotes the volume of convex sets in $\RR^d$ (\shortStyle{w.r.t.}\ the usual Lebesgue measure).
Note that the function $\vol(P^\circ(c))$ is differentiable in $c$, and so we can compute partial derivatives \shortStyle{w.r.t.}\ the components of $c$.
\begin{theorem}[Izmestiev \cite{izmestiev2010colin}, Theorem 2.4]\label{res:izmestiev}
Define a matrix $X\in\RR^{n\x n}$ with~compo\-nents
$$X_{ij}:=-\frac{\partial^2 \vol(P^\circ(c))}{\partial c_i\partial c_j}\Big|_{c=(1,...,1)}.$$
The matrix $X$ has the following properties:
\begin{myenumerate}
%
\item $X_{ij}< 0$ whenever $ij\in E(G_P)$,
\item $X_{ij}=0$ whenever $ij\not\in E(G_P)$,
\item $X\Psi=0$ (where $\Psi$ is the arrangement matrix of $P$),
\item $X$ has a unique negative eigenvalue, and this eigenvalue is simple,
\item $\dim\ker X=d$.
\end{myenumerate}
\end{theorem}
One can view the matrix $X$ as some kind of adjacency matrix of~a vertex- and edge-weighted version of $G_P$.
Part $(iii)$ states that $v$ satisfies a weighted form of the balancing condition \eqref{eq:balanced} with eigenvalue zero.
Since $\rank \Psi=d$, part $(v)$ states that $\Span \Psi$ is already the whole 0-eigenspace.
And part $(iv)$ states that zero is the second smallest eigenvalue of $X$.
\begin{theorem}\label{res:implies_spectral}
Let $X\in\RR^{n\x n}$ be the matrix defined in \cref{res:izmestiev}.
If we have
\begin{myenumerate}
\item $X_{ii}$ is independent of $i\in V(G_P)$, and
\item $X_{ij}$ is independent of $ij\in E(G_P)$,
\end{myenumerate}
then $P$ is $\theta_2$-spectral.
\begin{proof}
By assumption there are $\alpha,\beta\in\RR$, $\beta>0$, so that $X_{ii}=\alpha$ for all vertices~$i\in$ $V(G_P)$, and $X_{ij}=\beta<0$ for all edges $ij\in E(G_P)$ (we have $\beta<0$ by \cref{res:izmestiev} $(i)$).
We can write this as
$$X=\alpha \Id + \beta A\quad\implies\quad (*)\;A=\frac\alpha\beta \Id+\frac1\beta X,$$
where $A$ is the adjacency matrix of $G_P$.
By \cref{res:izmestiev} $(iv)$ and $(v)$, the matrix $X$ has second smallest eigenvalue zero of multiplicity $d$.
By \cref{res:izmestiev} $(iii)$, the columns of $M$ are the corresponding eigenvectors.
Since $\rank \Psi=d$ we find that these are all the eigenvectors and $\Span \Psi$ is the 0-eigenspace of $X$.
By $(*)$ the eigenvalues of $A$ are the eigenvalues of $X$, but scaled by $1/\beta$ and shifted by $\alpha/\beta$. Since $1/\beta <0$, the second-\emph{smallest} eigenvalue of $X$ gets mapped onto the second-\emph{largest} eigenvalue of $A$.
Therefore, $A$ (and also $G_P$) has second-largest eigenvalue $\theta_2=\alpha/\beta$ of multiplicity $d$, and $\Span \Psi$ is the corresponding eigenspace.
By definition, $P$ is then the $\theta_2$-eigenpolytope of $G_P$ and is therefore~$\theta_2$-spectral.
\end{proof}
\end{theorem}
It is unclear whether \cref{res:implies_spectral} already characterizes $\theta_2$-spectral polytopes, or even spectral polytopes in general (see also \cref{q:characterization}).
\section{Edge-transitive polytopes}
\label{sec:edge_transitive}
We apply \cref{res:implies_spectral} to edge-transitive polytopes, that is, to polytopes for~which the Euclidean symmetry group $\Aut(P)\subset\Ortho(\RR^d)$ acts transitively on the edge set $\F_1(P)$.
No classification of edge-transitive polytopes is known.
Some \mbox{edge-transitive} polytopes are listed in \cref{sec:classification}.
Despite the name of this section, we are actually going to address polytopes that are simultaneously vertex- and edge-transitive.
This is not a huge deviation from the title: as shown in \cite{winter2020polytopes}, edge-transitive polytopes in dimension $d\ge 4$ are always also vertex-transitive, and the exceptions in lower dimensions are few (a continuous family of $2n$-gons for each $n\ge 2$, and two exceptional polyhedra).
\cref{res:implies_spectral} can be directly applied to simultaneously vertex- and edge-transitive polytopes, and so we have
\begin{corollary}
\label{res:vertex_edge_transitive_cor}
A simultaneously vertex- and edge-transitive polytope is $\theta_2$-spectral.
\end{corollary}
We collect all the notable consequences in the following theorem:
\begin{theorem}\label{res:edge_vertex_transitive}
If $P\subset\RR^d$ is simultaneously vertex- and edge-transitive, then
\begin{myenumerate}
\item $\Aut(P)\subset\RR^d$ is irreducible as a matrix group.
\item $P$ is uniquely determined by its edge-graph up to scale and orientation.\footnote{This shows that $P$ is \emph{perfect}, \shortStyle{i.e.,}\ is the unique maximally symmetric realization of its combinatorial type. See \cite{gevay2002perfect} for an introduction to perfect polytopes.}
\item $P$ realizes all the symmetries of its edge-graph.
\item if $P$ has edge length $\ell$ and circumradius $r$, then
%
\begin{equation}
\label{eq:circumradius}
\frac{\ell}r = \sqrt{\frac{2\lambda_2}{\deg(G_P)}} = \sqrt{2\Big(1-\frac{\theta_2}{\deg(G_P)}\Big)},
\end{equation}
%
where $\deg(G_P)$ is the vertex degree of $G_P$, and $\lambda_2=\deg(G_P)-\theta_2$ denotes its second smallest Laplacian eigenvalue.
\item if $\alpha$ is the dihedral angle of the polar dual $P^\circ$, then
%
\begin{equation}
\label{eq:dihedral_angle}
\cos(\alpha)=-\frac{\theta_2}{\deg(G_P)}.
\end{equation}
\end{myenumerate}
\begin{proof}
The complete proof of $(i)$ and $(ii)$ has to be postponed until \cref{sec:rigidity} (see \cref{res:edge_transitive_rigid}).
Concerning $(ii)$, from \cref{res:vertex_edge_transitive_cor} and \cref{res:reconstruction} already follows that $P$ is determined by its edge-graph up to \emph{invertible linear transformations}, but not necessarily only up to scale and orientation.
Part $(iii)$ follows from \cref{res:symmetries}.
Part $(iv)$ and $(v)$ were proven (in a more general setting) in \cite[Proposition 4.3]{winter2020symmetric}.
This applies literally to $(iv)$.
For $(v)$, note the following: if $\sigma_i\in\F_{d-1}(P^\circ)$ is the facet of the polar dual $P^\circ$ that corresponds to the vertex $v_i\in\F_1(P)$, then the dihedral angle between $\sigma_i$ and $\sigma_j$ is $\pi-\angle(v_i,v_j)$.
The latter expression was proven in \cite{winter2020symmetric} to agree with \eqref{eq:dihedral_angle}.
\end{proof}
\end{theorem}
It is worth emphasizing that large parts of \cref{res:edge_vertex_transitive} do not apply to polytopes of a weaker symmetry, as \shortStyle{e.g.}\ vertex-transitive polytopes.
Prisms are counterexamples to both $(i)$ and $(ii)$.
There are vertex-transitive neighborly polytopes (other than simplices) and they are counterexamples to $(ii)$ and $(iii)$.
\iffalse
A simultaneously vertex- and edge-transitive polytope can neither deformed nor projected to keep its edge-graph and symmetry.
The fact that $P$ is perfect can already be proven
from classical rigidity results and thus can be formulated for a much larger class of polytopes:
\begin{theorem}[Alexandrov, \cite{...}]
\label{res:alexandrov_rigidity}
A polyope is uniquely determined (up to orientation) by its combinatorial type and the shape of its facets.
\end{theorem}
\begin{theorem}
An inscribed polytope with all edges of length $\ell$ is uniquely determined (up to orientation) by its combinatorial type.
\begin{proof}
The proof is by induction.
A 2-dimensional inscribed polytope with all edges of length $\ell$ ic clearly a regular polygon, hence of unique shape.
Now consider $P$ is a $d$-dimensional inscribed polytope with all edges of length $\ell$.
Then also each facet of $P$ is inscribed with this edge-length.
Also, the combinatorial type of each facet is determined by the combinatorial type of $P$.
By induction assumption, the shape of each facet is hence uniquely determined.
By \cref{res:alexandrov_rigidity} this uniquely determines the shape of $P$.
\end{proof}
\end{theorem}
\begin{corollary}
A simultaneously vertex- and edge-transitive polytope is perfect.
\end{corollary}
\fi
\begin{remark}
\label{rem:edge_not_vertex}
There are two edge-transitive polyhedra that are not vertex-transitive: the \emph{rhombic dodecahedron} and the \emph{rhombic triacontahedron} (see also \cref{fig:edge_transitive}).~Only the former is $\theta_2$-spectral, and the latter is not spectral for any eigenvalue (this was already mentioned in \cite{licata1986surprising}).
Since the rhombic dodecahedron is not vertex-transitive, nothing of this follows from \cref{res:vertex_edge_transitive_cor}.
However, this polytope satisfies the conditions of \cref{res:implies_spectral}, which seems purely accidental.
It is the only known spectral polytope that is not vertex-transitive.
\end{remark}
\subsection{Rigidity and irreducibility}
\label{sec:rigidity}
The goal of this section is to prove the missing part of \cref{res:edge_vertex_transitive}:
\begin{theorem}
\label{res:edge_transitive_rigid}
If $P\subset\RR^d$ is simultaneously vertex- and edge-transitive, then
\begin{myenumerate}
\item $\Aut(P)\subset\Ortho(\RR^d)$ is irreducible as a matrix group, and
\item $P$ is determined by its edge-graph up to scale and orientation.
\end{myenumerate}
\end{theorem}
To prove \cref{res:edge_transitive_rigid}, we make use of \emph{Cauchy's rigidity theorem} for polyhedra (with its beautiful proof listed in \cite[Section 12]{aigner2010proofs}).
It states that every~polyhedron is uniquely determined by its combinatorial type and the shape of its faces.
This was generalized by Alexandrov to general dimensions $d\ge 3$ (proven \shortStyle{e.g.}\ in \cite[Theorem 27.2]{pak2010lectures}):
\begin{theorem}[Alexandrov]
\label{res:alexandrov}
Let $P_1,P_2\subset\RR^d, d\ge 3$ be two polytopes, so that
\begin{myenumerate}
\item $P_1$ and $P_2$ are combinatorially equivalent via a face lattice~isomorphism~$\phi:\F(P_1)\to\F(P_2)$, and
\item each facet $\sigma\in\F_{d-1}(P_1)$ is congruent to the facet $\phi(\sigma)\in\F_{d-1}(P_2)$.
\end{myenumerate}
Then $P_1$ and $P_2$ are congruent, \shortStyle{i.e.,}\ are the same up to orientation.
\end{theorem}
\begin{proposition}
\label{res:regular_rigid}
Let $P_1,P_2\subset\RR^d$ be two combinatorially equivalent polytopes,~each of which has
\begin{myenumerate}
\item all vertices on a common sphere (\shortStyle{i.e.,}\ is inscribed), and
\item all edges of the same length $\ell_i$.
\end{myenumerate}
Then $P_1$ and $P_2$ are the same up to scale and orientation.
\begin{proof}[Proof.\!\!]\footnote{This proof was proposed by the user \emph{Fedor Petrov} on MathOverflow \cite{petrovMO}.}
W.l.o.g.\ assume that $P_1$ and $P_2$ have the same circumradius, otherwise~re\-scale $P_2$.
It then suffices to show that $P_1$ and $P_2$ are the same up to orientation.
We proceed with induction by the dimension $d$.
The induction base is given by $d=2$, which is trivial, since any two inscribed polygons with constant edge length are regular and thus completely determined (up to scale and orientation) by their number of vertices.
Suppose now that $P_1$ and $P_2$ are combinatorially equivalent polytopes of dimension $d\ge 3$ that satisfy $(i)$ and $(ii)$.
Let $\phi$ be the face lattice isomorphism between them.
Let $\sigma\in\F_{d-1}(P_1)$ be a facet of $P_1$, and $\phi(\sigma)$ the corresponding facet in $P_2$.
In particular, $\sigma$ and $\phi(\sigma)$ are combinatorially equivalent.
Furthermore, both $\sigma$ and $\phi(\sigma)$ are of dimension $d-1$ and satisfy $(i)$ and $(ii)$. This is obvious for $(ii)$, and for $(i)$ recall that facets of inscribed polytopes are also inscribed.
By induction hypothesis, $\sigma$ and $\phi(\sigma)$ are then congruent.
Since this holds for all facets $\sigma\in\F_{d-1}(P_1)$, \cref{res:alexandrov} tells us that $P_1$ and $P_2$ are congruent, that is, the same up to orientation.
\end{proof}
\end{proposition}
We can now prove the main theorem of this section:
\begin{proof}[Proof of \cref{res:edge_transitive_rigid}]
By \cref{res:edge_vertex_transitive} the combinatorial type of $P$ is determined by its edge-graph.
By vertex-transitivity, all vertices are on a sphere.
By edge-transitivity, all edges are of the same length.
We can then apply \cref{res:regular_rigid} to obtain that $P$ is unique up to scale and orientation.
This proves $(ii)$.
Suppose now, that $\Aut(P)$ is not irreducible, but that $\RR^d$ decomposes as $\RR^d=W_1\oplus W_2$ into non-trivial orthogonal $\Aut(P)$-invariant subspaces.
Let $T_\alpha\in\GL(\RR^d)$ be the linear map that acts as identity on $W_1$, but as $\alpha\Id$ on $W_2$ for some $\alpha >1$.
Then $T_\alpha P$ is a non-orthogonal linear transformation of $P$ (in particular, combinatorially equivalent), on which $\Aut(P)$ still acts vertex- and edge-transitively.
By $(ii)$, this cannot be. Hence $\Aut(P)$ must be irreducible, which proves $(i)$.
\end{proof}
\subsection{A word on classification}
\label{sec:classification}
Despite the simple appearance of the definition of an edge-transitive polytope,
no classification was obtained so far.
There exists a classification of the 3-dimension edge-transitive polyhedra: besides the Platonic solids, these are the ones shown in \cref{fig:edge_transitive} (nine in total).
\begin{figure}[h!]
\includegraphics[width=0.75\textwidth]{img/edge_transitive_polyhedra}
\caption{From left to right, these are: the cuboctahedron, the icosido\-decahedron, the rhombic dodecahedron, and the rhombic triacontahedron.}
\label{fig:edge_transitive}
\end{figure}
There are many known edge-transitive polytopes in dimension $d\ge 4$ (so we~are not talking about a class as restricted as the regular polytopes).
There are 15 known edge-transitive 4-polytopes (and an infinite family of duoprisms\footnote{The $(n,m)$-duoprism is the cartesian product of a regular $n$-gon and a regular $m$-gon. Those are edge-transitive if and only of $n=m$. Technically, the 4-cube is the $(4,4)$-duoprism but is usually not counted as such, because of its exceptionally large symmetry group.}), but already here, no classification is known.
It is known that the number of irreducible\footnote{Being not the cartesian product of lower dimensional edge-transitive polytopes.} edge-transitive polytopes grows at least linearly with the number of dimensions.
For example, there are $\lfloor d/2\rfloor$ \emph{hyper-simplices} in dimension $d$.
These are edge-transitive (even distance-transitive, see \cref{sec:distance_transitive}).
It is the hope of the author, that the classification of the edge-transitive polytopes can be obtained using their spectral properties.
Their classification can now be stated purely as a problem in spectral graph theory:
the classification of the edge-transitive polytopes (in dimension $d\ge 4$) is equivalent to the classification of $\theta_2$-spectral edge-transitive graphs, and since \cref{res:spectral_2}, we have a completely graph theoretic characterization of spectral graphs.
\begin{theorem}
\label{res:edge_transitive_spectral_graph}
Let $G$ be an edge-transitive graph. If $G$ is $\theta_k$-spectral, then
\begin{myenumerate}
\item $k=2$, and
\item if $G$ is \ul{not} vertex-transitive, then $G$ is the edge-graph of the rhombic dodecahedron (see \cref{fig:edge_transitive}).
\end{myenumerate}
\begin{proof}
We first prove $(ii)$.
As shown in \cite{winter2020polytopes} all edge-transitive polytopes in dimen\-sion $d\ge 4$ are vertex-transitive.
If $G$ is edge-transitive, not vertex-transitive and $\theta_k$-spectral, then its $\theta_k$-eigenpolytope is also edge-transitive but not vertex-transitive, hence of dimension $d\le 3$.
One checks that the 2-dimensional spectral polytopes are regular polygons, hence vertex-transitive.
The remaining polytopes are polyhedra, and we mentioned in \cref{rem:edge_not_vertex} that among these, only the rhombic dodecahedron is spectral, in fact $\theta_2$-spectral.
This proves $(ii)$.
Equivalently, if $G$ is vertex- and edge-transitive, then so is its eigenpolytope.
By \cref{res:vertex_edge_transitive_cor} this is a $\theta_2$-eigenpolytope.
Together with part $(ii)$, we find $k=2$~in~all cases, which proves $(i)$.
\end{proof}
\end{theorem}
\subsection{Arc- and half-transitive polytopes}
\label{sec:arc_transitive}
In a graph or polytope, an \emph{arc} is~an incident vertex-edge-pair.
A graph or polytope is called \emph{arc-transitive} if its symmetry group acts transitively on the arcs.
Being arc-transitive implies both, being vertex-transitive, and being edge-transitive.
In addition to that, in an arc-transitive graph, every edge can be mapped, not only onto every other edge, but also onto itself with flipped orientation.
There exist graphs that are simul\-taneously vertex- and edge-transitive, but not arc-transitive.
Those are called \emph{half-transitive} graphs, and are comparatively rare.
The smallest one has $27$ vertices and is known as the \emph{Holt graph} (see \cite{bouwer1970vertex,holt1981graph}).
For polytopes on the other hand, it is unknown whether there eixsts a distinction being arc-transitive and being simultaneously vertex- and edge-transitive.
No \emph{half-transitive polytope} is known.
Because of \cref{res:edge_vertex_transitive} $(i)$, we know that the edge-graph of a half-transitive polytope must itself be half-transitive.
Since such graphs are rare, the existence of half-transitive polytopes seems unlikely.
\begin{example}
The Holt graph is not the edge-graph of a half-transitive polytope: the Holt graph is of degree four, and its second-largest eigenvalue is of multiplicity six, giving rise to a 6-dimensional $\theta_2$-eigenpolytope.
But a 6-dimensional polytope must have an edge-graph of degree at least six, and so the Holt graph is not spectral.
\end{example}
The lack of examples of half-transitive polytopes means that all known edge-transitive polytopes in dimension $d\ge 4$ are in fact arc-transitive.
Likewise, a~classification of arc-transitive polytopes is not known.
\iffalse
\subsection{Half-transitive polytopes}
\label{sec:half_transitive}
Arc-transitivity implies vertex- and edge-transi\-ti\-vity, but for graphs, the converse is not true.
A graph (and we shall adopt this terminology for polytopes) is called \emph{half-transitive}, if it is both vertex- and edge-transitive, but not arc-transitive.
In a half-transitive graph, each edge can be mapped onto any other edge, but not onto itself with flipped orientation.
The existence of half-transitive graphs was proven first by Holt \cite{holt1981graph}.
Such graphs are relatively rare. The smallest half-transitive graph is now known as the Holt graph and has 27 vertices.
It is not known whether there are any \emph{half-transitive polytopes}.
However, by \cref{res:...} we can now conclude that they are at least as rare as the half-transitive graphs.
\begin{corollary}
The edge-graph of half-transitive polytope must be half-transitive.
\end{corollary}
It is known that the Holt graph is not the edge-graph of a half-transitive polytope.
\fi
\subsection{Distance-transitive polytopes}
\label{sec:distance_transitive}
Our previous results about edge-transitive polytopes already allow for a complete classification of a particular subclass, namely, the \emph{distance-transitive polytopes}, thereby also providing a list of examples of edge-transitive polytopes in higher dimensions.
The distance-transitive symmetry is usually only considered for graphs, and the distance-transitive graphs form a subclass of the distance-regular graphs.
The usual reference for these is the classic monograph by Brouwer, Cohen and Neumaier \cite{brouwer1989distance}.
For any two vertices $i,j\in V$ of a graph $G$, let $\dist(i,j)$ denote the graph-theoretic \emph{distance} between those vertices, that is, the length of the shortest path connecting them.
The \emph{diameter} $\diam(G)$ of $G$ is the largest distance between any two vertices in $G$.
\begin{definition}
A graph is called \emph{distance-transitive} if $\Aut(G)$ acts transitively on each of the sets
$$D_{\delta}:=\{(i,j)\in V\times V \mid \dist(i,j)=\delta\},\quad\text{for all $\delta\in\{0,...,\diam(G)\}$}.$$
Analogously, a polytope $P\subset\RR^d$ is said to be \emph{distance-transitive}, if its Euclidean symmetry group $\Aut(P)$ acts transitively on each of the sets
$$D_{\delta}:=\{(v_i,v_j)\in \F_0(P)\times \F_0(P) \mid \dist(i,j)=\delta\},\quad\text{for all $\delta\in\{0,...,\diam(G_P)\}$}.$$
Note that the distance between the vertices is still measured along the edge-graph rather than via the Euclidean distance.
\end{definition}
Being arc-transitive is equivalent to being transitive on the set $D_1$.
Hence, distance-transitivity implies arc-transitivity, thus edge-transitivity.
By our considerations in the previous sections, we know that the \mbox{classification} of distance-transitive polytopes is equivalent~to the classification of the $\theta_2$-spectral distance-transitive graphs.
Those where classified by Godsil (see \cref{res:spectral_distance_regular_graphs}).
In the following theorem we translated each such $\theta_2$-spectral distance-transitive graph into its respective eigenpolytope.
This gives a complete classification of the distance-transitive polytopes.
\begin{theorem}\label{res:distance_transitive_classification}
If $P\subset\RR^d$ is distance-transitive, then it is one of the following:
\begin{enumerate}[label=$(\text{\roman*}\,)$]
\item a regular polygon $(d=2)$,
\item the regular dodecahedron $(d=3)$,
\item the regular icosahedron $(d=3)$,
\item a cross-polytopes, that is, $\conv\{\pm e_1,...,\pm e_d\}$ where $\{e_1,...,e_d\}\subset\RR^d$ is the standard basis of $\RR^d$,
\item a hyper-simplex $\Delta(d,k)$, that is, the convex hull of all vectors $v\in\{0,1\}^{d+1}$ with exactly $k$ 1-entries,
\item a cartesian power of a regular simplex (also known as the Hamming polytopes; this includes regular simplices and hypercubes),
\item a demi-cube, that is, the convex hull of all vectors $v\in\{-1,1\}^d$ with~an~even number of 1-entries,
\item the $2_{21}$-polytope, also called Gosset-polytope $(d=6)$,
\item the $3_{21}$-polytope, also called Schläfli-polytope $(d=7)$.
\end{enumerate}
The ordering of the polytopes in this list agrees with the ordering of graphs in the list in \cref{res:spectral_distance_regular_graphs}.
The latter two polytopes where first constructed by Gosset in \cite{gosset1900regular}.
\end{theorem}
We observe that the list in \cref{res:distance_transitive_classification} contains many polytopes that are not regular, and contains all regular polytopes excluding the 4-dimensional exceptions, the 24-cell, 120-cell and 600-cell.
The distance-transitive polytopes thus form a distinct class of remarkably symmetric polytopes which is not immediately related to the class of regular polytopes.
Another noteworthy observation is that all the distance-transitive polytopes are \emph{Wythoffian poly\-topes}, that is, they are orbit polytopes of finite reflection groups.
\Cref{fig:distance_transitive_Coxeter} shows the Coxeter-Dynkin diagrams of these polytopes.
\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{img/distance_transitive_Coxeter}
\caption{Coxeter-Dynkin diagrams of distance-transitive polytopes.}
\label{fig:distance_transitive_Coxeter}
\end{figure}
\iffalse
Finally, we can show that \cref{res:distance_transitive_classification} is the final word on distance-transitive (actually distance-regular) graphs that are spectral: we can show that such a graph cannot be $\theta_k$-spectral for any $k\not=2$.
\begin{theorem}
\label{res:distance_regular_theta_k}
If $G$ is distance-transitive (actually distance-regular) and $\theta_k$-spectral, then $k=2$.
\end{theorem}
The proof uses some special properties of distance-transitive (actually distance-regular graphs) that we are citing in place.
In general, we recommend the chapters 11 and 13 of \cite{godsil1993algebraic}, the latter of which contains all the tools we need.
\begin{proof}[Proof of \cref{res:distance_regular_theta_k}]
The $\theta_1$-eigenpolytope of any graphs is a single point.
Since we generally assumed that our polytopes ar non-trivial, \shortStyle{i.e.,}\ of dimension $d\ge 2$, we from now on assume $k\ge 2$.
Let~$v_i\in$ $\RR^d$ be the coordinates assigned to the vertices $i\in V$ of $G$ as given~in~\cref{def:eigenpolytope}.
We know that the arrangement of the points $v_i$ in $\RR^d$ is as symmetric as the underlying graph, which in the case of distance-transitive graphs means that the value of $\<v_i,v_j\>$ depends only on the distance $\dist(i,j)$ (more generally, the same holds for distance-regular graphs).
We can then define the so-called \emph{cosine sequence} $u_\delta:=\<v_i,v_j\>$ whenever $\dist(i,j)$ $=\delta$.
A well-known result for distance-transitive (actually distance-regular) graphs is, that the sequence $u_0,...,u_{\diam(G)}$ has \emph{exactly} $k-1$ sign changes (see \cite{godsil1993algebraic}, Chapter 13, Lemma 2.1).
It remains to show, that if the $v_i$ embedd $G$ as the skeleton of a polytope, then there can be at most one signs change, hence $k=2$.
\begin{center}
\includegraphics[width=0.4\textwidth]{img/5_gon_cut}
\end{center}
This property is occasionally known as the \emph{tightness} of the skeleton of a polytope.
It says that the skeleton cannot be cut into more than two pieces by any hyperplane.
We give a formal proof below.
Suppose that there are at least two signs changes in $u_\delta$. This means, there exist $\delta>\delta'>0$ with $u_0u_{\delta'}<0$ and $u_{\delta'}u_{\delta}<0$, or equivalently $u_0 > u_\delta > 0 > u_{\delta'}$.
Suppose that the $v_i$ embedd $G$ as the skeleton of a polytope $P$.
Consider the linear optimization problem maximizing the linear functional $\<v_i,\free\>$ over $P$, which clearly attains its maximum in $v_i$.
Let $j\in V$ be a vertex with $\dist(i,j)=\delta$.
Applying the simplex algorithm yields a monotone path in the skeleton from $v_j$ to $v_i$.
But any such path must pass though a vertex $j'\in V$ with $\dist(i,j')=\delta'<\delta$. But we know that $\<v_i,v_{j'}\> = u_{\delta'} < 0 < u_{\delta}$.
So the path cannot have been monotone.
This is a contradiction, showing that there cannot have been two sign changes.
\end{proof}
Whether a similar argument is possible for larger classes of graphs is unknown.
\fi
\iffalse
\hrulefill
Let $G=(V,E)$ be a (simple, undirected) graph on the vertex set $V=\{1,...,n\}$.
From any eigenvalue $\theta\in\Spec(G)$ of $G$ (\shortStyle{i.e.,}\ of its adjacency matrix $A$) of multiplicity $d$, we can construct a so-called \emph{spectral realization} (or \emph{$\theta$-realization}) $v\: V\to\RR^d$ of $G$ as follows.
Let $\{u_1,...,u_d\}\subset\RR^n$ be an orthonormal basis of the $\theta$-eigenspace of $G$ and define the \emph{arrangement matrix} $M:=(u_1,...,u_d)\in\RR^{n\x d}$ in which the $u_i$ are the columns.
This matrix has one row per vertex, and so we can obtain a realization $v$ by assigning $i\in V$ to the $i$-th row of $M$:
$$M :=
\begin{pmatrix}
\;\rule[.5ex]{2.5ex}{0.4pt}\!\!\!\! & v_1^{\Tsymb} & \!\!\!\!\rule[.5ex]{2.5ex}{0.4pt}\;\; \\
& \vdots & \\[0.4ex]
\;\rule[.5ex]{2.5ex}{0.4pt}\!\!\!\! & v_n^{\Tsymb} & \!\!\!\!\rule[.5ex]{2.5ex}{0.4pt}\;\;
\end{pmatrix}\mathrlap{\in\RR^{n\times d}.}$$
The realization $V\ni i\mapsto v_i$ is called $\theta$-realization of $G$. The $\theta$-realization is unique up to re-orientation (depending only on the choice of the basis $\{u_1,...,u_d\}$).
\begin{definition}
The \emph{$\theta$-eigenpolytope} $P_G(\theta)$ of a graph $G$ is the convex hull of the $\theta$-realization $v$ of $G$:
$$P_G(\theta):=\conv\{v_i\mid i\in V\}.$$
\end{definition}
\fi
\iffalse
\section{The setting}
\label{sec:setting}
Throughout the paper, $P\subset\RR^d$ denotes a $d$-dimensional polytope of full dimen\-sion, that is, $P$ is the convex hull of finitely many points $v_1,...,v_n\in\RR^d$ (its \emph{vertices}), and $P$ is not contained in a proper affine subspace of $\RR^d$.
In particular, we assume the vertice to be labeled with the numbers $1$ to $n$.
By $\F_\delta(P)$, we denote the $\delta$-dimensional faces of $P$, in particular, its vertex set $\F_0(P)$, and its edge set $\F_1(P)$.
By $G_P=(V,E)$ we denote the \emph{edge-graph} of $P$, with vertex set $V=\{1,...,n\}$, so that $i\in V$ corresponds to the vertex $v_i\in \F_0(P)$, and each $ij\in E$ correspond to the edge $\conv\{v_i,v_j\}\in\F_1(P)$.
The term vertex (resp.\ edge) will be used for the elements of $V$ and $\F_0(P)$ (resp.\ $E$ and $\F_1(P)$) likewise, but no confusion is to be expected as these are in a one-to-one correspondence.
Finally, there are two matrices of major importance: for every polytope $P\subset\RR^d$ with labeled vertices $v_1,...,v_n$, the matrix
$$M :=
\begin{pmatrix}
\;\rule[.5ex]{2.5ex}{0.4pt}\!\!\!\! & v_1^{\Tsymb} & \!\!\!\!\rule[.5ex]{2.5ex}{0.4pt}\;\; \\
& \vdots & \\[0.4ex]
\;\rule[.5ex]{2.5ex}{0.4pt}\!\!\!\! & v_n^{\Tsymb} & \!\!\!\!\rule[.5ex]{2.5ex}{0.4pt}\;\;
\end{pmatrix}\mathrlap{\in\RR^{n\times d}}$$
is called \emph{arrangement matrix} of $P$ (the vertices $v_i$ are the rows of $M$).
The matrix $A\in\{0,1\}^{n\x n}$ with
$$A_{ij}=\begin{cases} 1 & \text{if $ij\in E$}\\0&\text{if $ij\not\in E$}\end{cases}$$
is the usual \emph{adjacency matrix} of $G_P$.
\section{Two open questions}
Most of this book talks about the relation between polytopes and their edge-graphs.
It is well-known, that not every graphs is the edge-graph of polytope.
Since Steinitz, we know that the edge-graphs of polyhedra (3-dimensional polytopes) are exactly the 3-connected planar graphs, and each such graph determines a single combinatorial type of polytopes.
In contrast, no characterization of polytopes in dimension $d\ge 4$ is known, and even worse, if a graph is an edge-graph, then in general, it tells us little about the polytope.
That means, there can be many combinatorially distinct polytopes with the same edge-graph, even of the same dimension.
As a light in the dark, we have the result of Blind \& Mani (later refined by Kalai), that every \emph{simple} polytope (that is, a $d$-dimensional polytope whose edge-graph is of degree $d$) is uniquely determined by its edge-graph.
Similar results are known for other classes of polytopes, \shortStyle{e.g.}\ zonotopes.
In this paper, we add another class of polyopes to that list: \emph{spectral polytopes}.
Given the definition of spectral polytope, this is not too surprising (it sounds like an algorithm to compute the polytope from its edge-graph), but the interesting bit is in the graphs that turn out to be spectral and which would not have been expected to be uniquely determined otherwise.
For example, consider highly symmetric polytopes.
Each regular polytopes is uniquely determined (among regular polytopes) by itd edge-graph. In other words, there are no two regular polytopes with the same edge-graph.
On the other hand, there are many distinct vertex-transitive polytopes with the same edge-graph (\shortStyle{e.g.}\ any two vertex-transitive $n$-vertex neighborly polytopes).
So regularity is too strong to be interesting, but vertex-transitivity is too weak.
In this paper, we show that edge-transitivity also already gives unique reconstruction from the edge-graph, since they are spectral.
\fi
\iffalse
\section{Balanced and spectral polytopes}
\label{sec:balanced}
We now come back to the curious observations, that sometimes, a polytope is the eigenpolytope of its edge-graph.
A polytope with this property will be called \emph{spectral}.
However, for this section, we decided to go a different path, and introduce spectral polytopes from a different perspective: via balanced polytopes.
The overall goal of this section is to investigate how to can find out whether a given polytope already is spectral.
The precursor to spectral polytopes are the balanced polytopes that we introduce now.
\begin{definition}
$P\subset\RR^d$ is called \emph{$\theta$-balanced} (or just \emph{balanced})~for~a~num\-ber $\theta\in\RR$ if
\begin{equation}\label{eq:balanced}
\sum_{\mathclap{j\in N(i)}} v_j=\theta v_i,\quad \text{for all $i\in V$}.
\end{equation}
The value $\theta$ is actually an eigenvalue of $G$ (see \cref{res:eigenvalue} below).
If the~multi\-plicity of this eigenvalue $\theta$ is exactly $d$, then $P$ is called \emph{$\theta$-spectral} (or just \emph{spectral}).
\end{definition}
\begin{example}
All neighborly polytopes are balanced (if the barycenter of its vertices is the origin).
We can see this as follows:
$$\sum_{\mathclap{j\in N(i)}} v_j = -v_i + \underbrace{\sum_{\mathclap{j\in V}} v_j}_{=0} = -v_i.$$
Note that the spectrum of $K_n$ is $\{(-1)^{n-1},(n-1)^1\}$, and so $-1$ is indeed the second largest eigenvalue.
\end{example}
\begin{example}
All regular polytope are $\theta_2$-spectral.
This was first shown by Licata and Powers \cite{licata1986surprising} for all regular polytopes excluding the 4-dimensional exceptions (the 24-cell, 120-cell and 600-cell).
For the remaining polytopes this was proven in \cite{winter...} (Theorem ??).
Alternatively, the same follows from \cref{res:edge_vertex_transitive}.
Other polytopes that were recognized as spectral by Licata and Powers were the
\emph{rhombic dodecahedron}, ... \msays{TODO}
\end{example}
\begin{example}
Probably all uniform polytopes have a realization that is spectral.
\end{example}
Note that all these polytopes are spectral to the \emph{second-largest eigenvalue $\theta_2$}.
This is plausible for different reasons, but no proof of this fact is currently known.
\begin{observation}
Nodal domains ...
\end{observation}
Being balanced can be interpreted in at least two different way.
One of which is in terms of rigidity theory (and applies more generally to graph realizations rather than just to skeleta of polytopes).
The other is purely in terms of spectral graph theory.
\begin{remark}
Being $\theta$-balanced can also be interpreted in the context of rigidity theory.
For an edge $ij\in E$, the vector $v_j-v_i$ is pointing from the vertex $v_i$ along the edge $ij$ and can be interpreted as a force pulling $v_i$ along this edge.
At each vertex, the sum of these forces behaves as follows:
\begin{equation}\label{eq:self_stress}
\sum_{\mathclap{j\in N(i)}} (v_j-v_i) = \sum_{\mathclap{j\in N(i)}} v_j - \deg(G)v_i =\big(\theta-\deg(G)\big) v_i.
\end{equation}
In the language of rigidity theory: if we put the ...
Equation \eqref{eq:self_stress} can also be written as $LM=\theta M$, where $L$ is the Laplacian of $G$.
\end{remark}
\begin{remark}
Let us call a graph \emph{$\theta$-spectral} if it is the edge-graph of a $\theta$-spectral polytope.
We can characterize such graphs purely in the language of spectral graph theory.
A graph is $\theta$-spectral if $i,j\in V$ are adjacent in $G$ if and only if there exists a $\theta$-eigenvector $u\in\RR^n$ that is maximized exactly on $i$ and $j$:
$$\Argmax_{k\in V} u_k = \{i,j\}.$$
\end{remark}
Now the central observation concerning balanced polytopes, paving the way to spectral polytopes, is the following:
\begin{observation}\label{res:eigenvalue}
The defining equation \eqref{eq:balanced} can be written as $AM=\theta M$, where $A\in\RR^{n\x n}$ (where the matrices $A$ and $M$ are defined as in \cref{sec:setting}).
In this form, it is apparent that $\theta$ is an eigenvalue of $A$, and the columns of $M$ are the corresponding eigenvectors.
\end{observation}
Since $P$ is assumed to have full dimension, the matrix $M$ has rank $d$, \shortStyle{i.e.,}\ its columns are linearly independent eigenvectors.
But they not necessarily span the whole eigenspace.
If they do, this is called a spectral polytope:
\begin{definition}
A $\theta$-balanced polytope $P\subset\RR^d$ is called \emph{$\theta$-spectral} (or just \emph{spectral}), if any of the following equivalent conditions is satisfied:
\begin{myenumerate}
\item $\theta\in\Spec(G_P)$ has multiplicity $d$ (the dimension of $P$),
\item the columns of $M$ span the $\theta$-eigenspace of $G_P$.
\end{myenumerate}
\end{definition}
\subsection{Applications}
There are at least two interesting facts about spectral polytopes.
\begin{corollary}
A spectral polytope realizes all the symmetries of its edge-graph.
\end{corollary}
\begin{corollary}
A $\theta_2$-spectral polytope is uniquely determined by its edge-graph (among $\theta_2$-spectral polytopes).
\end{corollary}
Of course, $\theta_2$ can be replaced by any $\theta_i$.
\fi
|
train/arxiv
|
BkiUdMM5qhLBAgyet4wH
| 5 | 1 |
\section{Introduction}
As targeted data rates exceed $400$~Gbps, simple but powerful hard-decision (HD) forward-error-correction (FEC) codes has gained renewed interest in optical transport networks (OTNs). Staircase codes (SCCs)~\cite{Smith2012}, which use iterative bounded-distance-decoding (BDD), is a family of HD-FEC codes that is particularly popular in optical communications. So far, SCCs have been recommended in several OTN standards, e.g., ITU-T G.709.2/Y.133.2~\cite{G709.2}, G.709.3/Y.133.3~\cite{G709.3} and OIF 400ZR Implementation Agreement~\cite{OIF400G}.
Although SCCs can achieve quite high coding gains (compared to other HD-FEC codes), there still exists a significant performance loss with respect to the achievable information rate (AIR) of HD-FEC as well as to the AIR of soft-decision (SD) FEC. For this reason, the design of improved SCC decoders is a hot topic in recent years, with a particular focus on good performance-complexity tradeoffs.
As each bit in SCCs is protected by two component codewords, one idea for improving the performance of SCCs is to prevent miscorrections by checking conflicts between the two component codewords~\cite{Christian1} or solving so-called \emph{stall patterns} via bit flipping (BF) through the intersections of nonzero-syndrome component words~\cite{Holzbaur2017}. Although these methods are simple as they only operate on binary messages, their gains are limited. To obtain a higher gain, an extreme solution is to completely replace the BDD component decoder with an SD decoder~\cite{Douxin_ISTC2018,CondoOFC2020}. However, this solution greatly increases decoding complexity.
Recently, a new class of decoding algorithms---often called soft-aided HD (SA-HD) decoders---have been shown to provide a good compromise between complexity and performance. The main idea is to \emph{assist} the HD decoding by using channel soft information, i.e., log-likelihood ratios (LLRs). For example,~\cite{Alireza} proposed to make a hard decision based on the weighted sum of the BDD output and the channel LLR, while~\cite{AlirezaarXiv2019} replaced BDD with generalized minimum distance decoding in which the erasures are determined via channel LLRs. Lately, an extension work of~\cite{Alireza}, called BEE-SCC, is reported in~\cite{AlirezaSCC2020}. BEE-SCC refined the reliability combining rule via density evolution and performed an extra decoding attempt based on error and erasure decoding of the component codes.
In~\cite{YiISTC2018,YiTCOM2019}, a soft-aided bit-marking (SABM) algorithm was proposed for SCCs. We will refer to this algorithm as SABM-SCC. Differently from the methods proposed in~\cite{Alireza,AlirezaarXiv2019,AlirezaSCC2020}, the soft information in SABM-SCC is only used to mark bits as highly reliable bits (HRBs) and highly unreliable bits (HUBs). The marked bits are then used to deal with both miscorrections and failures of BDD. Although the SABM-SCC decoder was shown to achieve considerable gains, its implementation is not straightforward, as sorting bits by reliability is required for every row of an SCC block for marking HUBs. With the idea of targeting hardware-friendly implementations, an improved SABM (iSABM) algorithm was recently introduced for SCCs (called iSABM-SCC)~\cite{Yi_iSABM2021}. The iSABM-SCC decoder uses two thresholds to classify the bits into HRBs, HUBs and uncertain bits (UBs). Combined with random selection of HUBs for bit flipping, gains up to $0.91$~dB with respect to standard SCCs were reported~\cite{Yi_iSABM2021}. The achieved gain is slightly higher than that of the state-of-the-art BEE-SCC decoder proposed in~\cite{AlirezaSCC2020}, however, a thorough parameter optimization and characterization of the iSABM-SCC decoder is yet to be performed.
In this paper, we start by reviewing the iSABM-SCC decoder. Then a detailed analysis on parameter optimization of the iSABM-SCC decoder is made based on numerical simulations over an additive white Gaussian noise (AWGN) channel. The gap between the optimized iSABM-SCC decoder and the AIRs of HD-FEC is also analyzed together with the achievable transmission reach over an optical fiber channel. The results show that the optimized iSABM-SCC decoder can decrease the gap of SCCs to the AIRs of HD-FEC to $0.26$--$1.02$~dB for code rates of $0.74$--$0.87$, and increase the fiber reach by up to $22\%$ for data rates of $401$~Gbps--$468$~Gbps.
\section{System Model, Staircase Codes and the iSABM-SCC Decoder}
\subsection{System Model}
Fig.~\ref{fig:model} shows the system model. Information bits are encoded into bits $b_{l,1},\ldots,b_{l,m}$ by an SCC encoder and then are mapped to symbols $x_{l}$ taken from an equally-spaced $M$-ary pulse amplitude modulation (PAM) constellation $\mathcal{S}=\{s_{1},s_{2},\ldots,s_{M}\}$ with $M=2^m$ points, where $l$ is the discrete time index, $l=0,1,2,\ldots$. The bit-to-symbol mapping is the binary reflected Gray code. After channel transmission, an HD-demapper is used to estimate the coded bits $\hat{b}_{l,1},\ldots,\hat{b}_{l,m}$, according to the received signal $y_l$. At the same time, the receiver calculates the LLR $\lambda_{l,k}$ for each bit $\hat{b}_{l,k}$, $k=1,\ldots,m$. Both the bits $\hat{b}_{l,1},\ldots,\hat{b}_{l,m}$ and the soft information $\lambda_{l,1},\ldots,\lambda_{l,m}$ are sent to the iSABM-SCC decoder to recover the information bits. In contrast, standard SCC decoder is only fed with the bits $\hat{b}_{l,1},\ldots,\hat{b}_{l,m}$.
In this paper, the performance of iSABM-SCC is analyzed for two channels. The first one is an AWGN channel, i.e., ${y_{l}}={x_{l}}+{z_{l}}$, where ${z_{l}} \sim \mathcal{CN}(0,N_0/2)$. The signal power is normalized, i.e., $E_s\triangleq\mathbb{E}[x^2_l]=(1/M)\sum^M_is^2_i=1$, where $\mathbb{E}[\cdot]$ denotes expectation. The signal-to-noise ratio (SNR) is defined as $\text{SNR}\triangleq E_s/N_0=1/N_0$. The LLR value $\lambda_{l,k}$ is defined as
\begin{equation}\label{LLR}
\lambda_{l,k}=\sum_{b \in \{0,1\}} (-1)^{\bar{b}} \log\sum_{i \in \mathcal{I}_{k,b}} \textrm{exp}\left(-\frac{(y_{l}-s_{i})^{2}}{N_0}\right),
\end{equation}
where $\bar{b}$ denotes bit negation and the set $\mathcal{I}_{k,b}$ enumerates all the constellation points in $\mathcal{S}$ whose $k$th bit is $b$.
The second channel we considered is an $N$-span optical fiber link. Each span is composed of $80$ km standard single-mode-fiber (SSMF) and an erbium-doped fiber amplifier (EDFA) with noise figure of $5$~dB. More details about the optical fiber channel will be given in Sec. III-B.
\vspace{-0.5em}
\subsection{Staircase Codes}
SCCs are built on simple algebraic component codes, e.g., Bose-Chaudhuri-Hocquenghem (BCH) codes. Let $\mathcal{C}$ be a BCH code with parameters of $(n_{c}, k_{c}, t)$, where $n_{c}$ is the codeword length, $k_{c}$ is the information length and $t$ is the error-correcting capability. An SCC based on $\mathcal{C}$ is defined as a sequence of staircase-like concatenated binary blocks $\boldsymbol{B}_{i}\in\{0,1\}^{w \times w}$, $i=0, 1, 2,\ldots$, with block size of $w=n_c/2$ and code rate of $R=2k_c/n_c-1$, where $\boldsymbol{B}_0$ is initialized to all zeros. The visualized structure of SCC is shown in~\cite[Fig.1(b)]{YiTCOM2019}. For $\forall i>1$, each row of the matrix $[\boldsymbol{B}^{T}_{i-1} \boldsymbol{B}_{i}]$ is a codeword in $\mathcal{C}$.
The decoding of SCCs is performed using a sliding window covering $L$ received blocks $\{\boldsymbol{Y}_{i}, \boldsymbol{Y}_{i+1},\ldots, \boldsymbol{Y}_{i+L-1}\}$, where $\boldsymbol{Y}_{i}$ corresponds to the transmitted block $\boldsymbol{B}_{i}$. Within the window, BDD is used to decode the rows in $[\boldsymbol{Y}^{T}_{i+L-2} \boldsymbol{Y}_{i+L-1}]$, $[\boldsymbol{Y}^{T}_{i+L-3} \boldsymbol{Y}_{i+L-2}]$,$\ldots$, $[\boldsymbol{Y}^{T}_{i} \boldsymbol{Y}_{i+1}]$ in turn, until the maximum number of iterations is reached. Then the decoder outputs $\boldsymbol{Y}_{i}$ and the window shifts to $\{\boldsymbol{Y}_{i+1}, \boldsymbol{Y}_{i+2},\ldots, \boldsymbol{Y}_{i+L}\}$ and repeats the decoding process in the last window.
To speed the decoding, $w$ BDDs can be used in parallel to decode the $w$ rows $r_1,r_2 \ldots, r_w$ of two neighbor SCC blocks. We thus treat the $w$ component decoders as a group and state that standard SCC performs $L-1$ groups of BDDs at each iteration.
Let $c_j$ be the transmitted component codeword that corresponds to the received sequence $r_j$, $j=1,\ldots,w$. The BDD output $\hat{c}_j$ is given by
\begin{subnumcases}{\hat{c}_j=}
c_j, & $\textrm{if~} d_{\textrm{H}}(r_j,c_j) \leq t $ \\
\tilde{c}_j, & $\textrm{if~} d_{\textrm{H}}(r_j,c_j) > t \textrm{~and~} d_{\textrm{H}}(r_j,\tilde{c}_j) \leq t$ \\
r_j, & \textrm {otherwise}
\end{subnumcases}
where $d_{\textrm{H}}(\cdot,\cdot)$ is the Hamming distance and $\tilde{c}_j$ is another codeword in $\mathcal{C}$. As (2a) shows, only when there are $t$ or less than $t$ errors, BDD can decode $r_j$ to $c_j$. Otherwise, either a miscorrection, i.e., (2b), or a failure, i.e., (2c), happens. BDD failure means that the decoder could not find a codeword in $\mathcal{C}$ that is within the Hamming distance of $t$ to $r_j$, and has to return the input $r_j$. Miscorrection is technically successful, but the output is $\tilde{c}_j$ rather than the transmitted $c_j$. Both failures and miscorrections limit the performance of BDD and thus the performance of SCCs.
\begin{figure}[!tb]
\includegraphics[width=0.49\textwidth]{./Figs/Fig1.pdf}
\vspace{-2.2em}
\caption{System model considered in this paper.}
\vspace{-1.35em}
\label{fig:model}
\end{figure}
\subsection{The iSABM-SCC Decoder}
To improve the performance of SCCs, the iSABM-SCC decoder was proposed in~\cite{Yi_iSABM2021}. iSABM-SCC marks bits with the help of channel LLRs to prevent miscorrections and to decode the BDD failures and miscorrections via BF. Furthermore, bit marking (BM) in iSABM-SCC uses two thresholds. Let $\delta_1$ and $\delta_2$ be the HRB threshold and HUB threshold ($\delta_1\geq\delta_2\geq0$), resp. According to the absolute value of $\lambda_{l,k}$, i.e., $|\lambda_{l,k}|$, the marking result for a bit $\hat{b}_{l,k}$ is given by
\setcounter{equation}{2}
\begin{equation}\label{2-bitMarking}
\begin{aligned}
\left\{
\begin{array}{lcl}
\text{HRB}, & & \text{if~} |\lambda_{l,k}| \geq \delta_1 \\
\text{UB}, & & \text{if~} \delta_2 \leq |\lambda_{l,k}| < \delta_1 \\
\text{HUB}, & & \text{if~} |\lambda_{l,k}| < \delta_2\\
\end{array}
\right.
\end{aligned}.
\end{equation}
A pair $(\delta_1,\delta_2)$ will be used to represent the thresholds in (\ref{2-bitMarking}).
With the marked information in the last $L-K$ blocks of a window, $K=0,\ldots,L-2$, iSABM-SCC performs $K$ groups of BDDs and $L-K-1$ groups of iSABMs at each iteration. This is shown in the left side of Fig.~\ref{fig:iSABM}.
\begin{figure}[!tb]
\centering
\includegraphics[width=0.5\textwidth]{./Figs/Fig2.pdf}
\vspace{-2em}
\caption{Flow chart (left) of the iSABM-SCC decoder in the $i$th window. The right figure shows the workflow of iSABM to decode a component word $r_j$, $j=1,\ldots,w$. $c'_j$ is the output of the iSABM component decoder.}
\vspace{-1.6em}
\label{fig:iSABM}
\end{figure}
The right side of Fig.~\ref{fig:iSABM} shows the workflow of iSABM to decode a received sequence $r_j$. Similar to standard SCC decoding, BDD is performed first. In the event of success, iSABM will check whether the BDD output $\hat{c}_j$ is a miscorrection or not. The criteria are that the detected errors should not be in conflict with (i) HRBs or (ii) any bit in a correctly decoded component codeword. Only when the two criteria are satisfied, the output of BDD will be accepted, i.e., $c'_j=\hat{c}_j$. Otherwise, $\hat{c}_j$ will be regarded as a miscorrection and be rejected.
For the miscorrections, the iSABM algorithm \emph{randomly} selects $d_{0}-w_{\text{H}}(e_j)-t$ HUBs in $r_j$ for flipping, where $d_0$ is the minimum Hamming distance between the codewords in $\mathcal{C}$ and $w_{\text{H}}(e_j)$ is the Hamming weight of the error pattern $e_j$ detected by BDD. In the case of BDD failure, the iSABM algorithm \emph{randomly} selects $1$ HUB in $r_j$ for flipping. The intuition here is that in some cases, BF will make the resulted sequence $\tilde{r}_j$ close enough to the transmitted codeword $c_j$, i.e., $d_\text{H}(\tilde{r}_j,c_j)=t$. Thus when BDD is excuted again, the residual errors in $\tilde{r}_j$ can be corrected. Similarly, successful BDD case follows a step of miscorrection detection in case of miscorrections caused by BF. In addition, in some cases the number of HUBs is less than the required number of bit flips ($1$ for failure recovery and $d_{0}-w_{\text{H}}(e_j)-t$ for miscorrection recovery). If
that happens, even if all the HUBs are flipped, BF can not bring $\tilde{r}_j$ close enough to $c_j$. Hence, the iSABM decoding will stop and output $r_j$, and then proceed to decode the next component word.
In the iSABM-SCC decoder, there is no need to store LLRs. iSABM-SCC only requires $2$ bits to represent the three groups: HRB, HUB and UB. The marked information is not updated either. To reduce the memory requirement, $1$-bit marking is analysed as well. Let $\delta_3$ be the $1$-bit marking threshold, the marking result for a bit $\hat{b}_{l,k}$ is given by
\begin{equation}\label{1-bitMarking}
\begin{aligned}
\left\{
\begin{array}{lcl}
\text{HRB}, & & \text{if~} |\lambda_{l,k}| \geq \delta_3 \\
\text{HUB}, & & \text{if~} |\lambda_{l,k}| < \delta_3\\
\end{array}
\right.
\end{aligned}.
\end{equation}
\section{Simulation Results}
\begin{figure}[!tb]
\centering
\includegraphics[width=0.45\textwidth]{./Figs/Fig3.pdf}
\vspace{-1em}
\caption{BER performance of iSABM-SCCs based on $\mathcal{C}_1$ under different $L-K$ values for 2-PAM.}
\label{fig:L-K_Optimization}
\vspace{-0.5em}
\end{figure}
Table~\ref{tab:SCCs} shows the SCC parameters we used. Let $\nu$ be an integer, the codeword length and information length of $\mathcal{C}$ are given by $n_c=2^{\nu}-1+u$ and $k_c=n-\nu t-u$, resp., where $u$ is the number of extended parity bits. The number of iterations is $\ell=7$, while the decoding window size is $L=9$.
\vspace{-1em}
\begin{table}[t]
\caption{SCC parameters we used in this paper}
\centering
{
\begin{tabular}{cccccccccc}
\hline
\hline
& $R$ & $\nu$ & $u$ & $t$ & $n_c$ & $k_c$ & $w$\\
\hline
\hline
$\mathcal{C}_1$ & 0.87 & 8 & 1 & 2 & 256 & 239 & 128\\
$\mathcal{C}_2$ & 0.80 & 8 & 1 & 3 & 256 & 231 & 128\\
$\mathcal{C}_3$ & 0.74 & 8 & 1 & 4 & 256 & 223 & 128\\
\hline
\hline
\end{tabular}
}
\label{tab:SCCs}
\vspace{-1em}
\end{table}
\vspace{0.5em}
\subsection{Parameter Optimization}
To maximize the performance of iSABM-SCC, it is key to carefully select the values of the parameters $L-K$ as well as $(\delta_1,\delta_2)$ for (3) or $\delta_3$ for (4). In what follows, we will discuss the optimization of the parameters $L-K$, $(\delta_1,\delta_2)$ and $\delta_3$ using numerical simulations over an AWGN channel.
Fig.~\ref{fig:L-K_Optimization} shows the performance of iSABM-SCCs based on $\mathcal{C}_1$ and 2-PAM for different $L-K$ values. BM uses (\ref{2-bitMarking}) with thresholds of $(10,2.5)$. Fig.~\ref{fig:L-K_Optimization} shows that for $L-K \leq 7$, the gain of iSABM-SCC with respect to standard SCC increases along with $L-K$, and achieves a maximum value of $0.68$~dB at a post-FEC bit-error ratio (BER) of $10^{-6}$ for $L-K=6, 7$.
Once $L-K>7$, an error floor appears. The reason for this is as follows. For the sliding window fashion, the blocks that are closer to the top left of the window are decoded more times. Hence, there is little probability of miscorrection for the BDDs related to the first two blocks at the top left of the window (due to the few errors). However, due to the inaccurate marked HRBs, not all miscorrections are identified. The mis-regarded correctly decoded component codewords and the inaccurate HRBs together tend to lead to the rejection of the correct BDDs related to the first two blocks of the window, and thus cause serious performance loss.
\begin{figure*}[!tb]
\centering
\includegraphics[width=0.945\textwidth]{./Figs/Fig4.pdf}
\vspace{-0.9em}
\caption{Optimization of $(\delta_1,\delta_2)$ for iSABM-SCC based on $\mathcal{C}_1$ for 2-PAM and SNRs of (a) $6.45$~dB, (b) $6.55$~dB and (a) $6.65$~dB. The rightmost figure (d) is the PDFs of $|\lambda_{l,k}|$ for SNRs of $6.45$~dB and $6.65$~dB. The optimal post-FEC BER is shown with $\varepsilon^*$. }
\vspace{-1.2em}
\label{fig:delta12Optimization}
\end{figure*}
Fig.~\ref{fig:delta12Optimization}(a)--(c) shows the optimization of $(\delta_1,\delta_2)$ for iSABM-SCC based on $\mathcal{C}_1$ with $L-K=7$ and (\ref{2-bitMarking}) for BM. They are tested for SNRs of $6.45$~dB, $6.55$~dB and $6.65$~dB, which are picked from the waterfall region shown in~Fig.~\ref{fig:L-K_Optimization}. In general, the cone-like shape indicates that whenever the values of $\delta_1$ and $\delta_2$ are too large or too small, the decoding performance of iSABM-SCC will degrade. If $\delta_1$ is too large, the HRB class may miss some highly reliable bits that weakens the ability of the decoder to prevent miscorrections. On the contrary, a small $\delta_1$ tends to mistakenly mark some errors as HRBs. As a result, some correct BDD outputs might be mistakenly interpreted as miscorrections and be rejected by the decoder. In the case of iSABM-SCC with a small $\delta_2$, the number of HUBs reduces, which increases the probability that there is no enough HUBs for flipping. On the contrary, a large $\delta_2$
may cause that some correct bits are mistakenly marked as HUBs. This increases the probability that the flipped bits are not errors and thus causes failures or miscorrections in the second BDD trial. All the mentioned above potentially give a performance deterioration.
Fig.~\ref{fig:delta12Optimization}(a)--(c) also shows that the optimal value of $(\delta_1,\delta_2)$, denoted by $(\delta^*_1,\delta^*_2)$, are $(10,2.5)$, $(10.5,3)$ and $(11,3)$ for SNRs of $6.45$~dB, $6.55$~dB and $6.65$~dB, resp. The corresponding optimal post-FEC BERs (denoted by $\varepsilon^*$) are $4.5\times10^{-3}$, $1.78\times10^{-4}$ and $5.11\times10^{-7}$. It is observed that the value of $(\delta^*_1,\delta^*_2)$ increases as SNR increases. Intuitively, this is due to that the decoder wants to resist the change of the proportion of HRBs, UBs and HUBs in the total bits, as the distribution of $|\lambda_{l,k}|$ shifts.
Fig.~\ref{fig:delta12Optimization}(d) shows how the probability density function (PDF) curve of $|\lambda_{l,k}|$ shifts to the right when SNR increases from $6.45$~dB to $6.65$~dB. The integrals between the PDF curve and the $x$-axis over the intervals $[0,\delta_2)$, $[\delta_2,\delta_1)$ and $[\delta_1,\infty)$ are the proportion of HUBs, UBs and HRBs in the bits, resp. Due to the right shift of the PDF curve, the proportion of HUBs and UBs decreases, while that of HRBs increases, if the value of $(\delta_1,\delta_2)$ is fixed, e.g., $(10,2.5)$. Increasing the thresholds to the optimal one, i.e., $(11,3)$ for SNR of $6.65$~dB, will increase the proportion of HUBs and UBs and decrease the proportion of HRBs. This shift of the optimal thresholds seems to want to keep the proportion of HRBs, UBs, and HUBs constant. To ensure the best performance for all SNRs, this may need to be considered in future work.
Fig.~\ref{fig:delta3Optimization} shows the optimization of $\delta_3$ for iSABM-SCC based on $\mathcal{C}_1$ using (\ref{1-bitMarking}) for BM and $L-K=7$. The three curves were studied for SNRs of $6.7$~dB, $6.8$~dB and $6.9$~dB, while the inset is the PDFs of $|\lambda_{l,k}|$ at SNRs of $6.7$~dB and $6.9$~dB. The corresponding optimal thresholds $\delta^*_3$ are found to be $8.5$, $9$ and $9.5$. On the one hand, the U-type curve indicates that the value of $\delta_3$ should not be too large or too small, either. On the other hand, a similar trend as that shown in Fig.~\ref{fig:delta12Optimization} is observed, i.e., the optimal marking threshold increases as SNR increases.
\begin{figure}[!tb]
\centering
\includegraphics[width=0.44\textwidth]{./Figs/Fig5.pdf}
\vspace{-1em}
\caption{Optimization of $\delta_3$ for iSABM-SCC based on $\mathcal{C}_1$ for 2-PAM. The inset figure is the PDFs of $|\lambda_{l,k}|$ for SNRs of $6.7$~dB and $6.9$~dB.}
\vspace{-1.5em}
\label{fig:delta3Optimization}
\end{figure}
\vspace{-0.35em}
\subsection{Performance Analysis}
\vspace{-0.32em}
Fig.~\ref{fig:AIR} shows the required SNRs for iSABM-SCC (with $L-K=7$), SABM-SCC and standard SCC to achieve a post-FEC BER of $10^{-6}$ ($x$-axis at the bottom). The rates are calculated as $I_\text{SCC}=4mR$ with $R\in\{0.74,0.80,0.87\}$, where the factor of $4$ represents four dimensions (4D) of dual-polarization and in-phase-quadrature (I/Q) of the optical signal. For iSABM-SCC using (\ref{2-bitMarking}), the marking thresholds we used are $(\delta_1,\delta_2)=(10,2.5)$, which are optimized for iSABM-SCC based on $\mathcal{C}_1$ at SNR of $6.45$~dB shown in Fig.~\ref{fig:delta12Optimization}(a). For iSABM-SCC using (\ref{1-bitMarking}), we simply set $\delta_3=\delta_1$, i.e., $\delta_3=10$, which gives a very close performance to iSABM-SCC with the optimal threshold at each tested SNR shown in Fig.~\ref{fig:delta3Optimization}. The thresholds we used in Fig.~\ref{fig:AIR} are fixed for all SNRs and code rates (this is also how it was done in~\cite{Yi_iSABM2021}). Fig.~\ref{fig:AIR} shows that iSABM-SCC can achieve $0.3$--$0.68$~dB ($0.42$--$0.89$~dB) extra gain for 2-PAM (8-PAM), when compared to standard SCCs. The performance of iSABM-SCC is significantly better than that of SABM-SCC, even with 1-bit marking.
The solid and dashed curves in Fig.~\ref{fig:AIR} are the maximum AIRs of HD-FEC and SD-FEC, resp. The AIR of SD-FEC is the generalized mutual information~\cite[eq.(14)]{AIR_AlexJLT2017}, while that of HD-FEC is given by $I_\text{HD} = 4m (1+p\log_2p+(1-p)\log_2(1-p))$, where $p$ is the average pre-FEC BER across the $m$ bits mapped into the selected single-dimensional modulation format. Fig.~\ref{fig:AIR} shows that due to the finite block length and limited complexity, there is a gap between the practical SNRs required for SCCs and the predicted SNRs via AIR. By replacing standard decoding with iSABM-SCC decoding, the gap of SCCs to the AIR preditions can be reduced to $0.29$--$0.82$~dB ($0.26$--$1.02$~dB) for 2-PAM (8-PAM). However, note that due to the use of soft information, the AIR of HD-FEC is not a true upperbound for the SA-HD decoder. An AIR upperbound for the SA-HD decoders is expected to be located between the solid and dashed curves. Accurately estimating such an upperbound is still an open problem.
\begin{figure}[!tb]
\centering
\includegraphics[width=0.455\textwidth]{./Figs/Fig6.pdf}
\vspace{-1em}
\caption{AIRs for HD-FEC and SD-FEC with dual-polarization I/Q-modulated (a) 2-PAM ($m=1$) and (b) 8-PAM ($m=3$). Markers show the results for iSABM-SCC, SABM-SCC and standard SCC.}
\vspace{-1.6em}
\label{fig:AIR}
\end{figure}
Fig.~\ref{fig:AIR} also shows the achievable rate and optical transmission reach for the SCC decoders ($x$-axis at the top). A dual-polarization wavelength-division-multiplexing coherent communication system is considered with $45$ GBaud symbol rate, $11$~channels, $50$~GHz channel spacing and $1550$~nm center wavelength. The optical fiber is with $0.2$ dB/km loss and $17$~ps/nm/km dispersion. The nonlinear coefficient of the fiber is $1.2$ (W$\cdot$km)$^{-1}$. The reach increase is evaluated based on the Gaussian noise model proposed in~\cite{GNmodel2014}.
Fig.~\ref{fig:AIR} shows that the achieved extra gains of iSABM-SCCs (with respect to standard SCCs) yields reach increases of up to $1896$~km ($17\%$) for data rates between $134$~Gb/s and $156$~Gb/s, and up to $154$~km ($22\%$) for data rates between $401$~Gb/s and $468$~Gb/s per wavelength.
\section{Conclusions}
In this paper, two aspects of the improved soft-aided bit-marking staircase decoder (iSABM-SCC) were analyzed. The first one is a parameter optimization for iSABM-SCC, while the second one is an analysis of the gap of iSABM-SCC to the AIR of HD-FEC in an AWGN channel and the reach enhancement in the optical fiber channel. It was shown that carefully optimizing the marking thresholds and the number of modified component decodings is key for iSABM-SCC to provide the best performance.
\bibliographystyle{IEEEtran}
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\section{Introduction}
Coloring problems are among the most studied problems in algorithmic graph theory. In its classical form, the $k$-coloring problem asks if there is an assignment of $k$ colors to the vertices of a graph in such a way that no edge is monochromatic. Many generalizations and variations of the classical coloring problem have been defined over the years. One of such variations is the clique $k$-coloring problem, in which the vertices are colored so that no maximal clique is monochromatic. In this article we study the star and biclique coloring problems, which are variations of the coloring problem similar to clique colorings. A biclique is a set of vertices that induce a complete bipartite graph $K_{n,m}$, while a star is a biclique inducing the graph $K_{1,m}$. In the star (biclique) $k$-coloring problem, the goal is to color the vertices with $k$ colors without generating monochromatic maximal stars (bicliques).
The clique coloring problem has been investigated for a long time, and it is still receiving a lot of attention. Recently, the clique $k$-coloring problem was proved to be \ensuremath{\Sigma^p_2}\xspace-complete~\cite{MarxTCS2011} for every $k \geq 2$, and it remains \ensuremath{\Sigma^p_2}\xspace-complete for $k=2$ even when the input is restricted to graphs with no odd holes~\cite{DefossezJGT2009}. The problem has been studied on many other classes of input graphs, for which it is was proved to be \NP-complete or to require polynomial time (e.g.~\cite{BacsoGravierGyarfasPreissmannSebHoSJDM2004,CerioliKorenchendler2009,DefossezJGT2006,GravierHoangMaffrayDM2003,KratochvilTuzaJA2002,MacedoMachadoFigueiredo2012}). Due to the close relation between cliques and bicliques, many problems on cliques have been translated in terms of bicliques (e.g.~\cite{AmilhastreVilaremJanssenDAM1998,PrisnerC2000a,TuzaC1984}). However, there are some classical problems on cliques whose biclique versions were not studied until recently~\cite{EguiaSoulignacDMTCS2012,GroshausMonteroJoGT2012,GroshausSzwarcfiterGC2007,GroshausSzwarcfiterJGT2010}. Clique colorings are examples of such problems; research on biclique colorings begun in 2010 in the Master Thesis of one of the authors~\cite{Terlisky2010} whose unpublished results are being extended in the present article. It is worth mentioning that, despite its youthfulness, at least two articles on biclique colorings were written: \cite{MacedoMachadoFigueiredo2012} develop a polynomial time algorithm for biclique coloring some unichord-free graphs, and \cite{MacedoDantasMachadoFigueiredo2012} determines the minimum number of colors required by biclique colorings of powers of paths and cycles.
The list coloring problem is a generalization of the coloring problem in which every vertex $v$ is associated with a list $L(v)$, and the goal is to color each vertex $v$ with an element of $L(v)$ in such a way that no edge is monochromatic. Function $L$ is called a list assignment, and it is a $k$-list assignment when $|L(v)| = k$ for every vertex $v$. A graph $G$ is said to be $k$-choosable when it admits an $L$-coloring with no monochromatic edges, for every $k$-list assignment $L$. The choosability problem asks whether a graph is $k$-choosable. In the same way as the coloring problem is generalized to the clique (star, biclique) coloring problem, the choosability problem is generalized to the clique (star, biclique) choosability problem. That is, a graph $G$ is clique (star, biclique) $k$-choosable when it admits an $L$-coloring generating no monochromatic maximal cliques (star, bicliques), for every $k$-list assignment $L$. The choosability problems seem harder than their coloring versions, because a universal quantifier on the list must be checked. This difficulty is reflected for the $k$-choosability and clique $k$-choosability problems in the facts that the former is \ensuremath{\Pi^p_2}\xspace-complete for every $k \geq 3$~\cite{GutnerTarsiDM2009}, whereas the latter is \ensuremath{\Pi^p_3}\xspace-complete for every $k \geq 2$~\cite{MarxTCS2011}. In~\cite{MoharvSkrekovskiEJC1999} it is proven that every planar graph is clique $4$-choosable. However, contrary to what happens with the clique coloring problem, there are not so many results regarding the complexity of the clique coloring problem for restricted classes of graphs.
In this paper we consider the star and biclique coloring and choosability problems, both for general graphs and for some restricted classes of graphs. The star and biclique coloring and choosability problems are defined in Section~\ref{sec:preliminaries}, where we introduce the terminology that will be used throughout the article. In Section~\ref{sec:general case}, we prove that the star $k$-coloring problem is \ensuremath{\Sigma^p_2}\xspace-complete for $k \geq 2$, and that it remains \ensuremath{\Sigma^p_2}\xspace-complete even when its input is restricted to $\{K_{2,2}, K_{k+2}\}$-free graphs. Clearly, every maximal biclique of a $K_{2,2}$-free graph is a star. Thus, we obtain as a corollary that the biclique $k$-coloring problem on $\{K_{2,2}, K_{k+2}\}$-free graphs is \ensuremath{\Sigma^p_2}\xspace-complete as well. The completeness proof follows some of the ideas by Marx~\cite{MarxTCS2011} for the clique coloring problem. In Section~\ref{sec:choosability} we show that the star $k$-choosability problem is \ensuremath{\Pi^p_3}\xspace-complete for $k \geq 2$, and that it remains \ensuremath{\Pi^p_3}\xspace-complete for $\{K_{2,2}, K_{k+2}\}$-free graphs. Again, the \ensuremath{\Pi^p_3}\xspace-completeness of the biclique $k$-coloring problem on $\{K_{2,2}, K_{k+2}\}$-free is obtained as a corollary. As in~\cite{MarxTCS2011}, we require a structure to force a color on a vertex. The remaining sections of the article study the star and biclique coloring problems on graphs with restricted inputs. These graphs are related to the graph $G$ that is generated to prove the \ensuremath{\Sigma^p_2}\xspace-completeness of the star coloring problem in Section~\ref{sec:general case}. The aim is to understand what structural properties can help make the problem simpler. In Section~\ref{sec:small forbiddens}, we discuss the star and biclique coloring and choosability problems on $K_3$-free, $P_3$-free, $\overline{P_3}$-free, and $\overline{K_3}$-free graphs. For $K_3$, $P_3$ and $\overline{P_3}$, the star coloring and star choosability problems are almost trivial and can be solved in linear time. On the other hand, both problems are as hard as they can be for $\overline{K_3}$-free graphs, even when the input is a co-bipartite graph. In Section~\ref{sec:diamond-free} we prove that the star coloring problem is \NP-complete for diamond-free graphs and that the star choosability problem is \ensuremath{\Pi^p_2}\xspace-complete for a superclass of diamond-free graphs. If no induced $K_{i,i}$ is allowed for a fixed $i$, then the biclique coloring and the biclique choosability problems are also \NP-complete and \ensuremath{\Pi^p_2}\xspace-complete. In Section~\ref{sec:split}, the star coloring and the star choosability problems on split graphs are proved to be \NP-complete and \ensuremath{\Pi^p_2}\xspace-complete, respectively. Finally, Sections \ref{sec:threshold}~and~\ref{sec:block} show that the star coloring and the star choosability problems are equivalent for both threshold and net-free block graphs, and both can be solved in linear time. Table~\ref{tab:results} sums up the results obtained in the article for the star coloring and star choosability problems.
\begin{table}
\centering
\begin{tabular}{|l|c|c|}
\hline
Graph class & star $k$-coloring & star $k$-choosability \\
\hline
$\{K_{2,2}, K_{k+2}\}$-free & \ensuremath{\Sigma^p_2}\xspace-complete & \ensuremath{\Pi^p_3}\xspace-complete \\
$K_3$-free, $P_3$-free, $\overline{P_3}$-free & $O(n+m)$ & $O(n+m)$ \\
$\overline{K_3}$-free & \NP-complete & \ensuremath{\Pi^p_2}\xspace-complete \\
co-bipartite & & \ensuremath{\Pi^p_2}\xspace-complete \\
\{$W_4$, gem, dart\}-free & \NP-complete & \ensuremath{\Pi^p_2}\xspace-complete \\
diamond-free & \NP-complete & \ensuremath{\Pi^p_2}\xspace \\
$\overline{C_4}$-free & \NP-complete & \ensuremath{\Pi^p_2}\xspace-complete \\
split & \NP-complete & \ensuremath{\Pi^p_2}\xspace-complete \\
threshold & $O(n+m)$ & $O(n+m)$ \\
net-free block & $O(n+m)$ & $O(n+m)$ \\
\hline
\end{tabular}
\caption{Complexity results obtained in this article.}\label{tab:results}
\end{table}
\section{Preliminaries}
\label{sec:preliminaries}
In this paper we work with simple graphs. The vertex and edge sets of a graph $G$ are denoted by $V(G)$ and $E(G)$, respectively. Write $vw$ to denote the edge of $G$ formed by vertices $v, w \in V(G)$. For the sake of simplicity, $E(G)$ is also considered as the family of subsets of $V(G)$ containing the set $\{v,w\}$ for each $vw \in E(G)$. For $v \in V(G)$, the \Definition{neighborhood} of $v$ is the set $N_G(v)$ of vertices adjacent to $v$, while the \Definition{closed neighborhood} of $v$ is $N_G[v] = N_G(v) \cup \{v\}$. A vertex $w$ \Definition{dominates} $v$, and $v$ is \Definition{dominated} by $w$, when $N_G[v] \subseteq N_G[w]$, while $w$ \Definition{false dominates} $v$, and $v$ is \Definition{false dominated} by $w$, when $N_G(v) \subseteq N_G(w)$. If $N_G[v] = N_G[w]$, then $v$ and $w$ are \Definition{twins}, and if $N_G(v) = N_G(w)$, then $v$ and $w$ are \Definition{false twins}. The \Definition{degree} of $v$ is $d_G(v) = |N_G(v)|$. A vertex is an \Definition{isolated vertex}, a \Definition{leaf}, and a \Definition{universal vertex} when $d_G(v)$ equals $0$, $1$, and $|V(G)|-1$, respectively. We omit the subscripts from $N$ and $d$ when no ambiguities arise.
The \Definition{complement} of $G$ is the graph $\overline{G}$ where $V(\overline{G}) = V(G)$ and $E(\overline{G}) = \{vw \mid vw \not\in E(G)\}$. For a graph $H$, the \Definition{union} of $G$ and $H$ is the graph $G \cup H$ where $V(G \cup H) = V(G) \cup V(H)$ and $E(G \cup H) = E(G) \cup E(H)$. Write $G = H$ to indicate that $G$ and $H$ are isomorphic. The \Definition{$n$-cycle} graph ($n \geq 3$), denoted by $C_n$, is the connected graph that has $n$ vertices of degree $2$. The \Definition{$n$-path} graph, denoted by $P_n$, is the graph obtained from $C_n$ by removing an edge. The \emph{$n$-wheel} graph ($n \geq 3$), denoted by $W_n$, is the graph obtained from $C_n$ by inserting a universal vertex. The \Definition{diamond}, \Definition{gem}, \Definition{dart}, and \Definition{net} are the graphs shown in Figure~\ref{fig:diam-dart-gem}. The \Definition{$n$-complete} graph, denoted by $K_n$, is the graph formed by $n$ pairwise adjacent vertices. An \Definition{independent set} is a subset of $V(G)$ formed by pairwise non-adjacent vertices. Graph $G$ is \Definition{bipartite} when $V(G)$ can be partitioned into two independent sets $S$ and $T$. In this case, the unordered pair $ST$ is called a \Definition{bipartition} of $G$. The \Definition{$(n,m)$-complete bipartite} graph ($n \geq 1$, $m \geq 1$), denoted by $K_{n,m}$, is the graph isomorphic to $\overline{K_n \cup K_m}$. Note that $K_{n,m}$ is a bipartite graph. The graph $K_{1,n}$ is also called the \Definition{$n$-star} graph. The universal vertices of $K_{1,n}$ are referred to as the \Definition{centers} of $K_{1,n}$, while $K_{1,n}$ is said to be \Definition{centered} at $v$. Note that $K_{1,1}$ has two centers.
\begin{figure}
\hfill\includegraphics{ diamond}\hfill\includegraphics{ dart}\hfill\includegraphics{ gem}\hfill{}\includegraphics{ net}\hfill{}
\caption{The diamond, dart, gem, and net graphs are shown from left to right.}\label{fig:diam-dart-gem}
\end{figure}
Let $W \subseteq V(G)$. We write $G[W]$ to denote the subgraph of $G$ induced by $W$, and $G \setminus W$ to denote $G[V(G) \setminus W]$. Set $W$ is said to be a \Definition{clique}, \Definition{biclique}, and \Definition{star} when $G[W]$ is isomorphic to a complete, bipartite complete, and star graph, respectively. For the sake of simplicity, we use the terms \Definition{clique} and \Definition{biclique} to refer to $G[W]$ as well. Moreover, we may refer to $ST$ as a biclique or star when $G[S \cup T]$ is a bipartite complete or star graph with bipartition $ST$. The family of maximal cliques, maximal bicliques, and maximal stars are denoted by $\ensuremath{\mathcal{C}}(G)$, $\ensuremath{\mathcal{B}}(G)$, and $\ensuremath{\mathcal{S}}(G)$, respectively.
A sequence of distinct vertices $P = v_1, \ldots, v_n$ is a \Definition{path} of \Definition{length} $n-1$ when $v_i$ is adjacent to $v_{i+1}$. If in addition $v_{n}$ is adjacent to $v_1$, then $P$ is a \Definition{cycle} of \Definition{length} $n$. A \Definition{tree} is a connected graph that contains no cycles. A \Definition{rooted tree} is a tree $T$ with a fixed vertex $r \in V(T)$ called the \Definition{root} of $T$. The \Definition{parent} of $v$ in $T$ is the neighbor of $v$ in its path to $r$. A path (resp.\ cycle) of $G$ is \Definition{chordless} when $G[P] = P_n$ (resp.\ $G[P] = C_n$). A \Definition{hole} is a chordless cycle of length at least $4$. A graph $G$ is said to be \Definition{$H$-free}, for some graph $H$, when no induced subgraph of $G$ is isomorphic to $H$. Similarly, $G$ is $\mathcal{F}$-free, for any family of graph $\mathcal{F}$, when $G$ is $H$-free for every $H \in \mathcal{F}$. A graph is \Definition{chordal} when it is $\{C_n\}_{n\geq4}$-free, i.e., chordal graphs have no holes.
A \Definition{coloring} of $G$ is a function $\rho$ that maps each vertex $v \in V(G)$ to a color $\rho(v) \in \mathbb{N}$. When $\rho(v) \leq k$ for every $v \in V(G)$, $\rho$ is called a \Definition{$k$-coloring}. We define $\rho(W) = \{\rho(v) \mid v \in W\}$ for any $W \subseteq V(G)$. Set $W$ is said to be \Definition{$\rho$-monochromatic} when $|\rho(W)| = 1$. When there is no ambiguity, we say that $W$ \Definition{monochromatic} instead of $\rho$-monochromatic. For a family $\mathcal{F}$ of subsets of $V(G)$, we say that $\rho$ is a \Definition{proper coloring} of $\mathcal{F}$ if no $W \in \mathcal{F}$ is monochromatic. Four kinds of families are considered in this article. A coloring $\rho$ is a \Definition{vertex}, \Definition{clique}, \Definition{biclique}, and \Definition{star coloring} when $\rho$ is a proper coloring of $E(G)$, $\ensuremath{\mathcal{C}}(G)$, $\ensuremath{\mathcal{B}}(G)$, and $\ensuremath{\mathcal{S}}(G)$, respectively. The problems of finding a proper coloring for these families are defined as follows.
\begin{Problem}
\problemName{\textsc{Vertex (resp.\ clique, biclique, star) $k$-coloring}}
\problemInput{A connected graph $G$ and a value $k \in \mathbb{N}$.}
\problemQuestion{Is there a vertex (resp.\ clique, biclique, star) $k$-coloring of $G$?}
\end{Problem}
List colorings are a generalization of colorings. A \Definition{list assignment} of $G$ is a function that maps each vertex $v \in V(G)$ to a set $L(v) \subseteq \mathbb{N}$. When $|L(v)| = k$ for every $v \in V(G)$, $L$ is called a \Definition{$k$-list assignment}. An $L$-coloring of $G$ is a coloring $\rho$ such that $\rho(v) \in L(v)$ for every $v \in V(G)$. Define $L(W) = \bigcup\{L(v) \mid v \in W\}$ for any $W \subseteq V(G)$. Given family $\mathcal{F}$ of subset of $V(G)$ and a number $k \in \mathbb{N}$, graph $G$ is said to be \Definition{$k$-choosable} with respect to $\mathcal{F}$ when there exists a proper $L$-coloring of $\mathcal{F}$ for every $k$-list assignment $L$ of $G$. Graph $G$ is \Definition{vertex}, \Definition{clique}, \Definition{biclique}, and \Definition{star $k$-choosable} when $G$ is $k$-choosable with respect to $E(G)$, $\ensuremath{\mathcal{C}}(G)$, $\ensuremath{\mathcal{B}}(G)$, and $\ensuremath{\mathcal{S}}(G)$, respectively. The problem of determining if $G$ is $k$-choosable is defined as follows.
\begin{Problem}
\problemName{\textsc{Vertex (resp.\ clique, biclique, star) $k$-choosability}}
\problemInput{A connected graph $G$ and a value $k \in \mathbb{N}$.}
\problemQuestion{Is $G$ vertex (resp.\ clique, biclique, star) $k$-choosable?}
\end{Problem}
The vertex (resp.\ clique, biclique, star) \Definition{chromatic number}, denoted by $\chi(G)$ (resp.\ $\chi_C(G)$, $\chi_B(G)$, and $\chi_S(G)$), is the minimum $k \in \mathbb{N}$ such that $G$ admits a vertex (resp.\ clique, biclique, star coloring) $k$-coloring. Similarly, the vertex (clique, biclique, star) \Definition{choice number}, denoted by $ch(G)$ (resp.\ $ch_C(G)$, $ch_B(G)$, $ch_S(G)$) is the minimum number $k \in \mathbb{N}$ such that $G$ is vertex (resp.\ clique, biclique, star coloring) $k$-choosable. By definition, $\chi(G) \leq ch(G)$ and $\chi_*(G) \leq ch_*(G)$ for $* \in \{C, B, S\}$.
For a function $f$ with domain $D$, the \Definition{restriction} of $f$ to $D' \subseteq D$ is the function $f'$ with domain $D'$ where $f'(x) = f(x)$ for $x \in D'$. In such case, $f$ is said to be an \Definition{extension} of $f'$ to $D$. A \Definition{leafed vertex} is a vertex adjacent to a leaf. For the sake of simplicity, whenever we state that $G$ \emph{contains a leafed vertex} $v$, we mean that $G$ contains $v$ and a leaf adjacent to $v$. It is well known that $\chi(G) \geq \chi(H)$ for every induced subgraph $H$ of $G$. Such a property is false for clique, biclique, and star colorings. In particular, for any graph $H$, it is possible to build a graph $G$ such that $ch_*(G) = 2$ and $G$ contains $H$ as an induced subgraph. For $* = C$, graph $G$ is built from $H$ by iteratively inserting a twin of each vertex of $H$. Similarly, $G$ is obtained by inserting false twins for $* = B$, while, by the next observation, $G$ is obtained by inserting a leaf adjacent to each vertex for $* = S$.
\begin{observation}\label{obs:leafed vertex}
Let $G$ be a graph with a list assignment $L$, $v$ be a leafed vertex of $G$, and $l$ be a leaf adjacent to $v$. Then, any $L$-coloring of $G \setminus l$ can be extended into an $L$-coloring of $G$ in such a way that there is no monochromatic maximal star with center in $v$.
\end{observation}
A \Definition{block} is a maximal set of twin vertices. If $v$ and $w$ are twin vertices, then $\{v\}\{w\}$ is both a maximal star and a maximal biclique, and thus $v$ and $w$ have different colors in any star or biclique $L$-coloring $\rho$. Consequently, $|\rho(W)| = W$ for any block $W$ of $G$. We record this fact in the following observation.
\begin{observation}\label{obs:block coloring}
Let $G$ be a graph with a list assignment $L$, and $v,w$ be twin vertices. Then, $\rho(v) \neq \rho(w)$ for any star or biclique $L$-coloring $\rho$ of $G$.
\end{observation}
\section{Complexity of star and biclique coloring}
\label{sec:general case}
In this section we establish the hardness of the star and biclique coloring problems by showing that both problems are \ensuremath{\Sigma^p_2}\xspace-complete. The main result of this section is that \stcol{k} is \ensuremath{\Sigma^p_2}\xspace-complete for every $k \geq 2$, even when its input is restricted to $\{C_4, K_{k+2}\}$-free graphs. Since all the bicliques of a $C_4$-free graph are stars, this immediately implies that \bccol{k} is also \ensuremath{\Sigma^p_2}\xspace-complete for $\{C_4, K_{k+2}\}$-free graphs. The hardness results is obtained by reducing instances of the \qsat{2} problem. The \qsat{h} problem is known to be $\Sigma_h^p$-complete for every $h$~\cite{Papadimitriou1994}, and is defined as follows.
\problem{\textbf{Quantified $3$-satisfiability with $h$ alternations} (\qsat{h})}
{A formula $\phi(\VEC x_1, \VEC x_2, \ldots, \VEC x_h)$ that is in $3$-CNF if $h$ is odd, while it is in $3$-DNF if $h$ is even.}
{Is $(\exists\VEC x_1)(\forall\VEC x_2)(\exists\VEC x_3)\ldots (Q_h\VEC x_h)\phi(\VEC x_1, \VEC x_2, \VEC x_3, \ldots, \VEC x_h)$ true? ($Q_h \in \{\exists, \forall\}$.)}
Recall that $\phi$ is in $3$-CNF if it is a conjunction of clauses where each clause is a disjunction with three literals. Similarly, $\phi$ is in $3$-DNF when it is a disjunction of clauses, each clause being a conjunction with three literals.
\subsection{Keepers, switchers, and clusters}
In this section we introduce the keeper, switcher, and cluster connections that are required for the reductions. The keeper connections are used to force the same color on a pair of vertices, in any star coloring. Conversely, the switcher connections force some vertices to have different colors. Finally, the cluster connections are used to represent the variables of a DNF formula. We begin defining the keeper connections.
\begin{defn}[$k$-keeper]\label{def:k-keeper connection}
Let $G$ be a graph and $v, w \in V(G)$. Say that $K \subset V(G)$ is a \Definition{$k$-keeper connecting $v, w$} ($k \geq 2$) when $K$ can be partitioned into a clique $D = \{d_1, \ldots, d_{k-1}\}$ and $k-1$ cliques $C_1, \ldots, C_{k-1}$ with $k$ vertices each in such a way that $D \cup \{v,w\}$ and $C_{i} \cup \{d_i\}$ are cliques for \range{i}{1}{k-1}, and there are no more edges incident to vertices in $K$.
\end{defn}
Figure~\ref{fig:keeper-switcher}~(a) shows a $k$-keeper connecting two vertices $v$ and $w$. The main properties of $k$-keepers are summarized in the following lemma.
\begin{lemma}\label{lem:keeper properties}
Let $G$ be a graph and $K$ be a $k$-keeper connecting $v,w \in V(G)$ ($k \geq 2$). Then,
\begin{enumerate}[(i)]
\item no induced hole or $K_{k+2}$ contains a vertex in $K$,\label{lem:keeper properties:forbidden}
\item $v$ and $w$ have the same color in any star $k$-coloring of $G$, and\label{lem:keeper properties:colors}
\item Any $k$-coloring $\rho$ of $G \setminus K$ in which $\rho(v) = \rho(w)$ can be extended into a $k$-coloring of $G$ in such a way that no monochromatic maximal star contains a vertex in $K$.\label{lem:keeper properties:extension}
\end{enumerate}
\end{lemma}
\begin{proof}
Let $C_1$, \ldots, $C_{k-1}$ and $D = \{d_1, \ldots, d_{k-1}\}$ be as in Definition~\ref{def:k-keeper connection}. (\ref{lem:keeper properties:forbidden}) is trivial.
(\ref{lem:keeper properties:colors}) Let $\rho$ be a star $k$-coloring of $G$ and fix $\range{i,j}{1}{k-1}$. Since $C_i$ is a block of size $k$, it contains vertices $c_i, c_j, c_v$ with colors $\rho(d_i)$, $\rho(d_j)$, and $\rho(v)$, respectively, by Observation~\ref{obs:block coloring}. Hence, $\rho(d_i) \neq \rho(v)$ since otherwise $\{d_i\}\{v, c_v\}$ would be a monochromatic maximal star, and, similarly, $\rho(d_i) \neq \rho(d_j)$ because $\{d_i\}\{d_j, c_j\}$ is not monochromatic. In other words, $|\rho(D)| = k-1$ and $\rho(v) \not\in \rho(D)$. Replacing $v$ and $w$ in the reasoning above, we can conclude that $\rho(w) \not\in \rho(D)$ as well. Therefore, $\rho(v) = \rho(w)$.
(\ref{lem:keeper properties:extension}) To extend $\rho$, set $\rho(D) = \{1, \ldots, k\} \setminus \{\rho(v)\}$, and $\rho(C_i) = \{1, \ldots, k\}$ for every \range{i}{1}{k-1}. It is not hard to see that no maximal star with a vertex in $K$ is monochromatic.
\end{proof}
\begin{figure}
\centering
\begin{tabular}{ccc}
\includegraphics{ k-keeper} & \includegraphics{ k-switcher} & \includegraphics{ long-k-switcher}\\
(a) & (b) & (c)
\end{tabular}
\caption{(a) A $k$-keeper connecting $v,w$. (b) A $k$-switcher connecting $\{u_1, \ldots, u_h\}$. (c) A long $k$-switcher connecting $\{w_1, \ldots, w_h\}$. In every figure, gray nodes represent leafed vertices, circular shapes represent cliques, and double lines represent a $k$-keeper connecting two vertices.}\label{fig:keeper-switcher}
\end{figure}
The \Definition{switcher connection}, whose purpose is to force a set of vertices to have at least two colors, is now defined (see Figure~\ref{fig:keeper-switcher}~(b)).
\begin{defn}[$k$-switcher]\label{def:switcher}
Let $G$ be a graph and $U = \{u_1, \ldots, u_h\}$ be an independent set of $G$ with $h \geq 2$. Say that $C \subset V(G)$ is a \Definition{$k$-switcher connecting $U$} ($k \geq 2$) when $|C| = k$, $C \cup \{u_i\}$ is a clique for \range{i}{1}{h}, and there are no more edges incident to vertices in $C$.
\end{defn}
The following result is the analogous of Lemma~\ref{lem:keeper properties} for switchers.
\begin{lemma}\label{lem:switcher properties}
Let $G$ be a graph and $C$ be a $k$-switcher connecting $U \subset V(G)$ ($k \geq 2$). Then,
\begin{enumerate}[(i)]
\item $|\rho(U)| \geq 2$ for any star $k$-coloring $\rho$ of $G$, and \label{lem:switcher properties:colors}
\item Any $k$-coloring $\rho$ of $G \setminus S$ in which $|\rho(U)| \geq 2$ can be extended into a $k$-coloring of $G$ in such a way that no monochromatic maximal star has its center in $C$.\label{lem:switcher properties:extension}
\end{enumerate}
\end{lemma}
As defined, switchers are not useful for proving the hardness of the star coloring problem for $C_4$-free graphs. The reason is that their connected vertices must have no common neighbors to avoid induced $C_4$'s, and our proof requires vertices with different colors and common neighbors. To solve this problem we extend switchers into long switchers by combining them with keepers (see Figure~\ref{fig:keeper-switcher}~(c)). We emphasize that the set of vertices connected by a long switcher need not be an independent set.
\begin{defn}[long $k$-switcher]\label{def:long switcher}
Let $G$ be a graph, and $W = \{w_1, \ldots, w_h\}$ be a set of vertices of $G$ with $h \geq 2$. Say that $S \subset V(G)$ is a \Definition{long $k$-switcher connecting $W$} ($k \geq 2$) when $S$ can be partitioned into an independent set of leafed vertices $U = \{u_1, \ldots, u_h\}$, a $k$-switcher $C$, and $k$-keepers $Q_1, \ldots, Q_h$ in such a way that $C$ connects $U$, $Q_i$ connects $w_i, u_i$ for \range{i}{1}{h}, and there are no more edges adjacent to vertices in $U$.
\end{defn}
The analogous of Lemma~\ref{lem:switcher properties} follows; its proof is a direct consequence of Observation~\ref{obs:leafed vertex} and Lemmas \ref{lem:keeper properties}~and~\ref{lem:switcher properties}.
\begin{lemma}\label{lem:long switcher properties}
Let $G$ be a graph and $S$ be a long $k$-switcher connecting $W \subset V(G)$ ($k \geq 2$). Then,
\begin{enumerate}[(i)]
\item no induced $C_4$ or $K_{k+2}$ of $G$ contains a vertex of $S$,
\item $|\rho(W)| \geq 2$ for any star $k$-coloring $\rho$ of $G$, and \label{lem:long switcher properties:colors}
\item Any $k$-coloring $\rho$ of $G \setminus S$ in which $|\rho(W)| \geq 2$ can be extended into a $k$-coloring of $G$ in such a way that no monochromatic maximal star has its center in $S$.\label{lem:long switcher properties:extension}
\end{enumerate}
\end{lemma}
The last type of connection that we require is the cluster, which is used to represent variables of DNF formulas. As switchers, clusters are used to connect several vertices at the same time. Specifically, clusters require two sets $X$, $-X$, and a vertex $s$. The purpose of the connection is to encode all the valuations of the variables in a formula $\phi$ by using monochromatic stars with center in $s$. Thus, if a variable $\VP x$ is being represented by $X$ and $-X$, then each monochromatic maximal star with center in $s$ contains all the vertices in $X$ and none of $-X$, when $\VP x$ is true, or it contains all the vertices of $-X$ and none of $X$, when $\VP x$ is false.
\begin{defn}[$\ell$-cluster]\label{def:cluster}
Let $G$ be a graph with vertices $s$, $X = \{x_1, \ldots, x_\ell\}$, and $-X = \{-x_1, \ldots, -x_\ell\}$ ($\ell \geq 2$). Say that $K \subseteq V(G)$ is an \Definition{$\ell$-cluster connecting $\langle s, X, -X\rangle$} when $K$ has $\ell$ leafed vertices $v_1, \ldots, v_\ell$, vertex $s$ is adjacent to all the vertices in $X \cup -X \cup K$, sequence $x_1, -x_1, v_1, \ldots, x_\ell, -x_\ell, v_\ell$ is a hole, and there are no more edges incident to vertices in $K$. For \range{i}{1}{\ell}, we write $X[i]$ and $-X[i]$ to refer to $x_i$ and $-x_i$, respectively.
\end{defn}
Note that if $K$ is an $\ell$-cluster connecting $\langle s, X, -X\rangle$, then the subgraph induced by $K \cup X \cup -X \cup \{s\}$ is isomorphic to a $3\ell$-wheel that has $s$ as its universal vertex. As mentioned, the main property of clusters is that its members can be colored in such a way that monochromatic stars represent valuations.
\begin{lemma}\label{lem:cluster properties}
Let $G$ be a graph and $K$ be an $\ell$-cluster connecting $\langle s, X, -X\rangle$ for $\{s\} \cup X \cup -X \subseteq V(G)$. Then,
\begin{enumerate}[(i)]
\item if $N(x) \cap N(x') \subseteq K \cup \{s\}$ for every $x, x' \in X \cup -X$, then no induced $K_4$ or $C_4$ of $G$ contains a vertex in $K$,
\item if $\{s\}S$ is a maximal star of $G$ with $S \cap K = \emptyset$, then $S \cap (X \cup -X)$ equals either $X$ or $-X$, and \label{lem:cluster properties:stars}
\item any $k$-coloring $\rho$ of $G \setminus K$ ($k \geq 2$) can be extended into a $k$-coloring of $G$ in such a way that $\rho(s) \not\in \rho(K)$, and no monochromatic maximal star of $G$ has its center in $K$. \label{lem:cluster properties:extension}
\end{enumerate}
\end{lemma}
\begin{proof}
Let $K = \{v_1, \ldots, v_\ell\}$, $x_1, \ldots, x_{\ell}$, and $-x_1, \ldots, -x_{\ell}$ be as in Definition~\ref{def:cluster}.
(i) For \range{i}{1}{\ell}, the non-leaf neighbors of $v_i$ are $s, -x_i, x_{i+1}$. Since $-x_i$ and $x_{i+1}$ are not adjacent and $N(-x_i) \cap N(x_{i+1}) = \{v_i, s\}$, it follows that $v_i$ belongs to no induced $K_4$ or $C_4$ of $G$.
(ii) Suppose $S \cap K = \emptyset$. If $x_i \in S$ for some \range{i}{1}{\ell}, then $-x_{i} \not\in S$ because $-x_i \in N(x_i)$. Then, since $v_i \not\in S$, it follows that $x_{i+1}$ belongs to $S$. Consequently, by induction on $i$, we obtain that $S \cap X$ equals either $X$ or $\emptyset$. In the former case, $S \cap -X = \emptyset$ because every vertex in $-X$ has a neighbor in $X$. In the latter case, $S \cap -X = -X$ because $S \cap K = \emptyset$ and at least one of $\{x_{i+1}, -x_i, v_i\}$ belongs to $S$ for every \range{i}{1}{\ell}.
(iii) Just extend $\rho$ so that $\rho(s) \not\in \rho(K)$, and color the leaves according to Observation~\ref{obs:leafed vertex}.
\end{proof}
\subsection{Hardness of the star-coloring problem}
\label{sec:general case:2-stp}
It is well known that a problem $P$ is \ensuremath{\Sigma^p_2}\xspace when the problem of authenticating a positive certificate of $P$ is $\coNP$~\cite{Papadimitriou1994}. For the star $k$-coloring problem, a $k$-coloring of $G$ can be taken as a positive certificate. Since it is $\coNP$ to authenticate that a $k$-coloring of $G$ is indeed a star coloring, we obtain that \stcol{k} is \ensuremath{\Sigma^p_2}\xspace. The following theorem states the \ensuremath{\Sigma^p_2}\xspace-hardness of the problem.
\begin{theorem}\label{thm:stcol-2}
\stcol{k} is \ensuremath{\Sigma^p_2}\xspace-complete and it remains \ensuremath{\Sigma^p_2}\xspace-complete even when its input is restricted to $\{C_4, K_{k+2}\}$-free graphs.
\end{theorem}
\begin{proof}
We already know that $\stcol{k}$ belongs to \ensuremath{\Sigma^p_2}\xspace. To prove its hardness, we show a polynomial time reduction from \qsat{2}. That is, for any $3$-DNF formula $\phi(\VEC x, \VEC y)$ with $\ell \geq 2$ clauses $\VP P_1, \ldots, \VP P_\ell$, and $n+m$ variables $\VEC x= \VP x_1, \ldots, \VP x_n$, $\VEC y = \VP y_1, \ldots, \VP y_m$, we build a graph $G$ that admits a star $k$-coloring if and only if $(\exists\VEC x)(\forall \VEC y)\phi(\VEC x, \VEC y)$ is true. For the sake of simplicity, in this proof we use $i$, $j$, $h$, and $q$ as indices that refer to values in $\{1, \ldots, n\}$, $\{1, \ldots, m\}$, $\{1, \ldots, \ell\}$, and $\{1, \ldots, k\}$.
Graph $G$ can be divided into connection, inner, and leaf vertices. \Definition{Connection} vertices are in turn divided into a set of \Definition{clause} vertices $P = \{p_1, \ldots, p_\ell\}$, a set of \Definition{$x$-vertices} $X = \{x_1, \ldots, x_n\}$, two sets of \Definition{$x$-literal} vertices $X_i$ and $-X_i$ with $\ell$ vertices each (for each $i$), two sets of \Definition{$y$-literal} vertices $Y_j$ and $-Y_j$ with $\ell$ vertices each (for each $j$), a set of \Definition{color} vertices $C = \{c_1, \ldots, c_k\}$, and two special vertices $s$ and $t$. Let $L_X = \bigcup_i(X_i \cup -X_i)$, and $L_Y = \bigcup_j^m(Y_j \cup -Y_j)$. \Definition{Inner} vertices are those vertices included in switchers, keepers, and clusters of $G$. Inner vertices and the edges between the connection vertices are given by the next rules.
\begin{description}
\item [Edges:] $s$ is adjacent to all the vertices in $P$, and if $\VP x_i$ (resp.\ $\overline{\VP x_i}$, $\VP y_j$, $\overline{\VP y_j}$) is a literal of $\VP P_h$, then $p_h$ is adjacent $-X_i[h]$ (resp.\ $X_i[h]$, $-Y_j[h]$, $Y_j[h]$).
\item [Keepers:] there is a $k$-keeper connecting $s$ with $t$.
\item [Long switchers:] there are long $k$-switchers connecting $\{x_i, -X_i[h]\}$ and $\{X_i[h], -X_i[h]\}$ (for every $i,h$), $\{c_1, p_h\}$ (for every $h$), $\{c_2, y\}$ for every $y \in L_Y$, $\{c_q, w\}$ for every $q > 2$ and every connection vertex $w \neq s$, $\{c_1, c_2\}$, and $\{c_2,t\}$.
\item [Variables:] there are $\ell$-clusters connecting $\langle s, X_i, -X_i\rangle$ and $\langle s, Y_j, -Y_j\rangle$.
\end{description}
Finally, each connection vertex other than $s$ is leafed. This ends up the construction of $G$, which can be easily computed from $\phi(\VEC x, \VEC y)$ in polynomial time. Figure~\ref{fig:coloring2} depicts a schema of the graph.
\begin{figure}
\centering
\includegraphics{ complexity-1-v2a}
\caption{Schema of the graph obtained from $\phi$ in Theorem~\ref{thm:stcol-2}. For the sake of simplicity, we omit $c_3, \ldots, c_k$ and the edges from $P$ to $L_X \cup L_Y$. Circular shapes drawn with dashes represent independent sets; marked edges between two vertices represent $k$-switchers connecting them; circular shapes with marks represent sets of vertices pairwise connected by $k$-switchers; and squares represent sets of vertices.}\label{fig:coloring2}
\end{figure}
Before dealing with the star $k$-coloring problem on $G$, we show that $G$ is $\{C_4, K_{k+2}\}$-free. By statement~(\ref{lem:keeper properties:forbidden}) of Lemmas \ref{lem:keeper properties}, \ref{lem:long switcher properties}~and~\ref{lem:cluster properties}, it suffices to prove that the subgraph $H$ induced by the connection vertices is $\{K_{k+2}, C_4\}$-free. For this, observe that any induced $C_4$ or $K_{k+2}$ must contain a vertex of $L_X \cup L_Y$ because $\{s\}P$ is an induced star of $H$ and $x_i$, $c_q$, and $t$ have degree at most $1$ in $H$. Now, $X_i[h]$ has at most three neighbors in $H$, namely $-X_i[h]$, $s$, and maybe $p_h$. Hence, since $N(-X_i[h]) = \{s, X_i[h]\}$ when $X_i[h]$ is adjacent to $p_h$, we obtain that $X_i[h]$ belongs to no induced $K_{k+2}$ nor $C_4$. A similar analysis is enough to conclude no vertex of $L_X \cup L_Y$ belongs to an induced $K_{k+2}$ nor $C_4$, thus $H$ is $\{C_4, K_{k+2}\}$-free.
Now we complete the proof by showing that $(\exists\VEC x)(\forall\VEC y)\phi(\VEC x, \VEC y)$ is true if and only if $G$ admits a star $k$-coloring. Suppose first that $(\forall\VEC y)\phi(\VEC x, \VEC y)$ is true for some valuation $\nu:\VEC x \to \{0,1\}$, and define $\rho$ as the $k$-coloring of $G$ that is obtained by the following two steps. First, set $\rho(c_q) = q$, $\rho(X_i) = \rho(x_i) = 2-\nu(\VP x_i)$, $\rho(-X_i) = 1 + \nu(\VP x_i)$, $\rho(p_k) = 2$, and $\rho(L_Y) = \rho(s) = \rho(t) = 1$. Next, iteratively set $\rho$ for the leaves and inner vertices according to Observation~\ref{obs:leafed vertex}, and statement~(\ref{lem:keeper properties:extension}) of Lemmas~\ref{lem:keeper properties}, \ref{lem:long switcher properties},~and~\ref{lem:cluster properties}. Observe that the second step is well defined because every pair of vertices connected by $k$-keepers have the same color, while every pair of vertices connected by long $k$-switchers have different colors.
We claim that $\rho$ is a star $k$-coloring of $G$. To see why, consider a maximal star $\{w\}S$ of $G$ and observe that $w$ cannot be a leaf. If $w \neq s$, then $\{w\}S$ is not monochromatic by Observation~\ref{obs:leafed vertex} and Lemmas~\ref{lem:keeper properties}, \ref{lem:long switcher properties}~and~\ref{lem:cluster properties}. Suppose, then, that $w = s$ and, moreover, that $S\setminus P$ is monochromatic. Then, no $\ell$-cluster intersects $S$ by statement (\ref{lem:cluster properties:extension}) of Lemma~\ref{lem:cluster properties}. Consequently, by statement (\ref{lem:cluster properties:stars}) of Lemma~\ref{lem:cluster properties}, $S \cap (X_i \cup -X_i)$ equals either $X_i$ or $-X_i$, while $S \cap (Y_j \cup -Y_j)$ equals either $Y_j$ or $-Y_j$ for every $i$ and every $j$. Extend $\nu$ to include $\VEC y$ in its domain, so that $\nu(\VP y_j) = 1$ if and only if $Y_j \subseteq S$. By hypothesis, $\nu(\phi(\VEC x, \VEC y)) = 1$, thus there is some clause $\VP P_h$ whose literals are all true according to $\nu$. If $p_h$ has some neighbor in $-X_i$, then $\nu(\VP x_i) = 1$, thus $-X_i \not\subset S$ because $\rho(-X_i) = 2$. Similarly, if $p_h$ has some neighbor in $X_i$ (resp.\ $-Y_j$, $Y_j$), then $X_i \not\subset S$ (resp.\ $-Y_j \not\subset S$, $Y_j\not\subset S$). Therefore, since $P$ is an independent set and $S \subset \{t\} \cup L_X \cup L_Y \cup P$, we obtain that $p_k \in S$, thus $\{s\}S$ is not monochromatic.
For the converse, let $\rho$ be a star $k$-coloring of $G$. Since there is are long $k$-switchers connecting $\{c_1, c_2\}$ and $\{c_q, w\}$ for every $q > 2$ and every connection vertex $w \neq s$, we obtain that $|\rho(C)| = k$ and $\rho(w) \in \{\rho(c_1), \rho(c_2)\}$ by statement~(\ref{lem:long switcher properties:colors}) of Lemma~\ref{lem:long switcher properties}. Define $\nu\colon \VEC x + \VEC y \to \{0,1\}$ as any valuation in which $\nu(\VP x_i) = 1$ if and only if $\rho(x_i) = \rho(c_1)$. Since $\nu(\VP y_j)$ can take any value from $\{0,1\}$, it is enough to prove that $\nu(\phi(\VEC x, \VEC y)) = 1$. Let $V_X = \bigcup_i((X_i \mid \nu(\VP x_i) = 1) \cup (-X_i \mid \nu(\VP x_i) = 0))$ and $V_Y = \bigcup_j((Y_j \mid \nu(\VP y_j) = 1) \cup (-Y_j \mid \nu(\VP y_j) = 0))$.
Let $S = V_X \cup V_Y \cup \{t\}$. By construction, $S$ is an independent set, thus $\{s\}S$ is a star of $G$. Recall that there are long $k$-switchers connecting $\{x_i, -X_i[h]\}$ and $\{X_i[h], -X_i[h]\}$. Hence $\rho(x_i) = \rho(X_i)$ and $\rho(-X_i) \neq \rho(x_i)$ by statement~(\ref{lem:long switcher properties:colors}) of Lemma~\ref{lem:long switcher properties}. This implies that $\rho(V_X) = \{1\}$. Similarly, there are long $k$-switchers connecting $\{c_2, y\}$ for each $y \in L_Y$, thus $\rho(V_Y) \subset \rho(L_Y) = 1$ as well. Finally, using Lemma~\ref{lem:keeper properties}, we obtain that $\rho(s) = \rho(t) = 1$ because there is a long $k$-switcher connecting $\{c_2,t\}$ and a $k$-keeper connecting $t, s$. So, by hypothesis, $\{s\}S$ is not a maximal star, which implies that $S \cup \{w\}$ is also an independent set for some $w \in N(s)$.
Since $t$ and $s$ have the same neighbors in the $k$-keeper $K$ connecting them, it follows that $w \not\in K$. Similarly, all the vertices in a cluster are adjacent to at least one vertex of $V_X \cup V_Y$. Finally, each vertex of $L_X \cup L_Y$ either belongs or has some neighbor in $V_X \cup V_Y$. Consequently, $w = p_h$, i.e., $w$ represents some clause $\VP P_h$. If $\VP x_i$ is a literal of $\VP P_h$, then $p_h$ is adjacent to $-X_i[h]$. Hence, since $V_X \cup V_Y \cup \{p_h\}$ is an independent set, it follows that $-X_i[h] \not\in V_X$. By the way $V_X$ is defined, this means that $\nu(\VP x_i) = 1$. Similar arguments can be used to conclude that if $\VP l$ is a literal of $\VP P_h$, then $\nu(\VP l) = 1$. That is, $\VP P_h$ is satisfied by $\nu$, thus $(\exists\VEC x)(\forall \VEC y)\phi(\VEC x, \VEC y)$ is true.
\end{proof}
\subsection{Graphs with no short holes and small forbidden subgraphs}
\label{sec:chordal}
Note that every hole $H$ of the graph $G$ defined in Theorem~\ref{thm:stcol-2} either 1. contains an edge $xy$ for vertices $x,y$ connected by a $k$-keeper, or 2.\ contains a path $x,v,-x$ for a vertex $v$ in a cluster $K$ connecting $\langle s, X, -X\rangle$ with $x \in X$ and $-x \in -X$. A slight modification of $G$ can be used in the proof of Theorem~\ref{thm:stcol-2} so as to enlarge the hole $H$. In case 1., $xy$ can be subdivided by inserting a vertex $z$ in such a way that $x,z$ and $z,y$ are connected by $k$-keepers. Similarly, in case 2., dummy vertices not adjacent to any $p_h$ can be inserted into $X$ and $-X$ so as to increase the distance between $x$ and $-x$ in $H$. Neither of these modifications generates a new hole in $G$. Thus, in Theorem~\ref{thm:stcol-2} we can use a iterative modification of $G$ whose induced holes have length at least $h$. The following corollary is then obtained.
\begin{corollary}
For every $h \in O(1)$, \stcol{k} is \ensuremath{\Sigma^p_2}\xspace-complete when the input is restricted to $K_{k+2}$-free graphs whose induced holes have length at least $h$.
\end{corollary}
An interesting open question is what happens when $h$ grows to infinity, i.e., what is the complexity of star $k$-coloring a chordal graph or a chordal $K_{k+2}$-free graph. In following sections we consider the star coloring and star choosability problems in some subclasses of chordal graphs, namely split, threshold, and block graphs.
Theorem~\ref{thm:stcol-2} also shows that the star $2$-coloring problem is hard for $\{K_4, C_4\}$-free graphs, which is a class of graphs defined by forbidding two small subgraphs. Thus, another interesting question posed by Theorem~\ref{thm:stcol-2} is what happens when other small graphs are forbidden, so as to understand what structural properties can simplify the problem. Following sections discuss the coloring problems from this perspective as well. In particular, we study the problem for: every $H$-free graphs where $H$ has three vertices; a superclass of diamond-free graphs; and split and threshold graphs. Before dealing with this restricted versions, we establish the complexity of the star $k$-choosability problem for $\{C_4, K_{k+2}\}$-free graphs.
\section{Complexity of the choosability problems}
\label{sec:choosability}
In this section we deal with the list version of the star and biclique-coloring problems. The goal is to show that \stchose{k} and \bcchose{k} are \ensuremath{\Pi^p_3}\xspace-complete problems even when their inputs are restricted to $\{C_4, K_{k+2}\}$-free graphs. Again, only one proof is required because the star and biclique-choosability problems coincide for $C_4$-free graphs. In This opportunity, however, the proof is by induction on $k$. That is, we first conclude that the \stchose{2} problem is \ensuremath{\Pi^p_3}\xspace-complete with a polynomial-time reduction from the \qsat{3}, and next we show that \stchose{k} can be reduced in polynomial time to the \stchose{(k+1)}. The proof for $k = 2$ is similar to the proof of Theorem~\ref{thm:stcol-2}; however, we did not find an easy way to generalize it for $k > 2$ because long switchers generate graphs that are not star $k$-choosable.
\subsection{Keepers, switchers, clusters and forcers}
For the case $k = 2$ we require the keeper and cluster connections once again, and a new version of the long switcher. We begin reviewing the main properties of keepers, switchers, and clusters with respect to the star choosability problem.
\begin{lemma}\label{lem:keeper choosability}
Let $G$ be a graph, $L$ be a $k$-list assignment of $G$, and $K$ be a $k$-keeper connecting $v,w \in V(G)$ ($k \geq 2$). Then, any $L$-coloring $\rho$ of $G \setminus (K \cup \{w\})$ can be extended into an $L$-coloring of $G$ in such a way that no monochromatic maximal star contains a vertex in $K$.
\end{lemma}
\begin{proof}
Let $C_1, \ldots, C_{k-1}$ and $D = \{d_1, \ldots, d_{k-1}\}$ be the vertices of $K$ as in Definition~\ref{def:k-keeper connection}. Extend $\rho$ into an $L$-coloring of $G$ such that $|\rho(D) \setminus \{\rho(v)\}| = k-1$, $|\rho(C_i)| = k$ for every \range{i}{1}{k-1}, and $\rho(w) \not\in \rho(D)$. Since $L$ is a $k$-list assignment, such an extension can always be obtained. Furthermore, no monochromatic maximal star has a vertex in $K$.
\end{proof}
\begin{lemma}\label{lem:switcher choosability}
Let $G$ be a graph, $L$ be a $k$-list assignment of $G$, and $C$ be a $k$-switcher connecting $U \subset V(G)$ ($k \geq 2$). Then, any $L$-coloring $\rho$ of $G \setminus C$ in which $|\rho(U)| \geq 2$ can be extended into an $L$-coloring of $G$ in such a way that no monochromatic maximal star has its center in $C$.
\end{lemma}
By the previous lemma, if $C$ is a $k$-switcher connecting $U$, then $G[U \cup C]$ is star $k$-choosable. This property does not hold for long switchers because it is no longer true that a $k$-keeper connects vertices of the same color. So, to avoid induced $C_4$'s, we need a new version of the long switcher. This new switcher is defined only for $2$-colorings, and it is star $2$-choosable as desired. We refer to this switcher as the \Definition{list switcher}. In short, the difference between the long $2$-switcher and the list switcher is that the latter has no leafed vertices and $2$-keepers are replaced by edges (see Figure~\ref{fig:switcher-forcer}~(a)). Its definition is as follows.
\begin{defn}[list switcher]\label{def:list switcher}
Let $G$ be a graph, and $W = \{w_1, \ldots, w_h\}$ be a set of vertices of $G$ with $h \geq 2$. Say that $S \subset V(G)$ is a \Definition{list switcher connecting $W$} when $S$ can be partitioned into an independent set $U = \{u_1, \ldots, u_h\}$ and a $2$-switcher $C$ in such a way that $C$ connects $U$, $w_iu_i \in E(G)$ for \range{i}{1}{h}, and there are no more edges adjacent to vertices in $U$.
\end{defn}
The following lemma is equivalent to Lemma~\ref{lem:long switcher properties} for list switchers.
\begin{lemma}\label{lem:list switcher properties}
Let $G$ be a graph and $S$ be a list switcher connecting $W \subset V(G)$. Then,
\begin{enumerate}[(i)]
\item no induced $C_4$ or $K_4$ of $G$ contains a vertex of $S$,
\item $|\rho(W)| \geq 2$ for any star $k$-coloring $\rho$ of $G$, and \label{lem:list switcher properties:colors}
\item for any $2$-list assignment $L$ of $G$, every $L$-coloring $\rho$ of $G \setminus S$ in which $|\rho(W)| \geq 2$ can be extended into an $L$-coloring of $G$ in such a way that no monochromatic maximal star contains a vertex in $S$.\label{lem:list switcher properties:extension}
\end{enumerate}
\end{lemma}
\begin{proof}
We only prove (\ref{lem:list switcher properties:extension}). Let $w_1, \ldots, w_h$, and $u_1, \ldots, u_h$ be as in Definition~\ref{def:list switcher}, and $\{x, y\}$ be the $2$-switcher connecting $\{u_1, \ldots, u_h\}$. Suppose, without loss of generality, that $\rho(w_1) \neq \rho(w_2)$, and observe that either $\rho(w_1) \not\in L(x)$ or $\rho(w_2) \not\in L(y)$ or $L(x) = L(y) = \{\rho(w_1),\rho(w_2)\}$. In this setting, extend $\rho$ to include $x$ and $y$ in such a way that $\rho(x) \neq \rho(y)$, $\rho(x) \neq \rho(w_1)$, and $\rho(y) \neq \rho(w_2)$. Following, extend $\rho$ into an $L$-coloring of $G$ such that $\rho(u_1) \neq \rho(y)$, $\rho(u_2) \neq \rho(x)$, and $\rho(u_i) \neq \rho(w_i)$ for \range{i}{3}{h}. It is not hard to see that no monochromatic maximal star contains a vertex in $S$.
\end{proof}
Finally, the proof of statement (\ref{lem:cluster properties:extension}) of Lemma~\ref{lem:cluster properties} implies the following lemma.
\begin{lemma}\label{lem:cluster choosability}
Let $G$ be a graph, $L$ be a $2$-list assignment of $G$, and $K$ be an $\ell$-cluster connecting $\langle s, X, -X\rangle$ for $\{s\} \cup X \cup -X \subseteq V(G)$. Then, any $L$-coloring $\rho$ of $G \setminus K$ can be extended into an $L$-coloring of $G$ in such a way that $\rho(s) \not\in \rho(K)$, and no monochromatic maximal star of $G$ has its center in $K$.
\end{lemma}
\begin{figure}
\centering
\begin{tabular}{c@{\hspace{1cm}}c}
\includegraphics{ list-switcher} & \includegraphics{ 2-forcer} \\
(a) & (b)
\end{tabular}
\caption{(a) A list switcher connecting $\{u_1, \ldots, u_h\}$. (b) A $2$-forcer connecting $v$ with a $2$-list assignment $L$; note that $1$ is the unique $L$-admissible color for $v$.}\label{fig:switcher-forcer}
\end{figure}
Besides the $2$-keepers, list switchers, and clusters, we use forth kind of connection that can be used to force the color of a given vertex when an appropriate list assignment is chosen. This connection is called the \emph{forcer} and, contrary to the other connections, it connects only one vertex.
\begin{defn}[$k$-forcer]\label{def:k-forcer connection}
Let $G$ be a graph and $v \in V(G)$. Say that $F \subset V(G)$ is a \emph{$k$-forcer connecting $v$} ($k \geq 2$) when $F$ can be partitioned into sets of leafed vertices $A$ and $B$, and cliques $C(a,b)$ for $a \in A \cup \{v\}$ and $b \in B$ in such a way that $|A| = k-1$, $|B|=k^{k}-1$, $C(a,b)$ is a $k$-switcher connecting $\{a,b\}$, and there are no more edges incident to vertices in $A \cup B$.
\end{defn}
Let $L$ be a $k$-list assignment of $G$ and $F$ be a $k$-forcer connecting $v \in V(G)$. We say that $c \in L(v)$ is \emph{$L$-admissible for $v$} when there is an $L$-coloring $\rho$ of $G$ in which $\rho(v) = c$ and no monochromatic maximal star has its center in $F$. Clearly, if $c$ is $L$-admissible for $v$, then any $L$-coloring $\rho$ of $G \setminus F$ in which $\rho(v) = c$ can be extended into an $L$-coloring of $G$ in such a way that no monochromatic maximal star has its center in $F$. The main properties of forcers are summed up in the following lemma (see Figure~\ref{fig:switcher-forcer}~(b)).
\begin{lemma}\label{lem:forcer properties}
Let $G$ be a graph and $F$ be a $k$-forcer connecting $v \in V(G)$ ($k \geq 2$). Then,
\begin{enumerate}[(i)]
\item no induced $K_{k+2}$ or $C_4$ of $G$ contains a vertex in $F$,\label{lem:forcer properties:forbidden}
\item for every $k$-list assignment $L$ of $G$ there is an $L$-admissible color for $v$, and\label{lem:forcer properties:extension}
\item every $k$-list assignment $L$ of $G \setminus F$ can be extended into a $k$-list assignment of $G$ in which $v$ has a unique $L$-admissible color.\label{lem:forcer properties:color}
\end{enumerate}
\end{lemma}
\begin{proof}
Let $A$, $B$, and $C(a,b)$ be as in Definition~\ref{def:k-forcer connection}, and define $A^* = A \cup \{v\}$. Statement (\ref{lem:forcer properties:forbidden}) follows from the fact that no pair of vertices in $A^* \cup B$ have a common neighbor.
(\ref{lem:forcer properties:extension}) Let $H$ be the complete bipartite graph with bipartition $\{h_a \mid a \in A^*\}\{h_b \mid b \in B\}$, and $M$ be a $k$-list assignment of $H$ where $M(h_a) = L(a)$ for every $a \in A^*\cup B$. In~\cite{MarxTCS2011} it is proven that $H$ admits a vertex $M$-coloring $\gamma$. Define $\rho$ as any $L$-coloring of $G$ in which $\rho(a) = \gamma(h_a)$ for $a \in A^* \cup B$, where $k$-switchers and leaves are colored according to Lemma~\ref{lem:switcher choosability} and Observation~\ref{obs:leafed vertex}, respectively. The coloring of the $k$-switchers is possible because $\gamma$ is a vertex $M$-coloring. By Observation~\ref{obs:leafed vertex}, no vertex in $A \cup B$ is the center of a maximal star, while by Lemma~\ref{lem:switcher choosability}, no vertex in $C(a,b)$ is the center of a maximal monochromatic star for $a \in A^*$ and $b \in B$. That is, $\rho(v)$ is $L$-admissible for $v$.
(\ref{lem:forcer properties:color}) Extend $L$ into a $k$-list assignment of $G$ such that 1.\ $L(a) \cap L(a') = \emptyset$ for every pair of vertices $a, a' \in A^*$, 2.\ $\mathcal{L}(B) = \{L(b) \mid b \in B\}$ is a family of different subsets included in $L(A^*)$ such that $|L(b) \cap L(a)| = 1$ for every $a \in A^*$ and $b \in B$, and 3.\ $L(C(a,b)) = L(b)$ for every $a \in A^*$, and $b \in B$. Define $H$ and $M$ as in statement (\ref{lem:forcer properties:extension}). By statement~(\ref{lem:forcer properties:extension}), there is an $L$-coloring $\rho$ of $G$ that contains no monochromatic maximal star with center in $F$. Since $C(a,b)$ is a block of $G$, it follows that $\rho(C(a,b)) = L(b)$ by Observation~\ref{obs:block coloring}. Then, since no maximal star with center in $C(a,b)$ is monochromatic, it follows that $\rho(a) \neq \rho(b)$ for every $a \in A^*$, $b \in B$. Thus, if $\gamma$ is the coloring such that $\gamma(h_a) = \rho(a)$ for every $a \in A^*\cup B$, then $\gamma$ is a vertex $M$-coloring of $H$. Consequently, as proven in~\cite{MarxTCS2011}, $\gamma(h_v) = \rho(v)$ is the unique color of $L(v)$ that belongs to the subset of $L(A^*) \not\in \mathcal{L}(B)$.
\end{proof}
\subsection{Hardness of the star choosability problem}
A problem $P$ is \ensuremath{\Pi^p_3}\xspace when the problem of authenticating a negative certificate of $P$ is \ensuremath{\Sigma^p_2}\xspace~\cite{Papadimitriou1994}. For the star $k$-choosability problem, a $k$-list assignment of $G$ can be taken as the negative certificate. Using arguments similar to those in Section~\ref{sec:general case:2-stp} for star $k$-colorings, it is not hard to see that it is a \ensuremath{\Sigma^p_2}\xspace problem to authenticate whether a graph $G$ admits no $L$-colorings for a given $k$-list assignment $L$. Therefore, \stchose{k} is \ensuremath{\Pi^p_3}\xspace. In this section we establish the hardness of \stchose{k}. For $k=2$ we reduce the complement of an instance of \qsat{3} into an instance of \stchose{2}. Then, we proceed by induction showing how to reduce an instance of \stchose{k} into an instance of \stchose{(k+1)} for every $k \geq 2$.
The proof for the case $k=2$ is, in some sense, an extension of Theorem~\ref{thm:stcol-2}. The goal is to force the true literals of $z$ variables to have the same color as $s$, so that a monochromatic maximal star centered at $s$ appears when the formula is false.
\begin{theorem}\label{thm:stchose-2}
\stchose{2} is \ensuremath{\Pi^p_3}\xspace-hard, and it remains \ensuremath{\Pi^p_3}\xspace-hard even when its input is restricted to $\{C_4, K_4\}$-free graphs.
\end{theorem}
\begin{proof}
The hardness of \stchose{2} is obtained by reducing the complement of \qsat{3}. That is, given a $3$-DNF formula $\phi(\VEC z, \VEC x, \VEC y)$ with $\ell$ clauses $\VP P_1, \ldots, \VP P_\ell$, and $n+m+o$ variables $\VEC x= \VP x_1, \ldots, \VP x_n$, $\VEC y = \VP y_1, \ldots, \VP y_m$, $\VEC z = \VP z_1, \ldots, \VP z_o$, we build a graph $G$ that is $2$-list-choosable if and only if $(\forall\VEC z)(\exists \VEC x)(\forall \VEC y)\phi(\VEC z, \VEC x, \VEC y)$ is true. For the sake of simplicity, in this proof we use $i$, $j$, $h$, and $f$ to refer to values in $\{1, \ldots, n\}$, $\{1, \ldots, m\}$, $\{1, \ldots, \ell\}$, and $\{1, \ldots, o\}$, respectively.
Graph $G$ is similar to the graph in Theorem~\ref{thm:stcol-2}. Its vertex set is again divided into connection, inner, and leaf vertices. In turn, \emph{connection} vertices are divided into a set $P = \{p_1, \ldots, p_\ell\}$, a set $X = \{x_1, \ldots, x_n\}$, sets $X_i$, $-X_i$, $Y_j$, $-Y_j$, $Z_f$, and $-Z_f$ with $\ell$ vertices each, and two vertices $s,t$. Let $L_X = \bigcup_i (X_i \cup (-X_i))$, $L_Y = \bigcup_j (Y_j \cup (-Y_j))$, and $L_Z = \bigcup_f (Z_f \cup (-Z_f))$.
\emph{Inner} vertices are those vertices included in $2$-keepers, list switcher, clusters and $2$-forcers. The following rules define inner vertices and the edges between connection vertices. \textbf{Edges:} $s$ is adjacent to all the vertices in $P$, and if $\VP x_i$ (resp.\ $\overline{\VP x_i}$, $\VP y_j$, $\overline{\VP y_j}$, $\VP z_f$, $\overline{\VP z_f}$) is a literal of $\VP P_h$, then $p_h$ is adjacent to $-X_i[h]$ (resp.\ $X_i[h]$, $-Y_j[h]$, $Y_j[h]$, $Z_f[h]$, $-Z_f[h]$). \textbf{Keepers:} $s$ and $t$ are connected by a $2$-keeper. \textbf{List switchers:} there are list switchers connecting \{$x_i$, $-X_i[h]$\} and \{$X_i[h]$, $-X_i[h]$\} (for every $i, h$), $\{Z_f[h], -Z_f[h]\}$ (for every $f, h$), and $\{p_h, t\}$ (for every $h$). \textbf{Clusters:} there are $\ell$-clusters connecting $\langle s, X_i, -X_i\rangle$, $\langle s, Y_j, -Y_j\rangle$, and $\langle s, Z_f, -Z_f\rangle$. \textbf{Forcers:} there are $2$-forcers connecting each vertex of $Z_f$ (for every $f$) and each vertex of $L_Y$.
Finally, every connection vertex other than $s$ is leafed. This ends up the construction of $G$ (see Figure~\ref{fig:choosing}), which can be easily computed from $\phi(\VEC z, \VEC x, \VEC y)$ in polynomial time. Arguments similar to those in Theorem~\ref{thm:stcol-2} are enough to conclude that $G$ is $\{C_4, K_4\}$-free.
\begin{figure}
\centering
\includegraphics{ complexity-2}
\caption{Schema of the graph obtained from $\phi$ in Theorem~\ref{thm:stchose-2}; for the sake of simplicity, we omit the edges from $P$ to $L_X \cup L_Y \cup L_Z$. Square vertices represent vertices connected to a $2$-forcer.}\label{fig:choosing}
\end{figure}
We now show that $(\forall\VEC z)(\exists\VEC x)(\forall\VEC y)\phi(\VEC z, \VEC x, \VEC y)$ is true if and only if $G$ is $2$-choosable. We first show that if $(\forall\VEC z)(\exists\VEC x)(\forall\VEC y)\phi(\VEC z, \VEC x, \VEC y)$ is true, then $G$ admits a star $L$-coloring $\rho$ for any $2$-list assignment $L$. The $L$-coloring $\rho$ is obtained by executing the following algorithm.
\begin{description}
\item[Step 1:] For every $w$ connected to a forcer, let $\rho(w)$ be $L$-admissible for $w$. Such a color always exists by statement~(\ref{lem:forcer properties:extension}) of Lemma~\ref{lem:forcer properties}. Suppose, w.l.o.g., that $\rho(t) = 1$ and let $\nu(\VEC z)$ be a valuation of $\VEC z$ such that $\nu(\VP z_f) = 1$ if and only if $\rho(Z_f) = \{1\}$.
\item[Step 2:] By hypothesis, $\nu$ can be extended to include $\VEC x$ so that $(\forall \VP y)\nu(\phi(\VEC z, \VEC x, \VEC y))$ is true. If $L(x_i) \neq L(-X_i[h])$ or $L(X_i[h]) \neq L(-X_i[h])$ for some $h$, then:
\begin{description}
\item [Step 2.1:] Let $\rho(X_i[h]) \neq \rho(-X_i[h])$ in such a way that $\rho(X_i[h]) = 1$ if and only if $1 \in L(X_i[h])$ and $\nu(\VP x_i) = 1$, while $\rho(-X_i[h]) = 1$ if and only if $1 \in L(-X_i[h])$ and $\nu(\VP x_i) = 0$.
\item [Step 2.2:] Let $\rho(x_i) \neq \rho(-X_i[h])$.
\item [Step 2.3:] Let $\rho(-X_i[k]) \neq \rho(x_i)$ and $\rho(X_i[k]) \neq \rho(-X_i[k])$ for every $k \neq h$.
\end{description}
If $L(x_i) = L(-X_i) = L(X_i)$, then:
\begin{description}
\item [Step 2.4:] Let $\rho(x_i) = \rho(X_i) = 1$ if and only if $1 \in L(x_i)$ and $\nu(\VP x_i) = 1$, and $\rho(-X_i) \neq \rho(x_i)$.
\end{description}
Note that, whichever case gets executed, $\rho(x_i) \not\in \rho(-X_i)$ and $\rho(-X_i[h]) \neq \rho(X_i[h])$ for every $h$.
\item[Step 3:] Let $\rho(-Z_f[h]) \neq \rho(Z_f[h])$ and $\rho(p_h) \neq 1$.
\item[Step 4:] Let $\rho$ for $s$, leaves, and inner vertices be as in Observation~\ref{obs:leafed vertex}, and Lemmas~\ref{lem:keeper choosability}, \ref{lem:list switcher properties}, \ref{lem:cluster choosability}~and~\ref{lem:forcer properties}. Observe that this is always possible. In particular, observe that every pair of vertices connected by a list switcher have different colors, while every vertex connected to a forcer has an $L$-admissible color.
\end{description}
We claim that $\rho$ is a star $L$-coloring of $G$. Let $\{w\}S$ be any maximal star of $G$. By Observation~\ref{obs:leafed vertex} and Lemmas~\ref{lem:keeper choosability}, \ref{lem:list switcher properties}, \ref{lem:cluster choosability}~and~\ref{lem:forcer properties}, $\{w\}S$ is not monochromatic when $w \neq s$. Suppose, for the rest of the proof, that $w = s$ and $\{s\}(S \setminus P)$ is monochromatic. By Lemma~\ref{lem:keeper choosability}, this implies that $t \in S$, thus $\rho(s) = \rho(t) = 1$. Also, by Lemma~\ref{lem:cluster choosability}, $S$ intersects no $k$-cluster, thus $S \subseteq \{t\} \cup L_X \cup L_Y \cup L_Z \cup P$. Moreover, by statement~(\ref{lem:cluster properties:stars}) of Lemma~\ref{lem:cluster properties}, $S \cap (X_i \cup -X_i)$ equals either $X_i$ or $-X_i$, $S \cap (Y_j \cup -Y_j)$ equals either $Y_j$ or $-Y_j$, and $S \cap (Z_f \cup -Z_f)$ is either $Z_f$ or $-Z_f$, for every $i$, $j$, and $f$. Extend $\nu$ to $\VEC y$ so that $\nu(\VP y_j) = 1$ if and only if $Y_j \subset S$. By hypothesis, $\nu(\phi(\VEC z, \VEC x, \VEC y)) = 1$, thus there is some clause $\VP P_h$ whose literals are all true according to $\nu$. If $p_h$ has some neighbor in $-Y_j$, then $\nu(\VP y_j) = 1$, thus $Y_j \subset S$ and $-Y_j \cap S = \emptyset$. If $p_h$ has some neighbor $-z_h \in -Z_f$, then $\nu(\VP z_f) = 1$ which means, by the way $\nu$ is defined for $\VEC z$ in Step~1, that $\rho(Z_f) = \{1\}$. Consequently, by Step~3, $\rho(-z_h) \neq 1$, i.e., $-z_h \not\in S$. Similarly, if $p_h$ has some neighbor in $Z_f$, then $\nu(\VP z_f) = 0$ which means that $\rho(Z_f) \neq \{1\}$. Thus, there must exist at least one vertex $z_f \in Z_f$ with $\rho(z_f) \neq 1$. Then, since $\rho(S) = 1$, it follows that $Z_f \not\subset S$. Finally, if $p_h$ has a neighbor in $-X_i$, then $\nu(\VP x_i) = 1$, thus $\rho(-X_i[h]) \neq 1$ for some $h$ by either Step~2.1 or Step~2.4. Hence, $\rho(-X_i) \neq \{1\}$, thus $-X_i \not\subset S$. Analogously, $p_h$ has no neighbors in $X_i \cap S$. Summing up, since $P \cup \{t\}$ is an independent set, it follows that $p_h$ has no neighbors in $S$, thus $p_h \in S$ and $\{s\}S$ is not monochromatic by Step~3.
For the converse, suppose $G$ is star $2$-choosable, and consider any valuation $\nu$ of $\VEC z$. Define $L$ to be a $2$-list assignment of $G$ so that $\nu(\VP z_f)$ is the unique color admissible for all the vertices in $Z_f$, $1$ is the unique color admissible for all the vertices in $L_Y \cup \{t\}$, and $L(w) = \{0,1\}$ for every vertex not connected to a $2$-forcer. By statement~(\ref{lem:forcer properties:color}) of Lemma~\ref{lem:forcer properties}, such list assignment $L$ always exists. Let $\rho$ be a star $L$-coloring of $G$ and extend $\nu$ to include $\VEC x + \VEC y$ in its domain so that $\nu(\VP x_i) = \rho(v_i)$. Note that $\nu(\VP y_j)$ can take any value from $\{0,1\}$, so it is enough to prove that $\nu(\phi(\VEC x, \VEC y, \VEC z)) = 1$. Define $V_X = \bigcup_i((X_i \mid \nu(\VP x_i) = 1) \cup (-X_i \mid \nu(\VP x_i) = 0))$, $V_Y = \bigcup_j((Y_j \mid \nu(\VP y_j) = 1) \cup (-Y_j \mid \nu(\VP y_j) = 0))$, and $V_Z = \bigcup_f((Z_f \mid \nu(\VP z_f) = 1) \cup (-Z_f \mid \nu(\VP z_f) = 0))$, and let $S = \{s\}(\{t\} \cup V_X \cup V_Y \cup V_Z)$. As in Theorem~\ref{thm:stcol-2}, it can be observed that (i) $S$ is a monochromatic star and (ii) every vertex in $N(s) \setminus P$ is either adjacent or equal to a vertex in $S$. Thus, since $\rho$ is a star $L$-coloring of $G$, there must be some vertex $p_h$ adjacent to no vertex in $V_X \cup V_Y \cup V_Z$. Moreover, such vertex $p_h$ corresponds to some clause $\VP P_h$ whose literals are all true by the way $\nu$ is defined.
\end{proof}
The proof for $k > 2$ is by induction, i.e., we reduce \stchose{k} into \stchose{(k+1)} for every $k \geq 2$. Roughly speaking, the idea of the reduction is to insert a vertex $z$ that forbids every vertex of the reduced graph to have the same color as $z$.
\begin{theorem} \label{thm:stchose-k}
\stchose{k} is \ensuremath{\Pi^p_3}\xspace-complete for every $k \geq 2$, and it remains \ensuremath{\Pi^p_3}\xspace-complete when the input is restricted to $\{C_4, K_{k+2}\}$-free graphs.
\end{theorem}
\begin{proof}
The proof is by induction on $k$. The base case $k = 2$ corresponds to Theorem~\ref{thm:stchose-2}. For the inductive step, we show how to transform a $\{C_4, K_{k+2}\}$-free graph $G_k$ into a $\{C_4, K_{k+3}\}$-free graph $G_{k+1}$ so that $G_k$ is star $k$-choosable if and only if $G_{k+1}$ is star $(k+1)$-choosable.
The vertices of $G_{k+1}$ are divided into connection and inner vertices. Connection vertices comprise a set $W$ inducing $G_{k}$ and a vertex $z$. Inner vertices are included in $(k+1)$-forcers or $(k+1)$-switchers connecting connection vertices. There is a $(k+1)$-forcer connecting $z$, and a $(k+1)$-switcher connecting $\{z,w\}$ for every $w \in W$. Let $C(w)$ be the $(k+1)$-switcher connecting $\{w,z\}$, i.e., $C(w)\cup\{w\}$ and $C(w) \cup \{z\}$ are cliques of $G_{k+1}$. By statement (\ref{lem:forcer properties:forbidden}) of Lemmas \ref{lem:switcher properties}~and~\ref{lem:forcer properties}, $G_{k+1}$ is $\{K_{k+3}, C_4\}$-free.
Suppose $G_k$ is star $k$-choosable. Let $L_{k+1}$ be a $(k+1)$-list assignment of $G_{k+1}$, and $c(z) \in L(z)$ be $L$-admissible for $z$. Recall that $c(z)$ always exists by statement~(\ref{lem:forcer properties:extension}) of Lemma~\ref{lem:forcer properties}. Define $L_k$ as a $k$-list assignment of $G_{k+1}[W]$ such that $L_k(w) \subseteq L_{k+1}(w) \setminus \{c(z)\}$ for $w \in W$. By hypothesis, there is a star $L_k$-coloring $\rho$ of $G_{k+1}[W]$. Define $\sigma$ to be the $L_{k+1}$-coloring of $G_{k+1}$ such that $\sigma(w)=\rho(w)$ for $w \in W$ and $\sigma(z) = c(z)$. Inner vertices are colored according to Lemma~\ref{lem:switcher choosability} and statement~(\ref{lem:forcer properties:extension}) of Lemma~\ref{lem:forcer properties}. Clearly, if $\{w\}S$ is a maximal star of $G_{k+1}$ and $w$ is a connection vertex, then either $w = z$ or $\{w\}S$ includes a maximal star of $G_{k}[W]$. Whichever the case, $\{w\}S$ is not monochromatic, i.e., $\sigma$ is a star coloring of $G_{k+1}$.
For the converse, let $L_k$ be an $k$-list assignment of $G_{k+1}[W]$ and take a color $c\not\in L(V(G_{k+1}))$. Define $L_{k+1}$ as any $(k+1)$-list assignment of $G_{k+1}$ such that $c$ is the unique $L_{k+1}$-admissible color for $z$, and $L_{k+1}(w) = L_{k+1}(C(w)) = L_{k}(w) \cup \{c\}$ for every $w \in W$. Such list assignment always exists by statement~(\ref{lem:forcer properties:color}) of Lemma~\ref{lem:forcer properties}. Let $\sigma$ be a star $L_{k+1}$-coloring of $G_{k+1}$. By construction, $\sigma(z) = \{c\}$, and by Lemma~\ref{lem:switcher properties}, $c \not\in \sigma(W)$. Hence, the restriction $\rho$ of $\sigma$ to $W$ is an $L_k$-coloring of $G_{k+1}[W]$. Moreover, if $\{w\}S$ is a maximal star of $G_{k+1}[W]$, then $\{w\}(S \cup \{x\})$ is a maximal star of $G_{k+1}$, for every $x \in C(w)$. Since $C(w)$ is a block of $G_{k+1}$ and $L(C(w)) = L(w)$, it follows that $\rho(w) = \sigma(w) \in \sigma(C(w))$. Hence, $\{w\}S$ is not monochromatic.
\end{proof}
\section{Forbidding graphs of order 3}
\label{sec:small forbiddens}
The previous sections dealt with time complexity of the star and biclique coloring and choosability problems. The remaining of the article is devoted to these problems in other restricted classes of graphs. As discussed in Section~\ref{sec:chordal}, we are interested in classes of graphs that are related to chordal graphs or can be defined by forbidding small induced subgraphs. In this section, we study the classes of $H$-free graphs, for every graph $H$ on three vertices.
There are four graphs with exactly three vertices, namely $K_3$, $P_3$, $\overline{P_3}$, and $\overline{K_3}$. The following theorem shows that $K_3$-free graphs are star $2$-choosable.
\begin{theorem}\label{thm:stcol triangle-free}
Every $K_3$-free graph is star $2$-choosable. Furthermore, for any $2$-list assignment, a star $L$-coloring can be obtained in linear time.
\end{theorem}
\begin{proof}
Let $L$ be a $2$-list assignment of a $K_3$-free graph $G$, $T$ be a rooted tree subgraph of $G$ with $V(T) = V(G)$, $r$ be the root of $T$, and $p(v)$ be the parent of $v$ in $T$ for each $v \in V(G) \setminus \{r\}$. Define $\rho$ to be an $L$-coloring of $G$ where $\rho(r) \in L(r)$ and $\rho(v) \in L(v) \setminus \{\rho(p(v))\}$ for every $v \in V(G) \setminus \{r\}$. Since $G$ is $K_3$-free, $\{v\}S$ is a maximal star of $G$ for $v \in V(G)$ only if $S = N(v)$, hence $\{v\}S$ is not monochromatic. Observe that a BFS traversal of $G$ is enough to compute $\rho$, thus $\rho$ is computed in linear time from $G$.
\end{proof}
As a corollary, we obtain that $\{C_4, K_3\}$-free graphs are biclique $2$-choosable also. However, this corollary can be easily strengthened so as to include those $K_3$-free graphs that are \emph{biclique-dominated}. A graph $G$ is \Definition{biclique-dominated} when every maximal biclique is either a star or has a false dominated vertex. Some interesting classes of graphs are $K_3$-free and biclique-dominated, including hereditary biclique-Helly graphs~\cite{EguiaSoulignacDMTCS2012}.
\begin{theorem}
Every $K_3$-free graph that is biclique-dominated is biclique $2$-choosable. Furthermore, for any $2$-list assignment, a biclique $L$-coloring can be computed in polynomial time.
\end{theorem}
\begin{proof}
Let $L$ be a $2$-list assignment of a $K_3$-free graph $G$ that is biclique-dominated. The algorithm for biclique $L$-coloring $G$ has two steps. First, apply Theorem~\ref{thm:stcol triangle-free} on $G$ so as to obtain a star $L$-coloring $\rho$ of $G$. Second, traverse each vertex $w$ and, for each $v$ that is false dominated by $w$, change $\rho(v)$ with any color in $L(v) \setminus \rho(w)$. (It is not important if $\rho(v)$ or $\rho(w)$ are later changed when other vertices are examined.) The coloring thus generated is a biclique $L$-coloring. Indeed, if a maximal biclique contains a false dominated vertex $v$, then it also contains the vertex $w$ such that $\rho(v)$ was last changed in the second step while traversing $w$. Since false domination is a transitive relation, it follows that $\rho(v) \neq \rho(w)$ when the second step is completed. On the other hand, if $S$ is a maximal biclique with no false dominated vertices, then $S$ is a star. Since the colors of the vertices of $S$ are not affected by the second step, we obtain that $S$ is not monochromatic. It is not hard to see that the algorithm requires polynomial time.
\end{proof}
Coloring a connected $P_3$-free graph is trivial because the unique connected $P_3$-free graph $G$ with $n$ vertices is $K_n$. Thus, $ch_B(G) = \chi_B(G) = \chi_S(G) = ch_S(G) = n$.
\begin{theorem}
If $G$ is a connected $P_3$-free graph with $n$ vertices, then $ch_B(G) = \chi_B(G) = \chi_S(G) = ch_S(G) = n$.
\end{theorem}
The case of $\overline{P_3}$-free graphs, examined in the next theorem, is not much harder.
\begin{theorem}
If $G$ is a $\overline{P_3}$-free graph with $k$ universal vertices, then $ch_B(G) = \chi_B(G) = \chi_S(G) = ch_S(G) = \max\{2,k\}$.
\end{theorem}
\begin{proof}
Let $K$ be the set of universal vertices of $G$. Clearly, $K$ is a block of $G$, thus $ch_B(G) \geq k$ and $ch_S(G) \geq k$ by Observation~\ref{obs:block coloring}. For the other bound, let $L$ be a $k$-list assignment of $G$, and $B_1, \ldots, B_n$ be the sets of vertices that induce components of $\overline{G} \setminus K$. Define $\rho$ as an $L$-coloring of $G$ such that $|\rho(K)| = k$ and $|\rho(B_i)| = 2$ for \range{i}{1}{n}. Note that $B_i$ is a set of false twin vertices (\range{i}{1}{j}) because $B_i$ is a clique of $\overline{G}$. Thus, every maximal star or biclique $S$ is formed by two vertices of $K$ or it contains a set $B_i$ for some \range{i}{1}{n}. Whichever the case, $S$ is not monochromatic, thus $\rho$ is a star and biclique $L$-coloring.
\end{proof}
The remaining class is the class of $\overline{K_3}$-free graphs. By definition, if $G$ is $\overline{K_3}$-free, then every maximal star and every maximal biclique of $G$ has $O(1)$ vertices. Thus, it takes polynomial time to determine if an $L$-coloring of $G$ is a star or biclique coloring, for any $k$-list assignment $L$. Hence, when restricted to $\overline{K_3}$-free graphs, the star and biclique $k$-coloring problems belong to \NP, while the star and biclique $k$-choosability problems belong to \ensuremath{\Pi^p_2}\xspace. The next theorem shows that, when $k \geq 3$, the choosability problems are \ensuremath{\Pi^p_2}\xspace-complete even when the input is further restricted to co-bipartite graphs.
\begin{theorem}\label{thm:co-bipartite choosability}
\stchose{k} and \bcchose{k} are \ensuremath{\Pi^p_2}\xspace-complete for every $k \geq 3$ when the input is restricted to co-bipartite graphs.
\end{theorem}
\begin{proof}
The proof is obtained by reducing the problem of determining if a connected bipartite graph with no false twins is vertex $k$-choosable, which is known to be \ensuremath{\Pi^p_2}\xspace-complete~\cite{GutnerTarsiDM2009}. Let $G$ be a connected bipartite graph with no false twins, $XY$ be a bipartition of $G$, and $k \in \mathbb{N}$. Define $H$ to be the bipartite graph obtained from $G$ by inserting, for every $vw \in E(G)$, the stars $\{a_i(vw)\}A_i(vw)$ (\range{i}{1}{4}) with $|A_i(vw)| = k-1$ and the edges $va_1(vw)$, $va_3(vw)$, $wa_2(vw)$, $wa_4(vw)$, $a_1(vw)a_2(vw)$, and $a_3(vw)a_4(vw)$ (see Figure~\ref{fig:co-bip choose}). We claim that $G$ is vertex $k$-choosable if and only if $\overline{H}$ is star (resp.\ biclique) $k$-choosable.
\begin{figure}
\centering\includegraphics{ demo-teo-20-v2}\caption{Transformation applied to $vw$ in Theorem~\ref{thm:co-bipartite choosability}; each independent set has $k-1$ vertices.}\label{fig:co-bip choose}
\end{figure}
Suppose first that $G$ is vertex $k$-choosable, and let $L$ be a $k$-list assignment of $H$ and $M$ be the restriction of $L$ to $V(G)$. By hypothesis, $G$ admits a vertex $M$-coloring $\rho$. Define $\sigma$ to be any vertex $L$-coloring of $H$ so that $\sigma(v) = \rho(v)$ for $v \in V(G)$, and $|\sigma(A_i(vw)\cup\{a_i(vw)\})| = k$ for every $vw \in E(G)$ and every \range{i}{1}{4}. It is not hard to see that such a coloring always exists. Clearly, every maximal star (resp.\ biclique) $S$ of $\overline{H}$ is formed by two twins of $\overline{H}$ or it contains two vertices that are adjacent in $H$. In the latter case $S$ is not $\sigma$-monochromatic because $\sigma$ is a vertex coloring of $H$, while in the former case $S$ is not $\sigma$-monochromatic because both of its vertices must belong to $A_i(vw)$, as $G$ has no false twins, for some $vw \in E(G)$ and some \range{i}{1}{4}.
For the converse, suppose $\overline{H}$ is star (resp.\ biclique) $k$-choosable, and let $M$ be a $k$-list assignment of $G$. Define $\sigma$ to be a star (resp.\ biclique) $L$-coloring of $\overline{H}$, for the $k$-list assignment $L$ of $H$ where $L(a) = L(v) = M(v)$ for every $vw \in E(G)$ with $v \in X$, and every $a \in A_i(vw) \cup \{a_i(vw)\}$ with \range{i}{1}{4}. Suppose, to obtain a contradiction, that $\sigma(v) = \sigma(w)$ for some $vw \in E(G)$ with $v \in X$ and $w \in Y$. Then, for every $a \in A_i(vw)$ (\range{i}{1}{4}), we obtain that $\sigma(a) \neq \sigma(v)$ because $\{a\}\{v, w\}$ is a maximal star (resp.\ biclique) of $\overline{H}$. Hence, since $A_i(vw)$ is a block of $\overline{H}$, we obtain by Observation~\ref{obs:block coloring} that $\sigma(A_i(vw)) = L(v) \setminus \{\sigma(v)\}$ for every $\range{i}{1}{4}$. Consequently, since $\{b\}\{a, a_1(vw)\}$ is a maximal star (resp.\ biclique) for every $b \in A_2(vw)$ and every $a \in A_1(vw)$, it follows that $\sigma(a_1(vw)) = \sigma(v)$. Analogously, $\sigma(a_i(vw)) = \sigma(v)$ for every \range{i}{1}{4}. But then, $\{a_i(vw) \mid 1 \leq i \leq 4\}$ is a monochromatic maximal biclique that contains a maximal star, a contradiction. Therefore, $\sigma(v) \neq \sigma(w)$ for every edge $vw$ of $V(G)$, which implies that the restriction of $\sigma$ to $V(G)$ is a vertex $M$-coloring of $G$.
\end{proof}
Let $G$ be a $K_3$-free graph with no false twins, and define $H$ as the $K_3$-free graph that is obtained from $G$ as in Theorem~\ref{thm:co-bipartite choosability}. By fixing the list assignment that maps each vertex to $\{1, \ldots, k\}$ in the proof of Theorem~\ref{thm:co-bipartite choosability}, it can be observed that $G$ admits a vertex $k$-coloring if and only if $\overline{H}$ admits a star (resp.\ biclique) $k$-coloring, for every $k \geq 3$. The problem of determining if a connected $K_3$-free graph with no false twins admits a vertex $k$-coloring is known to be \NP-complete~\cite{Lovasz1973,MaffrayPreissmannDM1996}. Hence, the star and biclique $k$-coloring problems are $\NP$-complete when restricted to $\overline{K_3}$-free graphs, for every $k \geq 3$.
\begin{theorem}\label{thm:co-k3 coloring}
\stcol{k} and \bccol{k} are \NP-complete for every $k \geq 3$ when the input is restricted to $\overline{K_3}$-free graphs.
\end{theorem}
\section{Graphs with restricted diamonds}
\label{sec:diamond-free}
The graph $G$ defined in Theorem~\ref{thm:stcol-2} contains a large number of induced diamonds. For instance, to force different colors on a pair of vertices $v$ and $w$, a $k$-switcher $C$ connecting $\{v, w\}$ is used. Such switcher contains $O(k^2)$ diamonds, one for each edge of $C$. An interesting question is, then, whether induced diamonds can be excluded from Theorem~\ref{thm:stcol-2}. The answer is no, as we prove in this section that the star coloring problem is \NP-complete for diamond-free graphs. By taking a deeper look at $G$, it can be noted that every diamond of $G \setminus \{s\}$ has a pair of twin vertices. In order to prove that the star coloring problem is \NP-complete for diamond-free graphs, we show that the problem is \NP even for the larger class of graphs in which every diamond has two twin vertices. This class corresponds to the class of \{$W_4$, dart, gem\}-free graphs (cf.\ below), and it is worth to note that its graphs may admit an exponential number of maximal stars. We also study the biclique coloring problem on this class, for which we prove that the problem is \NP when there are no induced $K_{i,i}$ for $i \in O(1)$. At the end of the section, we study the star and biclique choosability problems, which turn to be \ensuremath{\Pi^p_2}\xspace-hard for \{$C_4$, dart, gem\}-free graph.
Let $G$ be a graph. Say that $v \in V(G)$ is \Definition{block separable} if every pair of adjacent vertices $w,z\in N(v)$ not dominating $v$ are twins in $G[N(v)]$. The following lemma shows that \{$W_4$, dart, gem\}-free graphs are precisely those graphs in which every induced diamond has twin vertices, and they also correspond to those graphs is which every vertex is block separable. This last condition is crucial in the \NP coloring algorithms.
\begin{theorem}\label{thm:w4dartgemequivalence}
The following statements are equivalent for a graph $G$.
\begin{enumerate}[(i)]
\item $G$ is \{$W_4$, dart, gem\}-free.
\item Every induced diamond of $G$ contains a pair of twin vertices.
\item Every $v \in V(G)$ is block separable.
\end{enumerate}
\end{theorem}
\begin{proof}
(i) $\Longrightarrow$ (ii) If $D \subseteq V(G)$ induces a diamond with universal vertices $v,w$ and there exists $x \in N(v) \setminus N(w)$, then $D \cup \{x\}$ induces a $W_4$, a dart, or a gem in $G$ depending on the remaining adjacencies between $x$ and the vertices of $D$.
(ii) $\Longrightarrow$ (iii) Suppose $v \in V(G)$ is not block separable, thus $N[v]$ contains two adjacent vertices $w$ and $z$ not dominating $v$ that are not twins in $G[N[v]]$; say $d(z) \geq d(w)$. Then, $v$ and $z$ are the universal vertices of a diamond containing $w$ and a vertex in $N(z) \setminus N(w)$, i.e., $G$ contains an induced diamond with no twin vertices.
(iii) $\Longrightarrow$ (i) The $W_4$, dart, and gem graphs have a vertex of degree $4$ that is not block separable.
\end{proof}
Note that if $v$ is block separable, then $N[v]$ can be partitioned into sets $B_0, \ldots, B_\ell$ where $v \in B_0$ and each $B_i$ is a block of $G[N[v]]$. Moreover, no vertex in $B_i$ is adjacent to a vertex in $B_j$, for $1 \leq i < j \leq \ell$. We refer to $B_0, \ldots, B_\ell$ as the \Definition{block separation} of $v$. By definition, $\{v\}S$ is a maximal star of $G$ with $|S|> 1$ if and only if $\ell > 1$, $|S \cap B_0| = 0$ and $|S \cap B_i| = 1$ for \range{i}{1}{\ell}. By Theorem~\ref{thm:w4dartgemequivalence}, every vertex of a \{$W_4$, dart, gem\}-free graph admits a block separation, hence the next result follows.
\begin{lemma}
Let $G$ be a \{$W_4$, dart, gem\}-free graph with a coloring $\rho$. Then, $\rho$ is a star coloring of $G$ if and only if
\begin{itemize}
\item $|\rho(B)| = |B|$ for every block $B$ of $G$, and
\item for every $v \in V(G)$ with block separation $B_0, \ldots, B_\ell$, there exists $B_i$ such that $\rho(v) \not\in \rho(B_i)$.
\end{itemize}
\end{lemma}
It is well known that the blocks of a graph $G$ can be computed in $O(n+m)$ time. Hence, it takes $O(d(v)^2)$ time obtain the block separation of a block separable vertex $v$, and, consequently, the star $k$-coloring and the star $k$-choosability problems are in \NP and \ensuremath{\Pi^p_2}\xspace for \{$W_4$, dart, gem\}-free graphs, respectively.
\begin{theorem}\label{thm:np w4-dart-gem}
\stcol{k} is \NP when the input is restricted to \{$W_4$, dart, gem\}-free graphs.
\end{theorem}
\begin{theorem}\label{thm:ptp w4-dart-gem}
\stchose{k} is \ensuremath{\Pi^p_2}\xspace when the input is restricted to \{$W_4$, dart, gem\}-free graphs.
\end{theorem}
We now consider the biclique coloring problem. The algorithm for determining if a coloring $\rho$ is a biclique coloring of $G$ is divided in two steps. First, it checks that no monochromatic maximal star is a maximal biclique. Then, it checks that $G$ contains no monochromatic maximal biclique $K_{i,j}$ with $2 \leq i \leq j$.
For the first step, suppose $\rho$ is a coloring of $G$ where $\rho(B) = |B|$ for every block $B$ of $G$. Let $v$ be a vertex with a block separation $B_0, \ldots, B_\ell$. As discussed above, $\{v\}S$ is a maximal star if and only if $\ell > 1$, $|S \cap B_0| = 0$, and $|S \cap B_i| = 1$ for every \range{i}{1}{\ell}. If $\{v\}S$ is not a maximal biclique, then there exists $w \in V(G) \setminus N[v]$ adjacent to all the vertices in $S$. Observe that $w$ has at most one neighbor in $B_i$ with color $c$, for each color $c$. Otherwise, taking into account that twin vertices have different colors, $v,w,y,z$ would induce a diamond with no twin vertices, for $y,z \in N(v) \cap N(w)$. Therefore, at most one monochromatic maximal star with center $v$ is included in a biclique containing $w$, for each $w \in V(G) \setminus N(v)$. Thus, to check if there is a monochromatic maximal biclique containing $v$ we first check whether $\prod_{i=1}^\ell|\{z \in B_i \mid \rho(z) = \rho(v)\}| < n$. If negative, then $\rho$ is not a biclique coloring of $G$. Otherwise, all the monochromatic maximal stars with center in $v$ are generated in polynomial time, and for each such star $\{v\}S$ it is tested if there exists $w \in V(G) \setminus N[v]$ adjacent to all the vertices in $S$.
\begin{lemma}\label{lem:w4-dart-gem stars}
If a \{$W_4$, dart, gem\}-free graph $G$ and a coloring $\rho$ are given as input, then it takes polynomial time to determine if there exists a monochromatic maximal biclique $\{v\}S$ with $v \in V(G)$.
\end{lemma}
For the second step, suppose $S$ is an independent set with at least two vertices, and let $I = \bigcap_{v \in S}N(v)$. Note that if $w,z \in I$ are adjacent, then they are twins in $G$ because $w, z$ are the universal vertices of any induced diamond formed by taking a pair of vertices in $S$. Hence, $I$ can be partitioned into a collection $B_1, \ldots, B_\ell$ of blocks of $G$ where no vertex in $B_i$ is adjacent to a vertex in $B_j$, for $1 \leq i < j \leq \ell$. Thus, $ST$ is a maximal biclique of $G$ if and only if no vertex of $V(G) \setminus S$ is complete to $I$ and $|T \cap B_i| = 1$ for every \range{i}{1}{\ell}. That is, $G$ has a monochromatic maximal biclique $ST$ if and only if $|\rho(S)| = 1$, each block of $I$ has a vertex of color $\rho(S)$, and $\bigcap_{w\in I} N(w) = S$.
\begin{lemma}\label{lem:w4-dart-gem bicliques}
Let $G$ be a \{$W_4$, dart, gem\}-free graph. If an independent set $S$ and a coloring $\rho$ of $G$ are given as input, then it takes polynomial time to determine if $G$ has a monochromatic maximal biclique $ST$ with $T \subseteq V(G)$.
\end{lemma}
If $G$ is $K_{i,i}$-free for some constant $i$, then every biclique $ST$ of $G$ with $|S| \leq |T|$ has $|S| < i$. Thus, to determine if $\rho$ is a biclique coloring of $G$, it is enough traverse every independent set $S$ of $G$ with $O(i)$ vertices and to check that there exists no $T \subset V(G)$ such that $ST$ is a monochromatic maximal biclique. By Lemmas \ref{lem:w4-dart-gem stars}~and~\ref{lem:w4-dart-gem bicliques}, it takes polynomial time to determine if there exists $T$ such that $ST$ is a monochromatic maximal star. Since there are $n^{O(1)}$ independent sets with at most $i$ vertices, the algorithm requires polynomial time. We thus conclude that \bccol{k} and \bcchose{k} are respectively \NP and \ensuremath{\Pi^p_2}\xspace when the input is restricted to \{$K_{i,i}$, $W_4$, dart, gem\}-free graphs.
\begin{theorem}\label{thm:np w4-dart-gem-kii}
\bccol{k} is \NP when the input is restricted to \{$K_{i,i}$, $W_4$, dart, gem\}-free graphs, for $i\in O(1)$.
\end{theorem}
\begin{theorem}
\bcchose{k} is \ensuremath{\Pi^p_2}\xspace when the input is restricted to \{$K_{i,i}$, $W_4$, dart, gem\}-free graphs, for $i\in O(1)$.
\end{theorem}
In the rest of this section, we discuss the completeness of the star and biclique coloring and choosability problems. As in Sections \ref{sec:general case}~and~\ref{sec:choosability}, only one proof is used for each problem because $C_4$-free graphs are considered. For the reductions, two restricted satisfiability problems are required, namely \textsc{nae-sat}\xspace and \textsc{nae$\forall\exists$sat}\xspace. A valuation $\nu$ of a CNF formula $\phi$ is a \Definition{nae-valuation} when all the clauses of $\phi$ have a true and a false literal. The formula $(\exists \VEC x)\phi(\VEC x)$ is \Definition{nae-true} when $\phi(\VEC x)$ admits a nae-valuation, while $(\forall \VEC x)\phi(\VEC x)$ is \Definition{nae-true} when every valuation of $\phi(\VEC x)$ is a \Definition{nae-valuation}. \textsc{nae-sat}\xspace is the \NP-complete problem (see~\cite{Garey1979}) in which a CNF formula $\phi$ is given, and the goal is to determine if $\phi$ admits a nae-valuation. Analogously, \textsc{nae$\forall\exists$sat}\xspace is the \ensuremath{\Pi^p_2}\xspace-complete problem (see~\cite{EiterGottlob1995}) in which a CNF formula $\phi(\VEC x, \VEC y)$ is given, and the purpose is to determine if $(\forall \VEC x)(\exists \VEC y)\phi(\VEC x, \VEC y)$ is nae-true. We begin discussing the completeness of the star coloring problem. In order to avoid induced diamonds, we define a replacement of long switchers.
\begin{defn}[diamond $k$-switcher]\label{def:diamond k-switcher}
Let $G$ be a graph and $U = \{u_1, \ldots, u_h\}$ be an independent set of $G$ with $h \geq 2$. Say that $S \subset V(G)$ is a \Definition{diamond $k$-switcher connecting $U$} ($k \geq 2$) when $S$ can be partitioned into a vertex $w_1$, a set of leafed vertices $\{w_2, \ldots, w_h\}$, a family $Q_1, \ldots, Q_h$ of $k$-keepers, and a clique $C$ with $k$ vertices in such a way that $C \cup \{w_1\}$ is a clique, $\{w_1\}\{w_2, \ldots, w_h\}$ is a star, $Q_i$ connects $u_i, w_i$ for $\range{i}{1}{h}$, and there are no more edges adjacent to vertices in $S$.
\end{defn}
A diamond $k$-switcher is depicted in Figure~\ref{fig:diamond switcher}. The main properties of diamond switchers are given in the next lemma.
\begin{figure}[htb]
\centering
\includegraphics{ diam-switcher}
\caption{A diamond $k$-switcher connecting $\{u_1, \ldots, u_h\}$.}\label{fig:diamond switcher}
\end{figure}
\begin{lemma}\label{lem:diamond switcher properties}
Let $G$ be a graph and $S$ be a diamond $k$-switcher connecting $U \subset V(G)$ ($k \geq 2$). Then,
\begin{enumerate}[(i)]
\item no induced $C_4$, diamond, or $K_{k+2}$ of $G$ contains a vertex of $S$,\label{lem:diamond switcher properties:forbidden}
\item $|\rho(U)| \geq 2$ for any star $k$-coloring $\rho$ of $G$, and \label{lem:diamond switcher properties:colors}
\item Any $k$-coloring $\rho$ of $G \setminus S$ in which $|\rho(U)| \geq 2$ can be extended into a $k$-coloring of $G$ in such a way that no monochromatic maximal star has its center in $S$.\label{lem:diamond switcher properties:extension}
\end{enumerate}
\end{lemma}
\begin{proof}
Let $U = \{u_1, \ldots, u_h\}$, $w_1, \ldots, w_h$, $Q_1$, \ldots, $Q_{h}$, and $C$ be as in Definition~\ref{def:diamond k-switcher}. Statement (\ref{lem:diamond switcher properties:forbidden}) follows by statement~(\ref{lem:keeper properties:forbidden}) of Lemma~\ref{lem:keeper properties}, observing that $U$ is an independent set and that no induced diamond can contain a vertex in a $k$-keeper.
(\ref{lem:keeper properties:colors}) Let $\rho$ be a star $k$-coloring of $G$. Since $C$ is a block of size $k$, it contains a vertex $c$ with color $\rho(w_1)$ by Observation~\ref{obs:block coloring}. Then, taking into account that $\{w_1\}\{c, u_1, w_2, \ldots, w_h\}$ is a maximal star and $\rho(u_i) = \rho(w_i)$ by statement~(\ref{lem:keeper properties:colors}) of Lemma~\ref{lem:keeper properties} for $\range{i}{1}{h}$, it follows that $\rho(u_i) \neq \rho(u_1)$ for some $\range{i}{2}{h}$.
(\ref{lem:keeper properties:extension}) To extend $\rho$, first set $\rho(C) = \{1, \ldots, k\}$ and $\rho(w_i) = \rho(u_i)$ ($\range{i}{1}{h}$), and then iteratively extend $\rho$ to color the leaves and the $k$-keepers according to Observation~\ref{obs:leafed vertex} and statement~(\ref{lem:keeper properties:extension}) of Lemma~\ref{lem:keeper properties}.
\end{proof}
We are now ready to prove the \NP-completeness of the star-coloring problem.
\begin{theorem}\label{thm:w4dartgemnpc}
\stcol{k} is $\NP$-complete when the input is restricted to \{$C_4$, diamond, $K_{k+2}$\}-free graphs for every $k \geq 2$.
\end{theorem}
\begin{proof}
By Theorem~\ref{thm:np w4-dart-gem}, \stcol{k} is \NP for \{$C_4$, diamond, $K_{k+2}$\}-free graphs. For the hardness part, we show a polynomial time reduction from \textsc{nae-sat}\xspace. That is, given a CNF formula $\phi$ with $\ell$ clauses $\VP P_1,\ldots, \VP P_\ell$ and $n$ variables $\VP x_1, \ldots, \VP x_n$, we define a \{$C_4$, diamond, $K_{k+2}$\}-free graph $G$ such that $\phi$ admits a nae-valuation if and only if $G$ admits a star $k$-coloring.
The vertices of $G$ are divided into \Definition{connection} and \Definition{inner} vertices. For each \range{i}{1}{n} there are two connection vertices $x_i, -x_i$ \Definition{representing} the literals $\VP x_i$ and $\overline{\VP x_i}$, respectively. Also, there are $k-2$ connection vertices $y_3, \ldots, y_k$. Let $X = \{x_1, \ldots, x_n, -x_1, \ldots, -x_n\}$, $Y = \{y_3, \ldots, y_k\}$, and $P_h = \{x \in X \mid x \text{ represents a literal in } \VP P_h\}$ for \range{h}{1}{\ell}. Inner vertices are the vertices included in diamond $k$-switchers connecting connection vertices. For each $v \in X \cup Y$ and each $y \in Y$ there is a \Definition{color} diamond $k$-switcher connecting $\{v,y\}$. Also, for each $\range{i}{1}{n}$ there is a \Definition{valuation} diamond $k$-switcher connecting $\{x_i, -x_i\}$. Finally, there is a \Definition{clause} diamond $k$-switcher connecting $P_h$ for every \range{h}{1}{\ell}. Observe that $X \cup Y$ is an independent set of $G$. Thus, by statement~(\ref{lem:diamond switcher properties:forbidden}) of Lemma~\ref{lem:diamond switcher properties}, $G$ is \{$C_4$, diamond, $K_{k+2}$\}-free.
Suppose $\phi$ has a nae-valuation $\nu:\VEC x \to \{0,1\}$, and let $\rho$ be a $k$-coloring of the connection vertices such that $\rho(x_i) = 2-\nu(\VP x_i)$ and $\rho(y_j) = j$ for \range{i}{1}{n} and \range{j}{3}{k}. Clearly, every color or valuation $k$-switcher connects a pair of vertices that have different colors. Also, since $\nu$ is a nae-valuation, every set $P_h$ (\range{h}{1}{\ell}) has two vertices representing literals $\VP l_1$ and $\VP l_2$ of $\VP P_h$ with $\nu(l_1) \neq \nu(l_2)$. Hence, $P_h$ is not monochromatic, thus every clause diamond $k$-switcher connects a non-monochromatic set of vertices. Therefore, by statement~(\ref{lem:diamond switcher properties:extension}) of Lemma~\ref{lem:diamond switcher properties}, $\rho$ can be iteratively extended into a star $k$-coloring of $G$.
For the converse, suppose $G$ admits a star $k$-coloring $\rho$. By applying statement~(\ref{lem:diamond switcher properties:colors}) of Lemma~\ref{lem:diamond switcher properties} while considering the different kinds of diamond $k$-switchers, we observe the following facts. First, by the color diamond $k$-switchers, $|\rho(Y)| = k-2$ and $\rho(X) \cap \rho(Y) = \emptyset$. Then, we can assume that $\rho(X) \subseteq \{1,2\}$ and $\rho(Y) = \{3, \ldots, k\}$. Hence, by the valuation diamond $k$-switcher connections, we obtain that $\rho(x_i) \neq \rho(-x_i)$ for every $\range{i}{1}{n}$. Thus, the mapping $\nu:\VEC x \to \{0,1\}$ such that $\nu(\VP x_i) = 2-\rho(x_i)$ is a valuation. Moreover, by the clause diamond $k$-switcher connections, $P_h$ is not monochromatic for \range{h}{1}{\ell}. Consequently, $\nu$ is a nae-valuation of $\phi$.
\end{proof}
Observe that the graph $G$ defined in Theorem~\ref{thm:w4dartgemnpc} is not chordal. However, as discussed in Section~\ref{sec:chordal}, every edge $xy$ such that $x,y$ are connected by a $k$-keeper (inside the diamond $k$-switchers) can be subdivided so as to eliminate all the induced holes of length at most $i$, for every $i \in O(1)$.
We now deal with the star choosability problem. Recall that long switchers are not well suited for the star choosability problem because they contain keepers, and vertices connected by keepers need not have the same colors in every list coloring. Keepers are also present inside diamond switchers, thus it is not a surprise that diamond $k$-switchers are not star $k$-choosable. For this reason, as in Section~\ref{sec:choosability}, the proof is by induction, using list switchers for $k=2$. Since list switchers contain induced diamonds, the \ensuremath{\Sigma^p_2}\xspace-hardness will be obtained for \{$W_4$, gem, dart\}-free graphs, and not for diamond-free graphs. Unfortunately, we did not find a way to avoid these diamonds. Moreover, some kind of forcers are required as well; our forcers have induced diamonds that we were not able to remove either. The hardness proof for $k=2$ is, in some sense, a combination of the proofs of Theorems~\ref{thm:stchose-2}~and~\ref{thm:w4dartgemnpc}. Roughly speaking, the idea is to force the colors of the universal variables of the input formula as in Theorem~\ref{thm:stchose-2}, while a nae-valuation is encoded with colors as in Theorem~\ref{thm:w4dartgemnpc}.
\begin{theorem}\label{thm:w4-dart-gem ptp complete}
\stchose{2} is \ensuremath{\Pi^p_2}\xspace-hard when its input is restricted to \{$C_4$, dart, gem, $K_4$\}-free graphs.
\end{theorem}
\begin{proof}
The hardness of \stchose{2} is obtained by reducing \textsc{nae$\forall\exists$sat}\xspace. That is, given a CNF formula $\phi(\VEC z, \VEC x)$ with $\ell$ clauses $\VP P_1, \ldots, \VP P_\ell$, and $m+n$ variables $\VEC x= \VP x_1, \ldots, \VP x_n$, $\VEC z = \VP z_1, \ldots, \VP z_m$, we build a \{$C_4$, dart, gem, $K_4$\}-free graph $G$ that is star $2$-choosable if and only if $(\forall\VEC z)(\exists \VEC x)\phi(\VEC z, \VEC x)$ is nae-true. For the sake of simplicity, in this proof we use $i$, $f$, and $h$ as indices that refer to values in $\{1, \ldots, n\}$, $\{1, \ldots, m\}$, and $\{1, \ldots, \ell\}$.
Graph $G$ is an extension of the graph in Theorem~\ref{thm:w4dartgemnpc} for $k = 2$ (replacing diamond switcher with list switchers). It has a \Definition{connection} vertex $x_i$ (resp.\ $-x_i$, $z_f$, $-z_f$) \Definition{representing} $\VP x_i$ (resp.\ $\overline{\VP x_i}$, $\VP z_f$, $\overline{\VP z_f}$) for each $i$ (and each $f$), and two \Definition{connection} vertices $t$, $-t$. Let $L_X = \{x_i, -x_i \mid 1 \leq i \leq n\}$, $Z = \{z_1, \ldots, z_m\}$, $-Z = \{-z_1, \ldots, -z_m\}$, and $P_h = \{x \mid x \in X \cup Z \cup -Z \text{ represents a literal in } \VP P_h\}$. Graph $G$ also has \Definition{inner vertices} which are the vertices in list switchers and $2$-forcers connecting connection vertices. There are $2$-forcers connecting each vertex of $Z \cup \{t, -t\}$, and list switchers connecting: $\{t, x_i, -x_i\}$ and $\{-t, x_i, -x_i\}$ for each $i$; $\{z_f,-z_f\}$ for each $f$; and $P_h \cup \{t\}$ and $P_h \cup \{-t\}$ for each $h$.
Let $L$ be a $2$-list assignment of $G$, and suppose $(\forall\VEC z)(\exists\VEC x)\phi(\VEC z, \VEC x)$ is nae-true. Define $\rho$ as an $L$-coloring of the connection vertices satisfying the following conditions.
\begin{enumerate}[(i)]
\item $\rho(v)$ is any color $L$-admissible for $v \in Z \cup \{t,-t\}$. Such a color always exists by statement~(\ref{lem:forcer properties:extension}) of Lemma~\ref{lem:forcer properties}. Suppose, w.l.o.g., that $\rho(t) = 1$ and $\rho(-t) \in \{0,1\}$, and define $\nu(\VEC z)$ as a valuation of $\VEC z$ such that $\nu(\VP z_f) = 1$ if and only if $\rho(z_f) = 1$.
\item By hypothesis, $\nu$ can be extended into a nae-valuation of $\phi(\VEC z, \VEC x)$. If $\nu(\VP x_i) \in L(x_i)$, then $\rho(x_i) = \nu(\VP x_i)$. Otherwise, $\rho(x_i) \in L(x_i) \setminus \{1 - \nu(\VP x_i)\}$. Similarly, $\rho(-x_i) = 1-\nu(\VP x_i)$ if $1-\nu(\VP x_i) \in L(-x_i)$, while $\rho(-x_i) \in L(-x_i) \setminus \{\nu(\VP x_i)\}$ otherwise.
\item $\rho(-z_f) \in L(z_f) \setminus \{\rho(z_f)\}$.
\end{enumerate}
It is not hard to see that $\rho$ can always be obtained. Observe that $\rho(-z_f) \neq \rho(z_f)$, $\rho(x_i) = \rho(-x_i)$ only if $\rho(x_i), \rho(-x_i) \not\in \{\rho(t), \rho(-t)\}$, and $\rho(P_h)$ is monochromatic only if $\rho(P_h) \not\subseteq \{\rho(t), \rho(-t)\}$. Therefore, $\rho$ can be extended into an $L$-coloring of $G$ by statement~(\ref{lem:list switcher properties:extension}) of Lemma~\ref{lem:list switcher properties} and statement~(\ref{lem:forcer properties:extension}) of Lemma~\ref{lem:forcer properties}.
For the converse, suppose $G$ is star $2$-choosable, and consider any valuation $\nu$ of $\VEC z$. Define $L$ to be a $2$-list assignment of $G$ such that $1$ is the unique color admissible for $t$, $0$ is the unique color admissible for $-t$, $\nu(\VP z_f)$ is the unique color admissible for $z_f$, and $L(v) = \{0,1\}$ for $v \in V(G) \setminus (Z \cup \{t,-t\})$. By statement~(\ref{lem:forcer properties:color}) of Lemma~\ref{lem:forcer properties}, such a list assignment always exists. Let $\rho$ be a star $L$-coloring of $G$. By repeatedly applying statement~(\ref{lem:list switcher properties:colors}) of Lemma~\ref{lem:list switcher properties}, it can be observed that none of $\{t,x_i,-x_i\}$, $\{-t, x_i, -x_{i}\}$, $\{t\} \cup P_h$, and $\{-t\}$ are monochromatic. Therefore, $\nu$ is a nae-valuation of $\phi(\VEC z, \VEC x)$.
\end{proof}
Note that if $C$ is a $k$-switcher, then every induced diamond that contains a vertex $u \in C$ also contains a twin of $u$. Hence, by Theorem~\ref{thm:w4dartgemequivalence}, if $F$ is a $k$-forcer connecting $v$, then no vertex in $F$ belongs to an induced dart or gem. Consequently, if $G_k$ is a \{$C_4$, dart, gem, $K_{k+2}$\}-free graph, then the graph $G_{k+1}$ defined in the proof of Theorem~\ref{thm:stchose-k} is \{$C_4$, dart, gem, $K_{k+3}$\}-free. That is, a verbatim copy of the proof of Theorem~\ref{thm:stchose-k} can be used to conclude the following.
\begin{theorem}
\stchose{k} is \ensuremath{\Pi^p_2}\xspace-complete when its input is restricted to \{$C_4$, dart, gem, $K_{k+2}$\}-free graphs for every $k \geq 2$.
\end{theorem}
\section{Split graphs}
\label{sec:split}
In this section we consider the star coloring and star choosability problems restricted to split graphs. The reason for studying split graphs is that they form an important subclass of chordal graphs, and also correspond to the class of \{$2K_2$, $C_4$, $C_5$\}-free graphs~\cite{Golumbic2004}. A graph $G$ is \Definition{split} when its vertex set can be partitioned into an independent set $S(G)$ and a clique $Q(G)$. There are $O(d(v))$ maximal stars centered at $v \in V(G)$, namely $\{v\}(N(v) \cap S(G))$ and $\{v\}((N(v) \cup \{w\})\setminus N(w))$ for $w \in Q(G)$. Thus, the star coloring and star choosability problems on split graphs are \NP and \ensuremath{\Pi^p_2}\xspace, respectively. In this section we prove the completeness of both problems.
We begin observing that $G$ admits a star coloring with $\beta+1$ colors, where $\beta$ is the size of the maximum block. Indeed, each block $B$ of $Q(G)$ is colored with colors $\{1, \ldots, |B|\}$, while each vertex of $S(G)$ is colored with color $\beta+1$. We record this fact in the following observation.
\begin{observation}
If $G$ is a split graph whose blocks have size at most $\beta$, then $G$ admits a star coloring using $\beta+1$ colors. Furthermore, such a coloring can be obtained in linear time.
\end{observation}
Computing a star coloring of a split graph using $\beta+1$ colors is easy, but determining if $\beta$ colors suffice is an $\NP$-complete problem. The proof of hardness is almost identical to the one in Section~\ref{sec:diamond-free} for \{$C_4$, diamond, $K_{k+2}$\}-free graphs. That is, given a CNF formula $\phi$ we build a graph $G$ using \emph{split} switchers in such a way that $\phi$ admits a nae-valuation if and only if $G$ admits a star $k$-coloring. As switchers, \Definition{split} switchers force a set of vertices to have at least two colors in a star $k$-coloring. The difference is that split switchers do so in a split graph (see Figure~\ref{fig:split-switcher}).
\begin{figure}
\centering\includegraphics{ 2-split-switcher}
\caption{A split $k$-switcher connecting a set $W$.}\label{fig:split-switcher}
\end{figure}
\begin{defn}[split $k$-switcher]\label{def:nae connection}
Let $G$ be a split graph and $W \subseteq S(G)$ with $|W| \geq 2$. Say that $S \subset V(G) \setminus W$ is a \Definition{split $k$-switcher connecting $W$} when $S$ can be partitioned into two sets $X, Y \subseteq Q(G)$ and two vertices $a_1, a_2 \in S(G)$ in such a way that $|X| = |Y| = k$, $X \cup \{v\}$ is a clique for every $v \in W \cup \{a_1, a_2\}$, $Y \cup \{a_1\}$ and $Y \cup \{a_2\}$ are cliques, there are no more edges between vertices in $X \cup Y$ and vertices in $S(G)$, and there are no more edges incident to $a_1$ and $a_2$.
\end{defn}
The properties of split $k$-switchers are summarized in the following lemma.
\begin{lemma}\label{lem:split switcher properties}
Let $G$ be a split graph and $S$ be a split $k$-switcher connecting $W \subset S(G)$. Then,
\begin{enumerate}[(i)]
\item $|\rho(W)| > 1$ for every star $k$-coloring $\rho$ of $G$, and\label{lem:split switcher properties:colors}
\item For every $k$-list assignment of $G$, any $L$-coloring $\rho$ of $G \setminus S$ in which $|\rho(W)| > 1$ can be extended into an $L$-coloring of $G$ in such a way that no monochromatic maximal star has its center in $S$.\label{lem:split switcher properties:extension}
\end{enumerate}
\end{lemma}
\begin{proof}
Let $X$, $Y$, $a_1$, and $a_2$ be as in Definition~\ref{def:nae connection}.
(\ref{lem:split switcher properties:colors}) Let $\rho(G)$ be a star $k$-coloring of $G$. By definition, $X$ and $Y$ are blocks of $G$, thus $|\rho(X)| = |\rho(Y)| = k$ by Observation~\ref{obs:block coloring}. Let $x_c$ and $y_c$ be the vertices with color $c$ in $X$ and $Y$, respectively. Since the maximal star $\{x_c\}(W \cup \{y_c\})$ is not monochromatic, it follows that $\rho(W) \neq \{c\}$. Therefore, $|\rho(W)| > 1$.
(\ref{lem:split switcher properties:extension}) To extend $\rho$ to $S$, define $|\rho(X)| = |\rho(Y)| = k$, and $\rho(a_1) \neq \rho(a_2)$. Let $\{v\}V$ be a maximal star with $v \in S$. If either $v \in Y$ or $v \in X$ and $V \cap Y = \emptyset$, then $\{a_1, a_2\} \subseteq V$, thus $\{v\}V$ is not monochromatic. Otherwise, if $v \in X$ and $V \cap Y \neq \emptyset$, then $W \subseteq V$, thus $\{v\}V$ is not monochromatic as well.
\end{proof}
By replacing diamond $k$-switchers with split $k$-switchers in the proof of Theorem~\ref{thm:w4dartgemnpc}, the $\NP$-completeness of the star $k$-coloring problem for split graphs is obtained. For the sake of completeness, we sketch the proof, showing how to build the graph $G$ from the CNF formula.
\begin{theorem}\label{thm:splitnpc}
\stcol{k} restricted to split graphs is $\NP$-complete for every $k \geq 2$.
\end{theorem}
\begin{proof}
Split graphs have $O(n+m)$ maximal stars, hence \stcol{k} is $\NP$ for split graphs. For the hardness part, let $\phi$ be a CNF formula with $m$ clauses $\VP P_1,\ldots, \VP P_m$ and $n$ variables $\VP x_1, \ldots, \VP x_n$. Define $G$ as the split graphs with \Definition{connection} and \Definition{inner} vertices, as follows. For each \range{i}{1}{n} there are two connection vertices $x_i, -x_i$ \Definition{representing} the literals $\VP x_i$ and $\overline{\VP x_i}$, respectively. Also, there are $k-2$ connection vertices $y_3, \ldots, y_k$. Let $X = \{x_1, \ldots, x_n, -x_1, \ldots, -x_n\}$, $Y = \{y_3, \ldots, y_k\}$, and $P_h = \{x \in X \mid x \text{ represents a literal in } \VP P_h\}$ for \range{h}{1}{\ell}. Inner vertices form the split $k$-switchers connecting connection vertices. For each $v \in X \cup Y$ and each $y \in Y$ there is a \Definition{color} split $k$-switcher connecting $\{v,y\}$. Also, for each $\range{i}{1}{n}$ there is a \Definition{valuation} split $k$-switcher connecting $\{x_i, -x_i\}$. Finally, there is a \Definition{clause} split $k$-switcher connecting $P_h$ for every \range{h}{1}{\ell}. Clearly, $G$ is a split graph, and, as in the proof of Theorem~\ref{thm:w4dartgemnpc}, $G$ admits star $k$-star coloring if and only if $\phi$ admits a nae-valuation.
\end{proof}
For the star choosability problem of split graphs, the idea is to adapt the proof of Theorem~\ref{thm:w4-dart-gem ptp complete}, providing a new kind of forcer. This new forcer is called the \emph{split $k$-forcer} and it is just a $k$-forcer where its $k$-switchers form a clique. For the sake of completeness, we include its definition.
\begin{defn}[split $k$-forcer]\label{def:k-split-forcer connection}
Let $G$ be a split graph and $v \in S(G)$. Say that $F \subseteq V(G)$ is a \emph{split $k$-forcer connecting $v$} ($k \geq 2$) when $F$ can be partitioned into sets $A, B \subseteq S(G)$ and $C(a,b) \subseteq Q(G)$ for $a \in A\cup\{v\}$ and $b \in B$ in such a way that $|A| = k-1$, $|B| = k^k-1$, $C(a, b) \cup \{a\}$ and $C(a,b) \cup \{b\}$ are cliques, there are no more edges between vertices in $F \cap Q(G)$ and vertices in $S(G)$, and there are no more edges incident to vertices in $A \cup B$.
\end{defn}
Let $L$ be a $k$-list assignment of a split graph $G$ and $F$ be a split $k$-forcer connecting $v \in V(G)$. As in Section~\ref{sec:choosability}, we say that $c \in L(v)$ is \emph{$L$-admissible for $v$} when there is an $L$-coloring $\rho$ of $G$ such that $\rho(v) = c$ and no monochromatic maximal star has its center in $F$. The following lemma resembles Lemma~\ref{lem:forcer properties}.
\begin{lemma}\label{lem:split forcer properties}
Let $G$ be a split graph and $F$ be a split $k$-forcer connecting $v \in S(G)$. Then,
\begin{enumerate}[(i)]
\item for every $k$-list assignment $L$ of $G$ there is an $L$-admissible color for $v$, and\label{lem:split forcer properties:extension}
\item every $k$-list assignment $L$ of $G \setminus F$ can be extended into a $k$-list assignment of $G$ in which $v$ has a unique $L$-admissible color.
\end{enumerate}
\end{lemma}
\begin{proof}
The lemma can be proven with a verbatim copy of the proof of Lemma~\ref{lem:forcer properties}. In particular, observe that a split $k$-forcer can be obtained from a $k$-forcer by inserting the edges between $C(a,b)$ and $C(a', b')$, for every $a,a' \in A \cup \{v\}$ and $b,b' \in B$.
\end{proof}
The hardness of the star choosability problem is also obtained by adapting the proof of Theorem~\ref{thm:w4-dart-gem ptp complete}. We remark, however, that in this case no induction is required, because split $k$-switchers are $k$-choosable by statement~(\ref{lem:split switcher properties:extension}) of Lemma~\ref{lem:split switcher properties}. The proof is sketched in the following theorem.
\begin{theorem}\label{thm:split stchose}
\stchose{k} restricted to split graphs is \ensuremath{\Pi^p_2}\xspace-complete for every $k \geq 2$.
\end{theorem}
\begin{proof}
\stchose{k} is \ensuremath{\Pi^p_2}\xspace for split graphs because split graphs have a polynomial amount of maximal stars. Let $\phi(\VEC z, \VEC x)$ be a CNF formula with $\ell$ clauses $\VP P_1, \ldots, \VP P_\ell$, and $m+n$ variables $\VEC x= \VP x_1, \ldots, \VP x_n$, $\VEC z = \VP z_1, \ldots, \VP z_m$. Use $i$, $f$, $h$, and $q$ to denote indices in $\{1, \ldots, n\}$, $\{1, \ldots, m\}$, $\{1, \ldots, \ell\}$, and $\{3, \ldots, k\}$, respectively. Define $G$ as the split graph that has a \Definition{connection} vertex $x_i \in S(G)$ (resp.\ $-x_i$, $z_f$, $-z_f$) \Definition{representing} $\VP x_i$ (resp.\ $\overline{\VP x_i}$, $\VP z_f$, $\overline{\VP z_f}$), and $k$ \Definition{connection} vertices $t$, $-t$, $y_3, \ldots, y_{k}$. Let $X = \{x_i, -x_i \mid 1 \leq i \leq n\}$, $Y = \{y_3, \ldots, y_k\}$, $Z = \{z_1, \ldots, z_m\}$, $-Z = \{-z_1, \ldots, -z_m\}$, and $P_h = \{x \mid x \in X \cup Z \cup -Z \text{ represents a literal in } \VP P_h\}$. Graph $G$ also has \Definition{inner vertices} that are the vertices of split $k$-switchers and split $k$-forcers connecting connection vertices. There are split $k$-forcers connecting $z_f$, $t$, $-t$, and split $k$-switchers connecting: $\{t, x_i, -x_i\}$, $\{-t, x_i, -x_i\}$, $\{z_f, -z_f\}$, $P_k \cup \{t\}$, $P_k \cup \{-t\}$, and $\{v, y_q\}$ for every $v \in X \cup Y \cup -Z \cup \{t,-t\}$.
Following the proof of Theorem~\ref{thm:w4-dart-gem ptp complete}, it can be observed that, for $k = 2$, $G$ admits a star $L$-coloring, for a $k$-list assignment $L$ of $G$, if and only if $(\forall \VEC x)(\exists \VEC y)\phi(\VEC x, \VEC y)$ is nae-true. For $k > 2$, observe that if $(\forall \VEC x)(\exists \VEC y)\phi(\VEC x, \VEC y)$ is nae-true, then a star $L$-coloring $\rho$ is obtained if $\rho(Y)$ is taken so that $|\rho(Y)|= k-2$ and $\rho(t), \rho(-t) \not\in \rho(Y)$. Conversely, if $G$ admits a star $L$-coloring for the $k$-list assignment in which $L(v) = \{0, \ldots, k-1\}$ for every connecting vertex $v \not\in Z \cup \{t,-t\}$, then $\rho(X) = \{0,1\}$, thus a nae-valuation of $\phi(\VEC x, \VEC y)$ is obtained.
\end{proof}
To end this section, consider the more general class of $\overline{C_4}$-free graphs. By definition, $\{v\}S$ is a star of a graph $G$ if and only if $S$ is a maximal independent set of $G[N(v)]$. In~\cite{FarberDM1989}, it is proved that $G[N(v)]$ has $O(d(v)^2)$ maximal independent sets when $G$ is $\overline{C_4}$-free. Thus, $\overline{C_4}$-free graphs have $O(nm)$ maximal stars, which implies that the star coloring and star choosability problems on this class are \NP and \ensuremath{\Pi^p_2}\xspace, respectively.
\begin{theorem}\label{thm:2k2 coloring}
\stcol{k} and \stchose{k} are respectively \NP-complete and \ensuremath{\Pi^p_2}\xspace-complete for every $k \geq 2$ when the input is restricted to $\overline{C_4}$-free graphs.
\end{theorem}
\section{Threshold graphs}
\label{sec:threshold}
\newcommand{\AsVec}[1]{[#1]}
Threshold graphs form a well studied class of graphs which posses many definitions and characterizations~\cite{Golumbic2004,MahadevPeled1995}. The reason for studying them in this article is that threshold graphs are those split graphs with no induced $P_4$'s. Equivalently, a graph is a threshold graph if an only if it is $\{2K_2, P_4, C_4\}$-free.
In this section we develop a linear time algorithm for deciding if a threshold graph $G$ admits a star $k$-coloring. If affirmative, then a star $k$-coloring of $G$ can be obtained in linear time. If negative, then a certificate indicating why $G$ admits no coloring is obtained. We prove also that $G$ is star $k$-choosable if and only if $G$ admits a star $k$-coloring. Thus, deciding whether $G$ is star $k$-choosable takes linear time as well. It is worth noting that threshold graphs can be encoded with $O(n)$ bits using two sequences of natural numbers (cf.\ below). We begin this section with some definitions on such sequences.
Let $S = s_1, \ldots, s_r$ be a sequence of natural numbers. Each \range{i}{1}{r} is called an \Definition{index} of $S$. For $k \in \mathbb{N}$, we write $S = \AsVec{k}$ and $S \leq \AsVec{k}$ to respectively indicate that $s_i = k$ and $s_i \leq k$ for every index $i$. Similarly, we write $S > \AsVec{k}$ when $S \not\leq \AsVec{k}$, i.e., when $s_i > k$ for some index $i$. Note that $S$ could be empty; in such case, $S = \AsVec{k}$ and $S \leq \AsVec{k}$ for every $k \in \mathbb{N}$. For indices $i, j$, we use $S[i,j]$ to denote the sequence $s_i, \ldots, s_j$. If $i > j$, then $S[i,j] = \emptyset$. Similarly, we define $S(i, j] = S[i+1,j]$, $S[i, j) = S[i,j-1]$, and $S(i,j) = S[i+1, j-1]$.
A \Definition{threshold representation} is a pair $(Q, S)$ of sequences of natural numbers such that $|Q| = |S| + 1$. Let $Q = q_1, \ldots, q_{r+1}$ and $S = s_1, \ldots, s_r$. Each threshold representation defines a graph $G(Q, S)$ whose vertex set can be partitioned into $r+1$ blocks $Q_1, \ldots, Q_{r+1}$ with $|Q_1| = q_1, \ldots, |Q_{r+1}| = q_{r+1}$ and $r$ independent sets $S_1, \ldots, S_r$ with $|S_1| = s_1, \ldots, |S_r| = s_r$ such that, for $1 \leq i \leq j \leq r$, the vertices in $Q_i$ are adjacent to all the vertices in $S_j \cup Q_{j+1}$. It is well known that $G$ is a connected threshold graph if and only if it is isomorphic to $G(Q, S)$ for some threshold representation $(Q, S)$~\cite{Golumbic2004,MahadevPeled1995}. The following observation describes all the maximal stars of $G(Q, S)$.
\begin{observation}\label{obs:stars threshold}
Let $(Q, S)$ be a threshold representation and $v$ be a vertex of $G(Q, S)$. Then, $\{v\}W$ is a maximal star of $G(Q, S)$ if and only if there are indices $i \leq j$ of $Q$ such that $v \in Q_i$, and $W = \{w\} \cup \bigcup_{h=i}^{j-1} S_h$ for some vertex $w \in Q_j$.
\end{observation}
For $k \in \mathbb{N}$, we say that index $i$ of $Q$ is \Definition{$k$-forbidden} for the threshold representation $(Q, S)$ when either $q_i > k$ or $q_i = k$ and there exists some index $j > i$ such that $q_j = k$, $Q(i, j) = \AsVec{k-1}$ and $S[i, j) = \AsVec{1}$. The next theorem shows how to obtain a star $k$-coloring of $G$ when a threshold representation is provided.
\begin{theorem}\label{thm:threshold coloring}
The following statements are equivalent for a threshold representation $(Q, S)$.
\begin{enumerate}
\item $G(Q,S)$ is star $k$-choosable.
\item $G(Q,S)$ admits a star $k$-coloring.
\item No index of $Q$ is not $k$-forbidden for $(Q, S)$.
\end{enumerate}
\end{theorem}
\begin{proof}
(i) $\Longrightarrow$ (ii) is trivial.
(ii) $\Longrightarrow$ (iii). Suppose $G(Q, S)$ admits a star $k$-coloring $\rho$ and yet $Q$ contains some $k$-forbidden index $i$. Since $Q_i$ is a block of $G(Q, S)$, then $q_i \leq k$ by Observation~\ref{obs:block coloring}. Hence, $q_i = k$ and there exists an index $j > i$ such that $q_j = k$, $Q(i,j) = \AsVec{k-1}$, and $S[i,j) = \AsVec{1}$. Let $w_h$ be the unique vertex in $S_h$ for \range{h}{i}{j-1}. By Observation~\ref{obs:block coloring}, both $Q_i$ and $Q_j$ have at least one vertex of each color \range{c}{1}{k}, while for each index \range{h}{i+1}{j-1} there exists a color $c_h$ such that $\rho(Q_h) = \{1, \ldots, k\} \setminus \{c_h\}$. Consequently, there are indices $a < b$ in $\{i, \ldots, j\}$ such that $\rho(w_a) \in \rho(Q_a) \cap \rho(Q_b)$ and $c_h = \rho(w_h) = \rho(w_a)$ for every index \range{h}{a+1}{b-1}. Indeed, it is enough to take \range{a}{i}{j-1} as the maximum index with $\rho(w_a) \in \rho(Q_a)$ and \range{b}{a}{j} as the minimum index such that $\rho(w_a) \not\in \rho(Q_{a+1}) \cup \ldots \cup \rho(Q_{b-1})$. Therefore, if $v_a$ and $v_b$ are the vertices of $Q_a$ and $Q_b$ with color $c$, respectively, then $\{v_a\}\{v_b, w_{a+1}, \ldots, w_{b-1}\}$ is a monochromatic maximal star by Observation~\ref{obs:stars threshold}, a contradiction.
(iii) $\Longrightarrow$ (i). Let $L$ be a $k$-list assignment of $G(Q,S)$ and define $w_i$ as any vertex of $S_i$ for each index $i$ of $I$. For each index $i$ of $Q$, define \range{p(i)}{1}{i} as the minimum index such that $Q(p(i), i] < \AsVec{k}$. Let $\rho$ be an $L$-coloring of $G(Q, S)$ that satisfies all the following conditions for every index $i$ of $Q$:
\begin{enumerate}[(1)]
\item $|\rho(Q_i)| = q_i$,
\item if $q_i < k$ and $i \neq r+1$, then $\rho(w_{p(i)})$ and $\rho(w_i)$ do not belong to $\rho(Q_i)$,
\item if $q_i < k-1$ and $1 < i < r+1$, then $\rho(w_i) \neq \rho(w_{p(i)})$, and
\item if $s_i > 1$ and $i \neq r+1$, then $|\rho(S_i)| \geq 2$.
\end{enumerate}
A coloring satisfying all the above conditions can obtained iteratively, by coloring the vertices in $Q_i \cup S_i$ before coloring the vertices in $Q_j \cup S_j$ for every pair of indices $i < j$. We claim that $\rho$ is a star $L$-coloring of $G(Q, S)$. To see why, let $\{v\}W$ be a maximal star of $G(Q, S)$. By Observation~\ref{obs:stars threshold}, there are two indices $i \leq j$ of $Q$ such that $v \in Q_i$ and $W = \{w\} \cup S_i \cup \ldots \cup S_{j-1}$ for some $w \in Q_j$. If $i = j$, then $\rho(v) \neq \rho(w)$ by~(1). If $S[i, j) > \AsVec{1}$, then $S_h$ is not monochromatic by~(4). If $q_i < k$, then $\{v\} \cup S_i$ is not monochromatic by~(2). If $q_i = k$ and $q_j < k$, then \range{p(i)}{i}{j-1}, thus $\{v\} \cup S_{p(j)} \subset W$ is not monochromatic by~(2). Finally, if $q_i = q_j = k$ and $S[i, j) = \AsVec{1}$, then there exists index $h$ such that $q_h < k -1$; otherwise $i$ would be a $k$-forbidden index of $(Q, S)$. Then, by~(3), $\rho(w_h) \neq \rho(w_{p(h)})$ which implies that $W$ is not monochromatic. Summing up, we conclude that $G(Q,S)$ has no monochromatic maximal star.
\end{proof}
Theorem~\ref{thm:threshold coloring} has several algorithmic consequences for a threshold representation $(Q, S)$ of a graph $G$. As mentioned, $G$ is a split graph where $Q(G) = \bigcup Q_i$ and $S(G) = \bigcup S_i$, thus $\chi_S(G)$ is either $k$ or $k+1$, for $k = \max(Q)$. While deciding if $k$ colors suffice for a general split graph is an {\NP}-complete problem, only $O(|Q|)$ time is needed to decide whether $\chi_S(G) = k$ when $(Q, S)$ is given as input; it is enough to find a $k$-forbidden index of $Q$. Furthermore, if $k < \chi_S(G)$, then a $k$-forbidden index can be obtained in $O(|Q|)$ time as well. Also, if $\chi_S(G) = k$, then a star $k$-coloring $\rho$ of $G$ can be obtained in linear time by observing rules (1)--(4) of implication (iii) $\Longrightarrow$ (i). To obtain $\rho$, begin traversing $Q$ to find $p(i)$ for every index $i$ of $Q$. Then, color the vertices of each block $Q_i$ with colors $1, \ldots, q_i$. Following, color the vertices $w_1, \ldots, w_r$ in that order, taking the value of $q_i$ into account for each index $i$ of $S$. If $q_i \geq k-1$, then $\rho(w_i) = k$; otherwise, $\rho(w_i)$ is any value in $\{q_i+1, \ldots, k\} \setminus \rho(w_{p(i)})$. Finally, color the vertices in $S_i \setminus \{w_i\}$ with color $1$ for each index $i$ of $S$. To encode $\rho$ only two values are required for each index $i$ of $Q$, namely, $q_i$ and $\rho(w_i)$. Thus, $\rho$ can be obtained in $O(|Q|)$ time as well. Finally, $(Q, S)$ can be obtained in $O(n+m)$ time from $G$ when $G$ is encoded with adjacency lists~\cite{Golumbic2004,MahadevPeled1995}. Thus, all the above algorithms take $O(n+m)$ when applied to the adjacency list representation of $G$. We record all these observation in the theorem below.
\begin{theorem}
Let $(Q, S)$ be a threshold representation of a graph $G$. The problems of computing $\chi_S(G)$, $ch_S(G)$, a $k$-forbidden index of $Q$ for $k < \chi_S(G)$, and a $\chi_S(G)$-coloring of $G$ can be solved in $O(|Q|)$ time when $(Q, S)$ is given as input. In turn, all these problems take $O(n+m)$ time when an adjacency list representation of $G$ is given.
\end{theorem}
\section{Net-free block graphs}
\label{sec:block}
In this section we study the star coloring and star choosability problems on block graphs. A graph is a \Definition{block graph} if it is chordal and diamond-free. We develop a linear time algorithm for deciding if a net-free block graph $G$ admits a star $k$-coloring. The algorithm is certified; a star $k$-coloring of $G$ is obtained in the affirmative case, while a forbidden induced subgraph of $G$ is obtained in the negative case. As threshold graphs, net-free block graphs admit an $O(n)$ space representation using weighted trees (cf.\ below). The certificates provided by the algorithms are encoded in such trees, and can be easily transformed to certificates encoded in terms of $G$. We begin describing the weighted tree representation of net-free block graphs.
A \Definition{netblock representation} is a pair $(T, K)$ where $T$ is a tree and $K$ is a \Definition{weight function} from $E(T)$ to $\mathbb{N}$. The graph $G(T, K)$ \Definition{represented} by $(T, K)$ is obtained by inserting a clique $B(vw)$ with $K(vw)$ vertices adjacent to $v$ and $w$, for every $vw \in E(T)$. It is well known that a connected graph is a netblock graph if and only if it is isomorphic to $G(T, K)$ for some netblock representation $(T, K)$~\cite{BrandstadtLeSpinrad1999}. By definition, two vertices $x, y$ of $G(T, K)$ are twins if and only if 1.\ $xy \in B(vw)$ for some $vw \in E(T)$, or 2.\ $x$ is a leaf of $T$ adjacent to $v \in V(T)$ and $y \in B(vx)$, or 3.\ both $x$ and $y$ are leafs of $T$ (in which case $G(T, K)$ is a complete graph). The following observation describes the remaining maximal stars of $G(T, K)$.
\begin{observation}
\label{obs:blockbicliques}
Let $(T, K)$ be a netblock representation and $v$ be a vertex of $G(T, K)$. Then, $\{v\}S$ is a maximal star of $G(T, K)$ with $|S| > 1$ if and only if $v$ is an internal vertex of $T$ and $S$ contains exactly one vertex of $B(vw) \cup \{w\}$ for every $w \in N_T(v)$.
\end{observation}
Let $(T, K)$ be a netblock representation. For $k \in \mathbb{N}$, a \Definition{$k$-subtree} of $(T, K)$ is a subtree of $T$ formed by edges of weight $k$; a \Definition{maximal} $k$-subtree is a $k$-subtree whose edge set is not included in the edge set of another $k$-subtree. A \Definition{$k$-exit} vertex of $(T, K)$ is a vertex $v$ that has some neighbor $w \in V(T)$ such that $K(vw) < k$. The following theorem characterize those net-free block graphs that admit a star $k$-coloring.
\begin{theorem}\label{thm:block graph coloring}
The following statements are equivalent for a netblock representation $(T, K)$.
\begin{enumerate}[i.]
\item $G(T, K)$ is star $k$-choosable.
\item $G(T, K)$ admits a star $k$-coloring.
\item $K(e) \leq k$ for every $e \in E(T)$ and every maximal $(k-1)$-subtree of $(T, K)$ contains a $(k-1)$-exit vertex of $(T, K)$.
\end{enumerate}
\end{theorem}
\begin{proof}
(i) $\Longrightarrow$ (ii) is trivial.
(ii) $\Longrightarrow$ (iii). Suppose $G(T, K)$ admits a star $k$-coloring $\rho$. By Observation~\ref{obs:block coloring}, $K(e) \leq k$ for every $e \in E(T)$ and, moreover, $K(e) \leq k-1$ when $e$ incides in a leaf. Let $R$ be a maximal $(k-1)$-subtree of $(T, K)$, and consider a maximal path $P = v_1, \ldots, v_j$ of $R$ such that $\rho(v_i) \not\in \rho(B(v_iv_{i+1}) \cup \{v_{i+1}\})$ for $1 \leq i < j$. Note that if $j > 1$, then $|B(v_iv_{i+1})| = k-1$, thus (1) $\rho(v_j) \in \rho(B(v_{j-1}v_j))$. We claim that $v_j$ is a $(k-1)$-exit vertex. Indeed, if $v_j$ is a leaf of $T$ and $w$ is its unique neighbor in $T$, then $j = 1$ by (1) and the fact that $B(wv_j) \cup \{v_j\}$ is a block. Consequently, by the maximality of $P$, it follows that $K(v_jw) \neq k-1$, i.e., $v_j$ is a $(k-1)$-exit vertex. On the other case, if $v_j$ is an internal vertex of $T$, then, by Observation~\ref{obs:blockbicliques}, $\{v_j\}S$ is a maximal star of $G(T, K)$ for every $S$ that contains exactly one vertex of $K(v_jw)$ for each $w \in N_T(v)$. Therefore, there exists $w \in N_T(v_j)$ such that (2) $\rho(v_j) \not\in \rho(B(v_jw) \cup \{w\})$. By (1), $w \neq v_{j-1}$, thus $K(v_jw) \neq k-1$ by the maximality of $R$ and $P$. Moreover, $K(v_jw) \neq k$ by (2), thus $v_j$ is a $(k-1)$-exit vertex.
(iii) $\Longrightarrow$ (i). For every maximal $(k-1)$-subtree $R$ of $(T, K)$, let $z(R)$ be a $(k-1)$-exit vertex of $R$. For every $v \in V(R)$, define $z(v) = z(R)$. Observe this definition is correct, because maximal $(k-1)$-subtrees are vertex-disjoint. Let $L$ be a $k$-list assignment of $G(T,K)$, and define $\rho$ as an $L$-coloring of $G(T, K)$ satisfying the following properties for every $vw \in E(T)$.
\begin{enumerate}[(1)]
\item $\rho(v) \neq \rho(w)$,
\item $|\rho(B(vw))| = K(vw)$ and $\rho(B(vw) \cup \{v,w\}) = \min\{K(vw)+2, k\}$, and
\item if $K(vw) = k-1$ and $w$ belongs to the unique path from $v$ to $z(v)$ in $T$, then $\rho(v) \not\in \rho(B(vw))$.
\end{enumerate}
A coloring satisfying the above conditions can obtained by first coloring the vertices of $T$, and then coloring the vertices in $B(e)$ for every $e \in E(T)$. Observe, in particular, that if $K(vw) = k-1$, then $z(v) = z(w)$, thus (3) is always possible. We claim that $\rho$ is a star $L$-coloring of $G(T, K)$. Let $\{v\}S$ be a maximal star of $G(T, K)$. Suppose first that $S = \{x\}$, thus either $\{v, x\} \subseteq B(wz)$ for $wz \in V(T)$ or $v$ is a leaf of $T$ and $x \in B(vw) \cup \{w\}$ for some $w \in T$. In the former case $\rho(v) \neq \rho(x)$ by (2). In the latter case, by (2), either $\rho(v) \neq \rho(x)$ or $K(vw) \geq k-1$. If $K(vw) \geq k-1$, then $K(vw) = k-1$ and $w$ is not a leaf of $T$; otherwise one of $\{v\}$ and $\{vw\}$ would induce be a maximal $(k-1)$-subtree without $(k-1)$-exit vertices. Hence, $\rho(v) \neq \rho(x)$ by (3). Suppose now that $|S| > 1$, thus $v$ is an internal vertex of $T$ by Observation~\ref{obs:blockbicliques}. By hypothesis, $v$ has some neighbor $w \in V(T)$ such that either $K(vw) < k-1$ (when $v = z(v)$) or $w$ belongs to the path of $T$ between $v$ and $z(v)$ (when $v \neq z(v)$). By Observation~\ref{obs:blockbicliques}, $S$ contains a vertex $x$ of $B(vw)$. If $v = z(v)$, then $K(vw) < k-1$, thus $\rho(v) \neq \rho(x)$ by (1)~and~(2). On the other case, if $v \neq z(v)$, then $K(vw) = k-1$, thus $\rho(v) \neq \rho(z)$ by (3).
\end{proof}
The algorithmic consequences of Theorem~\ref{thm:block graph coloring} are analogous as those observed for threshold graph in Section~\ref{sec:threshold}. If $(T, K)$ is a netblock representation and $k$ is maximum among the sizes of the blocks of $G = G(T, K)$, then $\chi_S(G) = ch_S(G)$ equals either $k$ or $k+1$. When $(T, K)$ is given, it takes $O(|V(T)|)$ time to find all the $(k-1)$-subtrees and its $(k-1)$-exit vertices, if they exist. Thus, deciding if $\chi_S(G) = k$ takes $O(|V(T)|)$ time when $(T, K)$ is given as input. Furthermore, if $k < \chi_S(G)$, then it takes $O(|V(T)|)$ time to compute a maximal $(k-1)$-subtree of $(T, K)$ with no $(k-1)$-exit vertices. Such a subtree can be transformed into an induced subgraph of $G$ in $O(n+m)$ time if required. Also, a $\chi_S(G)$-star-coloring $\rho$ of $G$ can be obtained in linear with rules (1)--(3) of implication (iii) $\Longrightarrow$ (i). First apply a BFS traversal of $T$ to color the vertices of $T$ with rule (1), and then color the remaining vertices of $G$ following rules (2) and (3). Finally, observe that $(T, K)$ can be obtained in $O(n+m)$ time from $G$ when $G$ is encoded by adjacency lists. Thus, all the discussed algorithms take linear time when applied to the adjacency list representation of $G$. We record all these observation in the theorem below.
\begin{theorem}\label{thm:block graph complexity}
Let $(T, K)$ be a netblock representation of a graph $G$. The problems of computing $\chi_S(G)$, $ch_S(G)$, a maximal $(k-1)$-subtree of $(T, K)$ with no $(k-1)$-exit vertices for $k < \chi_S(G)$, and a $\chi_S(G)$-coloring of $G$ can be solved in $O(|V(T)|)$ time when $(T, K)$ is given as input. In turn, all these problems take $O(n+m)$ time when an adjacency list representation of $G$ is given.
\end{theorem}
\section{Further remarks and open problems}
In this paper we investigated the time complexity of the star and biclique coloring and choosability problems. In this section we discuss some open problems that follow from our research.
Theorem~\ref{thm:stcol-2} states that the star $k$-coloring problem is \ensuremath{\Sigma^p_2}\xspace-complete even when the input is restricted to \{$C_4$, $K_{k+2}$\}-free graphs. In Section~\ref{sec:chordal} we discussed how to generalize this theorem to include \{$C_i$, $K_{k+2}$\}-free graphs, for every $i \in O(1)$. An interesting question is what happens when $i$ grows to infinity, i.e., what happens when chordal graphs are considered. By Theorem~\ref{thm:splitnpc}, we know that the star $k$-coloring problem is at least \NP-hard on chordal graphs. Is it \NP-complete or not? Similarly, by Theorem~\ref{thm:split stchose}, the star $k$-choosability problem on chordal graphs is at least \ensuremath{\Pi^p_2}\xspace-hard; is it \ensuremath{\Pi^p_2}\xspace-complete?
\begin{openproblem}
Determine the time complexity of the star $k$-coloring ($k$-choosability) problem on chordal graphs and chordal $K_{k+2}$-free graphs.
\end{openproblem}
To prove the \ensuremath{\Sigma^p_2}\xspace-completeness of the star $k$-coloring problem we showed how to transform a formula $\phi(\VEC x, \VEC y)$ into a graph $G$. Graph $G$ has many connection vertices that are joined together by $k$-keepers and $k$-switchers. The purpose of the $k$-keepers is to force two vertices to have the same colors, while $k$-switchers are used to force different colors on a pair of vertices. Both $k$-keepers and $k$-switchers contain blocks of size $k$. By taking a close examination at Lemmas \ref{lem:keeper properties}~and~\ref{lem:switcher properties}, it can be seen that these blocks play an important role when colors need to be forced. An interesting question is whether these blocks can be avoided.
\begin{openproblem}
Determine the time complexity of the star $k$-coloring ($k$-choosability) problem on graph where every block has size at most $j$, for $j \leq k$.
\end{openproblem}
Keepers and switchers not only have blocks of size $k$; when a $k$-keeper or a $k$-switcher connects two vertices $v$ and $w$, a clique of size $k+1$ containing $v$ and $w$ is generated. We know that the star $k$-coloring problem is \ensuremath{\Sigma^p_2}\xspace-complete for $K_{k+2}$-free graphs, but what happens when $K_{k+1}$-free graphs are considered? The answer for the case $k=2$ is given by Theorem~\ref{thm:stcol triangle-free}, i.e., the star $2$-coloring problem is easy on $K_3$-free graphs. And for larger values of $k$?
\begin{openproblem}
Determine the time complexity of the star $k$-coloring ($k$-choosability) problem on $K_j$-free graphs, for $k > 2$ and $4 \leq j \leq k+1$.
\end{openproblem}
In Section~\ref{sec:choosability}, the \ensuremath{\Pi^p_3}\xspace-completeness of the star $k$-choosability problem on $C_4$-free graphs is proved by induction. For the case $k=2$, a graph $G_2$ is built from a DNF formula using the same ideas that we used to prove the hardness of the star $2$-coloring problem. Then, for the case $k>2$, the graph $G_{k-1}$ is transformed into a graph $G_{k}$. The reason for splitting the proof in two cases is that long $k$-switchers can no longer be included into $G_k$. So, to avoid the inclusion of induced $C_4$'s, list switchers are used to build $G_2$, while $k$-switchers are used to transform $G_{k-1}$ into $G_{k}$ for $k > 2$. This way, each generated graph $G_k$ is $C_4$-free. We did not find a way to extend the holes in $G_k$ as it is done in Section~\ref{sec:chordal} for the star $k$-coloring problem. Thus, $G_k$ contains $C_5$ as an induced subgraph for every $k > 2$, while $G_2$ contains $C_7$ as an induced subgraph. Is the problem simpler when such holes are avoided?
\begin{openproblem}
Determine the time complexity of the star $k$-choosability problem when all the holes in the input graph have length at least $i$, for $i \in O(1)$.
\end{openproblem}
The star coloring problem is easier when any of the graphs on three vertices does not appear as an induced subgraph, as discussed in Section~\ref{sec:small forbiddens}. Similarly, the biclique coloring problem is simpler when it contains no induced $P_3$, $\overline{P_3}$, or $\overline{K_3}$, or when it is $K_3$-free and biclique-dominated. Also, by Theorem~\ref{thm:np w4-dart-gem-kii}, the biclique coloring problem on $K_3$-free graphs is ``only'' \NP when it contains no induced $K_{i,i}$ for $i \in O(1)$.
\begin{openproblem}
Determine the time complexity of the biclique $k$-coloring ($k$-choosability) problem on $K_3$-free graphs.
\end{openproblem}
The star and biclique coloring problems are also simplified when diamonds are forbidden, as seen in Section~\ref{sec:diamond-free}. By Theorem~\ref{thm:np w4-dart-gem}, the star $k$-coloring problem is \NP for diamond-free graphs, while the biclique $k$-coloring problem is \NP for \{diamond, $K_{i,i}$\}-free graphs ($i \in O(1)$) by Theorem~\ref{thm:np w4-dart-gem-kii}. Additionally, if holes and nets are forbidden, then the star $k$-coloring problem can be solved easily by Theorem~\ref{thm:block graph complexity}. This leaves at least two interesting questions.
\begin{openproblem}
Determine the time complexity of the biclique $k$-coloring problems on diamond-free graphs.
\end{openproblem}
\begin{openproblem}
Determine the time complexity of the star $k$-choosability problems on diamond-free graphs.
\end{openproblem}
|
train/arxiv
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BkiUdOfxaKgT2YGEmwjG
| 5 | 1 |
\section{Introduction}\label{sec1}
In this paper, we assume that all graphs are loopless, while parallel edges are allowed.
For any graph $G$, let $V(G)$ and $E(G)$ be the vertex set
and edge set of $G$.
For any edge set $F\subseteq E(G)$,
let $G/F$ be the graph obtained from $G$ by
contracting all edges in $F$, and removing all loops.
Let $\T(G)$ denote the set of spanning trees of $G$.
For any positive integer $k$,
let $\brk{k}$ denote the set
$\{1,2,\cdots,k\}$.
Suppose $G$ is a weighted graph with weight function
$\omega: E(G)\rightarrow \mathbb{R}$.
For any $F\subseteq E(G)$ and any subgraph $H$ of $G$,
define $\omega(F)=\prod\limits_{e\in F} \omega(e)$ and $\omega(H)=\omega(E(H))$.
Let $\tau(G,\omega)=\sum\limits_{T\in \mathcal{T}(G)} \omega(T)$.
Sometimes we use $\tau(G)$ instead of $\tau(G,\omega)$ when there is no confusion.
It is obvious that for an unweighted graph $G$
(that is to say, a weighted graph with unit weight on
each edge), $\tau(G)=|\mathcal{T}(G)|$, i.e., the number of spanning trees of $G$.
Throughout this paper, any graph is assumed to be unweighted,
unless it is claimed.
Counting spanning trees in graphs is a very old topic in graph theory
having modern connections with many other fields in mathematics,
statistical physics and theoretical computer science, such as random walks,
the Ising model and Potts model, network reliability, parking functions,
knot/link determinants. See \cite{DGO, DY, Ehrenborg, GJ, Yan} for some recent
work on counting spanning trees.
For a subgraph $H$ of $G$,
let $\mathcal{T}_{H}(G)$ denote
the set of spanning trees $T\in \T(G)$
with $E(H)\subseteq E(T)$,
and let
$\tau_{H}(G)=\sum\limits_{T\in \mathcal{T}_{H}(G)} \omega(T)$.
For unweighted graph $G$, $\tau_{H}(G)=|\mathcal{T}_{H}(G)|$,
i.e., the number of spanning trees of $G$ containing all edges in $H$.
Note that usually the graph $H$ here is a forest or a tree,
because otherwise
$\mathcal{T}_{H}(G)=\emptyset$ and $\tau_{H}(G)=0$.
The celebrated Cayley's formula \cite{Cayley} states that $\tau(K_n)=n^{n-2}$.
In 1964, Moon generalized Cayley's formula by
obtaining a nice expression of $\tau_F(K_n)$ for any spanning forest $F$ of $K_n$.
\begin{theorem}[\cite{Moon}, also see Problem 4.4 in \cite{Lovasz}]\label{Moon}
For any spanning forest $F$ of $K_n$, if $c$ is the number of components
of $F$ and $n_1, n_2, \ldots, n_c$ are the orders of those components, then
$$\tau_{F}(K_n)=n^{c-2}\prod_{i=1}^c n_i.$$
\end{theorem}
It is easy to see that Cayley's formula is the special case
that $F$ is an empty graph.
It is also well known that $\tau(K_{m, n})=m^{n-1}n^{m-1}$
for any complete bipartite graph $K_{m, n}$
by Fiedler and Sedl\'{a}\v{c}ek \cite{FS}. So there is
a natural question: is there a bipartite analogue of
Moon's formula (Theorem \ref{Moon})?
That is to say,
for any given spanning forest $F$ in $K_{m,n}$,
what is the explicit expression of $\tau_F(K_{m, n})$?
It turns out that this question is much harder than the case of complete graphs.
In \cite{GD}, this question was partially answered for two special cases:
$F$ is a matching or a tree plus several possible isolated vertices.
\begin{theorem}[\cite{GD}]\label{matching}
For any matching $M$ of size $k$ in $K_{m, n}$,
$$\tau_M(K_{m, n})=(m+n)^{k-1}(m+n-k)m^{n-k-1}n^{m-k-1}.$$
\end{theorem}
\begin{theorem}[\cite{GD}]\label{tree}
For any tree $T$ of $K_{m, n}$,
$$
\tau_T(K_{m, n})=(sn+tm-st)m^{n-t-1}n^{m-s-1},
$$
where $s=|V(T)\cap X|$, $t=|V(T)\cap Y|$,
and $(X, Y)$ is the bipartition
of $K_{m, n}$ with $|X|=m$ and $|Y|=n$.
\end{theorem}
In this paper, we obtain an explicit expression
for $\tau_F(K_{m,n})$ for an arbitrary spanning forest $F$
of $K_{m,n}$.
\begin{theorem} \label{mainthm}
Let $(X, Y)$ be the bipartition
of $K_{m, n}$ with $|X|=m$ and $|Y|=n$.
For any spanning forest $F$ of $K_{m,n}$
with components $T_1,T_2,\ldots,T_k$,
\equ{main}
{
\tau_{F}(K_{m,n})=
\frac 1{mn}\left(\prod_{i=1}^k (m_in+n_im)\right)
\left (1-\sum_{i=1}^{k}\frac{m_in_i}{m_in+n_im}\right ),
}
where $m_i=|X\cap V(T_i)|$ and $n_i=|Y\cap V(T_i)|$
for all $i\in \brk{k}$.
\end{theorem}
\section{Preliminary results}\label{sec2}
This section provides some results which will be
applied in the next section for proving a key identity.
\begin{lemma}\label{le2-01}
For any set of $k$ pairs of real numbers
$\{a_i,b_i: i\in \brk{k} \}$, where $k\ge 1$,
if $a_iB+b_iA\ne 0$ for all
$i\in \brk{k}$
where $A=a_1+a_2+\cdots+a_k$ and $B=b_1+b_2+\cdots+b_k$, then \label{r2-3}
\equ{le2-01e1}
{
\left (
1-\sum_{i=1}^k\frac{a_ib_i}{a_iB+b_iA}
\right )^2
=\left(\sum_{i=1}^k\frac{a_i^2}{a_iB+b_iA}\right)
\cdot \left(\sum_{i=1}^k\frac{b_i^2}{a_iB+b_iA}\right).
}
\end{lemma}
\begin{proof}
Since $a_iB+b_iA\ne 0$, $A$ and $B$ cannot both be 0.
When $A=0$ and $B\neq 0$, equality (\ref{le2-01e1}) holds because
$$\left(1-\sum_{i=1}^k\frac{a_ib_i}{a_iB+b_iA}\right)^2=\left(1-\sum_{i=1}^k\frac{b_i}{B}\right)^2=0,$$
and
$$\left(\sum_{i=1}^k\frac{a_i^2}{a_iB+b_iA}\right)\cdot
\left(\sum_{i=1}^k\frac{b_i^2}{a_iB+b_iA}\right)=\frac{A}{B}\cdot\sum_{i=1}^k\frac{b_i^2}{a_iB}=0.$$
Similarly, equality (\ref{le2-01e1}) holds when $B=0$ and $A\neq 0$.
Let $W_i=a_iB+b_iA$ for $i\in \brk{k}$.
When $A\neq 0$ and $B\neq 0$, observe that
\aln{eq2-02}
{
A\sum_{i=1}^k\frac{a_ib_i}{W_i}
+B\sum_{i=1}^k\frac{a_i^2}{W_i}
=\sum_{i=1}^k\frac{a_i(Ab_i+Ba_i)}{W_i}
=\sum_{i=1}^ka_i=A,
}
implying that
\equ
{eq2-03}
{
\sum_{i=1}^k\frac{a_i^2}{W_i}
=\frac{A}{B}
\left (1-\sum_{i=1}^k\frac{a_ib_i}{W_i}
\right ).
}
Similarly,
\equ
{eq2-04}
{
\sum_{i=1}^k\frac{b_i^2}{W_i}
=\frac{B}{A}
\left (1-\sum_{i=1}^k\frac{a_ib_i}{W_i}
\right ).
}
Clearly, equality (\ref{le2-01e1}) follows from
(\ref{eq2-03}) and (\ref{eq2-04}).
\end{proof}
For a set $A$ of real numbers,
in the following, if $A$ is the empty set, we set
\equ{setting}
{
\prod_{a\in A}a=1 \quad \mbox{and}\quad
\sum_{a\in A}a=0.
}
\begin{lemma}\label{le2-0}
Let $S$ be a set of positive integers.
For any set of $2|S|$ real numbers $\{a_i,b_i: i\in S\}$,
\begin{equation} \label{le2-0-1}
\sum_{\emptyset \ne I\subseteq S}
\left (
\left(\prod_{j\in I}a_j\right)
\left(\prod_{r\in S\setminus I}b_r\right )
\right )
=\prod_{i\in S}(a_i+b_i)
-
\prod_{i\in S}b_i.
\end{equation}
\end{lemma}
\begin{proof}
The identity follows from the following fact:
\begin{equation} \label{le2-0-2}
\sum_{\emptyset\subseteq I\subseteq S}
\left (
\left(\prod_{j\in I}a_j\right )
\left(\prod_{r\in S\setminus I}b_r\right )
\right )
=\prod_{i\in S}(a_i+b_i).
\end{equation}
\end{proof}
\begin{lemma}\label{le2-1}
Let $S$ be a set of positive integers.
For any set of $3|S|$
real numbers $\{a_i,b_i,c_i: i\in S\}$,
\begin{equation} \label{le2-1-1}
\sum_{\emptyset \ne I\subseteq S}
\left (
\left (\sum_{i\in I}c_i \right )
\left(\prod_{j\in I}a_j\right )
\left (\prod_{r\in S\setminus I}b_r\right )
\right )
=\left(\prod_{j\in S}(a_j+b_j)\right)
\cdot
\sum_{i\in S}
\frac{c_ia_i}{a_i+b_i}.
\end{equation}
\end{lemma}
\begin{proof}
The identity follows from the following fact:
\begin{equation} \label{le2-1-2}
\sum_{\emptyset \ne I\subseteq S}
\left (
\left (\sum_{i\in I}c_i \right )
\left( \prod_{j\in I}a_j\right )
\left(\prod_{r\in S\setminus I}b_r\right )
\right )
=\sum_{i\in S} \left (
c_ia_i\prod_{j\in S\setminus \{i\}}(a_j+b_j)
\right ).
\end{equation}
\end{proof}
By Lemmas~\ref{le2-0} and~\ref{le2-1}, for an arbitrary real number $c$, we have
\begin{equation} \label{le2-1-1ex}
\sum_{\emptyset \ne I\subseteq S}
\left (
\left (
c+\sum_{i\in I}c_i \right )
\left(\prod_{j\in I}a_j\right)
\left(\prod_{r\in S\setminus I}b_r\right )
\right )
=\left (\prod_{j\in S}(a_j+b_j)\right )
\cdot
\left (c+
\sum_{i\in S}
\frac{c_ia_i}{a_i+b_i}
\right )
-c\prod_{j\in S}b_j.
\end{equation}
\label{r2-5}
\begin{lemma}\label{le2-11}
Let $S$ be a set of positive integers.
For any set of $3|S|$
real numbers $\{a_i,b_i,d_i: i\in S\}$,
\begin{equation} \label{le2-11-1}
\sum_{\emptyset \ne I\subseteq S}
\left (
\left (\sum_{i\in S\setminus I}d_i \right )
\left (\prod_{j\in I}a_j\right )
\left(\prod_{r\in S\setminus I}b_r\right )
\right )
=\left(\prod_{j\in S}(a_j+b_j)\right)
\cdot
\left(\sum_{i\in S}
\frac{d_ib_i}{a_i+b_i}\right )
-\left( \prod_{r\in S}b_r\right )
\cdot
\left( \sum_{i\in S}d_i\right ).
\end{equation}
\end{lemma}
\begin{proof}
It follows from Lemma~\ref{le2-1} and the following fact:
\begin{equation} \label{le2-11-2}
\sum_{\emptyset \ne I\subseteq S}
\left (
\left (\sum_{i\in S\setminus I}d_i \right )
\left (\prod_{j\in I}a_j\right )
\left(\prod_{r\in S\setminus I}b_r\right )
\right )
=\sum_{\emptyset \ne I\subseteq S}
\left (
\left (\sum_{i\in I}d_i \right )
\prod_{j\in S\setminus I}a_j
\prod_{r\in I}b_r
\right )-
\left( \prod_{r\in S}b_r\right )\cdot \left( \sum_{i\in S}d_i\right ).
\end{equation}
\end{proof}
\begin{lemma}\label{le2-2}
Let $S$ be a set of positive integers.
For any set of $4|S|$ real numbers
$\{a_i,b_i,c_i,d_i: i\in S\}$,
\begin{equation} \label{le2-2-1}
\sum_{\emptyset \ne I\subsetneq S}
\left (
\left (\sum_{i\in I}c_i \right )
\left (\sum_{q\in S\setminus I}d_q \right )
\left (\prod_{j\in I}a_j\right )
\left(\prod_{r\in S\setminus I}b_r\right )
\right )
=\left(\prod_{r\in S}(a_r+b_r)\right)
\cdot
\sum_{i,j\in S\atop i\ne j}
\frac{c_ia_id_jb_j}{(a_i+b_i)(a_j+b_j)}.
\end{equation}
\end{lemma}
\begin{proof}
The result follows from the following fact:
\begin{equation} \label{le2-2-2}
\sum_{\emptyset \ne I\subsetneq S}
\left (
\left (\sum_{i\in I}c_i \right )
\left (\sum_{q\in S\setminus I}d_q \right )
\left (\prod_{j\in I}a_j\right )
\left(\prod_{r\in S\setminus I}b_r\right )
\right )
=\sum_{i,j\in S\atop i\ne j}
\left (
c_ia_id_jb_j\prod_{r\in S\setminus \{i,j\}}(a_r+b_r)
\right ).
\end{equation}
\end{proof}
By Lemmas~\ref{le2-11} and~\ref{le2-2}, for an arbitrary real number $c$, we have
\aln{le2-2-1ex}
{
&\sum_{\emptyset \ne I\subseteq S}
\left (
\left (c+\sum_{i\in I}c_i \right )
\left (\sum_{q\in S\setminus I}d_q \right )
\left (\prod_{j\in I}a_j\right )
\left(\prod_{r\in S\setminus I}b_r\right )
\right )
\nonumber \\
=&\left(\prod_{r\in S}(a_r+b_r)\right)
\cdot \left (
\sum_{i,j\in S\atop i\ne j}
\frac{c_ia_id_jb_j}{(a_i+b_i)(a_j+b_j)}
+
c\sum_{i\in S}
\frac{d_ib_i}{a_i+b_i}
\right )
-c\left( \prod_{r\in S}b_r\right )\cdot \left( \sum_{i\in S}d_i\right ).
}
\section{An identity}\label{sec3}
Define a function $\phi$ on $2k$ variables
$x_1,x_2,\ldots,x_k$ and $y_1,y_2,\ldots,y_k$, where $k\ge 1$,
as follows:
\begin{equation} \label{e3-3}
\phi(x_1,y_1,x_2,y_2,\ldots,x_k,y_k)
=\frac{1}{XY}\left(\prod_{i=1}^k(x_iY+y_iX)\right)
\left (1 -\sum_{i=1}^k\frac{x_iy_i}{x_iY+y_iX}\right ),
\end{equation}
where $X=x_1+x_2+\cdots+x_k$ and $Y=y_1+y_2+\cdots+y_k$.
Observe that \label{r2-6}
$$\phi(x_1,y_1,x_2,y_2,\ldots,x_k,y_k)=\frac{1}{XY} \left(\left(\prod\limits_{i=1}^k(x_iY+y_iX)\right)
-\sum\limits_{i=1}^k \left(x_iy_i\prod\limits_{1\le j\le k\atop j\ne i}(x_jY+y_jX)\right)\right).$$
In the expansion of
$\left(\prod\limits_{i=1}^k(x_iY+y_iX)\right)-\sum\limits_{i=1}^k \left(x_iy_i\prod\limits_{1\le j\le k\atop j\ne i}(x_jY+y_jX)\right)$,
the expression consisting of all monomials not divisible by $XY$ is identically $0$, as
shown below:
\eqn{XY-non}
{
& &Y^k\prod_{i=1}^k x_i
+X^k\prod_{i=1}^k y_i
-\sum_{i=1}^k\left ( y_i
Y^{k-1}\prod_{j=1}^k x_j
\right )
-\sum_{i=1}^k \left (x_i
X^{k-1}\prod_{j=1}^k y_j
\right )\nonumber \\
&=&Y^k\prod_{j=1}^k x_j
+X^k\prod_{j=1}^k y_j
-Y^{k-1}\prod_{j=1}^k x_j
\cdot
\left (\sum_{i=1}^ky_i \right )
-X^{k-1}\prod_{j=1}^k y_j
\cdot
\left(\sum_{i=1}^k x_i\right) \nonumber \\
&=&0.
}
It follows that $\phi(x_1, y_1, x_2, y_2, \ldots, x_k, y_k)$ is a polynomial on $2k$ variables
$x_1, x_2, \ldots, x_k$ and $y_1, y_2,\ldots,y_k$.
For $k\le 3$,
\equ{phi}
{
\left \{
\begin{array}{l}
\phi(x_1,y_1)=1;\\
\phi(x_1,y_1,x_2,y_2)=z_{1,2};\\
\phi(x_1,y_1,x_2,y_2,x_3,y_3)=
z_{1,2}z_{1,3}+z_{1,2}z_{2,3}+z_{1,3}z_{2,3},
\end{array}
\right.
}
where $z_{i,j}=x_iy_j+x_jy_i$
for all $1\le i<j\le 3$.
For any $I\subseteq \brk{k}\setminus \{1\}$,
let
\begin{equation} \label{e3-2}
x_I=x_1+\sum_{i\in I}x_i,\quad
y_I=y_1+\sum_{i\in I}y_i.
\end{equation}
In this section, we shall establish the following identity,
which will be applied to prove the main result in the article.
\begin{theorem}\label{th3-1}
For any $2k$ real numbers $x_1,x_2,\ldots,x_k$
and $y_1,y_2,\ldots,y_k$, where $k\ge 2$,
\begin{equation} \label{e3-4}
\phi(x_1,y_1,x_2,y_2,\ldots,x_k,y_k)
=
\sum_{\emptyset\ne I\subseteq \brk{k}\setminus \{1\}}
\left (
(-1)^{|I|-1}
\left(\prod_{j\in I}(x_1y_j+x_jy_1)\right)
\phi(x_I, y_I, \underbrace{x_s,y_s}
_{\forall s\in \brk{k}\setminus (I\cup \{1\})})
\right ),
\end{equation}
where
$
\phi(x_I, y_I, \underbrace{x_s,y_s}
_{\forall s\in \brk{k}\setminus (I\cup \{1\})} )
=\phi(x_I,y_I,x_{i_1},y_{i_2},\ldots,x_{i_r},y_{i_r}),
$
$\{i_1,i_2,\ldots,i_r\}=\brk{k}\setminus (I\cup \{1\})$ and $r=k-1-|I|$.
\end{theorem}
\begin{proof}
Let $I$ be a non-empty subset of $\brk{k}\setminus \{1\}$. Then
\equ{th3-1-1}
{x_I+\sum_{s\in \brk{k}\setminus (I\cup \{1\})}x_s=\sum_{i=1}^kx_i=X
}
and
\equ
{th3-1-2}
{
y_I+\sum_{s\in \brk{k}\setminus (I\cup \{1\})}y_s=\sum_{i=1}^ky_i=Y.
}
In the remainder of the proof of Theorem~\ref{th3-1},
let $W_I=x_IY+y_IX$, and for each
$i\in \brk{k}$, let
$$
W_i=x_iY+y_iX\ \mbox{and}\
w_i=x_iy_1+y_ix_1.
$$
By the definition of the function $\phi$,
\aln{e3-5}
{
\phi(x_I, y_I, \underbrace{x_s,y_s}_
{\forall s\in \brk{k}\setminus (I\cup \{1\})} )
&=\frac{W_I}
{XY}\cdot
\left(\prod_{i\in \brk{k}\setminus (I\cup \{1\})}W_
\right)
\left (1 -\frac{x_Iy_I}{W_I
-\sum_{i\in \brk{k}\setminus (I\cup \{1\})}
\frac{x_iy_i}{W_i}
\right ).
}
Thus, the right-hand side of (\ref{e3-4})
can be expressed as
\equ{eq3-50}
{
\frac {1}{XY}\left (\Gamma_1-\Gamma_2-\Gamma_3 \right ),
}
where
\begin{eqnarray}
\Gamma_1&=&\sum_{\emptyset\ne I\subseteq \brk{k}\setminus \{1\}}
\left ((-1)^{|I|-1}
W_I
\left (\prod_{j\in I}
w_
\right )
\left (\prod_{i\in \brk{k}\setminus (I\cup \{1\})} W_
\right )
\right ), \label{eq3-51} \\
\Gamma_2&=&\sum_{\emptyset\ne I\subseteq \brk{k}\setminus \{1\}}
\left ( (-1)^{|I|-1}
x_Iy_I
\left (\prod_{j\in I}
w_j
\right )
\left (\prod_{i\in \brk{k}\setminus (I\cup \{1\})} W_
\right )
\right ), \label{eq3-52} \\
\Gamma_3&=&\sum\limits_{\emptyset\ne I\subseteq \brk{k}\setminus \{1\}}
\left ( (-1)^{|I|-1}
W_I
\left(\prod\limits_{j\in I}
w_
\right )
\left(\prod\limits_{i\in \brk{k}\setminus (I\cup \{1\})} W_i
\right )
\left ( \sum\limits_{i\in \brk{k}\setminus (I\cup \{1\})}\frac{x_iy_i}
{W_i}
\right )
\right ).\label{eq3-53}
\end{eqnarray}
In the following, we shall apply Lemmas~\ref{le2-01}--~\ref{le2-2} to simplify $\Gamma_1, \Gamma_2$ and $\Gamma_3$
in order to show that
\equ{gamma-sum}
{\Gamma_1-\Gamma_2-\Gamma_3=
\left(\prod_{i=1}^k(x_iY+y_iX)\right)
\left (1-\sum_{i=1}^k\frac{x_iy_i}
{x_iY+y_iX}\right).
}
Let
$X'=X-x_1$ and $Y'=Y-y_1$.
Also let
\equ{3-terms}
{
\left \{
\begin{array}{l}
Z=\prod\limits_{i=2}^k(x_iY'+y_iX'),\\
Z_1=\sum\limits_{i=2}^k\frac{x_iy_i}{x_iY'+y_iX'},
\quad
Z_2=\sum\limits_{i=2}^k\frac{x_i^2}{x_iY'+y_iX'},
\quad
Z_3=\sum\limits_{i=2}^k\frac{y_i^2}{x_iY'+y_iX'}.
\end{array}
\right.
}
Note that for any non-empty subset $I$ of $\brk{k}\setminus \{1\}$,
\equ{term12}
{
y_IX+x_IY-x_Iy_I=\left(y_1+\sum_{i\in I}y_i\right)X
+
\left (x_1+\sum_{i\in I}x_i\right )
\left (\sum_{i\in \brk{k}\setminus (I\cup \{1\})}y_i\right ).
}
In the remainder of this section, let $W'_i$ denote $x_iY'+y_iX'$ for each
$i\in \brk{k}$.
By applying identities \label{r2-8}
(\ref{le2-1-1ex}), (\ref{le2-2-1ex})
and (\ref{term12}),
\begin{eqnarray}\label{eq3-61}
& &\Gamma_1-\Gamma_2
\nonumber \\
&=&\sum_{\emptyset\ne I\subseteq \brk{k}\setminus \{1\}}
\left ((-1)^{|I|-1}
( W_I
-x_Iy_I)
\left (\prod_{j\in I}
w_j
\right )
\left (\prod_{i\in \brk{k}\setminus (I\cup \{1\})} W_i
\right )
\right ) \nonumber\\
&\overset{by\ (\ref{term12})}{=}&
X\sum_{\emptyset\ne I\subseteq \brk{k}\setminus \{1\}}
\left ( (-1)^{|I|-1}
\left (y_1+\sum_{i\in I}y_i\right )
\left (\prod_{j\in I}
w_j
\right )
\left (
\prod_{i\in \brk{k}\setminus (I\cup \{1\})}
W_i
\right )
\right )
\nonumber \\
& &+\sum_{\emptyset\ne I\subseteq \brk{k}\setminus \{1\}}
\left ( (-1)^{|I|-1}
\left (x_1+\sum_{i\in I}x_i\right )
\left (\sum_{i\in \brk{k}\setminus (I\cup \{1\})}y_i\right )
\left (\prod_{j\in I}
w_j
\right )
\left (
\prod_{i\in \brk{k}\setminus (I\cup \{1\})}
W_i
\right )
\right )
\nonumber \\
&=&-X\sum_{\emptyset\ne I\subseteq \brk{k}\setminus \{1\}}
\left (
\left (y_1+\sum_{i\in I}y_i\right )
\left (\prod_{j\in I}
(-w_j)
\right )
\left (
\prod_{i\in \brk{k}\setminus (I\cup \{1\})}
W_i
\right )
\right )
\nonumber \\
& &-\sum_{\emptyset\ne I\subseteq \brk{k}\setminus \{1\}}
\left (
\left (x_1+\sum_{i\in I}x_i\right )
\left (\sum_{i\in \brk{k}\setminus (I\cup \{1\})}y_i\right )
\left (\prod_{j\in I}
(-w_j)
\right )
\left (
\prod_{i\in \brk{k}\setminus (I\cup \{1\})}
W_i
\right )
\right )
\nonumber \\
&\overset{\small by\ (\ref{le2-1-1ex}), (\ref{le2-2-1ex})}
{=}&
Xy_1\prod_{i=2}^kW_i
-X\prod_{j=2}^k W'_j
\left (
y_1-\sum_{i=2}^k
\frac{y_i w_i
}
{W'_i}
\right )
+x_1\left(\prod_{i=2}^kW_i\right)
\left ( \sum_{j=2}^k y_j\right )
\nonumber \\
& &+\left ( \prod_{r=2}^k W'_r\right)
\left (
\sum_{2\le i,j\le k\atop i\ne j}
\frac{x_iy_j
w_i W_j
}{W'_iW'_j}
-x_1\sum_{i=2}^k
\frac{y_i W_i}{W'_i}
\right )
\nonumber \\
&=&\prod_{i=1}^kW_i
-x_1y_1\prod_{i=2}^k W_i
-XZ
\left (y_1-x_1Z_3-y_1Z_1\right )
+Z\left (
-x_1(YZ_1+XZ_3)+
\sum_{2\le i,j\le k\atop i\ne j}
\frac{x_iy_jw_i W_j
}
{W'_i W'_j}
\right )
\nonumber \\
&=&\prod_{i=1}^k W_i
-x_1y_1\prod_{i=2}^k
W_i
-Z\left (y_1X-y_1XZ_1+x_1YZ_1\right )
+Z\sum_{2\le i,j\le k\atop i\ne j}
\frac{x_iy_j
w_i W_j
}
{W'_i W'_j},
\end{eqnarray}
where in the second last equality, we combine $Xy_1\prod\limits_{i=2}^k W_i$
and $x_1\prod\limits_{i=2}^k W_i \left (\sum\limits_{j=2}^k y_j\right )$ to obtain
$\prod\limits_{i=1}^kW_i-x_1y_1\prod\limits_{i=2}^k W_i$.
\label{r2-7}
By applying identity (\ref{le2-2-1ex}),
we have
\begin{eqnarray}\label{eq3-63}
\Gamma_3&=&-\sum\limits_{\emptyset\ne I\subseteq \brk{k}\setminus \{1\}}
\left (
\left( W_1
+\sum\limits_{i\in I} W_i
\right)
\left(\sum\limits_{i\in \brk{k}\setminus (I\cup \{1\})}
\frac{x_iy_i}{W_i}
\right)
\left (\prod\limits_{j\in I}
(-w_j)
\right )
\left(\prod\limits_{i\in \brk{k}\setminus (I\cup \{1\})} W_i
\right)
\right )
\nonumber \\
&\overset{by\ (\ref{le2-2-1ex})}{=}&
W_1
\left(\prod\limits_{i=2}^k
W_i
\right)
\left (\sum_{i=2}^k\frac{x_iy_i}
{W_i}\right )
+\left(\prod_{i=2}^k
W'_i
\right)
\left ( -W_1
\sum_{i=2}^k\frac{x_iy_i}
{W'_i}
+\left. \sum_{2\le i,j\le k\atop i\ne j}
\frac{
w_i W_i x_jy_j
}
{W'_i W'_j
}
\right )
\right.
\nonumber \\
&=&\left(\prod\limits_{i=1}^k
W_i
\right)
\left (\sum_{i=2}^k
\frac{x_iy_i}{W_i}\right )
-W_1
ZZ_1
+ Z\sum_{2\le i,j\le k\atop i\ne j}
\frac{w_iW_ix_jy_j}
{W'_i W'_j
}.
\end{eqnarray}
Note that for any set $S$ of positive integers,
and real numbers $a_i,b_i$ for $i\in S$, we have
$$
\sum_{i,j\in S\atop i\ne j}(a_ib_j)
=\left (\sum_{i\in S}a_i\right )\cdot
\left( \sum_{i\in S}b_i \right )
-\sum_{i\in S}(a_ib_i).
$$
Thus,
\begin{eqnarray}\label{eq3-7-1}
& &\sum_{2\le i,j\le k\atop i\ne j}
\frac{x_iy_j w_iW_j}
{W'_i W'_j}
-\sum_{2\le i,j\le k\atop i\ne j}
\frac{w_iW_ix_jy_j}
{W'_iW'_j}
\nonumber \\
&=&\left( \sum_{i=2}^k
\frac{x_iw_i}
{W'_i}\right )
\cdot
\left(\sum_{i=2}^k \frac{y_iW_i}
{W'_i}\right )
-\sum_{i=2}^k \frac{x_iy_iw_iW_i}
{(W'_i)^2}
-\left(
\left (\sum_{i=2}^k
\frac{x_iy_i}{W'_i} \right )
\cdot
\left (\sum_{i=2}^k
\frac{w_iW_i}{W'_i} \right )
-\sum_{i=2}^k
\frac{x_iy_iw_iW_i}
{(W'_i)^2
\right)
\nonumber \\
&=&(x_1Z_1+y_1Z_2)(Y'+y_1Z_1+x_1Z_3)
-Z_1(x_1Y'+y_1X'+x_1^2Z_3+y_1^2Z_2+2x_1y_1Z_1)
\nonumber \\
&=&y_1(Y'Z_2-X'Z_1)
+x_1y_1(Z_2Z_3-Z_1^2)
\nonumber \\
&=&y_1(Y'Z_2-X'Z_1)
+x_1y_1(1-2Z_1)
\nonumber \\
&=&y_1(Y'Z_2+x_1-X'Z_1-2x_1Z_1),
\end{eqnarray}
where the second last equality follows from
the fact that $(Z_1-1)^2=Z_2Z_3$ by Lemma~\ref{le2-01}.
Thus, by (\ref{eq3-61}), (\ref{eq3-63})
and (\ref{eq3-7-1}),
\begin{eqnarray}\label{eq3-12}
& &\Gamma_1-\Gamma_2-\Gamma_3
-\left(\prod_{i=1}^k(x_iY+y_iX)\right)
\left (1-\sum_{i=1}^k\frac{x_iy_i}{x_iY+y_iX}\right )
\nonumber \\
&=&-Z\left (y_1X-y_1XZ_1+x_1YZ_1\right )
+ZZ_1(x_1Y+y_1X)+Zy_1(Y'Z_2+x_1-X'Z_1-2x_1Z_1)
\nonumber \\
&=&y_1Z(X'Z_1+Y'Z_2-X')
\nonumber \\
&=&y_1Z
\left (-X'+\sum_{i=2}^k
\frac{x_i(y_iX'+x_iY')}{x_iY'+y_iX'}\right )
\nonumber \\
&=&0.
\end{eqnarray}
Thus, (\ref{e3-4}) follows from (\ref{eq3-50}), (\ref{eq3-12})
and the definition of $\phi(x_1,y_1,x_2,y_2,\ldots,x_k,y_k)$
in (\ref{e3-3}).
\end{proof}
\section{Counting spanning trees in a
special type of multigraphs}
\label{sec4}
Let $V=\{v_i: i\in \brk{k}\}$ and
$E=\{v_iv_j: i,j\in \brk{k}~\text{and}~i\neq j\}$ be the vertex set and edge set of
the complete graph $K_k$, where $k\ge 1$.
Let $\omega$ be a weight function on $E$.
If $\omega(v_iv_j)$ is a nonnegative integer
for all $i,j$ with $1\le i<j\le k$,
then $\tau(K_k, \omega)$ is the number of spanning trees
of the multigraph with vertex set
$\{u_1,u_2,\ldots,u_k\}$ which contains
exactly $\omega(v_iv_j)$ parallel edges
joining $u_i$ and $u_j$
for all $i,j$ with $1\le i<j\le k$.
For any non-empty subset
$I$ of $\brk{k}\setminus \{1\}$,
let $G_I$ denote the complete graph of order $k-|I|$
with vertex set $\{v_I\}\cup \{v_i: i\in \brk{k}\setminus (I\cup \{1\})\}$
and weight function $\omega_I$ on the edge set
of $G_I$ defined as follows:
\equ{weight-f}
{
\left \{
\begin{array}{ll}
\omega_I(v_iv_j)=\omega(v_iv_j),
&\forall i,j\in \brk{k}\setminus (I\cup \{1\}),i\ne j;\\
\omega_I(v_Iv_j)=\sum\limits_{r\in I\cup \{1\}}\omega(v_rv_j),
\qquad
&\forall j\in \brk{k}\setminus (I\cup \{1\}).
\end{array}
\right.
}
Note that $G_I$ is actually the graph obtained from $K_k$ by identifying
all vertices in $\{v_i: i\in
I\cup \{1\}\}$ as one vertex.
By the inclusion-exclusion principle,
the following recursive relation on $\tau(K_k,\omega)$
can be obtained.\label{r2-9}
\begin{lemma}\label{le4-1}
For any weight function $\omega$ on the edge set $E$ of $K_k$, \label{r2-10}
\begin{equation}\label{le4-1-e1}
\tau(K_k,\omega)
=\sum_{\emptyset \ne I\subseteq \brk{k}\setminus \{1\}}
\left ( (-1)^{|I|-1}
\tau(G_I, \omega_I)
\prod_{i\in I}\omega(v_1v_i)\right ).
\end{equation}
\end{lemma}
\def \C {{\cal C}}
\def \F {{\cal F}}
\begin{proof} For any $i\in \brk{k}\setminus \{1\}$, let $A_i$
denote the set of members $T$ in $\mathcal{T}(K_k)$ with
$v_1v_i\in E(T)$.
Clearly,
\equ{le4-1-e2}
{
\mathcal{T}(K_k)=\bigcup_{i=2}^k A_i.
}
For any $T\in \mathcal{T}(K_k)$, by the inclusion-exclusion principle,
\equ{le4-1-e3}
{
\sum_{\emptyset\ne I\subseteq \brk{k}\setminus \{1\}}
\left ((-1)^{|I|-1}|\{T\}\cap \bigcap_{i\in I} A_i|
\right )=1.
}
Thus,
\equ{le4-1-e4}
{
\tau(K_k,\omega)=\sum_{T\in \bigcup_{2\le i\le k}A_i}
\omega(T)
=\sum_{\emptyset \ne I\subseteq \brk{k}\setminus \{1\}}
\left ( (-1)^{|I|-1}
\sum_{T\in \bigcap_{i\in I} A_i}
\omega(T)\right ).
}
Let $I$ be any non-empty subset of $\brk{k}\setminus \{1\}$ and
let $H_I$ denote the multiple graph
obtained from $K_k$ by
identifying all vertices
in the set $\{v_i: i\in I\cup \{1\}\}$
and removing all loops produced.
The vertex set of $H_I$ is
$(V(K_k)\setminus \{v_i:i\in I\cup \{1\}\})\cup v_I$.
Clearly, $H_I$ includes each edge
$v_iv_j$, where
$i,j\in \brk{k}\setminus (I\cup \{1\})$,
while each edge $v_iv_j$ in $K_k$,
where $i\in I\cup \{1\}$ and
$j\in \brk{k}\setminus (I\cup \{1\})$,
is changed to an edge of $H_I$
joining $v_I$ and $v_j$.
There are exactly $1+|I|$
parallel edges
in $H_I$ joining $v_I$ and $v_j$
for each $j\in \brk{k}\setminus (I\cup \{1\})$.
The weight function on $E(H_I)$
is the restriction of $\omega$ to $E(H_I)$ and parallel edges in
$H_I$ may have different weights.
For each $T\in \bigcap_{i\in I} A_i$,
let $T_I$ be the tree obtained from $T$ by identifying all vertices
in the set $\{v_i: i\in I\cup \{1\}\}$.
Clearly $\omega(T)$ and $\omega(T_I)$
have the following relation:
\equ{le4-1-e7}
{
\omega(T)=\prod_{i\in I}\omega(v_1v_i)
\cdot
\omega(T_I).
}
Moreover,
$T\rightarrow T_I$ is a bijection
from
$\bigcap_{i\in I} A_i$ to
$\mathcal{T}(H_I)$, implying that
\equ{le4-1-e5}
{
\sum_{T\in \bigcap_{i\in I} A_i}\omega(T)
=\prod_{i\in I}\omega(v_1v_i)
\cdot \sum_{T\in \bigcap_{i\in I} A_i}\omega(T_I)
=\tau(H_I, \omega)\cdot
\prod_{i\in I}\omega(v_1v_i).
}
Note that $G_T$ can be obtained from $H_I$ by merging all parallel edges
with ends $v_I$ and $v_j$ into one
for each
$j\in \brk{k}\setminus (I\cup \{1\})$.
By the definition of $\omega_I$,
$\tau(H_I, \omega)
=\tau(G_I, \omega_I)$.
Thus, (\ref{le4-1-e1}) follows from (\ref{le4-1-e4}) and (\ref{le4-1-e5}).
\label{r1-2}
\end{proof}
Recall the function $\phi$ defined in the previous
section. In the following, we shall show that
$\tau(K_k, \omega)$ can be expressed in terms of
$\phi$ when $\omega$ satisfies certain conditions.\label{r2-13}
\begin{theorem}\label{th3}
Let $V=\{v_1,v_2,\ldots,v_k\}$ be the
the vertex set of the complete graph $K_k$, where $k\ge 1$,
and $\omega$ be a weight function on the edge set $E$ of $K_k$.
If there exist $2k$ real numbers $x_1,x_2,\ldots,x_k$
and $y_1,y_2,\ldots,y_k$ such that
$\omega(v_iv_j)=x_iy_j+x_jy_i$ holds
for every pair $i$ and $j$ with $1\le i<j\le k$,
then,
\begin{equation}\label{th3-e1}
\tau(K_k, \omega)
=\phi(x_1,y_1,x_2,y_2,\ldots,x_k,y_k).
\end{equation}
\end{theorem}
\begin{proof} Note that for
$1\le k\le 3$,
\equ{eq4-0}
{
\tau(K_k, \omega)=
\left \{
\begin{array}{ll}
1, &k=1;\\
\omega(v_1v_2), &k=2;\\
\omega(v_1v_2)\omega(v_1v_3)
+\omega(v_1v_2)\omega(v_2v_3)
+\omega(v_1v_3)\omega(v_2v_3), &k=3.
\end{array}
\right.
}
As $\omega(v_iv_j)=x_iy_j+x_jy_i$,
(\ref{th3-e1}) follows from (\ref{phi}) and
(\ref{eq4-0}) when $k\le 3$.
Assume that the result holds for $k\le N$, where $N\ge 3$.
In the following, we assume that $k=N+1$
and show that it holds in this case by induction.\label{r2-15}
Recall that for any non-empty subset of $I$ of $\brk{k}\setminus \{1\}$,
$G_I$ is the complete graph
with vertex set $\{v_I\}\cup \{v_j: j\in \brk{k}\setminus (I\cup \{1\})\}$ and
weight function $\omega_I$ on its edge set
defined in (\ref{weight-f}), i.e.,
\equ{weight-f2}
{
\left \{
\begin{array}{ll}
\omega_I(v_iv_j)=\omega(v_iv_j)=x_iy_j+x_jy_i,
&\forall i,j\in \brk{k}\setminus (I\cup \{1\}),i\ne j;\\
\omega_I(v_Iv_j)
=\sum\limits_{r\in I\cup \{1\}}(x_ry_j+x_jy_r)
=x_Iy_j+y_Ix_j,
\quad
&\forall j\in \brk{k}\setminus (I\cup \{1\}).
\end{array}
\right.
}
where $x_I=x_1+\sum\limits_{r\in I}x_r$
and $y_I=y_1+\sum\limits_{r\in I}y_r$.
As $G_I$ is a complete graph of order $k-|I|<k$
with a weight function $\omega_I$ satisfying conditions
in (\ref{weight-f2}),
by inductive assumption,
(\ref{th3-e1}) holds for $G_I$, i.e.,
\begin{equation}\label{th3-e5}
\tau(G_I, \omega_I)
=\phi(x_I,y_I, \underbrace{x_i,y_i}
_{\forall i\in \brk{k}\setminus (I\cup \{1\})}).
\end{equation}
By Lemma~\ref{le4-1}, (\ref{th3-e5}) and Theorem~\ref{th3-1},
\eqn{th3-e6}
{
\tau(K_k, \omega)
&=&\sum_{\emptyset\ne I\subseteq \brk{k}\setminus \{1\}} (-1)^{|I|-1}
\tau(G_I, \omega_I)
\prod_{i\in I}\omega(v_1v_i)
\nonumber \\
&=&\sum_{\emptyset \ne I\subseteq \brk{k}\setminus \{1\}}
\left ( (-1)^{|I|-1}
\tau(G_I, \omega_I)
\prod_{i\in I}(x_iy_1+y_ix_1)
\right )
\nonumber \\
&=&\sum_{\emptyset \ne I\subseteq \brk{k}\setminus \{1\}}
\left ( (-1)^{|I|-1}
\prod_{i\in I}(x_iy_1+y_ix_1)
\cdot
\phi(x_I,y_I, \underbrace{x_i,y_i}
_{\forall i\in \brk{k}\setminus (I\cup \{1\})})
\right )
\nonumber \\
&=&\phi(x_1,y_1,x_2,y_2,\cdots,x_k,y_k).
}
Hence the result holds.
\end{proof}
\section{Spanning trees in $K_{m, n}$ containing a spanning forest
$F$}
\label{sec5}
Now we are ready to prove the main result.
\noindent
{\bf Proof of Theorem \ref{mainthm}.}
For any spanning forest $F$ of $K_{m,n}$
with components $T_1,T_2,\ldots,T_k$,
observe that
\equ{eq5-1}
{
\tau_{F}(K_{m,n})=\tau(K_{m,n}/F),
}
where $K_{m,n}$ is unweighted and $K_{m,n}/F$ is \label{r2-16}
the multigraph obtained from $K_{m,n}$
by contracting all edges in $F$.
Note that $K_{m,n}/F$ is a multigraph of order $k$
whose vertices correspond to components of $F$,
as $K_{m,n}/F$ can also be obtained from $K_{m,n}$
by identifying all vertices in $T_i$
for all $i\in \brk{k}$, and removing all loops.
Thus, we may assume that $K_{m,n}/F$ has vertices
$v_1,v_2,\ldots,v_k$ such that the number of parallel
edges joining $v_i$ and $v_j$ is equal to
the number of edges in $K_{m,n}$
with one end in $T_i$ and the other end in $T_j$.
As $|X\cap V(T_s)|=m_s$ and $|Y\cap V(T_s)|=n_s$
for all $s=1,2,\ldots,k$,
$K_{m,n}/F$ contains exactly $m_in_j+m_jn_i$ parallel
edges joining $v_i$ and $v_j$
for all $1\le i<j\le k$.
By Theorem~\ref{th3},
\equ{eq5-4}
{
\tau(K_{m,n}/F)=
\phi(m_1,n_1,m_2,n_2,\ldots,m_k,n_k).
}
Thus, by (\ref{eq5-1}) and the definition of $\phi(x_1,y_1,x_2,y_2,\ldots,x_k,y_k)$
in (\ref{e3-3}), the result holds.
{\hfill$\Box$}
\section{Remarks}
\label{sec6}
Another approach for proving the main result is to
establish results analogue to Lemma~\ref{le4-1} and
Theorem~\ref{th3-1}.
The following identity analogue to Lemma~\ref{le4-1}
can be obtained easily:
\begin{equation}\label{e6-1}
\tau(K_k, \omega)
=\sum_{\emptyset \ne I\subseteq \brk{k}\setminus \{1\}}
\left (
\tau(G_I, \omega'_I)
\prod_{i\in I}\omega(v_1v_i)\right ),
\end{equation}
where $\omega'_I$ is different from
$\omega_I$ defined in (\ref{weight-f}), as
for any $j\in \brk{k}\setminus (I\cup \{1\})$,
\equ{e6-2}
{
\omega'_I(v_Iv_j)=\sum_{r\in I}\omega(v_rv_j)
=\omega_I(v_Iv_j)-\omega(v_1v_j),
}
although $\omega'_I(v_iv_j)=\omega_I(v_iv_j)$
for all $i,j\in \brk{k}\setminus (I\cup \{1\})$ with $i\ne j$.
By Theorem~\ref{mainthm} and (\ref{e6-1}),
the following identity analogue to
Theorem~\ref{th3-1} holds:
\begin{equation} \label{e6-3}
\phi(x_1,y_1,x_2,y_2,\cdots,x_k,y_k)
=
\sum_{\emptyset\ne I\subseteq \brk{k}\setminus \{1\}}
\left (
\prod_{j\in I}(x_1y_j+x_jy_1)
\phi(x'_I, y'_I, \underbrace{x_s,y_s}
_{\forall s\in \brk{k}\setminus (I\cup \{1\})})
\right ),
\end{equation}
where $x'_I=\sum\limits_{i\in I}x_i=x_I-x_1$
and $y'_I=\sum\limits_{i\in I}y_i=y_I-y_1$.
However, it is quite challenging to prove (\ref{e6-3})
directly.
Note that for any $I$ with $\emptyset \ne I\subseteq \brk{k}\setminus \{1\}$,
\equ{e6-4}
{
\left \{
\begin{array}{l}
x'_I+\sum_{i\in I}x_i=x_2+x_3+\cdots+x_k=X-x_1,\\
y'_I+\sum_{i\in I}y_i=y_2+y_3+\cdots+y_k=Y-x_1.
\end{array}
\right.
}
By the definition of the function $\phi$,
\aln{e6-5}
{
\phi(x'_I, y'_I, \underbrace{x_s,y_s}
_{\forall s\in \brk{k}\setminus (I\cup \{1\})} )
&=\frac{x'_IY'+y'_IX'}{X'Y'}\cdot
\left(\prod_{i\in \brk{k}\setminus (I\cup \{1\})}(x_iY'+y_iX')\right)
\nonumber \\
& \cdot
\left (1 -\frac{x'_Iy'_I}{x'_IY'+y_I'X'}
-\sum_{i\in \brk{k}\setminus (I\cup \{1\})}\frac{x_iy_i}
{x_iY'+y_iX'}\right ),
}
where $X'=X-x_1$ and $Y'=Y-y_1$.
Observe that
the left-hand of (\ref{e6-3}) has a denominator $XY$,
while its right-hand side has a denominator $X'Y'$.
Clearly, the main result (i.e., Theorem~\ref{mainthm})
also follows from (\ref{e6-1}) and
(\ref{e6-3}).
In the end, we propose some problems.
\begin{problem}\label{prob1}
Find a bijective proof for Theorem~\ref{mainthm}.
\end{problem}
Another problem is to extend Theorem~\ref{mainthm} to complete
$k$-partite graphs, where $k\ge 3$.
\begin{problem}\label{prob2}
Let $K_{n_1,n_2,\cdots,n_k}$ be a complete $k$-partite graph
and $F$ be a spanning forest in $K_{n_1,n_2,\ldots,n_k}$,
where $k\ge 3$.
Find a formula for counting the number of spanning trees in
$K_{n_1,n_2,\ldots,n_k}$ which contain all edges in $F$.
\end{problem}
For $k=3$, we propose the following conjecture for
a lower bound of $\tau_F(K_{n_1,n_2,n_3})$.
\begin{conjecture}\label{con1}
Let $X_1,X_2$ and $X_3$ be the partite sets of
the complete tripartite graph
$K_{n_1,n_2,n_3}$, where $|X_i|=n_i$ for $i\in \brk{3}$.
For any spanning forest $F$ in $K_{n_1,n_2,n_3}$
with $k$ components $T_1,T_2,\cdots,T_k$,
\eqn{con-e1}
{
\tau_F(K_{n_1,n_2,n_3})
&\ge &
\frac 1{n_1n_2+n_1n_3+n_2n_3}
\left (
\prod_{i=1}^{k}\left(
(n-n_1)n_{1,i}+(n-n_2)n_{2,i}+(n-n_3)n_{3,i} \right )
\right )
\nonumber \\
& &\cdot \left (
1-\sum_{i=1}^{k}
\frac {n_{1,i}n_{2,i}+n_{1,i}n_{3,i}+n_{2,i}n_{3,i}}
{(n-n_1)n_{1,i}+(n-n_2)n_{2,i}+(n-n_3)n_{3,i}}
\right ),
}
where $n=n_1+n_2+n_3$ and $n_{s,i}=|X_s\cap V(T_i)|$ for
$s=1,2,3$ and $i\in \brk{k}$.
\end{conjecture}
It is trivial to verify that
the equality of (\ref{con-e1}) holds for $k\le 2$.
\section*{Acknowledgements}
Jun Ge is supported by NSFC (No. 11701401) and
the joint research project of Laurent Mathematics Center of Sichuan Normal University
and National-Local Joint Engineering Laboratory of System Credibility Automatic Verification.
The authors are grateful to two anonymous referees for their careful examination and constructive comments.
|
train/arxiv
|
BkiUbYU4eIZijWIz2VdC
| 5 | 1 |
\section{Introduction}
\noindent The dominant way of selling ad-impressions on ad-exchanges (AdXs) is now through real-time bidding (RTB) systems. These systems leverage auctions at scale to allocate user impressions arriving at various digital publishers to competing bidding advertisers or intermediaries such as demand side platforms (DSPs). Most RTB auctions on AdXs are single-unit auctions implemented typically via second-price auctions (SPAs) and, more recently, via first-price auctions (FPAs).\footnote{See \cite{Muthu09} and \cite{cmbl17} for an overview of the economics of AdXs and \cite{Sayedi2019} for the reasons behind the recent shift to FPAs.} The speed, scale and complexity of RTB transactions is astounding. There are billions of auctions for impressions on an AdX on any given day, each consummated on average in less than 100 milliseconds, and each transaction corresponds to different impression characteristics and possibly requires a unique bid and clearing price. The complexity and scale of available ad-inventory, the speed of the transaction that allows little time for deliberation and the complex nature of competition imply that advertisers participating in RTB auctions have significant uncertainty about the value of the impressions they are bidding for as well as the nature of competition they are bidding against. Developing accurate and reliable ways of measuring the value of advertising in this environment is therefore essential for the advertiser to profitably trade on the exchange and to ensure that acquired ad-impressions generate sufficient value. Measurement needs to deliver incremental effects of ads for different types of ad and impression characteristics and needs to be automated. Experimentation thus becomes attractive as a device to obtain credible estimates of such causal effects.
Motivated by this, the current paper presents an experimental design to perform causal inference on RTB advertising in both SPA and FPA settings. Our design enables learning heterogeneity in the inferred average causal effects across ad and impression characteristics. The novelty of the proposed experimental design is in addressing two principal challenges that confront developing a scalable experimental design for RTB advertising.
The first challenge is that measuring the average treatment effect ($ATE$) of advertising requires comparing outcomes of users who are exposed to ads with outcomes of users who are not. The complication of the RTB setting is that ad-exposure is not under complete control of the experimenter because it is determined by an auction. This precludes simple designs in which ad-exposure is randomized directly to users. Instead, we need a design in which the experimenter controls only an input to the auction (bids), but wishes to perform inference on the effect of a stochastic outcome induced by this input (ad-exposure).
The second challenge pertains to the cost of experimentation on the AdX. Obtaining experimental impressions is costly: one has to win the auction to see the outcome with ad-exposure, and one has to lose the auction to see the outcome without. When bidding is not optimized, the economic costs of both can be substantial. These economic costs arise when one bids too much to obtain an impression with ad-exposure (overbidding) or decides deliberately not to induce ad-exposure on what would have been a very valuable impression for the advertiser (underbidding/opportunity cost). With millions of auctions potentially in the experiment, suboptimal bidding can make the experiment unviable. Therefore, to be successful an effective experimental design has to deliver inference on the causal effect of ads while also judiciously managing the cost of experimentation by implementing a bidding policy that is optimized to the value of the impressions encountered.
It is not obvious how an experiment can be designed prior to actual implementation to address both challenges simultaneously. Optimal bidding requires knowing the value of each impression, which was the goal of experimentation in the first place. \textit{Online} methods, which introduce randomization to induce exploration of the value of advertising and combine it with concurrent exploitation of the information learned to optimize bidding, thus become highly attractive in such a setting.
At the core of online methods is the need to balance the goal of learning the expected effect of ad-exposure (henceforth called the advertiser's ``inference goal'') with the goal of learning the optimal bidding policy (henceforth called the advertiser's ``economic goal''). The tension is that finding the best bidding policy does not guarantee proper estimation of ad-effectiveness and vice versa. At one extreme, with a bidding policy that delivers on the economic goal, the advertiser could win most of the time, making it difficult to measure ad-effectiveness since outcomes with no ad-exposures would be scarcely observed. At the other extreme, with pure bid randomization the advertiser could estimate unconditional ad-effectiveness (the $ATE$) or how ad-effectiveness varies with observed heterogeneity (the $CATE$s) and deliver on the inference goal, but may end up incurring large economic losses in the process.
The contribution of this paper is to present a multi-armed bandit design (MAB) and statistical learning framework that address both these considerations. In our design, observed heterogeneity is summarized by a context, $x$, bids form arms, and the advertiser's monetary payoffs form the rewards, so that the best arm, or optimal bid, maximizes the advertiser's expected payoff from auction participation given $x$. Exploiting the economic structure of SPAs and FPAs, we derive, under conditions we outline, the relationship between the optimal bid at a given $x$ and the $CATE$ of the ad at the value $x$, or $CATE(x)$. For SPAs, we show that these two objects are equal, so that, in our experimental design, the twin tasks of learning the optimal bid and estimating ad-effectiveness not only can build off each other, but are perfectly aligned. For FPAs, we demonstrate that the two goals are closely related, though only imperfectly aligned. Leveraging this relationship improves the efficiency of learning and generates an online experimental design that delivers on both goals for the advertiser at minimal cost under both auction formats.
To implement the design, we present a Thompson Sampling (TS) algorithm customized to our auction environment trained online via a Markov Chain Monte Carlo (MCMC) method. The algorithm adaptively chooses bids across rounds based on current estimates of which arm is the optimal one. These estimates are updated on each round via MCMC through Gibbs sampling, which leverages data augmentation to impute the missing potential outcomes and the censored or missing competing bids in each round. Simulations show that the algorithm is able to recover treatment effect heterogeneity as represented by the $CATE$s of advertising and considerably reduces the expected costs of experimentation compared to non-adaptive ``A/B/n'' tests, explore-then-commit (ETC) strategies and a canonical off-the-shelf TS algorithm.
The rest of the paper discusses the relationship between the approach presented here with the existing literature and explains our contribution relative to existing work. The following section defines the design problem. Sections \ref{sec:balancing-obj} and \ref{sec:achieve} show how we leverage auction theory to balance the experimenting advertiser's objectives. Section \ref{sec:BITS} presents the modified TS algorithm we use to implement the experimental design. Section \ref{sec:exper} shows extensive simulations documenting the performance of the proposed algorithm and shows its advantages over competing designs. The last section concludes.
\subsection*{Related literature}
\noindent Our paper lies at the intersection of three broad fields of study: the literature on online learning in ad-auctions, the literature on experimentation in digital advertising, and the literature on causal inference with bandits. Given that each of these streams of literature is mature, we discuss only the most related papers for brevity, and the reader is referred to some of the cited papers for further reading.
\subsubsection*{Literature on online learning in ad-auctions}
\begin{comment}
The literature on how a bidder can adaptively learn to bid leveraging past experiences in an auction has a long history in the literature dating from \cite{Friedman56} (\cite{LAFFONT1997} section 2 reviews this literature).
\end{comment}
\noindent While the primary focus of this paper is on causal inference and experimentation, to the extent that we solve an online learning-to-bid problem during experimentation, this paper is also closely related to the literature on online learning in ad-auctions, in which learning strategies are suggested for use by auction participants to make good auction-related decisions. One feature of this literature is that the majority of studies adopt the seller's perspective, focusing on the problem of designing mechanisms that maximize the seller's expected profit (e.g., finding an optimal reserve price in SPAs when the distribution of valuations of the bidders is unknown \textit{a priori} to the auctioneer). Examples include \cite{cesa2014regret}, \cite{MohriMedina14}, \cite{RoughgardenWang16}, \cite{ZhaoChen20} and \cite{KanoriaNazerzadeh14}.
Our study, which addresses the problem of ad-experimentation from the advertiser's perspective, is more closely related to a smaller subliterature within this stream at the intersection of online learning and auction theory that has studied the problem of bidding from the perspective of the bidder. The key issue is that RTB bidders have significant uncertainty about the value of advertising they buy on AdXs. In turn, MAB polices are appealing devices for AdXs to learn which advertisers to pick as suggested in \cite{McAfee2011}, although neither a specific algorithm for advertisers to learn-to-bid nor a strategy for them to conduct online experimentation on AdXs is outlined.
\cite{bgmms2019} present an algorithm for contextual bandits with cross-learning across contexts and discuss an application to advertisers learning-to-bid in FPAs. Their insight is that if we consider the value of the ad to the advertiser, $v$, as the context, observing the reward from bidding at context $v$ yields information about the reward at another context $v'$ for the same bid, so there is cross-learning across contexts within each arm. They show that leveraging this information improves the efficiency of learning-to-bid algorithms.\footnote{This feature is shared by the bandit problem presented in this paper as well, with an added advantage that the existence of a shared payoff structure across arms implies that the problem displays \textit{cross-arm} learning in addition to the within-arm, cross-context learning pointed out by \cite{bgmms2019}.} However, the informational assumption in \cite{bgmms2019} $-$ that the bidder's value for the item (i.e., the context) is known to her prior to bidding $-$ implies that there is no scope for causal inference on this value to the bidder, unlike the situation we consider here. \cite{hanZhouWeismann2020} extend this analysis and present stochastic bandit algorithms for learning-to-bid in FPAs. \cite{hanZhouWeismannAdverserial2020} present an analogous algorithm for the adversarial case. They discuss ways to overcome a key impediment to resolving the uncertainty a learning advertiser has over competitors' bids, which is that the highest competing bid is not observed when she wins the auction, leading to censored feedback and a form of a winner's curse in learning. \cite{hanZhouWeismann2020} and \cite{hanZhouWeismannAdverserial2020} maintain the same assumption as \cite{bgmms2019}: bidders have uncertainty about the distribution of competing bids, but know their own valuation exactly prior to bidding. Therefore, again, there is no role for causal inference on the intrinsic value of advertising from the advertiser's perspective.\footnote{\cite{hanZhouWeismann2020}'s informational assumptions regarding competing bids are also different from ours: we assume the advertiser does not observe the highest competing bid in an FPA when winning or losing (full censoring); they assume the advertiser does not observe the highest competing bid if she wins, but observes it if she loses (partial censoring).}
\cite{weed16} and \cite{FengPodimataSyrgkanis2018} relax the assumption that advertisers know their private value and present learning-to-bid algorithms for SPAs. Like the above papers, their emphasis is on addressing the challenge associated with the censored ``win-only'' or ``outcome-based'' feedback that arises in SPAs. While these papers relax the assumption of exact knowledge of the ad's value by the advertiser, the way the informational assumptions are relaxed is different from ours, and implies a much more limited scope for causal inference on the value of the ad to the advertiser. \cite{weed16}'s assumption is that the bidder is uncertain about her valuation prior to bidding, but the value of the good is fully observed if the auction is won. In turn, \cite{FengPodimataSyrgkanis2018}'s assumption is that the bidder is uncertain about her valuation prior to bidding, but the value of the ad is fully observed if the auction is won and the ad is clicked on. This implies an implicit assumption that the impression generates no value to the advertiser when the auction is lost and no ad-exposure or ad-click occurs. In canonical auction-theoretic settings, it is assumed that the bidder gets no utility if she loses the auction. However, in the context of ads a user may have a non-zero propensity to buy the advertiser's products even in the absence of ad-exposure or ad-click, and winning the auction increments this propensity, so this assumption may not be as well suited.
Relaxing this assumption changes the implied inference problem substantively. In causal inference terms, if we call $Y(1)$ and $Y(0)$ respectively the potential outcomes to the advertiser with and without ad-exposure to the user, the value of the ad is $Y(1)-Y(0)$. The assumption above implies that $Y(0)\equiv0$. This assumes away a principal challenge in causally inferring the effect of ads addressed in this paper, that $Y(1)$ and $Y(0)$ are not observed together for the same impression because the auction outcome censors the potential outcomes (this is \cite{holland1986}'s ``fundamental problem'' of causal inference). Performing inference on the effect of ads while addressing this censoring problem is a major focus of the experimental design in this paper. Another difference is that when winning helps learn the valuation fully, the bandit's exploration motive prioritizes higher bids, because that increases the chance of winning. In our setting, learning the value of advertising necessarily requires losing the auction sometimes, because that allows observing $Y(0)$. Hence, this force for higher bidding in exploration is less pronounced, which in turns affects the nature of bidding and the induced cost of experimentation.
This paper is related to a series of recent studies that analyze market equilibrium when learning bidders interact in repeated RTB auctions. \cite{DikkalaTardos13} characterize an equilibrium credit provision strategy for an ad-publisher who faces bidders that are uncertain about their values from participating in SPAs for ads. The credit provided by the publisher to bidders incentivizes them to experiment with their bidding to resolve their uncertainty, and, when set optimally, improves the publisher's revenue. \cite{IyerJohariSundararajan14} adopt a mean-field approximation for large markets, in which bidders track only an aggregate and stationary representation of the competitors' bids to study repeated SPAs where bidders learn about their own private values over time. They apply it to the problem of selecting optimal reserve prices for the publisher. \cite{balseiroBesbesWeintraub2015repeated} characterize equilibrium strategic interactions between budget-constrained advertisers who face no uncertainty about their private valuations, but have uncertainty about the number, bids and budgets of their competitors. They also develop a fluid mean-field approximate equilibrium, and use their characterization to recommend optimal budget pacing strategies for advertisers and optimal reserve prices for the publisher. Finally, \cite{balseiroGur2019learning} and \cite{TunuguntlaHoban21} provide pacing algorithms and characterize equilibria in bidding when budget-constrained advertisers who observe their current private values before bidding (but are uncertain about their and competitors' future values and budgets) interact over time in repeated SPAs. \cite{TunuguntlaHoban21} also discuss augmenting their algorithm with bandit exploration when the advertiser's valuation has to be learned. Overall, the goals of these papers $-$ to characterize equilibria and to suggest equilibrium budget pacing strategies, credits or reserve prices $-$ are different from ours, which is to develop an experimental design from the advertiser's perspective for causal inference on ads bought via SPAs and FPAs. We note the methods in these studies could form the basis for extending the analysis in this paper to develop experimental designs for simultaneous experimentation by multiple advertisers on an AdX.
There is a smaller subliterature that uses more general reinforcement learning (RL) approaches beyond bandits to optimize RTB advertising policies (\citealp{cai2017real,wu2018multi, jin2018real}). These papers are not concerned with estimating ad-effects, and we depart from these approaches in that our goal is to perform causal inference. Further, we achieve this goal by leveraging key properties of the auction format, thus contributing to a nascent literature, to our knowledge, on direct applications of auction theory to enable causal inference. While several studies combined experimental designs with auction theory, their goals were to identify optimal policies such as bids as in the aforementioned studies, reserve prices (\citealp{asmt2016,ostrovsky2009reserve,pouget2018optimizing,razk2019}) or auction formats (\citealp{chn2016}), not to estimate the causal effect of an action such as advertising determined by the outcome of an auction.
\subsubsection*{Literature on experimentation in digital advertising}
\noindent This paper is related to the literature on pure experimental approaches to measure the effect of digital advertising.\footnote{For a more detailed review we refer the reader to \cite{GordonetalInefficienciesdigitalAds21}, who provide in their Section 1 a recent overview and critical discussion.} One feature that distinguishes this paper from several studies in this stream is its focus on developing an experimental design from the advertiser's perspective; in contrast, many of the proposed experimental designs, such as ``ghost-ads'' for display advertising (\citealp{johnson2017ghost}) or search ads (e.g., \citealp{simonovRao2018competition}), are designed from the publisher's perspective and require observation of the ad-auction logs or cooperation with the publisher for implementation. In RTB settings, the advertiser bidding on AdXs does not control the auction and does not have access to these logs, precluding such designs (the next section provides a more detailed discussion).
Existing experimental designs that have been proposed from the advertiser's perspective include geo-level randomization (e.g., \citealp{blakeNoskoTadelis2015consumer}) or inducing randomization of ad-intensities by manipulating ad campaign frequency caps on DSPs (e.g., \citealp{sahni2019experimental}). Unlike these papers, our design leverages bid randomization, is tailored to the RTB setting, is an online, rather than offline, inferential procedure, and leverages auction theory for inference.
\cite{lw2018} suggest using bid randomization as a device to infer the causal effects of RTB ads. Their method uses bids as an instrumental variable for ad-exposure and delivers estimates of the local average treatment effect of ads, unlike the experimental design proposed here, which leverages the link to auction theory to deliver $ATE$s for SPAs and FPAs. Also, unlike the experimental design outlined here, their method is not adaptive and involves only pure exploration. Therefore, it does not have the feature that the bid randomization policy also minimizes the costs from experimentation by concurrently exploiting the information learned during the test to optimize advertiser payoffs. Finally, adaptive experimental designs for picking the best creative for ad-campaigns are presented in \cite{scott2015multi}, \cite{schwartzetal2017}, \cite{Juetal2019} and \cite{GengLN20}. While related, the problem addressed in these papers of selecting a best performing variant from a set of candidates is conceptually distinct from the problem addressed in this paper of measuring the causal effect of an RTB ad-campaign.
Finally, while there are differences in implementation, our philosophy towards experimental design $-$ which aligns the goal of the design with the payoff-maximization objective of the advertiser $-$ is aligned with that of \cite{fb2018}, who advocate ad-experimental designs that are profit maximizing. The advantage of the bandit-based allocation here is that traffic is adaptively assigned to bid-arms in a way that respects the advertiser's profits, analogous to \cite{fb2018}'s setup. Some salient differences in implementation are that we adopt a many-period bandit-allocation, while \cite{fb2018} use a two-period setup, and that our design shows how to implement profit maximizing tests for outcomes over which the advertiser only has imperfect control.
\subsubsection*{Literature on causal inference with bandits}
\noindent There is an emerging literature that discusses online causal inference with bandits in various general settings.\footnote{Note that existing approaches to inference with bandits differ based on whether they pertain to the \textit{offline} setting, where pre-collected data is available to the analyst, or the \textit{online} setting, where data arrive sequentially, with the online literature being relatively more recent. Unlike online methods, offline methods are meant to be implemented \textit{ex-post}, which implies that the data collection, though done sequentially, is typically not made explicitly to facilitate inference upon its completion. Also, the methods are meant for retrospective application on logged data, which means that data collection does not explicitly reflect in real-time the progress made towards the inferential goal. This paper relates more closely to the online stream, so we discuss only papers related to online inference.} Performing causal inference with bandits is complicated by the adaptive nature of data collection, wherein future data collection depends on the data already collected. Although bandits possess attractive properties in finding the best arm, estimates of arm-specific expected rewards typically exhibit a bias often referred to as ``adaptive bias'' \citep{Xu:2013,Villar2015MultiarmedBM}. In particular, \cite{Nie2018} show that archetypal bandit algorithms such as Upper Confidence Bound (UCB) and TS compute estimates of arm-specific expected rewards that are biased downwards. Due to this problem, leveraging bandits for causal inference for RTB ads is complicated even in a simple case where assignment of users to ads is fully under the control of the advertiser. If we set up ad-exposure and no-ad-exposure as bandit arms, so that the difference in rewards between the two arms represents the causal effect of the ad, adaptive bias in estimating the respective arm-rewards contaminates this difference.
Online methods to find the best arm while correcting for such adaptive bias include \cite{goldenshluger2013}, \cite{Nie2018}, and \cite{BastaniBayati2015}, who suggest data-splitting by forced-sampling of arms at prescribed times, and \cite{dimakopoulou2018estimation} and \cite{hhzwa2020}, who correct for the bias via balancing and inverse probability weighting. Online methods to perform frequentist statistical inference that is valid for bandits, but which avoid issues of explicitly bias-correcting estimates of arm-specific expected rewards, are presented in \cite{yrjw2017}, \cite{jj2018} and \cite{Juetal2019}.
This paper has a different focus on inference compared to the above studies. Broadly speaking, the above methods aim to either find the best arm or learn without bias the expected reward associated with the best arm. In contrast, our goal is to obtain an unbiased estimate of the effect of an action (ad-exposure) that is \textit{imperfectly obtained by pulling arms} (placing bids). Therefore, in our setup the target treatment whose effect is to be learned is not an arm, but a shared stochastic outcome that arises from pulling arms. Hence, arms are more appropriately thought of as \textit{encouragements} for treatments, which makes our setup the online analogue of an offline encouragement design from the program evaluation literature (e.g., \citealp{ImbensRubin1997}). In addition to this difference, our bandit design, which treats bids and their associated payoffs as arms/rewards (rather than ad-exposure and the payoff of ad-exposure as arms/rewards), presents a different approach to avoiding the aforementioned adaptive bias. In our approach, we obtain the object we would like to estimate and perform inference on via the \textit{identity} of the best arm and the theoretical relationship between the two rather than the expected value of its reward. Since typical MAB algorithms recover the identity of the best arm without bias, we are able to leverage them for inference on ad-effects without bias in an online environment. Again, this is achieved by maintaining a close link to auction theory, which makes our approach different in spirit from the above, more theory-agnostic approaches.
Our setup also shares similarities with the instrument-armed bandit setup of \cite{Kallus2018IV}, in which there is a difference between the treatment-arm pulled and the treatment-arm applied due to the possibility of user non-compliance. However, the difference between the pulled and applied treatments, which is important to the design here, is not a feature of the design considered by \cite{Kallus2018IV}, because the pulled and applied treatments belong to the same set in his design. Also, unlike the setup in \cite{Kallus2018IV}, where exposure to a treatment is the outcome of a choice by the user to comply with the pulled arm, exposure here is obtained via a multi-agent game that is not directly affected by the user, thus characterizing a different exposure mechanism.
Bandits have been explicitly embedded within the \textit{structural causal framework} of \cite{Pearl2009} in a series of papers by \cite{Bareinboim2etal015}, \cite{Lattimoreetal2016} and \cite{forneyetal2017counterfactual}. Our paper is related to this stream as our application is a specific instance of a structural causal model tailored to the auction setting: we assume the existence of a probabilistic, microfounded generative process that is the common shared causal structure behind the distributions of the rewards for each arm. As this stream has emphasized, the link to the model in our application is helpful to making progress on the inference problem. This approach has also been followed by other papers in economics (see, for example, the references in \citealp{bv2008}) and marketing (\citealp{msa2019}) that study pricing problems where firms aim to learn the optimal price from a grid of prices, corresponding to arms, which share the same underlying demand function.
\begin{section}{Problem formulation}
\noindent Our goal is to develop an experimental design to measure the causal effect of the ads an advertiser buys programmatically on AdXs. To buy the ad, the advertiser (she) needs to participate in an auction ran by the AdX for each impression opportunity. Winning the auction allows the advertiser to display her ad to the user. The AdX's auction format can either be a SPA or a FPA.
Recall that we define the advertiser' goal of estimating the expected effect of displaying the ad to a user (he) as her \textit{inference goal}. To state this goal more precisely, let $Y(1)$ denote the revenue the advertiser receives when her ad is shown to the user and let $Y(0)$ denote the revenue she receives when her ad is not shown. $Y(1)$ and $Y(0)$ are potential outcomes to the advertiser expressed in monetary units and the causal effect of the ad is $Y(1)-Y(0)$. All the information the advertiser has about the user and impression opportunity is captured by a variable $x$ that can take $P$ different values, so that $x\in\mathbb{X}\equiv \left \{ x_1,...,x_P \right \}$.\footnote{In our MAB setup, $x$ is the \emph{context} of the auction. It can be obtained from a vector $Z$ of observable display opportunity variables that can include, for example, user and publisher characteristics, with $P$ being the total number of different combinations of values across all elements of $Z$. Consumer segmentation of user characteristics in this manner is common in the literature; see, for instance, \cite{msa2019}.}
The advertiser's inference goal specifically is to estimate a set of \textit{conditional average treatment effects} (\textit{CATE}s) in which exposure to the ad is the treatment, where $CATE(x)$ $=$ $\mathbb{E}_{1,0}\left[Y(1)-Y(0)\middle\vert x\right]$. The $CATE$s represent heterogeneity in the average treatment effect of the ad across subsets of users spanned by $x$. The subscripts ``$1,0$'' on the expectation operator in the $CATE$ explicitly indicate that the expectation is taken with respect to the conditional distribution of $Y(1)$ and $Y(0)$ on $x$, about which the advertiser has uncertainty. The advertiser needs to estimate this object because she does not have complete knowledge of the distribution of potential outcomes $Y(1)$ and $Y(0)$ conditional on $x$. Therefore, achieving the inference goal requires the collection of data informative of this distribution.
An experimental design that delivers on the advertiser's inference goal needs to address four issues, which we discuss in sequence below. All four are generated by the distinguishing feature of the AdX environment that the treatment $-$ ad-exposure $-$ can only be obtained by winning an RTB auction.
\subsection*{Issue 1: The advertiser cannot randomize treatment directly}
\noindent The first issue is that the existence of the mediating auction precludes typical experimental approaches to measuring treatment effects that involve collecting data while randomizing treatment and then using these data to estimate $CATE$s. The outcome of the RTB auction is not under the advertiser's complete control because she does not determine the highest competing bid on the AdX. This lack of control disallows her from randomizing the treatment, ad-exposure.
Although ad-exposure cannot be perfectly controlled, participation in auctions is fully under the advertiser's control. Therefore, a viable alternative design involves randomizing her participation across auctions. Without loss of generality, we can think of auction participation randomization as analogous to \textit{bid randomization}, with ``no participation'' corresponding to a bid of $0$, and ``participation'' corresponding to a positive bid. Consequently, we could consider bid randomization as an alternative identification strategy to recover $CATE$s in this environment.
\subsection*{Issue 2: Bid randomization alone is insufficient for identification}
\noindent This leads to the second issue: the auction environment constrains what can be learned from participation/bid randomization experimental designs. In particular, bid randomization is generally \textit{not} sufficient to yield identification of $CATE$s if the relationship between $Y(1)$, $Y(0) $ and competing bids remains completely unrestricted.
To see this, let the highest bid against which the experimenting advertiser competes in the auction be denoted by $B_{CP}$. For simplicity, assume that $P=1$ so that $x$ can only take one value and hence can be ignored. Let $D \equiv \mathbbm{1} \left \{ B_{CP} \leq b \right \}$ denote winning the auction (i.e., ad-exposure), where $b$ is the experimenting advertiser's bid. Let $Y\equiv D\times Y(1) + (1-D)\times Y(0)$, $Y(1)=\lambda_1+\eta_1$ and $Y(0)=\lambda_0+\eta_0$, where $\lambda_1$ and $\lambda_0$ are constants and $\mathbb{E}[\eta_1]=\mathbb{E}[\eta_0]=0$, so that $ATE= \lambda_1 - \lambda_0$.
Consider measuring the $ATE$ via a regression. A regression of $Y$ on $D$ using data collected with bid randomization corresponds to:
\begin{align}\label{eq:regr}
Y_i &=\lambda_0 + ATE \times D_i + D_i \eta_{1i} + (1-D_i) \eta_{0i},
\end{align}
where $i$ indexes an observation. For the OLS estimator of $ATE$ to be consistent the indicator $D_i$ has to be uncorrelated with the errors $\eta_{1i}$ and $\eta_{0i}$, and there are two potential sources of such correlation, $b_i$ and $B_{CP,i}$. Thus, even if $b_i$ is picked at random, a correlation can still exist through $B_{CP,i}$.
This consideration motivates Assumption \ref{ass:pv} below, which we maintain for the rest of the paper.
\begin{assum}\label{ass:pv} ``Private values'' / Conditional independence\hfill\break
$\left \{Y(1),Y(0) \right \} \independent B_{CP} \vert x $.
\end{assum}
One way to interpret Assumption \ref{ass:pv} is from the perspective of auction theory. Noting that the value of the ad-exposure to the advertiser depends on $Y(1)$ and $Y(0)$, Assumption \ref{ass:pv} implies that, conditional on $x$, knowledge of $B_{CP}$ has no effect on the experimenting bidder's assessment of $Y(1)$ and $Y(0)$, and, consequently, on her assessment of her own willingness-to-pay, or valuation. Therefore, we can think of Assumption \ref{ass:pv} as analogous to a typical private values condition.\footnote{We say that it is ``analogous to'' but not exactly a private values condition because the specific model of ad-auctions that we will present below is slightly different from the canonical model for auctions presented in the theory literature (e.g., \citealp{milgrom1982theory}). In the canonical model, a bidder obtains a signal about the item she is bidding on prior to the auction, and uses that to form her expectation of its value. Under a private values condition, it is then without loss of generality to normalize the bidder's valuation to be the signal itself (e.g., \citealp{ah2002}, pp. 2110--2112). If we apply that formulation here, this would imply that the signal would correspond to the treatment effect $Y(1)-Y(0)$ itself. This formulation simply does not fit our empirical setting in which the treatment effect is not known to the advertiser prior to bidding (this is the motivation for running the experiment in the first place). Reflecting this, the model we present does not have signals. As a consequence, it does not map exactly to the canonical dichotomy between private and interdependent values, which is framed in terms of bidders' signals.} This does not mean that there are no correlations between the values that competing advertisers assign to an auctioned impression. The maintained assumption is that these common components of bidder valuations are accommodated in the observed vector $x$. Part of the motivation arises from the programmatic bidding environment. When advertisers bid in AdXs, they match the id (cookie/mobile-identifier) of the impression with their own private data. If $x$ is large and encompasses what can be commonly known about the impression, each advertiser's private value after conditioning on $x$ would only be weakly correlated.
Another way to think of Assumption \ref{ass:pv} is in causal inference terms as an unconfoundedness assumption. Statistically, it simply is a conditional independence assumption, which is more likely to hold when $x$ is large and spans the common information set that auction participants have about the user impression. This is most likely to happen when the experimenter is a large advertiser or an intermediary such as a large DSP, which has access to large amounts of user data that can be matched to auctioned user impressions in real-time.
Along with bid randomization, Assumption \ref{ass:pv} yields identification of $CATE(x)$. For instance, from equation (\ref{eq:regr}) we see that OLS identifies $ATE$ consistently under this condition, because both $b$ and $B_{CP,i}$ are independent of $\eta_{1i},\eta_{0i}$.\footnote{Another advantage is that, under this assumption, the target estimand, $CATE(x)$, becomes a common component of the expected reward associated with all the bid-arms for a given $x$, thereby facilitating cross-arm learning of both this estimand and optimal bids. If potential outcomes were instead correlated with others' bids given $x$, the resulting expected reward associated with each bid-arm would be not be a function of the $CATE(x)$, but rather of a more complex expectation of the potential outcomes that varies across arms, precluding such cross-arm learning efficiencies. See also \cite{bgmms2019} who make a similar independence assumption and point out its usefulness for cross-context learning.}
\subsection*{Issue 3: High costs of experimentation}
\noindent While bid randomization combined with Assumption \ref{ass:pv} is sufficient for identification, a third issue to consider is the cost of experimentation. As mentioned in the introduction, inducing ad-exposure involves paying the winning price in the auction, and inducing no-exposure involves foregoing the value generated from ad-display. Therefore, collecting experimental units on the AdX involves costs.
These costs can be high under suboptimal bidding. Bidding higher than what is required to win the auction involves wastage of resources from overbidding, and bidding 0 involves possible opportunity costs from underbidding, especially on users to whom ad-exposure would have been beneficial. In a high frequency RTB environment, a typical experiment can involve millions of auctions, so that if bidding is not properly managed, the resulting economic losses can be substantial.\footnote{With a fixed budget, poor bidding also affects the quality of inference when wastage of experimental resources leads to reduced collection of experimental data, leading to smaller samples and reduced statistical precision.} These costs can form an impediment to implementation of the experiment in practice, precluding its use.
The key to controlling costs is to optimize the bidding, specifically by finding the optimal bid, $b^*$, to submit to the auction for each value of $x$. Henceforth, we refer to the advertiser's goal of obtaining the optimal bidding policy in the experiment, $b^*(x)$, as her \textit{economic goal}. As we discuss in more detail below, the advertiser's inference and economic goals are directly related, though not necessarily perfectly aligned.
To characterize the optimal bid more formally and relate it to the advertiser's inference goal, we first turn to the advertiser's optimization problem from auction participation. The advertiser's payoff as a function of her bid, $b$, is denoted $\pi(b,Y(1),Y(0),B_{CP})$.
In an SPA, $\pi(\cdot)$ is
\begin{align}\label{eq:revenueSPA}
\pi(b,Y(1),Y(0),B_{CP})&= \mathbbm{1} \left \{ B_{CP} \leq b \right \} \times \left [ Y(1) - B_{CP} \right ] + \mathbbm{1} \left \{ B_{CP} > b \right \} \times Y(0) \nonumber \\
&=\mathbbm{1} \left \{ B_{CP} \leq b \right \} \times \left \{ \left [ Y(1) - Y(0) \right ] - B_{CP} \right \} + Y(0),
\end{align}
while in an FPA, $\pi(\cdot)$ is
\begin{align}\label{eq:revenueFPA}
\pi(b,Y(1),Y(0),B_{CP})&= \mathbbm{1} \left \{ B_{CP} \leq b \right \} \times \left [ Y(1) - b \right ] + \mathbbm{1} \left \{ B_{CP} > b \right \} \times Y(0) \nonumber \\
&=\mathbbm{1} \left \{ B_{CP} \leq b \right \} \times \left \{ \left [ Y(1) - Y(0) \right ] - b \right \} + Y(0).
\end{align}
As an aside, notice the formulation of auction payoffs in equations (\ref{eq:revenueSPA}) and (\ref{eq:revenueFPA}) is different from typical set-ups. As mentioned previously, in most auction models the term $Y(0)$ is set to zero because it is assumed that a bidder only accrues utility when she wins the auction. However, this convention is not suitable to our setting given the interpretation of the terms $Y(1)$ and $Y(0)$. In particular, a consumer might have a baseline propensity to purchase the advertiser's product even if he is not exposed to her ad, which is associated with the term $Y(0)$. Exposure to the ad might affect this propensity, further implying that $Y(1)\neq Y(0)$.
\paragraph*{Optimal bidding} \hfill
\noindent The advertiser is assumed to be risk-neutral, and to look for a bid that maximizes the expected payoff from participating in the auction,
\begin{align}\label{eq:exp_opt}
\bar{\pi}(b|x) = \mathbb{E}_{1,0,CP}[\ensuremath{\pi(b,Y(1),Y(0),B_{CP})}|x],
\end{align}
\noindent where the subscripts ``$1,0, CP$'' on the expectation operator indicate that the expectation is taken with respect to the conditional distribution of $Y(1)$, $Y(0)$ and $B_{CP}$ on $x$. Analogous to the assumption that the advertiser does not fully know the distribution of $Y(1)$ and $Y(0)$, we assume that she also does not know the distribution of $B_{CP}$. This implies that the advertiser faces uncertainty over the joint conditional distribution of $Y(1)$, $Y(0)$ and $B_{CP}$ on $x$, which we denote by $F \left (\cdot, \cdot,\cdot \middle \vert x \right )$. It is important to note that the fact that we postulate that there exists a distribution for $B_{CP}$ does not imply that competitors are randomizing bids or following mixed strategies, although it does allow for it. As typical in game-theoretic approaches to auctions, a given bidder, in this case the advertiser, treats the actions taken by her competitors as random variables, which is why we treat $B_{CP}$ as being drawn from a probability distribution.\footnote{This modeling approach for competition, summarizing it by $B_{CP}$, is reduced-form. Typically $B_{CP}$ is more precisely defined because more information about the environment is available. For example, if the advertiser knew she was competing against $M$ competitors, then $B_{CP}$ would correspond to the highest order statistic out of the $M$ competing bids. If a reserve price was in place, then $B_{CP}$ would further be the maximum between this reserve price and the highest bid. This order statistic could be further characterized depending on what assumptions are made by the signals or information that competitors use to pick their bids, such as symmetry. If $M$ is unknown but the advertiser knows the probability distribution governing $M$, then $B_{CP}$ is the highest order statistic integrated against such distribution. The reason why we follow this reduced-form approach is twofold. The first is due to practical constraints: in settings such as ours, advertisers rarely have information about the number and identities of competitors they face, so conditioning on or incorporating it would be infeasible. The second is that we are focusing on the optimization problem faced by a single advertiser, who takes the actions of other agents as given. Since $B_{CP}$ can incorporate both a varying number of competitors and a reserve price, it is a convenient modeling device to solve this problem.}
Solving for $b^*(x)$ in the presence of uncertainty over $F \left (\cdot, \cdot,\cdot \middle \vert x \right )$ is a non-standard and highly non-trivial auction problem. The problem is non-standard because under the outlined circumstances, the advertiser faces two levels of uncertainty. The first, ``lower-order'' uncertainty is similar to the one faced by bidders in typical auction models: the advertiser is uncertain about what her competitors bid, which is encapsulated by $B_{CP}$. She is also uncertain about her own valuation. As mentioned above, under this formulation the advertiser's valuation corresponds to the treatment effect, $Y(1) - Y(0)$, which is never observed in practice.
The second, ''higher-order'' uncertainty is not present in standard auction models and is also the source of the inference goal. While in the majority of auction models a given bidder does not have complete knowledge of her valuation or of her competing bids, she does know the distributions from which these objects are drawn, which would correspond to the conditional joint distribution of $Y(1)$, $Y(0)$ and $B_{CP}$ on $x$ in our model. This is not the case here where the advertiser faces uncertainty regarding $F \left (\cdot, \cdot,\cdot \middle \vert x \right )$.
To see why this bidding problem is non-trivial, notice that without access to data informative of this distribution, the advertiser would have to integrate over $F \left (\cdot, \cdot,\cdot \middle \vert x \right )$ to construct the expected payoff from auction participation. The optimization problem she would then need to solve is:
\begin{align}\label{eq:full_opt}
\max_b \mathbb{E}_F \left \{ \mathbb{E}_{1,0,CP} \left [ \pi(b,Y(1),Y(0),B_{CP}) \middle \vert F, x \right ] \middle \vert x \right \}.
\end{align}
In equation (\ref{eq:full_opt}), the inner expectation is taken with respect to $Y(1)$, $Y(0)$ and $B_{CP}$ while keeping their joint distribution fixed, where $\pi(b,Y(1),Y(0),B_{CP})$ is given in equations (\ref{eq:revenueSPA}) and (\ref{eq:revenueFPA}). Thus, it reflects the aforementioned lower-order uncertainty. In turn, the outer expectation is taken with respect to $F \left (\cdot, \cdot,\cdot \middle \vert x \right )$, which is reflected in the subscript ``$F$'' and on the conditioning on $F$ in the inner expectation. At the most general level, the advertiser would consider all trivariate probability distribution functions whose support is the three-dimensional positive real line. As such, the optimization problem given in (\ref{eq:full_opt}) is neither standard nor tractable, and the solution to this problem is in all likelihood highly sensitive to the beliefs over distributions the advertiser can have.
Without access to data, the advertiser would have no choice but to try to solve the optimization problem in (\ref{eq:full_opt}). However, as we noted above, achieving the inference goal requires the collection of data, which can also be used to address her economic goal. Therefore, instead of tackling the optimization problem in (\ref{eq:full_opt}), one strategy for optimizing bidding would be to use the data collected in the experiment to construct updated estimates of $\mathbb{E}_{1,0,CP} \left [ \pi \left (b, Y(1),Y(0),B_{CP} \right ) \middle \vert F, x \right ]$, and to perform bid optimization with respect to this estimate as the experiment progresses. This way, the data generated from the experiment are used to address both the advertiser's inference goal and to optimize expected profits in order to address her economic goal, so that both goals are pursed simultaneously.
\subsection*{Issue 4: Aligning the inference and economic goals}
\noindent This leads to the final issue: how to balance the simultaneous pursuit of both the inference and economic goals in the experiment? This issue arises because typical strategies aimed at tackling one of the goals can possibly have negative impacts on accomplishing the other, suggesting an apparent tension between the two.
To see this, consider what would happen if the experiment focused only on the advertiser's inference goal by randomizing bids without any concurrent bid optimization. We already alluded to the consequences of this for the advertiser's economic goal in our previous discussion: pure bid randomization can hurt the advertiser's economic goal by inducing costs from suboptimal bidding.
Consider now what would happen if the experiment focused only on the advertiser's economic goal: learning $b^*(\cdot)$. Notice from equations (\ref{eq:revenueSPA}), (\ref{eq:revenueFPA}) and (\ref{eq:exp_opt}) that the payoffs from bidding $b$ are stochastic from the advertiser's perspective, and that the optimal bid, $b^*(\cdot)$, is the maximizer of the expected payoff from auction participation, $\bar{\pi}(b|x)$, which is an unknown objective function to the advertiser. Therefore, in this setup, pursuing the economic goal involves finding the best bid to play in an environment where payoffs are stochastic, and maximizing expected payoffs against a distribution which has to be learned by exploration. A MAB or RL approach is thus attractive in this situation because it can recover $b^*(\cdot)$ while minimizing the costs from suboptimal bidding, which pure randomization does not assess. By following this strategy the advertiser would adaptively collect data to learn a good bidding policy by continuously re-estimating and re-optimizing $\bar{\pi}(b|x)$.
In principle, these data could also be used to estimate $CATE(x)$ by running the regression in (\ref{eq:regr}) for each $x$, for example. However, the adaptive nature of the data collection procedure induces autocorrelation in the data, which can impact asymptotic statistical and econometric properties of estimators. Moreover, even if all desired properties hold, underlying properties of the data and the algorithm used can affect the estimator adversely. To see this, consider the following example. For simplicity, assume once again that $P=1$ so that $x$ can be ignored and further assume that $\Pr(B_{CP} \leq b^*)\approx 1$. A good algorithm would converge quickly, eventually yielding relatively few observations of $Y(0)$ compared to those of $Y(1)$, which would hinder the inference goal since typical estimators for $CATE$s based on such imbalanced data tend to be noisy.
This framing demonstrates that the inference and economic goals can possibly be in conflict. The advertiser's goals are clearly related since they both depend on knowledge about the distribution $F \left (\cdot,\cdot,\cdot\middle\vert\cdot \right )$. The challenge faced by the advertiser in accomplishing her goals is that this distribution is unknown to her. While there are known approaches to gather data and tackle each of the advertiser's goals in isolation, it is unclear whether they can perform satisfactorily in achieving both goals concurrently. Addressing this is the core remaining piece for experimental design, which is discussed next.
\end{section}
\begin{section}{Balancing the advertiser's objectives}\label{sec:balancing-obj}
\noindent The strategy we adopt to balance the two goals leverages the microfoundations of the experiment by recognizing that the goals are linked to each other by the economic logic of optimal bidding. Because the bidding logic depends on the auction format, the extent to which the two goals are balanced will also differ by auction format. In particular, we will show that, in SPAs, the inference and economic goals can be perfectly aligned, while in FPAs, leveraging this linkage is still helpful, but the goals can only be imperfectly aligned.
To characterize our approach, we consider the limiting outcome of maximizing the true expected profit function with respect to bids when the joint distribution $F \left (\cdot, \cdot,\cdot \middle \vert \cdot \right )$ is known to the advertiser. In what follows, we will use the expressions in equations (\ref{eq:revenueSPA}) and (\ref{eq:revenueFPA}) ignoring the second term, $Y(0)$, because it does not depend on the advertiser's bid, $b$. This expression also has the benefit of directly connecting the potential outcomes to this auction-theoretic setting, with the treatment effect $Y(1) - Y(0)$ taking the role of the advertiser's valuation.
We combine equations (\ref{eq:revenueSPA}) and (\ref{eq:exp_opt}) to write the object she aims to learn in an SPA as the maximizer with respect to $b$ of:
\begin{align}\label{eq:auc_epaySPA}
\bar{\pi}(b \vert x) &\equiv \mathbb{E} \left [ \pi(b,Y(1),Y(0),B_{CP}) \middle \vert x \right ] \nonumber \\
&= \Pr \left ( B_{CP} \leq b \middle \vert x \right ) \times \mathbb{E} \left \{ \left [ Y(1) - Y(0) \right ] - B_{CP} \middle \vert B_{CP} \leq b; x \right \} ,
\end{align}
In an FPA, we combine equations (\ref{eq:revenueFPA}) and (\ref{eq:exp_opt}) to write the object she aims to learn as the maximizer with respect to $b$ of:
\begin{align}\label{eq:auc_epayFPA}
\bar{\pi}(b \vert x) &\equiv \mathbb{E} \left [ \pi(b,Y(1),Y(0),B_{CP}) \middle \vert x \right ] \nonumber \\
&= \Pr \left ( B_{CP} \leq b \middle \vert x \right ) \times \mathbb{E} \left \{ \left [ Y(1) - Y(0) \right ] - b \middle \vert B_{CP} \leq b; x \right \}.
\end{align}
Once again, the expectation in equations (\ref{eq:auc_epaySPA}) and (\ref{eq:auc_epayFPA}) is taken with respect to $Y(1)$, $Y(0)$ and $B_{CP}$, but the subscript ``1,0,CP'' is omitted to ease notation. The conditioning on the distribution $F$ is also omitted since $\bar{\pi} \left ( \cdot \middle \vert x \right )$ is the true expected profit function, which therefore utilizes the true conditional distribution $F \left ( \cdot, \cdot, \cdot \middle \vert \cdot \right )$ to compute the relevant expectations and probabilities. We denote the maximizers of these respective expressions by $b^*(x)$.
To ensure that $b^*(x)$ is well-defined, we maintain the following weak technical assumption on $F \left ( \cdot, \cdot, \cdot \middle \vert \cdot \right )$.
\begin{assum}\label{ass:joint} {Well behaved $F\left ( \cdot, \cdot, \cdot \middle \vert \cdot \right )$} \hfill\break
\subasu The joint distribution $F(\cdot,\cdot,\cdot \vert x)$ admits a continuous density, $f(\cdot,\cdot,\cdot\vert x)$, over $\mathbb{R}_+^3$ for all $x$. \label{ass:dens} \\
\subasu $\mathbb{E} \left [ Y(1) \vert x \right ] < \infty$, $\mathbb{E} \left [ Y(0) \vert x \right ] < \infty$, and $\mathbb{E} \left [ B_{CP} \vert x \right ] < \infty$ for all $x$. \label{ass:fin_means} \\
\subasu The density of $B_{CP}$ given $x$, $f_{CP}\left ( \cdot \middle \vert x \right )$, is strictly positive in the interior of $\mathbb{R}_+$ for all $x$. \label{ass:pos_dens} \\
\subasu $\frac{f_{CP}\left(b_{CP}|x\right)}{F_{CP}\left(b_{CP}|x\right)}$ is decreasing in $b_{CP}$ for all $x$. \label{ass:rev_haz}
\end{assum}
Assumption \ref{ass:joint} not only is mild but also relatively common in auction models and is made solely for tractability. Assumptions \ref{ass:dens} and \ref{ass:fin_means} are minimal requirements. In causal inference terms, Assumption \ref{ass:pos_dens} is equivalent to an overlap assumption that $0 < P(D = 1|x) < 1$, where $P(D = 1|x)$ is the propensity score. Letting $D \equiv \mathbbm{1} \left \{ B_{CP} \leq b \right \}$ as before, so that $P(D = 1|x) \equiv P(B_{CP} \leq b|x)$, Assumption \ref{ass:pos_dens} implies $0 < P(B_{CP} \leq b|x) < 1$. In addition, Assumption \ref{ass:pos_dens} could in principle be relaxed as we will mention below. Finally, Assumption \ref{ass:rev_haz} is only required to determine $b^*(x)$ for FPAs. It states that the conditional distribution of $B_{CP}$ on $x$ has a decreasing reversed hazard rate. As argued by \cite{block_savits_singh_1998}, this property holds for several distributions, including all decreasing hazard rate distributions and increasing hazard rate Weibull, gamma and lognormal distributions.
We now investigate the relationship between $b^*(\cdot)$ and $CATE(\cdot)$ under Assumptions \ref{ass:pv} and \ref{ass:joint}. If the distribution $F \left (\cdot, \cdot, \cdot \middle \vert \cdot \right )$ was known, computing $CATE(\cdot)$ would be straightforward, as would be solving for $b^*(\cdot)$ by maximizing expression (\ref{eq:auc_epaySPA}) or expression (\ref{eq:auc_epayFPA}). The results below characterize this relationship first for SPAs and then for FPAs.
\begin{prop}Optimal bid in SPAs\label{prop:alignSPA}\hfill\break
If Assumptions \ref{ass:pv} and \ref{ass:joint} hold and the auction is an SPA, $b^*(x)= \max \left \{ 0, CATE(x) \right \}$.
\end{prop}
\begin{proof}{
To prove Proposition \ref{prop:alignSPA}, we first rewrite equation (\ref{eq:auc_epaySPA}):
\begin{align*}
\bar{\pi}(b\vert x)
&= \Pr \left \{ B_{CP} \leq b \middle\vert x \right \} \left \{ \mathbb{E} \left [ Y(1) -Y(0) \middle \vert B_{CP} \leq b ;x \right ] - \mathbb{E} \left [ B_{CP} \middle \vert B_{CP} \leq b ;x \right ] \right \} \\
&= \Pr \left \{ B_{CP} \leq b \middle\vert x \right \} \left \{ \mathbb{E} \left [ Y(1) -Y(0) \middle \vert x \right ] - \mathbb{E} \left [ B_{CP} \middle \vert B_{CP} \leq b ;x \right ] \right \} \\
&= \Pr \left \{ B_{CP} \leq b \middle\vert x \right \} \left \{ CATE(x) - \mathbb{E} \left [ B_{CP} \middle \vert B_{CP} \leq b;x \right ] \right \} \\
&=\int_0^b \left [CATE(x) - b_{CP} \right ]f_{CP}(b_{CP}|x)db_{CP} ,
\end{align*}
where the second equality follows from Assumption \ref{ass:pv}. Notice that $\bar{\pi}(b\vert x)$ becomes a bidder's expected payoff from a second-price sealed-bid auction in which the bidder's \textit{private value} equals $CATE(x)$. Because the advertiser cannot submit negative bids, when $CATE(x)\leq 0$ the optimal bid is $b^*(x)=0$ since the integrand is negative and $f_{CP}(\cdot|x)>0$ in the interior of $\mathbb{R}_+$ due to Assumption \ref{ass:pos_dens}. Otherwise, notice that the integrand is non-negative as long as $b(x) \leq CATE(x)$, which implies the optimal bid cannot be greater than $CATE(x)$. Once again, due to Assumption \ref{ass:pos_dens}, $f_{CP}(\cdot|x)>0$, so that $b^*(x)=CATE(x)$.\footnote{Notice that Assumption \ref{ass:pos_dens} could be relaxed to assuming that $f_{CP}(\cdot|x)$ is strictly positive on neighborhoods around 0 and $CATE(x)$.}}
\end{proof}
We now characterize the analogous relationship for FPAs.
\begin{prop}Optimal bid in FPAs\label{prop:alignFPA}\hfill\break
If Assumptions \ref{ass:pv} and \ref{ass:joint} hold and the auction is an FPA, $b^*(x)= \max \left\{0,\chi^{-1}\left [CATE(x) \right ] \right \}$, where $\chi\left(b\right)=b+\frac{F_{CP}\left(b|x\right)}{f_{CP}\left(b|x\right)}$.
\end{prop}
\begin{proof}{
To prove Proposition \ref{prop:alignFPA}, we proceed as above, by rewriting equation (\ref{eq:auc_epayFPA}):
\begin{align*}
\bar{\pi}(b\vert x)&=\Pr\left\{ B_{CP}\leq b\middle\vert x\right\} \left\{ \mathbb{E}\left[Y(1)-Y(0)-b\middle\vert B_{CP}\leq b;x\right]\right\} \\&=\Pr\left\{ B_{CP}\leq b\middle\vert x\right\} \left\{ \mathbb{E}\left[Y(1)-Y(0)\middle\vert x\right]-b\right\} \\&=F_{CP}\left(b|x\right)\left\{ CATE(x)-b\right\},
\end{align*}
where, once again, the second equality follows from Assumption \ref{ass:pv}. If $CATE(x)\leq 0$, it is straightforward to verify that $b^*(x)=0$ since the advertiser cannot submit negative bids. Consider now the case where $CATE(x)>0$. The first-order condition with respect to $b$ is given by
\begin{align}\label{eq:ATE-FPA}
CATE(x)=b+\frac{F_{CP}\left(b|x\right)}{f_{CP}\left(b|x\right)}\equiv \chi (b).
\end{align}
Assumption \ref{ass:pos_dens} ensures that $\chi(0)=0$. In addition, Assumption \ref{ass:rev_haz} implies that the right-hand side of equation (\ref{eq:ATE-FPA}) is monotonically increasing in $b$. Hence, $\chi(\cdot)$ is invertible, which yields a unique solution. Denoting its inverse by $\chi^{-1}(\cdot)$, the optimal bid is therefore given by $b^*(x) = \max \left\{0,\chi^{-1}\left [CATE(x) \right ] \right \}$.}
\end{proof}
\subsubsection*{Implications}
\noindent Propositions \ref{prop:alignSPA} and \ref{prop:alignFPA} have three implications. First, they show how the optimal bids, $b^*(x)$, are related to $CATE(x)$. In an SPA, Proposition \ref{prop:alignSPA} shows that whenever displaying the ad is beneficial, that is, when $CATE(x)\geq 0$, one should bid the $CATE(x)$. In an FPA, Proposition \ref{prop:alignFPA} shows that whenever displaying the ad is beneficial, one should bid less than the $CATE(x)$.\footnote{Equation (\ref{eq:ATE-FPA}) shows that $b^*(x)\neq CATE(x)$ whenever $CATE(x)\neq 0$ because the right-hand side consists of a sum of two non-negative terms. In particular, if $CATE(x)>0$ it follows that $b^*(x)<CATE(x)$ due to the typical bid shading in FPAs.} In turn, when ad-exposure is detrimental, that is, when $CATE(x)<0$, the propositions convert this qualitative fact into a clear economic policy as the advertiser would have no interest in displaying the ad in the first place, which can be guaranteed by a bid of zero. A consequence of these relationships is that the advertiser can follow a MAB or RL strategy in the experiment to learn $b^*(\cdot)$, and once they are learned, can obtain $CATE$s by leveraging these relationships. This will form the basis of the algorithm we propose in the next section.
Second, one sees from the propositions that, for both SPAs and FPAs, the object of inference, $CATE(x)$, is a common component of the expected payoff associated with the bids one could consider for a given $x$. Thus, if one thinks of bids as ``arms'' in the sense of a bandit, leveraging the economic structure of the problem induces cross-arm learning within each context (i.e., pulling each arm contributes to learning $CATE$s). This cross-arm learning, which is purely a consequence of maintaining a link to economic theory, helps the bandit-learner proposed in the next section infer the $CATE(x)$ more efficiently, which, in turn, also allows a more efficient learning of the optimal bidding policy for each context.
Third, the propositions show precisely how the inference and economic goals can be aligned in the experiment. For SPAs, Proposition \ref{prop:alignSPA} is powerful because it implies that whenever displaying the ad is beneficial, the advertiser's economic and inference goals are \textit{perfectly} aligned, as learning $b^*(x)$ and estimating $CATE(x)$ consist of the \textit{same} task. Our proposed experimental design for SPAs will be to find the best bid for each $x$, $b^*(x)$, and set that to be the $CATE(x)$. This design will have the feature that achieving good performance in learning the best bid does not come at the cost of reduced performance on measuring \textit{CATE}s or vice versa.
For FPAs, Proposition \ref{prop:alignFPA} is also helpful because it implies that whenever displaying the ad is beneficial, the advertiser's economic and inference goals reinforce each other, as learning $b^*(x)$ helps learning $CATE(x)$ in the experiment. Our proposed experimental design for FPAs will be to find the best bid for each $x$, $b^*(x)$, and set $\chi\left [ b^*(x) \right ]$ to be $CATE(x)$. However, because the inference and economic goals are not perfectly aligned, the design will have the feature that good performance in learning optimal bids may come at the cost of reduced performance in learning the \textit{CATE}s or vice versa.
\end{section}
\begin{section}{Accomplishing the advertiser's objectives}\label{sec:achieve}
\noindent We leverage Propositions \ref{prop:alignSPA} and \ref{prop:alignFPA} to develop an experimental design that concurrently accomplishes the advertiser's goals. Our proposal comprises an adaptive design that learns $b^*(\cdot)$ over a sequence of display opportunities. We begin by stating the following assumption, which we maintain throughout the analysis.
\begin{assum}\label{ass:dgp}Independent and identically distributed (i.i.d.) data\hfill\break
$\left \{Y_i(1),Y_i(0),B_{CP,i} \right \} \overset{\text{iid}}{\sim} F(\cdot,\cdot,\cdot \vert x_i)$ and $x_i \overset{\text{iid}}{\sim} F_x(\cdot)$.
\end{assum}
Assumption \ref{ass:dgp} is a typical assumption made in stochastic bandit problems, that the randomness in payoffs is i.i.d. across occurrences of play. It imposes restrictions on the data generating process (DGP). For instance, if the same user appeared more than once and if his potential outcomes $Y(1)$ and $Y(0)$ were serially correlated, this condition would not hold. In turn, if competing bidders solved a dynamic problem because of longer-term dependencies, budget or impression constraints, $B_{CP}$ could become serially correlated as a result, in which case this condition would also not hold.
Assumption \ref{ass:dgp} justifies casting the advertiser's problem as a MAB. In particular, when Assumptions \ref{ass:joint} and \ref{ass:dgp} hold, $b^*(x)$ is well-defined and common across auctions for every $x$ for SPAs and FPAs. Under these assumptions, it is natural to represent the advertiser's economic goal as a contextual MAB problem. In such setting, the advertiser considers a potentially context-specific set of $r_x=1,...,R_x$ arms, each of which associated with a bid, $b_{r_x}$. The advertiser's goal can be expressed as minimizing cumulative regret from potentially bidding suboptimally over a sequence of auctions while learning $b^*(\cdot)$. Hence, as customary, we implicitly assume that for each $x$ the grid contains the optimal bid, $b^*(x)$.
Algorithms used to solve MAB problems typically base the decision of which bid to play in round $t$, $b_t$, on a tradeoff between randomly picking a bid to obtain more information about its associated payoff (exploration) and the information gathered until then on the optimality of each bid (exploitation). The existing information at the beginning of round $t$ is a function of all data collected until then, which we denote by $W_{t-1}$. Each observation $i$ in these data, whose structure is displayed in Table \ref{tab:single} for SPAs and FPAs, is an ad display auction.
\begin{table}[H]
\begin{centering}
\begin{tabular}{c|cccccccc}
\hline \hline
$i$ & $b_i$ & $x_i$ & $D_i$ & $Y_i $ & $Y_i(1)$ & $Y_i(0)$ & \multicolumn{2}{c}{$B_{CP,i}$} \\
\hline\
& & & & & & & $SPA$ & $FPA$ \\
\hline \\
1 & $b_1$ & $x_1$ & 1 & $ y_1 $ & $y_1$ & $-$ & $b_{CP,1}$ & $\leq b_1$\\ \\
2 & $b_2$ & $x_2$ & 0 & $y_2$ & $-$ & $y_2$ & $\geq b_2$ & $\geq b_2$ \\
$\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ \\ \\
\hline \hline
\end{tabular}
\par\end{centering}
\caption{Snapshot of data structure}
\label{tab:single}
\end{table}
For the analysis of SPAs it will be useful to define the variable $\bar{B}_{CP}\equiv\min\{B_{CP},b\}$. Stacking the data presented in Table \ref{tab:single} across auctions for each round $\tau$, it follows that we can write $W_{t}=\left \{ b_{\tau},x_{\tau}, D_{\tau}, Y_{\tau}, \bar{B}_{CP,\tau}, \omega_{\tau} \right \}_{\tau=1}^{t}$ for SPAs and $W_{t}=\left \{ b_{\tau},x_{\tau}, D_{\tau}, Y_{\tau}, \omega_{\tau} \right \}_{\tau=1}^{t}$ for FPAs. In both cases, the $\omega$s are seeds, independent from all other variables, required for randomization depending on which algorithm is used.
\begin{comment}
\begin{figure}[ht]
\centering
\begin{tikzpicture}
[node distance=5mm and 30mm,
box/.style = {draw, minimum height=12mm, align=center},
sy+/.style = {yshift= 2mm},
sy-/.style = {yshift=-2mm},
every edge quotes/.style = {align=center}]
\node[box] at (0,0) {Advertiser};
\node[box] at (10,2) {Nature};
\node[box] at (5,-2) {Auction};
\node[box] at (5,-6) {Data};
\draw [thick][->] (9.1,2) -- (1.2,.5);
\node[align=center, above] at (5.1,1.5) {$x$ };
\draw [thick][->] (1.2,-.5) -- (4.3,-1.25);
\node[align=center, left] at (2.8,-1.4) {$b$ };
\draw [thick][->] (9.1,1.25) -- (5.2,-1.25);
\node[align=center, right] at (7.8,-0.3) {$\left \{ Y(1),Y(0),B_{CP} ,x \right \}$ };
\draw [thick][->] (5,-2.7) -- (5,-5.25);
\node[align=center, right] at (5.2,-4) {$\left \{ Y,D,\bar{B}_{CP},b,x \right \}$ }; \end{tikzpicture}
\caption{Data generating process (DGP)}
\label{fig:dgp}
\end{figure}
\end{comment}
\begin{comment}
\begin{figure}
\centering
\includegraphics[width=0.65\textwidth]{figures/DGP.png}
\caption{Data generating process (DGP)}
\label{fig:dgp}
\end{figure}
\end{comment}
Notice that these data suffer from two issues. The first, common to both auction formats, is the fundamental missing data problem in causal inference referenced before (\citealp{holland1986}): that $Y(1)$ and $Y(0)$ are not observed together at the same time. The second regards what we observe regarding $B_{CP}$ and differs across the two auction formats. For SPAs, we have a censoring problem related to the competitive environment: for SPAs, $B_{CP}$ is only observed when the advertiser wins the auction; otherwise, all she knows is that it was larger than the bid she submitted. Hence, the observed data have a similar structure to the one in the model defined by \cite{amemyia} as the Type 4 Tobit model. However, for FPAs this restriction is stronger: we never observe $B_{CP}$ and only have either a lower or upper bound on it depending on whether the advertiser wins the auction, so that the observed data has a Type 5 Tobit model structure.
There are two points of departure between this setup and a standard MAB problem. First, in the latter, each arm is associated with a different DGP, so it is commonly assumed that the reward draws are independent across arms. This is not true in our setting: given the economic structure of the problem, conditional on $x$ the values of $ \left \{ Y(1), Y(0), B_{CP}\right \}$ are the same regardless of which arm is pulled, which creates correlation between rewards across arms. In particular, this is a nonlinear stochastic bandit problem as defined by \cite{BubekCesa-Bianchi2012}. Second, on pulling an arm the advertiser observes three different forms of feedback: an indicator for whether she wins the auction and obtains treatment (ad-exposure), the highest competing bid conditional on winning for SPAs or a bound on it conditional on losing, or a bound on the highest competing bid for FPAs, and the reward. This contrasts with the canonical case in which the reward forms the only source of feedback, and fits into the class of ``complex online problems'' studied by \cite{gopalan2014thompson}.
\end{section}
\begin{section}{Bidding Thompson Sampling (BITS) algorithm}\label{sec:BITS}
\noindent We now propose a specific procedure to achieve the advertiser's goals, which is a version of the TS algorithm. Since it aims to learn the advertiser's optimal bid, we refer to it as Bidding Thompson Sampling (BITS).
\begin{subsection}{General procedure}
\noindent It is not our goal to solve for or implement the optimal learning policy that minimizes cumulative regret over a finite number of rounds of play. In fact, a general solution for MAB problems with correlated rewards across contexts and arms such as the one we consider is not yet known. What we require is an algorithm that performs ``well'' in terms of minimizing cumulative regret and that can easily accommodate and account for information shared across arms. Hence, we make use of the TS algorithm \citep{thompson1933}, which is a Bayesian heuristic to solve MAB problems.\footnote{See \cite{scott2015multi} for an application to computational advertising and \cite{russo2018} for an overview.} TS typically starts by parametrizing the distribution of rewards associated with each arm. Since our problem departs from standard MAB problems in that the DGP behind each of the arms, that is, the distribution $F(\cdot,\cdot,\cdot\vert\cdot)$, is the same, we choose to parametrize it instead and denote our vector of parameters of interest by $\theta$. Expected rewards depend on $\theta$, so we will often write $\bar{\pi}(\cdot \vert \cdot,\theta)$. This is the same approach followed by \cite{gopalan2014thompson}, who showed that, in a setting that has similarities to ours, the TS algorithm exhibits a logarithmic regret bound. This is the sense in which we consider it to perform ``well.''
The algorithm runs while a criterion, $c_t$, is below a threshold, $T$. After round $t$, the prior over $\theta$ is updated by the likelihood of all data gathered by the end of round $t$, $W_t$. We denote the number of observations gathered on round $t$ by $n_{t}$ and the total number of observations gathered by the end of round $t$ by $N_t=\sum_{\tau=1}^t n_{\tau}$. If $n_{t}=1$ for all $t$ the algorithm proceeds auction by auction. We present it in this way to accommodate batch updates. Given the posterior distribution of $\theta$ given $W_t$, we calculate
\begin{align}\label{eq:opt_prob}
\psi_t(b_{r_x}\vert x) \equiv \Pr \left (\text{arm }r_x \text{ is the best arm }\middle \vert W_t;x \right )
\end{align}
and update the criterion $c_t$. In round $t+1$, arm $r_x$ is pulled for each observation with context $x$ with probability $\psi_t(b_{r_x}\vert x)$. The generic structure of the TS algorithm is outlined below.
\RestyleAlgo{boxruled}
\SetAlgoNoLine
\LinesNotNumbered
\begin{algorithm}[H]
\caption{Thompson Sampling\label{alg:ts}}
\stepcounter{AlgoLine}\ShowLn Set priors, $\psi_0(\cdot\vert\cdot)$, $c_0$ and $T$.\hfill
\While{($c_t<T$)}{
\stepcounter{AlgoLine}\ShowLn Pull arms according to $\psi_{t-1}(\cdot\vert \cdot)$. \hfill
\stepcounter{AlgoLine}\ShowLn Combine new data with previously obtained data in $W_t$. \hfill
\stepcounter{AlgoLine}\ShowLn Update the posterior distribution of $\theta$ with $W_t$. \hfill
\stepcounter{AlgoLine}\ShowLn Compute $\psi_{t}(\cdot\vert \cdot)$, $c_{t}$ and $b_t^*(\cdot)$.}
\end{algorithm}
\end{subsection}
\begin{subsection}{Parametrizing distribution of rewards}
\noindent We now present the specific parametrization we use in our problem. Because of the structure of the data we described in Section \ref{sec:achieve} and because of the algorithm we use, our procedure requires reimplementing a Bayesian estimator to a Type 4 or Type 5 Tobit model on each round. Hence, the specific parametric structure we impose is chosen to make this estimator as simple as possible and, consequently, to speed up the implementation of the algorithm.
Let $X_i$ be the following $P-$dimensional vector of mutually exclusive dummies:
\begin{align}\label{eq:ctxt}
X_i \equiv \left [ \mathbbm{1} \left \{x_i = x_1 \right \}, \mathbbm{1} \left \{x_i = x_2 \right \},...,\mathbbm{1} \left \{x_i = x_P \right \} \right ]' .
\end{align}
Notice that there is a one-to-one correspondence between the vector $X_i$ and the variable $x_i$. Hence, we will use them interchangeably. We assume that
\begin{align}\label{eq:normal}
\begin{bmatrix}\log Y_i(1) \\ \log Y_i(0) \\ \log B_{CP,i} \end{bmatrix} \Big\vert X_i \overset{i.i.d.}{\sim} N \left ( \begin{bmatrix} X_i'\delta_1 \\ X_i'\delta_0 \\ X_i'\delta_{CP} \end{bmatrix}, \begin{bmatrix} \sigma_1^2 & \rho \sigma_1 \sigma_0 & 0 \\ \rho \sigma_1 \sigma_0 & \sigma_0^2 & 0 \\ 0 & 0 & \sigma_{CP}^2 \end{bmatrix} \right ) \equiv N \left ( \begin{bmatrix} \Delta'X_i \\ X_i'\delta_{CP} \end{bmatrix}, \begin{bmatrix} \Sigma & 0 \\ 0' & \sigma_{CP}^2 \end{bmatrix} \right ),
\end{align}
where $\Delta\equiv \left [\delta_1, \delta_0 \right ]$. We collect the parameters in $\theta \equiv \left [\delta_1',\delta_0', \delta_{CP}', \sigma_1^2, \sigma_0^2,\sigma_{CP}^2, \rho \right ]'$.
Notice that this parametrization directly imposes Assumption \ref{ass:pv} and that it implies that $CATE(X_i)=\exp\left\{X_i'\delta_1 + 0.5 \sigma_1^2 \right\} - \exp\left\{X_i'\delta_0 + 0.5 \sigma_0^2 \right\}$. In addition, since the potential outcomes are never observed simultaneously, $\rho$ is not point identified without further restrictions.\footnote{However, it is possible to exploit the positive semidefiniteness of $\Sigma$ to partially identify $\rho$. See, for example, \cite{vijve1993} and \cite{kp1997}.} Hence, since our interest is in $CATE(\cdot)$ and since it does not depend on $\rho$, we follow \cite{ch2000} and explicitly assume that $\rho=0$. This assumption has the benefit of simplifying the algorithm we present. A more general version that allows for $\rho\neq 0$ is given in Appendix \ref{App:Gibbs-with-rho}. Finally, notice that (\ref{eq:normal}) also implies that for SPAs the expected payoff is:
\begin{align}\label{eq:exp_auc_spa}
\begin{split}
\bar{\pi}(b\vert X_i,\theta)&= \Phi \left ( \frac{\log b - X_i'\delta_{CP}}{\sigma_{CP}} \right ) \times CATE(X_i) \\
&\qquad - \Phi \left ( \frac{\log b - X_i'\delta_{CP}}{\sigma_{CP}} - \sigma_{CP} \right ) \times \exp \left \{ X_i'\delta_{CP} + 0.5 \sigma_{CP}^2 \right \} ,
\end{split}
\end{align}
and for FPAs the expected payoff is:
\begin{align}\label{eq:exp_auc_fpa}
\begin{split}
\bar{\pi}(b\vert X_i,\theta)&= \Phi \left ( \frac{\log b - X_i'\delta_{CP}}{\sigma_{CP}} \right ) \times \left [ CATE(X_i) -b \right ] ,
\end{split}
\end{align}
where $\Phi(\cdot)$ is the cumulative distribution function of the standard normal distribution and where we omit the terms that do not depend on $b$ for brevity.
\end{subsection}
\begin{subsection}{Choice of priors}\label{sec:prior}
\noindent We choose independent normal-gamma priors, which are conjugate to the normal likelihood induced by the DGP in (\ref{eq:normal}). We choose these priors solely for convenience since they speed up the algorithm. For $k\in\left\{1,0,CP\right\}$, we set:
\begin{align}\label{eq:priors}
\begin{split}
& \sigma_k^{-2} \sim \Gamma \left (\alpha_k,\beta_k \right ) \\
&\delta_k \vert \sigma_k^2 \sim N \left (\mu_{\delta_k},\sigma_k^2A_k^{-1} \right ),
\end{split}
\end{align}
where $\left \{\alpha_k,\beta_k\right \}_{k\in\left\{1,0,CP\right\}} $ are non-negative scalars, $\left \{\mu_{\delta_k} \right \}_{k\in\left\{1,0,CP\right\}} $ are $P-$dimensional vectors and $\left \{A_k \right \}_{k\in\left\{1,0,CP\right\}} $ are $P$-by-$P$ matrices. For the gamma distribution, the parametrization is such that if $G\sim\Gamma(\alpha,\beta)$, then $\mathbb{E}[G] = \alpha/\beta.$ We discuss how to use historical data to choose the parameters of the prior distributions below.
\end{subsection}
\begin{subsection}{Drawing from posterior: Gibbs sampling}
\noindent Implementing the algorithm requires computing updated probabilities, $\psi_t(\cdot\vert\cdot)$, which cannot be done analytically because of the missingness and censoring in the feedback data. Nevertheless, it is still possible to exploit \textit{conditional} conjugacy via data augmentation and use Gibbs sampling to obtain draws from the posterior distribution of $\theta$ given $W_t$. Using these draws we can then estimate $\psi_t(\cdot \vert\cdot)$ via Monte Carlo integration. We first describe the steps of this estimation procedure for SPAs, which combines the methods introduced by \cite{chib1992} and \cite{kp1997} in a single Gibbs sampling algorithm with data augmentation, and then describe how it can be modified to accommodate FPAs. Bayesian data augmentation forms an elegant way to solve the censoring and missingness problems induced by the ad-auction. In each draw, we augment the Markov chain with the missing potential outcomes and competing bids, and then perform Bayesian inference on the required treatment effects conditioning on these augmented variables.
\subsubsection{Data augmentation}
\noindent The first step in our procedure is to draw the missing values from our data conditional on $\left (W_t, \theta \right )$. We begin by drawing the missing values $\left \{ \log B_{CP,i} \right \}_{i:D_i=0}$. Given $\left (W_t, \theta \right )$ and under (\ref{eq:normal}), it follows that:
\begin{align}\label{eq:miss1}
&\log B_{CP,i}^{miss} \Big\vert D_i=0, \log Y_i, \log \bar{B}_{CP,i} , \log b_i , X_i, \theta \overset{d}{=} \nonumber \\
& \log B_{CP,i}^{miss} \Big\vert D_i=0, \log b_i,X_i, \delta_{CP}, \sigma_{CP}^2 \sim TN \left ( X_i'\delta_{CP}, \sigma_{CP}^2,\log b_i,+\infty\right ),
\end{align}
where $\overset{d}{=}$ means equality in distribution and $TN \left (\delta_*, \sigma_*^2,l,u \right )$ denotes the truncated normal distribution with mean $\delta_*$, variance $\sigma_*^2$, lower truncation at $l$ and upper truncation at $u$.
Now we proceed to draw the missing values $\left \{ \log Y_i (1) \right \}_{i:D_i=0}$ and $\left \{ \log Y_i (0) \right \}_{i:D_i=1}$. Given $\left (W_t, \theta \right )$ and under (\ref{eq:normal}), it follows that:
\begin{align}\label{eq:miss5}
&\log Y_{i}^{miss}(1) \Big\vert D_i=0, \log Y_i, \log \bar{B}_{CP,i} , \log b_i , X_i, \theta \overset{d}{=} \nonumber \\
& \log Y_{i}^{miss}(1) \Big\vert D_i=0, X_i, \delta_1, \sigma_1^2 \sim N \left ( X_i'\delta_1 , \sigma_1^2 \right )
\end{align}
and,
\begin{align}\label{eq:miss6}
&\log Y_{i}^{miss}(0) \Big\vert D_i=1, \log Y_i, \log \bar{B}_{CP,i} , \log b_i , X_i,\theta \overset{d}{=} \nonumber \\
& \log Y_{i}^{miss}(0) \Big\vert D_i=1,X_i, \delta_0, \sigma_0^2 \sim N \left ( X_i'\delta_0 , \sigma_0^2 \right ).
\end{align}
Now, defining,
\begin{align}\label{eq:misses3}
\delta_i^{miss}= D_i \times X_i' \delta_0 + (1- D_i)\times X_i'\delta_1 \\
\sigma_i^{2,miss}= D_i \times \sigma_0^2 + (1 - D_i) \times \sigma_1^2 ,
\end{align}
we can combine (\ref{eq:miss5}) and (\ref{eq:miss6}) into:
\begin{align}\label{eq:miss4}
&\log Y_i^{miss} \Big \vert \log Y_i, D_i, \log \bar{B}_{CP,i} , \log b_i , X_i, \theta \overset{d}{=} \nonumber \\
&\log Y_i^{miss} \Big \vert D_i, X_i, \delta_1, \delta_0, \sigma_1^2 , \sigma_0^2 \sim N \left ( \delta_i^{miss}, \sigma_i^{2,miss} \right ).
\end{align}
\subsubsection{Creating the "complete" data}
\noindent Given a draw from the distributions given above, $ \left \{ \log Y_i^{miss}, \log B_{CP,i}^{miss} \right \}$, we can construct the "complete" data implied by that draw of the Markov chain:
\begin{align}\label{eq:var_til}
\begin{split}
&\log \tilde{Y}_i(1) = D_i \log Y_i + (1 - D_i) \log Y_i^{miss} \\
&\log \tilde{Y}_i(0) = D_i \log Y_i^{miss} + (1 - D_i) \log Y_i \\
&\log \tilde{B}_{CP,i} = D_i \log \bar{B} _{CP,i} + (1 - D_i) \log B_{CP,i}^{miss}.
\end{split}
\end{align}
\subsubsection{Drawing from posterior distribution}\label{sec:gibbs-posterior}
\noindent The last step is to draw new parameters from their full conditional distributions. Collect the parameters of the priors in $\theta_{\text{prior}}\equiv\left \{\mu_{\delta_k},A_{k},\alpha_{k},\beta_{k} \right \}_{k\in\left\{1,0,CP\right\}}$. For ease of notation, we stack all the ``complete'' data by the end of round $t$ in the following $N_t$-by-$1$ vectors: $\log \tilde{Y}_t(1)$, $\log \tilde{Y}_t(0)$, $\log \tilde{B}_{CP,t}$, $D_t$ and $\log b_t$. We also use the $N_t$-by-$P$ matrix $X_t$, whose $i^{\text{th}}$ row is the vector $X_i'$, and collect them all in the complete data set $\tilde{W}_t \equiv \left [\log \tilde{Y}_t(1), \log \tilde{Y}_t(0), \log \tilde{B}_{CP,t} \log b_t, D_t, X_t \right ]$. Finally, let the $(q-1)^{\text{th}}$ draw of the parameters be $\theta^{(q-1)}= \left [\delta_1'^{(q-1)},\delta_0'^{(q-1)},\delta_{CP}'^{(q-1)},\sigma_1^{2,(q-1)},\sigma_0^{2,(q-1)},\sigma_{CP}^{2,(q-1)} \right ]'$. Given the structure of the model, it then follows that the full conditional distributions of the parameters simplify in the following way:
\begin{align}\label{eq:full_cond_sigma}
\begin{split}
&\sigma_{CP}^{2,(q)} \Big \vert \theta^{(q-1)},\theta_{\text{prior}},\tilde{W}_t \: \overset{d}{=} \: \sigma_{CP}^{2,(q)} \Big \vert \log \tilde{B}_{CP,t}, X_t, \mu_{{\delta}_{CP}}, A_{CP}, \alpha_{CP}, \beta_{CP} \\
&\sigma_{1}^{2,(q)} \Big \vert \theta^{(q-1)},\theta_{\text{prior}},\tilde{W}_t \: \overset{d}{=} \: \sigma_{1}^{2,(q)} \Big \vert \log \tilde{Y}_{t}(1), X_t, \mu_{{\delta}_{1}}, A_{1}, \alpha_{1}, \beta_{1} \\
&\sigma_{0}^{2,(q)} \Big \vert \theta^{(q-1)},\theta_{\text{prior}},\tilde{W}_t \: \overset{d}{=} \: \sigma_{0}^{2,(q)} \Big \vert \log \tilde{Y}_{t}(0), X_t, \mu_{{\delta}_{0}}, A_{0}, \alpha_{0}, \beta_{0}
\end{split}
\end{align}
and, letting $\sigma^{2,(q)}\equiv \left [\sigma_1^{2,(q)},\sigma_0^{2,(q)},\sigma_{CP}^{2,(q)} \right ]'$,
\begin{align}\label{eq:full_cond_delta}
\begin{split}
&\delta_{CP}^{(q)} \Big \vert \sigma^{2,(q)}, \delta_1^{(q-1)}, \delta_0^{(q-1)}, \delta_{CP}^{(q-1)}, \theta_{\text{prior}},\tilde{W}_t \: \overset{d}{=} \: \delta_{CP}^{(q)} \Big \vert \sigma_{CP}^{2,(q)}, \log \tilde{B}_{CP,t}, X_t, \mu_{{\delta}_{CP}}, A_{CP} \\
&\delta_{1}^{(q)} \Big \vert \sigma^{2,(q)}, \delta_1^{(q-1)}, \delta_0^{(q-1)}, \delta_{CP}^{(q-1)}, \theta_{\text{prior}},\tilde{W}_t \: \overset{d}{=} \: \delta_{1}^{(q)} \Big \vert \sigma_{1}^{2,(q)}, \log \tilde{Y}_{t}(1), X_t, \mu_{{\delta}_{1}}, A_{1} \\
&\delta_{0}^{(q)} \Big \vert \sigma^{2,(q)}, \delta_1^{(q-1)}, \delta_0^{(q-1)}, \delta_{CP}^{(q-1)}, \theta_{\text{prior}},\tilde{W}_t \: \overset{d}{=} \: \delta_{0}^{(q)} \Big \vert \sigma_{0}^{2,(q)}, \log \tilde{Y}_{t}(0), X_t, \mu_{{\delta}_{0}}, A_{0}.
\end{split}
\end{align}
The specific forms of these full conditional distributions are presented in Appendix \ref{App:full-cond-gibbs}.
\subsubsection{Summary}
\noindent The full Gibbs sampling procedure is summarized below. If one wishes to allow for $\rho\neq 0$ the procedure has to be adjusted. We present this more general algorithm in Appendix \ref{App:Gibbs-with-rho}.
\RestyleAlgo{boxruled}
\SetAlgoNoLine
\LinesNotNumbered
\begin{algorithm}[ht]
\caption{Gibbs sampling\label{alg:gs}}
\stepcounter{AlgoLine}\ShowLn Set $\theta^{(0)}$ and $\theta_{\text{prior}}$. \break
\For{($q=1,...,Q$)}{
\stepcounter{AlgoLine}\ShowLn Draw $\left \{ \log Y_i^{miss,(q)}(1), \log Y_i^{miss,(q)}(0), \log B_{CP,i}^{miss,(q)} \right \}_{i=1}^{N_t}$ according to equations (\ref{eq:miss1})$-$(\ref{eq:miss4}). \\
\stepcounter{AlgoLine}\ShowLn Construct $\left \{ \log \tilde{Y}_i^{(q)}(1), \log \tilde{Y}_i^{(q)}(0), \log \tilde{B}_{CP,i}^{(q)} \right \}_{i=1}^{N_t}$ according to equation (\ref{eq:var_til}). \\
\stepcounter{AlgoLine}\ShowLn Draw $\theta^{(q)}$ according to equations (\ref{eq:full_cond_sigma})$-$(\ref{eq:full_cond_delta}). }
\end{algorithm}
\end{subsection}
\begin{subsection}{Adaptation to FPAs}
\noindent The procedure described above for SPAs can be used with minor adjustments to handle FPAs. Accommodating FPAs involves two substantive changes. The first is that we use the expected profit function for FPAs in (\ref{eq:exp_auc_fpa}) rather than the one for SPAs in equation (\ref{eq:exp_auc_spa}). The second change is a function of the different data the advertiser would have access to in an FPA. As noted before, under an FPA, $B_{CP}$ is always missing, which necessitates the normalization $\sigma_{CP}^2=1$. However, under Assumption \ref{ass:pv} $B_{CP}$ is conditionally independent from $Y(1)$ and $Y(0)$ given $x$, and therefore the augmentation step for $B_{CP}$ would not depend on observed outcomes given $x$. Therefore, instead of embedding the Bayesian approach to the Tobit model by \cite{chib1992} in the BITS algorithm, we use the Bayesian approach to a Probit model introduced by \cite{ac1993}.
More concretely, in addition to drawing the missing values $\left \{\log B_{CP,i} \right \}_{i:D_i=0}$ according to (\ref{eq:miss1}) with $\sigma_{CP}^2=1$, we also draw the missing values $\left \{\log B_{CP,i} \right \}_{i:D_i=1}$ according to,
\begin{align}\label{eq:miss10}
&\log B_{CP,i}^{miss} \Big\vert D_i=1, \log Y_i, \log \bar{B}_{CP,i} , \log b_i , X_i, \theta \overset{d}{=} \nonumber \\
& \log B_{CP,i}^{miss} \Big\vert D_i=1, \log b_i,X_i, \delta_{CP} \sim TN \left ( X_i'\delta_{CP}, 1,-\infty,\log b_i\right ).
\end{align}
Hence, instead of creating the variable $\log \tilde{B}_{CP,i}$ according to equation (\ref{eq:var_til}), we equal it to $\log B_{CP,i}^{miss}$ for all $i$. Other than these two modifications, the rest of the procedure remains unchanged.
\end{subsection}
\begin{subsection}{Estimating optimality probability of each arm and implied $\pmb{CATE}$s}
\noindent Once the draws from the posterior $\theta^{(q)}$ are obtained as above in round $t$, for each draw $\theta^{(q)}$, context $x$ and arm $b_{r_x}$, we can compute $\bar{\pi}(b_{r_x} \vert x,\theta^{(q)})$ via equation (\ref{eq:exp_auc_spa}) for SPAs or equation (\ref{eq:exp_auc_fpa}) for FPAs. The probability that an arm $b_{r_x}$ is best for context $x$ as of round $t$ is estimated by averaging over the $Q$ draws:
\begin{align}\label{eq:est_opt_prob}
\hat{\psi}_t(b_{r_x}\vert x)=\frac{1}{Q}\sum_{q=1}^Q \mathbbm{1} \left \{ \bar{\pi} \left (b_{r_x}\middle\vert x,\theta^{(q)} \right ) > \bar{\pi} \left (b_{r'_x}\middle \vert x,\theta^{(q)} \right )\text{ for all }r'_x\neq r_x \right \},
\end{align}
which is used then for allocation of traffic as outlined in Algorithm \ref{alg:ts}.
Given $\hat{\psi}_t(b_{r_x}\vert x)$, and leveraging Proposition \ref{prop:alignSPA}, the procedure implies that, for an SPA, $CATE(x)$ is
\begin{align}\label{eq:est_CATE_spa_dbn}
\begin{array}{ccc}
b_{r_{x}} & \text{with probability} & \hat{\psi}_{t}(b_{r_{x}}\vert x),
\end{array}
\end{align}
that is, we read off $CATE(x)$ as the label associated with what the procedure implies is the best bid-arm for that $x$. This leads to the following estimator of $CATE(x)$ in an SPA,
\begin{align}\label{eq:est_CATE_spa}
\widehat{CATE}_t(x)=\sum_{r_{x}=1}^{R_x}\hat{\psi}_{t}(b_{r_{x}}\vert x)\times b_{r_{x}}.
\end{align}
For an FPA, we can estimate the bid-adjustment in equation (\ref{eq:ATE-FPA}) for each bid-arm $b_{r_x}$ by averaging the inverse of the reversed hazard rate of $B_{CP}$ at that $x$ over the $q=1,...,Q$ draws of $\theta^{(q)}$. Leveraging Proposition \ref{prop:alignFPA}, the procedure implies that in an FPA,
\begin{align}\label{eq:est_CATE_fpa_dbn}
\hat{\chi}_t(b_{r_{x}})\equiv b_{r_{x}}+\frac{1}{Q}\sum_{q=1}^{Q}\frac{F_{CP}\left(b_{r_{x}}|x;\theta^{(q)}\right)}{f_{CP}\left(b_{r_{x}}|x;\theta^{(q)}\right)} \text{ with probability } \hat{\psi}_{t}(b_{r_{x}}\vert x),
\end{align}
that is, we read off $CATE(x)$ as the label associated with what the procedure implies is the best bid-arm for that $x$ plus the bid-adjustment term for that bid-arm. This leads to the following estimator of $CATE(x)$ for FPAs
\begin{align}\label{eq:est_CATE_fpa}
\widehat{CATE}_t(x)=\sum_{r_{x}=1}^{R_x}\hat{\psi}_{t}(b_{r_{x}}\vert x)\times\hat{\chi}_t(b_{r_{x}}).
\end{align}
In equations (\ref{eq:est_CATE_fpa_dbn}) and (\ref{eq:est_CATE_fpa}), $F_{CP}(\cdot|\cdot)$ and $f_{CP}(\cdot|\cdot)$ are, respectively, the conditional $CDF$ and $PDF$ of $B_{CP}$ on $x$ evaluated at the draws $\theta^{(q)}$, so that the bid-adjustment is the inverse of the reversed hazard rate of $B_{CP}$ averaged over the $Q$ draws from the updated posterior distribution. By assumption \ref{ass:rev_haz}, the inverse of the reversed hazard rate is monotonically increasing in the bid for each draw. The average over draws is therefore a weighted sum of monotonically increasing functions, which is also monotonically increasing. Therefore, we are guaranteed that the estimates $\hat{\chi}_t(b_{r_{x}})$ in equation (\ref{eq:est_CATE_fpa_dbn}) are monotonically increasing in $b_{r_{x}}$, so a well-defined mapping between the bid-value and $\widehat{CATE}_t(x)$ exists. For the specific parametrization chosen in equation (\ref{eq:normal}), $F_{CP}(\cdot|\cdot)$ and $f_{CP}(\cdot|\cdot)$ correspond to a lognormal distribution with parameters $X_i'\delta_{CP}$ and $\sigma_{CP}^2$, which makes it easy to compute the right-hand side of equation (\ref{eq:est_CATE_fpa}) at the end of each round.
\end{subsection}
\begin{subsection}{Stopping the experiment}\label{sec:stop}
\noindent The last piece required to complete the discussion of the experiment is a decision criterion for when to stop the experiment. Before outlining our suggested criterion, we note at the outset that an advantage of the adaptive cost control in the experiment is that the adverse profit consequences of allowing it to run without stopping are low: as the experiment proceeds, more traffic is assigned to the bid-arm that provides highest expected payoff from auction participation to the advertiser, so continuing to let the experiment run protects the advertiser's interests from continued auction participation. In this situation, the proposed experimental design can be viewed simply as an explore-exploit scheme for optimal bidding that also has an auxiliary benefit of delivering estimates of $CATE$s to the advertiser. In other use-cases, the proposed experimental design forms the basis of an explicit test the advertiser runs in order to measure $CATE$s, for which developing a principled approach to stoppage is useful (see \citealp{scott2015multi} and \citealp{GengLN20} for examples). With this latter perspective in mind, this section discusses stopping rules that terminate the experiment when the inference goal is achieved with reasonable precision.
The simplest stopping rule is to specify the total number of rounds the algorithm has to run through, in which case $c_t = t$ and $T$ is some exogenous threshold, which could map to a notion of time or budget available to run the experiment. A more nuanced stopping rule uses the data collected through the algorithm to inform the decision of when to stop the experiment. The stopping rule discussed below follows this approach. We motivate it first in a non-contextual setting to provide intuition, and then generalize it to the more complex contextual case.
\subsubsection{Non-contextual case}
\noindent Consider first a non-contextual MAB problem, that is, $x$ can take only one value. Thus, we omit $x$ for the remaining of this section to ease notation. The algorithm aims to identify the best bid-arm while minimizing the costs of experimentation. Therefore, we can leverage a stopping rule based on the confidence with which the optimal arm was found. More precisely, suppose we set $T=0.95$ and:
\begin{align}\label{eq:our_crit}
c_t = \max_r \hat{\psi}_t(b_r).
\end{align}
We can interpret this as a decision to stop when the posterior distribution of $\theta$ given $W_t$ leads us to believe that the arm with current highest probability of being the optimal arm is the true best arm with at least 95\% probability. By virtue of equation (\ref{eq:est_CATE_spa_dbn}) for SPAs, $\hat{\psi}_t(b_r)$ also represents the probability, based on the current posterior, with which we believe that bid-arm value $b_{r}$ is the true $ATE$. Analogously, by virtue of equation (\ref{eq:est_CATE_fpa_dbn}) for FPAs, $\hat{\psi}_t(b_r)$ represents the probability, based on the current posterior, with which we believe the adjusted bid-arm value $\hat{\chi}_t(b_{r})$ is the true $ATE$. Thus, we can also interpret the stopping rule in equation (\ref{eq:our_crit}) as a decision to stop when the posterior distribution of $\theta$ given $W_t$ leads us to believe that the $ATE$ value associated with the arm with current highest probability of being the true best arm is the true $ATE$ with at least 95\% probability.
This stopping rule has an attractive feature in that it has a well-defined interpretation in terms of Bayes factors, which are often used for Bayesian hypothesis testing. Let $\zeta_t(b_r)$ be the posterior odds ratio of arm $r$ being the optimal arm by the end of round $t$. Then,
\begin{align}\label{eq:por}
\zeta_t(b_r) &= \frac{\Pr_t(b_r\text { is the optimal bid})}{\Pr_t(b_r\text { is not the optimal bid})} \nonumber \\
&=\frac{\Pr_t(b_r\text { is the optimal bid})}{1-\Pr_t(b_r\text { is the optimal bid})} \nonumber \\
&=\frac{\psi_t(b_r)}{1-\psi_t(b_r)}.
\end{align}
Thus, $c_t$ can alternatively be constructed as $\max_r \hat{\zeta}_t(b_r)$, with corresponding threshold $T=19$, so that stopping is based off a threshold on the implied Bayes factor.\footnote{It is important to emphasize that following this stopping rule is not equivalent to conducting a sequential Bayesian hypothesis test. Such procedure would require us to establish a null hypothesis that one specific arm was the best and base the decision to stop solely on this arm's Bayes factor or posterior odds ratio. Instead, here we
remain agnostic as to which arm is the best, and base our decision to stop on which arm has strongest evidence in its favor.}
\subsubsection{Contextual case}
\noindent The contextual case is more complex because now there is not a single best arm, but $P$ best arms. Thus, a natural but conservative approach would be to require 95\% posterior probability over a list of $P$ arms as being the optimal ones. In this case, the threshold rule can be expressed by:
\begin{align}\label{eq:our_crit2}
c_t = \min_{x \in \mathbb{X}}\max_{r_x} \hat{\psi}_t(b_{r_x}\vert x),
\end{align}
while maintaining the requirement that $c_t>T=0.95$. Consequently, upon stoppage there would be at least 95\% posterior probability on the $CATE(x)$ value associated with the bid-arms for each $x$.
In some scenarios, the advertiser's inference objective may be to estimate and perform inference on the unconditional $ATE$. Under these circumstances, the stopping rule above is likely to be too stringent. The $ATE$ is the weighted average of $CATE(x)$ over the distribution of $x$. To achieve the goal of learning the weighted average of $CATE$s with a given level of precision, it is not necessary to learn every $CATE(x)$ with the same level of precision. We now present a slightly less demanding stoppage criterion that reflects this.
Recall that the context $x$ takes $P$ values, indexed by $p=1,...,P$, and that for each value $x_{p}$ we consider $R_{x_{p}}$ different bid-arms. Consequently, considering only the values from the grids the $ATE$ can take at most $R \equiv \prod_{p=1}^P R_{x_{p}} $ values because $ATE = \sum_{p=1}^P F_x(x_{p})\times CATE(x_{p})$ and because each $CATE(x_p)$ can take $R_{x_p}$ values. Consider the $R$ values that $ATE$ can take and select a grid composed of the $\Upsilon$ unique values among these $R$, which we denote $o_{\upsilon}$ for $\upsilon=1,...,\Upsilon$. An alternative criterion is to stop the experiment when the posterior at the end of a round implies with at least 95\% probability that the $ATE$ is equal to one of the $o_{\upsilon}$ values in this grid.
To make this criterion precise, consider, for each $o_{\upsilon}$, a sequence $s$ over the $P$ contexts such that the implied estimate of $ATE$ from these values equals $o_{\upsilon}$. In other words, $s$ is a sequence of values $\left \{ b^s_{r_{x_{p}}} \right \}_{p=1,...,P}$, each taken from one of the $R_{x_p}$ values in the grid of each context $x_p$, such that $o_{\upsilon} = \sum_{p=1}^P F_x(x_{p})\times b^s_{r_{x_{p}}}$ for SPAs and $o_{\upsilon} = \sum_{p=1}^P F_x(x_{p})\times \hat{\chi}_t \left (b^s_{r_{x_{p}}} \right )$ for FPAs. Let $S_{\upsilon}$ be the total number of such sequences. Then the alternative stopping criterion is given by:
\begin{align}\label{eq:our_crit3}
c_{t}=\max_{\upsilon\in\{1,...,\Upsilon\}} \left \{ \sum_{s=1}^{S_{\upsilon}} \sum_{p=1}^P F_x(x_p) \times \hat{\psi}_t \left ( b^s_{r_{x_{p}}} \middle \vert x_p \right ) \right \}
\end{align}
\noindent and we stop when $c_t>T=0.95$. Notice that while this decision rule depends on the confidence with which we believe to have found the true $ATE$ as implied by the posterior distribution of $\theta$ given $W_t$, traffic is still allocated to each arm according to (\ref{eq:est_opt_prob}). Hence, the decision to stop the experiment is aligned with the advertiser's inference goal, while the way it performs randomization is aligned with her economic goal.
Notice that this stopping criterion presupposes that the distribution from which contexts are drawn, $F_x(\cdot)$, is known to the researcher. When this is the case, the grid of values $\left \{ o_{\upsilon} \right \}_{\upsilon=1,...,\Upsilon}$ is fixed for SPAs, but it changes for FPAs because the values $\hat{\chi}_t \left (b_{r_x} \right )$ change over the rounds. When this is not the case, one could replace it with empirical frequencies estimated using data collected via the algorithm, in which case the grids will vary across rounds for both auction formats. While we expect the stopping rule given in (\ref{eq:our_crit3}) to shorten the duration of the experiment when compared to the one given in (\ref{eq:our_crit2}), we found in simulations that the difference between these two rules is minimal.\footnote{It is important to mention that the statistical implications of data-driven stoppage in sequential experiments is still being debated in the literature. Even though data-based stopping rules are known to interfere with frequentist inference, which motivated the development of new methods to explicitly account for this interference both for non-adaptive (\citealp{johari2015}) and adaptive (\citealp{yrjw2017, jj2018, Juetal2019}) data collection procedures, Bayesian inference has historically been viewed as \textit{immune} to optional stopping rules (\citealp{lindley1957statistical, edwards1963bayesian, savage1972, good1991}). Nevertheless, a recent debate has emerged concerning the effects of optional stopping on frequentist properties and interpretation of Bayes estimators (\citealp{yuet2014, sh2014, rouder, dienes2016, deng2016continuous, schon2017, wagen2019,tkvr2019, dhg2018,rouder2,hdhg2018}). We do not attempt to resolve this debate in this paper. In several simulations we ran, we found that the practical impact of optional stopping was minimal in our setting.}
\end{subsection}
\begin{subsection}{Practical considerations and extensions}\label{sec:discuss}
\noindent We conclude the experimental design discussing some practical considerations that arise in implementation and ways in which the design can be extended to accommodate variations in the experimentation environment and advertiser goals.
\subsubsection{Regret minimization versus best-arm identification}
\noindent We implement BITS under a regret minimization framework based on the viewpoint that the advertiser seeks to maximize her payoffs from auction participation during the experiment. We could alternatively cast the problem as one of pure best-arm identification as studied by \cite{Bubecketal2009}, for example. In the best-arm identification formulation, the problem is cast in terms of pure exploration, so the role of adaptive experimentation is to obtain information efficiently before committing to a final decision involving the best-arm identified with that information.\footnote{\cite{russo16} provides an adaptation of TS to best-arm identification. For an example of a study that adopts this approach to identify an optimal treatment assignment policy, see \cite{ks2019}.} To leverage Propositions \ref{prop:alignSPA} and \ref{prop:alignFPA}, what we need is a MAB framework to recover the arm with highest expected reward, so the core idea behind our proposed approach ports in a straightforward way to this alternative formulation of the experimental objective.
\subsubsection{Parametric assumptions and alternative algorithms}
\noindent More flexible parametric specifications can be used instead of (\ref{eq:normal}) and (\ref{eq:priors}).\footnote{For a discussion of how more flexible parameterizations can be used for Bayesian estimation of treatment effects, see, for example, \cite{heckman2014treatment}.} A more flexible distribution may be especially desirable for FPAs due to the explicit dependence of the $CATE$s on the distribution of $B_{CP}$ via equation (\ref{eq:ATE-FPA}). The cost of more flexibility is that the researcher has to employ more complex MCMC methods, which may be slower than the Gibbs sampling algorithm presented above. If the updating becomes slow, the induced latency may form an impediment to implementation in practical ad-tech settings. This is because conditional conjugacy is likely to fail under alternative parametric specifications. Furthermore, any algorithm with convergence guarantees could in theory be used instead of the proposed BITS algorithm if the practitioner is not comfortable with making specific distributional assumptions, which may not be required for these other methods.
\subsubsection{Obtaining draws from posterior distribution}
\noindent The method we presented requires the researcher to employ Gibbs sampling in each round, which becomes slower as the number of rounds increases. This is because the it requires the posterior to be updated conditioning on the data collected from the beginning of the experiment. The size of these data grows as we increase the rounds. If the procedure becomes too slow, Sequential Monte Carlo (SMC) or particle filtering methods could instead be used to speed up the sampling.\footnote{For an application of SMC methods to MAB problems, see \cite{cb2013}.} In SMC, one updates conditioning on the data collected in the most recent round, rather than from the beginning of the experiment. SMC is also attractive if the practitioner chooses to use more flexible parametric specifications.
\subsubsection{Using additional data on competing bids}
\noindent It is straightforward to adapt the BITS procedure if different types of auction data are made available. For SPAs, we have assumed the advertiser only observes $B_{CP}$ when she effectively has to pay this amount; otherwise, all she knows is that it is bounded below by $b$. On the other hand, for FPAs the advertiser never observes $B_{CP}$. These assumptions characterize the most stringent data limitations in these auction environments.
In some scenarios, the data limitations may be less stringent. For instance, if the transaction price from the auction is made public by the AdX to auction participants, the advertiser can possibly obtain a more precise lower bound on $B_{CP}$ whenever the transaction price is larger than $b$ in SPAs. This yields a new definition of $\bar{B}_{CP}$. Accommodating this does not require any modification to the BITS procedure, but does require us to assume that the transaction price is also independent from the potential outcomes conditional on $x$. For FPAs, disclosure of the transaction price would imply that the advertiser would observe $B_{CP}$ whenever she lost the auction, which would give rise to an analogous procedure as the one adopted above for SPAs.
Finally, in the event $B_{CP}$ itself is made public by the AdX, the algorithm simplifies further since the censoring problem vanishes. Hence, the practitioner can update the posterior distribution over the parameters $\delta_{CP}$ and $\sigma_{CP}^2$ analytically, without the need to use the Gibbs sampling procedures because of the exact conjugacy implied by (\ref{eq:normal}) and (\ref{eq:priors}).
\subsubsection{Budget constraints}
\noindent The current formulation does not explicitly incorporate budget constraints, which can be present and relevant in practice for running experiments. Nevertheless, budget constraints are implicitly considered in the design by the use of a stopping rule for the experiment as discussed in Section \ref{sec:stop}. While there are more general RL tools to perform bid optimization in the presence of budget constraints, such as \cite{cai2017real}, these methods do not address causal inference tasks. Part of the complication is that the optimal bidding policy becomes a function both of $x$ and of the remaining budget, and linking it to $CATE$s becomes non-trivial. Formal incorporation of budget constraints would therefore need extending the algorithm beyond its current scope, and is left as a topic for future research.
\subsubsection{Choosing priors to address the cold start problem}\label{sec:prior_cold}
\noindent While priors always play an important role in Bayesian inference, they can become even more important in the context of experimentation as a way to deal with a ``cold start'' problem. Well-informed priors might situate the algorithm at a good starting point, shortening the duration of the experiment and, consequently, decreasing its costs. On the other hand, poorly specified priors might have the opposite effect and become inferior even to diffuse, non-informative priors. We discuss briefly how historical data that may be available to the advertiser can be used to inform the choice of the prior parameters. We assume the experimenter has access to a historical data set $W_n = \left \{ b_i,X_i,D_i,Y_i, \bar{B}_{CP,i} \right \}_{i=1}^n$ for SPAs and $W_n = \left \{ b_i,X_i,D_i,Y_i \right \}_{i=1}^n$ for FPAs.
To leverage the historical data, we can equate the means and variances of the prior distributions to the approximate means and variances of estimators of $\delta_1$, $\delta_0$, $\delta_{CP}$, $\sigma_1^2$, $\sigma_0^2$ and $\sigma_{CP}^2$, where the last estimator is only required for SPAs. It is straightforward to develop a maximum likelihood estimator (MLE) for the prior parameters, which, in this case, are $\sqrt{n}-$consistent and asymptotically normal. The MLE sets, for $k \in \left \{ 1,0,CP \right \} $,
\begin{align}
\begin{split}
&\frac{\alpha_k}{\beta_k}=\hat{\sigma}_k^{-2} \\
&\frac{\alpha_k}{\beta_k^2}=\frac{1}{n}\hat{Avar} \left [ \sqrt{n} \left ( \hat{\sigma}_k^{-2} - \sigma_k^{-2} \right ) \right ]=\frac{\hat{\sigma}_k^8}{n}\hat{Avar} \left [ \sqrt{n} \left ( \hat{\sigma}_k^{2} - \sigma_k^{2} \right ) \right ] \\
&\mu_{\delta_k} = \hat{\delta}_k \\
& A_k = n \hat{\sigma}_k^2 \left \{ \hat{Avar} \left [ \sqrt{n} \left ( \hat{\delta}_k - \delta_k \right ) \right ] \right \}^{-1}.
\end{split}
\end{align}
Notice that for the parameters of the Gamma distributions this is equivalent to:
\begin{align}
\begin{split}
&\alpha_k = n \hat{\sigma}_k^{-4} \left \{ \hat{Avar} \left [ \sqrt{n} \left ( \hat{\sigma}_k^{-2} - \sigma_k^{-2} \right ) \right ] \right \}^{-1} = n \hat{\sigma}_k^4 \left \{ \hat{Avar} \left [ \sqrt{n} \left ( \hat{\sigma}_k^2 - \sigma_k^2 \right ) \right ] \right \}^{-1} \\
&\beta_k = n \hat{\sigma}_k^{-2} \left \{ \hat{Avar} \left [ \sqrt{n} \left ( \hat{\sigma}_k^{-2} - \sigma_k^{-2} \right ) \right ] \right \}^{-1} = n \hat{\sigma}_k^6 \left \{ \hat{Avar} \left [ \sqrt{n} \left ( \hat{\sigma}_k^2 - \sigma_k^2 \right ) \right ] \right \}^{-1}.
\end{split}
\end{align}
For the purposes of this estimation we assume that $D_i \independent Y_i(1), Y_i(0) \vert X_i$. Since we maintain Assumption \ref{ass:pv}, the only potential source of dependence between $D_i$ and the potential outcomes is $b_i$. The validity of this assumption is not a concern when the algorithm is implemented for actual experimentation since the bids $b_i$ are under the control of the experimenter. If the historical data come from an experiment in which bids were randomized, this condition is also satisfied. If the assumption is violated in the historical data, the resulting priors have some bias; however, the experimentation algorithm will still consistently recover the true $CATE$s. Further details about the estimators leveraging historical data and their computation are relegated to Appendix \ref{App:max-like-historical}.
\end{subsection}
\end{section}
\begin{section}{Simulation evaluations of proposed approach}\label{sec:exper}
\noindent This section provides simulation results documenting the performance of the proposed approach. We first discuss the results for SPAs, and then for FPAs. Within each, we first show that the algorithm works as intended in a non-contextual setup, and then demonstrate performance under the more involved contextual setup. Performance on the economic goal is assessed using cumulative pseudo-regret as the performance metric. Performance on the inference goal is assessed using the Mean Squared Error (\textit{MSE}) as the performance metric. The algorithm is compared to several alternative experimental designs for each scenario. Overall, the results show that the proposed approach performs well and is superior to the other considered alternatives.
\begin{subsection}{SPAs}
\subsubsection{Non-contextual case}
\noindent We begin by considering the non-contextual case, in which $P=1$ so we can ignore $x$. Our goal is to show that the BITS algorithm works in practice in recovering the true $ATE$. Additionally, we assess its performance in achieving both the inference and economic goals relative to alternative methods.
\paragraph{DGP, simulation details and grids of bids} \hfill
\noindent For these simulations, we assume that:
\begin{align}\label{eq:dgp_sim_nc}
\begin{bmatrix} \log Y_i(1) \\ \log Y_i(0) \\ \log B_{CP,i} \end{bmatrix} \overset{iid}{\sim} N \left ( \begin{bmatrix} 0.809 \\ 0.22 \\ 0.4 \end{bmatrix}, \begin{bmatrix} 0.49 & 0 & 0 \\ 0 & 0.81 & 0 \\ 0 & 0 & 0.25 \end{bmatrix} \right ),
\end{align}
where the value of $\delta_1$ is chosen so that $ATE=1$. These values are chosen for the sake of illustration.
We simulate 1,000 different epochs. Each epoch has $T=100$ rounds, each of which with 50 new observations, so that $n_t=50$ for $1 \leq t \leq 100$. We keep these values fixed and change the grid of bids we use to assess how the performance of our algorithm changes. In particular, we consider three different grids:
\begin{enumerate}
\item 3 arms: $b \in \{0.6,1.0,1.5 \}$
\item 5 arms: $b \in \{0.6,0.8,1.0,1.25,1.5 \}$
\item 10 arms: $b \in \{0.6,0.7,0.8,0.9,1.0,1.1,1.2,1.3,1.4,1.5 \}$
\end{enumerate}
Since the width of bids and the number of observations are kept fixed while the number of arms increases, we expect the performance of BITS to deteriorate with finer grids.
\paragraph{Approaches under consideration} \hfill
\noindent We consider the following experimental approaches to estimate the $ATE$:
\begin{enumerate}
\item \textit{A/B test (A/B)}: to randomize treatment, the A/B test simply randomizes with equal probability the bid placed from the same grid of bids used by $BITS$. In other words, the A/B test is a design that implements equal allocation of experimental traffic to various arms non-adaptively. Once the data are collected, the $ATE$ is estimated by running a regression of $Y$ on $D$ using the experimental sample. Under pure bid randomization, the estimated slope coefficient from this regression is consistent for the $ATE$ due to Assumption \ref{ass:pv}.
\item \textit{Explore-then-commit (ETC)}: this approach proceeds as an A/B test for the first half of the experiment, that is, for the first 50 rounds; it then collects all data from these 50 rounds and runs a regression of $Y$ on $D$ as above to estimate the $ATE$. In the second half of the experiment, this approach places this estimate as the bid for arriving impressions, thus committing to what was learned in the first part of the experiment to exploitation of that information in the latter part.
\item \textit{Off-the-shelf Thompson Sampling (TS)}: this is the basic implementation for the TS algorithm to this setting. It assumes that the rewards obtained from each arm are independent draws from arm-specific normal distributions. It then updates the mean and variance of each such distribution in the usual way employing only observations obtained from that arm. We use the conjugate normal-gamma priors with uninformative parameters for all arms. To estimate the probabilities that each arm is optimal, we take 1,000 draws from each normal distribution and use Monte Carlo integration.
\item \textit{Bidding Thompson Sampling (BITS)}: to implement our algorithm, we make use of non-informative priors for all parameters, that is, we set $\alpha_k=\beta_k=\mu_{\delta_k}=A_k=0$ for $k \in \{1,0,CP\}$. Every time we run Gibbs sampling, we set the initial values to $\delta_1^{(0)}=\delta_0^{(0)}=\delta_{CP}^{(0)}=0$ and $\sigma_1^{2,(0)}=\sigma_0^{2,(0)}=\sigma_{CP}^{2,(0)}=1$. We take $Q=$ 1,000 draws, drop the first half and use only the multiples of 10 (510, 520, etc.) to estimate the optimality probabilities to mitigate possible dependence between the draws.
\end{enumerate}
\paragraph{Criteria of comparison}\hfill
\noindent To compare the performance of the aforementioned methods we consider two metrics. The first, \textit{cumulative pseudo-regret}, represents the economic goal. Following \cite{BubekCesa-Bianchi2012}, the pseudo-regret from method $\iota$ on round $t$ is given by:
\begin{align}\label{eq:psr}
\text{Pseudo-regret}_{\iota,t}=\bar{\pi}(b^*)-\sum_{r=1}^R \text{Pr}_{\iota,t} \left (b_r\text{ is pulled} \right )\times \bar{\pi}(b_r),
\end{align}
and therefore cumulative pseudo-regret at round $t$ is given by:
\begin{align}\label{eq:cpsr}
\text{Cumulative pseudo-regret}_{\iota,t}&=\sum_{\tau=1}^t\text{Pseudo-regret}_{\iota,\tau} \nonumber \\
&=t\times\bar{\pi}(b^*)-\sum_{\tau=1}^t\sum_{r=1}^R \text{Pr}_{\iota,\tau} \left (b_r\text{ is pulled} \right )\times \bar{\pi}(b_r).
\end{align}
Notice that for the A/B test it follows that $\text{Pr}_{\iota,t} \left (b_r\text{ is pulled} \right )=1/R$ for all $t$, and therefore it exhibits constant pseudo-regret and linear cumulative pseudo-regret. Moreover, we note that for ETC the bid placed when $t>50$ does not belong to the original grid, but is rather the OLS estimate from the regression of $Y$ on $D$ using the data gathered on the first 50 rounds. Finally, we note that for both TS and BITS the probability $\text{Pr}_{\iota,t}(\cdot)$ is not known exactly; instead, it is estimated via Monte Carlo integration.
The second metric, \textit{MSE}, represents the inference goal. We define the estimated \textit{MSE} of the $ATE$ of method $\iota$ as:
\begin{align}\label{eq:mse_ate}
\widehat{MSE}_{\iota,ATE}=\frac{1}{E}\sum_{e=1}^E \left ( \widehat{ATE}_{\iota,e} - ATE \right )^2,
\end{align}
where $e$ indexes the epoch and $E$ is the total number of epochs, which, in our simulations, is 1,000. The term $\widehat{ATE}_{\iota,e}$ is the estimate of $ATE$ obtained at the end of epoch $e$ by method $\iota$. For the A/B test this estimate corresponds to the OLS estimate of the slope coefficient of the regression of $Y$ on $D$ using data from all 100 rounds, while for ETC it corresponds to the same object using data from the first 50 rounds. In turn, for TS and BITS we use the final optimality probabilities to average over all bids in the grid, that is, we have that $\widehat{ATE}_{\iota,e}=\sum_{r=1}^R \hat{\psi}_{\iota,T,e} \left (b_r \right )\times b_r$.
\paragraph{Results}\hfill
\noindent We now present the results from our simulation exercises. Before comparing BITS to the other methods, we first demonstrate it succeeds in recovering the optimal bid and, therefore, the $ATE$. To do so, for each round $t$ we create a boxplot of $\hat{\psi}_t(b^*)=\hat{\Pr}_t(b^*\text{ is optimal})$ across the 1,000 epochs, where the edges of the box correspond to the interquartile range and the darker stripe corresponds to the median. For the algorithm to work we require that $\lim_{t\to+\infty} \hat{\psi}_t(b^*)=1$. Figure \ref{fig:opt_prob_grids} displays results using each of the three grids of bids described above.
Looking at the figure, we see that the BITS algorithm converges to the true optimal bid as data accumulate, which is most easily seen in Figure \ref{fig:opt_prob_3arms} due to the simplicity of its design since it considered only three arms. Unsurprisingly, as the number of arms increases the speed of convergence of the algorithm diminishes, as seen in Figures \ref{fig:opt_prob_5arms} and especially \ref{fig:opt_prob_10arms}. Notice that this is not only a function of the number of arms, but also because we kept its width fixed. Making the grid finer implies that the arm-specific expected rewards are closer to one another, so that the algorithm requires more data to precisely estimate and distinguish them.
Having shown that the BITS algorithm consistently recovers the optimal bid, we now proceed to compare it to the aforementioned alternative methods. We begin by displaying the evolution of cumulative pseudo-regret averaged over the 1,000 epochs. Figure \ref{fig:cps_reg_grids} displays the results for each method separately for each grid of bids.
Looking at the figure, we see that the BITS algorithm dominates the alternative methods in terms of average cumulative pseudo-regret. This is because BITS incorporates regret minimization as an explicit goal and fully exploits the structure of the data. As explained above, cumulative pseudo-regret is linear for an A/B test. For ETC, it increases much more slowly after the first 50 rounds. This is because 50 rounds worth of data, which in this cases corresponds to 2,500 observations, is enough to obtain a precise estimate of $b^*=ATE$. It is interesting to note that the off-the-shelf TS algorithm converges much more slowly than BITS, illustrating the consequence of ignoring the structure of the data. Under our specification, the off-the-shelf TS algorithm does not even overcome the simple ETC policy in terms of average cumulative pseudo-regret. Finally, notice that adding more bids to the grid while keeping its width fixed has a beneficial effect for minimizing regret. This is because while having more bids slows down the convergence of the algorithm, it considers bids whose expected rewards are closer to the optimal one.
\begin{figure}
\centering
\begin{subfloat}[3 arms\label{fig:opt_prob_3arms}]
{\includegraphics[width=0.45\textwidth]{opt_prob_3arms.pdf}}
\end{subfloat}\\
\begin{subfloat}[5 arms\label{fig:opt_prob_5arms}]
{\includegraphics[width=0.45\textwidth]{opt_prob_5arms.pdf}}
\end{subfloat}\\
\begin{subfloat}[10 arms\label{fig:opt_prob_10arms}]
{\includegraphics[width=0.45\textwidth]{opt_prob_10arms.pdf}}
\end{subfloat}
\caption{Convergence of BITS algorithm for SPAs}
\label{fig:opt_prob_grids}%
\end{figure}
\begin{figure}
\centering
\begin{subfloat}[3 arms\label{fig:cps_reg_3arms}]
{\includegraphics[width=0.45\textwidth]{cps_reg_3arms.pdf}}
\end{subfloat}\\
\begin{subfloat}[5 arms\label{fig:cps_reg_5arms}]
{\includegraphics[width=0.45\textwidth]{cps_reg_5arms.pdf}}
\end{subfloat}\\
\begin{subfloat}[10 arms\label{fig:cps_reg_10arms}]
{\includegraphics[width=0.45\textwidth]{cps_reg_10arms.pdf}}
\end{subfloat}
\caption{Average cumulative pseudo-regret per method for SPAs}
\label{fig:cps_reg_grids}%
\end{figure}
To conclude this analysis, we now present the results for $\widehat{MSE}_{\iota,ATE}$ as defined above. Once again, we compute different results for each one of the three grids of bids we consider. Results are displayed in Table \ref{tab:mse}.
\begin{table}[h]
\begin{centering}
\begin{tabular}{c|ccc}
\hline \hline
Estimation method ($\iota$) & 3 arms & 5 arms & 10 arms \\
\hline \\
A/B & 0.007 & 0.007 & 0.007 \\ \\
ETC & 0.012 & 0.013 & 0.016 \\ \\
TS & 0.031 & 0.023 & 0.019 \\ \\
BITS & 0.002 & 0.005 & 0.006 \\ \\
\hline \hline
\end{tabular}
\par\end{centering}
\caption{$\widehat{MSE}_{\iota,ATE}$ for each estimation method for SPAs}
\label{tab:mse}
\end{table}
A few patterns become apparent. First, the ordering of performance does not change across the grids: BITS is the best at minimizing $\widehat{MSE}_{\iota,ATE}$, followed by A/B, ETC and then TS. Second, the performance of A/B does not seem to be altered by the inclusion of additional arms, while ETC is only slightly affected.
The effects of including additional arms is more interesting when we consider the adaptive methods, TS and BITS. Since the width of the arms is fixed, these results reflect the tradeoff between two elements: the addition of more arms with expected rewards that are closer to the optimal and the speed of convergence. For a slow algorithm such as TS, while the inclusion of additional arms further slows it down, it also forces the algorithm to consider options that are closer to the optimal one. As a result, at the end of 100 rounds the algorithm puts relatively more mass on closer-to-optimal arms, which diminishes its $\widehat{MSE}_{\iota,ATE}$. On the other hand, the end result is reversed for a fast algorithm such as BITS: the decrease in the speed of convergence offsets the consideration of more near-to-optimal alternatives, increasing the $\widehat{MSE}_{\iota,ATE}$.
\paragraph{Summary} \hfill
\noindent The results displayed above showcase the qualities of the BITS algorithm for SPAs. As expected, it provides a consistent approach to recovering the optimal bid as shown in Figure \ref{fig:opt_prob_grids} and it dominates typical alternative approaches to estimate the $ATE$ in terms of regret as shown in Figure \ref{fig:cps_reg_grids}. In addition, we also document that the algorithm performs well on the inference goal via the $\widehat{MSE}_{\iota,ATE}$, which is illustrated in Table \ref{tab:mse}. While these results are a function of the specification we chose for the simulation exercises, they indicate that BITS is an attractive option to achieve the advertiser's dual objectives.
\begin{subsubsection}{Contextual case}
\noindent Having established the validity of BITS for the non-contextual case, we now proceed to analyze its performance for the contextual case. For brevity, we do not compare BITS to alternative methods here (the qualitative nature of the comparisons remain unaltered). Instead, we simply document its performance in recovering multiple bids, and thus $CATE$s, at the same time.
\paragraph{DGP, simulation details and grids of bids}\hfill
\noindent For these simulations, we assume that:
\begin{align}\label{eq:dgp_sim_ctxt}
\begin{bmatrix}\log Y_i(1) \\ \log Y_i(0) \\ \log B_{CP,i} \end{bmatrix} \Big\vert X_i \overset{iid}{\sim} N \left ( \begin{bmatrix} 0.81 & 1.04 & 1.25 & 1.43 & 1.57 \\ 0.20 & 0.31 & 0.45 & 0.59 & 0.70 \\ 0.25 & 0.33 & 0.40 & 0.47 & 0.55 \end{bmatrix} X_i, \begin{bmatrix} 0.36 & 0 & 0 \\ 0 & 0.64 & 0 \\ 0 & 0 & 0.25 \end{bmatrix} \right ).
\end{align}
Hence, we have $P=5$ contexts. The vector $\delta_1$ is chosen so that the $ATE$s equals 1.00, 1.50, 2.00, 2.50 and 3.00. We assume that the contexts are equiprobable, so that the unconditional $ATE$ is 2.00.
We simulate 1,000 different epochs. Each epoch has $T=100$ rounds, each of which with 100 new observations, so that $n_t=100$ for $1 \leq t \leq 100$, divided equally across the 5 contexts. Each context has its specific grid of bids under consideration. In particular, we consider the following grids:
\begin{enumerate}
\item Context 1: $b \in \{0.25,0.50,1.00,1.25,1.50 \}$
\item Context 2: $b \in \{0.50,1.00,1.50,2.00,2.50 \}$
\item Context 3: $b \in \{1.00,1.50,2.00,2.50,3.00 \}$
\item Context 4: $b \in \{1.00,1.75,2.50,3.25,4.00 \}$
\item Context 5: $b \in \{1.00,2.00,3.00,4.00,5.00 \}$
\end{enumerate}
\paragraph{Results}\hfill
\noindent To demonstrate the validity of BITS to the contextual case we display the evolution of two objects. First, to show that the BITS consistently recovers $b^*(x)$ for all $x$, we plot the interquartiles and median across the 1,000 epochs of the lowest estimated optimality probability of the true optimal arm for each round across the different contexts, that is, $\min_{x\in\mathbb{X}} \hat{\psi}_t \left ( b^*(x) \middle \vert x \right )$. The convergence of this object to 1 implies that all best arms are concurrently recovered. Figure \ref{fig:main_opt_prob} shows that this is achieved.
Second, we show that the unconditional pseudo-regret per round, averaged over the different contexts, converges to 0. This is a consequence of BITS identifying the best arm for all contexts concurrently. Results are displayed in Figure \ref{fig:main_opt_reg}.
\begin{figure}
\centering
\begin{subfloat}[$\min_{x\in\mathbb{X}} \hat{\psi}_t \left (b^*(x)\middle \vert x \right )$\label{fig:main_opt_prob}]
{\includegraphics[width=0.7\textwidth]{main_opt_prob.pdf}}
\end{subfloat} \\
\begin{subfloat}[Unconditional pseudo-regret per round\label{fig:main_opt_reg}]
{\includegraphics[width=0.7\textwidth]{main_opt_reg.pdf}}
\end{subfloat}
\caption{Convergence of BITS algorithm for contextual case for SPAs}
\label{fig:main_res_ctxt}%
\end{figure}
\end{subsubsection}
\end{subsection}
\begin{subsection}{FPAs}
\subsubsection{Non-contextual case}
\noindent We again begin with the non-contextual case. The structure of this exercise is very similar to the one implemented for SPAs.
\paragraph{DGP, simulation details and grids of bids}\hfill
\noindent For these simulations, we assume that:
\begin{align}\label{eq:dgp_sim_fpa_nc}
\begin{bmatrix} \log Y_i(1) \\ \log Y_i(0) \\ \log B_{CP,i} \end{bmatrix} \overset{iid}{\sim} N \left ( \begin{bmatrix} 0.736 \\ 0.22 \\ 0.481 \end{bmatrix}, \begin{bmatrix} 0.49 & 0 & 0 \\ 0 & 0.81 & 0 \\ 0 & 0 & 1 \end{bmatrix} \right ),
\end{align}
where the value of $\delta_1$ is chosen so that $ATE=0.8$ and the value of $\delta_{CP}$ is chosen so that $b^*=0.5$. Again, these values are chosen for the sake of illustration.
We simulate 1,000 different epochs. Each epoch has $T=100$ rounds, each of which with 50 new observations, so that $n_t=50$ for $1 \leq t \leq 50$. We keep these values fixed and change the grid of bids we use to assess how the performance of our algorithm changes. In particular, we consider three different grids:
\begin{enumerate}
\item 3 arms: $b \in \{0.1,0.5,1.0 \}$
\item 5 arms: $b \in \{0.1,0.3,0.5,0.75,1.0 \}$
\item 10 arms: $b \in \{0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0 \}$
\end{enumerate}
As before, since the width of bids and the number of observations are kept fixed while the number of arms increases, we expect the performance of BITS to deteriorate as the number of arms increases.
\paragraph{Results}\hfill
\noindent To demonstrate that the BITS algorithm recovers the optimal bid for FPAs we replicate Figure \ref{fig:opt_prob_grids} in Figure \ref{fig:opt_prob_fpa_grids}. Once again, for the algorithm to work we require that $\lim_{t\to+\infty} \hat{\psi}_t(b^*)=1$, which is indeed obtained as seen from the figure.
We now compare the performance of the BITS algorithm to that of A/B, ETC and TS using the same criteria as above. We start by replicating Figure \ref{fig:cps_reg_grids}. Results are displayed below in Figure \ref{fig:cps_reg_grids_fpa}. They are qualitatively identical to the previous ones in that BITS performs best compared to the other methods. One qualitative difference from the SPA case is noticeable: now the off-the-shelf TS outperforms the ETC policy, which uses the collected data to choose the best bid as the one whose observations yield highest sample profit, at least for the duration of this experiment. We note that the gap between these two methods diminishes as more arms are added to the grid.
As above, we also compare these methods in terms of $\hat{MSE}$. However, we cannot perform this comparison with the off-the-shelf TS. Since this method does not exploit the link to auction payoffs, Proposition \ref{prop:alignFPA} no longer holds, so estimating the $ATE$ leveraging this relationship is not viable. In turn, the estimator of $ATE$ for BITS is $\widehat{ATE}_{\iota,e}=\sum_{r=1}^R \hat{\psi}_{\iota,T,e} \left (b_r \right )\times \hat{\chi}_T (b_r)$. Results comparing against the other methods are displayed in Table \ref{tab:mse_fpa}.
Unlike before, BITS is now dominated by A/B and ETC on the inference goal. This is not surprising. Since Proposition \ref{prop:alignSPA} no longer holds, the pursuit of the inference and the economic goals are not perfectly aligned for FPAs. Hence, pursuit of the economic goal (in which BITS beat the other methods) may come at the cost of performance of the economic goal. Another way to see the reduced performance on the inference goal in FPAs compared to SPAs is that in FPAs we have two sources of uncertainty in estimating the $ATE$, which lead to a higher $\widehat{MSE}$: the uncertainty over the optimal arm, as before in SPAs, plus the uncertainty arising from the MCMC draws used to construct the $ATE$. Nevertheless, we note that the $\widehat{MSE}$s are still low, demonstrating that BITS can perform well in estimating the $ATE$ for FPAs.
\begin{figure}
\centering
\begin{subfloat}[3 arms\label{fig:opt_prob_fpa_3arms}]
{\includegraphics[width=0.45\textwidth]{opt_prob_fpa_3arms.pdf}}
\end{subfloat}\\
\begin{subfloat}[5 arms\label{fig:opt_prob_fpa_5arms}]
{\includegraphics[width=0.45\textwidth]{opt_prob_fpa_5arms.pdf}}
\end{subfloat}\\
\begin{subfloat}[10 arms\label{fig:opt_prob_fpa_10arms}]
{\includegraphics[width=0.45\textwidth]{opt_prob_fpa_10arms.pdf}}
\end{subfloat}
\caption{Convergence of BITS algorithm for FPAs}
\label{fig:opt_prob_fpa_grids}%
\end{figure}
\begin{figure}
\centering
\begin{subfloat}[3 arms\label{fig:cps_reg_fpa_3arms}]
{\includegraphics[width=0.45\textwidth]{cps_reg_fpa_3arms.pdf}}
\end{subfloat}\\
\begin{subfloat}[5 arms\label{fig:cps_reg_fpa_5arms}]
{\includegraphics[width=0.45\textwidth]{cps_reg_fpa_5arms.pdf}}
\end{subfloat}\\
\begin{subfloat}[10 arms\label{fig:cps_reg_fpa_10arms}]
{\includegraphics[width=0.45\textwidth]{cps_reg_fpa_10arms.pdf}}
\end{subfloat}
\caption{Average cumulative pseudo-regret per method for FPAs}
\label{fig:cps_reg_grids_fpa}%
\end{figure}
\begin{table}
\begin{centering}
\begin{tabular}{c|ccc}
\hline \hline
Estimation method ($\iota$) & 3 arms & 5 arms & 10 arms \\
\hline \\
A/B & 0.004 & 0.003 & 0.003 \\ \\
ETC & 0.007 & 0.007 & 0.007 \\ \\
BITS & 0.010 & 0.023 & 0.011 \\ \\
\hline \hline
\end{tabular}
\par\end{centering}
\caption{$\widehat{MSE}_{\iota,ATE}$ for each estimation method for FPAs}
\label{tab:mse_fpa}
\end{table}
Finally, unlike in SPAs, the recovery by BITS of the true best bid in an FPA does not automatically imply it recovers the true $ATE$ without bias. While solving the MAB problem consistently recovers $b^*$, to estimate the $ATE$ in FPAs we also need to consistently estimate $\frac{F_{CP}(\cdot)}{f_{CP}(\cdot)}$ as seen in equation (\ref{eq:ATE-FPA}). One could wonder how the reversed hazard rate of $B_{CP}$ is identified with FPAs, when one never actually observes $B_{CP}$. The intuition for the identification is that in each FPA auction we participate in, we observe a bound on $B_{CP}$ corresponding to the bid we place. When we win an auction with a context $x$, the bid placed $b$ is an upper bound on $B_{CP}$. When we lose an auction with a context $x$ playing bid $b$, the bid placed $b$ is a lower bound on $B_{CP}$. Thus, each auction yields an observation on the upper and lower bounds on $B_{CP}$. Under Assumption \ref{ass:pv}, these observations are from the truncated above or below marginal distributions of $B_{CP}$, and under Assumption \ref{ass:joint}, there will be observations of upper and lower bounds of $B_{CP}$ for all $b$ within the bid-grid for all $x$. Hence, the distribution of $B_{CP}$ is identified for each $x$, as is its reversed hazard rate and, consequently, the $CATE$s.
To demonstrate that the $ATE$ is recovered without bias, Figure \ref{fig:ATE_FPA_unbiased} shows the estimated density across epochs of the $ATE$ estimated by BITS. Specifically, we estimate the $ATE$ via equation (\ref{eq:est_CATE_fpa}) for each epoch, and use a Gaussian kernel and Silverman's rule-of-thumb bandwidth. Looking at the figure, we see that the distribution is centered at $ATE=0.8$, the true value, for all grid sizes, although, as expected, convergence is quicker when the number of arms is smaller.
\begin{figure}[H]
\centering
\begin{subfloat}[3 arms\label{fig:ATE_FPA_3arms}]
{\includegraphics[width=0.45\textwidth]{dens_fpa_3arms_BITS.pdf}}
\end{subfloat}\\
\begin{subfloat}[5 arms\label{fig:ATE_FPA_5arms}]
{\includegraphics[width=0.45\textwidth]{dens_fpa_5arms_BITS.pdf}}
\end{subfloat}\\
\begin{subfloat}[10 arms\label{fig:ATE_FPA_10arms}]
{\includegraphics[width=0.45\textwidth]{dens_fpa_10arms_BITS.pdf}}
\end{subfloat}
\caption{BITS estimate of $ATE$ for FPAs}
\label{fig:ATE_FPA_unbiased}%
\end{figure}
\subsubsection{Contextual case}
\noindent We now replicate the results for the contextual case for FPAs.
\paragraph{DGP, simulation details and grids of bids}\hfill
\noindent For these simulations, we assume that:
\begin{align}\label{eq:dgp_sim_ctxt_fpa}
\begin{bmatrix}\log Y_i(1) \\ \log Y_i(0) \\ \log B_{CP,i} \end{bmatrix} \Big\vert X_i \overset{iid}{\sim} N \left ( \begin{bmatrix} 0.65 & 0.75 & 0.85 & 0.95 & 1.05 \\ 0.20 & 0.30 & 0.40 & 0.50 & 0.60 \\ -1.38 & -0.50 & 0.20 & 0.80 & 1.31 \end{bmatrix} X_i, \begin{bmatrix} 1.44 & 0 & 0 \\ 0 & 1.21 & 0 \\ 0 & 0 & 1.00 \end{bmatrix} \right ).
\end{align}
Hence, we have $P=5$ contexts. The vector $\delta_{CP}$ is chosen so that $b^*$ equals 0.50, 0.75, 1.00, 1.25 and 1.50. We assume the contexts are equiprobable.
We simulate 1,000 different epochs. Each epoch has $T=100$ rounds, each of which with 100 new observations, so that $n_t=100$ for $1 \leq t \leq 100$, divided equally across the 5 contexts. Each context has its specific grid of bids under consideration. In particular, we consider the following grids:
\begin{enumerate}
\item Context 1: $b \in \{0.25,0.375,0.50,0.625,0.75 \}$
\item Context 2: $b \in \{0.25,0.50,0.75,1.00,1.25 \}$
\item Context 3: $b \in \{0.40,0.70,1.00,1.30,1.60 \}$
\item Context 4: $b \in \{0.45,0.85,1.25,1.65,2.05 \}$
\item Context 5: $b \in \{0.50,1.00,1.50,2.00,2.50 \}$
\end{enumerate}
\paragraph{Results}\hfill
\noindent We replicate Figure \ref{fig:main_res_ctxt} obtained for SPAs, with analogous results displayed in Figure \ref{fig:main_res_fpa_ctxt}. Looking at the figure, we see we obtain the same qualitative patterns, demonstrating that the BITS algorithm succeeds in concurrently recovering the optimal bid for different contexts when the auction mechanism is an FPA.
\begin{figure}
\centering
\begin{subfloat}[$\min_x \hat{\psi}_t \left (b^*(x)\middle \vert x \right )$\label{fig:main_opt_prob_fpa}]
{\includegraphics[width=0.7\textwidth]{main_opt_prob_fpa.pdf}}
\end{subfloat} \\
\begin{subfloat}[Unconditional pseudo-regret per round\label{fig:main_opt_reg_fpa}]
{\includegraphics[width=0.7\textwidth]{main_opt_reg_fpa.pdf}}
\end{subfloat}
\caption{Convergence of BITS algorithm for contextual case}
\label{fig:main_res_fpa_ctxt}%
\end{figure}
\end{subsection}
\end{section}
\begin{section}{Concluding remarks}
\noindent An online experimental design for causal inference for RTB advertising is presented. The experimental design leverages the theory of optimal bidding under sealed-bid SPAs and FPAs to align the twin goals of obtaining economic payoff maximization and inference on the expected effect of advertising for varying subpopulations. The algorithm is framed as a contextual bandit implemented via a modified TS that is adaptively updated via MCMC. The SPA environment is historically the most popular auction format for RTB ads and the proposed experimental design perfectly aligns the economic and inference goals of the advertiser in this environment. Extensions to the more complex FPA environment, which has become more recently popular, are also presented.
Some broader implications of the experimental design beyond RTB advertising are worth mentioning. First, the ideas presented here can be useful in other situations outside of ad-auctions where there is a cost to obtaining experimental units and where managing these costs is critical for the viability of the experiment. Another implication is that, in business experiments, incorporating the firm's payoff or profit maximization goal into the allocation and acquisition of experimental units is helpful. Given the burgeoning utilization of experimentation by firms, we believe that leveraging this perspective in business experiments more broadly has value. Finally, another takeaway from the proposed approach is that it demonstrates the utility of embedding experimental design in the microfoundations of the problem, which enables leveraging economic theory to make progress on running experiments more efficiently. This aspect could be utilized in other settings where large-scale experimentation is feasible and where economic theory puts structure on the behavior of agents and associated outcomes.
\end{section}
\newpage
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\section{Introduction}
Unprecedented properties of whispering-gallery-mode resonators (WGMRs) \cite{BGII, Ultra_high_Q_crystalline_microcavities, WGM_microresonators_sensing_lasing, Nonlinear_and_quantum_optics, Optical_resonators_part1, Optical_resonators_part2}, including extraordinary quality factors and extremely small mode volume defined their use in numerous contemporary applications, such as photonics, nonlinear optics, and precise measurements \cite{Kilohertz_optical_resonances}. WGMRs are widely used to realize various effective detectors \cite{Acoustic_sensor, Temp_sensor, Humidity_sensor, fiberstrain}. Resonators made of electro-optical materials (e.g. $LiNbO_3$, $LiTaO_3$) allow precise resonant-frequency control by means of electric field application and are used as high-speed modulators \cite{electroopticalmodulation}. In contrast, WGMRs from magnetic field-sensitive materials are much less studied, though outstanding original sensors based on magnetostriction were demonstrated \cite{yu2016optomechanical}.
In this work, we aim to investigate the possibility of magneto-optical effects utilization for the purpose of WGMR-enhanced magnetic field sensing. The Terbium Gallium Garnet ($Tb_3 Ga_5 O_{12}$, TGG) has been chosen the WGMR material for two reasons. First, it shows a strong magneto-optical effect (Faraday polarization rotation), bearing an outstanding Verdet constant $\nu$ of $\SI[per-mode=symbol]{-111}{\radian\per\tesla\per\metre}$ at $\SI{670}{\nano\metre}$ wavelength \cite{TGG_pulsed, TGG_temp_wave}. For example, pure silica has $\nu=\SI[per-mode=symbol]{2.80}{\radian\per\tesla\per\metre}$ at the same wavelength \cite{SilicaFibreFaraday}. Second, it has low intrinsic losses ($\alpha <\SI{0.001}{\per\centi\metre}$) at the same wavelength, which is promising for a high $Q$-factor of a WGMR. This unique combination of useful properties defined its application in optical isolators. TGG is reported to have the figure of merit $\nu/\alpha\sim\SI[per-mode=symbol]{1100}{\radian\per\tesla}$ at $\SI{670}{\nano\metre}$ \cite{TGG_TSLAG_FOM}. Its close competitor, yttrium iron garnet (YIG), shows $\nu/\alpha=\SI[per-mode=symbol]{2.51}{\radian\per\tesla}$ at $\SI{780}{\nano\metre}$ \cite{YIG_FOM} due to the higher intrinsic losses, which limits WGMR performance.
In this study, we have predicted and observed experimentally the presence of the Faraday effect-induced eigenfrequency deviation -- the WGM resonant wavelength dependence on the external magnetic field magnitude. Also, we have implemented a two-port polarization control setup and confirmed that the polarization plane experiences path-dependent declination, which nullifies for a complete roundtrip.
\begin{figure}[bht]
\centering
\includegraphics[width=0.99\linewidth]{Geometries_4.jpg}
\caption{Light-magnetic field interaction geometries in an axisymmetric WGMR (optical path shown in red): (a) collinear configuration -- magnetic field $\boldsymbol{B}$ produced by current $\boldsymbol{I}$ is co-aligned with the wave vector along the light's path; (b) transverse configuration -- the magnetic field is uniform along the light's path.}
\label{fig:field_orientation}
\vspace{-16pt}
\end{figure}
\section{Model}
\label{sec:Model}
The Faraday effect is usually considered in the collinear geometry where light travels along the magnetic field force lines \cite{Zvedin}. In an initially isotropic optical medium, a simple relation for the angle of the plane of polarization rotation is easily obtained:
\begin{align}
\Theta_F &= \nu B_{\parallel} L,
\label{Faraday_rotation_angle_expression}
\end{align}
where $\Theta_F$ is the polarization plane rotation angle, $\nu$ is the Verdet constant, $B_{\parallel}$ is the magnetic field vector $\mathbf{B}$ projection onto the light propagation direction and $L$ is the optical path length. Such interaction geometry has been studied in application to WGMRs in \cite{80Tl}. It has been shown there that a polarization transmission spectrum of a system comprising a WGMR and a coupling element can be easily obtained using a recurrent formula
\begin{align}
\boldsymbol{E}_{j+1}=a\hat{M_r} \boldsymbol{E}_j, \label{recurrent}
\end{align}
where $\boldsymbol{E_j}$ is a Jones vector representing the $j-$th roundtrip polarization state, $a=\exp \left( -2n\pi ^{2}R/\lambda Q \right)$ is the extinction coefficient expressed in terms of the quality factor $Q$, $n$ is the refractive index of the medium, $\lambda$ is the light wavelength and $\hat{M}_r$ is the full roundtrip Jones matrix.
The output Jones vector $\boldsymbol{E}_{out}$ at a given wavelength can be calculated by summarizing the inputs of the waves that have passed through the cavity once, twice, thrice, and so on:
\begin{equation}
\begin{split}
\boldsymbol{E}_{out} = t\boldsymbol{E}_0 & +\kappa(-\kappa^*) a \hat{M_r}\boldsymbol{E}_0 +\kappa(-\kappa^*) t^* a^2 \hat{M_r}^2\boldsymbol{E}_0 +\dots \\
&=t\boldsymbol{E}_0-\kappa\kappa^* \frac{1}{t^*}\biggl(\sum_{m=1}^{\infty}\bigl(t^*a \hat{M_r}\bigr)^{m}\biggr) \boldsymbol{E}_0,
\label{equation:Esum}
\end{split}
\end{equation}
where $\boldsymbol{E}_0$ is the input light polarization state, $t, t^*, \kappa, \kappa^*$ are the coupling coefficients between the WGMR and the coupling element. Calculating $\boldsymbol{E}_{out}$ with \eqref{equation:Esum} for a range of wavelengths, one can construct the transmission spectrum of a WGMR. The eigenfrequency deviation amplitude is then easily deduced as a function of the applied magnetic field.
The full roundtrip Jones matrix for the collinear configuration can be found using the permittivity tensor \cite{Zvedin}:
\begin{align}
\hat{\varepsilon}/\varepsilon_0 &=
\begin{pmatrix}
\varepsilon_x & 0 & i\gamma \\
0 & \varepsilon_y & 0 \\
-i\gamma & 0 & \varepsilon_z
\end{pmatrix}.
\label{vareps}
\end{align}
Here the coordinate system is chosen so that the magnetic field vector $\boldsymbol{B}$ and the wavevector of light $\boldsymbol{k}$ are co-aligned, the non-diagonal imaginary terms $\gamma = n \nu \lambda B_{\parallel}/ \pi $, where $n$ is the refractive index, are the components of the gyration vector, representing the Faraday effect \cite{Zvedin}. The diagonal terms are not equal, which accounts for the fact that the eigenfrequencies of the TM (in our model - $x$-polarized) and TE ($z-$polarized) whispering-gallery-modes of the same order are different (Eq. 1 in Ref. \cite{Gorodetsky2}). This difference between the TE and TM modes' eigenfrequencies is calculated, for example, in Ref. \cite{Osada}:
\begin{align}
\Omega _{TM}-\Omega _{TE}=\frac{c}{Rn^2}\sqrt{n^{2}-1},
\label{osadas}
\end{align}
where $c$ is the speed of light, $R$ -- cavity radius, $\Omega_{TM, \; TE}=2\pi c n_{TM, \; TE}/\lambda$. This difference may be viewed as being caused by the so-called geometric birefringence that can be represented by the permittivity variation $\delta\varepsilon$:
\begin{align}
\varepsilon_x =& \bar{\varepsilon} + \delta\varepsilon, &
\varepsilon_y =& \bar{\varepsilon}, &
\varepsilon_z =& \bar{\varepsilon} - \delta\varepsilon, &
n_z-n_x =& \delta n,
\label{vareps_xyz}
\end{align}
where $\bar{\varepsilon}$ is simply $n^2$. Henceforth:
\begin{align}
n_{TM} &=\sqrt{n^2+\delta\varepsilon}, &
n_{TE} &=\sqrt{n^2-\delta\varepsilon}.
\label{n_vareps}
\end{align}
Combining \eqref{osadas} with \eqref{n_vareps}, one can obtain
\begin{align}
\delta\varepsilon & \cong \frac{\lambda}{2\pi R}\frac{\sqrt{n^2-1}}{n}.
\end{align}
As shown in Ref. ~\cite{Tabor}, the Jones matrix, for a medium with permittivity tensor $\hat{\varepsilon}$ takes the following form:
\begin{align}
\hat{M}=
\begin{pmatrix}
\cos{\phi}-i \cos{\chi}\sin{\phi} & -\sin{\chi}\sin{\phi} \\
\sin{\chi}\sin{\phi} & \cos{\phi}+i \cos{\chi}\sin{\phi} \\
\end{pmatrix}
\exp{(-i\psi)}, \label{MJones}
\end{align}
where
\begin{align}
\psi &=\bar{k} L, & \phi &= \delta k L,\\
\delta k &=\frac{k_{TE}-k_{TM}}{2}, & \bar{k} &= \frac{k_{TE}+k_{TM}}{2},\\
\cos{\chi} &= \frac{\delta\varepsilon}{\sqrt{\delta\varepsilon^2+\gamma^2}}, &
\sin{\chi} &= \frac{\gamma}{\sqrt{\delta\varepsilon^2+\gamma^2}},
\end{align}
\begin{align}
k_{TM,TE}^2 &= \bigg(\frac{2 \pi}{\lambda}\bigg)^2 \mu\bigg[\bar{\varepsilon}\pm\sqrt{(\delta \varepsilon)^2+\gamma^2}\bigg]. \label{kTETM}
\end{align}
Here, $L$ is the optical path length, $\mu=1$ due to TGG being paramagnetic. Jones matrix \eqref{MJones} can be applied in the collinear configuration by simply putting $L=2\pi R$, since the angle between the magnetic field vector $\boldsymbol{B}$ and the wavevector $\boldsymbol{k}$ equals zero throughout the whole light path, meaning that ${B}_{\parallel}$ remains constant.
The collinear geometry has been implemented in \cite{Thermally_tunable_WGM}
by passing an electric wire through the center (the symmetry axis) of the optical cavity, as shown in Fig. \ref{fig:field_orientation}(a). The WGM cavity has been formed through melting of a portion of Faraday-rotator material upon a metallic wire. However, our particular goal was to study the possibility of creating a WGMR-based magnetic field sensor, which is why we have turned to the transverse magneto-optic interaction geometry that corresponds to a magnetic field source placed outside the resonator, as shown in Fig. \ref{fig:field_orientation}(b). On the one hand, such configuration is more robust, allows for a stronger magnetic field, and is far easier to implement. On the other hand, the model, described by Eqs. \ref{vareps} to \ref{kTETM}, developed in Ref.~\cite{80Tl}, can not be readily applied in this case since the $B_{\parallel}$ component of the magnetic field $B_{\parallel}$ varies along the light's path. Assuming that the magnetic field is uniform, we can put ${B_{\parallel}}={B}\cos\theta$ for the transverse configuration (Fig. \ref{fig:field_orientation}(b)). Now we can divide the total circumference of the resonator into $N$ small sections (the upper limit for $N$ is actually the azimuthal number of a WGM, which is in the order of $\sim 10^5$), assuming the angle $\theta =\theta_j$ to be constant within each section. The full roundtrip Jones matrix $\hat{M_r}$ of a WGMR can then be obtained numerically through multiplication of the individual sections' Jones matrices $\hat{M}\big(\theta_j\big)$:
\begin{align}
\hat{M_r} &= \lim_{N\rightarrow\infty}\prod_{j=1}^{N} \hat{M}\big(\theta_j\big) = \lim_{N\rightarrow\infty}\prod_{j=1}^{N} \hat{M} \bigg(\frac{2\pi j}{N}\bigg). \label{MRound}
\end{align}
This matrix can now be used to calculate the WGMR transmission spectrum using Eq. (\ref{equation:Esum}). It is worth noting that the proposed approach is suitable for arbitrary configurations (e.g. inhomogeneous magnetic field distributions).
The corresponding calculation has been performed using the following set of parameters: $R=\SI{2.85}{\milli\metre}$, $\nu=\SI[per-mode=symbol]{-111}{\radian\per\tesla\per\metre}$ for $\lambda=\SI{670}{\nano\metre}$ wavelength, $n=1.95$.
The eigenfrequency deviation amplitude calculation results for both configurations are presented in Fig. \ref{fig:Shift}, along with the experimental data for the transverse configuration. The wavevectors' expressions $k_{TM,TE}$ given by \eqref{kTETM} can be expanded in a Taylor series, from which the frequency deviation quadratic dependence on the magnetic field magnitude ($\gamma \sim B_{\parallel}$) is clear:
\begin{align}
k_{TM,TE} &= \frac{2 \pi}{\lambda} \bigg( \sqrt{\bar{\varepsilon} \pm \delta \varepsilon}\pm
\frac{ \gamma ^2}{4 \delta \varepsilon \sqrt{\bar{\varepsilon} \pm \delta \varepsilon}} + O(\gamma^4) \bigg).
\end{align}
The resulting tuning rates are $\SI[per-mode=symbol]{1.25e-6}{\mega\hertz\per\gauss\squared}$ for the transverse configuration and $\SI[per-mode=symbol]{10.43e-6}{\mega\hertz\per\gauss\squared}$ for the collinear configuration. The utilized scheme (\ref{MRound}) was found to converge as fast as $1/N^2$.
\section{Experimental results}
\label{sec:experiment}
\subsection{Frequency deviation amplitude measurements}
The $Q$-factor of the $\SI{2.85}{\milli\metre}$-diameter WGMR fabricated of TGG (see Section 1 of Supplement for details on fabrication process) has been estimated to be $(1.45\pm 0.17) \times 10^8$ for the TE-modes using the technique developed in Ref. \cite{Matsko} (see Section 2 of Supplement), which can be considered as a record among TGG materials bearing high Verdet constant.
A schematic of the experimental setup is presented in Fig. \ref{setup_main}. The light from a tunable ($667 \leq \lambda \leq \SI{671}{\nano\metre}$) $\SI{35}{\milli\watt}$ external-cavity laser (ECL) is focused onto an inner surface of a rutile ($TiO_2$) prism via a lens. The refractive index for the extraordinary wave (TE mode) in the prism is $n_{c} = 2.95$ (the refractive index of TGG is $1.95$). A small portion of the light beam intensity has been coupled into a $\SI{89}{\mega\hertz}$ FSR free-space interferometer to provide control over the laser wavelength sweep (PD1).
Lastly, the light leaving the WGMR through the prism is focused onto the aperture of a photodetector (PD2).
\begin{figure}[bth]
\centering
\includegraphics[width=1\linewidth]{Setup_optimized_2.jpg}
\caption{Experimental setup: (1) -- oscilloscope; (2) -- lock-in amplifier; (3) -- stepper motor driver; (4) -- external cavity laser control unit; (5) -- diamond prism; PD 1-3 -- photodetectors; TGG -- microresonator; Analyser -- Glann prism polarization analyser.
\label{setup_main}}
\vspace{-10pt}
\end{figure}
A high-amplitude alternating magnetic field generator has been constructed (Fig. \ref{setup_main}) by placing ten permanent magnets on a rotating disk driven with a low mechanical noise stepper motor, each neighboring magnet facing the opposite pole to the resonator. The resulting alternating magnetic field signal was close to sinusoidal with a negligible harmonics coefficient. The magnetic field magnitude was set by adjusting the distance between the disk and the WGMR and measured with a Hall sensor placed in the proximity of the cavity. Dynamic measurements were performed at magnetic field alternation frequencies ranging from $20$ to $\SI{40}{\hertz}$.
The laser was tuned to the resonance dip slope (Fig. \ref{fig:Shift}, inset), which allowed to translate its side-to side movement caused by the eigenfrequency deviation into the PD2 signal amplitude variation. This signal was then fed to a lock-in amplifier (Stanford Research Systems SR560) along with the magnetic field generator driving signal, which allowed to drastically increase the setup sensitivity (see Section 3 of Supplement).
The resulting frequency deviation amplitude measured on the second harmonic of the driving signal is plotted against the magnetic field magnitude in Fig. \ref{fig:Shift}. The experimental result (dots) shows good correspondence with the theoretical predictions (black dashed line). The x-axis error comes from the Hall sensor positioning uncertainty, while the y-axis precision is limited by the slow thermal drift of the system (see Section 3 of Supplement). The frequency deviation amplitude was found to be of $\SI{1.6}{\mega\hertz}$ at the maximum achievable in this setup field strength of $\SI{1100}{G}$.
\begin{figure}[bth]
\centering
\includegraphics[width=1\linewidth]{Shift_with_modified_inset_2.jpg}
\caption{The TGG WGMR eigenmode frequency modulation amplitude on the second harmonic of the applied magnetic field. The inset shows the WGMR transmission spectrum obtained in the wavelength sweep mode of the laser. The traces are: yellow -- PD2 signal, magenta -- PD1 signal, green -- laser wavelength sweep signal. The markers highlight the slope section used in the experiment.
\vspace{-16pt}
\label{fig:Shift}}
\end{figure}
\subsection{Polarization rotation measurements}
Let us consider a hypothetical case of a WGMR that possesses no geometrical birefringence (the TE and TM modes frequencies are equal). In this case, the resulting plane of polarization rotation angle $\Theta_r$, accumulated throughout a full light roundtrip, is simply a line integral of the magnetic field vector $\boldsymbol{B}$ projection onto the light propagation direction.
\begin{align}
\Theta_r &= \nu R \int_{0}^{2\pi} \frac{(\boldsymbol{B} \cdot \boldsymbol{k})}{|\boldsymbol{k}|} d\theta.
\end{align}
This way, since the light path within the optical cavity is a closed-loop, in a non-birefringent WGMR the polarization rotation angle caused by the Faraday effect, will be precisely zero due to Ampere's circuital law in the transverse geometry.
Even in the real case where the Faraday rotation is hindered by the geometrical birefringence \cite{80Tl}, one can reasonably expect that the full roundtrip polarization plane rotation angle will also be zero, or, at least, close to zero. To test this hypothesis, polarization state measurements have been performed. An analyzer has been placed between the collimating lens and PD2 to probe the throughput light polarization state. The magnetic field was created by a single permanent magnet in this case. As expected, no polarization plane rotation (TE and TM modes mixing) has been observed. The accuracy of these measurements has been estimated to be $\sim \ang{0.01}$, limited by the minimum resolvable PD2 signal variation.
In order to further support our rotation cancellation hypothesis it was decided to probe the polarization state at some intermediate point by using an additional coupling element -- a diamond prism, which allowed us to assess the out-coupled light polarization state, as shown in Fig. \ref{setup_main}. The diamond prism output light intensity (PD3) is plotted against the analyzer rotation angle in Fig. \ref{fig:polarization_Rotation}.
It follows the well-known Malus law pattern for both cases (with and without magnetic field application).
The amount of polarization plane rotation acquired along the light path section ($\ang{89}$) between the rutile and the diamond prisms is the difference between the x-axis positions of the $\cos^2$ curves and it equals $\ang{1.59}$ declination from the initial state. A calculation performed via Eq. (\ref{equation:Esum}) gives us $\ang{1.53}$ declination, in good correspondence with the experimental results. We believe that this validates our previous result -- the absence of polarization plane rotation for a full roundtrip.
\begin{figure}[htb] \centering \includegraphics[width=1\linewidth]{PolRot2.png} \caption{
Measured intensity of the light at the resonant wavelength, out-coupled by the additional diamond prism located $\ang{89}$ away from the coupling prism, vs. polarizer angle. A polarization plane rotation of $\ang{1.59}$ is observed. }
\label{fig:polarization_Rotation}
\vspace{-16pt}
\end{figure}
\section{Discussion}
We have demonstrated to the best of our knowledge the record quality factor ($Q =1.45\times 10^8$) among materials with a high Verdet constant.
We have observed an eigenfrequency modulation, induced by a harmonic low-frequency transverse homogeneous magnetic field. The eigenfrequency deviation amplitude turned out to be proportional to the magnetic field magnitude squared.
The resulting frequency deviation amplitude is independent of the $Q$-factor. Nevertheless, a high $Q$-factor allows us to increase the measurement accuracy. A simple model based on the Jones matrices product calculation has been developed to describe this behavior.
The calculation results are in excellent agreement with the experimental data. It is important that the developed model is applicable for arbitrary magnetic field distributions.
As a weak magnetic field sensor, TGG WGMR is, obviously, inferior to many other devices, offering sensitivity below $\SI{1}{\femto \tesla}$ \cite{SERF} or superior field strength to frequency deviation amplitude conversion coefficient at low frequencies \cite{yu2016optomechanical}.
Nevertheless, TGG WGMR has its advantages, including higher frequency field detection capability, limited only by the optical mode relaxation time ($\SI{3}{\mega\hertz}$ in our case), and absence of saturation at any reachable magnetic field strengths. The possible utilization of the interaction between alternatively polarized (TE-like and TM-like) mode families, via a high-frequency alternating magnetic field in WGM resonators, manufactured from Faraday-rotating materials deserves further study.
\section*{Funding}
This work was supported by the Russian Science Foundation (project 20-12-00344).
\section*{Acknowledgments}
Verdet constant for the used TGG samples was measured at the VNIIOFI center of the shared use (VNIIOFI CSU). The authors thank Artem E. Shitikov for useful discussion.
\section*{Supplemental document}
See \href{https://osapublishing.figshare.com/s/d8de0a651a1f2f497cd5}{Supplement 1} for supporting content.
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train/arxiv
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BkiUeR7xK0iCl2n1_-h8
| 5 | 1 |
\section{Introduction}
All considered graphs in this paper are simple, finite and undirected. We follow~\cite{BonMur08} for all terminology and notation not defined here. A $k$-(vertex-)coloring of a graph $G$ is an assignment $\varphi: V(G)\to\{1,\ldots,k\}$. A coloring $\varphi$ is said to be \textit{proper} if every color class is an independent subset of the vertex set of $G$. A \textit{hypergraph} $\mathcal{H}=(V(\mathcal{H}),\mathcal{E}(\mathcal{H}))$ is a generalization of a graph, its (hyper-)edges are subsets of $V(\mathcal{H})$ of arbitrary positive size. There are various notions of (vertex-)coloring of hypergraphs, which when restricted to graphs coincide with proper graph coloring. One such notion was introduced by Even at al.~\cite{EveLotRonSmo03} (in a geometric setting) in connection with frequency assignment problems for cellular networks, as follows. A coloring of a hypergraph $\mathcal{H}$ is \textit{conflict-free} \textit{(CF)} if for every edge $e\in \mathcal{E}(\mathcal{H})$ there is a color $c$ that occurs exactly once on the vertices of $e$. The \textit{CF chromatic number} of $\mathcal{H}$ is the minimum $k$ for which $\mathcal{H}$ admits a CF $k$-coloring. For graphs, Cheilaris~\cite{Che09} studied the \textit{CF coloring with respect to neighborhoods}, that is, the coloring in which for every non-isolated vertex $x$ there is a color that occurs exactly once in the (open) neighborhood $N(x)$, and proved the upper bound $2\sqrt{n}$ for the CF chromatic number of a graph of order $n$. For more on CF colorings see, e.g., \cite{CheTot11, GleSzaTar14, KosKumLuc12, PacTar09, Smo13}.
A similar but considerably less studied notion (concerning a weaker requirement for the occurrence of a color) was introduced by Cheilaris et al.~\cite{CheKesPal13} as follows. An \textit{odd coloring} of a hypergraph $\mathcal{H}$ is a coloring such that for every edge $e\in \mathcal{E}(\mathcal{H})$ there is a color $c$ with an odd number of vertices of $e$ colored by $c$. Particular features of the same notion (under the name \textit{weak-parity coloring}) have been considered by Fabrici and G\"{o}ring~\cite{FabGor16} (in regard to face-hypergraphs of planar graphs) and also by Bunde et al.~\cite{BunMilWesWu07} (in regard to coloring of graphs with respect to paths, i.e., path-hypergraphs). For various edge colorings of graphs with parity condition required at the vertices we refer the reader to~\cite{AtaPetSkr16, Pet15, KanKatVar18, LuzPetSkr15, LuzPetSkr18, Pet18, PetSkr21}.
\medskip
In this paper we study certain aspects of odd colorings for graphs with respect to (open) neighborhoods, that is, the colorings of a graph $G$ such that for every non-isolated vertex $x$ there is a color that occurs an odd number of times in the neighborhood $N_G(x)$. Our focus is on colorings that are at the same time proper, and we mainly confine to planar graphs.
\smallskip
Let us denote by $\chi_o(G)$ the minimum number of colors in any proper coloring of a given graph $G$ that is odd with respect to neighborhoods, call this the \textit{odd chromatic number} of $G$. Note that the obvious inequality $\chi(G)\leq\chi_o(G)$ may be strict; e.g., $\chi(C_4)=2$ whereas $\chi_o(C_4)=4$. Similarly, $\chi(C_5)=3$ whereas $\chi_o(C_5)=5$. In fact, the difference $\chi_o(G)-\chi(G)$ can acquire arbitrarily large values. Indeed, let $G$ be obtained from $K_n$ $(n\geq2)$ by subdividing each edge once. Since $G$ is bipartite, $\chi(G)=2$. On the other hand, it is readily seen that $\chi_o(G)\geq n$. Note in passing another distinction between the chromatic number and the odd chromatic number. The former graph parameter is monotonic in regard to the `subgraph relation', that is, if $H\subseteq G$ then $\chi(H)\leq\chi(G)$. This nice monotonicity feature does not hold for the odd chromatic number in general. For example, $C_4$ is a subgraph of the kite $K_4-e$, but nevertheless we have $\chi_o(C_4)=4>3=\chi_o(K_4-e)$.
\medskip
The Four Color Theorem~\cite{AppHak77,RobSanSeyTho96} asserts the tight upper bound $\chi(G)\leq4$ for the chromatic number of any planar graph $G$. One naturally starts wondering about an analogous bound for the odd chromatic number of all planar graphs. Since $\chi_o(C_5)=5$, four colors no longer suffice. It is our belief that five colors always suffice.
\begin{conjecture}
\label{conj:1}
For every planar graph $G$ it holds that $\chi_o(G)\leq 5$.
\end{conjecture}
The bound in Conjecture~\ref{conj:1} is achieved for infinitely many planar graphs. Namely, let $\mathcal{F}=\{G: G$ is a non-trivial connected graph such that every block of $G$ is isomorphic to $C_5\}$.
\begin{proposition}
\label{prop:F}
If $G\in\mathcal{F}$ then $\chi_o(G)=5$.
\end{proposition}
\begin{proof}
Consider a counter-example $G$ with minimum number of blocks, $b(G)$. Since $G\neq C_5$, we have that $b(G)\geq2$. Let $B$ be an endblock of $G$, and let $v_1,v_2,v_3,v_4,v_5$ be the vertices of the $5$-cycle $B$ met on a circular traversing that starts at the unique cut vertex $v_1$ contained within $B$. Consider the graph $G'=G-\{v_2,v_3,v_4,v_5\}$. Since $G'\in\mathcal{F}$ and $b(G')=b(G)-1$, it holds that $\chi_o(G')=5$. Take an odd coloring of $G'$ with color set $\{1,2,3,4,5\}$ such that $v_1$ is colored by $1$ and the color $4$ has odd number of occurrences in $N_{G'}(v_1)$. Extend to $V(G)$ by assigning the color $i$ to $v_i$, for $i=2,3,4,5$. This clearly gives an odd $5$-coloring of $G$. Consequently, since we are supposing $\chi_o(G)\neq5$, there must exist an odd coloring $c$ of $G$ with color set $\{1,2,3,4\}$. Upon permutation of colors, we may assume that $c(v_i)=i$ for $1\le i\le4$. Hence, it must be that $c(v_5)=2$. Thus, any color which satisfies the odd-condition in $N_G(v_1)$ has an odd number of occurrences in $N_{G'}(v_1)$ as well. It follows that the restriction of $c$ to $G'$ is an odd $4$-coloring, contradicting the minimality choice of $G$.
\end{proof}
The main purpose of this paper is to provide the first support to Conjecture~\ref{conj:1} by proving the following.
\begin{theorem}
\label{mainresult}
For every planar graph $G$ it holds that $\chi_o(G)\leq 9$.
\end{theorem}
The rest of the article is organized as follows. In the next section we provide a proof of Theorem~\ref{mainresult}. And in Section~3 we briefly convey some of our ideas for possible further work on the topic of proper colorings of graphs that are odd with respect to neighborhoods.
\section{Proof of Theorem~1.3}
Throughout, we will refer to a coloring of this kind as a \textit{nice coloring}; that is, a nice coloring is a (proper) odd coloring which uses at most $9$ colors. Arguing by contradiction, let $G$ be a counter-example of minimum order $n=n(G)$. Clearly, $G$ is connected and has $n\geq10$ vertices. We proceed to exhibit several structural constraints of $G$.
\bigskip
\noindent \textbf{Claim 1.} \textit{The minimum degree $\delta(G)$ equals $5$.}
\medskip
\noindent Since $G$ is a connected planar simple graph with $n>2$ vertices, the inequality $\delta(G)\leq5$ is a consequence of Euler's formula. Consider a vertex $v$ of degree $d_G(v)=\delta(G)$. If $d_G(v)=1$ or $3$, then take a nice coloring of $G-v$. By forbidding at most six colors at $v$ (namely, at most three colors used for $v$'s neighbors and at most three additional colors in regard to oddness concerning the neighborhoods in $G-v$ of these vertices), the coloring extends to a nice coloring of $G$.
Suppose next that $d_G(v)=2$, and say $N_G(v)=\{x,y\}$. Construct $G'$ by removing $v$ from $G$ plus connecting $x$ and $y$ if they are not already adjacent. By minimality, $G'$ admits a nice coloring $c$. Say $c(x)=1$ and $c(y)=2$, and let color(s) $1'$ and $2'$, respectively, have odd number of occurrences in $N_{G'}(x)$ and $N_{G'}(y)$. If there are several possibilities for $1'$, then choose $1'\neq2$; do similarly for $2'$ in regard to $1$. Extend the coloring to $G$ by using for $v$ a color different from $1,2,1',2'$. The properness of the coloring is clearly preserved. As for the weak-oddness concerning neighborhoods, $v$ is fine because $1\neq2$. If $1'\neq2$ then $x$ is fine since $1'$ remains to be odd on $N_G(x)$. Contrarily, if $1'=2$ then $c(v)$ is odd on $N_G(x)$. Similarly, the vertex $y$ is also fine.
Finally, suppose $d_G(v)=4$ and let $w\in N_G(v)$. Remove $v$ and connect $w$ by an edge to every other non-adjacent neighbor of $v$. The obtained graph $G''$ is simple and planar. Indeed, it can be equivalently obtained from $G$ by contracting the edge $vw$ (if parallel edges arise through possible mutual neighbors of $v$ and $w$, then for each such adjacency a single edge is kept and its copies are deleted). Notice that under any nice coloring of $G''$, the color of $w$ occurs exactly once in $N_G(v)$. Therefore, any such coloring extends to a nice coloring of $G$ by forbidding at most eight colors at $v$.
We conclude that $d_G(v)=5$.\qed
\bigskip
We refer to any vertex of degree $d$ as a \textit{$d$-vertex}. Similarly, a vertex of degree at least $d$ is a \textit{$d^+$-vertex}. Analogous terminology applies to faces in regard to a planar embedding of $G$.
\bigskip
\noindent \textbf{Claim 2.} \textit{If $v$ is a $5$-vertex of $G$, then it has at most one neighbor of odd degree.}
\medskip
\noindent Suppose each of two vertices $u,w\in N_G(v)$ has an odd degree in $G$. Consider a nice coloring $c$ of $G-v$. We intend to extend $c$ to $G$. Since $N_G(u)$ and $N_G(w)$ are odd-sized, no color is blocked at $v$ in regard to oddness in the neighborhoods of $u$ and $w$. Moreover, as $N_G(v)$ is odd-sized as well, the oddness of a color in this particular neighborhood is guaranteed. Therefore, by forbidding at most eight colors at $v$ (all of $c(N_G(v))$ and at most $3$ additional colors concerning oddness in neighborhoods of the vertices forming $N_G(v)\backslash\{u,w\}$), the coloring $c$ extends to a nice coloring of $G$.\qed
\bigskip
Since the graph $G$ is connected, for an arbitrary planar embedding Euler's formula gives
$$|V(G)|-|E(G)|+|F(G)|=2\,,$$ where $F(G)$ is the set of faces.
Our proof is based on the discharging technique. We assign initial charges to the vertices and faces according to the left-hand side of the following equality (which immediately follows from Euler's formula):
$$\sum_{v\in V(G)}(d(v)-6)+\sum_{f\in F(G)}(2d(f)-6)=-12\,.$$
Thus, any vertex $v$ receives charge $d(v)-6$, and any face $f$ obtains charge $2d(f)-6$. As $G$ is simple, for every face $f$ it holds that $d(f)\geq3$, implying that its initial charge is non-negative. By Claim~1, the only vertices that have negative initial charge are the $5$-vertices of $G$ (each is assigned with charge $-1$).
\bigskip
\noindent \textbf{Claim 3.} \textit{Any $d$-face $f\in F(G)$ is incident with at most $\left\lfloor \frac{2d}{3} \right\rfloor$ $5$-vertices.}
\medskip
\noindent By Claim~2, on a facial walk of $f$ no three consecutive vertices are of degree $5$. From this, the stated upper bound for the number of $5$-vertices incident with $f$ follows immediately.\qed
\bigskip
We use the following discharging rules (cf. Figure~\ref{fig:rules}):
\begin{itemize}
\item[(R1)] Every $4^+$-face $f$ sends charge $1$ to any incident $5$-vertex $v$.
\item[(R2)] Every $8^+$-vertex $u$ sends charge to any adjacent $5$-vertex $v$ if both faces incident with the edge $uv$ are triangles, as follows: if one of these triangles has another $5$-vertex (besides $v$), then the sent charge equals $\frac{1}{3}$; otherwise, the sent charge equals $\frac{1}{2}$. (In view of Claim~2, it cannot be that both these triangles have a $5$-vertex $\neq v$.)
\item[(R3)] Every $4^+$-face $f$ sends through every incident edge $uw$ with $d(u)=d(w)=6$ charge $\frac{1}{2}$ to a $5$-vertex $v$ if $uvw$ is a triangular face and $f$ is not a $4$-face incident with two $5$-vertices.
\end{itemize}
\begin{figure}[ht!]
$$
\includegraphics[scale=0.8]{rules}
$$
\caption{In (R2), the numbers standing in brackets beside vertices or faces indicate their degrees. In (R3), $f$ is not a $4$-face incident with two $5$-vertices and $uvw$ is an adjacent triangular face.}
\label{fig:rules}
\end{figure}
\bigskip
\noindent \textbf{Claim 4.} \textit{No $8^+$-vertex becomes negatively charged by applying $(R2)$.}
\medskip
\noindent Suppose there is a $d$-vertex $u$ with $d\geq8$ that becomes negatively charged by applying Rule~2. Recall that its initial charge was $d-6$. Consider a circular ordering of $N_G(u)$ in regard to the embedding of $G$. Note that no three consecutive neighbors $v_i,v_{i+1},v_{i+2}\in N_G(u)$ have received charge from $u$ during the discharging process, for otherwise $v_{i+1}$ contradicts Claim~2. Therefore, $u$ gave charge to at most $\left\lfloor \frac{2d}{3} \right\rfloor$ of its neighbors, and at most $\frac{1}{2}$ of charge per neighbor. Consequently, $d-6<\frac{2d}{3}\cdot\frac{1}{2}$. Equivalently, $d<9$. So $u$ is an $8$-vertex. Moreover, since we are supposing that it is negatively charged after applying Rule~2, the vertex $v$ gave charge to precisely five $5$-vertices. However, then it is easily seen that the initial charge $2$ of $v$ is reduced by at most $\frac{11}{6}=\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{2}$, a contradiction.\qed
\bigskip
\noindent \textbf{Claim 5.} \textit{No $4^+$-face becomes negatively charged by applying $(R1)$ or $(R3)$.}
\medskip
\noindent Suppose there is a $d$-face $f$ that becomes negatively charged by applying Rules~1 or~3. Recall that its initial charge was $2d-6$. For the purposes of this proof, it is useful to think of the charge sent by $f$ according to Rule~3 as follows. Consider any maximal sequence of `consecutive' triangular faces that are adjacent to $f$, and split the sequence into $2$-tuples and a possible $1$-tuple. For every such $2$-tuple of triangular faces, the total charge $1=\frac{1}{2}+\frac{1}{2}$ first goes to their common $6$-vertex, and only afterwards splits evenly and reaches the targeted $5$-vertices (possibly the same). And for every such $1$-tuple the charge $\frac{1}{2}$ coming from $f$ to this face first splits evenly and goes to the pair of $6$-vertices shared with $f$, and only afterwards reaches the targeted $5$-vertex.
With this perspective in mind, consider a facial walk of $f$ and notice that during the discharging it gives away to each incident vertex charge of either $0,\frac{1}{4}$, or $1$; moreover, no three consecutive vertices receive from $f$ charge $1$ (according to Claim~2). Hence, on at most $\left\lfloor \frac{2d}{3} \right\rfloor$ occasions $f$ gives charge $1$, and on the other $\left\lfloor \frac{d+1}{3} \right\rfloor$ occasions $f$ gives charge at most $\frac{1}{4}$. Consequently, $2d-6<\frac{2d}{3}+\frac{d+1}{12}$. Equivalently, $d<\frac{73}{15}$. So $f$ is a $4$-face. However, it is easily seen that a $4$-face gives away at most $2$ of its initial charge (which also equals $2$), a contradiction. \qed
\bigskip
Since the total charge remains negative (it equals $-12$), from Claims~4 and 5 it follows that there is a $5$-vertex $v$ which remains negatively charged even after applying the discharging rules. By (R1) and (R2), we have the following.
\bigskip
\noindent \textbf{Claim 6.} \textit{The $5$-vertex $v$ has only $3$-faces around it, and it is a neighbor of at most two $8^+$-vertices. Moreover, if $v$ neighbors exactly two $8^+$-vertices, then these three vertices have another $5$-vertex as a common neighbor.}
\bigskip
Let $v_1,v_2,v_3,v_4,v_5$ be the neighbors of $v$ in a circular order regarding the considered plane embedding, i.e., such that $vv_1v_2,vv_2v_3,vv_3v_4,vv_4v_5,vv_5v_1$ are the five $3$-faces incident with $v$.
\bigskip
\noindent \textbf{Claim 7.} \textit{For some $j=1,2,\ldots,5$, the vertices $v_{j},v_{j+2}$ are not adjacent and their only common neighbors are $v,v_{j+1}$.}
\medskip
\noindent Arguing by contradiction, suppose that every pair of vertices $v_{j},v_{j+2}$ are either adjacent or have a common neighbor $\neq v,v_{j+1}$. This readily implies the existence of a vertex $x\neq v$ that is adjacent to all five vertices $v_1,v_2,v_3,v_4,v_5$.
Now take a nice coloring $c$ of $G-v$, and let $s=|c(N_G(v))|$. In view of the $5$-cycle $v_1v_2v_3v_4v_5$, it holds that $s\in\{3,4,5\}$. In case $s=3$, the coloring $c$ extends to a nice coloring of $G$ since at most $8$ colors are forbidden at $v$. These are the three colors used on $N_G(v)$ and at most five additional colors in regard to `oddness' in $N_{G-v}(v_i)$ for $i=1,2,\ldots,5$. Hence $s\in\{4,5\}$, and from this we are able to further deduce that there are precisely $s$ vertices $v_i (i=1,2,\ldots,5)$ such that $c(v_{i-1})\neq c(v_{i+1})$.
Unless the coloring $c$ extends to $G$, each of the nine available colors is blocked at the vertex $v$, either due to properness or to `oddness'. Consequently, there are at least $(9-s)$ vertices among the $v_i$'s each of which blocks at $v$ a separate color $\neq c(v_1),c(v_2),c(v_3),c(v_4),c(v_5)$ in regard to `oddness' in its neighborhood within $G-v$. Clearly, any such $v_i$ is of even degree in $G$ and any color from $\{c(v_{i-1}),c(v_{i+1})\}$ appears an even number of times in $N_{G-v}(v_i)$. Moreover, for at least four such vertices $v_i$, say $v_1,v_2,v_3,v_4$, it also holds that $c(v_{i-1})\neq c(v_{i+1})$. Noting that the color $c(x)$ occurs in every of the four neighborhoods $N_{G-v}(v_i)$ for $i=1,\ldots,4$, it follows that $c(x)$ has an odd number of appearances in such a neighborhood for at most one vertex $v_i$. Therefore, at least three $v_i$'s are of degree $\geq 2+2+2+1=7$ in $G-v$. (Namely, each of the colors $c(v_{i-1}),c(v_{i+1}),c(x)$ yields a summand of $2$, and the summand of $1$ refers to the color blocked by $v_i$ due to `oddness' in $N_{G-v}(v_i)$).
However, this yields at least three $8^+$-neighbors of $v$ in $G$, contradicting Claim~6.\qed
\bigskip
Let $v_j,v_{j+2}$ fulfil Claim~7. Consider the graph $(G-v)/\{v_j,v_{j+2}\}$ obtained from $G-v$ by identifying $v_j$ and $v_{j+2}$, and then deleting one of the arising two parallel edges (between the new vertex $\{v_j,v_{j+2}\}$ and the vertex $v_{j+1}$). Take a nice coloring of $(G-v)/\{v_j,v_{j+2}\}$, and look into the inherited coloring $c$ of $G-v$.
\bigskip
\noindent \textbf{Claim 8.} \textit{All $v_i$'s are of even degree in $G$, and the coloring $c$ is nice.}
\medskip
\noindent
Notice that besides the odd-condition at $v_{j+1}$, the coloring $c$ is nice on $G-v$. If $d_G(v_{j+1})$ is odd, then when adding $v$ to $G-v$ we do not need to care about odd-condition at $v_{j+1}$, as it will automatically hold. But then we need to avoid only $8$ colors when coloring $v$, which is doable, a contradiction. Hence $d_G(v_{j+1})$ is even. Consequently, $c$ is a nice coloring of $G-v$ (as $d_{G-v}(v_{j+1})$ is odd and the rest of the conditions are fine). Since $c$ does not extend to a nice coloring of $G$, by reasoning in the same manner as at the beginning of the proof of Claim~7, we are able to conclude that $|c(N_G(v))|\geq4$. On the other hand, by the construction of $c$ we clearly have $|c(N_G(v))|<5$. So it must be that exactly four colors appear in $N_G(v)$. Moreover, as $c$ is non-extendable, each of the vertices $v_1,v_2,v_3,v_4,v_5$ blocks at $v$ a separate color $\notin c(N_G(v))$ in regard to already (and uniquely) fulfilled `oddness' in its neighborhood in $G-v$. However, this readily implies that each $v_i$ is of even degree in $G$.\qed
\begin{figure}[ht!]
$$
\includegraphics[scale=0.6]{localfaces}
$$
\caption{Local color distribution, guaranteed `free hanging' colors and the faces adjacent with the local triangulation around $v$.}
\label{fig:localfaces}
\end{figure}
The proof of Claim~8 certifies that, upon permuting colors, the following color distribution occurs under $c$: the colors $1,2,1,3,4$ are used on $v_1,v_2,v_3,v_4,v_5$, respectively, and each of the colors $5,6,7,8,9$ happens to be the only color with an odd number of appearances on a separate neighborhood $N_{G-v}(v_i)$ (cf. Figure~\ref{fig:localfaces}(a)). Moreover, as colors $5,6,7,8,9$ are `exclusive' in regard to `oddness' on $N_{G-v}(v_1),N_{G-v}(v_2),N_{G-v}(v_3),N_{G-v}(v_4),N_{G-v}(v_5)$, respectively, any of the colors $1,2,3,4$ has an even (possibly $0$) number of occurrences in each set $N_{G-v}(v_i)$ $(i=1,\ldots,5)$. Consequently, the situation depicted in Figure~\ref{fig:localfaces}(b) is present, that is, at four of the $v_i$'s (namely, $v_1,v_2,v_4,v_5$) we are guaranteed two more `free hanging' colors per vertex.
Let $f_i$ $(i=1,\ldots,5)$ be the face incident with $v_iv_{i+1}$ but not with $v$ (cf. Figure~\ref{fig:localfaces}(b)). Note that $f_1,f_2,f_3,f_4,f_5$ need not be pairwise distinct. We say that $f_i$ is \textit{convenient} if it is a $4^+$-face and $v_i,v_{i+1}$ are $6$-vertices whose only common neighbor is $v$.
\bigskip
\noindent \textbf{Claim 9.} \textit{If all of $v_1,v_2,v_3,v_4,v_5$ are $6$-vertices in $G$, then at least two members of the list $f_1,f_2,f_3,f_4,f_5$ are convenient.}
\medskip
\noindent It suffices to observe the following: if a given $f_k$ is a $3$-face, then the vertices $v_k$ and $v_{k+1}$ must share a free hanging color. Since all the $v_i$'s are $6$-vertices, it follows that $v_3$ and $v_4$ do not have a common free hanging color. Consequently, $f_3$ is convenient. Quite similarly, $f_5$ is convenient.\qed
\bigskip
Following the same line of reasoning, we also conclude the following:
\bigskip
\noindent \textbf{Claim 10.} \textit{If precisely one of $v_1,v_2,v_3,v_4,v_5$ is an $8^+$-vertex in $G$, then some $f_i$ is convenient.}
\medskip
\noindent Indeed, we may assume that neither $v_1$ nor $v_5$ is the $8^+$-neighbor of $v$. Then $v_1,v_5$ are $6$-vertices without a common free hanging color. Consequently, $f_5$ is convenient.\qed
\bigskip
Our next (and final) claim assures that every convenient face $f_i$ sends charge $\frac{1}{2}$ to $v$, in accordance with (R3).
\bigskip
\noindent \textbf{Claim 11.} \textit{No convenient face is a quadrangle $ABCD$ where $A,B$ are $5$-vertices and $C,D$ are $6$-vertices.}
\medskip
\noindent Arguing by contradiction, we suppose that the situation depicted in Figure~\ref{fig:nice}(a) is present. Without loss of generality, we may assume that $B$ and $D$ are non-adjacent and $N_G(B)\cap N_G(D)=\{A,C\}$. Indeed, if each of the pairs of vertices $A,C$ and $B,D$ are either adjacent or share a common neighbor outside the set $\{A,B,C,D\}$, then by planarity, the situation depicted in Figure~\ref{fig:nice}(b) occurs. However, then $C$ and $D$ have a common neighbor $\neq v$, which contradicts that the considered face is convenient.
\begin{figure}[ht!]
$$
\includegraphics[scale=0.6]{nice}
$$
\caption{A convenient face next to the $5$-vertex $v$ that happens to be a quadrangle $ABCD$, where $A,B$ are $5$-vertices and $C,D$ are $6$-vertices.}
\label{fig:nice}
\end{figure}
With our assumption for the pair $B,D$, we look at the graph $G'=(G-\{v,A,C\})/\{B,D\}$, that is, $G'$ is obtained from $G-\{v,A,C\}$ by identifying $B$ and $D$. Since $G'$ is planar and smaller than $G$, it admits a nice coloring. Consider the inherited partial coloring $c$ of $G$. Note that vertices $B$ and $D$ are colored the same under $c$, and the only uncolored vertices are $v,A$ and $C$. We extend $c$ to a nice coloring of $G$ as follows.
First we color $v$, and for the time being we don't care about preserving oddness in $N_{G-A-C}(D)$. Apart from the color $c(D)$ and possibly a second color in regard to unique oddness in $N_{G-\{v,A\}}(C)$, at most $6=3\cdot 2$ more colors are forbidden at $v$
(by its three neighbors $\neq C,D$ and oddness in their respective neighborhoods within $G-\{v,A,C\}$). Thus there is a color which is available for $v$.
Next we color $C$. There are at most $8=1+1+3\cdot2$ colors forbidden at $C$ (namely, the color $c(B)=c(D)$, the color $c(v)$ and possibly six more colors concerning the other three neighbors of $C$ and oddness in their respective neighborhoods within $G-A$). So there is a color available for $C$. Once $C$ is assigned with an available color, note that there is a color with an odd number of appearances in $N_{G-A}(D)$ (because $d_{G-A}(D)=5$).
Finally, we color $A$. Since this is a $5$-vertex, it suffices to choose a color for $A$ so that the properness of the coloring and the oddness in $N_{G-A}(D)$ are preserved. As $B$ and $D$ are colored the same, and $B$ is a $5$-vertex, the number of forbidden colors at $A$ is at most $8=1+1+3\cdot2$ (the color $c(B)=c(D)$, possibly a second color in regard to unique oddness in $N_{G-A}(D)$, and six more colors concerning the other three neighbors of $A$ and oddness in their respective neighborhoods within $G-A$). We conclude that $c$ indeed extends to a nice coloring of $G$, a contradiction.\qed
\bigskip
From Claims~9-11 it follows that the vertex $v$ receives charge at least $1$ during the discharging process, hence it cannot remain to be negatively charged. This contradiction concludes our proof.
\section{Further work}
By Proposition~\ref{prop:F}, there exist plenty of planar graphs with girth $5$ that require five colors for an odd coloring. We are tempted to propose the following.
\begin{conjecture}
\label{odd4}
Every planar graph $G$ of girth at least $6$ has $\chi_o(G)\leq4$.
\end{conjecture}
If true, Conjecture~\ref{odd4} would imply the Four Color Theorem (4CT). Indeed, say $G$ is an arbitrary planar graph. Subdivide every edge of $G$ once, i.e., consider the complete subdivision $S(G)$. As girth $g(S(G))\ge6$, take an odd $4$-coloring $c$ of $S(G)$. The restriction of $c$ to $V(G)$ is a proper $4$-coloring of $G$.
\medskip
Although the problem of odd coloring is intriguing already in the class of plane graphs,
the concept can be naturally studied also for graphs embedded in higher surfaces.
For a surface $\Sigma$, we define the \textit{odd chromatic number} of $\Sigma$,
\[\chi_o(\Sigma) = \max_{G\hookrightarrow \Sigma}\chi_o(G),\]
as the maximum of $\chi_o(G)$ over all graphs $G$ embedded into $\Sigma$. Conjecture~\ref{conj:1} and Proposition~\ref{prop:F} combined purport that $\chi_o(S_0) = 5$, where $S_0$ is the sphere. It would be interesting
to have a similar characterization to the Heawood number for other surfaces of higher
genus.
\begin{problem}
Determine $\chi_o(\Sigma)$ for surfaces $\Sigma$ of higher genus.
\end{problem}
Possibly the odd chromatic number of higher surfaces coincides with the corresponding Heawood number.
\bigskip
\noindent
{\bf Acknowledgements.}
This work is partially supported by ARRS Program P1-0383 and ARRS Project J1-3002.
\bibliographystyle{plain}
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\section{Introduction}
Image completion, or image in-painting, is a fascinating and quickly growing area of research in machine learning. Given an image with some pixels removed, the image completion task is to correctly fill in those missing pixels. Therefore, the input to an image completion algorithm is an image with some pixels removed, and the algorithm outputs its estimate of the values of those missing pixels. This has various use cases. For instance, if an image is partially corrupted, the corrupted pixels can be removed, and an image completion algorithm can be used to fill in the missing pixels so that the image looks like the original, uncorrupted version. As well, object removal can be accomplished with image completion. The pixels corresponding to the object can be removed, and then an image completion algorithm can fill in the missing pixels values in accordance with the rest of the image.\\
This can be a challenging problem, because in some cases a large portion of the image is removed, and the algorithm must try and recover the missing information using only the remaining parts of the image. In fact, this is not a problem most humans perform well at, as it takes artistic abilities to make an image look natural. My problem will face further challenges, as I chose to use very small images which therefore contain less information in them. My approach was to remove a square from the center of each image using a mask and then apply a variety of different convolutional networks to the masked images, to see which could best predict the original images. I drew inspiration for my network architectures from some of the networks presented in the next section.
\section{Related work}
Current state of the art in image completion largely relies on GANs (Generative Adversarial Networks), which are able to learn a distribution for images, and then try to fill in the missing pixel values according to this learned distribution. These networks are able to perform incredibly well even when a large portion of the original image is removed. However, GANs, and in particular DCGANs (Deep Convolutional Generative Adversarial Networks), require huge amounts of data, are typically only trained and used on one class of images (e.g. faces, scenery, or cats), and are employed on larger images which contain more pixels and therefore more information [2]. \\
Prior to DCGANs appearance, most image completion algorithms did not rely on machine learning, and instead focused on mathematical methods for in-painting. One interesting algorithm performed in-painting by scouring a large corpus of images to find semantically similar ones, and then in-painted the corrupted image by taking fragments from these similar corpus images [5]. Other techniques tried to find similar patches within the same image that could fill in the missing pixels while satisfying certain constraints [4].\\
In researching prior work on image completion, I noticed parallels between image completion and image segmentation, where the input is an image and the output is a map of which pixels correspond to different objects in the image. Both of these tasks require high dimensional outputs, which is different from the most common computer vision task, classification. As well, if an algorithm can identify which pixels correspond to an object, then I theorized if some of those pixels are removed, it can more easily in-paint them by using information from the object. These ideas lead me to consider networks used in image segmentation. \\
The Fully Convolutional Network (FCN) used by Long and Shelhamer was one of the first networks to perform exceptionally well on image segmentation. Unlike previous work, their network only used convolutional layers and could therefore be used on inputs of arbitrary sizes [7]. The simplicity of their approach along with the high performance motivated me to try this model as my initial approach.\\
SegNet built on FCN by employing an encoder-decoder architecture where the image was first transformed into a low dimensional encoding, and then that low dimensional encoding was up-sampled to produce an output of the same size as the input [3]. The idea of using encoders and decoders has yielded astounding results in fields outside of image segmentation as well, most notably in machine translation.
\section{Dataset and Features}
\begin{wrapfigure}{r}{0.25\textwidth}
\centering
\includegraphics[width=0.25\textwidth]{All}
\caption{\small Sample Training Example}
\end{wrapfigure}
As mentioned, I wanted to see how image completion algorithms would fare on smaller datasets with both fewer training examples and smaller image sizes. As well, I wanted the dataset to include multiple object classes, to see how well an algorithm could perform under tougher conditions. I settled on the CIFAR-10 dataset because it met all of these conditions. CIFAR-10 is a subset of the Tiny Images dataset which contains 80 million 32x32 RGB images. CIFAR-10 includes just 10 object classes, with 6,000 images per class for a total of 60,000 32x32 RGB images. These classes vary widely, from airplanes to frogs to trucks [6]. The data was split into 50,000 train images, 5,000 development images, and 5,000 test images. Since CIFAR-10 was intended for object classification, some pre-processing was required to transform the dataset.\\
Instead of feeding each image to the network directly, a mask was applied that zeroed out the center 8x8 pixels. This 8x8 region was now what the network was trying to predict. Therefore, the true label for each image was the inverse mask applied to the image, resulting in an 8x8 RGB label. This label was flattened into a 192 (8x8x3) dimensional vector, so that it could be more easily fed into a loss function. The pixel values were also scaled from the usual [0,255] range down to the [0,1] range. An example of this process is shown in Figure 1.
\section{ Methods }
In total, 6 different networks were trained and compared. While there were 6 total networks, they fall into just 3 different categories: fully convolutional network, convolutional network with fully connected layers, and encoder-decoder convolutional network. Regardless of the algorithm used, each network was given a masked-out image and had to predict the 192-dimensional output. Mean squared error was used as the loss function for all of the algorithms. It computes the mean squared difference between each element in the predicted 192-vector and the label 192-vector. This means that it compares how close each channel value (red, green, and blue) is in the prediction versus the ground truth. By optimizing mean squared error, the algorithms learn to match the prediction to the true value for each pixel. The formula for mean squared error is as follows:
$$MSE = \frac{1}{192} \sum_{i=1}^{192}(y_i - \hat{y}_i)^2 $$
As well, all the networks were trained using mini-batch gradient descent with the Adam optimization algorithm with variable learning rates which were hand tuned. All layers (except max pooling and flattening layers) use rectified linear activations. Since the final per channel output should be in the range [0,1], sigmoid activations were used at first. However, they proved to work poorly and instead the rectified linear activation was used even for the final layer, and it was clipped to be in the range [0,1].\\
3 fully convolutional networks were trained: a shallow, deep, and super deep network. By only including convolutional and max pooling layers and no deconvolutions or fully connected layers, these networks did not have access to the same amount of global information as the other networks and were theorized to over-fit the training data less. The deeper networks not only have more layers, but they also have more filters per layer. This allows them to learn more features from the input image (e.g. texture patterns, lines, etc.). The kernels used along with the input size at each layer are as follows, where the part in brackets is the kernel applied and number of filters used is in parentheses. For example, 32x32x3 ->[5x5 Convolution (10)]-> 28x28x10 means that 10 5x5 filters were applied to the 32x32x3 input and resulted in a 28x28x10 output for that layer. Note: all convolutions are "valid" convolutions unless otherwise noted, and all max pooling layers are 2x2 filters with strides of 2.\\
Shallow Network: 32x32x3 ->[5x5 Convolution (10)]-> 28x28x10 ->[5x5 Convolution (20)]-> 24x24x20 ->[Max Pooling]-> 12x12x20 ->[5x5 Convolution (20)]-> 8x8x20 ->[1x1 Convolution (3)]-> 8x8x3 ->[Flatten]-> 192x1\\
Deep Network: 32x32x3 ->[5x5 Convolution (20)]-> 28x28x20 ->[5x5 Convolution (40)]-> 24x24x40 ->[Max Pooling]-> 12x12x40 ->[3x3 Convolution (60)]-> 10x10x60 [3x3 Convolution (80)]-> 8x8x80 ->[1x1 Convolution (3)]-> 8x8x3 ->[Flatten]-> 192x1\\
Super Deep Network: 32x32x3 ->[7x7 "Same" Convolution (40)]-> 32x32x40 ->[7x7 "Same" Convolution (40)]-> 32x32x40 ->[7x7 Convolution (40)]-> 26x26x40 ->[7x7 Convolution (60)]-> 20x20x60 ->[5x5 Convolution (60)]-> 16x16x60 ->[Max Pooling]-> 8x8x60 ->[5x5 Convolution (60)]-> 4x4x60 ->[4x4 Convolution (192)]-> 1x192\\
A convolutional network with fully connected layers was also trained. This model was based on the models typically used in object classification, where first convolutions are applied, and then the output is flattened and fed into a series of fully connected layers and finally a softmax layer. However, my network differed from this in that I did not have a softmax layer, and instead of having just one output, I had 192. Following the previous conventions, the network is as follows.\\
Fully Connected Network: 32x32x3 ->[5x5 Convolution (10)]-> 28x28x10 ->[5x5 Convolution (20)]-> 24x24x20 ->[Max Pooling]-> 12x12x20 ->[5x5 "Same" Convolution (30)]-> 12x12x30 ->[3x3 "Same" Convolution (40)]-> 12x12x40 ->[Flatten]-> 5,760x1 ->[Fully Connected Layer]-> 768x1 ->[Fully Connected Layer]-> 192x1\\
The last models tried were 2 encoder-decoder networks. These models first created a lower dimensional representation of the missing pixels, and then use deconvolutions to up-sample the representation and create the final output. By first creating a lower dimensional encoding of the missing pixels, encoder-decoder networks tend to over-fit less, as they can only store so much information in the lower dimension. The deconvolutions used were all 3x3 kernels with strides of 2. Following the previous conventions, the networks are as follows.\\
Encoder Network: 32x32x3 ->[5x5 Convolution (10)]-> 28x28x10 ->[5x5 Convolution (20)]-> 24x24x20 ->[Max Pooling]-> 12x12x20 ->[5x5 Convolution (30)]-> 8x8x30 ->[5x5 Convolution (128)-> 4x4x128 ->[3x3 Deconvolution (3)]-> 6x6x3 ->[3x3 Deconvolution (3)]-> 8x8x3 ->[Flatten]-> 192x1\\
Deep Encoder Network: 32x32x3 ->[5x5 Convolution (30)]-> 28x28x30 ->[5x5 Convolution (40)]-> 24x24x40 ->[Max Pooling]-> 12x12x40 ->[5x5 Convolution (40)]-> 8x8x40 ->->[5x5 Convolution (40)]-> 4x4x40 ->[Flatten]-> 1x640 ->[Fully Connected Layer]-> 1x768 ->[3x3 Deconvolution (40)]-> 2x2x40 ->[3x3 Deconvolution (40)]-> 4x4x40 ->[3x3 Deconvolution (3)]-> 8x8x3 ->[Flatten]-> 192x1\\
\section{Experiments/Results/Discussion}
Various hyper-parameters were experimented with over the course of this project, including the different model architectures, activation functions, and learning rates. I began with the idea to try at least one of each type of architecture: fully convolutional network, convolutional network with fully connected layer, and encoder-decoder network.\\
For the learning rate, I experimented with different values ranging from .005 to .00001. However, this was all done by looking at the development error and hand tuning the learning rate for each model. As well, I decreased the learning rate as the models trained.\\
The first model tested was the shallow fully convolutional network (Shallow Network). With this network, I tested the sigmoid verses rectified linear activation for the final layer, and quickly saw that the sigmoid had poor mean squared error compared to the rectified linear activation. So, for all subsequent models, I only used the rectified linear activation (clipped to be in the range [0,1]); although it is possible the sigmoid activation would have worked better for some of these networks, it would have doubled the number of models I had to test. Using the rectified linear activation, the algorithm achieved a train and development loss that were approximately equal, .0153 and .0154, respectively. Figure 2a shows the epoch vs loss graph for this model (the first 5 epochs are removed for all graphs). While I did not know what Baye's error would be for this task, after analyzing the images output by the network, they were not as good as I knew was possible. This lead me to believe that the algorithm had a bias problem and lead me to my second model.\\
\begin{figure}[h]
\vspace{-.25in}
\centering
\begin{subfigure}{.33\textwidth}
\includegraphics[width=\textwidth]{basic_601}
\caption{\small Shallow Network}
\end{subfigure}
\begin{subfigure}{.33\textwidth}
\includegraphics[width=\textwidth]{superDeep_321}
\caption{\small Super Deep Network}
\end{subfigure}
\begin{subfigure}{.33\textwidth}
\includegraphics[width=\textwidth]{deep_870}
\caption{\small Deep Network}
\end{subfigure}
\caption{\small Epoch vs Loss Graphs}
\vspace{-.1in}
\end{figure}
Super Deep Network had orders of magnitude more parameters due to having more layers and more filters per layer. This allowed the model to learn more complex features than Shallow Network. However, the network took up to a minute to run per epoch, compared to around 5 seconds for Shallow Network. As well, it became apparent that there was not enough data in the training set to feed this deeper network, as it was able to over-fit the training set with a train loss of .0161 versus a development loss of .0253 after 320 epochs. Figure 2b shows the epoch vs loss graph for this model. While the train loss would likely have continued to decline if I trained for more epochs, it was clear the model was overfitting the data and the development loss was increasing every epoch. I believe that more data likely would have reduced a lot of the variance, and I considered adding the CIFAR-100 dataset (60,000 additional tiny images) to the training set or performing data augmentation on CIFAR-10. However, since part of my goal was to see how these algorithms performed on limited data, I decided against it and instead tried a new network, which was somewhere between Super Deep Network and Shallow Network in complexity.\\
The next model, Deep Network, had 5 convolutional layers unlike Shallow Network's 4 and Super Deep Network's 7. As well, it had more filters than Shallow Network, which I believed to be the biggest reason Shallow Network could not fit the data. Deep Network was able to perform much better than either previous network, achieving train and development losses of 0.013620 and 0.014624, respectively, after 870 epochs. Figure 2c shows the epoch vs loss graph for Deep Network.\\
Next, I built a model similar to Shallow Network but with 2 fully connected layers at the end. This model, Fully Connected Network, was able to fit the training set exceptionally well. Figure 3a shows the epoch vs loss graph for this model. However, it was unable to achieve development loss as low as Deep Network. The network clearly had a huge variance problem and lead me to abandon exploring more fully connected networks. I believed the issue was that predicting a high dimensional output using fully connected layers was too ripe for over-fitting.\\
The final class of models, the encoder-decoder networks, both performed reasonably well. Encoder Network achieved minimum development loss after 440 epochs, with a train and development loss of 0.015688 and 0.016934, respectively. Figure 3b shows the epoch vs loss graph for this model. After training for another 200 epochs, the algorithms development loss increased slightly and its training loss decreased slightly; however, it appeared to have almost leveled off, signifying that the model likely was under-fitting the data.\\
The final model, Deep Encoder Network, however over-fit the data. It achieved a training loss as low as .010 and development loss of .018 after 440 epochs. Figure 3c shows the epoch vs loss graph for this model. This network utilized a fully connected layer to help make the encoding, which could have contributed to it overfitting the training data. \\
\begin{figure}[h]
\vspace{-.25in}
\centering
\begin{subfigure}{.33\textwidth}
\includegraphics[width=\textwidth]{FC_521}
\caption{\small Fully Connected Network}
\end{subfigure}
\begin{subfigure}{.33\textwidth}
\includegraphics[width=\textwidth]{encoder_600}
\caption{\small Encoder Network}
\end{subfigure}
\begin{subfigure}{.33\textwidth}
\includegraphics[width=\textwidth]{deepEnc_461}
\caption{\small Deep Encoder Network}
\end{subfigure}
\caption{\small Epoch vs Loss Graphs}
\vspace{-.1in}
\end{figure}
In order to better understand the mistakes the algorithms were making, I analyzed some of the training and development images output by the networks (Note: the algorithms only output the center square and I recombined this with the input image). In Figure 4, I compared the output of Deep Network versus the output of Fully Connected Network on two sample development set images. While I used means squared error as my loss function, my end goal was an algorithm that could fill in the missing pixels to create realistic looking images. Therefore, here I was looking to see whether a human could tell that the algorithms' outputs were not real images. In particular, I was trying to see if it was obvious that the 8x8 center region had been altered. Looking at 4b, this image appeared realistic enough to be an unmodified image. However, in 4c, the center 8x8 region clearly had been altered. In comparing 4b and 4c with the original image, 4a, it was easier to see that both images had been modified. However, the Deep Network's output was still pretty good.\\
One common area where all the algorithms failed was images with sharp edges. Looking at 4e and 4f, both algorithms blurred the edge between the truck's cab and trailer, while the original image 4d had a sharp line here. It is possible that adding more filters to the initial layers of the networks would allow the algorithms to better identify edges; however, this would likely have lead to the over-fitting problems that plagued the deeper networks. This also showcases how mean squared error can be a bad metric at times. If the algorithms had output a straight line of red pixels on one side and white pixels on the other, but shifted the line to the left or white of where it was originally, the algorithm would have incurred a large MSE, but the image would have looked realistic. Instead, by blurring the line somewhat, it ensures that the pixel values are not too far off what they should be, and thus incurs a lower MSE on average. I looked into alternative loss functions to mitigate this issue, but did not find one which I felt worked as well as MSE.\\
\begin{figure}[h]
\vspace{-.2in}
\centering
\begin{subfigure}{.2\textwidth}
\includegraphics[width=\textwidth]{deep_orig_img_515}
\caption{\small Original Image}
\end{subfigure}
\begin{subfigure}{.2\textwidth}
\includegraphics[width=\textwidth]{deep_pred_img_515}
\caption{\small Deep Network}
\end{subfigure}
\begin{subfigure}{.2\textwidth}
\includegraphics[width=\textwidth]{sig_pred_img_515}
\caption{\small Fully Connected Network}
\end{subfigure}
\begin{subfigure}{.2\textwidth}
\includegraphics[width=\textwidth]{deep_orig_img_4493}
\caption{\small Original Image}
\end{subfigure}
\begin{subfigure}{.2\textwidth}
\includegraphics[width=\textwidth]{deep_pred_img_4493}
\caption{\small Deep Network}
\end{subfigure}
\begin{subfigure}{.2\textwidth}
\includegraphics[width=\textwidth]{sig_pred_img_4493}
\caption{\small Fully Connected Network}
\end{subfigure}
\caption{\small Sample Network Outputs}
\vspace{-.2in}
\end{figure}
\section{Conclusion/Future Work }
The goal of this project was to determine whether machine learning image completion algorithms could be trained on small images and small datasets and perform well. Figure 5 shows the MSE for the different algorithms on the test set, where the models are tested at their lowest development loss. From this chart and the development errors listed previously, Deep Network was the best performing model, with a test MSE of .015.\\
\begin{wrapfigure}{r}{0.25\textwidth}
\centering
\includegraphics[width=0.25\textwidth]{chart}
\caption{\small Test Losses}
\end{wrapfigure}
After analyzing some of the test images output by Deep Network, many of them looked realistic. However, images with certain defining features were almost never predicted correctly (as shown in Figure 4). This problem plagued all the networks and is one of the key benefits of using GANs. For instance, if an image of a human face with the eyes removed was fed to Deep Network, it would likely fill in the missing pixels with normal skin, as it performs a type of weighted average of the pixels surrounding the removed values. However, a GAN may have learned the distribution of human faces and recognize that it needs eyes in those locations to look realistic. This issue is in part due to my networks being trained on multiple classes of images, while GANs are often only trained on one class.\\
While the Deep Network performed the best on this dataset, if I increased the amount of training data, the deeper, more complex models would have likely outperformed it. This highlights one of the key issues in deep learning: you often need more data if you want to train a deeper model.\\
Given more time, I would like to experiment with this theory and try my models on a larger dataset with more images and possibly larger images. ImageNet's 64x64 RGB dataset would be perfect for this task and I think the deeper models would perform very well. Using larger datasets, I would likely be able to have more filters without risking over-fitting my training set. These additional filters could hopefully learn more complex features such as sharp edges, so that the algorithms would not make the same kinds of mistakes as they did on CIFAR-10.\\
As well, I would like to experiment more with encoder-decoder networks. I was only able to experiment with two different sized encodings, but I would like to see how larger and small dimensional encoding affect performance. I theorize that smaller encodings would reduce over-fitting and lead to interesting results. As well, it would be interesting to build a model that included encodings and had a skip connection with the input image, so that the model would have an easier time maintaining important features like sharp lines.
\section*{References}
\medskip
\small
[1] M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen,
C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat,
I. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia,
R. Jozefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mane,
R. Monga, S. Moore, D. Murray, C. Olah, M. Schuster,
J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. Tucker,
V. Vanhoucke, V. Vasudevan, F. Viegas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng. 2015. TensorFlow: Large-scale machine learning on heterogeneous systems. www.tensorflow.org.\\
[2] Amos, B. 2018. Image Completion with Deep Learning in TensorFlow. bamos.github.io/2016/08/09/deep-completion/.\\
[3] V. Badrinarayanan, A. Kendall and R. Cipolla. 2017. SegNet: A Deep Convolutional Encoder-Decoder Architecture for Image Segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 39, no. 12, pp. 2481-2495, Dec. 1 2017. doi: 10.1109/TPAMI.2016.2644615.\\
[4] C. Barnes, E. Shechtman, A. Finkelstein, and D. Goldman. 2009.
Patchmatch: a randomized correspondence algorithm for
structural image editing. TOG, 2009.\\
[5] Hays, J., and Efros, A. A. 2007. Scene completion using
millions of photographs. ACM Trans. Graph (SIGGRAPH 2007)
26, 3.\\
[6] A. Krizhevsky. 2010. Convolutional Deep Belief Networks
on CIFAR-10. www.cs.toronto.edu/~kriz/cifar.html.\\
[7] Long, J., Shelhamer, E., and Darrell, T. 2014. Fully convolutional networks for semantic segmentation. CoRR,
abs/1411.4038.\\
\end{document}
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\section{Simulation details}
The scattering function $S(\mathbf{k})$ (or structure factor) shown in the insets of Fig.~3\ of the main text is calculated using a fast Fourier transform $\hat{\rho}_\text{inst}(\mathbf{k})$ of the \emph{instantaneous} density profile of the corresponding snapshot binned on a two-dimensional grid: $S_\text{inst}(\mathbf{k})=\lvert \hat{\rho}_\text{inst}(\mathbf{k}) \rvert^2/N$. The method of plotting the scattering profile results in a slight smoothing of the peaks. As a result, the peaks at $\mathbf{k}=(2\pi/a)\,(\pm 1,\pm 1)$, which are actually essentially point-like, obtain a finite width rendering them visible. For the results in Fig.~4(a) of the main text, a more precise calculation of the average scattering function was used:
\begin{equation
S(\mathbf{k}_R)=\frac{1}{N} \left\langle \left\lvert \sum_{n=1}^N e^{\mathrm{i} \mathbf{k}_R \cdot \mathbf{r}_n} \right\rvert^2 \right\rangle,
\label{eq:S_kR}
\end{equation
which was averaged over multiple configurations during the simulation and over the different symmetry related $\mathbf{k}_R$ vectors. The coloring of the particles in Fig.~3\ of the main text is performed as follows, the degree to which particle $i$ are orange and light blue is equal to $S_x^{(R)}(i)$ and $S_y^{(R)}(i)$ repectively, where
\begin{equation}
S_{x}^{(R)}(i)\equiv \max_{\sigma=\pm 1}\frac{1}{N_i} \sum_j \cos\left(\tfrac{2\pi}{a} (\hat{x} + \sigma \hat{y}/2 ) \cdot [\mathbf{r}_j-\mathbf{r}_i]\right)
\end{equation}
and $S_{y}^{(R)}(i)$ is given by the same expression with $x$ and $y$ interchanged. The sum over $j$ is limited to the $N_i$ particles that are in a square of width $5 a$ centered around the position $\mathbf{r}_i$ of particle $i$.
\section{Direct comparison of the results from simulations and theory}
In order to clarify the similarities of simulations and theory, first the density profiles obtained from DFT are computed for the state points marked by black stars in Fig.~2(a) and the results are shown in Fig.~\ref{fig:supmat_1}. Similar to the simulation snapshots shown in Fig.~3, particles in the square phase at high interaction strength $V_0$ are strongly pinned or in other words, the density peaks are very high (see Fig.~\ref{fig:supmat_1}(c)). At low $V_0$ (Fig.~\ref{fig:supmat_1}(a)), the square phase is a modulated fluid with much lower density peaks.
\begin{figure}[tb]
\includegraphics[width=0.9\textwidth]{figure_s1}
\caption{Density contour plots of 6x6 squares for fixed packing fraction $\eta=0.69$ and external field strength (a) $V_0/k_BT=1$, (b) $V_0/k_BT=2$ , and (c) $V_0/k_BT=4$, similar to those indicated by stars in Fig.~2.}
\label{fig:supmat_1}
\end{figure}
In DFT, the rhombic phase spans the whole system (see Fig.~\ref{fig:supmat_1}(b)), whereas a simulation snapshot consists of regions of rhombic and regions of square symmetry. As the state point for the rhombic phase is very close to the phase boundary, the rhombic structure is hardly visible but it can be confirmed by analyzing the profiles.
In analogy to the analysis of the results from our MC simulations, a structure factor related quantity can be calculated. The Fourier transform of the mean density distribution $\hat{\rho}({\bf k})$ is used as
\begin{equation}
S'({\bf k})=\frac{\lvert\hat{\rho}({\bf k})\rvert^2}{S_0},
\end{equation}
conatining a normalization factor $1/S_0$. Here, the average and the Fourier transform are taken in opposite order compared with Eq.~\ref{eq:S_kR}. The results are in good agreement with those presented in Fig.~3. There is an intermediate peak at ${\bf k}_R=\tfrac{2\pi}{a}(1/2,1)$ only in the rhombic phase. In the simulations, the shift of particles from the square to rhombic phase occurs in the $x$ as well as the $y$ direction forming different clusters. In contrast, DFT yields only one way of dislocation. Thus, the intermediate peaks can only occur in one of the two possible directions. In the inset of Fig.~4(a), we plot the peak height at ${\bf k}_R$ as a function of $V_0/k_BT$. All $S'({\bf k})$ are normalized with a factor $1/S_0$ such that the peak height obtained from DFT agrees with that of the black curve in Fig.~4(a) resulting from our simulations.
\begin{figure}[t]
\includegraphics[width=0.9\textwidth]{figure_s2}
\caption{Structure factor $S({\bf k})$ obtained from DFT at the state points indicated in Fig.~\ref{fig:supmat_1}}
\label{fig:supmat_2}
\end{figure}
Additional to the analysis of a square substrate, an external potential with rectangular symmetry can be induced for a system with on average one particle per substrate minimum. This setup also yields a rhombic preordering as shown in a density contour plot in Fig~\ref{fig:supmat_3}.
\begin{figure}[htb]
\includegraphics[width=0.5\textwidth]{figure_s3}
\caption{Rhombic phase at packing fraction $\eta=0.69$ on a rectangular pattern with size ratio 1:1.2 and external field strength $V_0/k_BT=1.5$ indicated by black crosses.}
\label{fig:supmat_3}
\end{figure}
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\section{Introduction}
With the first direct observations of incident gravitational waves
(GWs) expected in only a few years, an
exciting new window on the Universe is about to be opened. But our
interpretation of these observations will be limited by our
understanding of how information about the GW sources is encoded within the
waves themselves. The parameter space of likely sources is large, and
compact binary systems consisting of two black holes (BHs), two neutron stars,
and one of each are likely to be the most promising
sources. However, filling this parameter space with the corresponding
theoretical gravitational waveforms
through merger---when GWs are strongest and many binary parameters
are most distinguishable---will require a large number of
computationally expensive numerical relativity (NR) simulations.
At the heart of the computational challenge lies an arguably even
greater theoretical one, which seeks to find an optimal approach for
solving Einstein's equations on the computer. These approaches generally decompose
the intrinsically four-dimensional set of Einstein's field equations
into time-evolution and constraint equations---similar to
Maxwell's equations \cite{Baumgarte_2010}. Once the data on the initial 3D spatial
hypersurface are specified, the time-evolution equations are
repeatedly evaluated, gradually building the four-dimensional
spacetime one 3D hypersurface at a time.
Einstein's equations guarantee the freedom to choose coordinates in
the 4D spacetime, and these are specified via a set of gauge
evolution equations. Devising robust gauge conditions,
particularly when black holes inhabit the spacetime, has been a
problem at the forefront of NR for decades. In fact, discovery of gauge
conditions leading to stable evolutions in the presence of moving black holes played a large
role in the 2005 numerical relativity revolution, culminating in the
first successful binary black hole (BBH) inspiral and merger
calculations~\cite{Pretorius:2005gq,Campanelli:2005dd,Baker:2005vv}.
The most widely adopted formulation for compact binary evolutions
is the highly robust ``BSSN/moving puncture'' (hereafter BSSN/MP)
formulation \cite{Campanelli:2005dd,Baker:2005vv}, which combines the
Baumgarte-Shapiro-Shibata-Nakamura (BSSN) 3+1 decomposition of
Einstein's equations
\cite{Nakamura:1987zz,Shibata:1995we,Baumgarte:1998te} with the
``moving puncture'' (MP) gauge conditions that implement the {\tt 1+log}/$\Gamma$-freezing gauge
evolution equations \cite{Alcubierre:2002kk,vanMeter:2006vi}. It
has been about a decade since BSSN/MP was first developed, and it
remains in use by much of the NR community, largely unmodified. This
is a testament to the robustness of the MP gauge choice, as well as the
difficulties associated with devising stable techniques for evolving
spacetimes containing BHs.
The BSSN/MP equations are typically solved on the computer with
high-order finite-difference approaches on adaptive-mesh refinement
(AMR) numerical grids
\cite{Zlochower:2005bj,Baker:2006yw,Bruegmann:2006at,Campanelli:2007ew,Herrmann:2007ac,Sperhake:2006cy,Etienne:2007jg,Pollney:2007,Pollney:2009yz}. Such
grids are essential for computational efficiency, as the simulation
domain must be hundreds to thousands of times larger than the initial binary
separation to causally disconnect the outer boundary from the physical
domain of interest for the duration of the simulation. This effectively prevents errors linked to the
enforcement of approximate outer boundary conditions from propagating
inward and lowering the convergence order of our gravitational waveforms. In
addition, very high resolution must be used in the strongly curved
spacetime near the BHs, while resolution requirements are much lower far from
the binary, where GWs with wavelengths of order $10M$
must be accurately propagated ($M$ is the total ADM mass of the
system). Thus, AMR grids in MP calculations
typically resolve the strong-field region best, with progressively
lower resolutions away from this region to the outer boundary. Typical
gridspacings at the outer boundary can be $\sim 1000$ times larger
than in the strong-field region.
The BSSN/MP+AMR paradigm has been most thoroughly tested in the
context of BBH evolutions, and a large number of theoretical BBH
GWs have been produced with this paradigm that widely sample parameter
space (e.g., \cite{Lousto:2013wta,Pekowsky:2013ska,Hinder:2013oqa}).
However, when the BSSN/MP+AMR paradigm is pushed to very high
accuracies or to difficult regions of BBH parameter space, it has been
found (e.g., \cite{Zlochower:2012fk}) to yield inconsistent
gravitational waveform convergence with increasing resolution,
making error estimates difficult, and effectively limiting waveform
accuracy. In \cite{Zlochower:2012fk} it was hypothesized that these
convergence issues might be linked to noise from
high-frequency waves generated at grid refinement boundaries.
This is a compelling hypothesis, especially in light of
Fig.~\ref{Intro_Ham_constraint_bumps}, which shows
regularly-spaced spikes in L2 norm Hamiltonian constraint violations
on a logarithmic time scale. Interestingly, the onset of each spike
in this BBH evolution is timed almost perfectly to grid refinement
crossings of a wave that starts from the strong-field region at the
onset of the calculation and propagates outward {\it superluminally at
speed $\sqrt{2}c$} (vertical lines). This propagation speed is a
smoking gun, as linearized analyses \cite{Alcubierre:2002iq} indicate only one propagation mode
with that speed in the standard BSSN/MP formulation. This mode
primarily involves the lapse and is governed by the lapse evolution
gauge condition. The connection between spikes in Hamiltonian
constraint violations and the initial outgoing lapse wave has been
noted previously~\cite{Baker:2006yw}.
\begin{figure}
\includegraphics[angle=270,width=0.45\textwidth]{Ham_constraint__step0_with_AMR_bdries.eps}
\caption{L2 norm of Hamiltonian constraint violation
$||\mathcal{H}(t)||$ outside black hole horizons (Eq.~\ref{L2_Ham}),
versus time (solid red curve), for ``standard'' BSSN/MP+AMR BBH
calculation. Dashed blue vertical lines denote the times at which an
outgoing wave traveling at $\sqrt{2}c$---starting from the origin at
the onset of the calculation---encounters a grid refinement
boundary.}
\label{Intro_Ham_constraint_bumps}
\end{figure}
Expanding on this idea, we note that all BSSN evolution equations are
coupled to the lapse and its derivatives, so noise
from reflections in this {\it gauge} quantity can be easily converted
into noise in {\it physical} modes (e.g., noise in GWs) and even constraint
violations. Further,
differencing errors from the part of the wave that is transmitted into
the coarser grid are converted into constraint violations and errors in
non-gauge waves as well.
We expect improved resolution would mitigate this
problem, but even a factor of two increase---the
typical range in most BSSN/MP convergence studies---would enable
evolution of a sharp (short-wavelength) feature through only one
additional coarser level before it succumbs to dissipation through
under-resolution. Additionally, it is likely that zero- or low-speed
modes in Hamiltonian constraint violations (see, e.g.,
\cite{Gundlach:2004jp}) may exacerbate any errors generated as the
initial sharp lapse wave propagates into a coarse grid.
Since sharp features in the initial outgoing lapse wave are likely
responsible for spikes in Hamiltonian constraint violations, as well
as noise and other errors in physical quantities, we can conceive of
at least two strategies for mitigating this effect: (1) adjusting the
initial conditions to minimize gauge dynamics, in a similar vein
to~\cite{Hannam:PRD80:2009}, or (2) modifying the standard MP lapse
evolution equation so that the initial sharp outgoing lapse wave is
stretched and smoothed as it propagates outward from the binary. We
chose the latter strategy. Strikingly, we find our stretching
and smoothing modifications to the lapse evolution equation are
sufficient to reduce constraint violations by {\it more than an order
of magnitude}, confirming our hypothesis, {\it except near the
beginning of our calculations}, where early-time spikes in
constraint violations remain.
We remove these stubborn spikes by initially reducing the timestep on
the BSSN field equations by a factor of $\sim 10^3$, while the
gauge evolution equations are evolved according to the original time
coordinate. This time reparameterization technique increases the gauge
wave speeds to about 30 times the coordinate speed of light, allowing
the gauge evolution to very quickly relax near $t=0$ in response to
the physical fields. As the gauge fields relax, evolution of the BSSN
field variables is slowly accelerated so that their timestep matches
that of the MP gauge evolution, after about $10M$ of evolution in the
un-reparameterized time coordinate. We show this time
reparameterization is, on an analytic level, equivalent to a
time-dependent rescaling of the lapse and shift, leaving the BSSN
field equations unmodified in the reparameterized time coordinate.
When combined, these gauge improvements significantly reduce
constraint violations {\it from the onset of the
calculation through BBH merger}. For example, the L2 norm
Hamiltonian constraint violation outside the BHs is reduced on average
by a factor of 20 at $t>0$ at lowest resolution. Hamiltonian
constraint violations converge at a much higher order in the presence
of our improved gauge conditions, so this factor increases to $\sim
50$ at the highest resolutions attempted.
Turning to gravitational waveforms, we find that in the ``standard''
BSSN/MP+AMR paradigm, noise appears to dominate GW power at wave periods
$\sim 10M$, and this spike in power does not diminish with increasing
resolution. Meanwhile, our ``best'' gauge choice appears to
drop this noise-generated GW power spike by about an order of
magnitude, and its power spectrum at short wave periods {\it does}
drop with increasing resolution. In conjunction with this
removal of waveform noise---and perhaps most significantly---we find that
gravitational waveform convergence is far cleaner when using the
new techniques, enabling more reliable Richardson extrapolations and
possibly paving the way to higher accuracies than the standard
BSSN/MP+AMR paradigm might allow. In fact, our ``best''
gauge choice reduces Richardson-extrapolation-based error estimates of
GW phase and amplitude error during inspiral by about
40--50\%. However, we observe in these waveforms that only the
phase error is reduced significantly at merger.
Our gauge improvements are presented exclusively in the
challenging context of an 11-orbit, equal-mass BBH calculation in
which one BH is nonspinning and the other possesses a spin of
dimensionless magnitude 0.3, aligned with the orbital angular
momentum. Despite our focus on a single physical
scenario, we expect these new techniques will improve the accuracy and
reliability of simulations couple the MP+AMR
paradigm to BSSN and other $N$+1 NR formulations (e.g.,
\cite{Bona:2003fj,Gundlach:2005eh,Bernuzzi:2009ex,Zilhao:2011}),
not only for BBH and single-BH evolutions, but also for binary systems
with matter, such as neutron star--neutron star and black
hole--neutron star binaries. Finally, we anticipate that careful
reparameterization of the time coordinate when the gauge and physical
quantities change most significantly---both at the beginning of
evolutions {\it and during merger}---may help stabilize compact
binary simulations, making it easier to cover difficult regions of
parameter space.
The rest of the paper is organized as
follows. Section~\ref{Basic_Equations} summarizes our basic equations
and gauge improvements, Sec.~\ref{Num_Algs_Techs} outlines numerical
algorithms and techniques, Sec.~\ref{Results} details results from
these improved gauge conditions, and Sec.~\ref{Conclusions} summarizes
both our conclusions and plans for future improvements.
\section{Basic Equations}
\label{Basic_Equations}
In the following sections, all quantities are given in geometrized
units: $G=c=1$. Also, with the exception of Sec.~\ref{Num_Algs_Techs},
we designate quantities in terms of $M$, the total initial ADM mass of
our binary.
\subsection{Initial Data}
The Bowen-York initial data \cite{Bowen:1980yu,Brandt:1997tf}
prescription is used to generate an equal-mass, low-eccentricity BBH
system approximately 11 orbits prior to merger. One BH is nonspinning,
and the other is spinning with initial spin parameter $0.3$ aligned
with the orbital angular momentum. We choose this case because it (1)
avoids many of the symmetries of an equal-mass, aligned-spin system
(though is still symmetric about the orbital plane), (2) requires a
significantly long inspiral calculation to merger, and (3) is one of
the cases featured by the Numerical Relativity/Analytical
Relativity (NRAR) \cite{NRARwebsite,Hinder:2013oqa} collaboration
(NRAR case label {\tt U1+30+00} in \cite{Hinder:2013oqa}).
We adopt standard initial values for the lapse and shift, choosing
$\beta^i(t=0)=0$ and $\alpha(t=0)=\Psi^{-1}_\text{BL}$, where
$\Psi_\text{BL}$ is the Brill-Lindquist conformal factor~(Eq.~5
in~\cite{Bruegmann:2006at}). We devised other choices for initial
lapse and shift that more closely approximate their final, relaxed
values. These choices either had no effect or resulted in amplified
constraint violations. We attribute this to the fact that in the
process of settling, the gauge responds to early gravitational-field
dynamics, and \textit{vice versa}. Thus if one wishes to minimize early
gauge dynamics, a different strategy for specifying the initial gauge that
accounts for the early $t>0$ evolution of the spacetime might be
more effective.
\subsection{Evolution Equations and Diagnostics}
\subsubsection{Evolution Equations for Gravitational Fields}
We employ the standard set of BSSN equations for gravitational field
evolution \cite{Baumgarte:1998te}, except $W=e^{-2 \phi}$
\cite{vanMeter:2006g2n,Marronetti:2007wz} is chosen as the
evolution variable instead of the original BSSN conformal variable
$\phi$.
Such a choice or variant thereof (e.g., $\chi=W^2$; see \cite{Hinder:2013oqa} for
other possibilities) is motivated by the the fact that
finite-difference techniques generally approximate functions by
overlapping polynomials when taking derivatives, and $W$ (or $\chi$)
is better approximated by polynomials than $\phi$, particularly in the
strong-field region ($\phi\to \infty$ at black-hole punctures). Thus
choosing $W$ or $\chi$ as an evolution variable results in less
truncation error at a given resolution than using $\phi$.
\subsubsection{Reparameterization of the Time Coordinate}
\label{sec:TR}
We present a new modification of the BSSN/MP equations, originally
aimed at reducing an early spike of constraint violations in our BBH
evolutions. Understanding one source of these violations requires some
insights into how our AMR driver Carpet \cite{Schnetter:2003rb} works.
AMR drivers are generally faced with the problem of interpolating data
between grids at different resolutions. Carpet solves this problem by
interpolation ({\it prolongation}) in both space and time. Suppose we
wish to evolve data on a fine (high-resolution) AMR grid with spatial
resolution $\Delta x$ from time $t$ to
$t+\Delta t$. Also suppose data already exist on the surrounding coarser grid with
spatial resolution $2 \Delta x$ at $t$ and $t \pm2 \Delta t$. After the fine grid
has been updated, Carpet needs to fill the buffer cells between the
fine and coarser levels. However, data do not exist on the coarser
level at $t+\Delta t$, so coarse-grid data at $t$ and $t \pm2 \Delta t$ are
interpolated in space and time to provide data at $t+\Delta t$.
At the start of the calculation, the initial data solver provides data
at only a single time, $t=0$, but we have chosen second-order time
interpolation to fill buffer zones at grid refinement boundaries. Thus
we need data at three timesteps to proceed. Carpet offers two
algorithms for filling in these two missing timesteps. The first is
to simply copy the data from $t=0$, and the second evolves data at
$t=0$ backward and forward in time to fill them in. The latter
technique reduces truncation errors at a higher order of convergence
than the former, but constraint violations in either case just after
the evolution begins will generally be much larger than at $t=0$, as
the initial data are extremely accurate and are usually generated on
high-resolution spectral grids.
To mitigate the errors at the start of the calculation, we choose to
copy the data from $t=0$ {\it and} reparameterize the time coordinate
so that the timesteps near $t=0$ are incredibly small, compared to
our original time coordinate. The coordinate timestep then very
gradually and smoothly grows to the usual value. In particular, we
find the following prescription works well.
All the BSSN/MP evolution equations can be written in the following
form:
\begin{equation}
\partial_t \mathcal{E} = {\rm RHS},
\label{RHS}
\end{equation}
where $\mathcal{E}$ is any of the evolved variables,
$\partial_t$ denotes the usual partial time-derivative and RHS the
right-hand side of the evolution equation. Our first time
reparameterization scheme, which we call ``TR1'', simply multiplies
\textit{every} ``RHS'' by
\begin{equation}
\label{f_of_t}
f(t) = \Erf(t;10,5).
\end{equation}
Here, we define $\Erf(x;x_0,w)$ as follows
\begin{equation}
\label{Erf_of_x}
\Erf(x;x_0,w) = \frac{1}{2} \left[\erf\left(\frac{x-x_0}{w}\right)+1\right],
\end{equation}
where $\erf(x)$ is the standard, $C^{\infty}$ error
function, $x_0$ marks the center of the error function distribution and
$w$ the width.
Note that although $f(t)$ yields extremely small timesteps early in
the evolution, the timesteps return to the fiducial value
($f(t)\to 1$) smoothly and rapidly. The net effect of TR1 is an
increase in total wallclock time of only $0.5\%$ for our 11-orbit BBH
calculations, equivalent in cost to evolving without time
reparameterization an additional $11M$. Though our time
reparameterization techniques do appear to modify the RHS of the BSSN
field equations, we demonstrate in Appendix~\ref{TR_Appendix} that
Einstein's equations are still satisfied, as both of our time
reparameterization techniques are in fact equivalent to gauge
choices.
Using our ``TR1'' strategy, the initial timestep is $\sim 10^{-3}$
times the initial timestep without time reparameterization, and
we observe a significant reduction in evolution errors when compared
to copying the initial data alone without time reparameterization,
particularly in the momentum constraint. This finding is demonstrated in
Sec.~\ref{Results}.
Though our main objective in reparameterizing the time coordinate was
to reduce errors induced by copying data from $t=0$ to earlier times,
we find we can reduce errors at early times {\it significantly more}
by choosing {\it not} to reparameterize time in the gauge evolution
equations (i.e., the lapse and shift). We call this our ``second''
time reparameterization choice, or ``TR2''. TR2
increases the characteristic speeds of the lapse evolution equation,
of order 30 times the coordinate speed of light initially \footnote{This can be
seen by combining Eq.~\ref{trK_Evol_Eq} after time reparameterization
with Eq.~\ref{oldlapse}, then taking the principal part, as in
Sec.~\ref{basiceqs:gauge_evol_description}.}. The net
effect is accelerated evolution of the gauge relative to the
gravitational fields, causing the coordinates to relax much faster.
We apply two other major modifications to the gauge evolution equations,
including one aimed at stretching outgoing gauge waves and another
focused on adding dissipation terms that significantly reduce
short-wavelength noise. These modifications are described in the next
section.
\subsubsection{Gauge Evolution Equations}
\label{basiceqs:gauge_evol_description}
The usual MP gauge-evolution equations consist of the
``Gamma-driver'' shift condition \cite{Alcubierre:2002kk}, of which we choose the
first-order ``nonshifting-shift'' variant \cite{vanMeter:2006vi}, and the {\tt 1+log}
slicing condition for the lapse~\cite{Alcubierre:2002kk}.
\begin{eqnarray}
\label{gamma_driver_first_order}
\partial_t \beta^i &=& \frac{3}{4} \tilde{\Gamma}^i - \eta \beta^i,\ \ \ \eta=1.375 \\
\label{oldlapse}
\partial_t \alpha &=& \beta^i \partial_i \alpha - 2 \alpha K.
\end{eqnarray}
We refer to these gauge conditions as the ``OldGauge''.
The motivation behind this {\tt 1+log} slicing condition becomes
apparent when combined with the evolution equation for $K$, the trace
of the extrinsic curvature,
\begin{equation}
\label{trK_Evol_Eq}
\partial_t K = \beta^i \partial_i K - D^i D_i \alpha + \alpha \left(\tilde A_{ij}\tilde A^{ij} +\frac{1}{3}K^2\right).
\end{equation}
Taking a time-derivative of Eq.~\eqref{oldlapse}, combining with
Eq.~\eqref{trK_Evol_Eq}, and keeping only the principal parts yields
\begin{equation}
\label{opl_lapse_wave_eq}
\partial_t^2 \alpha \simeq 2 \alpha \gamma^{ij}\partial_i\partial_j \alpha,
\end{equation}
a wave equation with wave speed $\sqrt{2 \alpha}$ (see also
\cite{Alcubierre:2002iq}).
As mentioned previously, our strategy is based on the observation that
the sharp initial outgoing lapse wave appears to generate spikes in the
time evolution of the L2 norm of the Hamiltonian constraint violations when it crosses coarser and
coarser AMR grid-refinement boundaries. Thus we focus on modifying the
{\tt 1+log} slicing condition for the lapse to stretch this initial
wave packet, as well as damp short-wavelength noise in $\alpha$
generated by grid refinement boundary crossings. Gauge freedom
guarantees our ability to do this, and the job is made easier by its
interpretation as a wave equation (Eq.~\ref{opl_lapse_wave_eq}). We
review our improvements to the {\tt 1+log} slicing condition below.
Our improved {\tt 1+log} slicing condition is as follows
\begin{equation}
\label{lapse_master_eq}
\partial_t \alpha =
{\underbrace {\textstyle
f(t) \beta^i \partial_i \alpha
}_{\text{Term (1)}}}
-
{\underbrace {\textstyle
g(t,r) \alpha K
}_{\text{Term (2)}}}
+
{\underbrace {\textstyle
h(r) \nabla_{\rm flat}^2 \alpha
}_{\text{Term (3)}}}
+
\text{KO}.
\end{equation}
Here $r=\sqrt{x^2+y^2+z^2}$.
We now specify the unknown functions and explain each term.
\textbf{Term (1)}: $f(t)=f_0(t)\equiv\Erf(t;10,5)$ (consistent with
Eq.~(\ref{f_of_t})). {\it Note that this is the only term in the
lapse condition that is multiplied by $f(t)$ in the} TR2 {\it time
reparameterization, while all other RHS terms in
Eq.~\ref{lapse_master_eq} are multiplied by $f(t)$ as well in TR1.}
\textbf{Term (2)}: As in Eq.~(\ref{opl_lapse_wave_eq}), $\sqrt{g(t,r)
\alpha}$ provides the wave speed. We stretch outgoing lapse waves by
making $g(t,r)$ a monotonically increasing function of
coordinate radius $r$ (i.e., by accelerating the wave as it propagates
outward). In so doing, one must be careful to avoid exceeding the
maximum local speed allowed by the local Courant-Friedrichs-Lewy
(CFL) condition on each AMR grid.
In practice, however, this is not a highly restrictive constraint, as
the shift evolution equation's $\eta$ term
(\ref{gamma_driver_first_order}) also sets a stringent upper bound on
the timestep \cite{Schnetter:2010cz}, forcing us to
evolve the outermost AMR levels with a very small timestep
anyway. So without significant modification of current techniques,
we may already stably evolve waves in the outer regions of our grid
that propagate much faster than the local coordinate speed of
light. This enables us to choose the lapse acceleration function
$g(t,r)$ to be a steeply increasing
function of $r$, and we are careful to select a function that is
easily extensible to arbitrary binary separations,
choosing the following functional form:
\begin{eqnarray}
\label{g0_of_r}
g_0(r) &\equiv& v_i + (v_o-v_i) \Erf(r;x_1,w_1) \\
&&+ r e^{-4.42} \Erf(r;x_2,w_2), \nonumber
\end{eqnarray}
where we set inner wave speed $v_i=2$, outer wave speed
$v_o=8$, $x_1=30$, $w_1=10$, $x_2=130$, and
$w_2=15$. The variable $x_1$ may be adjusted proportionally up (down) for binaries
with larger (smaller) initial separations. Figure~\ref{Fig:g0_of_r}
shows how this function satisfies
the CFL condition $\sqrt{g(t,r)}<\Delta x/\Delta t$ on all
grids. Notice that this choice leads to a safety factor $\Delta
x/\Delta t/\sqrt{g(t,r)}$ of $\sim 2-50$ that reduces errors in time
and ensures that a moving AMR grid (tracking the trajectory of a BH)
does not cross into a CFL-unstable
region. Notice also that when the lapse wave hits the outer boundary,
its coordinate speed is nearly $10$. In principle, outer boundary
conditions should be adopted to accommodate this radially dependent
superluminal speed, though our outer boundary conditions for the lapse
were not adjusted for the runs presented here, with no
apparent ill effects.
The left panel of Fig.~\ref{Fig:lapse_stretch_demonstration} plots lapse along
the $x$-axis at different times, demonstrating that choosing
$g(r,t)=g_0(r)$ instead of the standard choice $g(r,t)=2$ stretches
the initial outgoing lapse wave. The red-solid curve shows the initial
lapse (the puncture is located at $x/M\approx6$), and the cyan dash-dotted
curve the final, settled lapse. The other curves demonstrate how the
initial lapse wave settles as it propagates away from the strong-field
region (direction of propagation shown by black arrow). In particular, we
find the outgoing lapse wave indeed possesses sharp features. Also, in
the OldGauge case, the wave front crosses $r=140M$ at $t\approx100$,
corresponding to a propagation speed of very nearly $\sqrt{2}$,
consistent with the linear analysis of the lapse evolution
equation. Notice that the choice $g(r,t)=g_0(r)$ does in fact
accelerate and stretch the initial outgoing lapse pulse, leading the
wavefront to the same location as the OldGauge case in about half the
time.
We explore another realization of $g(t,r)$, which, independent of
other terms in Eq.~(\ref{lapse_master_eq}),
introduces a time dependence:
\begin{eqnarray}
\label{g1_of_t_r}
g_1(t,r) &\equiv& [v_i(t) + (v_o-v_i(t)) \Erf(r;x_1,w_1)] \\
&&+ r e^{-4.42} \Erf(r;x_2,w_2). \nonumber
\end{eqnarray}
Here, all parameters are as in $g_0(r)$, except the inner wave speed
$v_i$ is chosen to be a function of time,
\begin{equation}
v_i(t)=0.5 + 1.5 \Erf(t;40,10).
\end{equation}
Notice that at early times, $v_i(t)\approx 0.5$, yielding sub-luminal
lapse speeds in the inner region, providing
a greater stretch, and at times $t \gtrsim 40$, $v_i(t)$ rapidly
increases to $2.0$, which brings $g_1(t,r)\to g_0(r)$. We have
attempted evolutions that maintain sub-luminal lapse speeds in the
inner region, and Hamiltonian constraint violations are significantly
reduced. However, maintaining sub-luminal lapse speeds also resulted
in a large increase in the BHs' irreducible masses as well as a very
strong spike in constraint violations near merger, perhaps due to a
gauge shock (see, e.g., \cite{Alcubierre:2002iq}).
We also attempted a much steeper function for $g(t,r)$ in the outer
regions, keeping a more constant safety factor, leading to wave
speeds $\sim 100$ at the outer boundary. This greatly reduced
constraint violations (Hamiltonian constraint in particular) through
much of the inspiral, but the constraints spiked after the
lapse wave returned from the outer boundary and crossed into the
strong-field region. No such reflection was observed with $g_0(r)$ or
$g_1(t,r)$, as we anticipate that the much slower lapse wave speeds
near the outer boundary enabled dissipation terms to act longer
and more effectively, resulting in significantly less reflection.
\begin{figure}
\includegraphics[angle=270,width=0.45\textwidth]{g0_of_r}
\caption{Local lapse speed parameter $\sqrt{g_0(r)}$ (solid red line),
as compared to the maximum allowed speed by the CFL condition at
each AMR level's most distant point from the origin, at $t=0$ (magenta
squares). Note that this curve is indistinguishable from
$g_1(t\gtrsim50,r)$, as $g_1(t,r) \to g_0(r)$ rapidly at
$t>40$. Note also that in cases for which time reparameterization
techniques are applied, $t$ refers to the reparameterized time
coordinate.}
\label{Fig:g0_of_r}
\end{figure}
\begin{figure*}
\includegraphics[angle=270,width=0.45\textwidth]{lapse_wave_stretch_demonstration_oldgauge_vs_astretch}
\includegraphics[angle=270,width=0.45\textwidth]{lapse_wave_stretch_demonstration_steps2vs7}
\caption{Effect of replacing the lapse speed factor of $2$ in the
``standard'' BSSN/MP {\tt 1+log} slicing condition (OldGauge,
Eq.~\ref{oldlapse}) with other functions. {\bf Left panel:}
Evolved OldGauge lapse along $x$-axis after $\approx 100M$ of evolution
(thick solid brown) is compared with the g0 lapse
(thick dashed magenta) as it crosses the same position, at $t\approx 55M$. For
comparison, initial lapse (thin solid red) is plotted alongside the
settled lapse of the merged BH remnant (thin dot-dashed blue). The black
arrow shows the direction of lapse wave propagation.
{\bf Right panel:} Same as left panel, except the g0
case (thick dashed magenta) is compared to our ``best'' gauge choice
BestGauge (g1+TR2+h+d5d7); thick solid
gray; see Table~\ref{Table_of_Runs} for full details).
}
\label{Fig:lapse_stretch_demonstration}
\end{figure*}
\textbf{Term (3)}: The flat-space Laplacian acts as a dissipative filter
that damps high-frequency oscillations. These oscillations include
the sharp features in the problematic wave pulse, and may include noise
resulting from reflections or interpolation errors at refinement
boundaries.
In constructing an $h(r)$ for our
AMR grids, care must be taken so that the following numerical
stability criterion for parabolic terms will be satisfied:
\begin{equation}
\label{parabolic_CFL_inequality}
h(r) \lesssim \frac{1}{6} \frac{(\Delta x)^2}{\Delta t},
\end{equation}
where $\Delta x=\Delta y=\Delta z$ is assumed, consistent with our
numerical grids. This is not a strict inequality, as it was
derived from a CFL analysis of the standard heat equation, using
centered second-order finite-differencing and Euler time
integration. Our numerical scheme is more sophisticated, so we reduce
$h(r)$ so that the inequality is comfortably satisfied. In addition,
$\Delta t/\Delta x$ is held fixed as we vary $\Delta x$ in our
resolution studies, so with sufficiently high resolution and
$h(r)>0$, the above inequality will always be violated for explicit
time integration. To protect against this, $h(r)$ is reduced
even more when higher resolution is deemed necessary. In general, we
choose $h(r)$ such that it is roughly $1/100$ the right-hand side of
the inequality (\ref{parabolic_CFL_inequality}) at all
radii. Specifically, we choose the following functional form for
$h(r)$:
\begin{equation}
\label{h_of_r}
h(r) = \exp\left(\sum_{n=0}^8 a_n \log(r)^n\right),
\end{equation}
where, in standard floating point notation:
\begin{eqnarray*}
a_i &=& \{ \text{\tt
-8.18859,
-2.43507e-3,
8.19733e-2,}\\ && \text{\tt
2.05629e-2,
6.78098e-4,
-2.97857e-4,}\\ && \text{\tt
1.69112e-5,
1.74269e-6,
-1.23471e-7}\},
\end{eqnarray*}
and $r=0$ is mapped to $r=0.01$. This function guarantees the above
inequality is comfortably satisfied at all resolutions in our
simulations. Figure~\ref{Fig:h_of_r} demonstrates how this function
has been adapted to our chosen AMR grid structure to ensure
inequality~(\ref{parabolic_CFL_inequality}) is satisfied. The gap between the
curves indicates the chosen safety factor of $\sim100$. Note that this
function can be adapted to other binary separations, though not as
easily as $g_0(r)$ and $g_1(t,r)$.
\begin{figure}
\includegraphics[angle=270,width=0.45\textwidth]{h_of_r}
\caption{Parabolic damping factor $h(r)$ (solid red line), as compared
to the maximum factor allowed by the CFL condition
(Eq.~\ref{h_of_r}) at each AMR level's most distant point from the
origin, at $t=0$ (magenta squares). $\Delta x$ corresponding to the
``hhr'' resolution (see Table~\ref{Table_of_Runs}) for which the
smallest grid spacing is $1/58.\bar{6}$, as the highest resolution
provides the most stringent constraint on the parabolic damping
strength.}
\label{Fig:h_of_r}
\end{figure}
\textbf{KO}: This refers to Kreiss-Oliger (KO) dissipation
terms on the lapse, which are high-order derivatives that behave as
an artificial viscosity, except unlike the parabolic term, the
strength of these terms drops to zero as
$\Delta x \to 0$. We choose the following form for the KO terms:
\begin{equation}
\text{KO terms} = d_5(t,r) \text{KO}_5 + d_7(t,r) \text{KO}_7.
\end{equation}
$\text{KO}_5$ and $\text{KO}_7$ are the standard fifth- and
seventh-order KO derivative operators, respectively \cite{Kreiss73}. The strength
factors $d_5(t,r)$ and $d_7(t,r)$ are normally set to values
$<1$. In fact, when using $i^{\rm th}$-order finite differencing
stencils for spatial derivatives, we typically choose $d_{i+1}=0.3$
and zero for all other $d_j$. We find that stronger, mixed-order KO
dissipation on the lapse reduces errors in our BBH calculations beyond
the typical choice. However, we find empirically that care must be
taken to avoid very strong, mixed-order KO dissipation at the puncture
or in the binary region. A significant reduction in constraint
violations is observed when $d_5(t,r)$ and $d_7(t,r)$ are set as
follows:
\begin{eqnarray}
\label{d5}
d_5(t,r)&=& 0.98 \Erf(t;25,5) \Erf(r;15,5) \\
\label{d7}
d_7(t,r)&=& 0.3 A(\phi) + 0.68 \Erf(t;25,5) \Erf(r;15,5),
\end{eqnarray}
where $\phi$ is the BSSN conformal variable, and $A(\phi)=e^{-10 (\phi/0.3)}$
for $\phi>0.3$ (the region just surrounding the apparent
horizons) and 1 otherwise.
In the right panel of
Fig.~\ref{Fig:lapse_stretch_demonstration} we plot the initial
outgoing wave pulse, comparing a case with all the
new gauge features just described (BestGauge) to a case that only
includes one new feature (g0); see Table~\ref{Table_of_Runs} for
details. The difference is dramatic, especially
when compared to the original gauge choice (left plot).
We conclude this section with two comments. First, a note on
well-posedness and hyperbolicity of the new terms. The introduction of
terms (1), (2) and (3) modify the principal part of the BSSN system
and in general a hyperbolicity analysis with terms (1) and (2) would
be needed to demonstrate that the underlying system admits a
well-posed initial value problem. Once strong hyperbolicity is
established with terms (1) and (2), well-posedness of the Cauchy
problem including term (3) can be studied using standard methods that
apply to mixed hyperbolic--parabolic PDEs (see,
e.g.,~\cite{Gustafsson:1995,Gundlach:2006}), because performing a
low-order reduction of Eq.~\ref{lapse_master_eq} yields a second-order
operator which has parabolic-like properties. However, a
well-posedness analysis of this system goes beyond the scope of our
paper. Nevertheless, we have indirect evidence to believe that
mathematical well-posedness can be established because our simulations
are stable, and convergent, and because the solutions obtained with
our new gauge are close to the solutions obtained with the old gauge,
for which the BSSN system becomes strongly hyperbolic and hence admits
a well-posed Cauchy problem.
Second, we note that $g_0(r)$, $g_1(t,r)$,
and $h(r)$ are $C^{\infty}$ for $r>0$, but all three functions possess
a small kink at $r=0$. In addition, $A(\phi)$ has a significant kink
just outside the apparent horizons. This lack of differentiability in
the evolution equations could potentially translate to non-convergent
errors in simulations. To determine if non-smoothness in $g_1(t,r)$
has any significant effect, we devised a ``smoothed'' version of
$g_1(t,r)$,
\begin{eqnarray}
g_{1s}(t,r)&=&g_0(t,r)-r\frac{\partial g_1}{\partial r}(t,0)\nonumber\\
&=&g_0(r)-r\frac{v_0-v_i(t)}{w_1\sqrt{\pi}}\exp\left(-\frac{x_1^2}{w_1^2}\right)
\end{eqnarray}
nearly identical to the
original function at $r/M \gg 1$ but differentiable at $r=0$, and
performed a run at ``lr'' (low) resolution with this new
function. Though our results were unchanged, we would generally
recommend choosing functions that are at least once differentiable
everywhere, including $r=0$.
\section{Numerical Algorithms and Techniques}
\label{Num_Algs_Techs}
To maintain round numbers in the following section, we specify all
quantities in code units. For conversion to physically
meaningful numbers, note that in code units the summed masses of the
BH apparent horizons is 1.0 and the total ADM mass $M$ of the
initial system is 0.99095, though we stress that these BBH
calculations are scale-free in terms of mass.
\subsection{Initial Data Parameters}
To generate BBH initial data, we use the {\tt TwoPunctures} code
\cite{Ansorg:2004ds}, with spectral grids of
resolution $(n_A,n_B,n_{\phi})=(64,64,44)$. These spectral data are
then mapped onto our finite-difference grids via the new spectral
interpolation scheme of \cite{Paschalidis:2013oya}. The full set of
{\tt TwoPunctures}/Bowen-York \cite{Bowen:1980yu}
parameters used to generate these BBH initial data are as specified
for case {\tt U1+30+00} in Table~A2 of \cite{Hinder:2013oqa}. Note that these
parameters were first generated by the AEI group (cf. case {\tt
A1+30+00} in the same table). With these parameters, the bare mass
of the \{spinning,nonspinning\} BH is found to be {\tt
\{0.4698442439908046,0.48811120218500131\}}, and these bare masses
yield ADM masses of \{0.5,0.5\} as measured at spatial infinity on the
individual punctures' interior worldsheets (see,
e.g.,~\cite{Baker:2002gm,Tichy:2003qi} for how this is computed).
\subsection{AMR Grid Parameters}
\label{nummethods:AMRgridparams}
To minimize computational expense, all grids impose symmetry across
the orbital plane ($z=0$), and only the upper-half plane ($z>0$) is
evolved. Evolution equations are evaluated on two nested sets of
Carpet-generated AMR grids, with one set tracking the centroid of each
apparent horizon with half-side-lengths $0.75 \times 2^n$, where
$n=\{0,\dots,3\},\{5,\dots,12\}$ ($n=4$ is skipped, so that the
region around the binary is better resolved). The lowest-resolution
grid (containing the outer boundary) has a half-side-length of 3072,
which is out of causal contact from all gravitational wave extraction
radii, for waves propagating at or below the speed of light, for the
duration of the simulation.
To reduce errors due to our low-order time-evolution (fourth-order)
and AMR time prolongation (second-order) below the level of
higher-order spatial finite differencing and prolongation errors, the
Courant-Friedrichs-Lewy (CFL) factor is set at most to 45\% of its
maximum value for BSSN evolutions (i.e., 0.225 instead of 0.5).
Specifically, the Carpet parameter {\tt
time\_refinement\_factors} is set to {\tt
"[1,1,1,1,1,1,1,2,4,8,16,32]"}, which corresponds to local CFL
factors $\Delta t/\Delta x$ of 0.225 for the highest six levels of
refinement, and $0.225/2^n$ for the $n$th level beyond that, where
$n=\{1,...,6\}$.
For each gauge choice, we perform simulations with four different
maximum grid resolutions, labeled \{lr,mr,hr,hhr\} in order of increasing resolution.
For \{lr,mr,hr,hhr\} runs, the most refined AMR grid (at each
BH) has resolution
$\Delta x=\{1/42.\bar{6},1/48,1/53.\bar{3},1/58.\bar{6}\}$,
corresponding to an average of
$\{37.5,42.1,46.8,51.5\}$ gridpoints across the average diameter of
the apparent horizons (at $t\approx800$), respectively. (Note that the
apparent horizon diameters of the two black holes differ by approximately 4\%). These grids
and resolutions have been carefully chosen so that at all
resolutions, the physical extent of each grid-refinement level is the
same. Unless grids and resolutions are carefully chosen, Carpet will
not respect the desired physical extent of grid-refinement levels,
instead rounding the physical size to the nearest gridpoint, which can
potentially render a convergence study inconsistent.
Spatial and temporal prolongation (i.e., interpolation between AMR
boundaries to fill buffer zones) are set to fifth- and second-order,
respectively. Also, the standard technique for reducing AMR buffer
zones as described in, e.g., \cite{Bruegmann:2006at}, is not applied
here, as there are indications that reducing AMR buffer zones may
result in inaccuracies \cite{Zlochower:2012fk}.
Thus, we require the use of 16 spatial buffer zones at AMR boundaries
(i.e., four ghostzones due to sixth-order upwinded finite-difference
stencils and seventh-order Kreiss-Oliger dissipation, multiplied by
four Runge-Kutta time evolution substeps).
\subsection{Evolution Parameters}
We use the Illinois group's finite-difference (FD) BSSN sector of
their AMR GRMHD code \cite{Etienne:2007jg} for spacetime evolutions,
with sixth-order-accurate upwinded FD stencils for all shift advection
terms, and sixth-order centered FD stencils on all other spatial
derivatives. Explicit fourth-order Runge-Kutta (RK4) timestepping is
used for time-evolution, and the CFL factors on each refinement level
are specified in Sec.~\ref{nummethods:AMRgridparams}.
Seventh-order Kreiss-Oliger dissipation with strength
parameter $\epsilon=0.3$ is applied to all BSSN gravitational field
and shift evolution equations. Kreiss-Oliger dissipation on the lapse
is applied on a case-by-case basis, as specified in
Table~\ref{Table_of_Runs}.
\subsection{Diagnostics}
Apparent horizon tracking and diagnostics (including the monitoring of
irreducible masses) are handled via the {\tt AHFinderDirect} thorn
\cite{Thornburg:2003sf} for all BHs in runs presented here.
We also employ an isolated horizon formalism
\cite{Ashtekar:1998sp,Dreyer:2002mx} diagnostic to monitor black hole
spins and masses throughout the evolutions.
Hamiltonian and momentum constraint violations are monitored via the
following L2-norms:
\begin{eqnarray}
\label{L2_Ham}
||\mathcal{H}|| &=& \sqrt{\int_\mathcal{V} \mathcal{H}^2\ d^3x},\\
\label{L2_Mom}
||\mathcal{M}^i|| &=& \sqrt{\int_\mathcal{V} \left(\mathcal{M}^i\right) ^2\ d^3x},
\end{eqnarray}
where $\mathcal{H}$ ($\mathcal{M}^i$) is the locally computed
Hamiltonian (momentum) constraint
violation (as defined in, e.g., \cite{Etienne:2007jg}) and $\mathcal{V}$
includes the entire simulation volume,
excluding spheres of radius $2.2M$ about each apparent horizon's
centroid. As a point of comparison, the maximum apparent horizon
radius of all BHs in these calculations varies between $\approx 0.23M$
and $0.77M$. Thus in all cases this integral excludes a region
both in and around each BH.
Gravitational waves are extracted at ten radii, equally spaced in $1/r$,
from $r=45.19M$ to $r=192.7M$. In particular, we compute the spin-weight
$-2$ spherical harmonic decomposition of the Newman-Penrose Weyl scalar
$\psi^4_{l,m}$ for all $(l,m)$ modes up to $l=4$. This scalar is
computed using our own heavily-modified version of the {\tt
PsiKadelia} thorn.
In this paper, we focus primarily on results for waveforms measured
at extraction radius $r=68.6M$, for the dominant ($l=m=2$) mode. We
find that for this dominant mode, our qualitative results are
unchanged at other extraction radii. The effect on other modes is
discussed in Sec.~\ref{Psi4noisereduction}.
The energy and angular-momentum ($z$-component) content of the non-radiative part of the
spacetime is estimated through ADM surface integrals at finite radius
\begin{eqnarray}
\label{M_ADM_def}
M(t) &=& \frac{1}{16\pi} \oint_r \left( \gamma_{ij,i} - \gamma_{ii,j} \right) dA^j,\\
\label{J_ADM_def}
J(t) &=& \frac{1}{16\pi} \varepsilon_{ijk} \oint_r \left( x^j K^k_m - x^k K^j_m \right) dA^m,
\end{eqnarray}
where $\gamma_{ij}$ and $K_{ij}$ are the three-metric and extrinsic
curvature, respectively.
\section{Results}
\label{Results}
Table~\ref{Table_of_Runs} lists all runs presented in this paper, and
this section is ordered as follows. First, Sec.~\ref{sec:stepbystep}
demonstrates how constraint violations are reduced as we add our
gauge/evolution modifications, one step at a time, at the lowest
(``lr'') resolution.
Next, Sec.~\ref{sec:noise_reduction} shows how our
improved gauge conditions reduce short-wavelength, high-frequency
noise both in GWs extracted from these calculations, and in the
ADM mass and angular momentum surface integral diagnostics.
Finally, we perform convergence studies at four resolutions, primarily
comparing results using the ``standard'' moving-puncture gauge
conditions (OldGauge) to our most-improved gauge choice
BestGauge (g1+TR2+h+d5d7). These
studies, presented in Sec.~\ref{sec:convergence}, focus on convergence
of BH irreducible masses, constraint violations, and
waveform amplitude, phase, and noise.
\begin{table*}[t]
\caption{Summary of runs performed. See Eq.~(\ref{lapse_master_eq})
and ensuing discussion for explanations of how functions $g(t,r)$,
$h(r)$, $d_5(t,r)$, and $d_7(t,r)$ are generated in our extensions
to the standard {\tt 1+log} slicing condition. ``TR Type''
refers to the chosen time reparameterization, where TR Type 0
denotes no reparameterization, TR Type 1 implies that the RHS of
{\it all} evolved variables is multiplied by $f(t)$
(Eq.~(\ref{f_of_t})), and TR Type 2 indicates that all RHS are
multiplied by $f(t)$ except for the shift RHS and most of the lapse
RHS (as specified in Eq.~(\ref{lapse_master_eq})). Resolutions
\{lr,mr,hr,hhr\} are as defined in
Sec.~(\ref{nummethods:AMRgridparams}).}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Case Name & $g(t,r)$ & TR Type & $h(r)$ & $d_5(t,r)$ & $d_7(t,r)$ & Resolutions \\
\hline\hline
OldGauge
& 2 & 0 & 0 & 0 & 0.3 & lr,mr,hr,hhr\\
\cline{1-2}\cline{7-7}
g0
& $g_0(r)$; Eq.~(\ref{g0_of_r}) & & & & & lr \\
\cline{1-1}\cline{3-3}\cline{7-7}
g0+TR1
& & 1 & & & & lr \\
\cline{1-1}\cline{3-3}\cline{7-7}
g0+TR2
& & 2 & & & & lr \\
\cline{1-1}\cline{4-4}\cline{7-7}
g0+TR2+h
& & & Eq.~(\ref{h_of_r})& & & lr,mr,hr,hhr \\
\cline{1-1}\cline{5-7}
g0+TR2+h+d5d7
& & & & Eq.~(\ref{d5}) & Eq.~(\ref{d7}) & lr,mr,hr\\
\cline{1-2}\cline{7-7}
g1+TR2+h+d5d7
& $g_1(t,r)$; Eq.~(\ref{g1_of_t_r})& & & & & lr,mr,hr,hhr\\
{\it alias:} {\bf BestGauge}
& & & & & & \\
\hline
\end{tabular}
\label{Table_of_Runs}
\end{table*}
\subsection{Step-by-Step Addition of New Features}
\label{sec:stepbystep}
This section adds our evolution and gauge changes step-by-step
at lowest (``lr'') resolution, unfolding the benefits of each
modification. Table~\ref{Table_of_Runs} contains a complete reference
of case names.
\subsubsection{Reduction of Constraint Violations}
\label{sec:stepbystep:constraints}
Figures~\ref{Fig:Ham_step_by_step} and~\ref{Fig:Momx_step_by_step}
show how Hamiltonian and momentum ($x$-component; $y$- and
$z$-components are similar) constraint violations are diminished as new
gauge and evolution techniques are added one at a time, at resolution
``lr''. Notice the most dramatic decrease in constraint violations
occurs between cases OldGauge and g0, except at early
times, where a very early spike in momentum constraint violations
remains (note that $\mathcal{M}^x(t=0)\sim 10^{-5}$). The
upper-left panel in Fig.~\ref{Fig:Momx_step_by_step} demonstrates that
this spike in early-time momentum constraint violation is dramatically
reduced when reparameterizing the time
coordinate such that {\it all} BSSN and gauge variables are evolved
very slowly at early times (i.e., {\it all} evolution equation RHSs
are multiplied by the $f(t)$ of Eq.~\ref{f_of_t}). Such a reduction
demonstrates that this early spike in momentum constraint violation is
likely due to copying data from $t=0$ as described in
Sec.~\ref{sec:TR}. However, there still exists a significant spike
slightly later, at
$t/M\approx 4$, in both Hamiltonian and momentum constraint violations
that was not reduced by multiplying {\it all} evolution equation RHSs
by $f(t)$.
This remaining spike might be related to the initial
settling of coordinates in the strong-field region, so we choose a time
reparameterization such that the lapse {\it and shift} evolution is
greatly accelerated initially (upper-right panels) with respect to the
evolution of all other variables, again as described in
Sec.~\ref{sec:TR}. In this way, the coordinates are able to settle
significantly faster than otherwise. Notice that although this
strategy significantly modifies the timestep only until about $t/M=10$
in these plots, the reduction in constraint violations is
longer-lasting, until $t/M\approx40$
(Fig.~\ref{Fig:Momx_step_by_step}), when apparently other
errors start to dominate and constraint violations match those of
the g0 technique alone. With just these improvements, the
momentum constraint violation maximum during inspiral is on par with
the {\it initial} momentum constraint peak in OldGauge.
About $150M$ after merger, at $t/M\gtrsim 2000$, a large spike in
momentum constraint violation appears, regardless of gauge choice. The
cause of this spike may be similar to that of the initial spikes, which
were significantly reduced by application of a time reparameterization
such that the lapse and shift evolution was significantly
accelerated relative to the BSSN field variables. So it is possible
this late-time spike might be reduced
via another time reparameterization at merger. Alternatively, this
spike may be caused by under-resolution of outgoing, short-wavelength
{\it physical} (i.e., non-gauge) waves generated in the strong-field
region during merger, common horizon formation, and ringdown.
The upper-right panel of Figs.~\ref{Fig:Ham_step_by_step}
and~\ref{Fig:Momx_step_by_step} demonstrate that addition of a
parabolic damping term ($h(r)$ as defined in Eq.~(\ref{h_of_r})
reduces Hamiltonian constraint violations from mid-inspiral to
merger by about 30\%, and momentum constraint violations midway
through evolutions up to 10\%. We anticipate that stronger damping
(corresponding to smaller safety factors for $h(r)$) may be possible
in the outer, weak-field regions of the grid, resulting in further
reductions of constraint violations.
The lower-left panel of Figs.~\ref{Fig:Ham_step_by_step}
and~\ref{Fig:Momx_step_by_step} show that addition of stronger
Kreiss-Oliger damping (as defined in Eqs.~\ref{d5} \& \ref{d7})
reduces momentum constraint violations over time by another 10\% at
most, but Hamiltonian constraint violations are reduced by up to a
factor of 2 early in the inspiral, and roughly 3\% for late inspiral
through merger ($t\gtrsim1000M$). Finally, we attempt an alternative
lapse acceleration function $g_1(t,r)$, which decreases the
lapse propagation speed significantly in the strong-field region
at early times. This change further reduces average constraint violations
during the early inspiral ($t\lesssim 300M$). It is conceivable that
residual constraint violations may be reduced beyond the levels we
were able to attain with gauge improvements alone,
either by parabolically smoothing the constraint-violating degrees of
freedom as in~\cite{Paschalidis:2007ey,Paschalidis:2007cp}, or by
adopting a constraint-damping formalism such as
Z4c~\cite{Bernuzzi:2009ex,Hilditch:2013} or CCZ4~\cite{Alic:2012}.
The overall reduction in constraint violations is shown in the
lower-right panel of Figs.~\ref{Fig:Ham_step_by_step}
and~\ref{Fig:Momx_step_by_step}. Averaged over time, Hamiltonian constraint
violations are reduced by about a factor of 20 and momentum constraint
violations by about a factor of 13.
We conclude this section by stressing that despite nearly two orders
of magnitude reduction in constraint violation, the addition of these
new features
has virtually no impact on computational expense. The next section
describes how these gauge improvements affect noise in important quantities,
such as gravitational waveforms and ADM (mass and angular momentum) surface
integrals.
\begin{figure*}
\includegraphics[angle=270,width=0.45\textwidth]{Ham_constraint__step_by_step__steps1-3}
\includegraphics[angle=270,width=0.45\textwidth]{Ham_constraint__step_by_step__steps3-5} \\
\vspace{0.3cm}
\includegraphics[angle=270,width=0.45\textwidth]{Ham_constraint__step_by_step__steps5-7}
\includegraphics[angle=270,width=0.45\textwidth]{Ham_constraint__step_by_step__steps1vs7}
\caption{Excised L2 norm of Hamiltonian constraint
violation vs time, $||\mathcal{H}||(t)$ (Eq.~\ref{L2_Ham}),
as new techniques are added one-by-one at fixed ``lr''
resolution. Cases are as defined in Table~\ref{Table_of_Runs}.
{\bf Upper-left panel}: $||\mathcal{H}||(t)$ for cases OldGauge (thick brown),
g0 (green dotted), and g0+TR1 (blue dashed).
{\bf Upper-right panel}: $||\mathcal{H}||(t)$ for cases OldGauge (thick
brown), g0+TR1 (blue
dashed), g0+TR2 (magenta solid),
g0+TR2+h (black dotted).
{\bf Lower-left panel}: $||\mathcal{H}||(t)$ for cases OldGauge (thick
brown), g0+TR2+h (black dotted),
g0+TR2+h+d5d7 (gray dashed), and
BestGauge (g1+TR2+h+d5d7; orange solid).
{\bf Lower-right panel}: $||\mathcal{H}||(t)$ for the
BestGauge case, divided by
$||\mathcal{H}||(t)$ for the OldGauge case
(solid red). This ratio is compared to its time-averaged value
($\approx 0.05$, green dashed) and unity (magenta dotted). Values
below unity indicate an improvement (reduction in
$||\mathcal{H}||(t)$).}
\label{Fig:Ham_step_by_step}
\end{figure*}
\begin{figure*}
\includegraphics[angle=270,width=0.45\textwidth]{Mom_constraint__step_by_step__steps1-3}
\includegraphics[angle=270,width=0.45\textwidth]{Mom_constraint__step_by_step__steps3-5} \\
\vspace{0.3cm}
\includegraphics[angle=270,width=0.45\textwidth]{Mom_constraint__step_by_step__steps5-7}
\includegraphics[angle=270,width=0.45\textwidth]{Mom_constraint__step_by_step__steps1vs7}
\caption{Excised L2 norm of $x$-component of Momentum constraint
violation vs time, $||\mathcal{M}^x||(t)$ (Eq.~\ref{L2_Mom}),
as new techniques are added one-by-one at fixed ``lr''
resolution. Cases are as defined in Table~\ref{Table_of_Runs}.
{\bf Upper-left panel}: $||\mathcal{M}^x||(t)$ for cases OldGauge (thick brown),
g0 (green dotted), and g0+TR1 (blue dashed).
{\bf Upper-right panel}: $||\mathcal{M}^x||(t)$ for cases OldGauge (thick
brown), g0+TR1 (blue dashed), g0+TR2
(magenta solid), g0+TR2+h (black dotted).
{\bf Lower-left panel}: $||\mathcal{M}^x||(t)$ for cases OldGauge (thick
brown), g0+TR2+h (black dotted),
g0+TR2+h+d5d7 (gray dashed), and
BestGauge (g1+TR2+h+d5d7; orange solid).
{\bf Lower-right panel}: $||\mathcal{M}^x||(t)$ for the
BestGauge case, divided by
$||\mathcal{M}^x||(t)$ for the OldGauge case
(solid red). This ratio is compared to its time-averaged value
($\approx 0.075$, green dashed) and unity (magenta dotted). Values
below unity indicate an improvement (reduction in
$||\mathcal{M}^x||(t)$).}
\label{Fig:Momx_step_by_step}
\end{figure*}
\subsection{Noise Reduction Features of New Gauge Conditions}
\label{sec:noise_reduction}
In addition to constraint-violation reductions, our new gauge
conditions also act to reduce short-wavelength noise in the GWs
($\psi^4(t)$) as well as the surface-integral representations of
ADM mass and angular momentum. We present these improvements here.
\subsubsection{Reduction of Noise in $\psi^4_{2,2}$}
\label{Psi4noisereduction}
The left panel of Fig.~\ref{Fig:Psi4_noise_reduction} shows how
short-period noise in $\psi^4_{2,2}(t-r)$ is significantly reduced
when we choose a gauge in which the lapse speed is accelerated
radially outward (g0), keeping all else identical to
a standard moving-puncture gauge choice (OldGauge). Notice that
the characteristic period for this noise is $P\sim10M$, which is
made clear in the right panel, which plots the power spectra of
Re($\psi^4_{2,2}(t)$) for cases OldGauge, g0, and
BestGauge (g1+TR2+h+d5d7) during early inspiral
($50\lesssim(t-r)/M\lesssim1160$), when the signal-to-noise ratio of
our numerical waveforms after junk radiation is lowest. Notice that
accelerating the outgoing lapse wave speed has the largest impact on
reducing high-frequency noise in Re($\psi^4_{2,2}(t)$), diminishing
the noise by nearly an order of magnitude at the peak noise frequency
(period of $\approx 10M$). Also, although additional tricks have some
noticeable impact on constraint violations, they appear to contribute
only a small amount to waveform noise at the shortest wavelengths.
Although we concentrate here on the dominant, ($l,m$)=(2,2), mode, we
find significant noise reduction in sub-dominant modes as well. As an
example, Fig.~\ref{Fig:Psi4_44_noise_reduction} demonstrates that in
OldGauge, the amplitude of the ($l,m$)=(4,4) mode of $\psi^4$ is
dominated by noise throughout a significant fraction of the
inspiral. However, in BestGauge the noise is mostly non-dominant. Based on
this and observations of other non-dominant modes, we conclude that
these gauge improvements stand to improve the usefulness of
sub-dominant modes from BSSN/MP evolutions.
\begin{figure*}
\includegraphics[angle=270,width=0.45\textwidth]{Psi4_inspiral_noise_demonstration}
\includegraphics[angle=270,width=0.45\textwidth]{Psi4_noise_reduction_plot}
\caption{Inspiral waveform noise analysis. {\bf Left panel}:
Re($\psi^4_{2,2}$) versus retarded time, for cases OldGauge
(dashed red) and g0 (solid blue). Inset
magnifies noisiest part of waveform, demonstrating that the dominant
noise frequency corresponds to a wave period of $P\approx 10M$ in
the OldGauge case.
%
{\bf Right panel}: Power spectrum of Re($\psi^4_{2,2}$) versus wave period
during inspiral, including data from $(t-r)/M\approx58$ to $1160$,
for cases OldGauge (thin green line: raw data, thick dashed red
line: B\'{e}zier-smoothed data; i.e., fit to a degree-$n$ B\'{e}zier
curve for $n$ data points), g0 (thin magenta line: raw data,
thick solid blue line: B\'{e}zier-smoothed data), and
BestGauge (g1+TR2+h+d5d7; thick dot-dashed cyan line:
B\'{e}zier-smoothed data). All data plotted in this figure are from
``lr'' resolution runs, and Re($\psi^4_{2,2}$) data are from extraction
radius $r=68.6M$. Re($\psi^4_{2,2}$) time series data were
multiplied by the tapering function $\Erf(t;200,50) \times
\Erf(-t;-1300,150)$ prior to the Fourier transform, to smoothly
suppress junk radiation before $\sim 100M$ and the high-frequency
late-inspiral/merger signals after $\sim 1500M$.}
\label{Fig:Psi4_noise_reduction}
\end{figure*}
\begin{figure}
\includegraphics[angle=270,width=0.45\textwidth]{Psi4_44_inspiral_noise_demonstration}
\caption{Amplitude of $\psi^4_{4,4}$ versus retarded time at ``hhr''
resolution, for cases OldGauge (dashed magenta) and BestGauge (solid blue).}
\label{Fig:Psi4_44_noise_reduction}
\end{figure}
\subsubsection{Reduction of Noise in ADM Surface Integrals}
\label{sec:ADMnoise}
It has been found that surface integral representations of ADM mass
$M(t)$ and angular momentum $J(t)$ (Eqs.~\ref{M_ADM_def} and \ref{J_ADM_def})
are noisy and suffer from large drifts in AMR moving-puncture BBH
evolutions~\cite{Marronetti:2007ya}. Though getting around the problem
is usually a matter of recasting the distant surface integral into a
sum of surface plus volume integrals, we find that this may be
unnecessary with suitable gauge
improvements. Figure~\ref{Fig:diag_integral_noise_reduction} plots
$M(t)$ and $J(t)$ surface integrals as measured at $r=114.2M$,
comparing our original moving-puncture gauge choice (OldGauge) with our most
sophisticated modification, g1+TR2+h+d5d7 (BestGauge),
all at ``hhr'' resolution. The overall pattern is a
drop in $M(t)$ and $J(t)$ as the GWs pass through the surface integral
radius. However, our new gauge choices reduce noise in these
surface integral diagnostics by a huge factor during inspiral,
particularly $J(t)$. This demonstrates that noise in the region far
from the BHs is not restricted to GWs, at least throughout the
inspiral. However, after merger a large amount of noise in $J(t)$
remains, despite our gauge improvements.
We believe the lack of noise reduction in $J(t)$ at late times may be
related to the spike in momentum constraint violation observed at
roughly the same (retarded) time (see
Sec.~\ref{sec:stepbystep:constraints}), which we hypothesized
may have something to do with either the rapid settling of the gauge
or the outgoing short-wavelength physical waves at and after merger.
\begin{figure*}
\includegraphics[angle=270,width=0.45\textwidth]{ADM_Mass_noise_reduction}
\includegraphics[angle=270,width=0.45\textwidth]{ADM_J_noise_reduction}
\caption{ADM surface integral noise analysis. {\bf Left panel}:
ADM mass $M(t)$ (Eq.~\ref{M_ADM_def}), as measured by a surface integral at
$r=114.2M$. Large blue points denote OldGauge case and thin cyan
line BestGauge (g1+TR2+h+d5d7). Note that all data are
taken from the ``hhr'' resolution runs. {\bf Right panel}: Same as
left plot, except ADM angular momentum $J(t)$ (Eq.~\ref{J_ADM_def}) is plotted.}
\label{Fig:diag_integral_noise_reduction}
\end{figure*}
\subsection{Convergence Study}
\label{sec:convergence}
In a mixed-order finite-difference code such as ours, in which some
components converge at different orders, we must be particularly
careful about error estimates, as lower-order errors can sometimes
be non-dominant, leading to higher-order convergence than otherwise
expected. In such a scheme, numerical results for
a quantity $Q_\text{numerical}$ at a given gridpoint, time, and grid
spacing $\Delta x$, are related to the exact quantity
$Q_\text{exact}$, via the following equation
\begin{equation}
\label{convergence_exact}
Q_\text{numerical}(\Delta x) = Q_\text{exact} + C \Delta x^n +
\mathcal{O}(\Delta x^{m})
\end{equation}
where $C$ is a constant, $n$ is the dominant convergence order,
$m \neq n$ is the next dominant convergence order, and $m<n$
is possible. Generally we would aim to have $C\Delta
x^n\gg\mathcal{O}(\Delta x^{m})$, so the $\mathcal{O}(\Delta x^{m})$
term can be neglected. When non-dominant convergence order terms are
neglected in this way, we call $Q_\text{exact}$ the {\it
Richardson-extrapolated} $Q$, $Q_\text{RE}$, as it
provides an estimate for $Q$ that removes the dominant error term,
similar to \cite{Richardson1911}:
\begin{equation}
\label{convergence}
Q_\text{RE} = Q_\text{numerical}(\Delta x) - C \Delta x^n.
\end{equation}
AMR prolongations in time and space are performed with second- and
fifth-order-accurate stencils, respectively, and temporal and spatial
derivatives via fourth- and sixth-order-accurate finite-difference
stencils, respectively. Thus the {\it dominant} convergence order $n$
is {\it a priori} unknown in our BBH calculations, as they are of
mixed order. As described in Sec.~\ref{nummethods:AMRgridparams}, we
have reduced the timestep on all grids well below the CFL limit to
prevent the relatively low-order temporal differencing and
interpolations from dominating our errors.
To determine the dominant convergence order $n$, we perform runs at
three resolutions, obtaining three values for
$Q_\text{numerical}(\Delta x)$. With these three knowns, we then
employ a bisection technique to solve Eq.~(\ref{convergence}) for
unknowns $n$, $Q_\text{RE}$ and $C$.
For quantities $Q_\text{numerical}(\Delta x)$ that are also
functions of time (e.g., $\psi^4(t)$), we obtain estimates for $n$ each
time $Q_\text{numerical}(\Delta x)$ is output by the code. We then
define the dominant convergence order as the average of all
bisection-calculated estimates for $n$, rounded to the nearest
integer. This averaged, rounded value for $n$, combined with
$Q_\text{numerical}(\Delta x)$ at the two lowest resolutions of the
three used to obtain $n$, are chosen as inputs to solve the linear
$2\times 2$ system for $C$ and $Q_\text{RE}$. We then find another
estimate for $Q_\text{RE}$ combining the same value for $n$ with the
two highest resolutions of the three $Q_\text{numerical}(\Delta x)$
used to estimate $n$. The difference between these two values for
$Q_\text{RE}$ is used as an error estimate, as deviations from zero
will be caused by variations in $n$ as well as higher-order terms
(cf. Eqs.~(\ref{convergence}) and~(\ref{convergence_exact})).
In the special case where $Q_\text{RE}\to0$, as is
expected for quantities that should converge to zero (e.g., constraint
violations), we may solve Eq.~(\ref{convergence}) for $n$ with only
two realizations of $Q_\text{numerical}(\Delta x)$. In this special
case, $n$ can be found analytically.
\subsubsection{Irreducible Mass Convergence}
\label{sec:Mirr_conv}
Figure~\ref{Fig:M_irr_conv_study} presents a convergence study of
$M_\text{irr}(t)$ for the spinning BH (i.e., the BH with initial spin
parameter$=0.3$) at various resolutions. For ease of comparison (it is
difficult to distinguish a dotted line from a dashed one if data are
exceedingly noisy), the ``mr'', ``hr'', and
``hhr'' data are B\'{e}zier-smoothed (i.e., fit to a degree-$n$
B\'{e}zier curve for $n$ data points). However, to show how the
high-frequency noise in $M_\text{irr}(t)$ is affected by gauge choice,
we do not smooth the ``lr'' run data. Notice how $M_\text{irr}(t)$
data are {\it significantly} less noisy in the improved gauge case
BestGauge (right panel) than in the OldGauge case
(left panel), demonstrating that gauge-generated noise affects the
strong-field region as well. We observe the same relative noise in the
higher-resolution runs prior to smoothing. We also note that our
gauge improvements have little impact on the secular drift in
$M_\text{irr}(t)$. As for the initially {\it nonspinning} BH, similar
results are observed, except the ``lr'' run is not as much an outlier.
To understand why the ``lr'' resolution run appears to be such a
significant outlier, we take a look at the isolated horizon formalism
spin \cite{Ashtekar:2004cn} of the initially spinning BH, shown in
Fig.~\ref{Fig:IHF_spin_bh1_oldgauge}. Notice the
secular increase in dimensionless spin parameter $(J/M^2)_\text{IHF}$ at ``lr'' resolution, which
completely dwarfs spin parameter drifts in the highest three
resolution runs. Figure~\ref{Fig:IHF_spin_bh1_oldgauge} shows data
from the OldGauge case only. For comparison, we report that at
``lr'' resolution, g0+TR2+h shares the same drift
as OldGauge to within a percent, and
g1+TR2+h+d5d7 (BestGauge) exhibits about
6\% less drift. Finally we note that for the initially nonspinning BH,
we observe no significant drift in spin parameter over time. Based on
these observations, we conclude that the ``lr'' run is at insufficient
resolution to properly resolve the spinning BH.
Throwing out data at the ``lr'' resolution, and using data at
resolutions \{mr,hr,hhr\}, Fig.~\ref{Fig:M_irr_conv_study_conv_order}
shows convergence order $n$ for $M_\text{irr}$ (Eq.~\ref{convergence}) versus time. This
panel compares observed convergence order $n$ for both BHs in all
three cases throughout much of the inspiral. Convergence is poor for
$t\gtrsim1100M$, as BH2 $M_\text{irr}$ data at all three resolutions
cross near $t\approx1300M$, regardless of gauge choice. Apparently
this clear non-convergence in BH2 negatively impacts BH1 convergence as
well. Outside of this non-convergent episode, we find average
convergence orders slightly higher than fifth order (our chosen
spatial prolongation order), independent of the gauge choice,
although the results using the BestGauge choice are clearly less
noisy.
In the next section, we show that this nonconvergence in irreducible
mass drift at low resolution does not translate to nonconvergence at
low resolution in constraint violations.
\begin{figure*}
\includegraphics[angle=270,width=0.45\textwidth]{M_irr_oldgauge_bh1}
\includegraphics[angle=270,width=0.45\textwidth]{M_irr_step7_bh1}
\caption{Convergence study: $M_\text{irr}(t)$.
{\bf Left panel}: $M_\text{irr}(t)-M_\text{irr,hhr,max}$ over time for BH1 (i.e., the
BH with initial spin parameter$=0.3$), where $M_\text{irr,hhr,max}$
is the maximum value of $M_\text{irr}$ in the ``hhr'' run. This
panel plots data for the OldGauge case at
resolutions ``lr'' (solid red), ``mr'' (medium-dashed blue), ``hr''
(long-dashed magenta), and ``hhr'' (dotted green).
{\bf Right panel}: Same as left panel, but for case
BestGauge (g1+TR2+h+d5d7).
``smoothed'' denotes that the $n$ data points for a given
dataset plotted in this figure have been fit to degree-$n$
B\'{e}zier curve, and the resulting B\'{e}zier curve is shown.}
\label{Fig:M_irr_conv_study}
\end{figure*}
\begin{figure}
\includegraphics[angle=270,width=0.45\textwidth]{IHF_spin_bh1_oldgauge.eps}
\caption{Spin parameter versus time for BH1 (i.e., ``the spinning
BH'', with initial spin parameter $\approx0.3$), as given by isolated horizon
formalism. Data are plotted at resolutions ``lr'' (thick solid red),
``mr'' (dotted blue), ``hr'' (dashed magenta), and
``hhr'' (thin solid gray). Note that the offset 0.30002 is chosen
for $(J/M^2)_\text{IHF}(t)$, which is the value all runs converge to
after a rapid initial settling. We attribute this settling to the
fact that our initial data are conformally flat. Note also that only
case OldGauge is shown, as all other cases share the same
qualitative behavior.}
\label{Fig:IHF_spin_bh1_oldgauge}
\end{figure}
\begin{figure}
\includegraphics[angle=270,width=0.43\textwidth]{M_irr_convergence_order}
\caption{Convergence study: Implied convergence order $n$
(Eq.~\ref{convergence}) for $M_\text{irr}(t)$, using
$M_\text{irr}(t)$ data at three highest resolutions \{mr,hr,hhr\}
for both BHs during inspiral. Convergence order for BH1 (initial
spin parameter$=0.3$; solid red) and BH2 (initially nonspinning;
dashed blue) are compared to $n=5$ (dot-dashed gray). The top plot
shows data from the OldGauge case the middle plot
g0+TR2+h, and the bottom BestGauge
(g1+TR2+h+d5d7). For ease of analysis, all raw $M_\text{irr}(t)$
data in this plot were first smoothed with a moving-window linear
least-squares algorithm prior to computing convergence order. Also,
data in the range $t/M<30$ and $t/M>1250$ have been removed, as
$M_\text{irr}(t)$ at all three resolutions overlaps at $t/M=0$ and
$t/M\approx1300$ in the nonspinning BH (BH2) data, so near these
points convergence order estimates for BH2 are unreliable.}
\label{Fig:M_irr_conv_study_conv_order}
\end{figure}
\subsubsection{Convergence to Zero of Constraint Violations}
\begin{figure*}[t]
\includegraphics[angle=270,width=0.45\textwidth]{Ham_constraint_convergence__step1}
\includegraphics[angle=270,width=0.45\textwidth]{Ham_constraint_convergence__step7}\\
\vspace{0.3cm}
\includegraphics[angle=270,width=0.45\textwidth]{Ham_constraint_convergence__distribution-hrhhr}
\includegraphics[angle=270,width=0.45\textwidth]{Ham_constraint_std_convergence__steps1and7.eps}
\caption{Convergence to zero of Hamiltonian constraint
violations.
{\bf Upper-left panel}: Top plot shows raw $||\mathcal{H}(t)||$ at
resolutions ``lr'' (solid red), ``mr'' (dotted green), ``hr''
(short-dashed blue), and ``hhr'' (long-dashed magenta) for OldGauge
case. Bottom plot shows implied convergence order $n$
(Eq.~\ref{convergence}) to zero for resolution pairs \{lr,mr\}
(solid red); \{mr,hr\} (dotted green); and \{hr,hhr\} (dashed
blue).
%
{\bf Upper-right panel:} Same as upper-left panel, except for
BestGauge case (g1+TR2+h+d5d7).
%
{\bf Lower-left panel:} Distribution of $||\mathcal{H}(t)||$
convergence order, using \{hr,hhr\} data for
OldGauge case (thick red),
g0+TR2+h (medium thickness magenta), and
BestGauge (g1+TR2+h+d5d7; thin black). Note
that the thick red line corresponds to the distribution of convergence
order, as shown by the red line in the bottom plot of the upper-left
panel.
%
{\bf Lower-right panel:} Top plot: $||\mathcal{H}(t)||$ for
OldGauge case, rescaled according to the observed average
convergence order to zero over all resolutions, $n=1.62$. Bottom
plot: same as top, except for BestGauge (g1+TR2+h+d5d7) case, where
the average convergence order to zero over all resolutions is found
to be $n=5.14$. All cases are rescaled to ``lr'' resolution.}
\label{Fig:Ham_constraint_order_description}
\end{figure*}
\begin{table*}
\caption{Convergence to zero of Hamiltonian and $x$-component of
Momentum constraint violations. ``Median'' (``Mean'') numbers denote
median (average) convergence order using an evenly-spaced sample of
integrated L2-norm of constraint violations in time. This
integral excludes regions around the BHs (Eq.~\ref{L2_Ham}), and for
$||\mathcal{H}_\text{inner}||$, the region $r>32.3M$ as well. Note
that for consistency, all data are truncated at time $t=2086.4M$,
which is the earliest time at which a run was stopped ($\approx
210M$ after merger).}
\begin{tabular}{|c|c|c|c|}
\hline
Case Name & $||\mathcal{H}||_\text{lr,mr}$ Median (Mean) &
$||\mathcal{H}||_\text{mr,hr}$ Median (Mean) &
$||\mathcal{H}||_\text{hr,hhr}$ Median (Mean) \\
\hline
OldGauge
& 1.43 (1.64) & 1.30 (1.62) & 1.46 (1.60) \\
\hline
g0+TR2+h
& 5.17 (4.70) & 5.97 (4.31) & 5.27 (4.24) \\
\hline
g0+TR2+h+d5d7
& 5.57 (5.27) & 5.86 (4.97) & No hhr \\
\hline
BestGauge (g1+TR2+h+d5d7)
& 5.63 (5.38) & 5.92 (5.16) & 5.41 (4.87) \\
\hline
\hline
& $||\mathcal{H}_\text{inner}||_\text{lr,mr}$ Median (Mean) &
$||\mathcal{H}_\text{inner}||_\text{mr,hr}$ Median (Mean) &
$||\mathcal{H}_\text{inner}||_\text{hr,hhr}$ Median (Mean) \\
\hline
OldGauge
&5.77 (5.75) &5.72 (5.46) &5.98 (5.87) \\
\hline
g0+TR2+h
&5.82 (5.80) &5.86 (5.67) &5.96 (5.74)\\
\hline
g0+TR2+h+d5d7
&5.78 (5.75) &5.61 (5.69) & No hhr \\
\hline
BestGauge (g1+TR2+h+d5d7)
&5.81 (5.79) &5.94 (5.77) &5.76 (5.71) \\
\hline
\hline
& $||\mathcal{M}^x||_\text{lr,mr}$ Median (Mean) &
$||\mathcal{M}^x||_\text{mr,hr}$ Median (Mean) &
$||\mathcal{M}^x||_\text{hr,hhr}$ Median (Mean) \\
\hline
OldGauge
& 1.37 (1.81) & 1.37 (1.62) & 1.29 (1.96) \\
\hline
g0+TR2+h
& 1.09 (1.71) & 1.00 (1.98) & 1.37 (2.00) \\
\hline
g0+TR2+h+d5d7
& 1.27 (1.86) & 1.31 (1.86) & No hhr \\
\hline
BestGauge (g1+TR2+h+d5d7)
& 1.24 (1.94) & 1.26 (1.84) & 1.78 (1.91) \\
\hline
\end{tabular}
\label{Convergence_order_table}
\end{table*}
Table~\ref{Convergence_order_table} presents the average and median order
at which Hamiltonian and ($x$-component) momentum constraints converge to zero for
various cases. To better understand how these numbers are generated,
Fig.~\ref{Fig:Ham_constraint_order_description} outlines the
procedure. The top left (right) plots in the upper panels show the raw
Hamiltonian constraint data $||\mathcal{H}(t)||$ (Eq.~\ref{L2_Ham})
for the OldGauge (BestGauge) case, and the bottom plots of the
upper panels use data at pairs of resolutions to solve for the implied
convergence order $n$ (Eq.~\ref{convergence}, where $Q_\text{RE}\to0$,
leaving only $C$ and $n$ as unknowns). Using many data points for $n$,
uniformly spaced in time, we can then construct histograms of
convergence order from this plot. The lower-left panel shows the
histogram of convergence order for ``hr'' and ``hhr'' resolutions,
where the thick red line corresponds to the solid red curve in the
bottom plot of the upper-left panel. This histogram is compared to the
same histogram for cases g0+TR2+h and BestGauge (g1+TR2+h+d5d7). We
point out two clear patterns: (1) the g0+TR2+h and BestGauge cases
possess significantly higher average convergence order than OldGauge,
(2) BestGauge appears to converge at a higher average convergence
order than g0+TR2+h. Notice the distribution widens with increasing
average convergence order. This is a reflection of the fact that after
t~300M, the OldGauge ||H|| converges below second-order and remains
there until merger. In BestGauge, convergence order drops
significantly as the sharp lapse wave crosses progressively
lower-resolution refinement boundaries. However, this subconvergent
spike in noise does not translate to sub-convergence later on in the
evolution. We attribute this to the lapse stretching and smoothing
features of BestGauge. Instead of plotting histograms for all cases at
all resolutions, we simply report the median and mean convergence
orders for all cases for which multiple resolution data are available
(Table~\ref{Convergence_order_table}).
According to Table~\ref{Convergence_order_table}, we expect
$||\mathcal{H}(t)||$ to converge to zero on average at order $n=1.62$
in the OldGauge case, and $n=5.14$ in the BestGauge
(g1+TR2+h+d5d7) case.
The lower-right panel of
Fig.~\ref{Fig:Ham_constraint_order_description} demonstrates how well
data at different resolutions overlap when rescaled according to these
observed mean convergence orders. Notice that although
convergence order drops to as low as second order at each
spike in the BestGauge $||\mathcal{H}(t)||$ (cf. upper-right and
lower-right panels), BestGauge convergence order equilibrates at
late times to $n\approx6$. In the OldGauge case,
$||\mathcal{H}(t)||$ convergence order steadily drops from the outset
of the calculation (upper-left panel) as the initial sharp lapse wave
crosses into progressively lower resolution AMR grid boundaries.
Recall that in Fig.~\ref{Fig:Ham_step_by_step}, we found that
$||\mathcal{H}(t)||$ is reduced by a factor of 20 on average in the
BestGauge case, compared to a standard moving-puncture gauge
choice (OldGauge). Table~\ref{Convergence_order_table}
demonstrates that the {\it convergence order} in $||\mathcal{H}(t)||$
is also significantly improved with our gauge improvements, increasing
from below second-order to nearly sixth-order convergence. Thus at
higher resolutions, we expect our gauge improvements to reduce
$||\mathcal{H}(t)||$ by an even higher factor than that observed
at ``lr'' resolution. In fact ``BestGauge, hhr'' drops
$||\mathcal{H}(t)||$ by a factor of approximately {\it 58}, as
compared to the ``OldGauge, hhr'' case.
When the L2 norm of $\mathcal{H}(t)$ (Eq.~\ref{L2_Ham}) excises
the region outside $r=32.3M$ (i.e., $||\mathcal{H}_\text{inner}||$ in
Table~\ref{Convergence_order_table}), we observe nearly sixth-order
convergence in {\it all cases}. Thus a bulk of sub-convergent
Hamiltonian constraint violations comes from the outer regions, where
gravitational waves are measured, and the under-resolution of the
spinning BH at ``lr'' resolution (as discussed in
Sec.~\ref{sec:Mirr_conv}) does not appear to translate to inconsistent
convergence at low resolutions in constraint violations.
We did not have such an ``inner region'' diagnostic for momentum
constraint violations (i.e., an $||\mathcal{M}^x_\text{inner}||$
diagnostic). Recall that at ``lr'' resolution, $||\mathcal{M}^x||$ has
been reduced on average by a factor of $\approx 13.4$ with our new
gauge choices (Fig.~\ref{Fig:Momx_step_by_step}). However, its {\it
convergence to zero} does not improve significantly and remains
between first and second-order regardless of gauge choice.
This may be due to the fact that even in our most advanced
gauge choice (BestGauge), some high-frequency noise remains and
further efforts may be required to tamp down this remaining noise.
So do drastic constraint violation reductions in the
outer regions of our grid translate to improvements in gravitational
waveform (i.e., $\psi^4$) convergence, or does the relatively large
drift in the spinning BH's spin parameter over time imply poor waveform
convergence at ``lr'' resolution? The next section examines this
question in detail.
\subsubsection{Waveform Convergence}
\begin{figure*}
\includegraphics[angle=270,width=0.45\textwidth]{Psi4_phase_diff_vs_time}
\includegraphics[angle=270,width=0.45\textwidth]{Psi4_amp_diff_vs_time} \\
\vspace{0.3cm}
\includegraphics[angle=270,width=0.45\textwidth]{Psi4_convergence_order_oldgauge_phase}
\includegraphics[angle=270,width=0.45\textwidth]{Psi4_convergence_order_oldgauge_amp} \\
\vspace{0.3cm}
\includegraphics[angle=270,width=0.45\textwidth]{Psi4_convergence_order_step7_phase}
\includegraphics[angle=270,width=0.45\textwidth]{Psi4_convergence_order_step7_amp}
\caption{$\psi^4_{2,2}$ convergence study, using {\it smoothed}
amplitude and phase data.
{\bf Upper-left (right) panel}:
phase (amplitude) difference between pairs of resolutions, for OldGauge $|$lr-mr$|$
(thick solid red), $|$mr-hr$|$ (thick dashed blue), and $|$hr-hhr$|$ (thick
dotted magenta), as well as BestGauge $|$hr-hhr$|$
(g1+TR2+h+d5d7, thin dashed black).
{\bf Middle-left (right) panel}: OldGauge phase (amplitude)
convergence order versus time, using results from \{lr,mr,hr\}
resolutions (raw data: thin dashed green, B\'{e}zier-smoothed data:
thick solid red) and \{mr,hr,hhr\}
resolutions (raw data: thin dashed magenta, B\'{e}zier-smoothed data:
thick dashed blue).
{\bf Lower-left (right) panel}: same as middle panels, except for
BestGauge (g1+TR2+h+d5d7).
Please note that for
ease of analysis, all raw amplitude and phase data in this figure
were first smoothed with
a moving-window quadratic least-squares fit method prior to computing
convergence order, and where specified, B\'{e}zier smoothing was
applied separately to the resulting convergence order data as well.
Also, data $(t-r)/M<500$ have been removed from
convergence order plots, as they are too noisy even after smoothing
to show a clear convergence order (see top panels).
$\psi^4_{2,2}$ data are measured at radius $r=68.6M$.}
\label{Fig:Psi4_conv_study}
\end{figure*}
\begin{figure*}[t]
\includegraphics[angle=270,width=0.45\textwidth]{Psi4_oldgauge_Richardson_phase}
\includegraphics[angle=270,width=0.45\textwidth]{Psi4_oldgauge_Richardson_amp} \\
\vspace{0.3cm}
\includegraphics[angle=270,width=0.45\textwidth]{Psi4_step7_Richardson_phase}
\includegraphics[angle=270,width=0.45\textwidth]{Psi4_step7_Richardson_amp} \\
\vspace{0.3cm}
\includegraphics[angle=270,width=0.45\textwidth]{Psi4_Richardson_mr_hr_hhr__difference_phase}
\includegraphics[angle=270,width=0.45\textwidth]{Psi4_Richardson_mr_hr_hhr__difference_amp}
\caption{$\psi^4_{2,2}$ convergence and error-analysis study, using
{\it raw, unsmoothed} phase and amplitude data.
{\bf Top panels}:
Upper-left (right) plot: Phase (Amplitude) versus retarded time, for
OldGauge case at resolutions ``lr'' (solid red), ``mr'' (long-dashed
green), ``hr'' (medium-dashed blue), and ``hhr'' (dotted
magenta). Richardson-extrapolated phase is overlaid,
using data from ``hr'' and ``hhr'' resolutions, and assuming
constant fifth-order convergence (dash-dotted black).
Lower-left (right) plot: simple (not absolute value of) difference
in phase (normalized amplitude) between resolutions.
{\bf Middle panels}: same as top panels, but for case
BestGauge (g1+TR2+h+d5d7).
{\bf Bottom panels}: left (right) plot shows
$|$RE(mr,hr)-RE(hr,hhr)$|$ for waveform phase (amplitude)
versus retarded time, assuming fifth-order convergence, where
RE(mr,hr) denotes the Richardson-extrapolated value combining phase
(amplitude) numerical data at resolutions \{mr,hr\} with the
assumption of fixed $n=5$ convergence order (Eq.~\ref{convergence}).
Amplitude data are normalized by the higher-resolution value.
Cases OldGauge (dotted magenta), g0+TR2+h
(long-dashed blue), and BestGauge
(g1+TR2+h+d5d7; solid red) are compared. Zoom inset on the left plot
shows accumulated phase difference near the end of the inspiral,
with vertical gray line denoting $M\omega_{2,2}=0.2$ (corresponding
to a retarded time of $(t-r)/M\approx1866$). All $\psi^4_{2,2}$ data
are measured at radius $r=68.6M$.
}
\label{Fig:Psi4_Rich_extrap_study}
\end{figure*}
\begin{figure*}[t]
\includegraphics[angle=270,width=0.45\textwidth]{Psi4_noise_reduction_convergence_plot_oldgauge}
\includegraphics[angle=270,width=0.45\textwidth]{Psi4_noise_reduction_convergence_plot_step7}
\caption{Re($\psi^4_{2,2}$) noise convergence properties, during
early inspiral [$58\lesssim(t-r)/M\lesssim1160$].
{\bf Left panel}:
B\'{e}zier-smoothed Power spectra of Re($\psi^4_{2,2}$) for case
OldGauge, at resolutions ``lr'' (dotted red), ``mr''
(short-dashed green), ``hr'' (dashed blue), and ``hhr'' (solid
black). These are compared to the same quantity at ``hhr''
resolution for case BestGauge
(g1+TR2+h+d5d7; thick magenta).
{\bf Right panel}:
B\'{e}zier-smoothed power spectra of Re($\psi^4_{2,2}$) for case
BestGauge, at resolutions ``lr'' (dotted red), ``mr''
(short-dashed green), ``hr'' (dashed blue), and ``hhr'' (solid
magenta). These are compared to the same quantity at ``hhr''
resolution for case OldGauge (thick black).
%
Note that all Re($\psi^4_{2,2}$) data plotted are measured at radius
$r=68.6M$, and for an example of unsmoothed data, see
Fig.~\ref{Fig:Psi4_noise_reduction}. Re($\psi^4_{2,2}$) time series data were
multiplied by the tapering function $\Erf(t;200,50) \times
\Erf(-t;-1300,150)$ prior to the Fourier transform.}
\label{Fig:Psi4_waveform_noise_convergence}
\end{figure*}
Figure~\ref{Fig:Psi4_conv_study} presents a gravitational waveform
convergence study, separating $\psi^4_{2,2}(t)$ into amplitude and
phase separately, as these quantities could be influenced differently
by truncation error~\cite{Husa:2008}.
The top left (right) panel of Fig.~\ref{Fig:Psi4_conv_study} shows
$|Q_\text{numerical}(\Delta x_1)-Q_\text{numerical}(\Delta x_2)|$,
where $Q$ is the {\it smoothed} phase (amplitude) of $\psi^4_{2,2}(t)$,
and \{$\Delta x_1,\Delta x_2$\} denotes data at two adjacent resolutions in the set
\{lr,mr,hr,hhr\}. Notice that the \{hr,hhr\} phase difference is
slightly smaller during inspiral in the BestGauge
(g1+TR2+h+d5d7) than the OldGauge case. This is one measure that
demonstrates our gauge improvements reduce phase errors.
The remaining left (right) panels of Fig.~\ref{Fig:Psi4_conv_study}
plot convergence order $n$ in $\psi^4_{2,2}$ phase (amplitude) as a
function of time, comparing the OldGauge (middle panels) to the
BestGauge (g1+TR2+h+d5d7; bottom panels)
cases. At the three lowest resolutions \{lr,mr,hr\}, inconsistent
convergence between amplitude and phase is observed, regardless of
gauge choice. We attribute this to the under-resolution of the
spinning BH (Sec.~\ref{sec:Mirr_conv}). Cleaner convergence is
observed when computing $n$ from data at the three highest resolutions
\{mr,hr,hhr\}. However, convergence order in the OldGauge case
oscillates significantly between $n=4$ and $n=5$ until $t\approx
1300M$, and then drops to $n\approx4$ until merger. The same data from
our BestGauge case indicate extremely steady $n=5$ convergence order
with almost no oscillation, until merger.
At merger ($t\gtrsim1750M$), loss in convergence is observed
regardless of gauge choice, pointing to a non-convergent effect
not addressed by our gauge modifications. However, there is some
chance that this loss of convergence may be restored with another
careful time reparameterization near merger, which will allow the
gauge to more quickly settle as the gravitational fields undergo the
extremely rapid changes associated with merger.
The consistent $n=5$ convergence observed in BestGauge is a highly
significant result, as it strongly indicates that our errors are
dominated by the fifth-order-accurate spatial interpolation errors at
grid refinement boundaries in BestGauge. We could have
reasonably expected this {\it a priori} given that the lower-order
temporal errors have been tamped down significantly by our choice of
timesteps well below the CFL limit. The convergence order
oscillations in OldGauge below $n=5$ are troubling, leading us to
question the reliability of Richardson-extrapolated estimates of
$\psi^4_{2,2}$ that assume a standard, fixed-in-time integer
convergence order when choosing OldGauge-like gauge conditions.
We believe that the OldGauge convergence order oscillations in
$\psi^4_{2,2}$ amplitude and phase may be related to the large amount of
power in $\psi^4_{2,2}$ noise, particularly at wave periods of
$P\sim 10M$ as shown in Fig.~\ref{Fig:Psi4_noise_reduction}, as this
noise has been largely removed in the BestGauge case. In
fact, in Sec.~\ref{sec:waveform_noise_conv} below, we show that
waveform noise at the dominant noise frequency ($P\approx 10M$) in
OldGauge {\it does not diminish with increasing resolution}, at the
resolutions chosen.
We end our discussion on Fig.~\ref{Fig:Psi4_conv_study} by cautioning
that our clean, fifth-order convergence in BestGauge has only been
demonstrated with the three highest resolutions, and that
higher-resolution studies will be useful in cementing our
conclusions. Although our gauge improvements have apparently
eliminated the non-convergent gravitational waveform noise (as shown
in Sec.~\ref{sec:waveform_noise_conv} below), some noise
does remain, and we anticipate that at high resolutions far beyond
that attempted here, further gauge improvements may be necessary to
maintain a uniform, integer convergence order.
We now turn our attention to the Richardson-extrapolated
waveforms. Given that our gauge improvements yield cleaner
gravitational-waveform convergence at higher resolutions, we may be able to
more precisely estimate the true value of $\psi^4_{2,2}$
through Richardson extrapolation. The Richardson-extrapolated value is
given by $Q_\text{RE}$ in Eq.~(\ref{convergence}), replacing $Q$ with
$\psi^4_{2,2}$ at all times $t$. Since we have established the
dominant waveform convergence order is cleanly $n=5$ for \{mr,hr,hhr\}
resolutions, we can compute $\psi^4_{2,2,\text{RE}}$ in
Eq.~(\ref{convergence}) by setting $n=5$ and combining numerical data
at two resolutions.
The top and middle panels of Fig.~\ref{Fig:Psi4_Rich_extrap_study} compare
{\it raw} $\psi^4_{2,2}$ amplitude and phase data at all four
resolutions to Richardson-extrapolated values at the highest two
resolutions \{hr,hhr\}.
At $t\lesssim800M$ ($t\lesssim1200M$) in the OldGauge (BestGauge),
simple differences between data at adjacent resolutions are too noisy
to easily distinguish convergence properties (top panels). This can be
somewhat mitigated by careful smoothing, as was shown in
Fig.~\ref{Fig:Psi4_conv_study}, but both phase and amplitude data at
$t\lesssim500$ are too noisy to determine convergence order cleanly in
OldGauge, even with smoothing. Due to the large noise reductions
of BestGauge, the situation is greatly improved (middle panels), though determining
convergence order in phase (amplitude) remains difficult for times
$t\lesssim 600M$ ($t\lesssim 900M$) without smoothing. Notice that the
difference between Richardson-extrapolated phase and amplitude data
(as computed at \{hr,hhr\} resolutions) and ``hhr'' resolution data
is larger than the difference between ``hr'' and ``hhr''
data.
The bottom left (right) panel of Fig.~\ref{Fig:Psi4_Rich_extrap_study}
shows differences in Richardson-extrapolated phase (amplitude) versus
time. One Richardson-extrapolated value combines data at \{mr,hr\}
resolutions---which we denote RE(mr,hr)---and the other combines
\{hr,hhr\}---denoted by RE(hr,hhr). Any nonzero value for
$|$RE(mr,hr)-RE(hr,hhr)$|$ may be caused by either deviations from
fifth-order convergence---which we have shown to be particularly
significant in OldGauge (Fig.~\ref{Fig:Psi4_conv_study})---or
truncation errors at non-dominant orders (cf. Eqs.~\ref{convergence}
and~\ref{convergence_exact}). Thus we consider
$|$RE(mr,hr)-RE(hr,hhr)$|$ as an error estimate for phase, and
$|$RE(mr,hr)-RE(hr,hhr)$|$/RE(hr,hhr) for amplitude.
Implied phase errors $|$RE(mr,hr)-RE(hr,hhr)$|$ increase
with time, but far more smoothly in the improved gauge cases, as
compared to the OldGauge case. Though phase errors are
roughly comparable in the OldGauge and g0+TR2+h cases, we find a
reduction in phase error of about $40\%$ in the BestGauge
(g1+TR2+h+d5d7) case throughout inspiral and
near merger. Normalized amplitude errors are much noisier, but appear
about 40--50\% smaller during inspiral in the BestGauge case as well,
as compared to the OldGauge and g0+TR2+h cases. However, during and
after merger, amplitude and phase errors for all three cases are large
and overlap. These large errors are expected, as the BBH merger
time changes as resolution is increased, and we do not account for
this effect by, e.g., shifting the waveforms so that their merger
times overlap.
We may use the two highest-resolution estimates for
$\psi^4_{2,2,\text{RE}}$ to compute accumulated amplitude and phase
errors at the end of inspiral, where we define the ``end of inspiral''
to be at the time in which gravitational wave frequency is
$M\omega=0.2$. An almost identical frequency was used in the NINJA
\cite{Ajith:2012az,Aasi:2014tra} and NRAR \cite{Hinder:2013oqa}
collaborations as the fiducial time at which to measure
waveform amplitude and phase errors, the only difference being that we
choose to normalize $\omega$ by total initial ADM mass instead of the
combined initial masses of the punctures. The difference in frequency
is less than one percent for runs presented here. We find accumulated
phase errors at $M\omega=0.2$ (vertical line in inset) of
approximately 0.046, 0.081, and 0.025 radians for OldGauge, g0+TR2+h,
and BestGauge cases, respectively. Thus the BestGauge gauge choice
reduces phase errors by about a factor of two. This is a rather
unfortunate time to measure phase error, as the OldGauge case phase
error does not increase as predictably as the BestGauge case, and just
happens to experience a local minimum in the rate of phase error
increase at $M\omega=0.2$. Had we measured phase errors slightly
earlier or perhaps at a higher resolution, we might have seen
further reduction of phase errors by BestGauge.
As pointed out earlier, normalized amplitude errors near merger are
comparable between the different cases, and we measure them at the
``end of inspiral'' to be 5.7\%, 6.0\%, and 5.8\% for OldGauge,
g0+TR2+h, and BestGauge cases, respectively. We conclude that
although amplitude errors are clearly smaller during inspiral in the
BestGauge case, they are comparable just prior to merger.
In Sec.~\ref{sec:noise_reduction}, we established that, when compared
to the OldGauge case, the BestGauge case significantly reduces noise
in gravitational waveforms, particularly at peak noise
frequencies. Further, we showed in the previous section that GWs
produced by BestGauge are very consistently fifth-order
convergent, while the OldGauge case experiences disturbing
oscillations in implied convergence order. We suspect that these
convergence order oscillations are related to the much noisier
waveforms in OldGauge, but we would na\"\i vely
expect that the
noise should reduce with increasing resolution. The next section
demonstrates that at peak noise frequencies, the noise in OldGauge
{\it does not} diminish with increasing resolution.
\subsubsection{Waveform Noise Convergence Properties}
\label{sec:waveform_noise_conv}
Figure~\ref{Fig:Psi4_waveform_noise_convergence} presents a power
spectrum analysis of waveform noise convergence. The left panel shows
that the largest noise-related spike in OldGauge Re($\psi^4_{2,2}$) power
does not appear to diminish with increased
resolution, lending support to the notion that
poor waveform convergence in OldGauge is due to high-frequency
noise. In fact, we find that the power attributable to noise in
OldGauge at $P\approx 10M$ is about 1\% of the {\it maximum} power,
corresponding to the physical GW signal (at GW periods of half the
binary orbital period). Next notice that this large, non-converging
noise-related spike in the power spectrum in OldGauge runs has been
reduced by about an order of magnitude in our BestGauge gauge choice. This
is consistent with the time-domain data of
Fig.~\ref{Fig:Psi4_noise_reduction}, in which waveform
noise at ``lr'' resolution is shown to be hugely reduced in the
BestGauge case, as compared to OldGauge. In BestGauge, we observe
three distinct bumps in the power spectrum, likely associated with
noise. Unlike the dominant
noise-related spike at $P\approx 10M$ in OldGauge, these three bumps
in the BestGauge power spectra appear to drop more cleanly with
increasing resolution than OldGauge, and as resolution is
increased, the bumps appear to decrease in amplitude and shift toward
shorter wave periods.
\section{Conclusions and Future Work}
\label{Conclusions}
It has been found that the standard BSSN/MP+AMR paradigm
yields inconsistent convergence in gravitational waveforms in BBH
evolutions. Without consistent convergence, it may be impossible to
produce reliable error estimates for these waveforms, particularly
when very high accuracies are needed. It has been hypothesized
\cite{Zlochower:2012fk} that this inconsistent convergence may be
related to short-wavelength waves being reflected from grid-refinement
boundaries, producing high-frequency noise and constraint violations.
We present a set of improvements to the moving-puncture gauge
conditions that, with {\it negligible} increase in computational
expense, greatly reduce noise and improve convergence
properties of gravitational waveforms from BBH inspiral and
mergers. These improvements are presented in the context of an NRAR
\cite{Hinder:2013oqa} BBH calculation that evolves a low-eccentricity BBH
system $\approx 11$ orbits to merger,
where one BH has an initial dimensionless spin of $0.3$ aligned with the
orbital angular momentum, and the other is nonspinning. Evolutions are performed at up to four resolutions with a
variety of gauge choices, starting with the ``standard'' moving
puncture gauge choice and adding improvements one by one until our
``best'' gauge choice is reached.
Our gauge improvements in part stem from the observation that
regularly-spaced spikes in Hamiltonian constraint
violations are timed precisely to the grid-refinement boundary
crossings of an early outgoing wave traveling at coordinate speed
$\sqrt{2}$. Based on linear analyses \cite{Alcubierre:2002iq}, we
know of only one propagation mode
with that speed in the standard BSSN/MP formulation, which primarily
involves the lapse and is governed by the lapse evolution
gauge condition.
The initial outgoing lapse wave pulse possesses sharp features, which
are problematic because, as \cite{Zlochower:2012fk}
points out, high-frequency waves crossing into a coarser AMR grid will
be partially reflected at the boundary, generating noise. Now the lapse
{\it and its derivatives} are strongly coupled to the gravitational
field evolution equations, so noise
generated by this sharp outgoing lapse pulse crossing refinement
boundaries can be easily converted to noise in gravitational field
variables (such as GWs) and constraint-violating modes
(cf. \cite{Etienne:2011re}).
Guided by this, we focus our efforts primarily on modifications to the lapse
evolution equation, aimed at stretching and smoothing the initial
outgoing lapse wave pulse. We stretch the lapse wave by
monotonically increasing its speed as it propagates outward so that
the front of the lapse pulse propagates slightly faster than
the back. We smooth the
lapse wave by adding both parabolic and stronger-than-usual
Kreiss-Oliger dissipation terms. The most significant improvements
spawn from the stretching of the initial lapse pulse.
Despite the stretching of the initial lapse pulse, early-time
spikes in the constraint violations are not reduced. We are able to
significantly tamp down these spikes via a time reparameterization
that greatly accelerates the initial evolution of the lapse and shift
relative to the BSSN gravitational field evolution
variables. Effectively, this modification makes lapse waves propagate
at speeds up to $\sim 30$ times the coordinate speed of
light initially, enabling the lapse to respond much more quickly to
the rapidly-settling gravitational fields in the strong-field region
during the early evolution.
Our ``best'' gauge condition (BestGauge) reduces Hamiltonian and
momentum constraint violations by factors of $\sim 20$ and $\sim 13$,
respectively. In addition, with this same gauge choice, noise in
the dominant gravitational-wave mode---$\psi^4_{2,2}(t)$---is reduced
by nearly an order of magnitude, particularly at peak noise
frequencies. Such numerical noise can be far more problematic in
sub-dominant [i.e., $(l,m)\neq$(2,2)] modes, at times completely
obscuring the signal. As an example, with OldGauge, the $(l,m)$=(4,4)
mode is noise-dominated throughout much of the inspiral, but with the
new gauge improvements, this mode is much cleaner and more amenable to
analysis. We observe improvements, generally of a lesser degree, in
other sub-dominant modes as well.
Finally, we observe a significant reduction in ADM mass and angular
momentum noise at large radius,
particularly during inspiral and merger. Regardless of gauge choice,
we observe a large amount of noise in ADM angular momentum just after
merger, corresponding to a spike in momentum constraint at about this
time. We are uncertain of the cause for this noise/spike, but suspect they may
be due to either rapid coordinate and spacetime evolution associated with
BH merger, or possibly high-frequency {\it physical} waves propagating
outward into less-refined numerical grids. This warrants further
investigation, as the former hypothesis can be tested through a
late-time time reparameterization and the latter through grid
structure adjustments.
In addition to analyses of how constraint violations and noise are
affected by our new gauge choices, we perform a suite of constraint-violation,
irreducible-mass, and waveform ($\psi^4_{2,2}$) convergence
tests with a set of three gauge choices, starting from the ``standard''
moving-puncture gauge choice (OldGauge) to our ``best'' gauge
choice (BestGauge).
Using the ``standard'' moving-puncture gauge choice (OldGauge), the L2 norm of
Hamiltonian constraint violations converges to zero at between first
and second order with increasing resolution. If this L2 norm integral
excludes the wavezone region far from the binary, convergence order
increases to between fifth and sixth order, indicating that
subconvergence is related to effects far from the binary, on distant,
low-resolution grids. Meanwhile, with our improved gauge choices, we
observe between fifth- and sixth-order convergence in this diagnostic
{\it even when the integral extends to the outer boundary}. Though
momentum constraint violations at a given resolution are significantly
reduced with our new gauge choices, we do not observe an
improvement in the convergence to zero of momentum constraint
violations. This may be due to the fact that even in our ``best''
gauge choice, some high-frequency noise remains in our BBH
calculations.
Based on our waveform convergence analyses, we find that
$\psi^4_{2,2}(t)$ data at \{mr,hr,hhr\} resolutions yield approximate
fifth-order convergence, though large fluctuations around fifth order
are observed in the OldGauge case. These fluctuations are
significantly reduced when improved gauge conditions are adopted,
particularly in the BestGauge case.
With such consistent fifth-order convergence observed in the highest
three resolutions in our best gauge choice (BestGauge), we then
analyze the difference between two Richardson-extrapolated
realizations of phase and normalized amplitude. One realization uses data at ``hr'' and ``hhr''
resolutions and the other ``mr'' and ``hr'' resolutions. Any nonzero
difference between these Richardson-extrapolated realizations of
phase and normalized amplitude can be attributed to fluctuations in
assumed convergence order $n$ from $n=5$ (which appears to be related
to noise) or to error from higher-order terms
(cf. Eqs.~\ref{convergence} and~\ref{convergence_exact}). Since these
differences in Richardson-extrapolated values are directly related to
errors, we use them as our error estimates for the amplitude and phase
(though it may be an unconventional choice;
cf. \cite{Hinder:2013oqa}). Comparing the ``standard'' moving-puncture gauge
choice to our ``best'' gauge, we find significant (factor of $\sim$ 2)
reductions in both amplitude and phase errors during inspiral. At the
end of the inspiral, when the frequency of the $\psi^4_{2,2}$ is
$M\omega=0.2$, we observe roughly a factor of two reduction in phase
errors in the BestGauge case but no significant improvement in
amplitude errors, as these are dominated near merger by small offsets
in the time of merger at different resolutions.
We then perform a GW noise analysis, comparing OldGauge and
BestGauge results at different resolutions
(see Fig.~\ref{Fig:Psi4_waveform_noise_convergence}). In the OldGauge case, the largest
spike in GW power directly related to numerical GW noise, at wave
periods $P\approx 10M$ (Fig.~\ref{Fig:Psi4_noise_reduction}), does not
drop with increasing resolution. Further, we find that
the noise-dominated power in this OldGauge spike is about 1\% of
the physical GW power maximum at $P=0.5P_\text{orb}$ throughout much
of the early inspiral. GW power at wave periods associated with noise
in OldGauge runs is reduced by nearly an order of
magnitude in the BestGauge runs, and unlike OldGauge, the power at
these wave periods appears to drop monotonically with
resolution, suggesting that noise in BestGauge converges away more
cleanly. Given the excellent amplitude- and phase-convergence
properties observed in the BestGauge case, we believe this GW power
spectrum analysis strongly indicates that poor convergence in the
``standard'' moving-puncture gauge conditions may indeed be related to
poorly-convergent GW noise generated by the initial sharp outgoing lapse
pulse.
Future work will examine remaining uncertainties in these
evolutions, including the nearly simultaneous large spike in momentum
constraint violations and large noise in the ADM angular momentum
surface integral at the end of our evolutions. Though we have found
that our gauge improvements greatly reduce noise in
many quantities apparently generated by the sharp initial outgoing
lapse pulse, we have not completely eliminated the noise, and it is
unknown how much more phase and amplitude errors can be driven
downward with the current improvements. Without doubt, higher
resolutions and higher-order evolutions will be helpful in determining
this.
Though these new gauge conditions and techniques have been
presented in the context of moving-puncture BBH evolutions only, we
fully expect them to be beneficial in a wide variety of NR contexts,
including compact binary systems with matter or even BBH evolutions
using other $N$+1 NR formulations. With the era of gravitational-wave
astronomy now upon us, we hope this work will spur others to join the
search for gauge conditions and methods optimal for generating high-quality
gravitational waveforms within the MP+AMR context, as each improvement
will accelerate our community toward its goals in this exciting time.
\acknowledgments This paper was supported in part by NSF Grants
PHY-0963136 and PHY-1300903 as well as NASA Grants NNX11AE11G and
NNX13AH44G at the University of Illinois at Urbana-Champaign. BJK and
JGB were supported by NASA grants 09-ATP09-0136 and 11-ATP-046. VP
gratefully acknowledges support from a Fortner Fellowship at
UIUC. This work used the Extreme Science and Engineering Discovery
Environment (XSEDE), which is supported by NSF grant number
OCI-1053575. A significant portion of the calculations presented here
were performed as part of the Blue Waters sustained-petascale
computing project, which is supported by the National Science
Foundation (award number OCI 07-25070) and the state of Illinois. Blue
Waters is a joint effort of the University of Illinois at
Urbana-Champaign and its National Center for Supercomputing
Applications.
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\section{Introduction}
Video background estimation and moving object detection is a classic problem in computer vision. Among several existing approaches, one of the most prevalent ones is to solve it in a matrix decomposition framework~\cite{Bouwmans201431,Bouwmans2016}. Let $A\in \mathbb{R}^{m\times n'}$ be a matrix encoding $n'$ video frames, each represented as a vector of size $m$. Our task is to decompose all frames of the video into background and foreground frames: $A = B+ F.$
As described above, the problem is ill-posed, and more information about the structure of the decomposition is needed. In practice, background videos are often static or close to static, which typically means that $B$ is of low rank~\cite{oliver1999}. On the other hand, foreground usually represents objects occasionally moving across the foreground, which typically means that $F$ is sparse. These and similar observations leads to the development of models of the form~\cite{Bouwmans2016, Bouwmans201431, APG,LinChenMa,duttaligongshah}:
\begin{equation}\label{eq:general}\min_{B} f_{\rm rank}(B) + f_{\rm spar}(A-B),\end{equation}
where $f_{\rm rank}$ is a suitable function that encourages the rank of $B$ to be low, and $f_{\rm spar}$ is a suitable function that encourages the foreground $F$ to be sparse.
Xin \etal~\cite{xin2015} recently proposed a background estimation model---generalized fused lasso~(GFL)---arising as a special case of \cite{golub} with the choice $f_{\rm rank}(B) = {\rm rank}(B)$ and $f_{\rm spar}(F) = \lambda \|F\|_{\rm GFL}$:
\begin{align}\label{gfl}
\min_{B}{\rm rank}(B)+\lambda\|A-B\|_{\rm GFL}.
\end{align}
In this model, $\|\cdot\|_{GFL}$ is the ``generalized fused lasso'' norm, which arises from the combination of the $\ell_1$ norm (to encourage sparsity) and a local spatial total variation norm (to encourage connectivity of the foreground).
\paragraph{Supervised background estimation.} In the modern world, {\em supervised} background estimation models play an important role in the analysis of the data captured from the surveillance cameras. As the name suggests, these models rely on prior availability of some ``training'' background frames, $B_1 \in \mathbb{R}^{m\times r}$. Without loss of generality, assume that the training background frames correspond to the first $r$ frames of $B$, i.e., $B = [B_1\;B_2]$, where $B_1\in \mathbb{R}^{m\times r}$ is known and $B_2\in \mathbb{R}^{m\times n}$ is to be determined, with $n'=r+n$. Let $A=[A_1\;A_2]$ be partitioned accordingly, and let $F_2 = A_2-B_2 \in \mathbb{R}^{m\times n}$. In this setting, \cite{xin2015} further specialized the model \eqref{gfl} by adding the extra assumption that ${\rm rank}(B)={\rm rank}(B_1)$
As a result, the columns of the unknown matrix $B_2$ can be written as a linear combinations of the columns of $B_1$. Specifically, $B_2$ can be written as $B_1S$, where $S\in \mathbb{R}^{r\times n}$ is a coefficient matrix. Thus, problem (\ref{gfl}) can be written in the form
\begin{align}\label{gfl_sfl}
&\min_{S'}{\rm rank}(B_1[I\;S'])+\lambda\|A_2 - B_1S'\|_{\rm GFL}.
\end{align}
While \eqref{gfl_golub} is the the problem Xin \etal~\cite{xin2015} {\em wanted} to solve, they did not tackle it directly and instead further assumed that $S$ is sparse, and solved the modified problem
\begin{align}\label{gfl_sfl_sparse}
&\min_{S'}\|S'\|_{1}+\lambda\|A_2 - B_1 S' \|_{\rm GFL},
\end{align}
where $\|\cdot\|_{1}$ denotes the $\ell_1$ norm of matrices.
\section{New Model}
In this paper we propose a new supervised background estimation model, one that we argue is much better than \eqref{gfl_sfl_sparse} in several aspects. Moreover, our model and the methods we propose significantly outperform other state-of-the-art methods.
\paragraph{L1 regression.} As in \eqref{gfl_sfl_sparse}, our model is also based on a modified version of \eqref{gfl_sfl}. We do not need to assume any sparsity on $S'$, and instead make the trivial observation that ${\rm rank}(B_1[I \;S']) = {\rm rank}(B_1)$. Since $B_1$ is known, the first term in the objective function \eqref{gfl_sfl} is constant, and hence does not contribute to the optimization problem. Hence we may drop it. Moreover, we suggest replacing the GFL norm by the $\ell_1$ norm. This leads to a very simple {\em L1 (robust) regression problem}:
\begin{equation} \min_{S' \in \mathbb{R}^{r\times n}} \|A_2-B_1 S'\|_1. \label{model:L1-version1}
\end{equation}
\paragraph{Dimension reduction.} The above model can be further simplified. It may be the case that the rank of $B_1 \in \mathbb{R}^{m\times r}$ is smaller\footnote{If this is not the case, it still may be the case that the column space of $B_1$ can be very well approximated by a space with less or much less than $r$ dimensions.} (or much smaller) than $r$. In such a situation, we can replace $B_1$ in \eqref{model:L1-version1} by a thinner matrix, which allows us to reduce the dimension of the optimization variable $S'$. In particular, let $B_1=QR$
be the QR decomposition of $B_1$, where $Q\in \mathbb{R}^{m\times k}$, $R\in \mathbb{R}^{k\times r}$, $k={\rm rank}(B_1)$, and $Q$ has orthonormal columns. Since the column space of $B_1$ is the same as the column space of $Q$, by using the substitution $B_1 S' = Q S $, we can reformulate \eqref{model:L1-version1} as the {\em lower-dimensional } L1 regression problem:
\begin{equation} \boxed{ \min_{S \in \mathbb{R}^{k\times n}} f(S) :=\|A_2-Q S\|_1} \label{dutta_richtarik_l1}
\end{equation}
\begin{figure}
\centering
\includegraphics[width=.5\textwidth]{Results/Compare_methods_ST_1-eps-converted-to.pdf}
\caption{\small{ROC curve to compare between our proposed $\ell_1$ regression algorithms on {\tt Basic} video, frame size $144\times176$.}}
\label{compare_methods}
\end{figure}
\paragraph{Decomposition.} Let $A_2 = [a_1,\dots,a_n]$ and $S = [s_1,\dots,s_n]$, where $a_i \in \mathbb{R}^m$, $s_i \in \mathbb{R}^t$ for all $i\in [n]:=\{1,2,\dots,n\}$. Our model \eqref{dutta_richtarik_l1} can be decomposed into $n$ parts, one for each frame:
\begin{equation}\label{dutta_richtarik_l1: decompose}
f(S)= \sum_{i=1}^n f_i(s_i), \quad f_i(s_i) := \|a_i - Q s_i\|_{1},
\end{equation}
where $\|\cdot\|_1$ is the vector $\ell_1$ norm. Therefore, \eqref{dutta_richtarik_l1} reduces to $n$ small ($k
$-dimensional) and independent {\em $\ell_1$ regression problems:}
\begin{equation}\label{dutta_richtarik_l1: decompose_problems}
\boxed{\min_{s_i\in \mathbb{R}^t} f_i(s_i), \qquad i \in [n]}
\end{equation}
\paragraph{Advantages of our model.}We now list some advantages of our model \eqref{dutta_richtarik_l1} as compared to \eqref{gfl_sfl_sparse}. We show that 1) our model does not involve the unnecessary sparsity inducing term $\|S'\|_1$, that 2) our model does not include the trade-off parameter $\lambda$ and hence issues with tuning this parameter disappear, that 3) our model involves a simple $\ell_1$ norm as opposed to the more complicated GFL norm, that 4) the dimension of $S$ is smaller (and possibly much smaller) than that of $S'$, that 5) our objective is separable across the $n$ columns of $S$ corresponding to frames, which means that we can solve for each column of $S$ in {\em parallel} (for instance on a GPU), and that 6) for the same reason, we can solve for each frame as it arrives, in an {\em online} fashion.
\paragraph{Further contributions.}
Our model works well with just a few training background frames (e.g., $r=10$). This should be compared with the 200 training frames in GFL model. We propose 5 methods for solving the model, out of which 4 can work online and all 5 can work in a batch mode. Our model solves all the following challenges: static and semi-static foreground, newly added static foreground, shadows that are already present in the background and newly created by moving foreground, occlusion and disocclusion of the static and dynamic foreground, the ghosting effect of the foreground in the background. To the best of our knowledge, no other algorithm can solve all the above challenges in a single framework.
\section{Scalable Algorithms for L1 Regression}
The separable (across frames) structure of our model allows us to devise both batch and online background estimation algorithms. To the best of our knowledge, this is the first formulation which can operate in both batch and online mode. Since our problem decomposes across frames $i\in [n]$, it suffices to describe algorithms for solving the $\ell_1$ regression problem \eqref{dutta_richtarik_l1: decompose_problems} for a single $i$. This problem has the form
\begin{equation}\label{dutta_richtarik_l1: simple}
\min_{x\in \mathbb{R}^t} \phi(x) := \|Q x - b\|_1 = \sum_{j=1}^m |q_j^\top x - b_j|,
\end{equation}
where $x \in \mathbb{R}^t$ corresponds to one of the reconstruction vectors $s_i$, and $b\in\mathbb{R}^m$ corresponds to the related frame $a_i$. We write $b = (b_1,\dots,b_m)\in \mathbb{R}^m$, and let $q_j \in \mathbb{R}^t$ be the $j$th row of $Q$ for $j\in [m]$.
\begin{figure}
\centering
\includegraphics[width=2.6in]{Results/IRLS_grasta_reprocs_rpca_stuttgart_ROC_1-eps-converted-to.pdf}
\caption{\small{ROC curve to compare between IRLS, iEALM, GRASTA, and ReProCS on {\tt Basic} video, frame size $144\times176$.}}
\label{ROC_methods}
\end{figure}
\paragraph{Five methods.} In this work we propose to solve \eqref{dutta_richtarik_l1: simple} via four algorithms:~(a)~iteratively reweighted least squares~(IRLS), (b)~homotopy method,~(c)~stochastic subgradient descent (variant 1), (d)~stochastic subgradient descent (variant 2), and (e) Augmented Lagrangian Method of Multipliers~(ALM)~(see Appendix 2).
The first four algorithms can be used in both batch and online setting and can deal with grayscale and color images. If we assume the camera is static, and assume constant illumination throughout the video sequence, then our online methods can provide a good estimate of the background. Moreover, all algorithms are robust to the intermittent object motion artifacts, that is, static foreground (whenever a foreground object stops moving for a few frames), which poses a big challenge to the state-of-the-art methods. Additionally, our online methods are fast as we perform neither conventional nor incremental principal component analysis~(PCA). In contrast, conventional PCA~\cite{pca} is an essential subproblem to numerically solve both RPCA and GFL problems. In these problems, each iteration involves computing PCA, which operates at a cost~$\mathcal{O}(mn^2)$ and is due to SVD on a $m\times n$ matrix. We also recall that the state-of-the-art online, semi-online, or batch incremental algorithms, such as the Grassmannian robust adaptive subspace estimation~(GRASTA)~\cite{grasta}, recursive projected compressive sensing algorithm~(ReProCS)~\cite{reprocs,pracreprocs,modified_cs}, or incremental principal component pursuit~(incPCP)~\cite{incpcp,matlab_pcp,inpcp_jitter}, use either thin or partial PCA as well.
\paragraph{The need for simpler solvers for $\ell_1$ regression.} It is natural to ask: why do we need a new set of algorithms to solve the classical $\ell_1$ regression problem when there are several well known solvers, for example, CVX~\cite{cvx,gb08}, $\ell_1$ magic~\cite{l1magic}, and SparseLab 2.1-core~\cite{sparselab}? It turns out that a high resolution video sequence (characterized by very large $m$) is computationally extremely expensive for the above mentioned classic solvers. Moreover, we do not need highly accurate solutions. Hence, simple and scalable methods are preferable to more involved and computationally demanding methods. The $\ell_1$ magic software, for example, in our experiments took 126 minutes (on a computer with Intel i7 Processor and 16 GB memory) to estimate the background on the {\tt Waving Tree} dataset with $A_2\in\mathbb{R}^{19,200\times 66}$. In contrast, our IRLS method took 0.59 seconds only for 66 frames.
\begin{figure}
\centering
\includegraphics[width=2.5in, height = 2.0 in]{Results/l1_grasta_reprocs_rpca_stuttgart_1-eps-converted-to.pdf}
\caption{\small{Comparison of Mean SSIM~(MSSIM) of IRLS, iEALM, GRASTA, and ReProCS on~{\tt Basic} video.~IRLS has the best MSSIM.~To process 600 frames each of size $144\times176$, iEALM takes 164.03 seconds, GRASTA takes 20.25 seconds, ReProCS takes \textcolor{blue}{14.20} seconds, and our IRLS takes~\textcolor{red}{7.51} seconds.}}
\label{ssim}
\end{figure}
\subsection{Iteratively Reweighted Least Squares~(IRLS)}\label{IRLS:sec}
In the past decade, IRLS has been used in various domains, ranging from reconstruction of sparse signals from underdetermined systems, to the low-rank and sparse matrix minimization problems in face clustering, motion segmentation, filter design, automatic target detection, to mention just a few applications~\cite{richtarik_thesis, Burrus_homotopy, Candes08, ingrid, osborne, lu_irls,iirls}. We find that the IRLS algorithm is a good fit to solve~\eqref{dutta_richtarik_l1: simple}. Also, each iteration of IRLS reduces to a single weighted $\ell_2$ regression problem for an over determined system. To the best of our knowledge, we are the first to use IRLS to propose a background estimation model.
We now briefly describe IRLS for solving \eqref{dutta_richtarik_l1: simple}. First note that the cost function $f$ in \eqref{dutta_richtarik_l1: simple} can be written in the form
\begin{equation}\label{dutta_richtarik_rls: decompose_problems}
\phi(x) = \sum_{j=1}^m | q_j^\top x - b_j | = \sum_{j=1}^m \frac{( q_j^\top x - b_j )^2 }{ | q_j^\top x - b_j| }.
\end{equation}
For $x\in \mathbb{R}^m$ and $\delta>0$ define a diagonal weight matrix via
$W_\delta(x):={\rm Diag}(1 / \max \{ | q_j^\top x - b_j | , \delta \} ).$ Given a current iterate $x_k$, we may fix the denominator in \eqref{dutta_richtarik_rls: decompose_problems} by substituting $x_k$ for $x$, which makes $\phi$ dependent on $x$ via $x$ appearing in the numerator only. The problem of minimizing the resulting function in $x$
is a {\em weighted least squares problem}. The normal equations for this problem have the form
\begin{equation}\label{dutta_richtarik_irls: 1}
Q^\top W_0(x_k) Q x=Q^\top W_0(x_k) b.
\end{equation}
IRLS is obtained by setting $x_{k+1}$ to be equal to the solution of \eqref{dutta_richtarik_irls: 1}. For stability purposes, however, we shall use weight matrices $W_\delta(x_k)$ for some threshold parameter $\delta>0$ instead. This leads to the IRLS method:
\begin{equation}\label{dutta_richtarik_irls: equation}
\boxed{x_{k+1}=(Q^\top W_\delta(x_k) Q)^{-1} Q^\top W_\delta (x_k ) b}
\end{equation}
Osborne \cite{osborne} and more recently~\cite{sigl} performed a comprehensive analysis of the performance of IRLS for $\ell_p$ minimization with $1<p<3$.
\subsection{Homotopy Method}
In this section we generalize the IRLS method \eqref{dutta_richtarik_irls: equation} by introducing a {\em homotopy}~\cite{Burrus_homotopy} parameter $1\leq p \leq 2$. We set $p_0=2$ and choose $x_0 \in \mathbb{R}^t$ (in our experiments, random initialization will do). Consider the function
\[
\phi_p(x,y) := \sum_{j=1}^m \frac{(q_j^\top x - b_j)^2}{|q_j^\top y - b_j|^{2-p}}.\]
Note that $\phi_1(x,x) $ is identical to the $\ell_1$ regression function $\phi$ appearing in \eqref{dutta_richtarik_rls: decompose_problems}. Given current iterate $x_k$, consider function $\phi_{p_k}(x,x_k)$. This is a weighted least squares function of $x$. Our homotopy method is defined by setting
\[ x_{k+1} = \arg\min_x\phi_{p_k} (x, x_k), \]
and subsequently decreasing the homotopy parameter as $p_{k+1} = \max\{p_k \eta, 1\}$, where $0<\eta< 1$ is a constant reduction factor.
As in the case of IRLS, the normal equations for the above problem have the form
\begin{equation}\label{dutta_richtarik_homotopy_matrix: decompose_problems}
Q^\top W_{0,p_k} (x_k) Q x= Q^\top W_{0,p_k} (x_k) b,
\end{equation}
where
$W_{\delta,p}(x):={\rm Diag}(1 / \max \{ | q_j^\top x - b_j |^{2-p} , \delta \} ).$
The (stabilized) solution of \eqref{dutta_richtarik_homotopy_matrix: decompose_problems} is given by
\begin{equation} \boxed{x_{k+1} = (Q^\top W_{\delta,p_k} (x_k) Q)^{-1} Q^\top W_{0,p_k} (x_k) b } \label{alg:homotopy}\end{equation}
As mentioned above, one step of the homotopy scheme \eqref{alg:homotopy} is identical to one step of IRLS~\eqref{dutta_richtarik_irls: 1} when $p_k=1.$ In practice, however, the homotopy method sometimes performs better~(see Figures~\ref{compare_methods},~\ref{Toscana}, and Table~\ref{sbi_quant}).
\begin{figure}
\centering
\includegraphics[width = 0.5\textwidth]{Results/Qualitative/Compare_methods_batch1.pdf}
\caption{\small{Background recovered on Stuttgart, Wallflower, and I2R dataset.~Comparing with the ground truth~(second column), IRLS recovers the best quality background.}}
\label{Qual_comparison}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width =\textwidth]{Results/Qualitative/Compare_GFL_inWLR_2.pdf}
\caption{\small{Qualitative and Quantitative comparison with supervised GFL and inWLR.~GFL and IRLS construct better backgrounds on the {\tt Waving Tree} video.~On {\tt Basic} video all the methods have similar performance. However, supervised GFL takes 117.11 seconds and 6.25 seconds on {\tt Waving Tree} and {\tt Basic} video, respectively, to process 1 frame; whereas inWLR takes \textcolor{blue}{3.39} seconds and \textcolor{blue}{17.83} seconds, respectively on those two sequences.~In contrast,~IRLS takes \textcolor{red}{0.59} seconds and \textcolor{red}{7.02} seconds, respectively and recovers the similar SSIM map.}}
\label{Qual_comparison:GFL_inWLR}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width = \textwidth]{Results/Color_online/SBI_3.pdf}
\caption{\small{Background and foreground recovered by online methods on SBI dataset.~The videos have static, semi-static foreground, newly added static foreground, shadows that already present in the background and newly created by moving foreground, and occlusion and disocclusion of static and dynamic foreground.~For a comprehensive review of the dataset we refer the readers to~\cite{sbi_a}.}}
\label{SBI}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width = \textwidth]{Results/Color_online/toscana3.pdf}
\caption{\small{Qualitative and Quantitative comparison on {\tt Toscana-HD} video.~Besides IRLS and Homotopy, the two best methods on {\tt Toscana}, that is, Photomontage~\cite{photomontage} and SOBS1~\cite{SOBS} have MSSIM 0.9616 and 0.9892 and CQM 50.2416 and 43.3002, respectively~\cite{boumans_taxonomy}.}}
\label{Toscana}
\end{figure*}
\subsection{Stochastic Subgradient Descent}
In this section we propose the use of {\em two variants} of stochastic subgradient descent (SGD) to solve~\eqref{dutta_richtarik_l1: simple}:
\begin{equation} \label{eq:i098sh09s}
\min_{x\in \mathbb{R}^t} \phi(x) := \frac{1}{m}\sum_{j=1}^m \phi_j(x),\end{equation}
where $\phi_j(x) := m |q_j^\top x - b_j|$. Functions $\phi_j$ are convex, but not differentiable. However, they are subdifferentiable. A classical result from convex analysis says that the subdifferential of a sum of convex functions is the sum of the subdifferentials. Therefore, the subddiferential $\partial \phi$ of $\phi$ is given by the formula
$\partial \phi(x) = \frac{1}{m} \sum_{j=1}^m \partial \phi_j(x).$
In particular, if we choose $j\in [m]$ uniformly at random, and pick $g_j(x) \in \partial \phi_j(x)$, then ${\rm E} [g_j(x)] \in \partial \phi(x)$. That is, $g_j(x)$ is an unbiased estimator of a subgradient of $\phi$ at $x$.
A generic SGD method applied to \eqref{eq:i098sh09s} (or, equivalently, to \eqref{dutta_richtarik_l1: simple}) has the form
\begin{equation}
\label{alg:SGD-generic} \boxed{x_{k+1} = x_k - \eta_k g_i(x_k)}
\end{equation}
An easy calculation using the chain rule for subdifferentials of convex functions gives the following formula for $\partial\phi_j(x)=m q_j \partial | q_j^\top x-b_j|$ (see, for instance, \cite{nesterov:book}):
\begin{eqnarray}\label{partial}
\partial\phi_j(x)&=&\left\{
\begin{array}{@{}ll@{}}
m q_j, & \text{if}\ q_j^\top x-b_j >0 \\
-m q_j, & \text{if}\ q_j^\top x- b_j <0 \\
0, & \text{otherwise}
\end{array}.\right.
\end{eqnarray}
When $q_j^\top x_k-b_j$ is nonzero, each iterate of SGD moves in the direction of either vector $q_j$ or $-q_j$, with an appropriate stepsize. The initialization of the method (i.e., choice of $x_0\in \mathbb{R}^t$ and the learning rate parameters $\eta_k$) plays an important role in the convergence of the method.
We consider two variants of SGD depending on the choice of $\eta_k$ and on the vector that we output.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
method & $\eta_k$ & output \\
\hline
SGD 1 & $\tfrac{R}{\sqrt{k} \|g_j(x_k)\|}$ & $x_k$ \\
\hline
SGD 2 & $\tfrac{B}{\rho \sqrt{K}}$ & $\hat{x}_K = \tfrac{1}{K}\sum_{k=0}^{K-1} x_k$ \\
\hline
\end{tabular}
\end{center}
\caption{\small Two variants of SGD.}
\label{tbl:SGD}
\end{table}
In SGD 1 we always normalize each stochastic subgradient, and multiply the resulting vector by $R/\sqrt{k}$, where $k$ is the iteration counter, for some constant $R>0$ which needs to be tuned. This method is a direct extension of the subgradient descent method in \cite{nesterov:book}. The output is the last iterate. While we provide no theoretical guarantees for this method, it performs well in our experiments.~On the other hand, SGD 2 is a more principled method. This arises as a special case of the SGD mehod described and analyzed in \cite{MLbook}. In this method, one needs to decide on the number of iterations $K$ to be performed in advance. The method ultimately outputs the average of the iterates. The stepsize $\eta_k$ is set to $B/\rho \sqrt{K}$, where $B>0$ and $\rho>0$ are parameters the value of which can be derived from the following iteration complexity result:
\begin{theorem}[\cite{MLbook}] Let $x_*$ be a solution of \eqref{eq:i098sh09s} and let $B>0$ be such that
$ \|x_*\|\leq B$. Further, assume that $\|g_j(x)\|\leq \rho$ for all $x\in \mathbb{R}^t$ and $j\in [m]$. If SGD 2 runs for $K$ iterations with $\eta =\frac{B}{\rho\sqrt{T}}$, then
$
{\rm E}[\phi(\hat{x}_K)]-\phi(x_*)\leq \frac{B\rho}{\sqrt{K}},
$
where $
\hat{x}_K$ is given as in Table~\ref{tbl:SGD}. Moreover, for any $\epsilon>0$ to achieve ${\rm E}[\phi(\hat{x}_K)]-\phi(x_*)\leq \epsilon$ it suffices to run SGD 2 for $K$ iterations where $K\geq \frac{B^2\rho^2}{\epsilon^2}.$
\end{theorem}
\begin{table*}
\footnotesize
\begin{center}
\begin{tabular}{|l|c|c|c|}
\hline
Dataset & Video & No. of frames & Resolution\\
\hline
Stuttgart~\cite{cvpr11brutzer} & {\tt Basic}~(Grayscale) & 600 & $144\times176$ \\
& {\tt Basic}~(RGB-HD) & 600 & $600\times 800$ \\
& {\tt Lightswitch}~(RGB-HD) & 600 & $600\times 800$ \\
SBI~\cite{sbi_a}& {\tt IBMTest2}~(RGB) &91& $320\times240$ \\
& {\tt Candela}~(RGB) &351 & $352\times 288$\\
& {\tt Caviar1}~(RGB) & 610&$384\times 288$ \\
& {\tt Caviar2}~(RGB) &461 &$384\times 288$ \\
&{\tt Cavignal}~(RGB)& 258 & $200\times 136$ \\
& {\tt HumanBody}~(RGB) & 741 & $320\times 240$ \\
& {\tt HallandMonitor}~(RGB)& 296& $352\times 240$ \\
& {\tt Highway1}~(RGB) & 440 & $320\times 240$ \\
& {\tt Highway2}~(RGB) & 500 & $320\times 240$ \\
&{\tt Toscana}(RGB-HD) &6 & $600\times 800$ \\
Wallflower~\cite{wallflower} &{\tt Waving Tree}(Grayscale) &66 & $120\times 160$ \\
&{\tt Camouflage} (Grayscale) &52 & $120\times 160$ \\
I2R/Li dataset~\cite{lidata} & {\tt Meeting Room}(Grayscale) &1209 & $64\times 80$ \\
&{\tt Watersurface} (Grayscale) &162 & $128\times 160$ \\
&{\tt Lightswitch}(Grayscale) &1430 & $120\times 160$ \\
&{\tt Lake}(Grayscale) &80 & $72\times 90$ \\
\hline
\end{tabular}
\end{center}
\caption{\small{Data used in this paper.}}\label{dataset}
\end{table*}
\begin{table*}[!h]
\footnotesize
\begin{center}
\begin{tabular}{|l|c|c|c|c|}
\hline
Algorithm & Abbreviation & Appearing in Experiment & Reference\\
\hline
Iterative Reweighted Laast Squares& IRLS & Figure \ref{compare_methods}--\ref{Mean_2},and Table\ref{sbi_quant},\ref{time}& This paper\\
Homotopy & Homotopy& Figure ~\ref{compare_methods},\ref{SBI}--\ref{Mean_2}, and Table \ref{sbi_quant}, \ref{time}& This paper\\
Stochastic Subgradient Descent 1& SGD 1 & Figure \ref{compare_methods} and Table \ref{tbl:SGD}& This paper\\
Stochastic Subgradient Descent 2& SGD 2 & Figure \ref{compare_methods} and Table \ref{tbl:SGD}& This paper\\
Inexact Augmented Lagrange Method of Multipliers& iEALM &Figure \ref{ROC_methods}-\ref{Qual_comparison}&\cite{LinChenMa} \\
Supervised Generalized Fused Lasso & GFL & Figure \ref{Qual_comparison:GFL_inWLR}, \ref{Stuttgart}& \cite{xin2015} \\
Grassmannian Robust Adaptive Subspace
Tracking Algorithm & GRASTA&Figure~\ref{ROC_methods}--\ref{Qual_comparison} & \cite{grasta} \\
Recurssive Projected Compressive Sensing & ReProCS &Figure~\ref{ROC_methods}--\ref{Qual_comparison} &\cite{reprocs,pracreprocs,modified_cs} \\
Incremental Weighted Low-Rank & inWLR&Figure \ref{Qual_comparison:GFL_inWLR}& \cite{inWLR}\\
Incremental Principal Component Pursuit & incPCP & Figure \ref{SBI}--\ref{Mean_2}, and Table \ref{sbi_quant}, \ref{time} &\cite{incpcp,matlab_pcp,inpcp_jitter}\\
Background estimated by weightless neural networks & BEWIS&Table \ref{sbi_quant}&\cite{BEWIS} \\
Independent Multimodal Background Subtraction Multi-Thread & IMBS-MT &Table \ref{sbi_quant}& \cite{IMBS} \\
RSL2011& - &Table \ref{sbi_quant}& \cite{RSL} \\
Color Median&-&Table \ref{sbi_quant}&\cite{colormedian}\\
Photomontage &-&Table \ref{sbi_quant}& \cite{photomontage} \\
Self-Organizing Background Subtraction1 & SOBS1 &Table \ref{sbi_quant}& \cite{SOBS}\\
\hline
\end{tabular}
\end{center}
\caption{\small{Algorithms compared in this paper.}}\label{algo}
\end{table*}
\section{Numerical Experiments}
To validate the robustness of our proposed algorithms, we tested them on some challenging real world and synthetic video sequences containing occlusion, dynamic background, static, and semi-static foreground.~For this purpose, we extensively use 19 gray scale and RGB videos from the Stuttgart, I2R, Wallflower, and the SBI dataset~\cite{cvpr11brutzer,lidata,sbi_a, SBI_web, wallflower}. .~We refer the readers to Table \ref{dataset} to get an overall idea of the number of frames of each video sequence used, video type, and resolution.
For quantitative measure, we use the receiver operating characteristic~(ROC) curve, recall and precision~(RP) curve, the structural similarity index (SSIM), SSIM map~\cite{mssim}, multi-scale structural similarity index (MSSSIM)~\cite{msssim}, and~color image quality measure~(CQM)~\cite{boumans_taxonomy,cqm}.~Due to the availability of ground truth~(GT) frames, we use the Stuttgart artificial dataset~(has foreground GT) and the SBI dataset~(have background GTs) to analyze the results quantitatively and qualitatively.~To calculate the average computational time we ran each algorithm five times on the same dataset and compute the average.
Throughout this section, the best and the $2^{\rm nd}$ best results are colored with \textcolor{red}{red} and \textcolor{blue}{blue}, respectively.
\subsection{Comparison between our proposed algorithms}
First we compare the performance of our proposed algorithms in batch mode on the {\tt Basic} scenario.~Figure~\ref{compare_methods} shows that all four algorithms are are very competitive and we note that IRLS has the least computational time.~We ran each of IRLS and Homotopy method for five iterations, and SGD 1 and SGD 2 for 5000 iterations.~IRLS takes \textcolor{red}{7.02} seconds, Homotopy takes \textcolor{blue}{8.47} seconds, SGD 1 takes 17.81 seconds, and SGD 2 takes 17.67 seconds.~We mention that the choice of $R$ in SGD 1 and $B$ and $\rho$ in SGD 2 are problem specific.~Due to computational efficiency, we compare IRLS $\ell_1$ with other batch methods in the next section.
\begin{table*}
\footnotesize
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline
Video & SOBS1 &RSL2011& IMBS-MT &BEWIS & Color Median &IRLS &Homotopy & incPCP\\
\hline
{\tt IBMTest2}&\textcolor{red}{\bf 0.9954} &0.9303&0.9721&0.9602& 0.9939&0.9950 &\textcolor{blue}{\bf 0.9953}&0.9670\\
{\tt Candela} &0.9775 &{0.9916} & 0.9893 & 0.9852& 0.9382& \textcolor{red}{\bf 0.9995} & \textcolor{blue}{\bf 0.9992} &0.9412\\
{\tt Caviar1}& 0.9781 &0.9947 &0.9967 &0.9813& 0.9918& \textcolor{red}{\bf 0.9994} & \textcolor{blue}{\bf 0.9993}&0.8649\\
{\tt Caviar2} & 0.9994 &0.9962 &0.9986 &0.9994 & 0.9994 & \textcolor{red}{\bf 0.9999} & \textcolor{blue}{\bf 0.9998} &0.9935\\
{\tt Cavignal} &0.9947 &0.9973 &0.9982 & \textcolor{blue}{\bf 0.9984} & 0.7984 & \textcolor{red}{\bf 0.9989} &0.9975 &0.8312\\
{\tt HumanBody}&0.9980 &0.9959 &0.9958 &0.9866 & 0.9970 & \textcolor{red}{\bf 0.9996} & \textcolor{blue}{\bf 0.9990} &0.9360\\
{\tt HallandMonitor} &0.9832 &0.9377 &0.9954 &0.9626 &0.9640 & \textcolor{blue}{\bf 0.9991} & \textcolor{red}{\bf 0.9992} &0.9355\\
{\tt Highway1} &0.9968 &0.9899 &0.9939 &0.9886 & 0.9924 & \textcolor{blue}{\bf 0.9980} & \textcolor{red}{\bf 0.9985} &0.8847\\
{\tt Highway2 }&0.9991 &0.9907 &0.9960 &0.9942 & 0.9961 &\textcolor{blue}{\bf 0.9994} &\textcolor{red}{\bf 0.9997} &0.9819\\
{\tt Toscana} & 0.9616 & 0.0662 & 0.8903 & 0.8878 &0.8707 & \textcolor{blue}{\bf 0.9853} & \textcolor{red}{\bf 0.9996} & 0.8416\\
\hline
{\bf Average} &0.9814 &0.9491 &0.9929 &0.9745 & 0.9542 &\textcolor{blue}{\bf 0.9975} &\textcolor{red}{\bf 0.9987} &0.9177\\
\hline
\end{tabular}
\end{center}
\caption{\small{Comparison of average MSSSIM of the different methods on SBI dataset. Source:~\cite{SBI_web, boumans_taxonomy,sbi_a}.}}\label{sbi_quant}
\end{table*}
\subsection{Comparison with RPCA,~GFL, and other state-of-the-art methods}
In this section we compare IRLS with other state-of-the-art batch background estimation methods, such as, iEALM~\cite{LinChenMa} of RPCA, GRASTA, and ReProCS on the {\tt Basic} scenario.~Figure~\ref{ROC_methods} shows that IRLS sweeps the maximum area under the ROC curve.~Additionally, in Figure~\ref{ssim} IRLS has the best mean SSIM~(MSSIM) among all other methods.~Moreover, in batch mode, IRLS takes the least computational time.
Next in Figure~\ref{Qual_comparison} we present the background recovered by each method on Stuttgart, Wallflower, and I2R dataset. The video sequences have occlusion, dynamic background, and static foreground. IRLS can detect the static foreground and also robust to sudden illumination changes.
\begin{figure*}
\centering
\includegraphics[width = \textwidth]{Results/Color_online/Stuttgart2.pdf}
\caption{\small{Qualitative comparison on {\tt Basic} scenario HD scene.~The {\bf SSIM}s are~(the $1^{\rm st}$ number indicates frame 200 and the $2^{\rm nd}$ number indicates frame 600):~incPCP 0.021 and 0.0173, IRLS 0.9089 and \textcolor{red}{0.9731}, homotopy \textcolor{blue}{0.9327} and \textcolor{blue}{0.9705}, GFL \textcolor{red}{0.9705} and 0.9310.~The {\bf MSSSIM}s are:~incPCP 0.6315 and 0.4208, IRLS 0.8777 and \textcolor{red}{0.9746}, homotopy \textcolor{blue}{0.9166} and \textcolor{blue}{0.9725}, GFL \textcolor{red}{0.9175} and 0.9645.}}
\label{Stuttgart}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width = \textwidth]{Results/Color_online/SBI_1A.pdf}
\caption{\small{Background and foreground recovered by online methods on SBI dataset. The videos have shadows that already present in the background and newly created by moving foreground, occlusion and disocclusion of dynamic foreground.}}
\label{SBI_2}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width = \textwidth]{Results/Color_online/Stuttgart_LS.pdf}
\caption{\small{Background and foreground recovered by our proposed online methods on {\tt Lightswitch} video. Both IRLS and homotopy captures the effect of change in illumination, irregular movements of the tree leaves, and reflections. Comparing with the No FG image both of our proposed method do pretty well.}}
\label{LS}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Results/Color_online/meanCQM_SBI_st-eps-converted-to.pdf}%
\caption{\small{Mean CQM of online methods on SBI dataset and {\tt Basic-HD} video.~The higher the CQM value, the better is the recovered image.}}%
\label{Mean_2}
\end{figure}
Finally, we compare our IRLS with the supervised GFL model of Xin \etal~\cite{xin2015} and inWLR of Dutta \etal.~\cite{inWLR}~(see Figure~\ref{Qual_comparison:GFL_inWLR}).~For {\tt Waving Tree} scenario, supervised GFL uses 200 training frames and it takes 117.11 seconds to compute the background and foreground from one training frame and the ssim of the FG is \textcolor{red}{0.9996}.~inWLR does not use any training frames and takes \textcolor{blue}{3.39} seconds to compute the background and foreground from the video sequence that consists of 66 frames and the MSSIM is \textcolor{blue}{0.9592}. In contrast, IRLS uses 15 training frames and takes \textcolor{red}{0.59} seconds to process the entire video with an MSSIM 0.9398. For {\tt Basic} scenario, supervised GFL again uses 200 training frames and takes 6.25 seconds to process one training frame and the ssim of the FG is {0.9462}.~inWLR does not require any training frame and takes \textcolor{blue}{17.83} seconds to process 600 frames in a batch-incremental mode and the MSSIM is \textcolor{blue}{0.9463}. In contrast, IRLS uses only 15 training frames and takes \textcolor{red}{7.02} seconds to process the entire video and the MSSIM is \textcolor{red}{0.9524}.
\subsection{Online implementation on RGB videos}
In this section we show the robustness of two of our algorithms on RGB videos in online mode.~Due to the space limitation we only provide results on IRLS and homotopy algorithm~(these two methods were also the fastest in the batch mode).~Primarily, we compare our results with incPCP and GFL~\cite{incpcp,matlab_pcp,inpcp_jitter,xin2015}.~We should mention that besides incPCP, probabilistic robust matrix factorization~(PRMF)~\cite{prmf} and RPCA bilinear projection~(RPCA-BL)~\cite{RPCA-BL} has online extensions.~However, PRMF uses the entire available data in its batch normalization step and there is no available implementation of online RPCA-BL.~To the best of our knowledge incPCP is the only state-of-the-art online method which deals with HD RGB videos in full online mode.~The incPCP code is downloaded from the author's website\footnote{https://sites.google.com/a/istec.net/prodrig/Home/en/pubs/incpcp}.~As mentioned in the software package we use the standard PCP~(fixed camera) mode for incPCP~\cite{incpcp,matlab_pcp} implementation.
\paragraph{Discussions.}We use {\tt Basic-HD} and the SBI dataset to provide extensive qualitative and quantitative comparison. The online mode of our algorithm only uses the available pure background frames to learn the basis $Q$ for each color channel and then operate on each test frame in a complete online mode.~Note that we only use 10 training frames and we strongly believe that one can use even less number of training frames to obtain almost the similar performance.~Homotopy uses less iterations than IRLS to produce a comparable background and hence it is faster than IRLS in online mode.~In Figure~\ref{SBI} and \ref{SBI_2} we compare IRLS and homotopy against incPCP on the SBI dataset.~Compare to the ghosting appearances in the incPCP backgrounds, our online methods construct a clean background for each video sequence.~We also removed the static foreground, occluded foreground, and the foreground shadows.~In Figure~\ref{Toscana} and~\ref{Stuttgart} we show our performance on HD video sequences.~In addition to incPCP, we compared with supervised GFL on the {\tt Basic-HD}~(see Figure~\ref{Stuttgart}).~Supervised GFL uses 200 training frames~(the average processing time of the training frames is 7.31 seconds) and takes 431.78 seconds to process each test frame and produce a comparable quantitative result as online IRLS and homotopy.~For computational time comparison with incPCP we refer to Table~\ref{time}.
Finally we provide the results of online IRLS and homotopy on one of the most challenging HD video sequences, that is, {\tt Lighswitch} of the Stuttgart dataset. This scenario is a nighttime scenario and has varying illumination effects throughout the video sequence. Starting from frame 125 the illumination suddenly changes. Additionally, it has reflections, traffic light change, and movements of the tree leaves. We used 10 daytime pure background frames for training purpose and by using them we estimated the nighttime scene. As expected in Figure~\ref{LS} both IRLS and homotopy perform pretty well with the changing illumination which can be verified from the pure {\tt Lightswitch} BG frame (Figure~\ref{LS} third column).
Additionally, we compare our quantitative results against other state-of-the-art algorithms, such as, the adaptive neural background algorithm aka Self-Organizing Background Subtraction1~(SOBS1)~\cite{SOBS}, Photomontage~\cite{photomontage}, Color Median,~RSL2011~\cite{RSL}, Independent Multimodal Background Subtraction Multi-Thread~(IMBS-MT)~\cite{IMBS}, background estimated by weightless neural networks~(BEWIS)~\cite{BEWIS} on SBI dataset.~We refer~Table~\ref{sbi_quant}~(Source:~\cite{boumans_taxonomy}) and~Figure~\ref{Toscana}.~Finally in Figure~\ref{Mean_2} we provide the mean CQM of the online methods on SBI dataset and {\tt Basic-HD} video. In online mode, IRLS and Homotopy outperform incPCP in mean CQM and mean MSSSIM in each video.
\section{Conclusion}
We proposed a novel and fast model for supervised video background estimation. Moreover, it is robust to several background estimation challenges.~We used the simple and well-known $\ell_1$ regression technique and provided several online and batch background estimation methods that can process high resolution videos accurately. Our extensive qualitative and quantitative comparison on real and synthetic video sequences demonstrated that our supervised model outperforms the state-of-the-art online and batch methods in almost all cases.
\section{Appendix 1: Historical Comments}
We start by making a connection between the supervised GFL model proposed by Xin \etal~\cite{xin2015} and the constrained low-rank approximation problem of Golub \etal~\cite{golub}.
\subsection{Golub's constrained low-rank approximation problem}
In 1987, Golub et al.~\cite{golub} formulated the following constrained low-rank approximation problem: Given $A=[A_1\;A_2]\in\mathbb{R}^{m\times n'}$ with $A_1\in\mathbb{R}^{m\times r}$ and $A_2\in\mathbb{R}^{m\times n}$, find $A_G=[\tilde{B}_1\;\tilde{B}_2]$ such that, $\tilde{B}_1 \in\mathbb{R}^{m\times r},\tilde{B}_2\in\mathbb{R}^{m\times n}$, solve:
\begin{eqnarray}\label{golub's problem}
[\tilde{B}_1\;\tilde{B}_2]=\arg\min_{\substack{B=[B_1\;B_2]\\B_1=A_1\\{\rm rank}(B)\le r}}\|A-B\|_F^2,
\end{eqnarray}
where $\|\cdot\|_F$ denotes the Frobenius norm of matrices.~Motivated by \cite{golub}, Dutta \etal recently proposed more general weighted low-rank~(WLR) approximation problems and showed their application in the background estimation problem \cite{duttali_acl,duttali,duttali_bg}.
\paragraph{Connection with \eqref{golub's problem}.}
Recall that the background estimation model the generalized fused lasso~(GFL) proposed by Xin \etal~\cite{xin2015} with the choice $f_{\rm rank}(B) = {\rm rank}(B)$ and $f_{\rm spar}(F) = \lambda \|F\|_{\rm GFL}$ can be written as:
\begin{align*}
\min_{B}{\rm rank}(B)+\lambda\|A-B\|_{\rm GFL}.
\end{align*}
In this model, $\|\cdot\|_{GFL}$ is the ``generalized fused lasso'' norm. With the extra assumption that ${\rm rank}(B)={\rm rank}(B_1)$ and by using the $\|\cdot\|_{\rm GFL}$ norm, problem~\eqref{gfl} is a constrained low rank approximation problem as in~\eqref{golub's problem} and can be written as follows:
\begin{equation*}\label{gfl_golub}
\boxed{\min_{B = [B_1\;B_2]}\{\|A-B\|_{\rm GFL} \; \text{subject to}\; {\rm rank}(B)\le r, B_1=A_1\}.}
\end{equation*}
\begin{table}
\footnotesize
\begin{center}
\begin{tabular}{|l|c|c|c|c|}
\hline
Video~(No. of frames) & IRLS & Homotopy & incPCP\\
\hline
{\tt IBMTest2}~(91)& 37.28 & \textcolor{red}{\bf 21.84} &22.45 \\
{\tt Candela}~(351) &163.80 &\textcolor{blue}{\bf 133.6} & \textcolor{red}{\bf 72.15} \\
{\tt Caviar1}~(610)& 279.99 & \textcolor{blue}{\bf 213.99} & \textcolor{red}{\bf 120.58} \\
{\tt Caviar2}~(461)&199.16 & \textcolor{blue}{\bf 158.1} & \textcolor{red}{\bf 85.68} \\
{\tt Cavignal}~(258)& 71.26 & \textcolor{blue}{\bf 70.77}& \textcolor{red}{\bf 39} \\
{\tt HumanBody}~(741)&261.94 & \textcolor{blue}{\bf227.25} & \textcolor{red}{\bf 134.83}\\
{\tt HallandMonitor}~(296)&116.86 & \textcolor{blue}{\bf 88.99} & \textcolor{red}{\bf 59.63}\\
{\tt Highway1}~(440)&155.84 &\textcolor{blue}{\bf 134.03} & \textcolor{red}{\bf 81.44}\\
{\tt Highway2}~(500)&181.85 &\textcolor{blue}{\bf 156.92} & \textcolor{red}{\bf 87} \\
{\tt Basic-HD}~(600)&599.06 & \textcolor{blue}{\bf 457.2464} & \textcolor{red}{\bf 382.41}\\
{\tt Toscana-HD}~(6)&7.73 & \textcolor{blue}{\bf 5.13} & \textcolor{red}{\bf 3.1}\\
\hline
\end{tabular}
\end{center}
\caption{\small{Computational time~(in seconds) comparison for online methods.}}\label{time}
\end{table}
\section{Appendix 2 : Augmented Lagrangian Method of Multipliers~(ALM)}
In this section, we demonstrate an additional background estimation method by using the decomposition model used in the main paper. This method was not described in the main paper. As mentioned in the {\bf Further contributions} Section, we device a batch background estimation model (\textcolor{red}{fifth method}) by using the augmented Lagrangian method of multipliers (ALM).
\subsection{The algorithm}
The Augmented Lagrangian method of multipliers are one of the most popular class of algorithms in convex programming.~In our setup, the proposed method does not provide an incremental algorithm. Instead it relies on fast batch processing of the video sequence.~We can write~\eqref{dutta_richtarik_l1} as an equality constrained problem by introducing the variable $F_2$ as follows:
\begin{eqnarray}\label{dutta_richtarik_l1_constrained}
&\min_{F_2, S}\|F_2\|_{\ell_1}\nonumber\\
&{\rm subject~to~} A_2=QS+F_2.
\end{eqnarray}
We now form the augmented Lagrangian of~\eqref{dutta_richtarik_l1_constrained}:
\begin{eqnarray}\label{lagrange}
L(S,F_2,Y,\mu) = \|F_2\|_{\ell_1}+\langle Y, A_2-QS-F_2\rangle\\
+\frac{\mu}{2}\|A_2-QS-F_2\|_F^2,\nonumber
\end{eqnarray}
where $Y\in\mathbb{R}^{m\times n}$ is the Lagrange multiplier, $\langle Y, X \rangle = {\rm Trace} (Y^\top X)$ is the trace inner product, and $\mu>0$ is a penalty parameter. Completing the square and keeping only the relevant terms in~(\ref{lagrange}), for the given iterates $\{S^{(k)},F_2^{(k)},Y^{(k)},\mu_k\}$~we have
\begin{eqnarray*}\label{lagrange1}
S^{(k+1)}&=&\arg\min_{S}L(S,F_2^{(k)},Y^{(k)},\mu_k)\\
&=&\arg\min_S\frac{\mu_k}{2}\left\|A_2-QS-F_2^{(k)}+\frac{1}{\mu_k}Y^{(k)}\right\|_F^2,\\
F_2^{(k+1)}&=&\arg\min_{F_2}L(S^{(k+1)},F_2,Y^{(k)},\mu_k)\\
&=&\arg\min_{F_2} \|F_2\|_{\ell_1}+\frac{\mu_k}{2}\left\|A_2-QS^{(k+1)}-F_2+\frac{1}{\mu_k}Y^{(k)}\right\|_F^2.
\end{eqnarray*}
\begin{algorithm}
\small{
\SetAlgoLined
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\SetKwInOut{Init}{Initialize}
\nl\Input{$A=[A_1\;\;A_2] \in\mathbb{R}^{m\times n'}$ (data matrix),~threshold $\epsilon>0,\rho>1, \mu_0>0$\;}
\nl\Init {$A_1=QR, Y^{(0)}= A_2/{\|A_2\|_{\infty}}, S^{(0)}, F_2^{(0)}$\;}
\nl \While{not converged}
{
\nl $S^{(k+1)}= Q^\top (A_2-F_2^{(k)}+\frac{1}{\mu_k}Y^{(k)})$\;
\nl $F_2^{(k+1)} = \mathcal{S}_{\frac{1}{\mu_k}}(A_2-QS^{(k+1)}+\frac{1}{\mu_k}Y^{(k)})$\;
\nl $Y^{(k+1)}=Y^{(k)} +\mu_k(A_2-QS^{(k+1)}-F_2^{(k+1)})$\;
\nl $\mu_{k+1}=\rho\mu_k$\;
\nl $k=k+1$\;
}
\nl \Output{$S^{(k)},F_2^{(k)}$}}
\caption{ALM}\label{slm}
\end{algorithm}
The solution to the first subproblem is obtained by setting the gradient of $L(S,F_2^{(k)},Y^{(k)},\mu_k)$ with respect to $S$ to 0, and using the fact that $Q^\top Q = I$:
\begin{eqnarray}\label{sk}
S^{(k+1)}= Q^\top \left(A_2-F_2^{(k)}+\frac{1}{\mu_k}Y^{(k)}\right).
\end{eqnarray}
The second subproblem is the classic sparse recovery problem and its solution is given by
\begin{eqnarray}\label{fk}
F_2^{(k+1)} = \mathcal{S}_{\frac{1}{\mu_k}}\left(A_2-QS^{(k+1)}+\frac{1}{\mu_k}Y^{(k)}\right),
\end{eqnarray}
where $\mathcal{S}_{\frac{1}{\mu_k}}(\cdot)$ is the elementwise shrinkage function~\cite{shrinkage,shrinkage2}.~We update $Y_k$ and $\mu_k$ via:
\begin{eqnarray}\label{yk}
\left\{\begin{array}{ll}
Y^{(k+1)}=Y^{(k)} +\mu_k(A_2-QS^{(k+1)}-F_2^{(k+1)})\\
\mu_{k+1}=\rho\mu_k
\end{array},\right.
\end{eqnarray}
for a fixed $\rho>1.$
\paragraph{Dual problem.} Next we formulate the Lagrangian dual of~\eqref{dutta_richtarik_l1} to get an insight into the choice of the Lagrange multiplier $Y$. Using standard arguments, we obtain
\begin{eqnarray}\label{lagrange_dual}
\min_{F_2,S \;:\; A_2 = QS + F_2} \|F_2\|_{\ell_1} &=& \min_{F_2,S} \sup_{Y} \|F_2\|_{\ell_1}\notag \\
&&+\langle Y, A_2 - QS - F_2 \rangle \notag \\
& \geq & \sup_{Y}\min_{F_2,S} \|F_2\|_{\ell_1} \notag \\
&&+\langle Y, A_2-QS-F_2\rangle \nonumber\\
&=& \sup_{Y \;:\; \|Y\|_{\infty}\le 1, \; Q^\top Y = 0} \langle Y, A_2\rangle. \notag\\
\end{eqnarray}
The last problem above is the dual of~\eqref{dutta_richtarik_l1} . Clearly, the dual is a linear program. Note that the constraint $Q^\top Y$ dictates that the columns of $Y$ be orthogonal to all columns of $Q$ . That is, the columns of $Y$ must be from the nullspace of $Q$. If we relax this constraint, the resulting problem has a simple closed form solution, namely
\[Y^{(0)} = A_2/{\|A_2\|_{\infty}}.\]
This is a good choice for the initial value of $Y$ in Algorithm~\ref{slm} .
\subsection{Grassmannian robust adaptive subspace estimation~(GRASTA)}\label{Grasta:sec}
Due to close connection with our ALM, we explain the Grassmannian robust adaptive subspace estimation~(GRASTA) in this section.
In 2012, He et al.~\cite{grasta} proposed GRASTA, a robust subspace tracking algorithm, and showed its application in background estimation problem. Unlike Robust PCA~\cite{LinChenMa,APG}, GRASTA is not a batch-video background estimation algorithm. GRASTA solves the background estimation problem in an incremental manner, considering one frame at a time.~At each time step $i$, it observes a subsampled video frame $a_{{i}_{\Omega_s}}$. That is, each video frame $a_i\in\mathbb{R}^{m}$ is subsampled over the index set $\Omega_s\subset\{1,2,\cdots,m\}$ to produce $a_{{i}_{\Omega_s}}$, where $s$ is the subsample percentage. Similarly, denote the foreground as $F_2 = (f_1,\dots,f_n)$. Therefore, $f_{{i}_{\Omega_s}}\in\mathbb{R}^{|{\Omega_s}|}$ is a vector whose entries are indexed by $\Omega_s$.~Considering each video frame $a_{{i}_{\Omega_s}}$ has a low rank (say, $r$) and sparse structure, GRASTA models the video frame as:
\begin{eqnarray*}
a_{{i}_{\Omega_s}}=U_{{\Omega_s}}x+f_{{i}_{\Omega_s}}+\epsilon_{{\Omega_s}},
\end{eqnarray*}
where $U\in\mathbb{R}^{{m}\times r}$ be an orthonormal basis of the low-dimensional subspace, $x\in\mathbb{R}^r$ is a weight vector, and $\epsilon_{\Omega_s}\in\mathbb{R}^{|{\Omega_s}|}$ is a Gaussian noise vector. The matrix $U_{\Omega_s}\in\mathbb{R}^{|{\Omega_s}|\times r}$ results from choosing the rows of $U$ corresponding to the index set ${\Omega_s}$.~With the notations above, at each time step $i$, GRASTA solves the following optimization problem:~For a given orthonormal basis $U_{\Omega_s}\in\mathbb{R}^{|{\Omega_s}|\times r}$ solve
\begin{eqnarray}\label{Grasta}
\min_x\|U_{\Omega_s}x-a_{{i}_{\Omega_s}}\|_{\ell_1}.
\end{eqnarray}
Problem~\eqref{Grasta} is the classic least absolute deviations problem similar to~\eqref{dutta_richtarik_l1: decompose} and can be rewritten as:
\begin{eqnarray}\label{Grasta_alm}
&\min_{f_{{i}_{\Omega_s}}}\|f_{{i}_{\Omega_s}}\|_{\ell_1}\nonumber\\
&{\rm subject~to~} U_{{\Omega_s}}x+f_{{i}_{\Omega_s}}-a_{{i}_{\Omega_s}}=0.
\end{eqnarray}
Problem~\eqref{Grasta_alm} can be solved by the use of the augmented Lagrangian multiplier method~(ALM)~\cite{boyd}.~In GRASTA, after updating $x$ and $f_{{i}_{\Omega_s}}$, one has to update the orthonormal basis $U_{\Omega_s}$ as well. The rank one $U_{\Omega_s}$ update step is done first by finding a gradient of the augmented Lagrange dual of~\eqref{Grasta_alm}, and then by using the classic gradient descent algorithm. In summary, at each time step $i$, given a $U^{(i)}\in\mathbb{R}^{{m}\times r}$ and $\Omega_s\subset\{1,2,\cdots,m\}$, GRASTA finds $x$ and $f_{{i}_{\Omega_s}}$ via \eqref{Grasta_alm} and then updates $U_{{\Omega_s}}^{(i+1)}$. This process continues until the video frames are exhausted.
\paragraph{Comparison between ALM and GRASTA.}
1. At each step of GRASTA, the background and the sparse foreground are given as $U_{{\Omega_s}}x$ and $a_{{i}_{\Omega_s}} -U_{{\Omega_s}}x$, respectively and then one has to update the basis $U_{{\Omega_s}}$. In contrast,~\eqref{dutta_richtarik_l1_constrained} solves a supervised batch video background estimation problem. In our model, once we obtain the basis set from the $QR$ decomposition of the background matrix $A_1$, we do not update the basis further. 2. GRASTA lacks a convergence analysis which is harder to obtain as the objective function~\eqref{Grasta} in their set-up is only convex in each component.~\cite{grasta}. Our objective function in~\eqref{dutta_richtarik_l1} and in \eqref{lagrange} are convex and therefore allow us to propose a thorough convergence analysis for ALM.
\subsection{Cost of One Iteration}\label{complexity:ALM}
We discuss the complexity of one iteration of Algorithm~\ref{slm} when $A_1$ is of full rank, that is, ${\rm rank}(A_1)=r$. The complexity of the $QR$ decomposition at the initialization step is $\mathcal{O}(2mr^2-\frac{2}{3}r^3)$. Because $r\le r_{{\rm max}}$, the maximum number of available training frames, the above cost can be controlled by the user. Next, the complexity of one iteration of Algorithm~\ref{slm} is $\mathcal{O}(mnr).$ In contrast, the cost of each iteration of GRASTA is $\mathcal{O}(|{\Omega_s}|r^3+Kr|{\Omega_s}|+mr^2),$ where $K$ is the number of inner iterations and $|{\Omega_s}|$ is the cardinality of the index set $\Omega_s\subset\{1,2,\cdots,m\}$ from which each video frame $a_i\in\mathbb{R}^{m}$ is subsampled at a percentage $s$~(see Section~\ref{Grasta:sec}).
\subsection{Stopping Criteria}
Define $L_{k}:=L(S^{(k)},F_2^{(k)},Y^{(k-1)},\mu_{k-1})$. With the notations above, for a given $\epsilon>0$, Algorithm~\ref{slm} converges if $\|A_2-QS^{(k)}-F_2^{(k)}\|_F/\|A_2\|_F<\epsilon$, or $|L_{k}-L_{k-1}|<\epsilon$, or if the maximum iteration is reached.
\begin{figure*}
\centering
\begin{subfigure}{0.48\textwidth}
\includegraphics[width=\textwidth]{Results/ALM_grasta_reprocs_rpca_stuttgart_ROC-eps-converted-to.pdf}
\caption{}
\label{ROC_methods}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\includegraphics[width = \textwidth]{Results/ALM_grasta_reprocs_rpca_stuttgart_ssim-eps-converted-to.pdf}
\caption{}
\label{alm_ssim}
\end{subfigure}
\caption{\small{(a) ROC curve to compare between ALM, iEALM, GRASTA, and ReProCS on {\tt Basic} video, frame size $144\times176$.} (b) \small{Comparison of Mean SSIM~(MSSIM) of ALM, iEALM, GRASTA, and ReProCS on~{\tt Basic} video. ALM has the best MSSIM.~To process 600 frames each of size $144\times176$, iEALM takes 164.03 seconds, GRASTA takes 20.25 seconds, ReProCS takes \textcolor{blue}{14.20} seconds, and ALM takes~\textcolor{red}{13.13} seconds.}}
\end{figure*}
\subsection{Remarks on the Behaviour of ALM}
In this section, we propose the convergence of Algorithm~\ref{slm}.
\begin{lemma}\label{bound:yk}
The sequence $\{{Y}^{(k)}\}$ is bounded.
\end{lemma}
\begin{proof}
By the optimality condition of $F_2^{(k+1)}$ we have,
$$
0\in\partial_{F_2}L(S^{(k+1)},F_2,Y^{(k)},\mu_k).
$$
Therefore,
$$
0\in\partial\|F_2^{(k+1)}\|_{\ell_1}-\mu_k(A_2-QS^{(k+1)}-F_2^{(k+1)}+\frac{1}{\mu_k}Y^{(k)}),
$$
which implies ${Y}^{(k+1)}\in\partial\|F_2^{(k+1)}\|_{\ell_1}.$~By using Theorem 4 in~\cite{LinChenMa}~(see also~\cite{watson}), we conclude that the sequence $\{Y^{(k)}\}$ is bounded by the dual norm of $\|\cdot\|_{\ell_1}$, that is, the $\|\cdot\|_{\infty}$ norm.
\end{proof}
\begin{theorem}\label{Th:1}
There is a constant $\gamma$ such that
$$\|A_2-QS^{(k)}-F_2^{(k)}\|\le \frac{\gamma}{\mu_k},~~k=1,2,\cdots.$$
\end{theorem}
\begin{proof}
By using~\eqref{yk} we have
\begin{eqnarray*}
A_2-QS^{(k)}-F_2^{(k)}=\frac{1}{\mu_{k-1}}(Y^{(k)}-Y^{(k-1)}).
\end{eqnarray*}
The result follows by applying Lemma~\ref{bound:yk}.
\end{proof}
\begin{figure}
\centering
\includegraphics[width = 0.4\textwidth]{Results/Qualitative/ALM_qual.pdf}
\caption{\small{Background and foreground recovered by ALM.~The videos have static foreground and dynamic background.}}
\label{alm}
\end{figure}
\begin{theorem}
The sequence $\{L_{k}\}$ is bounded above and
$$
L_{k+1}-L_{k}\le O\left(\frac{1}{\mu_{k-1}}\right),~~k=1,2,\cdots.
$$
\end{theorem}
\begin{proof}We have,
\begin{eqnarray}
L_{k+1} &=&L(S^{(k+1)},F_2^{(k+1)},Y^{(k)},\mu_{k})\nonumber\\
&\le& L(S^{(k+1)},F_2^{(k)},Y^{(k)},\mu_{k})\nonumber\\
&\le& L(S^{(k)},F_2^{(k)},Y^{(k)},\mu_{k})\nonumber\\
&=& \|F_2^{(k)}\|_{\ell_1}+\langle Y^{(k)}, A_2-QS^{(k)}-F_2^{(k)}\rangle\nonumber\\
&&+\frac{\mu_k}{2}\|A_2-QS^{(k)}-F_2^{(k)}\|_F^2\nonumber\\
&=& \|F_2^{(k)}\|_{\ell_1}+\langle Y^{(k-1)}, A_2-QS^{(k)}-F_2^{(k)}\rangle\nonumber\\
&&+\frac{\mu_{k-1}}{2}\|A_2-QS^{(k)}-F_2^{(k)}\|_F^2 \notag \\
&& \qquad +\langle Y^{(k)}-Y^{(k-1)}, A_2-QS^{(k)}-F_2^{(k)}\rangle\nonumber\\
&& +\frac{\mu_k-\mu_{k-1}}{2}\|A_2-QS^{(k)}-F_2^{(k)}\|_F^2\nonumber\\
&\overset{\text{(using~\eqref{yk})}}=&L_k+\mu_{k-1}\|A_2-QS^{(k)}-F_2^{(k)}\|_F^2\nonumber\\
&&+\frac{\mu_k-\mu_{k-1}}{2}\|A_2-QS^{(k)}-F_2^{(k)}\|_F^2\nonumber\\
&=&L_k+\frac{\mu_k+\mu_{k-1}}{2}\|A_2-QS^{(k)}-F_2^{(k)}\|_F^2.\nonumber
\end{eqnarray}
Therefore,
$$
L_{k+1}-L_{k}\le\frac{\mu_k+\mu_{k-1}}{2}\|A_2-QS^{(k)}-F_2^{(k)}\|_F^2,~~k=1,2,\cdots.
$$
By using \eqref{yk} we have for $k=1,2,\cdots$
\[
L_{k+1}-L_{k} \leq \frac{\mu_k+\mu_{k-1}}{\mu_{k-1}^2}\|Y^{(k)}-Y^{(k-1)}\|_F^2 = \frac{1+\rho}{\mu_{k-1}}\|Y^{(k)}-Y^{(k-1)}\|_F^2.
\]
Next by using the boundedness of $\{{Y}^{(k)}\}$ we find
\begin{eqnarray*}
L_{k+1}-L_{k}&\le&O\left(\frac{1}{\mu_{k-1}}\right),~~k=1,2,\cdots,
\end{eqnarray*}
which is what we set out to prove.
\end{proof}
\begin{theorem} We have
$$
f^*-\|F_2^{(k)}\|_{\ell_1}\le O\left(\frac{1}{\mu_k}\right),
$$
where $f^*=\displaystyle{\min_{A_2=QS+F_2}\|F_2\|_{\ell_1}}.$
\end{theorem}
\begin{proof}
By using the triangle inequality we have
\begin{eqnarray}
\|F_2^{(k)}\|_{\ell_1}&\ge&\|A_2-QS^{(k)}\|_{\ell_1}\nonumber\\
&&-\|A_2-QS^{(k)}-F_2^{(k)}\|_{\ell_1}\nonumber\\
&\overset{\text{(using~\eqref{yk})}}\ge&f^*-\frac{1}{\mu_{k-1}}\|Y^{(k)}-Y^{(k-1)}\|_{\ell_1}.\nonumber\\
\end{eqnarray}
The result follows by applying boundedness of the multipliers $Y^{(k)}$.
\end{proof}
\section{Smooth Optimization of $\ell_1$ Regression with Parallel Coordinate Descent Methods~\cite{Richtarik_Olivier}}
Imagine a situation when one processes a very low-resolution video sequence with a huge number of available training frames. That is, when there are more training frames $r$ than the number of pixels $m$, the method used in~\cite{Richtarik_Olivier} to solve~\eqref{dutta_richtarik_l1: decompose_problems} for each $i$ could be more effective. In this scenario we propose to solve each $\ell_1$ regression problem in~\eqref{dutta_richtarik_l1: decompose_problems} by using the parallel coordinate descent methods on their smooth variants~\cite{Richtarik_Olivier}. Note that each $f_i(s_i)$ is a non-smooth continuous convex function on a compact set $E_1$. By using Nesterov's smoothing technique~\cite{Nesterov} one can find a smooth approximation $f^{\mu}_i(s_i)$ of $f_i(s_i)$ for any $\mu>0.$~Fercoq et al.~\cite{Richtarik_Olivier} minimized $f^{\mu}_i(s_i)$ to approximately solve the original $\ell_1$ regression problem that contains $f_i(s_i)$.
\section{Additional numerical experiments demonstrating the effectiveness of ALM}
To demonstrate the robustness of the ALM in batch mode, we compare ALM with other state-of-the-art batch background estimation methods, such as, iEALM~\cite{LinChenMa} of RPCA, GRASTA~\cite{grasta}, and ReProCS~\cite{reprocs} on the {\tt Basic} scenario. We use 15 training frames for ALM. Figure~\ref{ROC_methods} shows that ALM covers the maximum area under the ROC curve.~Additionally, in Figure~\ref{alm_ssim}, our ALM has the best mean SSIM~(MSSIM) among all other methods.~Moreover, in batch mode, ALM takes the least computational time.~The background and foreground recovered by ALM in batch mode also shows its effectiveness in supervised background estimation~(see Figure~\ref{alm})
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\section{Introduction}\label{introduction}
\noindent One of the outstanding features of
two dimensional field theories is bosonization where a free massless fermionic field can be written as a bosonic filed.
This property
is rooted in the work of Jordan and Wigner \cite{jordan} where it was shown that the fermionic creation and
annihilation operators may be represented as the bosonic counterparts. On the other hand, the idea of
describing strong interaction process in terms of currents was proposed in \cite{gellmann,dashen,sugy}. In
this approach, the dynamical variables are taken to be the currents and the canonical formalism is abandoned.
In other words, each particle does not correspond to a field which satisfies the canonical commutation relation but Hilbert space is built upon current operators.
Accordingly, it was shown that the energy momentum-tensor of these theories can be expressed as a quadratic
function of the currents known as Sugawara construction \cite{sugy}. Later on, it was proved that the
symmetric energy-momentum tensor of the two-dimensional free massless fermionic theory is exactly equivalent
to the Sugawara energy-momentum tensor which is bilinear in fermionic currents \cite{gross}.\\
\noindent
Indeed, this equivalence confirmed the result of \cite{jordan}, equivalence of free massless fermions and
bosons, in an elegant way. Then generalization of this famous equivalence to the curved space-time was
performed in \cite{davies} where boson-fermion correspondence was shown for a general metric in two
dimensions.\\
\noindent
Our purpose is to study whether this equivalence is satisfied for noncommutative space, where the
nature of the space-time changes at very short distances \cite{connes, sheikh,douglas}, which is not a trivial extension.
The authors in \cite{banerjee} considered the noncommutative generalization of the Sugawara energy-momentum tensor and then used the Seiberg-Witten map. While in the present work, the correspondence between the noncommutative Sugawara
construction and the symmetric energy-momentum tensor for two dimensional free fermionic theory is
addressed, without employing the Seiberg-Witten map. Applying the techniques described in \cite{gross}, we demonstrate that the noncommutative
Sugawara energy-momentum tensor exactly leads to the symmetric energy-momentum tensor.
An interesting physical consequence of this equivalence is noncommutative bosonization
that exhibits the relationship between the fermionic and bosonic fields in noncommutative space, as will be
discussed in the last section.
\section{Equivalence of the symmetric energy-momentum tensor and Sugawara energy-momentum tensor in
noncommutative space}
\noindent
The first part of this section includes derivation of the symmetric energy-momentum tensor using
variation of the action with respect to a generic metric. In the second part, we extend the Sugawara
construction to noncommutative space and demonstrate that it will be equivalent to the symmetric
energy-momentum tensor.
\subsection{Symmetric energy-momentum tensor}
\noindent
Let us start from the noncommutative version of the free massless fermionic Lagrangian density, which is obtained
by replacing the ordinary product with the star-product
\begin{eqnarray}\label{lagrangian} {\cal
L}=\frac{i}{2}\left(\bar\psi\gamma^{\alpha}\star\partial_{\alpha}\psi
-\partial_{\alpha}\bar\psi\gamma^{\alpha}\star\psi\right),
\end{eqnarray}
where the star-product is defined as follows
\begin{eqnarray}\label{star-def} f(x)\star
g(x)\equiv \exp\left(\frac{i\theta_{\alpha\beta}}{2}\frac{\partial}{\partial
a_{\alpha}}\frac{\partial}{\partial b_{\beta}}\right) f(x+a)g(x+b)\Bigg|_{a,b=0},
\end{eqnarray}
here
$\theta_{\mu\nu}$ is an antisymmetric constant matrix. As is usual in two-dimensional field theory, we choose to work in Euclidean
signature. In the noncommutative version, this has the added virtue that the Euclidean theory does not have issues with unitarity.
The noncommutative symmetric
energy-momentum tensor $T_{\mu\nu}^{^{\star}}$ is achieved by variation of the action $S$ with respect to a
generic metric $g_{\mu\nu}$ and setting $g^{\mu\nu}=\delta^{\mu\nu}$ in the end \cite{ashok}:
\begin{eqnarray}\label{def ofem tensor}
T_{\mu\nu}^{^{\star}}=\frac{2}{\sqrt{g}}\frac{\delta S}{\delta
g^{\mu\nu}(x)}\bigg|_{g^{\mu\nu}=\delta^{\mu\nu}},
\end{eqnarray}
where $g$ indicates the determinant of the metric with signature $(+,+)$.
The variation of the action corresponding to the Lagrangian density (\ref{lagrangian}) can be written as
\begin{eqnarray}
\delta S&=&\frac{i}{8}\int
d^{2}y\sqrt{g}\star\left(\bar\psi\star\gamma_{\alpha}\delta g^{\alpha\beta}
\star\partial_{\beta}\psi+
\bar\psi\star\gamma_{\alpha}\delta g^{\beta\alpha}\star\partial_{\beta}\psi-
\partial_{\alpha}\bar\psi\star\gamma_{\beta}\delta g^{\alpha\beta}\star\psi
-\partial_{\alpha}\bar\psi\star\gamma_{\beta}\delta g^{\beta\alpha}\star\psi\right)\nonumber\\
&+&\frac{i}{2}\int
d^{2}y~\left(\delta\sqrt{g}\right)\star\left(\bar\psi\gamma^{\alpha}\star\partial_{\alpha}\psi
-\partial_{\alpha}\bar\psi\gamma^{\alpha}\star\psi\right).
\end{eqnarray}
Using the
relation (\ref{def ofem tensor}) and the cyclic property of the star product under the integral, we have
\begin{eqnarray}
T_{\mu\nu}^{^{\star}}&=&-\frac{i}{4} \left(\partial_{\nu}\psi_{\beta}\star
\bar\psi_{\alpha}(\gamma_{\mu})^{\alpha\beta}+\partial_{\mu}\psi_{\beta}\star
\bar\psi_{\alpha}(\gamma_{\nu})^{\alpha\beta}+\partial_{\mu}\bar\psi_{\alpha}\star\psi_{\beta}(\gamma_{\nu})^{\alpha\beta}
+\partial_{\nu}\bar\psi_{\alpha}\star\psi_{\beta}(\gamma_{\mu})^{\alpha\beta}\right)
\nonumber\\&-&\frac{i}{2}\delta^{\mu\nu}\left(
\bar\psi_{\alpha}\star\partial_{\lambda}\psi_{\beta}(\gamma^{\lambda})^{\alpha\beta}
-\partial_{\lambda}\bar\psi_{\alpha}\star\psi_{\beta}(\gamma^{\lambda})^{\alpha\beta}\right).
\end{eqnarray}
Applying the equation of motion for free massless fermions, we find the energy-momentum tensor as
\begin{eqnarray}\label{final-canonical}
T_{\mu\nu}^{^{\star}}&=&-\frac{i}{4} \left(\partial_{\nu}\psi_{\beta}\star
\bar\psi_{\alpha}(\gamma_{\mu})^{\alpha\beta}+\partial_{\mu}\psi_{\beta}\star
\bar\psi_{\alpha}(\gamma_{\nu})^{\alpha\beta}+\partial_{\mu}\bar\psi_{\alpha}\star\psi_{\beta}
(\gamma_{\nu})^{\alpha\beta}
+\partial_{\nu}\bar\psi_{\alpha}\star\psi_{\beta}(\gamma_{\mu})^{\alpha\beta}\right),
\end{eqnarray}
which is
completely symmetric under $\mu\leftrightarrow\nu$.
\subsection{Sugawara energy-momentum tensor}
\noindent
The equivalence of the Sugawara construction and the symmetric energy-momentum tensor in commutative space has been shown in \cite{gross} and is reviewed in
appendix A. In the present section, we construct the noncommutative version of the Sugawara energy-momentum
tensor to demonstrate that it is precisely equivalent to (\ref{final-canonical}).\\ \noindent The Lagrangian
(\ref{lagrangian}) is invariant under global U(1) transformation which yields two different Noether currents
\cite{Hayakawa}
\begin{eqnarray}\label{currents}
{\cal
J}_{\mu}(x)=:\bar\psi_{\alpha}(x)\star\psi_{\beta}(x):(\gamma_{\mu})^{\alpha\beta},~~~~
J_{\mu}(x)=:\psi_{\beta}(x)\star\bar\psi_{\alpha}(x):(\gamma_{\mu})^{\alpha\beta},
\end{eqnarray}
where $:$ $:$ denotes normal
ordering. Now, we extend the commutative Sugawara construction to the noncommutative one as a bilinear
function of $J_{\mu}(x)$ with inserting star product instead of ordinary product
\begin{eqnarray}\label{nc-sugawara}
T_{\mu\nu}^{s^{\star}}=\frac{1}{2c}\bigg(~J_{\mu}(x)\star J_{\nu}(x)+
J_{\nu}(x)\star J_{\mu}(x)-\delta_{\mu\nu} J^{\lambda}(x)\star J_{\lambda}(x)\bigg),
\end{eqnarray}
where $c$ is the Schwinger constant which appears in the equal-time commutator of currents. Since the mass
dimension of energy-momentum tensor and of the currents in two dimensions is equal to two and one respectively,
the coefficient $c$ should be dimensionless while in four dimensions, it is a dimensionful quantity
with dimension of a mass
square. The
detailed analysis of the current algebra in two dimensions shows that the value of $c$ in noncommutative
case
is the same as the commutative one, $c=\frac{1}{\pi}$ \cite{Ghezelbash}.
To prove that (\ref{nc-sugawara}) is exactly equivalent to (\ref{final-canonical}), we need to regularize the operator products in (\ref{nc-sugawara}).
To this end, we use
the point-splitting technique \cite{schwinger-algebra} and replace $J_{\mu}(x)\star
J_{\nu}(x)$
with
\begin{eqnarray} &&\lim_{\epsilon\rightarrow 0}~\bigg(J_{\mu}(x+\epsilon)\star J_{\nu}(x)-\langle
J_{\mu}(x+\epsilon)\star J_{\nu}(x)\rangle\bigg),
\end{eqnarray}
which leads to
\begin{eqnarray}\label{nc-sugawara-1}
T_{\mu\nu}^{s^{\star}}&=&\frac{\pi}{2}\lim_{\epsilon\rightarrow 0}\bigg(J_{\mu}(x+\epsilon)\star
J_{\nu}(x)+ J_{\nu}(x+\epsilon)\star J_{\mu}(x)- \delta_{\mu\nu} J^{\lambda}(x+\epsilon)\star
J_{\lambda}(x)\nonumber\\ &-&\langle J_{\mu}(x+\epsilon)\star J_{\nu}(x) \rangle-\langle J_{\nu}(x+\epsilon)\star J_{\mu}(x)
\rangle+\delta_{\mu\nu}\langle J^{\lambda}(x+\epsilon)\star J_{\lambda}(x) \rangle \bigg).
\end{eqnarray}
To perform
some algebraic manipulations on (\ref{nc-sugawara-1}), we employ the star product definition (\ref{star-def})
\begin{eqnarray}\label{operator-definition}
J_{\mu}(x+\epsilon)=
{\cal F}_{ab}:\psi_{\beta}(x+\epsilon+a)\bar\psi_{\alpha}(x+\epsilon+b):\Bigg|_{a,b=0}(\gamma_{\mu})^{\alpha\beta},
\end{eqnarray}
where ${\cal F}_{ab}$ is an abbreviated notation for the exponential operator appearing in (\ref{star-def}).
Accordingly, the first term of the equation (\ref{nc-sugawara-1}) can be written as
\begin{eqnarray}\label{abbreviated-form}
J_{\mu}(x+\epsilon)\star J_{\nu}(x)&=&
{\cal F}_{fg}{\cal F}_{ab}{\cal F}_{cd}:\psi_{\beta}(x+\epsilon+f+a)\bar\psi_{\alpha}(x+\epsilon+f+b)::\psi_{\sigma}
(x+g+c)\bar\psi_{\rho}(x+g+d):\nonumber\\ &\times
&(\gamma_{\mu})^{\alpha\beta}(\gamma_{\nu})^{\rho\sigma}\Bigg|_{f,g,a,b,c,d=0}.
\end{eqnarray}
Using Wick's theorem, (\ref{abbreviated-form}) changes into
\begin{eqnarray} \label{simple-form}
J_{\mu}(x+\epsilon)\star
J_{\nu}(x)&=&{\cal F}_{fg}{\cal F}_{ab}{\cal F}_{cd}\bigg(:\psi_{\beta}(x+\epsilon+f+a)\bar\psi_{\alpha}(x+\epsilon+f+b)\psi_{\sigma}(x+g+c)\bar\psi_{\rho}(x+g+d):\nonumber\\&-&
:\psi_{\sigma}(x+g+c)\langle\psi_{\beta}(x+\epsilon+f+a)\bar\psi_{\rho}(x+g+d)\rangle\bar\psi_{\alpha}(x+\epsilon+f+b):\nonumber\\&-&
:\psi_{\beta}(x+\epsilon+f+a)\langle\psi_{\sigma}(x+g+c)\bar\psi_{\alpha}(x+\epsilon+f+b) \rangle\bar\psi_{\rho}(x+g+d):
\nonumber\\&-& \langle\psi_{\beta}(x+\epsilon+f+a)\bar\psi_{\rho}(x+g+d)
\rangle\langle\psi_{\sigma}(x+g+c)\bar\psi_{\alpha}(x+\epsilon+f+b)\rangle\bigg)\nonumber\\ &\times
&(\gamma_{\mu})^{\alpha\beta}(\gamma_{\nu})^{\rho\sigma}\Bigg|_{f,g,a,b,c,d=0}.
\end{eqnarray}
Rewriting the
other terms of (\ref{nc-sugawara-1}) similar to (\ref{simple-form}) and substituting them again into
(\ref{nc-sugawara-1}), we obtain
\begin{eqnarray}\label{final form}
T_{\mu\nu}^{s^{\star}}=\frac{\pi}{2}\lim_{\epsilon\rightarrow 0}\bigg[{\cal Q}_{\mu\nu}(x,\epsilon)-{\cal
R}_{\mu\nu}(x,\epsilon)-{\cal R}_{\nu\mu}(x,\epsilon)-{\cal S}_{\nu\mu}(x,-\epsilon)-{\cal
S}_{\mu\nu}(x,-\epsilon)-\delta_{\mu\nu}[{\cal R}_{\lambda}^{\lambda}(x,\epsilon)+{\cal
S}_{\lambda}^{\lambda}(x,-\epsilon)]\bigg],\nonumber\\
\end{eqnarray}
with
\begin{eqnarray}\label{components-of-T} {\cal
Q}_{\mu\nu}(x,\epsilon)&=&{\cal F}_{fg}{\cal F}_{ab}{\cal F}_{cd}:\bigg(\psi_{\beta}(x+\epsilon+f+a)\bar\psi_{\alpha}(x+\epsilon+f+b)\psi_{\sigma}
(x+g+c)\bar\psi_{\rho}(x+g+d)(\gamma_{\mu})^{\alpha\beta}(\gamma_{\nu})^{\rho\sigma}\nonumber\\&+&
\psi_{\beta}(x+\epsilon+f+a)\bar\psi_{\alpha}(x+\epsilon+f+b)\psi_{\sigma}
(x+g+c)\bar\psi_{\rho}(x+g+d)(\gamma_{\nu})^{\alpha\beta}(\gamma_{\mu})^{\rho\sigma}\nonumber\\
&-&\psi_{\beta}(x+\epsilon+f+a)\bar\psi_{\alpha}(x+\epsilon+f+b)\psi_{\sigma}
(x+g+c)\bar\psi_{\rho}(x+g+d)\delta_{\mu\nu}\nonumber\\&\times&(\gamma_{\lambda})^{\alpha\beta}(\gamma^{\lambda})^{\rho\sigma}\bigg):\Bigg|_{f,g,a,b,c,d=0},
\nonumber\\
{\cal
R}_{\mu\nu}(x,\epsilon)&=&{\cal F}_{fg}{\cal F}_{ab}{\cal F}_{cd}:\psi_{\sigma}(x+g+c)\langle\psi_{\beta}(x+\epsilon+f+a)\bar\psi_{\rho}(x+g+d)
\rangle\bar\psi_{\alpha}(x+\epsilon+f+b):\nonumber\\
&\times&(\gamma_{\mu})^{\alpha\beta}(\gamma_{\nu})^{\rho\sigma}\Bigg|_{f,g,a,b,c,d=0},\nonumber\\
{\cal
S}_{\nu\mu}(x,-\epsilon)&=&{\cal F}_{fg}{\cal F}_{ab}{\cal F}_{cd}:\psi_{\beta}(x+\epsilon+f+a)\langle\psi_{\sigma}(x+g+c)\bar\psi_{\alpha}(x+\epsilon+f+b)
\rangle\bar\psi_{\rho}(x+g+d):\nonumber\\ &\times
&(\gamma_{\mu})^{\alpha\beta}(\gamma_{\nu})^{\rho\sigma}\Bigg|_{f,g,a,b,c,d=0}.
\end{eqnarray}
The field ordering appearing in the vacuum expectation value of the relation (\ref{components-of-T}) is
not time ordering and is defined as\footnote{ We follow the convention used in
\cite{gross}.}
\begin{eqnarray}
S^{(+)}(x-y)=\psi(x)\bar\psi(y)-:\psi(x)\bar\psi(y):=\langle\psi(x)\bar\psi(y)\rangle.
\end{eqnarray}
In
view of the above definition, the quantities ${\cal R}_{\mu\nu}(x,\epsilon)$ and ${\cal
S}_{\nu\mu}(x,-\epsilon)$ can be represented as
\begin{eqnarray}\label{R-S}
{\cal
R}_{\mu\nu}(x,\epsilon)&=&{\cal F}_{fg}{\cal F}_{ab}{\cal F}_{cd}:\psi_{\sigma}(x+g+c)
\bar\psi_{\alpha}(x+\epsilon+f+b):\nonumber\\
&\times&S^{(+)}_{\beta\rho}(\epsilon+f+a-g-d)(\gamma_{\mu})^{\alpha\beta}(\gamma_{\nu})^{\rho\sigma}\Bigg|_{f,g,a,b,c,d=0},\nonumber\\
{\cal
S}_{\nu\mu}(x,-\epsilon)&=&{\cal F}_{fg}{\cal F}_{ab}{\cal F}_{cd}:\psi_{\beta}(x+\epsilon+f+a)\bar\psi_{\rho}(x+g+d):\nonumber\\
&\times
&S^{(+)}_{\sigma\alpha}(g+c-\epsilon-f-b)(\gamma_{\mu})^{\alpha\beta}(\gamma_{\nu})^{\rho\sigma}\Bigg|_{f,g,a,b,c,d=0}.
\end{eqnarray}
Converting equation (\ref{R-S}) to the star product form, we obtain
\begin{eqnarray}
&&{\cal
R}_{\mu\nu}(x,\epsilon)=-\left(\gamma_{\mu}
S^{(+)}(\epsilon)\gamma_{\nu}\right)^{\alpha\beta}:\bar\psi_{\alpha}(x+\epsilon)
\star\psi_{\beta}(x):,\nonumber\\ &&{\cal S}_{\nu\mu}(x,-\epsilon)=\left(\gamma_{\nu}
S^{(+)}(-\epsilon)\gamma_{\mu}\right)^{\alpha\beta}:\psi_{\beta}(x+\epsilon) \star\bar\psi_{\alpha}(x):,
\end{eqnarray}
where
\begin{eqnarray} S^{(+)}(\epsilon)=
-\frac{i}{2\pi}\frac{\epsilon_{\xi}\gamma^{\xi}}{\epsilon^{2}}.
\end{eqnarray}
Note that the minus sign in ${\cal
R}_{\mu\nu}(x,\epsilon)$ comes from the odd permutation of the fermionic fields.\\ Expanding $\psi$ and
$\bar\psi$ up to the first order in $\epsilon$ yields
\begin{eqnarray} &&{\cal
R}_{\mu\nu}(x,\epsilon)=\frac{i\epsilon^{\xi}}{2\pi\epsilon^{2}}\left(\gamma_{\mu}\gamma_{\xi}
\gamma_{\nu}\right)^{\alpha\beta}
:[\bar\psi_{\alpha}(x)+\epsilon^{\eta}\partial_{\eta}\bar\psi_{\alpha}(x)+{\cal O}(\epsilon^{2})]
\star\psi_{\beta}(x):, \nonumber\\ &&{\cal S}_{\nu\mu}(x,-\epsilon)=\frac{i\epsilon^{\xi}}{2\pi\epsilon^{2}}
\left(\gamma_{\nu}\gamma_{\xi}\gamma_{\mu}\right)^{\alpha\beta}
:[\psi_{\beta}(x)+\epsilon^{\eta}\partial_{\eta}\psi_{\beta}(x)+{\cal O}(\epsilon^{2})]
\star\bar\psi_{\alpha}(x):,
\end{eqnarray}
and using the following symmetric limits
\begin{eqnarray}\label{sym-lim}
\lim_{\epsilon\rightarrow 0}~(\frac{\epsilon^{\alpha}}{\epsilon^{2}})=0,~~~~~~~
\lim_{\epsilon\rightarrow 0}~(\frac{\epsilon^{\alpha}\epsilon^{\beta}}{\epsilon^{2}})=\frac{1}{2}\delta_{\alpha\beta},
\end{eqnarray}
we conclude that
\begin{eqnarray}\label{final-value-R-S}
&&\lim_{\epsilon\rightarrow 0}{\cal R}_{\mu\nu}(x,\epsilon)=\frac{i}{4\pi}
\left(\gamma_{\mu}\gamma_{\xi}\gamma_{\nu}\right)^{\alpha\beta} :\partial^{\xi}\bar\psi_{\alpha}(x)
\star\psi_{\beta}(x):, \nonumber\\ &&\lim_{\epsilon\rightarrow 0}{\cal
S}_{\nu\mu}(x,-\epsilon)=\frac{i}{4\pi} \left(\gamma_{\nu}\gamma_{\xi}\gamma_{\mu}\right)^{\alpha\beta}
:\partial^{\xi}\psi_{\beta}(x) \star\bar\psi_{\alpha}(x):.
\end{eqnarray}
Inserting the result
(\ref{final-value-R-S}) in (\ref{final form}), we arrive at
\begin{eqnarray}\label{final-form-1}
T_{\mu\nu}^{s^{\star}}&=&\frac{\pi}{2}{\cal Q}_{\mu\nu}(x)-\frac{i}{8}:\bigg[
\left(\gamma_{\mu}\gamma_{\xi}\gamma_{\nu}+\gamma_{\nu}\gamma_{\xi}\gamma_{\mu}\right)^{\alpha\beta}
\left(\partial^{\xi}\bar\psi_{\alpha}(x) \star\psi_{\beta}(x)+ \partial^{\xi}\psi_{\beta}(x)
\star\bar\psi_{\alpha}(x)\right)\nonumber\\&+&\delta_{\mu\nu}(\gamma^{\lambda}
\gamma_{\xi}\gamma_{\lambda})^{\alpha\beta}\left(\partial^{\xi}\bar\psi_{\alpha}(x)
\star\psi_{\beta}(x)+\partial^{\xi}\psi_{\beta}\star\bar\psi_{\alpha}\right)\bigg]:.
\end{eqnarray}
The product
of gamma matrices in (\ref{final-form-1}) can be simplified using the Clifford algebra
\begin{eqnarray}\label{identity}
\gamma_{\mu}\gamma_{\xi}\gamma_{\nu}+\gamma_{\nu}\gamma_{\xi}\gamma_{\mu}=
2\left(\delta_{\mu\xi}\gamma_{\nu}+\delta_{\nu\xi}\gamma_{\mu}-\delta_{\mu\nu} \gamma_{\xi}\right)
,~~~~~~~~~ \gamma_{\lambda}\gamma_{\xi}\gamma^{\lambda}=(2-d)\gamma_{\xi}.
\end{eqnarray}
Substituting
(\ref{identity}) into (\ref{final-form-1}) and applying the equation of motion
$\gamma^{\xi}\partial_{\xi}\psi=0$ and $\partial_{\xi}\bar\psi\gamma^{\xi}=0$, we obtain the Sugawara energy-momentum tensor
\begin{eqnarray}\label{final-form-2}
T_{\mu\nu}^{s^{\star}}&=&\frac{\pi}{2}{\cal
Q}_{\mu\nu}(x)-\frac{i}{4}:\bigg[ \bigg(\partial_{\nu}\bar\psi_{\alpha}(x)
\star\psi_{\beta}(x)+\partial_{\nu}\psi_{\beta}(x)\star\bar\psi_{\alpha}(x)\bigg)(\gamma_{\mu})^{\alpha\beta}
\nonumber\\&+& \bigg(\partial_{\mu}\bar\psi_{\alpha}(x)\star\psi_{\beta}(x)+
\partial_{\mu}\psi_{\beta}(x)\star\bar\psi_{\alpha}(x)\bigg)(\gamma_{\nu})^{\alpha\beta}\bigg]:.
\end{eqnarray}
We notice that the last term
of (\ref{final-form-1}) vanishes in two dimensions as a result of the identity
$\gamma_{\lambda}\gamma_{\xi}\gamma^{\lambda}=(2-d)\gamma_{\xi}$.
In order to show that $T_{\mu\nu}^{^{\star}}=T_{\mu\nu}^{s^{\star}}$, it is enough to
demonstrate ${\cal{Q}}_{\mu\nu}=0$. For simplicity, we carry out computations
in
the light-cone coordinate system, $x_{\pm} = x_{1} \pm ix_{2}$.
The representation of the Euclidean gamma matrices
\begin{eqnarray}\label{Euclidean-gamma}
\gamma_{1}=\left(
\begin{array}{cc}
0 & -i \\
i & 0 \\
\end{array}
\right),~~~~\gamma_{2}=\left(
\begin{array}{cc}
0 & -1 \\
-1 & 0 \\
\end{array}\right),
\end{eqnarray}
in the light-cone coordinates is given by
\begin{eqnarray}
\gamma_{+}=\gamma_{1}+i\gamma_{2}= \left(
\begin{array}{cc}
0 & -2i\\
0 & 0 \\
\end{array}\right)
,~~~~ \gamma_{-}=\gamma_{1}-i\gamma_{2}= \left(
\begin{array}{cc}
0 & 0\\
2i & 0 \\
\end{array}\right),
\end{eqnarray}
with \begin{eqnarray} g^{\mu\nu}=\left(
\begin{array}{cc}
g^{++} & g^{+-}\\
g^{-+} & g^{--} \\
\end{array}\right)=
\left(
\begin{array}{cc}
0 & \frac{1}{2}\\
\frac{1}{2} & 0 \\
\end{array}\right).
\end{eqnarray}
The equation of motion for two-dimensional
massless fermions, which is described
by $i\gamma^{\mu}\partial_{\mu}\psi=0$ with
$ \psi=\left(
\begin{array}{c}
\psi_{1}\\
\psi_{2}\\
\end{array}
\right)$, in the
light-cone coordinate system reduces to
$\partial_{+}\psi_{1}=\partial_{-}\psi_{2}=0$. Thus
\begin{eqnarray}
&&\psi_{1}=\psi_{1}(x_{-}),~~~~~\psi_{2}=\psi_{2}(x_{+}),\nonumber\\&&
\bar\psi_{1}=\bar\psi_{1}(x_{+}),~~~~~\bar\psi_{2}=\bar\psi_{2}(x_{-}).
\end{eqnarray}
Using $[x_{+},x_{-}]=2\theta$, the
expression $\psi_{\beta}(x)\star\bar\psi_{\alpha}(x)(\gamma_{\mu})^{\alpha\beta}$ with on-shell Dirac fermions is rewritten as
\begin{eqnarray}\label{simp-lc}
&&\psi_{\beta}\star\bar\psi_{\alpha}(\gamma_{+})_{\alpha\beta}=
-2i~\psi_{2}(x_{+})\star\bar\psi_{1}(x_{+})=-2i~\psi_{2}(x_{+})\bar\psi_{1}(x_{+}),\nonumber\\
&&\psi_{\beta}\star\bar\psi_{\alpha}(\gamma_{-})_{\alpha\beta}=
+2i~\psi_{1}(x_{-})\star\bar\psi_{2}(x_{-})=+2i~\psi_{1}(x_{-})\bar\psi_{2}(x_{-}).
\end{eqnarray}
With equation (\ref{simp-lc}), it would be possible to find all the components of ${\cal Q}_{\mu\nu}$. Since
there is no singularity in ${\cal Q}_{\mu\nu}(x,\epsilon)$, we have
\begin{eqnarray}
\lim_{\epsilon\rightarrow 0}{\cal Q}_{\mu\nu}(x,\epsilon)={\cal Q}_{\mu\nu}(x),
\end{eqnarray}
and hence
\begin{eqnarray} {\cal
Q}_{\mu\nu}(x)&=&:\bigg(\psi_{\beta}(x)\star\bar\psi_{\alpha}(x)\bigg)\star\bigg(\psi_{\sigma}
(x)\star\bar\psi_{\rho}(x)\bigg):(\gamma_{\mu})^{\alpha\beta}(\gamma_{\nu})^{\rho\sigma}\nonumber\\&+&:
\bigg(\psi_{\beta}(x)\star\bar\psi_{\alpha}(x)\bigg)\star\bigg(\psi_{\sigma}
(x)\star\bar\psi_{\rho}(x)\bigg):(\gamma_{\nu})^{\alpha\beta}(\gamma_{\mu})^{\rho\sigma}\nonumber\\
&-&:\bigg(\psi_{\beta}(x)\star\bar\psi_{\alpha}(x)\bigg)\star\bigg(\psi_{\sigma}
(x)\star\bar\psi_{\rho}(x)\bigg):\delta_{\mu\nu}(\gamma_{\lambda})^{\alpha\beta}(\gamma^{\lambda})^{\rho\sigma}.
\end{eqnarray}
One may then readily show that
\begin{eqnarray}
{\cal
Q}_{++}&=&-8:\bigg(\psi_{2}(x_{+})\bar\psi_{1}(x_{+})\bigg)\star\bigg(\psi_{2}(x_{+})\bar\psi_{1}(x_{+})\bigg):\nonumber\\
&=&-8:\psi_{2}(x_{+})\bar\psi_{1}(x_{+})\psi_{2}(x_{+})\bar\psi_{1}(x_{+}):.
\end{eqnarray}
Performing some straightforward permutations, we get ${\cal Q}_{++}=0$. Similarly
\begin{eqnarray} {\cal
Q}_{--}&=&-8:\bigg(\psi_{1}(x_{-})\bar\psi_{2}(x_{-})\bigg)\star\bigg(\psi_{1}(x_{-})\bar\psi_{2}(x_{-})\bigg):\nonumber\\
&=&-8:\psi_{1}(x_{-})\bar\psi_{2}(x_{-})\psi_{1}(x_{-})\bar\psi_{2}(x_{-}):\nonumber\\
&=&0.
\end{eqnarray}
Also for off-diagonal components ${\cal Q}_{\pm\mp}$, we have
\begin{eqnarray}
{\cal Q}_{\pm\mp}(x)&=&:\bigg(\psi_{\beta}(x)\star\bar\psi_{\alpha}(x)\bigg)\star\bigg(\psi_{\sigma}
(x)\star\bar\psi_{\rho}(x)\bigg):(\gamma_{\pm})^{\alpha\beta}(\gamma_{\mp})^{\rho\sigma}\nonumber\\&+&:
\bigg(\psi_{\beta}(x)\star\bar\psi_{\alpha}(x)\bigg)\star\bigg(\psi_{\sigma}
(x)\star\bar\psi_{\rho}(x)\bigg):(\gamma_{\mp})^{\alpha\beta}(\gamma_{\pm})^{\rho\sigma}\nonumber\\
&-&:2\bigg(\psi_{\beta}(x)\star\bar\psi_{\alpha}(x)\bigg)\star\bigg(\psi_{\sigma}
(x)\star\bar\psi_{\rho}(x)\bigg):(\gamma_{\lambda})^{\alpha\beta}(\gamma^{\lambda})^{\rho\sigma}.
\end{eqnarray}
Inserting $\gamma^{\pm}=\frac{1}{2}\gamma_{\mp}$ results in
\begin{eqnarray}
{\cal Q}_{\pm\mp}(x)&=& :\bigg(\psi_{\beta}(x)\star\bar\psi_{\alpha}(x)\bigg)\star\bigg(\psi_{\sigma}
(x)\star\bar\psi_{\rho}(x)\bigg):(\gamma_{\pm})^{\alpha\beta}(\gamma_{\mp})^{\rho\sigma}\nonumber\\&+&:
\bigg(\psi_{\beta}(x)\star\bar\psi_{\alpha}(x)\bigg)\star\bigg(\psi_{\sigma}
(x)\star\bar\psi_{\rho}(x)\bigg):(\gamma_{\mp})^{\alpha\beta}(\gamma_{\pm})^{\rho\sigma}\nonumber\\
&-&:\bigg(\psi_{\beta}(x)\star\bar\psi_{\alpha}(x)\bigg)\star\bigg(\psi_{\sigma}
(x)\star\bar\psi_{\rho}(x)\bigg):(\gamma_{\pm})^{\alpha\beta}(\gamma_{\mp})^{\rho\sigma}\nonumber\\
&-&:\bigg(\psi_{\beta}(x)\star\bar\psi_{\alpha}(x)\bigg)\star\bigg(\psi_{\sigma}
(x)\star\bar\psi_{\rho}(x)\bigg):(\gamma_{\mp})^{\alpha\beta}(\gamma_{\pm})^{\rho\sigma}\nonumber\\&=&0.
\end{eqnarray}
Consequently $Q_{\mu\nu}=0$. This means that the equivalence of the Sugawara energy-momentum tensor and
energy-momentum tensor $T_{\mu\nu}^{s^{\star}}=T_{\mu\nu}^{^{\star}}$ in two-dimensional
noncommutative space for free massless fermions is still satisfied. Also this equivalence occurs for the Sugawara form in terms of the current
${\cal J}_{\mu}(x)$, as defined in (\ref{currents}). We have
\begin{eqnarray}\label{sugawara-with-second-current}
\widehat{T}_{\mu\nu}^{s^{\star}}=\frac{\pi}{2}\bigg(~{\cal J}_{\mu}(x)\star {\cal J}_{\nu}(x)+
{\cal J}_{\nu}(x)\star {\cal J}_{\mu}(x)-\delta_{\mu\nu} {\cal J}^{\lambda}(x)\star {\cal J}_{\lambda}(x)\bigg).
\end{eqnarray}
To show this, let us write first all the components of the currents $J_{\mu}(x)$ and ${\cal J}_{\mu}(x)$ using the representation of the gamma matrices (\ref{Euclidean-gamma}) as follows
\begin{eqnarray}\label{current-1}
&&J_{1}(x)=i:\bigg(
\psi_{1}(x_{-})\star\bar\psi_{2}(x_{-})-\psi_{2}(x_{+})\star\bar\psi_{1}(x_{+})\bigg):=i:\bigg(
\psi_{1}(x_{-})\bar\psi_{2}(x_{-})-\psi_{2}(x_{+})\bar\psi_{1}(x_{+})\bigg):,\nonumber\\
&&J_{2}(x)=-:\bigg(
\psi_{2}(x_{+})\star\bar\psi_{1}(x_{+})+\psi_{1}(x_{-})\star\bar\psi_{2}(x_{-})\bigg):=-:\bigg(
\psi_{2}(x_{+})\bar\psi_{1}(x_{+})+\psi_{1}(x_{-})\bar\psi_{2}(x_{-})\bigg):,\nonumber\\
\end{eqnarray}
as well as
\begin{eqnarray}\label{current-2}
&&{\cal
J}_{1}(x)=i:\bigg(
\bar\psi_{2}(x_{-})\star\psi_{1}(x_{-})-\bar\psi_{1}(x_{+})\star\psi_{2}(x_{+})\bigg):=i:\bigg(
\bar\psi_{2}(x_{-})\psi_{1}(x_{-})-\bar\psi_{1}(x_{+})\psi_{2}(x_{+})\bigg):,\nonumber\\
&&{\cal
J}_{2}(x)=-:\bigg(
\bar\psi_{1}(x_{+})\star\psi_{2}(x_{+})+\bar\psi_{2}(x_{-})\star\psi_{1}(x_{-})\bigg):=-:\bigg(
\bar\psi_{1}(x_{+})\psi_{2}(x_{+})+\bar\psi_{2}(x_{-})\psi_{1}(x_{-})\bigg):.\nonumber\\
\end{eqnarray}
We notice that the star product appearing in the noncommutative currents is removed.
Applying the permutation on the fermionic fields of the relation (\ref{current-1}), we obtain
\begin{eqnarray}\label{equality-of-two-currents}
{\cal J}_{\mu}(x)=-J_{\mu}(x).
\end{eqnarray}
This is an interesting result in two dimensions. Unlike the four-dimensional case, where $J_{\mu}(x)$ and ${\cal J}_{\mu}(x)$ correspond to each other by the charge conjugation transformation \cite{Jabbari}, which is not conserved, in two dimensions the charge conjugation, as well as the Lorentz invariance, remain the symmetry of the theory as in their commutative case.
Inserting (\ref{equality-of-two-currents}) in (\ref{sugawara-with-second-current}) then leads to
\begin{eqnarray}
\widehat{T}_{\mu\nu}^{s^{\star}}=T_{\mu\nu}^{s^{\star}}=T_{\mu\nu}^{^{\star}}~.
\end{eqnarray}
One of the physical consequences of this equivalence is noncommutative bosonization, which is obtained by writing the transformation of the field $\psi$ under the spatial translation
\begin{eqnarray}
\partial_{x_{1}}\psi(x)=i\bigg[P_{1},\psi(x)\bigg],~~~~~~~P_{1}=\int dx'_{1}T_{21}^{s^{\star}},
\end{eqnarray}
where $T_{21}^{s^{\star}}$ is the conserved current arising from translational invariance\footnote{In two-dimensional Euclidean space $x_{2}=ix_{0}$.}. We have
\begin{eqnarray}\label{nch}
\partial_{x_{1}}\psi(x)=i\bigg[\int dx_{1}'T_{21}^{s^{\star}},\psi(x)\bigg]_{x_{2}=x'_{2}},
\end{eqnarray}
and substituting the value of $T_{21}^{s^{\star}}$ from (\ref{nc-sugawara}) in (\ref{nch}) yields
\begin{eqnarray}\label{nc-Heisenberg}
\partial_{x_{1}}\psi(x)=\frac{i\pi}{2}\bigg[\int dx_{1}'\bigg(J_{1}(x')\star J_{2}(x')+J_{2}(x')\star J_{1}(x')\bigg),\psi(x)\bigg]_{x_{2}=x'_{2}}.
\end{eqnarray}
To simplify (\ref{nc-Heisenberg}),
we insert $\int dx'_{2}\delta(x_{2}-x'_{2})=1$ to use the trace property of the star product, which is given by
\begin{eqnarray}\label{trace-property}
\int dx'_{1}dx'_{2}~J_{2}(x')\star J_{1}(x')=\int dx'_{1}dx'_{2}~J_{1}(x')\star J_{2}(x').
\end{eqnarray}
Inserting (\ref{trace-property}) in (\ref{nc-Heisenberg}) and applying
the operator definition of the star product (\ref{operator-definition}), we have
\begin{eqnarray}\label{nc-Heisenberg-2}
\partial_{x_{1}}\psi(x)&=&i\pi\int dx'_{1}dx'_{2}~\delta(x_{2}-x'_{2}){\cal F}_{ab}\bigg[J_{1}(x'+a)J_{2}(x'+b),\psi(x)\bigg]\Bigg|_{a,b=0}\nonumber\\&=&
i\pi\int dx'_{1}dx'_{2}~\delta(x_{2}-x'_{2}){\cal F}_{ab}\bigg(J_{1}(x'+a)\bigg[J_{2}(x'+b),\psi(x)\bigg]+\bigg[J_{1}(x'+a),\psi(x)\bigg]J_{2}(x'+b)
\bigg)\Bigg|_{a,b=0}.\nonumber\\
\end{eqnarray}
The bracket terms appearing in the right-hand side of (\ref{nc-Heisenberg-2}) are derived by considering the quantization condition $\{\psi_{\alpha}(x),\psi^{\dagger}_{\beta}(x')\}=\delta_{\alpha\beta}\delta
(x_{1}-x'_{1})$ as follows
\begin{eqnarray}\label{commutator-1}
&&\bigg[J_{2}(x'+a),\psi(x)\bigg]_{x_{2}=x'_{2}}=\psi(x'+a)\delta(x_{1}-x'_{1}+a),\nonumber\\
&&\bigg[J_{1}(x'+a),\psi(x)\bigg]_{x_{2}=x'_{2}}=\gamma_{5}\psi(x'+a)\delta(x_{1}-x'_{1}+a),
\end{eqnarray}
with $\gamma_{5}=i\gamma_{1}\gamma_{2}$. Substituting (\ref{commutator-1}) into (\ref{nc-Heisenberg-2}) and then converting the result into the star product form, we have
\begin{eqnarray}
\partial_{x_{1}}\psi(x)=
i\pi\bigg(J_{1}(x)+J_{2}(x)\gamma_{5}\bigg)\star\psi(x).
\end{eqnarray}
The solution of this equation is represented by
\begin{eqnarray}
\psi(x)={\cal P}\bigg(e^{i\pi\int_{-\infty}^{x_{1}} dx'_{1}[J_{1}(x')+J_{2}(x')\gamma_{5}]}_{\star}\bigg)\psi_{0},
\end{eqnarray}
where ${\cal P}$ denotes the path-ordering operator and $\psi_{0}$ is a constant spinor in two dimensions.
\\Now, as a result of (\ref{current-1}), we can use the bosonized form of the commutative current, which is introduced in appendix A. Hence, we conclude
\begin{eqnarray}
\psi(x)={\cal P}\bigg(e^{-i\sqrt{\pi}~[\gamma_{5}\phi(x)-\int_{-\infty}^{x_{1}} dx'_{1}\dot{\phi}(x')]}_{\star}\bigg)\psi_{0},
\end{eqnarray}
here $\dot{\phi}=\partial_{x'_{2}}\phi$.
\section{Discussion}
\noindent
In this paper, we established the noncommutative extension of the Sugawara construction in bilinear form of the
currents for free massless fermions in two dimensions. It was shown that this construction
is precisely equivalent to the symmetric energy-momentum tensor.\\
\noindent
To prove the correctness of this equivalence, we determined the energy-momentum tensor in two separate methods. The first was the direct calculation using the symmetric definition of the energy-momentum tensor for on-shell Dirac fermions and
the second contained a detailed analysis of noncommutative Sugawara construction by applying the point-splitting regularization. Furthermore, for simplification in our calculation, we considered the light-cone system. In this coordinate, we realized that the currents $J_{\mu}$ and ${\cal J}_{\mu}$, apart from a minus sign, are actually the same in two dimensions which leads to the charge conjugation symmetry restoration.
\par
Eventually, we presented a physical consequence of this equivalence, named as noncommutative bosonization (e.g. see \cite{nunez,grisaru,blas}), that relates a fermionic field to a bosonic field through an exponential function and demonstrated that a free massless fermion theory with a global $U(1)$ symmetry in noncommutative space corresponds to a free massless boson theory. Also, the bosonized version of a theory with local $U(1)$ symmetry such as two-dimensional noncommutative QED (NC-QED$_{2}$) was addressed in \cite{masoumeh}, where it was proven that
the bosonized action contains a noncommutative Wess-Zumino-Witten (WZW) part, a gauge kinetic part and an interaction part between the WZW and gauge field. \newpage
\noindent
The physical significance of the bosonization procedure is that it specifies
a duality between the strong and weak couplings for particular interacting quantum field theories.
The most famous example of this duality is the equivalence of the massive Thirring model and sine-Gordon model \cite{Coleman,Mandelstam},
where the weak coupling $\beta$ of the bosonic theory, that is the sine-Gordon model, is related to the strong coupling $g$ of the fermionic theory, the massive Thirring model, through the bosonization rule described by $\frac{4\pi}{\beta^{2}}=1+\frac{g}{\pi}$. Moreover, the duality between the noncommutative version of these models was studied in \cite{nunez,grisaru,blas} where it was shown that the strong-weak duality is also preserved.
However, it is notable that the strong-weak duality does not appear in the case of NC-QED$_{2}$ and its bosonized version,
because of the appearance of the same couplings in two theories.
\section{Acknowledgments}
\noindent
I am grateful to M.M. Sheikh-Jabbari for numerous fruitful discussions and careful reading of the manuscript.
I would also like to thank M. Khorrami for his valuable comments and I appreciate the insightful and constructive remarks of M. Chaichian and the referee, which led to improvement of the manuscript.
Moreover, I acknowledge the School of Physics of Institute for research in fundamental sciences (IPM) for the research facilities and environment.
\par\noindent \begin{appendix}
\section{Commutative Sugawara Construction} \setcounter{equation}{0}\noindent
In this appendix, we present a detailed analysis on the proof of the relation $T_{\mu\nu}^{s}=T_{\mu\nu}$
in two dimensional commutative space for free massless fermions, as argued in \cite{gross}, and describe some
interesting consequences of this equivalence \cite{Freundlich-1,Freundlich-2,sommerfield}. The Lagrangian for
the massless fermions is given by
\begin{eqnarray}
{\cal
L}=\frac{i}{2}\left(\bar\psi\gamma^{\mu}\partial_{\mu}\psi-\partial_{\mu}\bar\psi\gamma^{\mu}\psi\right),
\end{eqnarray}
which is invariant under the global phase transformation $\psi\rightarrow e^{i\alpha}\psi$ and
$\bar\psi\rightarrow e^{-i\alpha}\bar\psi$ that gives the conserved current
\begin{eqnarray}
j_{\mu}(x)=:\bar\psi(x)\gamma_{\mu}\psi(x):.
\end{eqnarray}
For this theory, the symmetric
energy-momentum tensor is written as follows
\begin{eqnarray}\label{com-canonic}
T_{\mu\nu}=\frac{i}{4}:\left(\bar\psi\gamma_{\mu}\partial_{\nu}\psi+\bar\psi\gamma_{\nu}\partial_{\mu}\psi-\partial_{\mu}
\bar\psi\gamma_{\nu}\psi-\partial_{\nu}\bar\psi\gamma_{\mu}\psi\right):.
\end{eqnarray}
The energy-momentum tensor in Sugawara form is described by a bilinear function of the currents as
\begin{eqnarray}\label{commutative-sugawara}
T_{\mu\nu}^{s}=\frac{\pi}{2}\left(~j_{\mu}(x)j_{\nu}(x)+j_{\nu}(x)j_{\mu}(x)-g_{\mu\nu}
j^{\lambda}(x)j_{\lambda}(x)\right).
\end{eqnarray}
To show $T_{\mu\nu}^{s}=T_{\mu\nu}$, we start with
(\ref{commutative-sugawara}) and replace $j_{\mu}(x)j_{\nu}(x)$ with
\begin{eqnarray}\label{remove-divergency}
&&\lim_{\epsilon\rightarrow 0}~\bigg(j_{\mu}(x+\epsilon)j_{\nu}(x)-\langle j_{\mu}(x+\epsilon)j_{\nu}(x)\rangle\bigg).
\end{eqnarray}
Applying Wick's theorem on (\ref{remove-divergency})
\begin{eqnarray}
j_{\mu}(x+\epsilon)j_{\nu}(x)&=&:\bar\psi(x+\epsilon)\gamma_{\mu}\psi(x+\epsilon)::
\bar\psi(x)\gamma_{\nu}\psi(x):\nonumber\\&
=&:\bar\psi(x+\epsilon)\gamma_{\mu}\psi(x+\epsilon)\bar\psi(x)\gamma_{\nu}\psi(x):\nonumber\\&
+&:\bar\psi(x+\epsilon)\gamma_{\mu}\langle\psi(x+\epsilon)\bar\psi(x)\rangle\gamma_{\nu}\psi(x):\nonumber\\&
+&:\bar\psi(x)\gamma_{\nu}\langle\psi(x)\bar\psi(x+\epsilon)\rangle\gamma_{\mu}\psi(x+\epsilon):\nonumber\\&
-&tr\left(\gamma_{\mu}\langle\psi(x)\bar\psi(x+\epsilon)\rangle\gamma_{\nu}\langle\psi(x+\epsilon) \bar\psi(x)\rangle\right),
\end{eqnarray}
and implementing a similar analysis for the other terms of (\ref{commutative-sugawara}), we
arrive at
\begin{eqnarray}\label{sugawara in com} T_{\mu\nu}^{s}&=&\frac{\pi}{2}\lim_{\epsilon\rightarrow
0}\bigg[{\cal M}_{\mu\nu}(x,\epsilon)+{\cal{N}}_{\mu\nu}(x,\epsilon)+
{\cal{N}}_{\nu\mu}(x,\epsilon)+{\cal{N}}_{\mu\nu}(x,-\epsilon)+{\cal{N}}_{\nu\mu}(x,-\epsilon)\nonumber\\&-&g_{\mu\nu}[
{\cal{N}}_{\lambda}^{\lambda}(x,\epsilon)+{\cal{N}}_{\lambda}^{\lambda}(x,-\epsilon)]\bigg],\nonumber\\
\end{eqnarray}
where ${\cal M}_{\mu\nu}$ and ${\cal N}_{\mu\nu}$ are defined as
\begin{eqnarray} {\cal
M}_{\mu\nu}(x,\epsilon)&=&:\bar\psi(x+\epsilon)\gamma_{\mu}\psi(x+\epsilon)\bar\psi(x)\gamma_{\nu}\psi(x):\nonumber\\
&+& :\bar\psi(x+\epsilon)\gamma_{\nu}\psi(x+\epsilon)\bar\psi(x)\gamma_{\mu}\psi(x):\nonumber\\
&-&:\bar\psi(x+\epsilon)\gamma_{\lambda}\psi(x+\epsilon)\bar\psi(x)\gamma^{\lambda}\psi(x):g_{\mu\nu},
\nonumber\\ {\cal{N}}_{\mu\nu}(x,\epsilon)&=&:\bar\psi(x+\epsilon)\gamma_{\mu}S^{(+)}(\epsilon)
\gamma_{\nu}\psi(x):,
\end{eqnarray}
and we have
\begin{eqnarray}
S^{(+)}(\epsilon)=\langle\psi(x+\epsilon)\bar\psi(x)\rangle=
-(\frac{i}{2\pi})\frac{\epsilon_{\alpha}\gamma^{\alpha}}{\epsilon^{2}}.
\end{eqnarray}
First, we concentrate on determining the value of ${\cal M}_{\mu\nu}(x)$.
\begin{eqnarray}
{\cal M}_{00}={\cal M}_{11}=:(j_{0})^{2}+(j_{1})^{2}:,~~~~~{\cal
M}_{01}={\cal M}_{10}=:j_{0}j_{1}+j_{1}j_{0}:.
\end{eqnarray}
Choosing
$\gamma_{0}=\sigma_{z}$ and $\gamma_{1}=i\sigma_{y}$, we find
\begin{eqnarray}
j_{0}=\bar\psi_{1}\psi_{1}-\bar\psi_{2}\psi_{2},~~~~~j_{1}=\bar\psi_{1}\psi_{2}-\bar\psi_{2}\psi_{1}.
\end{eqnarray}
Therefore, all the components of ${\cal M}_{\mu\nu}$ in terms of the fermionic fields are given by
\begin{eqnarray} &&{\cal M}_{00}={\cal
M}_{11}=:(\bar\psi_{1}\psi_{1}-\bar\psi_{2}\psi_{2})^{2}+(\bar\psi_{1}
\psi_{2}-\bar\psi_{2}\psi_{1})^{2}:,\nonumber\\ &&{\cal M}_{01}={\cal
M}_{10}=:(\bar\psi_{1}\psi_{1}-\bar\psi_{2}\psi_{2})(\bar\psi_{1}\psi_{2}-\bar\psi_{2}\psi_{1})+(\bar\psi_{1}\psi_{2}-\bar\psi_{2}\psi_{1})
(\bar\psi_{1}\psi_{1}-\bar\psi_{2}\psi_{2}):.
\end{eqnarray}
Performing some permutations on fermionic fields yields ${\cal M}_{\mu\nu}=0$.
In the next step, our purpose is to obtain the value of
${\cal N}_{\mu\nu}$. To this end, let us start
with expansion of the fermionic fields up to the first order in $\epsilon$
\begin{eqnarray}
&&{\cal
N}_{\mu\nu}(x,\epsilon)=-\frac{i\epsilon^{\xi}}{2\pi\epsilon^{2}}
:[\bar\psi(x)+\epsilon^{\alpha}\partial_{\alpha}\bar\psi(x)+{\cal O}(\epsilon^{2})]\gamma_{\mu}\gamma_{\xi}
\gamma_{\nu} \psi(x):
\end{eqnarray}
Taking the symmetric limits (\ref{sym-lim}),
\begin{eqnarray}\label{final-J}
&&\lim_{\epsilon\rightarrow 0}{\cal N}_{\mu\nu}(x,\epsilon)=\frac{i}{4\pi}
:\partial^{\xi}\bar\psi(x)\gamma_{\mu}\gamma_{\xi} \gamma_{\nu}\psi(x):,
\end{eqnarray}
putting (\ref{final-J})
in (\ref{sugawara in com}) and using the identity (\ref{identity}) for on-shell fermions
in two dimensions, we find
\begin{eqnarray}
T_{\mu\nu}^{s}=\frac{i}{4}:(\bar\psi\gamma_{\mu}\partial_{\nu}\psi+\bar\psi\gamma_{\nu}\partial_{\mu}\psi
-\partial_{\mu}\bar\psi\gamma_{\nu}\psi-\partial_{\nu}\bar\psi\gamma_{\mu}\psi):,
\end{eqnarray}
which is
exactly equal to $T_{\mu\nu}$ mentioned in (\ref{com-canonic}). This equivalence suggests the existence
of a canonical massless pseudo scalar field, satisfying $[\phi(x,t),\dot{\phi}(y,t)]=i\delta(x-y)$, which is
related to the conserved current $j_{\mu}(x)$ through the following equation \cite{Freundlich-2}
\begin{eqnarray}\label{current-boson}
j_{\mu}(x)=\frac{1}{\sqrt{\pi}}~\epsilon_{\mu\nu}\partial^{\nu}\phi(x).
\end{eqnarray}
If we substitute (\ref{current-boson}) into the Sugawara form (\ref{commutative-sugawara}) and use the
identity $\epsilon_{\mu\alpha}\epsilon_{\nu\beta}=g_{\mu\beta}g_{\alpha\nu}-g_{\mu\nu}g_{\alpha\beta}$, it is
found that
\begin{eqnarray}
T_{\mu\nu}^{s}=\frac{1}{2}\left(\partial_{\mu}\phi\partial_{\nu}\phi+\partial_{\nu}\phi\partial_{\mu}\phi
-g_{\mu\nu}\partial_{\lambda}\phi\partial^{\lambda}\phi \right),
\end{eqnarray}
which describes the energy-momentum tensor for a free massless boson.
Another interesting result of
the Sugawara construction is that it is possible to find explicitly the fermionic filed in terms of the
bosonic field, as mentioned in the introduction part. To show this, we consider the equation describing the transformation of the field $\psi$ under the spatial translation \cite{sommerfield}
\begin{eqnarray}\label{noether}
\partial_{x_{1}}\psi(x)=i\bigg[\int
dx'~T_{01}^{s},\psi(x)\bigg],~~~~~P_{0}=\int dx'_{1}~T_{01}^{s},
\end{eqnarray}
where $T_{01}^{s}$ is the Noether current of translational symmetry.
Inserting $T_{01}^{s}$ from (\ref{commutative-sugawara}) into (\ref{noether}), we have
\begin{eqnarray}
\partial_{x_{1}}\psi(x)=\frac{i\pi}{2}\bigg[\int dx'_{1}~\bigg(j_{0}(x')j_{1}(x')+j_{1}(x')j_{0}(x')\bigg),\psi(x)\bigg]_{x_{0}=x'_{0}}.
\end{eqnarray}
Applying the equal-time commutation relations
\begin{eqnarray}
\bigg[j_{0}(x'),\psi(x)\bigg]_{x_{0}=x'_{0}}=-\psi(x)\delta(x_{1}-x'_{1}),~~~~~~~
\bigg[j_{1}(x'),\psi(x)\bigg]_{x_{0}=x'_{0}}=-\gamma_{5}\psi(x)\delta(x_{1}-x'_{1}),
\end{eqnarray}
with $\gamma_{5}=\gamma_{0}\gamma_{1}$, we arrive at
\begin{eqnarray}
\partial_{x_{1}}\psi(x)=-i\pi[j_{1}(x)+j_{0}\gamma_{5}(x)]\psi(x).
\end{eqnarray}
Solving this equation yields
\begin{eqnarray}\label{solution}
\psi(x)=e^{-i\pi\int_{-\infty}^{x_{1}} dx'(j_{1}(x')+j_{0}(x')\gamma_{5})}\psi_{0},
\end{eqnarray}
where $\psi_{0}$ is a constant spinor in space-time. In the final step, we put the
bosonized form of the currents from (\ref{current-boson}) in (\ref{solution})
\begin{eqnarray}
\psi(x)=e^{i\sqrt{\pi}~[\gamma_{5}\phi(x)+\int_{-\infty}^{x_{1}} dx'_{1} \dot{\phi}(x')]}\psi_{0}.
\end{eqnarray}
As we see, the spinor field $\psi$ is mapped to the bosonic field $\phi$.
\end{appendix}
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train/arxiv
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BkiUbWU25YjgKLaE8Iga
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\section{Introduction}
Many soft quantities in QCD can be successfully described using the ideas of Regge theory, in which scattering is
dominated not by the exchange of single particles but rather by infinite towers of resonances whose mass squared and spin are
linearly related as $J= \alpha(0) + \alpha' M^2$. This
is closely connected to the idea of a string dual of QCD at large $N_c$. At very large energies total cross sections are
dominated by the exchange of the trajectory with the largest intercept $\alpha(0)$, a.k.a. the Pomeron. From a modern point of view the Pomeron is the leading Regge trajectory containing the lightest spin two glueball. For an overview of this perspective, our conventions, and a more detailed list of references see \cite{pom}. The connection between the Pomeron, Regge theory, and AdS/QCD has also been explored in \cite{bpst,bdst}.
The spin two glueball field can be treated as a second-rank symmetric traceless tensor $q_{\mu \nu}$. General arguments as well as specific calculations indicate that this field should couple predominantly to the QCD stress tensor $T^{\mu \nu}$:
\beq
S_{\rm int}= \lambda_{\cal P} \int d^4x \ q_{\mu \nu} T^{\mu \nu}~.
\eeq
The coupling $\lambda_{\cal P}$ sets the scale of the total cross section for $p~p$ and $p ~ \bar p$ scattering. For example,
in the model of \cite{pom} the total cross section is given by
\beq
\sigma_{tot}= c \pi \lambda_{\cal P}^2 \left(\frac{\alpha'_c s}{2} \right)^{\alpha_c(0)-1}
\eeq
where $c$ is a constant of order one and the Pomeron Regge trajectory is $J= \alpha_c(0) + \alpha'_c M^2$ \footnote{This formula differs by a factor of $4$ from the original formula in \cite{pom} due to an algebraic error in \cite{pom}.}. Fits to data
give a value $\lambda_{{\cal P},{\rm fit}} \simeq 8.28 ~ {\rm GeV}^{-1}$.
In \cite{pom} we assumed single Pomeron exchange and fit to data to obtain $\alpha_c(0) \simeq 1.09$. This behavior
eventually violates the Froissart bound, although this does not happen until values of $s$ much above those that
are currently accessible. Other authors have argued that total cross sections are better fit by a $\log^2(s)$ behavior
\cite{Block:1998hu,Cudellpdg,Igigpdg,Blockpdg} at current values of $s$. If this is the case, multiple Pomeron exchange must already be important (corresponding to multiloop diagrams in the dual string description) and explicit calculations will be much more difficult. However,
it still seems likely that the overall scale of the cross section will be set by the coupling $\lambda_{\cal P}$.
In \cite{pom} $\lambda_{\cal P}$ was calculated in a very simple approximation in which the proton was treated as a Skyrmion
constructed out of the pion field with the result that $\lambda_{{\cal P},{\rm Skyrme}} \simeq 3.9~{\rm GeV}^{-1}$ which is significantly
smaller than the experimental value. However, the description of the proton in the dual model of \cite{Sakai:2004cn}
is known to be more complicated than this. In particular, the towers of vector and axial-vector mesons contribute significantly to the
solution, a fact that has played an important role in the determination of nucleon-meson coupling constants
\cite{Hata:2007mb,Hong:2007kx, Hong:2007ay,Hong:2007dq,Hashimoto:2008zw,Park:2008sp,Kim:2009sr}. In this paper we perform
a more detailed calculation of $\lambda_{\cal P}$ in the model of \cite{Sakai:2004cn} in the limit of large 't Hooft coupling, and obtain
$ \lambda_{{\cal P},{\rm SS}} \simeq 6.38 ~{\rm GeV}^{-1}$ which is within $\sim 23 \% $ of the experimental value.
In section two, we give a very brief review of the Sakai Sugimoto model and of the structure of baryons in this model. In the third section we describe the baryon as a Skyrmion, and in the fourth we introduce the effective fermion description of baryons and computed the lowest baryon wave functions. In the fifth section we deduce the coupling of the glueball to baryons and compute the leading contribution to $\lambda_{\cal P}$ at large 't Hooft coupling. We end, in the sixth section, by discussing the limitations of this work due to the assumptions about large 't Hooft coupling.
\section{Review of the Sakai-Sugimoto model}
In the Sakai-Sugimoto model \cite{Sakai:2004cn} we start with the dual of the pure glue sector of QCD constructed in \cite{Wittenqcd}. Here, we have a stack of $N_c$ $D4$-branes wrapping an $S^1$. Antiperiodic boundary conditions along this $S^1$ are imposed on the fermion fields to break supersymmetry.
The $D4$-branes source a metric given by
\beq
ds^2 = \left( \frac{U}{R} \right)^{3/2}\left(\eta_{\mu \nu} dx^\mu dx^\nu + f(U) d\tau^2 \right) + \left( \frac{R}{U} \right)^{3/2} \left( \frac{dU^2}{f(U)} + U^2 d \Omega_4^2 \right)
\eeq
where, in terms of the string length $\ell_s$, string coupling $g_s$, and number of colors $N_c$, $R^3 = \pi g_s N_c \ell_s^3$. The function
$f(U)$ is given by
\beq
f(U)= 1 - \frac{U_{KK}^3}{U^3}~
\eeq
where $U_{KK}$ is the minimal value of the radial coordinate $U\in [U_{KK},\infty]$.
The solution has topology $R^{3,1} \times D \times S^4$ with $(U, \tau)$ coordinates on the disk $D$ and
$\tau$ the (periodic) angular coordinate on the disk with identification
\beq
\tau \sim \tau + \frac{2 \pi}{M_{KK}}~.
\eeq
$M_{KK}$, which governs the mass scale of states in the theory, is related to the parameters $R, U_{KK}$
appearing in the metric via
\beq
M_{KK}= \frac{3 U_{KK}^{1/2}}{2 R^{3/2}}~.
\eeq
The dilaton is given by
\beq
e^{- \Phi} = \frac{1}{g_s} \left( \frac{R}{U} \right)^{3/4}
\eeq
and in addition the Ramond-Ramond field $F_4$ carries $N_c$ units of flux on the $S^4$.
Quark degrees of freedom are included through the addition of $N_f$ $D8$-branes \cite{Sakai:2004cn}. While different embeddings of the $D8$-branes, described by profiles $\tau(U)$, are possible,
we choose the embedding of the $D8$-branes for which the minimal value of $U$ is $U_0=U_{KK}$, for which $\tau(U)$ is constant. The $D8$-branes are then flat inside the 10-dimensional space. The induced metric
on the $D8$-brane is
\beq
ds_{8+1}^2 = ds_{4+1}^2 + ds_4^2
\eeq
where
\beq
ds_{4+1}^2 = \left( \frac{U}{R} \right)^{3/2} \left( \eta_{\mu \nu} dx^\mu dx^\nu \right) + \left( \frac{R}{U} \right)^{3/2} \frac{1}{f(U)} dU^2
\eeq
and
\beq
ds_4^2= \left( \frac{R}{U} \right)^{3/2} U^2 d \Omega_4^2
\eeq
with $d \Omega_4^2$ the metric on the unit $S^4$.
Define $h_4$ to be the determinant of the $S_4$ metric components and $h_{4+1}$ the determinant of the $4+1$ metric
components.
We also define
\beq
V_4 = \int d \Omega_4 \equiv \frac{8 \pi^2}{3}
\eeq
and
\beq
{\rm Vol}_{S_4} = \int \sqrt{h_4} d \Omega_4 = R^3 U \times V_4
\eeq
\subsection{Conformal coordinates and coordinate ranges}
It is convenient to make a change of variables from $U$ to $w$ so that the $4+1$ metric is conformal to flat space. This requires that
\beq
\left( \frac{R}{U} \right)^{3/2} \frac{1}{f(U)} dU^2 = \left( \frac{U}{R} \right)^{3/2} dw^2
\eeq
which gives
\beq
w(U) = \int_{U_{KK}}^U \frac{R^{3/2} dU'}{\sqrt{{U'}^3 - U_{KK}^3}} ~.
\eeq
Inverting this defines $U(w)$, and the induced metric on the $D8$-brane is then
\beq
ds_{8+1}^2 = \left( \frac{U(w)}{R} \right)^{3/2} \left( dw^2 + \eta_{\mu \nu} dx^\mu dx^\nu \right) + R^{3/2} U^{1/2} d \Omega_4^2
\eeq
Now, the coordinate $U$ only covers half the D8-brane, and $w(U)$, as defined above, does the same. Therefore, we extend the range to $w \in [-w_{max}, +w_{max}]$ to cover the whole $D8$-brane. Wave functions are either even or odd
in $w$, a fact directly related to the $C,P$ properties of the corresponding four-dimensional meson states. Glueballs arise as four-dimensional normalizable modes of the higher-dimensional metric perturbations and the lowest spin two glueball has a
wavefunction
$\tilde T(w)$ which is an even function of $w$ \cite{Brower,Constable}.
In what follows we write the relevant 5d metric in terms of $w$ as
\beq \label{metdef}
ds_{4+1}^2= H(w) \left( dw^2 + \eta_{\mu \nu} dx^\mu dx^\nu \right)
\eeq
with $H(w) \equiv (U(w)/R)^{3/2}$.
\subsection{Action for gauge fields}
The Dirac-Born-Infeld (DBI) action, expanded to quadratic order in the gauge fields leads to
\beq
S_{D8} = \frac{\mu_8 (2 \pi \alpha')^2}{4} \int d^9 \xi \ e^{-\Phi} \sqrt{-{\rm Det} g_{ab}} \ {\rm Tr} F_{ab}F^{ab}
\eeq
with $a,b=0,1,\dots 8$.
Using the conformal metric this leads to
\beq
S_{D8}= \frac{\mu_8 (2 \pi \alpha')^2 R^3 V_4}{g_s} \int d^4x dw \ \frac{U(w)}{4} \eta^{MN} \eta^{PQ} \ {\rm Tr} F_{MP} F_{NQ}
\eeq
with $M,N=0,1,2,3,w$.
In \cite{Kim:2009sr} this is written as
\beq
S_{D8} = \int d^4x dw \ \frac{1}{4 e^2(w)} \ \eta^{MN} \eta^{PQ} \ {\rm Tr} F_{MP} F_{NQ} ~.
\eeq
If the generators are normalized as
\beq
{\rm Tr} T^a T^b = \frac{1}{2} \delta^{ab}
\eeq
this leads to a nonstandard definition of the coupling. With the canonical factor of $1/4$ in front of each component of the
gauge field action the canonically defined coupling is $g_5(w)= \sqrt{2} e(w)$.
Using the previous definitions we
can write
\beq
\frac{1}{e^2(w)}= \frac{\lambda N_c M_{KK}U(w)}{108 \pi^3 U_{KK}}
\eeq
which agrees with the result quoted in \cite{Hong:2007kx,Hong:2007ay,Kim:2009sr}.
Introducing a dimensionless version of $U(w)$ as $u(w)=U(w)/U_{KK}$,
\beq
\frac{1}{e^2(w)}= \frac{\lambda N_c M_{KK}u(w)}{108 \pi^3 }~,
\eeq
which highlights the fact that all $\ell_s$ dependence drops out of the leading gauge theory action once quantities
are expressed in terms of the defining parameters of the dual field theory: $\lambda, M_{KK}, N_c$.
\section{Description of the Skyrmion/Instanton/Baryon}
The baryon is described by a charge one instanton of the $SU(2)$ gauge field on the flavor branes, which spans the
$(x,y,x,w)$ directions. The exact solution of the equations of motion is not known, but it is believed that one can take
the flat space instanton as a reasonable approximation to the full solution. This solution has moduli consisting of the $SU(2)$ orientation, the location of the instanton in $(x,y,z,w)$, and the scale size $\rho$. The scale size of the instanton is fixed by balancing the gauge action, which makes, it want to shrink,
and the coupling of the baryon current to the tower of $\omega$ mesons arising from the Chern-Simons term, which makes it want to grow. It becomes a spin $1/2$ object after quantizing the collective coordinates, as is familiar from the original
Skyrmion literature. For details see \cite{Hata:2007mb,Hong:2007kx}. The location of the instanton in $(x,y,z)$ is arbitrary as
a consequence of 3d translational invariance. The nontrivial metric dependence on $w$, on the other hand, results in
a $w$-dependent contribution to the energy with the minimum occurring at $w=0$.
The baryon mass quoted in the literature is
\beq
m_B^{cl}= \frac{\lambda N_c}{27 \pi} M_{KK} \left(1 + \frac{\sqrt{2 \cdot 3^5 \cdot \pi^2/5}}{\lambda} + \cdots \right)~.
\eeq
For very large $\lambda$, $m_B^{cl}>>M_{KK}$ and the second term is subleading. When the baryon/instanton is
displaced from $w=0$ one finds that to quadratic order
\beq
m_B(w)= m_B^{cl} \left(1 + \frac{1}{3} (w M_{KK})^2 + \cdots \right)
\eeq
While this will turn out to be the correct expression for the 4d mass, it is important to distinguish it from the
5d mass.
The instanton/baryon is a gauge field
configuration depending on $x^i,w$ but independent of $x^0$. It is localized in $x^i,w$ and after integrating
over these coordinates the action becomes
\beq
S= \int m_B^{cl} \ dx^0
\eeq
and the mass $m_B^{cl}$ is extracted from this expression (see e.g. eqn 3.18 in \cite{Hata:2007mb}).
However in 5d curved space the action for a static particle of mass $m_5$ is expressed in
terms of the proper time as
\beq
S = \int m_5 \ d \tau
\eeq
which leads to
\beq
m_5 = \frac{m_B^{cl}}{\sqrt{g_{00}}}~.
\eeq
For the metric we are using $g_{00}= (U/R)^{3/2}$ and so the 5d mass is actually
\beq
m_5 (U/R)^{3/4}=m_B^{cl}~.
\eeq
\section{Effective Fermion description of baryons}
Following the treatment
of \cite{Hong:2007kx,Hong:2007ay,Hong:2007dq,Park:2008sp,Kim:2009sr}, which we examine in detail below, we describe the Skyrmion as an effective fermion field living in the D8 world-volume.
The basic idea is that the Skyrmion is an instanton constructed out of the flavor gauge fields,
with a size that is much smaller than $M_{KK}$ at large 't Hooft coupling $\lambda$. Since the baryon has spin $1/2$, it
is reasonable to write its effective description in terms of a 5d fermion field. To compute what we call $\lambda_{\cal P}$, the
coupling of the spin $2$ glueball to the proton, we will need to work out the coupling of the 5d metric to the 5d fermion
and reduce this to a 4d coupling using mode expansions for the 5d metric and the 5d fermion field representing
the baryon.
We saw above that the pure gauge part of this action can be derived by starting from a general coordinate-invariant
action and reducing it using 5d conformal coordinates. We follow a similar procedure for the 5d fermions.
We begin with a general coordinate (and local Lorentz invariant) action in 5d, to which we can add metric perturbations and thus
extract the coupling to the glueball.
Let us start from a curved space action in 5d -- unlike the flat, 4d effective action posited in \cite{Hong:2007kx,Hong:2007ay,Hong:2007dq,Park:2008sp,Kim:2009sr}:
\beq
S_f[\bar \psi,\psi,g]= -i {\cal N} \int d^5x \ e^{-\Phi(w)} {\rm Vol}_{S^4}(w)\sqrt{h_{4+1}} \left[ \bar \psi e^m_{\hat a} \Gamma^{\hat a} D_m \psi + m_5(w) \bar \psi \psi \right]
\eeq
Here $m,n,p$ are 5d world indices, $\hat a, \hat b, \hat c$ are 5d tangent space indices, and the covariant
derivative is (ignoring the gauge fields which will not enter into the rest of our calculation)
\beq
D_m \psi = (\partial_m -\frac{i}{4} \omega_m^{\hat a \hat b} \sigma_{\hat a \hat b} ) \psi
\eeq
with
\beq
\sigma_{\hat a \hat b} = \frac{i}{2} [\Gamma_{\hat a}, \Gamma_{\hat b}]~.
\eeq
The $\Gamma_{\hat a}$ are tangent space gamma matrices obeying
\beq
\{\Gamma_{\hat a},\Gamma_{\hat b} \} = 2 \eta_{\hat a \hat b}~.
\eeq
The factors of the dilaton and the volume of the $S^4$ are as we would expect them
to emerge from the SS model after quantizing the collective coordinates of the instanton to obtain
a spin $1/2$ object.
Note that
\beq
e^{-\Phi}{\rm Vol}_{S_4} \sqrt{h_{4+1}}= \frac{V_4}{g_s} U(w)^4~.
\eeq
Any normalization factors are included in the overall prefactor
${\cal N}$, which will later be absorbed by a redefinition of the fermion field. Other then the $w$-dependent prefactors this is the standard curved space action for a spin $1/2$ fermion.
We can now evaluate the action for the particular background metric given in \eqref{metdef}.
We choose
\beq
e^{\hat a}= H(w)^{1/2} \delta^{\hat a}_m dx^m
\eeq
and find that the structure equation for the spin connection is solved (modulo local Lorentz transformations and diffeomorphisms) by the spin connection
\beq
{\omega^{\hat \mu}}_{\hat \nu}=0, \qquad {\omega^{\hat \mu}}_{\hat w} = \frac{1}{2} \partial_w \ln(H(w)) dx^\mu~,
\eeq
with
\beq
D_w \psi = \partial_w \psi
\eeq
and
\beq
- \frac{i}{4} {\omega_\mu}^{\hat a \hat b} \sigma_{\hat a \hat b}= \frac{1}{8} {\omega_\mu}^{\hat a \hat b} [\Gamma_{\hat a},\Gamma_{\hat b}]= \frac{1}{4} {\omega_\mu}^{\hat \mu \hat w} [\Gamma_{\hat \mu},\Gamma_{\hat w}]= \frac{\partial_w H}{4H(w)} \Gamma_{\hat \mu} \Gamma_{\hat w}~.
\eeq
We can take $\Gamma_{\hat \mu}=\gamma_\mu$ and $\Gamma_{\hat w}= \gamma_5$ where the flat space $\gamma$ matrices are defined as
\beq \gamma^0= \begin{pmatrix} 0 & -1 \cr 1 & 0 \cr \end{pmatrix}, \qquad \gamma^i = \begin{pmatrix} 0 & \sigma_i \cr \sigma_i & 0 \cr
\end{pmatrix}, \qquad \gamma^5= \begin{pmatrix} 1 & 0 \cr 0 & -1 \cr \end{pmatrix}~,
\eeq
and find
\beq
S= -i {\cal N} \int dw d^4x \left({\rm Vol}_{S^4}(w) e^{-\phi(w)} \sqrt{h_{4+1}}\right) H^{-1/2} \left( \bar \psi \left[ \gamma^\mu \partial_\mu +\gamma^5 \partial_w + \frac{\partial_w H}{H} \gamma^5 \right] \psi + m_5(w) H^{1/2} \bar \psi \psi \right)~.
\eeq
We can now reabsorb $w$-dependent factors into the fermion field $\psi$ to obtain a simplified form for the action. Taking
\beq
\psi (x,w)\equiv \frac{1}{H(w)} {\cal B}(x,w)~,
\eeq
the action becomes
\beq
S= -i {\cal N} \int dw d^4x \left({\rm Vol}_{S^4}(w) e^{-\phi(w)} \sqrt{h_{4+1}}H^{-5/2} \right) \left( \bar {\cal B} \left[ \gamma^\mu \partial_\mu +\gamma^5 \partial_w \right] {\cal B} + m_5(w) H^{1/2} \bar {\cal B} {\cal B} \right)~.
\eeq
The prefactor in round brackets is
\beq
\left({\rm Vol}_{S^4}(w) e^{-\phi(w)} \sqrt{h_{4+1}}H^{-5/2} \right) = \frac{V_4 R^{15/4}}{g_s} U^{1/4}(w) ~.
\eeq
If we approximate this factor by its value at $w=0$, and then absorb this constant as well as ${\cal N}$ into ${\cal B}$, then
we reproduce the action posited in \cite{Hong:2007kx,Kim:2009sr},
\beq \label{start}
S_B= -i \int d^4x dw \left[ \bar {\cal B} \gamma^m D_m {\cal B}+ m_{\cal B}(w) \bar {\cal B} {\cal B}+ \cdots \right] - \int d^4x dw \frac{1}{4 e^2(w)} {\rm tr} F_{mn} F^{mn} ~,
\eeq
with $m_{B}^{cl}(w)= H^{1/2}(w)m_5(w)$.
Note that the covariant derivative above
only involves the gauge fields
\beq
D_m = \partial_m -i A_m~,
\eeq
and that we have omitted higher terms coupling the baryon field to the gauge fields, though these are necessary for reproducing
various interactions as well as the long-range tail of the Skyrmion configuration.
\section{Coupling of the fermion to the glueball metric perturbation}
The glueball is described by a symmetric, traceless perturbation of the $3+1$ part of the metric
$g_{\mu \nu} \rightarrow g_{\mu \nu} - h_{\mu \nu}$. The effect of this perturbation is equivalent to writing
\beq
\bar {\cal B} \gamma^\mu \partial_\mu {\cal B} = \bar {\cal B} \eta_{\mu \nu} \gamma^\mu \partial^\nu {\cal B} = \bar {\cal B}H^{-1}(w) g_{\mu \nu} \gamma^\mu \partial^\nu {\cal B}
\eeq
and replacing $g_{\mu \nu} \rightarrow g_{\mu \nu} - h_{\mu \nu}$ to obtain the coupling
\beq
H^{-1}(w) h_{\mu \nu} {\cal B} \gamma^\mu \partial^\nu {\cal B} = H^{-1}(w) h_{\mu \nu} \frac{1}{2} {\cal B} (\gamma^\mu \partial^\nu + \gamma^\nu \partial^\mu ) {\cal B}
\eeq
which is equal to
\beq
H^{-1}(w) h_{\mu \nu} T^{\mu \nu}_{\cal B}
\eeq
where $T^{\mu \nu}_{\cal B}$ is the $w$-dependent fermion stress-energy tensor.
We are now ready to compute $\lambda_{\cal P}$
starting with the coupling we just derived,
\beq
\int dw d^4x H^{-1}(w) h_{\mu \nu} T_{\cal B}^{\mu \nu}~.
\eeq
We write the metric perturbation $h_{\mu\nu}$ in terms of the glueball wave function following the treatment in \cite{Brower} to obtain
\beq
h_{\mu \nu}= \left( \frac{U(w)}{R} \right)^{3/2} \tilde T(U(w)) q_{\mu \nu}
\eeq
where $q_{\mu \nu}$ is a canonically normalized spin two field in four dimensions. The lightest mode of $h_{\mu\nu}(x,w)$
has 4d mass eigenvalue $m_{glue}=(2/3) M_{KK} \sqrt{5.5} \simeq 1.47 ~ {\rm GeV}$.
We also write the 5d baryon in terms of $w$-dependent wavefunctions and 4d fermion fields.
The 5d Dirac equation which follows from the action \eqref{start} is
\beq
\gamma^\mu \partial_\mu {\cal B} + \gamma^5 \partial_w {\cal B} + m_{\cal B}(w) {\cal B}=0~.
\eeq
To solve this we write ${\cal B}(x,w)$ in terms of $\gamma^5$ eigenstates
\beq \label{bardef}
{\cal B}(x^\mu,w)= \sum_n \begin{pmatrix} B^{(n)}_+(x) f^{(n)}_+(w) \cr B^{(n)}_-(x) f^{(n)}_-(w) \end{pmatrix}~.
\eeq
As was the case for the glueball, we are only interested in the lowest $(n=1)$ mode, which represents the physical nucleon doublet. Dropping
the $(n)$ superscript we then have for the lowest eigenmode
\bea
f_-(w) \bar \sigma^\mu \partial_\mu B_-(x)+ \partial_w f_+(w) B_+(x) + m_{\cal B}(w) B_+(x) f_+(w)&=& 0 \cr
f_+(w) \sigma^\mu \partial_\mu B_+(x)- \partial_w f_-(w) B_-(x) + m_{\cal B}(w) B_-(x) f_-(w)&=& 0
\eea
Comparing this to the equation for a massive 4d spinor with mass $m_{\cal N}$ we see that
the eigenvalue equation is
\bea \label{dq}
(\partial_w +m_{\cal B}(w)) f_+ & = & m_{\cal N} f_- \cr
(-\partial_w + m_{\cal B}(w)) f_- & = & m_{\cal N} f_+
\eea
and to get a standard 4d action the wave functions should be normalized to
\beq
\int dw |f_\pm(w)|^2 = 1~.
\eeq
At large $\lambda$ the baryons become very heavy compared to the scale $M_{KK}$ and the width of the wave functions
scales like $1/\sqrt{\lambda}$. Thus at very large $\lambda$ we
can approximate
\beq
|f_+(w)|^2 = |f_-(w)|^2 = \delta(w)~.
\eeq
In this approximation the interaction takes the form
\beq
\lambda_{\cal P} \int d^4x q_{\mu \nu} T_{\cal B}^{\mu \nu}(x)
\eeq
where
\beq
{\cal B}(x)=\begin{pmatrix} B_+(x) \cr B_-(x) \end{pmatrix}
\eeq
is the 4d nucleon wave function and $T_{\cal B}(x)$ is the 4d energy-momentum tensor of the spin 1/2 object. The coupling
constant is simply
\beq
\lambda_{\cal P} = \tilde{T}(U(0))~.
\eeq
Evaluating this at $w=0$ gives
\beq
\lambda_{{\cal P}} = \tilde T(1)= \sqrt{\frac{3}{0.9 \pi^2}} \frac{1}{f_\pi}= 6.38 ~ {\rm GeV}^{-1}~.
\eeq
\section{Conclusions and Further Issues}
In this note we have computed $\lambda_{P}$ in the effective fermion picture from \cite{Hong:2007kx, Hong:2007ay,Hong:2007dq,Park:2008sp,Kim:2009sr}. We found good agreement with the data, however we must keep in mind the limitations of this treatment of baryons in the Sakai-Sugimoto model.
The first two terms in the expansion of the baryon mass in powers of $1/\lambda$ are
\beq
m_B^{classical}= \frac{\lambda N_c}{27 \pi} M_{KK} \left(1 + \frac{\sqrt{2 \cdot 3^5 \cdot \pi^2/5}}{\lambda} + \cdots \right)
\eeq
which comes about from balancing the Coulomb and ``Pontragin'' contributions to the energy. These are
the first terms in an expansion in the inverse 't Hooft coupling $\lambda= g_{YM}^2 N_c$.
For very large $\lambda$, $m_B^{classical}>>M_{KK}$.
However, the actual values of the parameters needed to fit the $\rho$ mass and $f_\pi$ in this model are
$M_{KK}= 0.94 ~ {\rm GeV}$ and $\lambda=17$. Note that the proton mass is $0.938 ~{\rm GeV}$ which is almost
exactly $M_{KK}$. Using these values one finds
\beq
m_B^{classical}= 0.6 M_{KK} \left(1 + 1.82 + \cdots \right)
\eeq
The first order term is not a very accurate estimate of the baryon mass, and the second term, which is supposed to be subleading, is actually larger than the first. This $1/\lambda$ expansion should thus be employed with some care.
On the other hand, the leading approximation in $\lambda$ to a variety of
meson-nucleon couplings in this model does agree quite well with experimental data \cite{Hashimoto:2008zw,Hong:2007kx,Hong:2007ay,Kim:2009sr}. From the above calculation it seems that the same is true for the coupling of the spin two glueball to the proton, a quantity which sets the scale of the
total cross section for proton-proton scattering.
\begin{acknowledgements}
This work was supported in part by
NSF Grant PHY 0855039. We would like to thank Piljin Yi for helpful conversations and the Aspen Center for Physics for
providing a congenial atmosphere at the start of this work.
\end{acknowledgements}
|
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BkiUeLo4ukPiEV8g6c2S
| 5 | 1 |
\section*{To do}
\section{Introduction and main results}
\subsection{Loewner energy and two minimization questions}
Consider a simple curve $\gamma = \gamma[0,T]$ in the upper half plane $\mathbb{H}$ starting from zero. For each $0 < t \leq T$, $\mathbb{H} {\backslash} \gamma[0,t]$ is simply connected, and we can thus ``map down'' the curve with a Riemann map $g_t:\mathbb{H}{\backslash} \gamma[0,t] \rightarrow \mathbb{H}$ which fixes $\infty$ and takes the tip $\gamma(t)$ to some point $\lambda(t) \in \mathbb{R}$. Since every other point $\gamma(s) \in \gamma[0,t)$ on the curve corresponds to two prime ends in $\mathbb{H} {\backslash} \gamma[0,t]$, $g_t$ ``cuts $\gamma(s)$ in half,'' sending it to two points on either side of $x$. Thus $g_t$ ``unzips'' $\gamma[0,t]$ to two intervals around $g_t(\gamma(t))$, as in Figure \ref{Fig:LoewnerFlowB}.
When we appropriately parametrize $\gamma$ and appropriately normalize $g_t$, this flow of conformal maps for the growing curves $\gamma[0,t]$ satisfies \emph{Loewner's equation}
\begin{align}\label{Eq:LoewnerEqIntro}
\frac{\partial g_t(z)}{\partial t} =: \dot{g}_t(z) = \frac{2}{g_t(z) -\lambda(t)}, \qquad g_0(z) = z,
\end{align}
where $\lambda(t) := g_t(\gamma(t))$ is the image of the tip, also known as the \emph{driving function} of $\gamma$ (see \S\ref{Sec:BackgroundLoewner} for details). This differential equation uniquely encodes $\gamma$ through the real-valued function $\lambda$, allowing one to analyze $\gamma$ by means of $\lambda$.
\begin{figure}
\centering
\includegraphics[scale=0.16]{Loewner_flow_B.jpg}
\caption{\small The ``mapping down'' function $g_t$ ``unzipping'' the curve $\gamma$.}
\label{Fig:LoewnerFlowB}
\end{figure}
For example, one can consider the \emph{Loewner energy} $I(\gamma)$ of $\gamma$, which is defined as the Dirichlet energy of the driving function,
\begin{align}\label{Eq:LoewnerEnergy1}
I(\gamma) := \frac{1}{2}\int_0^\infty \dot{\lambda}(t)^2dt,
\end{align}
when $\lambda$ is absolutely continuous, and $I(\gamma) = +\infty$ otherwise. One can think of the Loewner energy as a measurement of the deviation of $\gamma$ from a hyperbolic geodesic, as $I(\gamma) = 0$ if and only if $\gamma$ is a hyperbolic geodesic (see \S\ref{Sec:BackgroundLoewnerEnergy} for the trivial argument). Moreover, $I(\gamma)$ is a conformally-natural measurement, in the sense of being invariant under conformal automorphisms of $\mathbb{H}$, as shown by Wang \cite{WangReverse}. While it appears to be defined by means of the parametrization of $\lambda$, Wang also showed \cite{WangEquiv} it can be expressed purely in terms of Riemann maps $g_t$ for $\gamma$. The same work additionally shows that the loop version of Loewner energy (where the integral in \eqref{Eq:LoewnerEnergy1} becomes over all of $\mathbb{R}$ instead of just $\mathbb{R}_{\geq 0}$) characterizes the Weil-Petersson quasicircles of the universal Teichm\"{u}ller space $T_0(1)$. That is, finite-energy loops are the closure of smooth Jordan loops in the Weil-Petersson metric. Bishop \cite{Bishop} built off this to give a slew of $\ell^2$-type geometric characterizations of finite-energy curves, giving the further intuition that finite-energy $\gamma$ are precisely those possessing ``square-summable curvature over all positions and scales.''\footnote{Here ``curvature'' refers to functionals such as the $\beta$-numbers from geometric measure theory, and not to classical notions that require second derivatives; $\gamma$ need not be in $C^2$, rather, its arc-length parametrization is $H^{3/2}$ \cite{Bishop}.}
While we discuss further background of the Loewner energy in \S\ref{Sec:BackgroundLoewnerEnergy}, it is already evident that $I(\gamma)$ is a significant functional on curves. It is only natural, then, to ask what $\gamma$ minimize it. If there are no constraints, then the answer is immediate and uninteresting: $\gamma$ is a straight line orthogonal to $\mathbb{R}$, which uses zero energy. In this paper we study the next case beyond this, considering the nature of ``one-point'' minimizers in the sense of the following two questions:
\begin{enumerate}[$(i)$]
\item\label{Q:HMinimizer} What is the infimal energy among all $\gamma$ which pass through a given point $z=re^{i\theta} \in \mathbb{H}$, and what is the nature of minimizers, if they exist?
\item\label{Q:WeldingMinimizer} What is the infimal energy among all $\gamma$ which start from 0 and weld a given $x<0<y$ to their base, and what is the nature of minimizers, if they exist?
\end{enumerate}
As we will see, question $(\ref{Q:HMinimizer})$ has already been partially investigated, but we revisit it using entirely different techniques, yielding fresh proofs for what has been known and also obtaining further results. To our knowledge, the second question is thus far unexplored.
\subsection{Answering question (\ref{Q:HMinimizer}): the Wang minimizers}\label{Intro:Wang}
Wang considered curves from 0 to some $re^{i\theta} \in \mathbb{H}$, and showed the infimal energy in this family is
\begin{align}\label{Eq:Energy1}
I(\gamma) = -8\log(\sin(\theta)),
\end{align}
and that it is furthermore uniquely attained \cite[Prop. 3.1]{WangReverse} ($r$'s absence in this formula reflects scale-invariance of the energy). We call these minimizing curves the \emph{Wang minimizers}. Wang's argument was stochastic in nature, using Schramm's formula for the probability that SLE passes to the right of $re^{i\theta}$ \cite{SchrammPerc}, combined with a SLE$_{0+}$ large-deviation principle. While elegant, one might wonder if these results could be re-derived without resorting to the probability toolbox, as they are intrinsically deterministic in nature. We answer here in the affirmative, and deterministically derive several properties of the Wang minimizers. We collect these in our main result for question $(\ref{Q:HMinimizer})$, Theorem \ref{Thm:Wang}, which we proceed to summarize.
We start in Theorem \ref{Thm:Wang}$(\ref{Thm:WangEnergy})$ by deterministically re-deriving the minimal-energy formula \eqref{Eq:Energy1}. Using the resulting system of differential equations, we proceed in Theorem \ref{Thm:Wang}$(\ref{Thm:WangDriver})$ to explicitly compute the associated driving functions $\xi_\theta$.
\begin{figure}
\centering
\includegraphics[scale=0.13]{Loewner_flow.jpg}
\caption{\small The downwards and upwards Loewner flows for $\gamma = \gamma[0,T]$.}
\label{Fig:LoewnerFlow}
\end{figure}
To help explain a ``universality'' property of the $\xi_\theta$ that we highlight in Theorem \ref{Thm:Wang}$(\ref{Thm:WangUniversal})$, we note that it is more natural to work with the ``upwards'' driver for the Wang minimizers. By this we mean the function $\xi_\theta(t) := \lambda_\theta(T-t)$ arising from ``reversing the movie'' of the $g_t$ by considering the conformal maps $h_t := g_{T-t} \circ g_T^{-1}$, as in Figure \ref{Fig:LoewnerFlow}. By \eqref{Eq:LoewnerEqIntro} the $h_t$ satisfy
\begin{align}\label{Eq:LoewnerEqIntroUp}
\dot{h}_t(z) = \frac{-2}{h_t(z) - \xi(t)}, \qquad h_0(z) = 0,
\end{align}
and we call the dynamics given by \eqref{Eq:LoewnerEqIntro} and \eqref{Eq:LoewnerEqIntroUp} the \emph{forwards} and \emph{reversed} Loewner flows, respectively. More picturesquely, we can also label them as the \emph{downwards} and \emph{upwards} flows, respectively. We write $\lambda$ for \emph{forwards/downwards} driving functions appearing in \eqref{Eq:LoewnerEqIntro} and $\xi$ for \emph{reversed/upwards} driving functions appearing in \eqref{Eq:LoewnerEqIntroUp}.
Let $\xi_\theta$ be the upwards driver which generates the Wang minimizer $\gamma_\theta$ from 0 to $e^{i\theta}$ on the time interval $[0,\tau_\theta]$, where $0 < \theta< \pi/2$, say. Beyond explicitly computing $\xi_\theta$, we show in Theorem \ref{Thm:Wang}($\ref{Thm:WangUniversal}$) that $\xi_\theta$ not only generates $\gamma_\theta$, but also \emph{all} Wang minimizers for angles $0 < \alpha < \pi/2$, up to translation and scaling. That is, $\xi_\theta$ is well-defined for all $t \geq 0$, and the curve it generates on any $[0,t]$ is a Wang minimizer for some $e^{i\alpha_t}$ (after translation and scaling), where $\alpha_t$ is decreasing in $t$ and has the entire interval $(0,\pi/2)$ as its range. By symmetry, $-\xi_\theta$ generates all Wang minimizers for angles $\pi/2 < \alpha <\pi$ (up to scaling and translation), and so each $\xi_\theta$ is actually \emph{universal} for all the Wang minimizers. Building off calculations in \cite{MRW}, we also explicitly compute the conformal welding \eqref{Eq:WangWeld} generated by $\xi_\theta$ and show it has an analogous universality property.
In Theorem \ref{Thm:Wang}($\ref{Thm:Wang0+}$) we consider the behavior of the minimizers as $\theta$ tends to zero and show that taking the na\"{i}ve $\theta \rightarrow 0^+$ limit in our driving function formula yields the correct driver $\xi_0(t) = -\frac{8}{\sqrt{3}}\sqrt{t}$ for the limiting curve $\gamma_0$, and that furthermore $\gamma_0$ is a subset of an explicit cubic algebraic variety in $x$ and $y$. One subtlety here is that the limiting curve touches the real line at $x=1$ and thus has infinite energy, and so usual compactness tools for collections of curves of finite energy are not available. Beyond identifying the limit, we show that $\gamma_0$ and its reflection $\gamma_0^*$ across $\partial \mathbb{D}$ is a \emph{boundary geodesic pair}, in the sense that $\gamma_0$ is the hyperbolic geodesic from 0 to 1 in its component of $\mathbb{H} {\backslash} \gamma_0^*$, and $\gamma_0^*$ is the hyperbolic geodesic from 1 to $\infty$ in its component of $\mathbb{H} {\backslash} \gamma_0$ (see Figure \ref{Fig:Gamma0}). This sense of geodesic pair slightly extends that recently defined by Marshall, Rohde and Wang \cite{MRW}, in that the point of intersection $\zeta = \gamma_0 \cap \gamma_0^*$ between the two geodesics is on the boundary of the domain.\footnote{We note that such geodesic pairs have been considered by Krusell under the name of ``fused geodesic multichords'' \cite[\S 4]{Krusell}).} We collect what our considerations show about boundary geodesic pairs in general simply-connected domains in Corollary \ref{Cor:BoundaryGeodesic}.
\begin{figure}
\centering
\includegraphics[scale=0.1]{Gamma0.jpg}
\caption{\small The limiting curve $\gamma_0$ for the Wang minimizers and its reflection $\gamma_0^*$ across $\partial \mathbb{D}$ form a (generalized) geodesic pair: $\gamma_0$ is the hyperbolic geodesic from 0 to 1 in its component of $\mathbb{H} {\backslash} \gamma_0^*$, and $\gamma_0^*$ is a hyperbolic geodesic from 1 to $\infty$ in its component of $\mathbb{H} {\backslash} \gamma_0$.}
\label{Fig:Gamma0}
\end{figure}
\subsection{Answering question (\ref{Q:WeldingMinimizer}): the welding minimizers}\label{Intro:Welding}
Our deterministic techniques yield a similar scope of results for the second question. To clarify the question statement, recall that the conformal welding $\varphi_t$ of the curve $\gamma$ is the map (determined by $\gamma$ and normalized by $g_t$) which identifies the two images of the points $\gamma(s)$ ``split in half'' by $g_t$. That is, $\varphi_t$ satisfies $g_t^{-1}(x) = \gamma(s) = g_t^{-1}(\varphi_t(x))$, where $g_t^{-1}(x)$ is the extension of $g_t^{-1}$ to $x \in \mathbb{R}$. For example, $\varphi_t(g_t(\gamma(s)-)) = g_t(\gamma(s)+)$ in Figure \ref{Fig:LoewnerFlowB}, while similarly $\varphi_T(g_T(\gamma(t)-)) =g_T(\gamma(t)+)$ in Figure \ref{Fig:LoewnerFlow}. We typically center our weldings by considering the \emph{centered downwards flow} maps $G_t(z) := g_t(z) - \lambda_t$, in which case $\varphi_t$ maps an interval $[x_t,0]$ to the left of the origin homeomorphically to an interval $[0,y_t]$ on the right.
The precise formulation of the welding minimization question is thus as follows: for fixed $x<0<y$, what is the infimal energy among all curves $\gamma: [0,T] \rightarrow \mathbb{H}\cup \{0\}$ which satisfy $\gamma(0)=0$ and have conformal welding $\varphi_T:[x,0]\rightarrow [0,y]$ under $G_T$? Our main result for this second question, Theorem \ref{Thm:EMW}, answers in part ($\ref{Thm:EMWEnergy}$) that the infimum is
\begin{align}\label{Eq:Enegy2}
-8\log\Big(\frac{2 \sqrt{-xy}}{y-x}\Big) = -8\log\Big(\frac{2\sqrt{r}}{1+r} \Big),
\end{align}
where $r:= -x/y$, and that this infimum is uniquely attained by a curve which is smooth on $(0,T]$. We call these curves the \emph{energy minimizers for weldings (EMW)} family. In Theorem \ref{Thm:EMW}($\ref{Thm:EMWDriver}$) we explicitly compute their driving functions and give an implicit equation satisfied by their conformal weldings. We also prove a ``universality'' property in Theorem \ref{Thm:EMW}$(\ref{Thm:EMWSingleCurve})$: there exists a single curve $\Gamma:[0,T] \rightarrow \mathbb{H}\cup\{0\}$ such that for every ratio $r \in (0,1)$, $\Gamma[0,t_r]$ is a welding minimizer for points $x_r < 0 < y_r$ with $-x_r/y_r=r$. That is, up to scaling, $\Gamma$ single-handedly parametrizes all the welding minimizers with ratios between 0 and 1 (and its reflection $-\overline{\Gamma}$ across the imaginary axis does so for ratios $1<r<\infty$). See Figure \ref{Fig:UniversalEMW}. Furthermore, like the limiting Wang minimizer $\gamma_0$, $\Gamma$ is algebraic, and we explicitly identify the associated real quartic variety and show that its complex square $\Gamma^2$ is nothing other than a circle tangent to $\mathbb{R}$.
\begin{figure}
\centering
\includegraphics[scale=0.5]{EMWUnivPDF.pdf}
\caption{\small The universal curve $\Gamma$ for the welding minimizers with ratios $0 <r = -x/y <1$. Starting from closest to the imaginary axis and moving counterclockwise, the four points show where to truncate for EMW curves with welding endpoint ratios $r=0.75, 0.5, 0.25$ and $0.01$, respectively. The complex square of $\Gamma$ is a circle tangent at the origin to $\mathbb{R}$.}
\label{Fig:UniversalEMW}
\end{figure}
Another parallel between the EMW curves and Wang minimizers is that both turn out to be SLE$_0$ with forcing. Wang showed that her minimizer $\gamma_\theta$ is SLE$_0(-8)$ with the force point $V(t)$ starting at $V(0) = e^{i\theta}$ \cite[(3.2)]{WangReverse} (and driver starting at zero). Recall that this means that the downwards driver $\lambda$ for $\gamma_\theta$ evolves according to
\begin{align*}
\dot{\lambda}(t) = \Real\Big(\frac{8}{V(t) - \lambda(t)}\Big), \qquad \lambda(0) = 0,
\end{align*}
while all other points $z(t)$, including $V(t)$, satisfy the standard downwards Loewner equation \eqref{Eq:LoewnerEqIntro} determined by $\lambda$. We show in Theorem \ref{Thm:EMW}($\ref{Thm:EMWSLE}$) that the EMW family is upwards SLE$_0(-4,-4)$ with force points starting at $V^1(0)=x, V^2(0)=y$, the two points to be welded together. That is, the upwards driving function $\xi(t) := \lambda(T-t)-\lambda(T)$ satisfies
\begin{align*}
\dot{\xi}(t) = \frac{-4}{V^1(t) - \xi(t)} + \frac{-4}{V^2(t)-\xi(t)}, \qquad \xi(0) = 0,
\end{align*}
and all other points evolve according to \eqref{Eq:LoewnerEqIntroUp}. Another way to formulate this is downwards SLE$_0(-4,-4)$ with the two force points placed at the two prime ends of $\mathbb{H}{\backslash} \gamma$ corresponding to the base of $\gamma$. The precise meaning of this, though, is the above upwards-flow formulation.\footnote{Thus one should keep in mind that upwards SLE$_0$ and downwards SLE$_0$ generate the same curve $\gamma$ so long as one handles the forcing start points appropriately. We choose to use the ``upwards'' terminology to match the natural picture of ``mapping up'' with a conformal map $F:\mathbb{H} \rightarrow \mathbb{H}{\backslash} \gamma$, which welds points $x<0<y$ to the base of $\gamma$.}
We recall that downwards SLE$_\kappa$ with forcing, introduced in \cite{LSW}, frequently arose in the early days of SLE \cite{DubedatMart,DubedatComm, SchrammWilson}, and has continued to repeatedly surface (see \cite{AngHoldenSun} and \cite{MSImaginary} for recent examples, for instance). Its $\kappa \rightarrow 0$ limit, SLE$_0$, has also appeared elsewhere: \cite{Alberts}, for example, shows that the Loewner flow of a geodesic multichord in the sense of Peltola and Wang \cite{PeltWang} is downwards SLE$_0$ with forcing of $+2$ at the critical points and $-8$ at the poles in $\mathbb{H}$ of the associated rational function. Furthermore, many of deterministic drivers that provided fodder for early investigations into the chordal Loewner equation also turn out to be SLE$_0$ with forcing. For instance, Krusell \cite{KrusellComm} has shown that the drivers $\lambda_c(t) = c - c\sqrt{1-t}$, $c \in \mathbb{R}$, initially studied in \cite{LMR,KNK,Lind4,LoewnerCurvature}, all correspond to downwards SLE$_0$ with forcing for appropriate choices of $\rho$ and forcing start point. Upwards SLE$_0(\rho,\rho)$ has also appeared before, albeit under different guises. We show in the appendix that the straight line segments, driven by $\tilde{\lambda}_c(t) = c\sqrt{t}$, correspond to upwards SLE$_0(-2,-2)$, while the the orthogonal circular arc is upwards SLE$_0(-3,-3)$. Combined with the EMW curves, we thus have a cohesive picture of upwards SLE$_0(\rho,\rho)$ for $\rho \in \{-4,-3,-2\}$.
As an aside, we acknowledge that one might reasonably ask if there is much use in having explicit, complicated driver formulas, such as our calculation for the Wang minimizers \eqref{Eq:WangDriver} or our EMW formula \eqref{Eq:EMWDriverFormula}. One response is that a robust library of non-trivial driving functions is always beneficial for running experiments, testing conjectures, and gaining intuition. Another specific use of the Wang driver formula \eqref{Eq:WangDriver} appears in forthcoming work \cite{MesikeppSharp}, where we use it to show that the sharpness of the driver-curve regularity theorem of Carto Wong \cite{Wong}.
\subsection{Asymptotic energy comparisons}\label{Sec:IntroAsymp}
While we have identified two families of minimizers, it is not \emph{a priori} obvious that they are distinct. For instance, suppose we start with a Wang minimizer $\gamma$ for $e^{i\theta}$ and map it down with $G_T$, taking the two prime ends at the base of $\gamma$ to the endpoints $x_T<0<y_T$ of the welding interval. If we now map back up with the EMW curve $\tilde{\gamma}$ for $x_T$ and $y_T$, do we recover $\gamma$? In Theorem \ref{Thm:EMW}($\ref{Thm:DistinctFamilies}$) we answer in the negative, and prove the families are never the same except in the trivial case vertical line case $\theta=\pi/2$.
\def0.65{0.65}
\begin{figure}
\centering
\includegraphics[scale=0.13]{Same_ratio.jpg}
\caption{\small EMW curves and Wang minimizers in orange and blue, respectively, that weld the same black points $x_0<0<y_0$ to their base. Note the increase in argument of the tip when moving from the Wang to the welding minimizer, which is expected from the monotonicity of \eqref{Eq:Energy1}.}
\label{Fig:WangEMWSameWeld}
\end{figure}
Given that the two families are thus distinct (except in the trivial case), we would like to know how their energies compare. For fixed $\theta$, for instance, how far from optimal is the energy of the EMW curve with tip at $e^{i\theta}$? Similarly, for fixed $x<0<y$, how much more energy does the Wang minimizer need to weld $x$ to $y$ compared to the EMW curve? In Theorem \ref{Thm:EMWWang} we find that the asymptotic answers are the same,
\begin{align}\label{Lim:Asymptotics1}
\lim_{\theta \rightarrow \pi/2} \frac{I(\text{EMW $\gamma$ to $e^{i\theta}$})}{I(\text{Wang $\gamma$ to $e^{i\theta}$})} = \Big( \frac{4}{\pi} \Big)^2 = \lim_{-x/y \rightarrow 1} \frac{I(\text{Wang $\gamma$ welding $x$ to $y$})}{I(\text{EMW $\gamma$ welding $x$ to $y$})}.
\end{align}
This is not obviously the case, as we discuss in Remark \ref{Remark:Reciprocals}.
We might also wonder how close generic smooth curves $\gamma$ are to minimizing energy at small scales. For a small initial segment $\gamma[0,\delta]$, for instance, how does $I(\gamma[0,\delta])$ compare to the minimal energy to reach $\gamma(\delta)$? Similarly, if $u(\delta)<0<v(\delta)$ will be welded by a smooth upwards driver $\xi$ at time $\delta$, how close to energy-minimizing is $\xi$ on $[0,\delta]$? In Theorem \ref{Lemma:Inf98} we see that, as in \eqref{Lim:Asymptotics1}, both ratios are asymptotically the same,
\begin{align}\label{Lim:Asymptotics2}
\lim_{\delta \rightarrow 0^+} \frac{I(\gamma[0,\delta])}{I(\text{Wang minimizer to }\gamma(\delta))} = \frac{9}{8} = \lim_{\delta \rightarrow 0^+} \frac{I(\xi[0,\delta])}{I(\text{EMW minimizer for }u(\delta), v(\delta))},
\end{align}
provided $\dot{\lambda}(0), \dot{\xi}(0) \neq 0$ (i.e. both curves are not locally hyperbolic geodesic segments). Since this result translates to any time $t$, we see smooth curves which are using energy are never locally minimizing, with the factor of energy inefficiency constant.
\subsection{Methods}\label{Intro:Methods} Our paper is philosophically similar to \cite{KNK} in that we use symmetries of the family in question to obtain systems of ODE's, that, given sufficient patience, we can explicitly solve. The symmetries of each family also naturally yield the universality properties. We use tools from quasiconformal mappings to obtain compactness and hence existence of minimizers. In the case of the welding minimizers, we also need a recent result \cite{TVY} saying driver convergence implies welding convergence.
Two simple observations which we need for our deterministic proof strategy are ``even approach'' properties for finite-energy curves under the downwards and upwards Loewner flow. The downwards-flow version says that the argument of the image $G_t(\gamma(\tau))$ of the tip $\gamma(\tau)$ of a finite-energy curve under the centered downwards map $G_t$ approaches $\pi/2$ as $t \rightarrow \tau^-$. That is, the image of the tip tends to the imaginary axis. Compare, for instance, the blue tip in the left and center images of Figure \ref{Fig:LoewnerFlowB} (where one centers the middle diagram to have the image under $G_t$ instead of $g_t$). The upwards-flow version says that if $x_0 < 0 <y_0$ will be welded by a finite-energy driver $\xi$ at some time $\tau$, then their images $x_t, y_t$ under the centered upwards-flow maps $h_t(z)-\xi(t)$ satisfy $-\frac{x_y}{y_t} \rightarrow 1$ as $t \rightarrow \tau^-$. That is, the points to be welded ``evenly approach'' the origin. These properties enable us to see that both energy-minimization problems are purely local in nature, which yields ODE's that we can solve for the energy formulas. We show in Example \ref{Eg:NotEquiv} that these properties, perhaps surprisingly, are not equivalent.
For the asymptotic energy comparisons, we use expansions of the curve $\gamma$ in terms of $\dot{\lambda}(0)$ from \cite{LindTran}, and a new expansion \eqref{Eq:InfinitesimalWelding} of the welding in terms of $\dot{\xi}(0)$. These enable us to say that \eqref{Lim:Asymptotics2} holds whenever $\lambda, \xi \in C^{\frac{3}{2} + \epsilon}$.
\subsection{Outline of the paper}\label{Intro:Outline}
We proceed in \S\ref{Sec:Background} to recall relevant background information. In \S\ref{Sec:Wang} we state and prove our results for the Wang minimizers, culminating in Theorem \ref{Thm:Wang}. In \S\ref{Sec:EMW} we similarly prove properties of the EMW curves, with Theorem \ref{Thm:EMW} the main result. We close by proving the energy comparisons in \S\ref{Sec:Compare}, and connecting SLE$_0$ to the classical deterministic drivers in the appendix, \S\ref{Sec:Appendix}.
\subsection{Acknowledgements}
The author is grateful to Steffen Rohde for many illuminating discussions as well as for the idea to consider the welding minimizers and to deterministically re-derive Wang's formula \eqref{Eq:Energy1}. He also thanks Don Marshall and Yilin Wang for multiple instructive discussions, as well as Ellen Krusell for introducing him to SLE$_\kappa(\rho)$ and for the suggestion that the EMW drivers may be related to SLE$_0(-4,-4)$. He is grateful for feedback from Daniel Meyer and Joan Lind on an earlier draft.
This research was partially conducted while the author was at Mathematical Sciences Research Institute during the spring 2022 semester and is thus partially supported by the National Science Foundation under Grant No. DMS-1928930.
\section{Background and notation}\label{Sec:Background}
\subsection{The Loewner equation}\label{Sec:BackgroundLoewner}
We start with a very brief review of the Loewner equation. For other overviews, see, e.g., \cite{SLEGuide}, \cite{Lind4} or \cite[\S 2]{WangReverse}, and consulte \cite{Beliaev}, \cite{Kemppainen} or \cite{Lawler} for more thorough treatments with proofs.
A simple curve $\gamma$ from $0$ to $z$ in $\mathbb{H}$ is a continuous injective map $\gamma:[0,T] \rightarrow \mathbb{H}$ such that $\gamma(0) \in \mathbb{R}$, $\gamma(0,T] \in \mathbb{H}$ and $\gamma(T) = z$. (Note that we will often abuse notation with respect to $\gamma$ in two ways: first, by writing the image of an interval $I$ under $\gamma$ as $\gamma I$ instead of $\gamma(I)$, and secondly, by using $\gamma$ to denote its trace $\gamma([0,T])$.) For such a curve and fixed $0 < t \leq T$, $g_t: \mathbb{H}{\backslash} \gamma(0,t] \rightarrow \mathbb{H}$ is the unique conformal map satisfying
\begin{align}\label{Eq:LoewnerNormalize}
g_t(z) = z + \frac{\hcap(\gamma[0,t])}{z} + O\Big(\frac{1}{z^2}\Big), \qquad z \rightarrow \infty,
\end{align}
and $g_0(z):= z$. Here $\hcap(\gamma[0,t])$ is, by definition, the coefficient of $1/z$ in this expansion. All our curves $\gamma$ will be \emph{parametrized by capacity}, which means that
\begin{align*}
\hcap(\gamma[0,t]) =2t.
\end{align*}
When we refer to the ``capacity time'' or ``Loewner time'' of a curve segment $\gamma$, we will always mean the half-plane capacity of $\gamma$ is $2t$, as above.
Note that such a ``mapping down'' function $g_A$ as in \eqref{Eq:LoewnerNormalize} can be uniquely defined for a set $A$ whenever $\mathbb{H} {\backslash} A$ is a simply-connected domain, and the \emph{half-plane capacity} $\hcap(A)$ is similarly defined. We also recall that $\hcap$ is strictly monotone: if $A_1 \subsetneq A_2$ are such sets, then $\hcap(A_1) < \hcap(A_2)$ \cite[\S A.4]{SLEGuide}, and thus it is always possible to parametrize growing curves $\gamma$ by capacity.
It is known that, for fixed $z$, $g_t(z)$ satisfies \emph{Loewner's differential equation}
\begin{align}\label{Eq:BackgroundLoewnerEq}
\dot{g}_t(z) = \frac{2}{g_t(z) -\lambda(t)}, \qquad g_0(z) = z,
\end{align}
where $\lambda(t)$ is \emph{driving function} of $\gamma$, which is to say, the continuous, real-valued image of the tip $\gamma(t)$ by the (extension) of $g_t$. Each point $z \in \mathbb{H}$ has a supremal time $\tau_z$ such the flow of $z$ under \eqref{Eq:BackgroundLoewnerEq} is defined on $[0,\tau_z]$. We recover $\gamma(0,t]$ by taking the \emph{hull of the Loewner flow} $\{\, z \in \mathbb{H} \;: \; \tau_z \leq t\, \}$.
There are several useful variations of the Loewner flow. We write $G_t(z) := g_t(z) -\lambda(t)$ for the centered mapping-down function (where the tip $\gamma(t)$ always maps to zero), and define $f_t := g_t^{-1}$ and $F_t := G_t^{-1}$. We reverse the direction of the flow via the maps $h_t$ which satisfy
\begin{align}\label{Eq:LoewnerEqUp}
\dot{h}_t(z) = \frac{-2}{h_t(z) -\xi(t)}, \qquad h_0(z) = z.
\end{align}
If $\xi(t) = \lambda(T-t)$ is the reversal of $\lambda$, then it is easy to see that $h_t^{-1} \circ g_{T-t} \equiv g_T$; that is, $h_t = g_{T-t}\circ g_T^{-1}$ is the conformal map from $\mathbb{H}$ to $\mathbb{H}{\backslash} g_{T-t}(\gamma[T-t,T])$ satisfying $h_t(z) = z - 2t/z + O(1/z^2)$ as $z \rightarrow \infty$. We frequently use the reversed driver which is shifted to start at zero, $\xi(t) = \lambda(T-t) - \lambda(T)$.
If $\gamma$ is not simple or $\gamma(0,T] \cap \mathbb{R} \neq \emptyset$, then $g_t$ is defined as the map $\mathbb{H}{\backslash} \Fill(\gamma(0,t]) \rightarrow \mathbb{H}$ satisfying the same normalization \eqref{Eq:LoewnerNormalize}, where $\mathbb{H}{\backslash} \Fill(\gamma(0,T])$ is the unbounded connected component of the complement of $\gamma(0,T]$.
The simplest example of a driving function $\lambda(t)$ is for the vertical line segment $\gamma_y = [0,iy]$, $y>0$, where a map ${\mathbb H} {\backslash} \gamma_y \rightarrow {\mathbb H}$ which is the identity at infinity to first order is
\begin{align*}
\sqrt{z^2+y^2} = z+\frac{y^2/2}{z} + O(z^{-3}), \qquad z \rightarrow \infty.
\end{align*}
Hence the capacity parametrization for the imaginary axis is $\gamma(t) = 2i\sqrt{t}$, and the driving function is $\lambda(t) \equiv 0$, the image of $\gamma(t)$. For further examples of driving functions, see \cite{KNK}, \cite[\S4.1]{Lawler}, \cite{LoewnerCurvature}, or Theorems \ref{Thm:Wang}($\ref{Thm:WangDriver}$) and \ref{Thm:EMW}($\ref{Thm:EMWDriver}$) below.
If we replace $\gamma$ by a scaled copy $r\gamma$ of itself, $r>0$, then to maintain capacity parametrization $r\gamma$ must be parametrized as $t \mapsto r\gamma(t/r^2)$, as one can see from \eqref{Eq:LoewnerNormalize}. Thus the driving function of $r\gamma$ is
\begin{align}\label{Eq:ScaledDriver}
r\lambda(t/r^2).
\end{align}
\subsection{The Loewner energy}\label{Sec:BackgroundLoewnerEnergy}
The Loewner energy of a curve $\gamma$ is the Dirichlet energy of its driving function $\lambda$, which we formally define through the following difference quotient. Let $\Pi[0,T]$ be the collection of all partitions of $[0,T]$.
\begin{definition}
The \emph{Loewner energy} $I(\gamma)$ of a curve $\gamma$ on $[0,T]$ (or $[0,T)$ if $T = \infty$) with downwards driving function $\lambda$ is
\begin{align}\label{Def:LoewnerEnergy}
I(\gamma) := \sup_{\mathcal{P} \in \Pi[0,T]} \sum_{j=1}^n \frac{(\lambda(t_j)-\lambda(t_{j-1}))^2}{2(t_j-t_{j-1})}.
\end{align}
\end{definition}
\noindent We may alternatively write $I(\lambda)$ for $I(\gamma)$, or even $I(\xi)$, as it is evident that the supremum in \eqref{Def:LoewnerEnergy} is not changed if we replace $\lambda$ with its reversal $\xi(t) = \lambda(T-t) -\lambda(T)$. We write $I_T$ if we need to emphasize the interval $[0,T]$ under consideration.
The factor of 2 in the denominator in \eqref{Def:LoewnerEnergy} is a normalization choice by Wang \cite{WangReverse} in order to have the Loewner energy be the good-rate function for SLE$_\kappa$ as $\kappa \rightarrow 0^+$ \cite{PeltWang}. The supremum in \eqref{Def:LoewnerEnergy} is because the sum is monotonic in the partition, and from analysis we have that a driver $\lambda$ with $I_T(\lambda)<\infty$ belongs to the \emph{Dirichlet space} on $[0,T]$, which is to say, $\lambda$ is absolutely continuous, $\dot{\lambda}\in L^2([0,T])$ and
\begin{align}\label{Eq:LoewnerEnergy}
I_T(\lambda) = \frac{1}{2}\int_0^T \dot{\lambda}(t)^2dt.
\end{align}
See, for instance, \cite[\S 1.4]{Peres}. By absolute continuity, $I(\lambda) = 0$ if and only if $\lambda \equiv 0$, and thus we may also view the Loewner energy as a measurement of the deviation of the curve from a hyperbolic geodesic (i.e., the curve generated by the zero driver).
The Loewner energy enjoys a number of properties which will be of use to us, including:
\begin{enumerate}[$(i)$]
\item Conformal and anti-conformal invariance: if $r >0$ and $x \in \mathbb{R}$, then $I(r\gamma+x) = I(\gamma)$, as one readily sees from \eqref{Eq:ScaledDriver}. If $\gamma$ is a curve from 0 to $\infty$, then its image $-1/\gamma$ under the automorphism $z \mapsto -1/z$ of $\mathbb{H}$ also has the same energy, $I(\gamma) = I(-1/\gamma)$ \cite{WangReverse}. That is, $I$ is invariant under $PSL(2,\mathbb{R})$, the automorphism group of $\mathbb{H}$.
We also have $I(\gamma) = I(-\overline{\gamma})$, where $-\overline{\gamma}$ is the reflection of $\gamma$ across the imaginary axis, since if $\gamma$ is driven by $\lambda$, then $-\overline{\gamma}$ is driven by $-\lambda$. While trivial, this observation is still useful, in that by symmetry it allows us to only consider minimizers with argument tip $\theta \in (0,\pi/2]$, for instance.
\item $I(\cdot)$ is lower semi-continuous with respect to the sup norm on drivers: If $\lambda_n \rightarrow \lambda$ uniformly on $[0,T]$, then
\begin{align}\label{Ineq:LoewnerEnergyLSC}
\liminf_{n \rightarrow \infty} I_T(\lambda_n) \geq I_T(\lambda),
\end{align}
as follows from the difference quotient expression \eqref{Def:LoewnerEnergy} (see \cite[\S 2.2]{WangReverse}).
\item\label{LE:Compactness} The collection of drivers on $[0,T]$ with energy bounded by $C<\infty$ is compact. Indeed, for any such driver $\lambda$,
\begin{align}\label{Eq:LoewnerHolder}
|\lambda(t_2) - \lambda(t_1)| \leq \int_{t_1}^{t_2} |\dot{\lambda}(t)|dt \leq \sqrt{2C} \sqrt{t_2-t_1}
\end{align}
by the Cauchy-Schwarz inequality. In particular, the family is bounded and equicontinuous on $[0,T]$, and so by Arzela-Ascoli is precompact in the uniform norm on $[0,T]$ (recall another name for the embedding $W^{1,2}([0,T]) \hookrightarrow C^{1/2}([0,T])$ is Morrey's inequality \cite[\S4.5.3]{FineProp}). Lower semi-continuity \eqref{Ineq:LoewnerEnergyLSC} then yields compactness.
\item It follows that finite-energy drivers $\lambda$ generate curves $\gamma$ via the Loewner equation that are quasi-arcs that do not meet $\mathbb{R}$ tangentially. Indeed, from \eqref{Eq:LoewnerHolder} we see that the H\"{o}lder -1/2 norm of $\lambda$ is locally small and so by \cite[Proof of Thm. 4.1]{LMR}, $\lambda$ generates such quasi-arcs on small scales, and thus also on finite intervals. Wang then extended this argument to include infinite time intervals \cite[Prop. 2.1]{WangReverse}.
\item\label{Lemma:CurvesCompact} As quasi-arcs are compact, one can upgrade the compactness of drivers of bounded energy to compactness of \emph{curves}. That is, for any $M<\infty$ and $T<\infty$, the collection
\begin{align*}
\{ \gamma:[0,T] \rightarrow \mathbb{H}\cup\{0\} \, : \, \gamma(0)=0, \gamma((0,T]) \subset \mathbb{H}, I_T(\gamma)\leq M \}
\end{align*}
of capacity-parametrized curves is compact in $C([0,T])$, see \cite[Lemma 2.7]{PeltWang}.
\end{enumerate}
Much more could be said about the Loewner energy, and we close with several brief and incomplete remarks. Rohde and Wang \cite{RohdeWang} generalized it to loops via a limiting procedure, and this appears to be the most natural setting. Indeed, Wang subsequently showed \cite{WangEquiv} that the loop Loewner energy can be explicitly given by the square of the $L^2$-norm of the pre-Schwarzian derivative of the Riemann maps to other side of the loop. That description led to Bishop's ``square-summable curvature'' characterizations of finite-energy loops in \cite{Bishop}, and also to connections to Teichm\"{u}ller theory and geometry, as Takhtajan and Teo had earlier shown that the same expression involving the conformal maps is the K\"{a}hler potential for the Weil-Petersson metric on universal Teichm\"{u}ller space \cite{TT}. The finiteness of that expression for the energy also characterizes the Weil-Petersson quasicircles $T_0(1)$ within the universal Teichm\"{u}ller space $T(1)$ \cite{TT}. That is, finite-energy loops are precisely the closure of smooth loops in the Weil-Petersson metric. See \cite{WangEquiv} for details.
Initial interest in studying the energy was stochastic in nature; Wang and Peltola showed it is the large-deviations good-rate function for SLE$_\kappa$ as $\kappa \rightarrow 0^+$ \cite{PeltWang,WangReverse}, and Tran and Yuan \cite{Yizheng} also showed the closure of finite-energy curves is the topological support of SLE$_\kappa$. The Loewner energy, in other words, occupies a fascinating crossroads between probability theory, univalent function theory, Teichm\"{u}ller theory, geometric measure theory, and hyperbolic geometry.\footnote{See also \cite{Krusell, PeltWang} for generalizations to ensembles of multiple curves, and \cite{Krusell} for a generalization which, informally, is to single curves $\gamma$ ``conditioned to pass through a given point'' $z_0 \in \mathbb{H}$.}
\subsection{SLE$_\kappa(\rho_1,\ldots, \rho_n)$ and its reversal.}\label{Sec:BackgroundSLE}
We recall that (chordal, downwards) \emph{SLE with forcing SLE$_\kappa(\rho_1,\ldots, \rho_n)$ starting from $(\lambda_0, U^1_0, \ldots, U^n_0)$} is the Loewner flow generated by driver $\lambda_t$ whose motion is defined by Brownian motion and interactions from particles $U^1_t, \ldots, U^n_t \in \overline{\mathbb{H}}$, the closure of $\mathbb{H}$, via the system of SDE's
\begin{align}\label{Eq:SLErhoSDE}
d\lambda_t = \sqrt{\kappa}dB_t - \sum_{j=1}^n \frac{\rho_j}{2}\Real \frac{2}{U^j_t - \lambda_t}dt, \qquad dU^j_t = \frac{2}{U^j_t - \lambda_t} dt, \qquad j=1,\ldots, n,
\end{align}
with initial conditions given by the $(n+1)$-tuple $(\lambda_0, U^1_0,\ldots, U^n_0)$. Here $B_t$ is a standard Brownian motion, and the process is defined until the first time $\tau$ such that, for some $j$,
\begin{align*}
\inf_{0 \leq t <\tau} |\lambda_t - U^j_t| = 0.
\end{align*}
So SLE with forcing is just normal SLE but where the evolution of $\lambda_t$ is influenced by the location of the \emph{force points} $U^1_t, \ldots, U^n_t$. The connection to energy-minimizers is when $\kappa =0$, in which case this becomes a deterministic system of ODE's.
For our purposes it will be convenient to occasionally reverse the direction of the flow, and we define \emph{upwards SLE$_\kappa(\rho_1,\ldots, \rho_n)$ starting from $(\xi_0, V^1_0, \ldots, V^n_0)$} to be the process given by flow \eqref{Eq:LoewnerEqUp} where $\xi_t$ and the force points $V^1_t, \ldots, V^n_t$ satisfy
\begin{align*}
d\xi_t = \sqrt{\kappa}dB_t + \sum_{j=1}^n -\frac{\rho_j}{2}\Real \frac{-2}{V^j_t - \lambda_t}dt, \qquad dV^j_t = \frac{-2}{V^j_t - \lambda_t} dt, \qquad j=1,\ldots, n,
\end{align*}
with initial conditions $(\xi_0, V^1_0, \ldots, V^n_0) \in \mathbb{R} \times \overline{\mathbb{H}}^{n}$. If $(\lambda_t, U^1_t, \ldots, U^n_t)$ is a sample of downwards SLE$_\kappa(\rho_1,\ldots, \rho_n)$ starting from $(0, U^1_0,\ldots, U^n_0)$ on $[0,T]$, then the reversed, centered process defined by
\begin{align*}
\xi_t := \lambda_{T-t} - \lambda_T, \qquad V^j_t := U^j_{T-t} - \lambda_T, \qquad j=1,\ldots,n,
\end{align*}
has the distribution of upwards SLE$_\kappa(\rho_1,\ldots, \rho_n)$ starting from $(0, U^1_T-\lambda_T, \ldots, U^n_T-\lambda_T)$. In this paper we only consider deterministic $\kappa=0$ processes, and the upshot is that we can exchange the upwards and downwards points of view when convenient by reversing the direction of the flow and keeping track of the location of the forcing points.
\subsection{Quasiconformal mappings}
We recall several properties of two-dimensional quasiconformal mappings. For proofs and more details, see \cite{Astala}, \cite{LehtoUFT} or \cite{LehtoQC}. Let $\Omega_1, \Omega_2$ be domains in the Riemann sphere $\ensuremath{\hat{\mathbb{C}}}$. A \emph{$K$-quasiconformal map} $f: \Omega_1 \rightarrow \Omega_2$ is an orientation-preserving homeomorphism which is absolutely continuous on a.e. line parallel to the axes and differentiable at Lebesgue-almost every $z \in \Omega_1$, and whose ``complex directional derivatives''
\begin{align*}
\partial_\alpha f (z) := \lim_{r \rightarrow 0} \frac{f(z + re^{i \alpha})-f(z)}{re^{i\alpha}}
\end{align*}
satisfy
\begin{align}\label{Eq:QCDirDerivBounds}
\max_\alpha |\partial_\alpha f(z)| \leq K \min_\alpha |\partial_\alpha f(z)|
\end{align}
at points $z$ of differentiability. More succinctly, $f$ is $K$-quasiconformal if $f \in W^{1,2}_{\text{loc}}(\Omega_1)$ with derivatives satisfying \eqref{Eq:QCDirDerivBounds} almost everywhere. Conformal maps are $K$-quasiconformal with $K=1$.
Our interest in quasiconformal maps stems from the fact that finite-energy arcs $\gamma \subset \mathbb{H} \cup \{0\}$ are images $f([0,i])$ of the imaginary axis segment $[0,i]$ under a $K=K(I(\gamma))$-quasiconformal map $f:\mathbb{H} \rightarrow \mathbb{H}$ which fixes $0$ and $\infty$ \cite[Prop. 2.1]{WangReverse}. Furthermore, families of quasiconformal maps have nice compactness properties, as expressed in the following two propositions. We will use these in tandem with the lower semi-continuity of energy \eqref{Ineq:LoewnerEnergyLSC} to obtain Loewner-energy minimizers.
\begin{proposition}\cite[Thm. 2.1]{LehtoUFT}
A family $F$ of $K$-quasiconformal mappings of $\Omega_1 \subset \ensuremath{\hat{\mathbb{C}}}$ is normal in the spherical metric if there exists three distinct points $z_1,z_2,z_3 \in \Omega_1$ and $\epsilon >0$ such that for any $f \in F$, $d_{\hat{{\mathbb C}}}(f(z_j),f(z_k)) > \epsilon$, whenever $j \neq k$, $j,k=1,2,3$.
\end{proposition}
\noindent Furthermore, the subsequential locally-uniform limits are also either $K$-quasiconformal or constant, a generalization of the Hurwitz theorem for conformal mapping.
\begin{proposition}\cite[Thm. 2.2, 2.3]{LehtoUFT}
Let $f_n: \Omega_1 \rightarrow \Omega_2$ be a sequence of $K$-quasiconformal maps which converges locally uniformly to $f$. Then $f$ is either $K$-quasiconformal or $f$ maps all of $\Omega_1$ to a single boundary point of $\Omega_2$.
\end{proposition}
\section{The Wang minimizers}\label{Sec:Wang}
We begin by considering the minimization question ($\ref{Q:HMinimizer}$): what is the infimal energy needed for a curve to go from 0 to $r_0e^{i\theta_0}$ in $\mathbb{H}$, and what is the nature of $\gamma$ which achieve the minimum, if this is possible? Our deterministic answer is found below in Theorem \ref{Thm:Wang}, but is prefaced by the following two lemmas. The latter respectively state that minimizers exist, and that, under the centered downwards Loewner flow, the tip of a finite-energy curve tends towards the imaginary axis.
\begin{lemma}\label{Lemma:AngleEnergyMinimizerExists}
For any $z \in \mathbb{H}$, there exists a simple curve $\gamma_z$ from 0 to $z$ in $\mathbb{H}$ such that
\begin{align}\label{Eq:AngleEnergyInf}
I(\gamma_z) = \inf_{\gamma \in \Gamma_z} I(\gamma),
\end{align}
where $\Gamma_z$ is the collection of all curves in $\mathbb{H}$ from 0 to $z$.
\end{lemma}
The argument is a standard application of the compactness of quasiconformal mappings and the lower semi-continuity of energy \eqref{Ineq:LoewnerEnergyLSC}. We will show below in Theorem \ref{Thm:Wang}(i) that the minimizer is unique.
\begin{proof}
By scale invariance of energy we may assume $z = e^{i\theta}$. In Lemma \ref{Lemma:CircularArc} below we will see that the orthogonal circular arc segment from 0 to $e^{i\theta}$ has energy \begin{align}\label{Eq:CAEnergy}
-9\log(\sin(\theta)),
\end{align}
and so the infimum in \eqref{Eq:AngleEnergyInf} is not $+\infty$ and it suffices to consider those Jordan arcs in $\Gamma_z$ with energy bounded by \eqref{Eq:CAEnergy}. By \cite[Prop. 2.1]{WangReverse}, each such $\gamma$ is a $K$-quasislit halfplane for some fixed $K = K(\theta)$, and so there exists a $K$-quasiconformal self-map $q$ of $\mathbb{H}$ fixing $0$ and $\infty$ with $q([0,i]) = \gamma$.
Let $\{\gamma_n\}$ be a sequence such that $I(\gamma_n)$ tends to the infimum in \eqref{Eq:AngleEnergyInf}, and let $\{q_n\}$ be corresponding $K$-quasiconformal maps. The family $\{q_n\}$ is normal in the spherical metric, and any limiting function is either a constant in $\partial_{\ensuremath{\hat{\mathbb{C}}}}{\mathbb H}$ or a $K$-quasiconformal self-map of $\mathbb{H}$ \cite[\S2.2f]{LehtoUFT}. The former cannot happen because all $q_n$ map $i$ to $e^{i\theta}$, and hence, by moving to a subsequence, which we relabel as $q_n$ again, we have a locally uniform limit $q_n \rightarrow q$ in $\mathbb{H}$ and a limiting curve $\gamma:=q([0,i])$. In fact, the convergence is uniform on $[0,i]$ by Schwarz reflection of the $q_n$ and $q$. We argue that the convergence is also uniform in the capacity parametrizations of $\gamma_n$ and $\gamma$.
All the curves $\gamma, \gamma_n$ are uniformly bounded in $\mathbb{H}$, and so they are also bounded in half-plane capacity. By extending each $\tilde{\gamma} \in \{\gamma, \gamma_n\}$ by an appropriate-length segment of the hyperbolic geodesic from $e^{i \theta}$ to $\infty$ in ${\mathbb H} \backslash \tilde{\gamma}$, we can consider all the $\tilde{\gamma}$ to be defined on the same interval $[0,T]$ of capacity time. By the proof of Theorem 4.1 in \cite{LMR}, the modulus of continuity of a $K$-quasiarc in its capacity parametrization depends only upon $K$, and hence $\{\gamma, \gamma_n\}$ is a bounded and equicontinuous family.\footnote{We have changed the curves with the hyperbolic geodesic segments, but this does not change their Loewner energy, and so the augmented curves are still all $K$-quasiarcs.} Hence by Arzela-Ascoli we move to a further subsequence, if necessary, and obtain a uniform capacity-parametrization limit $\gamma_n \rightarrow \gamma'$ on $[0,T]$. But clearly $\gamma'$ must be $\gamma$: since $\mathbb{H}{\backslash} q_n([0,si]) \rightarrow \mathbb{H} {\backslash} q([0,si])$ in the Carath\'{e}odory sense for any $0 \leq s \leq 1$, $\hcap(q_n([0,si])) \rightarrow \hcap(q([0,si]))$, and it readily follows from the uniform convergence $q_n \rightarrow q$ that the two limits are identical.
Since $\gamma_n \rightarrow \gamma$ uniformly and all the curves are simple, we also have the uniform convergence $\lambda_n \rightarrow \lambda$ of their associated drivers on $[0,T]$ by \cite[Thm. 1.8]{YuanTop}. The lower semicontinuity of energy then yields
\begin{align*}
I(\gamma) \leq \liminf_{n \rightarrow \infty}I(\gamma_n) = \inf_{\gamma \in \Gamma_z} I(\gamma),
\end{align*}
and as $e^{i\theta} \in \gamma$, we have that $\gamma$ is a minimizer.
\end{proof}
\begin{lemma}[``Even angle approach'']\label{Lemma:WangEvenApproach}
Suppose $\lambda$ is the driver for the simple curve $\gamma:[0,T] \rightarrow {\mathbb H} \cup \{0\}$ with $I(\gamma) < \infty$. Then under the centered downward Loewner flow generated by $\lambda$, the image $z_t:=G_t(\gamma(T))$ of the tip of the curve satisfies
\begin{align}\label{Lim:FiniteEnergyAngle}
\lim_{t \rightarrow T^-} \arg(z_t) = \frac{\pi}{2}.
\end{align}
\end{lemma}
For example, note the increase of the argument of $g_t(\gamma(T)) - \lambda_t$ towards $\pi/2$ from the left to the middle image in Figure \ref{Fig:LoewnerFlow}. Of course, if $\gamma$ in its entirety has finite energy, then so does $\gamma([0,\tau])$ for any $\tau <T$, and so \eqref{Lim:FiniteEnergyAngle} also holds as $t \rightarrow \tau^-$ for any point $\gamma(\tau)$ on $\gamma$.
This lemma is very similar to \cite[Lemma B]{PeltWang}, but the difference here is that we do not assume the minimal energy formula for curves through a point $e^{i\theta} \in \mathbb{H}$. Indeed, we will use Lemma \ref{Lemma:WangEvenApproach} in our deterministic proof of this formula. While the lemma could be proven through direct analysis of the Loewner equation, it is also a simple consequence of Lemma \ref{Lemma:AngleEnergyMinimizerExists}. The author is grateful to Don Marshall for suggesting the following elementary proof, which replaced an earlier more complicated argument.
\begin{proof}
The function
\begin{align*}
m(\theta) := \min_{\gamma \in \Gamma_\theta} I(\gamma),
\end{align*}
where $\Gamma_\theta$ is the collection of all curves from 0 to $e^{i \theta}$ in $\mathbb{H}$, is well defined by the previous lemma. We first show that $m$ is non-increasing on $(0,\pi/2]$, and non-decreasing on $[\pi/2, \pi)$, and by symmetry it suffices to consider the former case. Indeed, fix $0 < \theta < \pi/2$ and a curve $\gamma_\theta$ such that $I(\gamma_\theta) = m(\theta)$. Note that if we start with $\gamma_\theta$ and flow \emph{up} with $\xi_t \equiv 0$, the argument of the image of the tip of $\gamma_\theta$ increases to $\frac{\pi}{2}^-$ as $t \rightarrow \infty$, as one can see from the explicit conformal map $h_t(z) = \sqrt{z^2-4t}$. Since the zero driver does not add energy, we thus see $m(\theta') \leq I(h_t(\gamma_\theta)) = m(\theta)$ for $\theta < \theta' < \pi/2$, and so $m$ is non-increasing on $(0,\pi/2)$.
We also see that $m(\theta) > 0$ if $\theta \neq \pi/2$, for if $m(\theta)$ vanished, then any minimizer $\gamma_{\theta}$ would be driven by the zero driver, which corresponds to the imaginary axis, not a curve through $e^{i \theta}$. Thus $m$ is non-increasing on $(0,\pi/2]$.
Now, if there are times $t_n \rightarrow T^-$ such that $|\arg(z_{t_n}) - \pi/2| > \epsilon$, then $\lambda$ must expel at least some $m(\pi/2-\epsilon) = m(\pi/2+\epsilon)>0$ amount of energy on each $[t_n,T]$ by the above, contradicting $\lim_{t \rightarrow T^-} \int_t^T \dot{\lambda}^2(s)ds =0$.
\end{proof}
Our main results about the Wang minimizers are in the following theorem. We recall that the statement of $(i)$ is from Wang and is included because we provide an alternative, deterministic proof.
\begin{theorem}\label{Thm:Wang}
Let $\theta \in (0,\pi)$.
\begin{enumerate}[$(i)$]
\item \label{Thm:WangEnergy}\cite[Proposition 3.1]{WangReverse} There exists a unique $\gamma_\theta$ from 0 to $e^{i\theta}$ in $\mathbb{H}$ which minimizes the Loewner energy among all such curves. Furthermore,
\begin{align}\label{Eq:WangEnergy}
I(\gamma_\theta) = -8\log(\sin(\theta)).
\end{align}
\item\label{Thm:WangDriver} The driving function $\lambda$ for $\gamma_\theta$ is monotonic and $C^\infty([0,\tau_\theta))$. For $0<\theta <\pi/2$, the upwards-flow driver $\xi_\theta(t) = \lambda(\tau-t) - \lambda(\tau)$ is explicitly
\begin{align}\label{Eq:WangDriver}
\xi_\theta(t) &= - \frac{4\sqrt{2}}{\sqrt{3}}\left( \sqrt[3]{\sqrt{\frac{\sin^6(\theta)}{36\cos^2(\theta)}+t^2}+t} -\sqrt[3]{\sqrt{\frac{\sin^6(\theta)}{36\cos^2(\theta)}+t^2}-t} \right)^{3/2}
\end{align}
for $0 \leq t \leq \frac{1}{6}(1 - \frac{1}{2}\cos(2\theta)) =:\tau = \tau_\theta$. In particular, $\xi(0)=0$ and $\xi(\tau) = -\frac{4}{3}\cos(\theta)$. For $\pi/2 < \theta < \pi$, $\xi_\theta = -\xi_{\pi-\theta}$.
For any $0 < \theta < \pi$, the conformal welding $\varphi_\theta$ corresponding to the Loewner-flow normalization is
\begin{align}\label{Eq:WangWeld}
\varphi_\theta(x) = \frac{-x}{\sqrt{1 + \pi \frac{\cos(\theta)}{\sin^3(\theta)}x^2}},
\end{align}
where $\varphi_\theta: [x_\theta,0] \rightarrow [0,y_\theta]$, with \begin{align}\label{Eq:WangWeldInterval}
x_\theta = \frac{-\sqrt{\sin^3(\theta)}}{\sqrt{\sin(\theta) - \theta \cos(\theta)}} \qquad \text{ and } \qquad y_\theta = \frac{\sqrt{\sin^3(\theta)}}{\sqrt{\sin(\theta) + (\pi-\theta)\cos(\theta)}}.
\end{align}
\item \label{Thm:WangUniversal} (Universality of driver and welding) For $\theta \neq \pi/2$, the driver \eqref{Eq:WangDriver} and welding \eqref{Eq:WangWeld} are universal, in the sense of generating all Wang minimizers up to scaling, translation and reflection in the imaginary axis.
More precisely, for every $t >0$, $\xi_\theta |_{[0,t]}$ is a scaled and translated Wang minimizer $r_t\gamma_{\alpha(t)} + \xi_\theta(t)$, where the range of $t \mapsto \alpha(t)$ is the connected subinterval of $(0,\pi){\backslash} \{\pi/2\}$ containing $\theta$. Explicitly, the curve generated by $\xi_\theta$ on the interval
\begin{align*}
[0,t_\alpha] := \Big[0, \frac{\sin^3(\theta)\cos(\alpha)}{6\cos(\theta)\sin^3(\alpha)}\Big( 1 - \frac{1}{2}\cos(2\alpha)\Big) \Big]
\end{align*}
is (a translation of) the scaled Wang minimizer $r \gamma_\alpha$, where
\begin{align}\label{Eq:WangUniversalr}
r = \sqrt{\frac{\sin^3(\theta)}{\cos(\theta)} \cdot \frac{\cos(\alpha)}{\sin^3(\alpha)}}.
\end{align}
Similarly, $\varphi_\theta|_{[u,0]}$ generates a scaled Wang minimizer $r_u \gamma_{\alpha(u)}$ for every $u <0$, where the range of $u \mapsto \alpha(u)$ is the connected subinterval of $(0,\pi){\backslash} \{\pi/2\}$ containing $\theta$. Explicitly, the curve welded by $\varphi_\theta$ by the upwards centered Loewner map on the interval
\begin{align*}
[u_\alpha,0] := \Big[ -\sqrt{\frac{\sin^3(\theta)}{\cos(\theta)} \cdot \frac{\cos(\alpha)}{\sin(\alpha)-\alpha \cos(\alpha)}},0 \Big]
\end{align*}
is the scaled Wang minimizer $r \gamma_\alpha$, with $r$ likewise given by \eqref{Eq:WangUniversalr}.
\item\label{Thm:Wang0+} (Limiting curve) As $\theta \rightarrow 0^+$, $\gamma_\theta$ converges pointwise to $\gamma_0$ for $0 \leq t \leq \tau_0 = 1/12$, where $\gamma_0$ is the curve with reversed driver
\begin{align*}
\xi_0(t) = -\frac{8}{\sqrt{3}}\sqrt{t}
\end{align*}
on $[0,\tau_0]$. Furthermore, the Loewner mapping-down functions $g_\theta:\mathbb{H}{\backslash} \gamma_\theta \rightarrow \mathbb{H}$ converge locally uniformly to $g_0: \mathbb{H} {\backslash} \Fill(\gamma_0) \rightarrow \mathbb{H}$ on $\mathbb{H} {\backslash} \Fill(\gamma_0)$. Also, $\gamma_0$ and its reflection $1/\overline{\gamma}_0 =:\gamma_0^*$ over $\partial \mathbb{D}$ form a boundary geodesic pair in $(\mathbb{H}; 0, \infty, 1)$ and are a subset of the algebraic variety
\begin{align}\label{Eq:Variety}
(4-3x)y^2 = 3x(x-1)^2.
\end{align}
In particular, $\gamma_0(t)$ meets $\mathbb{R}$ at $x=1$ when $t=1/12$, and the angle between $\mathbb{R}$ and $\gamma$ with respect to the bounded component of $\mathbb{H}{\backslash} \gamma_0$ is $\pi/3$.
\end{enumerate}
\end{theorem}
\noindent See Figure \ref{Fig:Gamma0} for an illustration of $\gamma_0$ and the geodesic pairing property of part $(\ref{Thm:Wang0+})$ of the theorem.
\begin{remark}\label{Remark:RemainingWang} We preface the proof with several comments.
\begin{figure}
\centering
\includegraphics[scale=0.1]{Universal_weld.jpg}
\caption{\small The welding $\varphi_\theta$ in \eqref{Eq:WangWeld} is the welding for the universal curve of \cite{MRW} in the sense of providing the welding for the bottom arrow of this diagram, for any $\alpha \in (0,\pi/2)$. Here $\alpha' = \frac{\pi}{2} - \alpha$.}
\label{Fig:UniversalCurveWang}
\end{figure}
\begin{enumerate}[$(a)$]
\item\label{Remark:RestOfWang} When we map down an initial portion $\gamma_\theta([0,t])$ of a Wang minimizer, the remainder $G_t(\gamma_\theta([t,\tau]))$ is the Wang minimizer for the angle of $G_t(e^{i\theta})$, since if not, we could replace $G_t(\gamma_\theta([t,\tau]))$ with the minimizer and lower the energy. In contrast, the initial portion $\gamma_\theta([0,t])$ mapped down is \emph{not} also a Wang minimizer: we see below in \eqref{Eq:WangSystem} that if $\theta \neq 0$, $\dot{\lambda}_\theta \neq 0$, and so part $(iv)$ in particular applies to the Wang minimizers themselves. Hence the symmetry of the Wang family is with respect to the ``top,'' or the portion remaining after mapping down, not with respect to the ``base'' or the portion mapped down, and it is therefore natural to express the driving function \eqref{Eq:WangDriver} in terms of the upwards flow $\xi$ rather than the downwards flow $\lambda$. That is, the curve $\eta[0,t]$ generated by the upwards flow under $\xi$ on $[0,t]$, $t < \tau$, is always a Wang minimizer.
This is the opposite of what we will see for the EMW family, where what we map down is an EMW curve, but what remains is not. See Remark \ref{Remark:DualFam}.
\item Note that all quantities inside roots in \eqref{Eq:WangDriver} are non-negative, and so there is no ambiguity about branch cuts.
\item\label{Remark:WangUniversalCurve} In the remark following Corollary 2.3 in \cite{MRW} a ``universal curve'' $\gamma$ in $\mathbb{D}$ was noted for the Wang minimizer family, which is similar in spirit to the universality properties of the welding and driver in part $(\ref{Thm:WangUniversal})$ of the theorem. The connection is that \eqref{Eq:WangWeld} welds the universal curve of \cite{MRW} in the following sense (the reader is encouraged to refer to Figure \ref{Fig:UniversalCurveWang} while reading the following). Let $\gamma_\alpha^*$ be the hyperbolic geodesic in $\mathbb{H} {\backslash} \gamma_\alpha$ from $e^{i\alpha}$ to $\infty$ and first note that $\gamma_\alpha \cup \gamma_\alpha^*$ is a geodesic pair in $(\mathbb{H}; 0, \infty, e^{i\alpha})$. Indeed, applying $z \mapsto -1/\bar{z}$ to $\gamma_\alpha \cup \gamma_\alpha^*$ gives a curve from 0 through $e^{i \alpha}$ with the same minimizing energy by invariance of energy under reversal and reflection \cite{WangReverse}. Hence by uniqueness of minimizers, $-1/\gamma_\alpha^* = \gamma_\alpha$, which shows $\gamma_\alpha$ is the hyperbolic geodesic from 0 to $e^{i \alpha}$ in $\mathbb{H} {\backslash} \gamma_\alpha^*$.
The M\"{o}bius map $M_\alpha:\mathbb{H} \rightarrow \mathbb{D}$ given by
\begin{align}\label{Eq:GeodesicMobius}
M_\alpha(z) = ie^{-i\alpha}\frac{z-e^{i\alpha}}{z-e^{-i\alpha}}
\end{align}
takes $\gamma_\alpha \cup \gamma_\alpha^*$ to a geodesic pair $\gamma_{1,\alpha'} \cup \gamma_{2,\alpha'}$ in $(\mathbb{D}; e^{i\alpha'}, -e^{-i \alpha'}, 0)$, where $\alpha' = \frac{\pi}{2}-\alpha$. The universal curve $\gamma$ is a Jordan curve $\gamma$ in $\mathbb{D} \cup \{i\}$ that has the property that for each $\alpha'$, there exists $y_{\alpha'}$ such that the conformal map $f_{\alpha'} : \mathbb{D} {\backslash} (\gamma \cap \{\Imag(z)\geq y_{\alpha'}\}) \rightarrow \mathbb{D}$ with $f_{\alpha'}(0) = 0$, $f_{\alpha'}'(0)>0$ satisfies $f_{\alpha'}(\gamma \cap \{ \Imag(z) < y_{\alpha'} \}) = \gamma_{1,\alpha'} \cup \gamma_{2,\alpha'}$. Now, $\varphi_\theta|_{[u_\alpha,0]}$ welds $r_u \gamma_\alpha$, the minimizer $\gamma_\alpha$ scaled by some $r_u >0$, whose image under $f_{\alpha'}^{-1} \circ M_\alpha \circ z/r_u$ is $\gamma \cap \{\Imag(z) < y_{\alpha'}\} \cap \{\Real(z) \leq 0\}$, and in this sense we say $\varphi$ is the welding for $\gamma$.
\item We see in part $(\ref{Thm:Wang0+})$ that the curve $\gamma_0 \cup \gamma_0^*$ obtained in the $\theta \rightarrow 0^+$ limit is algebraic. It is natural to ask if $\Gamma_\theta := \gamma_\theta \cup \gamma_\theta^*$ is also algebraic for other values of $\theta$. The conjectural answer is that $\Gamma_\theta$ is algebraic if and only if $\theta \in \{0, \pi/2, \pi\}$. Indeed, the downwards driver for $\Gamma_\theta$ is $\lambda_\theta = \xi_\theta(\tau_\theta-t) - \xi_\theta(\tau_\theta)$ on $[0,\tau_\theta]$, followed by the constant driver $\lambda_\theta(\tau_\theta)$ thereafter. The expansion of $\xi_\theta$ at $t=0^+$, corresponding to the intersection point between $\gamma_\theta$ and $\gamma_\theta^*$, is
\begin{align}\label{Eq:WangDriverExpansion}
\xi(t) = -\frac{32}{3} \frac{\cos(\theta)}{\sin^3(\theta)} t^{3/2} + O(t^{7/2}),
\end{align}
and so the driver has global $C^{3/2}$-regularity. If $\Gamma_\theta$ were algebraic (and non-singular at $e^{i\theta}$, which is natural), then it would be smooth at $e^{i\theta}$, say at least $C^{2+\epsilon}$. By the correspondence between curve regularity and driver regularity \cite{RohdeWang}, this would mean the driver would be globally $C^{3/2+\epsilon}$, a contradiction.
The reason this is not a proof is that the curve-driver regularity correspondence of \cite{RohdeWang} is only proven for curves up to $C^2$. One expects it to hold for higher regularity as well (see \cite[Comment 4.1]{RohdeWang}), in which case the above argument becomes a proof. Note that the ``if'' direction of the conjecture is proven by $(\ref{Thm:Wang0+})$ and the fact that $\gamma_{\pi/2}$ is a line segment orthogonal to $\mathbb{R}$.
\end{enumerate}
\end{remark}
\begin{proof}
Minimizing drivers exist by Lemma \ref{Lemma:AngleEnergyMinimizerExists}, and we proceed to show that any such $\lambda$ satisfies a certain differential equation at all points of differentiability, which will give uniqueness and the formula \eqref{Eq:WangEnergy}.
Indeed, let $\lambda$ be a minimizer, and let $z(t) = x(t) + iy(t) := G_t(e^{i\theta}) = g_t(e^{i\theta}) - \lambda(t)$ be the image of $e^{i\theta}$ under the centered downwards flow generated by $\lambda$. Note that by the Loewner equation \eqref{Eq:BackgroundLoewnerEq}, $G_t$ satisfies
\begin{align}\label{Eq:LoewnerEqCentered}
\dot{G}_t(z) = \frac{2}{G_t(z)} - \dot{\lambda}(t)
\end{align}
at points of differentiability of $\lambda$. Since $I(\lambda)<\infty$, $\lambda$ is absolutely continuous and so differentiable for a.e. $t$. We claim that for a.e. $t$,
\begin{align}\label{Eq:WangLambdaPrimeFormula}
\dot{\lambda}(t) = \frac{8x(t)}{|z(t)|^2}.
\end{align}
By scale invariance the energy depends only on the angle, and by Lemma \ref{Lemma:WangEvenApproach} the angle $\theta(t):= \Arg(z(t))$ must tend towards $\pi/2$ if $\lambda$ minimizes energy. And indeed, it must move strictly monotonically: if the angle ever decreases and then returns to the same value, energy is wasted, while if the angle is constant over some interval, then $\lambda$ cannot be constant and we have also wasted energy. Hence $z(t)$ should traverse through the angles as efficiently as possible; the change in angle to the change in energy, $d\theta/dI$, must be optimal. That is, $d\theta/dI$ must be maximized when $\theta <\pi/2$ and minimized when $\theta > \pi/2$.
Suppose first that $\lambda$ is right-differentiable at $t=0$ and that $t=0$ is a Lebesgue point for $\dot{\lambda}^2$. Then the energy expelled on a small interval $[0,\Delta t]$ is $\frac{1}{2}\dot{\lambda}^2(0)\Delta t + o(\Delta t)$. Furthermore, $\theta$ is right-differentiable at $t=0$, and
\begin{align*}
\Delta \theta = \dot{\theta}(0)\Delta t + o(\Delta t) = \Imag\Big(\frac{\dot{z}(0)}{z(0)}\Big)\Delta t + o(\Delta t) = (-4x(0)y(0)+\dot{\lambda}(0)y(0))\Delta t + o(\Delta t)
\end{align*}
by \eqref{Eq:LoewnerEqCentered} and the fact that $|z(0)|=1$. We thus have
\begin{align*}
\frac{\Delta \theta}{\Delta I} = \frac{(-4xy+\dot{\lambda}y)\Delta t + o(\Delta t)}{\frac{1}{2}\dot{\lambda}^2\Delta t + o(\Delta t)} \rightarrow \frac{-8xy}{\dot{\lambda}^2} + \frac{2y}{\dot{\lambda}}
\end{align*}
as $\Delta t \rightarrow 0$, where $x,y$ and $\dot{\lambda}$ are evaluated at $t=0$, and $\dot{\lambda} =\dot{\lambda}(0)$ is the right derivative of $\lambda$. This expression is optimized with respect to $\dot{\lambda}$ when $\dot{\lambda}=8x$, which yields a local max when $x>0$ and a local min when $x<0$, as needed. Thus any minimizer for which $\dot{\lambda}(0)$ exists and where $t=0$ is a Lebesgue point of $\dot{\lambda}^2$ must satisfy \eqref{Eq:WangLambdaPrimeFormula} at $t=0$ (recall $|z(0)|=1$).
More generally, let $t_0$ be a point of differentiability of $\lambda$ and a Lebesgue point of $\dot{\lambda}^2$. Note that the remaining curve $\tilde{\gamma}:=G_{t_0}(\gamma([t_0,\tau]))$ must be an energy minimizer through $z(t_0)$, as discussed above in Remark \ref{Remark:RemainingWang}. Thus $\tilde{\gamma}/|z(t_0)|$ is a minimizer as in the previous paragraph, and so its driver $\tilde{\lambda}$ has initial right derivative $8\tilde{x}(0) = 8x(t_0)/|z(t_0)|$. Recalling the scaling relation \eqref{Eq:ScaledDriver}, we therefore have
\begin{align*}
|z(t_0)|\dot{\lambda}(t_0) = \dot{\tilde{\lambda}}(0) = \frac{8x(t_0)}{|z(t_0)|},
\end{align*}
as in \eqref{Eq:WangLambdaPrimeFormula}. Since $\lambda$ is differentiable at a.e. $t$ and a.e. $t$ is a Lebesgue point of the integrable function $\dot{\lambda}^2$, \eqref{Eq:WangLambdaPrimeFormula} holds as claimed.
By \eqref{Eq:LoewnerEqCentered} we thus obtain the system of differential equations
\begin{align}\label{Eq:WangSystem}
\dot{\lambda}(t) = \frac{8x}{x^2 + y^2}, \qquad \dot{x}(t) = \frac{-6x}{x^2+y^2}, \qquad \dot{y}(t) = \frac{-2y}{x^2+y^2}
\end{align}
for which the triple $(\lambda,x,y)$ generated by $\lambda$ is an a.e.-$t$ solution, and where each component is absolutely continuous. Since we can thus recover each of $\lambda, x$ and $y$ through integration and the three right-hand sides in \eqref{Eq:WangSystem} are continuous, we have that \eqref{Eq:WangSystem} actually holds for \emph{all} $t$, and hence each of $\lambda, x$ and $y$ is $C^1$ on $[0,\tau)$. By bootstrapping in \eqref{Eq:WangSystem}, then, each is $C^2$, and continuing, each is $C^\infty([0,\tau))$.\footnote{We note that Wang \cite[equation (3.2)]{WangReverse} also obtained this ODE for $\dot{\lambda}$, but only by means of using the minimal-energy formula \eqref{Eq:WangEnergy}, whereas we go the opposite direction, using \eqref{Eq:WangSystem} to derive this formula. We also note that \eqref{Eq:WangSystem} immediately gives that the Wang minimizer is downwards SLE$_0(-8)$ starting from $(\lambda_0, V_0) = (0, z(0))$; compare \eqref{Eq:SLErhoSDE} and note that we are using the centered maps.}
Classical solutions to \eqref{Eq:WangSystem} are also unique: starting at any point $(x_0,y_0)$, both $x(t)$, $y(t)$ are bounded away from 0 on a small time interval, and so the function $f(t,\lambda,x,y) = \big(8x(x^2+y^2)^{-1}, -6x(x^2+y^2)^{-1}, -2y(x^2+y^2)^{-1}\big)$ is Lipschitz. Thus we have smoothness, uniqueness, as well as monotonicity of $\lambda$ from \eqref{Eq:WangSystem}.
For the energy formula \eqref{Eq:WangEnergy}, write $m(\theta)$ for energy of the curve through $e^{i\theta}$. If we flow down starting from a fixed $\theta_0 = \theta(0)$, we know the remaining curve is always the minimizer for the angle
\begin{align}\label{Eq:WangMinimizerAngle}
\theta(t) = \arg(z(t)),
\end{align}
the argument of the image of the tip (see the remark before the proof). Hence through the composition $m(\theta(t))$ we may regard $m$ as a function of $t$, and we find
\begin{align*}
\frac{dm}{d\theta} = \frac{\dot{m}}{\dot{\theta}} = \frac{-\frac{1}{2}\dot{\lambda}^2}{\frac{-4xy}{(x^2+y^2)^2} + \frac{\dot{\lambda}y}{x^2+y^2}} = -8\frac{x}{y} = -8\cot(\theta),
\end{align*}
and therefore
\begin{align*}
m(\pi/2) - m(\theta_0) = -m(\theta_0) = \int_{\theta_0}^{\pi/2} -8\cot(\theta)d\theta = 8\log(\sin(\theta_0)),
\end{align*}
as claimed, completing the proof of $(i)$.
\bigskip
Our formulas in $(\ref{Thm:WangDriver})$ for the driving function and the capacity time now are exercises in ODE. We note from \eqref{Eq:WangSystem} that $x$ is monotonically decreasing (recall we are assuming $0 < \theta < \pi/2$), and so we may reparametrize $\lambda$ as a function of $x$ and note $\frac{d\lambda}{dx} = -\frac{4}{3}$ from \eqref{Eq:WangSystem}, and hence
\begin{align}\label{Eq:WangLambdaStep1}
\lambda(t) = \lambda(x(t)) - \lambda(x(0)) = \frac{4}{3}\cos(\theta) - \frac{4}{3}x(t).
\end{align}
In particular,
\begin{align}\label{Eq:WangTerminalLambda}
\lambda(\tau) = \frac{4}{3}\cos(\theta).
\end{align}
To determine $x(t)$, we note from \eqref{Eq:WangSystem} that
\begin{align}\label{Eq:WangLambdaStep2}
\frac{dx}{dy} = 3 \frac{x}{y}, \qquad \text{implying} \qquad x(t) = \frac{\cos(\theta)}{\sin^3(\theta)}y(t)^3
\end{align}
since $(x(0),y(0)) = (\cos(\theta),\sin(\theta))$. Writing $b = b(\theta):=\frac{\cos(\theta)}{\sin^3(\theta)}$ and substituting back into the equation for $\dot{y}(t)$ yields
\begin{align*}
\dot{y}(t) = \frac{-2}{b^2y^5 + y}
\end{align*}
which has implicit solution
\begin{align*}
\frac{b^2}{6}y(t)^6 + \frac{1}{2}y(t)^2 = -2t + \frac{1}{6}\cos^2(\theta) + \frac{1}{2}\sin^2(\theta).
\end{align*}
We thus see that $u(t) := y(t)^2$ satisfies the cubic
\begin{align*}
0 = u^3 + 3\frac{\sin^6(\theta)}{\cos^2(\theta)}u + 6 \frac{\sin^6(\theta)}{\cos^2(\theta)}\big(2t - \frac{1}{6}\cos^2(\theta) -\frac{1}{2}\sin^2(\theta) \big) =: u^3 + pu + q.
\end{align*}
As the discriminant $4p^3 + 27q^2$ is manifestly positive, by Cardano's cubic formula the real root is
\begin{multline}\label{Eq:Wangy}
y(t)^2 = \sqrt[3]{\frac{\sin^6(\theta)}{\cos^2(\theta)}6(\tau-t) + \sqrt{\frac{\sin^{12}(\theta)}{\cos^4(\theta)}36(\tau-t)^2 + \frac{\sin^{18}(\theta)}{\cos^6(\theta)}}}\\
+ \sqrt[3]{\frac{\sin^6(\theta)}{\cos^2(\theta)}6(\tau-t) - \sqrt{\frac{\sin^{12}(\theta)}{\cos^4(\theta)}36(\tau-t)^2 + \frac{\sin^{18}(\theta)}{\cos^6(\theta)}}}.
\end{multline}
Pulling out the trig functions and substituting back into \eqref{Eq:WangLambdaStep2}, and then into \eqref{Eq:WangLambdaStep1}, yields
\begin{multline}\label{Eq:WangDriverDown}
\lambda_\theta (t) = \\\frac{4}{3}\cos(\theta) - \frac{4}{3}\left( \sqrt[3]{6(\tau-t)+\sqrt{36(\tau-t)^2+\frac{\sin^6(\theta)}{\cos^2(\theta)}}} +\sqrt[3]{6(\tau-t)-\sqrt{36(\tau-t)^2+\frac{\sin^6(\theta)}{\cos^2(\theta)}}} \right)^{3/2},
\end{multline}
and the claimed formula for $\xi_\theta$ then follows from reversal and \eqref{Eq:WangTerminalLambda}.
For the terminal Loewner-time formula
\begin{align}\label{Eq:WangMinimizerHcapTime}
\tau_\theta = \frac{1}{6}(1 - \frac{1}{2}\cos(2\theta)),
\end{align}
note that we can regard $\tau_\theta$ as a function of $t$ through the composition $\tau_{\theta(t)} = \frac{1}{2}\hcap(\gamma_{\theta(t)})$, with $\theta(t)$ given by \eqref{Eq:WangMinimizerAngle}. That is, $\tau_{\theta(t)}$ is the capacity time of the minimizing segment $\gamma_{\theta(t)}$ from $0$ to $e^{i\theta(t)}$ (not the capacity time of the remaining curve), where the 1/2-factor is because the half-plane capacity is twice the Loewner time. We claim that $\tau_{\theta(t)}$ is differentiable in $t$ and satisfies
\begin{align}\label{Eq:Wanghcapderiv}
\frac{d}{dt}\tau_{\theta(t)} = -1 + 4 \tau_{\theta(t)}(1+2 \cos^2(\theta)).
\end{align}
Indeed, if we map a small portion $\gamma_\theta([0,\Delta t])$ of $\gamma_\theta$ down, we have
\begin{align}\label{Eq:WangMinHtheta2}
\tau_{\theta(\Delta t)} = \frac{\tau_\theta -\Delta t}{|z(\Delta t)|^2}
\end{align}
because the remaining curve is the minimizer through $\theta(\Delta t)$; the denominator is because of the scaling property $\hcap(r\gamma) = r^2 \hcap(\gamma)$. From \eqref{Eq:WangSystem},
\begin{align*}
\frac{d}{dt}|z(t)|^2 = \frac{-12x^2-4y^2}{x^2+y^2} = -4(1 + 2 \cos^2(\theta)),
\end{align*}
and so $|z(\Delta t)|^2 = 1 -4(1 + 2 \cos^2(\theta))\Delta t + o(\Delta t)$ as $\Delta t \rightarrow 0$. Plugging this into \eqref{Eq:WangMinHtheta2} yields
\begin{align*}
\frac{\tau_{\theta(\Delta t)} - \tau_{\theta}}{\Delta t} = \frac{-\Delta t+4\tau_\theta(1 + 2 \cos^2(\theta))\Delta t + o(\Delta t)}{\Delta t \big(1 -(1 + 2 \cos^2(\theta))\Delta t + o(\Delta t)\big)},
\end{align*}
yielding \eqref{Eq:Wanghcapderiv} in the limit. Since $\theta(t)$ is differentiable with respect to $t$ with
\begin{align}\label{Eq:WangThetaDot}
\dot{\theta}(t) = \frac{d}{dt}\Imag(\log(z(t))) = \frac{4xy}{(x^2+y^2)^2} = 4\cos(\theta)\sin(\theta)
\end{align}
at $t=0$, we have
\begin{align*}
\frac{d\tau_\theta}{d\theta} = \frac{\dot{\tau}_\theta}{\dot{\theta}} = \frac{ -1 + 4 \tau_\theta(1+2 \cos^2(\theta))}{4\cos(\theta)\sin(\theta)}.
\end{align*}
This ODE has explicit solution
\begin{align*}
\tau_\theta = C \frac{\sin^3(\theta)}{\cos(\theta)} + \frac{1}{6}\big(1 - \frac{1}{2}\cos(2\theta) \big).
\end{align*}
In order to have $\tau_\theta$ bounded as $\theta \rightarrow \frac{\pi}{2}^-$, we must have $C=0$, which yields \eqref{Eq:WangMinimizerHcapTime}. Note that this formula also extends to $\theta=\pi/2$, where the minimizer $\gamma_{\pi/2} = [0,i]$ satisfies
\begin{align*}
\hcap([0,i]) = \frac{1}{2} = \frac{1}{3}\big(1 - \frac{1}{2}\cos(\pi) \big).
\end{align*}
The welding formula \eqref{Eq:WangWeld} follows from conjugating the welding $\omega$ on $\mathbb{R}$ constructed in \cite{MRW} for a smooth geodesic pair by the coordinate change to the chordal Loewner setting. We start by considering $\gamma_{\theta}$ for fixed $0 < \theta < \pi$. As noted above in Remark \ref{Remark:RemainingWang}($\ref{Remark:WangUniversalCurve}$), $\gamma_{\theta}^* := 1/\bar{\gamma}_{\theta}$ is the hyperbolic geodesic in $\mathbb{H}{\backslash} \gamma_{\theta}$ from $e^{i\theta}$ to $\infty$. We wish to say that $\Gamma = \Gamma_{\theta} := \gamma_{\theta} \cup \gamma_{\theta}^*$ is the conformal image of the $C^1$-geodesic pair $\gamma_{1,\theta'} \cup \gamma_{2,\theta'}$ in $(\mathbb{D}; e^{i\theta'}, -e^{-i\theta'}, 0)$ of \cite[Corollary 2.3]{MRW}, where $\theta' = \frac{\pi}{2}-\theta$.\footnote{So note $\gamma_{j,\beta}$ refer to the curves in $\mathbb{D}$, while $\gamma_{\alpha}$ to the curve in $\mathbb{H}$. We use the former notation to stay close to the nomenclature used in \cite{MRW}.} See Figure \ref{Fig:UniversalCurveWang}. By the uniqueness of smooth geodesic pairs \cite[Theorem 3.9]{MRW}, it suffices to show that $\Gamma$ is at least $C^1$ in its arc-length parametrization. Indeed, note that the downwards driving function for $\Gamma$ is
\begin{align*}
\lambda_\theta(t) = \begin{cases}
\xi_{\theta}(\tau_{\theta} -t) - \xi_{\theta}(\tau_{\theta}) & 0 \leq t \leq \tau_{\theta} \\
- \xi_\theta(\tau_\theta) & \tau_{\theta} < t.
\end{cases}
\end{align*}
In particular, $\lambda_\theta$ is smooth away from $t = \tau_{\theta}$, and furthermore has the same $C^{3/2}$-regularity at $t=\tau_{\theta}$ that $\xi_{\theta}$ does as $t=0$ (recall \eqref{Eq:WangDriverExpansion}). Hence by the correspondence between driver and curve regularity \cite{Wong}, $\Gamma$ is $C^{2-\epsilon}$ near $e^{i\theta}$ in its capacity parametrization.\footnote{In fact, a little more is true, as $\Gamma$ is weakly $C^{1,1}$, as we discuss in \cite{MesikeppSharp}. The point here, however, is that $\Gamma$ is at least $C^1$.} In particular, $\Gamma$'s unit tangent vector varies continuously, and so $\Gamma$ is the claimed image of the $C^1$ geodesic pair $\gamma_{1,\theta'} \cup \gamma_{2,\theta'}$ in $(\mathbb{D};e^{i\theta'}, -e^{-i\theta'},0)$ for some $\theta' \in (-\pi/2,\pi/2)$. Noting that the M\"{o}bius transformation $M_\alpha$ in \eqref{Eq:GeodesicMobius} with $\alpha=\theta$ sends the triple $(0, e^{i\theta}, \infty)$ to $(-e^{-i(\frac{\pi}{2}-\theta)},0,e^{i(\frac{\pi}{2}-\theta)})$, we see we may take $\theta' = \frac{\pi}{2}-\theta$, as claimed. In particular, $\gamma_{\theta} = \Gamma([0,\tau_\theta])$ corresponds to $\gamma_{2,\theta'}$ under $M_\theta$, as in Figure \ref{Fig:UniversalCurveWang} (with $\alpha$ replaced by $\theta$).
In what follows we use the notation of \cite[Figure 1]{MRW}. Set $B =
\sin(\theta) + (\frac{\pi}{2}-\theta)\cos(\theta)$ and post-compose by the unique conformal map $G:\mathbb{D} {\backslash} \gamma_{2,\theta'} \rightarrow \mathbb{C} {\backslash} (-\infty, B]$ which maps the triple $(-i, e^{i\theta'},0)$ to $(-\frac{\pi}{2}\cos(\theta), B, \infty)$. Thus $G \circ M_\theta$ sends the two sides of $\gamma_{\theta}$ to (portions of) two sides of the slit $(-\infty, B]$. By the explicit construction in \cite[Ex. 3.1]{MRW}, the welding in the latter slit-plane setting is simply the shift $\omega(x) = x + 2\pi \cos(\theta)$ with
\begin{align*}
\omega: \Big(-\infty, -\sin(\theta) - \Big(\frac{3\pi}{2}-\theta\Big)\cos(\theta) \Big] \rightarrow \Big(-\infty,-\sin(\theta) + \Big(\frac{\pi}{2} + \theta \Big)\cos(\theta)\Big].
\end{align*}
That is, $M_\theta^{-1} \circ G^{-1}(x) = M_\theta^{-1}\circ G^{-1}(\omega(x)) \in \gamma_\theta$ for $x\leq-\sin(\theta)-(3\pi/2 - \theta)\cos(\theta)$. As we seek the welding $\varphi_{\theta}:[a_{\theta}, 0] \rightarrow [0,b_{\theta}]$ giving the identifications generated by the chordal Loewner flow, we apply the conformal map $f: \mathbb{C} {\backslash} (-\infty, B] \rightarrow \mathbb{H}$ given by $f(z) = -1/(i\sqrt{z-B})$, where $\log(z)$ is chosen so that $-\pi/2 \leq \Arg(z) \leq \pi/2$, and thus obtain $\tilde{\varphi} = \tilde{\varphi}_{\theta} := f \circ \omega^{-1} \circ f^{-1}$,
\begin{align*}
\tilde{\varphi}(x) = \frac{-x}{\sqrt{1+2\pi \cos(\theta)x^2}},
\end{align*}
mapping $[-1/\sqrt{2 \sin(\theta) - 2\theta \cos(\theta)},0]$ to $[0,1/\sqrt{2\sin(\theta) + 2 (\pi-\theta) \cos(\theta)}]$. However, $\tilde{\varphi}$ is not the Loewner-normalized welding if $f \circ G \circ M_\theta$ is not $z + O(1)$ as $z \rightarrow \infty$, corresponding to the hydrodynamic normalization \eqref{Eq:LoewnerNormalize}. The map $G$ is obtained as the Schwarz reflection of the map $G_{\theta'}$ across the imaginary axis, where
\begin{align*}
G_{\theta'}(z) = \frac{1}{2}\Big(z + \frac{1}{z} \Big) - i \cos(\theta)\log(z)
\end{align*}
\cite[Lemma 2.2]{MRW}. By noting
\begin{align*}
G_{\theta'}'(z) = \frac{(z-e^{i\theta'})(z+e^{-i\theta'})}{2z^2},
\end{align*}
we see
\begin{align*}
G(z) = B + \frac{e^{i\theta'}+e^{-i\theta'}}{4e^{2i\theta'}}(z-e^{i\theta'})^2 + O(z-e^{i\theta'})^3, \qquad \mathbb{D} \ni z \rightarrow e^{i\theta'},
\end{align*}
and thus find
\begin{align*}
f \circ G \circ M_\theta(z) = \frac{z}{\sqrt{2\sin^3(\theta)}} + O(1), \qquad z \rightarrow \infty.
\end{align*}
Hence after post-composing by $z \mapsto \sqrt{2\sin^3(\theta)}\,z$, we arrive at
\begin{align*}
\varphi_{\theta}(z) := \sqrt{2\sin^3(\theta)}\, \tilde{\varphi}_{\theta}\big(x/\sqrt{2\sin^3(\theta)}\big) = \frac{-x}{\sqrt{1+\pi \frac{\cos(\theta)}{\sin^3(\theta)}x^2}},
\end{align*}
which maps $[-\sin^{3/2}(\theta)/\sqrt{\sin(\theta) - \theta \cos(\theta)},0]$ to $[0,\sin^{3/2}(\theta)/\sqrt{\sin(\theta) + (\pi-\theta) \cos(\theta)}]$.
\bigskip
We show part $(\ref{Thm:WangUniversal})$ for a welding and driver corresponding to fixed $0 < \theta < \pi/2$; the argument for $\pi/2 < \theta < \pi$ is similar. Note that the $\varphi_\theta$ in \eqref{Eq:WangWeld} is defined for all $x<0$, and we claim that for any $u<0$, $\varphi_\theta|_{[u,0]}$ welds a scaled Wang minimizer $r_u \gamma_\alpha$ for some angle $0 < \alpha = \alpha_u < \pi/2$ (here the scale factor $r_u$ corresponds to welding $\varphi_\theta|_{[u,0]}$ with the centered upwards Loewner flow map). Indeed, note that if we rescale $\gamma_\theta$ by $c_\theta := \sqrt{\pi\cos(\theta)/\sin^3(\theta)}$, the corresponding welding (in the Loewner normalization) is
\begin{align}\label{Eq:WangRescale}
c_\theta \varphi_\theta(x/c_\theta) = \frac{-x}{\sqrt{1+x^2}},
\end{align}
which is independent of $\theta$. Thus \eqref{Eq:WangRescale} is universal in the sense of the theorem statement for generating Wang minimizers with tip at angle $0 < \alpha <\pi/2$, and as \eqref{Eq:WangRescale} is a fixed rescaling of $\varphi_\theta$, we see that $\varphi_\theta$ is also universal. Hence $\varphi_\theta|_{[u,0]}$ generates a scaled Wang minimizer as claimed.
By \eqref{Eq:WangWeldInterval}, $\theta \mapsto -x_\theta/y_\theta$ is strictly monotonic, and so the $\alpha$ of the Wang minimizer $r_u \gamma_\alpha$ generated by $\varphi_\theta$ on $[u,0]$ is entirely determined by the ratio $-u/\varphi_\theta(u)$. Solving $-u/\varphi_\theta(u) = -x_\alpha/y_\alpha$ yields
\begin{align*}
u_\alpha = -\sqrt{\frac{\sin^3(\theta)}{\cos(\theta)} \cdot \frac{\cos(\alpha)}{\sin(\alpha)-\alpha \cos(\alpha)}},
\end{align*}
and we observe that the scale factor $r_u$ is determined by $u_\alpha = r_u x_\alpha$, yielding
\begin{align*}
r_u = \sqrt{\frac{\sin^3(\theta)}{\cos(\theta)} \cdot \frac{\cos(\alpha)}{\sin^3(\alpha)}},
\end{align*}
as claimed.
The same argument also applies to the driving function: $\xi_\theta$ is defined for all $t \geq 0$, and $c_\theta \xi_\theta(t/c_\theta^2)$ is independent of $\theta$, showing $\xi_\theta|_{[0,t]}$ generates a scaled (and translated) Wang minimizer $r_{u(t)}\gamma_\alpha + \xi_\theta(t)$ for any $t>0$ under its upwards Loewner flow. To determine the $t$ corresponding to a given $\alpha$, we note that $\varphi_\theta$ and $\xi_\theta$ generate the same curves, and that the Loewner time for $r_u \gamma_\alpha$ is
\begin{align*}
r_u^2 \tau_\alpha = \frac{\sin^3(\theta)\cos(\alpha)}{6\cos(\theta)\sin^3(\alpha)}\Big( 1 - \frac{1}{2}\cos(2\alpha) \Big) =t_\alpha.
\end{align*}
by the scaling relation for half-plane capacity and \eqref{Eq:WangMinimizerHcapTime}. Thus $\xi_\theta|_{[0,t_\alpha]}$ generates $r_u \gamma_\alpha + \xi_\theta(t_\alpha)$.
\bigskip
For part $(\ref{Thm:Wang0+})$, we start by proving uniform convergence of the drivers $\xi_\theta$ to
\begin{align}\label{Eq:theta0driver}
\xi_0(t) = -\frac{8}{\sqrt{3}}\sqrt{t}
\end{align}
as $\theta \rightarrow 0^+$. As we will need to attach hyperbolic geodesics from $\gamma_\theta(\tau_\theta)$ and $\gamma_0(\tau_0)$ to $\infty$, we actually show uniform convergence $\tilde{\xi}_{\theta,T} \rightarrow \tilde{\xi}_{0,T}$ as $\theta \rightarrow 0^+$, where, for $\alpha \geq 0$ and fixed $T > \tau_\alpha$,
\begin{align*}
\tilde{\xi}_{\alpha,T}(t) := \begin{cases}
0 & 0 \leq t \leq T-\tau_\alpha,\\
\xi_\alpha\big(t-(T-\tau_\alpha) \big) & T - \tau_\alpha < t \leq T.
\end{cases}
\end{align*}
Thus $\tilde{\xi}_{\alpha,T}$ is the upwards driver $\lambda_{\alpha}(T-t) - \lambda_\alpha(T)$ for the hull which is $\gamma_\alpha$ followed by $T-\tau_\alpha$ units of time of the hyperbolic geodesic from $\gamma_\alpha(\tau_\alpha)$ to $\infty$ in $\mathbb{H} {\backslash} \Fill(\gamma_\alpha)$. Now, since $\tau_\theta \rightarrow \tau_0$ and $\theta \mapsto \xi_\theta(t)$ is continuous at $\theta=0^+$, point-wise convergence $\tilde{\xi}_{\theta,T}(t) \rightarrow \tilde{\xi}_{0,T}(t)$ is clear. Furthermore, a calculation shows that
\begin{align*}
\partial_\theta \tilde{\xi}_{\theta,T}(t) = \frac{\partial \xi_\theta}{\partial \theta}(t - (T-\tau_\theta)) \tau_\theta' >0
\end{align*}
when $0 < \theta < \pi/2$, $T-\tau_\theta \leq t \leq T$, and hence the point-wise limit is monotone, allowing us to upgrade to the claimed uniform convergence on the compact interval $[0,T]$ by the classical Dini theorem
Write $\tilde{\gamma}_\alpha$ for the curve which is $\gamma_\alpha = \gamma_\alpha[0,\tau_\alpha]$ followed by the hyperbolic geodesic $\eta_\alpha$ from $\gamma_\alpha(\tau_\alpha)$ to $\infty$ in $\mathbb{H}{\backslash} \Fill(\gamma_\alpha[0,\tau_\alpha])$, and $g_\alpha(t,z)$ for $\tilde{\gamma}_\alpha$'s mapping-down function. By \cite[Prop. 4.7]{Lawler}, the above driver convergence yields that, for any $T>0$ and $\epsilon>0$, one has
\begin{align}\label{Conv:DoubleUniform}
g_\theta(t,z) \xrightarrow[\theta \rightarrow 0^+]{unif} g_0(t,z) \qquad \text{ on } \qquad [0,T] \times \{\, z \; : \; \dist\big(z, \Fill(\tilde{\gamma}_0[0,T])\big)>\epsilon \,\}.
\end{align}
We wish to say that $g_\theta(\tau_\theta, \cdot) \rightarrow g_0(\tau_0,\cdot)$ uniformly on compacts of $\mathbb{H} {\backslash} \Fill(\gamma_0)$, which easily follows (note that we are comparing the maps at the different times $\tau_\theta$ and $\tau_0$). Indeed, for a fixed compact $K$ of $\mathbb{H} {\backslash} \Fill(\gamma_0)$, $\tilde{\gamma}_0([\tau_0,\tau_\theta])\cap K = \emptyset$ when $\theta$ is small, and so $g_\theta(\tau_\theta, \cdot)$ is defined on $K$ for all sufficiently-small $\theta$ by \eqref{Conv:DoubleUniform}. For $z \in K$ and $\theta$ close to zero,
\begin{align}
|g_\theta(\tau_\theta, z) - g_0(\tau_0,z)| &\leq |g_\theta(\tau_\theta, z) - g_0(\tau_\theta,z)| + |g_0(\tau_\theta, z) - g_0(\tau_0,z)| \notag\\
&\leq \epsilon_1 + C\sqrt{\diam(\tilde{\gamma}_0[0,\tau_\theta])\text{osc}(\tilde{\gamma}_0,\tau_\theta-\tau_0,\tau_\theta)} \label{Ineq:OneEstimate}\\
&\leq \epsilon_1 + \epsilon_2\notag
\end{align}
for all small $\theta$, where the two estimates in \eqref{Ineq:OneEstimate} are by \eqref{Conv:DoubleUniform} and \cite[Lemma 4.1]{Lawler}, respectively, and $\text{osc}(\eta, \delta, T)$ is the $\delta$-modulus of continuity of the curve $\eta$ up to time $T$,
\begin{align*}
\text{osc}(\eta, \delta, T):= \sup\{\, |\eta(t)-\eta(s)| \; : \; 0 \leq s,t \leq T, \; |s-t| \leq \delta \,\}.
\end{align*}
We conclude that we have the claimed locally-uniform convergence, and hence also, recalling \eqref{Eq:WangTerminalLambda}, the locally-uniform convergence of the centered mapping-down functions $g_\theta(\tau_\theta, \cdot) - \lambda_\theta(\tau_\theta)$ to $g_0(\tau_0, \cdot) - \lambda_0(\tau_0)$. Call $F_\theta(\cdot)$ and $F_0(\cdot)$ the inverses of the latter two maps. Then $F_\theta \rightarrow F_0$ locally-uniformly on $\mathbb{H}$, and in particular, writing a given segment $\eta_0(I_0)$ of the hyperbolic geodesic $\eta_0$ from $1$ to $\infty$ in $\mathbb{H} {\backslash} \Fill(\gamma_0)$ as $F_0([iy_1,iy_2])$ for some $0 < y_1 < y_2$, we have the Hausdorff convergence
\begin{align}\label{Conv:HausSegment}
F_\theta([iy_1,iy_2]) \xrightarrow[]{Haus} F_0([iy_1,iy_2]),
\end{align}
where we observe that $F_\theta([iy_1,iy_2]) \subset \gamma_\theta^*$, the hyperbolic geodesic from $\gamma_\theta(\tau_\theta)$ to $\infty$ in $\mathbb{H}{\backslash} \gamma_\theta$, which, as we have seen, is the reflection $1/\overline{\gamma}_\theta$.
We claim that it follows that $\eta_0 = \gamma_0^*$, i.e. $\gamma_0$'s reflection in $\partial \mathbb{D}$ is the hyperbolic geodesic. To this end, we first show uniform convergence of the capacity-parametrized curves $\gamma_\theta$ to $\gamma_0$ on $[0,\tau_0-\epsilon]$ for any $\epsilon>0$. Indeed, consider the Loewner energy
\begin{align}\label{Eq:ChoppedEnergy}
I_\epsilon(\theta) := \frac{1}{2}\int_\epsilon^{\tau_0} \dot{\xi}_\theta(s)^2ds,
\end{align}
which we claim is continuous in $\theta \geq 0$. Noting that $\theta \mapsto \sin^6(\theta)/\cos^2(\theta)$ is increasing and that we have excised the singularity in $\dot{\xi}$ when $t$ and $\theta$ are both zero, through coarse bounds one readily obtains a function $g \in L^1([\epsilon,\tau_0])$ such that $\dot{\xi}_\theta(s)^2 \leq g(s)$ for all $0 \leq \theta \leq \pi/4$. Dominated convergence then shows that $I_\epsilon(\cdot)$ is continuous, and hence bounded, on $0\leq \theta \leq \pi/4$, say. As $I_\epsilon(\theta)$ is the energy of $\gamma_\theta([0,\tau_0-\epsilon])$, by property ($\ref{Lemma:CurvesCompact}$) of the Loewner energy in \S\ref{Sec:BackgroundLoewnerEnergy}, we thus see that for any sequence $\theta_n \rightarrow 0$, $\gamma_{\theta_n}$ has a uniform limit $\gamma = \gamma(\{\theta_n\})$ on $[0,\tau_0-\epsilon]$. As above for the drivers $\tilde{\xi}_{\theta,T}(t)$, we have
\begin{align*}
\lambda_\theta(t) = \xi_\theta(\tau_\theta -t) - \xi_\theta(\tau_\theta) \xrightarrow[]{unif} \xi_0(\tau_0-t) - \xi_0(\tau_0) = \lambda_0(t)
\end{align*}
on $[0,\tau_0-\epsilon]$, and so by \cite[Lemma 4.2]{LMR}, $\gamma = \gamma_0|_{[0,\tau_0-\epsilon]}$, and we conclude that the limit is unique and therefore that $\gamma_\theta \rightarrow \gamma_0$ uniformly on $[0,\tau_0-\epsilon]$, as claimed.
From \eqref{Conv:HausSegment} we know that the reflections
\begin{align*}
1/\overline{F_\theta([iy_1,iy_2])} \xrightarrow[]{Haus} 1/\overline{F_0([iy_1,iy_2])}
\end{align*}
for any $0 < y_1 < y_2$, while we now see from the uniform convergence that $1/\overline{F_\theta([iy_1,iy_2])} = \gamma_\theta([t_{1,\theta},t_{2,\theta}])$ also converges in the Hausdorff sense to a segment $\gamma_0([t_1,t_2])$ of $\gamma_0$. By uniqueness of the limit we conclude that $1/\overline{\gamma_0([t_1,t_2])} = F_0([iy_1,iy_2])$, and thus that $\eta_0 \subset \gamma_0^*$, and therefore that $\eta_0 = \gamma_0^*$, as claimed. Furthermore, since $\gamma_0$'s reflection $\gamma_0^*$ is a hyperbolic geodesic, $\gamma_0$ itself is the hyperbolic geodesic from 0 to 1 in its component of $\mathbb{H} {\backslash} \gamma_0^*$. That is, $\gamma_0 \cup \gamma_0^*$ is a boundary geodesic pair in $(\mathbb{D}; 0, \infty, 1)$.
We can now show the algebraic formula \eqref{Eq:Variety} by exhibiting $\gamma_0 \cup \gamma_0^*$ as part of the real locus of a specific rational function $R$. More precisely, we will have
\begin{align*}
R^{-1}(\hat{\mathbb{R}}) = \gamma_0 \cup \gamma_0^* \cup \hat{\mathbb{R}} \cup \overline{\gamma}_0 \cup \overline{\gamma}_0^*,
\end{align*}
where $\hat{\mathbb{R}} = \mathbb{R} \cup \{ \infty\}$ and the bar denotes complex conjugation, as usual. The argument is essentially identical to that for \cite[Prop. 4.1]{PeltWang}, although our context is slightly different than the ``geodesic multichord'' setting, as our two geodesics share a common boundary point (i.e. we are in the ``fused multichord'' setting of Krusell \cite{Krusell}). We sketch the details for the convenience of the reader.
Take a conformal map $R_0$ from the bounded component of $\mathbb{H} {\backslash} \gamma_0$ to $\mathbb{H}$, and write $\Omega_0$ for the component of $\mathbb{H}{\backslash} \gamma_0^*$ containing $\gamma_0$. Since $\gamma_0$ is the hyperbolic geodesic from $0$ to $1$ in $\Omega_0$, we can Schwarz-reflect $R_0$ across $\gamma_0$ and extend $R_0$ to map the unbounded component of $\Omega {\backslash} \gamma_0$ to $\overline{\mathbb{H}}$, the lower half plane. Since $\gamma_0^*$ is the hyperbolic geodesic in $\mathbb{H} {\backslash} \Fill(\gamma_0)$, we can again reflect this reflection to map the right-most component of $\mathbb{H} {\backslash} \gamma_0$ to $\mathbb{H}$. Altogether, we obtain $R_0 : \mathbb{H} \rightarrow \mathbb{C}$ which takes $\mathbb{R} \cup \gamma_0 \cup \gamma_0^*$ to $\mathbb{R}$, and which we can again Schwarz-reflect across $\mathbb{R}$ to obtain a map $R$ which is an analytic, degree-three branched cover of the sphere, and is hence rational. By post-composing with an element of $\Aut({\mathbb H})=PSL_2(\mathbb{R})$, we obtain a unique such map that fixes $0,1$ and $\infty$, and we also see from the construction that it must take values 0,1 and $\infty$ with orders 2, 3 and 2, respectively. We readily find that
\begin{align*}
R(z) &= \frac{z^2(z-3)}{1-3z} = \Real(R(z)) + i \frac{2y\big(y^2(4-3x)-3x(x-1)^2\big)}{(1-3x)^2+9y^2},
\end{align*}
and hence obtain \eqref{Eq:Variety} for the pre-image $R^{-1}(\mathbb{R}) = \{z \; : \; \Imag(R(z)) = 0\}$.
Lastly, we note that the $\pi/3$ intersection angle of $\gamma_0(\tau_0-)$ with $\mathbb{R}$ follows from either the explicit formula for the intersection angle $\pi \theta$ of such drivers in \cite[Prop. 3.2]{LMR} using $\kappa = 8/\sqrt{3}$, or from observing that $R(z)-1 = (z-1)^3(1+g(z))$ for a function $g$ analytic and vanishing near $z=1$, and hence the pullback of an interval around $x=1$ by $R$ looks like the pull back of an interval $(-\epsilon, \epsilon)$ around $x=0$ by $z \mapsto z^3$.
\end{proof}
\begin{remark}\label{Remark:FurtherComputations}
We note one other possible explicit calculation: for $0 \leq t \leq \tau$, the Loewner energy of $\gamma_\theta([0,t])$ is
\begin{align*}
I(\gamma_\theta([0,t])) = -4\log\Big(\sin^2(\theta) + \frac{\cos^2(\theta)}{\sin^4(\theta)} y(t)^4 \Big),
\end{align*}
where $y(t)^4 = \Imag\big(g_t(\gamma_\theta(\tau))-\lambda(t)\big)^4$ is the square of \eqref{Eq:Wangy}. To see this, recall that the curve remaining after mapping down $\gamma_\theta([0,t])$ is the minimizer through angle $\alpha(t) := \arctan(y(t)/x(t))$ (by symmetry we may assume that $0 < \theta < \pi/2$, and thus likewise that $\alpha \in (0,\pi/2)$), and so has energy $-8\log(\sin(\alpha(t)))$. Recalling \eqref{Eq:WangLambdaStep2}, we thus see the energy of the first portion $\gamma_\theta([0,t])$ of the curve is
\begin{align*}
-8\log(\sin(\theta)) + 8\log(\sin(\alpha(t))) &= -8\log(\sin(\theta)) - 4\log\Big(1 + \frac{\cos^2(\theta)}{\sin^6(\theta)}y(t)^4\Big),
\end{align*}
as claimed.
\end{remark}
\subsection{Corollary: boundary geodesic pairs}
Let $D \subset \mathbb{C}$ be a simply-connected domain with boundary prime ends $a$ and $b$ and let $\zeta \in D$. In \cite{MRW}, Marshall, Rohde and Wang defined a \emph{geodesic pair} in $(D; a,b,\zeta)$ as a simple curve $\gamma \subset D$, continuously parametrized on $(0,\infty)$, say, such that $\gamma(\tau)=\zeta$ for some $\tau \in (0,\infty)$, and where $\gamma(0,\tau)$ is the hyperbolic geodesic in $D{\backslash} \gamma[\tau,\infty)$ from $a$ to $\zeta$, and $\gamma(\tau,\infty)$ is the hyperbolic geodesic in $D{\backslash} \gamma(0,\tau]$ from $\zeta$ to $b$. In \cite{MRW}, $\zeta$ always lies in the interior of $D$, but as a corollary to Theorem \ref{Thm:Wang}$(\ref{Thm:Wang0+})$ we may extend this definition to include $\zeta \in \partial D$. Indeed, when $a_1, a_2$ and $\zeta$ are boundary prime ends of $D$, we define a \emph{boundary geodesic pair} in $(D;a_1,a_2,\zeta)$ to be two simple curves $\gamma_1, \gamma_2 \subset D$ which connect $a_1$ to $\zeta$ and $\zeta$ to $a_2$, respectively, do not intersect in $D$, and have the property that $\gamma_j$ is the hyperbolic geodesic from $a_j$ to $\zeta$ in its component of $D {\backslash} \gamma_{3-j}$, $j=1,2$. We require that $\zeta$ is distinct from the $a_j$ but allow $a_1=a_2$.
The following is the boundary geodesic pair version of \cite[Thm. 2.5]{MRW}.
\begin{corollary}\label{Cor:BoundaryGeodesic}
If $D \subset \mathbb{C}$ is a simply-connected domain with boundary prime ends $a_1,a_2$ and $\zeta$, where $\zeta \notin \{a_1,a_2\}$, there is a unique boundary geodesic pair $\gamma_1 \cup \gamma_2$ in $(D;a_1,a_2,\zeta)$. If $a_1 \neq a_2$ and $\partial D$ has a tangent at $a_j$, then $\gamma_j$ intersects $\partial D$ orthogonally at $a_j$. If $\partial D$ has a tangent at $\zeta$, then $\gamma_1$, $\gamma_2$ and $\partial D$ form three angles of $\pi/3$ at $\zeta$, and the hyperbolic geodesic $\eta_j$ in $D$ from $a_j$ to $\zeta$ bisects the angle between the $\gamma_j$. If $a_1=a_2$, then $\gamma_1$, $\gamma_2$ and $\partial D$ form three angles of $\pi/3$ when $\partial D$ has a tangent at either $a_1$ or $\zeta$.
\end{corollary}
In the case of $\zeta \in D$, the tangent at $\zeta$ to the geodesic pair bisects the angle between the hyperbolic geodesic segments $\eta_j$ from $a_j$ to $\zeta$ in $D$ \cite[Thm. 2.5]{MRW}. When $\zeta \in \partial D$ (and $\partial D$ has a tangent at $\zeta$), then $\eta_1$ and $\eta_2$ intersect tangentially at $\zeta$, and this no longer holds. Corollary \ref{Cor:BoundaryGeodesic} says that the roles flip and that each of the $\eta_j$ now bisect the angle between the $\gamma_j$. See Figure \ref{Fig:BoundaryGeodesics}.
\begin{figure}
\centering
\includegraphics[scale=0.1]{Boundary_geodesics.jpg}
\caption{\small When $\zeta$ lies in the interior of $D$, the smooth geodesic pair bisects the angle between the hyperbolic geodesics $\eta_j$ connecting the $a_j$ to $\zeta$. When $\zeta \in \partial D$, the roles are reversed.}
\label{Fig:BoundaryGeodesics}
\end{figure}
\begin{proof}
The proof of existence is nearly identical to that for \cite[Thm. 2.5]{MRW}; in the case $a_1 \neq a_2$, we simply apply conformal invariance to transport the geodesic pair $\gamma_0 \cup \gamma_0^*$ in $(\mathbb{H};0,\infty,1)$ constructed in Theorem \ref{Thm:Wang}$(\ref{Thm:Wang0+})$ to the domain in question. Uniqueness follows from the fact that, given a geodesic pair $(\eta_1 \cup \eta_2)$ in $(\mathbb{H};0,\infty,1)$, we can use it as in the proof of Theorm \ref{Thm:Wang}$(\ref{Thm:Wang0+})$ to build a degree-three analytic branched cover $R$ of $\ensuremath{\hat{\mathbb{C}}}$, resulting in the same variety \eqref{Eq:Variety} after post-composition by an element of $PSL(2,\mathbb{R})$.
When $a_1=a_2$, we can without loss of generality take $(D;a_1,a_1,\zeta) = (\mathbb{H};0,0,\infty)$, and we readily see that the two rays $\gamma_1 = \{\, re^{\pi i/3} \; : \; r>0 \,\}$ and $\gamma_2 = \{\, re^{2\pi i/3} \; : \; r>0 \,\}$ form a geodesic pair satisfying the stated properties. Given any geodesic pair in $(\mathbb{H}; 0,0,\infty)$, the rational function $R$ we construct is again a third-degree branched cover of $\ensuremath{\hat{\mathbb{C}}}$, with ramification only above $0$ and $\infty$, each with degree three. It follows that $R(z) =cz^3$ for some $c \in \mathbb{R} {\backslash} \{0\}$, and we obtain the same pair of rays $\gamma_1 \cup \gamma_2$.
\end{proof}
\section{The energy minimizer for weldings (EMW) family}\label{Sec:EMW}
We now turn our attention to the second question in the introduction: what is the infimal energy needed to weld a given $x<0<y$, and what is the nature of minimizing curves, if they exist? Our approach is entirely parallel to that for the Wang minimizers in \S\ref{Sec:Wang}; we first prove existence of minimizers (Lemma \ref{Lemma:WeldingEnergyMinimizerExists}), then an ``even approach'' property for finite-energy curves (Lemma \ref{Lemma:EvenApproach}), and then collect together our main results on the minimizers (Theorem \ref{Thm:EMW}).
\begin{lemma}\label{Lemma:WeldingEnergyMinimizerExists}
For any $x_0<0<y_0$, there exists a driver $\xi$ with $\xi(0)=0$ which welds $x_0$ to $y_0$ under its upwards Loewner flow and satisfies
\begin{align*}
I(\xi) = \inf_{\eta \in \mathcal{W}} I(\eta),
\end{align*}
where $\mathcal{W}$ is the family of all drivers $\eta$ that weld $x_0$ to $y_0$, have $\eta(0)=0$, and generate a simple curve $\gamma^\eta$.
\end{lemma}
Note that the infimum would be the same if we allowed drivers in $\mathcal{W}$ that did not generate simple curves, as these drivers have infinite energy (finite-energy curves are quasi-arcs and thus simple \cite{WangReverse}).
Conceptually, the proof is nearly identical to the argument for Lemma \ref{Lemma:AngleEnergyMinimizerExists}; we again use compactness and lower semicontinuity of energy. A slight twist is why a limiting driver $\xi \in \mathcal{W}$, for which we use a convergence result in \cite{TVY}. We proceed with the details for completeness.
\begin{proof}
The orthogonal circular arc segment which welds $x_0$ to $y_0$ to its base has finite energy, see Lemma \ref{Lemma:CircularArc}, and so the infimum is finite. For $\eta \in \mathcal{W}$, let $\tau_\eta$ be the ``hitting time'' of $x_0,y_0$ under $\eta$, where $x(t), y(t)$ are the images of $x_0,y_0$ under the upwards flow generated by $\eta$. That is, $\tau_\eta$ is the first time when $x(\tau_\eta)=\eta(\tau_\eta)=y(\tau_\eta)$. Note that the $\tau_\eta$ are uniformly bounded over $\mathcal{W}$: as any curve $\gamma^\eta$ generated by $\eta \in \mathcal{W}$ has $\diam(\gamma^\eta) \asymp y_0-x_0$ \cite[top of p.74]{Lawler}, there exists $R>0$ such that $\gamma^\eta \subset B_R(0) \cap \mathbb{H}$ for all $\eta$, implying
\begin{align*}
\tau_\eta = \hcap(\gamma^\eta) \leq \hcap(B_R(0)\cap {\mathbb H}) = R^2 \hcap(B_1(0) \cap {\mathbb H}) = R^2
\end{align*}
by the monotonicity of hcap and explicit calculation. Thus, if $\{\eta_n\}$ is a sequence such that
\begin{align}\label{Lim:MinimalEnergyEMW}
I(\eta_n) \rightarrow \inf_{\eta \in \mathcal{W}} I(\eta) =:L,
\end{align}
then by flowing upwards, if necessary, with the constant driver $\eta_n(\tau)$ from the moment $\tau = \tau(\eta_n)$ that $x_0$ and $y_0$ are welded together, we may assume that each $\eta_n$ is defined on the same interval $[0,T]$. Hence $\{\eta_n\}$ is a bounded subset of $W^{1,2}([0,T])$, and so is precompact in $C([0,T])$ (recall the discussion of Loewner energy in \S\ref{Sec:BackgroundLoewnerEnergy}). If $\eta_{n} \rightarrow \xi$ is any uniform sub-sequential limit, by lower semicontinuity
\begin{align*}
I(\xi)\leq \liminf_{n \rightarrow \infty} I(\eta_{n}) =:L,
\end{align*}
and so $I(\xi) = L$ if $\xi \in \mathcal{W}$. As noted above, $\xi$ generates a simple curve $\gamma$ since $I(\xi) < \infty$, and so we just have to ensure that $\xi$ welds $x_0$ to $y_0$. By flowing up with the constant driver $\xi(\tau)$, if necessary, we may assume that the welding $\varphi$ for $\gamma$ is defined for $x_0$. Similarly extending the $\xi_n$, we still have $\xi_n \rightarrow \xi$ uniformly on a fixed time interval $[0,T']$. Since uniform driver convergence implies welding convergence, \cite{TVY}, we have $y_0 = \varphi_{n}(x_0) \rightarrow \varphi(x_0)$.
\end{proof}
\begin{lemma}[``Even welding approach'']\label{Lemma:EvenApproach}
Let $\xi$ be a driver with $\xi(0)=0$ and finite energy which welds $x_0 < 0 <y_0$ together at time $\tau$ under its upwards Loewner flow. Then if $x(t)$ and $y(t)$ are the positions of $x_0$ and $y_0$ under the centered upwards flow,
\begin{align*}
\lim_{t \rightarrow \tau^-}\frac{y(t)}{y(t)-x(t)} = \frac{1}{2}.
\end{align*}
Equivalently, $r(t):=-x(t)/y(t) \rightarrow 1$.
\end{lemma}
\begin{proof}
We claim that there is a positive lower bound $m(r)$ for the energy to weld points with ratio $r=r(0) = -x_0/y_0 \neq 1$ that is non-decreasing in $|r-1|$. By symmetry we may suppose $r>1$.
Note that if we initially flow \emph{downwards} with the constantly-zero driver for time $t$, the images of $x_0,y_0$ have ratio $\sqrt{x_0^2+4t}/\sqrt{y_0^2+4t}$, which monotonically decreases to 1 as $t \rightarrow \infty$ (recall the explicit solution to the Loewner equation in this case, see \S\ref{Sec:BackgroundLoewner}). So we may weld $-\sqrt{x_0^2+4t}$ to $\sqrt{y_0^2+4t}$ in the upwards flow by starting with the zero driver for time $t$ followed by $\xi$, which uses the same amount energy as $\xi$, thus showing
\begin{align*}
m\Big( \frac{\sqrt{x_0^2+4t}}{\sqrt{y_0^2+4t}} \Big) \leq m\Big(\frac{-x_0}{y_0}\Big).
\end{align*}
We conclude $m(r)$ is non-decreasing for $r>1$. Furthermore, $m(r)>0$ when $r \neq 1$, since minimizers exist by Lemma \ref{Lemma:WeldingEnergyMinimizerExists}, and a minimizer with zero energy can only weld symmetric points. Hence if $\xi$ welds $x_0$ to $y_0$ with finite energy, as \begin{align*}
m(r(t)) \leq \frac{1}{2}\int_t^\tau \dot{\xi}(s)^2ds \rightarrow 0
\end{align*}
as $t \rightarrow \tau^-$, we have $r(t) \rightarrow 1$.
\end{proof}
We provide an exact formula for $m(r)$ in \eqref{Eq:EMWEnergyFormula} below.
As an aside, we note that Lemmas \ref{Lemma:WangEvenApproach} and \ref{Lemma:EvenApproach} state that points welded by finite-energy $\xi$ ``evenly'' approach $\xi$ and then move up into $\mathbb{H}$ perpendicularly from the real line. While both properties hold for finite-energy curves, it is instructive to note that they are not equivalent, as the following example shows.\footnote{The fact that both properties hold for finite-energy curves is reminiscent of the fact that $I(\gamma) = I(-1/\gamma)$, Wang's reversibility of the Loewner energy \cite{WangReverse}. Reversibility gives a global sense in which finite-energy curves ``look the same'' in both directions, and, informally speaking, lemmas \ref{Lemma:WangEvenApproach} and \ref{Lemma:EvenApproach} give a local sense in which this also holds.}
\begin{figure}
\centering
\subfloat[][]{
\includegraphics[width=0.2\textwidth]{MECEgAllPDF.pdf}
\label{Fig:SubfigA}
}\qquad
\subfloat[][]{
\includegraphics[width=0.2\textwidth]{MECEgMapDownPDF.pdf}
\label{Fig:SubfigB}
}
\caption{\small{A curve $\gamma$ in subfigure \ref{Fig:SubfigA} which shows the properties in lemmas \ref{Lemma:WangEvenApproach} and \ref{Lemma:EvenApproach} are not equivalent. Subfigure \ref{Fig:SubfigB}} shows the image of $\gamma$ after mapping down the vertical line segment. While $\gamma$ satisfies the property in Lemma \ref{Lemma:WangEvenApproach} for all $\tau$, it does not satisfy the property of Lemma \ref{Lemma:EvenApproach} at its corner.}
\label{Fig:NotEquiv}
\end{figure}
\begin{example}\label{Eg:NotEquiv}
For $c>0$, consider the curve $\gamma$ whose downward driver is
\begin{align*}
\lambda(t) = \begin{cases}
0 & 0 \leq t \leq 1,\\
c\sqrt{t} & 1 < t \leq 2.
\end{cases}
\end{align*}
That is, the base of $\gamma$ is a vertical line segment, and when this is mapped down, what remains is a line segment which meets $\mathbb{R}_{\geq 0}$ at angle $\alpha \pi \in (0,\pi/2)$ for some $\alpha = \alpha(c)$ \cite[Example 4.12]{Lawler}, see Figure \ref{Fig:NotEquiv}. We claim that $\gamma$ satisfies the ``even angle approach'' property of Lemma \ref{Lemma:WangEvenApproach}, but not the ``even welding approach'' property of Lemma \ref{Lemma:EvenApproach}.
With respect to the former, it is not hard to see that for each $0<\tau \leq 2$,
\begin{align*}
\lim_{t \rightarrow \tau^-} \Arg(G_t(\gamma(\tau)) = \frac{\pi}{2}.
\end{align*}
This is obvious for $0 < \tau \leq 1$. If $1 < \tau \leq 2$, set $s:=1+(\tau-1)/2$ and first map down with $G_{s}$. The remaining curve $\tilde{\gamma}(t) := G_{s}(\gamma(t + s))$ on $0 \leq t \leq 2-s$ has driver $\tilde{\lambda}(t) = c\sqrt{t + s} - c \sqrt{s}$, which has finite energy, and so by Lemma \ref{Lemma:WangEvenApproach}, the image of $\gamma(\tau)$ must approach the imaginary axis as we continue to flow down.
On the other hand, the property in Lemma \ref{Lemma:EvenApproach} does not hold for the points $x_0,y_0$ which weld under the upwards flow to $\gamma(1)$, the corner of the curve. To see this, map down the vertical line segment; since the remaining curve $G_1(\gamma([1,2]))$ is a line segment with angle $\alpha \pi$ to $\mathbb{R}$, we claim that further pulling down any small portion $G_1(\gamma([1,1+\epsilon]))$ with the centered mapping down function sends the base to points $x(\epsilon) < 0 < y(\epsilon)$ that satisfy
\begin{align}\label{Eq:RatioBad}
\frac{y(\epsilon)}{y(\epsilon) - x(\epsilon)} = \alpha \neq \frac{1}{2}.
\end{align}
One can see this explicitly; the conformal map $F:\mathbb{H} \rightarrow \mathbb{H} {\backslash} L_\alpha$,
\begin{align*}
L_\alpha: = \{ re^{i \alpha \pi} \, : \, 0 \leq r \leq \alpha^\alpha(1-\alpha)^{1-\alpha} \},
\end{align*}
which satisfies $F(z) = z +O(1)$ as $z \rightarrow \infty$ is $F(z) = (z-\alpha)^\alpha(z+1-\alpha)^{1-\alpha}$ (see the construction in \cite[\S1 ``The Slit Algorithm'']{MRZip}, for instance). This sends $\alpha-1$ and $\alpha$ to the base of the curve, which by scaling shows \eqref{Eq:RatioBad}. Alternatively, the ratio of the harmonic measures of either side of $G_1(\gamma([1,1+\epsilon]))$ as seen from $\infty$ is always $r \neq 1$, independent of $\epsilon$. So by conformal invariance of harmonic measure, the two intervals $[x(\epsilon),0]$, $[0, y(\epsilon)]$ one obtains upon mapping down also have the same ratio of lengths, which is likewise equivalent to \eqref{Eq:RatioBad}. Thus the corner in $\gamma$ breaks the ``even welding approach'' property.
\end{example}
We call the curve family whose existence is given by the follow theorem the \emph{energy minimizers for welding (EMW) family}.
\begin{theorem}\label{Thm:EMW}(Energy)
Fix $x_0<0<y_0$.
\begin{enumerate}[$(i)$]
\item \label{Thm:EMWEnergy} There exists a unique driving function $\xi$ with $\xi(0)=0$ which welds $x_0$ to $y_0$ and minimizes the Loewner energy among all such drivers. This driver is $C^\infty([0,\tau))$, monotonic and satisfies
\begin{align}\label{Eq:EMWEnergyFormula}
I(\xi) = \frac{1}{2}\int_0^\tau \dot{\xi}(t)^2dt = -8 \log(2\sqrt{\alpha(1-\alpha)}) = -8\log \Big( \frac{2\sqrt{r}}{1+r} \Big)
\end{align}
where $\alpha := \frac{y_0}{y_0-x_0}$, $r:=-x_0/y_0$, and $\tau$ is the time $\xi$ takes to weld $x_0$ to $y_0$.
\item \label{Thm:EMWDriver}(Driver and welding) When $y_0 \neq -x_0$, the downwards driving function $\lambda = \lambda_{x_0,y_0}$ for the minimizer is explicitly
\begin{align}\label{Eq:EMWDriverFormula}
\lambda(t) = -\frac{16}{\sqrt{3}}\, \frac{y_0+x_0}{|y_0+x_0|}\, t^{3/2}\left( \frac{(y_0-x_0)^2}{24^{2/3}|y_0+x_0|^{2/3}} + 2 \Real \Big( \sqrt{\frac{(y_0-x_0)^6}{576(y_0+x_0)^2}-t^2} +it \Big)^{2/3} \right)^{-3/2}
\end{align}
for $0 \leq t \leq \frac{1}{24}(x_0^2-4x_0y_0+y_0^2) =: \tau = \tau(x_0,y_0)$, where $z \mapsto z^{2/3}$ is defined with the principal branch of the logarithm.
In particular, $\lambda(0)=0$ and $\lambda(\tau) = -\frac{2}{3}(x_0+y_0)$.
When $r = -x_0/y_0 <1$, the conformal welding $\varphi:[x_0,0] \rightarrow [0,y_0]$ corresponding to $\lambda$ satisfies the implicit equation
\begin{align}\label{Eq:EMWWeldImplicit}
W(r,T \circ \varphi \circ T^{-1}(x)) = W(r,x)
\end{align}
for all $-\infty< x \leq y_0/x_0$, where
\begin{align*}
W(r,x) := rx + \frac{1}{x} + (1-r)\log(x),
\end{align*}
and $T(x) = T_{x_0,y_0}(x) := \frac{x-y_0}{x-x_0}$. When $r>1$, one obtains a similar implicit equation by reflecting across the imaginary axis and using $\varphi^{-1}$.
\item \label{Thm:EMWSLE}(SLE$_0$) The upwards driver $\xi(t) = \lambda(\tau - t) - \lambda(\tau)$ is upwards SLE$_0(-4,-4)$ starting from $(0,x_0,y_0)$.
\item \label{Thm:EMWSingleCurve}(Universal curve) The curve $\Gamma \subset \mathbb{H}\cup\{0\}$ satisfying
\begin{align}\label{Eq:EMWUniversalVariety}
(x^2+y^2)^2 = -4xy,
\end{align}
continuously parametrized $t \mapsto \Gamma(t)$ so that its base is perpendicular to $\mathbb{R}$, is universal for the EMW curves, in the sense that for each ratio $r\in(0,1)$, there exists $t_r$ such that the segment $\Gamma([0,t_r])$ is an EMW curve which welds points $x_r<0<y_r$ with ratio $r = -x_r/y_r$ to its base. $\Gamma$'s downwards driving function is
\begin{align*}
\lambda_\Gamma(t) = -\frac{16}{\sqrt{3}}\,t^{3/2}\Big((\pi/6)^{2/3} + 2 \Real \big(\sqrt{(\pi/6)^2 -t^2} + it\big)^{2/3} \,\Big)^{-3/2}, \qquad 0 \leq t \leq \pi/6.
\end{align*}
Furthermore, $\Gamma^2 = \{z^2 \, : \, z \in \Gamma\}$ is the circle $x^2 + (y+1)^2 =1$. The reflection $(x^2+y^2)^2 = 4xy$ of $\Gamma$ across the imaginary axis gives a similar universal curve $\Gamma'$ for ratios $r \in (1,\infty)$ with driver $\lambda_{\Gamma'} = -\lambda_{\Gamma}$.
\item\label{Thm:DistinctFamilies} The Wang minimizer $\gamma_\theta$, $0 < \theta < \pi$, coincides with an EMW curve if and only if $\theta =\pi/2$.
\end{enumerate}
\end{theorem}
\begin{remark}\label{Remark:DualFam}
We preceed the proof with two comments.
\begin{enumerate}[$(a)$]
\item\label{Remark:RemainingEMW} Fix $x_0 < 0 <y_0$ with $-x_0 \neq y_0$, and flow up with the corresponding EMW driver $\xi$ for some time $t<\tau$. Theorem \ref{Lemma:Inf98}($\ref{Thm:LocalCompareEMW}$) below shows that this initial ``Loewner portion'' of the curve is generally not itself an EMW curve, as we may take $\eta = \xi$. However, the remaining curve that $\xi$ generates on $[t,\tau]$ must be: if not, then we could replace $\xi$ with the corresponding EMW driver for $x(t)$ and $y(t)$ and lower the energy. Hence the symmetry of the EMW family is with respect to the ``base,'' or what remains to flow up, rather than with respect to the ``top,'' or the curve segment already flown up into $\mathbb{H}$. In terms of the downwards Loewner flow of an EMW curve $\gamma$, this says $\gamma[0,t]$ always is an EMW curve, whereas $G_t(\gamma[0,t])$ never is (which is the opposite of what we saw with the Wang minimizers in Remark \ref{Remark:RemainingWang}($\ref{Remark:RestOfWang}$)). From this point of view, it is natural to write the driver formula \eqref{Eq:EMWDriverFormula} in terms of the downwards driver $\lambda$ rather than its reversal $\xi$.
\item We note that the only complex number appearing in the driver formula \eqref{Eq:EMWDriverFormula} is the $it$ term within the $2/3$-power, as we show below in the discussion around \eqref{Ineq:AlwaysNegative} that $t \in [0,\tau]$ implies $t^2 \leq (y_0-x_0)^6/(576(y_0+x_0)^2)$. That is, $a \geq 0$ in the $\Real ((\sqrt{a}+it)^{2/3})$ term of the driver formula.
\end{enumerate}
\end{remark}
To simplify notational clutter, we will interchangeably use $\xi(t)$ and $\xi_t$ and similarly for other functions of $t$.
\begin{proof}
Since $I(\xi) =L < \infty$, $\xi$ is absolutely continuous. We first show
\begin{align}\label{Eq:EMWDeriv}
\dot{\xi}(t) = -4\Big( \frac{1}{x(t)} + \frac{1}{y(t)} \Big)
\end{align}
at all times $t$ where $\dot{\xi}$ exists and where $t$ is a Lebesgue point of $\dot{\xi}^2$, and where $x(t)$ and $y(t)$ are the images of $x_0$ and $y_0$ under the centered upwards Loewner flow maps $H_t(z) := h_t(z)-\xi(t)$. By \eqref{Eq:LoewnerEqUp}, we have that
\begin{align}\label{Eq:LoewnerEqUpCentered}
\dot{H}_t(z) = - \frac{2}{H_t(z)} - \dot{\xi}(t)
\end{align}
at points $t$ of differentiability of $\xi$. By Lemma \ref{Lemma:EvenApproach}, $r(t):=-x(t)/y(t) \rightarrow 1$, and since $\xi$ minimizes energy, $r$ must (strictly) monotonically approach 1. Indeed, if $r(t_0) = 1$ for some $t_0 < \tau$, then we must have $r(t) \equiv 1$ for all $t_0 \leq t \leq \tau$. If $1 \neq r(t_1)=r(t_2)$ for some $t_1 < t_2$, then $\xi$ cannot be constant on $[t_1,t_2]$, because a ratio distinct from 1 changes under the constant driver. So energy would be unnecessarily wasted on $[t_1,t_2]$, and we conclude that the claimed monotonicity holds.
To be a minimizer, $\xi$ must expel as little energy as possible to move the ratio to $1$. For $I(t) := \frac{1}{2}\int_0^t \dot{\xi}(s)^2ds$, we thus need to optimize $dr/dI$, in the sense of maximizing it when $r <1$ and minimizing it when $r>1$.
Suppose first that $\xi$ is right-differentiable at $t=0$ and that $t=0$ is a Lebesgue point for $\dot{\xi}^2$. Then the energy expelled on a small interval $[0,\Delta t]$ is $\frac{1}{2}\dot{\xi}^2(0)\Delta t + o(\Delta t)$, and $r$ is right-differentiable at $t=0$ with
\begin{align}\label{Lim:drdI}
\Delta r = \dot{r}(0)\Delta t + o(\Delta t) = \Big( \frac{2(y^2-x^2)}{xy^3} + \frac{\dot{\xi}}{y^2}(y-x) \Big)\Delta t + o(\Delta t)
\end{align}
by the Loewener equation \eqref{Eq:LoewnerEqUpCentered}, where $x,y$ and $\dot{\xi}$ are all evaluated at $t=0$, and $\dot{\xi}$ is the right derivative of $\xi$. We thus have
\begin{align*}
\frac{\Delta r}{\Delta I} \rightarrow \frac{4(y^2-x^2)}{\dot{\xi}^2 xy^3}+ \frac{2(y-x)}{\dot{\xi}y^2}
\end{align*}
as $\Delta t \rightarrow 0$. This expression is optimized with respect to $\dot{\xi}$ when $\dot{\xi}(0)$ satisfies \eqref{Eq:EMWDeriv} at $t=0$, in the sense of yielding a local max if $r(0)<1$ and a local min if $r(0)>1$. Thus any minimizer for which $\dot{\xi}(0)$ exists and where $t=0$ is a Lebesgue point of $\dot{\xi}^2$ must satisfy \eqref{Eq:EMWDeriv} at $t=0$.
More generally, let $t_0$ be a point of differentiability of $\xi$ and a Lebesgue point of $\dot{\xi}^2$. Note that the remaining driver $\xi|_{[t_0,\tau]}$ must be an energy minimizer for welding $x(t_0)$ to $y(t_0)$, as noted above in Remark \ref{Remark:DualFam}. By the previous paragraph, we see $\dot{\xi}(t_0)$ satisfies \eqref{Eq:EMWDeriv}. Since $\xi$ is differentiable at a.e. $t$ and a.e. $t$ is a Lebesgue point of the integrable function $\dot{\xi}^2$, we have \eqref{Eq:EMWDeriv} at a.e. $t$.
Plugging this formula into \eqref{Eq:LoewnerEqUpCentered} yields that $\xi(t),x(t)$ and $y(t)$ are absolutely continuous a.e. solutions to the system of differential equations
\begin{align}\label{EMWDiffEqSystem}
\dot{\xi}(t) = -\frac{4}{x(t)} - \frac{4}{y(t)}, \qquad \dot{x}(t) = \frac{2}{x(t)} + \frac{4}{y(t)}, \qquad \dot{y}(t) = \frac{4}{x(t)} + \frac{2}{y(t)}
\end{align}
with $\xi(0)= x(0)-x_0 = y(0)-y_0=0$, $0 \leq t < \tau$. Since each function is absolutely continuous, replacing the derivatives with these continuous expressions for all $t$ does not change the values of the functions, and we see that each function is actually $C^1$. Then each of the right-hand sides in \eqref{EMWDiffEqSystem} is $C^1$, and so each of the three functions is at least $C^2$. Continuing to bootstrap we see the functions are smooth on $[0,\tau)$.
Note also that classical solutions to \eqref{EMWDiffEqSystem} are unique: by scale invariance we may assume $\min\{-x,y\} =1$, and for small times $0 \leq t \leq t_0$, $|x(t)|$ and $|y(t)|$ are therefore both bounded below since $\xi$ is $\sqrt{L}$-H\"{o}lder -1/2 continuous. Hence the map $f(t,\xi,x,y) = (-4(x^{-1}+y^{-1}), 2x^{-1}+4y^{-1}, 4x^{-1}+2y^{-1})$ is Lipschitz in $(\xi,x,y)$ on $[0,t_0]$, giving uniqueness. Note that uniqueness also gives that if $r(0) \neq 1$, $r(t) \neq 1$ for all $t< \tau$, and so \eqref{Eq:EMWDeriv} also shows that $\xi$ is strictly monotone.
For the energy formula, we note from \eqref{Lim:drdI} and our formula \eqref{Eq:EMWDeriv} for $\dot{\xi}$ that we have the ODE
\begin{align*}
\frac{dI}{dr} = \frac{4y}{x}\cdot \frac{x+y}{x-y} = \frac{4}{r} \cdot\frac{1-r}{1+r} = \frac{4}{r} - \frac{8}{r+1},
\end{align*}
and therefore
\begin{align*}
I(r(t)) - 0 = 4\log\Big(\frac{r(t)}{r_0} \Big) - 8\log\Big( \frac{r(t)+1}{r_0+1} \Big).
\end{align*}
Since $r(t) \rightarrow 1$, sending $t \rightarrow \tau^-$ yields \eqref{Eq:EMWEnergyFormula}.
\bigskip
For the explicit formulas in $(ii)$, we first derive the hitting time $\tau$, which follows from observing from \eqref{EMWDiffEqSystem} that
\begin{align}\label{Eq:EMWTime}
\frac{d}{dt}((y_t-x_t)^2 -2x_ty_t) = -24
\end{align}
for $0 \leq t < \tau$. Hence integrating from $0$ to $\tau-\epsilon$ and sending $\epsilon \rightarrow 0$ yields
\begin{align*}
- (y_0-x_0)^2 +2x_0y_0 = -24\tau,
\end{align*}
as claimed (recall $x_\tau = y_\tau = 0$). Also, if $A_t = (x_t+y_t)/2$, we find $\dot{A}_t = -\frac{3}{4}\dot{\xi}_t$ and hence
\begin{align}\label{Eq:EMWEndPoint}
0 - \frac{1}{2}(x_0+y_0) = -\frac{3}{4}\xi_\tau -0 \qquad \text{or} \qquad \xi_\tau = \frac{2}{3}(x_0+y_0).
\end{align}
We next claim that, in addition to \eqref{Eq:EMWDeriv}, $\xi$ also satisfies the ODE
\begin{align}\label{Eq:EMWDeriv2}
\dot{\xi}_t = \frac{16(\xi_t - \xi_\tau)}{(\xi_t-\xi_\tau)^2 - \frac{32}{3}(\tau-t)}
\end{align}
for $0 \leq t < \tau$. To obtain this, we first observe that after flowing up for some time $t_0 < \tau$, the driver $\tilde{\xi}_t:=\xi_{t+t_0} - \xi_{t_0}$ on $0 \leq t \leq \tau-t_0$ is the EMW driver for welding $x_{t_0}$ to $y_{t_0}$, as discussed in Remark \ref{Remark:DualFam}($\ref{Remark:RemainingEMW}$). Thus, from \eqref{Eq:EMWEndPoint}, we have
\begin{align*}
\frac{2}{3}(x_{t_0}+y_{t_0}) = \tilde{\xi}_{\tau-t_0} = \xi_{\tau} - \xi_{t_0},
\end{align*}
or in other words
\begin{align*}
\xi_t -\xi_\tau = -\frac{2}{3}(x_t+y_t)
\end{align*}
for any $0 \leq t \leq \tau$. Note that by integrating \eqref{Eq:EMWTime} from $t$ to $\tau-\epsilon$ and sending $\epsilon \rightarrow 0$ we also have
\begin{align*}
\tau - t = \frac{1}{24}\big((y_t-x_t)^2-2x_ty_t \big) = \frac{1}{24}\big((y_t+x_t)^2-6x_ty_t \big)
\end{align*}
for $0 \leq t \leq \tau$, and plugging in these last two formulas into the right-hand side of \eqref{Eq:EMWDeriv2} yields
\begin{align*}
\frac{16(\xi_t - \xi_\tau)}{(\xi_t-\xi_\tau)^2 - \frac{32}{3}(\tau-t)} = \frac{-\frac{32}{3}(x_t+y_t)}{\frac{4}{9}(x_t+y_t)^2-\frac{4}{9}\big((x_t+y_t)^2-6x_ty_t \big)} = -4 \Big( \frac{1}{x_t}+\frac{1}{y_t} \Big),
\end{align*}
thus verifying \eqref{Eq:EMWDeriv2} in light of \eqref{EMWDiffEqSystem}.
We proceed to obtain $\xi$ by solving \eqref{Eq:EMWDeriv2}, and start by introducing the change of variables
\begin{align*}
\nu(t) := \xi\Big(\tau + \frac{3t}{32} \Big) - \frac{2}{3}(x_0 + y_0).
\end{align*}
We will momentarily assume that $-x <y$, so that $\nu$ is a non-positive, increasing function defined for $-\frac{32}{3}\tau \leq t \leq 0$, and which vanishes only at $t=0$. In terms of $\nu$, \eqref{Eq:EMWDeriv2} says
\begin{align*}
\dot{\nu} = \frac{\frac{3}{2}\nu}{\nu^2+t},
\end{align*}
and so if we define $\mu(t) := \nu(t)^{-2/3}$, where $w \mapsto w^{-2/3}$ is the precalculus function mapping $\mathbb{R}\backslash\{0\}$ to $(0,\infty)$, we see
\begin{align*}
\dot{\mu} = \frac{-\mu}{\mu^{-3}+t}, \qquad \text{implying} \qquad \frac{d}{dt}\big( -\frac{1}{2} \mu^{-2} + t\mu \big) = 0,
\end{align*}
and thus $-\frac{1}{2} \mu(t)^{-2} + t\mu(t) = C$. Evaluating at $t = -\frac{32}{3}\tau$ yields
\begin{align*}
C = -\Big( \frac{2}{3}\Big)^{1/3}\frac{(y_0-x_0)^2}{(y_0+x_0)^{2/3}},
\end{align*}
and we see that $\mu$ satisfies
\begin{align}\label{Eq:muCubic}
\mu^3 - \frac{C}{t} \mu^2 - \frac{1}{2t} =0.
\end{align}
The corresponding depressed cubic in $\tilde{\mu} := \mu - \frac{C}{3t}$ is
\begin{align}\label{Eq:tmuCubic}
\tilde{\mu}^3 - \frac{C^2}{3t^2}\tilde{\mu} - \frac{2C^3}{27t^3} - \frac{1}{2t} = 0,
\end{align}
which has discriminant
\begin{align*}
4\Big(\frac{C^2}{3t^2}\Big)^3-27 \Big(\frac{2C^3}{27t^3} + \frac{1}{2t}\Big)^2 = -\frac{2C^3}{t^4}- \frac{27}{4t^2}.
\end{align*}
Hence there are three real solutions $\tilde{\mu}$ to \eqref{Eq:tmuCubic} when $\frac{27}{8}t^2 < -C^3$, or
\begin{align*}
\frac{81}{16}t^2 < \frac{(y_0-x_0)^6}{(y_0+x_0)^2},
\end{align*}
and as $-\frac{32}{3}\tau \leq t < 0$, we have $81t^2/16 \leq 576 \tau^2 = (x_0^2-4x_0y_0+y_0^2)^2.$ The inequality
\begin{align}\label{Ineq:AlwaysNegative}
(x_0^2-4x_0y_0+y_0^2)^2 < \frac{(y_0-x_0)^6}{(y_0+x_0)^2}
\end{align}
is equivalent to $(3y_0-x_0)(y_0-3x_0)>0$, which is always true, and so \eqref{Eq:tmuCubic} always has three real roots. Using Cardano's method, we see that the zeros are
\begin{multline*}
\tilde{\mu}_k = e^{2\pi ik/3}\sqrt[3]{\frac{1}{27t^3}\left(C^3 + \frac{27}{4}t^2 + \frac{27}{12}t\sqrt{\frac{8}{3}C^3 + 9t^2} \right)}\\
+e^{-2\pi ik/3}\sqrt[3]{\frac{1}{27t^3}\left(C^3 + \frac{27}{4}t^2 - \frac{27}{12}t\sqrt{\frac{8}{3}C^3 + 9t^2} \right)},
\end{multline*}
$k =0,1,2$, where all roots are the principal branches. Recalling that inputs $t$ are negative for $\nu$ (and hence also for $\mu$ and $\tilde{\mu}$), and also that $\frac{8}{3}C^3 +9t^2<0$ by \eqref{Ineq:AlwaysNegative}, this simplifies to
\begin{align*}
\tilde{\mu}_k = -\frac{2}{3t}\Real\Bigg(e^{2\pi ik/3}\sqrt[3]{-C^3 - \frac{27}{4}t^2 - \frac{27}{12}ti\sqrt{-\frac{8}{3}C^3 - 9t^2} }\,\Bigg).
\end{align*}
Near $t=0^-$, the three solutions $\mu_k$ for $\mu(t)=\tilde{\mu}(t) + \frac{C}{3t}$ thus satisfy
\begin{align*}
\mu_k = \frac{2C}{3t}\Big(\cos\Big( \frac{2\pi k}{3} \Big) + \frac{1}{2} \Big) + O(1).
\end{align*}
Since $\mu(t) = \nu(t)^{-2/3} \rightarrow +\infty$ as $t \rightarrow 0^-$, our desired solution is when $k=0$, yielding
\begin{align*}
\mu(t) = \frac{C}{3t}-\frac{2}{3t}\Real\Bigg(\sqrt[3]{-C^3 - \frac{27}{4}t^2 - \frac{27}{12}ti\sqrt{-\frac{8}{3}C^3 - 9t^2} }\,\Bigg).
\end{align*}
(Since $\Big|\sqrt[3]{-C^3 - \frac{27}{4}t^2 - \frac{27}{12}ti\sqrt{-\frac{8}{3}C^3 - 9t^2} }\Big| = -C$, it is also easy to see that this solution satisfies $\mu(t)>0$, as needed.) Undoing our changes of variable, we have
\begin{align*}
\xi(t) = \frac{2}{3}(x_0+y_0) - \mu^{-3/2}\Big(\frac{32}{3}(t-\tau)\Big),
\end{align*}
and using \eqref{Eq:EMWEndPoint}, we thus see that the downwards driver is $\lambda(t) = \xi(\tau-t)-\xi(\tau) = -\mu^{-3/2}(-32t/3)$. Following arithmetic, we thus have
\begin{align*}
\lambda(t) &= -128\sqrt{3}\,t^{3/2}\Bigg(\frac{(y_0-x_0)^2}{(y_0+x_0)^{2/3}}+2\Real \sqrt[3]{\frac{(y_0-x_0)^6}{(y_0+x_0)^2}-1152t^2+48it\sqrt{\frac{(y_0-x_0)^6}{(y_0+x_0)^2}-576t^2}} \,\Bigg)^{-3/2}\\
&= -\frac{16}{\sqrt{3}}\,t^{3/2}\Bigg(\frac{(y_0-x_0)^2}{24^{2/3}(y_0+x_0)^{2/3}}+2\Real \sqrt[3]{\bigg( \sqrt{\frac{(y_0-x_0)^6}{576(y_0+x_0)^2}-t^2}+it \bigg)^2}\,\Bigg)^{-3/2}.
\end{align*}
Since the complex number $w=\sqrt{(y_0-x_0)^6/576(y_0+x_0)^2-t^2}+it$ is always in the first quadrant, we may interchange the order of the square and principal cube root and simply write $\sqrt[3]{w^2} = w^{2/3}$.
We assumed $-x_0<y_0$, and by symmetry the driver $\lambda_{x_0,y_0}(t)$ for $y_0<-x_0$ is $-\lambda_{-y_0,-x_0}(t)$. We conclude that the proposed formula \eqref{Eq:EMWDriverFormula} holds in either case.
We prove the implicit equation \eqref{Eq:EMWWeldImplicit} for the welding below at the end of our proof for part $(\ref{Thm:EMWSingleCurve})$.
\bigskip
For $(\ref{Thm:EMWSLE})$, we note the upwards SLE$_0(-4,-4)$ ODE's for driver $\eta_t$ and force points $u_t, v_t$ are
\begin{align}\label{Eq:SLE_0System}
\dot{\eta}_t = \frac{-4}{u_t - \eta_t} + \frac{-4}{v_t-\eta_t}, \qquad \dot{u}_t = \frac{-2}{u_t - \xi_t}, \qquad \dot{v}_t = \frac{-2}{v_t-\xi_t}.
\end{align}
If $x_t:=u_t-\xi_t$ and $y_t:=v_t - \xi_t$ are the flow of the centered force points, the first ODE becomes
\begin{align*}
\dot{\eta}_t = -\frac{4}{x_t} - \frac{4}{y_t},
\end{align*}
which is precisely what we showed for $\xi$ in \eqref{EMWDiffEqSystem}. (The latter two equations in \eqref{Eq:SLE_0System} also hold for our $x_t = u_t + \xi_t$, $y_t = v_t+\xi_t$, since they are simply the dynamics for the upwards Loewner flow under driver $\xi$.)
\bigskip
For $(\ref{Thm:EMWSingleCurve})$, we start by arguing for the existence of a (initially non-explicit) universal curve $\Gamma$ and compute its driver. Note that if we have two ratios $0 < r_1 < r_2<1$ and we flow up with an energy-minimizing driver $\xi_1$ for $r_1$ (defined uniquely up to scaling), by Lemma \ref{Lemma:EvenApproach} the ratio $-x(t)/y(t)$ of its welding endpoints will monotonically approach 1 and hence be $r_2$ at some time $t_2$. At this point, as explained in Remark \ref{Remark:DualFam}($\ref{Remark:RemainingEMW}$), the remaining curve $\gamma_{r_2}$ generated by $\xi_1$ on $[t_2,\tau_1]$ is the energy-minimizing curve for $r_2$ (again, defined uniquely up to scaling). Thus the resulting curve $\gamma_{r_1}$ for $r_1$ contains $\gamma_{r_2}$ as a subcurve. Similarly, by starting with lower ratios $0 < r < r_1$ and rescaling the curve $\gamma_r$ to some $c_r\gamma_r$ so that the subcurve of $c_r\gamma_r$ corresponding to ratio $r_1$ is $\gamma_{r_1}$, we see that we obtain a simple curve $\bigcup_{0<r<1} c_r\gamma_r$, where each $c_r\gamma_r$ is a curve generated by an EMW driver with welding endpoints of ratio $r$, and where
\begin{align}\label{Inclusion:UniversalEMW}
c_{r_2}\gamma_{r_2} \subset c_{r_1}\gamma_{r_1} \qquad \text{whenever} \qquad r_1 < r_2.
\end{align}
Hence such a curve $\Gamma$ exists, and is defined uniquely up to scaling. We proceed to pick the representative among all scaled versions that will yield the exact algebraic formula \eqref{Eq:EMWUniversalVariety}.
Note that the inclusion \eqref{Inclusion:UniversalEMW} shows that the downwards driving function $\lambda_{r_1}$ of $c_{r_1}\gamma_{r_1}$ is simply an extension of the driver $\lambda_{r_2}$ for $c_{r_2}\gamma_{r_2}$, and hence the driver $\lambda_{\Gamma}$ for $\Gamma$ is the limit of the $\lambda_r$'s. Begin by choosing the nested family $\{c_r\gamma_r\}_{0 < r < 1}$ such that right-end of the welding interval $y_r$ for $c_r\gamma_r$ satisfies
\begin{align}\label{Lim:EMWUNiversaly}
y_r \rightarrow 2\sqrt{\pi} \qquad \text{ as } \qquad r \rightarrow 0,
\end{align}
and set $\Gamma:= \bigcup_{0 < r <1}\gamma_r$.\footnote{Such a choice of $\gamma_r$ is possible. For instance, as $r$ decreases to zero, we may re-scale the curves so that the right endpoint $y_r$ is always $2\sqrt{\pi}$. In the end the curve does not degenerate to the origin, since the diameter of each curve is comparable to $2\sqrt{\pi}$ \cite[Prop.4.4]{LMR}. Thus any choice of $\Gamma$ is bounded, and we may choose the scale for the limiting curve such that \eqref{Lim:EMWUNiversaly} holds.} Writing $\lambda(t) = \lambda(x,y,t)$, we have
\begin{align}
\lambda_{\Gamma}(t) = \lim_{r \rightarrow \infty} \lambda(x_r,y_r,t) &= \lambda(0,2\sqrt{\pi},t) \notag \\
&= -\frac{16}{\sqrt{3}}\,t^{3/2}\Big((\pi/6)^{2/3} + 2 \Real \big(\sqrt{(\pi/6)^2 -t^2} + it\big)^{2/3} \,\Big)^{-3/2} \label{Eq:EMWUniversalDriver}
\end{align}
for $0 \leq t \leq \pi/6 = \tau(0,2\sqrt{\pi})$, where the second equality is by the continuity in \eqref{Eq:EMWDriverFormula} at $(x,y) = (0,2\sqrt{\pi})$. Thus the claimed driver formula holds.
Now let $\tilde{\Gamma}$ be the curve in $\mathbb{H}$ satisfying the claimed algebraic formula $(x^2+y^2)^2 = -4xy$, parametrized by half-plane capacity. We wish to show $\tilde{\Gamma}$ has this same driver \eqref{Eq:EMWUniversalDriver}. To do so, we explicitly compute the conformal map $F_t$ from $\mathbb{H}$ to the complement of a portion $\tilde{\Gamma}_t$ of $\tilde{\Gamma}$. Appropriately normalizing $F_t$ at infinity, we will read off its driver and see it is, indeed, the same as \eqref{Eq:EMWUniversalDriver}.
\begin{figure}
\centering
\includegraphics[scale=0.12]{EMW_map.jpg}
\caption{\small Constructing the conformal map for the EMW curves.}
\label{Fig:EMWUniversalConstruction}
\end{figure}
First, note that if $z = u+iv \sim (u,v) \in \tilde{\Gamma}$, then for $z^2 \sim (u^2 -v^2, 2uv) =: (x,y)$, we have
\begin{align*}
x^2 + (y+1)^2 = (u^2+v^2)^2+4uv + 1 = 1
\end{align*}
by the equation for $\tilde{\Gamma}$, and we thus see $\tilde{\Gamma}^2 \subset \{x^2+(y+1)^2 =1\}$. It follows by continuity that the two sets are identical. As circles are more tractable, we find $F_t$ by first computing the conformal map $\tilde{F}_t:\mathbb{H} \rightarrow \mathbb{C} {\backslash} C_\theta$, where for $\pi/2 < \theta < 5\pi/2$,
\begin{align*}
C_\theta := \{\, -i + e^{i \alpha} \; : \; \pi/2 < \alpha \leq \theta \,\} \cup \mathbb{R}_{\geq 0},
\end{align*}
and $\theta = \theta(t)$. See Figure \ref{Fig:EMWUniversalConstruction}. Note that $1/C_\theta$ consists of two half-infinite horizontal line segments,
\begin{align*}
\Big\{\, \frac{\cos(\alpha)}{2(1-\sin(\alpha))} + \frac{i}{2} \; : \; \frac{\pi}{2} < \alpha \leq \theta \,\Big\} \cup \mathbb{R}_{\geq 0}.
\end{align*}
For a parameter $s = s(\theta)<0$, consider the conformal map $f = f_\theta$ from $\mathbb{H}$ to $\mathbb{C} {\backslash} (1/C_\theta)$ which takes $(-\infty,s]$ to the top half of the upper horizontal line, $[s,0)$ to its lower side, and then $(0,1]$ and $[1,\infty)$ to the bottom and top halves of $\mathbb{R}_{\geq 0}$, respectively. That is, $f$ ``folds'' $(-\infty,s]\cup[s,0)$ to the upper horizontal line and $(0,1]\cup[1,\infty)$ to the lower. The idea behind the Schwarz-Christoffel method (see \cite{MarshallComplex}, for instance) allows us to compute $f$, as we observe that $f'(x)$ is defined and has constant argument on each of the four intervals $(-\infty,s), (s,0), (0,1), (1,\infty)$. Futhermore, as $x$ increases, $f'$ has changes of argument of $-\pi, 2\pi$ and $-\pi$, respectively, at the interfaces of the intervals (the $2\pi$ change is clear through considering small half circles centered at $z=0$). Hence by Lindel\"{o}f's maximum principle \cite{MarshallComplex},
\begin{align*}
\Imag \log f'(z) &= \arg(z-1) - 2 \arg(z) +\arg(z-s) =\Imag \log \frac{(z-1)(z-s)}{z^2} + C_1,
\end{align*}
and so $f'(z) = C_2(z-1)(z-s)/z^2$, whence integration yields $f(z) = C_2\big(z - \frac{s}{z} - (1+s)\log(z) \big) + C_3$. Here $\arg(z)$ corresponds to the principal branch of the logarithm. Since $f(1)=0$ (recall $f$ ``folds'' there), we see $C_3 = -C_2(1-s)$, and as $\Imag f(x) = 1/2$ for $x<0$, we find $C_2 = -1/(2\pi(1+s))$, thus yielding
\begin{align}\label{Eq:EMWUniversalf}
f(z) = \frac{-1}{2\pi(1+s)}\Big(z - \frac{s}{z} - (1+s)\log(z) \Big) + \frac{1-s}{2\pi(1+s)}.
\end{align}
Upon reciprocation back to $\mathbb{C} {\backslash} C_\theta$, we want a map from $\mathbb{H}$ which fixes infinity and sends zero to the tip $-i + e^{i\theta}$, and we thus consider
\begin{align}
\tilde{F}_t(z) :=& \frac{1}{f(z)}\circ \frac{z-1}{z} \circ \Big( z+ \frac{1}{1-s} \Big) \notag\\
=& 2\pi \frac{1+s}{1-s} \bigg(\frac{s-(1-s)^2z^2}{\big((1-s)z+s \big) \big((1-s)z+1 \big)}+ \frac{1+s}{1-s}
\log \big( \frac{(1-s)z+s}{(1-s)z+1} \big) + 1 \bigg)^{-1} \label{Eq:TildeFt}\\
=& -4\pi \frac{1+s}{1-s}z^2 - \frac{16\pi}{3} \Big(\frac{1+s}{1-s}\Big)^2 z - \frac{2\pi(1+s)(5s^2 + 28s+5)}{9(1-s)^3} + O(z^{-1}), \qquad z \rightarrow \infty. \notag
\end{align}
The expansion of the square $F_t^2$ of the centered mapping up function $F_t = g_t^{-1}(z+\lambda_t)$ for $\sqrt{C_\theta}$ is $z^2 + O(z)$ as $z \rightarrow \infty$ (here the square root for $\sqrt{C_\theta}$ has branch cut along $\mathbb{R}_{\geq 0}$, sending the semi-circle to $\tilde{\Gamma}_t$). Noting that $\tilde{F}_t$ maps $(-1/(1-s),0)$ to the inside of the circular arc and $(0,-s/(1-s))$ to the outside, we must have
\begin{align*}
\frac{1}{1-s} < \frac{-s}{1-s} \qquad \Leftrightarrow \qquad -1>s
\end{align*}
since the harmonic measure of the inside of the circle, as seen from infinity, is less than that of the outside. Thus $-4\pi(1+s)/(1-s)>0$, and we pre-compose with the appropriate dilation to obtain
\begin{align*}
F_t^2(z) &= \tilde{F}_t \circ \frac{1}{2\sqrt{\pi}} \sqrt{-\frac{1-s}{1+s}} \, z\\
&= z^2 - \frac{8 \sqrt{\pi}}{3}\Big(- \frac{1+s}{1-s} \Big)^{3/2} z - 2\pi \frac{1+s}{(1-s)^3}(5s^2 + 28s+5) + O(z^{-1}), \qquad z \rightarrow \infty.
\end{align*}
Taking the square root, we thus have that the centered mapping-up function $F_t:\mathbb{H} \rightarrow \mathbb{H} {\backslash} \tilde{\Gamma}_t$ is
\begin{align*}
F_t(z) = \sqrt{F_t^2(z)} = z - \frac{4\sqrt{\pi}}{3} \Big(- \frac{1+s}{1-s} \Big)^{3/2} + \frac{\pi}{3}\,\frac{1+s}{(1-s)^3}(s^2-4s+1) z^{-1} + O(z^{-2}), \qquad z \rightarrow \infty.
\end{align*}
Note that $F_t$ maps $0$ to the tip of the curve, and $x_t<0<y_t$ to the base, where
\begin{align*}
x_t = \frac{-2\sqrt{\pi}\sqrt{-(1+s)}}{(1-s)^{3/2}}, \qquad y_t = \frac{-2\sqrt{\pi} s \sqrt{-(1+s)}}{(1-s)^{3/2}},
\end{align*}
and hence the ratio $r = r_t := -x/y = -1/s<1$. The expansion in terms of this (more natural) parameter $r$ is
\begin{align*}
F_t(z) = z - \frac{4\sqrt{\pi}}{3} \Big( \frac{1-r}{1+r} \Big)^{3/2} - \frac{\pi}{3}\,\frac{1-r}{(1+r)^3}(r^2+4r+1) z^{-1} + O(z^{-2}),
\end{align*}
from which we see the Loewner time of $\tilde{\Gamma}_t$ is
\begin{align}\label{Eq:TildeTau}
t = t(r) = \frac{\pi}{6}\,\frac{1-r}{(1+r)^3}(r^2+4r+1) < \frac{\pi}{6},
\end{align}
and the ending driver position is
\begin{align}\label{Eq:TildeLambda}
\tilde{\lambda}(t) = - \frac{4\sqrt{\pi}}{3} \Big( \frac{1-r}{1+r} \Big)^{3/2}.
\end{align}
We proceed to solve for $r$ in terms of $t$ in \eqref{Eq:TildeTau} and then show $\tilde{\lambda}(r(t))$ is the same as our other formula \eqref{Eq:EMWUniversalDriver}.
From \eqref{Eq:TildeTau} we see that $r^3 + 3r^2 +3\alpha r + \alpha =0$ for $\alpha = \frac{t-\pi/6}{t+\pi/6}$, for which the corresponding depressed cubic in $q:=r+1$ is
\begin{align}\label{Eq:TimeCubic}
q^3-3\beta q + 2\beta =0, \qquad \beta = \frac{\pi/3}{t+\pi/6}
\end{align}
which has discriminant $108\beta^3 -108\beta^2 >0$ by \eqref{Eq:TildeTau}, and hence three real solutions. As $r \nearrow 1$, $t \searrow 0$, and at $t=0$ we have $q^3-3\beta q + 2\beta = (q-2)(q^2+2q-2)$, for which $q=2 \sim r=1$ is the largest solution. By continuity (of roots of $q$ with respect to $t$) we thus seek the largest solution to \eqref{Eq:TimeCubic}. By Cardano's method the solutions are
\begin{align*}
q = e^{2\pi i k/3} \sqrt[3]{-\beta + i\beta \sqrt{\beta-1}}+ e^{-2\pi i k/3}\sqrt[3]{-\beta -i\beta \sqrt{\beta-1}},
\end{align*}
$k =0,1,2$, where all roots are principal branches. From the $t=0$ case we see we need $k=0$, and thus find
\begin{align*}
r(t) &= -1 + \sqrt[3]{\frac{\pi/3}{\pi/6 + t}} \bigg( \sqrt[3]{-1 + i \sqrt{\frac{\pi/6 - t}{\pi/6 + t}}} + \sqrt[3]{-1 - i \sqrt{\frac{\pi/6 - t}{\pi/6 + t}}}\, \bigg)\\
&= -1 + \frac{\sqrt[3]{\pi/3}}{\sqrt{\pi/6+t}}\Big( \sqrt[3]{-\sqrt{\pi/6+t}+i\sqrt{\pi/6-t}} + \sqrt[3]{-\sqrt{\pi/6+t}-i\sqrt{\pi/6-t}} \, \Big).
\end{align*}
Writing $\mu_1 := \sqrt{\pi/6+t}$, $\mu_2:=\sqrt{\pi/6-t}$ and noting $\sqrt[3]{\pi/3} = \sqrt[3]{-\mu_1+i\mu_2}\sqrt[3]{-\mu_1-i\mu_2}$, we have that our expression \eqref{Eq:TildeLambda} for the driver becomes
\begin{align*}
\tilde{\lambda}(t) = - \frac{4}{\sqrt{3}}\Bigg( \frac{2\mu_1 - \sqrt[3]{-\mu_1+i\mu_2}^2\sqrt[3]{-\mu_1-i\mu_2}-\sqrt[3]{-\mu_1+i\mu_2}\sqrt[3]{-\mu_1-i\mu_2}^2}{\sqrt[3]{-\mu_1+i\mu_2} + \sqrt[3]{-\mu_1-i\mu_2}} \Bigg)^{3/2}.
\end{align*}
To show this is equivalent to \eqref{Eq:EMWUniversalDriver}, we first observe
\begin{align*}
\sqrt[3]{-\mu_1+i\mu_2}^2 &= e^{i2\pi/3}\sqrt[3]{(-\mu_1+i\mu_2)^2}\\
&= 2^{1/3}e^{i2\pi/3}\sqrt[3]{t-i\sqrt{(\pi/6)^2-t^2}}\\
&= 2^{1/3}i \sqrt[3]{\sqrt{(\pi/6)^2-t^2} +it},
\end{align*}
and so, by conjugating,
\begin{align*}
\sqrt[3]{-\mu_1-i\mu_2}^2 = -2^{1/3}i \sqrt[3]{\sqrt{(\pi/6)^2-t^2} -it}.
\end{align*}
The universal driver formula \eqref{Eq:EMWUniversalDriver} is thus
\begin{align*}
\lambda_{\Gamma}(t) &= -\frac{16}{\sqrt{3}}\,t^{3/2}\Big((\pi/6)^{2/3} + 2 \Real \big(\sqrt{(\pi/6)^2 -t^2} + it\big)^{2/3} \,\Big)^{-3/2}\\
&= -\frac{32}{\sqrt{3}}\,t^{3/2}\Big((\pi/3)^{2/3} -\sqrt[3]{-\mu_1-i\mu_2}^4 - \sqrt[3]{-\mu_1+i\mu_2}^4 \,\Big)^{-3/2}\\
&= -\frac{32}{\sqrt{3}}\,t^{3/2}\Big(\sqrt[3]{-\mu_1+i\mu_2}^2\sqrt[3]{-\mu_1-i\mu_2}^2 -\sqrt[3]{-\mu_1-i\mu_2}^4 - \sqrt[3]{-\mu_1+i\mu_2}^4 \,\Big)^{-3/2}
\end{align*}
Writing $\nu:= \sqrt[3]{-\mu_1+i\mu_2} = \sqrt[3]{-\sqrt{\pi/6 +t}+i\sqrt{\pi/6-t}}$, we thus wish to show
\begin{align*}
-\frac{32}{\sqrt{3}}\,t^{3/2}\big( \nu^2\Bar{\nu}^2 - \Bar{\nu}^4-\nu^4 \big)^{-3/2} = - \frac{4}{\sqrt{3}}\Big( \frac{2\mu_1 - \nu^2 \Bar{\nu} - \nu \Bar{\nu}^2}{\nu + \Bar{\nu}} \Big)^{3/2}
\end{align*}
or
\begin{align*}
8t^{3/2} = \Big(\frac{( \nu^2\Bar{\nu}^2 - \Bar{\nu}^4-\nu^4 )
(2\mu_1 - \nu^2 \Bar{\nu} - \nu \Bar{\nu}^2)}{\nu + \Bar{\nu}}\Big)^{3/2},
\end{align*}
which follows from cancellations after expanding the numerator on the right-hand side. Since the driving function uniquely identifies the curve, we have proven that the curve described by \eqref{Eq:EMWUniversalVariety} is the universal EMW curve.
\bigskip
With the conformal maps in hand, we return to prove the implicit relation \eqref{Eq:EMWWeldImplicit} for the welding. Indeed, from our computation \eqref{Eq:EMWUniversalf} above, we see that the welding $\tilde{\varphi}$ generated by $f$ for the EMW curve $\gamma$ satisfies $W(r, \tilde{\varphi}(x)) = W(r,x)$ for all $-\infty<x \leq s = y_0/x_0$ (the identifications established by $f$ do not change when we take the square root of the reciprocal of the image of $f$ to obtain $\gamma$, as these are conformal maps). See Figure \ref{Fig:EMWUniversalConstruction}. However, $\tilde{\varphi} = T \circ \varphi \circ T^{-1}$, where $T$ is the M\"{o}bius automorphism of $\mathbb{H}$ sending the triple $(x_0,0,y_0)$ to $(-\infty, s,0)$. In terms of our maps in Figure \ref{Fig:EMWUniversalConstruction}, we have
\begin{align*}
T(x) = \frac{x-1}{x} \circ \Big(x + \frac{1}{1-s} \Big) \circ \frac{x}{2\sqrt{\pi}}\sqrt{- \frac{1-s}{1+s}} \circ \frac{-2\sqrt{\pi}\, x \, \sqrt{-(1+s)}}{x_0(1-s)^{3/2}} = \frac{x-y_0}{x-x_0},
\end{align*}
as claimed.
\bigskip
Lastly, for part $(\ref{Thm:DistinctFamilies})$, it is clear that if $\theta=\pi/2$, then the Wang minimizer is the vertical line segment $[0,i]$, which is the same as the EMW curve for $x=-1,y=1$. Both use zero energy. For $\theta \neq 0$, we see from \eqref{Eq:EMWDriverFormula} that near $t=0$, the EMW driver satisfies $\lambda(t) = Ct^{3/2} + O(t^{5/2})$ for some $C \neq 0$. However, from \eqref{Eq:WangDriverDown} we see that the downwards Wang driver $\lambda_\theta$ is always smooth near $t=0$ (no radicands vanish at $t=0$), and so the drivers are never identical.
\end{proof}
\section{Energy comparisons}\label{Sec:Compare}
In this section we explore how the close sufficiently-smooth curves are to be locally energy minimizing, and how the energies of our two minimizing families compare.
\begin{theorem}\label{Lemma:Inf98}
Let $\epsilon>0$.
\begin{enumerate}[$(i)$]
\item \label{Thm:LocalCompareWang} If $\gamma$ is any curve driven by $\lambda \in C^{3/2+\epsilon}([0,T))$ with $\lambda(0)=0$, $\dot{\lambda}(0) \neq 0$, then
\begin{align*}
\lim_{\delta \rightarrow 0^+} \frac{I(\gamma[0,\delta])}{I(\text{Wang minimizer from 0 to }\gamma(\delta))} = \frac{9}{8}.
\end{align*}
\item \label{Thm:LocalCompareEMW} Let $\eta \in C^{3/2+\epsilon}([0,T))$ be a driver with $\eta(0)=0$, $\dot{\eta}(0) \neq 0$, and $\tilde{\gamma}[0,\delta]$ the curve generated by $\eta$ by means of the upwards Loewner flow on $[0,\delta]$. Let $u(\delta)<0<v(\delta)$ be the points welded to the base of $\tilde{\gamma}[0,\delta]$ at time $\delta$. Then
\begin{align*}
\lim_{\delta \rightarrow 0^+} \frac{I(\eta[0,\delta])}{I(\text{EMW minimizer for }u(\delta), v(\delta))} = \frac{9}{8}.
\end{align*}
\end{enumerate}
\end{theorem}
Our proof will use infinitesimal expansions for the parametrized curve and the points welded together. The former says that for a driver $\lambda \in C^{3/2+\epsilon}$, the associated curve $\gamma = \gamma^\lambda$ satisfies
\begin{align}\label{Eq:InfinitesimalCurve}
\gamma(\delta) = 2i\sqrt{\delta} + \frac{2}{3}\dot{\lambda}(0)\delta - \frac{i}{18}\dot{\lambda}(0)^2\delta^{3/2} + O(\delta^{3/2 + \epsilon})
\end{align}
as $\delta \rightarrow 0$ \cite[Prop. 6.2]{LindTran}. The latter is a similar expansion for the conformal welding, and says that for a driver $\eta \in C^{3/2+\epsilon}$, the points $x(\delta) < 0 < y(\delta)$ welded together at time $\delta$ by the upwards flow generated by $\eta$ are
\begin{align}\label{Eq:InfinitesimalWelding}
\begin{split}
x(\delta) &= -2 \sqrt{\delta} + \frac{2}{3}\dot{\xi}(0)\delta - \frac{1}{18}\dot{\xi}(0)^2 \delta^{3/2} + O(\delta^{3/2+\epsilon}),\\
y(\delta) &= 2 \sqrt{\delta} + \frac{2}{3}\dot{\xi}(0)\delta + \frac{1}{18}\dot{\xi}(0)^2 \delta^{3/2} + O(\delta^{3/2+\epsilon})
\end{split}
\end{align}
as $\delta \rightarrow 0$ \cite[Thm. 3.5]{Mesikepp}.
\begin{proof}
The energy used by $\gamma[0,\delta]$ is $\frac{1}{2}\dot{\lambda}(0)^2\delta + o(\delta)$ as $\delta \rightarrow 0$. By symmetry we may assume $\dot{\lambda}(0)>0$, and thus by \eqref{Eq:InfinitesimalCurve} the angle of the tip of the curve is
\begin{align*}
\Arg(\gamma(\delta)) &= \frac{\pi}{2} - \arctan \left( \frac{\frac{2}{3}\dot{\lambda}(0)\delta + O(\delta^{3/2+\epsilon})}{2\sqrt{\delta} - \frac{1}{18}\dot{\lambda}(0)^2 \delta^{3/2}+ O(\delta^{3/2+\epsilon})} \right) = \frac{\pi}{2}-\frac{1}{3}\dot{\lambda}(0)\sqrt{\delta} +O(\delta^{1+\epsilon}).
\end{align*}
Noting that $-8\log(\sin(\theta)) = 4(\theta-\pi/2)^2 + O(\theta - \pi/2)^4$, we see that the minimal-energy curve through this angle has energy
\begin{align*}
\frac{4}{9}\dot{\lambda}(0)^2\delta + O(\delta^{3/2 + \epsilon}),
\end{align*}
yielding the ratio
\begin{equation*}
\frac{\frac{1}{2}\dot{\lambda}(0)^2\delta + o(\delta)}{\frac{4}{9}\dot{\lambda}(0)^2\delta + O(\delta^{3/2+\epsilon})} \rightarrow \frac{9}{8}.
\end{equation*}
For the welding claim, we use the expansions in \eqref{Eq:InfinitesimalWelding} to see that the ratio $\alpha(\delta) := \frac{y(\delta)}{y(\delta)-x(\delta)}$ of the welded points $x(\delta)$ and $y(\delta)$ is
\begin{align*}
\alpha(\delta) = \frac{1}{2}+\frac{1}{6}\dot{\eta}(0)\sqrt{\delta} + O(\delta^{1+\epsilon}).
\end{align*}
Recalling from \eqref{Eq:EMWEnergyFormula} that the minimal energy for ratio $\alpha$ is
\begin{align*}
-4 \log(4\alpha(1-\alpha)) = 16(\alpha- 1/2)^2 + O(\alpha- 1/2)^4,
\end{align*}
we thus see the minimal energy to weld $x(\delta)$ and $y(\delta)$ is
\begin{align*}
\frac{4}{9}\dot{\xi}(0)^2\delta + O(\delta^{3/2+\epsilon}).
\end{align*}
As the actual energy is $\frac{1}{2}\dot{\xi}(0)^2\delta + o(\delta)$, we find the same asymptotic ratio.
\end{proof}
We also note that hyperbolic geodesics in $\mathbb{H}$ have a distinguished place in terms of energy, compared to both the Wang minimizers and the EMW curves.
\begin{lemma}\label{Lemma:CircularArc} If $\theta \neq \pi/2$, the orthogonal circular arc to $z_0 = re^{i \theta}$ uses exactly $9/8$th of the energy of the Wang minimizer to $z_0$. If $x<0<y$ with $-x \neq y$, the orthogonal circular arc which welds $x$ to $y$ uses exactly $9/8$th the energy of the EMW curve which welds these points.
\end{lemma}
Of course, if $\theta = \pi/2$ or $-x=y$, then all the curves in question are the same and use zero energy. The surprising aspect here, perhaps, is that there is instant (and constant) disagreement as soon as we deviate from the vertical line segment. Given this lemma, we can guess the constants in Theorem \ref{Lemma:Inf98}, because sufficiently-smooth curves should locally be like circular arcs.\footnote{One could make this precise by comparing the expansion of the circular arc from $0$ to $\gamma(\delta)$ to \eqref{Eq:InfinitesimalCurve}, and the expansion of the circular arc which welds $x(\delta)$ and $y(\delta)$ to \eqref{Eq:InfinitesimalWelding}. Theorem \ref{Lemma:Inf98} is saying the ``local similarity'' holds at least in the energy sense.}
\begin{proof}
The proof is via explicit computation. By reflection and scaling invariance it suffices to consider the orthogonal circular arc $\gamma_1$ from $0$ to $x=1$, whose downwards driving function $\lambda_1$ was computed by Lind \cite[p.5]{Lind4},
\begin{align}\label{Eq:CADriver1}
\lambda_1(t) = \frac{3}{2}(1-\sqrt{1-8t}), \qquad 0 \leq t \leq \frac{1}{8}.
\end{align}
Using the same conformal maps needed to determine $\lambda_1$, it is similarly possible (``although unpleasant,'' to quote Lind) to see that the capacity parametrization of the arc is explicitly
\begin{align*}
\gamma_1(t) = 1-\sqrt{1-8 t}+ i \sqrt[4]{1-8 t} \sqrt{1-\sqrt{1-8 t}}, \qquad 0 \leq t \leq \frac{1}{8},
\end{align*}
and hence
\begin{align*}
\Arg(\gamma_1(t)) =: \theta(t) = \arctan\Big(\frac{\sqrt[4]{1-8 t} \sqrt{1-\sqrt{1-8 t}}}{1-\sqrt{1-8 t}} \Big), \qquad 0 < t < \frac{1}{8}.
\end{align*}
Inverting, we find that the unique time needed to achieve a given argument $\theta \in (0, \pi/2)$ along $\gamma_1$ is
\begin{align}\label{Eq:CATForTheta}
t(\theta) = \frac{1}{8}\big(1-\sin^4(\theta)\big),
\end{align}
and hence the energy to angle $\theta$ is
\begin{align*}
\frac{1}{2} \int_0^{t(\theta)} \frac{36}{1-8s}ds = -9 \log(\sin(\theta)),
\end{align*}
as claimed (recall \eqref{Eq:WangEnergy}).
Fix $0 < t < 1/8$. Similar to the computation of \eqref{Eq:CADriver1}, one finds that the centered conformal welding $\varphi_t$ of the segment $\gamma_1[0,t]$ is explicitly $\varphi_t:[x_t,0] \rightarrow [0,y_t]$, where
\begin{align*}
\varphi_t(x) = - \frac{\sqrt{1-8t}\,x}{\sqrt{1-8t} -2x}
\end{align*}
and
\begin{align}\label{Eq:CAWeldEndpoints}
x_t = -1 + \sqrt{1-8t} - \sqrt{1-\sqrt{1-8t}} \quad \text{and} \quad y_t = -1 + \sqrt{1-8t} + \sqrt{1-\sqrt{1-8t}}.
\end{align}
By inspection the ratio $\alpha_t = \frac{y_t}{y_t-x_t}$ is monotonically decreasing, and inverting we have the unique time $t(\alpha)$ to obtain a given welding ratio $\alpha \in (0,1/2)$ is
\begin{align}\label{Eq:CATForAlpha}
t(\alpha) = \frac{1}{8} -2\alpha^2(1-\alpha)^2,
\end{align}
with corresponding energy
\begin{align*}
\frac{1}{2} \int_0^{t(\alpha)} \frac{36}{1-8s}ds = -\frac{9}{2}\log(4\alpha(1-\alpha)),
\end{align*}
exactly $9/8$ of \eqref{Eq:EMWEnergyFormula}.
\end{proof}
Lastly, we wish to compare the energy usage between the two minimizing families themselves. We have the following asymptotic results.
\begin{theorem}\label{Thm:EMWWang}
Let $x_r$ and $y_r$ satisfy $x_r<0<y_r$ with $-x_r/y_r=r$. Then
\begin{align*}
\lim_{r \rightarrow 1} \frac{I(\text{EMW $\gamma$ welding $x_r$ and $y_r$ to its base})}{I(\text{Wang $\gamma$ welding $x_r$ and $y_r$ to its base})} = \Big(\frac{\pi}{4}\Big)^2.
\end{align*}
Considering the tips of the curves, we have
\begin{align}\label{Lim:EnergySameTip}
\lim_{\theta \rightarrow \pi/2} \frac{I(\text{EMW $\gamma$ with tip at $e^{i\theta}$})}{I(\text{Wang $\gamma$ with tip at $e^{i\theta}$})} = \Big(\frac{4}{\pi}\Big)^2.
\end{align}
\end{theorem}
\noindent For discussion of the fact that these constants are reciprocals, see Remark \ref{Remark:Reciprocals} after the proof. We note in Remark \ref{Remark:EnergiesNotAlwaysSame} that these ratios do not always hold.
\begin{proof}
In Theorem \ref{Thm:Wang}($\ref{Thm:WangDriver}$), we saw that the welding $\varphi_\theta$ for the Wang minimizer to $e^{i\theta}$ satisfies $\varphi_\theta:[x_\theta,0] \rightarrow [0,y_\theta]$, where
\begin{align*}
x_\theta = -\sqrt{\frac{\sin ^3(\theta )}{\sin (\theta )-\theta \cos (\theta )}} \qquad \text{and} \qquad y_\theta = \sqrt{\frac{\sin ^3(\theta )}{\sin (\theta )-\theta \cos (\theta )+\pi \cos (\theta )}},
\end{align*}
and thus
\begin{align}\label{Eq:rFromThetaWang}
r(\theta) = -\frac{x_\theta}{y_\theta}= 1 - \frac{\pi}{2}\Big(\theta - \frac{\pi}{2}\Big) + O\Big(\theta - \frac{\pi}{2}\Big)^2, \qquad \theta \rightarrow \frac{\pi}{2},
\end{align}
which inverts as
\begin{align*}
\theta(r) = \frac{\pi}{2} - \frac{2}{\pi}(r - 1) + O(r - 1)^2, \qquad r \rightarrow 1.
\end{align*}
The associated energy is thus
\begin{align*}
-8\log(\sin(\theta(r))) = \frac{16}{\pi^2}(r- 1)^2 + O(r-1)^3.
\end{align*}
For the EMW curves, on the other hand, \eqref{Eq:EMWEnergyFormula} says the energy at ratio $r$ is
\begin{equation}\label{Eq:EMWEnergyTaylor}
-8 \log\Big( \frac{2\sqrt{r}}{1+r} \Big) = (r-1)^2 + O(r-1)^3
\end{equation}
as $r \rightarrow 1$, and so we find the claimed ratio.
For the second claim, we use the conformal map constructed in the proof of Theorem \ref{Thm:EMW}($\ref{Thm:EMWSingleCurve}$) to find the angle of the tip of the EMW curve associated to a given ratio $r =-x/y$. By symmetry it suffices to consider the case $r<1$, as there. Recall the map $\tilde{F}_t$ in \eqref{Eq:TildeFt} sends zero to $-i+e^{i\theta}$, where $\Arg(i+e^{i\theta}) = 2\beta$ is twice the argument $\beta$ of the tip $\Gamma(t_r)$ of the corresponding point on the universal curve $\Gamma([0,t_r]) = \sqrt{C_\theta}$. See Figure \ref{Fig:EMWUniversalConstruction}. We thus have
\begin{align}\label{Eq:EMWBeta}
2\beta = \frac{3\pi}{2} - \arctan\Big( \frac{\Real \tilde{F}_t(0)}{\Imag \tilde{F}_t(0)} \Big) = \frac{3\pi}{2} - \arctan\Big( \frac{2(1+r)+(1-r)\log(r)}{\pi(1-r)} \Big),
\end{align}
and so
\begin{align*}
\beta(r) = \frac{\pi}{2} - \frac{\pi}{8}(r-1) + O(r-1)^2
\end{align*}
as $r \rightarrow 1^-$, and inverting yields
\begin{align}\label{Eq:rFromThetaEMW}
r(\beta) = 1 - \frac{8}{\pi}\Big(\beta- \frac{\pi}{2} \Big) + O\Big(\beta- \frac{\pi}{2} \Big)^2, \qquad \beta \rightarrow \frac{\pi}{2}.
\end{align}
Hence from \eqref{Eq:EMWEnergyTaylor} the energy is
\begin{align*}
-8\log \Big( \frac{2\sqrt{r(\beta)}}{1+r(\beta)}\Big) = \frac{64}{\pi^2}\Big(\beta - \frac{\pi}{2}\Big)^2 + O\Big(\beta - \frac{\pi}{2}\Big)^3.
\end{align*}
The Wang minimizers $\gamma_\beta$, on the other hand, satisfy
\begin{align*}
I(\gamma_\beta) = -8\log(\sin(\beta)) = 4\Big(\beta - \frac{\pi}{2}\Big)^2 + O\Big(\beta - \frac{\pi}{2}\Big)^4
\end{align*}
as $\beta \rightarrow \pi/2$, yielding the claimed limiting ratio.
\end{proof}
\begin{remark}\label{Remark:Reciprocals}
It does not appear to be \emph{a priori} obvious that the two limits in Theorem \ref{Thm:EMWWang} should be reciprocals, as we proceed to explain. Note our minimal-energy formulas have expansions \eqref{Eq:EMWEnergyTaylor} and
\begin{align*}
-8\log(\sin(\theta)) &= 4 \left(\theta -\frac{\pi
}{2}\right)^2+O\left(\theta -\frac{\pi
}{2}\right)^4,
\qquad \theta \rightarrow \frac{\pi}{2}.
\end{align*}
Let $\theta$ be the argument of the tip of a curve welding $x$ to $y$, and suppose that as $\theta \rightarrow \pi/2$, the ratio $-x/y$ approached unity $C$ times as fast for the EMW curves as for the Wang minimizers. That is, suppose the Wang minimizers' ratio $r_W$ satisfied $r_W(\theta) = 1 + D(\theta-\pi/2) + O(\theta-\pi/2)^2$, while the EMW's $r_E$ satisfied $r_E(\theta) = 1 + CD(\theta-\pi/2) + O(\theta-\pi/2)^2$. Then the energy of the Wang minimizer as a function of $r$ would be
\begin{align*}
-8\log(\sin(\theta(r))) = \frac{4}{D^2}(r-1)^2 + O(r-1)^3,
\end{align*}
which combined with \eqref{Eq:EMWEnergyTaylor} shows the first ratio in Theorem \ref{Thm:EMWWang} would be $D^2/4$. Similarly, the second ratio would be $C^2D^2/4$, and hence these are reciprocals if and only if $C^2D^4=16$. From \eqref{Eq:rFromThetaWang} we see $D=-\pi/2$, and so the EMW curves' ratio would need to approach $1$ precisely $16/\pi^2$ as fast as for the Wang minimizers as $\theta \rightarrow \pi/2$. From \eqref{Eq:rFromThetaEMW} this is the case, but from the outset it is not obvious that it should be so.
\end{remark}
\begin{remark}\label{Remark:EnergiesNotAlwaysSame}
\begin{figure}
\centering
\includegraphics[scale=0.75]{EnergySameTipPDF.pdf}
\caption{\small Energies for the Wang and EMW curves with the tip $e^{i\theta}$ in blue and orange, respectively, where the $x$-axis is $\theta \in (\pi/2, \pi)$ and the $y$-axis the energy. The larger growth of the EMW energy is not surprising given the curves tangentially turn towards $\mathbb{R}$ as $\theta \rightarrow \pi^-$.}
\label{Fig:EnergySameTip}
\end{figure}
We also remark that the asymptotics of Theorem \ref{Thm:EMWWang} do not hold for all ratios $r$ and all angles $\theta$. With respect to \eqref{Lim:EnergySameTip}, for instance, it is easy to intuitively see that the energy of an EMW curve whose tip has angle $\theta$ close to $\pi^-$ (or $0^+$) is much larger than that for the corresponding Wang minimizer. This is because as $\theta \rightarrow \pi^-$, the Wang asymptotically hits the real line with angle $\pi/3$ (the picture is the reflection of the plots in Figure \ref{Fig:BoundaryGeodesics} over the imaginary axis), while the EMW curve asymptotically turns in and hits $\mathbb{R}$ tangentially with angle $\pi$ (see Figure \ref{Fig:UniversalEMW}), which intuitively requires vastly more energy. This is confirmed in Figure \ref{Fig:EnergySameTip}, where we plot the Wang minimizer energy $-8\log(\sin(\beta))$ and the composition of the energy formula $-8\log(2\sqrt{r}/(1+r))$ for the EMW curves with the numerical inverse $r(\beta)$ of the EMW angle formula $\beta(r)$ given by \eqref{Eq:EMWBeta}.
\end{remark}
\section{Appendix: other instances of upwards SLE$_0(\rho,\rho)$}\label{Sec:Appendix}
We have seen in Theorem \ref{Thm:EMW}($\ref{Thm:EMWSLE}$) that the welding minimizers are upwards SLE$_0(\rho,\rho)$ with $\rho=-4$ and forcing starting at the points $x<0<y$ to be welded together. If we instead change $\rho$ to $-2$ or $-3$, we obtain straight line segments or orthogonal circular arc segments, respectively.
\begin{lemma}\label{Lemma:UpwardsSLE}
Fix $x<0<y$.
\begin{enumerate}[$(i)$]
\item\label{Lemma:StraightSlitUpwardsSLE} The hull $\gamma_\tau$ generated by upwards SLE$_0(-2,-2)$ started from $(\xi(0), V_1(0), V_2(0)) = (0, x,y)$ on $[0,\tau]$, where $\tau$ is the first hitting time of $\xi$ and one of the $V_j$, is a (straight) line segment whose angle of intersection with the right-hand side of $\mathbb{R}$ is $\frac{\pi y}{y-x}$, and the images of the force points $V_1(\tau)$ and $V_2(\tau)$ are the two prime ends corresponding to the base of $\gamma_\tau$.
\item\label{Lemma:CAUpwardsSLE} The hull $\tilde{\gamma}_t$ generated by upwards SLE$_0(-3,-3)$ started from $(\xi(0), V_1(0), V_2(0)) = (0, x,y)$ is a circular arc which meets $\mathbb{R}$ orthogonally, where $t$ is any time $0 < t \leq \tau$, the first hitting time of $\xi$ and one of the $V_j$. At time $\tau$, $V_1(\tau)$ and $V_2(\tau)$ are the two prime ends corresponding to the base of $\tilde{\gamma}_\tau$.
\end{enumerate}
\end{lemma}
\begin{proof}
For $(\ref{Lemma:StraightSlitUpwardsSLE})$, let $\gamma:[0,\tau] \rightarrow \mathbb{H} \cup \{0\}$ be a line segment with angle $\pi \alpha := \frac{\pi y}{y-x}$ to $\mathbb{R}_{\geq 0}$. For some fixed $x_t < 0 < y_t$, we recall that the centered mapping-up function
\begin{align}\label{Eq:SlitMapUp}
\tilde{F}_{t'}(z) = (z-y_t)^{\frac{y_t}{y_t-x_t}}(z-x_t)^{\frac{-x_t}{y_t-x_t}} = z - (x_t+y_t) + \frac{x_ty_t}{2}\frac{1}{z} + O\Big( \frac{1}{z^2}\Big), \qquad z \rightarrow \infty,
\end{align}
takes $[x_t,0]$ and $[0,y_t]$ to two sides of a straight line segment $\tilde{\gamma}[0,t']$ that has base at the origin, angle
\begin{align}\label{Eq:SlitAngle}
\frac{\pi y_t}{y_t-x_t} = \pi\tilde{\alpha}
\end{align}
with $\mathbb{R}_{\geq 0}$, and capacity time $2t' = -x_ty_t/2$ (see \cite[``The Slit Algorithm'']{MRZip} for a derivation). Thus the centered mapping-up function $F_t$ for $\gamma$ is of this form, where $x_t = -\frac{1-\alpha}{\alpha} y_t$ by \eqref{Eq:SlitAngle}, and
\begin{align}\label{Eq:StraightSlitWeldEndpoints}
2t = -\frac{x_ty_t}{4} = \frac{1-\alpha}{4\alpha}y_t^2 \qquad \Rightarrow \qquad y_t = 2 \sqrt{\frac{\alpha}{1-\alpha}}\sqrt{t}, \quad x_t = -2\sqrt{\frac{1-\alpha}{\alpha}}\sqrt{t}.
\end{align}
From \eqref{Eq:SlitMapUp} we also see that
\begin{align*}
\lambda(t) = -(x_t+y_t) = \frac{2(1-2\alpha)}{\sqrt{\alpha(1-\alpha}}\sqrt{t}.
\end{align*}
The upwards SLE$_0$ claim is equivalent to showing that $\gamma$'s downward Loewner flow is downwards SLE$_0(-2,-2)$ with forcing starting at the two prime ends at the base of $\gamma$. By this we mean that, for any $t>0$, we have
\begin{align*}
\frac{1-2\alpha}{\sqrt{\alpha(1-\alpha)}\sqrt{t}} = \dot{\lambda}(t) = \frac{2}{g_t(0-) -\lambda(t)} + \frac{2}{g_t(0+)-\lambda(t)} = \frac{2}{x_t} + \frac{2}{y_t},
\end{align*}
which indeeds holds by \eqref{Eq:StraightSlitWeldEndpoints}.
For $(\ref{Lemma:CAUpwardsSLE})$ we start by considering the orthogonal circular arc growing from the origin towards $x=2\sqrt{2}$, which has downwards driver $\lambda(t) = 3\sqrt{2} - 3\sqrt{2}\sqrt{1-t}$ by \eqref{Eq:CADriver1} (scale by $r=2\sqrt{2}$ and recall \eqref{Eq:ScaledDriver}). The claim is equivalent to saying $\lambda$ is generated by downwards SLE$_0(-3,-3)$ with forcing starting at the two prime ends corresponding to the base. In other words, for any $t>0$, we claim
\begin{align}\label{Eq:UpwardsSLE33}
\dot{\lambda}(t) = \frac{3}{g_t(0-)-\lambda(t)} + \frac{3}{g_t(0+)-\lambda(t)}.
\end{align}
Using the explicit conformal maps one uses to compute the driver $\lambda(t)$ in the first place, one finds
\begin{align*}
g_t(0-) &= \sqrt{2} - \sqrt{2}\sqrt{1-t}-2\sqrt{2}\sqrt{1-\sqrt{1-t}},\\
g_t(0+) &= \sqrt{2} - \sqrt{2}\sqrt{1-t}+2\sqrt{2}\sqrt{1-\sqrt{1-t}},
\end{align*}
from which arithmetic shows that \eqref{Eq:UpwardsSLE33} holds. Since the ratio $\frac{\lambda(t)-V_1(t)}{V_2(t)-\lambda(t)}$ attains all values in the interval $(1,\infty)$ as $t$ varies in $(0,1)$, after scaling and reflection this case covers all possible initial conditions $(0,x,y)$.
\end{proof}
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\section{Introduction}
In the current age of internet, online social media sites have become available to all people, and people tend to post their personal experiences, current events, local and global news. For this reason, the daily usages of social media are growing up and making a large dataset that becomes an important source of data to make different types of research analysis. Moreover, the social media data are real time data and accessible to monitor. Therefore, several research works are conducted to perform different types real time predictions using the social media data, such as stock movement prediction \cite{nguyen2015sentiment}, relation extraction \cite{ritter2015weakly}, and natural disaster prediction \cite{yoo2018social,leung2020big}.
Twitter is such a social media site that can be accessed through people's laptops and smartphones. The rapid growth of smartphone or laptop usages enables people to share an emergency that they observe in real time. For this reason, many disaster relief organizations and news agencies are interested in monitoring Twitter data programmatically. However, unlike long articles, tweets are short length text, and they tend to have more challenges due to their shortness, sparsity (i.e., diverse word content) \cite{chen2011short}, velocity (rapid growth of short text like SMS and tweet) and misspelling \cite{alsmadi2019review}. For these reasons, it is very challenging to understand whether a person's words are announcing a disaster or not. For example, a tweet like this, "$\#oldBand$ $amazing$ $performance$! $light,$ $color,$ $fire$ $on$ $stage$! $lots$ $of$ $people$ $and$ $huge$ $chaos!$" tells us an experience of a person in a concert and we can say from that he enjoyed it, because of the word, "$amazing$". Even though it contains the word, "$fire$", it does not mean any danger or emergency; rather, it is used to describe the colorful decoration of the stage. Let's assume another tweet like this, "$California$ $Hwy.$ $20$ $closed$ $in$ $both$ $directions$ $due$ $to$ $Lake$ $Country$ $fire$". Here, the word ``$fire$" means disaster, and the tweet describes an emergency. The two examples show that one word could have multiple meanings based on its context. Therefore, understanding the context of words is important to analyze a tweet's sentiment.
Different researchers proposed different methods to understand the meaning of a word by representing it in embedding or vector \cite{pennington2014glove,DBLP:journals/corr/abs-1301-3781,bojanowski2016enriching}. Neural network-based methods such as Skip-gram \cite{DBLP:journals/corr/abs-1301-3781}, FastText \cite{bojanowski2016enriching} are popular for learning word embeddings from large word corpus and have been used for solving different types of NLP tasks. These methods are also used for sentiment analysis of Twitter data \cite{deho2018sentiment,poornima2020comparative}. However, those embedding learning methods provide static embedding for a single word in a document. Hence, the meaning of the word,``$fire$" would remain the same in the above two examples for these methods.
To handle this problem, the authors of \cite{devlin2018bert} proposed a contextual embedding learning model, Bidirectional Encoder Representations from Transformers (BERT), that provides embeddings of a word based on its context words. In different types of NLP tasks such as text classification \cite{sun2019fine}, text summarization \cite{liu2019text}, entity recognition \cite{hakala2019biomedical}, BERT model outperformed traditional embedding learning models. However, it is interesting to discover how the contextual embeddings could help to understand disaster-type texts. For this reason, we plan to analyze the disaster prediction task from Twitter data using both context-free and contextual embeddings in this study. We use traditional machine learning methods and neural network models for the prediction task where the word embeddings are used as input to the models. We show that contextual embeddings work better in predicting disaster-types tweets than the other word embeddings. Finally, we provide an extensive discussion to analyze the results.
The main contributions of this paper are summarized as follows.
\begin{enumerate}
\item We analyze a real-life natural language online social network dataset, Twitter data, to identify challenges in human sentiment analysis for disaster-type tweet prediction.
\item We apply both contextual and context-free embeddings in tweet representations for disaster prediction through machine learning methods and show that context-free embeddings (BERT) can improve the accuracy of disaster prediction compared with contextual embeddings.
\item We provide a detailed explanation of our method and results and share our codes publicly that will enable researchers to run our experiments and reproduce our results for future research work directions \footnote{\label{our_code} https://github.com/ashischanda/sentiment-analysis}.
\end{enumerate}
The rest of the paper is organized as follows. In section 2, some related works are introduced. The main methodology of this paper is elaborated in section 3. The dataset and the experiments are presented in section 4 and 5, respectively. Finally, the conclusion is drawn in section 6.
\section{Related Works}
Many research works analyzed Twitter data for understanding emergency situation and predicting disaster analysis \cite{karami2020twitter, zou2018mining, ashktorab2014tweedr, olteanu2014crisislex}. One group of researchers used text mining and statistical approaches to understand crises \cite{karami2020twitter, zou2018mining}, another group of researchers focused on clustering text data to identify a group of tweets that belong to disaster \cite{ashktorab2014tweedr, olteanu2014crisislex}. Later, different traditional machine learning models are used to analyze Twitter data and predict disaster or emergency situations where words of a tweet are represented as embeddings \cite{palshikar2018weakly,
algur2021classification, singh2019event}. For example, Palshikar et al. \cite{palshikar2018weakly} proposed a weakly supervised model where words are presented with a bag of words (BOW) model. Moreover, frequency-based word representation is used in \cite{algur2021classification} for disaster prediction from Twitter data using Naive Bayes, Logistic Regression, Random Forest, and SVM methods. The authors in \cite{singh2019event} used a Markov-based model to predict the location of Tweets during a disaster. In a recent work \cite{pota2021multilingual}, the authors proposed a pre-processing method for BERT-based sentiment analysis of tweets. However, it is interesting to explore the model performance on different word embeddings to observe how the context words help to predict a tweet as a disaster.
\section{Methodology}
In this section, we discuss our approach of leveraging word embedding for disaster prediction from Twitter data using machine learning methods. We consider three types of word embeddings, 1) bag of words (BOW), 2) context-free, and 3) contextual embeddings. The word embeddings are used in both traditional machine learning methods and deep learning models as input for disaster prediction.
\subsection{BOW embeddings}
The bag-of-words (BOW) model is a common approach for text representation of a word document. If there are $V$ words in a text vocabulary, then BOW is a binary vector or array of length $|V|$ where each index of the array is used to present one word of the vocabulary. If a word exists in a document, then the corresponding array index of the word becomes one; otherwise, it contains zero. We use BOW embeddings of Twitter data in three traditional machine learning methods such as decision tree, random forest, and logistic regression to predict the sentiment of a tweet.
Even though BOW is good for representing words of a document, it loses contextual information because the order of words is not recorded in the binary structure. However, contextual information is required to understand and analyze the sentiment of a text. For this reason, we also plan to use context-based embeddings for this sentiment analysis task.
\subsection{Context-free embeddings}
Many existing research works proposed to learn word embeddings based on the co-occurrences of word pairs in documents. GloVe \cite{pennington2014glove} is one common method for learning word embeddings from the co-occurrences of words in documents. However, neural network-based models such as Skip-gram \cite{DBLP:journals/corr/abs-1301-3781}, FastText \cite{bojanowski2016enriching} became popular recently to learn word representations from documents and used for sentiment analysis.
In our research study, we use the pre-trained embeddings of three context-free embedding models (GloVe, Skip-gram, FastText) in a neural network-based model to analyze the sentiment of tweet data and predict disaster types tweets. To represent a tweet in context-free embeddings, we took the average of word embeddings of a tweet following the same strategy of \cite{kenter2016siamese}. For the calculated vector of a tweet, we use softmax to predict the sentiment of the tweet. Let's suppose that the vector of a tweet is $v$, we have a set of labels, $L$ = $\{$"positive", "negative"$\}$ and $Z$ $\in$ $\mathbb{R}^{(|L| \times d)}$ is the weight matrix of softmax function. Then, the probability of the tweet to be positive or disaster is calculated as follows
\begin{equation*}
p(y(l_i)=1)=\frac{e^{Z_i \cdot v}}{\sum_{l_k \in L}e^{Z_k \cdot v}}
\end{equation*}
Recently, deep neural networks are also used for sentiment analysis. To observe how the context-free embeddings work on deep neural networks, we used a bidirectional recurrent neural network with LSTM gates \cite{hochreiter1997long}. The Bi-LSTM model processes the input words of a tweet from right to left and in reverse. The Bi-LSTM block is followed by a fully connected layer and sigmoid function to output as an activation function.
\subsection{Contextual embeddings}
Unlike the other word embeddings, BERT embeddings \cite{devlin2018bert} generates different vectors for the same word in different contexts. Recent advances in NLP have shown that BERT model have outperformed traditional embeddings in different NLP tasks, like entity extraction, next sentence prediction. In our study, we plan to investigate how well do contextual embeddings work better than traditional embeddings in sentiment analysis. For this purpose, we use the pre-trained embeddings of BERT models in the same neural network models to predict disaster types tweets.
\begin{table*}
\centering
\caption{Sample pre-processed tweets}
\begin{tabular}{l|l}
\hline
\textbf{Tweet (original)} & \textbf{Tweet (after preprocessing)}\\ \hline
$\#$RockyFire Update =$>$ California Hwy. 20 closed in both & rockyfire update california hwy 20 closed \\
directions due to Lake County fire - $\#$CAfire $\#$wildfires & directions due lake county fire cafire wildfires \\ \hline
$@$TheAtlantic That or they might be killed in an airplane & theatlantic might killed airplane accident
\\
accident in the night a car wreck$!$ & night car wreck \\ \hline
\end{tabular}
\label{tab:data_processing}
\end{table*}
\begin{table}
\centering
\caption{Training data statistic}
\begin{tabular}{l|l}
\hline
Total train data & 7,613\\ \hline
Total positive data (or disaster tweets) & 3,271\\ \hline
Total unique words & 21,940\\ \hline
Total unique words with frequency $>$ 1& 6,816\\ \hline
Avg. length of tweets & 12.5\\ \hline
Median length of tweets & 13\\ \hline
Maximum length of tweets & 29\\ \hline
Minimum length of tweets & 1\\ \hline
\end{tabular}
\label{tab:data_statistic}
\end{table}
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{picture/word_distribution.png}
\caption{Twitter length distribution in training data}
\label{fig:word_distribution}
\end{figure}
\section{Dataset}
For this study, we used a Twitter dataset from a recent Kaggle competition (Natural Language Processing with Disaster Tweets \footnote{\label{data_label} https://www.kaggle.com/c/nlp-getting-started}). Kaggle competition is a very well-known platform for machine learning researchers where many research agencies share their data to solve different types of research problems. For example, many researchers used data from Kaggle competitions to analyze real-life problems and propose models to solve the problems, such as sentiment analysis, feature detection, diagnosis prediction \cite{koumpouri2015evaluation, tolkachev2020deep, iglovikov2017satellite, yang2018deep, yang2020computational}.
In the selected Kaggle competition, a dataset of 10,876 tweets is given to predict which tweets are about real disasters and which ones aren't using machine learning model. This dataset has two separate files, train (7,613 tweets) and test (3,263 tweets) data, where each row of the train data contains id, natural language text or tweet, and label. The labels are manually annotated by humans. They labeled a tweet as positive or one if it is about real disaster, otherwise as negative or zero. On the other hand, the test data has D and natural language text but no label. The competition site stores the labels of test data privately and uses that to calculate test scores based on user's machine learning model predictions and create leader-board for the competition based on the test score. Moreover, this dataset was created by the figure-eight company and originally shared on their website \footnote{\label{data_source} https://appen.com/open-source-datasets/}.
We used the training data to train different machine learning models and predict test data labels using trained models. We reported both the train and test data score in our experiment. Note that our purpose is not to get a high score in the competition, rather use Twitter data to study our research goals.
\subsection{Data pre-processing} Since the twitter data is natural language text, and it contains different types of typos, punctuation, abbreviations, and numbers. For this reason, before training machine learning models on the natural language text, a text pre-processing step is required to remove stop words and word tokenization. Hence, we removed all the stopwords and punctuations from the training data and converted all the words into lower-case letters. Table \ref{tab:data_processing} shows some pre-processed tweets with the original tweets.
\subsection{Data analysis} Before running any machine learning methods on our data, we analyzed our dataset to obtain some insights about the data. Table \ref{tab:data_statistic} shows some statistical results on the training data after pre-processing the text. From the table, we find that there are 43$\%$ tweets that are annotated as real disasters and 57$\%$ are not. There are a total of 21,940 unique words, while only 6,816 words have frequency $>$1. The average length of tweets is 12.5. However, it is important to check the length of positive and negative tweets separately to verify whether they have common characteristics. Figure \ref{fig:word_distribution} shows word distribution for both the positive and negative tweets. The figure shows many negative tweets with small word lengths ($<$10), but most positive and negative tweets are in a word length of 10 to 20.
We also analyzed the word frequency for positive and negative tweets. Figure \ref{fig:word_cloud} shows the most frequent words in a word cloud where the high font of a word presents high frequency. We can find some common words in both types of tweets (i.e., https, t, co, people). However, Figure \ref{fig:word_cloud}(a) highlights many disaster-related words like storm, fire, bomber, death, and earthquake. On the other hand, Figure \ref{fig:word_cloud}(b) highlights daily used words such as think, good, love, now, time. From this figure, it is clear that the most frequent words are different in the two types of tweets, and understanding the meaning of words is important to classify them.
\begin{figure*}[h!]
\centering
\includegraphics[scale=0.4]{picture/word_cloud.png}
\caption{Showing the most frequent words in training data}
\label{fig:word_cloud}
\end{figure*}
\section{Experiments}
In our experimental study, we conduct several experiments based on the real Twitter data to predict disaster-types tweets. At first, we describe the experimental settings and model training procedures in this section. Then, we analyze the experimental results in detail.
\subsection{Experimental settings}
\subsubsection{Traditional ML models with BOW embeddings}
From the Table \ref{tab:data_statistic}, we find that the training data has 21,940 unique words where 6,816 words have frequency more than 1. To avoid infrequent words, we considered only the vocabulary of 6,816 words in our BOW representations. To represent a tweet in BOW embeddings, we took a binary array of 6,816 length where it had 1 if a word of tweet was present in the vocabulary, otherwise 0.
We used the BOW embeddings to predict the sentiment of a tweet using three traditional machine learning models, 1) decision tree, 2) random forest and 3) logistic regression. We used python Sklearn package \footnote{https://scikit-learn.org/stable/} and used all the default parameters to train the models on our train dataset. After training the model, we used the test data to get labels and submit that in Kaggle to have test score.
\subsubsection{Deep learning models with context-free embeddings}
For this experiment, we chose three context-free methods, 1) Skip-gram \cite{DBLP:journals/corr/abs-1301-3781}, 2) FastText \cite{bojanowski2016enriching}, and 2) GloVe \cite{pennington2014glove}. We used publicly available pre-trained embeddings of Skip-gram and fastText models that are trained on Wikipedia data \footnote{ https://nlp.stanford.edu/projects/glove/}. The pre-trained embeddings of FastText is collected from Mikolov et. al. 2018 \cite{mikolov2018advances}. The size of all the pre-trained embeddings or feature is 300.
The proposed softmax model is trained for 100 epochs using a stochastic gradient algorithm to minimize the categorical cross entropy loss function. We took 1$\%$ of training data as validation data and used the validation data to stop the training model if the loss value for the validation data didn't decrease in last ten epochs. Similarly to the softmax model, we also trained our Bi-LSTM model using batch gradient descent algorithm for 100 epochs to minimize the binary cross entropy loss function. We followed the same stop rule for this model.
\subsubsection{Deep learning models with contextual embeddings}
To obtain contextual embeddings, we downloaded publicly available pre-trained BERT model (Bert-base-uncased) \cite{devlin2018bert} from the official site of the authors \footnote{https://github.com/google-research/bert}. We gave tweets as inputs in the BERT model and took the hidden states of the [$CLS$] token of the last layer from the model as embeddings of the given tweets. Then, the embeddings is used in our sigmoid model to predict sentiment of tweets. The same setting is used in a previous paper \cite{ji2021does} to predict patient diagnosis from medical note words using pre-trained BERT model.
Moreover, we can find embeddings of each words of a tweet from the pre-trained BERT model. The BERT's pre-trained word embeddings are used as input to our Bi-LSTM model. The authors of \cite{lu2020vgcn} used the similar setting for the sentiment analysis of text data.
\subsection{Evaluation metric}
Three different metrics are used in our experiment to evaluate the performance of the machine learning models on the disaster prediction task such as, 1) accuracy, 2) F1 score, and 3) Area Under the Curve (AUC). In our experiment, we considered disaster tweets as 'positive class' and others as 'negative class'. Hence, True Positive (TP) means the actual disaster tweets that are predicted as disaster while False Positive (FP) shows the tweets that are actually false, but predicted as true. True Negative (TN) and False Negative (FN) imply in the same way. The accuracy is the number of correctly predicted tweets among all of the tweets and it is calculated as follows.
\begin{equation*}
\text{Accuracy (Acc)} =\frac{\text{TP+TN}}{\text{ TP+FP+TN+FN} }
\end{equation*}
F1 score is another popular metric to test predictive performance of a model. The F1 score is measured by the harmonic mean of recall and precision where recall means the number of true labels are predicted by a model among the total number of existing true labels and precision means the number of true labels are predicted by a model divided by the total number of labels are predicted by the model. The F1 score is calculated as follows.
\begin{equation*}
\text{Recall (R)} =\frac{\text{TP}}{\text{ TP+FN} }
\end{equation*}
\begin{equation*}
\text{Precision (P)} =\frac{\text{TP}}{\text{ TP+FP} }
\end{equation*}
\begin{equation*}
\text{F1 score (F1)} =\frac { \text{2 $\times$ (P $\times$ R)} }{ \text{(P+R)} }
\end{equation*}
On the other hand, AUC tells us how much a model is capable of distinguishing between classes. The higher score of the AUC means the model works better at predicting negative classes as zero and positive classes as one.
\subsection{Experimental results}
\subsubsection{Quantitative results}
Table \ref{tab:prediction_results} provides the results of all the machine learning models on the disaster prediction tasks for all three types of embeddings. The table shows results for both the training and test data. Since the test data results are collected from the Kaggle competition, we only can report the accuracy score.
The table shows that the logistic regression model has the best results for the BOW embeddings among the three traditional machine learning models. However, the results of neural network models for context-free embeddings are better than the traditional machine learning models that used context-free embeddings as inputs. Among the three context-free embeddings (Skip-gram, FastText, GloVe), the GloVe with Bi-LSTM model has the best train and test score for all the three evaluation metrics. Note that the results also show us that deep learning model like Bi-LSTM has better results than the shallow neural network model such as the softmax model.
Moreover, when we used the same shallow neural network and deep learning models for contextual embeddings such as BERT; we found that there are 2$\%$ improvements on AUC and Acc over the context-free embeddings. It means that contextual embeddings are helpful and have the best performance for the disaster prediction task.
\begin{table}
\centering
\caption{Performance of different machine learning models on disaster prediction for different types of word representations or embeddings}
\begin{tabular}{|l|l|l|l|l|}
\hline
\textbf{Model} & \multicolumn{3}{l|}{ \textbf{Train data} } & \textbf{Test data} \\
\hline
& AUC &F1 & Acc & Acc \\
\hline
\multicolumn{5}{|l|} { \textbf{BOW embeddings} } \\ \hline
Decision tree &0.6320 &0.5896 &0.6273 &0.6380\\ \hline
Random forest &0.8313 &0.7320 &0.7848 &0.7042\\ \hline
Logistic regression &\textbf{0.8660} &\textbf{0.7443} &\textbf{0.7927} &\textbf{0.7293}\\ \hline \hline
\multicolumn{5}{|l|} { \textbf{Context-free embeddings}} \\ \hline
Skip-gram+Softmax &0.8281 &0.7301 &0.7769 &0.7649\\ \hline
FastText+Softmax &0.8336 &0.7231 &0.7769 &0.7826\\ \hline
GloVe+Softmax &0.8246 &0.7323 &0.7717 &0.7827\\ \hline
Skip-gram+Bi-LSTM &0.8272 &0.7440 &0.7808 &0.7775\\ \hline
FastText+Bi-LSTM &0.8327 &0.7369 &0.7817 &0.7955\\ \hline
GloVe+Bi-LSTM &\textbf{0.8351} &\textbf{0.7500} &\textbf{0.7991} &\textbf{0.8093}\\ \hline \hline
\multicolumn{5}{|l|} { \textbf{Contextual embeddings} }\\ \hline
BERT+Softmax &0.8513 &0.8254 &0.8292 &0.8250\\ \hline
BERT+Bi-LSTM &\textbf{0.8578} &\textbf{0.8316} &\textbf{0.8351} &\textbf{0.8308}\\ \hline
\end{tabular}
\label{tab:prediction_results}
\end{table}
\subsubsection{Qualitative results}
Table \ref{tab:prediction_results} shows us quantitative results for the prediction of disaster tweets where the neural model with contextual embeddings outperformed the other models. However, it is difficult to understand from the result that when the contextual embeddings predict a disaster tweet successfully while context-free models fail. For this purpose, we observe the prediction results of the Bi-LSTM model for both the context-free (GloVe) and contextual embeddings (BERT). Table \ref{tab:sample_prediction} shows the model predictions with true labels for some sample tweets. From the table, we can find that the predictions for GloVe embeddings for the first two tweets are positive, maybe because of the word, ``$accident$'', in the tweets, but the true labels are negative for the two tweets. If we read the tweets, then we can understand that the tweets are not related to disaster or crisis. On the other hand, since the BERT model generates word embeddings based on the context words, it successfully predicts the tweets as negative.
The predictions for GloVe embeddings for the third and fourth tweets are false while they are true. Note that no disaster-related words are used in the two tweets, but the tweets described serious situations. The predictions for BERT embeddings are also correct for the third and fourth tweets. The predictions of GloVe and BERT embeddings for the fifth and sixth tweets of Table \ref{tab:prediction_results} are correct. Since there are some disaster-related words (i.e., suicide, bomber, bombing) in the tweets, both models successfully labeled them.
After analyzing the results of Table \ref{tab:sample_prediction}, it can be implied that the context-free embeddings are helpful to predict a tweet as a disaster if disaster-related words (i.e., accident, bomb) exist in the tweet. In contrast, contextual embeddings help to understand the context of a tweet that is challenging and important for the sentiment analysis task. Although every tweet has a short length text, contextual embeddings works efficiently to understand the sentiment of a tweet.
\begin{table*}
\centering
\caption{Showing sentiment predictions of Bi-LSTM model for pre-trained GloVe and BERT embeddings}
\begin{tabular}{l|l|l|l|l}
\hline
& \textbf{Sample tweets} & \multicolumn{2}{l|}{ \textbf{Prediction} } & \textbf{True}\\
& &\textbf{GloVe} & \textbf{BERT} & \textbf{label} \\ \hline
1 & I swear someone needs to take it away from me, cuase I'm just accident prone. & Yes & No &No\\ \hline
2 &$@$Dave if I say that I met her by accident this week- would you be& & & \\
&super jelly Dave? :p & Yes & No &No\\ \hline
3 &Schoolgirl attacked in Seaton Delaval park by 'pack of animals' & No & Yes &Yes\\ \hline
4 &Not sure how these fire-workers rush into burning buildings & & & \\
&but I'm grateful they do. $\#$TrueHeroes & No & Yes &Yes\\ \hline
5 & A suicide bomber has blown himself up at a mosque in the south & Yes & Yes &Yes\\ \hline
6 &Bombing of Hiroshima 1945 & Yes & Yes &Yes\\ \hline
\end{tabular}
\label{tab:sample_prediction}
\end{table*}
\section{Conclusion}
In this paper, we described an extensive analysis for predicting disaster from Twitter data using different types of word embeddings. Our experimental results show that contextual embeddings have the best result for predicting disaster from tweets. We also showed that deep neural network models outperformed traditional machine learning methods in the disaster prediction task. Advance deep neural network models such as multi-layer convolutional models can also be used for this prediction task to achieve higher accuracy.
\bibliographystyle{ACM-Reference-Format}
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\section{Introduction}
Hilbert schemes of points have a rich literature in algebraic geometry, commutative algebra, combinatorics,
representation theory, and approximation theory. Various aspects of them have been studied in many contexts.
In this paper we study the local equations and the singularities of $\text{Hilb}^n(\mathbb{C}^d)$. For a
general introduction to
the field, see \cite[Chapter 18]{MS:comb}.
In \cite{H:hil}, Haiman proved the remarkable result that the isospectral Hilbert scheme
of points in the plane is normal, Cohen-Macaulay and Gorenstein.
He also showed that this implies the $n!$ conjecture and the positivity conjecture for
the Kostka-Macdonald coefficients. In addition, he conjectured that
the isospectral Hilbert scheme over the principal component of
$\text{Hilb}^n(\mathbb{C}^d)$ is Cohen-Macaulay for any $d, n\geq 1$. In particular, his
conjecture implies that the principal component of $\text{Hilb}^n(\mathbb{C}^d)$ is
Cohen-Macaulay (see \cite[Section 5.2]{H:hil} and \cite[Conjecture 18.38]{MS:comb}).
We provide a counterexample to the conjecture. The idea is to look at the local neighborhood
near $\mathfrak{m}^2$ on the principal component of $\text{Hilb}^{9}(\mathbb{C}^8)$, which is an affine cone over a
certain projective variety.
We will see that its local equations contain generators of high degree. Then the geometry of
the projective variety implies that its affine cone is not Cohen-Macaulay. Our main result is the following:
\begin{theoremalpha}\label{mainthmprinhilb}
The principal component of $\text{Hilb}^{9}(\mathbb{C}^8)$ is not locally Cohen-Macaulay at $\mathfrak{m}^2$.
\end{theoremalpha}
Vakil showed that a number of important moduli spaces satisfy Murphy's law, and many others studied
badly-behaved moduli spaces of positive-dimensional objects (see \cite{V:bad} and the references therein). However very little is
known about how bad the singularities of the Hilbert scheme of points on a smooth variety of
dimension $>2$ can be. On the other hand, Haiman \cite[Proposition 2.6 and Remark.(2) in p.213]{MH:tq} showed that
a certain blow-up of $\text{Sym}^n(\mathbb{C}^d)$ is the principal component
of $\text{Hilb}^n(\mathbb{C}^d)$, and Ekedahl and Skjelnes \cite{ES:good} generalized it
to the case of quasi-projective schemes. If $d=2$ then the blow-up is a resolution of singularities, but
Theorem~\ref{mainthmprinhilb} implies that if $d,n \gg 0$ then the blow-up destroys the Cohen-Macaulayness
of $\text{Sym}^n(\mathbb{C}^d)$.
Turning to a more detailed description, we consider the Hilbert scheme $\text{Hilb}^{d+1}(\mathbb{C}^d)$ of
$(d+1)$ points in affine $d$-space $\mathbb{C}^d$, because it contains the squares $\mathfrak{m}^2$ of
maximal ideals. It parameterizes
the ideals $I$ of colength $(d+1)$ in
$\mathbb{C}[\mathbf{x}]=\mathbb{C}[x_1, ... ,x_d]$.
Let $V_d\subset \text{Hilb}^{d+1}(\mathbb{C}^d)$ denote the
affine open subscheme consisting of all ideals $I\in
\text{Hilb}^{d+1}(\mathbb{C}^d)$ such that $\{1,x_1, ... ,x_d\}$ is
a $\mathbb{C}$-basis of
$\mathbb{C}[\mathbf{x}]/I$. We will call $V_d$ \emph{the symmetric affine subscheme}.
We note that the square of any maximal
ideal in $\mathbb{C}[\mathbf{x}]$ belongs to the symmetric affine subscheme. One may think of $V_d$ as
a deformation space of $\mathfrak{m}^2$.
The following proposition is probably well-known to experts \cite[Section 6]{GLS:elem}, \cite{MEHu:elem}.
\begin{prop}
Let $d\geq 2$. Let $V_d$ be the symmetric affine open subscheme of
$\emph{Hilb}^{d+1}(\mathbb{C}^d)$. Then $V_d$ is isomorphic to
$$\mathbb{C}^d \times \emph{Spec} (R_d/I_d),$$
where $R_d$ is a $d({{d+1}\choose 2}-1)$-dimensional polynomial ring and
$I_d$ is a homogeneous ideal generated by certain quadratic
polynomials. $($When $d=2$, $I_2$ is the zero ideal $(0)$.$)$
\end{prop}
More precisely, since $V_d$ admits a natural action of $GL(d)$, we can describe the quotient ring in terms of Schur
functors.
\begin{thm}\label{mainthm1}
Let $d\geq 3$. Then $V_d$ is isomorphic to
$$\mathbb{C}^d \times\emph{Spec}
\frac{\emph{Sym}^{\bullet}(\mathbb{S}_{(3,1,1,\cdots,1,0)}W)}{<\mathbb{S}_{(4,3,2,\cdots,2,1)}W>},$$where
$W$ is a $d$-dimensional $\mathbb{C}$-vector space,
$(3,1,1,\cdots,1,0)$ is a partition of $(d+1)$ and
$(4,3,2,\cdots,2,1)$ is of $(2d+2)$.
\end{thm}
If $d\leq 6$ then $V_d$ is irreducible \cite{EI:zero}, \cite{S:def}, \cite{CEVV}.
However if $d\geq 7$ then $V_d$ is reducible, and there is a
distinguished component called a principal component. For any $d$, let $P_d$ denote the principal component
of $V_d$. Here we regard
it as its reduced structure.\footnote{It is not known whether $\text{Hilb}^n(\mathbb{C}^d)$ is reduced or
not, for $d\geq 3$.} The general elements in $P_d$ are
radical ideals defining $(d+1)$ distinct points whose linear span is
non-degenerate, i.e. there is no hyperplane passing through them in
$\mathbb{C}^d$. The most special elements in $P_d$ are $\mathfrak{m}^2$. It is clear that the dimension of the principal
component is $d(d+1)$.
Let $J_d$ denote the defining ideal of $P_d$, in other words,
$$
P_d \text{ }\cong \text{ }\mathbb{C}^d \times \text{Spec} (R_d/J_d),
$$
where $J_d$ is a reduced homogeneous ideal.
There has been some interest in trying to find the equations $P_d$ satisfy
(e.g. \cite[Problem 18.40]{MS:comb}, \cite[Remark 3.4]{S:limit}). But up to now
they have not been known to satisfy any other equations, besides the quadratic Pl\"{u}cker relations.
We present some new equations and obtain the following result.
\begin{thm}\label{eightp}
Let $d=8$ and let $P_8$ be the principal component of $V_8$. Then
$P_8$ is isomorphic to
$$\mathbb{C}^8 \times \emph{Spec} (R_8/J_8),$$
where $R_8$ is a $8({{9}\choose 2}-1)$-dimensional polynomial ring and a
set of the minimal homogeneous generators of $J_8$ contains certain
polynomials of degree $90$. In particular, the Castelnuovo-Mumford
regularity of $J_8$ is $\geq 90$, while the dimension of $\emph{Proj}
(R_8/J_8)$ is $63$.
\end{thm}
Again more precisely,
\begin{prop}\label{mainthm2}
The principal component $P_8$ is isomorphic to
$$\mathbb{C}^8 \times\emph{Spec}
\frac{\emph{Sym}^{\bullet}(\mathbb{S}_{(3,1,1,\cdots,1,0)}W)}{J_8},$$where
the vector space of the minimal homogeneous generators of $J_8$ contains
$$\mathbb{S}_{(133,130,126,122,119,60,60,60)}W.$$
\end{prop}
\begin{proof}[Proof of Theorem~\ref{mainthmprinhilb}]
Together with the following lemma and proposition, Theorem~\ref{eightp} implies that the principal
component $P_8$ is not Cohen-Macaulay. More concretely, if $P_8$ were Cohen-Macaulay,
then $\text{Proj} (R_8/J_8)$ would be arithmetically
Cohen-Macaulay, but then Lemma~\ref{rationalacm} and Proposition~\ref{rationx} would
imply $\text{reg}(\text{Proj} (R_8/J_8))\leq 64$,
which would contradict Theorem~\ref{eightp}.
\end{proof}
\begin{lem}\label{rationalacm}
Let $S \subset \mathbb{P}^N$ be a projective arithmetically
Cohen-Macaulay variety of dimension $n$. Suppose that there is a
smooth open set $\tilde{U} \subset S$ such that\\
\noindent $\bullet$ $\emph{codim}_S(S\setminus \tilde{U}) \geq 2$, and\\
\noindent $\bullet$ $\tilde{U}$ is
covered by rational proper curves, i.e., for any point $x\in
\tilde{U}$, there is a smooth irreducible rational proper curve
on $\tilde{U}$ passing through $x$.
Then $\emph{reg}(S)\leq n+1$.
\end{lem}
\begin{prop}\label{rationx}
Let $X=\emph{Proj} (R_d/J_d)$ for $2\leq d\leq 8$. Then there is a
smooth open set $\tilde{U}_d \subset X$ such that\\
\noindent $\bullet$ $\emph{codim}_X (X\setminus \tilde{U}_d) = 2$, and\\
\noindent $\bullet$ $\tilde{U}_d$ is
covered by rational proper curves.
\end{prop}
I am grateful to Professors Rob Lazarsfeld, David Eisenbud, William Fulton, Mark Haiman,
Anthony Iarrobino, Steve Kleiman, Ezra Miller, Mircea Musta\c{t}\u{a}, Bjorn Poonen, Mihnea Popa,
Boris Shekhtman, Roy Skjelnes,
Bernd Sturmfels, Ravi Vakil, and Dustin Cartwright for their valuable advices, suggestions, comments,
discussions and correspondence.
\section{Local equations of the Hilbert scheme of points}\label{proofmain1}
In this section we prove Theorem ~\ref{mainthm1}. In fact the defining ideal of $V_d$ will be obtained
by very concrete computations.
Before we begin the proof, let us explain the notation more precisely. By Lemma \ref{comput},
there is an injective homomorphism
$$j:\mathbb{S}_{(4,3,2,\cdots,2,1)}W \hookrightarrow \text{Sym}^{2}(\mathbb{S}_{(3,1,1,\cdots,1,0)}W)$$
of Schur modules. Then $j$ induces natural maps
$$\aligned &\mathbb{S}_{(4,3,2,\cdots,2,1)}W \otimes \text{Sym}^{r-2}(\mathbb{S}_{(3,1,1,\cdots,1,0)}W)\\
&\hookrightarrow
\text{Sym}^{2}(\mathbb{S}_{(3,1,1,\cdots,1,0)}W)\otimes
\text{Sym}^{r-2}(\mathbb{S}_{(3,1,1,\cdots,1,0)}W)\\
&\rightarrow \text{Sym}^{r}(\mathbb{S}_{(3,1,1,\cdots,1,0)}W),\text{
}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{
}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{
}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{
}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{
}\text{ }r \geq 2,\endaligned$$which define the quotient ring
$\frac{\text{Sym}^{\bullet}(\mathbb{S}_{(3,1,1,\cdots,1,0)}W)}{<\mathbb{S}_{(4,3,2,\cdots,2,1)}W>}$.
To ease notations and references, we introduce the notion of ideal
projectors(cf. \cite{B:algebra}, \cite{dB:ideal}, \cite{dB:limits},
\cite{S:limit}).
\begin{defn}
(cf. \cite{B:algebra}) A linear idempotent map $P$ on
$\mathbb{C}[\mathbf{x}]$ is called an \textbf{ideal projector} if
$\text{ker} P$ is an ideal in $\mathbb{C}[\mathbf{x}]$.
\end{defn}
We will use \emph{de Boor's formula}:
\begin{thm}\label{dB} \emph{(\cite{dB:ideal}, de Boor)} A linear mapping $P : \mathbb{C}[\mathbf{x}] \rightarrow \mathbb{C}[\mathbf{x}] $ is an
ideal projector if and only if the equality
\begin{equation}\label{deBoor}
P(gh) = P(gP(h))
\end{equation}
holds for all $g, h \in \mathbb{C}[\mathbf{x}]$.
\end{thm}
Let $\mathcal{P}$ be the space of ideal projectors onto span
$\{1,x_1,...,x_d\}$, in other words,
$$ \mathcal{P}:=\{P : \text{ideal projector } | \text{ ker} P \in
V_d\}.
$$ The space $\mathcal{P}$ is isomorphic to the symmetric affine
subscheme $V_d$ \cite[p3]{S:bivideal}. For the sake of simplicity,
we prefer to work on $\mathcal{P}$ in place of $V_d$.
First we consider the natural embedding of $\mathcal{P}$. Gustavsen,
Laksov and Skjelnes \cite{GLS:elem} gave more general description of
open affine coverings of Hilbert schemes of points.
\begin{lem}
The space $\mathcal{P}$ can be embedded into
$\mathbb{C}^{(d+1){{d+1}\choose 2}}$.
\end{lem}
\begin{proof}[Sketch of proof]
For each ideal projector $P\in \mathcal{P}$ and each pair $(i,j)$,
$1\leq i,j\leq d$, there is a collection $p_{0,ij},p_{1,ij}, \cdots
,p_{d,ij}$ of complex numbers such that
\begin{equation}\label{pequations}
P(x_i x_j) = p_{0,ij}+\sum_{m=1}^d p_{m,ij}x_m.
\end{equation}
As $(i,j)$ varies over $1\leq i,j\leq d$, each ideal projector $P\in
\mathcal{P}$ gives rise to a collection $p_{0,ij}, p_{r, st}$
$(1\leq i,j,r,s,t \leq d)$ of complex numbers. Of course
$p_{0,ij}=p_{0,ji}$ and $p_{r,st}=p_{r, ts}$. So we have a map $f:
\mathcal{P} \rightarrow\mathbb{C}^{(d+1){{d+1}\choose
2}}=\frac{\mathbb{C}[p_{0,ij},\text{ }\text{ } p_{r, st}]_{1\leq
i,j,r,s,t \leq d}}{(p_{0,ij}-p_{0,ji},\text{ }\text{ }p_{r,st}-p_{r,
ts})}$.
Here we only show that $f$ is one-to-one. It is proved in
\cite{GLS:elem} that $f$ is in fact a scheme-theoretic embedding.
We will show that if $P_1, P_2\in\mathcal{P}$ and if
$f(P_1)=f(P_2)$, i.e. $P_1(x_i x_j)=P_2(x_i x_j)$ for every $(i,j)$,
$1\leq i,j\leq d$, then $P_1=P_2$. Since $P_1$ and $P_2$ are linear
maps, it is enough to check that
$P_1(x_{i_1}...x_{i_r})=P_2(x_{i_1}...x_{i_r})$ for any monomial
$x_{i_1}...x_{i_r}$. This follows from de Boor's formula
(\ref{deBoor}):
$$\aligned P_1(x_{i_1}...x_{i_r})&=P_1(x_{i_1}P_1(x_{i_2}\cdots
P_1(x_{i_{r-1}}x_{i_r})\cdots))\\
&=P_2(x_{i_1}P_2(x_{i_2}\cdots
P_2(x_{i_{r-1}}x_{i_r})\cdots))=P_2(x_{i_1}...x_{i_r}),
\endaligned$$where we have used the property that $P(g)$ is a linear
combination of $1,x_1,\dots,x_d$ for any $g \in
\mathbb{C}[\mathbf{x}]$.
\end{proof}
Next we describe the ideal defining $\mathcal{P}$ in
$$\frac{\mathbb{C}[p_{0,ij},\text{ }\text{ } p_{r, st}]_{1\leq
i,j,r,s,t \leq d}}{(p_{0,ij}-p_{0,ji},\text{ }\text{ }p_{r,st}-p_{r,
ts})}=:R,$$where we keep the notations in the above proof. Let
$I_{\mathcal{P}}$ denote the ideal.
\begin{lem}\label{idealP}
Let $C(a;j,(i,k))\in R$ denote the coefficient of $x_a$ in
$$P(x_k P(x_i x_j)) - P(x_i P(x_k x_j)) \in R[x_1, \cdots,
x_d].$$ Then $I_{\mathcal{P}}$ is generated by $C(a;j,(i,k))$'s
$(0\leq a \leq d$,$\text{ }$ $1 \leq i,j,k\leq d)$. (We regard an
element in $R[x_1, \cdots, x_d]_0 \cong R$ as a coefficient of
$x_0$.)
\end{lem}
For example, if $a\neq j,i,k$ then
\begin{equation}\label{aijkaneqijk}
C(a;j,(i,k))=\sum_{m=1}^d (p_{m,ij}p_{a,km}-p_{m,kj}p_{a,im}).\end{equation}
If $a=k$ then
\begin{equation}C(k;j,(i,k))=p_{0,ij}+\sum_{m=1}^d (p_{m,ij}p_{k,km}-p_{m,kj}p_{k,im}).\end{equation}
\begin{proof}[Proof of Lemma ~\ref{idealP}]
The de Boor's formula (\ref{deBoor}) implies that $I_{\mathcal{P}}$
is generated by coefficients of $x_a$'s ($1\leq a \leq d$) in
$P(gP(h))-P(hP(g))$ (all $g,h\in \mathbb{C}[\mathbf{x}]$). But any
$P(gP(h))-P(hP(g))$ can be generated by $P(x_k P(x_i x_j)) - P(x_i
P(x_k x_j))$'s.
\end{proof}
We note that $C(a;j,(i,k))+C(a;j,(k,i))=0$ so from now on we
identify $C(a;j,(i,k))$ with $-C(a;j,(k,i))$.
\begin{lem}
In fact, $I_{\mathcal{P}}$ is generated by $C(a;j,(i,k))$'s $(1\leq
a \leq d$,$\text{ }$ $1 \leq i,j,k\leq d)$.
\end{lem}
\begin{proof}
It is enough to prove that for any $1 \leq i,j,k\leq d$, the
polynomial $C(0;j,(i,k))$ is generated by $C(a;b,(e,f))$'s $(1 \leq
a,b,e,f\leq d)$. Fix any $u$, $1 \leq u\leq d$. Then we have
$$\aligned C(0;j,&(i,k))
=\sum_{m=1}^d
(p_{m,ij}p_{0,km}-p_{m,kj}p_{0,im})\\
=-\sum_{m=1}^d \Big(&p_{m,ij} \sum_{t=1}^d
(p_{t,km}p_{u,tu}-p_{t,ku}p_{u,tm})
-p_{m,kj}\sum_{t=1}^d(p_{t,im}p_{u,tu}-p_{t,iu}p_{u,tm})\Big)\\
+\sum_{m=1}^d \Big(&p_{m,ij} C(u;k,(m,u))
-p_{m,kj}C(u;i,(m,u))\Big)
\endaligned$$
$$\aligned
=-\sum_{t=1}^d \Big(&p_{u,tu}\sum_{m=1}^d (p_{m,ij}p_{t,km}-p_{m,kj}p_{t,im})\\
-&p_{t,ku}\sum_{m=1}^d(p_{m,ij}p_{u,tm}-p_{m,it}p_{u,jm})
+p_{t,iu}\sum_{m=1}^d(p_{m,kj}p_{u,tm}-p_{m,kt}p_{u,jm})\Big)\\
+\sum_{m=1}^d
&p_{u,jm}\sum_{t=1}^d(p_{t,ku}p_{m,it}-p_{t,iu}p_{m,kt})\\
+\sum_{m=1}^d \Big(&p_{m,ij} C(u;k,(m,u))
-p_{m,kj}C(u;i,(m,u))\Big)
\endaligned$$
$$\aligned
=-\sum_{t=1}^d \Big(&p_{u,tu}C(t;j,(i,k))
-p_{t,ku}C(u;i,(j,t))
+p_{t,iu}C(u;k,(j,t))\Big)\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\\
+\sum_{m=1}^d
&p_{u,jm}C(m;u,(k,i))\\
+\sum_{m=1}^d \Big(&p_{m,ij} C(u;k,(m,u))
-p_{m,kj}C(u;i,(m,u))\Big).
\endaligned$$
\end{proof}
So the set of generators of $I_{\mathcal{P}}$ is
$$\{C(a;j,(i,k))\text{ } | \text{ }1 \leq a,i,j,k\leq d)\}.$$
We associate to this a representation of $GL(W)$, where $W$ is a $d$-dimensional vector space.
\begin{prop}\label{vector_space}
The $\mathbb{C}$-vector space $Y$ of generators
$$\frac{<C(a;j,(i,k))\text{ } | \text{ }1 \leq a,i,j,k\leq d)>}{C(a;j,(i,k))+C(a;j,(k,i))}$$
is canonically isomorphic to
$$
\mathbb{S}_{(3,2,1,\cdots,1,0)}W \bigoplus
\mathbb{S}_{(3,1,1,\cdots,1,1)}W$$ as $\mathbb{C}$-vector spaces,
where $W$ is a $d$-dimensional vector space and $
\mathbb{S}_{(3,2,1,\cdots,1,0)}$ $($resp.
$\mathbb{S}_{(3,1,1,\cdots,1,1)})$ is the Schur functor
corresponding to the partition $(3,2,1,\cdots,1,0)$ $($resp.
$(3,1,1,\cdots,1,1))$ of $(d+2)$.
\end{prop}
\begin{proof}
Let $W=\bigoplus_{i=1}^d \mathbb{C}v_i$. Define
$$\varphi:Y \longrightarrow \bigwedge^{d-1}W \otimes W
\otimes \bigwedge^2 W$$ by
$$\varphi:C(a;j,(i,k)) \mapsto (-1)^a(v_1\wedge \cdots \wedge
\hat{v_a}\wedge \cdots \wedge v_d)\otimes v_j\otimes(v_i \wedge
v_k). $$ Then it is clear that $\varphi$ is injective.
By Littlewood-Richardson rule, we have
$$\aligned&\bigwedge^{d-1}W \otimes W \otimes \bigwedge^2 W\\
&\cong \mathbb{S}_{(1,1,1,\cdots,1,0)}W \otimes W \otimes \mathbb{S}_{(1,1,0,\cdots,0,0)}
W\\
&\cong
\mathbb{S}_{(3,2,1,\cdots,1,0)}W \bigoplus
\mathbb{S}_{(3,1,1,\cdots,1,1)}W
\bigoplus (\mathbb{S}_{(2,2,1,\cdots,1,1)}W)^{\oplus
2}\bigoplus \mathbb{S}_{(2,2,2,1,\cdots,1,0)}W\\
&\cong \mathbb{S}_{(3,2,1,\cdots,1,0)}W \bigoplus
\mathbb{S}_{(3,1,1,\cdots,1,1)}W
\bigoplus\bigwedge^{d}W \otimes \bigwedge^2 W \bigoplus\bigwedge^{d-1}W \otimes
\bigwedge^3 W,\endaligned$$where each partition is of $(d+2)$. We
will show that the image of any element of $Y$ under $\varphi$ lies neither on
$\bigwedge^{d}W \otimes \bigwedge^2 W$ nor $\bigwedge^{d-1}W \otimes
\bigwedge^3 W$.
Since
$$
\sum_{j=1}^d (-1)^j(v_1\wedge \cdots \wedge \hat{v_j}\wedge \cdots
\wedge v_d)\otimes v_j\otimes(v_i \wedge v_k),\text{ }\text{ }\text{
}1\leq i <k\leq d,
$$
generate $\bigwedge^{d}W \otimes \bigwedge^2 W$, we need to show
that
\begin{equation}\label{eq1}
\sum_{j=1}^d C(j;j,(i,k)) =0.
\end{equation} But this is elementary
because
$$\sum_{j=1}^d C(j;j,(i,k))=\sum_{j=1}^d\sum_{m=1}^d (p_{m,ij}p_{j,km}-p_{m,kj}p_{j,im})=0.$$
Since
$$\aligned
&(v_1\wedge \cdots \wedge \hat{v_a}\wedge \cdots
\wedge v_d)\otimes v_j\otimes(v_i \wedge v_k)\\
&+(v_1\wedge \cdots \wedge \hat{v_a}\wedge \cdots
\wedge v_d)\otimes v_k\otimes(v_j \wedge v_i)\\
&+(v_1\wedge \cdots \wedge \hat{v_a}\wedge \cdots \wedge v_d)\otimes
v_i\otimes(v_k \wedge v_j),\text{ }\text{ }\text{ }1\leq a\leq
d,\text{ } 1\leq j<i<k\leq d,
\endaligned$$
generate $\bigwedge^{d-1}W \otimes \bigwedge^3 W$, we need to show
that
\begin{equation}\label{eq2}
C(a;j,(i,k))+C(a;k,(j,i))+C(a;i,(k,j)) =0.
\end{equation}
But this is again elementary because
$$
\aligned
&\sum_{m=1}^d (p_{m,ij}p_{a,km}-p_{m,kj}p_{a,im})\\
&+\sum_{m=1}^d (p_{m,jk}p_{a,im}-p_{m,ik}p_{a,jm})\\
&+\sum_{m=1}^d (p_{m,ki}p_{a,jm}-p_{m,ji}p_{a,km})=0.
\endaligned
$$
Therefore $\varphi(Y) \subset \mathbb{S}_{(3,2,1,\cdots,1,0)}W
\bigoplus \mathbb{S}_{(3,1,1,\cdots,1,1)}W$, in other words,
$$\varphi:Y \longrightarrow \mathbb{S}_{(3,2,1,\cdots,1,0)}W
\bigoplus \mathbb{S}_{(3,1,1,\cdots,1,1)}W$$ is injective.
The next lemma completes the proof.
\end{proof}
\begin{lem}
$\varphi:Y \longrightarrow \mathbb{S}_{(3,2,1,\cdots,1,0)}W
\bigoplus \mathbb{S}_{(3,1,1,\cdots,1,1)}W$ is surjective.
\end{lem}
\begin{proof}
It is enough to show that there are no nontrivial
$\mathbb{C}$-linear relations among $C(a;j,(i,k))$'s other than
$\mathbb{C}$-linear combinations of (\ref{eq1}) and (\ref{eq2}).
Suppose
\begin{equation}\label{eq3}
C(a;j,(i,k))+\sum_{u,b,e,f} c_{u,b,(e,f)}C(u;b,(e,f)) =0,\text{
}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{
}c_{u,b,(e,f)}\in \mathbb{C}.
\end{equation}
If $a\neq i,j,k$ then $C(a;j,(i,k))$ contains a term
$p_{m,ij}p_{a,km}$ and a term $p_{m,kj}p_{a,im}$. The term
$p_{m,ij}p_{a,km}$ appears only in $C(a;j,(i,k))$ and $C(a;i,(k,j))$
among all $C(u;b,(e,f))$, $1\leq u,b,e,f\leq d$. Similarly the term
$p_{m,kj}p_{a,im}$ appears only in $C(a;j,(i,k))$ and
$C(a;k,(j,i))$. So the left hand side of (\ref{eq3}) must be a
nontrivial linear combination of (\ref{eq2}) and other relations.
Similarly even if $a=i,j,$ or $k$, each term in $C(a;j,(i,k))$
appears only in the ones involved in (\ref{eq1}) or (\ref{eq2}). To
get cancelation among these, the left hand side of (\ref{eq3}) must
contain (\ref{eq1}) or (\ref{eq2}). Repeating the argument,
(\ref{eq3}) becomes a linear combination of (\ref{eq1}) and
(\ref{eq2}).
\end{proof}
The following decomposition of Schur functors will be used later.
\begin{lem}\label{comput}
We have $$\bigwedge^{d-1}W \otimes \emph{Sym}^2 W\cong
\mathbb{S}_{(2,1,1,\cdots,1,1)}W\oplus
\mathbb{S}_{(3,1,1,\cdots,1,0)}W,$$and
$$\aligned
\emph{Sym}^{2}(\mathbb{S}_{(3,1,1,\cdots,1,0)}W)\cong
&\mathbb{S}_{(6,2,2,\cdots,2,0)}W \oplus
\mathbb{S}_{(5,3,2,\cdots,1,1)}W \oplus
\mathbb{S}_{(5,2,2,\cdots,2,1)}W\\
& \oplus \mathbb{S}_{(4,4,2,\cdots,2,0)}W \oplus
\mathbb{S}_{(4,3,2,\cdots,2,1)}W \oplus
\mathbb{S}_{(4,2,2,\cdots,2,2)}W. \endaligned$$ (If $d=3$ then
$\mathbb{S}_{(5,3,2,\cdots,1,1)}W$ does not appear.)
\end{lem}
\begin{proof}
The first isomorphism follows from the Littlewood-Richardson rule.
The second isomorphism can be calculated by
\cite[pp.124--128]{R:comb}.
\end{proof}
\begin{lem}\label{firstanaly}
There is an injective homomorphism
$$j:\mathbb{S}_{(4,3,2,\cdots,2,1)}W \hookrightarrow \emph{Sym}^{2}\Big(\bigwedge^{d-1}W \otimes \emph{Sym}^2
W\Big)$$ such that $\mathcal{P}$ $($hence the symmetric affine open subscheme
$V_d )$ is isomorphic to
$$\emph{Spec}
\frac{\emph{Sym}^{\bullet}(\bigwedge^{d-1}W \otimes \emph{Sym}^2
W)}{<\mathbb{S}_{(4,3,2,\cdots,2,1)}W>},$$ where
$(4,3,2,\cdots,2,1)$ is a partition of $(2d+2)$.
\end{lem}
\begin{proof}
Consider a diagram
$$\begin{CD} \frac{\mathbb{C}[p'_{0,ij}, \text{ }p'_{r, st}]_{1\leq i,j,r,s,t
\leq d}}{(p'_{0,ij}-p'_{0,ji}, \text{ }p'_{r,st}-p'_{r,ts})} @<f<<
\frac{\mathbb{C}[p_{0,ij}, \text{ }p_{r, st}]_{1\leq
i,j,r,s,t \leq d}}{(p_{0,ij}-p_{0,ji}, \text{ }p_{r,st}-p_{r,ts})}=:R\\
@VVgV @.\\ T:=\frac{\mathbb{C}[p'_{r, st}]_{1\leq r,s,t \leq
d}}{(p'_{r,st}-p'_{r,ts})} @.
\end{CD}$$
where $g$ is the natural projection and $f^{-1}$ is defined by
$$\aligned
&p'_{0,ij} \mapsto C(i+1; j,(i,i+1)),\text{ }\text{ }\text{ }\text{
}\text{ }\text{ }\text{ }\text{ }\text{ }1\leq i\leq
j\leq d,\\
&\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{
}\text{ }\text{ }\text{
}\text{ }\text{ }\text{(if } i=d \text{ then }i+1:=1) \\
&p'_{r,st} \mapsto p_{r,st},\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{
}1\leq r, s, t\leq d.\endaligned$$ In fact $f$
is an isomorphism because $p_{0,ij}$ is a linear term in
$$C(i+1;j,(i,i+1))=p_{0,ij}+\sum_{m=1}^d
(p_{m,ij}p_{(i+1),(i+1)m}-p_{m,(i+1)j}p_{(i+1),im}).$$
Since $C(i+1;j,(i,i+1))\in I_{\mathcal{P}}$, we have an induced
isomorphism
\begin{equation}\label{equ1}\frac{R}{I_{\mathcal{P}}}\cong\frac{T}{I_{\mathcal{P}}T},\end{equation}
where $I_{\mathcal{P}}T$ is the expansion of $I_{\mathcal{P}}$ to
$T$. We note that in this construction $C(i+1; j,(i,i+1))$ can be
replaced by any $C(k;j,(i,k))$ or $C(k;i,(j,k))$ ($k\neq i,j$),
because the resulting $I_{\mathcal{P}}T$ does not depend on the
choice $C(k;j,(i,k))$ or $C(k;i,(j,k))$. In fact this construction
is natural in the sense that we eliminate all the linear terms
appearing in $C(a;j,(i,k))$ so that the ideal $I_{\mathcal{P}}T$ is
generated by quadratic equations.
Since $p'_{0,ij}$ are eliminated under passing $g$, the direct
summand $\mathbb{S}_{(3,1,1,\cdots,1,1)}W(\cong \text{Sym}^2 W)$ in
$W$ is eliminated. Then, by Proposition~\ref{vector_space}, the
vector space of generators of $I_{\mathcal{P}}T$ is canonically
isomorphic to $\mathbb{S}_{(3,2,1,\cdots,1,0)}W$ hence to
$$\bigwedge^{d}W \otimes\mathbb{S}_{(3,2,1,\cdots,1,0)}W \cong
\mathbb{S}_{(4,3,2,\cdots,2,1)}W \subset
\text{Sym}^2\Big(\bigwedge^{d-1}W \otimes \text{Sym}^2 W\Big),$$where the
last containment follows from Lemma~\ref{comput}.
The isomorphism of rings $$T =\frac{\mathbb{C}[p'_{r, st}]_{1\leq
r,s,t \leq d}}{(p'_{r,st}-p'_{r,ts})}\cong
\text{Sym}^{\bullet}\Big(\bigwedge^{d-1}W \otimes \text{Sym}^2 W\Big)$$
naturally induces the isomorphism of quotient rings
\begin{equation}\label{equ2}\frac{T}{I_{\mathcal{P}}T} \cong \frac{\text{Sym}^{\bullet}(\bigwedge^{d-1}W \otimes \text{Sym}^2
W)}{<\mathbb{S}_{(4,3,2,\cdots,2,1)}W>}.\end{equation}Combining this
with\begin{picture}(5,5)\put(4,0){(}\end{picture}~\ref{equ1}) gives
the desired result.
\end{proof}
\begin{thm}\label{mainthm}
$\mathcal{P}$ $($hence the symmetric affine open subscheme
$V_d )$ is isomorphic to
$$\mathbb{C}^d \times\emph{Spec}
\frac{\emph{Sym}^{\bullet}(\mathbb{S}_{(3,1,1,\cdots,1,0)}W)}{<\mathbb{S}_{(4,3,2,\cdots,2,1)}W>},$$where
$(3,1,1,\cdots,1,0)$ is a partition of $(d+1)$ and
$(4,3,2,\cdots,2,1)$ is of $(2d+2)$.
\end{thm}
\begin{proof}[Sketch of Proof]
Define an isomorphism of rings $$T=\frac{\mathbb{C}[p'_{r,
st}]_{1\leq r,s,t \leq d}}{(p'_{r,st}-p'_{r,ts})}
\overset{\cong}\longrightarrow \frac{\mathbb{C}[q_{r, st}]_{1\leq
r,s,t \leq
d}}{(q_{r,st}-q_{r,ts})}=:Q$$by\\
\begin{picture}(200,30)
\put(100,12){$p'_{r,st} \mapsto $}
\put(135,12){\Big{\{}}\put(145,23){$q_{r,sr} + q_{s,ss},\text{
}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ if }
r=t$}\put(145,2){$q_{r,st},\text{ }\text{ }\text{ }\text{ }\text{
}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{
}\text{ }\text{ }\text{ }\text{ }\text{ }\text{if } r\neq s,t.$}
\end{picture}
As a matter of fact this is a natural isomorphism, because the
square of any maximal ideal in $\mathbb{C}[\mathbf{x}]$ satisfies
$p'_{r,sr}-\frac{1}{2}p'_{s,ss}=0$ ($r\neq s$), i.e. $q_{r,sr}=0$.
It is straightforward to check that no element in minimal generators
of $I_{\mathcal{P}}Q$ contains terms involving $q_{s,ss}$, $1\leq
s\leq d$. For example, if $a,i,j,k$ are distinct, then
$$\aligned C(a;j,(i,k))&=\sum_{m=1}^d (p_{m,ij}p_{a,km}-p_{m,kj}p_{a,im})\\
&=\sum_{m\neq a,j,i,k} (p_{m,ij}p_{a,km}-p_{m,kj}p_{a,im})\\
&\text{ }\text{
}+(p_{j,ij}p_{a,kj}-p_{j,kj}p_{a,ij})+(p_{a,ij}p_{a,ka}-p_{a,kj}p_{a,ia})\\
&\text{ }\text{
}+(p_{i,ij}p_{a,ki}-p_{i,kj}p_{a,ii})+(p_{k,ij}p_{a,kk}-p_{k,kj}p_{a,ik})\endaligned$$
becomes
$$\aligned
&\sum_{m\neq a,j,i,k} (q_{m,ij}q_{a,km}-q_{m,kj}q_{a,im})\\
&\text{ }\text{
}+((q_{j,ij}+q_{i,ii})q_{a,kj}-(q_{j,kj}+q_{k,kk})q_{a,ij})+(q_{a,ij}(q_{a,ka}+q_{k,kk})-q_{a,kj}(q_{a,ia}+q_{i,ii}))\\
&\text{ }\text{
}+((q_{i,ij}+q_{j,jj})q_{a,ki}-q_{i,kj}q_{a,ii})+(q_{k,ij}q_{a,kk}-(q_{k,kj}+q_{j,jj})q_{a,ik})\\
&=\sum_{m\neq a,j,i,k} (q_{m,ij}q_{a,km}-q_{m,kj}q_{a,im})\\
&\text{ }\text{
}+(q_{j,ij}q_{a,kj}-q_{j,kj}q_{a,ij})+(q_{a,ij}q_{a,ka}-q_{a,kj}q_{a,ia})\\
&\text{ }\text{
}+(q_{i,ij}q_{a,ki}-q_{i,kj}q_{a,ii})+(q_{k,ij}q_{a,kk}-q_{k,kj}q_{a,ik}),
\endaligned$$
in which no term involves $q_{s,ss}$, $1\leq s\leq d$.
Therefore we get $$\frac{T}{I_{\mathcal{P}}T} \cong
\frac{Q}{I_{\mathcal{P}}Q} \cong \mathbb{C}[q_{s, ss}]_{1\leq s\leq
d} \otimes_{\mathbb{C}} {\frac{\mathbb{C}[q_{r, st}]_{1\leq r,s,t
\leq d,\text{ }\text{ } r\neq s\text{ or }t\neq
s}}{(q_{r,st}-q_{r,ts})}}\Big{/}{I_{\mathcal{P}}Q}.$$
On the other hand, Lemma~\ref{comput} implies
$$\text{Sym}^{\bullet}\Big(\bigwedge^{d-1}W \otimes \text{Sym}^2 W\Big)\cong
\text{Sym}^{\bullet}(\mathbb{S}_{(2,1,1,\cdots,1,1)}W\oplus
\mathbb{S}_{(3,1,1,\cdots,1,0)}W).$$ We may identify the basis of
$\mathbb{S}_{(2,1,1,\cdots,1,1)}W$ with $\{q_{s,ss} | 1\leq s\leq
d\}$. So,
by\begin{picture}(5,5)\put(4,0){(}\end{picture}~\ref{equ2}), we have
$$\aligned\frac{T}{I_{\mathcal{P}}T} &\cong
\mathbb{C}[q_{s, ss}]_{1\leq s\leq d} \otimes_{\mathbb{C}}
{\frac{\mathbb{C}[q_{r, st}]_{1\leq r,s,t \leq d,\text{ }\text{ }
r\neq s\text{ or }t\neq
s}}{(q_{r,st}-q_{r,ts})}}\Big{/}{I_{\mathcal{P}}Q}\\&\cong
\text{Sym}^{\bullet}(\mathbb{S}_{(2,1,1,\cdots,1,1)}W)\otimes
\frac{\text{Sym}^{\bullet}(\mathbb{S}_{(3,1,1,\cdots,1,0)}W)}{<\mathbb{S}_{(4,3,2,\cdots,2,1)}W>}.\endaligned$$Combining
this with\begin{picture}(5,5)\put(4,0){(}\end{picture}~\ref{equ1})
gives the desired result.
\end{proof}
\begin{exmp}\label{secondcor}
It is well known (\cite{K:desing}) that if $d=3$ then $V_d$ is
isomorphic to a cone over the Pl\"{u}cker embedding of the
Grassmannian $G(2,6)$ with a three-dimensional vertex. Let $W$ be a $3$-dimensional vector space and $W'$ a $6$-dimensional
vector space. Then
$$
\frac{\text{Sym}^{\bullet}(\mathbb{S}_{(3,1,0)}W)}{<\mathbb{S}_{(4,3,1)}W>}\text{ }\cong\text{ }
\frac{\text{Sym}^{\bullet}(\bigwedge^2 (\mathbb{S}_{(2,0,0)}W))}{<\bigwedge^4 (\mathbb{S}_{(2,0,0)}W)>}
\text{ }\cong\text{ }
\frac{\text{Sym}^{\bullet}(\bigwedge^2 W')}{<\bigwedge^4 W'>}.$$ \qed
\end{exmp}
\section{Local equations of the principal component of the Hilbert scheme of points}\label{proofmain2}
In this section, we prove Proposition~\ref{mainthm2}. We start by showing
that $J_d$ has a representation-theoretic expression.
\begin{lem}\label{directsumschur}
The $\mathbb{C}$-vector space of the minimal generators of $J_d$ is the direct sum of some irreducible Schur functors.
\end{lem}
\begin{proof}
We prove a more general statement : the vector space $(J_d)_{\leq n}:=\bigoplus_{i=0}^n (J_d)_i$ is
the direct sum of some irreducible Schur functors for every $n$.
It is enough to show that there is a group homomorphism from $GL(d)$ to $GL((J_d)_{\leq n})$ which is
comparable with the natural action of the symmetric group $S_d$.
First, there is a natural way of defining $g\cdot p_{r,st}$ for $g\in GL(d)$.
If $p_{r,st}$ is given by $I_{\{p_1,...,p_{d}\}}\in P_d$ as in (\ref{pequations}), then $g\cdot p_{r,st}$ is
given by $I_{\{g\cdot p_1,...,g\cdot p_{d}\}}$.
Next, we define a homomorphism $\rho$ from $GL(d)$ to $GL((J_d)_{\leq n})$ as follows. For every $f(p_{r,st})_{1\leq r,s,t\leq d} \in (J_d)_{\leq n}$, define
$$
g\cdot f(p_{r,st})_{1\leq r,s,t\leq d}
$$
by
$$
g\cdot f(p_{r,st})_{1\leq r,s,t\leq d}:=f(g\cdot p_{r,st})_{1\leq r,s,t\leq d}.
$$
Since any point in $P_d$ satisfies $f=0$, we have $f(g\cdot p_{r,st})\in (J_d)_{\leq n}$. It
is easy to check that $\rho$ is a homomorphism from $GL(d)$ to $GL((J_d)_{\leq n})$. It is obvious that this
is comparable with the natural action of $S_d$.
\end{proof}
\begin{lem}
The vector space of the minimal homogeneous generators of the ideal $J_8 \subset \emph{Sym}^{\bullet}(\mathbb{S}_{(3,1,1,\cdots,1,0)}W)$ contains
$$\mathbb{S}_{(133,130,126,122,119,60,60,60)}W.$$\end{lem}
\begin{proof}[Sketch of proof]
The idea is to observe that there are relations among
$$
\{p_{r,st}\}_{1\leq r\leq 3, \text{ }4\leq s,t \leq 8}.
$$
This is suggested by the fact that a general ideal having the Hilbert function of the type $(1,5,3)$ is
not contained in the principal component $P_8$ \cite{EI:zero}, \cite{S:def}, \cite{CEVV}. In particular,
if $
\{p_{r,st}\}_{1\leq r\leq 3, \text{ }4\leq s,t \leq 8}
$ are general complex numbers and if the other coordinates are $0$, then the colength 9 ideal determined by those
coordinates does not belong to $P_8$.
By the algorithm in \cite[pp.124--128]{R:comb}, one can check that $\mathbb{S}_{(133,130,126,122,119,60,60,60)}W$ appears in the decomposition of $\text{Sym}^{90}(\mathbb{S}_{(3,1,...,1,0)}W)$.
We find elements in $\mathbb{S}_{(133,130,126,122,119,60,60,60)}W$ in a very explicit way.
Recall from (\ref{aijkaneqijk}) that if $a\neq j,i,k$ then $$C(a;j,(i,k))=\sum_{m=1}^d (p_{m,ij}p_{a,km}-p_{m,kj}p_{a,im})$$
in the polynomial ring
$\frac{\mathbb{C}[p_{r, st}]_{1\leq r,s,t \leq d}}{(p_{r,st}-p_{r,ts})}$. The key fact is that
any term in any $C(a;j,(i,k))$ with
$1\leq a \leq 3, 4\leq i,j,k\leq d$ is a product of two coordinates, one of which is in
$\{p_{r,st}\}_{1\leq r \leq 3, 4\leq s,t,\leq d}$ and the other is not. We consider the following $90\times 1$ matrix each of whose entry is a polynomial of degree 2.
$$
\left(
\begin{array}{c}
C(1;4,(5,6)) \\
\vdots \\
C(a; j, (i,k)) \\
\vdots \\
C(3;6,(5,6) ) \\
\end{array}
\right),
$$
where $1\leq a \leq 3$, and $4\leq j<i<k\leq8$ or $4\leq k< j<i \leq 8$ or $4=i=j<k\leq 8$
or $4=i<j=k\leq 8$ or $5=i=j<k\leq 6$ or $5=i<j=k \leq 6$.
Then we can observe that there is a $90\times 115$ matrix $ \mathbf{M}$ such that each entry of $ \mathbf{M}$ is one of the elements in
$$
\{0, \pm p_{r,st} \}_{1\leq r\leq 3, \text{ }4\leq s,t \leq 8},
$$
and $\mathbf{M}$ fits into the following matrix factorization:
$$
\left(
\begin{array}{c}
C(1;4,(5,6)) \\
\vdots \\
C(a; j, (i,k)) \\
\vdots \\
C(3;6,(5,6) ) \\
\end{array}
\right)=
\mathbf{M}\cdot
\left(
\begin{array}{c}
\vdots \\
p_{r',s't'}- \delta_{r',t'} \frac{p_{s',s's'}}{2} - \delta_{r',s'} \frac{p_{t',t't'}}{2} \\
\vdots \\
\end{array}
\right),
$$
where $\delta$ denotes the Kronecker delta, and $1\leq r',s'\leq 3<t'\leq 8$ or $4\leq r'\leq 8$, $4\leq s'\leq t'\leq 8 \text{ but }r',s',t' \text{ are not all equal}$.
Then exhaustive computations show that the determinant of any $90\times 90$ minor
of $\mathbf{M}$ lies in $\mathbb{S}_{(133,130,126,122,119,60,60,60)}W$ and in $J_8$, and
that the determinant of some $90\times 90$ minor is nonzero.\footnote{
One can check whether or not a given polynomial belongs to $J_8$, since $P_8$ admits
an explicit rational parametrization. (For example, see \cite[Theorem 3.3]{S:limit} or
\cite[Proposition 2.6]{MH:tq}.)} Then, thanks to Lemma~\ref{directsumschur}, $J_8$
contains $\mathbb{S}_{(133,130,126,122,119,60,60,60)}W.$
It remains to show that $\mathbb{S}_{(133,130,126,122,119,60,60,60)}W$ is contained
in the minimal generators of $J_8$. If $\mathbb{S}_{(133,...)}W$ were not minimal, there would be
a partition $\lambda$ of $89\cdot 9$ such that
$$
\mathbb{S}_{\lambda} W \subset J_8\cap \text{Sym}^{89}(\mathbb{S}_{(3,1,...1,0)}W),
$$and $\mathbb{S}_{\lambda} W$ generates $\mathbb{S}_{(133,...)}W$. But we have to check
that there is no such $\lambda$.
To this end, we consider all the partitions $\lambda$ such that
$$
\mathbb{S}_{\lambda} W \subset (\mathbb{S}_{(3,1,...1,0)}W)^{\otimes 89}
$$
and
$$\mathbb{S}_{\lambda} W \otimes \mathbb{S}_{(3,1,...1,0)}W \supset \mathbb{S}_{(133,130,126,122,119,60,60,60)}W.
$$
There are 15 such partitions, and I checked that none of their embeddings into
$\text{Sym}^{89}(\mathbb{S}_{(3,1,...1,0)}W)$ are in $J_8$. We remark that each $89\times 89$ minor
of $\mathbf{M}$ belongs
to one of such $\mathbb{S}_{\lambda} W$.
\end{proof}
\begin{rem}
We note that the generator $\mathbb{S}_{(4,3,2,\cdots,2,1)}W$ of $I_8$ does not generate $\mathbb{S}_{(133,...)}W$, in other words,
$$
\mathbb{S}_{(133,130,126,122,119,60,60,60)}W \not\subset <\mathbb{S}_{(4,3,2,\cdots,2,1)}W>.
$$ It is an elementary consequence of the combinatorial Littlewood-Richardson rule(for example,
see \cite[p456]{FH:repre}). In fact any $\mathbb{S}_\lambda W$ ($\lambda=(\lambda_1, \lambda_2, \cdots, \lambda_8)$) appearing in the decomposition of $\mathbb{S}_{(4,3,2,\cdots,2,1)}W \otimes (\mathbb{S}_{(3,1,1,\cdots,1,0)}W)^{\otimes (r-2)}$ satisfies $\lambda_{8-k} +\cdots + \lambda_8\geq rk+1$, for any $r\geq 2$ and any $k=0,...,7$.
Concretely speaking, the ideal generated by $C(a; j, (i,k))$ does not contain
any nonzero determinants of $90\times 90$ minors of $\mathbf{M}$. It is easy to
prove this without using Schur functors, because for any term $\prod_i p_{r_i,s_it_i}$ in any
determinant of $90\times 90$ minors of $\mathbf{M}$, we have $r_i\neq s_j, t_j$ for all $i,j$.
\end{rem}
\section{Proof of Lemma~\ref{rationalacm}}
Lemma~\ref{rationalacm} can be considered as a standard fact. We do not claim any novelty for its proof.
\begin{proof}[Proof of Lemma~\ref{rationalacm}]
Recall that the regularity index of $S$, $r(S)$, is the minimum
degree in which the Hilbert function of $S$ agrees with the Hilbert
polynomial (see \cite{BH:CM} for more details). If $S \subset
\mathbb{P}^N$ is an aCM scheme of dimension $n$, then $r(S) =
\text{reg}(S)-n-1$ (this follows from \cite[Theorem 4.4.3
(b)]{BH:CM}). Hence it is enough to show that $r(S)\leq 0$.
The Hilbert function of $S$ is $H(S,t) = h^0(S,\mathcal{O}_S(t))$
and its Hilbert polynomial is $\chi(S,t) = h^0 (S,\mathcal{O}_S(t))
+ (-1)^n h^n(S,\mathcal{O}_S(t)) = H(S,t) + (-1)^n h^0 (S, \omega_S(-t))$
where $\omega_S$ denotes the dualizing sheaf.
The second condition on $\tilde{U}$ implies $H^0(\tilde{U}, \omega_{\tilde{U}})=0$
(see \cite[Chapter 4]{Kol:rational}). Since $S$ is Cohen-Macaulay
and $\text{codim } S\setminus \tilde{U}\geq 2$, we have $H^0(S, \omega_S)=0$. So we have $h^0(S, \omega_S(-t)) = 0$
for all $t\geq 0$ and this establishes the lemma.
\end{proof}
\section{Proof of Proposition~\ref{rationx}}\label{proofrationalx}
In this section we prove Proposition~\ref{rationx}. We first construct an open
subset $U_d$ of $\text{Spec} (R_d/J_d)$, where $\tilde{U}_d$ will be the projective
counterpart of $U_d$ in $\text{Proj} (R_d/J_d)$.
Let $U_d$ be the open subset of $P_d$ consisting of all ideals $I\in P_d$
such that the radical $\text{Rad}(I)$ of $I$ defines at least $d$
distinct points. Then $U_d$ is smooth and $\text{codim}_{P_d} (P_d\setminus
U_d)=2$.\footnote{For $2\leq d \leq 8$, it can be checked by the computer algebra system Macaulay 2.} We consider the Hilbert--Chow morphism on $U_d$,
$$
\rho : U_d \longrightarrow \text{Sym}^{d+1}(\mathbb{C}^d),
$$
and the averaging map
$$
\pi : \text{Sym}^{d+1}(\mathbb{C}^d) \longrightarrow \mathbb{C}^d
$$
given by
$$
\pi\big(\{(x_{1,1},...,x_{1,d}), ..., (x_{d+1,1},...,x_{d+1,d}) \}\big) =
\Big(\frac{x_{1,1}+...+x_{d+1,1}}{d+1}, ... ,
\frac{x_{1,d}+...+x_{d+1,d}}{d+1}\Big).
$$
Let $j$ be the natural morphism
$$
j : U_d \hookrightarrow P_d \cong \mathbb{C}^d \times \text{Spec} (R_d/J_d),
$$
and let $pr_1$ be the projection
$$
pr_1 : \mathbb{C}^d \times \text{Spec} (R_d/J_d) \longrightarrow
\mathbb{C}^d.
$$
\begin{lem}
$pr_1|_{j(U_d)}$ agrees with $\pi \circ \rho \circ
j^{-1}|_{j(U_d)}$, in other words, the following diagram
\begin{picture}(1,1)
\put(-168,-22){\tiny{$>$}}\put(-145,-66){\tiny{$>$}}
\put(-260,-20){\line(1,0){96}}\put(-260,-64){\line(1,0){120}}
\put(-215,-16){\scriptsize{$\rho$}}\put(-202,-60){\scriptsize{$pr$}}\put(-194,-62){\tiny{$1$}}
\end{picture}
$$\begin{CD}
\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }
\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }
\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }
\text{ }\text{ }\text{ }\text{ } U_d \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }
\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }
\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }
\text{ }\text{ }\text{ }\text{ } @. \emph{Sym}^{d+1}(\mathbb{C}^d)\\
@A{j^{-1}}AA @VV{\pi}V\\
\mathbb{C}^d \times \emph{Spec} (R_d/J_d)\supset j(U_d) \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }
\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }
\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }
\text{ }\text{ }\text{ }\text{ } @. \mathbb{C}^d @.\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }
\text{ }\text{ }\text{ }
\end{CD}$$
is commutative $($up to automorphisms of $\mathbb{C}^d \times \emph{Spec} (R_d/J_d))$.
\end{lem}
\begin{proof}
For each element $I$ in $U_d$, there is a corresponding ideal projector $P_I$, which gives rise to
$p_{r,sr}$, $1\leq r,s \leq d$ as in (\ref{pequations}). From the proof of Theorem~\ref{mainthm},
we may define $$pr_1 \circ j : U_d \longrightarrow \mathbb{C}^d$$ by
$$\aligned I\text{ } \mapsto\text{ } &\Big(\frac{\sum_{r=1}^d p_{r,1r}}{d+1}, \cdots, \frac{\sum_{r=1}^d p_{r,dr}}{d+1}\Big).
\endaligned$$ It is elementary to check that this map is the same as $\pi \circ \rho$.
\end{proof}
We identify the fiber $(pr_1)^{-1}(O)$ over the origin
$O=(0,...,0)\in \mathbb{C}^d$, with the affine cone over
$\text{Proj} (R_d/J_d)$. By construction, $j(U_d) \cap (pr_1)^{-1}(O)$ parameterizes the
ideals defining $d$ distinct points and one more (possibly
infinitely near) point, whose average (=center of mass) is the
origin $O$. Hence scaling distances from $O$ by any nonzero constant
preserves membership in $j(U_d) \cap (pr_1)^{-1}(O)$. In other words,
if an ideal defining $(d+1)$ points, say $p_1,...,p_{d+1}$, belongs
to $j(U_d) \cap (pr_1)^{-1}(O)$, then so does the ideal defining
$\lambda\cdot p_1,...,\lambda\cdot p_{d+1}$ for any $\lambda\neq 0
\in\mathbb{C}$. In fact we have $$\text{Proj} (R_d/J_d)=(\text{Spec} (R_d/J_d) \setminus O
)/\sim,$$ where the equivalence relation is given by
$I_{\{p_1,...,p_{d+1}\}}\sim I_{\{\lambda\cdot p_1,...,\lambda\cdot
p_{d+1}\}}$, $\lambda\neq 0$.
Therefore all told, $j(U_d) \cap (pr_1)^{-1}(O)$ is the affine cone
over a certain open subset of $\text{Proj} (R_d/J_d)$,
with the vertex of the cone removed. We denote the open subset by $\tilde{U}_d$. So we get
$$\tilde{U}_d\text{ }=\text{ }\left(j(U_d) \cap
(pr_1)^{-1}(O)\right)/\sim.$$ Of course, for any point $q\in \mathbb{C}^d$, we have
$$\tilde{U}_d\text{ }\cong\text{ }\left(j(U_d) \cap
(pr_1)^{-1}(q)\right)/\sim.$$
\begin{lem}
For any $d\geq 2$, $\tilde{U}_d$ is covered by rational proper curves, i.e., for any point
$x\in \tilde{U}_d$, there is a smooth irreducible rational
proper curve on $\tilde{U}_d$ passing through $x$.
\end{lem}
\begin{proof}
The idea of the proof is to find a smooth irreducible rational
proper curve on $\tilde{U}_d$, and to apply the $GL(d)$-action to the curve.
First we find a $\mathbb{P}^1$ on $\tilde{U}_d$ as follows. Consider the following $(d+1)$
points on $\mathbb{C}^d$ : $x_i=(x_{i,1},...,x_{i,d})$, $1\leq i\leq d+1$, where
\smallskip
\begin{picture}(200,30)
\put(100,12){$x_{i,j}\text{ }=$}
\put(135,12){\Big{\{}}\put(145,23){$-d,\text{
}\text{ }\text{ }\text{ }\text{ }\text{ }\text{
}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ if }
i=j$}\put(145,2){$1,\text{ }\text{ }\text{ }\text{ }\text{
}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{
}\text{ }\text{ }\text{ }\text{ }\text{ }\text{if } i\neq j.$}
\end{picture}
\noindent Note that the image of the ideal defining $x_1,...,x_{d+1}$ under $(pr_1 \circ j)$ is the origin $O$.
We fix $d$ distinct points $[a_j : b_j]$ $(1\leq j\leq d)$ on $\mathbb{P}^1\setminus [1:1]$, and
define a morphism
$$
\varphi : \mathbb{P}^1 \longrightarrow \tilde{U}_d
$$
by
$$
[\alpha : \beta]\text{ }\mapsto\text{ }\Big[\small{\text{the ideal vanishing
along }}\Big(\small{\frac{b_1\alpha -a_1\beta}{b_1-a_1}}x_{i,1},...,
\small{\frac{b_d\alpha -a_d\beta}{b_d-a_d}}x_{i,d}\Big), 1\leq i\leq d+1\Big],
$$ where $[$ ideal $]$ denotes the equivalence class of the ideal. For each $j$, we define $\varphi([a_j:b_j])$ by
the equivalence class of the non-radical ideal as a limit of radical ideals,
where two points collide. In other words,
$$
\varphi([a_j : b_j])\text{ }=\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{lim}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\varphi([\alpha : \beta]).
$$\begin{picture}(1,1) \put(215,10){\tiny{$[\alpha : \beta]\rightarrow [a_j : b_j]$}}
\put(1,-4){It is straightforward to check that $\varphi$ is well-defined and
that $\varphi(\mathbb{P}^1)$ is smooth and irreducible.}\end{picture}
Now we can prove the lemma. For any element $[I]\in \tilde{U}_d$, there is $g\in GL(d)$ such that
$$
[I]\in g\cdot (\varphi(\mathbb{P}^1)),
$$
where $\cdot$ is the natural action of $GL(d)$.
\end{proof}
|
train/arxiv
|
BkiUdEw5qYVBWXyirj3D
| 4 | 0.8 |
\section{Appendices}
\textbf{Notation 1:} $\lambda=(\lambda_1,\lambda_2,\lambda_3)$ is the barycentric coordinates relative to the vertices $v_1$, $v_2$ and $v_3$, $\nabla_i = \nabla (\lambda_i), \forall i$, and $\phi_p$ is the basis function which correspond to the degree of freedom $p$. We pose (i,j,k) an arbitrary permutation of $(1,2,3)$. \\
Then, $e_{ij}$ corresponds to a node on the edge $[v_i,v_j]$, which is not a vertex (in green on \ref{fig:mesh_elment2} and \ref{fig:cubature_mesh2}), and finally $G$ corresponds to a node in the triangle (in magenta on \ref{fig:mesh_elment2} and \ref{fig:cubature_mesh2}). \\
\textbf{Notation 2: } $w_v$, $w_\alpha$ and $w_\beta$ are weights of respectively vertices , on edges and for the centroid points. For each basis function $\phi$ relative to a DOF, the corresponding weight represents $\int_K \phi$ with $K$ the triangle of reference. It is as well the weight used in quadradrature formula.
\subsection{Basic triangular elements, definition of basis functions} \label{appendix_lag_bf}
Below are represented classical/basic finite elements:
\begin{figure}[H]
\centering
\includegraphics[width=12cm]{image_element/basic_mesh.png}
\caption{Standar elements $\mathbb{P}_1$ on the left, $\mathbb{P}_2$ on the center and $\mathbb{P}_3$ on the right}
\label{fig:mesh_elment2}
\end{figure}
\subsubsection*{$\P_1$ element}
The $\mathbb{P}_1$ element contains $3$ degree of freedom: at each vertices $v_1$ , $v_2$ and $v_3$.
\begin{equation*}
\quad w_v = \frac{1}{3}
\end{equation*}
with $w_v$ weight of vertices. \\
And then, corresponding basis functions are
\begin{itemize}
\item At each vertices $v_1$, $v_2$ and $v_3$.
\begin{equation*}
\hspace*{-1cm}
\phi_{v_i}(\lambda) = \lambda_i, \qquad \mbox{and} \qquad \nabla \phi_{v_i}(\lambda) = \nabla_i
\end{equation*}
\end{itemize}
\subsubsection*{$\P_2$ element}
The $\mathbb{P}_2$ element contains $6$ degree of freedom: $3$ at each vertices $v_1$, $v_2$ and $v_3$ and $3$ on edges $(e_{ij},e_{jk},e_{ki})$ (midpoint of edge $[v_i,v_j]$, etc). Respectively, at vertices and midpoints we have those following weights:
\begin{equation*}
\quad w_v = 0, \quad w_\alpha = \frac{1}{3}
\end{equation*}
And then, corresponding basis functions are
\begin{itemize}
\item At vertices of the triangle:
\begin{equation*}
\hspace*{-1cm}
\phi_{v_i}(\lambda) = \lambda_i ( 2\lambda_i-1) , \qquad \mbox{and} \qquad \nabla \phi_{v_i}(\lambda) = \nabla_i (4\lambda_i-1)
\end{equation*}
\item On edges:
\begin{equation*}
\hspace*{-1cm}
\phi_{e_{ij}}(\lambda) = 4 \lambda_i \lambda_j , \qquad \mbox{and} \qquad \nabla \phi_{e_{ij}}(\lambda) = 4(\nabla_j \lambda_i + \nabla_i \lambda_j)
\end{equation*}
\end{itemize}
\subsubsection*{$\P_3$ element}
The $\mathbb{P}_3$ element contains $10$ degree of freedom : $3$ vertices $v_1$, $v_2$ and $v_3$, $6$ on edges\\ $(e_{ji}(\alpha),e_{ij}(\alpha),e_{kj}(\alpha),e_{jk}(\alpha),e_{ik}(\alpha),e_{ki}(\alpha))$, with $e_{ij}(\alpha)$ such as $\lambda_i = \alpha, \lambda_j = (1-\alpha),\lambda_k = 0$ ($\alpha=\frac{1}{3}$) and a centroid point $G$, such as $\lambda_1 = \lambda_2 = \lambda_3 = \frac{1}{3}$.
\begin{equation*}
\quad w_v = \frac{1}{30}, \quad w_\alpha = \frac{3}{40}, \quad w_\beta = \frac{9}{20}
\end{equation*}
with $w_v$, $w_\alpha$ and $w_\beta$ weights of respectively vertices, on edges and for the centroid point. \\
And then, corresponding basis functions are
\begin{itemize}
\item At vertices of the triangle:
\begin{equation*}
\hspace*{-1cm}
\phi_{v_i}(\lambda) = \frac{1}{2} \lambda_i ( 3 \lambda_i-1)(3\lambda - 2) , \qquad \mbox{and} \qquad \nabla \phi_{v_i}(\lambda) = \frac{1}{2} \nabla_i (9\lambda_i(3\lambda_i-2)+2)
\end{equation*}
\item On edges:
\begin{equation*}
\hspace*{-1cm}
\phi_{e_{ij}(\alpha)}(\lambda) = \frac{9}{2} \lambda_i \lambda_j (3 \lambda_i-1) , \qquad \mbox{and} \qquad \nabla \phi_{e_{ij(\alpha)}}(\lambda) = \frac{9}{2} \left( \nabla_i \lambda_j (6\lambda_i-1)+\nabla_j \lambda_i(3\lambda_i-1) \right)
\end{equation*}
\item On the barycentric point:
\begin{equation*}
\phi_{G}(\lambda) = 27 \lambda_i \lambda_j \lambda_k, \qquad \mbox{and} \qquad \nabla \phi_{G}(\lambda) = 27 ( \nabla_i \lambda_j \lambda_k + \lambda_i \nabla_j \lambda_k +\lambda_i \lambda_j \nabla_k)
\end{equation*}
\end{itemize}
\subsection{Cubature elements, definition of basis functions} \label{appendix_cohen_bf}
Below are represented cubature finite elements:
\begin{figure}[H]
\centering
\hspace*{-1cm}
\includegraphics[width=12cm]{image_element/condensed_mesh.png}
\caption{Cohen elements $\Tilde{\mathbb{P}}_1$, $\Tilde{\mathbb{P}}_2$ and $\Tilde{\mathbb{P}}_3$}
\label{fig:cubature_mesh2}
\end{figure}
\subsubsection*{$\Tilde{\P}_1$ element}
The $\Tilde{\mathbb{P}}_1$ element contains $3$ degree of freedom: at each vertices $v_1$ , $v_2$ and $v_3$.
\begin{itemize}
\item At vertices of the triangle:
\begin{equation*}
\hspace*{-1cm}
\phi_{v_i}(\lambda) = \frac{1}{2} \lambda_i ( 3 \lambda_i-1)(3\lambda - 2) , \qquad \mbox{and} \qquad \nabla \phi_{v_i}(\lambda) = \frac{1}{2} \nabla_i (9\lambda_i(3\lambda_i-2)+2)
\end{equation*}
\end{itemize}
Corresponding weights are $w_{v_i} = \frac{1}{3}$. And we write $w_v$ the weight for vertices in the triangle.
\subsubsection*{$\Tilde{\P}_2$ element}
The $\Tilde{\mathbb{P}}_2$ element contains $7$ degree of freedom: $3$ at each vertices $v_1$, $v_2$ and $v_3$ and $3$ on edges $(e_{ij},e_{jk},e_{ki})$ (midpoint of edge $[v_i,v_j]$, etc) and 1 centroid point $G_\beta$. Respectively, we have those following basis functions and weights:
\begin{itemize}
\item At vertices of the triangle:
\begin{equation*}
\phi_{v_i}(\lambda) = \lambda_i(2\lambda_i-1)+3\lambda_i \lambda_j \lambda_k
\end{equation*
\begin{equation*}
\mbox{and} \quad \nabla \phi_{v_i} = \nabla_i (4\lambda_i-1) + 3( \nabla_i \lambda_j \lambda_k + \lambda_i \nabla_j \lambda_k +\lambda_i \lambda_j \nabla_k)
\end{equation*}
\item On edges:
\begin{equation*}
\phi_{e_{ij}}(\lambda) = 4 \lambda_i \lambda_j (1-3 \lambda_k), \quad \phi_{e_{jk}}(\lambda) = 4 \lambda_j \lambda_k (1-3 \lambda_i) , \quad \phi_{e_{ki}}(\lambda) = 4 \lambda_k \lambda_i (1-3 \lambda_j)
\end{equation*}
\begin{equation*}
\mbox{and} \quad \nabla \phi_{e_{ij}} = 4((\nabla_i \lambda_j + \lambda_j \nabla_j)(1-3\lambda_k)-3\nabla_k \lambda_i \lambda_j)
\end{equation*}
\item On the centroid point:
\begin{equation*}
\phi_{G_\beta}(\lambda) = 27 \lambda_i \lambda_j \lambda_k, \mbox{and} \qquad \nabla \phi_{G_\beta}(\lambda) = 27 ( \nabla_i \lambda_j \lambda_k + \lambda_i \nabla_j \lambda_k +\lambda_i \lambda_j \nabla_k)
\end{equation*}
\end{itemize}
with $e_{ij}$ define such as $\lambda_i = \frac{1}{2}$, $\lambda_j = \frac{1}{2}$ and $\lambda_k =0$, the midpoint between vertices $v_i$ and $v_j$, and $G_\beta = (\frac{1}{3} ,\frac{1}{3} ,\frac{1}{3} )$, the barycentre of the triangle. \\
We define $w_v$ the weight at vertices of the triangle, $w_\alpha$ the weight on edges, and $w_\beta $ the weight for barycentric point. \\
Corresponding weights are $w_v = \frac{1}{20}$, $w_\alpha = \frac{2}{15}$ and $w_\beta = \frac{9}{20}$
\subsubsection*{$\Tilde{\P}_3$ element}
The $\mathbb{P}_3$ element contains $12$ degree of freedom : $3$ vertices $v_1$, $v_2$ and $v_3$, $6$ on edges\\ $(e_{ji}(\alpha),e_{ij}(\alpha),e_{kj}(\alpha),e_{jk}(\alpha),e_{ik}(\alpha),e_{ki}(\alpha))$, with $e_{ij}(\alpha)$ such as $\lambda_i = \alpha, \lambda_j = (1-\alpha),\lambda_k = 0$ ($\alpha = \frac{-15\sqrt{7}-21+\sqrt{168+174\sqrt{7}}}{2(-15\sqrt{7}-21)}$) and three centroid points $(G_i(\beta),G_j(\beta),G_k(\beta))$, with $G_i(\beta)$ such as $\lambda_i = \beta , \lambda_j = \frac{1-\beta}{2} , \lambda_k = \frac{1-\beta}{2} $, $\beta=\frac{1}{3}+\frac{2\sqrt{7}}{21}$. \\
Respectively, we have those following basis functions and weights:
\begin{itemize}
\item At vertices of the triangle:
\begin{align*}
\phi_{v_i}(\lambda) = & \underbrace{ \lambda_i \left( \sum_l \lambda_l^2 - \frac{1-2\alpha + \alpha^2}{\alpha(1-\alpha)}\lambda_i(\lambda_j+\lambda_k)+ A_{02} \lambda_j \lambda_k \right) }_{p_{i}(\lambda)} \\
& - \frac{8}{\beta (1-\beta)^2 (3\beta-1)} \lambda_i \lambda_j \lambda_k \left( \sum_{j} b_l (\lambda_l - \frac{1-\beta }{2}) \right)
\end{align*}
with $b_l = p_{i}(G_l)$. \\
As $w_i = \frac{1}{|K|}\int_K \phi_i(\lambda) ds $, we obtain
\begin{equation*}
\hspace*{-1cm}
A_{02} = \left( w_s -\frac{1}{10}-\frac{1}{15} \left(1- \frac{1-2\alpha + \alpha^2}{\alpha(1-\alpha)} \right) - \frac{1}{90}\frac{8}{\beta (1-\beta)^2 (3\beta-1)}( p_{i}(G_i) + p_{i}(G_j) + p_{i}(G_k)) \right) \times \left( \frac{360}{6+\frac{8(1-\beta^2) }{\beta (1-\beta)^2 (3\beta-1)}} \right)
\end{equation*}
And
\begin{align*}
\hspace*{-1cm}
\nabla \phi_{v_i}(\lambda) = &\nabla_i \left(\sum_l \lambda_l^2 - \frac{1-2\alpha + \alpha^2}{\alpha(1-\alpha)}\lambda_i(\lambda_j+\lambda_k)+ A_{02} \lambda_j \lambda_k \right) \\
& + \lambda_i \left( \sum_l 2\lambda_l \nabla_l - \frac{1-2\alpha + \alpha^2}{\alpha(1-\alpha)} \nabla_i(1-2 \lambda_i) + A_{02} (\nabla_j \lambda_k + \lambda_j \nabla_k) \right) \\
\hspace*{-2.5cm}
& - \frac{8}{\beta (1-\beta)^2 (3\beta-1)} \left[ (\nabla_i \lambda_j \lambda_k + \lambda_i \nabla_j \lambda_k +\lambda_i \lambda_j \nabla_k) \left( \sum_{1\leq l \leq 3} b_l \left( \lambda_l -\frac{1-\beta}{2} \right) \right)+\lambda_i \lambda_j \lambda_k \nabla_i \left( \sum_{1\leq l \leq 3} b_l \nabla_l \right) \right]
\end{align*}
\item On edges: $(e_{ji}(\alpha),e_{ij}(\alpha),e_{kj}(\alpha),e_{jk}(\alpha),e_{ik}(\alpha),e_{ki}(\alpha))$, \\
with $e_{ij}(\alpha)$ such as $\lambda_i = \alpha, \lambda_j = (1-\alpha),\lambda_k = 0$, and $\alpha = \frac{-15\sqrt{7}-21+\sqrt{168+174\sqrt{7}}}{2(-15\sqrt{7}-21)}$
\begin{align*}
\phi_{e_{ij}(\alpha)}(\lambda) = & \underbrace{ \frac{1}{\alpha(1-\alpha)(2\alpha-1)} \lambda_i \lambda_j (\alpha \lambda_i - (1-\alpha)\lambda_j + (1-2\alpha) \lambda_k)}_{p_{ij}(\lambda)} \\
&- \frac{8}{\beta (1-\beta)^2 (3\beta-1)} \lambda_i \lambda_j \lambda_k \left( \sum_{j} b_l (\lambda_l - \frac{1-\beta }{2}) \right)
\end{align*}
with $b_l = p_{ij}(G_l)$, and
\begin{align*}
\hspace*{-2cm}
\nabla \phi_{e_{ij}(\alpha)}(\lambda) = \frac{1}{\alpha(1-\alpha)(2\alpha-1)} & \left[ (\nabla_i \lambda_j+\lambda_i \nabla_j)(\alpha \lambda_i-(1-\alpha)\lambda_j+(1-2\alpha)\lambda_k) + \lambda_i \lambda_j (\alpha \nabla_i - (1-\alpha) \nabla_j + (1-2\alpha) \nabla_k \right] \\
- \frac{8}{\beta (1-\beta)^2 (3\beta-1)} & \left[ (\nabla_i \lambda_j \lambda_k + \lambda_i \nabla_j \lambda_k +\lambda_i \lambda_j \nabla_k) \left( \sum_{1\leq l \leq 3} b_l \left( \lambda_l -\frac{1-\beta}{2} \right) \right)+\lambda_i \lambda_j \lambda_k \nabla_i \left( \sum_{1\leq l \leq 3} b_l \nabla_l \right) \right]
\end{align*}
\item On the barycentric points : $(G_i(\beta),G_j(\beta),G_k(\beta))=(a_{10},a_{11},a_{12})$, \\
with $G_i(\beta)$ such as $\lambda_i = \beta , \lambda_j = \frac{1-\beta}{2} , \lambda_k = \frac{1-\beta}{2} $, $\beta=\frac{1}{3}+\frac{2\sqrt{7}}{21}$.
\begin{equation*}
\phi_{G_i(\beta)}(\lambda) = \frac{8}{\beta(1-\beta)^2(3\beta-1)} \lambda_i \lambda_j \lambda_k (\lambda_i-\frac{1-\beta}{2})
\end{equation*}
\begin{equation*}
\mbox{and} \quad \nabla \phi_{G_\beta}(\lambda) = \frac{8}{\beta(1-\beta)^2(3\beta-1)} \left( (\nabla_i \lambda_j \lambda_k + \lambda_i \nabla_j \lambda_k +\lambda_i \lambda_j \nabla_k) \left(\lambda_i-\frac{1-\beta}{2}\right)+\lambda_i \lambda_j \lambda_k \nabla_i \right)
\end{equation*}
\end{itemize}
We define $w_v$ the weight at vertices of the triangle, $w_\alpha$ the weight on edges, and $w_\beta $ the weight for barycentric points. \\
Corresponding weight is $w_v = \frac{1369+767\sqrt{7}}{120(859+395\sqrt{7})}$, $w_\alpha = \frac{287+115\sqrt{7}}{40(173+49\sqrt{7})}$ and $w_\beta = \frac{21\sqrt{7}}{40(2\sqrt{7}+1)}$. \\
\section{Definition of \textit{Lagrangian} basis functions}
\textbf{Notation 1:} $\lambda=(\lambda_1,\lambda_2,\lambda_3)$ is the barycentric coordinates relative to the vertices $v_1$, $v_2$ and $v_3$, $\nabla_i = \nabla (\lambda_i), \forall i$, and $\phi_p$ is the basis function which correspond to the degree of freedom $p$. We pose (i,j,k) an arbitrary permutation of $(1,2,3)$. \\
Then, $e_{ij}$ corresponds to a node on the edge $[v_i,v_j]$, which is not a vertex (in green on \ref{fig:mesh_elment2} and \ref{fig:cubature_mesh2}), and finnaly $G$ corresponds to a node in the triangle (in magenta on \ref{fig:mesh_elment2} and \ref{fig:cubature_mesh2}). \\
\textbf{Notation 2: } $w_v$, $w_\alpha$ and $w_\beta$ are weights of respectively vertices , on edges and for the centroid points. For each basis function $\phi$ relative to a DOF, the corresponding weight represents $\int_K \phi$ with $K$ the triangle of reference. It is as well the weight used in quadradrature formula.
\subsection{Basic triangular elements, definition and construction} \label{sec:appendix_lag_bf}
We represent classical/basic finite elements \ref{fig:mesh_elment2}
\begin{figure}[H]
\centering
\includegraphics[width=0.5\linewidth]{image_element/P1_P2_P3.png}
\caption{Standar elements $\mathbb{P}_1$ on the left, $\mathbb{P}_2$ on the center and $\mathbb{P}_3$ on the right}
\label{fig:mesh_elment2}
\end{figure}
Classical finite elements are built by discretizing the space uniformly until obtain polynomial functions as basis function with a efficient degree. As example, for elements of degree $1$, Degree of Freedom (DOF) are at each vertices of triangles, or elements of degree 2, DOF are at each vertices and middle of edges, etc. Then, basis functions are defined such as at each DOF $\{x_i\}_i$, for any basis function $\phi_j$ which correspond to the node $j$, $\phi_j(x_i) = 1$ if $i = j$, $0$ else. Corresponding basis function for elements are respectively of degree 1, 2 and 3.
\subsubsection{Finite element of degree 1: $\P_1$ element}
The $\mathbb{P}_1$ element contains $3$ degree of freedom: at each vertices $v_1$ , $v_2$ and $v_3$.
\begin{equation*}
\quad w_v = \frac{1}{3}
\end{equation*}
with $w_v$ weight of vertices. \\
And then, corresponding basis functions are
\begin{itemize}
\item At each vertices $v_1$, $v_2$ and $v_3$.
\begin{equation*}
\hspace*{-1cm}
\phi_{v_i}(\lambda) = \lambda_i, \qquad \mbox{and} \qquad \nabla \phi_{v_i}(\lambda) = \nabla_i
\end{equation*}
\end{itemize}
\subsubsection{Finite element of degree 2: $\P_2$ element}
The $\mathbb{P}_2$ element contains $6$ degree of freedom: $3$ at each vertices $v_1$, $v_2$ and $v_3$ and $3$ on edges $(e_{ij},e_{jk},e_{ki})$ (midpoint of edge $[v_i,v_j]$, etc). Respectively, at vertices and midpoints we have those following weights:
\begin{equation*}
\quad w_v = 0, \quad w_\alpha = \frac{1}{3}
\end{equation*}
And then, corresponding basis functions are
\begin{itemize}
\item At vertices of the triangle:
\begin{equation*}
\hspace*{-1cm}
\phi_{v_i}(\lambda) = \lambda_i ( 2\lambda_i-1) , \qquad \mbox{and} \qquad \nabla \phi_{v_i}(\lambda) = \nabla_i (4\lambda_i-1)
\end{equation*}
\item On edges:
\begin{equation*}
\hspace*{-1cm}
\phi_{e_{ij}}(\lambda) = 4 \lambda_i \lambda_j , \qquad \mbox{and} \qquad \nabla \phi_{e_{ij}}(\lambda) = 4(\nabla_j \lambda_i + \nabla_i \lambda_j)
\end{equation*}
\end{itemize}
\subsubsection{Finite element of degree 3: $\P_3$ element}
The $\mathbb{P}_3$ element contains $10$ degree of freedom : $3$ vertices $v_1$, $v_2$ and $v_3$, $6$ on edges\\ $(e_{ji}(\alpha),e_{ij}(\alpha),e_{kj}(\alpha),e_{jk}(\alpha),e_{ik}(\alpha),e_{ki}(\alpha))$, with $e_{ij}(\alpha)$ such as $\lambda_i = \alpha, \lambda_j = (1-\alpha),\lambda_k = 0$ ($\alpha=\frac{1}{3}$) and a centroid point $G$, such as $\lambda_1 = \lambda_2 = \lambda_3 = \frac{1}{3}$.
\begin{equation*}
\quad w_v = \frac{1}{30}, \quad w_\alpha = \frac{3}{40}, \quad w_\beta = \frac{9}{20}
\end{equation*}
with $w_v$, $w_\alpha$ and $w_\beta$ weights of respectively vertices, on edges and for the centroid point. \\
And then, corresponding basis functions are
\begin{itemize}
\item At vertices of the triangle:
\begin{equation*}
\hspace*{-1cm}
\phi_{v_i}(\lambda) = \frac{1}{2} \lambda_i ( 3 \lambda_i-1)(3\lambda - 2) , \qquad \mbox{and} \qquad \nabla \phi_{v_i}(\lambda) = \frac{1}{2} \nabla_i (9\lambda_i(3\lambda_i-2)+2)
\end{equation*}
\item On edges:
\begin{equation*}
\hspace*{-1cm}
\phi_{e_{ij}(\alpha)}(\lambda) = \frac{9}{2} \lambda_i \lambda_j (3 \lambda_i-1) , \qquad \mbox{and} \qquad \nabla \phi_{e_{ij(\alpha)}}(\lambda) = \frac{9}{2} \left( \nabla_i \lambda_j (6\lambda_i-1)+\nabla_j \lambda_i(3\lambda_i-1) \right)
\end{equation*}
\item On the barycentric point:
\begin{equation*}
\phi_{G}(\lambda) = 27 \lambda_i \lambda_j \lambda_k, \qquad \mbox{and} \qquad \nabla \phi_{G}(\lambda) = 27 ( \nabla_i \lambda_j \lambda_k + \lambda_i \nabla_j \lambda_k +\lambda_i \lambda_j \nabla_k)
\end{equation*}
\end{itemize}
\end{comment}
\section{\textit{Cubature} elements, definition and construction} \label{sec:appendix_cohen_bf}
In this section we give a description of the \textit{Cubature} finite elements \cite{article_cubature_2006,article_cubature_2001}. In \cref{fig:cubature_mesh2} we show the $\TP_3$ example comparing the Lagrangian nodes of \textit{Basic} and \textit{Cubature} elements.
\begin{figure}
\centering
\hspace*{-1cm}
\includegraphics[width=0.4\linewidth]{image_element/P3-basicVScubature.png}
\caption{Comparison of two element of degree three: at left the classical one $\P_3$, at right the \textit{Cubature} one $\TP_3$.}
\label{fig:cubature_mesh2}
\end{figure}
As defined in \cref{sec:discretization_cubature2D}, there are several requirements and optimization procedures in order to obtain the \textit{Cubature} elements. These elements are very import in our study because they permit to obtain diagonal mass matrix, and so they decrease considerably the time of computation.
We describe for $p=1,2,3$ the basis functions of the \textit{Cubature} elements.
\subsection{\textit{Cubature} elements of degree 1}
The $\Tilde{\mathbb{P}}_1$ element contains $3$ degree of freedom. Their nodes are located at the vertices $v_1=(1,0,0)$, $v_2=(0,1,0)$ and $v_3=(0,0,1)$ of the triangle.
\begin{itemize}
\item At vertices of the triangle:
\begin{equation*}
\hspace*{-1cm}
\phi_{v_i}(\lambda) = \lambda_i
\qquad \mbox{ for }i=1,2,3.
\end{equation*}
\end{itemize}
Corresponding weights are $w_{v_i} = \frac{1}{3}$.
\subsection{\textit{Cubature} elements of degree 2}
The $\Tilde{\mathbb{P}}_2$ element contains $7$ degrees of freedom: three at the vertices $v_1$, $v_2$ and $v_3$ and three at the midpoint of the edges that we denote as $e_{ij}=\frac{v_i+v_j}{2}$ for $(i,j)\in \lbrace (1,2),(2,3),(3,1) \rbrace$ and one at the centroid point $G_\beta:=\frac{v_1+v_2+v_3}{3}$. Respectively, we have the following basis functions and weights:
\begin{itemize}
\item At vertices of the triangle
\begin{align*}
&\phi_{v_i}(\lambda) = \lambda_i(2\lambda_i-1)+3\lambda_1 \lambda_2 \lambda_3, \text{ for }i \in \llbracket 1,\dots, 3 \rrbracket,\\
&w_{v}= \frac{1}{20};
\end{align*}
\item At edge midpoints
\begin{align*}
&\phi_{e_{ij}}(\lambda) = 4 \lambda_i \lambda_j (1-3 \lambda_k), \text{ for all } i\neq j \neq k \neq i \in \llbracket 1,\dots, 3 \rrbracket,\\
&w_{e}=\frac{2}{15};
\end{align*}
\item At the centroid
\begin{align*}
&\phi_{G_\beta}(\lambda) = 27 \lambda_1 \lambda_2 \lambda_3, \\
&w_\beta = \frac{9}{20}.
\end{align*}
\end{itemize}
\subsection{\textit{Cubature} elements of degree 3}
Following \cite{article_cubature_2001,article_cubature_2006} we derive the definitions of all the basis functions and points of \textit{Cubature} elements $\TP_3$. The notations are not uniform among different works, so we use the following one which can be used with all the different elements we have used in this work.
The space $\TP_3$ contains $12$ degrees of freedom: $3$ vertices $v_1$, $v_2$ and $v_3$, $6$ on edges: $e_{ij}^\alpha$ for $i,j \in \llbracket 1,\dots, 3 \rrbracket$ with $i\neq j$ defined by \begin{align*}
&e_{ij}^\alpha = (\delta_{1i}\alpha +\delta_{1j}(1-\alpha),\delta_{2i}\alpha +\delta_{2j}(1-\alpha),\delta_{3i}\alpha +\delta_{3j}(1-\alpha))\\
\text{ with }&\alpha = \frac{-15\sqrt{7}-21+\sqrt{168+174\sqrt{7}}}{2(-15\sqrt{7}-21)},
\end{align*}
with $\delta_{ij}$ is the Kronecker delta and three internal points $G_i^\beta$ for $i \in \llbracket 1,\dots, 3 \rrbracket$, with
\begin{align*}
G_i^\beta = \left(\beta \delta_{i1} + \frac{1-\beta}{2}(1-\delta_{i1}),\beta \delta_{i2} + \frac{1-\beta}{2}(1-\delta_{i2}), \beta \delta_{i3} + \frac{1-\beta}{2}(1-\delta_{i3})\right) \text{ with }\beta=\frac{1}{3}+\frac{2\sqrt{7}}{21},
\end{align*}
where $\alpha$ and $\beta$ are found through an optimization process \cite{article_cubature_2001,article_cubature_2006}.
Let us start giving the definitions of the weights for the different types of points.
We have that $w_v = \frac{1369+767\sqrt{7}}{120(859+395\sqrt{7})}$ is the weight for the vertices of the triangle, $w_\alpha= \frac{287+115\sqrt{7}}{40(173+49\sqrt{7})}$ is the weight on edges points, and $w_\beta $ the weight for barycentric points. \\
The weights corresponding to these types of points are $w_v = \frac{1369+767\sqrt{7}}{120(859+395\sqrt{7})}$, $w_\alpha = \frac{287+115\sqrt{7}}{40(173+49\sqrt{7})}$ and $w_\beta = \frac{21\sqrt{7}}{40(2\sqrt{7}+1)}$.
In order to simply the formulation of the basis functions, let us introduce some polynomials:
\begin{equation}
p_i(\lambda) := \lambda_i \left( \sum_{l=1}^3 \lambda_l^2 - \frac{1-2\alpha + 2\alpha^2}{\alpha(1-\alpha)}\lambda_i(\lambda_j+\lambda_k)+ A_{i} \lambda_j \lambda_k \right), \qquad \text{with }j\neq i \neq k,
\end{equation}
with \begin{equation}
\begin{split}
A_{i} = &\left( w_v -\frac{1}{10}-\frac{1}{15} \left(1- \frac{1-2\alpha + 2 \alpha^2}{\alpha(1-\alpha)} \right) - \frac{1}{90}\frac{8}{\beta (1-\beta)^2 (3\beta-1)}\left(\sum_{l=1}^3 p_{i}(G_l)\right) \right) \frac{360}{6+\frac{8(1+\beta) }{\beta (1-\beta) (3\beta-1)}} ;
\end{split}
\end{equation}
\begin{equation}
p_{ij}(\lambda) := \frac{1}{\alpha(1-\alpha)(2\alpha-1)} \lambda_i \lambda_j (\alpha \lambda_i - (1-\alpha)\lambda_j + (1-2\alpha) \lambda_k), \text{ with } i\neq j \neq k \neq i.
\end{equation}
We can then write the definition of the basis functions:
\begin{itemize}
\item At vertices of the triangle
\begin{align*}
\phi_{v_i}(\lambda) = &p_{i}(\lambda)
-\frac{8}{\beta (1-\beta)^2 (3\beta-1)} \left( \sum_{l=1}^3 p_i(G_l) \left(\lambda_l - \frac{1-\beta }{2}\right) \right)\prod_{l=1}^3\lambda_l, \text{ for }i \in \llbracket 1,\dots ,3 \rrbracket;
\end{align*}
\item At the nodes on edges
\begin{align*}
\phi_{e_{ij}^\alpha}(\lambda) = & p_{ij}(\lambda)- \frac{8}{\beta (1-\beta)^2 (3\beta-1)} \left( \sum_{l=1}^3 p_{ij}(G_l) \left(\lambda_l - \frac{1-\beta }{2}\right) \right)\prod_{l=1}^3\lambda_l, \text{ for }i\neq j \in \llbracket 1,\dots ,3 \rrbracket;
\end{align*}
\item At the internal points
\begin{equation*}
\phi_{G_i^\beta}(\lambda) = \frac{8}{\beta(1-\beta)^2(3\beta-1)} \left(\lambda_i-\frac{1-\beta}{2}\right)\prod_{l=1}^3\lambda_l, \text{ for }i \in \llbracket 1,\dots ,3 \rrbracket.
\end{equation*}
\end{itemize}
\subsubsection{\textit{Basic} Lagrangian equispaced elements}
On triangles, we consider Lagrange polynomials with degrees at most $p$: $\mathbb P^p=\lbrace \sum_{\alpha+\beta\leq p } c_{\alpha,\beta}x^\alpha y^\beta \rbrace$. We define the barycentric coordinates $\lambda_i(x,y)$ which are affine functions on $\R^2$ verifying the following relations
\begin{equation}
\lambda_i(v_j)=\delta_{ij}, \quad \forall i,j=1,\dots,3,
\end{equation}
where $v_j=(x_j,y_j)$ are the vertexes of the triangle and, with an abuse of notation, they can be written in barycentric coordinates as $v_j=(\delta_{1j},\delta_{2j},\delta_{3j})$.
Using these coordinates, we can define the Lagrangian polynomials on equispaced points on triangles. The equispaced points are defined on the intersection of the lines $\lambda_j=\frac{k}{p}$ for $k=0,\dots,p$.
A way to define the basis functions corresponding to the point $(x_\alpha,y_\alpha)=(\alpha_1/p,\alpha_2/p,\alpha_3/p)$ in barycentric coordinates, with $\alpha_i\in \{ 0,\dots, p\}$ and $\sum_i \alpha_i =1$, is in \cref{algo:baryCoordsLagrange}.
\begin{algorithm}
\begin{algorithmic}
\Require Point $(x_\alpha,y_\alpha)=(\alpha_1/p,\alpha_2/p,\alpha_3/p)$ in barycentric coordinates
\State $\varphi_\alpha(x)\gets 1 $
\For{$i=1,2,3$}
\For{$z=0,\dots,a_i$}
\State $\varphi_\alpha(x,y) \gets \varphi_\alpha(x,y) \cdot (\lambda_i(x,y)-\frac{z}{p})$
\EndFor
\EndFor
\end{algorithmic}
\caption{Lagrangian basis function in barycentric coordinates\label{algo:baryCoordsLagrange}}
\end{algorithm}
The polynomials so defined in a triangle form a partition of unity, but they have also negative values. This leads to negative or zero values of their integrals. This is problematic for some time discretization and we will see why. We will use these polynomials in combination with exact Gauss--Lobatto quadrature formulae for such polynomials and we will refer to them as \textit{Basic} elements.
\subsubsection{Bernstein polynomials}
Bernstein polynomials are as well a basis of $\mathbb P^p$ but they are not Lagrangian polynomials, hence, there is not a unique correspondence between point values and coefficients of the polynomials. Anyway, there exist a geometrical identification with the Greville points $(x_\alpha,y_\alpha)= (\alpha_1/p,\alpha_2/p,\alpha_3/p)$. Given a triplet $\alpha\in \N^3$ with $\alpha_i\in \llbracket 0,\dots, p\rrbracket$ and $\sum_i \alpha_i=p$, the Bernstein polynomials are defined as
\begin{equation}\label{eq:BernsteinPoly}
\varphi_\alpha(x,y) = p!\prod_{i=1}^3 \frac{\lambda_i^{\alpha_i}(x,y)}{\alpha_i!}.
\end{equation}
Bernstein polynomials verify additional properties besides the one already cited for Lagrangian points. As before, they form a partition of unity, the basis functions are nonnegative in any point of the triangle, and so
their integrals are strictly positive. More precisely $$\int_K \varphi_\alpha = \frac{|K|}{S}, \qquad S= \# \left\lbrace \alpha \in \N^3 : |\alpha|_1 = p\right\rbrace. $$ These properties lead also to the fact that the value at each point is a convex combination of the coefficients of the polynomials, so that
it is easy to bound minimum and maximum of the function by the minimum and maximum of the coefficients. This has been used in different techniques to preserve positivity of the solution \cite{bacigaluppi2019posteriori,kuzmin2020subcellBernstein}. We will use these polynomials with corresponding high order accurate quadrature formulae. We will denote these elements with the symbol $\mathbb B^p$ and we refer to them as \textit{Bernstein} elements.
\subsubsection{\textit{Cubature} elements} \label{sec:discretization_cubature2D}
Contrary to the work done in 1D \cite{michel2021spectral}, the extension of Legendre--Gauss--Lobatto points which minimize the interpolation error do not exist for the triangle. They have to be computed numerically such as \textit{Fekete} points \cite{article_fekete1, article_fekete2_SHERWIN1995189, article_fekete}. The problem of this approach is that it requires as classical finite elements the inversion of a sparse global mass matrix.\\
\textit{Cubature} elements were introduced by G. Cohen and P. Joly in $2001$ \cite{article_cubature_2001} for the wave equation (second order hyperbolic equation), and are an extension of Lagrange polynomials with the goal of optimizing the underlying quadrature formula error. We will denote the with the symbol $\tilde {\mathbb P}^p$ and they will be contained in another larger space of Lagrange elements, i.e., $\mathbb P ^p \subseteq \tilde {\mathbb P}^p \subseteq \mathbb P^{p'}$, with $p'$ the smallest possible integer.
Similar techniques have been used to minimize the interpolation error \cite{article_fekete1, article_fekete2_SHERWIN1995189, article_fekete}. The objective of these polynomials is to use the points of the Lagrangian interpolation of the polynomials as quadrature points. This means that the obtained quadrature is $\int_K f(x,y) = \sum_{\alpha} \omega_\alpha f(x_\alpha,y_\alpha)$, where $\int_K \varphi_\alpha = \omega_\alpha$ and $\varphi_\alpha(x_\beta,y_\beta) = \delta_{\alpha \beta}$. This approach can be considered an extension of the Gauss--Lobatto quadrature in 1D for non Cartesian meshes. The biggest advantage of this approach is to obtain a diagonal mass matrix.
The drawback is that one needs to increase the number of basis function inside one element to obtain an accurate enough quadrature rule. In our work, we propose to extend this approach to first order hyperbolic equations. A successful extension to elliptic problem is proposed in \cite{pasquetti:hal-01954133}.
A comparison between the equispace repartition and the \textit{Cubature} repartition for elements of degree $p=3$ is shown in \cref{fig:P3-basicVScub}.
\begin{figure}[H]
\centering
\includegraphics[width=0.6\linewidth]{image_element/P3-basicVScubature.png}
\caption{Comparison of the equispace repartition at left and the cubature repartition at right for elements of degree $p=3$.} \label{fig:P3-basicVScub}
\end{figure}
{For completeness we detail further the construction of the basis functions.} The challenges of this approach are the following:
\begin{itemize}
\item Obtain a quadrature which is highly accurate, at least $p+p'-2$ order accurate \cite{book_ciarlet};
\item Obtain positive quadrature weights $\omega_\alpha>0$ for stability reasons \cite{tordj1995};
\item Minimize the number of basis functions of $\tilde{\mathbb{P}^p}$;
\item The set of quadrature points has to be $\tilde{\P}^p$-unisolvent;
\item The number of quadrature points of edges as to be sufficient ensure the conformity of the finite element.
\end{itemize}
The optimization procedure that lead to these elements consists of several steps where the different goals are optimized one by one. The optimization strategy exploits heavily the symmetry properties that the quadrature point must have.
For $p=1$ the \textit{Cubature} elements do not differ from the \textit{Basic} elements but in the quadrature formula. For $p=2$ the \textit{Cubature} elements introduce an other degree of freedom at the center of the triangle, leading to 7 quadrature points and basis functions per element. For $p=3$ the additional degree of freedom in the triangle are 3, leading to 13 basis functions per triangle.
All the details of such elements can be found in \cite{article_cubature_2001,jund2007arbitrary}. We provide
in \cref{sec:appendix_cohen_bf} the detailed expressions
of the polynomials used in this work. We will use the symbol $\tilde{\mathbb{P}}^p$ and the name \textit{Cubature} elements to refer to them. \\
Other elements such as \textit{Fekete-Gauss} points \cite{article_cubature_2006,pasquetti:hal-01589136} exist in the literature. They are optimized to interpolate and integrate with high accuracy. However, it is shown that they require more computing time to achieve similar results than cubature points for high order of accuracy.
\begin{comment}
\subsubsection{Extension to Fekete elements}
Fekete and Gauss–Lobatto points are determined by minimizing the interpolation error inside the elements \cite{article_fekete1, article_fekete2_SHERWIN1995189, article_fekete}. They are commonly used in numerical methods which rely on both accurate high-order polynomial interpolation and quadrature properties. However, Gauss–Lobatto quadrature points are only known for tensor-product domains such as the line and square, making it unclear how to extend a Gauss–Lobatto numerical method to non-tensor-product domains like the triangle. Since Fekete points are known to be the Gauss–Lobatto points on the line \cite{fejer1932bestimmung} and in the $d$-dimensional cube \cite{article_fekete_gll}, Fekete points are one possible generalization of Gauss–Lobatto points for the triangle. At present there is no known analytic formula for the location of the Fekete points in the triangle; therefor we will use Fekete points compute numerically in \cite{article_fekete}. Relative basis functions are \textit{Lagrange} basis functions defined above in \refp{eq:lagrange_bf}.
\end{comment}
\begin{comment}
\textbf{Note on \textit{Cubature} elements:} \\%\label{sec:cubature_element}
\textit{Cubature} elements were introduced by G. Cohen and P. Joly in $2001$ \cite{article_cubature_2001}. By convention, we write $\P$ for \textit{Basic}/standard elements and $\Tilde{\P}$ for cubature elements. \\
Those elements allow to obtain a diagonal mass matrix using a high order accurate quadrature formula that matches with the degrees of freedom in the reference triangle, in order to maintain the accuracy while solving the system \refp{eq_ad_2}. Weights of the quadrature formula on each node correspond to the integral over the element of the \textit{Lagrange basis function} relative to the node.
The condition on the quadrature formulas to have the good convergence order are given in \cite{book_ciarlet}: by modify $\P_k$ per $\tilde{\P}_k$ such as
\begin{equation*}
\mathbb{P}_k \subseteq \tilde{\mathbb{P}}_k \subseteq \P_{k'} \qquad k \leq k'
\end{equation*}
we do not loose accuracy if the quadrature formula is exact at the order $k+k'-2$, and
\begin{itemize}
\item the space $\tilde{\P}_k$ is as small as possible.
\item the set of quadrature points has to be $\tilde{\P}_k$-unisolvent.
\item the number of quadrature points of edges as to be sufficient ensure the conformity of the finite element.
\item the quadrature weights have to be strictly positive.
\end{itemize}
The first condition leads to minimize the number of degrees of freedom, and to choose $k'$ as small as possible. The second and the third conditions are necessary in the sense of degrees of freedom have to be exact quadrature points. And the fourth ensures the stability criteria: if a weight is null, the (diagonal) mass matrix will not be invertible, and strictly negative weights lead to a scheme unconditionally unstable \cite{tordj1995}. It shows that in \cite{article_cubature_2001} that in $\tilde{\P}_1$, $\tilde{\P}_2$ and $\tilde{\P}_3$ elements, the quadrature formula has to be exact for polynomial function of degree respectively $0$, $3$ and $5$. Or in others words, the convergence order has to be respectively $\geq 1$, $4$ and $6$.
The spatial discretization is defined as follow:
\begin{itemize}
\item For $\P_1$ elements, we consider 3 degree of freedom (DOF) at each vertex.
\item For $\P_2$ elements, we consider 6 DOF, 3 at each vertex, 3 as midpoint of vertices on each edge and 1 as the barycenter of the triangle.
\item For $\P_3$ elements, we consider 10 DOF, 3 at each vertex, 6 on edges (2 per edge) and 3 in the triangle (forming a \textit{"bubble"} around the barycenter of the triangle).
\end{itemize}
For more details about the construction, see \cite{article_cubature_2001}.\\
\textbf{NB:} in the mono-dimensional case, Fekete, GGL and cubature points coincide exactly. \\
\end{comment}
\subsection{Preliminaries and time continuous analysis} \label{sec:fourier}
In order to study the stability and the dispersion properties of the previously presented numerical schemes, we will perform a dispersion analysis on the linear advection problem with periodic boundary conditions:
\begin{equation}
\partial_t u(t,\mathbf{x}) + \mathbf{a} \cdot \nabla u(t,\mathbf{x}) = 0, \quad \mathbf{a}\in \R^2, \quad (t,\mathbf{x}) \in \R^+ \times \Omega,
\label{eq_disp_1}
\end{equation}
with $\Omega = [0,1]\times[0,1]$. For simplicity, we consider $\mathbf{a} = (\cos( \Phi), \sin ( \Phi))$ with $\Phi \in [0,2\pi]$.
We then introduce the ansatz
\begin{align}
& u_h(\mathbf{x} , t) = Ae^{i(\mathbf{k} \cdot \mathbf{x} - \xi t)} = Ae^{i(\mathbf{k}\cdot \mathbf{x}-\omega t)}e^{\epsilon t} \label{eq_disp_uex1} \\
\mbox{with} \quad & \xi = \omega + i \epsilon, \quad i=\sqrt{-1}, \quad \mathbf{k}=(k_x,k_y)^T.
\end{align}
Here, $\epsilon$ denotes the damping rate, while the wavenumbers are denoted by $\mathbf{k}=(k_x,k_y)$, with $k_x=2\pi/L_x$ and $k_y=2\pi/L_y$ with $L_x$ and $L_y$ the wavelengths in $x$ and $y$ directions respectively.
The phase velocity $\mathbf{c}$ can be defined from
\begin{equation}
\mathbf{c}\cdot\mathbf{k} = \omega
\label{eq_PDE_velocity}
\end{equation}
and represents the celerity with which waves propagate in space. It is in general a function of the wavenumber. Substituting
\eqref{eq_disp_uex1} in the advection equation \eqref{eq_disp_1} for an exact solution we obtain that
\begin{align}
\label{eq:Phi_def}
\omega = \mathbf{k}\cdot \mathbf{a} \,,\quad
\mathbf{c} = \mathbf{a} \quad \mbox{and} \quad \epsilon = 0.
\end{align}
In other words
\begin{align}
u_h(\mathbf{x} , t) = Ae^{i\mathbf{k} \cdot (\mathbf{x} - \mathbf{a} t)} \,.
\end{align}
The objective of the next sections is to provide the semi- and fully-discrete equivalents of the above relations for the finite element methods introduced earlier.
We will consider polynomial degrees up to 3, for all combinations of stabilization methods and time integration techniques. This will also allow to investigate the parametric stability with respect to
the time step (through the \CFL~number) and stabilization parameter $\delta$. In practice, for each choice we will evaluate the accuracy of the discrete
approximation of $\omega$ and $\epsilon$, and we will provide conditions for the non-positivity of the damping $\epsilon$, i.e., the von Neumann stability of the method
\subsection{The eigenvalue system}
\label{sec:fourier_space}
The Fourier analysis for numerical schemes on the periodic domain is based on a discrete Parseval theorem.
Thanks to this theorem, we can study the amplification and the dispersion of the basis functions of the Fourier space.
The key ingredient of this study is the repetition of the stencil of the scheme from one cell to another one. In particular, using the ansatz \eqref{eq_disp_uex1} we can write local equations coupling degrees of freedom belonging to neighbouring cells through a multiplication by factors $e^{i\theta_x}$ and $e^{i\theta_y}$ representing the shift in space along the oscillating solution. The dimensionless coefficient
\begin{equation}\label{eq_theta}
\theta_x:= k_x\Delta x\, \text{ and }\, \theta_y:= k_y\Delta y
\end{equation}
are the discrete reduced wave numbers which naturally appear all along the analysis. Here, $\Delta x$ and $\Delta y$ are defined by the size of the elementary periodic unit that is highlighted with a red square as an example in \cref{fig_Xmesh_P2}.
Formally replacing the ansatz in the scheme we end up with a dense algebraic problem of dimension $N_{dof}$, where $N_{dof}$ is the number of all the degrees of freedom in the mesh. The obtained system with dimension $N_{dof}$ in the time continuous case reads
\begin{equation}
\text{Equations \eqref{eq_disp_1} and \eqref{eq_disp_uex1}} \quad \Rightarrow \quad - i\xi \mass \mathbf{U} + \mathbf{a} \cdot ( \mathcal{K}_x \mathbf{U} ,\mathcal{K}_y \mathbf{U}) + \delta \mathbb{S}\mathbf{U} = 0 \label{eq:eigenvalueProblem}
\end{equation}
\begin{equation}
\hspace*{-1cm}
\mbox{with} \quad (\mass)_{ij} = \int_{\Omega} \phi_i \phi_j dx, \qquad (\mathcal{K}_{x})_{ij} = \int_{\Omega} \phi_i \partial_x \phi_j dx, \qquad (\mathcal{K}_{y})_{ij} = \int_{\Omega} \phi_i \partial_y \phi_j dx
\end{equation}
with $\phi_j$ being any finite element basis functions, $ \mathbf{U} $ the array of all the degrees of freedom and $\mathbb{S}$ being the stabilization matrix defined through one of the stabilization techniques of \cref{sec:stabilization}. Although system \eqref{eq:eigenvalueProblem} is in general a global eigenvalue problem, we can reduce its complexity by exploiting more explicitly the ansatz \eqref{eq_disp_uex1}.
The choice of the mesh is crucial in order to exploit the ansatz and to find a unit block that repeats periodically in space. Hence, we must consider structured periodic meshes and we will focus, in particular, on two types of meshes. The first one is the $X$-mesh that is depicted in \cref{fig_Xmesh_P2} and the second one is the $T$-mesh depicted in \cref{fig_Tmesh_P2}. In those pictures also the distribution of some $\P_2$ elements are represented as an example.
More precisely, as it is done in \cite{article_fekete2_SHERWIN1995189} we can introduce elemental vectors of unknowns $\widetilde{\mathbf{U}}_{Z_{ij}}$, where $Z_{ij}$ is the stencil denoted by the red square in \cref{fig_Xmesh_P2}, which repeats periodically on the domain. So that $\widetilde{\mathbf{U}}_{Z_{ij}}$, for continuous finite elements, is an array of $d$ degrees of freedom inside a periodic unitary block $Z_{ij}$, excluding two boundaries (one on the top and one on the right for example). This number depends on the chosen (periodic) mesh type and on the elements. As an example, in \cref{fig_Xmesh_P2} we display for the \textit{X }type mesh the periodic elementary unit (in the red square) with \textit{Basic} and cubature degrees of freedom with $p=2$.
In the \textit{X} mesh for \textit{Basic} elements $p=2$ we have $d=8$, while for \textit{Cubature} $p=2$ we have $d=12$.
Using the periodicity of the solution and the ansatz \eqref{eq_disp_uex1} and denoting by $Z_{i\pm1,j\pm1}$ the neighboring elementary units, we can write the neighboring degrees of freedom by
\begin{equation}
\widetilde{\mathbf{U}}_{Z_{i\pm 1,j}} = e^{\pm\theta_x}\widetilde{\mathbf{U}}_{Z_{i,j}}, \qquad \widetilde{\mathbf{U}}_{Z_{i,j\pm 1}} = e^{\pm\theta_y}\widetilde{\mathbf{U}}_{Z_{i,j}},
\label{eq_ad_fourier0}
\end{equation}
and by induction all other degrees of freedom of the mesh.
\begin{figure}
\centering
\includegraphics[width=0.3\linewidth]{image_dispersion_2D/meshX/gridX4_P2_lagrange.pdf}
\includegraphics[width=0.3\linewidth]{image_dispersion_2D/meshX/gridX4_P2_cohen.pdf}
\caption{The \textit{X} type triangular mesh. At left, the \textit{Basic} finite element discretisation with $\P_2$ elements. At right, the grid configuration for $\Tilde{\P}_2$ \textit{Cubature} elements. The red square represents the periodic elementary unit that contains the degrees of freedom of interest for the Fourier analysis} \label{fig_Xmesh_P2}
\end{figure}
\begin{figure}
\centering
\subfigure[\textit{Basic} $\P_2$ elements]{
\includegraphics[width=0.35\linewidth]{image_dispersion_2D/meshT/gridT4_P2_lagrange.pdf}}
\subfigure[\textit{Cubature} $\Tilde{\P}_2$ elements]{
\includegraphics[width=0.35\linewidth]{image_dispersion_2D/meshT/gridT4_P2_cohen.pdf}}
\caption{The \textit{T} type triangular mesh with degrees of freedom in blue and periodic unit in the red square for the Fourier analysis} \label{fig_Tmesh_P2}
\end{figure}
This allows to show that the system \eqref{eq:eigenvalueProblem} is equivalent to a compact system of dimension $d$ (we drop the subscript $_K$ as they system is equivalent for all cells)
\begin{equation}
-i\xi \widetilde{\mass} \widetilde{\mathbf{U}} + a_x \widetilde{\mathcal{K}}_x \widetilde{\mathbf{U} } +a_y \widetilde{\mathcal{K}}_y \widetilde{\mathbf{U} } + \delta \widetilde{\mathbb{S}} \widetilde{\mathbf{U} } =0,
\label{eq_ad_fourier}
\end{equation}
where the matrices $\widetilde{\mass}$, $ \widetilde{\mathcal{K}}_x$, $ \widetilde{\mathcal{K}}_y$ and $\widetilde{\mathbb{S}}$ are readily obtained from the elemental discretization matrices by using Equations \eqref{eq_ad_fourier0}.
For the discrete Parseval theorem, we know that the norm or the reduced variable $\widetilde{\mathbf{U}}$ is equivalent to the norm of the discrete vector $\mathbf{U}$. Hence, studying the amplification factor of the two is equivalent.
We apply the same analysis to stabilized methods. The interested reader can access all 2D dispersion plots online \cite{TorloMichel2021git}. From the plot we can see that the increase in polynomial degree
provides the expected large reduction in dispersion error, while retaining a small amount of numerical dissipation, which permits the damping of \textit{parasite} modes.
An example of dispersion curves is given in \cref{fig:disp_cohen_P2_CIP}. The method used \textit{Cubature} $\TP_2$ elements, the CIP stabilization technique, and a wave angle $\theta = 5 \pi / 4$. We here show all 12 \textit{parasite} modes (see \cref{fig_Xmesh_P2}). The \textit{principal} mode of this system is represented in green. This figure also show the complexity of the analysis because of the number of modes to consider.
\begin{figure}[H]
\centering
\includegraphics[width=0.9\textwidth]{image_dispersion_2D/meshX/spatial_discrete/disp_space_cohen_P2_CIP_phase_10piD8.pdf}
\caption{Dispersion curves related to the 12 modes of $\widetilde{\mathbf{U}}_{Z_{ij}}$ of the system given by \textit{Cubature} $\TP_2$ elements, the CIP stabilization technique, and a wave angle $\theta = 5 \pi / 4$ on an \textit{X} mesh.\\
Phases $\omega$ (left) and amplifications $\epsilon$ (right).}
\label{fig:disp_cohen_P2_CIP}
\end{figure}
We summarize the number of modes for the \textit{X} mesh in \ref{tab:number_of_modes_2D-meshX}. A representation of each mesh is done in \ref{sec:app_mesh4fourier} for element of degree $p=2$ and $3$.
\begin{table}[H]
\small
\begin{center}
\begin{tabular}{| c || c | c | c | }
\hline
Element & $\mathbb{P}_1$ & $\mathbb{P}_2$ & $\mathbb{P}_3$ \\ \hline \hline
Cub. & 2 & 12 & 26 \\ \hline
Basic. & 2 & 8 & 18 \\ \hline
Bern. & 2 & 8 & 18 \\ \hline
\end{tabular}
\end{center}
\caption{\textit{X} mesh: Summary table of number of modes per systems.} \label{tab:number_of_modes_2D-meshX}
\end{table}%
\subsection{The fully discrete analysis} \label{sec:spectralAnalysis}
We analyze now
the fully discrete schemes obtained using the RK, SSPRK and DeC time marching methods.
Let us consider as an example the SSPRK schemes.
If we define as $A:=\mass^{-1} (a_x\mathcal{K}_x+a_y\mathcal{K}_y+\delta {\mathbb{S}})$ we can write the schemes as follows
\begin{equation}
\left \{
\begin{array}{ll}
\mathbf{U}^{(0)}:= & \mathbf{U}^n \\
\mathbf{U}^{(s)} := & \sum_{j=0}^{s-1} \left( \gamma_{sj} \mathbf{U}^{(j)} + \Delta t \mu_{sj} A \mathbf{U}^{(j)} \right), \quad s \in \llbracket 1,S \rrbracket ,\\
\mathbf{U}^{n+1}:=&\mathbf{U}^{(S)}.
\end{array}
\right .
\label{eq:discreteSSPRK}
\end{equation}
Expanding all the stages, we can obtain the following representation of the final stage:
\begin{equation}
\mathbf{U}^{n+1} = \mathbf{U}^{(0)} + \sum_{j=1}^{S} \nu_{j} \Delta t^jA^j \mathbf{U}^{(0)} = \left(\mathcal{I} + \sum_{j=1}^{S} \nu_{j}\Delta t^j A^j \right) \mathbf{U}^n, \label{RK_gamma0}
\end{equation}
where coefficients $\nu_j$ in \cref{RK_gamma0} are obtained as combination of coefficient $\gamma_{sj}$ and $\mu_{sj}$ in \cref{eq:discreteSSPRK} and $\mathcal{I} $ is the identity matrix. For example, coefficients of the fourth order of accuracy scheme RK4 are $\nu_1=1$, $\nu_2 = 1/2$, $\nu_3=1/6$ and $\nu_4 = 1/24$.\\
We can now compress the problem proceeding as in the time continuous case. In particular, using Equations \eqref{eq_ad_fourier0}
one easily shows that the problem can be written in terms of the local $d\times d$ matrices $\widetilde{A}:= \widetilde{\mass}^{-1}\left( a_x\widetilde{\mathcal{K}_x}+a_y\widetilde{\mathcal{K}_y}+\delta \widetilde{\mathbb{S}} \right)$
and in particular that
\begin{equation}
\widetilde{\mathbf{U}}^{n+1} = G \widetilde{\mathbf{U}}^{n}\quad\text{with}\quad
G:= \left(\widetilde{\mathcal{I}} + \sum_{j=1}^{S} \nu_{j} \Delta t^j \widetilde{A}^j \right) = e^{\epsilon \Delta t } e^{-i\omega \Delta t} ,
\label{eq:ampliG}
\end{equation}
where $G\in \R^{d\times d}$ is the amplification matrix depending on $\theta,\,\delta,\, \Delta t,\, \Delta x$ and $ \Delta y$. Considering each eigenvalue $\lambda_i$ of $G$, we can write the following formulae for the corresponding phase $\omega_i$ and damping coefficient $\epsilon_i$
\begin{align*}
\begin{cases}
e^{\epsilon_i \Delta t } \cos(\omega_i \Delta t) = \text{Re}(\lambda_i) ,\\
- e^{\epsilon_i \Delta t } \sin(\omega_i \Delta t) = \text{Im}(\lambda_i),
\end{cases}
\Leftrightarrow \, \begin{cases}
\omega_i\Delta t = \arctan \left( \frac{-\text{Im}(\lambda_i)}{\text{Re}(\lambda_i)} \right) ,\\
(e^{\epsilon_i \Delta t })^2 = \text{Re}(\lambda)^2 + \text{Im}(\lambda)^2,
\end{cases} \Leftrightarrow \, \begin{cases}
\dfrac{\omega_i}{k} = \arctan \left( \frac{-\text{Im}(\lambda_i)}{\text{Re}(\lambda_i)} \right) \frac{1}{k \Delta t},\\
\epsilon_i = \log\left( | \lambda_i | \right) \frac{1}{\Delta t}.
\end{cases}
\end{align*}
For the DeC method we can proceed with the same analysis transforming also the other involved matrices into their Fourier equivalent ones.
Using \cref{eq:DeCasSSPRK} these terms would contribute to the construction of $G$ not only in the $\widetilde{A}$ matrix, but also in the coefficients $\nu_j$, which become matrices as well.
At the end we just study the final matrix $G$ and its eigenstructure, whatever process was needed to build it up.
The matrix $G$ describes one timestep evolution of the Fourier modes for all the $d$ different types of degrees of freedom. The damping coefficients $\epsilon_i$ tell if the modes are increasing or decreasing in amplitude and the phase coefficients $\omega_i$ describe the phases of such modes.
We remark that a necessary condition for stability of the scheme is that $ |\lambda_i | \leq 1$ or, equivalently, $\epsilon_i \leq 0 $ for all the eigenvalues. The goal of our study is to find the largest CFL number for which the stability condition is fulfilled and such that the dispersion error is \textit{not too large}.\\
For our analysis, we focus on the \textit{X} type triangular mesh in \cref{fig_Xmesh_P2}
with elements of degree $1$, $2$ and $3$. This \textit{X} type triangular mesh is also used in \cite{articleTaoLiu} for Fourier analysis of the acoustic wave propagation system.
\subsection{Methodology}
The methodology we explain in the following, will be applied to all the combination of schemes we presented above (in time: RK, SSPRK and DeC, discretisation in space: \textit{Basic}, \textit{Cubature} and \textit{Bernstein}, stabilization techniques: CIP, OSS and SUPG), in order to find the best coefficients (CFL, $\delta$), as in \cite{michel2021spectral}. \\
It must be remarked that the dispersion analysis must satisfy the Nyquist stability criterion, i.e., $\Delta x_{max} \leq \frac{L}{2}$ with $\Delta x_{max}$ the maximal distance between two nodes on edges. In other words, $k_{max} = \frac{2\pi}{L_{min}} = \frac{2\pi}{2 \Delta x_{max}}=\frac{\pi}{\Delta x_{max}}$. This tells us where $k$ should vary, i.e., $k \in ] 0,\pi / \Delta x_{max} ]$. \\
What we aim to do is an optimization process also on the stabilization parameter and the CFL number.
With the notation of \cite{michel2021spectral}, we will set for the different stabilizations
\begin{equation*}
\begin{split}
\quad \text{OSS :} \;\;& \tau_K =\delta \Delta x | a |,\\[5pt]
\quad\text{CIP :} \;\;& \tau_f = \delta\Delta x^2 | a |,\\[5pt]
\quad \text{SUPG :}\;\;& \tau_K =\delta \Delta x/|a|.
\end{split}
\end{equation*}
One of our objectives is to explore the space of parameters (CFL,$\delta$), and to propose criteria allowing to set these parameters to provide the most stable, least dispersive and least expensive methods.
A clear and natural criterion is to exclude all parameter values for which there exists at least a wavenumber $\theta$ or an angle $\Phi \in [0,2\pi]$ such that we obtain an amplification of the mode, i.e., $\epsilon(\theta)>10^{-12}$ (taking into account the machine precision errors that might occur). Doing so, we obtain what we will denote as \textit{stable area} in $(\CFL,\theta)$ space.
For all the other points we propose 3 strategies to minimize a combination of dispersion error and computational cost. \\
In the following we describe the strategy we adopt to find the best parameters couple (\CFL,$\delta$) that minimizes a global solution error, denoted by $\eta_u$, while maximizing the {\CFL} in the stable area. In particular, we start from the relative square error of $u$
\begin{align}
\left| \frac{u(t)-u_{ex}(t)}{u_{ex}(t)}\right|^2= &\left|e^{\epsilon t - i t(\omega-\omega_{ex})}-1\right|^2\\
=&\left[e^{\epsilon t}\cos(t(\omega-\omega_{ex}))-1\right]^2+\left[e^{\epsilon t}\sin(t(\omega-\omega_{ex}))\right]^2\\
=&e^{2\epsilon t} - 2 e^{\epsilon t} \cos (t(\omega-\omega_{ex})) +1.
\end{align}
Here, we denote with $\epsilon$ and $\omega$ the damping and phase of the \textit{principal }mode and with $\omega_{ex}=\mathbf{k} \cdot \mathbf{a}$ the exact phase.
For a small enough dispersion error $|\omega-\omega_{ex} |\ll 1$, we can expand the cosine in the previous formula in a truncated Taylor series as
\begin{align}
\left|\frac{u(t)-u_{ex}(t)}{u_{ex}(t)}\right|^2\approx&\underbrace{\left[e^{\epsilon t} -1\right]^2}_{\text{Damping error}} + \underbrace{e^{\epsilon t}t^2 \left[\omega- \omega_{ex}\right]^2}_{\text{Dispersion error}}.
\end{align}
We then compute an error at the final time $T=1$, over the whole phase domain, using at least 3 points per wave $0\leq k \Delta x_p \leq \frac{2\pi}{3}$, with $\Delta x_p=\frac{\Delta x}{p}$, and $p$ the degree of the polynomials.
We obtain the following $\mathbb{L}_2$ error definition,
\begin{equation}\label{eq:dispersionError}
\eta_u(\omega,\epsilon)^2:= \frac{3}{2\pi} \left[\int_{0}^{\frac{2\pi}{3}} (e^{\epsilon}-1 )^2 dk + \int_{0}^{\frac{2\pi}{3}} e^\epsilon(\omega-\omega_{ex})^2 dk \right].
\end{equation}
Recalling that $\epsilon=\epsilon(k\Delta x,\CFL,\delta, \Phi)$ and $\omega=\omega(k,\Delta x,\CFL,\delta, \Phi)$, we need to further set the parameter $\Delta x_p$. We choose it to be large $\Delta x_p=1$, with the hope that for finer grids the error will be smaller.
Moreover, we need to check that the stability condition holds for all the possible angles $\Phi\in [0,2\pi]$.
Finally, we seek for the couple $(\text{CFL}^*,\delta^*)$ such that
\begin{equation}
\label{cfl_d}
(\text{CFL}^*,\delta^*)=\arg \max_{\text{CFL}} \left\lbrace \eta (\omega, \epsilon, \Phi')< \mu \min_{\text{stable } (\text{CFL},\delta)} \max_{\Phi} \eta(\omega,\epsilon, \Phi), \quad \forall\, \Phi' \in [0,2\pi] \,\right\rbrace,
\end{equation}
where the dependence on $\Phi$ of $\eta$ is highlighted with an abuse of notation.
For this strategy, the parameter $\mu$ must be chosen in order to balance the requirements on stability and accuracy.
After having tried different values, we have set $\mu$ to $10$ providing a sufficient flexibility to obtain results of practical usefulness. Indeed, the found values will be tested in the numerical section.
To show the influence of the angle $\Phi$ on the optimization problem we show an example for the \textit{X} mesh. For a given couple of parameters (\CFL,$\delta$) = $(0.4,0.01)$ we compare the results for $\Phi =0$ and $\Phi=3\pi/16$. In \cref{fig:fourier_two_angles} we compare the phases $\omega_i$ and the damping coefficients $\epsilon_i$ for the two angles. It is clear that for the angle $\Phi=0$, on the left, there are some modes which are not stable $\epsilon_i>0$, while for $\Phi=3\pi/16$ all modes are stable.
\begin{figure}
\centering
\subfigure[$\Phi =0$]{\includegraphics[width=0.49\linewidth,trim={100 0 100 0}, clip]{image_dispersion_2D/meshX/influence_of_wave_angle/paper_P2_cubature_SSPRK3_0piD16_OSS_phase.pdf}}
\subfigure[$\Phi=3\pi / 16$]{\includegraphics[width=0.49\linewidth,trim={100 0 100 0}, clip]{image_dispersion_2D/meshX/influence_of_wave_angle/paper_P2_cubature_SSPRK3_3piD16_OSS_phase.pdf}}
\caption{Comparison of dispersion curves $\omega_i$ and damping coefficients $\epsilon_i$, for \textit{Cubature} $\TP_2$ elements, with SSPRK time discretization and OSS stabilization. $\Phi =0$ at the left and $\Phi=3\pi / 16$ at the right.} \label{fig:fourier_two_angles}
\end{figure}
The angle can widely influence the whole analysis as one can observe in the plot of $\max_i \epsilon_i$ in \cref{fig:fourier_ex_comparison_phi}, where we observe that for the only angle $\Phi = 3\pi/16$ we would obtain an optimal parameter in (\CFL,$\delta$) = $(0.4,0.01)$, while, using all angles, this value is not stable anymore.
\begin{figure}
\centering
\subfigure[$\Phi =0$]{\includegraphics[width=0.45\linewidth,trim={500 0 460 0}, clip]{image_dispersion_2D/meshX/cfl_vs_tau/paper_epsilon_contour_disp_cfl_tau_2D_cubature_SSPRK_OSS.pdf}}
\subfigure[$\forall \Phi \in {[} 0,2 \pi {]} $]{\includegraphics[width=0.45\linewidth,trim={500 0 460 0}, clip]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/paper_epsilon_contour_disp_cfl_tau_2D_cubature_SSPRK_OSS.pdf}}
\caption{Plot of $\log(\max_i \epsilon_i)$ for \textit{Cubature} $\TP_2$ elements, SSPRK time discretization and OSS stabilization. The blue and light blue region is the stable one. At the left only for $\Phi =3\pi/16$, at the right we plot the maximum over all $\Phi$. }\label{fig:fourier_ex_comparison_phi}
\end{figure}
\begin{remark}
To define the stable region, we should only consider configurations for which the damping is below machine accuracy.
In practice, this cannot be done due to the fact that the eigenvalue problem arising from \eqref{eq:ampliG} is only solved
approximately using the linear algebra package of \texttt{numpy}. This introduces some uncertainty in the definition
of the stability region as machine accuracy needs to be replaced
by some other finite threshold.
\end{remark}
\subsection{Results of the fourier analysis using the \textit{X} type mesh} \label{sec:fourier_globalresults}
In this section, we illustrate the result obtained with the methodology explained above.
For clarity not all the results are reported in this work, however we place all the plots for all possible combination of schemes in an online repository \cite{michel2021spectral}. We will provide some examples here and a summary of the main results that we obtained.
\begin{figure}
\centering
\subfigure[SUPG]{\includegraphics[width=0.3\linewidth,trim={1000 0 0 0}, clip]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/epsilon_contour_disp_cfl_tau_2D_bernstein_DeC_SUPG.pdf}}
\subfigure[OSS]{\includegraphics[width=0.3\linewidth,trim={1000 0 0 0}, clip]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/epsilon_contour_disp_cfl_tau_2D_bernstein_DeC_OSS.pdf}}
\subfigure[CIP]{\includegraphics[width=0.3\linewidth,trim={1000 0 0 0}, clip]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/epsilon_contour_disp_cfl_tau_2D_bernstein_DeC_CIP.pdf}}
\caption{Damping coefficients $\log(\max_i \epsilon_i)$ for $\B_3$ \textit{Bernstein} elements and the DeC method with, from left to right, SUPG, OSS and CIP stabilization. The red dot is the optimum according to \eqref{cfl_d}.} \label{fig:particularfourier_all}
\end{figure}
The first type of plot we introduce is helping us in understanding how we can define the stability region in the $(\CFL,\delta)$ plane.
So, for every $(\CFL,\delta)$ we plot the maximum of $\log(\epsilon_i)$ over all modes and angles $\Phi \in [0,2\pi]$ (thanks to the symmetry of the mesh we can reduce this interval). An example is given in the right plot of \cref{fig:fourier_ex_comparison_phi},
it is clear that the whole blue area is stable and the yellow/orange area is unstable. In other cases, this boundary is not so clear and setting a threshold to determine the stable area can be challenging.
In \cref{fig:particularfourier_all} we compare different stabilizations for DeC with $\B_3$ elements.
In the CIP stabilization case, we clearly see that there is no clear discontinuity between unstable values and stable ones, as in SUPG, because there is a transient region where $\max_i \epsilon_i$ varies between $10^{-7}$ and $10^{-4}$.
The second type of plot combines the chosen stability region with the error $\eta_u$. We plot on the $(\CFL,\delta)$ plane some black crosses on the unstable region, where there exists an $i$ and $\Phi$ such that $\epsilon_i > 10^{-7}$. The color represents $\log(\eta_u)$ and the best value according to the previously described method is marked with a red dot.
In \cref{fig:fourier_global_basic_SSPRK_OSSX,fig:fourier_global_basic_SSPRK_CIPX,fig:fourier_global_cub_DeC_OSSX,fig:fourier_global_cub_DeC_CIPX} we show some examples of these plots for some schemes, for different $p=1,2,3$. In \cref{fig:fourier_global_basic_SSPRK_OSSX,fig:fourier_global_basic_SSPRK_CIPX} we test the \textit{Basic} elements with the SSPRK time discretization, while in \cref{fig:fourier_global_cub_DeC_OSSX,fig:fourier_global_cub_DeC_CIPX} we use the \textit{Cubature} elements with DeC time discretization. We compare also different stabilization technique: in \cref{fig:fourier_global_basic_SSPRK_OSSX,fig:fourier_global_cub_DeC_OSSX} we use the OSS, while in \cref{fig:fourier_global_basic_SSPRK_CIPX,fig:fourier_global_cub_DeC_CIPX} the CIP.
One can observe many differences among the schemes. For instance, for $p=3$ we see a much wider stable area for SSPRK than with DeC and, in the \textit{Cubature} DeC case, we see that the CIP requires a reduction in the CFL number with respect to the OSS stabilization.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_lagrange_SSPRK_OSS.pdf}
\caption{$\log(\eta_u)$ values (blue scale) and stable area (unstable with black crosses), on $(\CFL,\delta)$ plane. The red dot denotes the optimal value. From left to right $\P_1$, $\P_2$, $\P_3$ \textit{Basic} elements with SSPRK scheme and OSS stabilization}\label{fig:fourier_global_basic_SSPRK_OSSX}
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_lagrange_SSPRK_CIP.pdf}
\caption{$\log(\eta_u)$ values (blue scale) and stable area (unstable with black crosses), on $(\CFL,\delta)$ plane. The red dot denotes the optimal value. From left to right $\P_1$, $\P_2$, $\P_3$ \textit{Basic} elements with SSPRK scheme and CIP stabilization }\label{fig:fourier_global_basic_SSPRK_CIPX}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_cubature_DeC_OSS.pdf}
\caption{$\log(\eta_u)$ values (blue scale) and stable area (unstable with black crosses), on $(\CFL,\delta)$ plane. The red dot denotes the optimal value. From left to right $\TP_1$, $\TP_2$, $\TP_3$ \textit{Cubature} elements with DeC scheme and OSS stabilization}\label{fig:fourier_global_cub_DeC_OSSX}
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_cubature_DeC_CIP.pdf}
\caption{$\log(\eta_u)$ values (blue scale) and stable area (unstable with black crosses), on $(\CFL,\delta)$ plane. The red dot denotes the optimal value. From left to right $\TP_1$, $\TP_2$, $\TP_3$ \textit{Cubature} elements with DeC scheme and CIP stabilization}\label{fig:fourier_global_cub_DeC_CIPX}
\end{figure}
\begin{comment}
In the following section we will compare all the methods with these error measures, in order to suggest the best possible schemes between the proposed ones. All comparison of schemes can be find in \ref{sec:fourier_results_appendix}. In this study, we consider $\Phi = 3\pi /16$. We compute the fully discrete fourier analysis and compute the best combination of parameters. Results obtain will be use in the next section for numerical test, using the same wave angle.
\begin{figure}
\title{Computation of optimal parameters according to errors $\eta_\omega$ and $\eta_u$. $(\CFL,\delta)$ plot of $\eta_u$ (blue scale) and instability area (black crosses) for lagrange elements SSPRK scheme. From left to right $\mathbb{P}_1$, $\mathbb{P}_2$, $\mathbb{P}_3$. The purple circle is the optimizer of $\eta_u$, the green cross is the optimizer of $\eta_\omega$, the red star is the maximum stable CFL.}
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau/error_disp_cfl_tau_2D_lagrange_SSPRK_OSS.pdf}
\caption{\textit{X} mesh using the OSS stabilization method.}
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau/error_disp_cfl_tau_2D_lagrange_SSPRK_OSS.pdf}
\caption{\textit{T} mesh using the OSS stabilization method. }
\end{figure}
\begin{figure}
\title{Computation of optimal parameters according to errors $\eta_\omega$ and $\eta_u$. $(\CFL,\delta)$ plot of $\eta_u$ (blue scale) and instability area (black crosses) for \textit{Cubature} elements SSPRK scheme. From left to right $\TP_1$, $\TP_2$, $\TP_3$. The purple circle is the optimizer of $\eta_u$, the green cross is the optimizer of $\eta_\omega$, the red star is the maximum stable CFL.}
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau/error_disp_cfl_tau_2D_cubature_SSPRK_CIP.pdf}
\caption{\textit{X} mesh using the CIP stabilization method. }
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau/error_disp_cfl_tau_2D_cubature_SSPRK_CIP.pdf}
\caption{\textit{T} mesh using the CIP stabilization method. }
\end{figure}
\begin{figure}
\title{Computation of instability area regarding the maximum value of the damping $\epsilon$ for \textit{Cubature} elements SSPRK scheme and the CIP stabilization method. From left to right $\TP_1$, $\TP_2$, $\TP_3$. The purple circle is the optimizer of $\eta_u$, the green cross is the optimizer of $\eta_\omega$, the red star is the maximum stable CFL.\\ }
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau/epsilon_contour_disp_cfl_tau_2D_cubature_SSPRK_CIP.pdf}
\caption{\textit{T} mesh using the CIP stabilization method. } \label{fig:meshT_epsilon_contour_cub_SSP_CIP}
\end{figure}
\end{comment}
We summarize the results obtained by the optimization strategy in \cref{tab:dispersion_cfl_LinearAdvection_meshX-RES} for all the combinations of spatial, time and stabilization discretization. The CFL and $\delta$ presented there are optimal values obtained by the process above described, which we aim to use in simulations to obtain stable and efficient schemes. Unfortunately, as already mentioned above, for some schemes the stability area is not so well defined for several reasons. One of these reasons is the \textit{"shape"} of the stability area as for one-dimensional problems, see \cite{michel2021spectral}. Other issues that affect this analysis are the numerical precision, see Section \ref{sec:fourier_rocket}, and the mesh configuration, see Section \ref{sec:fourier_mesh_pattern}. In the following we study more in details these cases and how one can find better values.
\input{CFL_TAU_tab_LinearAdvection_2D_meshX-RES}
\subsection{Comparison with a space-time split stability analysis} \label{sec:fourier_rocket}
\begin{figure}
\centering
\includegraphics[width=0.5\linewidth,trim={1000 0 0 0}, clip]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/epsilon_contour_disp_cfl_tau_2D_cubature_SSPRK_SUPG.pdf}
\caption{Logarithm of the amplification coefficient $\log(\max_i (\varepsilon_i))$ for SUPG stabilization with $\TP_3$ \textit{Cubature} elements and the SSPRK method. Unstable region in yellow, the red dot is the optimal parameter according to \eqref{cfl_d}} \label{fig:particularfourier_cub_supg_ssp4}
\end{figure}
In this section, we show another stability analysis to slightly improve the results obtained above. Indeed, the solution of the eigenvalue problem \eqref{eq:ampliG} is only obtained within some approximation from the \texttt{numpy} numerical library. In some cases,
the threshold used to define the stability region is defined in a somewhat heuristic manner. So to confirm the results, we use
independently another criterion. To this end we
treat independently the temporal and spatial
discretizations as in the
method of lines. We then study only the spectral properties of the spatial discretization alone, computing the eigenvalues of the corresponding matrix $A$ (cf. \eqref{eq:discreteSSPRK}). With this information, we then
check whether they belong to the stability area of the time discretization
In particular, following \cite{book_stab_analysis_ch2}, we write the time discretization for Dahlquist's equation
\begin{equation}
\partial_t u - \lambda u = 0 \label{eq:stab_rocket_eq},
\end{equation}
in this example, we consider the SSPRK discretization \eqref{eq:discreteSSPRK}.
From \cref{RK_gamma0} we can write the amplification coefficient $\Gamma(\lambda)$, i.e.,
\begin{equation}
\mathbf{U}^{n+1} = \mathbf{U}^{(0)} + \sum_{j=1}^{S} \nu_{j} \Delta t^j\lambda^j \mathbf{U}^{(0)} = \underbrace{ \left(\mathcal{I} + \sum_{j=1}^{S} \nu_{j}\Delta t^j \lambda^j \right)}_{\Gamma(\lambda)} \mathbf{U}^n. \label{RK_gamma02}
\end{equation}
The stability condition for this SSPRK scheme is given by $\Gamma(\lambda) \leq 1$.
Now, when we substitute the Fourier transform of the spatial semidiscretization $\widetilde{A}$ to the coefficient $\lambda$ and we diagonalize the system (or we put it in Jordan's form), we obtain a condition on the eigenvalues of $\widetilde{A}$.
Then, using the parameters provided by the previous analysis (\CFL,$\delta$)=$(0.234 ,0.011)$, in Table~\ref{tab:dispersion_cfl_LinearAdvection_meshX-RES}, we plot the eigenvalues of $\widetilde{A}$ and the stability region of the SSPRK scheme for different $\theta \in [ 0, \pi ]$. We notice that for some values of $\theta$ some of the eigenvalues
fall slightly outside the
stable area, see Figure~\ref{fig:unstableSSPRKSUPG_rocket}. There are, indeed, few eigenvalues dangerously close to the imaginary axis and some of them have actually positive real part (blue dots).
\begin{figure}
\centering
\subfigure[(\CFL,$\delta$)=$(0.234 ,0.011)$ \label{fig:unstableSSPRKSUPG_rocket}]{\includegraphics[width=0.4\linewidth,trim={0 0 0 40}, clip]{image_dispersion_2D/meshX/rocket/paper_rocket_cubature_SSPRK_SUPG_P3_unstable.pdf}
}\;\;\;\;\;
\subfigure[(\CFL,$\delta$)=$(0.18 ,0.04)$ \label{fig:stableSSPRKSUPG_rocket}]{\includegraphics[width=0.4\linewidth,trim={0 0 0 40}, clip]{image_dispersion_2D/meshX/rocket/paper_rocket_cubature_SSPRK_SUPG_P3_stable.pdf}}
\caption{Eigenvalues of $\widetilde{A}$ using cubature discretization and the SUPG stabilization (varying $k$) and stability area of the SSPRK method. In red the stable eigenvalues, in blue the unstable ones.}
\label{fig:rocket_meshX_cub_SSPRK_SUPG_P3_unstable}
\end{figure}
As suggested before, if we decrease the CFL and increase $\delta$, we move towards a safer region, so considering (\CFL,$\delta$)=$(0.18 ,0.04)$ with the same $\theta$, we obtain all stable eigenvalues, as shown in Figure~\ref{fig:stableSSPRKSUPG_rocket}.
The summary of the optimal parameters of Table~\ref{tab:dispersion_cfl_LinearAdvection_meshX-RES} updated taking into account also a larger safety region in the (CFL, $\delta$) plane (as explained in this section) can be found in Table~\ref{tab:new_param_meshX_LinearAdvection-2D-RES} in Appendix~\ref{sec:app_fourier_param}.
\subsection{Different mesh patterns} \label{sec:fourier_mesh_pattern}
Another important aspect about this stability analysis is the influence of the mesh structure on the results. As an example, we use the \textit{T}-mesh, another regular and structured mesh type depicted in Figure~\ref{fig_Tmesh_P2}. In Figure~\ref{fig_Tmesh_P2} we plot also the degrees of freedom for elements of degree 2 and the periodic elementary unit that we take into consideration for the Fourier analysis. The number of modes in the periodic unit for this mesh type are summarized in Table~\ref{tab:number_of_modes_2D-meshT}. The elements of degree 3 can be found in Figure~\ref{fig:meshDofsP3} in Appendix~\ref{sec:app_mesh4fourier}.
\begin{table}
\small
\begin{center}
\begin{tabular}{| c || c | c | c | }
\hline
Element & $\mathbb{P}_1$ & $\mathbb{P}_2$ & $\mathbb{P}_3$ \\ \hline \hline
Cub. & 1 & 6 & 13 \\ \hline
Basic. & 1 & 4 & 9 \\ \hline
Bern. & 1 & 4 & 9 \\ \hline
\end{tabular}
\end{center}
\caption{Number of modes in the periodic unit for different elements in the \textit{T} mesh.} \label{tab:number_of_modes_2D-meshT}
\end{table}%
Even if for several methods we observe comparable results for the two mesh types, for some of them the analyses are quite different. An example is given by the \textit{Basic} elements with SSPRK schemes and CIP stabilization. For this method, we plot the dispersion error \eqref{eq:dispersionError} and the stability area in Figure~\ref{fig:fourier_global_basic_SSPRK_CIPXbis} for the \textit{X }mesh and in Figure~\ref{fig:fourier_global_basic_SSPRK_CIPT} for the \textit{T} mesh. We see huge differences in $\P_2$ and $\P_3$ where in the former a wide region becomes unstable for $\delta_L\leq \delta\leq \delta_R$ and for the latter we have to decrease a lot the value of $\delta$ to obtain stable schemes.
In the case of \textit{Cubature} elements with the OSS stabilization and SSPRK time integration, we have already seen in the previous section that the optimal parameters found were in a dangerous area. Repeating the stability analysis for the \textit{T} mesh we see that the situation is even more complicated. In Figure~\ref{fig:fourier_global_cubature_SSPRK_OSSXbis} we plot the analysis for the \textit{X} mesh and in Figure~\ref{fig:fourier_global_cubature_SSPRK_OSST} the one for the \textit{T} mesh. $\TP_3$ elements, though being stable for some parameters for the \textit{X} mesh, are never stable on the \textit{T} mesh. This means, that, when searching general parameters for the schemes, we have to keep in mind that different meshes leads to different results.
\begin{figure}
\centering
\subfigure[\textit{X} mesh \label{fig:fourier_global_basic_SSPRK_CIPXbis}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_lagrange_SSPRK_CIP.pdf}}
\subfigure[\textit{T} mesh\label{fig:fourier_global_basic_SSPRK_CIPT}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_lagrange_SSPRK_CIP.pdf}}
\caption{$\log(\eta_u)$ values (blue scale) and stable area (unstable with black crosses), on $(\CFL,\delta)$ plane. The red dot denotes the optimal value. From left to right $\P_1$, $\P_2$, $\P_3$ \textit{Basic} elements with SSPRK scheme and CIP stabilization}
\end{figure}
\begin{figure}
\subfigure[\textit{X} mesh \label{fig:fourier_global_cubature_SSPRK_OSSXbis}]{ \includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_cubature_SSPRK_OSS.pdf}}
\subfigure[\textit{T} mesh\label{fig:fourier_global_cubature_SSPRK_OSST}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_cubature_SSPRK_OSS.pdf}}
\caption{$\log(\eta_u)$ values (blue scale) and stable area (unstable with black crosses), on $(\CFL,\delta)$ plane. The red dot denotes the optimal value. From left to right $\TP_1$, $\TP_2$, $\TP_3$ \textit{Cubature} elements with SSPRK scheme and OSS stabilization }
\end{figure}
For completeness, we present the optimal parameters also for the \textit{T} mesh in Table~\ref{tab:new_param_meshT_LinearAdvection-2D-RES} in Appendix~\ref{sec:app_fourier_param}.
In general, it is important to consider more mesh types when doing this analysis. In practice, we will use the two presented above (\textit{X} and \textit{T} meshes). In the following, we will consider the stability region as the intersection of stability regions of both meshes.
\subsection{Final results of the stability analysis} \label{sec:fourier2D_final_results}
\begin{figure}
\centering
\subfigure[SSPRK with OSS]{\includegraphics[height=0.2\linewidth,trim={1000 0 80 0}, clip]{image_dispersion_2D/combined_mesh/cfl_vs_tau_angleLoop/epsilon_contour_disp_cfl_tau_2D_cubature_SSPRK_OSS.pdf}}
\subfigure[SSPRK with CIP]{\includegraphics[height=0.2\linewidth,trim={1000 0 80 0}, clip]{image_dispersion_2D/combined_mesh/cfl_vs_tau_angleLoop/epsilon_contour_disp_cfl_tau_2D_cubature_SSPRK_CIP.pdf}}
\subfigure[DeC with OSS]{\includegraphics[height=0.2\linewidth,trim={1000 0 80 0}, clip]{image_dispersion_2D/combined_mesh/cfl_vs_tau_angleLoop/epsilon_contour_disp_cfl_tau_2D_cubature_DeC_OSS.pdf}}
\subfigure[DeC with CIP]{\includegraphics[height=0.2\linewidth,trim={1000 0 0 0}, clip]{image_dispersion_2D/combined_mesh/cfl_vs_tau_angleLoop/epsilon_contour_disp_cfl_tau_2D_cubature_DeC_CIP.pdf}}
\caption{Maximum logarithm of the amplification coefficient $\log(\max_i (\varepsilon_i))$ for $\TP_3$ \textit{Cubature} elements on the \textit{X} and \textit{T} meshes} \label{fig:fourier_combined_cubP3_problem}
\end{figure}
\begin{figure}
\centering
\subfigure[SSPRK with OSS]{\includegraphics[height=0.2\linewidth,trim={1000 0 80 0}, clip]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/epsilon_contour_disp_cfl_tau_2D_cubature_SSPRK_OSS.pdf}}
\subfigure[SSPRK with CIP]{\includegraphics[height=0.2\linewidth,trim={1000 0 80 0}, clip]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/epsilon_contour_disp_cfl_tau_2D_cubature_SSPRK_CIP.pdf}}
\subfigure[DeC with OSS]{\includegraphics[height=0.2\linewidth,trim={1000 0 80 0}, clip]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/epsilon_contour_disp_cfl_tau_2D_cubature_DeC_OSS.pdf}}
\subfigure[DeC with CIP]{\includegraphics[height=0.2\linewidth,trim={1000 0 0 0}, clip]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/epsilon_contour_disp_cfl_tau_2D_cubature_DeC_CIP.pdf}}
\caption{Logarithm of the amplification coefficient $\log(\max_i (\varepsilon_i))$ for $\TP_3$ \textit{Cubature} elements on the \textit{X} mesh} \label{fig:fourier_Xmesh_cubP3_problem}
\end{figure}
Taking into consideration all the aspects seen in the previous sections, it is important to have a comprehensive result, which tells which parameters can be used in the majority of the situations. A summary of the parameters obtained for the \textit{X} and \textit{T} mesh is available in Appendix~\ref{sec:app_fourier_param}.
In Table~\ref{tab:restrictive_param_LinearAdvection-2D-RES}, instead, we present parameters obtained using the most restrictive case among different meshes and that insure an enough big area of stability around them, as explained in Section~\ref{sec:fourier_rocket}. These parameters can be safely used in many cases and we will validate them in the numerical sections, where, first, we validate the results of the \textit{X} mesh on a linear problem on an \textit{X} mesh, then we used the more general parameters in Table~\ref{tab:restrictive_param_LinearAdvection-2D-RES} for nonlinear problems on unstructured meshes. \\
\input{paramtab_meshcombined_2D-RES}
A special remark must be done for \textit{Cubature} $\TP_3$ elements combined with the OSS and the CIP stabilizations. In Figure~\ref{fig:fourier_combined_cubP3_problem} we see how the amplification coefficient $\max_i \varepsilon_i$ has always values far away from zero.
For the CIP stabilization this is always true and even for the $\TP_2$ elements the stability region is very thin.
As suggested in \cite{Burman2020ACutFEmethodForAModelPressure,larson2019stabilizationHighOrderCut} higher order derivatives jump stabilization terms might fix this problem, but it introduces more parameters. This has not been considered here. Another remark is that the \textit{T} configuration is very peculiar and, as we will see, on classical Delauney triangulations the issue seem to not affect the results. Moreover, the use of additional discontinuity capturing operators may alleviate this issue as some additional, albeit small, dissipation is explicitly introduced in smooth regions.
In \cref{sec:app_fourier_visco}, we propose to add an additional stabilization term for these unstable schemes, i.e., \textit{Cubature} $\TP_3$ elements and OSS or CIP stabilization techniques. This term is based on viscous term \cite{abgrallEntropyConservative,2011JCoPh.230.4248G,KUZMIN2020104742,inbookLlobell2020} and allows to stabilize numerical schemes for any mesh configuration.\\
For the OSS stabilization we observe a similar behavior in Figure~\ref{fig:fourier_combined_cubP3_problem}. The stability that we see in that plot are only due to the the $T$ mesh. Indeed, for the OSS stabilization on the \textit{X} mesh there exists a corridor of stable values, which turn out to be unstable for the \textit{T} mesh, see Figure~\ref{fig:fourier_Xmesh_cubP3_problem}.
In practice, also on unstructured grids we have not noticed instabilities when running with the parameters found with the \textit{X} mesh.
Hence, we suggest anyway some values of CFL and $\delta$ for these schemes, which are valid for the \textit{X} mesh, noting that they might be dangerous for very simple structured meshes.
The validation on unstructured meshes also for more complicated problems will be done in the next sections. \\
Overall, Table~\ref{tab:restrictive_param_LinearAdvection-2D-RES} gives some insight on the efficiency of the schemes. We remind that, in general, we prefer matrix free schemes, so this aspect must be kept in mind while evaluating the efficiency of the schemes. All the SUPG schemes, except when with DeC, and all the \textit{Basic} element schemes have a mass matrix that must be inverted. Among the others we see that for first degree polynomials schemes the DeC with \textit{Bernstein} polynomials and SUPG stabilization gives one of the largest CFL result, while for second degree polynomials the OSS \textit{Cubature} SSPRK scheme seems the one with best performance and, for fourth order schemes, again the \textit{Bernstein} DeC SUPG is one of the best.
In conclusion of this section, there are important points to highlight:
\begin{itemize}
\item The extension of the Fourier analysis to the two-dimensional space leads to significantly different results with respect to the one-dimensional one. Both in terms of global stability of the schemes, and in terms of optimal parameters. Moreover, in opposition to \cite{michel2021spectral}, \textit{Bernstein} elements with SUPG stabilization technique lead to stable and efficient schemes. \textit{Cubature} elements, which were the most efficient in one-dimensional problems, have stability issues on the two-dimensional mesh topologies studied.
\item The complexity of the analysis in two-dimensional space is increased. This not only implies a larger number of degrees of freedom, but also more parameters to keep into account, including the angle of the advection term and the possible different configuration of the mesh.
The visualization of the stability region of the time scheme as shown in Figure~\ref{fig:rocket_meshX_cub_SSPRK_SUPG_P3_unstable} with the eigenvalues of the semi-discretization operators helps in understanding the effect of CFL and penalty coefficient on the stability of the scheme, only for methods of lines. This helps in choosing and optimizing the couple of parameters.
\end{itemize}
\begin{remark}
Another possibility to characterize the linear stability of numerical method is proposed by J. Miller \cite{articleMiller1971}. This method is based on the study of the characteristic polynomial of the amplification matrix $G$.
However, this method does not provide information about the phase $\omega$, since it does not compute eigenvalues of $G$. For this reason, we choose the eigenanalysis.
\end{remark}
\section{Fourier analysis} \label{sec:appFourier}
In this section we collect all the plots and results that are essential to show the results of this work, but for structural reasons were not put in the main text.
\subsection{Mesh types and degrees of freedom}\label{sec:app_mesh4fourier}
We represent in Figure~\ref{fig:meshDofsP3} the mesh configurations used in the Fourier analysis and the degrees of freedom of the elements of degree 3. The red square represents the periodic elementary unit that contains the degrees of freedom of interest for the Fourier analysis.
\begin{figure}
\centering
\subfigure[\textit{X} mesh, \textit{Basic} elements \label{fig_Xmesh_P3}]{ \includegraphics[width=0.35\linewidth]{image_dispersion_2D/meshX/gridX4_P3_lagrange.pdf}}
\subfigure[\textit{X} mesh, \textit{Cubature} elements]{
\includegraphics[width=0.35\linewidth]{image_dispersion_2D/meshX/gridX4_P3_cohen.pdf}
}
\subfigure[\textit{T} mesh, \textit{Cubature} elements]{
\includegraphics[width=0.35\linewidth]{image_dispersion_2D/meshT/gridT4_P3_lagrange.pdf}}
\subfigure[\textit{T} mesh, \textit{Cubature} elements]{
\includegraphics[width=0.35\linewidth]{image_dispersion_2D/meshT/gridT4_P3_cohen.pdf}}
\caption{Degrees of freedom and periodic unit for different mesh patterns and elements of degree 3 \label{fig:meshDofsP3}}
\end{figure}
\subsection{Fourier analysis results - Optimal Parameters} \label{sec:app_fourier_param}
In this section, we put the optimal values of the stability analysis of Section~\ref{sec:fourier_globalresults} after the modification proposed in Section~\ref{sec:fourier_rocket}. In Table~\ref{tab:new_param_meshX_LinearAdvection-2D-RES} we show the parameters for the \textit{X} mesh and in Table~\ref{tab:new_param_meshT_LinearAdvection-2D-RES} we show the parameters for the \textit{T} mesh.
\input{paramtab_meshX_2D-RES}
\input{paramtab_meshT_2D-RES}
\subsection{Fourier analysis results - stability area} \label{sec:fourier_results_appendix}
Finally, we present a comparison of stability area between the \textit{T} and the \textit{X} mesh. This comparison if perform as before, for all wave angles $\theta$. We choose as example the comparison using \textit{Basic} element, SSPRK time integration method and the OSS stabilization technique in \cref{fig:App_fourier_global_basic_SSPRK_OSS}
The interested reader can access to results for all methods online \cite{TorloMichel2021git}.
\begin{figure}
\centering
\subfigure[\textit{X} mesh \label{fig:App_fourier_global_basic_SSPRK_OSSX}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_lagrange_SSPRK_OSS.pdf}}
\subfigure[\textit{T} mesh\label{fig:App_fourier_global_basic_SSPRK_OSST}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_lagrange_SSPRK_OSS.pdf}}
\caption{$\log(\eta_u)$ values (blue scale) and stable area (unstable with black crosses), on $(\CFL,\delta)$ plane. The red dot denotes the optimal value. From left to right $\P_1$, $\P_2$, $\P_3$ \textit{Basic} elements with SSPRK scheme and OSS stabilization.} \label{fig:App_fourier_global_basic_SSPRK_OSS}
\end{figure}
\begin{comment}
\begin{figure}
\centering
\subfigure[\textit{X} mesh \label{fig:App_fourier_global_basic_SSPRK_CIPX}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_lagrange_SSPRK_CIP.pdf}}
\subfigure[\textit{T} mesh\label{fig:App_fourier_global_basic_SSPRK_CIPT}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_lagrange_SSPRK_CIP.pdf}}
\caption{$\log(\eta_u)$ values (blue scale) and stable area (unstable with black crosses), on $(\CFL,\delta)$ plane. The red dot denotes the optimal value. From left to right $\P_1$, $\P_2$, $\P_3$ \textit{Basic} elements with SSPRK scheme and CIP stabilization.} \label{fig:App_fourier_global_basic_SSPRK_CIP}
\end{figure}
\begin{figure}
\centering
\subfigure[\textit{X} mesh \label{fig:App_fourier_global_basic_SSPRK_SUPGX}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_lagrange_SSPRK_SUPG.pdf}}
\subfigure[\textit{T} mesh\label{fig:App_fourier_global_basic_SSPRK_SUPGT}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_lagrange_SSPRK_SUPG.pdf}}
\caption{$\log(\eta_u)$ values (blue scale) and stable area (unstable with black crosses), on $(\CFL,\delta)$ plane. The red dot denotes the optimal value. From left to right $\P_1$, $\P_2$, $\P_3$ \textit{Basic} elements with SSPRK scheme and SUPG stabilization.} \label{fig:App_fourier_global_basic_SSPRK_SUPG}
\end{figure}
The comparison using \textit{Cubature} element, SSPRK time integration method and all stabilization techniques are represented in \cref{fig:App_fourier_global_cubature_SSPRK_OSS,fig:App_fourier_global_cubature_SSPRK_CIP,fig:App_fourier_global_cubature_SSPRK_SUPG}.
\begin{figure}
\centering
\subfigure[\textit{X} mesh \label{fig:App_fourier_global_cubature_SSPRK_OSSX}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_cubature_SSPRK_OSS.pdf}}
\subfigure[\textit{T} mesh\label{fig:App_fourier_global_cubature_SSPRK_OSST}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_cubature_SSPRK_OSS.pdf}}
\caption{$\log(\eta_u)$ values (blue scale) and stable area (unstable with black crosses), on $(\CFL,\delta)$ plane. The red dot denotes the optimal value. From left to right $\TP_1$, $\TP_2$, $\TP_3$ \textit{Cubature} elements with SSPRK scheme and OSS stabilization.} \label{fig:App_fourier_global_cubature_SSPRK_OSS}
\end{figure}
\begin{figure}
\centering
\subfigure[\textit{X} mesh \label{fig:App_fourier_global_cubature_SSPRK_CIPX}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_cubature_SSPRK_CIP.pdf}}
\subfigure[\textit{T} mesh\label{fig:App_fourier_global_cubature_SSPRK_CIPT}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_cubature_SSPRK_CIP.pdf}}
\caption{$\log(\eta_u)$ values (blue scale) and stable area (unstable with black crosses), on $(\CFL,\delta)$ plane. The red dot denotes the optimal value. From left to right $\TP_1$, $\TP_2$, $\TP_3$ \textit{Cubature} elements with SSPRK scheme and CIP stabilization.} \label{fig:App_fourier_global_cubature_SSPRK_CIP}
\end{figure}
\begin{figure}
\centering
\subfigure[\textit{X} mesh \label{fig:App_fourier_global_cubature_SSPRK_SUPGX}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_cubature_SSPRK_SUPG.pdf}}
\subfigure[\textit{T} mesh\label{fig:App_fourier_global_cubature_SSPRK_SUPGT}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_cubature_SSPRK_SUPG.pdf}}
\caption{$\log(\eta_u)$ values (blue scale) and stable area (unstable with black crosses), on $(\CFL,\delta)$ plane. The red dot denotes the optimal value. From left to right $\TP_1$, $\TP_2$, $\TP_3$ \textit{Cubature} elements with SSPRK scheme and SUPG stabilization.} \label{fig:App_fourier_global_cubature_SSPRK_SUPG}
\end{figure}
The comparison using \textit{Cubature} element, DeC time integration method and all stabilization techniques are represented in \cref{fig:App_fourier_global_cubature_DeC_OSS,fig:App_fourier_global_cubature_DeC_CIP,fig:App_fourier_global_cubature_DeC_SUPG}.
\begin{figure}
\centering
\subfigure[\textit{X} mesh \label{fig:App_fourier_global_cubature_DeC_OSSX}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_cubature_DeC_OSS.pdf}}
\subfigure[\textit{T} mesh\label{fig:App_fourier_global_cubature_DeC_OSST}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_cubature_DeC_OSS.pdf}}
\caption{$\log(\eta_u)$ values (blue scale) and stable area (unstable with black crosses), on $(\CFL,\delta)$ plane. The red dot denotes the optimal value. From left to right $\TP_1$, $\TP_2$, $\TP_3$ \textit{Cubature} elements with DeC scheme and OSS stabilization.} \label{fig:App_fourier_global_cubature_DeC_OSS}
\end{figure}
\begin{figure}
\centering
\subfigure[\textit{X} mesh \label{fig:App_fourier_global_cubature_DeC_CIPX}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_cubature_DeC_CIP.pdf}}
\subfigure[\textit{T} mesh\label{fig:App_fourier_global_cubature_DeC_CIPT}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_cubature_DeC_CIP.pdf}}
\caption{$\log(\eta_u)$ values (blue scale) and stable area (unstable with black crosses), on $(\CFL,\delta)$ plane. The red dot denotes the optimal value. From left to right $\TP_1$, $\TP_2$, $\TP_3$ \textit{Cubature} elements with DeC scheme and CIP stabilization.} \label{fig:App_fourier_global_cubature_DeC_CIP}
\end{figure}
\begin{figure}
\centering
\subfigure[\textit{X} mesh \label{fig:App_fourier_global_cubature_DeC_SUPGX}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_cubature_DeC_SUPG.pdf}}
\subfigure[\textit{T} mesh\label{fig:App_fourier_global_cubature_DeC_SUPGT}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_cubature_DeC_SUPG.pdf}}
\caption{$\log(\eta_u)$ values (blue scale) and stable area (unstable with black crosses), on $(\CFL,\delta)$ plane. The red dot denotes the optimal value. From left to right $\TP_1$, $\TP_2$, $\TP_3$ \textit{Cubature} elements with DeC scheme and SUPG stabilization.} \label{fig:App_fourier_global_cubature_DeC_SUPG}
\end{figure}
The comparison using \textit{Bernstein} element, DeC time integration method and all stabilization techniques are represented in \cref{fig:App_fourier_global_bezier_DeC_OSS,fig:App_fourier_global_bezier_DeC_CIP,fig:App_fourier_global_bezier_DeC_SUPG}.
\begin{figure}
\centering
\subfigure[\textit{X} mesh \label{fig:App_fourier_global_bezier_DeC_OSSX}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_bernstein_DeC_OSS.pdf}}
\subfigure[\textit{T} mesh\label{fig:App_fourier_global_bezier_DeC_OSST}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_bernstein_DeC_OSS.pdf}}
\caption{$\log(\eta_u)$ values (blue scale) and stable area (unstable with black crosses), on $(\CFL,\delta)$ plane. The red dot denotes the optimal value. From left to right $\P_1$, $\P_2$, $\P_3$ \textit{Bernstein} elements with DeC scheme and OSS stabilization.} \label{fig:App_fourier_global_bezier_DeC_OSS}
\end{figure}
\begin{figure}
\centering
\subfigure[\textit{X} mesh \label{fig:App_fourier_global_bezier_DeC_CIPX}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_bernstein_DeC_CIP.pdf}}
\subfigure[\textit{T} mesh\label{fig:App_fourier_global_bezier_DeC_CIPT}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_bernstein_DeC_CIP.pdf}}
\caption{$\log(\eta_u)$ values (blue scale) and stable area (unstable with black crosses), on $(\CFL,\delta)$ plane. The red dot denotes the optimal value. From left to right $\P_1$, $\P_2$, $\P_3$ \textit{Bernstein} elements with DeC scheme and CIP stabilization.} \label{fig:App_fourier_global_bezier_DeC_CIP}
\end{figure}
\begin{figure}
\centering
\subfigure[\textit{X} mesh \label{fig:App_fourier_global_bezier_DeC_SUPGX}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_bernstein_DeC_SUPG.pdf}}
\subfigure[\textit{T} mesh\label{fig:App_fourier_global_bezier_DeC_SUPG}]{
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_bernstein_DeC_SUPG.pdf}}
\caption{$\log(\eta_u)$ values (blue scale) and stable area (unstable with black crosses), on $(\CFL,\delta)$ plane. The red dot denotes the optimal value. From left to right $\P_1$, $\P_2$, $\P_3$ \textit{Bernstein} elements with DeC scheme and SUPG stabilization.} \label{fig:App_fourier_global_bezier_DeC_SUPG}
\end{figure}
\end{comment}
\begin{comment}
\begin{figure}
\title{Computation of optimal parameters according to errors $\eta_\omega$ and $\eta_u$. $(\CFL,\delta)$ plot of $\eta_u$ (blue scale) and instability area (black crosses) for lagrange elements SSPRK scheme. From left to right $\mathbb{P}_1$, $\mathbb{P}_2$, $\mathbb{P}_3$. The purple circle is the optimizer of $\eta_u$, the green cross is the optimizer of $\eta_\omega$, the red star is the maximum stable CFL.}
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_lagrange_SSPRK_OSS.pdf}
\caption{\textit{X} mesh using the OSS stabilization method.}
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_lagrange_SSPRK_OSS.pdf}
\caption{\textit{T} mesh using the OSS stabilization method. }
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_lagrange_SSPRK_CIP.pdf}
\caption{\textit{X} mesh using the CIP stabilization method. }
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_lagrange_SSPRK_CIP.pdf}
\caption{\textit{T} mesh using the CIP stabilization method. }
\end{figure}
\begin{figure}
\title{Computation of optimal parameters according to errors $\eta_\omega$ and $\eta_u$. $(\CFL,\delta)$ plot of $\eta_u$ (blue scale) and instability area (black crosses) for \textit{Cubature} elements SSPRK scheme. From left to right $\TP_1$, $\TP_2$, $\TP_3$. The purple circle is the optimizer of $\eta_u$, the green cross is the optimizer of $\eta_\omega$, the red star is the maximum stable CFL.}
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_cubature_SSPRK_OSS.pdf}
\caption{\textit{X} mesh using the OSS stabilization method.}
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_cubature_SSPRK_OSS.pdf}
\caption{\textit{T} mesh using the OSS stabilization method. }
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_cubature_SSPRK_CIP.pdf}
\caption{\textit{X} mesh using the CIP stabilization method. }
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_cubature_SSPRK_CIP.pdf}
\caption{\textit{T} mesh using the CIP stabilization method. }
\end{figure}
\begin{figure}
\title{Computation of optimal parameters according to errors $\eta_\omega$ and $\eta_u$. $(\CFL,\delta)$ plot of $\eta_u$ (blue scale) and instability area (black crosses) for \textit{Cubature} elements DeC scheme. From left to right $\TP_1$, $\TP_2$, $\TP_3$. The purple circle is the optimizer of $\eta_u$, the green cross is the optimizer of $\eta_\omega$, the red star is the maximum stable CFL.}
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_cubature_DeC_OSS.pdf}
\caption{\textit{X} mesh using the OSS stabilization method.}
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_cubature_DeC_OSS.pdf}
\caption{\textit{T} mesh using the OSS stabilization method. }
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_cubature_DeC_CIP.pdf}
\caption{\textit{X} mesh using the CIP stabilization method. }
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_cubature_DeC_CIP.pdf}
\caption{\textit{T} mesh using the CIP stabilization method. }
\end{figure}
\begin{figure}
\title{Computation of optimal parameters according to errors $\eta_\omega$ and $\eta_u$. $(\CFL,\delta)$ plot of $\eta_u$ (blue scale) and instability area (black crosses) for \textit{Bernstein} elements DeC scheme. From left to right $\mathbb{P}_1$, $\mathbb{P}_2$, $\mathbb{P}_3$. The purple circle is the optimizer of $\eta_u$, the green cross is the optimizer of $\eta_\omega$, the red star is the maximum stable CFL.}
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_bernstein_DeC_OSS.pdf}
\caption{\textit{X} mesh using the OSS stabilization method.}
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_bernstein_DeC_OSS.pdf}
\caption{\textit{T} mesh using the OSS stabilization method. }
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshX/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_bernstein_DeC_CIP.pdf}
\caption{\textit{X} mesh using the CIP stabilization method. }
\centering
\includegraphics[width=\linewidth]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/error_disp_cfl_tau_2D_bernstein_DeC_CIP.pdf}
\caption{\textit{T} mesh using the CIP stabilization method. }
\end{figure}
\end{comment}
\subsection{Complementary analysis using viscosity terms}
\subsection{Accounting for discontinuity capturing corrections }\label{sec:app_fourier_visco}
The stabilization terms accounted for so far are linear stabilization operators. For more challenging simulations,
additional non-linear stabilization techniques might be added to control the numerical solution in vicinity of strong non-linear fronts
and/or discontinuities. We consider here the effect of adding an extra viscosity term, as in
the entropy stabilization formulations proposed e.g. in
\cite{abgrallEntropyConservative,kuzmin2020entropyDG,2011JCoPh.230.4248G,KUZMIN2020104742,inbookLlobell2020}.
We in particular look at the approach proposed in \cite{2011JCoPh.230.4248G}, and used for shallow water waves
in \cite{10.1007/978-3-319-19800-2_36,inbookLlobell2020} and in \cite{ARPAIA2022101915,filippini:hal-01824108}.
In this approach the viscosity is designed to provide a first order correction $\mu_K=\mathcal{O}(h)$ close to
discontinuities, while for smooth enough solutions $\mu_K = c h^{p+1}$.
Our idea is to embed this high order correction explicitly in the analysis of the previous section to provide a heuristic
characterization of the fully discrete stability of the resulting stabilized formulation:
find $u_h\in V_h^p$ that satisfies for any $v_h\in W_h$
\begin{equation}
\int_{\Omega} v_h ( \partial_t u_h + \nabla \cdot f(u_h)) dx + \underbrace{ S(v_h,u_h)}_{\mbox{Diffusive term}}
+ \underbrace{\sum_K \int_K \mu_K(u_h) \nabla v_h \cdot \nabla u_h}_{\mbox{Viscosity term}} =0. \label{eq:FV-visco_app}
\end{equation}
\subsubsection{Note on the stability of the method}
As it is done for previous stabilization terms in \cref{sec:stabilization}, we can characterize the accuracy of this method estimating the truncation error for a polynomial approximation of degree $p$. Considering the smooth exact solution $u^e(t,x)$ of \eqref{eq:FV-visco_app}, for all functions $\psi$ of class at least $\mathcal{C}^1(\Omega)$ of which $\psi_h$ denotes the finite element projection, we obtain
\begin{equation}
\begin{split}
\epsilon(\psi_h) := \Big|
\int_{\Omega_h} &\psi_h \partial_t (u_h^e - u^e) \; dx - \int_{\Omega_h} \nabla \psi_h \cdot (f(u_h^e)-f(u^e))\; dx \\
+& \sum\limits_{K\in \Omega_h}\mu_K \int\limits_{K} \nabla \psi_h \cdot \nabla( u^e_h - u^e ) dx
\Big| \le C h^{p+1},
\end{split}
\label{eq_entrop2}
\end{equation}
with $C$ a constant independent of $h$. The estimate can be derived from standard approximation results applied to $u_h^e-u^e$ and to its derivatives, knowing that $\mu_K = \mathcal{O}(h^{p+1})$.
Then, for a linear flux, periodic boundaries and taking $\mu_K=\mu$ constant along the mesh, we can test with $v_h=u_h$ in \eqref{eq:FV-visco_app}, we get
\begin{equation}
\begin{split}
\int\limits_{\Omega_h} d_t \frac{ u^2_h}{2} = - \sum\limits_{K} \int\limits_{K} \mu ( \nabla u_h )^2 \leq 0,
\end{split}
\label{eq_entrop3}
\end{equation}
which can be integrated in time to obtain a bound on the $\mathbb{L}_2$ norm of the solution.
\subsubsection{The von Neumann analysis}
As we saw in \cref{sec:fourier2D_final_results}, the \textit{T} mesh configuration has stability issues. In particular, the numerical schemes using \textit{Cubature} $\TP_3$ elements, SSPRK and DeC time integration methods, and the OSS and the CIP stabilization techniques are unstable. We propose to evaluate these schemes adding the viscosity term in \eqref{eq:FV-visco_app}. For the von Neumann analysis, we use $\mu_K(u) = c h_K^{p+1}$ in \eqref{eq:FV-visco_app}, with $c \in \R^+$, $h_K$ the cell diameter and $p$ the degree of polynomial approximation. We show the plot of $\max_i \epsilon_i$ to understand how the stability region behaves with respect to $c$ using \textit{Cubature} $\TP_3$ elements. In \cref{fig:fourier_cubP3_SSPRK_OSS_viso} the maximum amplification factor $\epsilon$ is represented for varying $c$, using the OSS stabilization technique and the SSPRK time integration method. We note that the same behaviour is observed with CIP and DeC. Plots are available online \cite{TorloMichel2021git}.
\begin{figure}[h]
\centering
\subfigure[$\mu = 0$ \label{fig:fourier_cubP3_SSPRK_OSS_viso0}]{\includegraphics[height=0.25\linewidth,trim={950 0 80 0}, clip]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/epsilon_contour_disp_cfl_tau_2D_cubature_SSPRK_OSS.pdf}} \hfill
\subfigure[$\mu = 0.005 h_K^{p+1}$ \label{fig:fourier_cubP3_SSPRK_OSS_viso0005}]{\includegraphics[height=0.25\linewidth,trim={0 0 80 0}, clip]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/epsilon_contour_disp_cfl_tau_2D_cubature_SSPRK_OSS_visco-0005.pdf}} \hfill
\subfigure[$\mu = 0.05 h_K^{p+1}$ \label{fig:fourier_cubP3_SSPRK_OSS_viso005}]{\includegraphics[height=0.25\linewidth,trim={0 0 80 0}, clip]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/epsilon_contour_disp_cfl_tau_2D_cubature_SSPRK_OSS_visco-005.pdf}} \hfill \\
\subfigure[$\mu = 0.5 h_K^{p+1}$ \label{fig:fourier_cubP3_SSPRK_OSS_viso05}]{\includegraphics[height=0.25\linewidth,trim={0 0 80 0}, clip]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/epsilon_contour_disp_cfl_tau_2D_cubature_SSPRK_OSS_visco-05.pdf}}
\subfigure[$\mu = 5 h_K^{p+1}$ \label{fig:fourier_cubP3_SSPRK_OSS_viso5}]{\includegraphics[height=0.25\linewidth,trim={0 0 0 0}, clip]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/epsilon_contour_disp_cfl_tau_2D_cubature_SSPRK_OSS_visco-50.pdf}}
\caption{\textit{T} mesh - Von Neumann analysis using an additional viscosity term (see \eqref{eq:FV-visco_app}). \textit{Cubature} $\TP_3$ elements with SSPRK and OSS. Comparison of different $\mu$.} \label{fig:fourier_cubP3_SSPRK_OSS_viso}
\end{figure}
We can observe two main results. First, increasing the parameter $c$ up to around 0.1 allows to expand the stability region. Second, when the viscosity coefficients reaches too high values, it is necessary to decrease the CFL (see \cref{fig:fourier_cubP3_SSPRK_OSS_viso005} with $\mu=0.05$ and \cref{fig:fourier_cubP3_SSPRK_OSS_viso05} with $\mu=0.5$ as an example).
\begin{comment}
\subsubsection{Some convergence results using the viscosity term}
In this paragraph, we propose to evaluate the convergence rate for a linear case using the \textit{T} mesh configuration, i.e., the one with more restrictive CFL conditions. This test is performed using \textit{Cubature} $\TP_3$ elements, SSPRK and DeC time integration methods, and the OSS and the CIP stabilization techniques. The numerical test is \eqref{lin_adv_2D} presented in \cref{sec:LA2D_results-meshX}.
Using formulation \eqref{eq:FV-visco_app} containing the viscosity term
\begin{equation}
\sum_K \int_K \mu_K(u_h) \nabla v_h \cdot \nabla u_h, \quad \mbox{with} \: \: \mu_K(u_h) = c h^{p+1}, \label{eq:FV-visco_term0}
\end{equation}
and tuning stability coefficient $\delta$, CFL and viscosity coefficient $c$ found in \cref{fig:fourier_cubP3_SSPRK_OSS_viso,fig:fourier_cubP3_SSPRK_CIP_viso,}, we obtain fourth order accurate schemes. These tuned coefficients, and the corresponding convergence orders are summarized in \cref{tab:conv_order_meshTvisco_P3cub_LinearAdvection-2D-def_init}.
\begin{table}[H]
\small
\begin{center}
\begin{tabular}{| c | c || c | c | c || c | c | c | }
\hline
\multicolumn{2}{|c||}{Element $\&$ } & \multicolumn{3}{|c||}{\textit{Cubature} $\TP_3$ OSS} & \multicolumn{3}{|c|}{\textit{Cubature} $\TP_3$ CIP} \\ \hline
\multicolumn{2}{|c||}{ Time scheme } & \CFL \,($\delta$) & $c$ & order & \CFL\,($\delta$) & $c$ & order \\ \hline \hline
\multirow{2}{*}{ Cub.} & SSPRK & 0.15 (0.02) & 0.05 & 4.08 & 0.12 (0.0004) & 0.5 & 3.60 \\
& DeC & 0.15 (0.02) & 0.05 & 4.09 &0.08 (0.001) & 0.2 & 3.76 \\ \hline
\end{tabular}
\end{center}
\caption{Convergence order of methods using \textit{Cubature} $\TP_3$ elements and viscosity term \eqref{eq:FV-visco_term0} with tuned parameters} \label{tab:conv_order_meshTvisco_P3cub_LinearAdvection-2D-def_init}
\end{table}%
Many other formulations of viscosity terms exist in literature and can ensure convergent methods of order $p+1$ (using $\P_p$ elements) \cite{2011JCoPh.230.4248G,KUZMIN2020104742,inbookLlobell2020}. The majority use a nonlinear evaluation of the parameter $\mu_K$, based on the local entropy production. \\
\begin{remark}[Stability of entropy viscosity]
In our formulation the coefficient $\mu_K = c h^{p+1}$ leads to nongeneralizable conclusions and, hence, it is not recommended for general cases.
Indeed, with this formulation, the amplification matrix $G$ depends on \CFL, on $c$, on $\delta$, and also on $h^p$. This means that changing $h$, keeping all the other coefficients fixed might lead to unstable schemes. A more reliable formulation, which can be used for general cases, is given in \textit{J.L. Guermond et al.} \cite{2011JCoPh.230.4248G}. The viscous term is non-linear and it is based on the local entropy production.
{\color{red}
The intuitive value of $\mu_K$ should be $\mathcal{O}( h)$, with $c \in \R$. So, decrease the mesh size per $10$ (i.e. $h_{new} = 0.1 h_{init}$ should give the same stability than using $h_{init}$ with $c$ multiply by $0.1^p$. In this sense, we show the stability area without any classical stabilization technique, only the viscosity term using $\mu_K = c h^{p+1} $ in \cref{fig:fourier_cubP3_SSPRK_nostab_viso}, with $h_0 \approx 0.5$, and then using $h_1=2h_0$ and $h_2=0.1h_0$ (with the SSPRK method). We expect that $h_1=2h_0$ shifts the stability area to the left by $1/2^3=0.125$, and $h_2=0.1 h_0$ shifts the stability area to the right by $1/0.1^3=10^3$.}
\begin{figure}[h]
\centering
\subfigure[$\TP_3$ and SSPRK, $h_1=2h_0$]{\includegraphics[height=0.20\linewidth,trim={0 0 80 0}, clip]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/epsilon_contour_disp_cfl_tau_2D_cubature_SSPRK_any_viscoh2-50}}
\subfigure[$\TP_3$ and SSPRK, $h_0 \approx 0.5$]{\includegraphics[height=0.20\linewidth,trim={0 0 80 0}, clip]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/epsilon_contour_disp_cfl_tau_2D_cubature_SSPRK_any_visco-50}}
\subfigure[$\TP_3$ and SSPRK, $h_2 =0.1 h_0$]{\includegraphics[height=0.20\linewidth,trim={0 0 80 0}, clip]{image_dispersion_2D/meshT/cfl_vs_tau_angleLoop/epsilon_contour_disp_cfl_tau_2D_cubature_SSPRK_any_viscoh01-50}}
\caption{\textit{T} mesh - Von Neumann analysis on (\CFL,$c$) plane. Use of an the additional viscosity term $\mu_K = c h^{p+1} $. \textit{Cubature} $\TP_3$ elements with SSPRK and different mesh size.} \label{fig:fourier_cubP3_SSPRK_nostab_viso}
\end{figure}
\end{remark}
\end{comment}
\begin{comment}
We now introduce the definition of the viscous parameter define by \textit{J.L. Guermond et al.} \cite{2011JCoPh.230.4248G}. As we said before, the viscous term is non-linear, and based on the local entropy production. The definition follows \cite{2011JCoPh.230.4248G} and references therein.
\begin{enumerate}
\item We define the unique entropy pair $(E(u), \F(u))$ solution of the Cauchy problem \cref{eq:conservation_law2D}, which satiesfies
\begin{equation}
\partial_t E(u) + \nabla \cdot \F(u) \leq 0,
\end{equation}
$E$ convex ans $\F(u)=\int E'(u)f'(u)$, $f$ the nuerical flux in \cref{eq:conservation_law2D}. $E$ is the entropy and $\F$ the associated entropy flux. \\
For the problem \cref{eq:app_lin_adv_2D}, $E(u)=u^2 /2$ and $F=\ba u^2 /2$..
\item We define the entropy residual
\begin{equation}
D_h = \partial_t E(u_h) + \nabla \cdot \F(u_h),
\end{equation}
\item We define the entropy parameter $\mu_K$ considering in $K\in \Omega_h$ the entropy $\mu_E$ as
\begin{equation}
\mu_E := c_E h_K^2 \frac{ r_E(D_h |_K) }{\parallel E(u_h) - \bar{E}(u_h) \parallel_{\infty, \Omega}},
\end{equation}
with $\bar{E} $ the space-averaged value of the entropy, $c_E$ a tunable constant in $\R$ and $r_E :D \rightarrow \parallel D \parallel_{\infty,K}$. And the upper bound to the entropy viscosity
\begin{equation}
\mu_{max} = c_{max} h_{max} \max_{V_K} |f'(u)|
\end{equation}
with $V_K$ the neighborhood of $K$ and $c_{max}$ a tunable constant. Then
\begin{equation}
\mu_K (u) := \min (\mu_{max},\mu_E).
\end{equation}
As in \cite{2011JCoPh.230.4248G} we choose $c_{max}=0.1/p$ and $c_E = 0.5$
\end{enumerate}
Note that we evaluate $\partial_t E$ by a $(p+1)^{th}$ order BDF time scheme (Backward Difference Formula), We place in \cref{app:BDF_scheme} all BDF coefficients, $p=1,2,3$. $(r_E)_K$ is computed once per time step and used for each intermediate sub-time step. Finally, the variational formulation read
\begin{equation}
\int_{\Omega} v_h ( \partial_t u_h + \nabla \cdot f(u_h)) dx + \sum_K \mu_K(u_h) \int_K \nabla v_h \cdot \nabla u_h =0, \qquad \forall \: v_h\in W_h. \label{eq:FV-visco_app0}
\end{equation}
\begin{table}[H]
\small
\begin{center}
\begin{tabular}{| c | c || c | c | c || c | c | c | }
\hline
\multicolumn{2}{|c||}{Element $\&$ } & \multicolumn{3}{|c||}{$\TP_3$ - OSS} & \multicolumn{3}{|c|}{$\TP_3$ CIP} \\ \hline
\multicolumn{2}{|c||}{ Time scheme } & \CFL ($\delta$) & $c$ & conv. order & \CFL ($\delta$) & $c$ & conv. order \\ \hline \hline
\multirow{2}{*}{ Cub.} & SSPRK & cfl () & c & ... & cfl() & c & ... \\
& DeC & cfl () & c & ... & cfl() & c & ... \\ \hline
\end{tabular}
\end{center}
\caption{Convergence order of methods using viscosity term from \textit{J.L. Guermond et al.} \cite{2011JCoPh.230.4248G}, and corresponding parameters.} \label{tab:conv_order_meshTvisco_P3cub_LinearAdvection-2D-guermond}
\end{table}%
\textbf{Note about the formulation $\mu = c h^{p+1}$: }For smooth solution,
$| \partial_t E - \partial_t E_{ex}| =\mathcal{O}( h^{p+1})$,
$| \nabla \cdot \F - \nabla \cdot \F_{ex}| =\mathcal{O}( h^p)$ and
$\partial_t E_{ex}+\nabla \cdot \F_{ex}=0$. \\
So $r(D_h) = |\partial_t E + \nabla \cdot \F | = |(\partial_t E - \partial_t E_{ex}) + (\nabla \cdot \F - \nabla \cdot \F_{ex})| =\mathcal{O}( h^p)$. \\
Then $\mu_e = c_E h^2 \frac{ r(D_h |_K) }{\parallel E(u_h) - \bar{E}(u_h) \parallel_{\infty, \Omega}} = h^2 \mathcal{O}( h^{p-1})= \mathcal{O}(h^{p+1})$. \\
\subsection{Backward Difference Formula} \label{app:BDF_scheme}
In this appendix we introduce the time integration coefficients of the Backward Difference Formula (BDF) used for the partial derivative evaluation of the entropy dissipation at the order $k = 1,2,3,4$.
The BDF is a family of implicit imt integration methods. Considering the linear system \cref{eq:linear_system}:
\begin{equation}
d_t U = \rhs(t,U)
\end{equation}
\begin{itemize}
\item BDF1 also called the backward Euler method:
\begin{equation}
U_{n+1} - U_n = \Delta t \rhs(t_{n+1},U_{n+1})
\end{equation}
\item BDF2
\begin{equation}
U_{n+2} - \frac{4}{3} U_{n+1} + \frac{1}{3} U_n = \frac{2}{3} \Delta t \rhs(t_{n+2},U_{n+2})
\end{equation}
\item BDF3
\begin{equation}
U_{n+3} - \frac{18}{11} U_{n+2} + \frac{9}{11} U_{n+1} -\frac{2}{11} U_n = \frac{6}{11} \Delta t \rhs(t_{n+3},U_{n+3})
\end{equation}
\item BDF4
\begin{equation}
U_{n+4} - \frac{48}{25} U_{n+3} + \frac{36}{25} U_{n+2} - \frac{16}{25} U_{n+1} - \frac{3}{25} U_n =\frac{12}{25} \Delta t \rhs(t_{n+4},U_{n+4})
\end{equation}
\end{itemize}
\end{comment}
\section{Introduction}\label{sec:intro}
\input{introduction}
\section{Numerical discretization}\label{sec:numericalDiscretization}
\input{num_discretization}
\section{Fourier analysis}\label{sec:fourierAnalysis}
\input{fourier}
\input{fourier_visco_2D}
\section{Numerical verification} \label{sec:validation_num-meshX}
\input{numerical_meshX}
\section{Simulations on unstructured meshes}\label{sec:numericalSimulations}
\input{numerical}
\section{Conclusion}\label{sec:conclusion2D}
\input{conclusion}
\section*{Acknowledgment}
This work was performed within the Ph.D. project of Sixtine Michel: ``Finite element method for shallow water equations'', supported by INRIA and the BRGM, co-funded by in INRIA--Bordeaux Sud--Ouest and the Conseil R\'egional de la Nouvelle Aquitaine.
Davide Torlo has been funded by a postdoctoral fellowship in the team CARDAMOM in INRIA--Bordeaux Sud--Ouest and by a postdoctoral fellowship in SISSA.
R\'emi Abgrall has been supported by the Swiss National Foundation grant ``Solving advection dominated problems with high order schemes with polygonal meshes: application to compressible and incompressible flow problems'' under Grant Agreement No 200020\_175784.
\section*{Declarations}
\textbf{Funding}
Sixtine Michel was funded by in INRIA--Bordeaux Sud--Ouest and the Conseil R\'egional de la Nouvelle Aquitaine.
Davide Torlo has been funded by a postdoctoral fellowship in the team CARDAMOM in INRIA--Bordeaux Sud--Ouest and by a postdoctoral fellowship in SISSA
R\'emi Abgrall has been supported by the Swiss National Foundation grant ``Solving advection dominated problems with high order schemes with polygonal meshes: application to compressible and incompressible flow problems'' under Grant Agreement No 200020\_175784. \\
\textbf{Conflicts of interest/Competing interests }
The authors certify that there is no actual or potential conflict of interest in relation to this article.\\
\textbf{Availability of data and material}
The images for all the parameters of the stability analysis and convergence plots are available at \cite{TorloMichel2021git}.\\
\subsection{Stabilization Terms}\label{sec:stabilization}
\input{stabilization}
\subsection{Finite Element Spaces and Quadrature Rules} \label{sec:discretization2D}
\input{elements}
\subsection{Time integration}
\input{time}
\subsection{Linear advection test} \label{sec:LA2D_results-unstructure_mesh}
\input{convergence_tab_meshcombined_LinearAdvection_2D-RES}
We use the same test case of \cref{sec:LA2D_results-meshX}.
Convergence orders for all schemes are summarized in \cref{tab:conv_order_mesh_combined_LinearAdvection-2D-RES}.
We observe that all $\P_1$ discretizations provide the proper convergence order. For $\P_2$ discretization we spot a slight reduction of the order of accuracy, which lays for most of the schemes between $2$ and $\approx 2.5$ instead of being $3$. For polynomials of degree $3$, we observe an order reduction to 2 for the same schemes that lost the right order of accuracy also for \textit{X} mesh in the previous section.
In particular, we have that \textit{Bernstein} $\B_3$ polynomials with the \textit{DeC} result in an order of accuracy of $\approx 2$ instead of $4$, as well as the $\TP_3$ discretization with the combination DeC and SUPG stabilization.
As for the X mesh, the \textit{Basic} $\P_3$ discretization reach order of accuracy $\approx 4$ for all stabilization techniques, as well as \textit{Cubature} $\TP_3$ with SUPG and OSS stabilizations. \\
Also in this case, the results obtained with $\TP_3$ \textit{Cubature} elements and OSS stabilization are stable as we can see from the convergence analysis. This might mean that just few unfortunate mesh configurations, as the $T$ one, result in an unstable scheme and that, most of the time, the parameters found in \cref{tab:restrictive_param_LinearAdvection-2D-RES} are reliable for this scheme.
On the other hand, the combination $\TP_3$ and CIP gives an unstable scheme.
\begin{figure}[h!]
\begin{center}
\subfigure[\textit{Cubature} elements with DeC]{
\includegraphics[height=0.18\textheight,trim={0 0 64mm 0}, clip]{images_time_conv/timeVSerror/2D_meshcombined_timeError_LinearAdvection_DeC_cohen.pdf}} \qquad
\subfigure[\textit{Cubature} elements with SSPRK]{
\includegraphics[height=0.18\textheight,trim={10mm 0 0 0}, clip]{images_time_conv/timeVSerror/2D_meshcombined_timeError_LinearAdvection_SSPRK_cohen.pdf}}\\
\subfigure[\textit{Bernstein} elements with DeC]{ \includegraphics[height=0.18\textheight,trim={0 0 64mm 0}, clip]{images_time_conv/timeVSerror/2D_meshcombined_timeError_LinearAdvection_DeC_bezier.pdf}} \qquad
\subfigure[\textit{Basic} elements with SSPRK] {\includegraphics[height=0.18\textheight,trim={10mm 0 0 0}, clip]{images_time_conv/timeVSerror/2D_meshcombined_timeError_LinearAdvection_SSPRK_lagrange.pdf}}
\end{center}
\caption{Error for linear advection problem \eqref{lin_adv_2D} with respect to computational time for all elements and stabilization techniques}
\label{fig:2D_timeVsErrorLinAdv_meshcombined}
\end{figure}
We compare error and computational time for all methods presented above in \cref{fig:2D_timeVsErrorLinAdv_meshcombined}.
Looking at $\P_2$ and the $\P_3$ discretizations, as expected, the mass-matrix free combination, i.e., \textit{Cubature} elements with SSPRK and OSS, gives smaller computational costs than other combinations with \textit{Basic} elements. Conversely, the SUPG technique increase the computational costs with respect to all other stabilizations for all schemes. That is why we will not use it for the next test.
The plots and all the errors are available at the repository \cite{TorloMichel2021git}.
\begin{remark}[Entropy viscosity]
As remarked in \cref{sec:app_fourier_visco}, we can improve the stability of some schemes (\textit{Cubature} OSS) with extra entropy viscosity. Here, we test the convergence rate on the \textit{T} mesh configuration, i.e., the one with more restrictive CFL conditions and most unstable. This test is performed using \textit{Cubature} $\TP_3$ elements, SSPRK and DeC time integration methods, and the OSS and the CIP stabilization techniques. We solve again problem \eqref{lin_adv_2D}.
Using formulation \eqref{eq:FV-visco_app}
and tuning stability coefficient $\delta$, CFL and viscosity coefficient $c$ found in \cref{fig:fourier_cubP3_SSPRK_OSS_viso}, we obtain fourth order accurate schemes. These tuned coefficients, and the corresponding convergence orders are summarized in \cref{tab:conv_order_meshTvisco_P3cub_LinearAdvection-2D-def_init}.
\begin{table}[H]
\small
\begin{center}
\begin{tabular}{| c | c || c | c | c || c | c | c | }
\hline
\multicolumn{2}{|c||}{Element $\&$ } & \multicolumn{3}{|c||}{\textit{Cubature} $\TP_3$ OSS} & \multicolumn{3}{|c|}{\textit{Cubature} $\TP_3$ CIP} \\ \hline
\multicolumn{2}{|c||}{ Time scheme } & \CFL \,($\delta$) & $c$ & order & \CFL\,($\delta$) & $c$ & order \\ \hline \hline
\multirow{2}{*}{ Cub.} & SSPRK & 0.15 (0.02) & 0.05 & 4.08 & 0.12 (0.0004) & 0.5 & 3.60 \\
& DeC & 0.15 (0.02) & 0.05 & 4.09 &0.08 (0.001) & 0.2 & 3.76 \\ \hline
\end{tabular}
\end{center}
\caption{Convergence order of methods using \textit{Cubature} $\TP_3$ elements and viscosity term \eqref{eq:FV-visco_app} with tuned parameters} \label{tab:conv_order_meshTvisco_P3cub_LinearAdvection-2D-def_init}
\end{table}%
Many other formulations of viscosity terms exist in literature and can ensure convergent methods of order $p+1$ (using $\P_p$ elements) \cite{2011JCoPh.230.4248G,KUZMIN2020104742,inbookLlobell2020}. The majority use a nonlinear evaluation of the parameter $\mu_K$, based on the local entropy production. \\
\end{remark}
\subsection{Shallow water equations}
\input{convergence_tab_meshcombined_nlsw_2D-RES}
In this section we test the proposed schemes on the test case of \cref{sec:nlsw2D_results-meshX} with the unstructured mesh in \cref{fig:2d_mesh}.
Convergence orders are summarized in \cref{tab:conv_order_mesh_combined_nlsw-2D-RES}.
\begin{figure}[h!]
\begin{center}
\subfigure[\textit{Cubature} elements with DeC]{
\includegraphics[height=0.18\textheight,trim={0 0 64mm 0}, clip]{images_time_conv/timeVSerror/2D_meshcombined_timeError_nlsw_DeC_cohen.pdf}} \qquad
\subfigure[\textit{Cubature} elements with SSPRK]{
\includegraphics[height=0.18\textheight,trim={10mm 0 0 0}, clip]{images_time_conv/timeVSerror/2D_meshcombined_timeError_nlsw_SSPRK_cohen.pdf}}\\
\subfigure[\textit{Bernstein} elements with DeC]{
\includegraphics[height=0.18\textheight,trim={0 0 64mm 0}, clip]{images_time_conv/timeVSerror/2D_meshcombined_timeError_nlsw_DeC_bezier.pdf}} \qquad
\subfigure[\textit{Basic} elements with SSPRK] {
\includegraphics[height=0.18\textheight,trim={10mm 0 0 0}, clip]{images_time_conv/timeVSerror/2D_meshcombined_timeError_nlsw_SSPRK_lagrange.pdf}}
\end{center}
\caption{Error for shallow water problem \eqref{eq:num_test_SW2} with respect to computational time for all elements and stabilization techniques}
\label{fig:2D_timeVsErrorSW}
\end{figure}
Also for the shallow water equations, we have results that resemble the ones of the structured mesh. There are small differences in the order of accuracy in both directions in different schemes.
Comparing also the computational time of all the schemes in \cref{fig:2D_timeVsErrorSW}, we can choose what we consider the best numerical method for these test cases: \textit{Cubature} discretization with the OSS stabilization technique.
This performance seems fully provided by the free mass-matrix inversion, as the CFLs for the OSS technique (with SSPRK scheme) is approximately the same between \textit{Basic} and \textit{Cubature} elements (see \cref{tab:restrictive_param_LinearAdvection-2D-RES}).
The plots and all the errors are available at the repository \cite{TorloMichel2021git}.
\subsection{Remark on the steady vortex case} \label{sec:numerical_nlsw2D-stedyVSunsteady}
For completeness we consider now a steady vortex, similarly to what reported in \cite{paola_svetlana} for the isentropic Euler equations.
So, we consider again
the traveling vortex proposed in \cref{sec:nlsw2D_results-meshX} with $t_f =0.1s$. We compare the convergence orders between $u_c=0$ (steady case) and $u_c=0.6$ (unsteady case) in
\cref{tab:conv_order_nlsw_2D-steady} and \cref{tab:conv_order_nlsw_2D-unsteady}. As we can see, in the steady case we obtain, \emph{without any additional viscous stabilization}, the expected convergence order for all schemes, in particular for the DeC with Bernstein polynomial function. These results agree with the ones in \cite{paola_svetlana}. Comparing with the unsteady case, all the other schemes reach similar order of accuracy as obtained in \cref{tab:conv_order_mesh_combined_nlsw-2D-RES}. Running the test with additional corrections in DeC scheme, as often suggested in \cite{paola_svetlana,DeC_2017}, does not improve the convergence order in the unsteady case (even with $K=50$).
\begin{minipage}{0.54\textwidth}
\input{convergence_tab_meshcombined_nlsw_2D-steady_case_tf01}
\vspace*{3mm}
\end{minipage}\hfill
\begin{minipage}{0.44\textwidth}
\input{convergence_tab_meshcombined_nlsw_2D-unsteady_case_tf01}
\vspace*{3mm}
\end{minipage}
These results show that a numerical error appears in the spatio-temporal integration part of the solution \eqref{eq:L2}, which might be related to the fact that the high order derivatives are never penalized in our stabilizations and might produce some small oscillations.
\subsection{Long time simulation - shallow water system}
For a last numerical test, we consider again the non linear shallow water benchmark, i.e. the one define in the section \ref{sec:nlsw2D_results-meshX}, but on a domain $\Omega = [0,20] \times [0,1]$ and $t_f =25s$. Which correspond to a travelling vortex in a long free surface flow (see \ref{fig:travelling_vortex} for $\Omega = [0,2] \times [0,1]$ and $t_f=1s$). An extension of this test case was developed by M. Ricchiuto and D. Torlo in \cite{ricchiuto2021analytical} for Euler equations. \\
\begin{figure}[H]
\centering
\includegraphics[width=1\linewidth]{image_element/travelling_vortex2Davide_longbathy.png}
\caption{Initial water elevation - Travelling vortex test case, $\Omega = [0,20] \times [0,1]$ and $t_f =25s$.} \label{fig:travelling_vortex}
\end{figure}
We will compare the most relevant numerical methods that we obtained in the last section in term of accuracy and cpu time.
We consider:
\begin{itemize}
\item $\TP_2$ and $\TP_3$ \textit{cubature} discretizations with the OSS stabilization technique and SSPRK time scheme.
\item $\P_2$ and $\P_3$ \textit{basic} discretizations with the OSS stabilization technique and SSPRK time scheme.
\end{itemize}
\begin{table}[H]
\small
\begin{center}
\begin{tabular}{| c | c || c | c | c | c | }
\hline
\multicolumn{2}{|c||}{Element $\&$ } & \multirow{2}{*}{number of nodes} & \multirow{2}{*}{CFL} & \multirow{2}{*}{Error norm $L_1$} & \multirow{2}{*}{cpu-time $(s)$} \\
\multicolumn{2}{|c||}{ Time scheme } & & & & \\ \hline \hline
%
\parbox[t]{2mm}{\multirow{4}{*}{\rotatebox[origin=c]{90}{\centering Basic}}} \parbox[t]{2mm}{\multirow{4}{*}{\rotatebox[origin=c]{90}{\centering SSPRK}}} & \multirow{2}{*}{$\P_2$} & \multirow{2}{*}{10917} & \multirow{2}{*}{0.2 } & \multirow{2}{*}{4.575e-4} & \multirow{2}{*}{24783 } \\
& & & & & \\ \cline{3-6}
%
& \multirow{2}{*}{$\P_3$} & \multirow{2}{*}{11386} & \multirow{2}{*}{0.22 } & \multirow{2}{*}{3.514e-4 } & \multirow{2}{*}{19220 } \\
& & & & & \\ \hline \hline
%
\parbox[t]{2mm}{\multirow{4}{*}{\rotatebox[origin=c]{90}{\centering Cub.}}} \parbox[t]{2mm}{\multirow{4}{*}{\rotatebox[origin=c]{90}{\centering SSPRK}}} & \multirow{2}{*}{$\TP_2$} & \multirow{2}{*}{16165} & \multirow{2}{*}{0.2 } & \multirow{2}{*}{4.107e-4} & \multirow{2}{*}{10966 } \\
& & & & & \\ \cline{3-6}
%
& \multirow{2}{*}{$\TP_3$} & \multirow{2}{*}{16258} & \multirow{2}{*}{ 0.025} &\multirow{2}{*}{ 4.876e-4} & \multirow{2}{*}{40417 } \\
& & & & & \\ \hline %
\end{tabular}
\end{center}
\caption{Summary tab of numerical tests. At left: the numerical schemes, at right: results.}
\end{table}%
For the same mesh between \textit{Basic} and \textit{Cubature} elements, and the approximately same number of nodes between elements of order 2 and 3, there are two important things to notice. First of all, the error is approximately the same for the fourth methods. \\
Secondly, regarding the cpu-time, we observe that $\TP_3$ elements is much more expensive than others methods, which can be explain by the very low CFL number (ten times lower). Then, for approximatly the same CFL number, the $\TP_2$ discretization is the best choice for this numerical test. Indeed, it is twice more cheaper in term of time of computation, which can be explain by the mass matrix inversion which does not appear in this method.
\subsection{Linear advection equation test} \label{sec:LA2D_results-meshX}
We start with the linear advection equation \ref{eq:conservation_law2D} on the domain $\Omega = [0,2]\times[0,1]$ using Dirichlet inlet boundary conditions:
\begin{equation}\label{lin_adv_2D}
\begin{cases}
\partial_t u (t,\mathbf{x}) + \ba \cdot \nabla u (t,\mathbf{x}) = 0, \qquad & \quad (t,\mathbf{x}) \in [t_0,t_f] \times \Omega, \quad \ba = (a_x,a_y)^T \in \mathbb{R}^2, \\
u (0,\mathbf{x}) = u_0(\mathbf{x}), & \\
u (t,\mathbf{x_D}) = u_{ex} (t,\mathbf{x_D}), & \quad \mathbf{x_D} \in \Gamma_D = \{ (x,y)\in \R^2 ,x \in \{0,2\} \mbox{ or } y \in \{0,1\} \},
\end{cases}
\end{equation}
where $u_0((x,y)^T) = 0.1 \cos(2\pi\, r(x,y) )$, with $r(x,y)=\cos(\theta)x+\sin(\theta)y$ the rotation by an angle $\theta$ around $(0,0)$, $\mathbf{a}=(a_x,a_y)^T=(\cos(\theta), \sin(\theta) )^T$ and $\theta=3\pi/16$. The final time of the simulation is $t_f=2s$.
The exact solution is $u_{ex}(\mathbf{x},t)=u_0(x-a_x\, t,y-a_y\, t )$ for all $\mathbf{x}=(x,y)\in\Omega$ and $t\in\R^+$. The initial conditions are displayed in \cref{fig:ICcosLA}.
We discretize the domain with
the \textit{X} mesh pattern, see \cref{fig:2d_meshX}.
To have approximately the same number of degrees of freedom for different degrees $p$,
we use different mesh sizes for each order of accuracy: $\Delta x_1 = \{ 0.1, 0.05, 0.025 \}$ for $\mathbb{P}_1$, $\Delta x_2 = 2\Delta x_1 $ for $\mathbb{P}_2$, and $\Delta x_3 = 3 \Delta x_1 $ for $\mathbb{P}_3$ elements.
\begin{figure}
\centering
\subfigure[\textit{X} mesh on $\Omega=\left(0,2\right) \times \left( 0,1\right)$ \label{fig:2d_meshX}]{\includegraphics[width=0.46\textwidth]{image_element/mesh_X02.png}}
\hfill \subfigure[Cosinus test case with $\theta=3\pi/16$ \label{fig:ICcosLA}]{ \includegraphics[width=0.46\textwidth]{image_element/cos2D.png}}
\caption{Linear advection simulation on the \textit{X} mesh}
\end{figure}
\begin{figure}
\centering
\subfigure[\textit{Basic} elements \label{fig_err_2d_EDP-SSPRK_lag-OSS} ]{ \includegraphics[width=0.43\textwidth]{images_time_conv/meshX/2D_LinearAdvection_SSPRK_lagrange_OSS-RES.pdf}}
\hfill
\subfigure[\textit{Cubature} elements \label{fig_err_2d_EDP-SSPRK_cohen-OSS} ]{\includegraphics[width=0.41\textwidth]{images_time_conv/meshX/2D_LinearAdvection_SSPRK_cohen_OSS-RES.pdf}}
\caption{Error decay for linear advection problem with different elements and OSS stabilization and SSPRK time discretization: $\mathbb{P}_1$ in blue, $\mathbb{P}_2$ in green and $\mathbb{P}_3$ in red}
\end{figure}
A representative result is provided in \cref{fig_err_2d_EDP-SSPRK_lag-OSS,fig_err_2d_EDP-SSPRK_cohen-OSS}: it shows a comparison between \textit{Cubature} and \textit{Basic} elements with OSS stabilization and SSPRK time integration. As we can see, the two schemes have very similar errors except for $\P^1$ where the larger CFL increases the error. The \textit{Basic} elements require stricter CFL conditions, see \cref{tab:new_param_meshX_LinearAdvection-2D-RES}, and have larger computational costs because of the inversion of the mass matrix.
To show the main benefit of using the \textit{Cubature} elements (diagonal mass matrix), we plot in \cref{fig:2D_timeVsErrorLinAdv_meshX} the computational time of \textit{Basic} and \textit{Cubature} elements for the SSPRK time scheme and all stabilization techniques
\begin{figure}
\begin{center}
\subfigure[\textit{Cubature} elements]{
\includegraphics[height=0.19\textheight,trim={0 0 64mm 0}, clip]{images_time_conv/timeVSerror/2D_meshX_timeError_LinearAdvection_SSPRK_cohen.pdf}}
\subfigure[\textit{Basic }elements]{
\includegraphics[height=0.19\textheight,trim={8mm 0 0 0}, clip]{images_time_conv/timeVSerror/2D_meshX_timeError_LinearAdvection_SSPRK_lagrange.pdf}}
\caption{Error for linear advection problem \eqref{lin_adv_2D} with respect to computational time for SSPRK time discretization, comparing \textit{Basic }and \textit{Cubature }elements and all stabilization techniques}
\label{fig:2D_timeVsErrorLinAdv_meshX}
\end{center}
\end{figure}
As a first interesting result of numerical test, looking at the \cref{fig:2D_timeVsErrorLinAdv_meshX}, we can clearly see that, for a fixed accuracy, \textit{Cubature} elements obtain better computational times with respect to \textit{Basic} elements. Moreover, as expected, the SUPG stabilization technique requires more computational time as it requires the inversion of a mass matrix, even in the case where the CFL used in is larger than the ones for OSS or CIP stabilization, see \cref{tab:new_param_meshX_LinearAdvection-2D-RES}.
The order of accuracy reached by each simulations is shown in \cref{tab:conv_order_meshX_LinearAdvection-2D-RES}. The plots and all the errors are available at the repository \cite{TorloMichel2021git}.
\input{convergence_tab_meshX_LinearAdvection_2D-RES}
Looking at the table \ref{tab:conv_order_meshX_LinearAdvection-2D-RES}, we observe that almost all the stabilized schemes provide the expected order of accuracy. Exception to this rule are several $\P_2$ discretization which reach an order of accuracy of $\approx 2.5$, and all \textit{Bernstein} $\B_3$ polynomials with the \textit{DeC} which reach an order of accuracy of $2$. This result is very disappointing and it does not improve even adding more corrections, as suggested in \cite{paola_svetlana,DeC_2017}. Moreover, it has been independently verified that also in Fourier space the accuracy of DeC with Bernstein polynomials of degree 3 is only of order 2. This problem do not show up for steady problems, as there only the spatial discretization determines the order of accuracy. We will show it in \cref{sec:numerical_nlsw2D-stedyVSunsteady}, where we study also some steady vortexes. The authors still do not understand why the optimal order of accuracy is not reached. This opens doors to further research on this family of schemes. \\
Note that we do not show results for \textit{Bernstein} elements with \textit{SSPRK} technique because they are identical to \textit{Basic} elements, but are more expensive because of the projection in the \textit{Bernstein} element space and the interpolation in the quadrature points. \\
More comparisons on different grids (unstructured) will be done in \cref{sec:numericalSimulations}.
\subsection{Shallow water equations} \label{sec:nlsw2D_results-meshX}
We consider the non linear shallow water equations (no friction and constant topography):
\begin{equation}
\left \{
\begin{array}{lll}
\partial_t h + \partial_x (hu) + \partial_y (hv) & = 0, \qquad \quad & x\in \Omega = [0,2]\times[0,1], \\
\partial_t (hu) + \partial_x (hu^2 +g\frac{h^2}{2} ) + \partial_y (huv) & =0, & t \in [0,t_f]\\
\partial_t (hv) + \partial_x (huv) + \partial_y (hv^2 +g\frac{h^2}{2} ) & =0, & t_f =1s. \label{eq:num_test_SW2}
\end{array}
\right .
\end{equation}
An analytical solution of this system is given by travelling vortexes \cite{ricchiuto2021analytical}. We use here a vortex with compact support and in $\mathcal{C}^6(\Omega)$ described by
\begin{align}\label{eq_sol2_SW2}
\begin{pmatrix}
h(x,t)\\u(x,t)\\v(x,t)
\end{pmatrix}=
\begin{cases}
\begin{pmatrix}
h_c + \frac{1}{g} \frac{\Gamma^2}{\omega^2} \cdot \left( \lambda(\omega \mathcal{R}( \mathbf{x},t) ) - \lambda (\pi) \right) , \\
u_c + \Gamma(1+\cos (\omega \mathcal{R}( \mathbf{x},t)))^2 \cdot (- \mathcal{I}(\mathbf{x},t)_y), \\
v_c + \Gamma(1+\cos (\omega \mathcal{R}( \mathbf{x},t)))^2 \cdot ( \mathcal{I}(\mathbf{x},t)_x),
\end{pmatrix}, &\mbox{if } \omega \mathcal{R}( \mathbf{x},t) \leq \pi,\\[10pt]
\begin{pmatrix}
h_c & u_c & v_c
\end{pmatrix}^T, &\mbox{else,}
\end{cases}
\end{align}
with
\begin{align*}
\lambda(r) = &\frac{20\cos(r)}{3} + \frac{27\cos(r)^2}{16} + \frac{4\cos(r)^3}{9} + \frac{\cos(r)^4}{16} + \frac{20r\sin(r)}{3} \\
&+ \frac{35r^2}{16} + \frac{27r\cos(r)\sin(r)}{8} + \frac{4r\cos(r)^2 \sin(r)}{3} + \frac{r\cos(r)^3 \sin(r)}{4}.
\end{align*}
where $\mathbf{X_c} = (0.5,0.5)$ is the initial vortex center, $(h_c,\, u_c,\, v_c)=(1.,\,0.6,\,0)$ is the far field state, $ r_0 = 0.45$ is the vortex radius,
$\Delta h = 0.1$ is the vortex amplitude, and the remaining paramters are defined as
\begin{equation}
\left \{
\begin{array}{ll}
\omega = \pi / r_0 \qquad & \mbox{ angular wave frequency}, \\
\Gamma = \frac{12 \pi \sqrt{g \Delta h } }{r_0 \sqrt{315 \pi^2-2048}} \qquad & \mbox{ vortex intensity parameter}, \\
\mathcal{I}(\mathbf{x},t) = \mathbf{x} - \mathbf{X_c} - (u_c t,v_c t)^T \qquad & \mbox{ coordinates with respect to the vortex center}, \\
\mathcal{R}( \mathbf{x},t) = \| \mathcal{I}(\mathbf{x},t) \| \qquad & \mbox{ distance from the vortex center}. \label{eq_param_SW2}
\end{array}
\right .
\end{equation}
\begin{comment}
\begin{figure}[H]
\centering
\includegraphics[width=0.7\linewidth]{image_element/travelling_vortex2Davide.png}
\caption{Initial water elevation and velocity field (red arrows) - Travelling vortex test case, $\Omega = [0,2] \times [0,1]$.} \label{fig:travelling_vortex}
\end{figure}
\end{comment}
We discretize the mesh with uniform square intervals of length $\Delta x$ (see figure \ref{fig:2d_meshX}), and as before we perform a grid convergence by respecting the constraint $\Delta x_2 = 2\Delta x_1 $ for $\mathbb{P}_2$ elements and $\Delta x_3 = 3 \Delta x_1 $ for $\mathbb{P}_3$ elements. Because of the high cost of the SUPG technique, we only compare the OSS and the CIP stabilization techniques.
As an example of results, we again show the benefit of using \textit{Cubature} elements in \ref{fig:2D_timeVsErrorSW_meshX}. We can see that since the dimension of the discretized system is even larger than before (three times larger), the differences between \textit{Cubature} and \textit{Basic} elements are even more highlighted in the error-computational time plot.
\begin{figure}[h!]
\begin{center}
\includegraphics[height=0.18\textheight,trim={0 0 64mm 0}, clip]{images_time_conv/timeVSerror/2D_meshX_timeError_nlsw_SSPRK_cohen.pdf} \;\;
\includegraphics[height=0.18\textheight,trim={8mm 0 0 0}, clip]{images_time_conv/timeVSerror/2D_meshX_timeError_nlsw_SSPRK_lagrange.pdf}\\\vspace{2mm}
\end{center}
\caption{Error for shallow water system \eqref{eq:num_test_SW2} with respect to computational time for SSPRK method with \textit{Cubature} (left) and \textit{Basic} (right) elements and CIP and OSS stabilizations.}
\label{fig:2D_timeVsErrorSW_meshX}
\end{figure}
In \cref{tab:conv_order_meshX_nlsw-2D-RES} we show the convergence orders for this shallow water problem with the CFL and $\delta$ coefficients found in \cref{tab:new_param_meshX_LinearAdvection-2D-RES}.
\input{convergence_tab_meshX_nlsw_2D-RES}
The results obtained are similar to those of the \textit{linear advection} case.
We can also notice the $\P_2$ discretization reaching the proper convergence order, i.e., $3$, and \textit{Bernstein} $\B_3$ elements reaching an order of accuracy of $\approx 3$ which is more satisfying than the results obtained for the linear advection test, but still disappointing knowing that we were expecting $4$.
\subsubsection{Streamline-Upwind/Petrov-Galerkin - SUPG} \label{SUPG}
The SUPG method was introduced in \cite{article_supg01} (see also \cite{hst10, article_supg2} and references therein) and is strongly consistent in the sense that it vanishes when replacing the discrete solution with the exact one.
It can be written as a Petrov-Galerkin method replacing $v_h$ in \cref{remi:1} with a test function belonging to the space
\begin{equation}
W_h := \{ w_h:\quad w_h=v_h+\tau_K \nabla_u f(u_h) \cdot \nabla v_h; \quad v_h \in V_h^p \} . \label{eq_supg0}
\end{equation}
Here, $\nabla_u f(u_h) \in \R^{D\times D \times 2}$ is the Jacobian of the flux, $D$ the dimensions of the system, $\tau_K$ denotes a positive definite stabilization parameter with the dimensions of $D\times D$
that we will assume to be constant for every element.
Although other definitions are possible, here we will evaluate this parameter as
\begin{equation}
\tau_K = \delta h_K ( J_K ) ^{-1}
\end{equation}
where $h_K$ is the cell diameter and $J_K$ represents the norm of the flux Jacobian on a reference value of the element $K$. In the scalar case, $J_K = ||\nabla_u f(u) ||_K$.
The final stabilized variational formulation of \eqref{remi:2} read
\begin{equation}
\int_{\Omega} v_h \partial_t u_h \; dx + \int_{\Omega} v_h \nabla \cdot f(u_h) \; dx + \underbrace{\sum_{K \in \Omega} \int_{K} \big( \nabla_u f(u_h) \cdot \nabla v_h \big) \tau_K
\left(\partial_t u_h + \nabla \cdot f(u_h) \right)\; dx}_{S(v_h,u_h)} = 0.
\label{eq_supg1}
\end{equation}
The main problem of this stabilization method is that it depends on the time derivative of $u$ and, hence, it does not maintain the structure of the mass matrix in most cases.
To characterize the accuracy of the method, we can use the consistency analysis discussed \textit{inter alia} in \cite[\S3.1.1 and \S3.2]{AR:17}. In particular, of a finite element polynomial approximation of degree $p$ we can
easily show that given a smooth exact solution $u^e(t,x)$, replacing formally $u_h$ by the projection of $u^e$ on the finite element space,
we can write
\begin{equation}
\begin{split}
\epsilon(\psi_h) :=& \Big|
\int_{\Omega} \psi_h \partial_t (u_h^e - u^e) \; dx - \int_{\Omega} \nabla \psi_h \cdot (\nabla f(u_h^e)-\nabla f(u^e))\; dx \\ + &
\sum_{K \in \Omega}\sum\limits_{l,m \in K} \dfrac{\psi_l - \psi_m}{k+1} \int_{K} \big (\nabla_u f(u_h) \cdot \nabla \varphi_i) \tau_K
\left(\partial_t (u_h^e - u^e) + \nabla \cdot ( f(u_h^e) -f(u^e)) \right)\; dx
\Big| \le C h^{p+1},
\end{split}
\label{eq_supg2}
\end{equation}
with $C$ a constant independent of $h$, for all functions $\psi$ of class at least $\mathcal{C}^1(\Omega)$, of which $\psi_h$ denotes the finite element projection.
A key point in this estimate is the strong consistency of the method allowing to subtract its formal application to the exact solution (thus subtracting zero),
and obtaining the above expression featuring differences between the exact solution/flux and its evaluation on the finite element space.
Preserving this error estimate precludes the possibility of lumping the mass matrix, and in particular the entries associated to the stabilization term.
This makes the scheme relatively inefficient when using standard explicit time stepping.
As a final note, for a linear flux \cref{eq:conservation_law1},
exact integration, with $\tau_K = \tau$ and in the time continuous case,
a classical result is obtained for homogeneous boundary conditions by testing with $v_h =u_h + \tau\, \partial_t u_h$ \cite{article_supg2}:
\begin{equation}
\begin{split}
\int\limits_{\Omega_h}\partial_t\left(\dfrac{u^2_h}{2}+\tau^2\dfrac{(\mathbf{a} \cdot \nabla u_h)^2}{2}\right) +
\int\limits_{\Omega_h} \mathbf{a} \cdot \nabla \left( \dfrac{u^2_h}{2}+\tau^2\dfrac{( \partial_t u_h)^2}{2}\right) = -\int\limits_{\Omega_h}\tau (\partial_tu_h+\mathbf{a} \cdot \nabla u_h)^2.
\end{split}
\label{eq_supg3}
\end{equation}
For periodic, or homogeneous boundary conditions, this
shows that the norm $|||u|||^2 :=\int_{\Omega_h} \dfrac{u^2_h}{2}+\tau^2\dfrac{(\mathbf{a} \cdot \nabla u_h)^2}{2} dx$ is non-increasing.
The interested reader can refer to \cite{article_supg2} for the analysis of some (implicit) fully discrete schemes.
\subsubsection{Note on the SUPG technique applied to non scalar problems}
\input{SW_SUPG}
\subsubsection{Continuous Interior Penalty - CIP} \label{CIP}
Another stabilization technique, which maintains sparsity and symmetry of the Galerkin matrix, is the continuous interior penalty (CIP) method. It was developed by Burman and Hansbo originally in \cite{article_cip6} and then in a series of works \cite{article_cip5,article_cip3,article_cip4}. It can also be seen as a variation of the method proposed by Douglas and Dupont \cite{inbook_cip7}.
The method stabilizes the Galerkin formulation by adding
edge penalty terms
proportional to the jump of the gradient of the derivatives of the solution across the cell interfaces. The CIP introduces high order viscosity to the formulation, allowing the solution to tend to the vanishing viscosity limit. This term is
does not affect the structure of the mass
matrix. The method reads
\begin{equation}
\int_{\Omega_h} v_h \partial_t u_h \; dx + \int_{\Omega_h} v_h \nabla \cdot f(u_h)\; dx+ \underbrace{ \sum_{{\sf f} \in\mathcal{F}_h} \int_{\sf f} \tau_{\sf f} [\![n_{\sf f} \cdot \nabla v_h]\!] \cdot [\![n_{\sf f} \cdot \nabla u_h]\!] \; d\Gamma}_{S(v_h,u_h)} = 0,
\label{eq_cip0}
\end{equation}
where $[\![\cdot]\!]$ denotes the jump of a quantity across a face $\sf f$, $n_{\sf f}$ is a normal to the face $\sf f$ and where $\mathcal{F}_h$ is the collection of internal boundaries, and ${\sf f}$ are its elements.
Although other definitions are possible, we evaluate the scaling parameter in the stabilization as
\begin{equation}
\tau_{\sf f} = \delta \,h_{\sf f}^2 \| \nabla_uf\|_{\sf f},
\label{eq_cip1}
\end{equation}
where $\| \nabla_uf\|_{\sf f}$ a reference value of the norm of the flux Jacobian on ${\sf f}$ and $h_{\sf f}$ a characteristic size of the mesh neighboring $\sf f$.
As stated above, a clear advantage of CIP is that it does not modify the mass matrix, allowing to obtain efficient schemes if a mass lumping strategy can be devised.
On the other side, the stencil of the scheme increases as the jump of a degree of freedom interacts with cells which are not next to the degree of freedom itself (up to 2 cells distance).
Note that for higher order approximations \cite{Burman2020ACutFEmethodForAModelPressure,larson2019stabilizationHighOrderCut} suggest the use of jumps in higher derivatives to improve the stability of the method. However, here we consider the jump in the first derivatives in order to be able to apply the stability analysis and to study the influence of $\delta$ on the stability of the method. Some results might be definitely improved adding these stabilizations on higher derivatives.
The accuracy of CIP can be assessed with a consistency analysis as discussed in \cite[\S3.1.1 and \S3.2]{AR:17}.
This consists in, formally substituting $u_h$ by the projection onto the finite element polynomial of degree $p$ space of $u^e$, a given smooth exact solution $u^e(t,x)$, we can show that for all functions $\psi$ of class at least $\mathcal{C}^1(\Omega)$, of which $\psi_h$ denotes the finite element projection, we have the truncation error estimate
\begin{equation}
\begin{split}
\epsilon(\psi_h) := \Big|
\int_{\Omega} \psi_h \partial_t (u_h^e - u^e) \; dx -& \int_{\Omega} \nabla \psi_h \cdot ( f(u_h^e)- f(u^e))\; dx \\
+& \sum\limits_{{\sf f}\in\mathcal{F}_h} \int\limits_{\sf f}\tau_{\sf f} [\![n_f \cdot \nabla \psi_h]\!] \cdot [\![n_f \cdot \nabla (u_h^e-u^e)]\!]
\Big| \le C h^{p+1},
\end{split}
\label{eq_cip2}
\end{equation}
with $C$ a constant independent of $h$. The estimate can be derived from standard approximation results applied to $u^e_h-u^e$ and to its derivatives, noting that $\tau_{\sf f}$ is an $\mathcal{O}(h^2)$, which allows to obtain the estimation with the right order.
The symmetry of the stabilization allows to easily derive an energy stability estimate for the space discretized scheme only. In particular, for periodic boundary conditions and a linear flux
we can easily show that
\begin{equation}
\begin{split}
\int\limits_{\Omega_h}\partial_t\dfrac{u^2_h}{2}= - \sum\limits_{{\sf f}\in\mathcal{F}_h}\int\limits_{\sf f} \tau_{\sf f} [\![n_f \cdot \nabla u_h]\!]^2,
\end{split}
\label{eq_cip3}
\end{equation}
which gives a bound in time on the $\mathbb{L}_2$ norm of the solution. \\
Note that for higher than second order it may be relevant to consider additional penalty terms based on higher derivatives (see e.g. \cite{Burman2020ACutFEmethodForAModelPressure,burman2021weighted,paola_svetlana}). We did not do this in this work.
\subsubsection{Orthogonal Subscale Stabilization - OSS} \label{OSS}
Another symmetric stabilization approach is the Orthogonal Subscale Stabilization (OSS) method.
Originally introduced as Pressure Gradient Projection (GPS) in \cite{CODINA1997373} for Stokes equations,
it was extended to the OSS method in \cite{CODINA20001579,OSSCodinaBadia} for different problems with numerical instabilities, such as convection--diffusion--reaction problems.
This stabilization penalizes the fluctuations of the gradient of the solution with a projection of the gradient onto the finite element space. The method applied to \cref{remi:1} reads: find $u_h\in V_h^p$ such that $\forall v_h \in V_h^p$
\begin{equation}
\left \{
\begin{array}{ll}
& \int_{\Omega_h} v_h \partial_t u_h \; dx + \int_{\Omega_h} v_h \nabla \cdot f(u_h) \; dx + \underbrace{\sum_{K \in \Omega_h} \int\limits_{K} \tau_K \nabla v_h \cdot(\nabla u_h - w_h) \; dx}_{S(v_h ,u_h)}= 0, \\
& \int_{\Omega_h} v_h w_h\; dx - \int_{\Omega_h} v_h \nabla u_h\; dx = 0.
\end{array}
\right .\label{eq_oss0}
\end{equation}
For this method, the stabilization parameter is evaluated as
\begin{equation}
\label{eq_oss1}
\tau_K = \delta h_K \|\nabla_u f\|_K .
\end{equation}
The drawback of this method, with respect to CIP, is the requirement of a matrix inversion to project the gradient of the solution in the second equation of \eqref{eq_oss0}. This cost can be alleviated by the choice of elements and quadrature rules if they result in a diagonal mass matrix, as it will be the case for \textit{Cubature} elements that we will describe below.
As before we can easily characterize the accuracy of this method. The truncation error estimate for a polynomial approximation of degree $p$ reads in this case
\begin{equation}
\begin{split}
\epsilon(\psi_h) := \Big|
\int_{\Omega_h} &\psi_h \partial_t (u_h^e - u^e) \; dx - \int_{\Omega_h} \nabla \psi_h \cdot (f(u_h^e)-f(u^e))\; dx \\
+& \sum\limits_{K\in \Omega_h}\tau_K \int\limits_{K} \nabla \psi_h \cdot \nabla( u^e_h - u^e )
+ \sum\limits_{K \in\Omega_h} \tau_K\int\limits_{K} \nabla \psi_h \cdot ( \nabla u^e - w_h^e )
\Big| \le C h^{p+1},
\end{split}
\label{eq_oss2}
\end{equation}
where the last term is readily estimated using the projection error and the boundness of $\psi_h$ as
$$
\int_{\Omega_h} \psi_h ( w^e_h-\nabla u^e)\; dx = \int_{\Omega_h} \psi_h (\nabla u_h^e - \nabla u^e ) = \mathcal{O}(h^p).
$$
Finally, for a linear flux, periodic boundaries and taking $\tau_K=\tau$ constant along the mesh, we can test with $v_h=u_h$ in the first equation of \eqref{eq_oss0}, and with $v_h=\tau w_h$ in the second one and sum up the result to get
\begin{equation}
\begin{split}
\int\limits_{\Omega_h}\partial_t\dfrac{u^2_h}{2}= - \sum\limits_{K} \int\limits_{K} \tau_K ( \nabla u_h - w_h)^2,
\end{split}
\label{eq_oss3}
\end{equation}
which can be integrated in time to obtain a bound on the $\mathbb{L}_2$ norm of the solution. \\
The truncation consistency error analysis presented above for the three stabilization terms is completely formal and it does not comprehend an entire classical error analysis. These estimations tell us that the stabilization terms that we introduced are of the wanted order of accuracy and that they are usable to aim at the prescribed order of accuracy. This type of analysis has been already done for multidimensional problems inter alia in \cite{abgrallEntropyConservative}. More rigorous proof of error bounds with $h^{p+\frac12}$ estimates can be found in \cite{burman2021weighted} for the CIP.
We did not consider in this work projection stabilizations involving higher derivatives.
\subsubsection{Explicit Runge--Kutta and Strong Stability Preserving Runge--Kutta schemes}
Runge--Kutta time integration methods are one step methods consisting in $S$ stages defined by
\begin{equation}
\begin{split}
&U^{(0)}:=U^n,\\
&U^{(s)}:=U^n + \Delta t \sum_{j=0}^{s-1}\alpha_j^s \mass^{-1} \rhs(U^{(j)})\quad s=1,\dots, S,\\
&U^{n+1}:= U^n + \Delta t \sum_{s=0}^S \beta_s \mass^{-1} \rhs(U^{(s)}).
\end{split}\label{eq:RK}
\end{equation}
Here, we use for the solution the superscript $n$ to indicate the timestep and the superscript in brackets $(s)$ to denote the stage of the method. The coefficients $\alpha_j^s$ and $\beta_j^s$ can be defined in many different ways.
In particular, we will refer to Heun's method with RK2, to Kutta's method with RK3 and the original Runge--Kutta fourth order method as RK4. The respective Butcher tables can be found in \cref{sec:timeCoefficients} in \cref{tab:Butcher}, see \cite{butcher08numODE}.
A subset of the RK methods are the SSPRK introduced in \cite{shu-1988}. They consist in convex combinations of forward Euler steps, and can be rewritten as follows
\begin{equation}\label{eq:SSPRKformula}
\begin{split}
&U^{(0)}:=U^n,\\
&U^{(s)}:=\sum_{j=0}^{s-1} \left( \gamma_j^s U^{(j)} + \Delta t \mu_j^s \mass^{-1} \rhs(U^{(j)}) \right) \quad s=1,\dots, S,\\
&U^{n+1}:= U^{(S)} ,
\end{split}
\end{equation}
with $\gamma_j^s, \mu_j^s\geq 0$ for all $j,s=1,\dots, S$.
We will consider here the second order 3 stages SSPRK(3,2) presented by Shu and Osher in \cite{shu-1988}, the third order SSPRK(4,3) presented in \cite[Page 189]{Ruuth-2006}, and the fourth order SSPRK(5,4) defined in \cite[Table 3]{Ruuth-2006}.
For complete reproducibility of the results, we put all their Butcher's tableaux in \cref{sec:timeCoefficients} in \cref{tab:ButcherSSPRK}.
\subsubsection{The \textit{Deferred Correction} scheme}
Deferred Correction methods were introduced in \cite{ecdd148d37dd401ca8d415ae18d3ecf2} as explicit time integration methods for ODEs, but soon implicit \cite{DeC_ODE}, linearly implicit positivity preserving \cite{oeffner_torlo_2019_DeCPatankar} versions and extensions to PDE solvers \cite{DeC_2017} were studied.
In particular, in \cite{DeC_2017,DeC_AT,paola_svetlana,abgrall2019reinterpretation} the DeC is used in a different formulation for finite element methods and it introduces two operator through which it is possible to use a diagonal mass matrix without losing the accuracy order.
This is only achievable when the lumped matrix (defined as the sum on the rows of the full mass matrix) has only positive values on its diagonal. Hence, the use of \textit{Bernstein} polynomials is recommended in \cite{DeC_2017}, but also \textit{Cubature} elements can serve the purpose.
\begin{figure}[h]
\small \centering
\begin{tikzpicture}
\draw [thick] (0,0) -- (6,0) node [right=2mm]{};
\fill[black] (0,0) circle (1mm) node[below=2mm] {$t^n=t^{n,0} \,\, \quad$} node[above=2mm] {$U^{n,0}$}
(1,0) circle (0.7mm) node[below=2mm] {$t^{n,1}$} node[above=2mm] {$U^{n,1}$}
(2,0) circle (0.7mm) node[below=2mm] {}
(3,0) circle (0.7mm) node[below=2mm] {$t^{n,m}$} node[above=2mm] {$U^{n,m}$}
(4,0) circle (0.7mm) node[below=2mm] {}
(5,0) circle (0.7mm) node[below=2mm] {}
(6,0) circle (1mm) node[below=2mm] {$\qquad t^{n,M}=t^{n+1}$} node[above=2mm] {$U^{n,M}$};
\end{tikzpicture}
\caption{Subtimesteps inside the time step $[t^n,t^{n+1}]$} \label{fig_DeC-time_disc}
\end{figure}
Consider a discretization of each timestep into $M$ subtimesteps
as in \cref{fig_DeC-time_disc}.
For each subtimestep we define a high order approximation of the integral form of the ODE \eqref{eq:linear_system} from $t^{n,0}$ to $t^{n,m}$, i.e.,
\begin{equation}\label{eq:L2}
\mass \left( U^{n,m} -U^{n,0} \right) - \int_{t^{n,0}}^{t^{n,m}} \rhs (U(s)) ds \approx \L^2(\UU)^m:= \mass \left( U^{n,m} -U^{n,0} \right) - \Delta t \sum_{z \in \llbracket 0, M \rrbracket } \rho_{z}^m \rhs (U^{n,z}) = 0,
\end{equation}
with $\UU=\left( U^{n,0},\dots, U^{n,M} \right)$. Moreover, the quadrature rule in time uses the subtimesteps $t^{n,m}$ as quadrature points. The corresponding weights $\rho^{m}_z$ for every different subinterval are defined by Lagrangian basis functions in these subtimesteps (see \cite{DeC_2017,DeC_AT,paola_svetlana} for details).
The algebraic system $\L^2(\UU^*)=0$ is in general implicit and nonlinear
and, in order not to recast to nonlinear solvers, the DeC procedure approximates the solution of $\L^2(\UU^*)=0$ by successive iterations relying on a low order easy--to--invert operator $\L^1$. This operator is typically a first order forward Euler approximation with a lumped mass matrix, i.e.,
\begin{equation}\label{eq:L1}
\mass \left( U^{n,m} -U^{n,0} \right) - \int_{t^{n,0}}^{t^{n,m}} \rhs (U(s)) ds \approx \L^1(\UU)^m:= \diag \left( U^{n,m} -U^{n,0} \right) - \Delta t \beta^m \rhs (U^{n,0}) = 0.
\end{equation}
Here, $\diag$ denotes a diagonal matrix obtained from the lumping of $\mass$, i.e., $\diag_{ii}:=\sum_{j} \mass_{ij}$, and $\beta^m:= \frac{t^{n,m}-t^{n,0}}{t^{n+1}-t^n}$. The values of the coefficients $\beta^m$ and $\rho^m_z$ for equispaced subtimesteps can be found in \cref{sec:timeCoefficients}.
Denoting with the superscript $(k)$ index the iteration step, we describe the DeC algorithm as
\begin{subequations}
\begin{align}
&U^{n,m,(0)}:=U^n & m=0,\dots,M,\\
&U^{n,0,(k)}:=U^n & k=0,\dots, K,\\
&\L^1(\UU^{(k)})=\L^1(\UU^{(k-1)})-\L^2(\UU^{(k-1)})& k=1,\dots, K, \label{eq:DeCUpdate}\\
&U^{n+1}:=U^{n,M,(K)}.&
\end{align}
\end{subequations}
It has been proven \cite{DeC_2017} that if $\L^1$ is coercive, $\L^1-\L^2$ is Lipschitz with a constant $\alpha_1 \Delta t >0$ and the solution of $\L^2(\UU^*)=0$ exists and is unique, then, the method converges with an error of $\mathcal{O}(\Delta t^K)$. Hence, choosing $K=M+1$ we obtain a $K$-th order accurate scheme.
Relying only on the inversion of the low order operator, the method gets rid of the computational costs of the solution of the linear systems, leaving in the right hand side the mass matrix of the $\L^2$ operator, that should not be inverted.
The only requirement that is necessary for the DeC approach is the invertibility of the lumped mass matrix $\diag$,
which limits its application to spatial elements which possess this property.\textit{Basic} Lagrange polynomials do not guarantee such constraint already for degree 2. Hence, only other polynomials as \textit{Bernstein} and \textit{Cubature} can be used in combination with DeC.
Finally, for the following analysis we note that the DeC method can be cast in a form similar to a Runge--Kutta method by rewriting \cref{eq:DeCUpdate} as
\begin{equation}\label{eq:DeCasSSPRK}
U^{n,m,(k+1)}=U^{n,m,(k)} - \diag^{-1} \mass \left(U^{n,m,(k)}-U^{n,0,(k)}\right) +\sum_{j=0}^M \Delta t \rho_{j}^m \diag^{-1}\rhs(U^{n,j,(k)}).
\end{equation}
Comparing with the system of equations \eqref{eq:SSPRKformula}, we can immediately define the SSPRK coefficients associated to DeC as $\gamma^{m,(k+1)}_{m,(k)}=\mathbb{I}-\diag ^{-1} \mass$ with $\mathbb{I}$ the identity matrix, $\gamma^{m,(k+1)}_{0,(0)}=\diag ^{-1} \mass$, $\mu^{m,(k+1)}_{r,(k)}=\rho^m_r$ for $m,r=0,\dots,M$ and $k=0,\dots,K-1$ and instead of the mass matrix, we use the diagonal one.
\begin{remark}[DeC with SUPG]
The iterative procedure of the DeC method allows even to overcome the difficulties that some implicit stabilization as the SUPG has. Indeed, the SUPG stabilization term can be added only to the $\L^2$ operator, keeping the high order accuracy of this operator. Since the $\L^2$ operator is applied to the previously computed iteration, all the terms of the SUPG, included the time derivative of $u$ in \cref{eq_supg1}, can be explicitly computed on $U^{(k-1)}$, keeping then the diagonal mass matrix for the whole scheme.
\end{remark}
\section{Time discretization coefficients}\label{sec:timeCoefficients}
In this appendix we introduce the time integration coefficients used in this work, to make the study fully reproducible. In \cref{tab:Butcher} there are the RK coefficients, in \cref{tab:ButcherSSPRK} the SSPRK coefficients and in \cref{tab:DeCcoeff} the DeC coefficients.
\begingroup
\setlength{\tabcolsep}{6pt}
\renewcommand{\arraystretch}{1.2}
\begin{table}[h!]
\centering
\begin{tabular}{ c c c }
\begin{tabular}{| c | c c | }
\hline
\multicolumn{3}{|c|}{\textit{RK2}} \\ \hline
$\alpha$ & 1 & \\ \hline
$\beta$ & $\frac{1}{2}$ & $\frac{1}{2}$ \\ \hline
\end{tabular} &
\begin{tabular}{|c| c c c | }
\hline
\multicolumn{4}{|c|}{\textit{RK3}} \\ \hline
$\alpha$ & $\frac{1}{2}$ & & \\
& -1 & 2 & \\ \hline
$\beta$ & $\frac{1}{6}$ & $\frac{2}{3}$ & $\frac{1}{6}$ \\ \hline
\end{tabular} &
\begin{tabular}{| c | c c c c | }
\hline
\multicolumn{5}{|c|}{\textit{RK4}} \\ \hline
$\alpha$ & $\frac{1}{2}$ & & & \\
& 0 & $\frac{1}{2}$ & & \\
& 0 & 0 & 1 & \\ \hline
$\beta$ & $\frac{1}{6}$ & $\frac{1}{3}$ & $\frac{1}{3}$ & $\frac{1}{6}$ \\ \hline
\end{tabular}
\end{tabular}
\caption{Butcher Tableau of RK methods}\label{tab:Butcher}
\end{table}
\begin{table}[h!]
\begin{center}
\begin{tabular}{| c c c |c c c | }
\hline
\multicolumn{6}{|c|}{\textit{SSPRK(3,2)} by \cite{shu-1988}} \\ \hline
\multicolumn{3}{|c|}{$\gamma$} & \multicolumn{3}{|c|}{$\mu$} \\ \hline
1 & & & $\frac{1}{2}$ & & \\
0 & 1 & & 0 & $\frac{1}{2}$ & \\
$\frac{1}{3}$ & 0 & $\frac{2}{3}$ & 0 & 0 & $\frac{1}{3}$ \\ \hline
\multicolumn{6}{|c|}{CFL = 2.} \\ \hline
\end{tabular} \qquad
\begin{tabular}{| c c c c | c c c c | }
\hline
\multicolumn{8}{|c|}{\textit{SSPRK(4,3)} by \cite[Page 189]{Ruuth-2006}} \\ \hline
\multicolumn{4}{|c|}{$\gamma$} & \multicolumn{4}{|c|}{$\mu$} \\ \hline
1 & & & & $\frac{1}{2}$ & & & \\
0 & 1 & & & 0 & $\frac{1}{2}$ & & \\
$\frac{2}{3}$ & 0 & $\frac{1}{3}$ & & 0 & 0 & $\frac{1}{6}$ & \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & $\frac{1}{2}$ \\ \hline
\multicolumn{8}{|c|}{CFL = 2.} \\ \hline
\end{tabular}\\
\begin{tabular}{| c c c c c |}
\hline
\multicolumn{5}{|c|}{\textit{SSPRK(5,4)} by \cite[Table 3]{Ruuth-2006} } \\ \hline
\multicolumn{5}{|c|}{$\gamma$} \\ \hline
1 & & & & \\
0.444370493651235 & 0.555629506348765 & & & \\
0.620101851488403 & 0 & 0.379898148511597 & & \\
0.178079954393132 & 0 & 0 & 0.821920045606868 & \\
0 & 0 & 0.517231671970585 & 0.096059710526147 & 0.386708617503269 \\ \hline
\multicolumn{5}{|c|}{$\mu$} \\ \hline
0.391752226571890 & & & & \\
0 & 0.368410593050371 & & & \\
0 & 0 & 0.251891774271694 & &\\
0 & 0 & 0 & 0.544974750228521 &\\
0 & 0 & 0 & 0.063692468666290 & 0.226007483236906 \\ \hline
\multicolumn{5}{|c|}{CFL = 1.50818004918983} \\ \hline
\end{tabular}
\caption{Butcher Tableau of SSPRK methods}\label{tab:ButcherSSPRK}
\end{center}
\end{table}
\begin{table}[h!]
\begin{center}
\begin{tabular}{ |c|c|c c |}\hline
\multicolumn{4}{|c|}{Order 2}\\ \hline
m & $\beta^m$ & \multicolumn{2}{|c|}{$\rho^{m}_z$}\\ \hline
1 & 1 & $\frac{1}{2}$ & $\frac{1}{2}$ \\ \hline
\end{tabular}
\quad
\begin{tabular}{ |c|c|c c c |}\hline
\multicolumn{5}{|c|}{Order 3}\\ \hline
m & $\beta^m$ & \multicolumn{3}{|c|}{$\rho^{m}_z$}\\ \hline
1 & $\frac{1}{2}$ & $\frac{5}{24}$ & $\frac{1}{3}$ & $-\frac{1}{24}$ \\
2 & 1 & $\frac{1}{6}$ & $\frac{2}{3}$ & $\frac{1}{3}$\\ \hline
\end{tabular}
\quad
\begin{tabular}{ |c|c|c c c c|}\hline
\multicolumn{6}{|c|}{Order 4}\\ \hline
m & $\beta^m$ & \multicolumn{4}{|c|}{$\rho^{m}_z$}\\ \hline
1 & $\frac{1}{3}$ & $\frac{1}{8}$ & $\frac{19}{72}$ & $-\frac{5}{72}$ & $\frac{1}{72}$ \\
2 & $\frac{2}{3}$ & $\frac{1}{9}$ & $\frac{4}{9}$ & $\frac{1}{9}$ & 0\\
3 & 1 & $\frac{1}{8}$ & $\frac{3}{8}$ & $\frac{3}{8}$ & $\frac{1}{8}$\\ \hline
\end{tabular}
\caption{DeC coefficients for equispaced subtimesteps. }\label{tab:DeCcoeff}
\end{center}
\end{table}
\endgroup
|
train/arxiv
|
BkiUe0W6NNjgB1scZ_kD
| 5 | 1 |
\chapter{Introduction and Summary}
A manifestly background independent field theory of strings should define the
conceptual framework for string theory and should allow the
precise definition and explicit computation of nonperturbative
effects. The present version of quantum closed string field theory,
developed explicitly only for the case of bosonic strings, while not
manifestly background independent, was proven to be background
independent for the case of nearby backgrounds
[\senzwiebach,\senzwiebachtwo,\senzwiebachnew].
The proof indeed uncovered structures that are expected to be
relevant to the conceptual foundation of string theory.
At the computational level one can ask if present day string field theory
allows one to do new computations, in particular computations that are
not defined in first quantization. While efficient computation may require
the manifestly background independent formulation not yet available, it is
of interest to attempt new computations with present day tools. This is
the main purpose of the present paper.
Off-shell amplitudes are not
naturally defined without a field theory. Indeed, while the basic definition
of an off-shell string amplitude {\it is} given in first quantization,
off-shell
string amplitudes are only interesting if they obey additional properties
such as
permutation symmetry and consistent factorization. These properties are
automatically incorporated when the off-shell amplitudes arise from a
covariant string field theory [\sonodazwiebach].
Off-shell string amplitudes are obtained by integrating
over the relevant moduli space of Riemann surfaces differential forms
that correspond to the correlators of vertex operators inserted at the
punctures of the surfaces and antighost line integrals. The vertex operators
correspond to non-primary fields of the conformal field theory. In contrast,
in on-shell string amplitudes the vertex operators are always
primary fields.
In order to insert non-primary fields in a punctured Riemann surface we
must choose an analytic local coordinate at every puncture.
The moduli space of Riemann surfaces of genus $g$ and $N$ punctures is
denoted as $\overline{\cal M}_{g,N}$, and the moduli space of such surfaces
with choices of local coordinates at the punctures is denoted as
$\widehat{\cal P}_{g,N}\,$.\foot{The local coordinate
at each puncture is defined only up to a constant phase.}
An off-shell amplitude is just an
integral over a subspace of $\widehat{\cal P}_{g,N}$. Typically, the relevant
subspaces of $\widehat{\cal P}_{g,N}$ are sections over $\overline{\cal M}_{g,N}$.
Such sections are obtained by making a choice of local coordinates
at every puncture of each surface in $\overline{\cal M}_{g,N}$.
In closed string field theory, the use of minimal area metrics
allows one to construct these sections using
the vertices of the theory and the propagator.
Off-shell amplitudes arising in open string field
theory have been studied by Samuel [\samuelone,\samueltwo].
While interesting in their own right, off-shell amplitudes
are not physical observables. More relevant is the evaluation
of the string action for any choice of an off-shell string
field. This computation would be necessary in evaluating string
instanton effects. The string action, apart for the kinetic term, is the
sum of string interactions each of which is defined by a {\it string vertex},
namely, a subspace ${\cal V}_{g,N}$ of $\widehat{\cal P}_{g,N}$. Typically ${\cal V}_{g,N}$ is
a section over a compact subspace of $\overline{\cal M}_{g,N}$.
Therefore, given an off-shell string field, the contribution to the string
action arising from a particular interaction corresponds to a {\it partially
integrated} off-shell amplitude. The classical potential of a field theory
in flat Minkowski space is a simple example of the above considerations;
it amounts to the evaluation of the action for
field configurations that are spacetime constants.
Ideally we would like to compute, for the case of bosonic strings formulated
around the twenty-six dimensional Minkowski space, the complete classical
potential for the string field. This may be eventually possible but we
address here the computation of the classical potential for some string
modes. In particular we focus in the case of the tachyon of the closed bosonic
string.
For the case of open strings some interesting
results have been obtained concerning the classical {\it effective} potential
for the tachyon [\kosteleckysamuel]. This potential takes into account the
effect of all other fields at the classical level.
In the context of
closed string field theory only the cubic term in the tachyon potential
is known [\kosteleckysamueltwo]. The possible effects of this term have
been considered in Refs.[\kosteleckyperry,\raiten,\mahapatra].
Our interest in the computation of the closed string tachyon potential
was stimulated by G. Moore [\moore]
who derived the following formula for the potential $V(\tau)$ for the
tachyon field $\tau(x)$
$$V (\tau)= -\tau^2-\sum_{n=3}^\infty \,{\rm v}\{\cal N}\, {\tau^N\over N!}\, ,\quad\quad
{\rm where} \quad {\rm v}\{\cal N}\, \sim \,\int_{{\cal V}_{0,N}} \bigl(\prod_{I=1}^{N-3}
{{\rm d}}^2 h\{\cal I}\,\bigr) \prod_{I=1}^N
\left|h\{\cal I}'(0)\right|^{-2}\, .\eqn\moore$$
This potential is the tachyon potential with all other fields set to zero.
It is not an effective potential. It is fully nonpolynomial, and starts with
a negative sign quadratic term (the symbols appearing in the
expression for ${\rm v}\{\cal N}$ will be defined in sect.2).
The calculation of the tachyon potential
amounts to the calculation of the constant coefficients ${\rm v}\{\cal N}$ for
$N\geq 3$. For the cubic term, since ${\cal V}_{0,3}$ is
a point, the integral is actually not there, and the evaluation of the
coefficient of ${\rm v}_3$ is relatively straightforward.
The higher coefficients are difficult to compute since they involve
integrals over the pieces of moduli spaces ${\cal V}_{0,N}$.
We will rewrite \moore\ in
${\rm PSL}(2,\IC)$ invariant form in order to understand the
geometrical significance of the coefficients ${\rm v}\{\cal N}$ and to set up
a convenient computation scheme.
Moreover, we will obtain a generalization of eqn.\moore\ valid for
any component
field of the string field theory. The expression will be given in the
operator formalism and will be ${\rm PSL}(2,\IC)$ invariant.
\noindent
\underbar{Extremal Property.}
We will show that the
polyhedral vertices of closed string field theory are the solution to
the problem of minimizing recursively the coefficients in the expansion
defining the tachyon potential. That is, the choice of the Witten vertex,
among all possible choices of cubic string vertices\foot{Vertices are
defined by coordinate curves surrounding the punctures and defining
disks. The disks should not have finite intersection}
minimizes ${\rm v}\\3$. Once the three string vertex is chosen, the region
of moduli space to be covered by the four string vertex is fixed. The
choice of the standard tetrahedron for the four string vertex,
among all possible choices of four string vertex filling the required region,
will minimize the value of ${\rm v}\\4$. This continues to be the case for the
complete series defining the classical closed string field theory.
This fact strikes to us as the string field theory doing its best to
obtain a convergent series for the tachyon potential. It is also
interesting that a simple demand, that of minimizing recursively
the coefficients of the tachyon potential, leads uniquely to the polyhedral
vertices of classical closed string field theory. It has been clear
that the consistency of closed string field theory simply depends on having
a choice of string vertices giving a cover of moduli space.
The off-shell behavior, however, is completely
dependent on the choice of vertices,
and one intuitively feels there are choices that are better than other.
We see here nice off-shell behavior arising from polyhedra.
\noindent
\underbar{A Minimum in the Potential?}
In calculating the tachyon potential we must be very careful about sign
factors. The relative signs of the expansion coefficients are essential
to the behavior of the series. We find that all the even terms in the
tachyon potential, including the quadratic term, come with a negative
sign, and all the odd terms come with a positive sign. It then follows,
by a simple sign change in the definition of the tachyon field, that all
the terms in the potential have negative coefficients.
This implies that there is no global minimum in the potential since
the potential is not bounded from below.
Moreover, there is no local minimum that can be identified without detailed
knowledge of the complete series defining the tachyon
potential. If the series defining the potential
has no suitable radius of convergence further complications arise in
attempting to extract physical conclusions. We were not able
to settle the issue of convergence, but present some work that goes
in this direction.
In estimating the coefficient ${\rm v}\{\cal N}$ we must perform an integral of
the tachyon off-shell amplitude over ${\cal V}_{0,N}$. In this region the tachyon
amplitude varies strongly. In the middle region the amplitude is lowest, and
if this were the dominating region, we would get convergence. In some
corners of ${\cal V}_{0,N}$ the amplitude is so big that, if those
corners dominate, there would be no radius of convergence.
It is important to emphasize that only the tachyon effective potential
(or the full string field potential) is a significant object.
The tachyon potential
is not by itself sufficient to make physical statements. A stable
critical point of this potential may not even be a critical point
of the complete string field potential. The effects of the infinite number
of massive scalar fields must be taken into account.
Our results, making unlikely the existence of a stable critical point
reinforce the sigma-model arguments that suggest
that bosonic strings do not have time independent stable vacua [\banks],
but are not conclusive.
(See also Ref.[\tseytlin] for a discussion of tachyonic ambiguities in
the sigma model approach to the string effective action.)
The calculation of the full string field potential, or the tachyon effective
potential is clearly desirable. We discuss in Appendix A string field
redefinitions, and argue that it does not seem possible to bring the string
action into a form (such as one having a purely quadratic tachyon potential)
where one can easily rule out the existence of a local minimum.
\noindent
\underbar{General Off-Shell Amplitudes.}
Since general off-shell computations do not have some of the
simplifying circumstances that are present for the tachyon
(such as being primary, even off-shell), we derive a general formula
useful to compute arbitrary off-shell amplitudes. This formula,
written in the operator formalism, gives the integrand for generic
string amplitudes as a differential form in $\widehat{\cal P}_{0,N}$.
The only delicate point here is the construction of the
antighost insertions for Schiffer variations representing
arbitrary families of local coordinates (local coordinates at the
punctures as a function of the position of the punctures on the sphere).
Particular cases of this formula
have appeared in the literature. If the family of local coordinates
happens to arise from a metric, the required antighost insertions were given in
Ref.[\polchinski]. Antighost insertions necessary for zero-momentum dilaton
insertions were calculated in Refs.[\nelson].
\noindent
\underbar{Organization of the Contents of this Paper.}
We now give a brief summary of the contents of the present paper.
In sect.2 we explain what needs to be calculated to extract the
tachyon potential, set up our conventions, and summarize all our
results on the tachyon potential.
In sect.3 we prepare the grounds for the geometrical understanding
of the off-shell amplitudes. We review the definition of the mapping
radius of punctured disks and study its behavior under
${\rm PSL}(2,\IC)$ transformations (the conformal maps of the Riemann sphere to
itself). We show how to construct ${\rm PSL}(2,\IC)$ invariants for
punctured spheres equipped with coordinate disks, by using
the mapping radii of the punctured disks and coordinate differences between
punctures, both
computed using an arbitrary uniformizer. We review the extremal
properties of Jenkins-Strebel quadratic differentials [\strebel],
and show how our ${\rm PSL}(2,\IC)$ invariants, in addition to having extremal properties,
provide interesting (and seemingly new) functions on the moduli spaces
$\overline{\cal M}_{0,N}$.\foot{These functions are analogous to the function
that assigns to an unpunctured Riemann surface the area of the minimal
area metric on that surface.} In sect.4 we compute the off-shell amplitude
for scattering of $N$ tachyons at arbitrary momentum, and give the answer in
terms of integrals of ${\rm PSL}(2,\IC)$ invariants. This formula
is the off-shell extension of the Koba-Nielsen formula. At zero momentum
and partially integrated over moduli space, it gives us, for each $N$, the
coefficient ${\rm v}\{\cal N}$ of the tachyon potential. We show why these coefficients
are minimized recursively by the string vertices defined by Strebel
differentials. In sect.5 we do large$-N$ estimates for the coefficients ${\rm v}\{\cal N}$
of the tachyon potential in an attempt
to establish the existence of a radius of convergence for the series.
The measure of integration is computed exactly for corners in
${\cal V}_{0,N}$ representing a planar configuration for the
tachyon punctures on the sphere. We are also able to estimate
the measure of integration for a uniform distribution of
punctures on the sphere. In sect.6 we give the operator formalism
construction for general differential forms on $\widehat{\cal P}_{0,N}$
labelled by arbitrary off-shell states.
\chapter{String Action and the Tachyon Potential}
In this section we will show what must be calculated in
order to obtain the tachyon potential. This will help put in perspective
the work that will be done in the next few sections.
We will also give some of the necessary conventions, and we will comment
on the significance of the tachyon potential and the limitations of
our results. All of our results concerning the tachyon potential will be
summarized here.
The full string field action is a non-polynomial functional of the
infinite number of fields, and from the component viewpoint, a non-polynomial
function of an infinite number of spacetime fields.
Here we consider only the part of it which contains the tachyon field
$\tau(x)$.
We will call this part the tachyonic action $S^{{\rm tach}}(\tau)$. It is a
nonpolynomial, non-local functional of the tachyonic field $\tau(x)$.
In order to introduce the string field configuration associated to the
tachyon field $\tau(x)$ we first Fourier transform
$$\tau (p) = \int {{\rm d}^Dx} \,\, \tau (x)\, e^{-ipx} \, , \eqn\tachft$$
and use $\tau(p)$ to define the tachyon string field $\ket{T}$ as follows
$$\ket{T}=\int{{\rm d}^Dp\over (2\pi)^D}\,\tau(p)\, c_1\bar c_1\ket{{\bf 1},p}.
\eqn\tachfield$$
In the conformal field theory representing the bosonic string, the tachyon
vertex operator is given by $T_p = c\bar c e^{ipX}$
and is of conformal dimension
$(L_0 , \overline L_0) =(-1+ p^2/2 , -1 + p^2/2)$. The conformal field theory
state associated to this field is $\,T_p(0)\ket{{\bf 1}} =
c_1\bar c_1 \ket{{\bf 1},p}$. This state is BRST
invariant when we satisfy the on-shell condition
$L_0=\overline L_0 =0$, which requires $p^2=2= -M^2$ (this is the problematic
negative mass squared of the tachyon). The above representative $T_p$ for the
cohomology class of the physical tachyon is particularly nice, because
this tachyon operator remains a primary
field even off-shell ($p^2 \not= 2)$.
The tachyonic action is then given by evaluating the string field action
$S(\ket{\Psi})$ for $\ket{\Psi} = \ket{T}$:
$$S^{{\rm tach}}(\tau)= S\, \left(\, \ket{\Psi} = \ket{T}\, \right) , \eqn\tachact$$
where
$$S(\Psi)={1\over2}\bra{\Psi}c_0^-Q\ket{\Psi}+
\sum_{N=3}^\infty{\kappa^{N-2}\over N!}\{\Psi^N\}_{{\cal V}_{0,N}},\eqn\action$$
and $\kappa$ is the closed string field coupling constant (see
[\zwiebachlong]).
This action
satisfies the classical master equation $ \{S,S\}=0 $ when the string vertices
${\cal V}_0=\sum_{N\geq3}{\cal V}_{0,N}$ are chosen to satisfy the recursion relations
$\partial{\cal V}_0+{1\over2}\{{\cal V}_0,{\cal V}_0\}=0$ (see Ref.[\senzwiebachnew])
Let us verify that the above definitions lead to the correctly normalized
tachyon kinetic term
$$S^{{\rm tach}}_{{\rm kin}}=\, {1\over2}\,\bra{T}c_0^-Q\ket{T}\,.\eqn\tkin$$
Recall that the BRST operator $Q$ is of the form
$Q=c_0L_0+\bar c_0\bar L_0+\cdots$,
where the dots denote the terms which annihilate $\ket{T}$. Moreover, acting
on the state $c_1\bar c_1 \ket{{\bf 1},p}$ the operators $L_0$
and $\bar L_0$ both have eigenvalue $p^2/2-1$. We then find
$$S^{{\rm tach}}_{{\rm kin}}=
{1\over2}\int{{\rm d}^Dp\over (2\pi)^D}\int{{\rm d}^Dp'\over (2\pi)^D}
\bra{-p',{\bf 1}}c_{-1}\bar c_{-1}c_0^-c_0^+c_1\bar
c_1\ket{p,{\bf 1}}\,\,\tau(p')\,(p^2-2)\,\tau(p).\eqn\simpta$$
We follow the conventions of Ref.[\zwiebachlong] where
$$\bra{-p',{\bf 1}}c_{-1}\bar c_{-1}c_0^-c_0^+c_1\bar
c_1\ket{p,{\bf 1}}\equiv (2\pi)^D\,\braket{-p',{\bf 1}^c}{p,{\bf
1}}=(2\pi)^D\delta^D(p'+p)\, ,\eqn\innprod$$
and $c_0^\pm = {1\over 2}(c_0 \pm \bar c_0)$. Using this we finally find
$$S^{{\rm tach}}_{\rm kin}=
-{1\over2}\int{{\rm d}^Dp\over (2\pi)^D}\tau(-p)(p^2-2)\tau(p)\, ,\eqn\tchykntc$$
which is indeed the correctly normalized kinetic term.\foot{We work
in euclidean space with positive signature, and the action $S$ should
be inserted in the path integral as $\exp (S/\hbar)$, which is
a convenient convention in string field theory. The euclidean action
$S$ is of the form $S =-\int {\rm d}^Dx (K +V)$, where $K$ and $V$ stand for
kinetic and potential terms respectively.}
The $N$-th term in the expansion of the tachyonic action requires the
evaluation of string multilinear functions
$$S^{{\rm tach}}_{0,N}(\tau)={\kappa^{N-2}\over N!}\{T^N\}_{{\cal V}_{0,N}}\,,\eqn\termtach$$
and this will be one of the main endeavors in this paper. The answer will be of
the form
$$\{T^N\}_{{\cal V}_{0,N}} =
\int \prod\{\cal I} {{\rm d} p\{\cal I}\over (2\pi)^D}\,\, (2\pi)^D
\delta \left( \sum p\{\cal I} \right)\cdot V\{\cal N}(p_1, \cdots p\{\cal N})\, \tau(p_1)\cdots
\tau(p\{\cal N}) \, ,\eqn\fftor$$
where $V$, the function we will be calculating, is well defined
up to terms that vanish upon use of momentum conservation.
To extract from this the tachyon potential we evaluate the above term
in the action for spacetime constant tachyons $\tau(x) = \tau_0$, which
gives $\tau(p) = \tau_0 (2\pi)^D \delta(p)$, and as a consequence
$$S^{{\rm tach}}_{0,N}(\tau_0)= {\kappa^{N-2}\over N!} V_N({\bf 0})\, \tau_0^N
\cdot \,(2\pi)^D\delta({\bf 0}) \, .\eqn\fftorr$$
Since the infinite $(2\pi)^D\delta({\bf 0})$ factor just corresponds to the
spacetime volume, the tachyon potential will read
$$V (\tau)= -\tau^2 - \sum_{N\geq 3} {\kappa^{N-2}\over N!} \,{\rm v}_{{}\{\cal N}}\,\,
\tau^N \, ,\eqn\fftorr$$
where we have used the fact that the potential appears in the action
with a minus sign.
Here the expansion coefficients ${\rm v}_{{}\{\cal N}}$ are given by
$${\rm v}_{{}\{\cal N}}\equiv V\{\cal N}({\bf 0})\, . \eqn\littlev$$
We will see that the coefficient ${\rm v}\\3$ is given by\foot{The value quoted
here agrees with that quoted in Ref.[\kosteleckyperry] after adjusting
for a factor of two difference in the definition of the dimensionless
coupling constant.}
$${\rm v}_3 = - {3^9\over 2^{11}} \approx - 9.61\,. \eqn\firstermp$$
Analytic work, together with numerical evaluation gives [\belopolsky]
$${\rm v}_{{}_4} = 72.39\, \pm 0.01 \, .\eqn\numwork$$
Therefore, to this order the tachyon potential reads
$$V(\tau) = -\tau^2 + 1.602 \kappa \tau^3 - 3.016 \kappa^2 \tau^4 +\cdots\,\,,
\eqn\sofartt$$
and gives no local minimum for the tachyon.
The general form for ${\rm v}_{{}\{\cal N}}$ will be shown to be given by
$${\rm v}_{{}\{\cal N}} =(-)^N {2\over \pi^{N-3}}
\int_{{\cal V}_{0,N}} \prod_{I=1}^{N-3}
{{\rm d} x\{\cal I} d y\{\cal I}\over \rho\{\cal I}^2}\,\,
{1\over \rho\\{N-2}^2(0) \rho\\{N-1}^2(1) \rho\{\cal N}^2(\infty)} \, ,\eqn\coeffs$$
where the quantities $\rho\{\cal I}$, called mapping radii, will be discussed
in the next section. Since the integrand is manifestly positive, ${\rm v}\{\cal N}$
will be positive for even $N$ and negative for odd $N$.
Note that by a sign redefinition of the tachyon field
we can make all terms in the tachyon potential negative.
Therefore the tachyon potential is unbounded from below and
cannot have a global minimum. A local minimum may or
may not exist. Even these statements should be qualified if the
series defining the tachyon potential
has no suitable radius of convergence.
We will study the large-$N$ behavior of the coefficients ${\rm v}\{\cal N}$ in sect.5,
but we will not be able to reach a definite conclusion as far as the
radius of convergence goes.
Even if one could establish the existence of a local minimum for the
tachyon potential, the question remains whether it represents a vacuum
for the whole string field theory. One way to address this question
would be to compute the effective potential for the tachyon.
For a complete understanding of the string field
potential we should actually examine all zero-momentum Lorentz
scalar fields appearing
in the theory. This would include
physical scalars, unphysical scalars and trivial scalars.
Since even the number of physical scalars
at each mass level grows spectacularly fast [\goldstonethorn],
a more stringy way to discuss the
string field potential is clearly desirable.
\chapter{Geometrical Preliminaries}
In the present section we will begin by reviewing the definition of mapping
radius of a punctured disk. While this object requires a choice of
local coordinate at the puncture, it is possible to use it to construct
conformal invariants of spheres with punctured disks {\it without} having
to make choices of local coordinates at the punctures. We will discuss in
detail those invariants. We review the extremal properties of
the Strebel quadratic differentials and explain how to calculate mapping
radii from them. The invariants relevant to the computation of tachyon
amplitudes are shown to have extremal properties as well.
\section{Reduced Modulus and ${\rm PSL}(2,\IC)$ Invariants}
Given a punctured disk $D$, equipped with a chosen local coordinate $z$
vanishing at the puncture, one can define a conformal invariant
called the mapping radius $\rho_{{}_D}$ of the disk. It is calculated by
mapping conformally the disk $D$ to the unit disk
$|w|\leq 1$, with the puncture going to $w=0$. One then defines
$$ \rho_{{}_D} \equiv \biggl|{dz\over dw}\biggr|_{w=0}\,\,. \eqn\maprad$$
Alternatively one may map $D$ to a round disk $|\xi|\leq \rho\{\cal D}$, with
the puncture going to $\xi=0$, so that $|dz/d\xi|_0 = 1$.
The reduced modulus $M_D$ of the disk $D$ is defined to be
$$M_D \equiv {1\over 2\pi} \ln \rho_{{}_D}\,\,.\eqn\redmod$$
Clearly, both the mapping radius and the reduced modulus depend on the
chosen coordinate. If we change the local coordinate from $z$ to $z'$,
also vanishing at the puncture, we see using \maprad\ that the new
mapping radius ${\rho'}_{{}_D}$ is given by
$${\rho'}_{{}_D} = {\rho}_{{}_D}\, \biggl|{dz'\over dz}\biggr|_{z=0} \,,
\quad\quad\to \quad {\rho_{{}_D}\over |dz|} = \,{\rm invariant}.\eqn\mrtrans$$
Thus the mapping radius transforms like the inverse of a conformal metric $g$,
for which the length element $g |dz|$ is invariant. For the reduced modulus
we have
$$M'_D=M_D+{1\over 2\pi}\ln\biggl|{dz'\over dz}\biggr|_{z=0}\, .\eqn\mrmtr$$
\noindent
\underbar{${\rm PSL}(2,\IC)$ Invariants}
It should be noted that the above transformation property \mrtrans\ is not
in contradiction with the conformal invariance of the mapping radius. Conformal
invariance just states that if we map a disk, {\it and}
carry along the chosen local
coordinate at the puncture, the mapping radius does not change. This brings
us to a point that will be quite important. Throughout this paper
we will be dealing with punctured disks on the Riemann sphere. How will we
choose local coordinates at the punctures? It will be done as follows:
we will choose a global uniformizer $z$ on the sphere and keep it fixed.
If a punctured disk $D$ has its puncture at $z=z_0$, then the local
coordinate at the puncture will be taken to be $(z-z_0)$ (the case when
$z_0=\infty$ will be discussed later). Consider now an arbitrary
${\rm PSL}(2,\IC)$ map taking the sphere into itself
$$z \to f(z) = {az+b\over cz+d}\, ,\eqn\mapsltwoc$$
Under this map a disk $D$ centered at $z=z_0$ will be taken to a disk
$f(D)$ centered at $z= f(z_0)$. According to our conventions, the local
coordinate for $D$ is $z-z_0$ and the local coordinate for $f(D)$ is
$z-f(z_0)$. This latter coordinate is not the image of the original
local coordinate under the map. Therefore the mapping radius
{\it will transform},
and we can use \mrtrans\ to find
$$\rho_{{}_{f(D)}} = \rho_{{}_D} \biggl| {df\over dz}\biggr|_{z_0},\quad\quad
\to \quad \rho_{{}_D} = |cz_0 + d\,|^2 \, \rho_{{}_{f(D)}}\,.\eqn\mrsltwo$$
The off-shell amplitudes will involve the mapping radii of various disks.
Moreover, they must be ${\rm PSL}(2,\IC)$ invariant. How can that be achieved given
that we do not have a ${\rm PSL}(2,\IC)$ invariant definition of the mapping
radius? Let us first examine the case when we have two punctured disks
on a sphere. The data is simply a sphere with two marked points and two
closed Jordan curves each surrounding one of the points. We will associate
a ${\rm PSL}(2,\IC)$ invariant to this sphere. The invariant is calculated
using a uniformizer, but is independent of this choice.
Choose any uniformizer $z$ on the
sphere, and denote the disks by $D_1(z_1)$ and $D_2(z_2)$, where $z_1$ and
$z_2$ are the positions of the punctures. We now claim that
$${\chi_{}}_{ 12} \equiv {|z_1-z_2|^2\over
\rho_{{}_{D_1(z_1)}}\,\rho_{{}_{D_2(z_2)}}} \,, \eqn\twopmi$$
is a ${\rm PSL}(2,\IC)$ invariant (in other words, it is independent of the
uniformizer, or, it is a conformal invariant of the sphere with two punctured
disks). Indeed, under the ${\rm PSL}(2,\IC)$ transformation
given in \mapsltwoc\ we have that
$$ |z_1 - z_2 | = |cz_1 + d\,| \cdot\,| cz_2 + d\, | \cdot
\, | f(z_1) - f(z_2)|\,,\eqn\lftr$$
and it follows immediately from this equation and \mrsltwo\ that
$${|z_1-z_2|^2\over
\rho_{{}_{D_1(z_1)}}\,\rho_{{}_{D_2(z_2)}}}= {|f(z_1)-f(z_2)|^2\over
\rho_{{}_{f( D_1(z_1))}}\,\rho_{{}_{f(D_2(z_2))}}}\,,\eqn\verinv$$
which verifies the claim of invariance of the object $\chi_{12}$. It seems
plausible that any ${\rm PSL}(2,\IC)$ invariant built
from mapping radii of two disks must
be a function of $\chi_{12}$. It is not hard to construct in the same fashion
an ${\rm PSL}(2,\IC)$ invariant of three punctured disks. Indeed, we have
$$\chi_{123} \equiv {|z_1-z_2| \,|z_1-z_3| \, |z_2 -z_3|\over
\rho_{{}_{D_1(z_1)}}\,\rho_{{}_{D_2(z_2)}}\rho_{{}_{D_3(z_3)}} }\,,\eqn\threp$$
which is easily verified to be a conformal invariant. This invariant can
be written in terms of the invariant associated to two disks, one sees that
$$\chi_{123} = ( \,
\chi_{12}\,\chi_{13}\,\chi_{23}\,)^{1/2}\,. \eqn\notnew$$
This shows the invariant $\chi_{123}$ of three disks is not really new.
It is also clear that we can now construct many invariants of three disks.
We can form linear combinations of complicated functions build using
the invariants associated to all possible choices of two disks from the three
available ones. Nevertheless, the particular invariant $\chi_{123}$ given above
will be of relevance to us later on. Let us finally consider briefly the
case of four punctured disks, and concentrate on invariants having a
product of all mapping radii in the denominator. Let
$$\chi_{1234} \equiv {|z_1-z_2| \,|z_2-z_3| \, |z_3 -z_4|
|z_4 -z_1|\over
\rho_{{}_{D_1(z_1)}}\,\rho_{{}_{D_2(z_2)}}\rho_{{}_{D_3(z_3)}}
\rho_{{}_{D_4(z_4)}}}\,,\eqn\fhrep$$
and, as the reader will have noticed, the only requisite for invariance
is that, as it happens above,
every $z\{\cal I}$ appear twice in the numerator. This can be done in
many different ways; for example, we could have written
$$\chi'_{1234} \equiv {|z_1-z_2|^2 \, |z_3 -z_4|^2
\over\rho_{{}_{D_1(z_1)}}\,\rho_{{}_{D_2(z_2)}}\rho_{{}_{D_3(z_3)}}
\rho_{{}_{D_4(z_4)}}}\,,\eqn\fhrepe$$
and the ratio of the two invariants is
$$ {\chi'_4\over \chi_4} = |\lambda|\,,\quad
{\rm with} \quad \lambda = {(z_1-z_2) \, (z_3 -z_4)\over
(z_1 -z_4)(z_3-z_2) \, }= \{z_1,z_2;z_3,z_4\}\,, \eqn\crratio$$
which being independent of the mapping radii, and, by construction
a conformal invariant of a four-punctured sphere, necessarily has
to equal the cross-ratio of the four points (or a function of the
cross-ratio). The cross-ratio, as customary, will be denoted by $\lambda$.
It is the point where $z_1$ lands when $z_2,z_3$ and $z_4$ are mapped to
zero, one and infinity, respectively.
\noindent
\underbar{Letting one puncture go to infinity.}
It is sufficient to consider the behavior of the invariant $\chi_{12}$, given
by
$$\chi_{12} \equiv {|z_1-z_2|^2\over
\rho_{{}_{D_1(z_1)}}\,\rho_{{}_{D_2(z_2)}}}\, , \eqn\beginf$$
which we have seen is independent of the chosen uniformizer.
We must examine what happens as we change the uniformizer in such a way
that $z_2 \to \infty$. Given one uniformizer $z$ there is another one
$w=1/z$ that is well defined at $z=\infty$, the only point where
$z$ fails to define a local coordinate. This is why there is no
naive limit to $\chi_2$ as $z_2\to \infty$.
Using \mrtrans\ we express the mapping radius of the second disk in terms
of the mapping radius as viewed using the uniformizer induced by $w$. We have
$\rho_{{}_{D_2(z_2)}}= \Bigl| {dz\over dw}\Bigr|_{w_2}\rho_{{}_{D_2(w_2)}}=
|z_2|^2\rho_{{}_{D_2(w_2)}}$,
and substituting into the expression for $\chi_{12}$ we find
$$\chi_{12} = {|1-z_1/z_2|^2\over
\rho_{{}_{D_1(z_1)}}\,\rho_{{}_{D_2(w_2)}}}\, ,\eqn\tinve$$
and we can now take the limit as $z_2\to \infty$ without difficulty.
Writing, for convenience,
$\rho_{{}_{D_2(w_2=0)}}\equiv\rho_{{}_{D_2(\infty)}}$,
we get
$\chi_{12} = \rho_{{}_{D_1(z_1)}}^{-1}\,\rho_{{}_{D_2(\infty)}}^{-1}\,.$
The apparent dependence of $\chi_{12}$ on the choice of point
$z_1$ is fictitious. Any change
of uniformizer $z\to az+ b$ which changes $z_1$ leaving the point at infinity
fixed, will change the uniformizer at infinity, and the product
of mapping radii will remain
invariant. The point $z_1$ can therefore be chosen to be at the origin, and
we write our final expression for $\chi_2$
$$\chi_{12} = {1\over
\rho_{{}_{D_1(0)}}\,\rho_{{}_{D_2(\infty)}}}\,.\eqn\tiinve$$
Following exactly the same steps with $\chi_3$ and $\chi_4$ we obtain
$$\chi_{123} \equiv {|z_1-z_2| \over
\rho_{{}_{D_1(z_1)}}\,\rho_{{}_{D_2(z_2)}}\rho_{{}_{D_3(\infty)}} }\,,
\eqn\threpp$$
$$\chi_{1234} \equiv {|z_1-z_2| \,|z_2-z_3| \, \over
\rho_{{}_{D_1(z_1)}}\,\rho_{{}_{D_2(z_2)}}\rho_{{}_{D_3(z_3)}}
\rho_{{}_{D_4(\infty)}}}\,.\eqn\fhrepp$$
One could certainly take $z_1=0$ and $z_2=1$ for $\chi_{123}$, and,
$z_2=0$ and $z_3=1$ for $\chi_{1234}$. It should be remembered that whenever
a disk is centered at infinity, the local coordinate used is the inverse of
the chosen uniformizer on the rest of the sphere.
\section{Mapping Radii and Quadratic Differentials}
In this subsection we will review how one uses the Strebel
quadratic differential on a punctured sphere to define punctured disks.
These disks, called coordinate disks, define the local coordinates
used to insert the off-shell states. We will show how one can
use the quadratic differential to calculate the explicit form of the
local coordinates, and the mapping radii of the coordinate disks.
We will review the extremal properties of the Strebel
quadratic differentials and then discuss the extremal properties of
the ${\rm PSL}(2,\IC)$ invariants.
We will concentrate on the Strebel quadratic differentials relevant for
the restricted polyhedra of closed string field theory. The reader
unfamiliar with these objects may consult Refs.[\strebel,\saadizwiebach].
The Strebel quadratic differential for a sphere with $N$ punctures
in ${\cal V}_{0,N}$ induces a metric where the surface can be constructed
by gluing $N$ semiinfinite cylinders of circumference $2\pi$ across
their open boundaries. The gluing pattern is described by a restricted
polyhedron, which is a polyhedron having $N$ faces, each of perimeter
$2\pi$ and, in addition, having all nontrivial closed paths longer than
or equal to $2\pi$. Each semiinfinite cylinder defines a punctured
disk with a local coordinate $w$. The boundary $|w|=1$
corresponds to the edge of the cylinder, to be glued to the polyhedron,
and the puncture corresponds to $w=0$.
The Strebel quadratic differential on the sphere is usually expressed
as $\varphi = \phi(z) (dz)^2$, where $z$ is a uniformizer in the sphere.
At the punctures it has second order poles; if there is a puncture at
$z=z\{\cal I}$ the quadratic differential near $z\{\cal I}$ reads
$$\varphi = \Bigl( - {1\over (z-z\{\cal I})^2} + {b_{-1}\over z-z\{\cal I}}\, +\, b_0
\,+\, b_1(z-z\{\cal I}) \, + \cdots \Bigr)
(dz)^2 \,.\eqn\qdiff$$
Moreover, as mentioned above, the quadratic differential defines a disk
$D\{\cal I}$ on the sphere, with the puncture at $z\{\cal I}$. A local coordinate $w\{\cal I}$
on $D\{\cal I}$, such that $D\{\cal I}$ becomes a round disk can be found as follows.
We set
$$ z= \rho\{\cal I}\, w\{\cal I} + c_1w\{\cal I}^2 + c_2 w\{\cal I}^3 + \cdots \, , \eqn\ansatz$$
where $\rho\{\cal I}, c_1,c_2\cdots$ are constants to be determined. We have
written $\rho\{\cal I}$ for the coefficient of $w\{\cal I}$ on purpose. If we can
make the $D\{\cal I}$ disk correspond to the disk $|w\{\cal I}|\leq 1$, then $\rho\{\cal I}$
is by definition the mapping radius of the disk $D\{\cal I}$, since it is the
value of $|d(z-z\{\cal I})/dw\{\cal I}|$ at $w\{\cal I}=0$ (recall \maprad). We will actually
use the notation
$$ z = h\{\cal I} (w\{\cal I}), \quad\to \quad\rho\{\cal I} = |h\{\cal I}'(0)|\,.\eqn\stanform$$
Note that as explained in the previous subsection we are using the
local uniformizer on the sphere to define the mapping radius.
Back to our problem of defining the $w\{\cal I}$ coordinate, we demand that the
quadratic differential, expressed in $w\{\cal I}$ coordinates take the form
$$\varphi = - {1\over w\{\cal I}^2} (dw\{\cal I})^2 .\eqn\stanform$$
Since the above form is invariant under a change of scale, $w\{\cal I}\to aw\{\cal I}$,
we cannot determine by this procedure the constant $\rho\{\cal I}\,$. If $\rho\{\cal I}$
is fixed, the procedure will fix uniquely the higher coefficients
$c_1,c_2,\cdots$. While for general off-shell states the knowledge of the
coefficients $c\{\cal I}$ is necessary, for tachyons we only need the mapping
radius. This radius can be determined by the following method. Given
a quadratic differential one must find an arbitrary point $P$ lying
on the boundary of the punctured disk $D\{\cal I}$ defined by the quadratic
differential. Possibly, the simplest way to do this is to identify
the zeroes of the quadratic differential and then sketch the critical
trajectories to identify the various punctured disks and ring domains.
One can then pick $P$ to be a zero lying on the nearest
critical trajectory surrounding the puncture.
We now require $w\{\cal I} (P) =1$, and this will fix both the scale and the
phase of the local coordinate. This requirement is satisfied by taking
$$w\{\cal I} (z) = \exp \Bigl( i \int_{z(P)}^ z \sqrt{\phi(\xi)}\, d\xi \, \Bigr)\,,
\eqn\fixnmormm$$
where we take the positive branch for the square root. If the integral
can be done explicitly then the mapping radius is easily calculated
by taking a derivative $\rho\{\cal I} = |{dw\{\cal I}\over dz}|_{z\{\cal I}}^{-1}$. If the
integral cannot be done explicitly one can calculate the mapping radius
by a limiting procedure. One computes
$\rho\{\cal I} = \lim_{\epsilon\to 0} |{dw\{\cal I}\over dz}|_{z\{\cal I}+\epsilon}^{-1}$.
This leads, using \qdiff\ to the following result
$$\ln \rho\{\cal I} = \lim_{\epsilon\to 0} \, \biggl(
\Im{\rm m}\hskip-8pt\int_{z\{\cal I}+\epsilon}^{z(P)}
\hskip-8pt\sqrt{\phi(\xi)} d\xi\,\,+
\ln \epsilon \,\biggr)\, . \eqn\calcmr$$
The integration path is some curve in the disk $D\{\cal I}$, and using contour
deformation one can verify that the imaginary part of the integrand
does not depend on the choice of $P$ as long as $P$ is on the boundary
of $D\{\cal I}$. When using equation \calcmr\ one must choose a branch
for the square root, and keep the integration path away from the branch cut.
The sign is fixed by the condition that the limit exist.
Equation \calcmr\ and the recursive procedure
indicated above allow us,
in principle, to calculate the function $h\{\cal I}(w\{\cal I})$ if we know explicitly
the quadratic differential.
\noindent
\underbar{Extremal Properties.}
Imagine having an $N$ punctured Riemann sphere and label the punctures
as $P_1,P_2,\cdots P\{\cal N}$. Fix completely arbitrary local analytic
coordinates at these punctures. Now consider drawing closed Jordan curves
surrounding the punctures and defining punctured disks $D\{\cal I}$, in such
a way that the disks do not overlap (even though they might touch each other).
Given this data we can evaluate the functional
$${\cal F} = M_{D_1} + M_{D_2} + \cdots + M_{D\{\cal N}} \,,\eqn\summod$$
which is simply the sum of the reduced moduli of the various disks.
This functional, of course depends on the shape of the disks we have
chosen, and is well defined since we have picked some specific local
coordinates at the punctures. We may try now to vary the shape of the
disks in order to maximize ${\cal F}$. Suppose there is a choice of
disks that maximizes ${\cal F}$, then, it will maximize ${\cal F}$ whatever
choice of local coordinates we make at the punctures. This follows because
upon change of local coordinates the reduced modulus of a disk changes by a
constant which is independent of the disk itself (see \mrmtr). The interesting
fact is that the Strebel differential defines the disks that maximize
${\cal F}$ [\strebel].
Using the relation between reduced modulus and mapping radius
we see that the functional
$$(\rho_{{}_{D_1}}\cdots\rho_{{}_{D\{\cal N}}})^{-1}=\exp(-2\pi{\cal F})\,,\eqn\nfu$$
consisting of the inverse of the product of all the mapping radii, is
actually minimized by the choice of disks made by the Strebel quadratic
differential. This property will be of use to us shortly.
It is worth pausing here to note that the above definition of the functional
${\cal F}$ allows us to compare choices of disks given a {\it fixed} Riemann
sphere. Since we have chosen arbitrarily the local coordinates at the
punctures there is no reasonable way to compare the maximal values of
${\cal F}$ for two {\it different} spheres. It is therefore hard to think
of ${\rm Max}({\cal F})$ as a function on $\overline{\cal M}_{0,N}$.
This is reminiscent of the fact
that while for higher genus surfaces {\it without} punctures we can think of
the area of the minimal area metric as a function on moduli space,
it is not clear how to
do this for punctured surfaces. The difficulty again is due to the
regularization needed to render the area finite, this requires a choice
of local coordinates at the punctures, and there is no simple way to
compare the choices for different punctured surfaces. We now wish to
emphasize that our earlier discussion teaches us how
to define functions on $\overline{\cal M}_{0,N}$. These functions are
interesting because they are simple modifications of
$\exp[-2\pi \hbox{Max}({\cal F})]$ that
turn out to be functions on $\overline{\cal M}_{0,N}$.
Indeed, consider the invariant $\chi_{1234}$ that was defined as
$$\chi_{1234} \equiv {|z_1-z_2| \,|z_2-z_3| \, |z_3 -z_4|
|z_4 -z_1|\over
\rho_{{}_{D_1(z_1)}}\,\rho_{{}_{D_2(z_2)}}\rho_{{}_{D_3(z_3)}}
\rho_{{}_{D_4(z_4)}}}\,.\eqn\fhrepw$$
Recall that the mapping
radii entering in the definition of $\chi_{1234}$, as well as the
coordinate differences, are computed using the global uniformizer, and
the invariance of $\chi_{1234}$ just means independence of the result on
the choice of uniformizer. No choice is required to evaluate $\chi_{1234}$.
We obtain a function $f$ on $\overline {\cal M}_{0,4}$ by giving a number
for each four punctured sphere $R_4$ as follows. We equip the sphere $R_4$
with the
Strebel quadratic differential $\varphi_S (R_4)$ and we evaluate the invariant
$\chi_{1234}$ using the disks $D^I[\varphi_S (R_4)]$ determined by the
differential. In writing
$$f(R_4) \equiv \chi_{1234}\, \bigl(\, D^I\,[\varphi_S (R_4)]\,\bigr)\, .
\eqn\funcmod$$
We claim that $f(R_4)$ actually is the lowest value that the invariant
$\chi_{1234}$ can take for any choice of nonoverlapping disks in $R_4$
$$\chi_{1234}\, \bigl(\, D^I\,[\varphi_S (R_4)]\,\bigr) \leq
\chi_{1234}\, \bigl(\, D^I\,[ R_4]\,\bigr) \, .\eqn\claim$$
To see this, fix a uniformizer such that three of the punctures lie at
three points (say, $z=-1,0,1$) and the fourth puncture will lie at some
fixed point, which depends on the choice of four punctured sphere. This
fixes completely the numerator of $\chi$ and fixes the local coordinates
at the punctures, necessary to compute the mapping radii. Therefore
$$\chi_{1234} \propto
(\rho_{{}_{D_1(z_1)}}\,\rho_{{}_{D_2(z_2)}}\rho_{{}_{D_3(z_3)}}
\rho_{{}_{D_4(z_4)}} )^{-1}\, = \exp (-2\pi {\cal F})\,,\eqn\fhrepwx$$
where we recognize that, up to a fixed constant, the invariant is
simply related to the value of ${\cal F}$ evaluated with the chosen coordinates
at the punctures. As we now vary the disks around the punctures, ${\cal F}$ will
be maximized by the quadratic differential. This verifies that $\chi$
is minimized by the disks chosen by the quadratic differential (Eqn.\claim.)
We expect the function $f(R_4)$
to have a minimum for the most symmetric surface
in ${\cal M}_{0,4}$, namely, for the regular tetrahedron
[$\lambda = (1+i\sqrt{3})/2$]. We have not proven this, but
the intuition is that for the most symmetric surface we can get the
disks of largest mapping radii. There is, of course the issue of the
numerator of $\chi$ with the coordinate differences, which also varies
as we move in moduli space. Still, one can convince oneself that
the function $f(R_4)$ grows without bound as $R_4$ approaches degeneration.
\noindent
\underbar{Estimating Mapping Radii.} As we have mentioned earlier,
in defining the mapping radius of a punctured disk on the sphere
we use a local coordinate at the puncture which is obtained from
a chosen uniformizer on the sphere. While this mapping radius depends
on the uniformizer, we are typically interested in functions, such
as the $\chi$ functions, which are constructed out of mapping radii
and coordinate differences, and are independent of the chosen uniformizer.
Consider now the sphere as the complex plane $z$ together with the point
at infinity. The two following facts are useful tools to estimate the
mapping radius of a punctured disk centered at $z_0$.
\noindent
(i) If the disk $D$ is actually a round disk $|z-z_0| \leq R$, then
the mapping radius $\rho_{{}_D}$
is precisely given by the radius of the disk: $\rho_{{}_D} = R$. This
is clear since $w=(z-z_0)/R$ is the exact conformal map of $D$ to a unit disk.
\noindent
(ii) If the disk $D$ is not round but it is contained between two round
disks centered at $z_0$ with radii $R_1$ and $R_2$, with $R_1< R_2$, then
$R_1 < \rho_{{}_D} < R_2$. This property follows from the superadditivity
of the reduced modulus (see Ref.[\zwiebachma] Eqn. (2.2.25)).
Given an $N$ punctured
sphere, the Strebel quadratic differential will maximize the
product of the $N$ mapping radii. We can obtain easily a bound
$\rho_1\rho_2\cdots\rho\{\cal N} \geq R_1R_2\cdots R\{\cal N}$, where the $R\{\cal I}$ are
the radii of non-overlapping round disks centered at the punctures
with the sphere represented as the complex plane together with the point at
infinity.
\chapter{Off-Shell Amplitudes for Tachyons}
In this section we compute off-shell amplitudes for tachyons at
arbitrary momentum. We first discuss the case of three tachyons and
then the case of $N\geq 4$ tachyons which requires integration
over moduli space. We examine the results for the case of zero-momentum
tachyons obtaining in this way the coefficients
${\rm v}\{\cal N}$ of the tachyon potential. We explain why the choice of polyhedra
for the string
vertices, minimizes recursively the coefficients of the nonpolynomial
tachyon potential.
\section{Three Point Couplings}
We will now examine the cubic term in the string field potential.
Assume we are now given a three punctured sphere, and we want to
calculate the general off-shell amplitude for three tachyons. We
then must compute the correlator
$$A_{p_1p_2p_3}=
\Bigl\langle c\bar c e^{ip_1X} (w_1=0)\,\,c\bar c e^{ip_2X} (w_2=0)\,\,
c\bar c e^{ip_3X} (w_3=0) \Bigr\rangle\, .\eqn\correlcx$$
We must transform these operators from the local coordinates $w\{\cal I}$
to some uniformizer $z$. Let $w\{\cal I}=0$ correspond to $z=z\{\cal I}$. We then have
from the transformation law of a primary field
$$ c\bar c e^{ip\\IX} (w\{\cal I}=0) = c\bar c e^{ip\\IX} (z=z\{\cal I})\Bigl|
{dz\over dw\{\cal I}}
\Bigr|^{p\{\cal I}^2-2}_{w\{\cal I}=0}=c\bar ce^{ip\\IX} (z\{\cal I})\,\,
\rho_1^{p\{\cal I}^2-2}\,,\eqn\trtach$$
where in the last step we have recognized the appearance of the mapping
radius for the disk $D\{\cal I}$. The correlator then becomes
$$\eqalign{
A_{p_1p_2p_3}& =
\Bigl\langle c\bar c e^{ipX} (z_1)\,\,c\bar c e^{ipX} (z_2)\,\,
c\bar c e^{ipX} (z_3) \Bigr\rangle\,{1\over
\rho_1^{2-p_1^2} \rho_2^{2-p_2^2}\rho_3^{2-p_3^2} }\, ,\cr
&={ |z_1-z_2|^{2+2p_1p_2}|z_2-z_3|^{2+2p_2p_3}|z_1-z_3|^{2+2p_1p_3}\over
\rho_1^{2-p_1^2} \rho_2^{2-p_2^2}\rho_3^{2-p_3^2} \, }
\cdot [-2(2\pi)^D \delta^D (\sum p\{\cal I})\,]\,,\cr} \eqn\correlc$$
where we made use of \innprod, which introduces an extra factor of
$-2$ (shown in brackets) due to our convention
$c_0^\pm = (c_0 \pm \bar c_0)/2$.
In order to construct a manifestly ${\rm PSL}(2,\IC)$ invariant expression we
use momentum conservation in the denominator to write
$$\eqalign{
A_{p_1p_2p_3}&={ |z_1-z_2|^{2+2p_1p_2}\over (\rho_1\rho_2)^{1+p_1p_2}} \,
{|z_2-z_3|^{2+2p_2p_3}\over (\rho_2\rho_3)^{1+p_2p_3}}\,
{|z_1-z_3|^{2+2p_1p_3}\over
(\rho_1\rho_3)^{1+p_1p_3}}\cdot [-2(2\pi)^D \delta^D (\sum p\{\cal I})\,]\,,\cr
&= [-2(2\pi)^D \delta^D (\sum p\{\cal I})\,]\cdot
\prod_{I<J}^3 \,[\chi\\{IJ}]^{1+p\\Ip\\J}\, ,\cr}\eqn\minvf$$
which is the manifestly ${\rm PSL}(2,\IC)$ invariant description of the off-shell
amplitude.
We can now use the above result to extract the cubic coefficient of the
tachyon potential. By definition, the operator formalism bra
$\bra{V^{(3)}_{123}}$ representing
the three punctured sphere must satisfy
$$\bra{V^{(3)}_{123}} \left( c_1\bar c_1 \ket{{\bf 1},p_1}\right)^{(1)}
\left( c_1\bar c_1 \ket{{\bf 1},p_2}\right)^{(2)}
\left( c_1\bar c_1 \ket{{\bf 1},p_3}\right)^{(3)}\,
= A_{p_1p_2p_3} \,.\eqn\connect$$
Moreover, the multilinear function representing the cubic interaction
is given by
$$\eqalign{
\{ T\}_{{\cal V}_{0,3}} &\equiv \bra{V^{(3)}_{123}} T\rangle^{(1)} \ket{T}^{(2)}
\ket{T}^{(3)} \,,\cr
&= \int {{\rm d} p_1\over (2\pi)^D}{{\rm d} p_2\over (2\pi)^D}{{\rm d} p_3\over (2\pi)^D}
\,A_{p_1p_2p_3}\, \tau(p_1) \tau(p_2)\tau(p_3)\, ,\cr
&=\int \prod_{I=1}^3{{\rm d} p\{\cal I}\over (2\pi)^D}\,
(2\pi)^D \delta^D (\sum p\{\cal I})\,
\cdot (-2)\prod_{I<J}^3 \,[\chi\\{IJ}]^{1+p\\Ip\\J}\,\cdot
\tau(p_1) \tau(p_2)\tau(p_3) \,,\cr} \eqn\seehow$$
where use was made of the definition of the string tachyon field in
\tachfield, of \connect, and of \minvf.
Comparison with \fftor, and use of \littlev\ now gives
$${\rm v}\\3 = -2\cdot \prod_{I<J}^3 \,[\chi\\{IJ}] = -2\cdot
{ |z_1-z_2|^2|z_2-z_3|^2|z_1-z_3|^2\over
\rho_1^2 \rho_2^2 \rho_3^2}
= -2\cdot \chi_{123}^2\, , \eqn\tprjk$$
in terms of the ${\rm PSL}(2,\IC)$ invariant $\chi_{123}$. It follows from the extremal
properties discussed earlier that the minimum value possible for ${\rm v}\\3$
is achieved for the Strebel quadratic differential defining the Witten
vertex. We will calculate the minimum possible value for ${\rm v}\\3$ in
sect.5.1.
\section{The Off-Shell Koba-Nielsen Formula}
We now derive a formula for the off-shell scattering amplitude
for $N$ closed string tachyons at arbitrary momentum. The final
result will be a manifestly ${\rm PSL}(2,\IC)$ invariant expression.
The computation is simplified because the tachyon vertex operator
is primary even off-shell, and because its ghost structure is
essentially trivial.
We work in the $z$-plane and fix the position of the three last
insertions at $z\\{N-2}$, $z\\{N-1}$ and $z\{\cal N}$. The positions of the
first $N-3$ punctures will be denoted as $z_1,z_2,\cdots z\\{N-3}$.
We must integrate over the positions of these $N-3$ punctures.
Each will therefore
give a factor
$${\rm d} x\{\cal I}\wedge {\rm d} y\{\cal I}\,\, b({\partial\over \partial x})
\,b({\partial\over \partial y}) = {\rm d} x\{\cal I}\wedge {\rm d} y\{\cal I}\,
\,\, 2i\,\bar b_{-1} b_{-1} = - {\rm d} z\{\cal I}\wedge {\rm d}\bar z\{\cal I}\, \bar b_{-1} b_{-1}
\,, \eqn\fmea$$
where $z\{\cal I} = x\{\cal I} + iy\{\cal I}$.
There is a subtlety here, each of the antighost oscillators refers to
the $z$ plane, while the ghost oscillators in each tachyon insertion
$c_1^{w\{\cal I}}\bar c_1^{w\{\cal I}}\ket{0,p}^{w\{\cal I}}$ refer to the local coordinate
$w\{\cal I}$, where
$z= h\{\cal I}(w\{\cal I})$. Transforming the antighost oscillators we
obtain $b_{-1} = [h\{\cal I}'(0)]^{-1} b^{w\{\cal I}}_{-1}+\cdots$, where the dots indicate
antighost oscillators $b_{n\geq 0}^{w\{\cal I}}$ that annihilate the tachyon state.
For the antiholomorphic oscillator we have
$\bar b_{-1} = \left[\overline{h\{\cal I}'(0)}\right]^{~-1}\bar
b^{w\{\cal I}}_{-1}+\cdots$.
Therefore
each of the integrals will be represented by
$$2i\,{\rm d} x\{\cal I}\wedge {\rm d} y\{\cal I}\,\,{1\over \rho\{\cal I}^2}\,\ket{0,p}^{w\{\cal I}}=
2i\,{\rm d} x\{\cal I}\wedge {\rm d} y\{\cal I}\,{1\over \rho\{\cal I}^{2-p\{\cal I}^2}}\,\ket{0,p}\,,\eqn\mfrn$$
where $\rho\{\cal I}=|h\{\cal I}'(0)|$ is the mapping radius of the $I$-th disk.
The Koba-Nielsen amplitude will therefore be given by
$$\eqalign{
A_{p_1\cdots p_N} & = \left({i\over 2\pi}\right)^{N-3} (2i)^{N-3}
\int \prod_{I=1}^{N-3}
{{\rm d} x\{\cal I}{\rm d} y\{\cal I}\,\over \rho\{\cal I}^{2-p\{\cal I}^2} }
{1\over \rho\\{N-2}^{2-p\\{N-2}^2}
\rho\\{N-1}^{2-p\\{N-1}^2}\rho\\{N-2}^{2-p\\{N-2}^2}}\cr
&\quad \cdot \Bigl\langle
e^{ip_1X(z_1)}\cdots
e^{ip\\{N-3}X(z\\{N-3})}\,\,c\bar ce^{ip\\{N-2}X(z\\{N-2})}
\,c\bar ce^{ip\\{N-1}X(z\\{N-1})}\,
c\bar ce^{ip\\{N}X(z\\{N})}\Bigr\rangle \, ,\cr}\eqn\knoff$$
where the correlator is a free-field correlator in the complex plane.
We will not include in the amplitude the coupling constant factor
$\kappa^{N-3}$.
The extra factor $(i/2\pi)^{N-3}$ included in the formula above
is well-known to be necessary for
consistent factorization, and has been derived in closed string field theory.
\foot{The value used here appears in Ref.[\senzwiebachtwo], where
a sign mistake of Ref. [\zwiebachlong] was corrected.}
We then have
$$\eqalign{
A_{p_1\cdots p_N} &= (-)^N \cdot
{2\over \pi^{N-3}}
\int \prod_{I=1}^{N-3}
{{\rm d} x\{\cal I}{\rm d} y\{\cal I}\over \rho\{\cal I}^{2-p\{\cal I}^2} }\,\,
{|(z\\{N-2}-z\\{N-1})\,(z\\{N-2}-z\{\cal N})\,(z\\{N-1}-z\{\cal N})|^2
\over \rho\\{N-2}^{2-p\\{N-2}^2}
\rho\\{N-1}^{2-p\\{N-1}^2}\rho\\{N}^{2-p\\{N}^2}}\cr
&\quad \cdot \prod_{I<J}^N |z\{\cal I}-z\\J|^{2p\\Ip\\J}
\,\cdot \left[ (2\pi)^D \delta \left( \sum p\{\cal I} \right) \right] \,.\cr}
\eqn\knofff$$
Using momentum conservation, and the definition of the invariants
$\chi\\{IJ}$ and $\chi\\{IJK}$, we can write the above as
$$A_{p_1\cdots p_N} =(-)^N
{2\over \pi^{N-3}}\int \prod_{I=1}^{N-3}
{{\rm d} x\{\cal I} {\rm d} y\{\cal I}\over \rho\{\cal I}^2}\,\,\chi\\{N-2,N-1,N}^2
\, \cdot \prod_{I<J}^N \chi\\{IJ}^{p\\Ip\\J}\,
\,\cdot \left[ (2\pi)^D \delta \left( \sum p\{\cal I} \right)
\right] , \eqn\knofffx$$
It follows immediately from the transformation law for the mapping radius
that the measure $dz\{\cal I}\wedge d\bar z\{\cal I} / \rho\{\cal I}^2$ is ${\rm PSL}(2,\IC)$ invariant.
Therefore the above result is a manifestly ${\rm PSL}(2,\IC)$ invariant off-shell
generalization of the Koba-Nielsen formula.
For the case of four tachyons it reduces to an off-shell version of the
Virasoro-Shapiro amplitude
$$A_{p_1\cdots p_4} = {2 \over \pi}\int
{{\rm d} x_1 {\rm d} y_1\over \rho_1^2}\,\,
\chi_{234}^2
\, \cdot \prod_{I<J}^4 \chi\\{IJ}^{p\\Ip\\J}
\,\cdot \left[ (2\pi)^D \delta \left( \sum p\{\cal I}
\right) \right] \,.\eqn\knofffx$$
If we choose to place the second, third, and fourth punctures at zero, one
and infinity respectively, we end with
$$A_{p_1\cdots p_4} = {2 \over \pi}\int
{\rm d} x\, {\rm d} y \,{ |z|^{2p_1p_2} |z-1|^{2p_1p_3} \over \rho_1^{2-p_1^2}
\rho_2^{2-p_2^2}\rho_3^{2-p_3^2}\rho_4^{2-p_4^2}}
\,\cdot \left[ (2\pi)^D \delta \left( \sum p\{\cal I} \right) \right] \,.\eqn\fxyu$$
Another expression can be found where the variables of integration
are cross-ratios. We define
the cross ratio
$$\lambda\{\cal I} \equiv \{z\{\cal I}, z\\{N-2};z\\{N-1},z\{\cal N}\} = {(
z\{\cal I}-z\\{N-2})\, (z\\{N-1}-z\{\cal N})\over
\,(z\{\cal I}-z\{\cal N})\,\, (z\\{N-1}-z\\{N-2}) }\,,\eqn\cratio$$
and it follows that
$${\rm d} z\{\cal I}\wedge {\rm d}\bar z\{\cal I} = {{\rm d}\lambda\{\cal I}\wedge {\rm d}\bar \lambda\{\cal I}
\over |\lambda\{\cal I}|^2}\,
{ |z\{\cal I}-z\{\cal N}|^2 |z\{\cal I}-z\\{N-2}|^2\over |z\{\cal N}-z\\{N-2}|^2 }\,, \eqn\sdd$$
leading to
$$\eqalign{
A_{p_1\cdots p_N} &= 2(-)^{N}\left( {i\over 2\pi} \right)^{N-3}
\int \prod_{I=1}^{N-3}
\Bigl[ {{\rm d}\lambda\{\cal I}\wedge {\rm d}\bar \lambda\{\cal I}\over |\lambda\{\cal I}|^2}
\Bigl( {\chi\\{I,N-2,N-1,N}
\over \chi\\{N-2,N-1,N} }\Bigr)^2 \Bigr]
\, \cdot \chi^2\\{N-2,N-1,N}\cr
&\quad\cdot \prod_{I<J}^N \chi\\{IJ}^{p\\Ip\\J}
\,\cdot \left[ (2\pi)^D \delta \left( \sum p\{\cal I} \right) \right] \,. \cr}
\eqn\kknofff$$
For the case of four tachyons the above result reduces to
$$A_{p_1\cdots p_4} = {i \over \pi}\int
{{\rm d}\lambda \wedge{\rm d}\bar \lambda\over |\lambda|^2}\, \chi_{1234}^2
\,\cdot \prod_{I<J}^4 \chi\\{IJ}^{p\\Ip\\J}
\,\cdot \left[ (2\pi)^D \delta \left( \sum p\{\cal I} \right) \right] \,.
\eqn\kknofff$$
In the above expressions the $\lambda$ integrals extend over the whole sphere.
Having obtained manifestly ${\rm PSL}(2,\IC)$ invariant expressions valid for arbitrary
momenta, we now go back to our particular case of interest, which is the
case when all the momenta are zero. It is simplest to go back to \knoff\
to obtain
$$A_{1\cdots N} = (-)^N {2\over \pi^{N-3}}
\int \prod_{I=1}^{N-3}
{{\rm d} x\{\cal I} {\rm d} y\{\cal I}\over \rho\{\cal I}^2}\,\,
{|z\\{N-2}-z\\{N-1}|^2|z\\{N-2}-z\{\cal N}|^2 |z\\{N-1}-z\{\cal N}|^2
\over \rho\\{N-2}^2 \rho\\{N-1}^2 \rho\{\cal N}^2}
[(2\pi)^D \delta ({\bf 0})]\,,\eqn\wknoofffx$$
$$A_{1\cdots 4} = {2\over \pi}\int
{{\rm d} x_1\, {\rm d} y_1\over \rho_1^2}\,\,
{|z_{2}-z_{3}|^2|z_{2}-z_4|^2 |z_{3}-z_4|^2
\over \rho_{2}^2 \rho_{3}^2 \rho_4^2}
[(2\pi)^D \delta ({\bf 0})]\,,\eqn\wknoqfffx$$
These are the expressions we shall be trying to estimate. If we set the
three special points appearing in the above expressions to zero, one and
infinity, we find (see \s3.3)
$$A_{1\cdots N} = (-)^N {2\over \pi^{N-3}}
\int \prod_{I=1}^{N-3}
{{\rm d} x\{\cal I} d y\{\cal I}\over \rho\{\cal I}^2}\,\,
{1\over \rho\\{N-2}^2(0) \rho\\{N-1}^2(1) \rho\{\cal N}^2(\infty)}
\,[(2\pi)^D \delta ({\bf 0})]\,,\eqn\wfx$$
$$A_{1\cdots 4} = {2 \over \pi}\int
{{\rm d} x_1\, {\rm d} y_1\over \rho_1^2}\,\,
{1\over \rho_{2}^2(0) \rho_{3}^2(1) \rho_4^2(\infty)}
\, [(2\pi)^D \delta ({\bf 0})]\, .\eqn\wkx$$
Let us now use the above results to extract the quartic and higher order
coefficient of the tachyon potential.
By definition, the operator formalism bra representing
the collection of $N$-punctured spheres must satisfy
$$\int_{{\cal V}_{0,N}} \bra{\Omega_N^{(0)0,N}}
\left( c_1\bar c_1 \ket{{\bf 1},p_1}\right)^{(1)}
\cdots \left( c_1\bar c_1 \ket{{\bf 1},p_N}\right)^{(N)}\,
= A_{p_1\cdots p_N}({\cal V}_{0,N}) \,,\eqn\conct$$
where the ${\cal V}_{0,N}$ argument of $A_{p_1\cdots p_N}({\cal V}_{0,N})$ indicates
that the off-shell amplitude has only been partially integrated over the
subspace ${\cal V}_{0,N}$.
The corresponding multilinear function
is given by
$$\{ T\}_{{\cal V}_{0,N}} \equiv \int_{{\cal V}_{0,N}} \bra{\Omega_N^{(0)0,N}}
T\rangle^{(1)}\cdots
\ket{T}^{(N)} \,
= \int \prod_{I=1}^N{{\rm d} p\{\cal I}\over (2\pi)^D}\,
\,A_{p_1\cdots p_N}({\cal V}_{0,N})\, \tau(p_1) \cdots\tau(p_N)\,, \eqn\seehow$$
where we made use of the definition of the string tachyon field in
\tachfield, and of \conct.
Reading the value of the amplitude at zero momentum, and by virtue
of \fftor\ and \littlev\ we get
$$\eqalign{
{\rm v}\{\cal N} &= (-)^N{2\over \pi^{N-3}}
\int_{{\cal V}_{0,N}} \prod_{I=1}^{N-3}
{{\rm d} x\{\cal I} {\rm d} y\{\cal I}\over \rho\{\cal I}^2}\,\,
{|z\\{N-2}-z\\{N-1}|^2|z\\{N-2}-z\{\cal N}|^2 |z\\{N-1}-z\{\cal N}|^2
\over \rho\\{N-2}^2 \rho\\{N-1}^2 \rho\{\cal N}^2 }\, , \cr
& = (-)^N {2\over \pi^{N-3}}
\int_{{\cal V}_{0,N}} \prod_{I=1}^{N-3}
{{\rm d} x\{\cal I} d y\{\cal I}\over \rho\{\cal I}^2}\,\,
{1\over \rho\\{N-2}^2(0) \rho\\{N-1}^2(1) \rho\{\cal N}^2(\infty)}\,.\cr}\eqn\trjk$$
Note here the pattern of signs. All ${\rm v}\{\cal N}$ for $N$ even come with positive
sign, and all ${\rm v}\{\cal N}$ for $N$ odd come with a negative sign. Including
the overall minus sign in passing from the action to the potential
(Eqn.\fftorr), and the
sign redefinition $\tau \to -\tau$, all coefficients
of the tachyon potential become negative.
It is worthwhile to pause and reflect about the above pattern of signs.
In particular, since ${\rm v}\\4$ turned out to be positive, the quartic
term in the tachyon potential is negative, as the quadratic term is.
While the calculations leading to the sign factors are quite subtle,
we believe that the result should have been expected. In closed
string field theory, the elementary four point interaction
changes if we include stubs in the three string vertex.
Both the original interaction and the one for the case of stubs
must have the same sign, because they only differ by the region of
integration over moduli space, and the integrand, as we
have seen, has a definite sign.
On the other hand, the interaction arising from the stubbed theory
would equal the original interaction plus a collection of Feynman graphs
with two three-string vertices and with one propagator whose proper
time is only partially integrated. Such terms, for completely integrated
propagators and massive intermediate fields would give a contribution
leading to a potential unbounded below. For partially integrated
propagators they also contribute such kind of terms, both if the
field is truly massive, or if it is tachyonic. This indicates that
one should have expected an unbounded below elementary interaction.
It is now simple to explain why the choice of restricted
polyhedra (polyhedra with all nontrivial closed paths longer
than or equal to $2\pi$ [\kugokunitomo]) for closed
string vertices minimizes recursively the expansion coefficients of the
tachyon potential. We have seen that ${\rm v}\\3$ is minimized by
the Witten vertex. At the four point level we then have a missing region
${\cal V}_{0,4}$. In the parametrization given by the final form in \trjk\ the
region of integration corresponding to ${\cal V}_{0,4}$ is fixed. At each
point in this region, the integrand, up to a constant, is given by
$1/\prod_i \rho_i^2 = \exp (-4\pi {\cal F})$, and as explained around Eqn.\nfu,
this quantity is minimized by the choice of coordinate disks determined
by the Strebel differential. Since the integrand is positive definite
throughout the region of integration, and, at every point is minimized
by the use of the Strebel differential, it follows that the integral is
minimized by the choice of the Strebel differential for the string vertex.
That is precisely the choice that defines the restricted polyhedron
corresponding to the standard four closed string vertex. It is clear that
the above considerations hold for any ${\cal V}_{0,N}$. The minimum value for
${\rm v}\{\cal N}$ is obtained by using polyhedra throughout the region of integration.
Therefore, starting with the three string vertex, we are led recursively
by the minimization procedure to the restricted polyhedra of closed
string field theory.
\chapter{Estimates for the Tachyon Potential}
The present section is devoted to estimates of the tachyon potential.
While more analytic work on the evaluation of the tachyon potential
may be desirable, here we will get some intuitive feeling for the
growth of the coefficients ${\rm v}\{\cal N}$ for large $N$. The aim is to
find if the tachyon potential has a radius of convergence. We will
not be able to decide on this point, but we will obtain a series of
results that go in this direction.
For every number of punctures $N$, there is a
configuration of these punctures on the sphere for which
we can evaluate exactly
the measure of integration. This is the configuration where the punctures
are ``equally separated" in a planar arrangement. These configurations
appear as a finite number of points in the boundary of ${\cal V}_{0,N}$, and
in some sense are the most problematic. The large $N$ behavior of the
measure at those points is such that if the whole integrand were to be
dominated by these points the tachyon potential would seem to have no
radius of convergence. The shape of ${\cal V}_{0,N}$ around those points, however, is
such that the contributions might be suppressed.
In each ${\cal V}_{0,N}$ there are configurations where the punctures are
distributed most symmetrically. It is intuitively clear that at these
configurations the measure is in some sense lowest. It is possible to
estimate this measure for large $N$, and conclude that, if dominated
by this contribution, the tachyon potential should have some radius
of convergence.
The behavior of the measure for the tachyon potential is such that
the measure grows as we approach degeneration, and if ${\cal V}_{0,4}$, for example,
was to extend over all of $\overline{\cal M}_{0,4}$ the naive integral would be
infinite. This infinity is not physical, because we do not expect
infinite amplitude for the scattering of four zero-momentum tachyons.
We explain how analytic continuation of the contribution from the
Feynman graphs removes this apparent contradiction.
We begin by presenting several exact results pertaining two, three, and
four-punctured spheres.
\section{Evaluation of Invariants}
Consider first the invariant $\chi_{12}$ of a sphere with two punctured
disks (Eqn.\twopmi). The disks may touch but they are assumed not to overlap.
Since the mapping radii can be as small as desired, the invariant
$\chi_{12}$ is not bounded above. It is actually bounded below, by
the value attained when we have a Strebel quadratic differential.
We can take the sphere punctured at zero and infinity, and the quadratic
differential to be $\varphi= -(dz)^2/z^2$. While this differential
does not determine a critical trajectory, we can take it to be
any closed horizontal trajectory, say $|z|=1$. This divides the $z$-sphere
into two disks, one punctured at $z=0$ with unit disk $|z|\leq 1$ and
the other punctured at $z= \infty$, or $w=0$, with $w=1/z$, and with
unit disk $|w| \leq 1$. It follows that their mapping radii are both
equal to one. We therefore have, using \tiinve\
$$\widehat\chi_{12}\equiv {\rm Min}\, (\chi_{12}) = {\rm Min} \Bigl( {1\over
\rho_{{}_{D_1(0)}}\,\rho_{{}_{D_2(\infty)}}}\Bigr) = 1\,.\eqn\tiinve$$
We now consider three punctured spheres and try to evaluate the minimum
value of the invariant $\chi_{123}$ defined in Eqn.\threp.
This minimum is achieved for the
three punctured sphere corresponding to the Witten vertex. In this vertex
we can represent the sphere by the $z$ plane, with the punctures at
$e^{\pm i\pi/3}, -1$. The disk surrounding the puncture at $z=e^{i \pi /3}$
is the wedge domain $0\leq {\rm Arg}(z) \leq 2\pi/3$. This domain is mapped to
a unit disk $|w|\leq 1$ by the transformations
$$t = z^{3/2} \,,\quad w = {t-i\over t+1}\, .\eqn\mapsnes$$
One readily finds that the mapping radius $\rho$ of the disk is given
by $\rho = |{dz\over dw}|_{w=0} = 4/3$. Furthermore, the distance
$|z\{\cal I}-z\\J|$ between any of the punctures is equal to $\sqrt{3}$. Therefore
back in \threp\ we obtain
$$\widehat\chi_{123} \equiv
{\rm Min}\, (\chi_{123})= \Bigl( {3\sqrt{3}\over 4} \Bigr)^3\, . $$
This result implies that the minimum possible value for
$|{\rm v}\\3|$ is realized with ${\rm v}\\3 = -3^9/2^{11}$ (see Eqn.\tprjk).
Another computation that is of interest is that of the most symmetric
four punctured sphere, a sphere where the punctures are at $z=0,1,\infty$,
and at $z =\rho= e^{i\pi/3}$. The Strebel quadratic differential for this
sphere can be found to be
$$\varphi = { (2z+ 1)(2z- \rho) (2z-\bar\rho )
\over (z-1)^2 (z-\rho^2)^2(z-\bar\rho^2)^2 }\cdot {(dz)^2
\over z^2 }\, .\eqn\qdmss$$
Here the poles are located at the points $z=0, \,1,\, \rho^2$, and
$\bar\rho^2$.
The zeroes
are located at $z= -\half,\,\half \rho, \,\half\bar\rho$, and $\infty$.
One can use this expression
for a calculation of the mapping radii.
\section{The Measure at the Planar Configuration}
In each ${\cal V}_{0,N}$, for $N\geq 4$ there is a set of symmetric
planar configurations
for the punctures. They correspond to the surfaces obtained by Feynman
diagrams constructed using only the three string vertex, and with all
the propagators collapsed with zero twist angle. We will consider the
case of $N$ punctures and give an exact evaluation for the measure.
This will be done in the frame where three punctures are mapped to the
standard points $z=0,1,\infty$, and the rest of the punctures will
be mapped to the points $z_1,z_2,\cdots z\\{N-3}$ lying on the real line
in between $z=0$ and $z=1$. Consequently, we will take $z\\{N-2}=1,
z\\{N-1}=\infty$ and $z\{\cal N}=0$. The measure we will calculate will be defined as
$$d\mu\{\cal N} \equiv \prod_{I=1}^{N-3}
{{\rm d} x\{\cal I} {\rm d} y\{\cal I}\over \rho\{\cal I}^2}\,\,
{|z\\{N-2}-z\\{N-1}|^2|z\\{N-2}-z\{\cal N}|^2 |z\\{N-1}-z\{\cal N}|^2
\over \rho\\{N-2}^2 \rho\\{N-1}^2 \rho\{\cal N}^2}\, ,\eqn\demeaw$$
which, up to constants, is the measure that appears in \wknoofffx.
The result will be of the form
$d\mu\{\cal N} = f\{\cal N} \prod_{k=1}^{N-3} {\rm d} x_k {\rm d} y_k$,
where $f\{\cal N}$ is a number depending on the number of punctures.
We begin the computation by using a $\xi$ plane where we place all the $N$
punctures equally spaced on the unit circle $|\xi|=1$. We thus let $\xi_k$
be the position of the $k$-th puncture, with
$$\xi_k = \exp \bigl( (2k-1) i\pi/N)\,,\quad k=1,2,\cdots N\,.\eqn\listp$$
\if y\figcount
\midinsert
\centerline{\epsffile{xiplane.eps}}
\nobreak
\narrower
\singlespace
\noindent
Figure 1. Planar configuration of punctures on the sphere. Shown are
the maps from the ring domain associated with a specific puncture to
the unit disk.
\medskip
\endinsert
\else\fi
In this presentation the ring domain surrounding a
puncture, say the first one, is the wedge domain $0\leq {\rm Arg}
(\xi) \leq 2\pi/N$ (see Figure 1) The mapping radius can be computed
exactly by mapping the wedge to the unit disk $|w| \leq 1$, via $t=
\xi^{N/2}$ and $w = {t-i\over t+1}$. The result is $\rho= 4/N$, and
picking the three special punctures to be $\xi\\{N-2},\xi\\{N-1}$ and
$\xi\{\cal N}$, we find
$${\rm d}\mu\{\cal N} = 64 \cdot \sin^4 \Bigl({\pi\over N}\Bigr)\cdot
\sin^2 \Bigl({4\pi\over N}\Bigr)\cdot \Bigl({N\over 4}\Bigr)^{2N}\cdot
\prod_{k=1}^{N-3} {\rm d}^2\xi_k \,, \eqn\measup$$
where ${\rm d}^2\xi_k = {\rm d}\hbox{Re}\, \xi_k \,{\rm d}\hbox{Im} \,\xi_k $.
This is the measure, but in the $\xi$ plane. In order to transform it to
the $z$-plane we need the ${\rm PSL}(2,\IC)$ transformation that will satisfy
$z(\xi\{\cal N}) =0, z(\xi\\{N-1}) = 1$, and $z(\xi\\{N-2}) = \infty$. The desired
transformation is
$$ z = {\xi - e^{-i\pi/N} \over \xi - e^{-3i\pi/N}}\cdot \beta\,,\quad
{\rm with} \quad
|\beta| = {\sin (\pi/N)\over \sin (2\pi/N) }\,, \eqn\heudk$$
and a small calculation gives
$${\rm d}^2\xi_k = \Bigl|{dz\over d\xi}\Bigr|^{-2}_{\xi_k}
\,{\rm d} x_k {\rm d} y_k= 4\cdot { \sin^2(2\pi/N)\over \sin^4 (\pi/N)}\cdot
\sin^4 \Bigl( {(k+1)\pi\over N}\Bigr)\cdot {\rm d} x_k {\rm d} y_k\,. \eqn\amst$$
This expression, used in \measup\ gives us the desired expression for
the measure
$$\eqalign{
{\rm d}\mu\{\cal N} &= \,\,4^{2N-3}\cdot \sin^4 \Bigl({\pi\over N}\Bigr)\cdot
\sin^2 \Bigl({4\pi\over N}\Bigr)\cdot \Bigl({N\over 4}\Bigr)^{2N}\cdot
\Bigl({ \cos(\pi/N)\over \sin (\pi/N)}\Bigr)^{2N-6}\,\cr
&\quad\cdot\Bigl[\,\, \prod_{k=1}^{N-3}\sin^4
\Bigl( {(k+1)\pi\over N}\Bigr) \,\Bigr]
\cdot\prod_{k=1}^{N-3} {\rm d} x_k \,{\rm d} y_k \,.\cr}\eqn\finmeas$$
This is an exact result, valid for all $N\geq 4$. For the case of
$N=4$ it gives ${\rm d}\mu\\4 = 256 {\rm d} x {\rm d} y$.\foot{For $N=4$ the measure
can also be calculated exactly for the configuration with cross ratio
equal to $(1+i\sqrt{3})/2$. One finds [\belopolsky] ${\rm d}\mu\\4 = {2^{11}\over
3^4\root 3 \of 2 }\,{\rm d} x\,{\rm d} y \approx 20.07 \,{\rm d} x\,{\rm d} y$. This correponds
to the measure at the ``center'' of ${\cal V}_{0,4}$ and is indeed much smaller
than the measure $256\,{\rm d} x\,{\rm d} y$ at the corners of ${\cal V}_{0,4}$.}
Let us now consider
the leading behavior of this measure as $N\to \infty$.
The only term that requires some calculation is the product
$a\{\cal N} \equiv\prod_{k=1}^{N-3}\sin^4
( {(k+1)\pi/ N})$. One readily finds that as $N\to \infty$
$$\ln a\{\cal N} \,\sim\, 4\cdot {N\over \pi}
\int_0^\pi d\theta \ln (\sin \theta) = -4N \ln (2) \,\quad
\Rightarrow \quad a\{\cal N} \sim 4^{-2N} ,\eqn\estimp$$
and using this result, we find the large $N$ behaviour of the planar
measure
$${\rm d}\mu\{\cal N} \,\sim\, 4^5\cdot\pi^6\cdot
\Bigl[ {N^2\over 4\pi} \Bigr]^{2N-6}
\cdot\prod_{k=1}^{N-3} {\rm d} x_k {\rm d} y_k \,. \eqn\lnb$$
This was the result we were after. We see that this measure grows
like $N^{4N}$. This growth is so fast that presents an obstruction
to a simple proof of convergence for the series defining the tachyon
potential. Indeed, a very naive estimation would not yield convergence.
Let us see this next.
Let us assume that this planar uniform configuration is
indeed the point in ${\cal V}_{0,N}$ for which the
measure is the largest. This statement requires explanation, since
the numerical coefficient appearing in front of a measure can be
changed by ${\rm PSL}(2,\IC)$ transformations. Thus given any other configuration
in ${\cal V}_{0,N}$
with a puncture at $0,1$ and $\infty$ we do a transformation $z\to az$ with
$a= 1/z_{\rm max}$, where $z_{\rm max}$ is the position of the puncture
farther away from the origin. In this way we obtain a configuration with
all the punctures in the unit disk, the same two punctures at zero and
infinity, and some puncture at one. At this point the measures can
be compared and we expect the planar one to be larger. It is now clear
from the construction that the full ${\cal M}_{0,N}$ is overcounted if we fix
two punctures, one at zero, the other at infinity, and among the rest we
pick one at a time to be put at one, while the others are integrated all
over the inside of the unit disk. If we estimate this integral
using our value for the measure in the worst configuration we get
that each coefficient ${\rm v}\{\cal N} < [(N-2)\pi^{N-3}] N^{4N-12}$, where
the prefactor in brackets arises from the above described integrals
(this prefactor
does not really affect the issue of convergence).
The growth $N^{4N}$ rules out
the possibility of convergence. This
bound is quite naive, but raises the possibility that there
may be no radius of convergence for the tachyon potential.
\section{The Most Uniform Distribution of Punctures}
The corners of ${\cal V}_{0,N}$ turned out to be problematic. Since we expect
the measure for the coefficients of the tachyon potential to be lowest
at the most symmetric surfaces, we now estimate the measure at this point
in ${\cal V}_{0,N}$ for large $N$. The estimates we find are
consistent with some radius of convergence for the tachyon potential if
the integrals are dominated by these configurations.
It is possible to do a very simple estimate. To this end consider
the $z$ plane and place one
puncture at infinity with $|z|\geq 1$ its unit disk. In this way its
mapping radius is just one. All other punctures will be distributed
uniformly inside the disk. Because of area constraint we can imagine
that each puncture will then carry a little disk of radius $r$, with
$N\pi r^2 \sim \pi$ fixing the radius to be $r\sim 1/\sqrt{N}$.
The mapping
radius of each of these disks will be $r$. Another of the disks
will be fixed at $0$, and another to $2r$. We can now
estimate the measure, which is the integrand in \wknoofffx, where in
dealing with the three special punctures we make use of \threpp. We have then
$$ d\mu_{\rm sym} \sim {(2r)^2
\over r^2\cdot r^2 \cdot 1}\prod_{I=1}^{N-3}
{{\rm d} x\{\cal I} {\rm d} y\{\cal I}\over r^2}\,\sim N^{N-2}\prod_{I=1}^{N-3}
{\rm d} x\{\cal I} {\rm d} y\{\cal I}\,.
\eqn\wmeffx$$
Since all the punctures, except for the one at infinity, are
inside the unit disk, we can compare the measure given above
with the measure in the planar
configuration. In that case the measure coefficient went like $N^{4N}$ and
now it essentially goes like $N^N$, which is much smaller, as we
expected.\foot{Notice that if the punctures in the planar configuration
had remained in the boundary of the unit disk then the measure would
have only diverged like $N^{2N}$. The conformal map that brought them
all to the real line between $0$ and $1$ introduced an extra factor of
$N^{2N}$. This suggests that the divergence may actually not be as
strong as it seems at first sight.} We can also repeat the
estimate we did for the integration
over moduli space for the planar configuration, and again, we just get an extra
multiplicative factor of $N$, which is irrelevant. Therefore, if we
assume this configuration dominates we find ${\rm v}\{\cal N}\sim N^N$ and
the tachyon potential $\sum {{\rm v}\{\cal N}\over N!}\tau^N$ would have some radius
of convergence.
\section{Analytic Continuation and Divergences}
Here we want to discuss what happens if we do string field theory
with stubs of length $l$. It is well-known that as the length of
the stubs goes to infinity $l\to \infty$ then the region of moduli
space corresponding to ${\cal V}_{0,4}$ approaches the full ${\cal M}_{0,4}$.
In this case the coefficient of the quartic
term in the potential will go into the full off-shell amplitude for
scattering of four tachyons at zero momentum.
We will examine the measure of integration for the tachyon potential
as we approach the boundaries of moduli
space and see that we would get a divergence corresponding to a tachyon
of zero momentum propagating over long times. We believe this divergence
is unphysical, and that the correct approach is to define the amplitude
by analytic continuation from a region in the parameter space of the
external momenta where the amplitude converges. When the full off-shell
amplitude is built from the vertex contribution and the Feynman diagram
contribution, analytic continuation is necessary for the Feynman part.
We therefore examine the off-shell formula for the evaluation of the
four string vertex for general off-shell tachyons. What we need is the
expression given in Eqn.\fxyu\ integrated over ${\cal V}_{0,4}$
$$A_{p_1\cdots p_4} = {2\over \pi}\int_{{\cal V}_{0,4}}
{\rm d} x\,{\rm d} y \,{ |z|^{2p_1p_2} |z-1|^{2p_1p_3} \over \rho_z^{2-p_1^2}
\rho_0^{2-p_2^2}\rho_1^{2-p_3^2}\rho_\infty^{2-p_4^2}}\,, \eqn\fxyuu$$
where we have added subscripts to the mapping radii in order to
indicate the position of the punctures. Let us now examine what happens as
we attempt to integrate with $z\to 0$, corresponding to a degeneration where
punctures one and two collide. In this region $\rho_1$ and $\rho_\infty$
behave as constants, and we have that
$$A_{p_1\cdots p_4} \sim \int_{|z|< c}
{\rm d} x \,{\rm d} y \,{ |z|^{2p_1p_2} \over \rho_z^{2-p_1^2}
\rho_0^{2-p_2^2}}\,, \eqn\fxyuv$$
As the puncture at $z$ is getting close to the puncture at zero it is
intuitively clear that the mapping radii $\rho_z\sim \rho_0 \sim |z|/2$
as these are the radii of the ``largest'' nonintersecting disks surrounding
the punctures. Therefore
$$A_{p_1\cdots p_4} \sim \int_{|z|< c}
\,{ {\rm d} x \,{\rm d} y \over |z|^{4-p_1^2- p_2^2-2p_1p_2} }\,
= \int_{|z|< c}
\,{ {\rm d} x \,{\rm d} y \over |z|^{4-(p_1+ p_2)^2}} \,,\eqn\fxyyuv$$
and we notice that the divergence is indeed controlled by the momentum
in the intermediate channel. If all the momenta were set to zero before
integration, we get a divergence
of the form $\int {\rm d} r/r^3$. But the way to proceed is to do the integral
in a momentum space region where we have no divergence
$$A_{p_1\cdots p_4} \sim \int_{|z|< c}
\,{ {\rm d} r \over r^{3-(p_1+ p_2)^2}} \sim {1\over [2-(p_1+ p_2)^2] }
\,,\eqn\txyyuv$$
and the final result does not show a divergence for $p_1=p_2=0$.
Notice also that the denominator in the result is nothing else than
$L_0+\bar L_0$ for the intermediate tachyon, if that tachyon were
on shell, we would get a divergence due to it.
\chapter{A Formula for General Off-Shell Amplitudes on the Sphere}
In this section we will derive a general formula for off-shell
amplitudes on the sphere. In string field theory those amplitudes
are defined as integrals over subspaces ${\cal A}_N$ of the moduli space
$\widehat\P_{0,N}$ of $N$-punctured spheres with local coordinates, up to phases,
at the punctures. We will assume a real parameterization of ${\cal A}_N$
and derive an operator expression which
expresses the amplitude as a multiple integral over these parameters.
The new point here is the explicit description of the
relevant antighost insertions necessary to obtain the integrand, and
the discussion explaining why the result does not depend neither on the
parameterization of
the subspace nor on the choice of a global
uniformizer on the sphere.
We also give a formula for the case when the space ${\cal A}_N$ is parametrized
by complex coordinates. For this case we will emphasize the
analogy between the ${\bf b}$--insertions for moduli and the ${\bf b}$--insertions
necessary to have ${\rm PSL}(2,\IC)$ invariance.
Finally, we will show how the general formula works by re-deriving the
off--shell Koba-Nielsen amplitude considered earlier.
\section{An Operator Formula for N--String Forms on the Sphere}
Recall that the state space $\widehat{\cal H}$
of closed string theory consists of the states
in the conformal theory that are annihilated both by $L_0-\bar L_0$ and by
$b_0-\bar b_0$.
Following [\zwiebachlong] we now
assign an $N$--linear function on $\widehat{\cal H}$
to any subspace ${\cal A}$ of $\widehat{\cal P}_{0,N}$. The multilinear function is defined
as an integral over ${\cal A}$ of a canonical differential form
$$\{\Psi_1,\Psi_2,\dots,\Psi\{\cal N}\}_{\cal A}=
\int_{\cal A}{\Omega_{}^{}}^{(\dim{\cal A}-\dim{\cal M}_{0,N})0,N}_{\Psi_1\Psi_2\dots\Psi\{\cal N}}\,.
\eqn\general$$
One constructs the forms by verifying that suitable forms
in ${\cal P}_{0,N}$ do lead to well defined forms in $\widehat{\cal P}_{0,N}$. This is
the origin of the restriction of the CFT state space to $\widehat{\cal H}$.
The canonical $2(N-3)+k$--form $\Omega^{(k)0,N}$ on $\widehat{\cal P}_{0,N}$
is defined by its
action on $2(N-3)+k$ tangent vectors $V\{\cal I}\in T_{\Sigma_P}\widehat{\cal P}_{0,N}$ as
$$\Omega^{(k)0,N}_{\Psi_1,\Psi_2,\dots\Psi\{\cal N}}(V_1,\dots,V_{2(N-3)+k})=
\left({i\over2\pi}\right)^{N-3}
\bra{\Sigma_P}{\bf b}({\bf v}_1)\cdots{\bf b}({\bf v}_{2(N-3)+k})\ket{\Psi^N}\,.\eqn\bins$$
Here the surface state $\bra{\Sigma_P}$ is a bra living in
$({\cal H}^*)^{\otimes N}$ and represents the punctured Riemann surface $\Sigma_P$.
The symbol ${\bf v}\{\cal I}$ denotes a Schiffer variation representing the tangent
$V\{\cal I}$, and
$${\bf b}({\bf v})=\sum_{I=1}^N
\oint_{w\{\cal I}=0}{{\rm d} w\{\cal I}\over 2\pi i}v^{(I)}(w\{\cal I})b^{(I)}(w\{\cal I})+
\oint_{\bar w\{\cal I}=0}{{\rm d} \bar w\{\cal I}\over 2\pi i}
\bar v^{(I)}(\bar w\{\cal I})\bar b^{(I)}(\bar w\{\cal I})\,.\eqn\bfb$$
Recall that a Schiffer variation for an $N$-punctured surface (in our present
case a sphere) is an $N$--tuple of vector fields
${\bf v}=\left(v^{(1)},v^{(2)},\cdots,v^{(N)}\right)$
where the vector $v^{(k)}$ is a vector field defined in
the coordinate patch around the $k$--th puncture.\foot{In
general the vector fields $v^{(k)}$ are defined on some annuli
around the punctures and do not extend holomorphically to the whole
coordinate disk, in order to represent the change of modulus of the
underlying non-punctured surface (see [\zwiebachlong]). For $g=0$
the underlying surface is the Riemann sphere and has no moduli. Therefore, the
Schiffer vectors can be chosen to be extend throughout the coordinate disk.}
Let $w_k$ be the local coordinate around the $k$--th puncture $P_k$
($w_k(P_k)=0$). The variation defines a new $N$--punctured Riemann sphere
with a new chosen local coordinates $w_k' = w_k+{\epsilon} v^{(k)}(w_k)$.
The new $k$-th puncture is defined to be at the point $P_k'$ such that
$w_k'(P_k') =0$. It follows that the $k$--th puncture is shifted
by $-{\epsilon} v^{(k)}(P_k)$. For any tangent $V\in T_P\widehat{\cal P}_{0,N}$
there is a corresponding Schiffer vector. Schiffer vectors are unique
up to the addition of vectors that arise from the restriction of
holomorphic vectors on the surface minus the punctures.
Note that the insertions ${\bf b}({\bf v})$ are invariantly defined, they only
depend on the Schiffer vector, and do not depend on the local
coordinates. Indeed $b$ is a primary field of conformal dimension $2$
or a holomorphic $2$--tensor. Being multiplied by a holomorphic
vector field $v$ it produces a
holomorphic $1$--form, whose integral
$\oint_{w=0}{{\rm d} w\over 2\pi i}v(w)b(w)$
is well-defined and independent of the contour of integration.
In order to evaluate \general\ we choose some real
coordinates $\lambda_1,\dots,\lambda_{\dim{\cal A}}$ on ${\cal A}$. Let $\{V_{\lambda_k}\}=
\partial/\partial \lambda_k$ be the corresponding tangent
vectors, and let $\{{\rm d}\lambda_k\}$ be the corresponding dual one-forms,
\ie\ ${\rm d}\lambda_k(V_{\lambda_l})=\delta_{k,l}$.
Using $\{V_{\lambda_k}\}$ we can rewrite \general\ as
$$\{\Psi_1,\cdots,\Psi\{\cal N}\}_{\cal A}=\left({i\over2\pi}\right)^{N-3}
\int{\rm d}\lambda_1\cdots{\rm d}\lambda_{{\dim{\cal A}}}
\bra{\Sigma_P}{\bf b}({\bf v}_{\lambda_1})\cdots{\bf b}({\bf v}_{\lambda_{\dim{\cal A}}})
\ket{\Psi_1}\cdots\ket{\Psi\{\cal N}}\,.\eqn\lint$$
In order to continue we must parameterize ${\cal A}$ as it sits in the moduli space
$\widehat{\cal P}_{0,N}$. Let $w\{\cal I}$ be a local coordinate around the $I$--th puncture.
Given a global uniformizer $z$ on the Riemann sphere we can represent
$w\{\cal I}$ by as an invertible analytic function $w\{\cal I}(z)$ defined on some disk
in the $z$--plane which maps the disk to a standard unit disk $|w\{\cal I}|<1$. The
inverse map $h\{\cal I}=w\{\cal I}^{-1}$ is therefore an analytic function on a unit
disk. Therefore, $N$ functions $h\{\cal I}(w\{\cal I})$ define a point in $\widehat{\cal P}_{0,N}$,
the sphere with $N$ punctures at $h\{\cal I}(0)$ and local coordinates given by
$w\{\cal I}(z)=h\{\cal I}^{-1}(z)$. The embedding of ${\cal A}$ in $\widehat{\cal P}_{0,N}$
is then represented by a set of $N$ holomorphic functions
parameterized by the real coordinates $\lambda_k$ on ${\cal A}$:
$\{h_1(\{\lambda_k\};w_1),\cdots,h\{\cal N}(\{\lambda_k\};w\{\cal N})\}$.
It is well known how to write the state $\bra{\Sigma_P}$ in terms of
$h\{\cal I}$'s (see [\leclair]).
$$\eqalign{
\bra{\Sigma_P}=\,\, &2\cdot\int\prod_{I=1}^N {\rm d} p\{\cal I}\,
(2\pi)^D\delta^D\left(\sum p\{\cal I}\right)
\bigotimes_{I=1}^N\bra{{\bf 1}^c,p\{\cal I}}
\int{\rm d}^2\zeta^1{\rm d}^2\zeta^2{\rm d}^2\zeta^3\cr
&\cdot\displaystyle\exp\Big(E(\alpha)+F(b,c)
-\sum_{n=1}^3\sum_{m\geq -1}\left(
\zeta^n M^{nJ}_{m}b^{(J)}_m -
\bar\zeta^n\overline{{M}^{nJ}_{m}}\bar b^{(J)}_m
\right)\Big).}\eqn\sigmap$$
where repeated uppercase indices $I, J \cdots $, are summed over the $N$
values they take. In here
$$\eqalign{
\displaystyle E(\alpha)&=-{1\over2}\sum_{n,m\geq0}\left(
\alpha^{(I)}_n N^{IJ}_{nm}\alpha^{(J)}_m+
\bar\alpha^{(I)}_n \overline{N^{IJ}_{nm}}\bar\alpha^{(J)}_m
\right)\,,\cr
\displaystyle F(b,c)&=\sum_{n\geq2\atop m\geq -1}\left(
c^{(I)}_n\tilde N^{IJ}_{nm}b^{(J)}_m+
\bar c^{(I)}_n\overline{\tilde{N}^{IJ}_{nm}}\bar b^{(J)}_m
\right)\,.\cr}\eqn\efdef$$
The vacua satisfy $\braket{{\bf 1}^c,p}{{\bf 1},q}=\delta(p-q)$, and
the odd Grassmann variables $\zeta^n$ are integrated using
$$\int{\rm d}^2\zeta^1{\rm d}^2\zeta^2{\rm d}^2\zeta^3~\,\zeta^1\bar\zeta^1
~\zeta^2\bar\zeta^2 ~\zeta^3\bar\zeta^3 \equiv 1\,.\eqn\grint$$
Note that the effect of this integration is to give the product of
six antighost insertions coming from the last term in the exponential in
Eqn.\sigmap. The minus sign in front of this term is actually irrelevant
(as the reader can check) but it was included for later convenience.
A bar over a number means complex conjugate while a bar
over an operator is used in order to distinguish the left--moving
modes from right--moving ones.
The {\em Neumann coefficients} $ N_{mn}^{IJ}$ and
$\tilde{N}^{IJ}_{mn}$ are given by the following formulae:
$$
\eqalign{
N_{00}^{IJ}&=\displaystyle\left\{ {\log(h'\{\cal I}(0)),\hfill I=J\atop
\log(h\{\cal I}(0)-h\\J(0)),\quad I\neq J}\right.\cr
N_{0n}^{IJ}&=\displaystyle{1\over n}\oint_{w=0} {{\rm d} w\over 2\pi i} w^{-n}
h'\\J(w){-1\over h\{\cal I}(0)-h\\J(w)}\,,\cr
N_{mn}^{IJ}&=\displaystyle{1\over m}\oint_{z=0} {{\rm d} z\over2\pi i}z^{-m} h'\{\cal I}(z)
{1\over n}\oint_{w=0}{{\rm d} w\over2\pi i} w^{-n}h'\\J(w)
{1\over (h\{\cal I}(z)-h\\J(w))^2}\,,\cr
\tilde{N}_{mn}^{IJ}&=\displaystyle{1\over m}\oint_{z=0}{{\rm d} z\over2\pi i} z^{-m+1}
(h'\{\cal I}(z))^2
{1\over n}\oint_{w=0}{{\rm d} w\over2\pi i} w^{-n-2} (h'\\J(w))^{-1}
{1\over h\{\cal I}(z)-h\\J(w)}\,.\cr}
\eqn\neumann$$
Moreover,
$$M_{m}^{nJ} =\displaystyle\oint_{w=0}{{\rm d} w\over 2\pi i} w^{-m-2} (h'\\J(w))^{-1}
[h\\J(w)]^{n-1},\quad n=1,2,3\,. \eqn\mneu$$
\noindent
\underbar{Antighost Insertions.}
Now let us show how to take the ${\bf b}$ insertions
into account. In order to calculate the ${\bf b}$ insertion associated to a tangent
vector $V \in T\\\Sigma\widehat{\cal P}_{0,N}$, we must find
the Schiffer vector (field) that realizes the deformation of the surface
$\Sigma$ specified by $V$.
Consider a line $c(t)$ in $\widehat{\cal P}_{0,N}$ parameterized by the real
parameter $t$: $c:[0,1]\to\widehat{\cal P}_{0,N} $, such that $\Sigma = c(0)$, and
the tangent vector to the curve
is $V=c_*\left({{\rm d}\over{\rm d} t}\right)$. We will see now how one can use this
setup to define in a natural way a vector field on the neighborhood of
the punctures of the Riemann surface $\Sigma$. This vector field is
the Schiffer vector.
\if y\figcount
\midinsert
\epsfxsize\hsize
\centerline{\epsffile{tanvec.eps}}
\nobreak
\narrower
\singlespace
\noindent
Figure 2. We show how to obtain a Schiffer vector field associated to
a tangent vector in $\widehat{\cal P}_{0,N}$. Shown are the Riemann surface $\Sigma$
and the local coordinate plane $w$.
\medskip
\endinsert
\else\fi
We can represent the curve $c(t)$ by $N$
functions $h\{\cal I}(t;w)$, holomorphic in $w$, and parameterized by $t$.
Choose a fixed value $w_0$ of the $w$ disk.
We now define a map $f^{w_0}: c(t)\to \Sigma$ from the curve $c(t)$
to a curve on the surface $\Sigma$.
The function $f^{w_0}$ takes
$c(t)$ to $z\{\cal I}(t)= h\{\cal I}(t,w_0)$ for each value of $t$.
We can now use the map $f^{w_0}$ to produce a push-forward map of vectors
$f_*^{w_0}: Tc\to T_{h\{\cal I}(0,w_0)}\Sigma$. In this way we can produce the
vector $f_*^{w_0} V\in T_{z\{\cal I}(0,w_0)}\Sigma$. By varying the value of
$w_0$ we obtain a vector {\it field} on the neighborhood of the $I$-th
puncture. We claim that this vector field, with a minus sign,
is the Schiffer vector.
In components, and with an extra minus sign, the pullback gives
$$v^{(I)}(z)=-{\partial h\{\cal I}\over\partial t}\left(t;w\{\cal I}(z)\right)\,,\eqn\derschiffer$$
It is useful to refer the Schiffer vector to the local coordinate $w\{\cal I}$.
We then find, by pushing the vector further
$$v^{(I)}(w\{\cal I})=- \left({\partial h\{\cal I}\over \partial w\{\cal I}}\right)^{-1}
\cdot{\partial h\{\cal I}(t;w\{\cal I})\over \partial t}.\eqn\tan$$
By definition, the Schiffer
vector ${\bf v}(V)$ corresponding to the vector $V$ is given by the
collection of vector fields
${\bf v}=\left(v^{(1)}(w_1),\cdots,v^{(N)}(w\{\cal N})\right)$.
If we define the vector $V_{\lambda_k}\in T\widehat{\cal P}_{0,N}$ to be the
tangent associated to the coordinate
curve parameterized by $\lambda_k$, we then write the Schiffer variation for
$V_{\lambda_k}$ as:
$$v_{\lambda_k}^{(I)}(w\{\cal I})=-\,{1\over h\{\cal I}'(w\{\cal I})}{\partial h\{\cal I}(\lambda;w\{\cal I})\over \partial\lambda_k}\,,
\eqn\svk$$
where $h\{\cal I}'(w\{\cal I}) \equiv (\partial h\{\cal I}/\partial w\{\cal I})$.
Before proceeding any further, let us confirm that the `natural' vector
we have obtained is indeed the Schiffer vector. This is easily done.
Let $p$ denote a point in the Riemann surface $\Sigma$ and let $w\{\cal I}(p)$
denote its local coordinate. By definition, the Schiffer vector defines
a new coordinate $w\{\cal I}'(p)$ as $w\{\cal I}'(p) = w(p) + dt \, v^I(w(p))$, where
$t$ is again a parameter for the deformation. Since the $z$-coordinate
of the point $p$ does not change under the deformation, we must have that
$h\{\cal I} (t+ dt , w\{\cal I}'(p)) = h\{\cal I}(t , w\{\cal I}(p))$. Upon expansion of this last
relation one immediately recovers Eqn.\tan.
One more comment is in order. What happened to the usual ambiguity
in choosing Schiffer vectors? Schiffer vectors are ambiguous since
there are nonvanishing $N$-tuples that do not induce any deformation.
This happens when the $N$-tuples can be used to define a holomorphic
vector on the surface minus the punctures. In our case the ambiguity is due
to the fact that the functions $h\{\cal I}(\lambda , w)$ can be composed with any
${\rm PSL}(2,\IC)$ transformation $S$ in the form $S\circ h\{\cal I}(\lambda , w)$. We
will come back to this point later.
Using Eqn.\bfb\ we can now write ${\bf b}({\bf v}^\lambda)$ as
$${\bf b}({\bf v}_{\lambda_k})=-\sum_{m\geq -1}
( B^{kJ}_m b^{(J)}_m+\overline{B^{kJ}_m}\bar b^{(J)}_m)\,,\eqn\tyu$$
where, as usual, the repeated index $J$ is summed over the number of punctures,
and
$$
B^{kJ}_{m}=\oint_{w=0}{{\rm d} w\over 2\pi i}\, w^{-m-2}
\,{1\over h'\\J(w)}\, {\partial h\\J(\lambda ,w)\over \partial\lambda_k} \,.\eqn\MNeumann$$
The range $m\geq -1$ has been obtained because the Schiffer vectors
can be chosen to be holomorphic and not to have poles at the punctures
(this will not be the case for higher genus surfaces).
Let us now treat the $b$
insertions in a way similar to that used for the zero modes
in \sigmap. Let $\zeta\{\cal I}$ and $\eta\{\cal I}$ be anti-commuting
variables, then
$$\int
{\rm d}\xi_1\dots{\rm d}\xi_n{\rm e}^{\xi_1\eta_1+\cdots+\xi_n\eta_n}=
\int
{\rm d}\xi_1{\rm e}^{\xi_1\eta_1}\cdots
\int{\rm d}\xi_n{\rm e}^{\xi_n\eta_n}=
\eta_1\cdots\eta_n.
\eqn\trick$$
This observation allows us to represent the product of ${\bf b}$ insertions in
\bins\ as an integral of an exponent.
$${\bf b}({\bf v}_{\lambda_1})\cdots{\bf b}({\bf v}_{\lambda_{\dim{\cal A}}})=
\int\prod_{k=1}^{\dim{\cal A}}{\rm d}\xi^k\exp\Bigl(
-\sum_{k=1}^{\dim{\cal A}}\sum_{m\geq-1}\xi^k\left(B^{kJ}_m b^{(J)}_m+\overline{
B^{kJ}_m}\bar b^{(J)}_m
\right)\,\Bigr)\,, \eqn\tricky$$
where the $\xi^n$'s are real Grassmann odd variables. The multi-linear
product \general\ now assumes the form
$$
\eqalign{\displaystyle
\{\Psi^N\}_{\cal A}&=2\left( {i\over 2\pi} \right)^{N-3}\int
\prod_{I=1}^N {\rm d}^D p\{\cal I}
(2\pi )^D\delta^D\left(\sum_{I=1}^Np\{\cal I}\right)
\bigotimes_{I=1}^N\bra{{\bf 1}^c,p\{\cal I}}
\int\prod_{k=1}^{\dim{\cal A}}{\rm d}\lambda_k
\prod_{k=1}^{\dim{\cal A}}{\rm d}\xi^k\cr
&\int{\rm d}^2\zeta^1{\rm d}^2\zeta^2{\rm d}^2\zeta^3
\displaystyle\exp\Bigg( E(\alpha) + F(b,c)
\displaystyle-\sum_{k=1}^3\sum_{m\geq -1}\left( \zeta^k M^{kJ}_{m}b^{(J)}_m
-\bar\zeta^k\overline{M^{kJ}_{m}}\bar b^{(J)}_m \right)\cr
& \qquad\qquad\qquad\qquad\qquad
\qquad\qquad\qquad\displaystyle-\sum_{k=1}^{\dim{\cal A}}
\sum_{m\geq -1}\xi^k\left(
B^{kJ}_{m}b^{(J)}_m+
\overline{B^{kJ}_{m}}\bar b^{(J)}_m
\right)\Bigg)\ket{\Psi^N},} \eqn\aterm$$
The above formula together with \efdef,\neumann, \mneu\ and \MNeumann\
gives a closed expression for a multi-linear form associated with
${\cal A}\subset\widehat{\cal P}_{0,N}$.
The resemblance of the last two terms appearing in the exponential
is not a coincidence. While the last term arose from the antighost
insertions for moduli, the first term, appearing already in the description
of the surface state $\bra{\Sigma}$, can be thought as the antighost
insertions due to the Schiffer vectors that represent ${\rm PSL}(2,\IC)$
transformations. This is readily verified. Consider the sphere with
uniformizer $z$. The six globally defined vector fields are given by
$v_k(z) = z^k$, and $v_k'(z) = iz^k$ with $k=0,1,2$. Referring them
to the local coordinates one sees that $v_k^I(w) = [h\{\cal I}(w)]^k/h\{\cal I}'(w)$
and ${v'}_k^I(w) = i[h\{\cal I}(w)]^k/h\{\cal I}'(w)$. As a consequence
$${\bf b}(v_k) = \sum_{m\geq -1} \left( M^{kJ}_m\, b_m + \overline{M^{kJ}_m}
\,\bar b_m \right) \,,\eqn\sltwoc$$
$${\bf b}(v_k') = i\sum_{m\geq -1} \left( M^{kJ}_m\, b_m - \overline{M^{kJ}_m}
\,\bar b_m \right) \,,\eqn\slytwoc$$
where the $M$ coefficients were defined in Eqn.\mneu. It is clear that the
product of the six insertions precisely reproduces the effect of the first
sum in the exponential of Eqn.\aterm.
In order to be used in \aterm\ the subspace
${\cal A}$ is parametrized by some
coordinates $\lambda_k$. The expression for the multilinear form is independent
of the choice of coordinates; it is a well-defined form on ${\cal A}$.\foot{This
is easily verified explicitly. Under coordinate transformations,
the product $\prod {\rm d}\lambda_i$ transforms with a Jacobian, and the product
of antighost insertions, as a consequence of \MNeumann\ transforms with
the inverse Jacobian.} Once the parametrization is chosen, the space ${\cal A}$
has to be described by the $N$ functions $h\{\cal I}(\lambda ,w)$.
These functions, as we move on ${\cal A}$ are defined up to a
{\it local} linear fractional
transformation. At every point in moduli space we are free to change
the uniformizer. Let us see why \aterm\ has local ${\rm PSL}(2,\IC)$ invariance.
The bosonic Neumann coefficients $N_{nm}^{IJ}$ are
${\rm PSL}(2,\IC)$ invariant by themselves for $n,m\geq1$. The sums
$\sum_{IJ} \alpha^{(I)}_0 N^{IJ}_{00}\alpha^{(J)}_0=
\sum_{IJ}N^{IJ}_{00}p\\Ip\\J$ and $\sum_{J} N^{IJ}_{m0}\alpha^{(J)}_0=
\sum_{J}N^{IJ}_{m0}p\\J$ can be shown to be invariant due to
momentum conservation $\sum p\\J=0$. A detailed analysis of
${\rm PSL}(2,\IC)$ properties of the ghost part of a surface state
have been presented by LeClair {\it et al.} in [\leclair]
for open strings. Their arguments can be readily generalized to our
case. The only truly new part that appears in \aterm\
is the last term in the exponential.
Under global ${\rm PSL}(2,\IC)$ transformations, namely transformations of the form
$h\to (ah+b)/(ch+d)$, with $a,b,c$ and $d$ independent of $\lambda$,
this term is invariant
because so is every coefficient $B^{kJ}_m$. Since a general local
${\rm PSL}(2,\IC)$ tranformation can be written locally as a global transformation
plus an infinitesimal local one, we must now show invariance under
infinitesimal local transformations. These are transformations of the form
$$\widetilde h\{\cal I} = h\{\cal I} + a(\lambda) + b(\lambda)h\{\cal I} + c(\lambda) h\{\cal I}^2 \, ,\eqn\localtr$$
for $a,b$, and $c$ small.
A short calculation shows that
$$-{1\over \widetilde h\\J'}{\partial \widetilde h\{\cal I}\over \partial \lambda_k}
\,=\, -{1\over h\\J'}{\partial h\{\cal I}\over \partial \lambda_k}\, - \,
{1\over h\\J'} \Bigl[ {\partial a\over \partial \lambda_k} +
{\partial b\over \partial \lambda_k}h\{\cal I} +
{\partial c\over \partial \lambda_k}h\{\cal I}^2 \Bigr] \, . \eqn\trlocal$$
On the left hand side we have the new Schiffer vector, and the first
term the right hand side is the old Schiffer vector. We see that they
differ by a linear superposition of the Schiffer
vectors $v_k^I$ and ${v'}_k^I$ introduced earlier in our discussion of
${\rm PSL}(2,\IC)$ transformations (immediately above Eqn.\sltwoc.) It follows that
the extra contributions they make to the antighost insertions vanish
when included in the multilinear form because the multilinear form already
includes the antighost insertions corresponding to the Schiffer vectors
generating ${\rm PSL}(2,\IC)$.
\noindent
\underbar{Complex Coordinates}
In some applications the subspace ${\cal A}$ has even dimension and can be
equipped with complex coordinates. Let $\dim^c{\cal A}$ denote the complex
dimension of ${\cal A}$
and let $\{\lambda_1,\cdots,\lambda_{\dim^c{\cal A}}\}$ be a set of complex coordinates.
The subspace ${\cal A}$ can now be represented by
the collection of functions
$\{h\{\cal I}(\{\lambda_k\},\{\bar\lambda_k\};w\{\cal I})\}$ with $I=1,\cdots ,N$. In
order to derive a formula for this case we simply take the earlier
result for two real insertions and pass to complex coordinates. We
thus consider
$$ \Omega^2 = d\lambda_1\wedge d\lambda_2 \,\, d\xi^1 d\xi^2 \, \exp
\Bigl[\, -\sum_{k=1}^2 \xi^k\left(
B^{kJ}_{m}b^{(J)}_m+
\overline{B^{kJ}_{m}}\bar b^{(J)}_m\right) \Bigr]\, , \eqn\trans$$
Using complex coordinates $\lambda_1' = \lambda_1+ i\lambda_2$ and ${\xi'}^1 = \xi^1+ i
\xi^2$,
and letting $\int d^2 {\xi'}\, \xi' \bar \xi' \equiv 1$, we can write
the above as
$$ \Omega^2 = d\lambda'_1\wedge d\bar\lambda'_1 \,\, d^2 \xi'_1 \,
\exp \Bigl[\, -{\xi'}^1\left(
B^{1 J}_{m}b^{(J)}_m+
\overline{B^{\bar 1J}_{m}}\bar b^{(J)}_m\right)
+\bar{\xi'}^1\left(
B^{\bar 1J}_{m}b^{(J)}_m+
\overline{B^{1J}_{m}}\bar b^{(J)}_m\right)\Bigr]\, , \eqn\tras$$
where we have defined
$$\eqalign{B^{ kJ}_{m}&=\displaystyle
\oint_{w=0}{{\rm d} w\over2\pi i}w^{-m-2}
{1\over h'\\J(w) } {{\partial h\\J(\lambda,\bar\lambda;w)\over\partial \lambda_k}},\cr
B^{\bar k J}_{m}&=\displaystyle
\oint_{w=0}{{\rm d} w\over2\pi i}w^{-m-2}
{1\over h'\\J(w) } {{\partial h\\J(\lambda,\bar\lambda;w)\over\partial\bar\lambda_k}}.\cr} \eqn\sdff$$
Note that the $B^{\bar k J}_{m}$ coefficients do not vanish because
the embedding of ${\cal A}$ in $\widehat{\cal P}_{0,N}$ need
not be holomorphic and, as a consequence, the derivatives
$\partial h\{\cal I}/\partial\bar\lambda_k$ need not vanish. Using this
result we can now rewrite \aterm\ for the case of complex coordinates
for moduli
$$\eqalign{\displaystyle
\{\Psi^N\}_{{\cal A}}=2&\left({i\over2\pi}\right)^{N-3}
\hskip-6pt \int\prod_{I=1}^N{\rm d}^Dp\{\cal I}(2\pi)^D\delta^D\left(\sum p\{\cal I}\right)
\bigotimes_{I=1}^N\bra{{\bf 1}^c,p\{\cal I}}
\int_{\cal A}\prod_{k=1}^{\dim^c{\cal A}}{\rm d}\lambda_k\wedge{\rm d}\bar\lambda_k
\prod_{k=1}^{\dim^c{\cal A}}{\rm d}^2\xi^k \cr
&\prod_{k=1}^3 \hbox{d}^2\zeta^k \,\displaystyle
\exp\Bigg(E(\alpha)+F(b,c)
-\displaystyle\sum_{k=1}^3\Big(
\zeta^kM^{kJ}_{m}b^{(J)}_m -
\bar\zeta^k\overline{M^{kJ}_{m}}\bar b^{(J)}_m\Big)\cr
& \qquad -\displaystyle\sum_{k=1}^{\dim^c{\cal A}}\Big[ \xi^k\Big(
B^{ kJ}_{m}b^{(J)}_m+
\overline{B^{\bar k J}_{m}}\bar b^{(J)}_m\Big)
-\displaystyle\bar\xi^k\Big(
B^{\bar kJ}_{m}b^{(J)}_m+
\overline{B^{ k J}_{m}}\bar b^{(J)}_m
\Big)\Big] \Bigg)\ket{\Psi^N},}\eqn\atermc$$
where $m\geq -1$ for the implicit oscillator sum.
It is useful to bring out the similarity between the antighost
insertions for ${\rm PSL}(2,\IC)$ and those for moduli. In order to achieve
this goal we introduce
$\xi^{\dim^c{\cal A}+ k}\equiv \zeta^k$ for $k=1,2,3$.
$$\eqalign{\displaystyle
\{\Psi^N\}_{{\cal A}}=2&\left({i\over2\pi}\right)^{N-3}
\hskip-6pt \int\prod_{I=1}^N{\rm d}^Dp\{\cal I}(2\pi)^D\delta^D\left(\sum p\{\cal I}\right)
\bigotimes_{I=1}^N\bra{{\bf 1}^c,p\{\cal I}}
\int_{\cal A}\prod_{k=1}^{\dim^c{\cal A}}{\rm d}\lambda_k\wedge{\rm d}\bar\lambda_k \cr
&\prod_{k=1}^{\dim^c{\cal A}+ 3}\hskip-8pt{\rm d}^2\xi^k \cdot\displaystyle
\exp\Bigg(E(\alpha)+F(b,c)
-\displaystyle\hskip-8pt\sum_{k=1}^{\dim^c{\cal A}+ 3}\Big(
\xi^k{\cal B}^{kJ}_{m}b^{(J)}_m -
\bar\xi^k\overline{{\cal B}^{kJ}_{m}}\bar b^{(J)}_m\Big)\cr
& \qquad -\displaystyle\sum_{k=1}^{\dim^c{\cal A}}\Big( \xi^k\,
\overline{B^{\bar k J}_{m}}\,\bar b^{(J)}_m
-\displaystyle\bar\xi^k
B^{\bar kJ}_{m}b^{(J)}_m
\Big) \Bigg)\ket{\Psi^N},}\eqn\atermcc$$
where the script style ${\cal B}$ matrix elements are defined as
$$
{\cal B}^{kJ}_{m}=
\cases{B^{kJ}_m\, , & for $k\leq\dim^c{\cal A}$ \cr
M^{(k-\dim^c{\cal A})J}_m\, , &
for $k-\dim^c{\cal A}=1,2,3$. \cr}\eqn\ems$$
This concludes our construction of off-shell amplitudes as
forms on moduli spaces of punctured spheres.
\section{Application to Off-Shell Tachyons}
Let us see how the formulae derived above work
for the case of
$N$ tachyons with arbitrary momenta. This particular example
allows us to confirm our earlier calculation of off-shell tachyon
amplitudes. Specifically, we are going to evaluate the multilinear function
$\{\tau_{p_1},\cdots,\tau_{p\{\cal N}}\}$
where $\ket{\tau_{p_i}}=c_{1}\bar{c}_{1}\ket{{\bf 1}, p_i}$. In this
case the state to be contracted with the bra representing the multilinear
function is
$\ket{\tau^N}=\otimes_{I=1}^Nc_1^{(I)}\bar c_1^{(I)}\ket{{\bf 1},p\{\cal I}}$.
Upon contraction with this state we will only get contributions
from $b^{(I)}_{-1}$, $\bar{b}^{(I)}_{-1}$, and
the matter zero modes $\alpha_0^{(I)}=\bar\alpha_0^{(I)}=i p\{\cal I}$.
We will use as moduli the complex coordinates $z_1, \cdots , z\\{N-3}$
representing the position of the first $(N-3)$ punctures. Therefore,
$\lambda_k = z_k$, for $k=1, \ldots , N-3$, and we must use Eqn.\atermcc\
to calculate the multilinear function. Our setting of the $z$-coordinates
as moduli implies that the functions $h\\J$ take the form
$$h\\J (z,\bar z , w) = z\\J + a(z ,\bar z) w + \cdots \, .\eqn\basicform$$
It then follows that
$$M_{-1}^{kJ} =\displaystyle\oint_{w=0}{{\rm d} w\over 2\pi i}\, {1\over w} (h'\\J(w))^{-1}
[h\\J(w)]^{k-1} \, = \,{z\\J^{k-1} \over {h'}\\J(0) }\, , \eqn\xneue$$
and furthermore
$$B^{kJ}_{-1}=\oint_{w=0}{{\rm d} w\over 2\pi i}\, {1\over w}
\,{1\over h'\\J(w)}\, {\partial h\\J\over \partial\lambda_k}\,=\,
{\delta^{kJ}\over{h'}\\J(0) } \,,\eqn\xeumann$$
while $B^{\bar k J}_m = 0$, since $\partial h\\J/\partial \bar z_k = 0$.
With this information, back in \atermcc\ we find
$$\eqalign{\displaystyle\{\tau_{p_1},\cdots,\tau_{p\{\cal N}}\}_{{\cal A}}=2&\left({i\over2\pi}\right)^{N-3}
\hskip-6pt (2\pi)^D\delta^D ({\bf 0} )
\int\prod_{k=1}^{N-3}{\rm d} z_k\wedge{\rm d}\bar z_k\,\,\prod_{k=1}^N
{\rm d}^2\xi^k \cr
&\cdot\bra{{\bf 1}^c}
\exp\Bigg(E(\alpha)
-\displaystyle\sum_{k=1}^N\Big(
\xi^k{\cal B}^{kJ}_{-1}b^{(J)}_{-1} -
\bar\xi^k\overline{{\cal B}^{kJ}_{-1}}\bar b^{(J)}_{-1}\Big)
\Bigg)\prod_{I=1}^N c_1^{(I)}\bar c_1^{(I)}\ket{{\bf 1}}\,. \cr}\eqn\atermtt$$
We can now calculate the bosonic contribution from $E(\alpha)$
$$\eqalign{
\exp (E(\alpha)) &=
\exp\Big(-{1\over2}\sum_{I,J=1}^N
\left(\alpha_0^{(I)}N_{00}^{IJ}\alpha_0^{(J)}+
\bar\alpha_0^{(I)}\overline{N_{00}^{IJ}}\bar\alpha_0^{(J)}\right)\Big)\cr
& = \exp\Big(-{1\over2}\sum_{I,J=1}^N
\left(N_{00}^{IJ}+\overline{N_{00}^{IJ}}\right) p\{\cal I} p\\J\Big)\cr
&=\prod_{I<J}\left(
{|h\{\cal I}(0)-h\\J(0)|^2\over |h'\{\cal I}(0)|\cdot|h'\\J(0)|}
\right)^{p\\Ip\\J} = \prod_{I<J} \, \chi\\{IJ}^{p\\Ip\\J}\,, \cr}
\eqn\evalalpha$$
where we have used the expression for Neumann coefficients \neumann,
momentum conservation, and \twopmi. We thus obtain
$$\eqalign{\displaystyle\{\tau_{p_1},\cdots,\tau_{p\{\cal N}}\}_{{\cal A}}=\,
&{2\over \pi^{N-3}}\, (2\pi)^D\delta^D ({\bf 0} )
\int\prod_{k=1}^{N-3}{\rm d} x_k\, {\rm d} y_k\,
\prod_{I<J} \, \chi\\{IJ}^{p\\Ip\\J}\, \cr
&\cdot\prod_{k=1}^{N}{\rm d}^2\xi^k \bra{{\bf 1}^c} \exp\Big(
-\displaystyle\sum_{k=1}^N\Big(
\xi^k{\cal B}^{kJ}_{-1}b^{(J)}_{-1} -
\bar\xi^k\overline{{\cal B}^{kJ}_{-1}}\bar b^{(J)}_{-1}\Big)
\Big)\prod_{I=1}^N c_1^{(I)}\bar c_1^{(I)}\ket{{\bf 1}}\,. \cr}\eqn\atermttt$$
Consider the second line of the above equation. The effect of the
ghost state is to select the term in the exponential proportional
to the product of all antighosts. Since
$$\bra{{\bf 1}^c}
\prod_{I=1}^N b_{-1}^{(I)}\bar b_{-1}^{(I)}
\prod_{I=1}^N c_1^{(I)}\bar c_1^{(I)}\ket{{\bf 1}}= (-)^N\, ,\eqn\auxil$$
we can write the second line of \atermtt\ using a second set of
Grasmmann variables $\eta^k$
$$(-)^N\int\prod_{k=1}^{N}{\rm d}^2\xi^k \,{\rm d}^2\eta^k
\exp\Big[
\displaystyle\sum_{k,p=1}^N\Big(
-\xi^k\,{\cal B}^{kp}_{-1}\,\eta^p +
\bar\xi^k\,\overline{{\cal B}^{kp}_{-1}}\,\bar \eta^p\Big)\Big]
\,=\,(-)^N\, \det ({\cal B}) \, \det (\bar {\cal B} )\,, \eqn\detprop$$
where in the last step we used a standard formula in Grassmann integration.
We can now use Eqns. \xneue, \xeumann, and \ems\ to calculate $|\det {\cal B} \,|^2$.
We find
$$\eqalign{
|\det {\cal B} |^2 &= |(z\{\cal N} -z\\{N-2}) (z\{\cal N} -z\\{N-1}) (z\\{N-2} -z\\{N-1})|^2
\prod_{I=1}^N {1\over |{h'}\{\cal I} (0)|^2}\, , \cr &
= \chi^2\\{N-2,N-1,N}\prod_{I=1}^{N-3} {1\over \rho\{\cal I}^2} \, , \cr}\eqn\ambv$$
where we made use of the definition of the mapping radius and of
Eqn.\threp. We can now assemble the final form of the tachyon
multilinear function. Back in \atermttt\ we have
$$\{\tau_{p_1},\cdots,\tau_{p\{\cal N}}\}_{\cal A}=(-)^N{2\over\pi^{N-3}}\int
\prod_{I=1}^{N-3}{{\rm d} x\{\cal I} {\rm d} y\{\cal I} \over\rho\{\cal I}^2}
\, \chi^2_{N-2,N-1,N}\prod_{I<J}\chi\\{IJ}^{q\\Iq\\J}\cdot
(2\pi)^D \delta ({\bf 0})\,. \eqn\finalf$$
which agrees precisely with the off-shell Koba-Nielsen formula \knofffx.
\ack We are grateful to G. Moore who stimulated our interest in the
computation of the tachyon potential, and gave us his result as
quoted in Eqn.\moore. We also wish to acknowledge instructive discussions
with T. Banks, A. Sen, H. Sonoda and C. Thorn.
\APPENDIX{A}{A: The tachyon potential and string field redefinitions.}
Here we wish to discuss whether it is possible to make a field redefinition
of the string field tachyon such that the string action is brought to a form
where one could rule out the existence of a local minimum.
Even at the level of open string field theory this seems hard to achieve.
The tachyon potential is of the form $V\sim -\tau^2 + g \tau^3$. The cubic
term produces a local minimum with a nonzero vacuum expectation value
for $\tau$. We are
not allowed to just redefine $\tau$ to absorb the cubic term in the
quadratic one; this is
a non-invertible field redefinition. Using the massive fields is no help
since the transformations must preserve the kinetic terms, and therefore
should be of the form $\tau\to \tau+ f(\phi_i , \tau)$ and
$\phi_i \to \phi_i + g( \phi_i ,\tau )$, with $f$ and $g$ functions that
must start quadratic in the fields. Such transformations cannot eliminate
the cubic term in the tachyon potential.
Let us examine the question of field redefinitions in a more stringy way.
Assume it is possible to write the string action as
$$S = \int {\rm d} x \, \bigl[ {\cal L} (\nabla \tau , \phi_i) + \tau^2 (1+ f(\phi_i))
\bigr]\, , \eqn\tachleth$$
namely, that one can separate out a term just depending on derivatives of
the tachyon field, and all other fields, and a quadratic term for the
tachyon potential. The
term $f(\phi_i)$ was included to represent couplings to fields like
dilatons or background metric. If the above were true we would expect
no perturbatively stable minimum for the tachyon (the factor
$(1+ f(\phi_i))$ is expected to be nonvanishing). We will now argue
that the string action {\it cannot} be put in the form described
in Eqn.\tachleth\ by means of a string field redefinition. It is therefore
not possible to rule out a local minimum by such simple means.\foot{It is
not clear to us whether this result is in contradiction with that of
Ref.[\banks], where presumably the relevant action is the effective
action obtained after integrating out classically the massive fields.}
If Eqn.\tachleth\ holds, a constant infinitesimal shift of the tachyon
field $\tau \to \tau + \epsilon$ would have the effect of shifting the
action as
$$S \to S + 2\epsilon \int {\rm d} x \, \tau (1+ f(\phi_i)) + {\cal O} (\epsilon ^2)
\, .\eqn\tachses$$
We should be able to prove such ``low-energy tachyon theorem'' with
string field theory. For this we must find the change in the string
action as we shift the string field as follows
$$\ket{\Psi}_1 \to \ket{\Psi}_1 + \epsilon \ket{T_0}_1 + \epsilon
\bra{h^{(2)}_{23}}\Psi\rangle_2\ket{{\cal S}_{13}}+ \cdots \, ,\eqn\tryform$$
where $\ket{T_0}=c_1\bar c_1 \ket{{\bf 1}}$,
the dots indicate quadratic and higher terms in the string field,
$\bra{h^{(2)}_{23}}$ is a symmetric bra, and $\ket{{\cal S}_{13}}$ is the
sewing ket [\senzwiebachtwo]. Indeed the transformation of the string field
cannot be expected to be a simple shift along the zero-momentum tachyon,
since the string field tachyon
should differ from the tachyon appearing in \tachleth. If we now vary the
string action \action\ we find
$$S \,\to \, S + \, \,\epsilon\bra{\Psi} c_0^-Q\ket{T_0}\, +{1\over 2}\epsilon
\left[ \, \bra{V^{(3)}_{123}}T_0\rangle_3 +\,
\bra{h^{(2)}_{12}}(Q_1+Q_2) \right] \ket{\Psi}_1\ket{\Psi}_2
+\cdots\,. \eqn\varacttr$$
We must now see that by a suitable choice of $\bra{h^{(2)}_{12}}$ the variation
of the action takes the form required by \tachses. Indeed, the term
$\epsilon\,\tau$ arises from the $\epsilon\bra{\Psi} c_0^-Q\ket{T_0}$
term in \varacttr\ since $ c_0^-Q\ket{T_0}$ can only couple to the tachyon
field in $\Psi$. Assume the function $f(\phi_i)$ in \tachses\ is zero,
in that case there is no extra variation in the action, and we must
require that
$$ \bra{V^{(3)}_{123}}T_0\rangle_3 +\,
\bra{h^{(2)}_{12}}(Q_1+Q_2) = 0 \, .\eqn\musthave$$
This equation cannot have solutions; acting once more with $(Q_1+ Q_2)$ we
find that \musthave\ requires that
$\bra{V^{(3)}_{123}}Q_3\ket{T_0}_3 =0$ which cannot hold (recall
$Q\ket{T_0}\not= 0$). Even if $f(\phi_i)$ is not zero, we do not expect
solutions to exist. In this case we still must have that \musthave\ should
be zero contracted with any arbitrary two states, except when one of them is
a zero momentum tachyon. Again using states of the form
$(Q_1+Q_2)\ket{a}_1\ket{b}_1$ where neither $\ket{a}$ nor $\ket{b}$ is
a zero momentum tachyon, we see that again the equation cannot be satisfied.
This shows that there is no simple ``low-energy tachyon theorem'' that rules
out
a local minimum.
\refout
\end
|
train/arxiv
|
BkiUdso5qg5A51O-MCS2
| 5 | 1 |
\section{Introduction}
The nearly simultaneous detection of
gravitational waves GW170817
and the $\gamma$-ray burst GRB 170817A~\cite{TheLIGOScientific:2017qsa,%
Monitor:2017mdv,GBM:2017lvd}
places a very tight constraint on the
speed of gravitational waves, $c_{\rm GW}$.
The deviation of $c_{\rm GW}$ from the speed of light
is less than 1 part in $10^{15}$.
This can be translated to constraints on
modified gravity such as scalar-tensor theories,
vector-tensor theories, massive gravity,
and Ho\v{r}ava gravity~\cite{Creminelli:2017sry,%
Sakstein:2017xjx,Ezquiaga:2017ekz,Baker:2017hug,Gumrukcuoglu:2017ijh,%
Oost:2018tcv}.
In particular, in the context of the Horndeski theory
(the most general scalar-tensor theory
having second-order equations of motion\footnote{The original
form of the Horndeski action is different from its modern expression
obtained by extending the Galileon theory~\cite{Deffayet:2011gz}.
The equivalence of the two apparently different theories
was shown in~\cite{Kobayashi:2011nu}.})~\cite{Horndeski:1974wa},
two of the four free functions in the action are strongly constrained,
leaving only the simple, traditional form of
nonminimal coupling of the scalar degree of freedom to
the Ricci scalar, {\em i.e.,} the ``$f(\phi){\cal R}$''-type coupling.
However, it has been pointed out that
there still remains an interesting, nontrivial class of scalar-tensor theories
beyond Horndeski
that can evade the gravitational wave constraint
as well as solar-system tests~\cite{Creminelli:2017sry,Ezquiaga:2017ekz,Sakstein:2017xjx,%
Crisostomi:2017lbg,Langlois:2017dyl,Babichev:2017lmw,%
Dima:2017pwp,Crisostomi:2017pjs,Bartolo:2017ibw}.
Such theories have higher-order equations of motion as they are
more general than the Horndeski theory,
but the system is degenerate and hence
is free from the dangerous extra degree of freedom that
causes Ostrogradski instability.
Earlier examples of degenerate higher-order scalar-tensor (DHOST) theory
are found in~\cite{Zumalacarregui:2013pma,Gleyzes:2014dya},
and the degeneracy conditions are systematically
studied and classified
at quadratic order in second derivatives of the scalar field
in~\cite{Langlois:2015cwa,Crisostomi:2016czh}
and at cubic order in~\cite{BenAchour:2016fzp}.
Degenerate theories can also be generated from
nondegenerate ones via noninvertible disformal
transformation~\cite{Takahashi:2017pje,Langlois:2018jdg}.
One of the most interesting phenomenologies of DHOST theories is
efficient Vainshtein screening outside matter sources
and its partial breaking in the inside~\cite{Kobayashi:2014ida}.
The partial breaking of screening modifies,
for instance, stellar structure~\cite{Koyama:2015oma,Saito:2015fza}.
This fact was used to test DHOST theories,
or, more specifically, the Gleyzes-Langlois-Piazza-Vernizzi (GLPV)
subclass~\cite{Gleyzes:2014dya},
against astrophysical observations~\cite{Sakstein:2015zoa,%
Sakstein:2015aac,Jain:2015edg,Sakstein:2016ggl,Sakstein:2016lyj,Salzano:2017qac}.
Going beyond the weak-field approximation,
relativistic stars in the GLPV theory have been studied
in~\cite{Babichev:2016jom,Sakstein:2016oel}.
In this paper, we consider relativistic stars in
DHOST theories that are more general than the GLPV theory
but evade the constraint on the speed of gravitational
waves~\cite{Creminelli:2017sry,Sakstein:2017xjx,Ezquiaga:2017ekz}.
So far, this class of theories have been investigated
by employing the weak-field
approximation~\cite{Crisostomi:2017lbg,Langlois:2017dyl,Dima:2017pwp}
and in a cosmological context~\cite{Crisostomi:2017pjs}.
Very recently, compact objects including relativistic stars
in the GLPV theory with $c_{\rm GW}=1$
have been explored in Ref.~\cite{Chagoya:2018lmv}.
This paper is organized as follows.
In the next section, we introduce the DHOST theories
with $c_{\rm GW}=1$ and
derive the basic equations describing a spherically symmetric
relativistic star. To check the consistency with the previous results,
we linearize the equations and see the gravitational potential
in the weak-field approximation in Sec.~III.
Then, in Sec.~IV, we give boundary conditions
imposed at the stellar center and in the exterior region.
Our numerical results are presented in Sec.~V.
We draw our conclusions in Sec.~VI.
Since some of the equations are quite messy,
their explicit expression is shown in Appendix~A.
\section{Field equations}
The action of the quadratic DHOST theory we study is given by
\begin{align}
S=\int {\rm d}^4x\sqrt{-g}\left[f(X){\cal R}+\sum_{I=1}^5{\cal L}_I+{\cal L}_{\rm m}\right],
\label{eq:Lagrangian}
\end{align}
where ${\cal R}$ is the Ricci scalar, $X:=\phi_\mu \phi^\mu$, and
\begin{align}
{\cal L}_1&:=A_1(X)\phi_{\mu\nu}\phi^{\mu\nu},
\\
{\cal L}_2&:=A_2(X)(\Box\phi)^2,
\\
{\cal L}_3&:=A_3(X)\Box\phi\phi^\mu\phi_{\mu\nu}\phi^\nu,
\\
{\cal L}_4&:=A_4(X)\phi^\mu\phi_{\mu\rho}\phi^{\rho \nu}\phi_\nu,
\\
{\cal L}_5&:=A_5(X)(\phi^\mu\phi_{\mu\nu}\phi^\nu)^2,
\end{align}
with $\phi_\mu=\nabla_\mu\phi$
and $\phi_{\mu\nu}=\nabla_\mu\nabla_\nu\phi$.
The functions $A_I(X)$ must be subject to certain conditions
in order for the theory to be degenerate and satisfy $c_{\rm GW}=1$,
as explained shortly.
Here, shift symmetry is assumed and
the other possible terms
of the form $G_2(X)$ and $G_3(X)\Box\phi$ are omitted.
In particular, we do not include the usual kinetic term $-X/2$
in this paper.\footnote{The inclusion of the usual kinetic term $-X/2$
gives rise to an interesting branch of asymptotically locally flat solutions
with a deficit angle, as has been recently explored in Ref.~\cite{Chagoya:2018lmv}.
We leave this possibility for future research.
The appearance of a deficit angle in the GLPV theory was
pointed out earlier in Refs.~\cite{DeFelice:2015sya,Kase:2015gxi}.
As we will see, our ansatz for the scalar field
is different from that assumed in Refs.~\cite{DeFelice:2015sya,Kase:2015gxi}
(see Eq.~(\ref{eq:vt}) below).
Probably this is the reason why we can obtain spherically symmetric solutions
that are regular at the center.}
These assumptions are nothing to do with
the degeneracy conditions and the $c_{\rm GW}=1$ constraint
to be imposed below.
However,
with this simplification one can concentrate on
the effect of Vainshtein breaking.
Note in passing that the DHOST theories
are equivalent to the Horndeski theory with disformally coupled
matter. Therefore, the setup we are considering
is in some sense similar to that explored in Ref.~\cite{Minamitsuji:2016hkk}.
However, the crucial difference is that, in contrast to
Ref.~\cite{Minamitsuji:2016hkk},
our theory corresponds to the case where the conformal
and disformal coupling functions depend on the first derivative of the
scalar field.
We require that
the speed of gravitational waves, $c_{\rm GW}$, is equal to
the speed of light~\cite{Monitor:2017mdv}.
In our theory
$c_{\rm GW}$ is given by $c_{\rm GW}^2=f/(f-XA_1)$~\cite{deRham:2016wji},
so that
\begin{align}
A_1=0.
\end{align}
The degeneracy conditions read
\begin{align}
A_2&=-A_1=0,
\\
A_4&=-\frac{1}{8f}\left(8A_3f-48f_X^2-8A_3f_XX+A_3^2X^2\right),\label{eq:degn1}
\\
A_5&=\frac{A_3}{2f}\left(4f_X+A_3X\right).\label{eq:degn2}
\end{align}
We thus have two free functions, $A_3$ and $f$, in the
quadratic DHOST sector with $c_{\rm GW}=1$.
This theory has been explored recently in
Refs.~\cite{Creminelli:2017sry,Sakstein:2017xjx,Ezquiaga:2017ekz,%
Baker:2017hug,Crisostomi:2017lbg,Langlois:2017dyl,Dima:2017pwp,Crisostomi:2017pjs}.
(See Ref.~\cite{Bartolo:2017ibw} for a different subclass
with $f=\;$const and $A_2=-A_1\neq 0$.)
Following Ref.~\cite{Crisostomi:2017pjs}, we introduce
\begin{align}
B_1:=\frac{X}{4f}(4f_X+XA_3),
\end{align}
and use this instead of $A_3$.
In the special case with $B_1=0=A_5$, the action
reduces to that of the GLPV theory.
We consider a static and spherically symmetric metric,
\begin{align}
{\rm d} s^2=-e^{\nu(r)}{\rm d} t^2+e^{\lambda(r)}{\rm d} r^2 +r^2{\rm d}\Omega^2.
\end{align}
The scalar field is taken to be
\begin{align}
\phi(t,r)=vt+\psi(r),\label{eq:vt}
\end{align}
where $v \,(\neq0)$ is a constant.
Even though $\phi$ is linearly dependent on
the time coordinate, it is consistent with the static spacetime
because the action~(\ref{eq:Lagrangian}) possess a shift symmetry,
$\phi\to\phi+c$, and $\phi$ without derivatives does not appear
in the field equations.
This ansatz was also used to obtain black hole solutions
in the Galileon and Horndeski theories in
Refs.~\cite{Babichev:2013cya,Kobayashi:2014eva,Babichev:2016rlq,%
Babichev:2016fbg,Babichev:2017guv}.
The field equations are given by
\begin{align}
{\cal E}_\mu^\nu&=T_\mu^\nu,\label{eq:grav-eq}
\\
\nabla_\mu J^\mu& =0,\label{eq:dJ=0}
\end{align}
where
${\cal E}_{\mu\nu}$ is obtained by varying the action with respect to the metric
and
$J^\mu$ is the shift current defined by
$\sqrt{-g}J^\mu=\delta S/\delta\phi_\mu$.
The energy-momentum tensor is of the form
\begin{align}
T_\mu^\nu={\rm diag}(-\rho, P,P,P).
\end{align}
The radial component of the conservation equations,
$\nabla_\mu T^{\mu}_\nu=0$, reads
\begin{align}
P'=-\frac{\nu'}{2}(\rho+P),\label{eq:fluid}
\end{align}
where ${}':={\rm d}/{\rm d} r$.
With direct calculation we find that
\begin{align}
J^r\propto {\cal E}_{tr}.
\end{align}
Therefore, the gravitational field equation ${\cal E}_{tr}=0$
requires that $J^r$ vanishes. Then,
the field equation for the scalar field~(\ref{eq:dJ=0})
is automatically satisfied.
To write Eq.~(\ref{eq:grav-eq}) and $J^r=0$ explicitly,
it is more convenient to use $X=-e^{-\nu}v^2+e^{-\lambda}\psi'^2$
instead of $\psi$.
In terms of $X$, we have
\begin{align}
{\cal E}_t^t&=b_1 \nu''+b_2X''+\widetilde{{\cal E}}_t(\nu,\nu',\lambda, \lambda', X, X'),
\label{eq:Ett}
\\
{\cal E}_r^r&=c_1 \nu''+c_2X''+\widetilde{{\cal E}}_r(\nu,\nu',\lambda, \lambda', X, X'),
\label{eq:Err}
\\
\psi' J^r&=c_1 \nu''+c_2X''+\widetilde{{\cal E}}_J(\nu,\nu',\lambda, \lambda', X, X'),
\label{eq:Jr}
\end{align}
where
\begin{align}
b_1&=2fB_1e^{-\lambda}\left(\frac{e^{-\nu}v^2}{X}\right),
\\
b_2&=b_1
\left[\frac{e^\nu}{v^2}+\frac{B_1(3X+4e^{-\nu}v^2)}{X^2} -\frac{4e^{-\nu}v^2f_X}{Xf}\right],
\\
c_1&=-2fB_1e^{-\lambda}\left(
\frac{e^{-\nu}v^2+X}{X}\right),
\\
c_2&=c_1
\left[ \frac{B_1(3X+4e^{-\nu}v^2)}{X^2} -\frac{4e^{-\nu}v^2f_X}{Xf} \right],
\end{align}
but the explicit expression of $\widetilde{{\cal E}}_t$,
$\widetilde{{\cal E}}_r$, and $\widetilde{{\cal E}}_J$ are messy.
We see that ${\cal E}_r^r$ and $\psi' J^r$
have the same coefficients $c_1$ and $c_2$.
Moreover, we find by an explicit computation
that ${\cal E}_r^r$ and $\psi' J^r$
are linearly dependent on $\lambda'$ and their coefficients are
also the same. Therefore, by taking the combination
${\cal E}_r^r-\psi' J^r$ one can remove $\nu''$, $X''$, and $\lambda'$.
Then, the field equation ${\cal E}_r^r-\psi'J^r=P$
can be solved for $\lambda$ to give
\begin{align}
e^\lambda = {\cal F}_\lambda (\nu,\nu',X,X',P),
\label{eq:elambda}
\end{align}
where
\begin{align}
{\cal F}_\lambda=\frac{2X+B_1rX'}{2X^3(2f+r^2P)}
\bigl\{4e^{-\nu}v^2(fB_1-Xf_X)rX'&
\notag \\
+Xf\left[3B_1rX'+2X(1+r\nu')\right]
\bigr\}.&
\end{align}
Using Eq.~(\ref{eq:elambda}), we can eliminate $\lambda$ and $\lambda'$
from Eqs.~(\ref{eq:Ett}) and~(\ref{eq:Jr}).
In doing so we replace $P'$ with $\nu'$, $\rho$, and $P$
by using Eq.~(\ref{eq:fluid}).
We then obtain
\begin{align}
\psi' J^r=k_1\nu''+k_2X''+{\cal J}_1(\nu,\nu',X,X',\rho,P)=0,
\label{eq:cJ}
\end{align}
where $k_{1,2}=k_{1,2}(\nu,\nu',X,X',P)$.
The field equation ${\cal E}_t^t+\rho=0$
can also be written in the form
\begin{align}
k_1\nu''+k_2X''+{\cal J}_2(\nu,\nu',X,X',\rho,P)=0.
\end{align}
Note that we have the same coefficients $k_1$ and $k_2$.
This is due to the degeneracy conditions.
We thus arrive at a first-order equation,
${\cal J}_1={\cal J}_2$, which can be solved for $X'$ as
\begin{align}
X'={\cal F}_1(\nu,X,\rho, P)\nu'+\frac{{\cal F}_2(\nu,X,\rho,P)}{r},
\label{eq:dX}
\end{align}
where ${\cal F}_1$ and ${\cal F}_2$ are complicated.
Their explicit form is presented in Appendix~\ref{App:f1f2f3}.
Finally, we use Eq.~(\ref{eq:dX})
to eliminate $X'$ and $X''$ from Eq.~(\ref{eq:cJ}).
This manipulation also removes $\nu''$, as it should be
because the theory is degenerate.
We thus arrive at
\begin{align}
\nu'={\cal F}_3(\nu,X,\rho,\rho',P),\label{eq:dnu}
\end{align}
where the explicit expression of ${\cal F}_3$ is extremely long
and is presented in Appendix~\ref{App:f1f2f3}.
We have thus obtained our basic equations
describing
the Tolman-Oppenheimer-Volkoff system
in DHOST theories.
Given the equation of state relating $\rho$ and $P$,
one can integrate Eqs.~(\ref{eq:fluid}), (\ref{eq:dX}), and~(\ref{eq:dnu})
to determine $P=P(r)$, $\nu=\nu(r)$, and $X=X(r)$.
Equation~(\ref{eq:elambda}) can then be used to obtain $\lambda=\lambda(r)$.
\section{Nonrelativistic, weak-field limit}
Since our procedure to obtain spherically symmetric solutions
is different from that of previous works~\cite{Crisostomi:2017lbg,%
Langlois:2017dyl,Dima:2017pwp,Crisostomi:2017pjs},
it is a good exercise to check here
that one can reproduce the previous result
in a nonrelativistic, weak-field limit.
We write
\begin{align}
\nu=\delta\nu,\quad \lambda = \delta \lambda,\quad X=-v^2+\delta X,
\end{align}
and derive linearized equations for a nonrelativistic source
with $P=0$.
It is straightfoward to derive
\begin{align}
{\cal F}_\lambda &\simeq 1+r\left(\delta\nu'+\frac{2f_X}{f} \delta X'\right),
\\
{\cal F}_1&\simeq -\frac{v^2 f}{2v^2f_X+fB_1},
\\
{\cal F}_2&\simeq \frac{f(\delta X-v^2\delta\nu)}{2v^2f_X+fB_1}
-\left(\frac{v^2}{v^2f_X+fB_1}\right)\frac{r^2\rho}{8},
\\
{\cal F}_3&\simeq \frac{\delta X-v^2\delta\nu}{rv^2}-4\pi G_Nr\rho
\notag \\ &\quad +2\pi G_N\left[-\frac{12v^2f_X}{f}
+\frac{(1-3B_1)B_1f}{v^2f_X+fB_1}
\right]r\rho
\notag \\ &\quad
+2\pi G_N \Upsilon_1 r^2\rho',
\end{align}
where we introduced
\begin{align}
8\pi G_N:=\left.\frac{1}{2f(1-3B_1)+4Xf_X}\right|_{X=-v^2},\label{def:GNinf}
\end{align}
and
\begin{align}
\Upsilon_1:=\left.
-\frac{(-2Xf_X+fB_1)^2}{f(-Xf_X +fB_1)}\right|_{X=-v^2}.\label{def:Up1}
\end{align}
We will see below that $G_N$ can indeed be regarded as
the Newton constant.
We then solve the following set of equations:
\begin{align}
\delta \lambda&={\cal F}_\lambda -1,\label{eq:Newton1}
\\
\delta X'&={\cal F}_1\delta\nu'+\frac{{\cal F}_2}{r},\label{eq:Newton2}
\\
\delta \nu'&={\cal F}_3.\label{eq:Newton3}
\end{align}
Combining Eqs.~(\ref{eq:Newton2}) and~(\ref{eq:Newton3}),
the following second-order equation for $\delta\nu$ can be derived,
\begin{align}
\delta\nu''+\frac{2}{r}\delta\nu'=
2G_N\left[
\frac{M'}{r^2}+\Upsilon_1\left(\frac{M''}{2r}+\frac{M'''}{4}\right)\right],
\label{eq:ddnu}
\end{align}
where $M(r)$ is the enclosed mass defined as
\begin{align}
M(r):=4\pi\int^r\rho(s) s^2{\rm d} s.
\end{align}
Equation~(\ref{eq:ddnu}) can be integrated to give
\begin{align}
\delta\nu'=\frac{C_0}{r^2}
+2G_N\left(\frac{M}{r^2}+\frac{\Upsilon_1}{4}M''\right),
\end{align}
where $C_0$ is an integration constant.
Combining Eqs.~(\ref{eq:Newton2}) and~(\ref{eq:Newton3}) again,
we obtain
\begin{align}
\delta X &=v^2\delta \nu+ \frac{v^2C_0}{r}+\frac{2v^2G_N M}{r}
\notag \\ &\quad +
v^2G_N\left[ 1+\frac{2v^2f_X}{f}-\frac{(1-B_1)fB_1}{2(v^2f_X+fB_1)}
\right]M'.
\end{align}
Finally, we use Eq.~(\ref{eq:Newton1}) to get
\begin{align}
\delta \lambda = \frac{C_0}{r}+2G_Nr\left(
\frac{M}{r^2}-\frac{5\Upsilon_2}{4}\frac{M'}{r}
+\Upsilon_3 M''
\right),
\end{align}
where
\begin{align}
\Upsilon_2&:=\left.\frac{8Xf_X}{5f}\right|_{X=-v^2},\label{def:Up2}
\\
\Upsilon_3&:=\left.-\frac{B_1}{4}\left(
\frac{-2Xf_X+fB_1}{-Xf_X +fB_1}\right)\right|_{X=-v^2}.\label{def:Up3}
\end{align}
Imposing regularity at the center, we take $C_0=0$.
We may set $M'=0$ and $M''=0$ outside the source,
and then we have
$\delta\nu= - \delta\lambda = -2G_NM/r$,
which coincides with the solution in general relativity
if $G_N$ is identified as the Newton constant.
Gravity is modified only inside the matter source, and
we have seen that
there are three parameters, $\Upsilon_{1,2,3}$, that characterize
the deviation from the standard result.
They are subject to
\begin{align}
2\Upsilon_1^2-5\Upsilon_1\Upsilon_2-32\Upsilon_3^2=0,
\end{align}
so that actually only two of them are independent.
Note that in the case of the GLPV theory, one has $B_1=0$,
and hence $\Upsilon_2=2\Upsilon_1/5$ and $\Upsilon_3=0$.
To see that the previous result is correctly reproduced,
we perform a small coordinate transformation
\begin{align}
\varrho=r\left[1+\frac{1}{2}\int^r\frac{\delta\lambda(r')}{r'}{\rm d} r'\right].
\end{align}
The metric then takes the form
\begin{align}
{\rm d} s^2=-(1+2\Phi){\rm d} t^2 +
(1-2\Psi)\left({\rm d}\varrho^2+\varrho^2{\rm d}\Omega^2\right),
\end{align}
where
\begin{align}
\Phi =\frac{\delta\nu}{2},
\quad
\Psi = \frac{1}{2}\int^r\frac{\delta\lambda(r')}{r'}{\rm d} r'.
\label{eq:wfpsiphi}
\end{align}
Thus, we can confirm that Eq.~(\ref{eq:wfpsiphi}) reproduces
the previous result found in the
literature~\cite{Crisostomi:2017lbg,Langlois:2017dyl,Dima:2017pwp,Crisostomi:2017pjs}.
Constraints on the $\Upsilon$ parameters have been obtained from
astrophysical observations in the case of
$\Upsilon_3=0$~\cite{Koyama:2015oma,%
Sakstein:2015zoa,Sakstein:2015aac,Jain:2015edg,%
Sakstein:2016ggl,Sakstein:2016lyj,Salzano:2017qac,Babichev:2016jom,Sakstein:2016oel}.
For example, the mass-radius relation of white dwarfs yields the constraint
$-0.18<\Upsilon_1<0.27$~\cite{Jain:2015edg}.
This is valid even in the case of $\Upsilon_3\neq 0$, because
the constraint comes only from $\Phi$ in such nonrelativistic systems.
To probe $\Psi$, one needs nonrelativistic systems and/or
observations based on propagation of light rays such as
gravitational lensing, and a tighter constraint has been
imposed on $\Upsilon_1$
combining the information on $\Psi$~\cite{Sakstein:2016ggl}.
However, this relies on the assumption that $\Upsilon_3=0$.
For this reason, it is important to study
relativistic stars in theories with $\Upsilon_3\neq 0$.
Another constraint can be obtained from the Hulse-Taylor pulsar,
which limits the difference between $G_N$ and the effective
gravitational coupling for gravitational waves,
$G_{\rm GW}=1/16\pi f(-v^2)$~\cite{Jimenez:2015bwa}.
The constraint reads~\cite{Dima:2017pwp}
\begin{align}
\frac{G_{\rm GW}}{G_N}-1 \left.=
\frac{2Xf_X}{f}-3B_1\right|_{X=-v^2}
<{\cal O}(10^{-3}).\label{Ggwconstraint}
\end{align}
Note, however, that this constraint is
based on the assumption that the scalar radiation
does not contribute to the energy loss, whose validity must be
ascertained in the Vainshtein-breaking theories.
\section{Boundary conditions}
\subsection{Boundary conditions at the center}\label{subsec:center}
To derive the boundary conditions at the center of a star,
we expand
\begin{align}
&\nu=\nu_c+\frac{\nu_2}{2}r^2+\cdots,
\quad
X=X_c\left(1+\frac{X_2}{2}r^2+\cdots\right),
\notag
\\
&\rho=\rho_c+\frac{\rho_2}{2}r^2+\cdots,
\quad
P=P_c+\frac{P_2}{2}r^2+\cdots,
\end{align}
where
\begin{align}
X_c:=-e^{-\nu_c}v^2.\label{eq:Xc}
\end{align}
In deriving the relation~(\ref{eq:Xc}) we used regularity
at the center, $\phi'(t,0)=\psi'(0)=0$.
We then expand ${\cal F}_\lambda$, ${\cal F}_1$, ${\cal F}_2$,
and ${\cal F}_3$ around $r=0$ to obtain
\begin{align}
&{\cal F}_\lambda\simeq
1+a_\lambda r^2,
\quad
{\cal F}_1\simeq a_1,
\notag \\
&
{\cal F}_2\simeq a_2 r^2,\quad {\cal F}_3\simeq a_3r,
\end{align}
where
\begin{align}
a_\lambda&:=\nu_2+\frac{4X_cX_2f_X(X_c)-P_c}{2f(X_c)},
\\
a_1&:=\frac{X_cf(X_c)}{f(X_c)B_1(X_c)-2X_cf_X(X_c)},
\end{align}
while $a_2$ and
$a_3$ are similar but slightly more messy.
Equation~(\ref{eq:elambda}) implies
$e^\lambda\simeq 1+a_\lambda r^2$,
so that we find
\begin{align}
\lambda=a_\lambda r^2+\cdots.
\end{align}
Equations~(\ref{eq:dX}) and~(\ref{eq:dnu}) reduce to
the following algebraic equations
\begin{align}
X_cX_2=a_1 \nu_2+a_2,\quad \nu_2=a_3,
\end{align}
leading to
\begin{align}
\nu_2&=\frac{8\pi G_c}{3}\left(\rho_c+3P_c\right)
\notag \\
&\quad +4\pi G_c\left[\eta_1 \rho_c+\left(5\eta_2+12\eta_3\right)
P_c\right],
\\
X_2&=-8\pi G_c
\left[
2\rho_c+3P_c-\frac{4\eta_3}{\eta_1+4\eta_3}(\rho_c-3P_c)
\right],
\end{align}
where
\begin{align}
\eta_1&:=\left.-\frac{(-2Xf_X+fB_1)^2}{f(-Xf_X+fB_1)}\right|_{X=X_c},
\\
\eta_2&:=\left.\frac{8Xf_X}{5f}\right|_{X=X_c},
\\
\eta_3&:=\left.-\frac{B_1}{4}\left(
\frac{-2Xf_X+fB_1}{-Xf_X +fB_1}\right)\right|_{X=X_c},
\end{align}
and we defined the effective gravitational constant
at the center as
\begin{align}
8\pi G_c:=\left.\frac{1}{2f(1-3B_1)+4Xf_X}\right|_{X=X_c}.
\end{align}
The above quantities are defined following
Eqs.~(\ref{def:GNinf}), (\ref{def:Up1}), (\ref{def:Up2}), and~(\ref{def:Up3}),
but now they are evaluated at the center, $X=X_c$.
If gravity is sufficiently weak and the Newtonian approximation is valid,
we have $|\nu_c|\ll 1$ and hence $X_c\simeq -v^2$, leading to
$G_c\simeq G_N$ and $\eta_i\simeq \Upsilon_i$.
In this case, corrections to the standard expression for the metric
near $r=0$ are small as long as $\Upsilon_i\ll 1$.
However, if the system is in the strong gravity regime,
$X_c$ may differ significantly from $-v^2$ and therefore
the internal structure of relativistic stars may be modified
notably even in theories where only a tiny correction is expected
in the Newtonian regime.
Finally,
from Eq.~(\ref{eq:fluid}) we get
\begin{align}
P_2=-\frac{\nu_2}{2}(\rho_c+P_c).
\end{align}
Now,
given $\nu_c$ and $\rho_c$ (or $P_c$),
Eqs.~(\ref{eq:fluid}), (\ref{eq:dX}), and~(\ref{eq:dnu})
can be integrated from the center outward.
Let us move to the boundary conditions outside the star.
\subsection{Exterior solution}
For $\rho=P=0$, Eqs.~(\ref{eq:dX}) and~(\ref{eq:dnu})
reduce to the following simple set of equations:
\begin{align}
X'=0,\quad \nu'=-\frac{X+e^{-\nu}v^2}{rX}.
\end{align}
This can be integrated to give
\begin{align}
X=-v^2,\quad e^{\nu}=1-\frac{{\cal C}}{r},
\end{align}
where ${\cal C}$ is an integration constant and
we imposed that
$\psi'\to 0$ as $r\to \infty$.
We then have
\begin{align}
e^\lambda &= {\cal F}_\lambda(\nu,\nu',-v^2,0,0)=1+\nu' r
\notag \\
&=\left(1-\frac{{\cal C}}{r}
\right)^{-1}.
\end{align}
The exterior metric is thus obtained exactly
without linearizing the equations,
which coincides with the Schwarzschild solution in general relativity.
The stellar interior must be matched to this exterior solution.
It will be convenient to write
\begin{align}
{\cal C}=2G_N\mu,
\end{align}
because then $\mu$ is regarded as the mass of the star.
\subsection{Matching at the stellar surface}
\label{subsec:match}
The stellar surface, $r=R$, is defined by $P(R)=0$.
The induced metric is required to be continuous across the surface,
so that $\nu$ must be continuous there.
Since $X=\phi_\mu\phi^\mu$ is a spacetime scalar,
it is reasonable to assume that
this quantity is also continuous across the surface.
We thus have the two conditions imposed at $r=R$:
\begin{align}
e^{\nu(R)}&= 1-\frac{2G_N\mu}{R},\label{jc:2}
\\
X(R)&=-v^2.\label{jc:1}
\end{align}
We tune the central value $\nu_c$ in order for the solution
to satisfy Eq.~(\ref{jc:1}). The second condition~(\ref{jc:2})
is used to determine the integration constant $\mu$.
As we have seen in the previous section,
$\mu=M(R)$ in the nonrelativistic, weak-field limit.
In the present case, however, $\mu$ does not necessarily
coincide with $M(R)$ because the nonrelativistic and weak-field
approximations are not justified
in general for our interior solutions.
Note that in general we may have $\rho_-':=\rho'(R_-)\neq 0$
while $\rho'(R_+)=0$, where
$R_\pm:=\lim_{\varepsilon\to 0} R(1\pm\varepsilon)$.
As a particular feature of the DHOST theories
with partial breaking of the Vainshtein mechanism,
the right-hand side of Eq.~(\ref{eq:dnu}) depends on
$\rho'$~\cite{Babichev:2016jom}. This implies that $\nu'$ is discontinuous
across the stellar surface.
Then, from Eq.~(\ref{eq:dX}) we see that
$X'$ is also discontinuous across the surface.
Furthermore, since the right hand side of Eq.~(\ref{eq:elambda})
depends on $\nu'$ and $X'$,
$\lambda$ is also discontinuous there in general.
With some manipulation we see that
\begin{align}
&1-e^{[\lambda(R_-)-\lambda(R_+)]/2}
\notag \\
&=\pi G_N\rho_-'R^3 B_1\left.\left[2e^{-\nu(R)}-\frac{B_1f}{B_1f-Xf_X}\right]
\right|_{X=-v^2},
\end{align}
which shows that $\lambda$ is nevertheless continuous in theories with $B_1=0$.
However, it is found that
\begin{align}
&X'(R_+)-X'(R_-)
\notag \\ &
=
2\pi G_N\rho_-'R^2X
\left.\left[2e^{-\nu(R)}-\frac{B_1f}{B_1f-Xf_X}\right]
\right|_{X=-v^2},
\end{align}
and therefore $X'$ is discontinuous even if $B_1=0$.
This is also the case for $\nu'$.
In the next section,
we will show our numerical results in which one can find these
discontinuities.
\section{Numerical results}
As a specific example, we study the model of Ref.~\cite{Crisostomi:2017pjs}:
\begin{align}
f=\frac{M_{\rm Pl}^2}{2}+\alpha X^2,\quad A_3=-8\alpha-\beta,
\end{align}
where $\alpha$ and $\beta$ are constants.
Note that we are using the notation such that $8\pi G_N\neq M_{\rm Pl}^{-2}$.
We have
\begin{align}
B_1=-\frac{\beta X^2}{2(M_{\rm Pl}^2+2\alpha X^2)},
\end{align}
and hence
the model with $\beta\neq 0$ is more general than the GLPV theory.
For this choice of the functions the degeneracy conditions~(\ref{eq:degn1})
and~(\ref{eq:degn2}) leads to
\begin{align}
A_4&=\frac{M_{\rm Pl}^2(8\alpha+\beta)+(16\alpha^2-6\alpha\beta-\beta^2/4)X^2}{M_{\rm Pl}^2+2\alpha X^2},
\\
A_5&=\frac{\beta(8\alpha+\beta)X}{M_{\rm Pl}^2+2\alpha X^2}.
\end{align}
This model, with the addition of the lower order Horndeski terms,
admits viable self-accelerating cosmological solutions~\cite{Crisostomi:2017pjs},
and therefore is interesting.
Hereafter we will use the dimensionless parameters
defined as
\begin{align}
\overline{\alpha}:=\frac{\alpha v^4}{M_{\rm Pl}^2},\quad
\overline{\beta}:=\frac{\beta v^4}{M_{\rm Pl}^2}.
\end{align}
The parameters
that characterize Vainshtein breaking in the Newtonian regime,
$\Upsilon_i$, can then be estimated as
\begin{align}
\Upsilon_i \sim \overline{\alpha},\overline{\beta}.
\end{align}
From Eq.~(\ref{Ggwconstraint}) we also estimate
\begin{align}
\frac{G_{\rm GW}}{G_N}-1\sim \overline{\alpha},\overline{\beta}.
\end{align}
Therefore, by taking sufficiently small $\overline{\alpha}$ and
$\overline{\beta}$ (say, ${\cal O}(10^{-3})$),
current constraints can be evaded.
In the following numerical calculations,
we will employ such small values of the parameters.
The equation of state we use is given by
\begin{align}
\rho=\left(\frac{P}{K}\right)^{1/2}+P,
\end{align}
with $K=7.73\times 10^{-3}\, (8\pi G_N)^3M_{\odot}^2$
($K=123\,M_{\odot}^2$ in the units where $G_N=1$),
which has been used frequently in the modifed gravity
literature~\cite{Cisterna:2015yla,Cisterna:2016vdx,Maselli:2016gxk,Silva:2016smx,%
Babichev:2016jom}.
With this simple equation of state we focus on
the qualitative nature of the solutions.
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=80mm]{plot_valpha_rhoM.eps}
\end{center}
\caption{The mass ($\mu$) versus central density
diagram for $\overline{\beta}=0$.
The parameters are given by
$\overline{\alpha} = 2\times 10^{-3}, 10^{-3}, 4\times 10^{-4},%
2\times 10^{-4}, 10^{-4}, 0$ (GR) (solid lines, from top to bottom), and
$\overline{\alpha}=-10^{-4}, -2\times 10^{-4}, -4\times 10^{-4}, -10^{-3}, -2\times 10^{-3}$
(dashed lines, from top to bottom).
\label{fig:plot_valpha_rhoM.eps}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=80mm]{plot_valpha_RM.eps}
\end{center}
\caption{The mass ($\mu$) versus radius diagram for $\overline{\beta}=0$.
The parameters are given by
$\overline{\alpha} = 2\times 10^{-3}, 10^{-3}, 4\times 10^{-4},%
2\times 10^{-4}, 10^{-4}, 0$ (GR) (solid lines, from right to left), and
$\overline{\alpha}=-10^{-4}, -2\times 10^{-4}, -4\times 10^{-4}, -10^{-3}, -2\times 10^{-3}$
(dashed lines, from top to bottom).}%
\label{fig:plot_valpha_RM.eps}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=80mm]{plot_vbeta_rhoM.eps}
\end{center}
\caption{The mass ($\mu$) versus central density
diagram for $\overline{\alpha}=0$.
The parameters are given by
$\overline{\beta} = 2\times 10^{-3}, 10^{-3}, 4\times 10^{-4},%
2\times 10^{-4}, 10^{-4}, 0$ (GR) (solid lines, from top to bottom), and
$\overline{\beta}=-10^{-4}, -2\times 10^{-4}, -4\times 10^{-4}, -10^{-3}, -2\times 10^{-3}$
(dashed lines, from top to bottom).
\label{fig:plot_vbeta_rhoM.eps}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=80mm]{plot_vbeta_RM.eps}
\end{center}
\caption{The mass ($\mu$) versus radius
diagram for $\overline{\alpha}=0$.
The parameters are given by
$\overline{\beta} = 2\times 10^{-3}, 10^{-3}, 4\times 10^{-4},%
2\times 10^{-4}, 10^{-4}, 0$ (GR) (solid lines, from right to left), and
$\overline{\beta}=-10^{-4}, -2\times 10^{-4}, -4\times 10^{-4}, -10^{-3}, -2\times 10^{-3}$
(dashed lines, from top to bottom).}%
\label{fig:plot_vbeta_RM.eps}
\end{figure}
We start with the theories with $\overline{\beta}=0$
and focus on the effect of $\overline{\alpha}$.
Figures~\ref{fig:plot_valpha_rhoM.eps}
and~\ref{fig:plot_valpha_RM.eps}
show the mass ($\mu$) versus central density relation and
the mass versus radius relation, respectively.
In all cases $\overline{\alpha}$ is taken to be very small so that
the Vainshtein-breaking effect is not significant in the Newtonian regime.
It can be seen that
for fixed $\rho_c$ or $R$ the mass is larger (smaller)
for $\overline{\alpha}>0$ ($\overline{\alpha}<0$)
than in the case of general relativity (GR).
Interestingly, there is a maximum central density, $\rho_{c,{\rm max}}$,
above which no solution can be found.
This property is similar to what was found
in Ref.~\cite{Cisterna:2015yla},
where the subclass of the Horndeski theory
having derivative coupling to the Einstein tensor was studied.
For $\overline{\alpha}\lesssim 2\times 10^{-4}$,
we see that $\mu<\infty$ as $\rho_c\to\rho_{c,{\rm max}}$,
but for $\overline{\alpha}\gtrsim 2\times 10^{-4}$
we find that $\mu\to \infty$ and $R\to\infty$
as $\rho_c\to\rho_{c,{\rm max}}$.
Therefore, in the latter case there are solutions at high densities
that are very different from relativistic stars in GR.
Note that this occurs even for a tiny value of $\overline{\alpha}$.
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=80mm]{plot_valpha_metric.eps}
\end{center}
\caption{Metric components $e^{\nu}$ and $e^{-\lambda}$ as a function of $r$.
The parameters are given by $\overline{\alpha}=2\times 10^{-3}, \overline{\beta}=0$
(dashed lines)
and $\overline{\alpha}=-2\times 10^{-3}, \overline{\beta}=0$
(dotted lines). Solid lines represent the result of GR.}%
\label{fig:plot_valpha_metric.eps}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=80mm]{plot_valpha_x.eps}
\end{center}
\caption{$X/v^2$ as a function of $r$.
The parameters are given by $\overline{\alpha}=2\times 10^{-3}, \overline{\beta}=0$
(dashed line)
and $\overline{\alpha}=-2\times 10^{-3}, \overline{\beta}=0$
(dotted line).}%
\label{fig:plot_valpha_x.eps}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=80mm]{plot_vbeta_metric.eps}
\end{center}
\caption{Metric components $e^{\nu}$ and $e^{-\lambda}$ as a function of $r$.
The parameters are given by $\overline{\alpha}=0, \overline{\beta}=2\times 10^{-2}$
(dashed lines)
and $\overline{\alpha}=0, \overline{\beta}=-2\times 10^{-2}$
(dotted lines). Solid lines represent the result of GR.}%
\label{fig:plot_vbeta_metric.eps}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=80mm]{plot_vbeta_x.eps}
\end{center}
\caption{$X/v^2$ as a function of $r$.
The parameters are given by $\overline{\alpha}=0, \overline{\beta}=2\times 10^{-2}$
(dashed line)
and $\overline{\alpha}=0, \overline{\beta}=-2\times 10^{-2}$
(dotted line).}%
\label{fig:plot_vbeta_x}
\end{figure}
Next, we fix $\overline{\alpha}=0$ and
draw the same diagrams for different (small)
values of $\overline{\beta}$.
The results are presented in Figs.~\ref{fig:plot_vbeta_rhoM.eps}
and~\ref{fig:plot_vbeta_RM.eps}, which are
seen to be qualitatively similar to
Figs.~\ref{fig:plot_valpha_rhoM.eps}
and~\ref{fig:plot_valpha_RM.eps}, respectively.
Therefore, although $\overline{\beta}$ is supposed to
signal the ``beyond GLPV'' effects, they are not manifest
and the roles of the two parameters
$\overline{\alpha}$ and $\overline{\beta}$
in relativistic stars
are qualitatively similar.
We have performed numerical calculations
for more general cases with
$\overline{\alpha}\neq 0$ and $\overline{\beta}\neq 0$,
and confirmed that they also
lead to qualitatively similar results.
Just for reference
some examples of radial profiles of
the metric and $X$ in the stellar interior are presented in
Figs.~\ref{fig:plot_valpha_metric.eps} and~\ref{fig:plot_valpha_x.eps}
for $(\overline{\alpha},\overline{\beta})=(\pm 2\times 10^{-3},0)$
and in Figs.~\ref{fig:plot_vbeta_metric.eps} and~\ref{fig:plot_vbeta_x}
for $(\overline{\alpha},\overline{\beta})=(0,\pm 2\times 10^{-3})$.
As mentioned in Sec.~\ref{subsec:match},
one can find that $X'$ is
nonvanishing at the surface, leading to the discontinuity of $X'$, and
that $e^{-\lambda}$ does not agree with $e^{\nu}$ there
in Fig.~\ref{fig:plot_vbeta_metric.eps}, indicating the discontinuity
of $e^{-\lambda}$ since $e^{\nu}$ must be continuous.
\section{Conclusions}
In this paper, we have studied the Tolman-Oppenheimer-Volkoff system in
degenerate higher-order scalar-tensor (DHOST) theory
that is consistent with the GW170817 constraint on
the speed of gravitational waves.
Although the field equations are apparently of higher order,
we have reduced them to a first-order system
by combining the different components.
This is possible because the theory we are considering is degenerate.
In DHOST theories that are more general than Horndeski,
breaking of the Vainshtein screening mechanism occurs
inside matter~\cite{Kobayashi:2014ida,Crisostomi:2017lbg,Langlois:2017dyl,Dima:2017pwp},
which would modify the interior structure of stars.
Assuming a simple concrete model of DHOST theory with two parameters
and the equation of state, we have solved numerically the field equations.
The parameters were chosen so that the Vainshtein-breaking effect
in the Newtonian regime is suppressed by
the factor $\Upsilon_i\lesssim 10^{-3}$.
Nevertheless, we have found a possible large modification
in the mass-radius relation. This is significant in particular
at densities as high as the maximum above which no solutions
can be obtained.
In this paper, we have focused on the rather qualitative nature
of relativistic stars in the DHOST theory, but
it would be important to employ more realistic equations of state
for testing the theory against astrophysical observations.
This is left for further study.
It would be also interesting to explore to what extent
the modification to the stellar structure depends on
the concrete form of the DHOST Lagrangian.
We hope to come back to this question in a future study.
\acknowledgments
This work was supported in part by
MEXT KAKENHI Grant Nos.~JP15H05888, JP16H01102 and JP17H06359 (T.K.);
JSPS KAKENHI Grant Nos.~JP16K17695 (T.H.) and JP16K17707 (T.K);
and MEXT-Supported Program for the Strategic Research Foundation at Private Universities,
2014-2018 (S1411024) (T.H. and T.K.).
\clearpage
\begin{widetext}
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train/arxiv
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\section{Introduction}
In non-relativistic topological matter, effectively quasi-relativistic description of low-energy quasiparticles with linear spectrum phenomena may emerge \cite{Volovik2003, Horava05}. In particular, in three spatial dimensions at a generic (two-fold) degenerate fermion band crossing at momentum $\mathbf{p}_W$, the Hamiltonian is of the Weyl form \cite{Herring1937, Abrikosov1971, NielsenNinomiya83, Volovik2003}
\begin{align}
H_{W} = \sigma^a e^{i}_a (p- p_W)_i + \cdots \label{eq:WeylHamiltonian}
\end{align}
where the $e^{i}_a = \partial_{p_i} H(\mathbf{p})\vert_{p_W}$ are the linear coefficients of the Hermitean Pauli matrices $\sigma^a$, $a=1,2,3$, close to the Weyl node(s) at $\mathbf{p}_W$. The net chirality {$\sum_{\{\mathbf{p}_W\}} \textrm{sgn}(\det e^{i}_a)$ vanishes \cite{NielsenNinomiya83}. For slowly varying parameters in the operator $\sigma^ae^{\mu}_a i \partial_{\mu}\equiv i\partial_t -H_W$, the semi-classical fields $e^{\mu}_a(x) = \{e^0_{a}, e^i_a\}$ are promoted to background spacetime tetrad fields with dimensions of unity for temporal indices $e_a^0$ and velocity for spatial $e^i_a$. At the level of the Hamiltonian, the shift of the Weyl node $\mathbf{p}_W$ acts as an emergent (axial) gauge field with emergent Lorentz symmetry to the linear order but the linear expansion \eqref{eq:WeylHamiltonian} at $\vek{p}_W$ is valid at much lower scales, however. If the fermions are charged, the fermions can in addition couple to the electromagnetic vector potential via minimal coupling.
These background fields imply the chiral anomaly for the low-energy massless quasiparticles. For the applications of the chiral anomaly in Weyl semimetal and Weyl superfluids/superconductors, see e.g. \cite{NielsenNinomiya83, Volovik1986a, Volovik2003, ZyuzinBurkov12}. In particular, the non-trivial coordinate dependence (torsion) related to the tetrads $e_a^{\mu}(x)$ in \eqref{eq:WeylHamiltonian} can lead to the gravitational Nieh-Yan (NY) anomaly
\cite{NiehYan1982a,NiehYan1982b,Nieh2007,Yajima96,ChandiaZanelli97a,ChandiaZanelli97b,ChandiaZanelli98, Parrikar2014, FerreirosEtAl19,Nissinen2019}. Nevertheless, the NY contribution to the anomaly has remained contentious and subtle due to presence of a dimensionful ultra-violet (UV) scale $\Lambda^2$ with canonical dimensions of momentum, as required by the canonical dimensions of $e^a_{\mu}$. See, however, Ref. \onlinecite{Nissinen2019} and the discussion below.
Here we discuss the temperature corrections to the gravitational NY anomaly and different finite temperature terms in the hydrodynamic momentum transport of chiral Weyl superfluids. The results apply for the chiral superconductors as well, when the electromagnetic potential is added $\mathbf{v}_s \to \vek{v}_s-e\vek{A}/m$ in some convenient gauge. We show that these corrections to the momentum currents include, among various other similar terms, a term originating from the torsional Nieh-Yan anomaly. For all such low-energy temperature corrections proportional to $T^2$ in the free energy, the prefactors in the corresponding terms are dimensionless, i.e. do not depend on the details of the microscopic physics, but are fully determined by geometry, topology and the number of effective fermionic species \cite{VolovikZelnikov2003,Volovik2003}. Specifically, we compare the finite temperature corrections to the lowest order gradient terms in free-energy of the chiral superfluid, and identify the contribution from the chiral Nieh-Yan anomaly in the low-energy quasirelativistic Weyl superfluid, which leads us to conjecture that the finite temperature NY anomaly term can be similarly universal in general Weyl systems. Finally we compare these results to relativistic Weyl fermions with positive and negative branches at zero momentum.
\section{Torsional anomaly}
For spacetimes with torsion (and curvature), Nieh and Yan \cite{NiehYan1982a,NiehYan1982b,Nieh2007} introduced the 4-dimensional invariant
\begin{equation}
N=\mathcal{T}^a \wedge \mathcal{T}_a - e^a \wedge e^b\wedge R_{ab} \,
\label{N}
\end{equation}
where $e^a = e^a_{\mu} dx^{\mu}$ is the local tetrad 1-form field and $\mathcal{T}^a = de^a + \omega^a_{\ b} \wedge e^b$ and $R^{ab}=d\omega^a_{b} + \omega^a_{\ c} \wedge \omega^c_{\ b}$, in terms of the tetrad and spin-connection $\omega^a_{\ b} = \omega^a_{\mu b} dx^{\mu}$. As usual, the spacetime metric follows as $g_{\mu\nu} = e^{a}_{\mu} e^{b}_{\nu}\eta_{ab}$, where $\eta_{ab}$ is the local orthonormal (Lorentz) metric. This invariant can be written, using the associated Bianchi identities, as
\begin{equation}
N=dQ \,\,, \,\, Q=e^a\wedge {\cal T}_a \,.
\label{N2}
\end{equation}
$N$ is a locally exact 4-from independent from $\textrm{tr} (R \wedge R)$ and the dual of the scalar curvature $\sqrt{g}\mathcal{R}$ in the presence of non-zero torsion. It can be associated with a difference of two topological terms, albeit in terms of an embedding to five dimensions \cite{ChandiaZanelli97a,ChandiaZanelli97b,ChandiaZanelli98}. In terms of four-dimensional chiral fermions on such a spacetime, it has been suggested that this invariant contributes to the axial anomaly, i.e. the anomalous production of the chiral current:
\begin{equation}
\partial_\mu (e j_5^\mu) = \frac{\Lambda^2}{4\pi^2} N({\bf r},t) \,, \label{eq:NYterm}
\end{equation}
where $e=\det e^a_{\mu}$ and $e j^{\mu}_5$ is the axial current (pseudotensor) density and the non-universal parameter $\Lambda$ has dimension of relativistic momentum (mass) $[\Lambda]=[1/L] = [M]$ and is determined by some ultraviolet (UV) energy scale.
Given the anomaly term \eqref{eq:NYterm}, there has been several attempts to consider the Nieh-Yan anomaly in condensed matter systems with Weyl fermions like \eqref{eq:WeylHamiltonian}, see e.g. \cite{Parrikar2014, SunWan2014, FerreirosEtAl19, Nissinen2019}. However, in non-relativistic systems the relativistic high-energy cut-off $\Lambda$ is not a well defined parameter and, moreover, can be anisotropic. The complete UV theory is, of course, non-Lorentz invariant and the linear, quasirelativistic Weyl regime is valid at much lower scales. Moreover, the anomalous hydrodynamics of superfluid $^3$He at zero temperature suggests that the chiral anomaly is either completely exhausted by the emergent axial gauge field corresponding to the shift of the node or, conversely, the gravitational NY anomaly term arising from the tetrad and spin-connection for local Lorentz invariance along the uniaxial symmetry direction.
It was recently shown in Ref. \onlinecite{Nissinen2019} that the low-energy theory satisfies the symmetries and conservations laws related to an emergent quasirelativistic spacetime with torsion and $\Lambda$ is determined from the UV-scale where the linear Weyl approximation breaks down, as dictated by the underlying $p$-wave BCS Fermi superfluid. Here we will consider the temperature corrections to the anomaly and the hydrodynamic free-energy in the chiral Weyl superfluid, with the expectation that $\Lambda = T$, $k_B=1$, with some dimensionless prefactor $\gamma$, in units where the tetrads have canonical dimensions of velocity, or the relevant ``speed of light" is set unity.
In terms of effective low-energy actions, the fully relativistic analogs work unambiguously only for terms in the effective action with dimensionless coefficients. Perhaps the most well-known example being the 2+1-dimensional topological Chern-Simons (CS) terms describing the Quantum Hall effect. Gravitational Chern-Simons terms can be similarly quantized in terms of chiral central charge which has relation to thermal transport and the boundary conformal field theory \cite{Volovik90, ReadGreen01}. The CS action was recently generalized to 3+1d (and higher odd space dimensions) crystalline topological insulators, using so-called dimensionful elasticity tetrads $E$ with dimension $[E]=[1/L]=[M]$. Topological polarization terms can also be written down \cite{SongEtAl2019} by dimensional descent. The ensuing higher dimensional Chern-Simons and polarization terms are expressed as the mixed responses $E\wedge A\wedge d A$ and $E \wedge E \wedge dA$ with quantized dimensionless coefficients \cite{NissinenVolovik2018b, NissinenVolovik2018, SongEtAl2019}.
Another such example is the temperature correction to curvature effects, with $\delta S_{\rm eff}$ = $\int T^2{\cal R}$ in the low-energy action \cite{VolovikZelnikov2003}. This represents the analog of the gravitational coupling (inverse Newton constant) in the low-energy action, where the curvature scalar ${\cal R}$ is the effective scalar spacetime curvature. Since $[T]^2[{\cal R}]=[M]^4$, the coefficient of this term is dimensionless, and can be given in terms of universal constants: it is fully determined by the number of the fermionic and bosonic species in the effective theory on flat background, and thus works both in relativistic and non-relativistic systems \cite{VolovikZelnikov2003}. The same universal behavior takes place with the terms describing the chiral magnetic and chiral vortical effects in Weyl superfluid $^3$He-A, where the coefficients are dimensionless \cite{VolovikVilenkin2000,Volovik2003, BasarEtAl14}. Similarly, it has been observed that the coefficient of the $\textrm{tr}(R\wedge R)$ graviational anomaly in chiral Weyl systems affects the thermal transport coefficients in flat space \cite{LandsteinerEtAl11, LoganayagamSurowka12, JensenEtAl13, Landsteiner2014, Lucas2016, GoothEtAl17, StoneKim18}. These coefficients are fundamental, being determined by the underlying degrees of freedom in addition to symmetry, topology and geometry. From this perspective especially, since the NY form is second order in derivatives and can be computed from linear response, our findings are very interesting and warrant further research.
Our goal in the present paper is to separate different $T^2$ contributions in the hydrodynamic free-energy of Weyl superfluids in order to identify the terms responsible for different relativistic phenomena, including a term from the thermal Nieh-Yan anomaly, as well as gravitational terms of the form $\int T^2({\cal R} + \mu^2)$, where $\mu$ is the chiral chemical potential $\mu \ll T$.
\section{Temperature correction to the Nieh-Yan term} \label{sec:TempNY}
The relativistic zero-temperature anomaly term in the axial current production $\Lambda^2({\cal T}^a \wedge {\cal T}_a+ e^a\wedge e^b\wedge R_{ab})$ is still not confirmed in general, see however Ref. \onlinecite{Nissinen2019}. On one hand the UV cut-off parameter $\Lambda$ is not well-defined in relativistic field theory with fundamental chiral fermions. On the other hand, such a cut-off is not in general available in non-relativistic matter with quasi-relativistic low-energy chiral fermions and can be anisotropic \cite{Nissinen2019} or even zero. However, the term of the form $\gamma T^2({\cal T}^a\wedge {\cal T}_a+ e^a\wedge e^b\wedge R_{ab})$ has the proper dimensionality $[M]^{4}$, and its prefactor $\gamma$ could be a universal constant in canonical units, being expressed via some invariant related to the degrees of freedom.
For concreteness, we focus on the finite temperature Nieh-Yan anomaly in chiral $p$-wave Weyl superfluid (such as $^3$He-A) with
\begin{equation}
\partial_\mu (ej_5^\mu) = \gamma T^2 N({\bf r},t) \,,
\label{T2NiehYan}
\end{equation}
and check whether the dimensionless parameter $\gamma$ can be universal. We now use the result obtained by Khaidukov and Zubkov \cite{Zubkov2018} and Imaki and Yamamoto \cite{Imaki2019} for the finite temperature contribution to the chiral current. For a single (complex) chiral fermion, one has for the chiral current
\begin{equation}
j_5^k= - \frac{T^2}{24} \epsilon^{kij}\mathcal{T}^0_{ij} \,.
\label{j5}
\end{equation}
We assume that this current can be covariantly generalized to the 4-current:
\begin{equation}
e j^\mu_5= - \frac{T^2}{24} \epsilon^{\mu\nu\alpha\beta} e_{\nu a}\mathcal{T}^a_{\alpha\beta} \,.
\label{j5general}
\end{equation}
Then one obtains the divergence
\begin{equation}
\partial_{\mu} (e j^\mu_5)= - \frac{T^2}{48} \epsilon^{\mu\nu\alpha\beta} \mathcal{T}_{a\mu \nu}\mathcal{T}^a_{\alpha\beta}
\,.
\label{j5nonconservation}
\end{equation}
In the presence of curvature $R(\omega)$, this becomes the temperature correction to the full Nieh-Yan term in Eq. (\ref{T2NiehYan}), where now the non-universal cut-off $\Lambda$ is substituted by the well defined temperature $T$, and the dimensionless parameter $\gamma=1/12$:
\begin{equation}
\partial_\mu (e j^\mu_5) =- \frac{T^2}{12} N({\bf r},t) \,.
\label{j5nonconservationR}
\end{equation}
Note it is possible that the local relativistic (Tolman) temperature $T=T_0/\vert e^0_t \vert$ enters the local anomaly, while the constant $T_0$ is the global equilibrium temperature of the condensed matter system \cite{VolovikZelnikov2003}. In \eqref{eq:WeylHamiltonian} we have simply $e^0_t = -1$.
\section{From relativistic physics to chiral Weyl superfluid} \label{sec:superfluid_LL}
In the presence of a finite Weyl node $p_{W}\neq 0$, the chiral anomaly for the chiral current leads to the anomalous production of the linear momentum \cite{Volovik2003, Nissinen2019}. Even though chiral current is not well-defined at high-energy, the spectral flow of chiral quasiparticles is accompanied by the spectral flow of the linear momentum ${\bf p}_W$ of the Weyl point. In $^3$He-A there are two (spin-degenerate) Weyl points with opposite chirality and opposite momenta, ${\bf p}_{W\pm}=\pm p_F{\hat{\bf l}}$, where $\hat{\bf l}$ is the unit vector of the orbital
momentum of the superfluid. In particular, the anomalous production of the linear momentum density $n_{\pm}$ at nodes of the two opposite Weyl points sum-up leading to
\begin{equation}
\dot {\bf P}_{\rm anom} = -p_F \hat{\bf l}\, (\partial_t n_5),
\label{anomalousProductionP}
\end{equation}
where $n_5=n_+-n_-$ is the chiral density. The corresponding quasirelativistic momentum density of the Weyl fermions, valid in the vicinity of the node, is\cite{Nissinen2019}
\begin{equation}
{\bf P}_{\textrm{NY-node}} =- p_F \hat{\bf l} e j^0_5\,.
\label{NYmomentum}
\end{equation}
Thus Eq. \eqref{j5nonconservation} gives the temperature correction to this
anomalous momentum production, leading to a mass current due to thermal Nieh-Yan anomaly. Next we discuss this in detail for the quasiparticles and the superfluid vacuum in the non-relativistic chiral Weyl $p$-wave superfluid, using a Landau level model for the currents \cite{Volovik85,Nissinen2019}. For relativistic Weyl fermions, see \cite{Stone2019}. For considerations of similar temperature effects in chiral Weyl superfluid, see Ref. \cite{KobayashiEtAl18}
\subsection{Nieh-Yan term from the hydrodynamics of chiral Weyl superfluid}
It is known that the hydrodynamics of chiral gapless $^3$He-A experiences momentum anomalies related to the spectral flow through the Weyl points. Let us express Eqs. \eqref{anomalousProductionP}, \eqref{NYmomentum} in terms of the hydrodynamic variables and quasiparticles of the chiral superfluid. The Weyl fermions arise from the BdG Hamiltonian close to the nodes,
\begin{align}
H_{\rm BdG}(-i \partial) = \left( \begin{matrix} \epsilon(-i\partial) & \frac{1}{2i}\{\partial_i,\Delta^i \} \\ \frac{1}{2i}\{\partial_i,\Delta^{*i} \} & -\epsilon(i \vek{\partial}) \end{matrix}\right)
\end{align}
Here $\epsilon(\mathbf{p}) = \frac{p^2-p_F^2}{2m^*}$ is the normal state dispersion minus the Fermi level $\mu_F$; $\Delta_i = c_\bot(\hat{\bf m}+i\hat{\bf n})$ is the order parameter in the $p+ip$ chiral superfluid;
$\hat{\bf l}=\hat{\bf m}\times\hat{\bf n}$ is the unit vector in the direction of the orbital angular momentum of Cooper pairs; anisotropic node velocities are $c_\parallel=v_F$ and $c_\bot =\Delta_0/p_F$, constituting effective speeds of light in Weyl equation along $\hat{\bf l}$
and in transverse directions, respectively. In the weak coupling BCS theory $c_\bot \ll c_\parallel$, in $^3$He-A their ratio is of order $10^{-3}$. In a chiral superconductor we in addition perform the minimal coupling $\epsilon(-i\partial_i) \to \epsilon(i D_i) +eA_0$, where $D_i=\partial_i-eA_i$ is the gauge covariant derivative and $A_\mu$ is the electromagnetic potential. Equivalently this amounts to $\mathbf{v_s} \to \mathbf{v}_s-\frac{e\vek{A}}{m}$ in the free-energy.} In what follows we ignore the superfluid velocity (giving rise to spin-connection in addition to tetrads) in the anomaly considerations. Then the only hydrodynamic variable appearing in the torsion is the unit vector of the orbital momentum $\hat{\bf l}$ \cite{Nissinen2019}. Our results can afterwards then generalized to include the superfluid velocity appearing in the free-energy.
The Weyl nodes are at $\mathbf{p}_{W\pm} = \pm p_F \unitvec{l}$. Expanding the Hamiltonian as $H_{\rm BdG} \simeq \sigma^a e^i_a (\hat{p}-p_W)_i$, the vierbein ${\bf e}^i_a$ in the vicinity of the Weyl point take the form:
\begin{eqnarray}
e^{\mu}_a = \{e^t_{a},{\bf e}^i_a\} &=&
\,\left(\begin{array}{cc}1 & 0\\
0 & c_\bot \hat{\bf m} \\ 0 & c_\bot \hat{\bf n} \\ 0 &c_\parallel \hat{\bf l}\end{array} \right) , \quad a = 0,1,2,3.
\end{eqnarray}
For the inverse vierbein ${\bf e}^i_a {\bf e}^a_j = \delta^i_j$ we have
\begin{eqnarray}
e^a_{\mu} = \{e^a_t, {\bf e}^a_i \} &=&
\,\left(\begin{array}{cccc}1 & 0 & 0&0 \\
0 & \frac{1}{c_\bot}\hat{\bf m} & \frac{1}{c_\bot}\hat{\bf n} & \frac{1}{c_\parallel}\hat{\bf l}\end{array} \right). \label{Eu}
\end{eqnarray}
In order to compute the thermal contribution from the quasiparticles in the presence of torsion, we assume that the tetrad gives rise to a constant torsion via $\unitvec{m} = \unitvec{x}$, $\unitvec{n} = \unitvec{y} - T_B x \unitvec{z}$, where $T_B \ll 1$ is a perturbation \cite{Volovik85, Nissinen2019}. In this case, the 3d spectrum organizes into one-dimensional states on 2d Landau levels (LLs) \cite{Volovik85,Volovik2003,Parrikar2014, Stone2019, Laurila20}, where the relevant spatial torsion
\begin{align}
\frac{1}{2}\epsilon^{ijk}\mathcal{T}^3_{jk} e_3^i = c_{\parallel} \unitvec{l}\cdot ( \nabla \times \frac{\unitvec{l}}{c_{\parallel}}) \equiv T_B,
\end{align}
where $T_B = \unitvec{l}\cdot\nabla\times\unitvec{l} = \mathcal{T}^z_{xy}$, is playing the role of effective magnetic field. Only the gapless lowest LLs are relevant, while the gapped levels, with gap $\sim c_\perp \sqrt{p_FT_B}$ are particle-hole symmetric and cancel out \cite{Volovik85, BalatskiiEtAl86}. We approximate $\epsilon(\mathbf{p}) = \frac{p^2-p_F^2}{2m} \simeq \frac{p_z^2-p_F^2}{2m} = \epsilon(p_z)$, strictly valid when $p_{\bot} \ll m c_{\perp}$ which coincides with the linear Weyl regime $\epsilon_{p_z} = \pm v_F(p_z-p_F)$. However, we can fix the anisotropic dispersion to $\epsilon(p_z)$ for all momenta to obtain a convenient UV-complete model, for which the following analysis for the total vacuum current is valid, as long as the correct cutoff for the linear Weyl regime is maintained for $^3$He-A, see Appenix \ref{Appendix2}. The dispersion of the lowest LL becomes $E_{n=0} = -\textrm{sgn}(p_z T_B)\epsilon(p_z)$, while density of states per lowest LL per three-momentum $p_z$ becomes $n_{\rm LL}(p_z) = \frac{\abs{p_z T_B}}{4\pi^2}$. The lowest LL is particle-like (hole-like) for $p_W = \mp p_F\unitvec{l}$. For more on this non-relativistic model with torsion, see \cite{Volovik85, BalatskiiEtAl86,Nissinen2019, Laurila20}
The total (chiral) quasiparticle current along $\unitvec{l}=\frac{e^i_3}{v_F}\approx \unitvec{z}$ becomes
\begin{align}
\mathbf{P}^{\rm qp}\cdot\unitvec{l} &= -2 \int_{0}^{\infty} N_{\rm LL}(p_z) dp_z~ p_z n_{F}(\epsilon_{p_z}-\mu_F) \nonumber\\
&=-\left(\frac{p_F^3}{6\pi^2}+\frac{p_F T^2}{6 c_{\bot}^2} \right) \unitvec{l}\cdot\nabla\times\unitvec{l} \label{eq:LL_current} \\
&= \vek{j}^{\rm vac}_{\rm anom\parallel} + \vek{j}^{\rm qp}_{\rm anom\parallel}(T) \nonumber.
\end{align}
where $n_{F}(x) =(e^{\beta x} +1)^{-1}$ is the quasiparticle distribution function and a factor of two comes from the spin-degeneracy. In order to compute the integral,
we have used $T\ll\Delta_0 \ll \mu_F$ as well as the linear expansion $\epsilon(\Delta p_z) \approx v_F(\Delta p_z c_{\perp}/v_F) \approx T$ with anisotropic momentum scaling, where close to the nodes $\Delta p_z \sim \frac{T}{\Delta_0}p_F = \frac{T}{c_{\perp}}$ for both $T, \epsilon \ll mc_\perp^2$, the cutoff of the linear Weyl regime in $^3$He-A. See Appendix \ref{Appendix2} and \cite{Nissinen2019,Laurila20} for more details.
From the above we conclude that $\vek{j}^{\rm vac}_{\rm anom}$ is the anomalous superfluid current from filled quasiparticle states, see Eq. \eqref{eq:j-current} below,
\begin{align}
\vek{j}^{\rm vac}_{\rm anom\parallel} = -\frac{p_F^3}{6\pi^2}\unitvec{l}(\unitvec{l}\cdot \nabla \times \unitvec{l}) = -\frac{C_0}{2}\unitvec{l}(\unitvec{l}\cdot \nabla \times \unitvec{l}), \label{eq:vacuum_j}
\end{align}
whereas the finite temperature contribution to the quasiparticle momentum is
\begin{align}
\vek{j}^{\rm qp}_{\rm anom\parallel}(T) = -\frac{p_F T^2}{6 c_{\perp}^2}\unitvec{l}(\unitvec{l}\cdot \nabla \times \unitvec{l}) \label{eq:Tcorrection}.
\end{align}
and arises due to thermal normal component close to the nodes. These contributions to the anomalous vacuum current from the perspective of the superfluid are analyzed in the next subsection.
The relativistic anomaly results in Sec. \ref{sec:TempNY} and the Landau level argument for the chiral superfluid suggest in addition the following temperature correction to the anomalous momentum in $^3$He-A, in the vicinity of the node, from the thermal Nieh-Yan anomaly \cite{Nissinen2019}:
\begin{align}
{\bf P}_{\textrm{NY-node}}(T) =-p_F \hat{\bf l}\, n_5(T)= \frac{p_F T^2 }{12 c_\bot^2}\hat{\bf l}(\hat{\bf l}\cdot(\nabla\times \hat{\bf l})).
\label{deltaPanomalous}
\end{align}
With the Landau level approximation, the quasiparticle density $n_5(T)$ between the two Weyl nodes is computed from a similar integral as Eq. \eqref{eq:LL_current},
\begin{align}
n_5(T) &= -2 \int_{0}^{\infty} N_{\rm LL}(p_z) dp_z \bigg[n_F(\epsilon_{p_z}-\mu_F) -\Theta(\mu-\epsilon_{p_z})\bigg] \nonumber\\
&= -\frac{T_B}{2\pi^2} \int_0^{\infty} dx ~x n_{F}(x) \nonumber\\
&= - \frac{T^2}{12c_{\perp}^2} (\unitvec{l}\cdot\nabla\times\unitvec{l}). \label{eq:LL_density}
\end{align}
Similar to the total current, this is the contribution from the linear spectrum close to the nodes with $\Delta p_{\parallel}\sim(\frac{c_{\perp}}{c_{\parallel}})\frac{T}{c_{\bot}}$ due to thermal fluctuations, which renormalizes the velocity coefficient and requires the cutoff of the Weyl regime relevant for the full dispersion, such as $^3$He-A. The thermal fluctuations for the superfluid are discussed in Appendix \ref{Appendix1}.
What have been calculated above are actually the temperature dependence of the anomalous torsional conductivities of the superfluid, i.e. then chiral momentum-density and number densities in response to $T_B$, see e.g. \cite{Landsteiner2014}. Note in particular that the leading torsional contribution from the NY anomaly in Eqs. \eqref{eq:LL_current}, \eqref{eq:LL_density} from thermal fluctuations is suppressed by the the factor $c_{\bot}^{-2}$, in contrast to the antisymmetric torsion $S = \frac{1}{2}\epsilon^{ijk} e^3_i \mathcal{T}^3_{jk} = \frac{T_B}{c_\parallel^2}$ from \eqref{Eu}, as was found in Ref. \cite{Nissinen2019} without the Landau level approximation to the anomaly. It is also intriguing to see that the non-relativistic thermal integrals in Eqs. \eqref{eq:LL_current},\eqref{eq:LL_density} with explicit non-relativistic vacuum regulatization due to the filled quasiparticle states coincide with similar expression for relativistic Weyl fermions with positive and negative branches \cite{LoganayagamSurowka12, JensenEtAl13, BasarEtAl14, Landsteiner2014, Zubkov2018, Imaki2019}, see Appendix \ref{Appendix3}.
\subsection{Anomalous vacuum current}
The hydrodynamic anomaly in momentum conservation arises between the quasiparticles and vacuum mass current $\vek{P} = \vek{j}$ of the superfluid, which at $T=0$ has the following general form \cite{Cross1975, VollhardtWolfle, Volovik2003}:
\begin{align}
\mathbf{j} = \rho \mathbf{v}_s + \frac{1}{4m} \nabla \times (\rho \unitvec{l}) - \frac{C_0}{2} \unitvec{l} (\unitvec{l} \cdot (\nabla \times \unitvec{l}) )\,,
\label{eq:j-current}
\end{align}
where the last term is anomalous with the parameter $C_0$ from the combined orbital-gauge symmetry of the superfluid \cite{VolovikMineev81}, that fully determines the axial anomaly in the system due to the Weyl quasiparticles of the superfluid \cite{Volovik2003}
\begin{align}
C_0(T=0) = p_F^3/3\pi^2 = \rho,
\end{align}
where we ignore corrections of the order of $(\Delta_0^2/E_F^2) = (c_{\perp}/c_{\parallel})^2 \ll 1$ at zero temperature to the density from pairing ($\rho-C_0 \sim \rho (\Delta_0^2/E_F^2)$). Notably, this term exists only on the weak coupling side of the topological BEC-BCS Lifshitz transition, where the pair of the Weyl points with ${\bf p}_{W\pm}=\pm p_F{\hat{\bf l}}$ appears in the quasiparticle spectrum \cite{VolovikMineev81, Volovik2003}.
Here we are interested in the temperature correction to the anomaly, which may come from the Nieh-Yan term.
The extension of the anomalous current to nonzero temperature gives
\begin{equation}
{\bf P}_{\rm anom}(T) = -\frac{1}{2}C_0(T) \hat{\bf l}(\hat{\bf l}\cdot(\nabla\times \hat{\bf l})
\,,
\label{Cross}
\end{equation}
where according to Cross \cite{Cross1975,VollhardtWolfle}, the anomalous parameter $C_0(T)$
has the following temperature dependence at low $T\ll T_c$:
\begin{equation}
C_0(T) =C_0(0) - T^2 \frac{p_F}{6 c_\bot^2} \left(1+ \frac{m^*}{m} \right)\,.
\label{C}
\end{equation}
Here $m^*$ is the effective mass of quasiparticles in the normal Fermi liquid, which differs from the bare mass $m$ of the $^3$He atom due to the Fermi liquid corrections. In Eqs. \eqref{eq:vacuum_j} and \eqref{eq:Tcorrection} we have neglected the Fermi-liquid corrections due to interactions: An additional factor $\frac{1}{2}(\frac{m^*}{m}-1)(\rho_n^{(0)}/\rho)$ arises from the reduced quasiparticle momentum flow, due to Galilean invariance. The current becomes \cite{Cross1975}
\begin{align}
\vek{j}_{\rm anom} = \frac{1}{1+\frac{1}{3} F^s_1(\rho^{(0)}_{n\parallel}/\rho)} \frac{m^*}{m} \vek{j}^{(0)}_{\rm anom} \label{eq:backflow}
\end{align}
where the Landau parameter $\frac{1}{3}F_1^s = \frac{m^*}{m}-1$ and $\rho_{n\parallel, \perp}^{(0)}(T)$ are the bare thermal quasiparticle densities without Fermi-liquid corrections along and perpendicular to $\unitvec{l}$. They are given by, see the Appendix \ref{Appendix1},
\begin{align}
\rho^{(0)}_{n\parallel} = \frac{\pi^2 T^2}{\Delta_0^2}\rho, \quad \rho^{(0)}_{n\bot} = \frac{7\pi^4T^4}{15 \Delta_0^4}\rho, \label{eq:rhoT}
\end{align}
where by Galilean invariance, $\rho\delta_{ij} = \rho_{s ij} + \rho_{n ij}$ at all temperatures. From Eqs. \eqref{eq:backflow}, \eqref{eq:rhoT} we gather
\begin{align}
\frac{C_0(T)}{2} = \frac{p_F^3}{6\pi^2} - \frac{p_F T^2}{6c_{\bot}^2} -\frac{p_F T^2}{12 c_{\perp}^2}\left(\frac{m^*}{m}-1\right) + O(T^4), \label{eq:NY_Fermi_corrections}
\end{align}
which is the Fermi-liquid corrected result Eq. \eqref{eq:backflow} and corresponds to the result \eqref{eq:Tcorrection} from the reduction of superfluid density when the Fermi-liquid corrections are ignored.
We summarize these findings as follows. While Eq. \eqref{Cross} with temperature corrections Eq. \eqref{C} looks similar to Eq. \eqref{deltaPanomalous}, $\vek{P}_{\rm anom}(T)$ represents the consistent anomaly vacuum momentum density from all filled states which depends on the non-fundamental parameters $m^*$ and $m$ via the Fermi-liquid corrections, while we expect that the $T^2$-contribution to the
(covariant) NY anomaly arise from contributions close to the linear Weyl nodes and should contain fundamental and universal prefactors.
Concerning the vacuum momentum current of the superfluid, one reason is that there are several different $T^2$ contributions to the current and free energy in the chiral superfluid, and they correspond to phenomemena with different origins and scales. Although of similar form in terms of the low-energy Goldstone variables of the superfluid, they can expressed in relativistic form with fundamental parameters only when carefully keeping track of each individual contribution to avoid double counting. In particular, in the next section we shall see that the non-fundamental parameters $m^*$ and $m$ do not enter the free energy or the final results, when experessed in terms of the correct relativistic variables valid in the quasirelativistic low-energy theory.
\section{Relativistic corrections to free-energy}
Let us consider the $T^2$-corrections in the free energy which are second order in derivatives containing combinations of $(\hat{\bf l}\cdot {\bf v}_s)$ and $(\hat{\bf l}\cdot (\nabla\times\hat{\bf l}))$, neglecting all higher order $O(T^4, \partial^3)$ terms. These terms can be distributed into three groups, which have different dependence on $m^*$ and $m$:
\begin{align}
F&=F_1+F_2+F_3 \,,
\label{123}
\\
F_1[\unitvec{l},\vek{v}_s]&=\frac{p_F}{12} \frac{T^2}{c_\perp^2}(\hat{\bf l}\cdot {\bf v}_s)(\hat{\bf l}\cdot (\nabla\times\hat{\bf l})),
\label{1}
\\
F_2[\unitvec{l},\vek{v}_s]&=- \frac{p_Fm^*}{96m^2} \frac{T^2}{c_\perp^2}
\left(4m(\hat{\bf l}\cdot {\bf v}_s)-(\hat{\bf l}\cdot (\nabla\times\hat{\bf l}))\right)^2,
\label{2}
\\
F_3[\unitvec{l}]&=- \frac{p_F}{288m^*} \frac{T^2}{c_\perp^2}(\hat{\bf l}\cdot (\nabla\times\hat{\bf l}))^2,
\label{3}
\end{align}
The relativistic form of each of these free-energy contributions arises separately as follows.
The term $F_3$ in Eq. \eqref{3} describes the universal temperature correction to the Newton gravitational coupling, which depends on the number of fermionic species \cite{VolovikZelnikov2003}:
\begin{equation}
F_3[\mathcal{R},T]=- \frac{v_F}{288} \frac{T^2}{c_\perp^2}(\hat{\bf l}\cdot (\nabla\times\hat{\bf l}))^2
=\frac{T^2}{144}\sqrt{-g} \cal{R}\,.
\label{NewtonG}
\end{equation}
Note that being exressed in terms of the scalar curvature and metric determinant $\sqrt{-g} =1/v_Fc_\perp^2$, this term becomes universal: it does not contain the microscopic parameters of the system: $p_F$, $m$ and $m^*$. The prefactor is fully determined by the number of fermionic species.
The term $F_2$ in Eq. \eqref{2} is expressed in terms of the combination $\unitvec{l}\cdot \vek{v} = \hat{\bf l}\cdot {\bf v}_s-\frac{1}{4m}\hat{\bf l}\cdot (\nabla\times\hat{\bf l})$, proportional to the ground state current which does not receive corrections due to Galilean invariance. Here the velocity ${\bf v}={\bf j}/\rho$ is the velocity of the ``total quantum vacuum", where ${\bf j}\equiv {\bf j}(T=0)$ is the total vacuum current in Eq. \eqref{eq:j-current} at $T=0$.
We conclude that the $F_2$ contribution gives the temperature $T$ and chemical potential $\mu$ correction to the free energy of the gas of chiral fermionic particles in the limit $|\mu| \ll T$ (see Eqs. (9.12) and (10.42) in Ref. \cite{Volovik2003}):
\begin{eqnarray}
F_2[\mu,T] =- \frac{p_Fm^*}{96m^2} \frac{T^2}{c_\perp^2}
\left(4m\,\hat{\bf l}\cdot {\bf v}_s-\hat{\bf l}\cdot (\nabla\times\hat{\bf l})\right)^2
=\nonumber
\\
=-\frac{T^2}{6} \sqrt{-g} \mu^2 \,,
\label{A0}
\end{eqnarray}
where the chiral chemical potential of the superfluid is determined by the Doppler shift
\begin{equation}
\mu_R=-\mu_L =p_F {\bf v}\cdot \hat{\bf l}= p_F \hat{\bf l}\cdot {\bf v}_s-(p_F/4m)\hat{\bf l}\cdot (\nabla\times\hat{\bf l})
\,.
\label{A0mu}
\end{equation}
The final version of Eq. \eqref{A0} does not contain microscopic parameters. The Eq. \eqref{A0} also gives rise to the mass of ``photon" in $^3$He-A \cite{Volovik1998}. In principle, the chiral chemical potential may also serve as the $\Lambda$ parameter in the Nieh-Yan term, see however Appendix \ref{Appendix3}.
Finally $F_1$ in Eq.(\ref{1}) is the term in free energy, which gives rise to the vacuum current without any factors of $\frac{m^*}{m}$ with contribution from the Nieh-Yan thermal anomaly in Eqs. \eqref{deltaPanomalous}, \eqref{C}, \eqref{eq:NY_Fermi_corrections}:
\begin{align}
{\bf P}_{\rm anom}(T) &= \frac{\delta F_1}{\delta {\bf v}_s}=\frac{p_F}{12} \frac{T^2}{c_\perp^2}\hat{\bf l}(\hat{\bf l}\cdot (\nabla\times\hat{\bf l})) \label{J}
\\
&\simeq -p_F n_{5}(T)\hat{\bf l}, \nonumber
\end{align}
where the last equality was derived in Eq. \eqref{eq:LL_density}. We stress however that $\vek{P}_{\rm anom}(T)$ represents the total anomalous superfluid current, whereas the right-hand side is equal to the quasiparticle momentum density due to thermal fluctuations close to the node.
\section{Conclusions}
We discussed the possibility of thermal Nieh-Yan anomaly where the role of the non-universal dimensionful UV cut-off $\Lambda$ is played by the temperature IR scale, and the dimensionless prefactor $\gamma$ in the anomaly is universal. We identified a contribution from this anomaly in the known low-temperature corrections of non-relativistic chiral $p$-wave Weyl superfluid (or superconductor). In this system, the anomaly results from thermal effects of the linear Weyl spectrum at finite momentum in the presence of an explicit vacuum of filled quasiparticles, although the end results are similar to relativistic fermions when interpreted carefully in terms of the anisotropy and cutoff of the quasirelativistic Weyl regime.
What we calculated, via the anisotropic Landau level model with non-relativistic symmetries in Sec. \ref{sec:superfluid_LL}, are actually the temperature dependence of the anomalous torsional conductivities $\sigma_{\mathcal{T}^a}$ of the quasiparticle system, i.e. the chiral momentum-density and number densities (at the nodes at finite momenta $\pm p_F$) in response to the spacelike torsion $T_B= \frac{1}{2}\epsilon^{ijk}e^3_i \mathcal{T}^3_{jk}$. Namely, for example,
\begin{align}
\langle P^a \rangle =\langle e^a_i T^{0i} \rangle = e^a_i \frac{\sigma_{\mathcal{T}^b}(T)}{2} \epsilon^{0ijk} T^b_{jk} = \delta^{a3}\sigma_{T_B} T_B,
\end{align}
$P^i = T^{0i}$, where $T^{\mu\nu}$ is the stress-tensor, which are of second order in derivates and can be calculated in linear response, e.g. Kubo formulas, to the background tetrads \cite{LandsteinerEtAl11, Landsteiner2014, BradlynRead15, Gromov15}. A similar momentum anomaly in anisotropic system with non-relativistic symmetries was also considered in \cite{Copetti20}. For the zero temperature case, see \cite{Nissinen2019,Laurila20} as well as \cite{Stone2019, Stone2019b} for a relativistic model related to Weyl semi-metals. We, however, stress that the universal gravitational NY anomaly and thermal physics we discussed arise in flat space from the geometric background fields in the low-energy quasiparticle Hamiltonian. These tetrads arise universally in all Weyl systems \eqref{eq:WeylHamiltonian} and couple to the momentum \cite{Parrikar2014, ShapourianEtAl15} as in gravity. The connection of our results and the relation of the gravitational NY anomaly, with the coefficient $\gamma$, to thermal transport in Weyl system should be further elucidated \cite{Luttinger64, LandsteinerEtAl11, LoganayagamSurowka12, JensenEtAl13, Lucas2016, GoothEtAl17, KobayashiEtAl18}.
Detailed consideration of the temperature dependent anomaly terms in the hydrodynamics of the non-relativistic $p$-wave chiral superfluid with quasirelativistic Weyl fermions demonstrates that in the hydrodynamics of this liquid there are several $T^2$ terms, which can be assigned to different emergent relativistic phenomena, both anomalous and non-anomalous. In particular, we identified and discussed the term in the vacuum momentum corresponding to the (consistent) thermal Nieh-Yan anomaly. As expected this term originates from thermal fluctuations close to the linear nodes with emergent quasirelativistic torsion with anisotropy. Note that in terms of the superfluid and Weyl fermions, the $T=0$ anomalous vacuum contribution to the current can be assigned only to a non-local action \cite{Volovik1986c}. We showed how the various $T^2$ low-temperature corrections can be written aslow-energy relativistic terms with dimensionless prefactors, which do not seem to depend on microscopic physics, but are fully determined by geometry, topology and the number of fermionic and bosonic quantum fields. Detailed comparison of the finite temperature superfluid hydrodynamics with Fermi-liquid corrections to the anomalous quasiparticle axial current production in the presence of arbitrary textures and superfluid velocity remains to be identified \cite{Volovik85, Combescot86} . See however Ref. \onlinecite{Nissinen2019} for the zero temperature case.
\emph{Note added:} After the initial submission of this manuscript as a preprint, arXiv:1909.08936v1, with the predicted $T^2$-contribution to the NY anomaly, Eqs. \eqref{j5nonconservationR}, \eqref{J}, the recent preprints \cite{Stone2019, Stone2019b, Ojanen2019} discussing related torsional anomaly phenomena at finite temperatures appeared. General aspects of the temperature anomaly in Weyl materials were further discussed in the short paper \cite{NissinenVolovik2019}. In particular, the result Eq. \eqref{j5nonconservationR} has been confirmed in Ref. \onlinecite{Stone2019} by a direct calculation of the spectral flow of relativistic Landau levels in the presence of a constant torsional magnetic field $T^3_{\mu\nu}$ \cite{Volovik85, BalatskiiEtAl86, Parrikar2014} at finite temperature. Here similar computations for the non-relativistic Weyl superfluid in Eqs. \eqref{eq:LL_current} and \eqref{eq:LL_density} give corresponding results. While the current manuscript was being finalized, also the paper \cite{Imaki20} appeared discussing the anomaly for relativistic fermion at finite temperature and chemical potential.
\emph{Acknowledgements:} GEV thanks Mike Zubkov for discussions. JN thanks Z.-M. Huang for correspondence and T. Ojanen for discussions.
This work has been supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant Agreement No. 694248).
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\section{Introduction}
\label{sec:intro}
Omnidirectional (360$^\circ$) video content recently got very popular in the media industry, because it
allows the viewer to experience the content in an immersive and interactive way.
A critical factor for the long term success of 360$^\circ$ video content is the availability of convenient tools for producing and editing 360$^\circ$ video content for a multitude of platforms (like mobile devices or VR Headsets). In order to address these shortcomings, in the ongoing Hyper360 \footnote{\url{https://www.hyper360.eu/}} project an innovative toolset for 360$^\circ$ video is proposed.
Specifically, within the Hyper360 project an innovative complete solution for the capturing, production, enhancement, delivery and consumption of an free viewpoint video to the OTT \footnote{An over-the-top (OTT) media service is a streaming media service offered directly to viewers via the Internet} media sectors is developed and validated through the production of pilots with these tools and the assessment of the pilots by the audience. By augmenting 360$^\circ$ video with 3D content acting as a \emph
{Mentor}, novel and powerful storytelling opportunities are added in the production process. Additionally, by extracting and adapting to the viewer preferences, a personalized consumption experience can be provided.
The whole Hyper360 pipeline consists of four layers:
\begin{itemize}
\item The \textbf{Capturing layer} contains the \emph{OmniCap} tool (described in section \ref{subsec:omnicap}) which is respsonsible for capturing 360$^\circ$ video with a variety of cameras (multi camera arrays, tiny fisheye lense devices, ...) and ensuring high quality content via the integrated quality analysis component.
\item The \textbf{Production layer} comprises two post-processing tools for enriching 360$^\circ$ videos and offering an enhanced viewpoint experience. Specifically, the \emph{OmniConnect} tool (section \ref{subsec:omniconnect}) manages the annotation of 360$^\circ$ media with hyperlinks to other media and content metadata. Additionally, the \emph{CapTion} tool (section \ref{subsec:caption}) is responsible for capturing the performance of human narrators in 3D and embeddding the generated 3D content in the 360$^\circ$ video.
\item The \textbf{Delivery layer} comprises cloud services that facilitate the delivery of produced, integrated media under seamless, personalised 360$^\circ$ experiences. Within this scope, Hyper360's \emph{OmniCloud} includes user profiling and recommendation services for personalized consumption (section \ref{subsec:profiling}) and the automatic camera path generator tool for generating a 2D video for playout on a conventional TV set (section \ref{subsec:autocmam}).
\item The \textbf{Presentation layer} contains the various players (section \ref{subsec:players}) required for the playback of the enriched 360$^\circ$ content on multiple platforms.
\end{itemize}
The user partners in the Hyper360 project, the broadcasters Rundfunk Berlin-Brandenburg and Mediaset, have produced pilots (see section \ref{sec:pilots}) using the first prototype of the Hyper360 toolset. Furthermore, they have carried out dedicated audience assessment sessions for the produced pilots.
\section{Tools}
\label{sec:tools}
\subsection{OmniCap}
\label{subsec:omnicap}
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{omnicap_image_from_journal_paper_barnabas.png}
\caption{OmniCap tool for a mixed 360$^\circ$ and regular cameras setup.}
\label{fig:omnicap}
\end{figure}
The OmniCap tool was designed with the purpose of managing multiple 360$^\circ$ and regular camera types for professional productions. It employs broadcast hardware (4K resolution, 3D stereoscopic capture, Black Magic Design MultiView Units and SDI+Ethernet camera connectivity) and provides a pipeline to manage up to 36 cameras simultaneously placed on a set. The system supports multiple camera types and configurations in a broad range of mounting and imaging options including shader-based live stitching, safe-zones, camera parameter controls, real-time effects, virtual camera movements and integration of 3D elements into the live scene as required. In Figure \ref{fig:omnicap} its usage for a mixed setup of 360$^\circ$ and regular cameras is shown. In order to support high quality on-set capture as well as post production workflows, the OmniCap system also includes a 360 desktop player with integrated tracking, object detection and quality control. For the quality control, existing algorithms for detection of defects like signal clipping, blurriness, flicker or noise level on conventional video have been adapted to the specifics of 360$^\circ$ video (especially the equirectangular projection) and extended in order to provide localized defect information. The adaptions for the blurriness algorithm are described in \cite{Fassold2019Blurriness}. General strategies how to adapt computer vision algorithms to the specifics of 360$^\circ$ video (like the equirectangular projection) are given in \cite{Fassold2019Adapting}.
\subsection{OmniConnect}
\label{subsec:omniconnect}
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{omniconnect_image_rbb.png}
\caption{OmniConnect tool for enriching 360$^\circ$ video}
\label{fig:omniconnect}
\end{figure}
OmniConnect offers many features to enrich 360$^\circ$ videos with different types of hotspots such as shapes, 2D videos, audio fragments, html pages, multiline texts, images and metadata. To do that OmniConnect employs a user-friendly web interface (see Figure \ref{fig:omniconnect}) compatible with any browser supporting the HTML5 and WebGL standards. It includes an integrated player for previewing purposes. In addition to that, OmniConnect also offers the possibility to publish the enriched 360$^\circ$ video file by specifying one or more target device classes (iOS, Android, PC and HbbTV).
\begin{figure}[t]
\centering
\includegraphics[width = 0.48\textwidth]{img/CapTion.png}
\caption{Caption software toolset. From top to bottom: The Volumetric Capture, the Performance Capture and the 360 Fusion applications.}
\label{fig:CapTion}
\end{figure}
\subsection{CapTion \& 3D Mentor}
\label{subsec:caption}
The CapTion toolkit comprises a set of tools designed for mixed reality media productions that aim to combine two types of Free-Viewpoint Video (FVV), namely 360$^\circ$ and 3D content. The first system of the tool-chain is an affordable and portable volumetric capturing hardware system with its corresponding software application \cite{sterzentsenko2018low}, which gives the ability to record spatially calibrated \cite{papachristou2018markerless} and temporally aligned RGB-D videos. It is followed by the Performance Capture application which produces dynamic (i.e. animated) 3D textured meshes of human performances \cite{alexiadis2018fast}. The exported 3D content can be embedded in omnidirectional media through the 360$^\circ$ Fusion tool that utilizes neural networks for 360$^\circ$ scene understanding \cite{Zioulis_2018_ECCV, karakottas2018360d, zioulis2019spherical, karakottas2019360} to allow for seamless
placement and realistic rendering of the 3D content inside omnidirectional media.
Figure \ref{fig:CapTion} presents sample views of the Volumetric Capture, Performance Capture and 360$^\circ$ Fusion applications.
\subsection{Profiling and Recommendation Engine}
\label{subsec:profiling}
The personalisation mechanism in Hyper360 relies on the holistic representation of content annotation, in order to minimise loss of information and characterise it with enhanced metadata. To this end, through Hyper360's \emph{Semantic Interpreter}, content annotation is construed to a set of fuzzy concept instances, based on an expanded version of the LUMO ontology \cite{tsatsou2014lumo}. This interpretation builds upon learned lexico-syntactic relations between free-form, domain-specific metadata. Subsequently, Hyper360's \emph{Profiling Engine} harvests the novel opportunities that omnidirectional video offers for implicitly capturing the viewers' preferences. The viewpoint choice allows the \emph{Profiling Engine} to capture spatio-temporal and other behavioural information (which part of the video the viewer focuses on and for how long, which objects in a scene they interact with, etc.), which are combined with appropriate semantic meaning of the content. Lastly, the \emph{Recommendation Engine} semantically matches learned user profiles with semantic content metadata, through an extension of the LiFR fuzzy reasoner \cite{tsatsou2014lifr}. The application of recommendation is three-fold within Hyper360, offering personalised navigation within the 360$^\circ$ media, which is manifested both as cues for personalised camera path(s), as well as cues for personalised 3D Mentor narratives, while also achieving targeted embedded objects (hotspots, hyperlinks, nested media) delivery, as seen in Figure \ref{fig:Recom}.
\begin{figure}[t]
\centering
\includegraphics[width = 0.48\textwidth]{img/Recommendation_Engine.png}
\caption{Example of embedded objects recommendation in omnidirectional video.}
\label{fig:Recom}
\end{figure}
\subsection{Automatic Camera Path Generator}
\label{subsec:autocmam}
The goal of automatic camera path generation (automatic cinematography / virtual director) is to calculate automatically a visually interesting camera path from a 360$^\circ$ video, in order to provide a traditional TV-like consumption experience. This is necessary for consumption of a 360$^\circ$ video on older TV sets, which do not provide any kind of interactive players for 360$^\circ$ video. Furthermore, even on devices capable of consuming 360$^\circ$ videos interactively, an user might prefer a lean-back mode, without the need to navigate around actively to explore the content. The initial prototype of the automatic camera path generator (more details can be found in \cite{Fassold2019Autocam}) is based on the information about the scene objects (persons, animals, ...), which is extracted with the method given in \cite{Fassold2019Omnitrack}.
For each scene object, a saliency score is calculated based on several influencing factors (object class and size, motion magnitude, neighbours of object), which indicates the "interestingness" of the object. From the calculated saliency scores for the scene objects, an automatic camera path is generated for the current shot by tracking the object with the highest saliency score.
\subsection{OmniPlay}
\label{subsec:players}
The Hyper360 player technology is a suite of players for multiple platforms delivering the backbone of the Hyper360 playback environment by supporting all standard player functionalities, hotspots (images, text, video) as well as a number of innovative functionalities based on the free viewpoint media's characteristics (like fusion of 360$^\circ$ video and 3D content, automatic camera path, personalisation multi-viewport coalescing and nested media graphs). Players have been developed for the following platforms: Windows, web browser, smartphones (Android / iOS), VR Headsets like HTC Vive or Oculus Go, SmartTV (HbbTV / AppleTV) and traditional TV. For the implementation of the player, proper software frameworks have been used for the specific platform (mainly Unity, JavaScript / WebGL and XCode). As not every platform supports the same user interaction modality (e.g. SmartTV supports only key-based navigation), platform-specific adaptions to the UI have been done.
\section{Pilots}
\label{sec:pilots}
The user partners in the Hyper360 project, the broadcasters Rundfunk Berlin-Brandenburg (RBB) and Mediaset (RTI), employed the first prototypes of the developed tools in order to produce pilot content. For this, RBB focused on the \emph{Immersive Journalism} scenario, which allows first person experience of the events or situations described in news reports and documentary film.
The focus of RTI was on the \emph{Targeted Advertising} scenario, where entertainment or infotainment content is enriched with advertising objects and hyperlinks. For both scenarios, first concepts for potential pilots have been developed during joint workshops with the broadcasters production department. The most promising concepts have been refined subsequently and transformed into storyboards, which formed the base for the actual pilot production. Two pilots have been produced by RBB: \emph{T\"ater Opfer Polizei} and \emph{Fontane360}. The first one is a CSI-like crime scene investigation with gamification elements, whereas the second one is more calm and follows Fontane through one of his walks through the Mark Brandenburg. RTI produced also two pilots named \emph{TechnoGym} and \emph{Universita Cattolica of Milan}. The former explains the proper use of gym equipment, whereas the latter has the aim of helping new students in choosing the right course. The pilots were used subsequently to asses the whole immersive experience as well as its major components (like the players, 3D mentor, personalisation, automatic camera path) by test consumers. The gathered feedback from the test consumers was generally positive.
\section{Conclusion}
The work done so far in the Hyper360 project on tools for capturing, production, enhancement, delivery and consumption of enriched 360$^\circ$ video content has been presented. Furthermore, the first pilots which have produced with these tools have been described.
\section*{Acknowledgment}
This work has received funding from the European Union's Horizon 2020 research and innovation programme, grant n$^\circ$ 761934, Hyper360 (``Enriching 360 media with 3D storytelling and personalisation elements''). Thanks to Rundfunk Berlin-Brandenburg and Mediaset for providing the 360\textbf{$^\circ$}~video content.
\balance
\bibliographystyle{IEEEbib}
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\section{Thermal correlators in 2d CFTs}
\label{appendix2dthermal}
\subsection{Modes in a BTZ black hole}
In our discussion above we frequently referred to modes in the black
hole background. In this appendix, we explore these modes in the
background of a BTZ black hole. This will allow us to check several
claims including their behaviour near the horizon and their growth for
large spacelike momenta that we made above.
Let us start with the usual BTZ coordinates
\[
ds^2 = -(r^2-r_h^2)dt^2 + (r^2-r_h^2)^{-1}dr^2 + r^2 dx^2.
\]
The horizon is at $r_h$ and $x$ runs from $-\infty$ to $+\infty$ i.e. we are looking at a planar BTZ black hole. The temperature of the black hole is
\[
\beta = {2\pi \over r_h}
\]
We consider a massive scalar whose dual operator has dimension $\Delta$, obeying the equation
\[
(\Box -m^2)\phi=0
\]
Solving the KG equation with an ansatz of the form $e^{-i\omega t}e^{i k x}\psi(r)$ we find two linearly independent solutions
\[
\psi_1(r)=\left({r^2\over r_h^2}\right)^a\left({r^2\over r_h^2}-1\right)^b \, _2F_1\left(1+a+b-{\Delta \over 2},a+b+{\Delta \over 2},1+2b,1-{r^2\over r_h^2}\right)
\]
\[
\psi_2(r) =\left({r^2\over r_h^2}\right)^{a}\left({r^2\over r_h^2}-1\right)^{-b}\, _2F_1\left(1+a-b-{\Delta
\over 2},a-b+{\Delta \over 2},1-2b,1-{r^2 \over r_h^2} \right)
\]
where
\[
a = {i k \over 2 r_h}\qquad b= {i \omega \over 2r_h}
\]
For any given $\omega,k$ only a specific linear combination of these modes is normalizable at infinity. Using hypergeometric identities we find that the normalizable combination is
\[
\psi_1(r) -\left[{\Gamma(1+2b)\Gamma(a-b+{\Delta\over 2})
\Gamma(-a-b+{\Delta\over 2})\over \Gamma(1-2b) \Gamma(-a+b+{\Delta \over 2})
\Gamma(a+b+{\Delta \over 2})}\right]\psi_2(r)
\]
Using some hypergeometric identities and fixing the overall normalization we can rewrite the normalizable mode as
\[
\hat{\psi}_{\omega,k}(r) = {1\over \Gamma(\Delta)\sqrt{r_h}}\sqrt{\Gamma(a+b+\Delta/2)\Gamma(a-b+\Delta/2)
\Gamma(-a+b+\Delta/2)\Gamma(-a-b+\Delta/2)\over \Gamma(2b)\Gamma(-2b)}
\]
\[
\times \left({r^2\over r_h^2}\right)^a \left({r^2\over r_h^2}-1 \right)^{-a-{\Delta\over 2}}
{ \, _2 F_1\left(a-b+\Delta/2,a+b+\Delta/2,\Delta,{r_h^2 \over r_h^2-r^2}\right)}
\]
The overall normalization was fixed so that the modes are canonically normalized with respect to the Klein Gordon norm, which is why we use the hatted notation --- as discussed in section
\ref{modesbrane}. In particular, near the horizon we have the expansion\footnote{The overall factor of $r_h^{-1/2}$ in the near horizon normalization, is automatically fixed if we
require the modes defined by \eqref{psinearhapp}, \eqref{btzmodes} to have canonical Klein-Gordon norm. }
\begin{equation}
\label{psinearhapp}
\hat{\psi}_{\omega,k}(r) = r_h^{-1/2} \left(e^{i\delta_{\omega,k}}e^{i\omega r_*} + e^{-i \delta_{\omega,k}} e^{-i\omega r_*}\right)
\end{equation}
where $r_*$ is the tortoise coordinate (in which horizon is at $r_*\rightarrow -\infty$)
\[
r_* = {1\over 2 r_h}\log\left(r-r_h\over r+r_h\right)
\]
and the ``phase shift'' is
\[
e^{i \delta_{\omega,k}} = 4^{b} \sqrt{\Gamma(-2b) \,\Gamma(a+b+\Delta/2)\,\Gamma(-a+b+\Delta/2)\over\Gamma(2b)\,
\Gamma(-a-b+\Delta/2)\,\Gamma(a-b+\Delta/2)}
\]
To be more precise the complete bulk mode is
\begin{equation}
\label{btzmodes}
\hat{f}_{\omega,k}(t,x,r) = \hat{\psi}_{\omega,k}(r) e^{-i\omega t} e^{i k x}
\end{equation}
here both the frequency $\omega$ and the momentum $k$ are continuous.
For the bulk operator field we have
\begin{equation}
\label{bulkfieldbtz}
\phi(t,x,r) =
\int_{\omega>0} {d\omega\,dk \over (2 \pi)^2} {1\over \sqrt{2\omega}}
\left(a_{\omega,k}\hat{f}_{\omega,k}(t,x,r) + {\rm h.c.}\right)
\end{equation}
with
\[
[a_{\omega,k},a^\dagger_{\omega',k'}] = \delta(\omega-\omega')\delta(k-k')
\]
In the AdS Hartle-Hawking state we have
\[
\begin{split}
&\langle a_{\omega,k} \, a^\dagger_{\omega',k'}\rangle_{\rm HH} = {e^{\beta \omega}\over e^{\beta \omega}-1} \delta(\omega-\omega')\delta(k-k')\\
&\langle a_{\omega,k}^\dagger \,a_{\omega',k'}\rangle_{\rm HH} = {1\over e^{\beta \omega}-1} \delta(\omega-\omega')\delta(k-k')
\end{split}
\]
Using these formulas we can compute the bulk 2-point function and then by taking the bulk points to the boundary, we can recover the boundary 2-point function
\[
G_\beta(t,x;t',x') = \lim_{r,r'\rightarrow \infty} \left[\Upsilon^2 r^{\Delta}(r')^{\Delta} \langle \phi(t,x,r) \phi(t',x',r')\rangle_{\rm HH}\right],
\]
where $\Upsilon$ is given in \eqref{normfactdef}.
In momentum space, and using time- and space-translational invariance we have
\[
G_\beta(\omega,k) = \int dt\,dx\, e^{i\omega t - i k x} G_\beta(t,x;0,0)
\]
From the previous results we find
\begin{equation}
\label{btzfourp}
G_\beta(\omega,k) =
\Upsilon^2 {1\over 2 \pi (2\omega)}{ e^{\beta \omega} \over e^{\beta \omega}-1}\left({2\pi \over \beta}\right)^{2\Delta-1}\left|{\Gamma\left(i{\beta(\omega+k)\over 4\pi}+{\Delta\over 2}\right)\Gamma\left(i{\beta(\omega-k)\over 4\pi}+{\Delta\over 2}\right)
\over \Gamma(\Delta)\Gamma\left(i{\beta\omega\over 2\pi}\right)}\right|^2\qquad,\qquad \omega>0
\end{equation}
and
\begin{equation}
\label{btzfourn}
G_\beta(-\omega,k) =
\Upsilon^2 {1\over 2 \pi (2\omega)}{ 1 \over e^{\beta \omega}-1}\left({2\pi \over \beta}\right)^{2\Delta-1}\left|{\Gamma\left(i{\beta(\omega+k)\over 4\pi}+{\Delta\over 2}\right)\Gamma\left(i{\beta(\omega-k)\over 4\pi}+{\Delta\over 2}\right)
\over \Gamma(\Delta)\Gamma\left(i{\beta\omega\over 2\pi}\right)}\right|^2\quad,\qquad \omega>0
\end{equation}
Actually although the two forms above are useful to see the qualitative
properties of the solution we can rewrite them in a simpler form using
\begin{equation}
\left| \Gamma\left(i{\beta\omega\over 2\pi}\right) \right|^2 = {2 \pi^2 \over \beta |\omega|} {1 \over e^{\beta |\omega| \over 2} - e^{-\beta |\omega| \over 2}}
\end{equation}
This leads to the expression
\begin{equation}
G_{\beta}(\omega, k) = {\Upsilon^2 \over (2 \pi)^2} e^{\beta \omega \over 2}
\left(2 \pi \over \beta \right)^{2 \Delta - 2} \left|{\Gamma\left(i{\beta(\omega+k)\over 4\pi}+{\Delta\over 2}\right)\Gamma\left(i{\beta(\omega-k)\over 4\pi}+{\Delta\over 2}\right) \over \Gamma(\Delta)}\right|^2,
\end{equation}
and this expression is valid for $\omega$ both positive and negative.
In the next subsection we will rederive these expressions from the boundary CFT, using the constraints from 2d conformal invariance. Here we notice that these results manifestly
satisfy the general properties we mentioned in section \ref{sec:outside}:
i) It is obvious from the expressions above that the KMS condition is satisfied
\[
G(-\omega,k) = e^{-\beta \omega} G(\omega,k)
\]
ii) It can be checked that the 2-point function $G(\omega,k)$ is exponentially suppressed for large spacelike momenta, that is for fixed $\omega$ and large $k$ we have
\[
G(\omega,k) \underset{|k| \rightarrow \infty}{ \lessapprox} e^{-{\beta k \over 2}}
\]
as expected from our general arguments in section \ref{subsec:thermalfourier}.
iii) It can be checked that in the limit of low temperature ($\beta\rightarrow \infty$) the thermal Wightman function $G_\beta(\omega,k)$ reduces to the zero-temperature one
that we found in section \ref{sec:emptyads}, that is
$$
G_\beta(\omega,k) \approx N_\Delta \,\theta(\omega)\,\theta(\omega^2-k^2)\, (\omega^2 - k^2)^{\Delta-1} ,\qquad {\rm for}\quad \beta\rightarrow \infty
$$
\subsection{Boundary correlators in 2 dimensions}
The two
dimensional CFT correlator at finite temperature, can be completely fixed using conformal invariance and we will rederive the results \eqref{btzfourp}, \eqref{btzfourn} directly from the boundary.
We start with the Euclidean correlator. We put the CFT on ${\mathbb R}^1\times {\mathbb S}^1$ where the perimeter of the circle is $\beta$. We then have
\begin{equation}
\label{exact2dcorrelator}
\langle {\cal O}(\tau,x) {\cal O}(0,0)\rangle_{\beta} = \left({2\pi \over \beta} \right)^{2\Delta} \left[2 \cosh\left({2\pi x \over \beta}\right)- 2 \cos\left({2\pi \tau \over \beta}\right)\right]^{-\Delta}
\end{equation}
By taking the short distance expansion we can check that the normalization is correct.
However, we can now use modular invariance and understand this to be
the thermal correlator of a CFT on flat space at an inverse
temperature $\beta$.
Now, let us continue to Lorentzian space, following the same logic as in section \ref{sec:emptyads}. (See also \cite{Skenderis:2008dg}.)
We find that
the time-ordered correlator is given by
\[
\langle T\left\{{\cal O}(t,x), {\cal O}(0,0) \right\} \rangle_{\beta} = \left({2 \pi \over \beta} \right)^{2 \Delta} \left[2 \cosh \left({2 \pi x \over \beta} \right) -
2 \cosh \left( {2 \pi (1 - i \epsilon) t \over \beta} \right) \right]^{-\Delta}
\]
and for the Wightman correlator, we have
\[
\begin{split}
&\langle {\cal O}(t,x), {\cal O}(0,0) \rangle_{\beta} = \left({2 \pi \over \beta} \right)^{2 \Delta} \left[2 \cosh \left({2 \pi x \over \beta} \right) -
2 \cosh \left( {2 \pi (t - i \epsilon) \over \beta} \right) \right]^{-\Delta} \\
&= \left({2 \pi \over \beta} \right)^{2 \Delta} 2^{-2 \Delta} \left[ \sinh {\pi (u - i \epsilon) \over \beta} \sinh{\pi (v - i \epsilon) \over \beta} \right]^{-\Delta}
\end{split}
\]
where we defined $u=t-x,\,v=t+x$. So, the basic integral that we are interested in is
\begin{equation}
\label{ibetadef}
I_{\beta}(k_{+}) = \left({2 \pi \over \beta} \right)^{ \Delta} 2^{- \Delta} \int_{-\infty}^{\infty} e^{i k_{+} u} \left(\sinh{\pi (u - i \epsilon) \over \beta} \right)^{-\Delta} d u
\end{equation}
Let us understand the qualitative properties of this integral. We see that the integral has branch cuts running along the segments $(-\infty + i n \beta + i \epsilon, i n \beta + i \epsilon)$ for integer $n$. Now, if $k_{+} > 0$, then we can move the $u$ contour down to the first branch cut, which is at ${\rm Im}~u = - i \beta$. So, for negative $k_{+}$, we get a contribution proportional to
$e^{-k_+ \beta}$ i.e. the integral is exponentially damped for large negative $k_{+}$. On the other hand, for positive $k_{+}$, there is no such damping.
To evaluate the integral precisely, we write it as
\begin{equation}
\begin{split}
I_{\beta}(k_{+}) &= \left({2 \pi \over \beta} \right)^{\Delta} 2^{-\Delta} \left[e^{i \pi \Delta} \int_{-\infty}^{0} d u \, e^{i k_{+} u} \left| \sinh {\pi u \over \beta} \right|^{-\Delta} + \int_0^{\infty} d u \, e^{i k_{+} u} \left| \sinh {\pi u \over \beta} \right|^{-\Delta} \right],\\
\end{split}
\end{equation}
where the phase factors come from the $i \epsilon$ prescription explained in section \ref{sec:emptyads}.
This integral can be performed analytically to obtain
\begin{equation}
\begin{split}
I_{\beta}(k_{+})
&= \left({2 \pi \over \beta} \right)^{ \Delta} {\pi^2 \beta \over 4} {\csc (\pi \Delta ) \over \Gamma (\Delta )} \left[ e^{i \pi \Delta} \frac{ \Gamma \left(\frac{1}{2} \left(\frac{i k_{+} \beta}{\pi }+\Delta \right)\right)}{\Gamma \left(\frac{i k_{+}
\beta }{2 \pi }-\frac{\Delta }{2}+1\right) }
+ \frac{ \Gamma \left(\frac{1}{2} \left(\Delta -\frac{i k_{+} \beta }{\pi }\right)\right)}{\Gamma \left(-\frac{i k_{+} \beta }{2 \pi }-\frac{\Delta }{2}+1\right)} \right] \\
&= \left({2 \pi \over \beta} \right)^{ \Delta} {\pi^3 \beta \over 4} \frac{ \left(\coth \left(\frac{1}{2} (k_{+} \beta +i \pi \Delta )\right)+1\right) \text{csch}\left(\frac{1}{2} (k_{+} \beta -i \pi \Delta )\right)}{\Gamma \left(-\frac{i k_{+} \beta }{2 \pi }-\frac{\Delta }{2}+1\right) \Gamma \left(\frac{i k_{+}\beta }{2 \pi }-\frac{\Delta }{2}+1\right) \Gamma (\Delta )} \\
&= \left({2 \pi \over \beta} \right)^{ \Delta - 1} {\pi^2 \over 2} \frac{ e^{i \pi \Delta + k_{+} \beta \over 2} \Gamma \left(\frac{\Delta }{2} + \frac{i k_{+} \beta }{2 \pi } \right) \Gamma \left(\frac{\Delta }{2} - \frac{i k_{+}\beta }{2 \pi }\right)}{ \Gamma (\Delta )}.
\end{split}
\end{equation}
Since the full answer for the Green function is given by
\begin{equation}
G_{\beta}(\omega, k) = I_{\beta}({k + \omega \over 2}) I_{\beta}({k - \omega \over 2}),
\end{equation}
and since, in this case $\Upsilon^2 = 2 \pi (2 \pi)^2$,
we see that our boundary calculation matches precisely with the answer
from the bulk up to a momentum and temperature independent pre-factor.
\section{Black holes in AdS: setup and review}
\label{semiblack}
Just as we did for the
CFT in its vacuum state, we would now like to re-organize the degrees of
freedom of the CFT, in a heavy typical pure state, into degrees of freedom
that resemble perturbative fields propagating on an AdS black hole
background.
As we have explained in the introduction, our setup is that we start with
the CFT in a pure state and then allow it to ``settle down'' so that
it resembles a thermal state more and more. When this happens, we
find that the CFT describes fields propagating in, what is called, the
``eternal AdS black hole.''
Maldacena \cite{Maldacena:2001kr} explained that the eternal AdS black hole has a holographic description in terms of two copies of a CFT in a specific entangled quantum state. In this paper, the eternal black hole geometry will emerge as an auxiliary device for computations done {\it in a single} CFT!
In fact, this is not so surprising from the naive semi-classical perspective since it is indeed true that quantum fields
on a collapsing star start behaving like those in an eternal black hole background, if we probe the
geometry ``late enough.'' We review these ideas from semi-classical General Relativity in some detail below and we also review the basic formalism of quantizing fields in a black hole background. The reader who is familiar with
these topics, or is willing to accept our claims, can jump directly to section \ref{sec:outside}.
We wish to emphasize an important logical point. In our construction in sections \ref{sec:outside} and \ref{sec:behind}, we will {\em not assume} any of the claims that we are making in this section. Rather one of the points of our paper is that we independently find a picture in the conformal field theory which is consistent with the
expectations of conventional semi-classical physics. Our review below is meant to (a) remind the reader what these expectations are and (b) serve as a guide --- although not as a logical crutch --- for our later construction.
\subsection{Collapsing stars and eternal black holes}
In the first part of this section, we review the semi-classical expectation that the details of a collapsing star cease to matter both in front and behind the horizon for ``late enough'' times.
More specifically, and referring to figure \ref{collapseads}
we have
\vskip10pt
\noindent {\bf Semi-Classical Expectation:} {\it Late time bulk correlators in region A and region B of the collapsing star geometry, can be well approximated by correlators in region $\rm I$ and region $\rm II$ respectively of the eternal black hole.}
\vskip10pt
\begin{figure}
\begin{center}
\begin{subfigure}[t]{6cm}
\psfrag{Aads}{\rm A}
\psfrag{Bads}{\rm B}
\psfrag{tnot}{$t_0$}
\includegraphics[width=6cm]{collapseads.eps}
\caption{\parbox{5cm}{Collapse of a star (red) to form a black hole in AdS. The matter is injected from the boundary at some time $t_0$. A local observer (black line) dives in much later.} \label{fig:collapsinggeometry}}
\end{subfigure}
\qquad
\qquad
\begin{subfigure}[t]{6cm}
\psfrag{frontads}{$\rm I$}
\psfrag{backads}{$\rm III$}
\psfrag{blackads}{$\rm II$}
\psfrag{whiteads}{\rm IV}
\includegraphics[width=6cm]{kruskalads.eps}
\caption{\parbox{5cm}{AdS eternal black hole , reproduces the measurements of the observer at late times. The quantum fields are placed in the AdS-Hartle-Hawking vacuum.}\label{fig:eternalads}}
\end{subfigure}
\caption{Collapse vs eternal black hole in AdS}
\label{collapseads}
\end{center}
\end{figure}
Here when we say late, we mean late enough so that all the matter has fallen into the black hole and that the fluctuations of the horizon (quasi-normal modes) have mostly decayed away, but not so late that quantum mechanical effects become important. The two timescales have different parametric dependence on $N$ (or $\hbar$) so they can be clearly separated.
For asymptotically flat black holes this claim holds when the eternal black hole is taken in the Unruh vacuum. For black holes formed in AdS --- for instance, by throwing in matter from infinity as in figure \ref{fig:collapsinggeometry} ---, the claim holds when the AdS eternal black hole is taken in the Hartle-Hawking vacuum.\footnote{
The boundary of AdS acts like a reflecting wall, so the radiation coming out of the black hole eventually turns around and falls back in. The Hartle-Hawking state describes an equilibrium configuration where there is no net flux of energy. There is no AdS analogue of the Unruh vacuum.} These results are of course very well known in the context of flat space \cite{Unruh:1976db,birrell1984quantum,Hayden:2007cs} and we believe that they extend naturally
to the case of AdS.
The claim that the geometry outside the black-hole ``settles down'' is probably familiar to most readers; the claim that, at late times, we can replace the entire history of the collapse in region B in figure \ref{fig:collapsinggeometry} by an effective Kruskal geometry is probably less familiar.
However, the intuitive justification for these statements is the same and can be seen in the Kruskal diagram in figure \ref{fig:kruskal}.
Within geometric optics, an early-time observer (Observer E) can influence the late-time observer (Observer L) only by emitting a photon or another particle that travels along a trajectory that intersects the world-line of L. However, as L dives in later and later, the window and the solid angle within which E must send his signal becomes smaller and smaller.\footnote{Among other places, this fact was discussed in the works \cite{Susskind:1993mu,Hayden:2007cs,Sekino:2008he} in relation to the consistency between black hole complementarity and the absence of quantum cloning.}
This geometric-optics observation can easily be extrapolated to classical wave mechanics. If we consider a source that emits energy within some solid angle then if we keep the power of the source fixed, its influence at late times diminishes. So, perturbatively, it is clear that a disturbance in the Kruskal geometry at early times cannot influence the late time physics either in front or behind the horizon. The argument that the details of the collapse itself do not matter at late times is an extrapolation from this perturbative argument.
\begin{figure}
\begin{center}
\psfrag{early}{E}
\psfrag{late}{L}
\includegraphics[width=5cm]{kruskaldiagram.eps}
\caption{Kruskal diagram of AdS eternal black hole. As the observer (L) dives in later, it gets increasingly difficult for any signal from an earlier observer (E) to reach him. }
\label{fig:kruskal}
\end{center}
\end{figure}
Notice that if we fix the time at which the observer L falls in, it is always possible to turn on some matter-excitations which influences the observer. However, these configurations --- while they may be solutions to the equations of motion --- are not relevant as approximations to the geometry of the collapsing black hole. In our case, as we show in the figure, we create a black hole by turning on a source on the boundary, wait a ``sufficient'' time, and throw the observer L in. By the arguments above, the semi-classical expectation is that this observer should just perceive the geometry of an eternal black hole all along his world line.
One subtle point is that the existence of region $\rm III$ cannot be neglected in region $\rm II$. This is because, while classically no influence can propagate from region $\rm III$ to region $\rm II$ and influence the late time observer in figure \ref{fig:eternalads}, when we {\em quantize} the field in region $\rm II$ then, even at late times, it has both ``left-moving modes'' that are analytic continuations of modes from region $\rm I$ and ``right moving modes'' that are analytic continuations of modes from region $\rm III$. We return to this in section \ref{modesbrane}.
We emphasize again that in our construction, we will not assume either this feature of the quantum mechanics on the eternal AdS spacetime, or the semi-classical general relativity expectation above. Rather we will find both independently in the dual conformal field theory.
\subsection{Classical Properties of the AdS eternal black hole \label{secadseternal}}
The AdS eternal black hole that we have drawn in figure \ref{fig:eternalads} is a maximal continuation of the AdS-Schwarzschild black hole, just like the Kruskal geometry is a continuation of the Schwarzschild geometry. In this section, we review the metric and geometry of this space in some more detail.
We will work with the planar version of AdS black holes (i.e. branes). It is straightforward to rewrite everything in terms of AdS black holes with spherical event horizons --- this is even necessary if we wish to address questions related to Poincare recurrence, and other finite volume effects. In this paper, we did not do so since it was more convenient to work with momenta ${\bf k}$ rather than spherical harmonics.
The metric of the eternal AdS black brane is given by
\begin{equation}
\label{threebrane}
ds^2 = {\ell^2 \over z^2} \left[-h(z) dt^2 + {1 \over h(z)} dz^2 + d\vect{x}^2 \right],
\end{equation}
where \[
h(z) = 1-{z^d \over z_0^d}.
\]
The horizon is at $z=z_0$, the boundary at $z=0$ and $\vect{x}$ is a $(d-1)$-dimensional vector. There is no flux turned on here. The metric \eqref{threebrane} is a solution to the equations of motion for the action
\[
S = {-1 \over 16 \pi G_N} \int \sqrt{-g} \left[R + {d(d-1) \over \ell^2} \right].
\]
We have displayed the AdS radius $\ell$ explicitly here because it will make a brief appearance in our discussion of the temperature below. However, in what follows, and everywhere else, we will set $\ell = 1$.
We introduce the tortoise coordinate defined by
$
{dz_* \over dz} =- h^{-1}(z)
$. Now the horizon is at $z_*\rightarrow-\infty$. We fix the overall additive ambiguity in the definition of $z_*$ by requiring $z_*\rightarrow 0$ as $z\rightarrow 0$. The metric takes the form
\[
ds^2 = {h(z) \over z^2}(-dt^2 +dz_*^2) +{d\vect{x}^2 \over z^2}
\]
We go to lightcone coordinates
\begin{equation}
\label{lightconec}
u= t-z_*\qquad,\quad v= t+z_*
\end{equation}
\[
ds^2 = -{h(z)\over z^2} du dv + {d\vect{x}^2 \over z^2}
\]
Here $z$ is defined implicitly via $u,v$ by the previous changes of coordinates. Finally we define
\begin{equation}
\label{kruskalc}
U = -e^{-{d u \over 2 z_0}}\qquad,\qquad V = e^{{d v \over 2 z_0}}
\end{equation}
to get
\begin{equation}
\label{kruskal}
ds^2 = {4 h(z)\over d^2 U V} \left({z_0 \over z}\right)^2
dU dV +{d\vect{x}^2 \over z^2}
\end{equation}
This metric is originally defined in the region $U<0,V>0$. The future horizon is at $U\rightarrow 0, V={\rm constant}$ and the past horizon at $V\rightarrow 0, U={\rm constant}$. In the form \eqref{kruskal}, it is clear that the metric is smooth at both these horizons and can be smoothly extended past them.\footnote{Near the horizons we have $h \approx {d\over z_0}(z_0-z)$. The tortoise coordinate is $z_* \approx {z_0 \over d}\log\left({z_0-z \over z_0}\right)$. From the change of coordinates we find $h \approx d e^{d z_* \over z_0} = d U V.$}
On the other hand there is a specific positive value of $UV$ for which $h$ blows up (in our conventions for $z_*$ this happens at $UV = e^{-\pi}$). These points represent the future and past singularities.
\paragraph{How we take the Large $N$ Limit \\}
Let us pause briefly to specify precisely what we mean by taking the large $N$ limit in the context of this bulk geometry. Although this is a point that seems to cause confusion at times, what we are doing is perfectly conventional. In taking the large $N$ limit, we {\em keep the solution \eqref{threebrane} fixed}. This means that the temperature of the gauge theory also remains fixed and does not scale with $N$. Indeed, the temperature of the black-brane solution \eqref{threebrane} can easily be calculated to be (with all factors restored)
\[
T = {\hbar c d \over 4 \pi z_0 k_B},
\]
where $c$ is the speed of light and $k_B$ is the Boltzmann constant. Note that $G_N$ does not appear here. On the other hand, with this fixed temperature, the ADM mass-density of the black-brane will contain a factor of ${1 \over G_N}$. So the energy density of the boundary CFT does scale with $N$, as we take the large $N$ limit.
We should point out that $\ell$ also does not appear in the formula for the temperature. This is because we are considering the black-brane solution in AdS. If we were to instead consider the AdS-Schwarzschild solution, then the temperature would depend on $\ell$. Instead of thinking of a black-brane, the reader may instead prefer to think of a {\em big black hole} in AdS i.e. one where the horizon size is larger than the AdS radius. Such a black-hole is thermodynamically favoured, since the corresponding temperature is higher than the Hawking-Page transition temperature. So, for the conformal field theory on a sphere on radius $R$, our analysis is valid when we take the temperature to be any number larger than the phase transition temperature in units of ${1 \over R}$, as long as the temperature {\em does not scale with $N$}.
\subsection{Quantization in an eternal AdS black hole \label{modesbrane}}
We also need to remind the reader how to quantize a field on the background of an eternal AdS black hole.
In figure \ref{penroseadsb} we see a Cauchy slice for the entire spacetime.
\begin{figure}
\begin{center}
\psfrag{frontads}{$\rm I$}
\psfrag{backads}{$\rm III$}
\psfrag{blackads}{$\rm II$}
\psfrag{whiteads}{$\rm IV$}
\psfrag{uzero}{U=0}
\psfrag{vzero}{V=0}
\psfrag{s1}{$\Sigma_{\rm I}$}
\psfrag{s2}{$\Sigma_{\rm III}$}
\includegraphics[width=10cm]{kruskalads2.eps}
\caption{Cauchy slice for the eternal AdS black brane geometry.}
\label{penroseadsb}
\end{center}
\end{figure}
It can be thought of as the union of two smaller slices, denoted as $\Sigma_{\rm I}$ and $\Sigma_{\rm III}$. The slice $\Sigma_{\rm I}$ is a complete Cauchy slice if we restrict ourselves to events taking place in region $\rm I$ and the slice $\Sigma_{\rm III}$ for events in region $\rm III$. However in order to describe regions $\rm II$ and $\rm IV$ we need the entire slice $\Sigma_{\rm I} \oplus \Sigma_{\rm III}$. When we quantize the field in AdS we impose normalizable boundary conditions at infinity. This means that only the subleading mode (``the vev'') of the field can be turned on. For simplicity we avoid discussing the window of masses where two alternative quantizations are acceptable.
One way to find a complete set of solutions to be used for the quantization, is to first work with the wedge $\rm I$, whose Cauchy slice is $\Sigma_{\rm I}$ , and then with the wedge $\rm III$ and then put them together (the same thing as we do for the quantization of Rindler space or the flat space Schwarzschild solution). So we start with region $\rm I$.
We consider the black hole solution \eqref{threebrane} and a scalar field obeying
\[
(\Box - m^2) \phi =0
\]
we consider a solution of the form
\[
f_{\omega,\vect{k}}(t,\vect{x},z) = e^{-i \omega t + i \vect{k} \cdot \vect{x}} \psi_{\omega,\vect{k}}(z)
\]
Plugging into the Klein-Gordon equation we get a second order ordinary differential equation for $\psi_{\omega,\vect{k}}(z)$.
It has two linearly independent solutions. We impose normalizability of the solution near the boundary i.e.
\[
\psi_{\omega, \vect{k}} (z) \underset{z \rightarrow 0}{\longrightarrow} \Upsilon^{-1}\,z^{\Delta},
\]
which eliminates one linear combination of the solutions. So, for each choice of $(\omega,\vect{k})$ we have a unique normalizable solution.
We do not impose any boundary conditions at the horizon. Had we imposed ingoing boundary conditions we would have found solutions only for complex $\omega$ i.e.
the quasinormal frequencies. For the quantization of the field we need to find a complete set of solutions of the wave equation without any restriction on the horizon.
The solutions that we found are linear combinations of ingoing and outgoing modes. By an appropriate choice of the overall phase, the modes can be taken to be real.
If we introduce the tortoise radial coordinate $z_*$, in which the horizon is at $z_*\rightarrow - \infty$ we find that the modes behave like
\begin{equation}
\label{nearhpsi}
\psi_{\omega,\vect{k}}\underset{z \rightarrow z_0}{\longrightarrow} c(\omega,\vect{k})\left(e^{-i\delta_{\omega,\vect{k}}}e^{-i\omega z_*} + e^{i\delta_{\omega,\vect{k}}}e^{i\omega z_*}\right)
\end{equation}
where $c(\omega,\vect{k})$ is a positive real constant.
The relative phase difference $e^{i\delta_{\omega,\vect{k}}}$ is physically meaningful and cannot be removed by changing the conventions.
It is also possible to normalize the modes, so that they are canonical
with respect to the Klein-Gordon norm. In this second normalization, we write
\[
\hat{f}_{\omega,\vect{k}}(t,\vect{x},z) = e^{-i \omega t + i \vect{k} \cdot \vect{x}} \hat{\psi}_{\omega,\vect{k}}(z),
\]
where
\[
\begin{split}
&\hat{\psi}_{\omega, \vect{k}} \underset{z \rightarrow z_0}{\longrightarrow} z_0^{d-1 \over 2} \times (e^{-i\delta_{\omega,\vect{k}}}e^{-i\omega z_*} + e^{i\delta_{\omega,\vect{k}}}e^{ i\omega z_* }),\\
&\hat{\psi}_{\omega, \vect{k}} \underset{z \rightarrow 0}{\longrightarrow} c(\omega,\vect{k})^{-1} z_0^{{d-1 \over 2}} z^{\Delta} {1 \over \Upsilon}.
\end{split}
\]
This notation, where modes that go like ``1'' near the boundary are denoted by $f_{\omega,\vect{k}}(t,\vect{x},z)$ and those that go like ``1'' near the horizon are denoted by $\hat{f}_{\omega,\vect{k}}(t,\vect{x},z)$ will be used below.
In any case, we have found a complete set of solutions of the Klein-Gordon equation for region $\rm I$ which can be used to expand the quantum field $\phi$ in region $\rm I$ in creation and annihilation modes
\[
\phi(t,\vect{x},z) = \int_{\omega>0} {d\omega d^{d-1}\vect{k} \over (2 \pi)^d}\,\,{1\over \sqrt{2\omega}}\left[a_{\omega,\vect{k}} \hat{f}_{\omega,\vect{k}}(t,\vect{x},z) + \,\,{\rm h.c.}\right]
\]
The modes satisfy the standard commutation relations
\[
[a_{\omega,\vect{k}},a_{\omega',\vect{k}'}^\dagger] = \delta(\omega-\omega')\delta^{d-1}(\vect{k}-\vect{k}')\]
with all other commutators vanishing.
Notice that the spectrum in $\omega$ is continuous. This does not have to do with the non-compactness of the spatial directions, even if we consider a black hole in global AdS we still
find a continuous spectrum in $\omega$ --- even though the boundary theory lives on a compact space ${\mathbb S}^{d-1}$. The continuum in $\omega$ is a large $N$ artifact and related
to the approximately continuous spectrum of the dual large $N$ gauge theory in the deconfined phase, see \cite{Festuccia:2005pi, Festuccia:2006sa} for more details.
Following the same analysis in region $\rm III$ we get another set of creation and annihilation modes that we denote by $\widetilde{a}_{\omega,\vect{k}}$. These modes satisfy an identical-oscillator type algebra among themselves, and commute with all the modes $a_{\omega, \vect{k}}$. If we have the expansion of the field in a complete basis both in regions I and III, it is straightforward to extend it to regions II and IV.
While it should be obvious from figure \ref{penroseadsb}, we would like to emphasize again that in order to describe a local field in region II (inside the black hole) it is necessary to use {\it both} the operators $a_{\omega,\vect{k}}$ which are visible outside the horizon {\it and} the operators $\widetilde{a}_{\omega,\vect{k}}$ which seem to come from region III.
Finally let us mention that the natural vacuum for a quantum field in AdS in the presence of a big AdS black hole is the analogue of the Hartle-Hawking vacuum. The Hawking radiation from the black holes
is reflected by the AdS potential and an equilibrium state like that of the flat-space Hartle Hawking state is reached (there is no analogue of the Unruh vacuum). In terms of our oscillators
the Hartle Hawking state is characterized by thermal occupation levels
\begin{equation}
\label{hhocup}
\langle a_{\omega,\vect{k}} \,a^\dagger_{\omega',\vect{k}'}\rangle_{\rm HH} = {e^{\beta \omega}\over e^{\beta \omega}-1} \delta(\omega-\omega')\delta^{d-1}(\vect{k}-\vect{k}')
\end{equation}
\begin{equation}
\label{hhocupb}
\langle a_{\omega,\vect{k}}^\dagger \,a_{\omega',\vect{k}'}\rangle_{\rm HH} = {1\over e^{\beta \omega}-1} \delta(\omega-\omega')\delta^{d-1}(\vect{k}-\vect{k}')
\end{equation}
and similar for the modes $\widetilde{a}_{\omega,\vect{k}}$. Here $\beta$ is the Hawking temperature of the black hole \eqref{threebrane}.
\section{Acknowledgments}
We would like to thank Ofer Aharony, Luis Alvarez-Gaume, Nima
Arkani-Hamed, Jan de Boer, Borun Chowdhury, Sheer El-Showk, Ben
Freivogel, Rajesh Gopakumar, Monica Guica, Dan Harlow, Gary Horowitz,
Veronika Hubeny, Norihiro Iizuka, Tom Hartman, Elias Kiritsis, Neil
Lambert, Klaus Larjo, Gilad Lifschytz, Loganayagam, David Lowe, Juan Maldacena, Don
Marolf, Paul McFadden, Shiraz Minwalla, Boris Pioline, Joe Polchinski,
Mukund Rangamani, Ashoke Sen, Masaki Shigemori, Joan Simon, Sandip
Trivedi, Erik Verlinde, Spenta Wadia, Edward Witten and Xi Yin for helpful discussions. We would
also like to thank all the participants of the discussion meeting on
string theory at ICTS, Bangalore, in June 2012 and members of the quantum information group at HRI. KP would like to thank the organizers and participants
of the ``Bits, Branes and Black Holes'' workshop at KITP 2012, the Cosmology and Complexity 2012 meeting in Hydra,
the Amsterdam summer string workshop 2012, the ``Holographic Way'' workshop in
Nordita and the Crete Center for Theoretical Physics for hospitality. SR is partially supported by a
Ramanujan fellowship of the Department of Science and Technology
(India). SR gratefully acknowledges the hospitality of Brown
University, where part of this work was done.
\section{Quantization in Rindler space \label{rindlerq}}
\subsection{Expansion in Rindler modes}
We start with $d+1$ dimensional Minkowski space
\[
ds^2 = -dt^2 + dz^2 + d\vect{x}^2
\]
and consider a massless scalar field obeying
\[
\Box \phi = 0
\]
\subsubsection{Region I}
We first expand the field in modes in region I. In that region the Rindler coordinates $(\tau,\sigma,\vect{x})$ are defined by $t= \sigma \sinh \tau,\, z = \sigma \cosh t$. The metric looks like
\[
ds^2 = -\sigma^2 d\tau^2 + d\sigma^2 + d\vect{x}^2
\]
The field in region I has the expansion
\[
\phi(\tau,\sigma,\vect{x}) = \int_{\omega>0}{d\omega d^{d-1}\vect{k} \over (2 \pi)^d} \left[{1\over \sqrt{2\omega}} a_{\omega,\vect{k}} e^{-i\omega \tau+i \vect{k} \vect{x}}{2 K_{i\omega}(|\vect{k}| \sigma)
\over |\Gamma(i\omega)|} + {\rm h.c.}
\right]
\]
Notice that the Bessel function $K_{i\omega}(|\vect{k}| \sigma)$ is real.
We define the lightcone coordinates in Minkowski space
\[
U = t-z
\]
\[
V= t+z
\]
In region I we have $U<0,V>0$. In terms of the Rindler coordinates in region I we have
\[
U = - \sigma e^{-\tau}\]
\[
V = \sigma e^{\tau}
\]
Considering the field expansion near the horizon between regions I and II we find
\[
\phi \underset{U \rightarrow 0}{\approx} \int_{\omega>0}{d\omega d^{d-1}\vect{k} \over (2 \pi)^d}{1\over \sqrt{2\omega}} a_{\omega\vect{k}} e^{i \vect{k} \vect{x}} \left[V^{-i\omega} e^{-i\delta} + (-U)^{i\omega}e^{i\delta}\right]
\]
\[
+ {1\over \sqrt{2\omega}} a_{\omega,\vect{k}}^\dagger e^{-i \vect{k} \vect{x}} \left[V^{i\omega} e^{i\delta} + (-U)^{-i\omega}e^{-i\delta}\right]
\]
where the phase shift is
\begin{equation}
\label{phase}
e^{i\delta} = \left({|\vect{k}|\over 2}\right)^{i\omega} {\Gamma(-i\omega) \over |\Gamma(i\omega)|}
\end{equation}
Due to the Riemann-Lebesgue lemma, when $U\rightarrow 0$ terms like $U^{-i \omega}$ can be discarded. So what is important is to keep the terms that depend on $V$ only, and we have
\begin{equation}
\label{boundaryIandII}
\phi \underset{U \rightarrow 0}{\approx}\int_{\omega>0}{d\omega d^{d-1}\vect{k} \over (2 \pi)^d} {1\over \sqrt{2\omega}} \left[a_{\omega,\vect{k}} e^{i \vect{k} \vect{x}} V^{-i\omega} e^{-i\delta} +
a_{\omega,\vect{k}}^\dagger e^{-i \vect{k} \vect{x}} V^{i\omega} e^{i\delta} \right]
\end{equation}
\subsubsection{Region III}
Here the coordinates are $t= -\sigma \sinh \tau,\,\,z=-\sigma \cosh \tau$. We notice that the Rindler time $\tau$ runs opposite of the Minkowski time $t$. Also,
notice that due to our parametrization we still have $\sigma>0$ even though $z<0$ in region III. We expand the field as
\[
\phi(\tau,\sigma,\vect{x}) = \int_{\omega>0}{d\omega d^{d-1}\vect{k} \over (2 \pi)^d} \left[{1\over \sqrt{2\omega}} \widetilde{a}_{\omega,\vect{k}} e^{i\omega \tau-i \vect{k} \vect{x}}{2 K_{i\omega}(|\vect{k}| \sigma)
\over |\Gamma(i\omega)|} + {\rm h.c.} \right]
\]
Notice that we have defined the modes $\widetilde{a}_{\omega,\vect{k}}$ to multiply the function which is the conjugate of the one before.
We want to expand the field near the horizon between regions III and II. Again in Minkowski lightcone coordinates we have
\[
U = \sigma e^{-\tau}
\]
\[
V = -\sigma e^\tau
\]
The horizon is now at $V\rightarrow 0$. Following the same steps as for region I we find that near the horizon the non-vanishing terms are
\begin{equation}
\label{boundaryIIandIII}
\phi \underset{V \rightarrow 0}{\approx} \int_{\omega>0}{d\omega d^{d-1}\vect{k} \over (2 \pi)^d} {1\over \sqrt{2\omega}}\left[ \widetilde{a}_{\omega,\vect{k}} e^{-i \vect{k} \vect{x}} U^{-i\omega} e^{-i\delta}
+\widetilde{a}^\dagger_{\omega,\vect{k}} e^{i\vect{k}\vect{x}} U^{i\omega}e^{i\delta} \right]
\end{equation}
the phase factor is again given by \eqref{phase}.
\subsubsection{Region II}
Here we choose the Rindler coordinates as $t =\sigma \cosh \tau,\,\, z = \sigma \sinh \tau$. The horizon between I and II is at $\tau\rightarrow +\infty$. In terms of the lightcone coordinates
we have
\[
U = \sigma e^{-\tau}
\]
\[
V = \sigma e^{\tau}
\]
We write the general expansion
as
\[
\phi = \int_{\omega>0}{d\omega d^{d-1}\vect{k} \over (2 \pi)^d} {1\over \sqrt{2\omega}}\left[A_{\omega,\vect{k}}e^{-i \omega \tau + i \vect{k}\vect{x}} J_{i\omega}(|\vect{k}|\sigma) +
B_{\omega,\vect{k}} e^{-i \omega t + i \vect{k} \vect{x}} J_{-i \omega}(|\vect{k}|\sigma) + {\rm h.c.} \right]
\]
For the $J$ Bessel functions as $\sigma\rightarrow 0$ we have
\[
J_{i\omega}(|\vect{k}|\sigma) \approx {1\over \Gamma(1+i\omega)} \left({|\vect{k}|\sigma \over 2}\right)^{i\omega} +\ldots
\]
Now we are looking at the expansion closed to the horizon between I and II. There we have $U\rightarrow 0$,
so we keep only the terms which depend on $V$ and we find
\[
\phi \underset{U \rightarrow 0}{\approx} \int_{\omega>0}{d\omega d^{d-1}\vect{k} \over (2 \pi)^d} {1\over \sqrt{2\omega}}
\left[
{B_{\omega,\vect{k}}\over \Gamma(1-i\omega)} e^{i\vect{k} \vect{x}}\left({|\vect{k}|\over 2}\right)^{-i\omega} V^{-i\omega}+{\rm h.c.}
\right]
\]
Comparing with the expansion \eqref{boundaryIandII} we find that
\[
B_{\omega,\vect{k}} = {\Gamma(1-i\omega) \Gamma(i\omega) \over |\Gamma(i\omega)|}a_{\omega,\vect{k}}= -i\sqrt{\pi \omega \over \sinh(\pi \omega)} a_{\omega,\vect{k}}
\]
While, looking at the horizon between II and III we find
\[
\phi \underset{V \rightarrow 0}{\approx} \int_{\omega>0} {d\omega d^{d-1} \vect{k} \over (2 \pi)^d} {1\over \sqrt{2\omega}} \left[{A_{\omega,\vect{k}}\over \Gamma(1+i\omega)}e^{i\vect{k}\vect{x}}
\left({|\vect{k}|\over 2}\right)^{i\omega}U^{i\omega} + {\rm h.c.}
\right]
\]
Comparing with \eqref{boundaryIIandIII} we find
\[
A_{\omega,\vect{k}} = {\Gamma(1+i\omega)\Gamma(-i\omega) \over |\Gamma(i\omega)|}\widetilde{a}_{\omega,\vect{k}}^\dagger
=i\sqrt{\pi \omega \over \sinh(\pi \omega)} \widetilde{a}_{\omega,\vect{k}}^\dagger
\]
Putting everything together we find the expansion in region II
\begin{equation}
\label{regionIIrindler}
\phi(\tau,\sigma,\vect{x}) = \int_{\omega>0}{d\omega d^{d-1}\vect{k} \over (2 \pi)^d} {1\over \sqrt{2\omega}}\sqrt{\pi \omega\over \sinh(\pi \omega)} \left[i\,\widetilde{a}^\dagger_{\omega,\vect{k}}
\,e^{-i\omega \tau+ i \vect{k} \vect{x}}\, J_{i\omega}(|\vect{k}|\sigma)
- i a_{\omega,\vect{k}} e^{-i\omega \tau + i \vect{k} \vect{x}} J_{-i\omega}(|\vect{k}|\sigma)+ {\rm h.c.}\right]
\end{equation}
\subsubsection{Bogoliubov transformation}
Now we express the Rindler modes in terms of the Unruh modes $d^{1,2}_{\omega,\vect{k}}$. We have
\begin{equation}
\label{unruh}
d^1_{\omega,\vect{k}} = {a_{\omega,\vect{k}} - e^{-\pi \omega}\, \widetilde{a}_{\omega,\vect{k}}^\dagger \over \sqrt{1-e^{-2\pi \omega}}}\qquad,\qquad
d^2_{\omega,\vect{k}} = {\widetilde{a}_{\omega,-\vect{k}} - e^{-\pi \omega}\, a_{\omega,-\vect{k}}^\dagger \over \sqrt{1-e^{-2\pi \omega}}}
\end{equation}
or inverting
\begin{equation}
\label{unruhinv}
a_{\omega,\vect{k}} = {d^1_{\omega,\vect{k}} + e^{-\pi \omega}(d^2_{\omega,-\vect{k}})^\dagger\over \sqrt{1-e^{-2\pi \omega}}}\qquad,\qquad
\widetilde{a}_{\omega,\vect{k}} = {d^2_{\omega,-\vect{k}}+ e^{-\pi \omega} (d^1_{\omega,\vect{k}})^\dagger \over \sqrt{1-e^{-2\pi \omega}}}
\end{equation}
The Minkowski vacuum $|0\rangle$ is defined by $d^{1,2}_{\omega,\vect{k}}|0\rangle =0$. The Rindler mode occupation levels are
\[
\langle 0| a_{\omega,\vect{k}} a^\dagger_{\omega',\vect{k}'}|0\rangle ={e^{2\pi \omega} \over e^{2\pi \omega}-1}\delta(\omega-\omega')\delta^{d-1}(\vect{k}-\vect{k}')\,\,,\,\,
\langle 0| a_{\omega,\vect{k}}^\dagger a_{\omega',\vect{k}'}|0\rangle ={1 \over e^{2\pi \omega}-1}\delta(\omega-\omega')\delta^{d-1}(\vect{k}-\vect{k}')
\]
and similarly for the $\widetilde{a}_{\omega,\vect{k}}$ modes.
\subsection{2-point function in terms of Rindler modes}
\subsubsection{Region I}
We can now write the usual Wightman 2-point function of a scalar field in terms of the Rindler modes. For points in region I we have
\[
\langle 0| \phi(\tau_1,\sigma_1,\vect{x}_1) \phi(\tau_2,\sigma_2,\vect{x}_2)|0\rangle = (2\pi)^d \int_{\omega>0}{d\omega d^{d-1}\vect{k} \over (2 \pi)^d}\]
\[{1\over 2\omega}
\Bigg[{e^{2\pi \omega}\over e^{2\pi \omega}-1} {4K_{i\omega}(|\vect{k}|\sigma_1)K_{i\omega}(|\vect{k}|\sigma_2)
\over |\Gamma(i\omega)|^2}e^{-i\omega \tau_{12}+i\vect{k}\,\vect{x}_{12}}+{1\over e^{2\pi \omega}-1} {4K_{i\omega}(|\vect{k}|\sigma_1)K_{i\omega}(|\vect{k}|\sigma_2)
\over |\Gamma(i\omega)|^2}e^{i\omega \tau_{12}-i\vect{k}\,\vect{x}_{12}}\Bigg]\]
We are interested in the convergence of this integral in the region $\omega=0$. We have the explicit factor of ${1\over 2\omega}$ in front, the thermal occupation factors
give another factor of ${1\over \omega}$. The Bessel function $K_{i\omega}(z)$ goes to a non-zero constant as $\omega$ goes to zero, for fixed $z$. Finally the factor ${1\over |\Gamma(i\omega)|^2}$
goes like $\omega^2$ for small $\omega$. All in all, the integrand goes like $\omega^0$ for small $\omega$ and hence the integral converges when $\omega\rightarrow 0$.
\subsubsection{Region II}
Now let us consider two points behind the Rindler horizon i.e. in region II, using the expansion \eqref{regionIIrindler}. We have contributions of several bilinears made out of $a_{\omega,\vect{k}}$ and $\widetilde{a}_{\omega,\vect{k}}$. If focus on only the contributions from the bilinears $\langle 0|a_{\omega,\vect{k}}
a_{\omega',\vect{k}'}^\dagger|0\rangle$ and $\langle 0|a_{\omega,\vect{k}}^\dagger
a_{\omega',\vect{k}'}|0\rangle$ of the non-tilde operators, we find the terms
\[
\begin{split}
(2\pi)^d \int_{\omega>0}{d\omega d^{d-1}\vect{k} \over (2 \pi)^d} {\pi \over \sinh(\pi \omega)}\Bigg[ &{e^{2\pi \omega} \over e^{2\pi \omega}-1} J_{i\omega}(|\vect{k}|\sigma_1)
J_{-i\omega}(|\vect{k}|\sigma_2)e^{-i\omega \tau_{12}+i \vect{k}\,\vect{x}_{12}}\\
& + {1 \over e^{2\pi \omega}-1} J_{-i\omega}(|\vect{k}|\sigma_1)
J_{i\omega}(|\vect{k}|\sigma_2)e^{i\omega \tau_{12}-i \vect{k}\,\vect{x}_{12}}\Bigg]
\end{split}
\]
For small $\omega$ (and fixed $\sigma_1,\sigma_2$) the Bessel functions go to nonzero constants, hence the integrand goes like ${1\over \omega^2}$ and the integral
diverges as
\[
\int_{\omega>0} {d\omega \over \omega^2}
\]
This seems to suggest that the small $\omega$ region has a very large contribution to the 2-point function, however we know that this cannot be the correct answer.
After all, the 2-point function that we are considering the the standard Wightman function of a massless scalar field, which is obviously finite for two points in region II. Hence it must be that, while the contributions from these two terms mentioned above are formally divergent, the total contribution from all terms --- that is from the non-tildes and from cross terms--- must be finite. There must be cancellations between the terms that we have considered and the remaining terms.
In other words, if we regroup the terms correctly {\it before} doing the $\omega$ integral, the resulting expression must be manifestly finite as we integrate down to $\omega=0$. It turns out that regrouping the $a_{\omega,\vect{k}}$ and $\widetilde{a}_{\omega,\vect{k}}$ into the ``Unruh combinations'' \eqref{unruh} makes the integral manifestly convergent. Indeed, substituting from \eqref{unruhinv} into \eqref{regionIIrindler} we find that the field in region II can be written as
\[
\phi(\tau,\sigma,\vect{x})= \int_{\omega>0} {d\omega d^{d-1}\vect{k} \over (2 \pi)^d} {1\over \sqrt{2\omega}}\sqrt{\pi \omega\over \sinh(\pi \omega)}\left[e^{-i\omega \tau + i \vect{k} \, \vect{x}}\,i\, \left({
e^{-\pi \omega}J_{i\omega}(|\vect{k}|z) - J_{-i\omega}(|\vect{k}|\sigma)\over \sqrt{1-e^{-2\pi \omega}}}\right)d^1_{\omega,\vect{k}} + {\rm h.c.} \right]
\]
\[
+{\rm terms\,\,involving\,\,}d^2_{\omega,\vect{k}}
\]
We have that
\[
e^{-\pi \omega}J_{i\omega}(z) - J_{-i\omega}(z) = -\sinh (\pi \omega)H^2_{i\omega}(z)
\]
Hence we find
\[
\phi(\tau,\sigma,\vect{x}) = \int_{\omega>0} {d\omega d^{d-1}\vect{k} \over (2 \pi)^d} {\sqrt{\pi}\over 2}e^{\pi \omega\over 2}\left[-\,i \,e^{-i\omega \tau + i \vect{k} \,\vect{x}}
H^2_{i\omega}(|\vect{k}|\sigma) d^1_{\omega,\vect{k}}
+ {\rm h.c.} \right]
\]
\[
+{\rm terms\,\,involving\,\,}d^2_{\omega,\vect{k}}
\]
On the Minkowski vacuum we have $d^{1,2}_{\omega,k} |0\rangle = 0$. So the 2-point function for points in region II becomes
\[
\langle 0| \phi(\tau_1,\sigma_1,\vect{x}_1) \phi(\tau_2,\sigma_2,\vect{x}_2) |0\rangle = (2\pi)\int_{\omega>0}{d\omega d^{d-1} \vect{k} \over (2 \pi)^d} {\pi \over 4} e^{\pi \omega}
e^{-i\omega t_{12}+i \vect{k}\, \vect{x}_{12}} H^2_{i \omega}(|\vect{k}|\sigma_1)\,H^2_{i\omega}(|\vect{k}|\sigma_2)^* +\]
\[
+{\rm terms\,\,involving\,\,}d^2_{\omega,\vect{k}}
\]
In this form we notice that the contribution from
$d^1_{\omega,\vect{k}}$ is manifestly finite when integrating all the
way down to $\omega=0$. The same is true for the contribution from $d^2_{\omega,\vect{k}}$.
\section{Introduction}
Even though quantum gravity has attracted theoretical interest for decades, several aspects of the theory continue to be actively debated. This includes, among other questions, the issue of how to define ``local operators'' in a theory of quantum gravity. Where black holes are involved, the situation seems to be even more puzzling. What is the nature of spacetime behind the horizon of the black hole? What about the horizon itself? Even though the principle of equivalence suggests that there is nothing special about the horizon of a large black hole, there have been several speculations that quantum gravity effects cause the interior to be modified into a fuzzball \cite{Mathur:2012np,Mathur:2009hf,Mathur:2008nj,Lunin:2002bj,Lunin:2001fv}, and more recently, that the horizon of an ``old black hole''
is replaced by a firewall \cite{Almheiri:2012rt}. (See also \cite{Bousso:2012as, Nomura:2012sw, Mathur:2012jk, Susskind:2012rm, Bena:2012zi, Giveon:2012kp, Banks:2012nn, Ori:2012jx, Brustein:2012jn, Susskind:2012uw, Marolf:2012xe, Hossenfelder:2012mr, Nomura:2012cx, Hwang:2012nn, Larjo:2012jt,braunstein2009v1, Avery:2012tf, Chowdhury:2012vd}.) These latter proposals originate, not from direct calculations in quantum gravity, but rather in arguments that the information
paradox (in various incarnations) cannot be solved without modifying the geometry at, or behind the horizon.
While quantum gravity is a mysterious subject, the AdS/CFT correspondence \cite{Maldacena:1997re} provides us with a setting where we can examine these ideas within a perfectly well defined theory. To this end, in this paper we would like to examine the following conceptual questions:
\begin{itemize}
\item
Is it possible to describe the results of local experiments in AdS, at least within perturbation theory in ${1 \over N}$, using the boundary field theory?
\item
In the presence of a black hole, can we also describe local experiments {\em behind} the black hole horizon in the boundary theory?
\item
Does our construction of the degrees of freedom behind the horizon shed any light on the information puzzle?
\end{itemize}
More precisely, we will imagine that within anti-de Sitter space, we start with some matter that then collapses to form a large black hole. We will then try and reconstruct local operators outside this black hole, and behind the horizon.
In fact, if we imagine an observer who lives outside this black hole for a while and then dives in, then we would require answers to all the questions above to describe his experience; this is the reason for the title of our paper.
We will consider some strongly coupled CFT in $d$ dimensions, which has the properties that would allow it to have a bulk dual, without restricting ourselves to any specific example of the AdS/CFT correspondence. We will place the CFT in a pure initial state that thermalizes after a while i.e. it evolves to a state that is almost indistinguishable from a thermal state. In this state, we will then show how to reorganize all the operators that are accessible in the CFT to a low-energy observer, into fields
that are labeled by points in the semi-classical geometry of a big black hole in \ads[d+1]. We emphasize that these are still CFT operators, although rather than being labeled by a boundary point, they are labeled by a bulk point. We will further show, to lowest order in the ${1 \over N}$ expansion and argue to higher orders, that the correlators of the operators that we have constructed are the same as the correlators of perturbative fields on this geometry. Second, we will
push this construction {\em past the horizon} and show how to construct perturbative fields behind the horizon in terms of CFT operators. This will give us important clues about how to resolve the firewall paradox, as we explore in section \ref{sec:applications}.
Our construction follows important work by several other authors \cite{Banks:1998dd, Balasubramanian:1999ri, Bena:1999jv, Hamilton:2006fh,Hamilton:2006az,Hamilton:2005ju,Hamilton:2007wj,VanRaamsdonk:2009ar,VanRaamsdonk:2010pw,VanRaamsdonk:2011zz,Czech:2012bh,Heemskerk:2012mn}. However, as we describe in more detail below, we have been able to make some technical improvements on the construction of holographic operators outside the horizon. Our construction of the black hole interior in terms of CFT data is new, and to our knowledge has not been explored before.
We now quickly summarize our results. Before considering the black hole, we start by considering empty AdS. To construct local operators in this space, we consider a generalized free-field \cite{ElShowk:2011ag} ${\cal O}(t, \vect{x})$ when the CFT is in the vacuum. Working with the modes ${\cal O}_{\omega,\vect{k}}$ of ${\cal O}(t, \vect{x})$ in momentum space, we are able to write a CFT operator that is labeled by a point in the bulk of the AdS Poincare patch
\begin{equation}
\label{cftpure}
\phi_{\text{CFT}} (t,\vect{x}, z) = \int_{\omega>0} {d \omega d^{d-1} \vect{k} \over (2 \pi)^d} \,\left[ {\cal O}_{\omega, \vect{k}} \xi_{\omega, \vect{k}}(t, \vect{x}, z) + \text{h.c}.\right]
\end{equation}
When the mode functions $\xi$ are appropriately chosen, the operator on the left has the same correlators as a free-field propagating in \ads[d+1]: for example, its commutator at two points that are spacelike separated in \ads[d+1] vanishes. We show how these operators can be continued beyond the Poincare patch onto all of global AdS. This serves as a warm up for our next task of looking beyond the black hole horizon.
We then consider generalized free-fields ${\cal O}(t,\vect{x})$, but in a CFT state that, although pure, is ``close'' to the thermal state. We will refer to this state as $\state{\Psi}$ below. Now, we find that we have to write
\begin{equation}
\label{cftthermal}
\phi_{\text{CFT}} (t,\vect{x}, z) = \int_{\omega>0} {d \omega d^{d-1} \vect{k} \over (2 \pi)^d}\,\left[ {\cal O}_{\omega, \vect{k}} f_{\omega, \vect{k}}(t, \vect{x}, z) + \text{h.c}.\right]
\end{equation}
Although \eqref{cftthermal} looks deceptively similar to \eqref{cftpure}, there are several differences. First, the mode functions $f$ are different from the ones that we encountered above. Another important difference is that, while in \eqref{cftpure}, we can set the mode functions for all cases where $\omega^2 < \vect{k}^2$ to zero, we cannot do so in \eqref{cftthermal}. Nevertheless, with $f$ chosen appropriately, \eqref{cftthermal} gives a good and local description of fields in front of the black hole horizon.
Next, we point out that in this pure state $\state{\Psi}$, after it has settled down, for each such operator ${\cal O}$, there must necessarily exist operators $\widetilde{\cal O}$ that have the properties that they (a) commute with ${\cal O}$ and (b) that, in the state $\state{\Psi}$, measurements of $\widetilde{\cal O}$ are completely (anti)-correlated with measurements of ${\cal O}$. (We make this more precise in section \ref{sec:behind}.) For us, these operators $\widetilde{\cal O}$ play the role that operators in the ``second copy'' of the CFT would have played, had we been dealing with an eternal black hole. Using these operators $\widetilde{\cal O}$ we now construct operators behind the horizon:
\[
\phi_{\text{CFT}}(t,\vect{x},z) =
\int_{\omega>0} {d\omega d^{d-1}\vect{k} \over (2 \pi)^d}\, \left[ {\cal O}_{\omega,\vect{k}} g_{\omega,\vect{k}}^{(1)}(t,\vect{x},z) + \widetilde{\cal O}_{\omega,\vect{k}} g_{\omega,\vect{k}}^{(2)}(t,\vect{x},z)+ \text{h.c.}
\right]
\]
where $g^{(1)}$ and $g^{(2)}$ are again functions that are chosen to make
this operator local.
Our construction is perfectly regular as we cross the horizon. This appears to be in contradiction with both the fuzzball and the firewall proposals. In section \ref{sec:applications} we first show that it is not possible to pinpoint the microstate of the CFT by measuring correlators of light operators to any given fixed order in the ${1 \over N}$ expansion. Our construction then implies that by doing experiments that are limited to some finite order in ${1 \over N}$, either at low or high energies, the bulk observer cannot distinguish the microstates of the black hole. Since the fuzzball proposal solves the information paradox by postulating that the observer can detect the microstate by doing ``low energy experiments'', our proposal appears to be inconsistent with this resolution.
Turning to the firewall paradox, in this paper we provide only indirect evidence for the absence of firewalls at the horizon. This is because our construction works in detail for a big black hole that does not evaporate (except over the Poincare recurrence time), and so this leaves us with the theoretical possibility that small black holes in AdS could have firewalls near the horizon. However, our description of the degrees of freedom in the interior of the black hole also provides us with several lessons that we can use to understand the information paradox that appear to make firewalls superfluous.
In particular, our construction can, roughly, be interpreted as showing how, if we expand our space of observables to include operators that give us finer and finer information about the CFT microstate, then eventually we reach a stage where the operators behind the horizon are no longer independent of
the operators in front of the horizon. Such a description provides a natural realization of black-hole ``complementarity'' \cite{'tHooft:1990fr,Susskind:1993if} and, as we explore below, removes the necessity of firewalls.
We should also point out that while this might naively imply a violation of causality, this causality violation is visible only when we measure very high point correlators of operators (where the number of insertions scales with $N$) or equivalently measure a low point correlator to exponential accuracy in $N$. At this level of accuracy, we do not believe that a semi-classical spacetime description makes sense at all and so this putative causality-violation is not a cause for concern.
There have been many earlier attempts to study the interior of the black hole using AdS/CFT and the literature on the information paradox is vast. Besides the papers mentioned above, a very incomplete list of references which were relevant to our work include \cite{Kiem:1995iy,Balasubramanian:1999zv,Giddings:2001pt,Maldacena:2001kr,Hubeny:2002dg,Kraus:2002iv,Levi:2003cx,Fidkowski:2003nf,Barbon:2003aq,Kaplan:2004qe,Balasubramanian:2004zu,Festuccia:2005pi,Balasubramanian:2005mg,
Balasubramanian:2005qu,Festuccia:2006sa,Balasubramanian:2007qv,Marolf:2008mf,Balasubramanian:2008da,deBoer:2009un,Horowitz:2009wm,Balasubramanian:2011dm,Avery:2011nb,Simon:2011zza}.
A brief overview of this paper is as follows. In section \ref{sec:emptyads}, we describe the construction of local operators in empty AdS using boundary operators in a flat space CFT in a pure state. We work in the Poincare patch and then show how our construction can be continued past the Poincare horizon into global AdS. In section \ref{sec:outside}, we repeat this construction in front of the horizon of a big black hole in AdS. In section \ref{sec:behind}, we push this construction past the black-hole horizon and write down local bulk operators behind the horizon in terms of CFT operators. In section \ref{sec:applications}, we discuss the implications of these results, with a particular view to the information paradox and various unconventional proposals for resolving it --- including the ``firewall'' paradox and the fuzzball proposal. We also discuss a qubit toy model that is surprisingly effective at realizing many of these ideas. In section \ref{sec:subtleties}, we address some common worries
and
show that our construction is not destabilized by tiny effects like
Poincare recurrence on the boundary. We also describe schematically how it may be extended to higher orders in the ${1 \over N}$ expansion, and discuss some other subtleties. We conclude in section \ref{sec:conclusion}. The appendices provide some technical details and also examine the specific case of two-dimensional thermal correlators and the BTZ-black-hole background.
\section{Reconstructing local bulk observables in empty AdS \label{sec:emptyads}}
In this section, we develop the methodology of constructing local
operators from boundary operators in empty AdS.
Consider an \ads[d+1]/\cft[d] duality, where we have some {\em generalized free fields} ${\cal O}$ that live on the boundary. By {\em generalized free fields}, we mean that the correlators of ${\cal O}$ factorize to leading order in some parameter, which we denote by ${1 \over N}$.\footnote{In this paper, by $N$ we are not referring to the rank of the gauge-group on the boundary. The reader may prefer to think of $N$ as the coefficient of the two-point function of the stress tensor
(which actually scales like $K^2$ in the ${{\mathcal N} = 4}$ SYM theory with gauge group SU(K)) but more generally, $N$ can be taken
to any parameter that controls the factorization of correlators.}
\begin{equation}
\label{factorization}
\langle 0 | {\cal O}(\vect{x_1}) \ldots {\cal O}(\vect{x_{2n}}) |0 \rangle= {1 \over 2^n} \sum_{\pi} \langle 0 |{\cal O}(\vect{x_{\pi_1}}) {\cal O}(\vect{x_{\pi_2}}) |0 \rangle \ldots \langle 0 | {\cal O}(\vect{x_{\pi_{2n-1}}}) {\cal O}(\vect{x_{\pi_{2n}}})|0 \rangle + \ldots,
\end{equation}
where $\pi$ runs over all permutations and the dots denote terms subleading in $1/N$.
Even though the operators ${\cal O}$ look deceptively simple, they are, in fact, rather complicated as Heisenberg operators. For example, as we will discuss below, the same Heisenberg operators have rather different properties about a thermal state: for example, their
2-point function will look very different from the 2-point function about the vacuum. This is because there are ${1 \over N}$ effects that we have neglected in \eqref{factorization} that become important in a thermal state where the energy density scales with some power of $N$.
However, for now, we turn to the properties of these operators in the vacuum of the CFT, since that is what is pertinent to empty AdS.
\subsection{Properties of generalized free field modes about the vacuum}
\label{subsec:freevac}
It will be convenient for us to work in Fourier space, as opposed to the position space prescriptions of \cite{Banks:1998dd, Bena:1999jv, Hamilton:2005ju, Hamilton:2006fh,Hamilton:2006az,Hamilton:2007wj,Heemskerk:2012mn,Bousso:2012mh}.
It will also be convenient to work with correlators where we choose a prior ordering, as opposed to the more commonly considered time-ordered correlators for reasons that will become apparent shortly.
We define the Fourier modes of ${\cal O}(x)$ as usual
\begin{equation}
\label{okdef}
{\cal O}_{\omega,\vect{k}} = \int dt d^{d-1}
\vect{x} \,\,{\cal O}(t,\vect{x}) \,\,e^{i\omega t - i\vect{k} \cdot \vect{x}} .
\end{equation}
Here by the boldface $\vect{k},\vect{x}$ we denote the $(d-1)$-dimensional spatial components of the corresponding $d$-vectors. We are working in signature mostly plus. Let us determine a few properties of these modes. Since the only correlators at leading order in ${1 \over N}$ are 2-point functions, we can analyze the properties of these Fourier modes by studying the Wightman functions.
Let us first point out the advantage of Wightman functions --- we use
this term synonymously with correlators where we pick an ordering ahead of time --- over time-ordered correlators. The time-ordered correlator is defined by
\[
\langle 0 | T\{{\cal O}(t, \vect{x}) {\cal O}(t', \vect{x'}\}) |0 \rangle = \theta(t - t') \langle 0 | {\cal O}(t, \vect{x}) {\cal O}(t', \vect{x'}) |0 \rangle + \theta(t' - t) \langle 0 | {\cal O}(t', \vect{x'}) {\cal O}(t, \vect{x}) |0 \rangle.
\]
When we Fourier transform this expression, apart from the ${\cal O}_{\omega,\vect{k}}$ defined above, we also get a contribution from the Fourier transform of the $\theta$ function. So, to study the properties of ${\cal O}_{\omega,\vect{k}}$, it is simpler to consider the Wightman function.
\paragraph{An Analysis of the Wightman 2-Point Function}
We now proceed to analyze the 2-point Wightman function to study the properties of ${\cal O}_{\omega,\vect{k}}$. To compute this object, we start in {\em Euclidean space}, where the 2-point correlator is given by
\[
\langle {\cal O}(\tau, \vect{x}) {\cal O}(0,\vect{0}) \rangle = \left({1 \over \tau^2 + \vect{x}^2}\right)^{\Delta}.
\]
Here $\Delta$ is the conformal dimension of ${\cal O}$. This form is fixed by scale invariance.
Now, let us continue to Lorentzian space. When we continue to Lorentzian space, to get the {\em time-ordered correlator} we should take $\tau = i (1 - i \epsilon) t$. The logic of this prescription is that we want to rotate the time contour but not rotate it all the way. In particular, as Euclidean time runs from $-i \infty$ to $i \infty$, we want
Lorentzian time to run from $-\infty (1 + i \epsilon)$ to $\infty(1 + i \epsilon)$. With this prescription, we find that the time-ordered correlator is given by
\[
\langle 0 | T\left\{{\cal O}(t,\vect{x}), {\cal O}(0,\vect{0}) \right\} |0 \rangle = \left({-1 \over (1 - i \epsilon)^2 t^2-\vect{x}^2} \right)^{\Delta} = \left({-1 \over t^2 - \vect{x}^2 - i \epsilon}\right)^{\Delta},
\]
The time-ordered function coincides with the Wightman function for $t > 0$. If we assume that ${\cal O}$ is a Hermitian operator then the Wightman function for $t < 0$ is simply the complex conjugate of the time-ordered correlator.
\[
\langle 0 | {\cal O}(t,\vect{x}) {\cal O}(0,\vect{0}) |0 \rangle^* = \langle 0 | {\cal O}(0,\vect{0}) {\cal O}(t,\vect{x}) |0 \rangle = \langle 0 | T\{{\cal O}(0,\vect{0}) {\cal O}(t,\vect{x}) \} |0 \rangle, \quad \text{for} \quad t < 0,
\]
This leads to
\begin{equation}
\label{wightmanfuncvacuum}
\langle 0 | {\cal O}(t,\vect{x}), {\cal O}(0,\vect{0}) |0 \rangle = \left({-1 \over t^2 - \vect{x}^2 - i \epsilon t}\right)^{\Delta} = \left({-1 \over (t - i \epsilon)^2 - \vect{x}^2} \right)^{\Delta}.
\end{equation}
Let us understand this expression. We choose the branch cut of the function $f(x) = x^{\Delta}$ to lie along the negative $x$ axis and $f$ to be real for positive real $x$. Then, the $i \epsilon$ prescription tells us
how to pick up the phase of the answer. For any point on the $(t,\vect{x})$ plane let us define the following real number
\[
\xi \equiv \left|t^2 - \vect{x}^2\right|^{\Delta} .
\]
Notice that this is an unambiguously defined positive real number (or zero, on the lightcone). The Wightman function is
\begin{equation}
\label{wightman}
\langle 0 | {\cal O}(t,\vect{x}) {\cal O}(0,\vect{0})|0 \rangle = {e^{i {\cal Q}}\over \xi}
\end{equation}
where the phase--${\cal Q}$ is as follows
\[
{\cal Q} = \left\{\begin{array}{ll}
0&\text{for}~\vect{x}^2 - t^2 > 0,\\
-\pi \Delta&\text{for}~t^2 - \vect{x}^2 > 0,~{\rm and}~t>0\\
\pi \Delta&\text{for}~t^2 - \vect{x}^2 > 0,~{\rm and}~t<0\\
\end{array}\right.
\]
We should emphasize that this is a perfectly Lorentz invariant prescription. For $\vect{x}^2 > t^2$, there is no ambiguity in the choice of phase. For $t^2 > \vect{x}^2$, a Lorentz transformation cannot change the sign of $t$.
\paragraph{Fourier Transform}
Now, we will Fourier transform the expression above to return to the properties of ${\cal O}_{\omega,\vect{k}}$. The expression \eqref{wightmanfuncvacuum} is
valid in any number of dimensions. We define the quantity $G(\omega,\vect{k})$ by
\[
\langle 0 | {\cal O}_{\omega,\vect{k}} {\cal O}_{\omega',\vect{k}'}|0 \rangle = G(\omega,\vect{k}) \delta(\omega+\omega')\delta^{d-1}(\vect{k}+\vect{k}')
\]
where we used the time and space translational invariance of the 2-point function \eqref{wightman}.
We have
\begin{equation}
\label{wvacuumdef}
G(\omega, \vect{k})= \int dt d^{d-1}\vect{x}\,\, \left({-1 \over (t - i \epsilon)^2 - \vect{x}^2} \right)^{\Delta} e^{i \omega t - i \vect{k} \cdot \vect{x}}
\end{equation}
\begin{figure}
\label{wightmanfuncfig}
\begin{center}
\begin{subfigure}[b]{6cm}
\includegraphics[width=6cm]{wightmananalyticfig.eps}
\caption{Analytic Structure}
\label{analyticwightman}
\end{subfigure}
\qquad
\qquad
\begin{subfigure}[b]{6cm}
\includegraphics[width=6cm]{wightmananalyticcontourfig.eps}
\caption{Contour of Integration}
\label{wvacontour}
\end{subfigure}
\caption{Wightman Function}
\end{center}
\end{figure}
The analytic structure of the integrand in the $t$-plane is shown in Figure \ref{analyticwightman}. Note that there are two branch cuts, both of which lie in the upper half plane.
Let us do the $t$ integral first. Note that this integral can be taken to have a branch cut running from $(i \epsilon + \norm{x}, i \epsilon + \infty)$ and another one from $(i \epsilon - \infty, i \epsilon - \norm{x})$. In the case where $\omega < 0$, we can close the $t$-contour in the {\em lower half plane}, and since there are no singularities in that region, we just get $0$. Now, by the remark above, \eqref{wvacuumdef} is invariant under Lorentz transformations that are continuously connected to the identity. If $\vect{k}^2 - \omega^2 > 0$, then by using such a transformation, we can make the time component of the momentum $d$-vector negative. This immediately tells us that we need to consider \eqref{wvacuumdef} only for vectors that have
\[
\omega > 0, \quad \text{and} \quad \omega^2 - \vect{k}^2 > 0
\]
We can now transform the $t$-integral in \eqref{wvacuumdef} by deforming the original contour, which runs along the real axis to the contour shown in \ref{wvacontour}. Considering the various phases along the legs of this contour carefully, we see that this is just
\begin{equation}
\label{wightmanfuncvacuumb}
G(\omega, \vect{k} ) = 2 \left( e^{i \pi \Delta} - e^{-i \pi \Delta} \right) \int d^{d-1} \vect{x} \int_{t=|\vect{x}|}^{\infty} {e^{i \omega t - i \vect{k} \cdot \vect{x}} \over |t^2 - \vect{x}^2|^{\Delta}} dt,
\end{equation}
We now change coordinates to: $\rho^2 = t^2 - \vect{x}^2$, and write $t = \rho \cosh \zeta, |\vect{x}| = \rho \sinh \zeta$. We also choose a frame where
the vector $(\omega, \vect{k}) \rightarrow (\sqrt{\omega^2 - \vect{k}^2}, 0, 0, \ldots)$. We can then rewrite \eqref{wightmanfuncvacuumb} as
\[
\begin{split}
G(\omega, \vect{k}) &= \theta(\omega) \theta(\omega^2 - \vect{k}^2) 2 \left( e^{i \pi \Delta} - e^{-i \pi \Delta} \right) {V_{d-2}} \int {e^{i \sqrt{\omega^2 - \vect{k}^2} \rho \cosh{\zeta}} \over \rho^{2 \Delta}} \, \rho^{d-1} d\rho \, \sinh(\zeta)^{d-1} d \zeta \\
&\equiv N_{\Delta,d} \,\theta(\omega) \theta(\omega^2 - \vect{k}^2) (\omega^2 - \vect{k}^2) ^{\Delta - d/2}
\end{split}
\]
where $V_{d-2}$ is the volume of the $d-2$-sphere and $N_{\Delta,d}$
is an irrelevant non-zero numerical constant that comes from the
integral over $\zeta$ and $\rho$.
Notice that this implies that the modes ${\cal O}_{\omega,\vect{k}}$ with positive $\omega$ annihilate the vacuum, while the modes ${\cal O}_{-\omega,-\vect{k}}$ and $\omega>0$, when acting on the vacuum, create excitations of energy $\omega$ and momentum $\vect{k}$. Moreover, {\em in the vacuum}, we have for the commutator
\[
\langle 0 |[{\cal O}_{\omega, \vect{k}}, {\cal O}_{\omega', \vect{k}'}]
|0 \rangle =
G(|\omega|, \vect{k})\, \text{sgn}(\omega) \,\delta(\omega+\omega') \,\delta^{d-1}(\vect{k} + \vect{k}'),
\]
In fact, large $N$ factorization \eqref{factorization} implies that the right hand side of the equation above is unchanged if replace the vacuum by any normalized state $|{\cal S} \rangle$ that is created by the action of only a finite number of insertions of ${\cal O}$.
At subleading orders in $N$, the spacelike modes of ${\cal O}$ are relevant, and they do not satisfy the algebra above. This implies that if we consider the Wightman two-point function in a state with an energy that scales with $N$ (like a big black hole), then these relations stop holding as we will find below.
However, the calculation above tells us that
{\it
\begin{quote}
At leading order in ${1 \over N}$, while computing finite-point correlators of ${\cal O}_{\omega,\vect{k}}$ about the vacuum, we can neglect the spacelike modes of ${\cal O}_{\omega,\vect{k}}.$
\end{quote}}
\noindent Moreover, if we define the operators (for $\omega > 0$ and
$\norm{k}^2 > 0$)
\[
\begin{split}
\hat{\cal O}_{\omega, \vect{k}} &= {{\cal O}_{\omega, \vect{k}} \over
\sqrt{G(\omega,\vect{k})}},
\\
\hat{\cal O}^{\dagger}_{\omega, \vect{k}} &= {{\cal O}_{-\omega, -\vect{k}} \over
\sqrt{G(\omega,\vect{k})}},
\end{split}
\]
then these operators (inserted between states $|{\cal S}\rangle$ made out of finite number of insertions of ${\cal O}$) just satisfy the algebra of free oscillators
\[
[\hat{O}_{\omega, \vect{k}}, \hat{O}_{\omega', \vect{k}'}^{\dagger}] = \delta(\omega-\omega') \delta^{d-1}(\vect{k} - \vect{k}').
\]
Physically this means that the excitations created by the action of the generalized free field ${\cal O}$ have the structure of a freely generated Fock space. However, these excitations are qualitatively different from those created by an ordinary free field. In the case of an ordinary free field $\phi$ on the boundary, the excitations are simply labeled by the $d-1$ components of their spatial momentum $\vect{k}$, while their energy is determined by $\omega = \sqrt{\vect{k}^2+m^2}$. This dispersion relation follows from the {\it equation of motion} that the ordinary free field satisfies (for example $\Box\phi = m^2\phi$). In contrast, the excitations created by a generalized free field ${\cal O}$ are labeled by $d$ independent numbers, namely $\vect{k}$ {\it and} $\omega$. Except for the condition that $\omega^2 > \vect{k}^2$, there is no constraint among them, i.e. no dispersion relation, because the generalized free field ${\cal O}$ {\it does not satisfy any wave equation on the boundary}.
In a sense that we will make precise in the next subsection, the excitations created by a generalized free field can be reorganized as the excitations of an ordinary free field living in a higher dimensional (AdS) spacetime.
From now on, we will take $\omega$ to be positive and will always work with positive frequency modes, that is, instead of writing ${\cal O}_{-\omega,-\vect{k}}$ we will write ${\cal O}_{\omega,\vect{k}}^\dagger$.
\subsection{Local operators on the Poincare patch}
Now, consider AdS$_{d+1}$ in the Poincare patch with geometry
\begin{equation}
\label{poincarepatch}
ds^2 = {-dt^2 + d\vect{x}^2 + dz^2 \over z^2}.
\end{equation}
Notice that we are working in units where the AdS radius is set to
one. We consider a free massive scalar field propagating on this background.
The equation of motion of this field is
\begin{equation}
\label{poincareeom}
(\Box -m^2)\phi=0
\end{equation}
It is quite easy to solve \eqref{poincareeom}. The standard solution
is in terms of Bessel functions and the {\em normalizable mode} is
\begin{equation}
\label{normalizablemomentum}
\xi_{\omega, \vect{k}}(t,\vect{x},z) = e^{-i\omega t + i \vect{k} \cdot
\vect{x}} {\Gamma(1+\nu)\over \Upsilon}\left({2\over \sqrt{\omega^2 - \vect{k}^2} }\right)^\nu z^{d/2}J_{\nu}(\sqrt{\omega^2 - \vect{k}^2} z) ,
\end{equation}
where $\nu = \sqrt{m^2 + d^2/4}$.
We have chosen the overall normalization of the mode such that it behaves
like $\Upsilon^{-1}\times z^\Delta \times e^{-i\omega t + i \vect{k} \cdot \vect{x}}$ near the boundary $z=0$, and we have $\Delta= \nu + d/2$. We will also take
\begin{equation}
\label{normfactdef}
\Upsilon = (2 \pi)^{d \over 2} \sqrt{2 {\Gamma(\Delta - {d \over 2} + 1) \pi^{d \over 2} \over \Gamma(\Delta)}},
\end{equation}
for later convenience.
Notice that it is possible to find normalizable modes which do not blow up at the Poincare horizon only if $\omega^2\geq \vect{k}^2$ i.e. there are no normalizable modes with spacelike momentum along the boundary directions. This is consistent with the fact that at large $N$ the spacelike Fourier modes ${\cal O}_{\omega,\vect{k}}$ can not create any excitations when acting on the vacuum, as we found above.
We can now easily write down a (nonlocal) CFT operator, which behaves like a {\em local field} in the Poincare patch. This is simply given by
\begin{equation}
\label{finalpoincare}
\boxed{
\phi_{\text{CFT}}(t, \vect{x}, z) = \int_{\omega>0} {d\omega d^{d-1} \vect{k} \over (2 \pi)^d}\left[{\cal O}_{\omega,\vect{k}}\,\, \xi_{\omega,\vect{k}}(t,\vect{x},z) + {\cal O}^{\dagger}_{\omega,\vect{k}}\,\, \xi^*_{\omega,\vect{k}}(t,\vect{x},z) \right]}
\end{equation}
We use the subscript CFT to emphasize that while $\phi_{\text{CFT}}$ seems to depend on all AdS coordinates $(t,\vect{x},z)$ it is still an operator in the conformal field theory, though clearly non-local. From the point of view of the CFT, the coordinate $z$ is in a sense ``auxiliary''. It simply parameterizes how exactly we have smeared the boundary operator ${\cal O}(t,\vect{x})$ to reproduce the nonlocal operator $\phi_{\text{CFT}}(t,\vect{x},z)$.
The main point here is that since $\phi_{\text{CFT}}$ has exactly the same expansion as that of a free massive field in AdS, it necessarily behaves --- at large $N$ --- like a local field in the ``emergent'' AdS space, which is constructed by the boundary coordinates $t,\vect{x}$ together with the parameter $z$ and equipped with the metric \eqref{poincarepatch}. For example, with the choice in \eqref{normfactdef}, we have\footnote{Our unusual choice for the normalization of this field has to do with the fact that we wanted to remove factors of $2 \pi$ from our momentum mode commutators.}
\begin{equation}
[\phi_{\text{CFT}}(t, \vect{x}, z), \dot{\phi}_{\text{CFT}}(t, \vect{x'}, z')] = {i \over (2 \pi)^d} \delta^{d-1}(\vect{x} - \vect{x'}) \delta(z - z') z^{d-1}.
\end{equation}
Moreover, the action of the conformal transformations on ${\cal O}(t,\vect{x})$ generates just an action of the isometries of AdS on $\phi_{\text{CFT}}(t,\vect{x},z)$.
\subsection{Local operators behind the Poincare horizon}
So far what we have done is not surprising, and was already done in the works cited at the beginning of the subsection \ref{subsec:freevac}. We now show how it is possible to construct local fields on {\em global AdS} using these modes.
Figure \ref{poincaretoglobal} shows two ways in which this may be done.
\begin{figure}[!h]
\begin{center}
\begin{subfigure}[b]{6cm}
\includegraphics[width=6cm, height=5cm]{pointoglobal1}
\caption{Global time slices \label{globaltimecoord}}
\end{subfigure}
\qquad
\qquad
\begin{subfigure}[b]{6cm}
\includegraphics[width=6cm, height=5cm]{pointoglobal2}
\caption{Poincare time slices \label{pointimecoord}}
\end{subfigure}
\caption{Local operators in global AdS using the boundary of the Poincare patch \label{poincaretoglobal}}
\end{center}
\end{figure}
In the method indicated in \ref{globaltimecoord}, we first construct local operators on a Cauchy slice that lies entirely within the Poincare patch. We then use the equations of motion to evolve these operators forward in global AdS time beyond the Poincare patch. In practice, to write down an explicit local field like \eqref{finalpoincare}, we would need to do a Bogoliubov transform from the solutions to the wave equation given in \eqref{normalizablemomentum} that have a well defined energy with respect to Poincare time to solutions that have a well defined energy with respect to global AdS time.
It is simpler to write down explicit operators using the method indicated
in figure \ref{pointimecoord}. We can quantize a field in global AdS
using the sequence of spacelike slices shown there. There is a natural
continuation of these slices beyond the future horizon. We now demonstrate this construction more precisely.
Consider global AdS written in coordinates
\[
ds^2 = -\cosh^2 \rho \,d t^2 + d \rho^2 + \sinh^2 \rho \,d \Omega_{d-1}^2
\]
The $d-1$ sphere that enters here can be parameterized in terms of $d-1$ angles and embedded in $R^d$ through $(\Omega_1, \Omega_2, \ldots \Omega_d) = (\cos \theta_1, \sin \theta_1 \cos \theta_2, \ldots \sin \theta_1 \sin \theta_2 ... \sin \theta_{d-1})$. (If $d$ is even, then the last entry has a $\cos \theta_d$ instead but this detail is not relevant below.)
To {\em motivate our construction} of operators in this space, we write
\begin{equation}
\label{globaltopoincare}
\begin{split}
t &= \frac{\cosh \rho \sin \tau }{\cos \tau \cosh \rho + \cos \theta_1 \sinh \rho} \\
x_i &= \frac{\Omega_{i+1} \sinh\rho }{\cos \tau \cosh \rho + \cos \theta_1 \sinh\rho}, \quad i = 1 \ldots d-1 \\
z &= {1 \over \cos \tau\cosh \rho + \cos \theta_1 \sinh \rho}
\end{split}
\end{equation}
The boundary, which is at $z = 0$ is clearly at $\rho = \infty$. The horizon, which is $z = \infty$ is defined by the equation
\[
\cos \tau\cosh \rho + \sin \theta_1 \sinh \rho=0.
\]
At the boundary, the coordinate change between $t,\vect{x}$ and $\tau, \Omega_i$ is
\[
t = {\sin \tau \over \cos \tau + \sin \theta_1}, \quad x_i = {\Omega_{i+1} \over \cos \tau + \sin \theta_1 }.
\]
Now we analytically continue our solutions from the Poincare patch to all of global AdS. We take the solutions given in \eqref{normalizablemomentum} and substitute the coordinate transformation \eqref{globaltopoincare}. So, in global coordinates we write the solution
\[
\xi_{\omega,\vect{k}}(\tau, \rho, \Omega_i) = e^{-i \omega t(\tau, \rho, \Omega_i)} e^{i \vect{k} \cdot \vect{x}(\tau, \rho, \Omega_i)}{\Gamma(1+\nu) \over \Upsilon}
\left({2\over \sqrt{\omega^2 - \vect{k}^2}}\right)^\nu z(\tau, \rho, \Omega_i))^{d \over 2} J_{\nu}(\sqrt{\omega^2 - \vect{k}^2} z(\tau, \rho, \Omega_i)),
\]
where the functions $t(\rho, \tau, \Omega_i), \vect{x}(\rho, \tau, \Omega_i), z(\rho, \tau, \Omega_i)$ are given by \eqref{globaltopoincare}. There is a small subtlety regarding a phase that we need to be careful about.
When we cross $z = 0$, and go to negative $z$, we pick up a phase. This is because
\[
J_{\nu}(-x) = e^{i \pi \nu} J_{\nu}(x).
\]
As we cross the next horizon, we should then pick up a net phase of $e^{2 \pi i \nu}$.
With this convention, we can write down local operators in global AdS
simply as
\begin{equation}
\label{finalglobal}
\phi_{\text{CFT}}(\tau, \rho, \Omega_i) = \int_{\omega>0} {d\omega d^{d-1} \vect{k} \over (2 \pi)^d} \left[{\cal O}_{\omega,\vect{k}}\,\, \xi_{\omega,\vect{k}}(\tau, \rho, \Omega_i) + {\cal O}^{\dagger}_{\omega,\vect{k}}\,\, \xi^*_{\omega,\vect{k}}(\tau, \rho, \Omega_i) \right]
\end{equation}
The fields that we get by this construction are clearly periodic in the Poincare patch up to a phase. However, this is consistent with the fact that the equations of motion require $\phi$ to be periodic in global time, up to a phase.
It is also easy to see that if we take the limit of this operator
\[
\lim_{\rho \rightarrow \infty} \left[\Upsilon\, e^{\rho \Delta}\,\phi_{\text{CFT}}(\tau, \rho, \Omega_i)\right] = {\cal O}^g(\tau, \Omega_i)
\]
where ${\cal O}^g$ is nothing but the continuation of the operator ${\cal O}$ from Minkowski space to its infinite sheeted covering that is conformal to ${\mathbb S}^{d-1} \times {\mathbb R}$ and we remind the reader that
the normalization factor $\Upsilon$ is given in \eqref{normfactdef}. In fact, many years ago Luscher and Mack \cite{Luscher:1974ez} showed that the correlation functions of the CFT on ${\mathbb R}^{d-1,1}$ could be continued to ${\mathbb S}^{d-1} \times {\mathbb R}$. This continuation can also be done, by first continuing to Euclidean space, and then continuing back to get the space ${\mathbb S}^{d-1} \times {\mathbb R}$.
\paragraph{Continuing the CFT from ${\mathbb R}^{d-1,1}$ to ${\mathbb S}^{d-1} \times {\mathbb R}$}
For the benefit of the reader, we briefly review how CFTs can be
continued from Minkowski space to ${\mathbb S}^{d-1} \times {\mathbb R}$. The reader who is
already familiar with this topic can skip to the next subsection,
which has a ``discussion'' of the implications of our construction.
This extension is described very clearly in \cite{Aharony:1999ti}. (see pp. 37--39.). We focus on the time coordinate $t$ and the spatial radial coordinate $r$ of Minkowski space. Let us compactify it down to the triangle by writing $r\pm t = \tan(\theta \pm \tau)$. This maps a triangle in the $\tau,\theta$ patch to the full $t,r$ plane. Writing $u_{\pm} = r \pm t$, we find that
\[
{\partial \over \partial \tau} = {1 \over 2} \left[(1 + u_{+}^2) {\partial \over \partial u_{+}} + (1 + u_{-}^2) {\partial \over \partial u_{-}} \right]]= {1 \over 2} \left(P_0 + K_0\right),
\]
If we take our Hamiltonian to be ${1 \over 2}\left(P_0 + K_0 \right)$, then this Hamiltonian generates translations in $\tau$. In the $\tau, \theta$ plane we can continue correlators past the edges of the triangle
onto the whole cylinder.
There is another way to understand why it is this quantity that should
be used as the Hamiltonian when we go from ${\mathbb R}^{d-1,1}$ to ${\mathbb S}^{d-1} \times {\mathbb R}$. The basic point is that we can understand the action of the conformal algebra on the Hilbert space of the {\em Lorentzian theory}, by cutting up the Euclidean path integral that defines the theory in different ways. The boundary Euclidean path integral can be performed on ${\mathbb R}^d$ (or after adding a point at infinity on ${\mathbb S}^d$). We can cut this space either as ${\mathbb R}^{d-1} \times {\mathbb R}$ or as ${\mathbb S}^{d-1} \times {\mathbb R}$.\footnote{Note that both Minkowski space and the space ${\mathbb S}^{d-1} \times {\mathbb R}$ give the same space upon Euclidean continuation.}
The Hilbert space that we get in these two ways is different, although
the theories are isomorphic. In the ${\mathbb R}^{d-1} \times {\mathbb
R}$ slicing, states are defined by inserting an operator at
Euclidean time $-\infty$ and doing the path integral up to $0$. In the
${\mathbb S}^{d-1} \times {\mathbb R}$ slicing, states are defined by
inserting an operator at the origin and doing the path integral out to
unit radius.
The dual of a state is defined, in the flat-space slicing, by inserting an operator at Euclidean time $+\infty$ and doing the path integral back to $0$; in the ${\mathbb S}^{d-1} \times {\mathbb R}$ slicing by inserting an operator at $\infty$ and do the path integral back to the unit sphere.
Consequently, the definition of the adjoint is different as well. In the flat-space slicing, the adjoint operation clearly leads to the hermiticity conditions
\[
P_i^\dagger = P_i; \quad P_0^{\dagger} = -P_0; \quad K_i^\dagger =
K_i; \quad K_0^{\dagger} = -K_0.
\]
The minus sign for the time component comes, in this language, because the mapping from bras to kets involves taking $t_{\text{euclidean}} \rightarrow -t_{\text{euclidean}}$. Here, the $i$ component runs only over spatial indices. The algebra that we get in this manner from the Euclidean path integral is isomorphic to the group $SO(d,2)$, with the usual hermiticity conditions.
On the other hand, in the ${\mathbb S}^{d-1} \times {\mathbb R}$ slicing, the mapping from bras to kets involves taking $|x| \rightarrow {1 \over |x|}$. So, the adjoint of a translation involves an inversion followed by a translation and another inversion. This leads to the Hermiticity conditions
\[
P_{\mu}^\dagger = K_{\mu}.
\]
Hence, we get two different relations for the adjoint by changing the inner-product.
Now, instead of changing the inner product we can instead do a
similarity transform on the operators, and keep the inner product fixed. This similarity transform is given in detail in \cite{Minwalla:1997ka}. (See section 2.1.) This gives rise to the relation that the dilatation operator in the ${\mathbb S}^{d-1} \times {\mathbb R}$ slicing is related to the operators in the ${\mathbb R}^{d-1} \times {\mathbb R}$ slicing through
\[
D^{{\mathbb S}^{d-1}} = {i \over 2} (P_0^{{\mathbb R}^{d-1}} + K_0^{{\mathbb R}^{d-1}}),
\]
This operator is anti-hermitian, just like $P_0$ above but when we Wick rotate, we get the hermitian Hamiltonian above.
\paragraph{Discussion}
Note that, from the point of a view of a CFT that lives on the boundary of Poincare AdS, this construction seems a little surprising. A common, but incorrect, belief
is that this CFT has only partial information about global AdS. To the contrary, as we have shown above, the CFT gives us {\em all bulk correlators} in global AdS.
We have also encountered the objection that ``if someone were to turn on a source of change the Hamiltonian somewhere in global AdS beyond the Poincare patch, how would
the flat-space CFT know about this''? Indeed, the CFT would not be
able to account for such a ``divine intervention'' but we emphasize
that this objection could also have been raised in the original
Poincare patch. If we were to change the Hamiltonian somewhere in the bulk, this would simply change the theory away from the original CFT and without further information, we would not be able to compute these new correlators.
So, our flat-space CFT describes a {\em specific} theory on the Poincare patch, and a {\em specific} theory on global AdS. It can also describe small deformations away from this theory, such as those obtained by turning on sources in the bulk since the effect of these sources can be captured by a power series expansion in the original correlators. It cannot account for arbitrary changes to the Hamiltonian but this is not unusual and holds for all examples of the AdS/CFT correspondence.
\section{Reconstructing the black hole from the boundary, outside the horizon \label{sec:outside}}
We will now try to reconstruct the black hole geometry from the boundary.
In AdS/CFT we have the following identifications between the boundary CFT and the bulk
\[
\begin{split}
{\rm Thermalization\,\,of\,\,pure\,\,state} &\qquad\Leftrightarrow\qquad
{\rm Black\,\, hole\,\, formation\,\, by\,\, gravitational\,\, collapse}\cr
{\rm Thermal\,\, density\,\, matrix} & \qquad\Leftrightarrow\qquad {\rm Eternal\,\, black\,\, hole}
\end{split}
\]
In the same way that the collapsing star can be approximated {\it for certain questions} by the eternal black hole, we expect that late time\footnote{As in the bulk, by late we mean after the thermalization has occurred, but not too late so that Poincare recurrences and other finite $N$ effects become important.} CFT correlation functions on a heavy pure state can be approximated by correlation functions on a thermal density matrix.
Hence, we will first focus on thermal correlators and explain how to represent local bulk operators in the case where the boundary theory is in a thermal density matrix. We will then carry over this definition of the operator (at large $N$) to the case where we have a typical pure state. We will discuss the validity of this approach and possible non-trivial sensitivity on the specific pure state in section \ref{sec:subtleties}.
\subsection{Boundary thermal correlators}
\subsubsection{Large $N$ factorization at finite $T$}
\label{factorcav}
Let us consider a scalar conformal primary operator ${\cal O}$. For simplicity we assume that its thermal 1-point function ${\rm Tr}(\rho \,{\cal O})$ vanishes,
where $\rho = e^{-\beta H}$. This can be ensured by considering a theory with a ${\mathbb Z}_2$ symmetry, under which ${\cal O}$ is odd.
A central assumption in what follows is that large $N$ factorization holds for thermal correlation functions i.e. that we have the analogue of \eqref{factorization}
\[
{\rm Tr}(\rho \,{\cal O}(x_1)...{\cal O}(x_{2n})) = {1\over 2^n} \sum_\pi
{\rm Tr}(\rho \,{\cal O}(x_{\pi_1}) {\cal O}(x_{\pi_2}))...{\rm Tr}(\rho \,{\cal O}(x_{\pi_{2n-1}}) {\cal O}(x_{\pi_{2n}})) + \ldots,
\]
where the dots are terms that are subleading in ${1 \over N}$.
Of course the 2-point function ${\rm Tr}(\rho\, {\cal O}(x_1)\, {\cal O}(x_2))$ in which the thermal correlators factorize into is completely different
from the zero temperature 2-point function $\langle 0 | {\cal O}(x_1) {\cal O}(x_2) |0 \rangle$. Also, we need to stress that ---as in the zero temperature case---
there are several caveats about the validity of factorization. We should be careful to {\it first} fix all other parameters/numbers that enter the correlator and {\it then} take $N$ to infinity. For example, factorization can fail if
\begin{itemize}
\item We scale the number of operators (i.e. $n$ in the formula above) with some power of $N$. Scaling the number of external legs with powers of $N$ invalidates the naive 't Hooft counting and a priori we have no reason to expect the correlator to still factorize.
\item We scale the conformal dimension of ${\cal O}$ in an $N$-dependent way --- the same comments as above apply.
\item We scale some of the distances $|x_i-x_k|$ to be too small, in an $N$-dependent way (this can be thought of as a high energy scattering, where higher and higher loops in $1/N$ become important).
\item We scale the temperature in a $N$-dependent way. (See the paragraph in section \ref{secadseternal} for more details.)
\item We scale (some of) the distances $|x_i-x_j|$ to {\it increase} in an $N$-dependent fashion. This will be particularly important when we consider the operators evaluated on
typical pure states vs the thermal ensemble.
\end{itemize}
Of course this list is not supposed to be exhaustive. In general we have to be careful about how various factors in the problem scale, when taking the large $N$ limit.
\subsubsection{Analytic structure of thermal 2-point functions}
\label{therman}
We now discuss general properties on finite temperature Wightman functions. Since correlators at large $N$ factorize to products of 2-point functions, we will focus on the latter.
Consider two local operators ${\cal O}_1,{\cal O}_2$. From time and space translational invariance of the thermal ensemble we only need
\[
F_{12}(t) \equiv Z_{\beta}^{-1}{\rm Tr}\left(e^{-\beta H} {\cal O}_1(t,\vect{x}) \, {\cal O}_2(0,\vect{0})\right),
\]
where we did not indicate the $\vect{x}$ dependence on the LHS. Here
\[
Z_{\beta} = {\rm Tr}\left[e^{-\beta H}\right],
\]
is the partition function which will appear frequently below. Since the operators do not generally commute, we also have a related function
\[
F_{21}(t) \equiv Z_{\beta}^{-1}{\rm Tr}\left(e^{-\beta H} {\cal O}_2(0,\vect{0}) \, {\cal O}_1(t,\vect{x})\right).
\]
Let us keep $\vect{x}$ fixed and analytically continue $t$ to complex values. By inserting a complete set of states and using the positivity of the energy spectrum we find that the analytically continued function $F_{12}(t)$ is meromorphic in the strip
\[
-\beta<{\rm Im}\,t <0.
\]
However, in general, it cannot be analytically continued to positive values of ${\rm Im}\,t$. The function $F_{21}(t)$ can be analytically continued to positive imaginary values of $t$ and is meromorphic in the strip
\[
0<{\rm Im} \,t<\beta.
\]
The value of the function $F_{12}(t)$ for real $t$ is equal to the limit of the meromorphic function in the strip $-\beta<{\rm Im}t <0$, as we take ${\rm Im} t \rightarrow 0^-$ from below. As we approach the other end of the strip from above (i.e. ${\rm Im}t \rightarrow -\beta^-$) and by inserting a complete set of states and using the cyclicity of the trace we derive the so called KMS condition
\begin{equation}
\label{kms}
F_{12}(t-i\beta^-) = F_{21}(t).
\end{equation}
So the correlator is periodic in imaginary time, up to an exchange in the order of the insertion of operators.
The discussion so far has been very general and would apply to any quantum system at finite temperature, even in non-relativistic quantum mechanics. In relativistic QFT we have an important additional property: local operators at spacelike separation commute. This means that the functions $F_{12}(t)$ and $F_{21}(t)$ must have the same value along the segment of the real axis $-|\vect{x}|<t<|\vect{x}|$ and hence one must be the analytic continuation of the other! For $t>|\vect{x}|$ we generally have $\lim_{{\rm Im}\, t\rightarrow 0^-} F_{12}(t) \neq \lim_{{\rm Im}\, t\rightarrow 0^+} F_{21}(t)$ and this discontinuity is proportional to the (thermal expectation value of the) commutator of the two operators. Because of the KMS condition this discontinuity appears periodically along all semi-infinite lines $t = \pm (|\vect{x}| + {\cal R}^+) + i m \beta$ where ${\cal R}^+$ denotes positive real numbers and $m \in \mathbb{Z}$.
Hence, by combining the KMS condition with the spacelike commutativity of local fields we have arrived at the following important conclusion: consider the domain ${\cal D}$ of the complexified $t$-plane defined as $\mathbb{C}$ minus ``cuts'' starting from $\pm|\vect{x}| + i m \beta$ and extending all the way to infinity parallel to the real axis, as depicted in figure \ref{analyticity}. This domain ${\cal D}$ is a simply-connected domain. Then, there is a {\it holomorphic} function ${\cal F}(z)$ defined in the domain ${\cal D}$ with the property that
\[
{\cal F}(z+i\beta) = {\cal F}(z) ,
\]
for all $z\in {\cal D}$ and also
\begin{align}
\nonumber
\lim_{\epsilon \rightarrow 0^+} {\cal F}(t-i \epsilon) = F_{12}(t), \\
\nonumber
\lim_{\epsilon \rightarrow 0^+} {\cal F}(t+i \epsilon) = F_{21}(t).
\end{align}
The holomorphic function ${\cal F}$ contains all the information about the 2-point function of operators ${\cal O}_1,{\cal O}_2$ for both possible orderings.
\begin{figure}
\begin{center}
\psfrag{beta}{$\beta$}
\psfrag{beta2}{$2\beta$}
\psfrag{beta3}{$-\beta$}
\psfrag{beta4}{$-2\beta$}
\psfrag{Ret}{${\rm Re}(t)$}
\psfrag{zero}{$0$}
\psfrag{xplus}{$|\vect{x}|$}
\psfrag{xminus}{$-|\vect{x}|$}
\psfrag{Imt}{${\rm Im}(t)$}
\includegraphics[width=9cm]{analyt.eps}
\caption{Domain of analyticity of a thermal 2-point function of the two local operators ${\cal O}_1(t,\vect{x})$ and ${\cal O}_2(0,\vect{0})$. The domain ${\cal D}$ is defined as the complex plane with the indicated branch cuts removed. On this domain we have a holomorphic function ${\cal F}(z)$ which satisfies ${\cal F}(z+i\beta)= {\cal F}(z)$. The function ${\cal F}$ contains all information about the thermal 2-point function. For example, we have $\lim_{\epsilon\rightarrow 0^+}{\cal F}(t-i\epsilon) = Z_{\beta}^{-1}{\rm Tr}\left(e^{-\beta H} {\cal O}_1(t,\vect{x}) {\cal O}_2(0,\vect{0})\right)$ and also
$\lim_{\epsilon\rightarrow 0^+}{\cal F}(t+i\epsilon) = Z_{\beta}^{-1}{\rm Tr}\left(e^{-\beta H} {\cal O}_2(0,\vect{0}){\cal O}_1(t,\vect{x}) \right)$
. The discontinuity along the branch cut is proportional to the thermal expectation value of the commutator of the two operators.}
\label{analyticity}
\end{center}
\end{figure}
We mentioned that generally $F_{12}(t)$ cannot be analytically continued to complex $t$ with positive imaginary part by going through the part of the positive real axis with $t>|\vect{x}|$ (i.e. the analytic continuation is possible only by going around, and through the ``spacelike segment'' $-|\vect{x}|<t<|\vect{x}|$). While this is indeed generic, it is not always true: there are quantum systems (for example a free QFT) whose thermal 2-point function {\it can} be analytically continued up through the part of the positive real axis with $t>|\vect{x}|$. However even in that case, the analytic continuation of the function through what used to be the ``cut'' of the domain ${\cal D}$ {\it does not lead to the same value} as the one that we would get by going through the ``spacelike segment'' which lies purely within the domain ${\cal D}$. In other words the correlation functions analytically continued to complex times are generically multivalued functions --- and we have to specify the ``sheet'' on
which we are evaluating them.
We will define the ``principal sheet'' of the analytically continued thermal correlators, as the one defined by doing the continuation purely within the domain ${\cal D}$ described above. This will be very important when we discuss the analytic continuation behind the horizon. Let us emphasize this once more: whenever we write down a thermal correlator where some of the time arguments have been given an imaginary value, the {\it definition} of this analytically continued correlator is that one has to start with the correlator defined in ``real time'' and then continue it to the complex values by working only in the domain ${\cal D}$ and going ``around'' the cuts, through the Euclidean (spacelike) region.
After these generalities, let us return to the case where we are looking at the thermal 2-point function of a conformal primary ${\cal O}$. From translational invariance in space and time we only need the function
\[
G_\beta(t,\vect{x}) = Z_{\beta}^{-1}{\rm Tr}\left(e^{-\beta H}{\cal O}(t,\vect{x})\,{\cal O}(0,\vect{0})\right),
\]
we will also consider its Fourier transform
\[
G_\beta(\omega,\vect{k}) = \int dt d^{d-1}\vect{x} \,\, e^{i \omega t - i \vect{k}
\vect{x}}\,G_{\beta}(t,\vect{x}).
\]
We have the following general properties. First, rotational invariance implies
\[
G_\beta(t,\vect{x}) = G_\beta(t,-\vect{x}).
\]
Second the KMS condition for the thermal trace implies
\[
G_{\beta}(t-i\beta,\vect{x}) = G_\beta(-t,-\vect{x}).
\]
In Fourier space the corresponding statements are
\[
G_\beta(\omega,\vect{k}) = G_\beta(\omega,-\vect{k}),
\label{rotfourier}
\]
\[
G_\beta(-\omega,\vect{k}) = e^{-\beta \omega} G_\beta(\omega,\vect{k}).
\label{kmsfourier}
\]
These are exact properties which hold in all dimensions and arbitrary coupling.\footnote{Clearly the discussion in this section was also independent of large $N$.} We can explicitly verify them in the case of 2d CFT thermal real-time correlation functions, which can be explicitly computed, as presented in appendix \ref{appendix2dthermal}.
\subsubsection{Mode expansion of thermal 2-point function}
Now we proceed by expanding the boundary field in its Fourier modes. For simplicity we take the CFT to live in ${\mathbb R}^{d-1,1}$ at finite temperature.\footnote{We could also do a similar analysis for the finite temperature CFT on ${\mathbb S}^{d-1}\times {\rm time}$, where we would have to replace the ${\bf k}$ momenta with discrete spherical harmonic modes.} We have
\begin{equation}
\label{modesa}
{\cal O}(t,\vect{x}) = \int_{\omega>0} {d\omega d^{d-1}\vect{k} \over (2 \pi)^d} \,\,\left( {\cal O}_{\omega,\vect{k}} e^{-i\omega t +i \vect{k} \vect{x}} + {\cal O}_{\omega,\vect{k}}^\dagger e^{i\omega t -i \vect{k} \vect{x}}\right)
\end{equation}
As before, this is the {\it definition} of the (non-local) operators ${\cal O}_{\omega,\vect{k}}$.
We can now consider the thermal 2-point function of ${\cal O}$ expanded in terms of its modes. From translational invariance we immediately conclude that
\[
Z_{\beta}^{-1}{\rm Tr}\left(e^{-\beta H} {\cal O}_{\omega,\vect{k}} {\cal O}_{\omega',\vect{k}'} \right) =
Z_{\beta}^{-1}{\rm Tr}\left(e^{-\beta H} {\cal O}_{\omega,\vect{k}}^\dagger {\cal O}_{\omega',\vect{k}'}^\dagger \right) =0,
\]
and for the other combinations we have
\begin{align}
\nonumber
&Z_{\beta}^{-1}{\rm Tr}\left(e^{-\beta H} {\cal O}_{\omega,\vect{k}} {\cal O}_{\omega',\vect{k}'}^\dagger \right)
= G_\beta(\omega,\vect{k}) \delta(\omega-\omega') \delta^{d-1}(\vect{k}-\vect{k}'), \\
\nonumber
&Z_{\beta}^{-1}{\rm Tr}\left(e^{-\beta H} {\cal O}_{\omega,\vect{k}}^\dagger {\cal O}_{\omega',\vect{k}'} \right)
= G_\beta(-\omega,-\vect{k}) \delta(\omega-\omega') \delta^{d-1}(\vect{k}-\vect{k}').
\end{align}
So for the thermal expectation value of the commutator we have
\[
Z_{\beta}^{-1}{\rm Tr}\left(e^{-\beta H} [{\cal O}_{\omega,\vect{k}}\,,\, {\cal O}_{\omega',\vect{k}'}^\dagger] \right)
= \left(G_\beta(\omega,\vect{k})- G_\beta(-\omega,-\vect{k}) \right)\delta(\omega-\omega') \,\delta^{d-1}(\vect{k}-\vect{k}').
\]
Actually, given large $N$ factorization of the thermal correlators we can derive a more general statement: the relations above hold even in the presence of other operators ${\cal T}_m$ even if they are ``composite'', as long as they are ``light'' and their scaling dimension does not scale with $N$:
\begin{align}
\nonumber
{\rm Tr}\left(e^{-\beta H} {\cal T}_1 [{\cal O}_{\omega,\vect{k}}\,,\, {\cal O}_{\omega',\vect{k}'}] {\cal T}_2\right) =
&{\rm Tr}\left(e^{-\beta H} {\cal T}_1[{\cal O}_{\omega,\vect{k}}^\dagger\,,\, {\cal O}_{\omega',\vect{k}'}^\dagger] {\cal T}_2\right)=0, \\
\label{thermalcoma}
{\rm Tr}\left(e^{-\beta H} {\cal T}_1[{\cal O}_{\omega,\vect{k}}\,,\, {\cal O}_{\omega',\vect{k}'}^\dagger] {\cal T}_2\right)
= &\left(G_\beta(\omega,\vect{k})- G_\beta(-\omega,-\vect{k}) \right)\delta(\omega-\omega') \,\delta^{d-1}(\vect{k}-\vect{k}') \\ \nonumber &\times {\rm Tr}\left(e^{-\beta H} {\cal T}_1 {\cal T}_2 \right).
\end{align}
The commutator \eqref{thermalcoma} implies that the operators ${\cal O}_{\omega,\vect{k}}$ behave like the creation and annihilation operators of harmonic oscillators, though in an unconventional normalization. We can rescale them and define operators
\begin{equation}
\label{rescaled}
\hatbb{\cal O}_{\omega,\vect{k}} = {{\cal O}_{\omega,\vect{k}} \over \left(
G_\beta(\omega,\vect{k})- G_\beta(-\omega,-\vect{k})\right)^{1\over 2}},
\end{equation}
which have canonical commutation relations. If we compute the thermal expectation value of the ``occupation level'' $n_{\omega,\vect{k}}=\hatbb{\cal O}_{\omega,\vect{k}}^\dagger
\hatbb{\cal O}_{\omega,\vect{k}}
$ for each of these oscillators we have
\[
Z_{\beta}^{-1}{\rm Tr}\left( e^{-\beta H} \hatbb{\cal O}_{\omega,\vect{k}}^\dagger
\hatbb{\cal O}_{\omega',\vect{k}'}\right) = { G_\beta(-\omega,
-\vect{k})\over G_\beta(\omega,\vect{k})- G_\beta(-\omega,-\vect{k})} \delta(\omega-\omega')\delta^{d-1}(\vect{k}-\vect{k}')
\]
and we find using \eqref{rotfourier} and \eqref{kmsfourier} that
\begin{equation}
\label{thermaloca}
Z_{\beta}^{-1}{\rm Tr}\left( e^{-\beta H} \hatbb{\cal O}_{\omega,\vect{k}}^\dagger
\hatbb{\cal O}_{\omega',\vect{k}'}\right)= {1\over e^{\beta \omega}-1} \delta(\omega-\omega')\delta^{d-1}(\vect{k}-\vect{k}'),
\end{equation}
and similarly
\begin{equation}
\label{thermalocab}
Z_{\beta}^{-1}{\rm Tr}\left( e^{-\beta H} \hatbb{\cal O}_{\omega,\vect{k}}
\hatbb{\cal O}_{\omega',\vect{k}'}^\dagger \right)= {e^{\beta \omega}\over e^{\beta \omega}-1} \delta(\omega-\omega')\delta^{d-1}(\vect{k}-\vect{k}'),
\end{equation}
which is the standard occupation level for a harmonic oscillator of frequency $\omega$ when placed at temperature $\beta$.
The physical interpretation of these results is that the modes \eqref{thermalcoma} are the CFT analogue of the AdS-Schwarzschild modes of a scalar field around a black hole in AdS that we called $a_{\omega,\vect{k}}$ in section \ref{modesbrane} hence we have the natural identification
\vskip5pt
\[
\hatbb{\cal O}_{\omega,\vect{k}}\qquad\Leftrightarrow \qquad a_{\omega,\vect{k}}
\]
\vskip5pt
The CFT modes $\hatbb{\cal O}_{\omega,\vect{k}}$ seem to be thermally populated at the Hawking temperature of the black hole $\beta$, as we see in \eqref{thermaloca},\eqref{thermalocab}. This is the CFT analogue of the ``thermal atmosphere'' of the black hole that we discussed in the previous sections, for example in equations \eqref{hhocup}, \eqref{hhocupb}. In a sense, they can be thought of as a thermally excited gas of glueballs, hovering around the quark-gluon plasma --- though the interpretation
may be not fully accurate. More technically it means that, as expected, we get the occupation levels determined by the AdS-Hartle-Hawking vacuum for the dual scalar field. It is interesting that this result follows simply from general properties of thermal field theories {\it together with large $N$ factorization} and does not need any other assumptions about the coupling. In particular the same conclusion would hold for the regime of small $\lambda$ where
the spacetime and the dual black hole would be highly stringy. On the other hand, at finite $N$ there is no sense in which the excitations of ${\cal O}_{\omega,\vect{k}}$ behave like a freely generated Fock space.
Also notice that, since these modes are ---in a sense--- fluctuations around a non Lorentz invariant background we have no reason to expect that they will exist only for timelike $(\omega,\vect{k})$. Indeed, as we see in appendix \ref{appendix2dthermal} for the special case of 2d CFTs where we can compute $G_\beta(\omega,\vect{k})$ analytically, we see that the modes exist for all values of $(\omega,\vect{k})$, even spacelike ones.
\subsection{Properties of thermal Fourier modes}
\label{subsec:thermalfourier}
As we mentioned above, in a thermal state, the spacelike Fourier modes of our generalized free fields do not vanish. The reader might be surprised by this, given that we argued in section \ref{sec:emptyads}, using nothing but large $N$ factorization and conformal invariance, that correlators involving the spacelike modes ${\cal O}_{\omega,\vect{k}}$ should vanish at leading order in ${1 \over N}$. The resolution is the energy densities in the thermal state are $\Or[N^2]$, and so the previous argument breaks down. Nevertheless, as we will show below, even in the thermal state, it is true that
{\em correlators involving operators with large spacelike momenta die off exponentially.}
To see this, we note that
\[
\begin{split}
&G_{\beta}(\omega, \vect{k}) = \int dt\,d^{d-1} \vect{x} \, e^{i \omega t - i \vect{k} \cdot x} G_{\beta}(t,\vect{x})\\
&= V_{d-3} \int dt d\theta d\norm{x} \, e^{i \omega t} G_{\beta}(t,\vect{x}) \norm{x}^{d-2} e^{i \norm{k} \norm{x} \cos \theta} (\sin(\theta))^{d - 3} ,
\end{split}
\]
where $V_{d-3}$ is the volume of the $(d-3)$-sphere. Doing the $\theta$ integral, and using the rotational invariance of $G_{\beta}(t,\vect{x})$, we find that
\begin{equation}
\label{thermalgfourier1}
G_{\beta}(\omega, \vect{k}) = \frac{\pi 2^{\frac{5}{2}-\frac{d}{2}} \Gamma (d-3) (\norm{k})^{\frac{3}{2}-\frac{d}{2}}
}{\Gamma \left(\frac{d-3}{2}\right)}V_{d-3}\int dt d\norm{x}\,e^{i \omega t} G_{\beta}(t,\norm{x}) \norm{x}^{d-1 \over 2} J_{d - 3 \over 2}(\norm{k} \norm{x}) .
\end{equation}
Using the identity
\[
\int_{0}^{\infty} J_{\nu} (p x) J_{\nu}(p y) p d p = {1 \over x} \delta(x - y),
\]
we can rewrite \eqref{thermalgfourier1} as
\begin{equation}
\label{thermalgfourier2}
\frac{\pi 2^{\frac{5}{2}-\frac{d}{2}} \Gamma (d-3)
}{\Gamma \left(\frac{d-3}{2}\right)}{V_{d-3}} \norm{x}^{d-3 \over 2} G_{\beta}(\omega, \norm{x}) = \int_0^{\infty} G_{\beta}(\omega, \norm{k}) J_{d - 3 \over 2}(\norm{k} \norm{x}) \norm{k}^{d - 1 \over 2} d \norm{k}.
\end{equation}
We will now show below that if we consider the function $G_{\beta}(\omega, \norm{x})$ then we can continue $\norm{x}$ into the upper half plane till $\text{Im}(\norm{x}) = {\beta \over 2}$. Since, for large $\norm{k}$ and $\text{Im}(\norm{x}) = {\beta \over 2}$ the Bessel function grows like $e^{\beta \norm{k} \over 2}$, we see that the integral in \eqref{thermalgfourier2} can only converge if $G_{\beta} (\omega, \norm{k})$ dies off as $e^{-{\beta \norm{k} \over 2}}$ for large $\norm{k}$ and fixed $\omega$.
To see this analytic property, we write the thermal Green function as a sum over a complete set of states (indexed both by $|m\rangle$ and $|n\rangle$ below)
\begin{equation}
\label{thermalgspectrumdecomp}
\begin{split}
G_{\beta}(t,\norm{x}) &= \sum_{m, n} \langle m| e^{-\beta H} {\cal O}(t,\vect{x}) |n\rangle \langle n| {\cal O} (0,\vect{0})|m\rangle \\
&= \sum_{m,n} e^{-\beta E_m} e^{-i (E_n-E_m)t + i (\vect{P}_n-\vect{P}_m) \cdot \vect{x} }|\langle m|{\cal O}|n\rangle|^2,
\end{split}
\end{equation}
where we know, ahead of time, that the left hand side depends only on $\norm{x}$ although individual terms on the right hand side depend on $\vect{x}$.
By Fourier transforming this expression in time, we find that
\[
G_\beta(\omega,\norm{x}) = \int dt e^{i\omega t} G_{\beta}(t,\vect{x}) = \sum_{m,n} e^{-\beta E_m} \delta(\omega-E_n+E_m)e^{i (\vect{P}_n-\vect{P}_m) \cdot \vect{x}}|\langle m|{\cal O}|n\rangle|^2.
\]
Using the delta function we can rewrite this as
\begin{equation}
\label{thermalgusedelta}
\begin{split}
&G_\beta(\omega,\norm{x}) =e^{\beta \omega /2} \sum_{m,n} \delta(\omega-E_n+E_m) e^{-\beta E_m/2} e^{-\beta E_n/2}
e^{i (\vect{P}_n-\vect{P}_m) \cdot \vect{x}}|\langle m|{\cal O}|n\rangle|^2 \\
&= e^{\beta \omega /2} \sum_{m,n} \delta(\omega-E_n+E_m) e^{-\beta E_m/2} e^{-\beta E_n/2}
e^{i |\vect{P}_n-\vect{P}_m| \norm{x} \cos \theta_{n m}} |\langle m|{\cal O}|n\rangle|^2,
\end{split}
\end{equation}
where $\theta_{n m}$ is the angle between $\vect{P}_n - \vect{P}_m$ and $\vect{x}$.
Now, in a relativistic QFT, we have the ``spectrum condition'': in addition to $E\geq 0$ we also have that $E\geq |\vect{P}|$ for all states in the theory. Since
by the triangle inequality, $ |\vect{P}_m + \vect{P}_n| \leq |\vect{P}_m| + |\vect{P}_n|$, we see from the expansion \eqref{thermalgusedelta} that, viewed as a function of $\norm{x}$, the function $G_{\beta}(\omega, \norm{x})$ can be continued in the positive imaginary direction till $\text{Im}(\norm{x}) = {\beta \over 2}$.
This concludes our proof that for large $\norm{k}$ and fixed $\omega$, we have
\[
G_{\beta}(\omega, \norm{k}) \underset{\norm{k} \rightarrow \infty}{\longrightarrow} e^{-{\beta \norm{k} \over 2}}.
\]
In fact, this observation is very important in reconstructing local bulk observables in the presence of a black hole as discuss in subsection \ref{comparelit}.
\subsection{Uplifting of fields and local bulk observables}
We are now ready to construct local bulk observables outside the horizon of the black hole. Let us consider bulk modes in the background of an AdS black hole (black brane) that we defined in section \ref{modesbrane} and which have the form $f_{\omega,\vect{k}}(t,\vect{x},z) = e^{-i\omega t+i\vect{k}\vect{x}} \psi_{\omega,\vect{k}}(z)$ and normalized so that $\psi_{\omega,\vect{k}} \sim {1 \over \Upsilon} z^{\Delta}$ as $z\rightarrow 0$. Then we construct the operator
\begin{equation}
\label{uplifta}
\boxed{
\phi_{\text{CFT}}(t,\vect{x},z) = \int_{\omega>0}{d\omega d^{d-1}\vect{k} \over (2 \pi)^d} \, \left[{\cal O}_{\omega,\vect{k}} \, f_{\omega,\vect{k}}(t,\vect{x},z) + {\cal O}_{\omega,\vect{k}}^\dagger \, f_{\omega,\vect{k}}^*(t,\vect{x},z)\right]}
\end{equation}
We emphasize once more, that this is an operator in the CFT. Notice that, if instead of the modes $f_{\omega,\vect{k}}(t,\vect{x},z)$ which go to $z^{\Delta}$ near the boundary, we work with the modes $\hatbb{f}_{\omega,\vect{k}}(t,\vect{x},z) $ which multiply canonically normalized oscillators in the bulk, then we can rewrite the expansion as
\begin{equation}
\label{upliftb}
\phi_{\text{CFT}}(t,\vect{x},z) = \int_{\omega>0} {d\omega d^{d-1}\vect{k} \over (2 \pi)^d} \, \left[\hatbb{{\cal O}}_{\omega,\vect{k}} \, \hatbb{f}_{\omega,\vect{k}}(t,\vect{x},z) + \hatbb{{\cal O}}_{\omega,\vect{k}}^\dagger \, \hatbb{f}_{\omega,\vect{k}}^*(t,\vect{x},z) \right],
\end{equation}
where the operators $\hatbb{\cal O}_{\omega,\vect{k}}$ are precisely the ones defined by equation \eqref{rescaled}. The point is that the normalization factor relating ${\cal O}_{\omega,\vect{k}}$ to $\hatbb{\cal O}_{\omega,\vect{k}}$ is precisely the inverse of that relating $f_{\omega,\vect{k}}(z)$ to $\hatbb{f}_{\omega,\vect{k}}$, hence the two expansions \eqref{uplifta} and \eqref{upliftb} are consistent.
Written in the form \eqref{upliftb}, it should be clear that at large $N$ the operator $\phi_{\rm CFT}(t,\vect{x},z)$ behaves like a free local field on the black hole background. In particular we have
\begin{equation}
\label{bulkrecon}
Z_{\beta}^{-1}{\rm Tr}\left[e^{-\beta H} \phi_{\text{CFT}}(t_1,\vect{x}_1,z_1).....
\phi_{\text{CFT}}(t_n,\vect{x}_n,z_n)\right]_{\text{CFT}} = \langle \phi_{bulk}(t_1,\vect{x}_1,z_1)...\phi_{bulk}(t_n,\vect{x}_n,z_n)\rangle_{\rm HH},
\end{equation}
where the RHS is the correlator of a free scalar field in the black hole background (region $\rm I$) which is evaluated in the Hartle-Hawking vacuum.
Given this equality, we are entitled to identify the operator $\phi_{\rm CFT}(t,\vect{x},z)$ that we constructed with the local bulk operator in region $\rm I$ of
the black hole.
In particular, we have
\begin{equation}
\label{localityout}
[\phi_{\rm CFT}(t_1,\vect{x}_1,z_1)\,,\,\phi_{\rm CFT}(t_2,\vect{x}_2,z_2)] = 0
\end{equation}
for two point $(t_1,\vect{x}_1,z_1)\,,\,(t_2,\vect{x}_2,z_2)$ which are spacelike separated {\it with respect to the metric} \eqref{threebrane}. This holds as an operator equation
inserted inside the thermal trace, possibly together with the insertion of an $N^0$ number of other operators of the form $\phi_{\rm CFT}$. More generally, it holds as an operator
equation modulo caveats like those mentioned in section \ref{factorcav}.
\subsection{Comparison with previous studies \label{comparelit}}
Before we conclude this section, it is useful to compare our construction with that of \cite{Hamilton:2005ju}. The authors of that paper were working directly in position space, and attempted to write the bulk field as
\begin{equation}
\label{bhkernel}
\phi_{\text{CFT}}(t,\vect{x},z) = \int dt' d^{d-1}\vect{x}' {\cal O}(t', \vect{x}') K(t,\vect{x},z;,t', \vect{x}')
\end{equation}
for an appropriate choice of ``kernel'' $K$ . However, they found
that attempting to compute $K$ directly led to a divergence, and so they were forced to complexify the boundary.
From the discussion above, we can see the origin of this divergence. Consider a mode in the bulk, with a given frequency $\omega$, and momentum $\vect{k}$ that solves the wave equation. Near the horizon we have the expansion
\begin{equation}
\label{psinearh}
\hat{f}_{\omega,\vect{k}} \rightarrow e^{-i \omega t+i\vect{k}\vect{x}} \left( e^{i\delta_{\omega,k}}e^{i\omega z_*} + e^{-i \delta_{\omega,k}} e^{-i\omega z_*}\right),
\end{equation}
where $z_*$ is the tortoise coordinate, in which horizon is at $z_*\rightarrow -\infty$. (For a precise definition of the tortoise coordinate, we refer the reader to section \ref{secadseternal}.)
If we write the bulk field as
\begin{equation}
\label{nearbhop}
\phi_{\text{CFT}}(t,\vect{x}, z) = \int_{\omega>0} {d \omega d^{d-1} \over (2 \pi)^d} \vect{k} {1 \over \sqrt{2\omega}}
\left( a_{\omega,\vect{k}}\hat{f}_{\omega,\vect{k}}(t,\vect{x},z) + {\rm c.c.}\right),
\end{equation}
then merely the fact that $\phi_{\text{CFT}}$ must satisfy the canonical commutation relations near the horizon tells us that we must have
\begin{equation}
\label{nearbhcommut}
[a_{\omega,\vect{k}},a^\dagger_{\omega',\vect{k}'}] = \delta(\omega-\omega')\delta^{d-1}(\vect{k}-\vect{k}').
\end{equation}
Our analysis above then tells us that for $\beta \norm{k} >> 1$ and $\norm{k} >> \omega$, the relation between ${\cal O}_{k, \omega}$ with $a_{k, \omega}$ must asymptote to
\[
a_{\omega, \vect{k}} = {\cal O}_{\omega,\vect{k}} e^{\beta \norm{k} \over 4}
\]
However, if apart from doing this, we also try and Fourier transform this
expression to write $\phi_{\text{CFT}}(t,\vect{x},z)$ as an integral transform of ${\cal O}(t, \vect{x})$ as in \eqref{bhkernel}, it is clear that we will get a divergence. This is because, in the large-spacelike momenta region above, this Fourier transform
will look like
\[
\int {\cal O}(\vect{x'},t') e^{-i \vect{k} \cdot \vect{x'} + i \omega t'} d t d^{d-1} \vect{x} e^{\beta \norm{k} \over 4} \hat{f}_{\omega, \vect{k}}(t,\vect{x},z) d^{d -1} \vect{k} d \omega
\]
Clearly, the integral over $\vect{k}$ diverges at least near the horizon of the black hole where it is clear that $\hat{f}_{\omega, \vect{k}}$ has no compensatory decaying exponential factor.
However, this divergence is just fictitious. The problem, of course, is that in doing this integral we have no way of taking into account the fact that the operator ${\cal O}_{\omega, \vect{k}}$ has a ``natural norm'' that is exponentially suppressed for large spacelike momenta. In momentum space, we do not have to deal with this problem.
It is worth making one more comment on this exponentially suppressed norm. The claim above (a) regarding spacelike modes in a thermal state, (b) the near-horizon expansion \eqref{psinearh}, (c) the commutation relations \eqref{nearbhcommut}, and finally (d) the claim that boundary correlators are limits of bulk correlators
\[
\langle {\cal O}(t_1, \vect{x}_1) {\cal O}(t_2, \vect{x}_2) \rangle_\beta = \lim_{z_1, z_2 \rightarrow 0}\left[\Upsilon^2 z_1^{-\Delta} z_2^{-\Delta} \langle \phi_{\text{CFT}}(t_1,\vect{x}_1, z_1) \phi_{\text{CFT}}(t_2,\vect{x}_2, z_2)\rangle_\beta\right],
\]
If we consider the modes $\hat{f}_{\omega, k}(t, \vect{x}, z)$ normalized so that their near horizon expansion in \eqref{psinearh}, then near the boundary for large spacelike momenta, they must behave like
\begin{equation}
\hat{f}_{\omega, k}(t, \vect{x}, z) \underset{z \rightarrow 0}{\longrightarrow} z^{\Delta} e^{-{\beta \norm{k} \over 4}} e^{-i \omega t+ i \vect{k} \cdot \vect{x}}
\end{equation}
These ideas, and this result, are examined and verified in a very concrete setting in the BTZ black hole background in Appendix \ref{appendix2dthermal}.
\section{Looking beyond the black hole horizon \label{sec:behind}}
In the previous section we reviewed the boundary construction of local bulk observables which lie outside the horizon of the black hole. What about points behind the horizon?
For the reconstruction of local bulk observables in region $\rm I$ it was important to identify the modes of the scalar field in region $\rm I$ with the Fourier modes of the boundary operator. In section \ref{semiblack} we explained that in order to write operators in region $\rm II$ (inside the black hole) we need modes {\it both} from region $\rm I$ and region $\rm III$. This raises the question: what should play the role of the modes of region $\rm III$ in the boundary field theory?
In fact the reader can easily persuade herself that, without these modes, it is not possible to write down a local field operator behind the horizon. Heuristically, the reason for this is as follows. If we try and extend the modes that we have constructed for region I, past the horizon, it is only the ``left moving'' modes that can be smoothly analytically continued past the horizon. To obtain a local field theory, we need independent right-movers behind the horizon, and these must come from some analogue of region $\rm III$.\footnote{The very perceptive reader might note that our condition of imposing normalizability at the boundary actually led to a constraint between left and right movers. This is true, but our condition was imposed on a time-like boundary. Behind the horizon, if we were to impose such a constraint, it would set constraints on a space-like boundary leading to a loss of locality.}
Now, as we have pointed out above, at late times, the geometry of a collapsing black hole is well approximated by the geometry of an eternal black hole. It is also known that an eternal black hole in anti-de Sitter space can be described by {\em two} CFTs --- one on each boundary of this geometry --- that are placed in a particular entangled state. This construction was first explored in \cite{Maldacena:2001kr}. In this picture, it is clear what plays the role of the required right-moving modes in the eternal black-hole. These are simply the right-moving modes that are constructed in region $\rm III$ and come from the second copy of the CFT. This story is consistent with the fact that, at finite temperature, the CFT is well described in the thermofield doubled formalism of Takahashi and Umezawa \cite{takahashi1996thermo}.
In this thermofield doubled formalism, we double the Hilbert space of the theory and consider the pure state
\begin{equation}
\label{thermostate}
|\Psi_{\text{tfd}}\rangle = {1\over \sqrt{Z_\beta}}\sum e^{-{\beta E \over 2}} |E\rangle \otimes |E\rangle,
\end{equation}
where the states of the original theory are labeled by the energy $E$ (and perhaps other quantum numbers). Corresponding to this ``doubling'' of the Hilbert space, every operator ${\cal O}$ in the original Hilbert space can be given a partner $\widetilde{\cal O}$ that acts on the second copy of the space. The point of this construction is that thermal
expectation values of operators in the original (single-copy) Hilbert space, can equivalently be computed as standard expectation values {\it on the pure state} $|\Psi_{\text{tfd}}\rangle$ in the doubled Hilbert space
\[
Z_{\beta}^{-1}{\rm Tr}\left[e^{-\beta H} {\cal O}(t_1,\vect{x}_1)...{\cal O}(t_n,\vect{x}_n) \right] = \langle \Psi_{\text{tfd}}| {\cal O}(t_1,\vect{x}_1)...{\cal O}(t_n,\vect{x}_n)|\Psi_{\text{tfd}}\rangle
\]
Moreover, correlators involving both ${\cal O}$ and $\widetilde{\cal O}$ in the doubled Hilbert space can be related to correlators
in the thermal, single-copy Hilbert space, with the replacement
\[\widetilde{\cal O}\left(t,\vect{x})\rightarrow {\cal O}(t+i\beta/2,\vect{x}\right)\]
where the analytic continuation in complex time is defined along the ``principal sheet'' as described in section \ref{therman}. More precisely, we have
\[
Z_{\beta}^{-1}{\rm Tr}\left[e^{-\beta H} {\cal O}(t_1,\vect{x}_1)...{\cal O}\left(t_k+i\beta /2,\vect{x}_k\right)...\right] =
\langle \Psi_{\text{tfd}}| {\cal O}(t_1,\vect{x}_1)...\widetilde{\cal O}(t_k,\vect{x}_k)...|\Psi_{\text{tfd}}\rangle
\]
It is an easy exercise to verify that this identification of $\widetilde{\cal O}(t,\vect{x})$ in the thermofield doubled Hilbert space, with ${\cal O}\left(t+i\beta/2,\vect{x}\right)$ inside the thermal correlators of the single Hilbert space, is consistent with the commutator $[{\cal O},\widetilde{\cal O}]=0$ in the doubled Hilbert space.
\vskip10pt
To summarize, it is well known that if we have a single theory {\it placed in a thermal density matrix}, we can map the thermal correlators to correlators in a {\it doubled theory} placed in the specific entangled pure state \eqref{thermostate}. And moreover, operators acting on the ``second copy'' of the doubled theory correspond to analytically continued versions of the thermal correlators of the original theory.
However, besides this story about {\it thermal correlators}, we know that in the original single theory there is a large number of {\it pure states} which approximate the thermal density matrix to an excellent approximation. For these pure states, what is the meaning of the thermofield doubled formalism and of the second copy of operators $\widetilde{\cal O}$? We answer this question in the next subsection.
\subsection{Approximating a pure state as a thermofield doubled state}
While the thermofield formalism is used quite frequently, and quite possibly the answer to our question above is clear to experts, we were surprised not to find it stated clearly in the literature. We believe that the correct answer is as follows.
Consider a pure-state $|\Psi\rangle$ in a theory with many degrees of freedom that thermalizes. We try to ``coarse-grain'' this state and identify some degrees of freedom that are easily accessible to measurement, and others that are not. The fact that the observables of the theory admit a division into ``easily accessible to measurements'' (coarse-grained observables) and those that are not (fine-grained observables), is a {\it prerequisite} to be able to talk about ``thermalization of a pure state''. If we start with a closed quantum system, we can talk about ``thermalization of a pure state'' if there is some sense that we can divide the full system into two parts, one of which plays the role of the
``environment/heat-bath'' and the other of the ``system in thermal equilibrium'' and if moreover the dynamics is such that under time evolution the pure state $|\Psi\rangle$, {\it when projected on the small system}, eventually looks approximately like a thermal density matrix. Clearly this is an approximate notion, which can only be made precise if there is some parameter which controls the validity of these approximations. In the case relevant to us, the full quantum system is the order $O(N^2)$ degrees of freedom of the large $N$ gauge theory, which may be in some specific pure state $|\Psi\rangle$, while the role of the ``sub-system'' that reaches thermal equilibrium is the $O(1)$ degrees of freedom corresponding to light gauge invariant operators. The parameter which controls the validity of the splitting into coarse- and fine-grained spaces is $N$.
For example, in the case that we have been considering above, the observer may be able to measure the eigenvalue of the ``occupation level'' $n_{\omega,\vect{k}} = \hat{\cal O}^{\dagger}_{\omega, \vect{k}} \hat{\cal O}_{\omega, \vect{k}}$ of the modes \eqref{rescaled} but this does not completely specify the state of the system; for that, the observer would need to measure several other operators including those that correspond to ``stringy'' and trans-Planckian degrees of freedom in the bulk. Using this intuition, we divide the Hilbert space of the CFT into a direct-product of a ``coarse-grained'' part and a ``fine-grained'' part
\begin{equation}
\label{hcoarsefine}
{\cal H} = {\cal H}_{\text{coarse}} \otimes {\cal H}_{\text{fine}}.
\end{equation}
There are many such possible divisions of the gauge theory into coarse- and fine-grained degrees of freedom. However, as long as the coarse-grained degrees of freedom do not probe too much information --- we quantify this phrase below --- about the state we will see that our interpretation below holds.
We now consider some operator in the theory. For any generalized field ${\cal O}(t, \vect{x})$, what our low energy observer really measures is
\begin{equation}
\label{coarseprojector}
{\cal O}_{c}(t,\vect{x}) = {\cal P}_{\text{coarse}} {\cal O}(t,\vect{x}),
\end{equation}
where ${\cal P}_{\text{coarse}}$ is the ``projector'' onto the coarse-grained Hilbert space which traces out the fine grained degrees of freedom. We should mention that a division of the Hilbert space into ``coarse'' and ``fine'' degrees of freedom is often
performed in studies of ``dissipation'' in quantum systems. As far as we know, this was first attempted by von Neumann \cite{von2010proof} and our ``coarse'' operators are precisely the ``macroscopic'' operators of that paper.
Now, let us turn to the pure state that we are considering, which is close to a thermal state. Corresponding to the Hilbert space decomposition above, this state can be written as
\begin{equation}
\label{entangledcoarsefine}
| \Psi \rangle = \sum_{i,j} \alpha_{ij} | \Psi^{c}_i \rangle \otimes | \Psi^{f}_j \rangle,
\end{equation}
i.e. as an entangled state between the coarse- and fine-grained parts of the Hilbert space. Here the states $|\Psi^{c}_i\rangle$ run over an orthonormal basis of states in the coarse-grained Hilbert space, and $|\Psi^{f}_j\rangle$ run over
an orthonormal basis in the fine-grained Hilbert space.
In fact, what the low-energy observer sees directly is not
this entangled state, but rather a density matrix. Since the pure state is close to being thermal, we even know the eigenvalues of this density matrix. In some basis, the coarse-grained density matrix can be written
\begin{equation}
\label{thermaldensity}
\rho_{c} = {1 \over Z^c_{\beta}}\text{diag}\left(e^{-\beta E_i} \right)
\end{equation}
where $E_i$ are the eigenvalues of the coarse-grained Hamiltonian and $Z^c_{\beta} = \sum_i e^{-\beta E_i}$ is the coarse-grained partition function.\footnote{
In in any given pure state $|\Psi \rangle$, the exact coarse-grained density matrix may be different from the exactly thermal one above, but these differences are unimportant in what follows, so we ignore them.}
What is the form of the density matrix in the fine-grained Hilbert space? In fact, as is well known, we can find a basis for the {\em fine-grained} Hilbert space so that the density matrix looks exactly like \eqref{thermaldensity} with zeroes in the other places.
The argument is very simple, and we repeat it here for the reader's convenience. We need to ``diagonalize'' the matrix $\alpha_{i j}$ in \eqref{entangledcoarsefine}. This process, called a ``singular value decomposition'', involves writing $\alpha_{i j}$ as
\[
\alpha_{i j} = \sum_m U_{i m} D_{m m} V_{m j}
\]
where $U$ is a unitary matrix that acts in ${\cal H}_{\text{coarse}}$ and $V$ is a unitary matrix in ${\cal H}_{\text{fine}}$, while $D$ is a rectangular-diagonal matrix i.e. a matrix of the form
\begin{equation}
\label{rectangularmat}
D = \begin{pmatrix}
D_{11} & 0 & 0 & 0 &\ldots \\
0 & D_{22} & 0 & 0 & \ldots \\
0 & 0 & D_{33} & 0 & \ldots
\end{pmatrix}
\end{equation}
This means that we can re-write \eqref{entangledcoarsefine} as
\begin{equation}
\label{maximalentanglement}
| \Psi \rangle = \sum_i D_{i i} | \hat{\Psi}_i^{c} \rangle \otimes | \hat{\Psi}_i^{f} \rangle
\end{equation}
where now, both $|\hat{\Psi}_i^{c}\rangle$ and $|\hat{\Psi}_i^f\rangle$ are orthonormal. In fact from \eqref{thermaldensity}, we already know that $D_{i i} = {1 \over \sqrt{Z^c_{\beta}}} e^{-{\beta E_i \over 2}}$.
Now, \eqref{maximalentanglement} immediately tells us that the fine-grained density matrix must also look like
\[
\rho_f = \sum_i D_{i i} |\hat{\Psi}_i^{f} \rangle \langle \hat{\Psi}_i^f |.
\]
Let us return to the generalized free field ${\cal O}(t,\vect{x})$ that we have been discussing. As we mentioned, the low energy observer can only measure ${\cal O}_{c}(t,x)$ defined in \eqref{coarseprojector}. However, in the coarse-grained Hilbert space, we should be able
to write\footnote{Note, that we have kept the time evolution of the coarse operator arbitrary. In particular, the division \eqref{hcoarsefine} does not correspond to an analogous separation of the Hamiltonian, which may well couple the coarse and fine degrees of freedom. In studies of quantum dissipation, this coupling is sometimes quantified by means of an ``influence functional.'' It would be interesting
to understand this in detail for the boundary theory.}
\[
{\cal O}_{c}(t,\vect{x}) = \sum_{i_1, i_2} {\omega}_{i_1 i_2}(t,\vect{x})|\hat{\Psi}^c_{i_1} \rangle \langle \hat{\Psi}_{i_2}^c|.
\]
Now, we {\em define} an operator that acts in the same way on the fine-grained Hilbert space
\begin{equation}
\label{tildeodef}
\widetilde{\cal O}(t,\vect{x}) = \sum_{i_1, i_2} \omega^*_{i_1 i_2}(t,\vect{x})|\hat{\Psi}^f_{i_1} \rangle \langle \hat{\Psi}_{i_2}^f|.
\end{equation}
The complex conjugation on $\omega$ is consistent with the original conventions of \cite{takahashi1996thermo}, and implies that the map between ${\cal O}$ and $\widetilde{\cal O}$ is {\em anti-linear.} This is necessary to ensure that the map is invariant under a change of basis that leaves \eqref{maximalentanglement} invariant.
What we have shown is that, even in a pure state for the gauge theory, for each generalized free-field ${\cal O}(t,\vect{x})$ and its corresponding modes ${\cal O}_{\omega, \vect{k}}$, we spontaneously obtain doubled operators $\widetilde{\cal O}(t,\vect{x})$, with corresponding modes $\widetilde{\cal O}_{\omega, \vect{k}}$! We explore this emergence further in section \ref{sec:applications} in a concrete example.\footnote{The coarse-fine splitting of the Hilbert space described in this section holds to an excellent approximation in the CFT at large-N. The construction described here, also
provides very important intuition including the ``state dependence''
of the $\widetilde{\cal O}(t,\vect{x})$ operators. However, we refer
the reader to \cite{Papadodimas:2013,Papadodimas:2013b} for a somewhat more
refined construction.}
These doubled operators are precisely what we need to construct local operators in region \rm II. However, before we attack that question we first want to mention a relation between the correlators of these operators and analytically continued correlators of ordinary operators.
From this point in this paper, we will drop the subscript ${\cal O}_c(t, \vect{x})$; it is understood that correlators now refer to the coarse grained operators only.
\subsection{Relation to analytically continued thermal correlators}
In this subsection, we will point out below that $\widetilde{\cal O}$ operators can---in a sense that we make precise ---be thought of as analytically continued versions of the ordinary operators.
Let us say that we wish to compute a two-point correlator involving both ${\cal O}$ and $\widetilde{\cal O}$ in the state $| \Psi \rangle$ above
\begin{equation}
\label{o1o2rel}
\begin{split}
&\langle \Psi | {\cal O}(t_1, \vect{x}_1) \widetilde{\cal O}(t_2, \vect{x}_2) | \Psi \rangle = (Z^c_{\beta})^{-1}\sum_{i_1, i_2} e^{-{\beta (E_{i_1} + E_{i_2}) \over 2}} \langle \hat{\Psi}^{c}_{i_1} | {\cal O}(t_1, \vect{x}_1) | \hat{\Psi}^{c}_{i_2} \rangle \langle \hat{\Psi}^{f}_{i_1} | \widetilde{\cal O}(t_2, \vect{x}_2) | \hat{\Psi}^{f}_{i_2} \rangle \\
&= (Z^c_{\beta})^{-1} \sum_{i_1, i_2} e^{-{\beta (E_{i_1} + E_{i_2}) \over 2}} \langle \hat{\Psi}^{c}_{i_1} | {\cal O}(t_1, \vect{x}_1) | \hat{\Psi}^{c}_{i_2} \rangle \langle \hat{\Psi}^{c}_{i_1} | {\cal O}(t_2, \vect{x}_2) | \hat{\Psi}^{c}_{i_2} \rangle^* \\
&= (Z^c_{\beta})^{-1} \sum_{i_1, i_2} e^{-{\beta (E_{i_1} + E_{i_2}) \over 2}} \langle \hat{\Psi}^{c}_{i_1} | {\cal O}(t_1, \vect{x}_1) | \hat{\Psi}^{c}_{i_2} \rangle \langle \hat{\Psi}^{c}_{i_2} | {\cal O}(t_2, \vect{x}_2) | \hat{\Psi}^{c}_{i_1} \rangle \\
&= (Z^c_{\beta})^{-1} \sum_{i_1, i_2} e^{-\beta E_{i_1} } \langle \hat{\Psi}^{c}_{i_1} | {\cal O}(t_1, \vect{x}_1) | \hat{\Psi}^{c}_{i_2} \rangle \langle \hat{\Psi}^{c}_{i_2} | {\cal O}(t_2 + {i \beta \over 2}, \vect{x}_2) | \hat{\Psi}^{c}_{i_1} \rangle \\
&= (Z^c_{\beta})^{-1}{\rm Tr}_c\left[e^{-\beta H} {\cal O}(t_1, \vect{x}_1) {\cal O}(t_2 + {i \beta \over 2}, \vect{x}_2) \right],
\end{split}
\end{equation}
where the last trace is over the coarse grained Hilbert space.
Note that the operator ordering is important. The analytically continued operator always appears on the right. As we discussed above, when we discussed thermal correlators, if this operator had appeared on the left, the correlator above would naively have diverged and we would have had to define it by giving a prescription for how to cross the branch cut. In fact, the ordering in \eqref{o1o2rel} corresponds to always remaining in the {\em principal branch} of the function ${\cal F}$ described in section \ref{therman}.
\subsection{Reconstructing the full eternal black hole from a single CFT}
We now describe how to reconstruct, from the CFT, local bulk operators behind the horizon. We emphasize that we are working in a {\em single CFT} in a {\em pure state.}
As explained in the previous subsection, for every local operator ${\cal O}(t,\vect{x})$ we define the operator
\[
{\cal O}(t, \vect{x}) \rightarrow \widetilde{\cal O} (t,\vect{x}).
\]
and its Fourier modes
\[
\widetilde{\cal O}_{\omega,\vect{k}} = \int dt d^{d-1}\vect{x} \,e^{i\omega t-i\vect{k} \vect{x}}\, \widetilde{\cal O}(t,\vect{x})
\]
The complex conjugation in \eqref{tildeodef} implies that the modes $\widetilde{\cal O}_{\omega,\vect{k}}$ generate an algebra that is identical to
that of the modes ${\cal O}^{\dagger}_{\omega,\vect{k}}$, while at the same time these two sets of modes commute with each other. In the previous
section we identified the gauge theory modes ${\cal O}_{\omega,\vect{k}}$ with the Schwarzschild modes $a_{\omega,\vect{k}}$ in region I of the eternal black hole. It is clear that the modes $\widetilde{\cal O}_{\omega,\vect{k}}$ and $\widetilde{\cal O}_{-\omega,\vect{k}}$ --- which we emphasize act on the same Hilbert space, of the single CFT --- play the role of the bulk modes $\widetilde{a}^{\dagger}_{\omega,\vect{k}}$ and $\widetilde{a}_{\omega,\vect{k}}$ in region III of the eternal black hole. This means that we identify
\[
\begin{split}
\hatbb{\widetilde{\cal O}}_{\omega,\vect{k}}\qquad&\Leftrightarrow \qquad a^{\dagger}_{\omega,\vect{k}}, \\
\hatbb{\widetilde{\cal O}}_{-\omega,\vect{k}}\qquad&\Leftrightarrow \qquad a_{\omega,\vect{k}},
\end{split}
\]
where both identifications assume $\omega > 0$ and $\hatbb{\widetilde{\cal O}}$ is defined in analogy to \eqref{rescaled}.
Having established this identification, it is now straightforward to write down local bulk operators of all four regions of the eternal black hole. For region I, we already did that in the previous section and we had
\[
\phi_{\text{CFT}}^{\rm I}(t,\vect{x},z) = \int_{\omega>0}{d\omega d^{d-1}\vect{k} \over (2 \pi)^d}\,\left[{\cal O}_{\omega,\vect{k}} f_{\omega,\vect{k}}(t,\vect{x},z) + {\cal O}_{\omega,\vect{k}}^\dagger f^*_{\omega,\vect{k}}(t,\vect{x},z)\right]
\]
For region III, we have a similar expansion. We define the analogue of the AdS-Schwarzschild coordinates $t,\vect{x},z$ for region III, in terms of the Kruskal coordinates $U,V$ by the relations
\begin{equation}
\label{kruskalcb}
\begin{split}
& u = t-z_*\qquad,\qquad v=t+z_* \\ &
U = e^{-{d u \over 2 z_0}}\qquad,\qquad V = -e^{{d v \over 2 z_0}}
\end{split}
\end{equation}
with $z_*$ defined in terms of $z$ exactly as above equation \eqref{lightconec}. Then the CFT operator
\begin{equation}
\label{finalregionother}
\phi_{\text{CFT}}^{\rm III}(t,\vect{x},z) = \int_{\omega>0}{d\omega d^{d-1}\vect{k} \over (2 \pi)^d}\,\left[\widetilde{\cal O}_{\omega,\vect{k}} f_{\omega,\vect{k}}(t,\vect{x},z) + \widetilde{\cal O}_{\omega,\vect{k}}^\dagger f^*_{\omega,\vect{k}}(t,\vect{x},z)\right]
\end{equation}
plays the role of the local bulk field in region $\rm III$. Notice that while we parameterize the points in region III by the same set of coordinates $t,\vect{x},z$, they are obviously distinct points from those in region I, since their $U,V$ values are different.
Notice that the relations \eqref{kruskalcb} for region $\rm III$ are similar, but not quite the same, as the relations \eqref{kruskalc} for region $\rm I$. In particular, the different signs in the second line imply that as Kruskal time increases, $t$ decreases in region $\rm III$. Related to this, we have the fact that although \eqref{finalregionother} and \eqref{uplifta} look similar, $\widetilde{O}_{\omega, k}$ is actually a ``creation operator.'' (See appendix \ref{rindlerq} for a similar situation in the simpler case of Rindler space quantization.)
It is now straightforward to use the expansions in regions I and III to define a local field in regions II and IV. We focus on region II which is our main interest.
We can parametrize region $\rm II$ by $t,\vect{x},z$, with $z_0<z<\infty$ and $-\infty<t<\infty$, which is now a spacelike coordinate. Remember that in these coordinates the singularity is at $z\rightarrow \infty$. The time $t$ in this region increases as we approach the horizon between $\rm I$ and $\rm II$.
We look for solutions of the Klein Gordon equation of the form
\[
g_{\omega,\vect{k}}(t,\vect{x},z) = e^{-i\omega t+i\vect{k}\vect{x}} \chi_{\omega,\vect{k}}(z)
\]
We get a 2nd order ODE for $\chi_{\omega,\vect{k}}(z)$. What is important is that in region II there are no boundary conditions that we need to impose (unlike what happened in region $\rm I$ or $\rm III$ where we imposed the ``normalizable'' boundary conditions at infinity). So it seems that in region $\rm II$ we have twice the number of modes as in region $\rm I$ (or $\rm III$). Of course this was to be expected since region $\rm II$ lies in the causal future of both regions $\rm I$ and $\rm III$ and the information of both regions $\rm I$ and $\rm III$ eventually enters region $\rm II$. So it is normal to have twice as many modes in $\rm II$.
Hence for any choice of $\omega,\vect{k}$ we get two linearly independent solutions $\chi_{\omega,\vect{k}}^{(1,2)}(z)$. We can choose the linear combination of these solutions so that as $z\rightarrow z_0$ (from larger values, i.e. from inside the black hole) we have
\[
\chi_{\omega,\vect{k}}^{(1)}(z) \sim c(\omega, \vect{k}) (z-z_0)^{-i \omega},\qquad \chi_{\omega,\vect{k}}^{(2)}(z) \sim c(-\omega, \vect{k})(z-z_0)^{i \omega}\quad {\rm for}\quad z\rightarrow z_0^+,
\]
where $c(\omega, \vect{k})$ also appeared in \eqref{nearhpsi}.
Now imposing the continuity of the field
at the horizon between $\rm I$ and $\rm II$ and also between $\rm III$ and $\rm II$ we find that the field in region $\rm II$ has the expansion
\begin{equation}
\boxed{\label{finalbehind}
\phi^{\rm II}_{\text{CFT}}(t,\vect{x},z) =
\int_{\omega>0} {d\omega d^{d-1}\vect{k} \over (2 \pi)^d}\left[ {\cal O}_{\omega,\vect{k}}\, g_{\omega,\vect{k}}^{(1)}(t,\vect{x},z) + \widetilde{\cal O}_{\omega,\vect{k}} \,g_{\omega,\vect{k}}^{(2)}(t,\vect{x},z)+ {\rm h.c.}
\right]}
\end{equation}
where $g^{(1,2)}_{\omega,\vect{k}}(t,\vect{x},z) = e^{-i\omega t+i \vect{k}\vect{x}}\chi^{(1,2)}(z)$. A similar construction can be performed for the field in region $\rm IV$.
To summarize, using the boundary modes ${\cal O}_{\omega,\vect{k}}$ and $\widetilde{\cal O}_{\omega,\vect{k}}$ which were described above, we can reconstruct the local quantum field $\phi$ everywhere in the Kruskal extension and that at large $N$ the correlators of these operators satisfy the analogue of \eqref{bulkrecon}, but now operators can be inside or outside the horizon
\begin{equation}
\label{bulkreconb}
Z_{\beta}^{-1}{\rm Tr}\left(e^{-\beta H} \phi_{\text{CFT}}(t_1,\vect{x}_1,z_1) \ldots \phi_{\text{CFT}}(t_n,\vect{x}_n,z_n)\right)_{\text{CFT}} = \langle \phi_{\text{bulk}}(t_1,\vect{x}_1,z_1)...\phi_{\text{bulk}}(t_n,\vect{x}_n,z_n)\rangle_{\rm HH}
\end{equation}
The expansions \eqref{finalbehind} can be written more explicitly for the case of the BTZ black hole, where the mode wave functions $g_{\omega,\vect{k}}^{(1,2)}$ can be found analytically.
In particular, for any two points $P_1,P_2$ in any of the two regions I,II we have
\begin{equation}
\label{localityb}
[\phi_{\rm CFT}(P_1)\,,\,\phi_{\rm CFT}(P_2)] = 0
\end{equation}
when $P_1,P_2$ are spacelike separated with respect to the causal structure of the AdS-Kruskal diagram. Here, to write the operator $\phi_{\rm CFT}(P)$, we use either \eqref{uplifta} or \eqref{finalbehind} depending
on whether $P$ is in region I or II. Also, as before, this equation holds as an operator equation inside the thermal trace/or on a heavy
typical pure state, to leading order at large $N$ and modulo the caveats in section \ref{factorcav}.
Notice, that while it naively seems so, the claim \eqref{localityb} is not in conflict with the idea from black hole complementarity, that the Hilbert space of the interior of the black hole is not completely
independent from the Hilbert space outside. The point is that \eqref{localityb} holds only in the
large $N$ limit and up to the aforementioned caveats. So while the commutator of {\it two simple} measurements of the scalar field, inside and outside the horizon, is zero in
the large $N$ limit, if we consider the commutator of an operator inside the horizon with a very complicated measurement outside (which effectively corresponds to measuring $N^{a}\,\,\text{with}~ a>0$ instances of $\phi$), we have no reason to expect the commutator to vanish --- {\it not even in the large $N$ limit}. Hence the Hilbert
spaces inside and outside are not completely independent. These issues will be discussed in more detail in section \ref{sec:applications}.
\section{Applications \label{sec:applications}}
Our construction above tells us that bulk correlators can be written in terms of boundary correlators. This has a significant advantage: it translates questions about quantum gravity in the bulk, which are mysterious, to questions about conformal field theory correlators that are well defined.
There has been significant recent discussion of the information paradox. In an important paper, Mathur \cite{Mathur:2009hf} sharpened the contradiction between semi-classical evolution and unitarity in quantum gravity, by using the strong subadditivity of entropy. This argument was recently re-emphasized in \cite{Almheiri:2012rt}. Both papers pointed out that one way to resolve the contradiction would be to abandon the standard assumption that the horizon of the black hole is featureless. We can use our construction to obtain hints about the nature of the horizon, and also to the resolution of the information paradox. We do this in turn below.
\subsection{Nature of the horizon \label{subsechorizonnat}}
First, let us address the issue of what happens to the infalling observer when he crosses the horizon. This is the same as the question of what happens to correlators of $\phi_{\text{CFT}}$ for points near the horizon in a typical heavy pure state. If the theory has a good large $N$ expansion in the thermal state then, as we show below, these correlators are the same (to leading order in ${1 \over N}$) to those in the thermal ensemble. In the thermal ensemble, it is clear that correlators of $\phi_{\text{CFT}}$ are the same as those predicted by
semi-classical GR.
The reader should not be concerned about the fact that on one side of the horizon we have both the left and right moving modes of a field constructed just from ${\cal O}$, while on the other side the modes of ${\widetilde{\cal{O}}}$ appear.
This is analogous to the fact that when we quantize a field in Minkowski space using Rindler coordinates (see Appendix \ref{rindlerq} for notation and a review), in region I only the modes $a_{\omega,k}$ appear, while in region II the modes $\widetilde{a}_{\omega,k}$ also appear. Nevertheless, in the state corresponding to the Minkowski vacuum correlation functions are perfectly regular across the horizon.
In
precisely the same fashion, the definition \eqref{tildeodef} ensures that correlators of $\widetilde{\cal{O}}$, in a typical pure state close to the thermal state, have the same properties as those of ${\cal{O}}$, which we discussed in detail in section \ref{subsec:thermalfourier}. From here, it is easy to show that, in our construction, correlation functions are regular across the horizon.
So, we will conclude below that
\begin{quote}
For AdS duals of conformal field theories with a good ${1 \over N}$ expansion at finite temperature, our construction within AdS/CFT predicts that an observer falling through the horizon of a big black hole will measure the correlators predicted by semi-classical GR. In particular, he will not observe anything special about the horizon by doing either high energy\footnote{But not too high. Here we are talking about experiments at energy scale $E$, measured in the local frame of the infalling observer, where $E$ can be anything, as long as it does not scale with $N$ (or $\lambda$ --- in the case of the ${\cal N}=4$ SYM).} or low energy experiments.
\end{quote}
This conclusion should be contrasted with the fuzzball proposal, which makes a different claim about the nature of the horizon. We are aware that there are different perspectives on the fuzzball proposal. So, for the sake of precision, we will consider the following statement:
\begin{quote}
{\it (Extrapolated) Fuzzball Proposal:} The geometry is modified near the region where one expects to find the horizon, and it is this modification which contains information about the microstate that resolves the information paradox. It is possible to see these geometrical differences by doing experiments at the scale $T$--- the temperature of the black hole.\footnote{Regardless of the ability of the infalling observer to actually conduct experiments at this energy scale in his lifetime, this is a statement about correlation functions that we can examine.}
\end{quote}
It appears to us that this interpretation of the proposal is necessary if we are to resolve the information paradox using this proposal: the fuzzball proposal resolves the Hawking paradox by finding a mechanism to encode information about the black hole microstate in the Hawking radiation. This is also the statement that we seem to find in the recent literature \cite{Mathur:2009hf,Mathur:2012np, Mathur:2011uj, Avery:2012tf,Chowdhury:2012vd}.
Since the geometry can be measured by correlators of the metric, and of other {\em light fields}, the fuzzball proposal must imply the correlators of light operators differ between the different states of the ensemble, so that if we do our construction in one state, we get one geometry whereas in another state we get another geometry. More precisely, we take a typical pure state $|\Psi\rangle$ in the CFT and evaluate the correlator
\begin{equation}
\label{finalcorrelator}
\langle \Psi| \phi_{\text{CFT}}(P_1)...\phi_{\text{CFT}}(P_n) |\Psi\rangle
\end{equation}
where the points $P_1,...P_n$ correspond to bulk points along the trajectory of an infalling observer, then the proposal seems to be that this correlator gives us information about the microstate $|\Psi \rangle$.
Our construction tells us that (a) $\phi_{\text{CFT}}$ can be written in terms of the boundary operator ${\cal O}$ and (b) to accurately evaluate
correlators involving insertions of $\phi_{\text{CFT}}$ with finite momentum and frequency, we do not need a parametric enhancement of this frequency
on the boundary. The latter property is discussed in more detail in section \ref{sec:subtleties}. So, our construction allows us to translate the extrapolated fuzzball proposal into the following statement in the CFT:
\begin{quote}
{\it Implication of the (Extrapolated) Fuzzball Proposal}: By measuring low point correlators of light operators, we can distinguish between the various microstates of the CFT that constitute a black hole.
\end{quote}
We will now show that this is impossible if the CFT admits a large $N$ expansion for thermal correlators. If such a large $N$ expansion exists--- as we believe it does for, say, ${{\mathcal N} = 4}$ Super Yang-Mills at strong coupling--- then our construction, or the idea that bulk correlators can be written as non-singular transforms of CFT correlators is in contradiction with the fuzzball proposal.
Let us now consider a 2-point function for simplicity (higher-point functions can be considered similarly)
\[
\langle \Psi | {\cal O}(x_1) {\cal O}(x_2) | \Psi \rangle.
\]
The question is whether we can use this 2-point information to extract some information about the state $|\Psi\rangle$.
Let us say that we can measure this correlator to order ${1 \over N^{\alpha}}$, where $\alpha$ is some fixed power (that does not scale like $N$). Then we have
\begin{align}
\label{corrfourpt} \langle \Psi | {\cal O}(x_1) {\cal O}(x_2) | \Psi \rangle &= \langle \Psi(\infty) {\cal O}(x_1) {\cal O}(x_2)
\Psi(0) \rangle \\ \label{corrope} &= \sum_Q C_{{\cal O }{\cal O}}^Q \langle \Psi(\infty) {Q(x_2) \over |x_1 - x_2|^{2 \Delta_{\cal O} - \Delta_Q}} \Psi(0) \rangle \\ \label{corrthreept1}
&= \sum_Q C_{{\cal O}{\cal O}}^Q C_{\Psi \Psi Q} {1 \over |x_1 - x_2|^{2 \Delta_{\cal O} - \Delta_Q} |x_2|^{\Delta_Q}} \\ \label{corrsumq}
&\equiv \sum_Q G_Q(x_1, x_2).
\end{align}
Here in \eqref{corrfourpt} we have written the 2-point correlator in the state $|\Psi\rangle$ as a 4-point correlator in the vacuum. In \eqref{corrope} we have used the OPE expansion to write the operator product of the two ${\cal O}$ operators as a sum over all other operators $Q$ in the theory, with coefficients that are fixed by conformal invariance up to 3-point coefficients, which are pure numbers, denoted by $C_{{\cal O}{\cal O}}^Q$ above.\footnote{More precisely, this is true only for scalar operators. We have suppressed tensor operators only to lighten the notation. The inclusion of these does not alter the argument in any way.} We have then used conformal invariance again to evaluate the remaining 3-point function, and finally $G_Q$ is merely a short-hand for the contribution of the operator $Q$ to the initial correlator.
Now the key point is that at any given value of $x_1, x_2$, if this correlator has a large $N$ expansion, and if we are measuring it to an accuracy ${1 \over N^{\alpha}}$, then the number of
operators $Q$ that can contribute to this order must itself be bounded by $N^{\alpha}$. One might have hoped to disentangle the contribution of different operators $Q$ by separating the points $x_1$ and $x_2$ by some order $N$ distance, but as we saw in our construction above we do not need to do so to reconstruct local operators near the horizon if we are measuring bulk correlators that are separated by $O(1)$ near the horizon.
So the correlator really can, at most, contain information about the product $C_{{\cal O}{\cal O}}^Q C_{\Psi \Psi Q}$ for some $\kappa N^{\alpha}$ operators, where $\kappa$ is some order $1$ number. However, the black hole consists of $e^{S}$ states. Since $S \propto N$ (we remind the reader that, in our notation, $N$ is a measure of the central charge and not the rank of the boundary gauge group) and the temperature and the leading constants do not scale with $N$ in our setup, and so we can loosely say that we need $e^{N}$ pieces of information to identify the black hole microstate. Clearly this is impossible with the information we can glean from the correlator. To distinguish the different microstates of the black hole, we would need to measure correlators to an accuracy $e^{-N}$. At this level it is not clear (and probably not true) that the correlators have a geometric interpretation.
It is useful to consider the example of 2-charge solutions \cite{Lunin:2001fv,Lunin:2002bj} which can be identified with the Ramond ground states
of the $D1-D5$ CFT. In that case also, most of the classical solutions are string scale. In the AdS$_5$ case, to obtain information about string scale objects, we would need to measure correlators to an accuracy $e^{-\lambda}$. However, what our construction here tells us is that to recover information about a {\it big} black hole, we will need to measure correlators to an accuracy that scales exponentially with $N$, not just with $\lambda$. While string scale geometries might possibly make sense, by switching duality frames, the degeneracy of big black holes looks like it might come from Planck scale geometries. We do not understand how to discern geometries at the Planck scale.
\subsection{Recovering Information, small corrections, firewalls and complementarity}
We now turn to the issue of the information paradox itself. Indeed, this is the key issue in all the recent discussions on whether the nature of the horizon needs to be modified. For example, several regular solutions, with the same charges as the black hole, have been found in several theories (see \cite{Skenderis:2008qn} for a review and references to the very extended literature). These solutions are extremely interesting. However, by themselves, they are not enough to justify the extrapolated fuzzball proposal that we considered above. Instead, Mathur \cite{Mathur:2009hf} provided an indirect argument for why the nature of the horizon had to be
modified to resolve the information paradox.
A very similar argument has recently been made in \cite{Almheiri:2012rt}. However, starting with the same argument as Mathur, Almheiri et al. stopped at the (weaker) conclusion that the nature of the horizon must be modified. Since any such modification would generically cause the infalling observer to burn up, these authors spoke of ``firewalls'' rather than fuzzballs.
Since our conclusion suggests the opposite --- that the nature of the horizon is {\em not} modified --- we must explain how this is consistent with
the preservation of information. We briefly review the information loss arguments below. We then describe why we disagree with these arguments and furthermore how our proposal suggests a natural resolution to the information paradox.
\subsubsection{Strong Subadditivity and the Information Paradox}
\paragraph{Review of the ``Hawking Theorem''}
The argument that the information paradox cannot be resolved if we retain
the hypothesis that the horizon is featureless was termed the ``Hawking theorem'' in \cite{Mathur:2009hf}.
Consider a toy model of the Hawking process where a pair of entangled qubits are produced at the horizon at each step; one of them is emitted into the radiation outside and the other falls into the black hole. So, at each
stage we produce the pair
\begin{equation}
\label{pairproduction}
{1 \over \sqrt{2}} \left( |0 \rangle |0 \rangle + | 1 \rangle | 1 \rangle \right).
\end{equation}
Of these, the second bit goes out of the black hole, whereas the first bit falls inside. We will call the bit that goes outside $B$, while the bit that falls inside is $C$. Then, from \eqref{pairproduction}, we can abstract the lesson that $B$ and $C$ are maximally entangled up to small corrections
\begin{equation}
\label{bcpure}
S_{BC} \approx 0.
\end{equation}
This more general conclusion also follows from the semi-classical
observation that, in a certain frame, the state of the black-hole is merely the vacuum, and Hawking radiation is obtained by performing a Bogoliubov transformation on this state.
If we consider the density matrix of the bit outside, in this case, it is given by
\begin{equation}
\label{rhosingle}
\rho_B = {1 \over 2} \begin{pmatrix}1&0\\0&1 \end{pmatrix} \equiv {1 \over 2} I.
\end{equation}
The entanglement entropy of the bit with the black hole is given by
\[
S_B = -{\rm Tr} \left( \rho_B \log \rho_B \right) = \ln 2.
\]
However, in general we can abstract away the rule that
\begin{equation}
\label{bthermal}
S_{B} > 0, \quad S_{C} > 0.
\end{equation}
Since $S_{BC} \approx 0$, we have $S_{C} \approx S_{B}$.
Apart from the systems $B$ and $C$ designated above, let us also consider the system $A$, which comprises all the radiation that has been emitted by the black hole up to the step under consideration. If we assume that the black hole started in a pure state, then {\em eventually} the entanglement entropy of this radiation with the black hole should start decreasing.
So, for a very old black hole, it must be true that
\begin{equation}
\label{BpurifiesA}
S_{AB} < S_A.
\end{equation}
This is the statement that when the bit $B$ goes and joins the radiation $A$, it decreases its entropy.
Mathur \cite{Mathur:2009hf} pointed out that strong subadditivity tells us that \eqref{BpurifiesA}, \eqref{bcpure} and \eqref{bthermal} appear to be in contradiction. For three {\em separate} systems $A, B, C$, it is possible to show that
\begin{equation}
\label{strongsubadditivitym}
S_{A} + S_{C} \leq S_{AB} + S_{BC}
\end{equation}
Since we have $S_{A} > S_{AB}$ and $S_{C} > S_{BC}$, we seem to have a contradiction.
The same argument was made in a slightly different way by \cite{Almheiri:2012rt}. They instead used the identity
\begin{equation}
\label{strongsubadditivityp}
S_{ABC} + S_{B} \leq S_{AB} + S_{BC},
\end{equation}
which is equivalent to \eqref{strongsubadditivitym}. Since $S_{BC} = 0$ (see \eqref{bcpure}), we have $S_{ABC} = S_{A}$. Then from \eqref{BpurifiesA}, we get
\begin{equation}
S_{A} + S_{B} < S_{A} \Rightarrow S_{B} < 0 \, ?
\end{equation}
This clearly contradicts \eqref{bthermal}.
\paragraph{The Resolution\\}
The resolution to this paradox is automatically provided by our construction of the black hole interior in section \ref{sec:behind}:
\begin{quote}
Our construction makes it clear that for the black hole it is incorrect to apply \eqref{strongsubadditivitym} or \eqref{strongsubadditivityp} since $A, B, C$ {\em cannot} be treated as separate subsystems at the level of
accuracy where \eqref{BpurifiesA} is true.
\end{quote}
Note that bit $C$ that falls in, is an excitation of the $\widetilde{\cal O}$ operators. However, as we explained in detail above, the $\widetilde{\cal O}$ operators arise because we coarse-grained our initial Hilbert space.
Now, at the start of black hole evaporation this is a perfectly good description since we can easily accommodate the
outgoing radiation $A$ in our coarse grained Hilbert space.\footnote{We should mention that, in a strict sense, our construction works for a big black hole in AdS. With standard reflecting boundary conditions, this black hole never evaporates. To make it evaporate, we need to couple the boundary
theory to some other system which collects the ``glueballs'' as they form; such a system could mimic a boundary, which absorbs the radiation that reaches it. However, the reader who does not want to think of such a construction should note that the moral --- obtaining more and more fine details
of the system causes the semi-classical description in terms of a spacetime to break down --- is very robust and should carry over to flat space black holes.}
As more and more bits emerge, we need to enlarge our coarse-grained Hilbert space to
describe them.
As we do this, beyond a point, our construction of the $\widetilde{\cal O}$ operators breaks down. In a discrete Hilbert space, this happens when the number of rows in the matrix in \eqref{rectangularmat} (the dimension of ${\cal H}_{\text{coarse}}$) become larger than the number of columns (the dimension of ${\cal H}_{\text{fine}}).$
Indeed, it is precisely at this point --- when the density matrix of the radiation outside is of the same dimensionality as the rest of the system --- that we expect \eqref{BpurifiesA} to start being true.
So, our construction points out that if we wish to make our description of particles outside the black hole so precise that it can keep track of the quantum state of all the particles that have been emitted by an old black hole then, at this level of precision, the semi-classical picture of spacetime is invalid.
This {\em does not} imply a breakdown of effective field theory. In fact an infalling observer using effective field theory would measure the operator that we constructed in \eqref{finalbehind} and its correlators. We can rephrase the ``strong subadditivity'' paradox in terms of correlation functions. However, it is clear that the number of operators in these correlation functions would have to scale with $N$. It is for such correlators, with $O(N)$ insertions that we cannot --- and should not expect to be able to --- use effective field theory.
We should mention that a variant of our ideas was discussed in \cite{Bousso:2012as,Susskind:2012uw}, where it was written in the form $A = R_B$ indicating that the interior of the black-hole was, in some sense, a ``scrambled'' version of the exterior. The authors of \cite{Bousso:2012as,Susskind:2012uw} decided that this led to problems involving ``quantum cloning'' (quantum information appears to be duplicated), and with ``chronology protection''. Our idea is subtly different: when the CFT is obsered at a coarse-grained level, it is possible to write down a semi-classical spacetime that reproduces these observations, with a very specific combination of the fine-grained degrees of freedom playing the part of the interior of the black hole. Both these regions are rewritings of parts of the same CFT Hilbert space. If we insist on a higher level of accuracy, then it is not quantum mechanics that breaks down but rather the {\em interpretation of CFT correlators} in terms of a semi-classical and local spacetime.
\subsubsection{Can small correlations unitarize Hawking radiation?}
Once we have resolved the strong subadditivity paradox, there is still
apparently an information puzzle that makes no reference to the ingoing bit (i.e. to system $C$) : how can the large number of apparently
thermal bits that are emitted by the black hole lead to a pure state. In fact the resolution to this has
been understood for a long time \cite{Page:1993df}. We review this puzzle here and its resolution also. The reader who is already familiar with this
can skip straight to the example in \ref{sec:toy}.
\paragraph{The Paradox\\}
Naively if we assume that the {\em same process} of Hawking radiation repeats $K$ times (where $K$ is the number of bits emitted), then the density matrix of the radiation outside will look like
\[
\rho_K = (\rho_B)^K,
\]
where, by $\rho_B^K$ we really mean
\[
\rho_0^K \equiv \rho \otimes \rho \otimes \rho \ldots {\text{K~times}},
\]
i.e. it is a $2^K \times 2^K$ dimensional matrix.
The entropy of this density matrix is
\begin{equation}
\label{entangleK}
S_{\text{hawk}} = -{\rm Tr}(\rho_K \ln \rho_K) = K \ln 2.
\end{equation}
If we modify the density matrix \eqref{rhosingle} by a small amount --- such corrections would be expected through quantum and stringy effects --- but continue to assume that each Hawking emission is {\em exactly independent} then the entanglement entropy computed above does not change appreciably.
More precisely, if we take each individual density matrix to be
\[
\rho_{\text{str}} = \rho_1 + \epsilon \rho_{\text{corr}},
\]
then
\[
S_{\text{hawk}} - \left[- {\rm Tr}(\rho_{\text{str}}) \ln (\rho_{\text{str}})\right] \sim O(\epsilon).
\]
This conclusion continues to hold even if we allow $\rho_{\text{corr}}$ at each step to be different but {\em uncorrelated}.
The paper \cite{Mathur:2012np} explains that ordinary objects like coal or burning paper avoid this paradox since in those cases successive emissions are not uncorrelated. So, for example, if the computer on which we are typing this were to go up in flames then successive portions of the computer would emit distinct and identifiable thermal emissions. First, the keyboard would burn, and by collecting that radiation, an observer could recover some information about the keyboard, even if the keyboard were only a negligible fraction of the mass of the full computer.
However, this is simply the statement that small subsystems of everyday objects are {\em not} maximally entangled with the rest of the object. As we will show in a toy example below, by introducing very small off-diagonal elements (which, however, may link the first bit emitted to the last bit emitted), we can unitarize Hawking radiation.
\paragraph{The Resolution\\}
Just to be specific, let us consider a 4-dimensional asymptotically flat black hole of Schwarzschild radius $R_{bh}$ and take $l_{\rm p}$ to be the Planck scale. In the lifetime of this black hole, it will emit about $N \approx \left({ R_{\text{bh}} \over l_{\text{p}}}\right)^2$ quanta. Clearly, for the first $K << N$ of these quanta, the correction is unimportant and the density matrix is well approximated by the Hawking matrix.
Let us imagine that the density matrix that is produced at each step is actually given by
\begin{equation}
\label{unitarizingcorr}
\rho_{\text{exact}} = \rho_{\text{hawk}} + 2^{-N} \rho_{\text{corr}},
\end{equation}
where $\rho_{\text{corr}}$ is a density matrix that, in the natural basis of observables, has elements that are $O(1)$. This correction is exponentially suppressed and can have several origins. Moreover, since it is exponentially suppressed in ${R_{\text{bh}} \over l_{\text{p}}}$, it cannot be detected at any order in perturbation theory: this is an inherently non-perturbative correction.
This correction is reminiscent of the ``second saddle point'' discussed in \cite{Maldacena:2001kr}. It would be nice to see if such a semi-classical correction can be identified directly in Lorentzian space. However, even if such a semi-classical perspective does not exist, and this correction is visible in the gauge theory but cannot be interpreted in terms of a simple geometric process, that would not be a contradiction.
Now, it is clear from \eqref{unitarizingcorr} that for the first $K$ emissions, where $K << N$, the density matrix of the radiation is given by the Hawking matrix to an excellent approximation. However, as $K$ grows large and becomes comparable to $N$, we see that the individual elements of the Hawking density matrix become so small, that they are comparable to the size of the corrections. At this point, the corrections can no longer be neglected.
In fact, with the numerical pre-factors that we have inserted in front, the corrections are precisely of the correct order to unitarize the process. Note that after $N$ steps, the Hawking density matrix looks like
\[
\rho_{\text{hawk}} = {1 \over 2^N} I_{2^N \times 2^N},
\]
where $I_{2^N \times 2^N}$ is the identity matrix in $2^N$ dimensions.
For the full density matrix to be unitary, we must have, {\em in some basis},
\[
\rho_{\text{exact}} = \begin{pmatrix}1&0&0&\ldots&0\\
0&0&0&\ldots&0\\
\ldots&\ldots&\ldots&\ldots&\ldots \\
0&0&0&\ldots&0\\
\end{pmatrix}.
\]
So, in this basis---which may be quite unnatural from the point of observables accessible to an observer at low energies---the correction matrix must look like
\begin{equation}
\label{rhocorrunusualbasis}
\rho_{\text{corr}}
= \begin{pmatrix}2^N-1&0&0&\ldots&0\\
0&-1&0&\ldots&0\\
\ldots&\ldots&\ldots&\ldots&\ldots \\
0&0&0&\ldots&-1\\
\end{pmatrix}.
\end{equation}
We now show that this is not inconsistent with the statement below \eqref{unitarizingcorr}--- that $\rho_{\text{corr}}$ has all elements of $O(1)$ in a basis of natural observables.
For \eqref{rhocorrunusualbasis} to hold, we must have
\[
{\rm Tr}(\rho_{\text{corr}}^2) = 2^{2 N}.
\]
However, even if all the elements of $\rho_{\text{corr}}$ in an ordinary basis are $O(1)$, we have
\[
{\rm Tr}(\rho_{\text{corr}}^2)= \sum_{i j} \rho_{\text{corr}}^{i j} \rho_{\text{corr}}^{i j} = \Or[2^{2N}],
\]
since the sum over $i,j$ runs over $2^{2N}$ elements.
\subsubsection{An Example \label{sec:toy}}
Let us now give an example where all the ideas are realized. In particular:
\begin{enumerate}
\item
We will construct a toy model that describes the emission of ``spins''; each emitted spin has an almost exactly thermal density matrix.
\item
We will show how small corrections can unitarize this process so that after more than half the system has ``evaporated'', the density matrix that describes all the emitted spins together starts becoming pure.
\item
We will show how, for each spin that is emitted, we can also identify (within an effective description), a corresponding ``spin'' that remains within the system and is perfectly anti-correlated with the emitted spin. This is the analogue of the bit ``C'' that falls inside the black hole. However, we will show that
this effective description breaks down at {\em precisely} in time to avoid any contradiction with the strong subadditivity of entropy.
\end{enumerate}
Consider a system of $N$ spins, each of which can be in two states--- $|0\rangle$, or $|1 \rangle$. The Hilbert space of the system is spanned by $2^N$ basis vectors, each of which can be represented by a single binary number between $0$ and $2^{N-1}$. We can use this as a convenient representation of our basis
\[
|0 \rangle_{b} \equiv |000\ldots00 \rangle, |1 \rangle_{b} \equiv |000\ldots01\rangle, |2 \rangle_{b} \equiv |000\ldots10\rangle, |3 \rangle_{b} \equiv |000\ldots11\rangle, \ldots,
\]
where we have placed a $b$ in the subscript of the new basis to distinguish it from the old one.
Now, consider the state
\begin{equation}
\label{typicalstate}
|\Psi\rangle = {1 \over 2^{N/2}} \sum_{j=0}^{2^N-1} p_j | j \rangle_b,
\end{equation}
where $p_j$ is a number that can be either $1$ or $-1$. There are $2^{2^N-1}$ such states (since there are $2^N$ coefficients, but states that are the same up to an overall minus sign are equivalent) but let us consider a ``typical state'' in this set --- where we pick the $p_j$ coefficients with an equal probability to be either $1$ or $-1$.
In doing so, we are not considering the ensemble of states formed by the various choices of the $p_j$. We make some choice, and then consider the pure state $|\Psi\rangle$ corresponding to this choice. In speaking of a ``random'' choice above, we are merely pointing out that for our purposes the precise choice of the sequence $p_j$ is not important, and most choices will work.
Now, if we consider the reduced density matrix corresponding to the first spin, we find that we have
\begin{align}
\rho_1 &= {1 \over 2^{N}} \left(\sum_{j=0}^{2^{N-1}-1} p_{2 j}^2 |0 \rangle \langle 0| + p_{2 j+1}^2 |1 \rangle \langle 1| + p_{2 j} p_{2 j +1}\Big(
|0 \rangle \langle 1| +|1 \rangle \langle 0| \Big)\right) \\
&= {1 \over 2} \Bigg(|0 \rangle \langle 0| + |1 \rangle \langle 1| + \Or[2^{-{N\over2}}] \Big(|0 \rangle \langle 1|+|1 \rangle \langle 0| \Big) \Bigg).
\end{align}
Here, in the last step we have used the fact that successive $p_j$ coefficients are uncorrelated. Depending on the precise choice of the $p_j$ coefficients, that we started out with, we will get different values for ${1 \over 2^N} \sum p_{2 j} p_{2 j + 1}$. However, for a typical state, we expect this value to be of the order $2^{-{N \over 2}}$.\footnote{For a very few states, this value can be close to $1$, but these states are not ``typical'' and do not meet our purpose.}
Thus we see that for ``typical states'', the density matrix of the first spin is very close to the density matrix predicted by Hawking. The full density matrix is also of the form \eqref{unitarizingcorr}. However, the corrections become important once we start considering (in this example) ${N \over 2}$ spins.
Thus if the dynamics of this spin chain naturally leads it to one of these typical states, then its evaporation will appear much like Hawking radiation. By a basis change, we can transform $\rho_{\text{corr}}$ into precisely the form \eqref{rhocorrunusualbasis}.
However, this is not sufficient. Indeed, as we pointed out above, a key feature of Hawking radiation is that for each emitted photon, there is also an ``ingoing photon'' that can perhaps be observed by an observer inside the black hole. The two photons are maximally entangled; each of them individually looks thermal but taken together they form a pure state. We referred to this feature in \eqref{pairproduction} above.
From our construction of the $\widetilde{O}$ operators above, we know how to construct the ingoing bit. To make this more precise, let us assume that in a ``coarse-grained'' description we can observe the first $p$ bits of the $N$ qubits in the toy model above. We assume that the other $N-p$ bits are much harder to observe and enter only in the ``fine grained'' description of the system. In the same notation as above, we can write down a basis for both of these spaces
\[
\begin{split}
\text{coarse}: \quad |i\rangle_{\text{c}}, ~ \text{with}~ 0 \leq i \leq 2^p - 1\\
\text{fine}:\quad |j\rangle_{\text{f}}, ~\text{with}~ 0 \leq j \leq 2^{N-p} - 1.
\end{split}
\]
The subscripts $c$ and $f$ emphasize that these are in a different vector space, from the space we used to describe the full system above.
Now the state $|\Psi\rangle$ can clearly be written as
\[
| \Psi \rangle = \sum_{i, j} a_{i j} |i, j \rangle = \sum_i |i \rangle_{\text{c}} \otimes | \phi_i \rangle_{\text{f}},
\]
where $|i, j\rangle \equiv |i\rangle_{\text{c}} \otimes |j\rangle_{\text{f}}$ and
\[
|\phi_i \rangle_{\text{f}} \equiv \sum_j a_{i j} | j \rangle_{\text{f}}.
\]
Let us call $S_1$ the operator acting on the first spin as the Pauli matrix $\sigma_3 = \begin{pmatrix} & 1 & 0 \\ & 0 & -1\end{pmatrix}$.
With this notation, the operator $\widetilde{S}_1$ in this basis can be written as
\begin{equation}
\label{tildes1}
\widetilde{S}_1 = {\cal I}_{\text{c}} \otimes \left( \sum_{i=2^{p-1}}^{2^p - 1} | \phi_i \rangle \langle \phi_i | - \sum_{i = 0}^{2^{p-1} - 1} | \phi_i \rangle \langle \phi_i | \right),
\end{equation}
where ${\cal I}_{\text{c}}$ is the identity on the coarse-grained space.
Note that the definition of $\widetilde{S}_1$ depends not only on the state $| \Psi \rangle$ but also on our division of the space into a coarse-grained and a fine-grained part. This operator $\widetilde{S}_1$ commutes with all operators in the coarse-grained space as is evident from \eqref{tildes1}
\[
[\widetilde{S}_1, S_m] = 0, ~\text{for}~1 \leq m \leq p.
\]
Moreover, in the state $|\Psi \rangle$ measurements of $S_1$ are precisely anti-correlated with measurements of $\widetilde{S}_1$. Similarly, we can define operators $\widetilde{S}_2, \ldots \widetilde{S}_p$ that are anti-correlated with $S_2 \ldots S_p$ respectively, and also commute with all the ordinary operators.
It is also important to recognize where this process breaks down. Once we expand our description of the coarse-grained system to $p = {N \over 2}$, then the construction above is no longer possible; there is simply not ``enough space'' in the fine-grained space.
Before we conclude this section, we would like to emphasize that this toy model holds important lessons for what happens in the real black hole. First it shows us how exponentially suppressed corrections can conspire, in an equally large Hilbert space, to restore unitarity to a seemingly thermal density matrix. Second, this system also gives us a toy model of black hole complementarity. As long as less than half the spins have been measured, it is possible to pair each emitted spin with a perfectly anti-correlated and independent degree of freedom in the remaining spins. This is analogous to the statement that as long as we are measuring a suitably limited number of observables, there is no difficulty in describing the interior of the black hole as a separate space where semi-classical dynamics holds. However, once we start probing the system very finely by going either to very high energies, or with very heavy operators, the semi-classical spacetime description starts to break down. This is precisely what
happens
in the spin model above.
\section{Conclusions and further directions \label{sec:conclusion}}
Let us summarize the implications of our construction. Since the paper is already quite long we will be as brief as possible.
In what follows it is important to keep in mind
that our precise results apply to a black brane or a big black hole\footnote{Although, for notational reasons, we presented the results for a black brane it should be clear that similar local bulk observables
can be defined for a big black hole in global AdS, by replacing the integrals over $\vect{k}$ with sums over spherical harmonics.} in AdS, neither of which evaporate. We leave it to the reader to decide how relevant the following conclusions are for black holes in asymptotically flat space.
\vskip5pt
\noindent {\bf Reconstructing local bulk observables outside the black hole:}
We revisited the construction of local operators near a big AdS black hole. At large $N$ it is straightforward to write CFT operators which reconstruct local bulk fields. We pointed out that a divergence which appears when attempting to write a transfer function in position space, is a harmless artifact --- effectively ``regularized'' by the general behavior of boundary thermal correlators --- and can be evaded by working with the operators in momentum space. In particular, there is no need for an analytic continuation along the spacelike directions of the boundary. The construction of local bulk observables outside the black hole seems to be robust under the inclusion of $1/N$ corrections.
\vskip10pt
\noindent {\bf Reconstructing the region behind the horizon:} We argued that in certain theories, including large $N$ gauge theories, there is a natural splitting of the Hilbert space into coarse- and fine-grained components. Typical pure states have the property that they ``self-thermalize'', i.e. the reduced density matrix of the coarse-grained Hilbert space is automatically driven by the dynamics very close to the thermal one. In such a situation, for every operator acting on the coarse-grained Hilbert space, it is natural to identify a ``partner'' operator acting on the fine-grained Hilbert space. These partner operators obey an identical algebra as the
coarse-grained operators, and can be identified with the ``tilde'' operators in the ``thermofield doubled Hilbert space'' formalism. Using these partner operators we can reconstruct local bulk observables behind the horizon of the black hole. We argue that, despite what one might naively expect, local bulk observables behind the horizon are
essentially supported over finite time scales on the boundary and hence --- in the large $N$ limit --- they are not too sensitive on the specific microstate. This implies that the ``interior geometry'' (as probed by low-energy experiments) looks the same for all pure states, in contradiction with (some versions of) the fuzzball proposal.
\vskip10pt
\noindent {\bf Fate of the infalling observer:} From these results it follows that a semi-classical observer falling cross the horizon of a big black hole, will not measure anything special. In particular he will not see a firewall or a fuzzball. Our precise prediction is the following. If we first fix: i) the size of the black hole in AdS units, ii) the trajectory of the infalling observer and the points at which he will measure the local fields, iii) the number of measurements he can make and the accuracy he has iv) the type (mass) of fields that he can measure {\it and then} send $N$ to infinity (i.e. we do not scale any of the previous quantities with $N$), then the observer will not notice any difference from semi-classical GR. Whether this is consistent or not with the fuzzball proposal depends on its precise definition. However, to the extent that the fuzzball proposal posits a departure from
semi-classical GR at energy scales of the order of the temperature of the black hole, our construction seems to contradict it.
\vskip10pt
\noindent {\bf Lessons about the information paradox:} We argued that, contrary to the claims of \cite{Mathur:2012np} and \cite{Almheiri:2012rt}, {\it small corrections to the Hawking computation can restore
unitarity}. The corrections suggested by our construction are small in the sense that low point correlators of perturbative fields are almost exactly the same as those computed by semi-classical quantum gravity. The intuitive reason that this is consistent with the unitarity of the Hawking process is that the number of emitted Hawking particles is large, so even small correlations between them can carry away the information.
The works of \cite{Mathur:2012np} and \cite{Almheiri:2012rt}, claimed that this is not possible by invoking the strong subadditivity theorem applied to particles located {\it both inside and outside} the black hole.
Our construction of local bulk observables, explicitly demonstrates that the semi-classical Hilbert spaces corresponding to the interior and exterior of the black hole {\it are not independent}
and hence it is not permissible to use the strong subadditivity theorem.
We discussed this in the context of a simple qubit toy model. Our construction suggests a form of ``black hole complementarity'' from the point of view of the boundary conformal field theory, which would be interesting to understand in more detail.
\vskip10pt
\noindent {\bf Future work:} Clearly there are many questions that need to be explored in more detail. An obvious one is to understand how to extend the reconstruction of the bulk in perturbation theory in $1/N$. We would like to further investigate the properties of the tilde operators, the splitting of the Hilbert space into coarse- and fine-grained components and the conditions under which an isolated quantum system can undergo thermalization. We hope to revisit this question in future work.
It would be nice to find a more realistic dynamical toy-model that would capture the essential features of complementarity. Finally, given that we have constructed local bulk observables behind the horizon, it would obviously be interesting to study what happens when the bulk point approaches the black hole singularity.
\section{Various technicalities \label{sec:subtleties}}
In this section, we discuss some subtle points of our construction.
\subsection{Stability of the reconstructed bulk operators/``spread of the transfer function''}
A possible objection to our construction is the following: in Schwarzschild coordinates, an infalling observer never crosses the horizon, since the point of crossing formally has Schwarzschild time $t\rightarrow \infty$. Moreover, the ``time'' of the gauge theory is naturally identified with the Schwarzschild time in the bulk. These statements together may be taken as an indication that in order to reconstruct local operators at and behind the horizon, we need information from the gauge theory {\it for all times}. If true, this could be a serious problem since we expect the late time behavior of pure states to have significant variance due to Poincare recurrences and other kinds of statistical fluctuations, including ${1 \over N}$
corrections that might pile up over time.
The same conclusion seems to follow if one (wrongly) assumes that the support of the operator $\phi_{\rm CFT}(t,\vect{x},z)$ is mostly localized on the intersection of the boundary of AdS with the spacelike cone emanating from the point $(t,\vect{x},z)$. This (wrong) assumption seems to suggest that as the point moves close to the horizon, the support of the operator spreads over the entire boundary and moreover it becomes difficult to understand what happens when the point moves behind the horizon.
Fortunately these arguments are not really correct and we will explain that the reconstructed bulk observables $\phi_{\rm CFT}(t,\vect{x},z)$ only require knowledge of the boundary fields for (approximately) a finite interval in boundary time, which can be made parametrically smaller, in the large $N$ limit, from time scales related to fluctuations and recurrences.
We believe that our momentum space formulation helps us get a sharp sense of the accuracy of our construction. In particular, the question above can be converted to the following questions:
\begin{enumerate}
\item
From a measurement of boundary correlators, with a given accuracy and over a certain time-scale, how accurately can we extract the modes ${\cal O}_{\omega,\vect{k}}$ that we require above.
\item
Can a small inaccuracy in the ${\cal O}_{\omega,\vect{k}}$ modes get blown up to a large inaccuracy in the reconstructed bulk fields $\phi_{\rm CFT}$.
\end{enumerate}
Let us start with the first question. Let us say that we measure the correlator on the boundary for a {\em finite time interval} $[0,T]$. (It is trivial to do a similar analysis for the spatial coordinates, but for simplicity, we focus on time here.) Moreover, the observer samples this correlator at short-time intervals $t_{uv}, 2 t_{u v}, \ldots$ for a total number of samples ${T \over t_{uv}}$. (We will take this to be an integer to lighten the notation.) Using this procedure how accurately can we discern the various frequencies?
Let us say that the original correlator is $C(t) = \int C_{\omega} e^{-i \omega t}$. By doing a discrete Fourier transform using the sample points above, we can measure the frequency modes
\begin{equation}
\label{measuredc}
\hat{C}\left({2 \pi \nu \over T} \right) = {t_{u v} \over T} \sum_{j = 0}^{{T \over t_{u v}} - 1} \int C_{\omega} e^{-i \omega j \over T} e^{2 \pi i \nu j \over T} d \omega = \int C_{\omega} R_{\omega} d \omega
\end{equation}
with
\begin{equation}
R_{\omega} = {t_{uv} \over T} \frac{e^{i T (2 \pi \nu -\omega )} -1}{e^{ i t_{uv} (2 \pi \nu - \omega)}-1 }.
\end{equation}
We remind the reader (see figure \ref{spreadfuncplot}) that if $\omega t << 1$ and $\omega T >> 1$, then $R_{\omega}$ is a sharply peaked function with a maximum of $1$ at $\omega = 2 \pi \nu$.
\begin{figure}
\begin{center}
\includegraphics[height=5cm]{spreadfunc.eps}
\caption{$R_{\omega}$ with $\nu = 1$ \label{spreadfuncplot}}
\end{center}
\end{figure}
On the other hand, if $\omega$ is a very low frequency --- signals at ultra-low frequencies can be produced by phenomena like Poincare recurrence --- then \eqref{measuredc} is negligible except at $\nu = 0$. So these frequencies do not contaminate the measurement of Fourier modes with higher frequencies.
Similarly, \eqref{measuredc} also tells us that if we wish to measure
the Fourier mode to some accuracy ${1 \over N^{\alpha}}$, then we can do
that by measuring the real-time signal to the same accuracy --- we do not
require any parametric enhancement of accuracy at low frequencies.
So we reach the expected conclusion it is possible measure the Fourier modes
of the operator ${\cal O}_{\omega, \vect{k}}$ between some IR-cutoff ${1 \over T}$ and some UV-cutoff ${1 \over t_{uv}}$ to any given accuracy
by sampling position correlators at intervals smaller than $t_{uv}$ and for a length of time larger than $T$, with the same accuracy.
Having now established that the Fourier modes ${\cal O}_{\omega, \vect{k}}$
can be measured in principle, the next question has to do with whether
low frequency modes become very important near the horizon. In order to do that we consider the bulk 2-point function of our local operators $\phi_{\rm CFT}(t,\vect{x},z)$.
\subsubsection{Outside the horizon}
First we start with points outside the horizon. We have
\begin{align*}
&{1 \over Z_{\beta}} {\rm Tr}\Big(e^{-\beta H} \phi_{\rm CFT}(t_1,\vect{x}_1,z_1) \phi_{\rm CFT}(t_2,\vect{x}_2,z_2)\Big) \\
&= (2\pi)^d \int_{\omega>0} {d\omega \over (2 \pi) 2 \omega} {d^{d-1}\vect{k} \over (2 \pi)^{d-1}}\, \begin{aligned}[t] \Bigg[&\hat{f}_{\omega,\vect{k}}(t_1,\vect{x}_1,z_1) \hat{f}^*_{\omega,\vect{k}}(t_2,\vect{x}_2,z_2) {e^{\beta \omega}
\over e^{\beta \omega}-1} \\ & +
\hat{f}^*_{\omega,\vect{k}}(t_1,\vect{x}_1,z_1) \hat{f}_{\omega,\vect{k}}(t_2,\vect{x}_2,z_2) {1
\over e^{\beta \omega}-1} \Bigg]
\end{aligned}
\end{align*}
Here we used the definition of the bulk operators \eqref{upliftb} and the thermal expectation values of the Fourier modes $\hat{\cal O}_{\omega,\vect{k}}$ that we discussed around equation \eqref{thermaloca}. Remember that the hatted modes $\hat{f}_{\omega,\vect{k}}(t,\vect{x},z)$ are those which are canonically normalized with respect to the Klein Gordon norm.
The question about the late-time sensitivity is a question about the region of this integral around $\omega=0$. First we notice that we have an explicit factor of ${1\over \omega}$. Second, the thermal factors ${e^{\beta\omega}\over e^{\beta\omega}-1}$ and ${1\over e^{\beta\omega}-1}$ both go like ${1\over \beta \omega}$ for low $\omega$. Third, the modes have the property that for fixed $(t,\vect{x},z)$ we have that $\hat{f}_{\omega,\vect{k}}(t,\vect{x},z)$ goes to zero linearly with $\omega$ for small $\omega$. This can be verified explicitly in the case of the modes on a BTZ background (see appendix \ref{appendix2dthermal}) but is also true for other black holes. All in all we find that if we keep the points fixed and consider low $\omega$, the integrand goes like $\omega^0$ and hence the integral converges.
Since the integral is convergent in the limit $\omega=0$, it means that the sensitivity of the operator $\phi_{\rm CFT}(t,\vect{x},z)$ to the region of small $\omega$ (or equivalently ``late-time'' in position space) is actually small. In order to demonstrate this let us consider the order of various limits a bit more carefully. Let us consider the reconstructed local bulk operator \eqref{uplifta}, but with the inclusion of an IR cutoff $\delta$ in frequency space
\[
\phi_{\rm CFT}^{\delta}(t,\vect{x},z) \equiv \int_{\omega>\delta}{d\omega d^{d-1}\vect{k} \over (2\pi)^d}\, \left[{\cal O}_{\omega,\vect{k}} \, f_{\omega,\vect{k}}(t,\vect{x},z) + {\cal O}_{\omega,\vect{k}}^\dagger \, f_{\omega,\vect{k}}^*(t,\vect{x},z)\right]
\]
The operator $\phi_{\rm CFT}^{\delta}$ is not exactly the same as our original operator $\phi_{\rm CFT}(t,\vect{x},z) =\phi^{\delta=0}_{\rm CFT}$ defined in \eqref{uplifta}. However, since the integral is convergent in the region of $\omega=0$, the difference between correlation functions of $\phi^\delta_{\rm CFT}$ and correlation functions of $\phi^{\delta=0}_{\rm CFT}$ goes to zero as $\delta\rightarrow 0$.
Coming back to the question about the sensitivity to the details of the pure state: suppose we want to reconstruct the bulk with some ``resolution'' $\epsilon$ (which we take to be $N$-independent). For any given $\epsilon$ there is an IR cutoff $\delta$, such that the correlators of $\phi_{\rm CFT}^\delta(t,\vect{x},z)$ reproduce those of a local bulk field up to the accuracy $\epsilon$, {\it if the boundary correlators agreed with the thermal ones, down to the IR frequency cutoff $\delta$}. But for any given and fixed IR cutoff $\delta$ in frequency space, we can take $N$ to be large enough, so that the boundary correlators on a typical pure state agree with those of the thermal ensemble down to $\omega \approx \delta$.
Putting everything together, we find that for any desired ``resolution $\epsilon$'', it is possible to take $N$ to be large enough and to ensure that the details of the typical pure state become unimportant.
\subsubsection{Inside the horizon}
In order to reach the same conclusion for points behind the horizon, we need to demonstrate that the integral over $\omega$ in the 2-point function for points behind the horizon, is well-behaved around $\omega=0$. Here the situation is more interesting. The reconstructed bulk operator is given by \eqref{finalbehind}, that is
\[
\phi^{\rm II}_{\text{CFT}}(t,\vect{x},z) =
\int_{\omega>0} {d\omega d^{d-1}\vect{k} \over (2 \pi)^d} \left[ {\cal O}_{\omega,\vect{k}}\, g_{\omega,\vect{k}}^{(1)}(t,\vect{x},z) + \widetilde{\cal O}_{\omega,\vect{k}} \,g_{\omega,\vect{k}}^{(2)}(t,\vect{x},z)+ {\rm h.c.}
\right]
\]
When computing the 2-point function of this operator we find several contributions coming from the 2-point functions of the ``usual'' modes ${\cal O}_{\omega,\vect{k}}$, from the 2-point functions of the ``tilde'' modes $\widetilde{\cal O}_{\omega,\vect{k}}$, as well as from cross-terms between the two.
If we focus on the contribution to the 2-point function coming from the terms $\langle {\cal O}_{\omega,\vect{k}}
{\cal O}^\dagger_{\omega',\vect{k}'}\rangle_\beta$ and $\langle {\cal O}_{\omega,\vect{k}}^\dagger {\cal O}_{\omega',\vect{k}'}\rangle_\beta$ we find that the integral over $\omega$ is divergent from the $\omega=0$ region. This is the Fourier-space manifestation of the arguments mentioned in the first two paragraphs of this section. It might seem that this small $\omega$ divergence would invalidate our claims.
However, instead of focusing only on {\it some} of the terms contributing to the 2-point function, if we consider the contribution from all terms together, i.e. including the ``tilde'' modes (and the cross terms) we find that the integral over small $\omega$ becomes convergent! The divergence encountered above when considering the modes ${\cal O}_{\omega,\vect{k}}$ alone, disappears when we consider both ${\cal O}_{\omega,\vect{k}}$ and $\widetilde{\cal O}_{\omega,\vect{k}}$ together!
This means that there is a specific way to reorganize the terms in the integral over $\omega$, so that the integral becomes manifestly convergent for small $\omega$. It turns out that if we group together the modes ${\cal O}_{\omega,\vect{k}}$ with the $\widetilde{\cal O}_{\omega,\vect{k}}$ in the natural ``Kruskal'' combinations
\[
{\cal K}^{(1)}_{\omega, \vect{k}} = {{\cal O}_{\omega,\vect{k}} - e^{-{\beta \omega\over 2}} \widetilde{\cal O}_{\omega,\vect{k}} \over \sqrt{1-e^{-\beta \omega}}}\qquad,\qquad
{\cal K}^{(2)}_{\omega, \vect{k}} = {\widetilde{\cal O}_{\omega,\vect{k}}^\dagger - e^{-{\beta \omega\over 2}}{\cal O}^{\dagger}_{\omega,\vect{k}} \over \sqrt{1-e^{-\beta \omega}}}
\]
then the integral is manifestly convergent for small $\omega$. The Kruskal creation operators are, of course, just given by the Hermitian conjugates
of the relations above.
As a matter of fact, this situation can also be studied in the case of Rindler space, where the modes are easy to write down explicitly. We present the analysis in appendix \ref{rindlerq}. In Rindler space, when computing the 2-point function for points behind the ``Rindler horizon'', we find the same small $\omega$ divergence, when considering the contribution from only one set of modes.
We can check explicitly that the divergence disappears if the modes are grouped together in the ``Unruh modes''. The relation between the Unruh and the Rindler modes given in \eqref{unruh} is precisely the same as the relation between the Kruskal and the AdS-Schwarzschild modes above.
To summarize, for points behind the horizon the integral over small $\omega$ is still convergent, which means that the sensitivity of the bulk 2-point function to the low $\omega$ (or late time) boundary correlators is small. The argument about the order of limits mentioned in the previous subsection can be repeated
in identical form, and we conclude that the reconstructed black hole region II is --- in the large $N$ limit --- insensitive to the details of the specific pure state.
\subsubsection{What happens exactly on the horizon?}
At the horizon, the mode functions do not diverge but they start oscillating very rapidly. We can see this from the formula \eqref{psinearh}. Near the horizon, in terms of the coordinates $U$ and $V$ defined in section \ref{secadseternal}, the modes are simply
\[
\hat{f}_{\omega, \vect{k}} = e^{i \vect{k} \cdot \vect{x}} \left(e^{i \delta_{\omega, \vect{k}}} e^{-2 \pi z_0 \omega \over d} U^{2 i z_0 \omega \over d} + e^{-i \delta_{\omega, \vect{k}}} V^{-2 i z_0 \omega \over d} \right)
\]
Let us say that we have been able to measure boundary modes accurately up to some frequency ${2 d \over z_0 N^{\alpha}}$. ($z_0$ is a natural scale since it also sets the temperature.) Near the future horizon, which is at $U = 0$, modes below
this frequency become important only for
\begin{equation}
\label{breakdownregion}
\ln(U) \sim -N^{\alpha} \Rightarrow U \sim e^{-N^{\alpha}}.
\end{equation}
It is only in this region that our construction starts to develop inaccuracies.
In fact the proper time taken by the infalling observer to cross the region \eqref{breakdownregion} is itself exponentially suppressed in $N$. It is unclear to us whether it is possible even, in principle, to speak of the experience of the infalling observer over such a short time scale. If we {\em average} over the experience of the observer as he crosses the black hole, over some time scale that is larger than this --- even a time scale that is power law suppressed in $N$ and scales like $\Or[{1 \over N^{\alpha}}]$ --- then we see that the tiny frequencies that we have neglected on the boundary do not cause any difficulties.
\subsection{Pure vs thermal states}
A tacit assumption in our analysis in sections \ref{sec:behind} and \ref{sec:outside} was that correlators of the operator $\phi_{\rm CFT}(t,\vect{x},z)$ in a ``typical'' heavy pure state are the same as its correlators in
the thermal ensemble. This is consistent with general intuition from statistical mechanics and it was on this basis that we concluded that an infalling observer will not notice anything special as he falls through the horizon, when the black hole is in a pure state.
The argument from the beginning of section \ref{subsechorizonnat} provides further weight to this expectation. In that section, we argued that if
correlators of light operators factorize on a typical heavy pure state
then these correlators cannot be used to distinguish between various typical pure states $|\Psi \rangle$ in the thermal ensemble. More precisely, our arguments suggest that by measuring these correlators to an accuracy ${1 \over N^{\alpha}}$, we could divide these pure states into $\Or[N^{\alpha}]$ different classes but still not pinpoint which one we are in since there are $\Or[e^{-N}]$ different states in the ensemble.
But, how do we know that factorization holds in a pure state? Even if
factorization holds for thermal correlators --- and we might be able
to make an argument for this by applying the usual power counting
arguments to the Feynman diagram expansion of these correlators in the
Schwinger-Keldysh formalism --- it is not obvious that this carries over to pure states.
However, the argument for this is simple. The key point is that the question of how different an observable $A$ is, on a typical pure state, from the same observable on the thermal ensemble, is a question about the {\it variance} of the observable across the ensemble of all pure states. This can be estimated from computing the {\it thermal expectation value of the square of the observable}.
In other words the variance of $A$ across different pure states can be related to the following quantity {\it which can be computed within the thermal ensemble}
\begin{equation}
\label{variance}
{1 \over Z_{\beta}} {\rm Tr}(e^{-\beta H} A^2) - {1 \over Z_{\beta}^2} {\rm Tr}(e^{-\beta H}A)^2
\end{equation}
Let us apply this measure to the observable
\[
A = {\cal O}_{\omega_1, \vect{k_1}} {\cal O}_{\omega_2, \vect{k_2}} {\cal O}_{\omega_3, \vect{k_3}} {\cal O}_{\omega_4, \vect{k_4}} - {1 \over Z_{\beta}^2} {\rm Tr} \left(e^{-\beta H} {\cal O}_{\omega_1, \vect{k_1}} {\cal O}_{\omega_2, \vect{k_2}} \right) {\rm Tr} \left(e^{-\beta H} {\cal O}_{\omega_3, \vect{k_3}} {\cal O}_{\omega_4, \vect{k_4}} \right)- \ldots,
\]
where $\ldots$ indicate the other
products of two point functions that enter here. Then the fact that
the expectation value of $A$ and its powers vanishes in the thermal ensemble up to ${1 \over N}$ means that no ``appreciable'' class of pure states can have non-vanishing $A$.
One might also wonder about the significance of $\Or[1]$ variance of
the operators given in \eqref{thermaloca} and \eqref{thermalocab}. In fact
this variance has a natural semi-classical interpretation: it just indicates the statistical fluctuations in the thermal gas of particles that
surround the black hole. These fluctuations do not modify the leading order geometry, since they are an $O(1)$ and not $O(N^2)$ effect and thus cannot backreact in the large $N$ limit. Moreover the amount of information
contained in them, as we argued before, is parametrically smaller than that of the entropy of the black hole itself.
\subsection{Definition of tilde operators}
As the reader will have noticed, the construction of the $\widetilde{\cal O}$ operators in section \ref{sec:behind} suggests that the precise Heisenberg operators $\widetilde{\cal O}$ that an observer will encounter may differ quite significantly depending on which pure state the black hole is in. The important point is that this non-uniqueness is unimportant from an operational point of view. The bulk observer lives in a particular black hole microstate. By measuring correlators that involve a finite number of bulk fields $\phi$, the observer can only infer that the $\widetilde{\cal O}$ operators {\em effectively} satisfy the same algebra as the ${\cal O}$ operators. The reader may wish to consult the toy model in section \ref{sec:toy} for an explicit construction of the $\widetilde{\cal O}$ operators, where this dependence on the state and its operational insignificance can both be seen.
The $\widetilde{\cal O}$ operators also depend on the specific division of the CFT Hilbert space into a coarse and a fine grained part. Once again, this dependence is not operationally significant. This property can also be seen in the toy model of section \ref{sec:toy}.
\subsection{Do our operators describe the ``real'' infalling observer?}
A question that we have commonly encountered is: ``how do we know that this description corresponds to the `real' experience of the infalling observer.''
Before we address this, let us briefly emphasize a philosophical point, which is uncontroversial and even seemingly banal. Let us say that we are given a quantum system ${\cal Q}$ and that in some approximation the accessible observables in this system can be re-organized into observables $\phi^{(i)}(\vect{x})$, where ${\vect{x}}$ are points on some manifold ${\cal M}$ and, and $i$ is some index that labels the operator in question. Moreover if the correlators of these operators are the same as the correlators of perturbative fields propagating on ${\cal M}$, then the system ${\cal Q}$ is indistinguishable from the system describing perturbative fields on ${\cal M}$.
In this paper, we have shown that the natural observables of a CFT with large $N$ factorization can be reorganized, at leading order, into the correlators of a non-local CFT operator $\phi_{\text{CFT}}(t, \vect{x}, z)$, which is labeled by points in AdS (or an AdS black hole). All the accessible dynamical processes of the CFT can be given a description in terms of these perturbative fields propagating on AdS (black hole).
Ultimately, ontological questions cannot really be settled by physics. However, we hold that such a situation is indistinguishable from a ``true'' perturbative field propagating in this spacetime. So our operators do describe the ``real'' infalling observer.
\subsection{Uniqueness of our construction and interactions}
We now turn to the issue of the uniqueness of our construction, which is something that we glossed over above. In fact, even at \eqref{finalpoincare}, the reader could have asked: ``what impels us to multiply the creation and annihilation operators with the modes inside anti-de Sitter space. Why can't we choose modes from some other spacetime.'' Of course, the AdS/CFT correspondence tells us that any other spacetime will not work. Already at this level, we see that if we would like the bulk theory to realize the symmetries of the boundary in a natural way (as isometries), then we should choose modes from anti-de Sitter space.
Furthermore, we believe that if we choose a different spacetime then we will not be able to correct our prescription at subleading orders in ${1 \over N}$ consistent with locality. We can see the difficulty immediately. While writing down \eqref{finalpoincare}, we emphasized that one reason it worked was because we did not have to worry about the ``spacelike'' modes ${\cal O}_{\omega,\vect{k}}$ at leading order in perturbation theory. However, if we go even to ${1 \over N}$, we cannot consistently neglect these modes and, in fact, their presence will lead to a conflict with locality as was explored in \cite{Kabat:2011rz}. Let us briefly describe how this problem can be fixed and how we can extend our construction to higher orders in ${1 \over N}$.
Our argument is somewhat indirect. It is believed that if we take a consistent interacting CFT with various conditions on its operator spectrum (such as the presence of only a small number of operators at low dimensions) then we should be able to write down a bulk interaction in anti-de Sitter space that reproduces any set of boundary correlators \cite{Heemskerk:2009pn, Heemskerk:2010ty,
Fitzpatrick:2010zm,Fitzpatrick:2012cg,ElShowk:2011ag}. Of course, implementing this procedure in practice is quite difficult but schematically let us say that we have constructed the bulk Lagrangian that reproduces the boundary correlators. For the operator ${\cal O}$ under consideration, let us say that it is of the form:
\[
L_{\text{bulk}} = \int \sqrt{-g} \left[\partial_{\mu} \phi \partial^{\mu} \phi - V(\phi) \right],
\]
where $V(\phi)$ has a perturbative expansion in powers of ${1 \over N}$.
Then, as was discussed in \cite{Kabat:2011rz,Kabat:2012hp,Heemskerk:2012mn} it is possible to correct our prescriptions above perturbatively in the interaction. We now simply start solving the Heisenberg equations of motion:
\[
\Box \phi = V(\phi).
\]
with the zeroth order solution $\phi_0$ taken to be \eqref{finalpoincare} or, in the black hole background, to be \eqref{finalbehind}.
So, with the bulk Green function $G(x,y)$ where $x,y$ are bulk points, at first order we have:
\[
\phi_1(y) = \int G(x,y) V(\phi_0) d y,
\]
where, of course, at this order only the lowest order terms in ${1 \over N}$ in $V(\phi_0)$ contribute. We can, of course, keep track of the higher order terms
and use the solution $\phi_1$ to repeat this procedure at any order in perturbation theory. The Heisenberg field that we obtain in this way is manifestly local.
The point, of course, is that if we are not in AdS we cannot write down a bulk interaction consistent with boundary correlators and so we do not have an algorithm for extending our construction to higher orders in ${1 \over N}$. This is, of course, not a proof but is highly suggestive. We would like to explore this further in future work.\footnote{In fact the example of conformal gravity \cite{Maldacena:2011mk} is already subtle. In this example, it was shown that it is possible to reproduce correlators in AdS$_4$/CFT$_3$ with ordinary Hilbert-Einstein gravity in the bulk by consider another theory on flat space (cut off in one direction at $z = 0$) but with the Lagrangian of conformal gravity. We expect that this is only a tree-level coincidence and cannot be extended to higher orders in ${1 \over N}$}.
|
train/arxiv
|
BkiUdU45qoaAwdii6Tun
| 5 | 1 |
\section{Introduction}
Fractional calculus has turned out to be a powerful analytical tool in various disciplines: It has been recognized that
especially in more recently emerging fields dealing with complex, chaotic, turbulent, critical, fractal and anomalous transport phenomena, problems can appropriately be described by equations
which involve fractional operators. A broad overview on applications of the fractional approach can be found in the review articles of Metzler and Klafter \cite{metzler,metzler2014}.
There exist various definitions (Riemann, Liouville, Caputo, Gr\"unwald-Letnikow, Marchaud, Weyl, Riesz, Feller, and others) for fractional derivatives and integrals, e.g. \cite{hilfer-2008,metzler,samko,samko2003,podlubny} among many others. This diversity of definitions is due to the fact that fractional operators take different kernel representations in different function spaces which is a consequence of the non-local character of fractional kernels.
To apply fractional calculus to a physical problem, it must be carefully analyzed which fractional operator has to be chosen.
Often discrete models which yield in a continuum limit the correct fractional operators are most helpful.
As an example for such a procedure may serve our recently developed discrete self-similar spring model leading to dispersion relations and
to discrete self-similar Laplacian operators in the form of Weierstrass-Mandelbrot fractal functions \cite{michel}. A continuum limit of this discrete self-similar Weierstrass-Mandelbrot type Laplacian operator can be defined which yields the 1D infinite space {\it fractional Laplacian} (Riesz fractional derivative)\footnote{We use in this paper synonymously the terms ``fractional Laplacian" and ``Riesz fractional derivative".} \cite{michel14,michel-fcaa}.
Especially phenomena of anomalous diffusion are described by diffusion equations where instead of the conventional Laplacian $\Delta$
the fractional Laplacian $-(-\Delta)^{\frac{\alpha}{2}}$ occurs, leading to L\'evy $\alpha$-stable distribution-solutions. L\'evy $\alpha$-stable distributions were introduced by Paul L\'evy as a generalization of gaussian distributions \cite{levy}.
The associated microscopic random motions to L\'evy distributions are the so called L\'evy flights. L\'evy flights are the natural generalization of Brownian motion, the latter being described by gaussian distributions. L\'evy distributions contain as special case gaussian distributions and Brownian motion is a special case of L\'evy flight, and the (nonlocal) fractional Laplacian contains the (local) conventional Laplacian as a special case. The role of the conventional Laplacian as the
generator for gaussian distributions is the same as the role of the fractional Laplacian as the generator of $\alpha$-stable L\'evy distributions.
So we always may ask whether a "gaussian phenomenon" has a "L\'evy phenomenon" generalization, governed by equations where the Laplacian is generalized by the fractional Laplacian. Indeed Laskin demonstrated in a series of seminal papers
\cite{Laskin2000,Laskin,Laskin2010} (and see also the references therein)
that a fractional generalization of Quantum Mechanics and Statistical Mechanics can be introduced\footnote{for which Laskin coined the notions fractional Quantum Mechanics and fractional Statistical Mechanics, respectively.}: Fractional Quantum Mechanics is based on" Laskin path integrals" on L\'evy flyer paths which generalize the Feynman path integrals taken over gaussian paths. As a result the Laplacian of the kinetic part in the Schr\"odinger equation occurs in a fractionally generalized manner in the form of fractional Laplacian. It needs to be emphasized that this is not a simple replacement of operators, but well based on the statistical deduction by means of path integrals.
Last but not least the fractional approach seems to be the appropriate language to capture especially fractal aspects of phenomena. An interesting application to problems in turbulence is presented in the paper of Chen
\cite{chen-fract-turb}. Further due to the nonlocal characteristics of fractional operators, it
has turned out to be appropriate to capture certain kind of nonlocal material behavior \cite{Carpinteri11,Challamel13},
when the elastic interaction is scale free \cite{michelc}.
Ortigueira developed in a seminal paper a discrete fractional approach and demonstrated its interlink to the
continuous Riesz fractional kernels \cite{riesz2}: Starting ad-hoc with a fractional centered differences formulation, Ortigueira developed a fractional generalization of Cauchy's integral formula for complex functions, leading in the continuum limit to the well known standard fractional Laplacian and
their inverse operator kernels, the Riesz potentials, on the 1D infinite space.
In the present paper we deduce a discrete fractional model from a harmonic potential
defined on the periodic chain of arbitrary particle numbers $N$.
Our model recovers in the infinite chain limit $N\rightarrow \infty$ the Ortigueira model of \cite{riesz2},
and leads to the continuum limit kernels of the fractional Laplacian (Riesz fractional derivative)
for the infinite and finite $L$-periodic strings, respectively.
Zoia et al also analyze some aspects of the infinite chain model, discussing applications
to L\'evi flights \cite{Zoia2007}, however, without directly analyzing continuum limits of the discrete fractional approach.
In the present paper our goal is the development of a fractional generalization of classical harmonic Montroll-Potts lattice
models for periodic especially finite 1D lattices (chains) and their continuum limits. The classical harmonic approximations of lattices (e.g.
\cite{Born54},\cite{Maradudin63}) describe harmonic inter-particle interactions by inter-particle springs
connecting close neighbor particles resulting in Born-von-Karman models involving the discrete second-order centered differences operator (Born-von-Karman Laplacian).
In the fractional approach to be developed in the present paper,
instead of the classical discrete Born-von-Karman Laplacian, its fractional generalization occurs
in the form of a power law operator function of the Born-von-Karman Laplacian.
The present paper is organized as follows.
We deduce from ``fractional harmonic lattice potentials" on the cyclically closed linear chain a discrete fractional Laplacian matrix.
We do so by applying our recent approach to generate
nonlocal lattice models by matrix functions where the generator operator is the discrete centered Born von
Karman Laplacian \cite{michel-collet}. First we obtain the discrete fractional Laplacian
in explicit form for the infinite chain for particle numbers $N\rightarrow \infty$,
being in accordance with the fractional centered difference models of Ortiguiera \cite{riesz2}
and Zoia et al. \cite{Zoia2007}.
Utilizing the discrete infinite chain fractional Laplacian matrix we construct an
explicit representation for the {\it fractional Laplacian matrix on the $N$-periodic finite chain} where the particle number
$N$ can be arbitrary not necessarily large. To the best of our knowledge the fractional Laplacian matrix
on the finite $N$-periodic chain, so far has not been reported in the literature.
Then we analyse continuum limits of the discrete fractional model:
The infinite space continuum limit of the fractional Laplacian matrix yields the well known infinite space kernel
of the standard fractional Laplacian. The periodic string continuum
limit yields an explicit representation for the kernel of the fractional Laplacian (Riesz fractional derivative) which fulfills periodic boundary conditions and is defined on the finite $L$-periodic string.
The periodic string fractional Laplacian is obtained by convolution of the infinite space kernel with the periodic unity projection operator ($L$-periodic Dirac's $\delta$-function). The $L$-periodic string fractional Laplacian kernel represents the periodic string continuum limit expression of the $N$-periodic finite chain fractional Laplacian matrix.
To the best of our knowledge the fractional Laplacian on the $L$-periodic string, as developed in this paper so far is not
given in the literature.
\section{Preliminaries}
We consider a periodic, cyclically closed linear chain with equidistant lattice points $p=0,..,N-1$ consisting of $N$ identical particles. Each particle has the same mass $\mu$ and each mass point $p$ has equilibrium position at $0\leq x_p=ph\leq L$ ($p=0,..,N-1$) where $L$ denotes the length of the chain and $h$ the interparticle distance (lattice constant). Let us denote by $u_p=u(x_p)$ ($0 \leq x_p=ph < L=Nh$) the displacement field of particle $p$.
Further we impose periodicity (cyclic closure of the chain)
\begin{equation}
\label{period}
u_p=u_{p+N} ,\hspace{2cm} u(x_p)= u(x_p+L)
\end{equation}
for the displacement field for which we use the equivalent notations $u_p=u(x_p)$. We utilize for the cyclically closed chain the cyclic index convention, namely $p \rightarrow p \,\,\, mod\, (N) \in \{0,1,..,N-1\}$. We can imagine the cyclic chain as a closed ring of $N$ identical particles
without boundary.
The starting point of our approach is to propose
harmonic potentials which lead by Hamilton's variational principle to discrete {\it fractional Laplacian operators} being {\it power law matrix functions} of the discrete centered second difference operator (discrete Born von Karman Laplacian) as generator. We refer these potentials to as fractional harmonic potentials.
Following our recently proposed general approach to generate nonlocal constitutive behavior by matrix functions \cite{michel-collet}, we can write any elastic potential on the 1D cyclic chain in the harmonic approximation in the following compact form \cite{michel-collet}
\begin{equation}
\label{compfo}
V_f = \frac{\mu}{2}\sum_{p=0}^{N-1}u_p^*f(2{\hat 1}-D-D^{\dagger})u_p = -\frac{1}{2} \sum_{p=0}^{N-1}\sum_{q=0}^{N-1} u_q^*
\Delta_f(|p-q|)u_p
\end{equation}
where $\Delta_f(|p-q|)$ indicates the (negative-semidefinite) Laplacian $N\times N$-matrix (discrete Laplacian operator) of the problem which fulfills also the periodicity condition $\Delta_f(|p-q|)=\Delta_f(|p-q+N|)$.
We have introduced in (\ref{compfo}) the shift operator operator $D(\tau)$
which is defined by its action $D(\tau)u(x)= u(x+\tau)$. When we utilize in this paper the notation $D$ by skipping the argument, we mean the right hand sided next-neighbor particle shift operator $Du_p=u_{p+1}$, i.e. we utilize synonymously $D=:D(h)$
and its adjoint operator $D^{\dagger}=:D(-h)$ which indicates the left hand sided next-neighbor shift operator $D^{\dagger}u_p=u_{p-1}$.
The operators
$D=D(h)$ and $D^{\dagger}=D(-h)$ are adjoint (hermitian conjugate) to each other, where shift operators are unitary which is expressed by $DD^{\dagger} =D^{\dagger}D={\hat 1}$ with unity operator $D(0)={\hat 1}$. Shift operators on the cyclically
closed chain are $N$-periodic including the unity operator\footnote{
${\hat 1}=D(0)=D^{Nn}=D(nNh)=D(nL)$ ($n \in {\bf \Z}_0$)}.
Further we introduced in (\ref{compfo}) the ``characteristic function" $f$ which is specified in below equation (\ref{charfu}). We utilize the equivalent notations
\begin{equation}
\label{shift}
Du_p = u_{p+1} , \hspace{0.5cm} D^{\dagger}u_p = u_{p-1} ,\hspace{0.5cm} D(h)u(x_p)=u(x_p+h)=u(x_{p+1}), \hspace{0.5cm} D(-h)u(x_p)=u(x_p-h)=u(x_{p-1})
\end{equation}
with the matrix elements\footnote{We utilize often the more simple notation $2-D-D^{\dagger}$ synonymously for
$2{\hat 1}-D(h)-D(-h)$.}
\begin{equation}
\label{matrixrep}
D_{pq} =\delta_{p+1,q} , \hspace{0.5cm} D^{\dagger}_{pq} =\delta_{p-1,q} ,
\hspace{0.5cm} [2{\hat 1}-D-D^{\dagger}]_{pq}= 2\delta_{pq}-\delta_{p+1,q}-\delta_{p-1,q}
\end{equation}
where cyclic index convention is always assumed including for the Kronecker symbol $\delta_{ij}={\hat 1}_{ij}$.
Assuming that $u(x_p)$ is sufficiently smooth we can put $D(\pm h)=\exp{(\pm h\frac{d}{dx})}$. The central symmetric second difference operator (discrete Born von Karman Laplacian) then is $D(h)+D(-h) -2 = (D(\frac{h}{2})-D(-\frac{h}{2}))^2 = 4\sinh^2{\frac{h}{2}\frac{d}{dx}}$ which is a useful representation especially for the determination of
the eigenvalues (dispersion relation).
The function $f$ which we introduced in (\ref{compfo}) as operator function ($N\times N$-matrix function), we refer to as {\it characteristic function}\footnote{The characteristic function itself is defined as a scalar function.}. $\mu f$ contains the entire constitutive information (the material constants) of the harmonic system. The characteristic function has to fulfill the following physically necessary properties \cite{michel-collet}
\begin{equation}
\label{charfu}
f(\lambda) >0 ,\hspace{1cm} 0<\lambda \leq 4 ,\hspace{1cm} f(\lambda=0)=0
\end{equation}
The positiveness for $0<\lambda \leq 4$ is equivalent to elastic stability of the chain, and $f(\lambda=0)=0$ reflects translational invariance (zero elastic energy for uniform translations).
The characteristic function has the dimension $[f] = sec^{-2}$ and the {\it dispersion relation} of the harmonic system (\ref{compfo}) is simply determined by \cite{michel-collet}
\begin{equation}
\label{disprel}
\omega_f^2(\kappa_l) = f(\lambda = 4 \sin^2{\frac{\kappa_l}{2}})
\end{equation}
where $-\pi \leq \kappa_l=\frac{2\pi}{N}l \leq \pi$ ($-\frac{N}{2} \leq l \leq \frac{N}{2}$, $l\in {\bf Z}_0$) denote the $N$ distinct {\it non-dimensional} wave numbers within the first Brioullin zone. It follows from (\ref{charfu}) and (\ref{disprel}) that the characteristic matrix function $f(2-D-D^{\dagger})$ is a positive (semi-) definite $N\times N$ and self-adjoint (symmetric) Toeplitz type matrix, i.e. of the form $f_{pq}= f(|p-q|)$.
The matrix elements fulfill periodicity and reflection-symmetry with respect to $\frac{N}{2}$, namely\footnote{This symmetry is easily seen by putting $p=\frac{N}{2}+\chi$ and by accounting for
$f(|\frac{N}{2}+\chi|)=f(|-\chi-\frac{N}{2}+N|)=f(|\frac{N}{2}-\chi|)$.}
\begin{equation}
\label{percharfu}
\begin{array}{l}
f(|p|) = f(|p+nN|) ,\hspace{1cm} n \in {\bf \Z}_0 ,\hspace{1cm} 0\leq p \leq N-1\nonumber \\ \nonumber \\
f(|\frac{N}{2}+\chi| = f(|\frac{N}{2}-\chi|) = f(|nN+\frac{N}{2}-\chi|) ,\hspace{1cm} \frac{N}{2}\pm \chi \in {\bf \Z}_0 , \hspace{0.5cm} n\in {\bf \Z}_0
\end{array}
\end{equation}
Due to periodicity, the points of reflection-symmetry repeat periodically being
located at $p_n=\frac{N}{2} +nN$ ($n \in {\bf Z}_0$). Corresponding symmetries to (\ref{percharfu}) also exist
for the dispersion relation (\ref{disprel}) in the reciprocal $\kappa$-space.
\newline\newline
In order to analyze continuum limits (long-wave limits) subsequently, it is convenient to introduce the dimensional wave number $k_l=\frac{\kappa_l}{h}=\frac{2\pi}{L}l$ ($L=Nh$)
having dimension $cm^{-1}$.
Dispersion relation (\ref{disprel}) gives the $N$ eigenvalues of the $N\times N$ characteristic matrix function $f(2-D-D^{\dagger})$ acting in the $N$-dimensional ($N$-periodic) space of particle displacement vectors ${\bf u} = (u_p)$ on the chain.
The dispersion relation (\ref{disprel}) is obtained by
\begin{equation}
\label{detrel}
f(2-D-D^{\dagger}) \, \frac{e^{i\kappa_l p}}{\sqrt{N}} = f(4 \sin^2{\frac{\kappa_l}{2}}) \,\frac{e^{i\kappa_l p}}{\sqrt N}
\end{equation}
where $4 \sin^2{\frac{\kappa_l}{2}}$ is the eigenvalue determined by $(2-D-D^{\dagger})e^{i\kappa_l p}=4\sin^2{\frac{\kappa_l}{2}} \, e^{i\kappa_l p} \, $.
Further we introduce the generalized $N\times N$ {\it Laplacian matrix}
\begin{equation}
\label{laplamat}
\Delta_f = -\mu f(2-D-D^{\dagger}) ,\hspace{1cm} (\Delta_f)_{pq}= -\frac{\partial^2}{\partial u_p\partial u_q}V_f
\end{equation}
We utilize frequently in this paper synonymously the terms $\Gamma$-function and generalized factorial function defined by \cite{abramo}
\begin{equation}
\label{gammafu}
\beta !=: \Gamma(\beta+1) = \int_0^{\infty}\tau^{\beta} e^{-\tau}{\rm d}\tau , \hspace{0.5cm} \Re (\beta)>-1
\end{equation}
If $\beta$ is an integer, (\ref{gammafu}) recovers the usual definition of the factorial. Integral (\ref{gammafu})
exists for complex $\beta$ with\footnote{$\Re(..)$ indicates the real part of a complex number $(..)$.} $\Re(\beta) >-1$ where in this paper we deal only with real $\beta$. The main definition of the $\Gamma$-function (\ref{gammafu}) can be extended (analytically continued) to any complex arguments including $\Re (\beta+1) < 0$ (except arguments of negative integers and zero) by the recursion
\begin{equation}
\label{recursiongamma}
\Gamma(\beta-n+1) = (\beta-n) ! = \beta ! \prod_{s=0}^{n-1}\frac{1}{(\beta-s)}
\end{equation}
This recursion defines the analytical continuation of the $\Gamma$-function and defines it for any complex arguments except
negative integers and zero where the analytically continued $\Gamma$-function has singularities.
Further we employ the so-called {\it ceiling}-function $ceil(\xi)$ which may be defined by \cite{abramo}
\begin{equation}
\label{ceiling}
ceil(\xi)=: min(n\in {\bf \Z}_0| n\geq \xi)
\end{equation}
indicating the smallest integer greater or equal to $\xi$, and by ${\bf \Z}_0$ we denote the complete set of integers (positive, negative, and zero).
\section{Fractional discrete chain model}
In the framework of this simple matrix function approach defined on the cyclically closed (periodic) linear chain,
our goal is now, starting with a {\it characteristic function of power law form}, to deduce the
fractional discrete Laplacian matrices for the infinite and finite $N$-periodic linear chain, respectively.
We assume the characteristic function (\ref{charfu}) in power law form
\begin{equation}
\label{powerlaw}
f^{(\alpha)}(\lambda) = \Omega_{\alpha}^2 \,\lambda^{\frac{\alpha}{2}} , \hspace{1cm} \alpha >0
\end{equation}
where $\frac{\alpha}{2}$ denotes a positive, real (non-integer or integer) scaling exponent where we focus especially on the {\it fractional} cases where $\frac{\alpha}{2}$ is non-integer. Note that positiveness of $\alpha$ is a consequence of the requirement that uniform translations of the chain do not contribute to the elastic energy. As a consequence (see relation (\ref{charfu})), the trivial value
$\alpha=0$ physically is a forbidden case \cite{michel-collet}.
The positiveness of $\alpha$ guarantees that the problem has physically ''good" properties.
$\Omega_{\alpha}^2$ is a positive dimensional constant where $\Omega_{\alpha}$ has physical
dimension of a frequency $sec^{-1}$ so that the characteristic function (\ref{powerlaw}) has dimension $sec^{-2}$.
The fractional elastic potential can then with (\ref{compfo}) and (\ref{powerlaw}) be written as
\begin{equation}
\label{Valpha}
V_{\alpha} = \frac{\mu\Omega_{\alpha}^2}{2} \sum_{p=0}^{N-1}u_p^*(2-D-D^{\dagger})^{\frac{\alpha}{2}}u_p =: \frac{\mu}{2} \sum_{p=0}^{N-1}\sum_{q=0}^{N-1}u_q^*f^{(\alpha)}(|p-q|)u_p
\end{equation}
with the matrix elements $f^{(\alpha)}(|p-q|) =[(2-D-D^{\dagger})^{\frac{\alpha}{2}}]_{|p-q|}$
of the {\it fractional characteristic matrix function} which are to be determined in explicit form,
first for the infinite chain ($N\rightarrow \infty$), and then for the finite $N$-periodic chain for
arbitrary not necessarily large $N$.
First from relation
(\ref{disprel}) we obtain the dispersion relation
\begin{equation}
\label{disprelat}
\omega_{\alpha}^2(\kappa_l) = f^{(\alpha)}\left(\lambda=4\sin^2{(\frac{\kappa_l}{2})}\right) = \Omega_{\alpha}^2\,2^{\alpha}\,|\sin{(\frac{\kappa_l}{2}})|^{\alpha} ,\hspace{0.5cm} \kappa_l=\frac{2\pi}{N}l , \hspace{0.15cm} 0 \leq l \leq N-1
\end{equation}
with the only zero value for $l=0$ reflecting translational invariance of (\ref{Valpha}), and
$N-1$ positive values for $1\leq l \leq N-1$. The case $\alpha=2$ corresponds to the
classical Born von Karman chain with next neighbor particle springs, where (\ref{disprelat})
then recovers the well known classical dispersion of $\omega_{\alpha}^2(\kappa_l) =4\Omega_2^2\sin^2{(\frac{\kappa_l}{2}})$.
\newpage
\begin{figure*}[H]
\hskip3.5cm
\includegraphics[scale=0.5]{DispersionFract1.pdf}
\caption{$\frac{\omega_{\alpha}(\kappa)}{\omega_0} =0.5 \times 2^{\frac{\alpha}{2}}|\sin{(\frac{\kappa}{2}})|^{\frac{\alpha}{2}}$ (normalized with $\omega_0=2\Omega_{\alpha}$) of (\ref{disprelat}).}
\label{fig:1}
\end{figure*}
{\bf Figure 1}
\newline
{\it In figure 1 the non-dimensional $\frac{\omega_{\alpha}(\kappa)}{\omega_0} = 0.5\times 2^{\frac{\alpha}{2}}|\sin{(\frac{\kappa}{2}})|^{\frac{\alpha}{2}}$ (normalized with $\omega_0=2\Omega_{\alpha}$) of (\ref{disprelat}) is plotted over $0\leq \kappa \leq \pi$
and $0\leq \alpha \leq 2$ (plot on the left) and further for $2\leq \alpha \leq 4$ (plot on the right).
One can see that for small $\alpha\rightarrow 0$ the eigenfrequencies approach the value $\frac{\omega_{\alpha}(\kappa)}{\omega_0}\rightarrow 1 \times 0.5 $. When $\alpha$ increases, $\frac{\omega_{\alpha}(\kappa)}{\omega_0}$ becomes more and more narrowly distributed around $\kappa=\pi$ taking there its maximum value $0.5\times 2^{\frac{\alpha}{2}}$. For $\alpha=2$ one can see in these plots that the classical Born von Karman dispersion relation $\frac{\omega_{2}(\kappa)}{\omega_0}
= |\sin{(\frac{\kappa_l}{2}})|$ is recovered where the normalization is chosen such that maximum value $1$ is taken in the classical Born von Karman case $\frac{\alpha}{2}=1$.}
\newline\newline
The positive semi-definite $N\times N$ {\it fractional characteristic matrix function} $f^{(\alpha)}$ can be written as
\begin{equation}
\label{alphacarfou}
f^{(\alpha)}(2-D-D^{\dagger}) = \Omega_{\alpha}^2\, (2-D(h)-D(-h))^{\frac{\alpha}{2}}= \Omega_{\alpha}^2\, (-1)^{\frac{\alpha}{2}}\left\{D(\frac{h}{2})-D(-\frac{h}{2})\right\}^{\alpha}= \Omega_{\alpha}^2\left(-4\sinh^2{\frac{h}{2}\frac{d}{dx}}\right)^{\frac{\alpha}{2}}
\end{equation}
The $N\times N$ {\it fractional Laplacian matrix} is then by using (\ref{laplamat}) given by
\begin{equation}
\label{laplacianalpha}
\begin{array}{l}
\displaystyle \Delta_{\alpha} = -\mu\Omega_{\alpha}^2(2-D(h)-D(-h))^{\frac{\alpha}{2}} = -\mu f^{(\alpha)}(2-D-D^{\dagger})\nonumber \\ \nonumber \\
\displaystyle \Delta_{\alpha}(|p-q|) = \Delta(|x_p-x_q|) = -\frac{\mu}{N} \Omega_{\alpha}^2\sum_{l=0}^{N-1} e^{i\kappa_l(p-q)} \left(4\sin^2{(\frac{\kappa_l}{2}})\right)^{\frac{\alpha}{2}}
\end{array}
\end{equation}
where $x_p-x_q= h(p-q)$ and $k_lx_p=\kappa_l \,p$. Note that (\ref{laplacianalpha}) depends symmetrically on $D+D^{\dagger}$ which makes the fractional Laplacian matrix self-adjoint.
Hence it should exist a representation for (\ref{alphacarfou}) in the form
\begin{equation}
\label{charfualpha}
\begin{array}{l}
\displaystyle \Omega_{\alpha}^2(2-D(h)-D(-h))^{\frac{\alpha}{2}} = \sum_{p=-\infty}^{\infty}f^{(\alpha)}(|p|)\, D(hp)
= f^{(\alpha)}(0)+\sum_{p=1}^{\infty}f^{(\alpha)}(|p|)\left\{D(ph)+D(-ph)\right\} \nonumber \\ \nonumber \\
\displaystyle \Omega_{\alpha}^2(2-D-D^{\dagger})^{\frac{\alpha}{2}} u_n = f^{(\alpha)}(0)u_n+\sum_{p=1}^{\infty}f^{(\alpha)}(|p|)\left\{u_{n-p}+u_{n+p}\right\}
\end{array}
\end{equation}
where this series is written for the infinite chain $N\rightarrow \infty$.
First of all we notice that
for integers $\frac{\alpha}{2}=m \in {\bf \N}$ we obtain the matrix elements directly from the binomial expansion
\begin{equation}
\label{binomi}
\begin{array}{l}
\displaystyle f^{(2m)} =\Omega_{2m}^2\left\{-(D(\frac{h}{2})-D(-\frac{h}{2}))^2\right\}^{m} \nonumber \\ \nonumber \\ \displaystyle \hspace{1cm} =\Omega_{2m}^2(-1)^m\left(D(\frac{h}{2})-D(-\frac{h}{2})\right)^{2m} = \Omega_{2m}^2\sum_{p=-m}^{m} (-1)^{p} \frac{(2m)!}{(m+p)!(m-p)!}D(ph)
\end{array}
\end{equation}
which gives the matrix elements in terms of (centered) binomial coefficients
\begin{equation}
\label{binomialco}
f^{(2m)}(|p|) = \Omega_{2m}^2\, (-1)^{p} \frac{(2m)!}{(m+p)!(m-p)!}
\end{equation}
which are non-zero only for $|p|\leq m$.
Let us now determine explicitly the matrix elements $f^{(\alpha)}(|p|) $ for any especially also {\it non-integer} $\frac{\alpha}{2} >0$ first for the infinite, and then for $N$-periodic finite chain. With ''infinite chain" we refer to as the limiting case that the particle number $N \rightarrow \infty$ (where the length $L=Nh$ of the chain may either remain finite or tend to infinity). The spectral representation (\ref{laplacianalpha}) can then asymptotically be written as an integral\footnote{Since $\sum_{l=-\frac{N}{2}}^{\frac{N}{2}} \frac{G(\kappa_l)}{N} \sim \frac{1}{2\pi} \int_{-\pi}^{\pi}G(\kappa){\rm d}\kappa $ with (${\rm d}\kappa \sim
\kappa_{l+1}-\kappa_l \sim \frac{2\pi}{N}$)} and leads for $N\rightarrow \infty$ to
\begin{equation}
\label{fractlattice}
f^{(\alpha)}(|p|) = \frac{\Omega_{\alpha}^2}{2\pi}\int_{-\pi}^{\pi}e^{i\kappa p}\left(4\sin^2{\frac{\kappa}{2}}\right)^{\frac{\alpha}{2}} {\rm d}\kappa =\Omega_{\alpha}^2 \frac{2^{\alpha + 1}}{\pi}\int_{0}^{\frac{\pi}{2}}
\sin^{\alpha}(\varphi)\cos{(2p\varphi}){\rm d}\varphi
\end{equation}
The matrix elements of the fractional Laplacian matrix (\ref{laplacianalpha}) are
\begin{equation}
\label{fractla}
\Delta_{\alpha}(|p-q| =-\mu f^{(\alpha)}(|p-q|)
\end{equation}
We emphasize the opposite sign of the negative semi-definite fractional Laplacian matrix (\ref{fractla}) and the positive semi-definite fractional characteristic matrix function $f^{(\alpha)}$
(\ref{fractlattice}).
After some manipulations which are outlined in appendix I, we arrive at the representation\footnote{where we put from now on in all formulas $p=|p|$.}
\begin{equation}
\label{fractalat}
f^{(\alpha)}(|p|) =\Omega_{\alpha}^2\,\frac{2^{\alpha}}{\sqrt{\pi}}\frac{1}{(p-\frac{1}{2})!} \int_0^1\xi^{\frac{\alpha}{2}}\frac{d^p}{d\xi^p}\left\{\xi(1-\xi)\right\}^{p-\frac{1}{2}}{\rm d}\xi
\end{equation}
Now consider first the diagonal element ($p=0$) which yields
\begin{equation}
\label{diag}
f^{(\alpha)}(0) = \Omega_{\alpha}^2\,\frac{2^{\alpha}}{\pi}\int_0^1 \xi^{\frac{\alpha-1}{2}}(1-\xi)^{-\frac{1}{2}}{\rm d}\xi = \Omega_{\alpha}^2\, \frac{2^{\alpha}}{\pi} \frac{(\frac{\alpha-1}{2})!(-\frac{1}{2})!}{\frac{\alpha}{2}!} =
\Omega_{\alpha}^2\,\frac{\alpha !}{\frac{\alpha}{2}!\frac{\alpha}{2}!} >0
\end{equation}
necessarily being positive since related with the trace $\frac{1}{N}Tr(f^{\alpha})= f^{(\alpha)}(0)$ of the positive semi-definite $f^{(\alpha)}$.
For the $p^{th}$ element (where $p=|p|$) after multiple partial integrations which are performed in the appendix I, we obtain
\begin{equation}
\label{matrixelei}
f^{(\alpha)}(|p|) = \Omega_{\alpha}^2\,\frac{\alpha!}{\frac{\alpha}{2}!(\frac{\alpha}{2}+|p|)!}(-1)^p\prod_{s=0}^{|p|-1}(\frac{\alpha}{2}-s) ,\hspace{1cm} p \in {\bf \Z}_0
\end{equation}
We observe that for integers $\frac{\alpha}{2}=m \in {\bf \N}$ expression (\ref{matrixelei}) coincides with the binomial coefficients of
(\ref{binomialco}). This can be seen from the obvious relation
\begin{equation}
\label{relationfacyuly}
\prod_{s=0}^{p-1}(m-s) = \frac{m!}{(m-p)!} ,\hspace{1cm} 0\leq p\leq m
\end{equation}
This relation can be extended to non-integers holding for any $\frac{\alpha}{2} >0$, namely ($p=|p|$)
\begin{equation}
\label{generalisationalph2}
\prod_{s=0}^{p-1}(\frac{\alpha}{2}-s) = \frac{\frac{\alpha}{2}!}{(\frac{\alpha}{2}-p)!} ,\hspace{1cm} \frac{\alpha}{2}-p+1>0
\end{equation}
and holds when we employ the main definition of the $\Gamma$-function (\ref{gammafu}) as long $(\frac{\alpha}{2}-p)!=\Gamma(\frac{\alpha}{2}-p+1)$ is well defined, i.e. for $\frac{\alpha}{2}-p+1>0$.
This is the case for
$0\leq p \leq p_0$ where $p_0=ceil(\frac{\alpha}{2})$ with
the ceiling-function defined by (\ref{ceiling}).
Then with (\ref{generalisationalph2}) the matrix elements with $0\leq |p| \leq ceil(\frac{\alpha}{2})$ can be expressed
by a ``generalized centered binomial coefficient", namely
\begin{equation}
\label{generalizationalf}
f^{(\alpha)}(|p|) = \Omega_{\alpha}^2 \,(-1)^p\, \frac{\alpha!}{(\frac{\alpha}{2}-p)!(\frac{\alpha}{2}+p)!} ,
\hspace{0.5cm} 0\leq |p| \leq ceil(\frac{\alpha}{2})
\end{equation}
Representation (\ref{matrixelei}) shows that if $\frac{\alpha}{2} = m \in {\bf N}$ is an integer, matrix elements $f^{2m}(p>m=\frac{\alpha}{2}) =0$ with $p>m$ are vanishing and the characteristic matrix remains ''localized" where (\ref{binomialco}) is recovered by (\ref{matrixelei}). For non-integer $\frac{\alpha}{2} \notin {\bf \N}$, all elements $f^{\alpha}(|p|)$ are non-vanishing, reflecting non-locality of the fractional characteristic matrix, and (\ref{matrixelei}) can be written in the form (appendices I and II)
\begin{equation}
\label{matrixformii}
f^{(\alpha)}(|p|) = -\Omega_{\alpha}^2\frac{\Gamma(\alpha+1)}{\pi}\sin{(\frac{\alpha\pi}{2})}\frac{\Gamma(p-\frac{\alpha}{2})}{\Gamma(\frac{\alpha}{2}+p+1)} = -\Omega_{\alpha}^2\frac{\alpha!}{\pi}\sin{(\frac{\alpha\pi}{2})}\frac{(p-\frac{\alpha}{2}-1)!}{(\frac{\alpha}{2}+p)!} ,\hspace{0.5cm} p =|p| > \frac{\alpha}{2}
\end{equation}
When $\frac{\alpha}{2} \rightarrow m+0 $ ($m\in {\bf \N}$) approaches integers, then the asymptotic value of the $p_0^{th}$ ($p_0=ceil(\frac{\alpha}{2})$) element of (\ref{matrixformii})
tends asymptotically versus $f^{(\alpha)}(|p_0|=ceil(\frac{\alpha}{2}) \rightarrow (-1)^{m}\Omega_{2m}^2$ which is the value given by (\ref{generalizationalf}) for $p=m$ and the elements with $f^{(2m)}(|p|>m)$ are vanishing (reflected by the vanishing of $\sin{(\frac{\alpha\pi}{2})}$ for integers $\frac{\alpha}{2} = m \in {\bf \N}$ in (\ref{matrixformii})). This reflects again the localization of the fractional matrix for integer orders $\frac{\alpha}{2} \in {\bf \N}$ where only the $m+1$ elements (\ref{generalizationalf}) are non-zero taking the integer binomial form (\ref{binomialco}).
In order to give a more compact representation for the matrix elements $f^{(\alpha)}(|p|)$, we introduce the following definition of generalized {\it centered binomial coefficients} (where $\alpha >0 \in {\bf \R}$, $p=|p|$)
\begin{equation}
\label{cgenerlizedbinomi}
\left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\frac{\alpha}{2} +p
\end{array}\right) =: \left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\frac{\alpha}{2} -p
\end{array}\right)
=
\frac{\alpha!}{\frac{\alpha}{2}!(\frac{\alpha}{2}+|p|)!}\prod_{s=0}^{|p|-1}(\frac{\alpha}{2}-s)
\left\{\begin{array}{l} \displaystyle \frac{\alpha!}{(\frac{\alpha}{2}+p)!((\frac{\alpha}{2}-p)!} ,\hspace{0.5cm} 0\leq |p| \leq ceil(\frac{\alpha}{2}) \nonumber \\ \nonumber \\ \displaystyle (-1)^{p+1}\frac{\alpha!}{\pi}\sin{(\frac{\alpha\pi}{2})}\frac{(|p|-\frac{\alpha}{2}-1)!}{(\frac{\alpha}{2}+|p|)!}
, \hspace{0.5cm}|p| > \frac{\alpha}{2}
\end{array} \right.
\end{equation}
We have written (\ref{cgenerlizedbinomi}) such that all arguments of $\Gamma$-functions are positive,
so that they are defined through the main integral definition (\ref{gammafu}).
When we use the analytically continued recursive definition
for the $\Gamma$-function (\ref{recursiongamma}), we can write (\ref{cgenerlizedbinomi}) by utilizing
the Euler relation (\ref{eulergen}) $\forall p$ in unified way for non-integer\footnote{Since for
$\frac{\alpha}{2} \notin {\bf \N}$ no singularities of the
analytically continued $\Gamma$-function occur.} $\frac{\alpha}{2} \notin {\bf \N}$ alternatively
either by expression (\ref{cgenerlizedbinomi})$_1$ or expression by (\ref{cgenerlizedbinomi})$_2$.
The generalized centered binomial coefficients (\ref{cgenerlizedbinomi}) are per definition symmetric with respect to
$p \leftrightarrow -p$.
They fulfill the recursion relation
\begin{equation}
\label{recursion}
\left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\frac{\alpha}{2} +p+1
\end{array}\right) = \frac{(\frac{\alpha}{2}-p)}{(\frac{\alpha}{2}+p+1)} \left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\frac{\alpha}{2} +p
\end{array}\right)
\end{equation}
and as an important consequence of (\ref{recursion}) reflecting the properties of power functions, they fulfill the addition rule
\begin{equation}
\label{additionrule}
\left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\frac{\alpha}{2} +p
\end{array}\right) +
\left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\frac{\alpha}{2} +p-1
\end{array}\right)
= \frac{(\alpha+1)!}{(\frac{\alpha}{2}+p)!((\frac{\alpha}{2}-p+1)!} =
\left(\begin{array}{l}
\hspace{0.2cm} \alpha+1 \nonumber \\
\frac{\alpha}{2} +p
\end{array}\right) = \left(\begin{array}{l}
\hspace{0.2cm} \alpha+1 \nonumber \\
\frac{\alpha+1}{2} +p-\frac{1}{2}
\end{array}\right)
\end{equation}
where the last term in this equation is written in ``centered" representation of the generalized binomial coefficients.
Generalized binomial coefficients we define as $\left(\begin{array}{l} \xi \nonumber \\ \xi_1 \end{array}\right) = \left(\begin{array}{l} \xi \nonumber \\ \xi_2 \end{array}\right) = \frac{\xi!}{\xi_1 !\xi_2 !} ,\hspace{0.3cm} \xi=\xi_1+\xi_2$
where in general the analytically continued definition of the $\Gamma$-function (\ref{recursiongamma}) is assumed.
The properties (\ref{recursion}) and (\ref{additionrule})
are in full correspondence with the recursion- and addition rules of the usual binomial coefficients. Note that the centered generalized binomial coefficients (\ref{cgenerlizedbinomi}) maintain their sign $+1$
as long as $0\leq p \leq p_0=ceil(\frac{\alpha}{2})$. At $p=ceil(\frac{\alpha}{2})+1$ their sign switches to $(-1)$ and alternates for all $p>p_0$ as $(-1)^{p-p_0}$ (see also appendix I).
Then with definition (\ref{cgenerlizedbinomi}) the matrix elements $f^{\alpha}(|p|)$ can $\forall p$ be written in compact form\footnote{The notation $f^{\alpha}_{\infty}(|p|)$ with subscipt $_{\infty}$ indicates that
this expression holds for $N\rightarrow \infty$, and will be used for clarity in the subsequent paragraph.}
\begin{equation}
\label{compactanalmogy}
f^{\alpha}(|p|) = f^{\alpha}_{\infty}(|p|) = \Omega_{\alpha}^2 (-1)^p \, \left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\frac{\alpha}{2} + p
\end{array}\right) , \hspace{1cm} p\in {\bf Z}_0
\end{equation}
where for $\frac{\alpha}{2}= m \in {\bf N}_0$ (\ref{compactanalmogy}) recovers the integer case (\ref{binomialco}).
The elements of the {\it fractional Laplacian matrix} are connected with (\ref{compactanalmogy}) by
\begin{equation}
\label{laplamatel}
\Delta_{\alpha}(|p|) = -\mu f^{\alpha}(|p|) = \mu\Omega_{\alpha}^2 (-1)^{p+1} \, \left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\frac{\alpha}{2} +p
\end{array}\right) , \hspace{1cm} p\in {\bf Z}_0
\end{equation}
The fractional characteristic matrix elements (\ref{compactanalmogy}) coincide with the centered differences representation introduced as starting point of the fractional centered differences model of Ortiguera (eq. (3.8) in \cite{riesz2}). A fractional Laplacian matrix for the infinite chain in accordance with expression (\ref{matrixformii}) also
was reported by Zoia et al \cite{Zoia2007}.
\newline\newline
The representation of the fractional Laplacian matrix by means of the generalized binomial coefficients is a highly convenient notation generalizing the integer binomial expansion to non-integer cases.
In contrast to the binomial coefficients (\ref{cgenerlizedbinomi}), the matrix elements (\ref{compactanalmogy}) due to their additional prefactor $(-1)^p$ alternate in sign for $0\leq p\leq p_0$ and maintain their
sign\footnote{$\frac{\alpha}{2}=p_0-\chi$ ($0<\chi<1$ and $sign(\sin{\pi\chi})=1$ thus $-sign(\sin{\frac{\alpha\pi}{2}})= -sign(\sin{\pi(p_0-\chi)}) = (-1)^{p_0}$}
$(-1)^{p_0} =-sign(\sin{\frac{\alpha\pi}{2}})$ for $p> p_0$.
We can write the fractional characteristic matrix (\ref{alphacarfou}) as centered fractional differences series
\begin{equation}
\label{copactchar}
\begin{array}{l}
\displaystyle f^{(\alpha)}(2-D-D^{\dagger}) = \Omega_{\alpha}^2(2-D(h)-D(-h))^{\frac{\alpha}{2}} =
\Omega_{\alpha}^2\sum_{p=-\infty}^{\infty}(-1)^p\left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\frac{\alpha}{2} + p
\end{array}\right)D(hp) \nonumber \\ \nonumber \\ \nonumber \\ {\rm with} \nonumber \\
\displaystyle \hspace{2.5cm}
(-1)^{\frac{\alpha}{2}}\left(D(\frac{h}{2})-D(-\frac{h}{2})\right)^{\alpha}
= \left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\hspace{0.2cm} \frac{\alpha}{2}
\end{array}\right) +\sum_{p=1}^{\infty} (-1)^p \left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\frac{\alpha}{2} + p
\end{array}\right) \left(D(hp)+D(-hp)\right)
\end{array}
\end{equation}
For non-integer $\frac{\alpha}{2} \notin {\bf \N}$ (\ref{copactchar}) is an infinite series reflecting the non-locality of the fractional operator,
whilst for $\frac{\alpha}{2} \in {\bf \N}$ it takes the representation of finite centered binomial expansions (\ref{binomi}) where all terms with
$|p|>\frac{\alpha}{2}$ are vanishing, reflecting ''locality" in the integer cases.
We emphasize that (\ref{laplamatel}) with (\ref{compactanalmogy})
are asymptotic infinite chain limit expressions ($N\rightarrow \infty$).
In the subsequent subsection we construct explicitly the discrete fractional Laplacian matrix for the $N$-periodic chain where $N$ can be any arbitrary not necessarily large integer.
We can establish with (\ref{compactanalmogy}) and (\ref{copactchar}) the following interesting relation which {\it holds only on the unit circle $|z|=1$}, namely
\begin{equation}
\label{converge}
\begin{array}{l}
\displaystyle (2-z-z^{-1})^{\frac{\alpha}{2}} = \sum_{p=-\infty}^{\infty}(-1)^p\left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\frac{\alpha}{2} +p
\end{array}\right)
z^{p} = \left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\hspace{0.2cm} \frac{\alpha}{2}
\end{array}\right) +\sum_{p=1}^{\infty}(-1)^p\left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\frac{\alpha}{2} +p
\end{array}\right) (z^{p}+z^{-p}),\hspace{0.5cm} |z|=1 \nonumber \\ \nonumber \\
\displaystyle \left\{4\sin^2{\frac{\varphi}{2}}\right\}^{\frac{\alpha}{2}} =\left\{2(1-\cos{\varphi})\right\}^{\frac{\alpha}{2}} = \left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\hspace{0.2cm} \frac{\alpha}{2}
\end{array}\right) +2\sum_{p=1}^{\infty} (-1)^p \left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\frac{\alpha}{2} +p
\end{array}\right) \cos{(p\varphi)}
\end{array}
\end{equation}
This relation diverges everywhere in the complex $z$-plane {\it except on the unit circle $|z|=1$} and
determines the dispersion relation
of the infinite chain by putting $z=e^{i\varphi}$, where $-\pi \leq \varphi \leq \pi$ indicates the (for the infinite chain limit $N\rightarrow \infty$)
(quasi-) continuous dimensionless wave number within the first Brillouin zone. Translational invariance
(corresponding to $\varphi=0$ in (\ref{converge})) yields
the zero eigenvalue
\begin{equation}
\label{transla}
\left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\hspace{0.2cm} \frac{\alpha}{2}
\end{array}\right) +2\sum_{p=1}^{\infty} (-1)^p \left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\frac{\alpha}{2} +p
\end{array}\right) = 0
\end{equation}
and further relations of interest for special values of $\varphi$ can be established. We will show in the subsequent paragraph, that the infinite chain relation (\ref{converge}) is especially useful to construct the discrete fractional Laplacian matrix on the {\it finite $N$-periodic chain}.
\subsection{Fractional Laplacian matrix on the finite periodic chain}
In this paragraph we develop in explicit form the fractional characteristic matrix (\ref{alphacarfou}) for the discrete finite $N$-periodic
chain, where the particle number $N$ is not necessarily large.
To this end we reconsider the infinite chain relation (\ref{converge}) for $z=e^{i\kappa_l}$ where
$\varphi =\kappa_l=\frac{2\pi}{N}l$ now is a Bloch wave number of the {\it finite} $N$-periodic chain
\begin{equation}
\label{finitechain}
\left\{4\sin^2{\frac{\kappa_l}{2}}\right\}^{\frac{\alpha}{2}} =
\sum_{p=-\infty}^{\infty}(-1)^p\left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\frac{\alpha}{2} +p
\end{array}\right)
e^{i\kappa_lp} = \left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\hspace{0.2cm} \frac{\alpha}{2}
\end{array}\right) +2\sum_{p=1}^{\infty} (-1)^p \left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\frac{\alpha}{2} +p
\end{array}\right) \cos{(p\kappa_l)}
\end{equation}
We observe that the left hand side of this relation determines dispersion relation (\ref{disprelat}) of the {\it finite chain}.
Now evaluate this series by taking into account the $N$-periodicity of the Bloch phase
factors $e^{i\kappa_lp} =e^{i\kappa_l(p+N)}$ by collecting all terms with same cyclic indices $0\leq p\leq N-1$, i.e. with the same phase factors into the coefficients $f_N^{(\alpha)}(|p|)$,
where only $N$ distinguished phase factors for $0\leq p\leq N-1$ occur. So we get a representation of the form
\begin{equation}
\label{collect}
\Omega_{\alpha}^2
\left\{4\sin^2{\frac{\kappa_l}{2}}\right\}^{\frac{\alpha}{2}} = \sum_{p=0}^{N-1}e^{i\kappa_lp}f^{(\alpha)}_N(|p|) ,\hspace{1cm} \kappa_l=\frac{2\pi}{N} l , \hspace{0.5cm} (l=0,..,N-1)
\end{equation}
which indeed is the discrete dispersion relation (\ref{disprelat}) (the $N$ discrete eigenvalues) of the fractional Laplacian matrix of the {\it finite chain} of $N$ particles. The coefficients $f_N^{\alpha}(|p|)$ are given in below expression (\ref{finiteelem}).
Inverting relation (\ref{collect}) confirms that the coefficients $f_N^{(\alpha)}(|p|)$ are the
matrix elements of the fractional characteristic matrix $\Omega_{\alpha}(2-D-D^{\dagger})^{\frac{\alpha}{2}}$ for finite $N$, namely
\begin{equation}
\label{invert}
f_N^{(\alpha)}(|p-q|) = \Omega_{\alpha}^2\sum_{l=0}^{N-1}\frac{e^{i\kappa_l(p-q)}}{N}\left\{4\sin^2{\frac{\kappa_l}{2}}\right\}^{\frac{\alpha}{2}}
\end{equation}
which indeed is consistent with the spectral representation given in (\ref{laplacianalpha})$_2$ for the fractional Laplacian matrix for finite $N$.
The fractional characteristic function matrix elements {\it $f^{\alpha}_N(|p|)$ of the finite $N$-periodic chain} are obtained as
\begin{equation}
\label{finiteelem}
\begin{array}{l}
\displaystyle f^{(\alpha)}_N(|p|) = \sum_{n=-\infty}^{\infty}f^{(\alpha)}_{\infty}(|p-nN|) = \displaystyle f^{(\alpha)}_{\infty}(|p|)+\sum_{n=1}^{\infty}
\left(f^{(\alpha)}_{\infty}(|p+nN|)+ f^{(\alpha)}_{\infty}(|p-nN|)\right) ,\hspace{1cm} 0\leq p \leq N-1 \nonumber \\ \nonumber \\
\displaystyle \hspace{1.5cm} = \Omega_{\alpha}^2 (-1)^p \, \left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\frac{\alpha}{2} +p
\end{array}\right) + \Omega_{\alpha}^2\sum_{n=1}^{\infty}(-1)^{p+nN}\left(\left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\frac{\alpha}{2} +p +nN
\end{array}\right)+\left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\frac{\alpha}{2} +p -nN
\end{array}\right) \right) \nonumber \\ \nonumber \\
\displaystyle \hspace{1.5cm} = \Omega_{\alpha}^2\, \frac{2^{\alpha}}{N}\sum_{l=0}^{N-1}e^{i\kappa_lp}|\sin{\frac{\kappa_l}{2}}|^{\alpha} , \hspace{1cm} \kappa_l=\frac{2\pi}{N}l
\nonumber \\ \nonumber \\
\displaystyle \hspace{1.5cm}= \Omega_{\alpha}^2\left[(2-D-D^{\dagger})^{\frac{\alpha}{2}}\right]_{|p|} ,\hspace{1cm}
D^{p}=D^{p+nN} ,\hspace{1cm} n\in {\bf Z}_0
\end{array}
\end{equation}
In this relation $f^{(\alpha)}_{\infty}(|p|)$ indicate the matrix elements (\ref{compactanalmogy}) of the infinite chain.
We observe that (\ref{finiteelem}) contains an interesting evaluation of the spectral sum (\ref{finiteelem})$_3$ by means of the generalized centered binomial coefficients (\ref{cgenerlizedbinomi}).
The fractional characteristic matrix function elements of the finite chain (\ref{finiteelem}) obey further the general symmetries (\ref{percharfu}), and
the fractional Laplacian matrix of the $N$-periodic finite chain is
\begin{equation}
\label{laplafini}
\Delta_{\alpha,N}(|p|) = -\mu f^{(\alpha)}_N(|p|) ,\hspace{1cm} 0\leq p \leq N-1
\end{equation}
Relation (\ref{finiteelem}) is also obtained when considering the infinite series (\ref{copactchar}) and application of the cyclic index convention (equivalent to the periodicity of the shift operators on the finite chain).
We emphasize that relation (\ref{finiteelem}) is an exact representation for the $N$-periodic fractional characteristic matrix function, and (\ref{laplafini}) the corresponding fractional Laplacian matrix for the finite chain of $N$ particles where $N$ can be any integer not necessarily large. Especially for computational purposes (\ref{finiteelem}) seems to be useful. By utilizing a truncated part of the series
(\ref{finiteelem}) allows to represent the matrix elements in a desired accuracy.
Note that analogous representations as (\ref{finiteelem})$_1$ relating infinite chain Laplacian matrices with finite periodic chain Laplacian matrices can be estabished for any (including non-fractional) admissible characteristic function $f$.
We observe further that in the infinite chain limit
\begin{equation}
\label{obvious}
\lim_{N\rightarrow\infty} f^{(\alpha)}_N(|p|) = f^{(\alpha)}_{\infty}(|p|)
\end{equation}
the infinite chain characteristic matrix (\ref{compactanalmogy})
is recovered by (\ref{finiteelem}). We will see in the subsequent section that analogous expressions to (\ref{finiteelem})
exist for the continuum limit kernels of the fractional Laplacian on the finite $L$-periodic string. The fractional
Laplacian kernel of the $L$-periodic string deduced subsequently represents the (periodic-string-)
continuum limit kernel expression of (\ref{laplafini}) with (\ref{finiteelem}).
Relations between finite and infinite chain fractional characteristic matrix functions
such as established in this section
appear to be useful to develop discrete fractional calculus on finite lattices.
\section{Continuum limits of the discrete fractional chain model: \\1D Fractional Laplacians for the infinite and $L$-periodic string}
The aim of this section is to deduce the continuum limit kernels of the fractional characteristic matrices of previous section. The continuum limit kernels to be deduced can be considered as definitions of the fractional Laplacian (Riesz fractional derivatives) on the infinite and periodic string, respectively.
Generally we can define two kinds of continuum limits: \newline\noindent (i) The periodic string continuum limit where the length of the chain
$L=Nh$ is kept finite and $h\rightarrow 0$ (.i.e. $N(h)= L h^{-1} \rightarrow \infty$). \newline\noindent (ii) The infinite space limit where $h\rightarrow 0$, however, the length of the chain\footnote{which we refer to as {\it string} in the continuum limit} tends to infinity $N(h)h=L(h) \rightarrow \infty$\footnote{which can be realized for instance by chosing by $N(h)\sim h^{-\delta}$ where $\delta > 1$.}. The kernels of the infinite space limit can be recovered from those of
the periodic string limit by letting $L\rightarrow \infty$. The central aim of this section is the deduction of the $L$-periodic string continuum limit (i) for the fractional Laplacian, which is to the best of our knowledge, so far is not reported the
literature.
Following our recent approach \cite{michel-collet} we require in the continuum limit that extensive physical quantities, i.e. quantities
which scale with the length of the 1D system, such as the total mass $N\mu=M$ and the total elastic energy of the chain remain finite when $L$ is kept finite\footnote{In the case of infinite string $L\rightarrow \infty$ we require the mass per unit length and elastic energy per unit length to remain finite.}, i.e. neither vanish nor diverge. From the finiteness of the total mass of the chain, it follows that the particle mass $\mu=\frac{M}{N}=\frac{M}{L} h = \rho_0 h$ scales $\sim h$. Then by employing expression (\ref{Valpha}) for the fractional elastic potential,
the total continuum limit elastic energy ${\tilde V}_{\alpha}$ can be defined by
\begin{equation}
\label{elasten}
{\tilde V}_{\alpha} = \lim_{h\rightarrow 0+} V_{\alpha} = \frac{\mu\Omega_{\alpha}^2}{2}\sum_{p=0}^{N-1} u^*(x_p)\left(-4\sinh^2{\frac{h}{2}\frac{d}{dx}}\right)^{\frac{\alpha}{2}}u(x_p)
\end{equation}
Accounting for $2-D(h)-D(-h) =-4\sinh^2{\frac{h}{2}\frac{d}{dx}} \approx -h^2\frac{d^2}{dx^2} + O(h^4)$ so we get
\begin{equation}
\label{fraclap}
\lim_{h\rightarrow 0} \left(-4\sinh^2{\frac{h}{2}\frac{d}{dx}}\right)^{\frac{\alpha}{2}} = h^{\alpha} (-\frac{d^2}{dx^2})^{\frac{\alpha}{2}}
\end{equation}
Since (\ref{fraclap}) tends to
zero with leading order $h^{\alpha}$, (\ref{elasten}) can only remain finite if the constant $\Omega_{\alpha}^2$ and particle mass $\mu$ scale for $h\rightarrow 0$ asymptotically as
\begin{equation}
\label{scaling}
\Omega_{\alpha}^2(h) =A_{\alpha} h^{-\alpha} , \hspace{1cm} \mu(h) = \rho_0 h ,\hspace{1cm} A_{\alpha}, \rho_0 >0
\end{equation}
where $\rho_0$ denotes the mass density with dimension $g \times cm^{-1}$ and $A_{\alpha}$ denotes a positive dimensional constant
of dimension $ sec^{-2}\times cm^{\alpha}$, where the new constants $\rho_0, A_{\alpha}$ are independent of $h$. Note that the dimensional
constant $A_{\alpha}$ is only defined up to a non-dimensional positive
scaling factor as its absolute value does not matter due to the scale-freeness of the power function.
We obtain then as continuum limit of the elastic energy\footnote{by accounting for $\sum_{p=0}^{N-1}h G(x_p) \rightarrow \int_0^L G(x){\rm d}x$ and $h\rightarrow dx$, $x_p\rightarrow x$.}
\begin{equation}
\label{contilimielasten}
\begin{array}{l}
\displaystyle
{\tilde V}_{\alpha} = \lim_{h\rightarrow 0} \frac{\mu(h)}{2}
\sum_{q=0}^{N-1}\sum_{p=0}^{N-1}u_q^* f^{(\alpha)}_N(|p-q|)u_p \nonumber \\ \nonumber \\
\displaystyle
{\tilde V}_{\alpha} = \frac{\rho_0 A_{\alpha}}{2}
\int_0^Lu^*(x)\left(-\frac{d^2}{dx^2}\right)^{\frac{\alpha}{2}}u(x)\,{\rm d}x
=: -\frac{1}{2} \int_0^L\int_0^Lu^*(x'){\tilde \Delta}_{\alpha}(|x-x'|)u(x){\rm d}x
\end{array}
\end{equation}
The continuum limit Laplacian kernel ${\tilde \Delta}_{\alpha}(|x-x'|)$ can then formally be represented by the distributional representation\footnote{In the distributional sense of generalized functions \cite{gelfand}.}
\begin{equation}
\label{contilimlap}
{\tilde \Delta}_{\alpha,L}(|x-x'|) = -\rho_0A_{\alpha}\left(-\frac{d^2}{dx^2}\right)^{\frac{\alpha}{2}}\delta_L(x-x')
\end{equation}
where this relation contains as special case the distributional representation
of integer orders $m=\frac{\alpha}{2}\in {\bf \N}$ which we analyzed earlier \cite{michel-collet}). In (\ref{contilimlap}) $\delta_L(x-x')$ indicates the $L$-periodic Dirac's $\delta$-function.
From this consideration it follows that the continuum limit kernel indeed is to be conceived as the distributional representation of the {\it fractional Laplacian}\footnote{having due to the distributional definition dimension $[-\left(-\frac{d^2}{dx^2}\right)^{\frac{\alpha}{2}}\delta_L(x-x')]$ namely $cm^{-\alpha-1}$ as the dimension of the $\delta$-function $cm^{-1}$ is included}
(Riesz fractional derivative) $-\left(-\frac{d^2}{dx^2}\right)^{\frac{\alpha}{2}}\delta_L(x-x')$, indicating the (negative semi-definite) distributional representation of fractional powers of the 1D-Laplacian $\frac{d^2}{dx^2}$ (see for instance \cite{riesz,samko,gorenflo,michel-fcaa} and references therein). We notice that (\ref{fraclap}) is just a (non-distributional) formal definition of the fractional Laplacian. Its explicit spatial distributional representations for the infinite and especially for the finite $L$- periodic string are to be constructed subsequently.
\newline\newline
The periodic string fractional Laplacian kernel can be explicitly obtained from (\ref{contilimielasten})$_1$
by employing scaling relations (\ref{scaling}) and asymptotic expressions for the $N$-periodic finite chain fractional characteristic matrix
$f^{(\alpha)}_N(|p-q|)$ of (\ref{finiteelem}).
Before we construct the fractional Laplacian kernel on the $L$-periodic string explicitly, let us briefly evoke the general connection between
an infinite space kernel $g_{\infty}(|x|)$ defined over $-\infty < x < \infty$ and its $L$-periodic counterpart $g_L(|x|)$ defined on a finite principal interval\footnote{which
also can be chosen as $-\frac{L}{2} \leq x \leq \frac{L}{2}$ as a consequence of the symmetries (\ref{periodmirror})} of length $L$
for instance
$0\leq x\leq L$. We have in full correspondence to the symmetry properties of the discrete case (\ref{percharfu}), these symmetries also in the continuous case, namely
for the $L$-periodic kernel
\begin{equation}
\label{periodmirror}
\begin{array}{l}
g_L(|x|)=g_L(|x+nL|), \hspace{1cm} n \in {\bf \Z}_0 \nonumber \\ \nonumber \\
g_L(|\frac{L}{2}+\xi|)= g_L(|\frac{L}{2}-\xi|) =g_L(|nL+\frac{L}{2}-\xi|) ,\hspace{1cm} n\in {\bf \Z}_0
\end{array}
\end{equation}
(\ref{periodmirror})$_1$ is the periodicity condition and (\ref{periodmirror})$_2$ the reflection-symmetry condition, where the principal axes of reflection symmetry are periodically repeating at $\xi_n=nL+\frac{L}{2}$ ($n\in {\bf \Z}_0$).
\newline\newline
If the infinite space kernel $g_{\infty}$ is known we can construct by the following
projection convolution the $L$-periodic kernel
\begin{equation}
\label{projectionop}
g_L(|x|) = \int_{-\infty}^{\infty} \delta_L(x-x')\, g_{\infty}(|x'|)\,{\rm d}x' = \sum_{l=-\infty}^{\infty}{\hat g}(k_l)\Phi_l(x)\Phi_l^*(x')
\end{equation}
where the periodic kernel $g_L(|x|)$ has a discrete eigenvalue spectrum $\{{\hat g}_L(k_l)\}$ at the discrete set of Bloch wave numbers $k_l=\frac{2\pi}{L}l$ with $l\in {\bf \Z}_0$. Now consider the
infinite space kernel which has the spectral representation
\begin{equation}
\label{kernelinfinitspace}
g_{\infty}(|x|) = \frac{1}{2\pi}\int_{-\infty}^{\infty} {\hat g}_{\infty}(k) e^{ikx}{\rm d}k
\end{equation}
with the continuous spectrum $\{{\hat g}_{\infty}(k)\}$ ($-\infty < k < \infty$). Then account for
the $L$-periodic $\delta$-function $\delta_L(x-x')$ (unity operator in the space of $L$-periodic functions)
which can be written as
\begin{equation}
\label{periodicdelta}
\delta_L(x-x')= \sum_{n-\infty}^{\infty} \delta_{\infty}(x-x'-nL) = \sum_{l=-\infty}^{\infty} \Phi_l(x)\Phi_l^*(x') , \hspace{0.25cm} \Phi_l(\xi)=\frac{e^{ik_l\xi}}{\sqrt{L}} ,\hspace{0.25cm} k_l =\frac{2\pi}{L}l ,\hspace{0.25cm} l \in {\bf Z}_0
\end{equation}
where $\delta_{\infty}(\xi)$ denotes the infinite space Dirac's $\delta$-function. The spectral representation of the $L$-periodic $\delta$-function $\delta_L(\xi)$ is expressed by the $L$-periodic Bloch eigenfunctions $\Phi_l$. Performing convolution (\ref{projectionop}) yields with (\ref{periodicdelta})
\begin{equation}
\label{convolper}
g_L(|x|) = \sum_{n=-\infty}^{\infty} g_{\infty}(|x-nL|) = g_{\infty}(|x|) +\sum_{n=1}^{\infty}
\left(g_{\infty}(|x-nL|)+g_{\infty}(|x+nL|)\right) ,\hspace{0.3cm} 0\leq x\leq L
\end{equation}
This relation is the generalization of the so called {\it Ewald sum} which appears in the context of a Coulomb potential in a periodic cell \cite{Ewald21}.
Relation (\ref{convolper}) allows to represent the $L$-periodic kernel $g_L$ if the infinite space kernel $g_{\infty}$ is known.
Expression (\ref{projectionop}) for $g_L(x)$ is defined on its principal interval $0\leq x \leq L$ and is in full analogy with representation (\ref{finiteelem}) of the fractional $N$-periodic characteristic matrix function of the finite chain.
The series (\ref{convolper}) contains the infinite space kernel $g_{\infty}(|x|)$ plus image terms. The periodic kernel $g_L$ of (\ref{convolper}) and the infinite space kernel $g_{\infty}$ are generally connected by
\begin{equation}
\label{connect}
g_{\infty}(|x|) = \lim_{L\rightarrow \infty} g_L(|x|)
\end{equation}
where the image terms are vanishing for $L\rightarrow \infty$ due to
$|\int_L^{\infty}g_{\infty}(\xi){\rm d}\xi| \rightarrow 0$.
Relation (\ref{connect}) connects infinite space kernel with periodic string kernel and allows to recover
the infinite space kernel if only the $L$-periodic string kernel is known.
It is easy to see that the series (\ref{convolper}) indeed fulfills the symmetries (\ref{periodmirror}) and converges as good as $|\int_{-\infty}^{\infty} g_{\infty}(|\xi|){\rm d\xi}| < \infty$. Let us briefly consider the relation of the Fourier transforms of both kernels.
(\ref{convolper}) has a spectral representation
\begin{equation}
\label{fourierser}
g_L|x-x'|)=\sum_{l=-\infty}^{\infty} {\hat g}_L(k_l)\Phi_l(x)\Phi_l^*(x'),\hspace{1cm} \Phi_l(x)=\frac{e^{ik_lx}}
{\sqrt{L}} ,\hspace{0.15cm} k_l=\frac{2\pi}{L}l ,\hspace{0.15cm} l\in {\bf \Z}_0
\end{equation}
with the {\it discrete eigenvalue spectrum ${\hat g}(k_l)$} and $\Phi_l(x)$ are the ortho-normal Bloch-eigenfunctions\footnote{i.e. $\int_0^L\Phi_i(x)^*\Phi_l(x){\rm d}x =\delta_{il}$}. The eigenvalues ${\hat g}(k_l)$ are obtained utilizing
(\ref{convolper}) as piecewise integrations over the entire infinite space
\begin{equation}
\label{fourier}
\begin{array}{l}
\displaystyle {\hat g}_L(k_l)\Phi_l(x) = \int_0^L g_L(|x-x'|)\Phi_l(x'){\rm d}x'= \Phi_l(x)
\sum_{n=-\infty}^{\infty} \int_{nL}^{(n+1)L}g_{\infty}(|\xi|)e^{-ik_l\xi }{\rm d}\xi \nonumber \\ \nonumber \\
\displaystyle {\hat g}_L(k_l) = \int_{-\infty}^{\infty} g_{\infty}(|\xi|)e^{-ik_l\xi }\, {\rm d}\xi = {\hat g}_{\infty}(k=k_l)
\end{array}
\end{equation}
where (\ref{fourier})$_2$ is the inversion of (\ref{kernelinfinitspace}) for $k=k_l$.
The discrete eigenvalues ${\hat g}_L(k_l)={\hat g}_{\infty}(k=k_l)$ are a discrete subset taken at the distinct values of Bloch wave numbers $k=k_l$ from the continuous spectrum of eigenvalues
${\hat g}_{\infty}(k)$ of the infinite space kernel.
All required information to construct the periodic kernel $g_L$ is available if the infinite space kernel $g_{\infty}$ and the length of the string $L$ are known. Therefore, in order to get the fractional Laplacian for the finite $L$-periodic string, let us first consider the infinite space continuum limit of the fractional characteristic function matrix.
\newline\newline
{\bf \noindent Infinite space continuum limit} \newline Let us consider the continuum limit of the fractional matrix elements (\ref{compactanalmogy}) for $p=\frac{x}{h} >>1$ and $0\leq x \leq L \rightarrow \infty$ which is obtained from the asymptotics of (\ref{matrixformii}). Taking into account Sterling's asymptotic formula \cite{abramo} which holds for sufficiently large $\beta >>1$, namely $\displaystyle \beta! \sim \sqrt{2\pi\beta} \,\,\frac{\beta^{\beta}}{{e^{\beta}}}$, we get for $a,b$ finite and constant a power law behavior
\begin{equation}
\label{asympbet}
\frac{(\beta +a)!}{(\beta+b)!} \sim \beta^{a-b} \hspace{1cm} \beta >>1
\end{equation}
Then we obtain for $p>>1$ for the matrix elements (\ref{compactanalmogy})
the asymptotic representations
\begin{equation}
\label{asympmat}
f^{(\alpha)}(|p|)=\Omega_{\alpha}^2(-1)^p \left(\begin{array}{l}
\hspace{0.2cm} \alpha \nonumber \\
\frac{\alpha}{2} +p
\end{array}\right)= -\Omega_{\alpha}^2\frac{\alpha!}{\pi}\sin{(\frac{\alpha\pi}{2})}\frac{(p-\frac{\alpha}{2}-1)!}{(\frac{\alpha}{2}+p)!}
\sim -\Omega_{\alpha}^2\,\,\frac{\alpha!}{\pi}\sin{(\frac{\alpha\pi}{2})} \,\, p^{-\alpha-1}
\end{equation}
leading to the limit kernel of the fractional Laplacian matrix (\ref{laplamatel}) $\frac{|x|}{h} >>1$\footnote{The additional prefactor $h^{-2}$ comes into play as in the continuum limit a double sum $\sum_p\sum_q \rightarrow \int \int\frac{{\rm d}x{\rm d}x'}{h^2}$.}
\begin{equation}
\label{asymp}
{\tilde \Delta}_{\alpha}(|x|) = \lim_{h\rightarrow 0+} -\mu(h) \frac{1}{h^2}f^{(\alpha)}\left(\frac{|x|}{h}\right) = \rho_0A_{\alpha}
\frac{\alpha!}{\pi}\sin{(\frac{\alpha\pi}{2})} \,\, |x|^{-\alpha-1} ,\hspace{0.5cm} 0< |x| < \infty
\end{equation}
The continuum limit kernel (\ref{asymp}) has per construction the physical dimension $g\times sec^{-2}\times cm^{-2}$.
Comparison of (\ref{asymp}) with (\ref{contilimlap}) yields for the (negative-semi-definite) {\it fractional Laplacian kernel} (Riesz fractional derivative)
of the infinite space
\begin{equation}
\label{fractlaplkernel}
-\left(-\frac{d^2}{dx^2}\right)^{\frac{\alpha}{2}}\delta_{\infty}(x) =
\frac{\alpha !}{\pi}\frac{\sin{(\frac{\alpha\pi}{2})}}{ |x|^{\alpha+1}} ,\hspace{0.5cm} 0< |x| < \infty
\end{equation}
We emphasize that we are considering the infinite space and $\delta_{\infty}(x)$ indicates the infinite space Dirac's $\delta$-function.
(\ref{fractlaplkernel}) is the well known expression for the kernel of infinite space fractional Laplacian
(e.g. \cite{riesz2,gorenflo,michelb,michel-ima2014})\footnote{Eq. (4.12) in \cite{riesz2}
uses a different sign convention. The expression there denotes the positive (semi-definite) operator $\left(-\frac{d^2}{dx^2}\right)^{\frac{\alpha}{2}}$ which is referred in that paper to as ``fractional Laplacian''.}. The hypersingular behavior at $x=0$ of (\ref{fractlaplkernel}) can be removed by introducing a regularization in the distributional sense \cite{michel-ima2014}
\begin{equation}
\label{distri}
-\left(-\frac{d^2}{dx^2}\right)^{\frac{\alpha}{2}}\delta_{\infty}(x) = - \lim_{\epsilon\rightarrow 0+}\frac{\alpha !}{\pi}{\Re
\large\{\frac{i^{\alpha+1}}{(x+i\epsilon)^{\alpha+1}}}\large\}
\end{equation}
For $x\neq 0$ (\ref{distri}) coincides with (\ref{fractlaplkernel}), however, when integrating over $x=0$ (\ref{distri}) extends (\ref{fractlaplkernel}) such that regularity is achieved:
The regularization in (\ref{distri}) mimics the alternating behavior of the discrete fractional Laplacian (\ref{laplamatel}) at $0\leq p \leq ceil(\frac{\alpha}{2})$ by oscillations taking place when $x\rightarrow 0$
where the phase of the complex kernel (\ref{distri}) jumps from zero (at $x=0$) to $\frac{\pi(\alpha+1)}{2}$ (at $x=0\pm \delta x$). As a consequence the fractional Laplacian (\ref{distri}) applied to a constant field yields zero \cite{michel-ima2014}, which reflects relation (\ref{transla}). It is demonstrated in \cite{michel-ima2014} that the regularized representation for the fractional infinite space Laplacian (\ref{distri}) indeed is valid for any positive $\alpha >0$ as it removes the hypersingularity at $x=0$ of the non-regularized kernel (\ref{fractlaplkernel}) for all $\alpha >0$.
Further for integer $m=\frac{\alpha}{2}$, relation (\ref{distri}) takes indeed the distributional representation of integer order-Laplacian
\begin{equation}
\label{integerorder}
-\left(-\frac{d^2}{dx^2}\right)^{\frac{\alpha}{2}=m}\delta_{\infty}(x) = (-1)^{m+1}\frac{d^{2m}}{dx^{2m}} \lim_{\epsilon\rightarrow 0+}\frac{1}{\pi}\Re\{ \frac{i}{(x+i\epsilon)} \} = (-1)^{m+1}\frac{d^{2m}}{dx^{2m}} \lim_{\epsilon\rightarrow 0+}\frac{1}{\pi}\frac{\epsilon}{(x^2+\epsilon^2)} ,\hspace{1cm} m\in {\bf \N_0}
\end{equation}
where $\displaystyle \delta_{\infty}(x)=\lim_{\epsilon\rightarrow 0+}\frac{1}{\pi}\frac{\epsilon}{(x^2+\epsilon^2)}$ is Dirac's infinite space $\delta$-function.
We devote the subsequent paragraph to the analysis of the periodic string limit and the construction of the fractional Laplacian defined on the $L$-periodic string.
\newline\newline
{\bf \noindent Periodic string continuum limit: Fractional Laplacian on $L$-periodic string}
\newline\newline
With the above general consideration (relations (\ref{periodmirror})-(\ref{fourier})), we can directly pass
to the explicit representation of the $L$-periodic fractional Laplacian of (\ref{contilimlap}). The $L$-periodic fractional Laplacian kernel which we denote by $K_L^{(\alpha)}(|x|)$ takes the form
\begin{equation}
\label{contilimlapfinal}
\begin{array}{l}
\displaystyle -\left(-\frac{d^2}{dx^2}\right)^{\frac{\alpha}{2}}\delta_L(x) = K_L^{(\alpha)}(|x|) = \frac{\alpha ! \sin{(\frac{\alpha\pi}{2})}}{\pi} \sum_{n=-\infty}^{\infty}\frac{1}{|x-nL|^{\alpha+1}} \nonumber \\ \nonumber \\
\displaystyle \hspace{3cm}K_L^{(\alpha)}(|x|) = \frac{\alpha !\sin{(\frac{\alpha\pi}{2})}}{\pi L^{\alpha+1}}\left\{-\frac{1}{|\xi|^{\alpha+1}}+
{\tilde \zeta}(\alpha +1,\xi) + {\tilde \zeta}(\alpha +1,-\xi) \right\} ,\hspace{1.5cm} \xi=\frac{x}{L} \nonumber \\ \nonumber \\
\displaystyle \hspace{3cm}K_L^{(\alpha)}(|x|) = -\frac{\alpha !}{\pi} \lim_{\epsilon\rightarrow 0+} \Re\left\{\sum_{n=-\infty}^{\infty}
\frac{i^{\alpha+1}}{(x-nL+i\epsilon)^{\alpha+1}}\right\} \nonumber \\ \nonumber \\ \displaystyle \hspace{3cm} K_L^{(\alpha)}(|x|)=\frac{\alpha !}{\pi L^{\alpha+1}} \lim_{\epsilon\rightarrow 0+}
\Re \left\{\ i^{\alpha+1} \left( \frac{1}{(\xi+i\epsilon)^{\alpha+1}}
-\zeta(\alpha+1,\xi+i\epsilon)-\zeta(\alpha+1,-\xi+i\epsilon) \right)\right\}
\end{array}
\end{equation}
by accounting for (\ref{convolper}) and above representations for the infinite space fractional Laplacian
(\ref{fractlaplkernel}) and (\ref{distri}).
The fractional Laplacian kernel (\ref{contilimlapfinal}) is defined on the principal interval $0 \leq \xi = \frac{x}{L} \leq 1$ over the length of the string.
(\ref{contilimlapfinal}) represents the periodic string continuum limit $h\rightarrow 0$ with $Nh=L$ finite, of the finite chain fractional Laplacian matrix (\ref{laplafini}).
In (\ref{contilimlapfinal})$_2$ for the (hyper-) singular representation of the kernel, we have introduced a slightly modified version of Hurwitz $\zeta$-functions ${\tilde\zeta}(..)$\footnote{implemented as ``HurwitzZeta" in {\it Mathematica}
Eq. (2)
at http://mathworld.wolfram.com/HurwitzZetaFunction.html }. The distributional representation (\ref{contilimlapfinal})$_4$ is expressed by standard Hurwitz $\zeta$-functions denoted by $\zeta(..)$. These two variants of $\zeta$- functions are defined by \cite{whitaker}
\begin{equation}
\label{hurwitz}
{\tilde \zeta}(\beta,x)= \sum_{n=0}^{\infty}\frac{1}{|x+n|^{\beta}} \, ,\hspace{2cm} \zeta(\beta,x)=\sum_{n=0}^{\infty}
\frac{1}{(x+n)^{\beta}} ,\hspace{1.5cm} \Re\, \beta >1
\end{equation}
We see for $\alpha>0$ and $x\neq 0$ that the series in (\ref{contilimlapfinal}) is absolutely convergent as good as the power function integral $\int_1^{\infty}\xi^{-\alpha-1}{\rm d}\xi$. The expressions (\ref{contilimlapfinal})$_{3,4}$ represent the regularized distributional representation of the kernel where
(\ref{distri}) has been taken into account. In (\ref{contilimlapfinal})$_{3,4}$ we take into account that $\Re \frac{i^{\alpha+1}}{(\xi+i\epsilon)^{\alpha+1}} = \Re \frac{i^{\alpha+1}}{(-\xi+i\epsilon)^{\alpha+1}} $ is for $\epsilon\rightarrow 0+$ an even distribution with respect to $\xi$.
In full correspondence with the symmetries of the matrix elements of the finite chain fractional matrix, the periodic string fractional Laplacian kernel (\ref{contilimlapfinal}) fulfills the symmetries (\ref{periodmirror}), i.e. $L$-periodicity and reflection-symmetry with respect to the axes $x_n= \frac{L}{2}+nL$ ($n\in {\bf \Z}_0$).
We see in view of the hyper-singular representations (\ref{contilimlapfinal})$_{1,2}$ that $K_L^{(\alpha)}$ has periodically repeating singularities at $x=nL$ ($n\in {\bf \Z}_0$).
In the limiting case when $\frac{\alpha}{2}=m\in {\bf \N}$ takes integers, kernel (\ref{contilimlapfinal})
is vanishing everywhere except at the singularities (due to $\sin{\frac{\pi\alpha}{2}}=0$ for $\frac{\alpha}{2} \in {\bf \N}$).
In view of the regularized representation (\ref{contilimlapfinal})$_{3,4}$ and by accounting for (\ref{integerorder}) we notice
that (\ref{contilimlapfinal}) takes then the form
\begin{equation}
\label{integerperiodic}
\begin{array}{l}
\displaystyle K_L^{(\alpha=2m)}(|x|)= (-1)^{m+1}\frac{d^{2m}}{dx^{2m}} \sum_{n=-\infty}^{\infty} \lim_{\epsilon\rightarrow 0+}\frac{1}{\pi}\frac{\epsilon}{((x-nL)^2+\epsilon^2)} ,\hspace{1cm} \frac{\alpha}{2} = m\in {\bf \N_0}\nonumber \\ \nonumber \\
\displaystyle \hspace{3cm} =
(-1)^{m+1}\frac{d^{2m}}{dx^{2m}} \sum_{n=-\infty}^{\infty}\delta_{\infty}(x-nL) = -\left(-\frac{d^2}{dx^2}\right)^{\frac{\alpha}{2}=m}\delta_L(x)
\end{array}
\end{equation}
which indeed is, by accounting for the representation of (\ref{periodicdelta})
for the $L$-periodic $\delta_L$-function, the distributional representation of the {\it integer-order
Laplacian on the $L$-periodic space}. This includes when $\alpha$ approaches the (forbidden value) zero $\frac{\alpha}{2} \rightarrow 0+0$ the (negative) $L$-periodic unity operator $-\delta_L(x)$.
The integer cases (\ref{integerperiodic}) contain for $m=\frac{\alpha}{2}=1$ the continuum limit of the classical $L$-periodic Born von Karman chain, leading to classical elasticity governed by the Laplacian kernel $K_L^{(2)}(|x|)=\frac{d^2}{dx^2}\delta_L(x)$.
One can further see that in the limiting case of an infinitely long string $L\rightarrow\infty$ (\ref{contilimlapfinal})
recovers the infinite space fractional Laplacian of (\ref{fractlaplkernel}) and (\ref{distri})
where the series of the image terms \newline $\displaystyle \sim \sum_{n=1}^{\infty}
\left\{(|x\pm nL|^{-\alpha-1} + |x\pm nL|^{-\alpha-1}\right\} \sim L^{-\alpha}$
tends to zero as $L^{-\alpha}$.
\newline\newline
\begin{figure*}[H]
\hskip3.5cm
\includegraphics[scale=0.5]{Kernel1.pdf}
\caption{$L$-periodic fractional Laplacian kernel $K_L^{(\alpha)}$ of (\ref{contilimlapfinal}) for $L=1$.}
\label{fig:2}
\end{figure*}
{\bf Figure 2}
\newline
{\it In figure 2 the $L$-periodic fractional Laplacian kernel $K_L^{(\alpha)}$ of (\ref{contilimlapfinal}) is plotted for $L=1$
over two periods $0< x < 2$ for $0< \alpha < 4$. Due to the $\sin{(\frac{\pi\alpha}{2})}$-multiplyer, the kernel is positive the interval $0< \alpha< 2$, vanishing at $\alpha=2$ and taking negative values for $2<\alpha<4$. When approaching the boundaries $x_n=0,1,2,..$, the kernel quickly takes huge values absolute values (positive values for $0\leq \alpha \leq 2$, and negative values for $2\leq \alpha < 4$).}
\newline\newline
Further, we see from the general relation (\ref{fourier}) when applying it to (\ref{contilimlapfinal})
that
\begin{equation}
\label{eigeneq}
\begin{array}{l}
\displaystyle \int_0^LK_L^{(\alpha)}(|x-x'|)\Phi_l(x') {\rm d}x'\cdot =
-\Phi_l(x)\frac{\alpha !}{\pi}
\Re\int_{-\infty}^{\infty}\frac{e^{ik_lx}}{(\epsilon-ix)^{\alpha+1}}{\rm d}x \nonumber \\ \nonumber \\ \displaystyle \hspace{4cm} = -\Phi_l(x)\frac{1}{2\pi}\Re \int_{-\infty}^{\infty}{\rm d}\tau |\tau|^{\alpha} \int_{-\infty}^{\infty} e^{i x(k_l+\tau)}{\rm d}x \nonumber \\ \nonumber \\ \nonumber \\
\displaystyle \hspace{4cm} = - \Phi_l(x)\Re \int_{-\infty}^{\infty} |\tau|^{\alpha}\delta(\tau+k_l) {\rm d}\tau \nonumber \\ \nonumber \\
\displaystyle \hspace{4cm} = - |k_l|^{\alpha} \Phi_l(x)
\end{array}
\end{equation}
This convolution with the Bloch eigenfunctions $\Phi_l$ indeed yields the discrete eigenvalues $-|k_l|^{\alpha} = -(\frac{2\pi }{L})^{\alpha}|l|^{\alpha}$
($l \in {\bf \Z}_0$), especially with the lowest eigenvalue $k_0^{\alpha}=0$ (since $\alpha >0$) which means that application of the $L$-periodic fractional Laplacian to a constant is zero.
Indeed we have with (\ref{fourierser}) and (\ref{eigeneq}) the spectral representation of the $L$-periodic fractional Laplacian
\begin{equation}
\label{fourierserfractalperi}
-\left(-\frac{d^2}{dx^2}\right)^{\frac{\alpha}{2}}\delta_L(x-x')
= - \sum_{l=-\infty}^{\infty} |k_l|^{\alpha} \Phi_l(x)\Phi_l^*(x'),\hspace{0.5cm} \Phi_l(x)=\frac{e^{ik_lx}}
{\sqrt{L}} ,\hspace{0.3cm} k_l=\frac{2\pi}{L}l ,\hspace{0.3cm} l \in {\bf \Z}_0
\end{equation}
\noindent {\it Zero dimensional string (point) continuum limit.}
A further observation is mention worthy, namely in the limiting case $L\rightarrow 0$ when the length of the string shrinks to a point (limiting case of a zero-dimensional string), the summation over $n$ in the series (\ref{contilimlapfinal}) takes the form of an integral over the quasi-continuous variable $\xi=nL$ where the resulting fractional Laplacian kernel for the zero-dimensional string yields due to
\begin{equation}
\lim_{\epsilon\rightarrow 0+}\Re\int_{-\infty}^{\infty}
\frac{i^{\alpha+1}}{(x+i\epsilon)^{\alpha+1}}{\rm d}\xi = 0
\end{equation}
a vanishing constant.
\section{Conclusions}
In this paper we have deduced the fractional discrete Laplacian matrices in explicit forms for the infinite chain ($N\rightarrow \infty$),
and the $N$-periodic finite chain where the particle number $N$ is arbitrary and not necessarily large. The $N$-periodic finite chain fractional Laplacian matrix is obtained in explicit form in terms of an infinite series of the
infinite chain fractional Laplacian matrix.
These results appear to be useful to be generalized to define discrete
fractional operators on finite lattices in $n=1,2,3,..$ dimensions of the physical space.
Further we analyze continuum limits of the chain fractional Laplacians: The infinite space continuum limit yields the distributional representations of the well known 1D fractional Laplacian kernel (Riesz fractional derivative). The infinite space representation is utilized to deduce
the $L$-periodic kernel of the fractional Laplacian on the $L$-periodic string. For all these cases, infinite space and periodic string, respectively, the fractional Laplacians take in the special case of integer exponents $\frac{\alpha}{2} \in {\bf \N}_0$ the distributional representations of the corresponding
integer-orders of the Laplacian.
The $L$-periodic string fractional Laplacian kernel represents the periodic string continuum limit of the discrete fractional Laplacian matrix of the finite $N$-periodic chain. The exact representation of the periodic string fractional Laplacian kernel is expressed in compact form
by Hurwitz type $\zeta$-functions. Especially this representation appears to be useful for
computational purposes.
The present model has the potential to be extended to construct the fractional Laplacians in finite domains when boundary conditions are imposed. This could be done by constructing an appropriate projection formalism, where a unit operator (projector kernel) has to be constructed which spans the space of functions fulfilling the prescribed boundary conditions.
Generalizations of the discrete fractional matrices deduced in this paper can be extended to
nD periodic unit-cells (nD cyclically closed tori). In this way a discrete fractional
calculus on finite $nD$ lattices can be developed.
There is a vast field of open problems in anomalous diffusion (L\'evy flights), wave propagation, fractional Quantum Mechanics, turbulence and many other areas where the exact representations for the discrete and continuous fractional Laplacian as deduced in this paper can be useful.
Especially a challenging task could be the development of formulations of
discrete fractional Quantum Mechanics by utilizing the 1D discrete fractional operators of the present paper and their $nD$
generalizations.
\section{Acknowldgements}
Fruitful discussions with G\'erard Maugin are greatfully acknowledged.
\section{Appendix I}
In the derivations to follow, always the main definition (\ref{gammafu}) of the $\Gamma$-function is assumed, i.e. the arguments of $\Gamma$-functions are positive.
Those cases where the analytically continued definition (\ref{recursiongamma}) is employed are explicitly marked.
\newline\newline
Let us evaluate in details the important integral (\ref{fractlattice}) for the matrix elements of the fractional Laplacian matrix
\begin{equation}
\label{fractlatticeb}
f^{(\alpha)}(|p|) = \frac{\Omega_{\alpha}^2}{2\pi}\int_{-\pi}^{\pi}e^{i\kappa p}\left(4\sin^2{\frac{\kappa}{2}}\right)^{\frac{\alpha}{2}} {\rm d}\kappa =\Omega_{\alpha}^2 \frac{2^{\alpha + 1}}{\pi}\int_{0}^{\frac{\pi}{2}}
\sin^{\alpha}(\varphi)\cos{(2p\varphi}){\rm d}\varphi \hspace{0.5cm} \alpha >0 ,\hspace{0.5cm} p\in {\bf \Z}_0
\end{equation}
Introducing $\xi=\sin^2(\varphi)$ (${\rm d}\varphi =2^{-1}\left[\xi(1-\xi)\right]^{-\frac{1}{2}}{\rm d}\xi$) with $0 \leq \xi \leq 1$ and $\cos{\varphi} = \sqrt{1-\xi} \geq 0,\sin{\varphi}=\sqrt{\xi} \geq 0$ where $0\leq \varphi \leq \frac{\pi}{2}$.
Further let us put in the following deduction $p=|p|$. Then (\ref{fractlatticeb}) writes as
\begin{equation}
\label{matelwrites}
f^{(\alpha)}(|p|) = \Omega_{\alpha}^2\frac{2^{\alpha}}{\pi}\Re \int_{0}^{1} \xi^{\frac{\alpha}{2}-\frac{1}{2}} (1-\xi)^{-\frac{1}{2}}\left(\sqrt{1-\xi}+i\sqrt{\xi} \right)^{2p} {\rm d}\xi
\end{equation}
Then we utilize
\begin{equation}
\displaystyle \cos{(2p\varphi}) = \Re\{(\sqrt{1-\xi}+i \sqrt{\xi})^{2p} \} =\sum_{s=0}^p \frac{(2p)!}{(2s)!(2p-2s)!}(-1)^s\xi^s(1-\xi)^{p-s}
\end{equation}
where $\Re(..)$ denotes the real part of $(..)$. Further we account for
\begin{equation}
\label{factorials}
(2n)! = 2^{2n}n!\frac{(n-\frac{1}{2})!}{(-\frac{1}{2})!} \hspace{1cm} n\in {\bf \N}_0
\end{equation}
so that (\ref{matelwrites}) can be written as
\begin{equation}
\label{easytosee}
f^{(\alpha)}(|p|) = \Omega_{\alpha}^2 \frac{2^{\alpha}}{\sqrt{\pi}} \frac{(2p)!}{2^{2p}!p!}\times
\int_{0}^{1}\xi^{\frac{\alpha}{2}}\sum_{s=0}^p\frac{p!}{s!(p-s)!}(-1)^s
\frac{\xi^{s-\frac{1}{2}}}{(s-\frac{1}{2})!}\frac{(1-\xi)^{p-s-\frac{1}{2}}}{(p-s-\frac{1}{2})!}{\rm d}\xi
\end{equation}
where
\begin{equation}
\label{productderivative}
\sum_{s=0}^p\frac{p!}{s!(p-s)!}(-1)^s
\frac{\xi^{s-\frac{1}{2}}}{(s-\frac{1}{2})!}\frac{(1-\xi)^{p-s-\frac{1}{2}}}{(p-s-\frac{1}{2})!} = \frac{1}{(p-\frac{1}{2})!(p-\frac{1}{2})!}\frac{d^p}{d\xi^p}\left\{\xi(1-\xi) \right\}^{p-\frac{1}{2}}
\end{equation}
Thus we obtain (\ref{fractalat})
\begin{equation}
\label{fractalatb}
f^{(\alpha)}(|p|) = \Omega_{\alpha}^2 \frac{2^{\alpha}}{\sqrt{\pi}}\frac{1}{(p-\frac{1}{2})!} \int_0^1\xi^{\frac{\alpha}{2}}\frac{d^p}{d\xi^p}\left\{\xi(1-\xi)\right\}^{p-\frac{1}{2}}{\rm d}\xi
\end{equation}
where $(-\frac{1}{2})!=\Gamma(\frac{1}{2})=\sqrt{\pi}$. Note that for the (in our model) forbidden exponent $\alpha=0$ the matrix element (\ref{fractalatb}) yields $f^{(\alpha=0)}(|p|) = \Omega_0^2\,\delta_{p0}$ in accordance with ${\hat 1}\Omega_{0}^2$ of
relation (\ref{alphacarfou}) which also can directly be seen from (\ref{fractlatticeb}).
Now consider
\begin{equation}
\label{considerintehral}
\int_0^1\xi^{\frac{\alpha}{2}}\frac{d^p}{d\xi^p}\left\{\xi(1-\xi)\right\}^{p-\frac{1}{2}}{\rm d}\xi = \left\{\xi^{\frac{\alpha}{2}}\frac{d^{p-1}}{d\xi^{p-1}}\left\{\xi(1-\xi)\right\}^{p-\frac{1}{2}} \right\}_0^1 -\frac{\alpha}{2}\int_0^1\xi^{\frac{\alpha}{2}-1}\frac{d^{p-1}}{d\xi^{p-1}}\left\{\xi(1-\xi)\right\}^{p-\frac{1}{2}}{\rm d}\xi
\end{equation}
The lowest relevant orders of the boundary term $\{..\}$ is behaving as $\sim \xi^{\frac{\alpha+1}{2}}(1-\xi)^{\frac{1}{2}} $ + higher order terms and is hence vanishing at the boundary $\xi=0,1$. Performing $n \leq p$ times partial integration yields boundary terms with
lowest powers in $\xi$ and $1-\xi$ of the form
\begin{equation}
\label{boundaryterms}
\sim \xi^{\frac{\alpha}{2}-(n-1)}\frac{d^{p-n}}{d\xi^{p-n}}\left\{\xi(1-\xi)\right\}^{p-\frac{1}{2}}\sim
\xi^{\frac{\alpha+1}{2}}(1-\xi)^{n-\frac{1}{2}} ,\hspace{1cm} 1\leq n\leq p
\end{equation}
which are vanishing at the boundary $\xi=0,1$. Performing this procedure $p$ times yields then
\begin{equation}
\label{ntimes}
\begin{array}{l}
\displaystyle \int_0^1\xi^{\frac{\alpha}{2}}\frac{d^p}{d\xi^p}\left\{\xi(1-\xi)\right\}^{p-\frac{1}{2}}{\rm d}\xi = \int_0^1\xi^{\frac{\alpha}{2}-p}\left\{\xi(1-\xi)\right\}^{p-\frac{1}{2}}{\rm d}\xi \times\, (-1)^p \prod_{s=0}^{p-1}(\frac{\alpha}{2}-s)\nonumber \\ \nonumber\\
\displaystyle = \int_0^1\xi^{\frac{\alpha -1}{2}}\left\{(1-\xi)\right\}^{p-\frac{1}{2}}{\rm d}\xi \times\, (-1)^p \prod_{s=0}^{p-1}(\frac{\alpha}{2}-s)\nonumber \\ \nonumber \\
\displaystyle = \frac{\frac{\alpha-1}{2}!(p-\frac{1}{2})!}{(\frac{\alpha}{2}+p)!}\times\, (-1)^p \prod_{s=0}^{p-1}(\frac{\alpha}{2}-s)
\end{array}
\end{equation}
where in the last line we have used
\begin{equation}
\label{Bfunction}
\int_0^1\xi^{\beta_1}(1-\xi)^{\beta_2}{\rm d}\xi =\frac{\beta_1!\beta_2!}{(\beta_1+\beta_2+1)!},\hspace{1cm} \Re\, \beta_i >-1
\end{equation}
With (\ref{ntimes})$_3$ we get for the matrix element (\ref{fractalatb}) where always $p=|p|$
\begin{equation}
\label{matrixele}
f^{(\alpha)}(|p|) = \Omega_{\alpha}^2\frac{2^{\alpha}}{\sqrt{\pi}}\frac{\frac{(\alpha-1)}{2}!}{(\frac{\alpha}{2}+p)!}(-1)^p\prod_{s=0}^{p-1}(\frac{\alpha}{2}-s)
\end{equation}
To get this into a more convenient form we consider (\ref{Bfunction}) for $\beta_1=\beta_2=\frac{\alpha-1}{2}$
and by introducing $\xi=\frac{1}{2}(1+\sqrt{\eta})$ (${\rm d}\xi= 2^{-2}\eta^{-\frac{1}{2}}{\rm d}\eta$).
Then we have
\begin{equation}
\label{Bfualp}
\frac{(\frac{(\alpha-1)}{2})!(\frac{(\alpha-1)}{2})!}{\alpha !} =
2^{-\alpha}\int_0^1(1-\eta)^{\frac{(\alpha-1)}{2}}\eta^{-\frac{1}{2}}{\rm d}\eta = 2^{-\alpha} \frac{(\frac{(\alpha-1)}{2})!}{\frac{\alpha}{2}!}(-\frac{1}{2})!
\end{equation}
which is known as duplication formula \cite{abramo}. It follows with $(-\frac{1}{2})!=\sqrt{\pi}$ that
\begin{equation}
\label{2alphapi}
\frac{\alpha !}{\frac{\alpha}{2}!} = \frac{2^{\alpha}}{\sqrt{\pi}}(\frac{(\alpha-1)}{2})!
\end{equation}
Plugging (\ref{2alphapi}) into(\ref{matrixele}) yields a more illuminating representation, namely
\begin{equation}
\label{matrixeleibb}
f^{(\alpha)}(|p|) = \Omega_{\alpha}^2\frac{\alpha!}{\frac{\alpha}{2}!(\frac{\alpha}{2}+p)!}(-1)^p\prod_{s=0}^{p-1}(\frac{\alpha}{2}-s)
\end{equation}
We can distinguish two cases, namely (i) $ p \leq p_0=ceil(\frac{\alpha}{2}) $, and (ii) $ p \geq p_0=ceil(\frac{\alpha}{2})$ where (ii) is only relevant for non-integer $\frac{\alpha}{2}$ and then always equivalent to $p>\frac{\alpha}{2}$.
\newline\newline
{\bf (i) Case $ p \leq p_0=ceil(\frac{\alpha}{2}) $}:
\newline
Now we observe that we can write
\begin{equation}
\label{observation}
\prod_{s=0}^{p-1}(\frac{\alpha}{2}-s) = \frac{\frac{\alpha}{2}!}{(\frac{\alpha}{2}-p)!}
\end{equation}
since $(\frac{\alpha}{2}-p)!$ is well defined in case (i). Thus we can write
\begin{equation}
\label{matrixelecasei}
f^{(\alpha)}(|p|) = \Omega_{\alpha}^2(-1)^p\frac{\alpha!}{(\frac{\alpha}{2}-p!)(\frac{\alpha}{2}+p)!}
\end{equation}
which obviously is a generalization of the binomial coefficients including integer and non-integer $\alpha$.
Note this expression obviously contains the cases of {\it integers $\frac{\alpha}{2} = m \in {\bf \N}$} with the binomial coefficients of
(\ref{binomialco}). In that case all matrix elements (\ref{matrixelecasei}) with $p\geq \frac{\alpha}{2}+1$ are vanishing.
When $\frac{\alpha}{2} \notin {\bf \N}$ is not an integer, expression (\ref{matrixelecasei}) covers all
$p$ with $0\leq p \leq p_0={\rm ceil}(\frac{\alpha}{2})$ where $p_0={\rm ceil}(\frac{\alpha}{2})$ indicates the smallest integer greater $\frac{\alpha}{2}$ ($0<p_0-\frac{\alpha}{2}<1$).
\newline\newline
{\bf (ii) Case $ p \geq p_0 =ceil(\frac{\alpha}{2}) $ \hspace{0.25cm} ($p>\frac{\alpha}{2}$)}:
\newline
This case is only relevant when $\frac{\alpha}{2}$ is not integer, i.e. $p_0-\frac{\alpha}{2} >0 $.
In this case where $(\frac{\alpha}{2}-p)!$ is not well defined for $p>p_0$ when the main definition of the $\Gamma$-function (\ref{gammafu}) is assumed. It is convenient now to write the product in different manner, namely
\begin{equation}
\label{caseii}
\prod_{s=0}^{p-1}(\frac{\alpha}{2}-s) = \prod_{s=0}^{p_0-1}(\frac{\alpha}{2}-s)\prod_{s=p_0}^{p-1}(\frac{\alpha}{2}-s) =
\frac{\frac{\alpha}{2}!}{(\frac{\alpha}{2}-p_0)!}\prod_{s=p_0}^{p-1}(\frac{\alpha}{2}-s)
,\hspace{1cm} p_0={\rm ceil}(\frac{\alpha}{2})
\end{equation}
Taking into account that for $\frac{\alpha}{2}\notin {\bf \N}$ it holds $0 < p_0-\frac{\alpha}{2} < 1$, i.e. $\Gamma(p_0-\frac{\alpha}{2})$ is well defined, so that we can write for the second product in (\ref{caseii})
\begin{equation}
\label{furtherel}
\prod_{s=p_0}^{p-1}(\frac{\alpha}{2}-s) =(-1)^{p-p_0}\frac{(p-1-\frac{\alpha}{2})!}{(p_0-1-\frac{\alpha}{2})!} =(-1)^{p-p_0}
\frac{\Gamma(p-\frac{\alpha}{2})}{\Gamma(p_0-\frac{\alpha}{2})}
\end{equation}
We then can apply the Euler relation (e.g. \cite{abramo,michelb} briefly deduced in appendix II)
\begin{equation}
\label{Euler}
\Gamma(p_0-\frac{\alpha}{2})=\frac{1}{\Gamma(1-(p_0-\frac{\alpha}{2}))}\frac{\pi}{\sin{(\pi(p_0-\frac{\alpha}{2})}}
=(-1)^{p_0+1}\frac{\pi}{(\frac{\alpha}{2}-p_0)!\sin{\frac{\alpha\pi}{2}}}
\end{equation}
where all arguments of $\Gamma$-functions are positive since $0<p_0-\frac{\alpha}{2}< 1$ {\it and} $0<1-(p_0-\frac{\alpha}{2})<1$.
Utilizing (\ref{caseii}) with (\ref{Euler}) and (\ref{observation}) for $p=p_0$ we get
for the entire product
\begin{equation}
\label{takeacount}
\prod_{s=0}^{p-1}(\frac{\alpha}{2}-s) = \frac{(-1)^{p+1}}{\pi}(\frac{\alpha}{2})! \sin{(\frac{\alpha\pi}{2})}
\,\,\Gamma(p-\frac{\alpha}{2}) ,\hspace{1cm} p\geq p_0=ceil(\frac{\alpha}{2})
\end{equation}
The matrix elements (\ref{matrixelei}) finally assume the form (where always $p=|p|$)
\begin{equation}
\label{matrixform}
f^{(\alpha)}(|p|) = -\Omega_{\alpha}^2\frac{\Gamma(\alpha+1)}{\pi}\sin{(\frac{\alpha\pi}{2})}\frac{\Gamma(p-\frac{\alpha}{2})}{\Gamma(\frac{\alpha}{2}+p+1)} = -\Omega_{\alpha}^2 \frac{\alpha!}{\pi}\sin{(\frac{\alpha\pi}{2})}\frac{(p-\frac{\alpha}{2}-1)!}{(\frac{\alpha}{2}+p)!} ,\hspace{1cm} p > \frac{\alpha}{2}
\end{equation}
We see that this relation vanishes in the integer cases $\frac{\alpha}{2} \in {\bf \N}$ for $p>\frac{\alpha}{2}$ (since then $\sin{(\frac{\alpha\pi}{2})}=0$), reflecting the localization of
$f^{\alpha}(|p|)$ where only the elements (\ref{matrixelecasei}) assuming then (\ref{binomialco}) for $0\leq p \leq m$ ($m=\frac{\alpha}{2} \in {\bf \N}$) are non-vanishing.
Hence (\ref{matrixform}) holds for $p>\frac{\alpha}{2}$ for any integer or non-integer $\frac{\alpha}{2} > 0$.
Its validity can extended to all $p\in {\bf Z}_0$ when instead of the main
definition of the $\Gamma$-function (\ref{gammafu}), the analytically continued
definition of the $\Gamma$-function (\ref{recursiongamma}) is assumed.
\subsection{Appendix II}
Let us briefly deduce the Euler relation. To this end consider the Fourier integral $0< \mu <1$
\begin{equation}
\label{fourierintmu}
\begin{array}{l}
\displaystyle \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{ikx}|k|^{-\mu}{\rm d}k =\frac{1}{\pi}\Re\lim_{\epsilon\rightarrow 0+}\int_0^{\infty}e^{-k(\epsilon-ix)}k^{-\mu}{\rm d}k ,\hspace{1cm} 0\leq \mu<1 \nonumber \\ \nonumber \\
\displaystyle =\lim_{\epsilon\rightarrow 0+} \frac{1}{\pi}\Re\,(\epsilon-ix)^{\mu-1}\int_0^{\infty}e^{-\tau}\tau^{-\mu}{\rm d}\tau = \lim_{\epsilon\rightarrow 0+}\frac{\Gamma(1-\mu)}{\pi} \Re\,(\epsilon-ix)^{\mu-1} \nonumber \\ \nonumber \\
\displaystyle = \frac{|x|^{\mu-1}}{\pi} \Gamma(1-\mu)\sin{(\frac{\mu\pi}{2})} \hspace{1cm} x \neq 0
\end{array}
\end{equation}
which is a well-defined integral in the range $0 < \mu<1$. The inverse Fourier transformation of (\ref{fourierintmu})
gives then
\begin{equation}
\label{inverse}
\begin{array}{l}
\displaystyle |k|^{-\mu} = \frac{\Gamma(1-\mu)}{\pi}\sin{(\frac{\mu\pi}{2})}\int_{-\infty}^{\infty} e^{-ikx} |x|^{\mu-1}{\rm d}x
\hspace{0.5cm} \nonumber \\ \nonumber \\
\displaystyle |k|^{-\mu} = \frac{2}{\pi}\Gamma(1-\mu)\sin{(\frac{\mu\pi}{2})}
\lim_{\epsilon\rightarrow 0+}\Re \int_0^{\infty} e^{-x(\epsilon+ik)}x^{\mu-1}{\rm d}x = \frac{2}{\pi}\Gamma(1-\mu)\sin{(\frac{\mu\pi}{2})}\lim_{\epsilon\rightarrow 0+}\Re(\epsilon+ik)^{-\mu} \int_0^{\infty} e^{-\tau}\tau^{\mu-1}{\rm d}\tau \nonumber \\ \nonumber \\
\displaystyle 1 = \, \frac{2}{\pi}\cos{(\frac{\pi\mu}{2})}\sin{(\frac{\pi\mu}{2})} \Gamma(1-\mu)\Gamma(\mu) = \frac{\sin{(\pi\mu)}}{\pi}\Gamma(\mu)\,\,\Gamma(1-\mu)
\end{array}
\end{equation}
where the last relation (\ref{inverse})$_3$ is the Euler relation, also referred to as Euler reflection formula \cite{abramo} employed in (\ref{Euler}) (put there $\mu=p_0-\frac{\alpha}{2}$ and $p_0=ceil(\frac{\alpha}{2})$).
So far we still are restricted to $0< \mu < 1$. Let us consider now arbitrary arguments of the $\Gamma$-functions (except negative integers and zero) by utilizing the analytically continued recursive definition of the $\Gamma$-function (\ref{recursiongamma}).
Then we observe for $n\in {\bf \N}_0$
\begin{equation}
\label{extend}
\begin{array}{l}
\displaystyle \Gamma(1-\mu) = (-\mu)! = (-1)^n\Gamma(1-\mu-n)\,\prod_{s=0}^{n-1}(\mu+s) \nonumber \\ \nonumber \\
\displaystyle \Gamma(\mu) = (\mu-1)! = \Gamma(\mu+n)\,\prod_{s=0}^{n-1}\frac{1}{(\mu+s)}
\end{array}
\end{equation}
and obtain the identity
\begin{equation}
\label{gammagen}
\Gamma(1-\mu)\Gamma(\mu)=(-1)^n\Gamma(1-\mu-n)\Gamma(\mu+n) = \frac{\pi}{\sin{(\pi\mu)}}
\end{equation}
and by taking into account $(-1)^n\sin{(\pi\mu)}= \sin{\pi(\mu+n)}$, then (\ref{gammagen}) takes the form
\begin{equation}
\label{eulergen}
\Gamma(1-\mu-n)\Gamma(\mu+n) = \frac{\pi}{\sin{(\pi(\mu+n))}}
\end{equation}
which is Euler's reflection formula for arbitrary including negative non-integer real
arguments of the analytically continued $\Gamma$-function, where arguments of negative integers and zero are to be excluded due to the singularities at these points of the $\Gamma$-function.
|
train/arxiv
|
BkiUdhXxaKgTz-oDsOPy
| 5 | 1 |
\section{Conclusion}
We present a compositional 3D generative model of objects, where different semantically meaningful parts of objects are represented as independent 3D models.
Different parts can be independently sampled to create photorealistic images.
The disentanglement of a global model into parts is achieved using losses in image space, defined with the help of segmentation masks.
Unlike many existing approaches, our parts are semantically well-defined.
We validate our method using several experiments on portrait images and cat face images, and compare to several baseline and competing methods.
Compositional models have a lot of applications in Computer Vision and Computer Graphics, and we hope that our method inspires follow-up work in this area.
\section{Introduction}
3D GANs are generative models capable of producing 3D representations as output that can then be rendered into RGB images.
These models produce high-quality real images with explicit control over the camera parameters.
While these models reason about the 3D scene as a whole, objects in the real world is made up of many different semantic components composed together.
Compositional reasoning is thus very useful for scene understanding and synthesis.
Compositional models allow for independent control of the different parts, while keeping rest of the sample fixed.
For example, even though real world portrait datasets mostly contain images of male subjects with short hair, compositional reasoning of the face and hair regions allows for generalization to images of male subjects with long hair.
While compositional models have been studied in 2D ~\cite{zhu2015learning,azadi2020compositional,lin2018st,tsai2017deep}, they have been underexplored for 3D learning.
The approach of Yang~\etal~\cite{yang2021objectnerf} learns compositional object-centric NeRF models from supervised multi-view videos of a scene.
While this approach is scene specific, our goal instead is to learn from monocular and static image collections.
GIRAFFE~\cite{Niemeyer2020GIRAFFE} trains a compositional 3D GAN which uses a NeRF-like volumetric representation to model the scene and separates the scene into multiple components in an unsupervised manner.
However, GIRAFFE only models composition of multiple intact objects, but cannot model semantic parts within each object, which is our main focus in this work.
Thus, we present gCoRF, Generative Compositional Radiance Fields for synthesizing
3D volumes of objects represented as a composition of semantic parts.
We train our models on datasets of just monocular in-the-wild image collections.
In addition, we use an automatically computed segmentation map for each image to define the semantic parts.
Thus, unlike existing methods like GIRAFFE~\cite{Niemeyer2020GIRAFFE}, the definition of parts is well-defined in our case.
Our method learns to represent each object instance in 3D using multiple volumetric models, each representing one semantic part. %
The parts are composited using volumetric blending.
We initialize all the part models with a global object model, and then learn to decouple it in a supervised manner.
We also train a blending network which learns to align these sampled parts into a coherent scene, which can be rendered from any viewpoint.
At test time, each semantic part can be independently changed while keeping rest of the volume fixed.
In summary, we make the following contributions:
\begin{enumerate}
\item A method to learn decoupled semantic part 3D GANs of an object category using monocular image collections and corresponding semantic segmentation maps.
\item Given decoupled part 3D GANs, we learn a blending network which can composite independently sampled parts to synthesize a coherent object volume.
\item We show applications such as volumetric editing of various face parts (including eyes, eyebrows, nose and hair style), keeping rest of the face fixed. Here, we process both human faces and cat faces.
\end{enumerate}
\section{Method}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{images/pipeline2.pdf}
\caption{Overview of our method. We first decompose a global GAN into several part models using supervised losses guided by segmentation masks.
We then train a blending network which learns to composite the independently sampled parts into a coherent scene.
This network modifies the intermediate latent vectors of the part models.
At test time, our model enables independent sampling of the different parts in order to create photorealistic images.
}
\label{fig:pipeline}
\end{figure*}
Our method learns a compositional GAN model from just monocular in-the-wild images.
The compositional model is part-based, where each part is represented using its own MLP network as shown in Fig.~\ref{fig:pipeline}.
This enables novel-view synthesis, as well as independent control over different parts of the object.
Our method consists of three stages.
First, we learn a non-compositional global-scene GAN following existing methods~\cite{piGAN2021}.
Second, we learn to disentangle this global model into multiple models each representing a different part (Stage 1 of Fig.~\ref{fig:pipeline}).
Finally, we learn a blending network that blends the different parts into a coherent scene for synthesis (Stage 2 of Fig.~\ref{fig:pipeline}).
\subsection{Compositional Model}
\label{sec:compositional}
Our compositional model consists of several parts, each represented using a different 3D volume.
Thus, we have MLPs $N_i$, $i=1,\dots,P$, corresponding to $P$ parts in the object.
The MLPs are defined as $N_i \colon (\*z_i, \*x) \to {\sigma_i, \*c_i}$.
Here, $\*z_i$ are the latent vectors for each part, which can be independently randomly sampled from a Gaussian distribution.
We use volumetric integration to render each part independently as $R(\*z_i, \*C)$ for a camera with parameters $\*C$.
The output of this function is the rendered image for the part.
For each pixel, we cast a camera ray $\mathbf{r}(t) = \mathbf{o}+ t\mathbf{d}$ with near and far bounds $t_n$ and $t_f$, origin $\mathbf{o}$ and direction $\mathbf{d}$, and compute the pixel color as:
\begin{align}
\mathbf{C}(\mathbf{r}) &= \int_{t_n}^{t_f} T(t) \sigma_i(\mathbf{r}(t)) c_i(\mathbf{r}(t)) dt \nonumber \\
\text{where} \quad & T(t) = \text{exp}(-\int_{t_n}^{t}\sigma_i(\mathbf{r}(s)) ds).
\label{eq:nerf}
\end{align}
In practice, we use a discretized version of this equation, as explained in NeRF~\cite{mildenhall2020nerf}.
A complete image of the scene can be rendered as a composition of all the individual parts.
We first define the global volume as a composition of the individual parts, denoted with function $\mathrm{comp} \colon (\sigma_1,\dots,\sigma_P,\*c_1,\dots,\*c_P) \to (\sigma, \*c)$.
The density $\sigma$ at each point of the global volume is a simple summation of the densities of each individual part.
The color $\*c$ at point $\*x$ is computed as a weighted linear interpolation of the colors $\*c_i$, where the weights are the coefficients ($T(t) \sigma_i (\mathbf{r}(t))$ for the ray $\mathbf{r}(t)$) at point $\*x$ computed in the rendering function $R(\*z_i, \*C)$, see eq.~\ref{eq:nerf}.
These coefficients are normalized over all parts.
We can render the composite volume as $R(\mathrm{comp}(z_1,\dots,z_P, c_1,\dots,c_P), C)$.
\subsection{Decoupling Parts}
\label{sec:decoupleparts}
\paragraph{Learning a Global Model}
We use the $\pi$-GAN~\cite{piGAN2021} framework to learn the global GAN.
Here, a single MLP network takes a randomly sampled latent code $\*z$ as input along with 3D coordinates $\*x$.
The output of the network is a scalar density $\sigma$ and color $c \in \mathbb{R}^3$ at point $\mathbf{x}$ of object instance $\mathbf{z}$.
An image can be rendered from this 3D representation using volumetric rendering.
During training, images from the model are synthesized from a random distribution of cameras around the object.
The network is trained in a generative adversarial framework using a discriminator, and a non-saturating adversarial loss~\cite{piGAN2021}.
\begin{align}
\mathcal{L_\text{adv}} &= f \Big( D(R (\*z, C) \Big)
&+ f(-D(\mathbf{I})) + \lambda \|\nabla D(\mathbf{I})\|^2 .
\label{eq:gan}
\end{align}
Here, $R(\cdot)$
is the rendering function as explained in eq.~\ref{eq:nerf}, D is the discriminator,
$\mathbf{I}$ are real images sampled from the training dataset, $f(u) = -\log(1 + \exp(-u))$, and $\lambda$ is the coefficient for $R_1$ regularization.
\paragraph{Generator Network}
The generator networks in our case are parameterized as MLPs.
The randomly sampled latent vector $\*z$ is first mapped to an intermediate space $\*w$ using a ReLU MLP, following~\cite{Karras_2019_CVPR,piGAN2021}.
We use a SIREN MLP with FiLM conditioning on $\*w$ and linear conditioning on the input points to compute the density and color values at the point.
The same design is used for all part-based networks, where $\*z_i$ is first mapped into $\*w_i$.
\paragraph{Training} The second stage of our training is supervised on segmentation maps for the different parts, computed using off-the-shelf segmentation models~\cite{Yu-ECCV-BiSeNet-2018,zhang21}.
The weights of each part network $N_i$ is initialized with the pretrained global network $N$.
The mapping network that maps $\*z_i$ to $\*w_i$ for each part $i$ is fixed during training, and only the SIREN MLP is finetuned.
We sample a dataset of latents $\*z$, map them into $\*w$, and compute images $\*I$ from the global model using randomly distributed cameras.
Additionally, we compute segmentation maps $\*S_i$, $i=1,\dots,P$, where $\*S_i$ is one for pixels where part $i$ is present in the image $\*I$, and zero elsewhere.
Then, we fix $\*z_i = \*z$, $\forall i$.
The goal of this stage is to learn to separate the global model into the different parts.
We use the following loss function for training:
\begin{align*}
\mathcal{L}_\text{part} &= \sum_{i=0}^P || \*S_i \odot (R_i(\*z, \*C) - \*I) ||^2 \\
&+ \sum_{i=0}^P || \*S_i \odot (R(\mathrm{comp}(\*z,\dots,\*z, \*c'_1,\dots,\*c'_P), \*C) - \*I) ||^2 \\ &+ || R(\mathrm{comp}(\*z,\dots,\*z, \*c_1,\dots,\*c_P), \*C) - \*I) ||^2\,.
\end{align*}
Here, the first term compares the image difference of the rendered parts to the ground truth image in the segmented region for the part. %
The second term also compares each part to the image, while each part is rendered to also take the occlusions from other parts into account.
This is done by setting the color of all other parts to white, i.e., for part $i$, $\*c_j' = \*1$, $\forall j \not= i$, and $\*c_i' = \*c_i$.
This encourages the density predictions for the other parts to be zero.
The third term evaluates the rendering of the composited volume with the complete input image.
This training strategy allows us to decouple the different parts of the object in a supervised manner.
We use hyperparameters to balance the different terms. %
Please refer to supplementary document for ablative analysis of each loss term.
\subsection{Blending Network}
\label{sec:blendnetwork}
The decoupling stage allows us to train the individual part generative models.
However, so far, the networks were trained with coupled latent vectors, i.e., identical latent vectors for each part.
Independently sampling the latent vectors at test time can lead to artifacts at this point.
For example, for a model with independent face region and hair parts, the face shape learned in the second stage could be dependent on the hairstyle. %
To enable independent sampling with photorealistic output, we introduce a blending network as $B \colon (\*w_1,\dots,\*w_P) \to (\*d_1,\dots,\*d_P)$.
The output vectors are computed as $\*w'_i = \*w_i + \*d_i, \forall i$.
This network is parameterized as a ReLU MLP.
The goal is to update the latent vectors of the individual parts in order to compose a realistic and coherent global volume.
We use the following loss function to train the blending network:
\begin{align*}
\mathcal{L}_\text{blend} &= \mathcal{L}_\text{adv} + \sum_{i=0}^P || \*d_i ||^2 \\
&+ || R_w(\mathrm{comp}_w(\*w_1',\dots,\*w_P', \*c_1,\dots,\*c_P), \*C) - \\ & R_w(\mathrm{comp}_w(\*w_1,\dots,\*w_P, \*c_1,\dots,\*c_P), \*C)) ||^2\,.
\end{align*}
Here, $\mathcal{L}_\text{adv}$ is the adversarial loss introduced in eq.~\ref{eq:gan}, and the second term encourages the latent differences to be small.
The third term is similar to the second term, where $R_w (\cdot)$ and $\textrm{comp}_w (\cdot)$ are the equivalent functions of $R(\cdot)$ and $\textrm{comp}(\cdot)$ where the input latents are the output of the mapping network.
This term encourages the blending network to not change the latents too drastically.
We use hyperparameters to balance the different terms. %
With this, our method enables independent sampling of the different parts of an object while being able to generate coherent volumes.
\section{Related Works}
\subsection{3D Generative Adversarial Networks}
Generative Adversarial Networks (GANs)~\cite{goodfellow2014generative} have witnessed great success in synthesizing 2D images ~\cite{Karras_2019_CVPR,karras2020analyzing,Karras2021}, but cannot fully model the 3D nature of the visual world.
Recently, to enable 3D-aware image synthesis that allows explicit viewpoint control, there is a surge of interest to develop 3D GANs.
While some works use 3D supervision~\cite{wu2016learning,chen2021decor,Gao19,yenamandra2021i3dmm}, we focus on approaches trained on monocular image collection only as images are easy to collect in large scale.
The methods~\cite{nguyen2019hologan,nguyen2020blockgan,liao2020towards} combine voxelized feature grid with 2D convolutional neural renderer for 3D-aware image synthesis.
While achieving promising results, the 3D consistency is limited.
Henzler \etal~\cite{henzler2019escaping} and Szabo \etal~\cite{szabo2019unsupervised} learn to directly generate voxels and meshes respectively, but show artifacts due to the difficulty in training.
Recently, the prosperous of NeRF~\cite{mildenhall2020nerf} has motivate researchers to use coordinate-based implicit functions as the representation for 3D GANs.
In particular, GRAF~\cite{Schwarz2020NEURIPS} and pi-GAN~\cite{piGAN2021} have shown large potential of NeRF-based GAN for 3D aware image synthesis.
They are then extended to learn more accurate 3D object shapes~\cite{pan2021shadegan,xu2021generative,or2021stylesdf,chan2021efficient} and more photorealistic image synthesis~\cite{Niemeyer2020GIRAFFE,gu2022stylenerf,or2021stylesdf,chan2021efficient}.
These advances have largely pushed forward the boundary of 3D-aware image synthesis.
Among these works, some of them also study compositional image synthesis~\cite{Niemeyer2020GIRAFFE,nguyen2020blockgan,liao2020towards}.
All these works model composition in the object level (\eg, shifting or inserting an intact object) but cannot edit a part of an object without changing other parts.
In contrast, our method can model the compositionality within an object, allowing 3D object synthesis with more fine-grained control.
For instance, for any face volume generated by our model, we can edit one part like hair or eyes while keeping other parts fixed.
Recently proposed FENeRF~\cite{sun2021fenerf} is the closest related concurrent work to our method. While this method enables editing of parts, this requires a 2D segmentation mask for the edit.
In contrast, our method enables one to choose any sample from the part generator. And it is also not possible to have different texture for parts using FENeRF as they have single color latent vector to define the whole face. For example, they cannot have different hair color for the edited hair.
While there are many 2D face editing methods~\cite{Tan20, saha2021LOHO, Zhu21} exist in the literature, none of the methods work without segmentation mask as input at test time with a trained model.
Also, these methods expect the segmentation mask to align well with the input image with similar pose.
Since we model objects in 3D, we can account for these variations better.
\subsection{Compositional Scene Representation}
Many works have studied compositional model in the 2D image level~\cite{zhu2015learning,azadi2020compositional,lin2018st,tsai2017deep,johnson2018image,locatello2020object}.
Some works learn to composite images in a harmonized way~\cite{zhu2015learning,azadi2020compositional,lin2018st,tsai2017deep} while others sythesize images from given scene graphs~\cite{johnson2018image}.
Locatello \etal ~\cite{locatello2020object} proposed a slot attention architecture that can learn object composition in an unsupervised manner.
However, these 2D approaches cannot model complex occlusions in the 3D space.
Recently, several attempts have been made to model compositional scene in the 3D space using volumetric neural fields~\cite{mildenhall2020nerf}.
Specifically, Guo \etal \cite{guo2020osf} learns the NeRF for each object independently, which naturally allows composition.
The approaches of Ost~\etal~\cite{Ost_2021_CVPR} and Yang~\etal~\cite{yang2021objectnerf} can decomposite a scene into objects given the supervision of object masks or bounding boxes.
uORF~\cite{yu2021unsupervised} is an unsupervised approach to discover individual objects in a scene, but is only applied to synthetic data.
Wang \etal \cite{Wang21} proposed a compositional representation that models coarse structure with voxel grid and fine detials with NeRF.
Stelzner \etal \cite{stelzner2021decomposing} proposed a method which decomposes a scene into a set of NeRFs, with each NeRF corresponding to a different object.
All these method requires either videos, RGB-D or multi-view images as supervision.
Besides, they are designed for reconstructing scenes from existing images or videos, but cannot synthesize new objects.
In this work, we propose a compositional 3D generative model that learns from only monocular images and their corresponding segmentation maps.
Our approach not only allows editing of existing images, but also can synthesize new samples with fine-grained control over object parts.
\section{Results}
We evaluate our method on human portraits~\cite{karras2017progressive} and cats~\cite{zhang2008cat} datasets.
We model different semantic parts of the portraits like hair, eyes, eyebrows and nose using our method.
We used BiSeNet~\cite{Yu-ECCV-BiSeNet-2018} trained on CelebAHQMask~\cite{CelebAMask-HQ} dataset to obtain semantic map for portrait images, and DatasetGAN~\cite{zhang21} to obtain semantic map for Cats category.
We compare to several baselines and related approaches, both qualitatively and quantitatively, and demonstrate the advantages of our method.
We provide detailed explanation of network architecture, training curriculum, hyperparamters in the supplementary document.
As our method requires separate Generator backbone for each part, because of memory and computational expense, we train our networks at $64 \times 64$ resolution, following the settings of $\pi$-GAN~\cite{piGAN2021}.
All results are rendered at $128 \times 128$ resolution.
Please refer to supplementary document for more results and ablative analysis.
\subsection{Datasets}
We use the CelebAHQ~\cite{karras2017progressive} dataset for portrait images and Cats dataset~\cite{zhang2008cat} for cat faces.
Human portrait images are very suitable for the proposed task of
compositional learning with semantically well-defined parts such as hair, eyes, eyebrows and nose.
For cat faces, we show how we can learn independent models of cat eyes and the rest of the face.
As our main objective is to learn independent models for each of the face parts, we do not model the background and train on foreground segmented images.
\subsection{Qualitative Results}
\begin{figure}
\centering
\begin{minipage}[t]{.48\textwidth}
\centering
\includegraphics[width=\linewidth]{images/hair_4.png}
\caption{Independent sampling of hair and face region. Each row shows a fixed face region sample, while each column shows a fixed hair sample. We can independently sample from these models to create photorealistic images.
%
We also have explicit control over the head pose because of the underlying 3D representation.
%
}
\label{fig:result_hair}
\end{minipage}%
\hspace{0.1cm}
\begin{minipage}[t]{.48\textwidth}
\centering
\includegraphics[width=\linewidth]{images/eyes_4.png}
\caption{Independent sampling of eyes and the rest of the person. Each row shows a fixed non-eyes sample, while each column shows a fixed eyes sample. We can independently sample from these models to create photorealistic images.
}
\label{fig:result_eyes}
\end{minipage}%
\end{figure}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
\centering
\includegraphics[width=\linewidth]{images/eyebrows_4.png}
\caption{Independent sampling of eyebrows and the non-eyebrow region. Each row shows a fixed non-eyebrow sample, while each column shows a fixed eyebrows sample. We can independently sample from these models to create photorealistic images. }
\label{fig:result_eyebrows}
\end{minipage}%
\hspace{0.1cm}
\begin{minipage}{.48\textwidth}
\centering
\includegraphics[width=\linewidth]{images/nose_4.png}
\caption{Independent sampling of nose and the non-nose region. Each row shows a fixed non-nose sample, while each column shows a fixed nose sample. We can independently sample from these models to create photorealistic images. }
\label{fig:result_nose}
\end{minipage}%
\end{figure}
\begin{figure}
\centering
\centering
\includegraphics[width=\linewidth]{images/cats_4.png}
\caption{Independent sampling of eyes and the non-eyes region. Each row shows a fixed non-eyes sample, while each column shows a fixed eyes sample. We can independently sample from these models to create photorealistic images. }
\label{fig:results_cats}
\end{figure}
We show results of independent hair and face region models for portrait images in Fig.~\ref{fig:result_hair}.
Each row shows samples of the model with the same face but different hair in different poses.
Each column shows samples with the same hair but different faces in a fixed pose.
Please note the diversity of hairstyles we can synthesize for any face.
We show disentangled eye and the rest of the person in Fig.~\ref{fig:result_eyes}, eyebrows and the rest of the person in Fig.~\ref{fig:result_eyebrows}, and nose and the rest of the person in Fig.~\ref{fig:result_nose}.
Further, we show results of eyes vs rest of the scene disentanglement for cats in Fig.~\ref{fig:results_cats}.
These results demonstrate that our method works across different parts with different levels of occlusions and sizes.
Note that we have explicit control over the head pose as our method produces a 3D representation as output. %
We request the reader to check our supplemental video, where we show renderings of smooth interpolation of view points, part samples for many cases.
Control over the different parts of the object can be very useful for various applications ranging from image editing to image analysis.
For example, a facial recognition system trained on images could learn to distinguish between male and female subjects by looking at the hairstyle.
Our method could potentially enable data augmentations which, while being photorealistic, ensures a balanced distribution of hairstyles for training regardless of gender.
Please note that the contributions of this paper are to enable a rich set of control in generative models. We leave explorations in downstream tasks for future work.
\paragraph{Comparisons to Baselines}
\begin{figure}
\centering
\includegraphics[width=0.95\linewidth]{images/baseline_3dv.png}
\caption{Results of baseline with the models for each part trained independently from each other. Without the constraints between parts, such a baseline leads to artifacts due to incorrect separation.}
\label{fig:baseline}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{images/fenerf_comp.png}
\caption{Here we show editing results in comparison with FENeRF~\cite{sun2021fenerf}. Our method can edit not just the shape of the part, but also color independently of other parts. In contrast, FENeRF can edit the shape of the part, while keeping the part's color fixed (Geo). Editing the part color also changes the color of the whole face, as it relies on single appearance latent vector for the whole face (Geo+App). Here, "Geo" refers to "Geometry" and "App" refers to "Appearance".}
\label{fig:fenerf_comp}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{images/bn_abl.png}
\caption{Top row shows results without using the blending network. Without the blending network, the rendered results can have artifacts around the boundary between the parts.}
\label{fig:ablationblend}
\end{figure}
\begin{table*}[]
\centering
\begin{tabular}{llllll}
\hline
Metric & $\pi$-GAN & Hair & Eyes & Eyebrows & Nose \\ \hline
FID $\downarrow$ & 16.246 & 14.8738 & 14.0873 & 13.9845 & 14.2577 \\
KID $\downarrow$ & 0.012 & 0.0095 & 0.0091 & 0.0093 & 0.0094 \\
IS $\uparrow$ & 2.479 & 2.5371 & 2.5068 & 2.4455 & 2.4778 \\
\end{tabular}
\caption{
Here we show quantitative comparison with $\pi$-GAN for portrait datset~\cite{karras2017progressive}. To have fair comparison, we tie the sampling vectors for all the parts to synthesize images for evaluation, as $\pi$-GAN is not capable of synthesizing different semantic parts.}
\label{tab:coupled_sampling}
\end{table*}
\begin{table*}[]
\centering
\begin{tabular}{lllllllllll}
\hline
& $\pi$-GAN & Hair & Eyes & Eyebrows & Nose \\ \hline
& & & $\pi$-GAN-ind & & \\ \hline
FID $\downarrow$ & x & 143.184 & 87.6181 & 52.924 & 86.131 \\
KID $\downarrow$ & x & 0.14530 & 0.10107 & 0.0580 & 0.0981 \\
IS $\uparrow$ & x & 3.689 & 2.19169 & 2.4125 & 2.334 \\ \hline
& & & Ours(w/o BN) & & \\ \hline
FID $\downarrow$ & x & 20.8340 & 15.2308 & 15.1184 & 18.2041 \\
KID $\downarrow$ & x & 0.0145 & 0.0100 & 0.0099 & 0.01284 \\
IS $\uparrow$ & x & 2.1991 & 2.3435 & 2.3080 & 2.3069 \\ \hline
& x & & Ours & & \\ \hline
FID $\downarrow$ & x & 18.3422 & 13.9916 & 13.9293 & 15.1478 \\
KID $\downarrow$ & x & 0.01280 & 0.0092 & 0.0093 & 0.0102 \\
IS $\uparrow$ & x & 2.3293 & 2.4680 & 2.4069 & 2.4436 \\ \hline
\\
\end{tabular}
\caption{Here we show quantitative comparison with baseline ($\pi$-GAN-ind) and ablative study of blending network for portrait datset~\cite{karras2017progressive}. As the baseline models do not explicitly account for occlusion by other parts in the object, they end up learning some parts of the rest of the object with segmented color. As a result, the composited volume of different parts result in artifacts, which are reflected in bad numbers. }
\label{tab:random_sampling}
\end{table*}
\begin{table*}[]
\centering
\begin{tabular}{lllllll}
\hline
Metric & $\pi$-GAN & Ours (tied) & \vline $\pi$-GAN-ind & Ours(w/o BN) & Ours \\ \hline
FID $\downarrow$ & 13.3387 & 13.6265 & \vline 47.384 & 16.0410 & 14.6521 \\
KID $\downarrow$ & 0.0079 & 0.0066 & \vline 0.0420 & 0.0084 & 0.0075 \\
IS $\uparrow$ & 2.0481 & 2.0354 & \vline 2.0534 & 2.0537 & 2.0457 \\
\end{tabular}
\caption{Here we show results for Cats dataset. The first two quantitative columns (before vertical line) show results for samples with tiedt latent vectors between parts. The last 3 columns show numbers for samples with independent part sampling.}
\label{tab:cats_quant}
\end{table*}
\begin{figure*}
\centering
\includegraphics[width=0.8\linewidth]{images/realedits_4.png}
\caption{Results on real images. We embed the input images into the latent space of our model. This enables editing of semantic parts by sampling different components independently. }
\label{fig:result_real_edit}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=0.55\linewidth]{images/limitations2.PNG}
\caption{Limitations of our method. In some cases, the independently sampled parts can be very different, which introduces artifacts in the global rendering. }
\label{fig:limitations}
\end{figure}
We design a baseline where each part model is independently trained from scratch, i.e., we train a $\pi$-GAN model for each part on the corresponding segmented part images.
Once trained, we define a model which simply composite the independently sampled volumes using the compositional operation defined in Section~\ref{sec:compositional}, denoted as $\pi$-GAN-ind.
Fig.~\ref{fig:baseline} shows the results of the composited volumes.
This baseline does not learn a reasonable volume, as the constraints between the different parts are not modeled.
For example, the hair model can learn arbitrary depth for the face region with the color of the background.
This geometry can be in front of the sampled face region, and can thus occlude the face.
Our design models occlusions between the different parts and thus does not face this limitation.
We also compare to a baseline without using a blending network in Fig.~\ref{fig:ablationblend}.
We note that, blending network helps in synthesizing texture of the part similar to that of rest of the face, while keeping the shape of the sampled part (Nose column in Fig.~\ref{fig:ablationblend}).
Without the blending network (top row), there can often be artifacts in the boundary between the different parts (check the artifact between hair and face region in first row, first column in Fig.~\ref{fig:ablationblend} ).
\paragraph{Comparison to FENeRF}
We show qualitative comparison with FENeRF~\cite{sun2021fenerf} in Fig~\ref{fig:fenerf_comp}.
For a given input image, we can edit both the shape and color of the part independent of rest of the face. In the Fig~\ref{fig:fenerf_comp}, we take the fit our model to input image and take the corresponding part latent code from reference image to edit individual parts. FENeRF can only edit the shape while keeping the part color fixed. Editing the color of the part also changes the color of the rest of the face as they have only a global appearance vector.
\subsection{Quantitative Results}
We evaluate our model quantitatively using FID, KID, and IS scores, commonly used to evaluate generative models.
In Table ~\ref{tab:coupled_sampling}, we compare against $\pi$-GAN on both face and cats datasets.
Since $\pi$-GAN does not model independent sampling of parts, to have a fair comparison to our method, we tie the latent vectors (i.e., latents for all parts are identical) when we randomly sample images.
This evaluation also show that our training (Sec.~\ref{sec:decoupleparts}, Sec.~\ref{sec:blendnetwork}) does not negatively impact the quality of the model.
Our decoupled model performs almost similar or slightly better to $\pi$-GAN for all categories.
As the baseline models ($\pi$-GAN-ind) are trained independently, it can not be used for this evaluation.
In Table ~\ref{tab:random_sampling},~\ref{tab:cats_quant}, we compare our method against the $\pi$-GAN-ind baseline. In this table, we do not tie the latent vectors of the parts, but rather independently sample the latents.
As can be seen qualitatively as well in Fig.~\ref{fig:baseline}, naively training independent part models can not be used to composite multiple parts together.
Even though the missing parts in the image set is segmented with white color (same as back-ground), the network accounts it as part of the model and paints that region the segmented color.
Therefore, when composited with other parts, using our composite operator, it results in such white artifacts.
This can be observed quantitatively for all the categories.
Note that, the quantitative results for our method with random sampling and coupled sampling are in the similar range.
In Table ~\ref{tab:random_sampling},~\ref{tab:cats_quant}, we provide ablative analysis of our blending network.
In our experiments, we used blending network for all part categories, including hair, nose, eyes and eyebrows.
The blending network helps in overcoming certain artifacts pertaining to texture harmonization and artifacts near the boundary of composited parts in some cases, leading to better quantitative scores as verified in Table~\ref{tab:random_sampling},~\ref{tab:cats_quant}.
\section{Application - Editing Real Images}
Our model can enable many exciting applications.
Here, we demonstrate the application of editing real images.
This requires us to embed the real images into the latent space of the learned GAN.
Projecting real images onto the latent space of GANs is a well-studied problem~\cite{xia2021gan}.
Methods either use optimization-based techniques~\cite{tewari2020pie,abdal2019image2stylegan}, or learn an encoder to regress the latent vectors~\cite{alaluf2021restyle,richardson2021encoding}.
We design an optimization-based technique, since it tends to result in higher-quality images, however, at the cost of the speed of projection.
We use several energy terms to optimize for the latent vector.
We use a combination of L1 loss, perceptual loss, and facial recognition loss on the complete image.
In addition, we use an L1 loss between the renderings of each part with the corresponding segmented out part of the input image.
Finally, we use regularizers encouraging the optimized latent vectors to be close to their average values via L2 losses.
Please refer to the supplemental for more details on the optimization and for all hyperparameters used.
The real image editing results are shown in Fig.~\ref{fig:result_real_edit}, where several semantic parts of the portrait images are edited independently.
\section{Limitations}
While we demonstrated high-quality results for our compositional scene representation, several limitations remain.
Fig.~\ref{fig:limitations} show that when composing independently sampled hair and face regions which are quite apart, our methods sometimes struggles to synthesize coherent volume at the boundaries.
Similar to existing 3D GANs, we can only learn our models at relatively low resolutions. %
Concurrent work has shown ways to improve the resolution and quality of 3D GANs~\cite{gu2022stylenerf,chan2021efficient}, and our framework could potentially be used with newer 3D GANs. This is left this exploration for future work.
Although our method works well for the object categories as shown in this paper, we believe it might not hold true for other categories like articulated objects (For example, human bodies, hands etc.). But our method could be extended by explicitly accounting for geometric articulation of the parts, as we can already decouple part in 3D explicitly. We leave this for future work.
|
train/arxiv
|
BkiUdbM4uzlhdExj6yBe
| 5 | 1 |
\section{Introduction}\label{intro}
The physical properties, such as energy spectra, susceptibility,
etc., of magnetic clusters at the nanoscale depend on their size,
shape and the presence of different bondings among the
constituent chemical elements and thus the distribution of ligands
between the magnetic ions (for more details see
\cite{roduner2006,chamati_theory_2013,sellmyer_novel_2015,sieklucka_molecular_2017}
and references therein).
Some prominent examples are Mn based magnetic compounds
\cite{goodenough_1955,defotis_1990,law_2000,han_2004,perks_2012,gupta_2016,hanninen_2018}
and spin clusters with Ni magnetic ions
\cite{loose_2008,panja_2017,das_2017,woods_2017}.
Magnetic molecules possess unique properties that can be
characterized with great accuracy both experimentally and
theoretically. Thus they are ideal tools
to gain useful insights into the underlying coupling mechanisms.
The study of basic units like tetramers
\cite{klemm_single-ion_2008,hubner_2017} prove the importance
of analytical methods in revealing the role of electrons correlations
underpinning molecular magnetism.
On the experimental side, Inelastic Neutron Scattering (INS)
\cite{lovesey_1986,malcolm_neutron_1989,furrer_neutron_2009,toperverg_neutron_2015}
plays a central role in determining the relevant magnetic spectra.
In complement to different magnetic measurement methods, INS technique appears
to be essential, and in the past decades it has been widely
applied to explore the properties of spin clusters.
Experiments on the spin dimer
[Ni$_2$(ND$_2$C$_2$H$_4$ND$_2$)$_4$Br$_2$]Br$_2$
have demonstrated the important contribution of neutron spectroscopy \cite{stebler_intra_1982}.
INS measurements were obtained for different magnetic clusters, such as:
The trimer La$_4$Cu$_3$MoO$_{12}$, with strong intratrimer antiferromagnetic
interactions, where the copper ions form an isolated triangle
\cite{azuma_antiferromagnetic_2000}, the Fe based molecular wheel with eighteen
spin-$\tfrac52$ ions \cite{ummethum_2012}, the dimer
SrCu$_2$(BO$_3$)$_2$ with observed multiplet excitations
\cite{kageyama_direct_2000,gaulin_high-resolution_2004}, the
polyoxomolybdate Mn$_{72}$Fe$_{32}$
\cite{garlea_probing_2006}, and the magnetic molecule Fe$_9$ in the presence
of an external magnetic field \cite{vaknin_magnetic_2014}.
In the present work we propose an approach based on the assumption that
in a molecular magnet with multiple exchange bridges between any two magnetic
centers (ions) the distribution of unpaired valence electrons is not unique.
Thus, the electron's density distribution might vary among the existing exchange bridges
affecting the transition energy, and consequently leading to either a broadened excitation
width or splitting in the energy spectrum.
Accordingly the number of all energy levels form a set
that can utterly identify the most relevant spin bonds, despite being
indistinguishable.
To this end, we introduce a modified microscopic spin Hamiltonian with discrete couplings
that incorporates distinct spin coupling mechanisms among equivalent spins
allowing one to identify the different exchange paths.
The proposed Hamiltonian leads to an unperturbed energy structure that distinguish the
relevant magnetic features from those arising due the magnetic
anisotropy. It is worth mentioning that the present approach was successfully applied to the study
of the magnetic excitations in the trimers A$_3$Cu$_3$(PO$_4$)$_4$
with (A = Ca, Sr, Pb) \cite{georgiev_trimer_2019}.
Here, the present method will be validated by reproducing
the experimentally
obtained INS spectrum of a molecular magnet that has generated a great
deal of interest among researchers both on the theoretical, as well as
the experimental sides. This is the magnetic molecule
[Mo$_{12}$O$_{30}$($\mu_2$-OH)$_{10}$H$_2$(Ni(H$_2$O)$_3$)$_4$], denoted by
Ni$_4$Mo$_{12}$, where four spin-1 Ni$^{2+}$ ions are
sitting on the vertices of a distorted tetrahedron \cite{muller_2000}.
We would like to point out that previous studies
\cite{nehrkorn_inelastic_2010,furrer_magnetic_PRB_2010} did not
succeed to obtain the main peaks and the broadening in the INS
spectrum, see FIG. \ref{fig:NiExp}, despite including
single-ion anisotropy or higher-order terms in the
Heisenberg model with nearest-neighbor interaction or even
using the Hubbard model.
\begin{figure}
\includegraphics{INEXP.pdf}
\caption{Background corrected data for the INS spectrum of the polycrystalline
Ni$_4$Mo$_{12}$ taken from Ref. \cite{nehrkorn_inelastic_2010}.
}
\label{fig:NiExp}
\end{figure}
\section{The model and the method}\label{sec:fundamprinc}
In order to identify the experimentally observed magnetic peaks, see FIG. \ref{fig:NiExp}, one has
to calculate the scattering intensities $I_{n'n}(\mathbf{q})$,
of the existing transitions and analyse their
dependence on the temperature and the magnitude of the neutron scattering vector.
For identical magnetic ions, we have
\cite{lovesey_1986,malcolm_neutron_1989,furrer_neutron_2009,toperverg_neutron_2015}
\begin{equation}\label{eq:ScatteringIntensities}
I_{n'n}(\mathbf{q}) \propto F^2(\mathbf{q}) \sum_{\alpha, \beta} \varTheta^{\alpha \beta}
S^{\alpha \beta} (\mathbf{q},\omega_{n'n}).
\end{equation}
Here $\omega_{n'n}$ is the transition frequency,
$\mathbf{q}$ is the scattering vector, $F(\mathbf{q})$ is the
spin magnetic form factor \cite{jensen_rare_1991},
$\varTheta^{\alpha \beta}$ are the elements of polarization factor,
$S^{\alpha \beta} (\mathbf{q},\omega_{n'n})$ are the magnetic
scattering functions and $\alpha,\beta,\gamma\in\{x,y,z\}$.
To determine
the energy level structure and the transitions corresponding to
the experimentally observed magnetic spectra one needs a minimal
number of parameters to account for all couplings in the system.
The principal assumption is that
the magnetic excitations of spin clusters obtained via INS are mainly governed by the
exchange of electrons between
effective spin magnetic centers. Then, the experimental data are interpreted in terms
of a well defined microscopic spin model.
In the absence of anisotropy, i.e. negligible spin-orbit coupling,
the exchange interaction in molecular magnets can be described by
the Heisenberg model
\begin{equation}\label{eq:HeisenbergHamiltonian}
\hat{H} = \sum\limits_{i \ne j}^{} J_{ij}
\hat{\mathbf{s}}_i \cdot \hat{\mathbf{s}}_j,
\end{equation}
where $J_{ij} = J_{ji}$ is the exchange coupling that effectively accounts
for the exchange
interaction between the $i$th and $j$th ions.
However, to distinguish all magnetic excitations one has to use an appropriate
spin model leading to an energy sequence such that the
scattering functions in \eqref{eq:ScatteringIntensities}
identifies the relevant spin bonds.
Therefore, even with a selected \textit{a priori} spin coupling scheme
for a cluster with complex intermediate structure Hamiltonian
\eqref{eq:HeisenbergHamiltonian} may not be fully adequate to
obtain the correct energy structure.
\subsection{The effective spin Hamiltonian} \label{sec:spinmodel}
In order to characterize uniquely
each pair of magnetic centers in a magnetic cluster,
bonded via more than one intermediate bridge,
we propose the following Hamiltonian
\begin{equation}\label{eq:AddHamiltonian}
\hat{\mathcal{H}} =
\sum\limits_{i \ne j}^{} J_{ij}
\hat{\boldsymbol{\sigma}}_i \cdot \hat{\mathbf{s}}_j,
\end{equation}
where the couplings $J_{ij} = J_{ji}$
are effective exchange constants and the operator
$\hat{\boldsymbol{\sigma}}_i =
(\hat{\sigma}^x_i, \hat{\sigma}^y_i, \hat{\sigma}^z_i)$
accounts for the variation in valence electron's distribution
with respect to the $i$th magnetic center.
A detailed derivation of the
effective Hamiltonian \eqref{eq:AddHamiltonian} is very lengthy and falls
beyond the scope the this study (the case of magnetic dimers is
published in Ref. \cite{georgiev_2018a}). In the following we will give
a brief account of the main steps leading to construct the ensuing
Hamiltonian \eqref{eq:AddHamiltonian}. Let us point out that our computations are based on the Molecular Orbital
Theory \cite{fleming_molecular_2009} in terms of the
complete active space self consistent field method
\cite{roos_complete_2007,szalay_multicon_2012}.
We assume that each exchange bridge connecting two magnetic centers possess
a number of paired valence electrons, nuclei and thus favor a particular
spatial distributions of electrons.
The canonical Hamiltonian leading to \eqref{eq:AddHamiltonian} accounts for the
kinetic energy, electron-electron and electron-nuclei interactions of all
valence electrons within the adiabatic approximation.
The electrons are considered as delocalized, occupying molecular orbitals
$\phi_{k,m_i}(\mathbf{r}_i)$, $k\in\mathbb{N}$,
given by linear combinations of atomic orbitals
$\psi^{\eta}_{\mu_{\eta i},m_i}(\mathbf{r}_i)$, where $\mathbf{r}_i$ are the coordinates of the
$i$th electron, $\mu_{\eta i}$ label the electronic shell and subshell with respect to the
$i$th electron and $\eta$ nucleus, $m_i$ is the spin magnetic quantum number of the $i$th electron.
The state function of each electron configuration is given by a linear combination of
Slater determinants (of the orbitals
$\phi_{k,m_i}(\mathbf{r}_i)${}), where the
symmetrization is performed with respect to the spin quantum numbers $s_{ij}$
of all electron pairs. The corresponding basis set gives the number of all
probable electrons distributions along all exchange bridges.
In the simplest case of $N$ valence electrons with $N-2$ pairs, one of the
basis states is written as
\begin{align}\label{eq:TwoOrbital}
\Psi^{k,k'}_{S,M}(\mathbf{r}_1,\ldots,\mathbf{r}_{N})\equiv
&
\!\!\!
\sum_{P_{\mathbf{r}_1\ldots\mathbf{r}_{N}}}
\!\!\!
c_{\mathbf{r}_{N-1},\mathbf{r}_{N}}
\!\!\!
\prod_{i}^{\frac{N}{2}-1}
\!\!
\frac{
\Phi^i_{s_{2i-1,2i}}(\mathbf{r}_{2i-1},\mathbf{r}_{2i})
}{\sqrt{2^{\frac{N}{2}}N!}}
\nonumber \\
& \times
\Psi^{k,k'}_{s_{N-1,N}}(\mathbf{r}_{N-1},\mathbf{r}_{N})
\lvert S,M\rangle,
\end{align}
where the sum runs over all permutations on the set of
coordinates $\mathbf{r}_1\ldots\mathbf{r}_{N}$.
Both unpaired electrons occupy $k$th and $k'$th molecular orbitals with
\begin{align}
\Psi^{k,k'}_{s_{N-1,N}}(\mathbf{r}_{N-1},\mathbf{r}_{N})=
&
\tfrac{1}{\sqrt{2}}
\big[
\phi_{k}(\mathbf{r}_{N-1})\phi_{k'}(\mathbf{r}_{N})+
\nonumber \\ \nonumber
& (-1)^{s_{N-1,N}}
\phi_{k'}(\mathbf{r}_{N-1})\phi_{k}(\mathbf{r}_{N})
\big].
\end{align}
The remaining, paired electrons, are described by the states
\begin{align}
\Phi^i_{s_{2i-1,2i}}(\mathbf{r}_{2i-1},\mathbf{r}_{2i})=
&
\tfrac12
\big[
\phi_i(\mathbf{r}_{2i-1})\phi_i(\mathbf{r}_{2i})+
\nonumber \\ \nonumber
& (-1)^{s_{2i-1,2i}}
\phi_i(\mathbf{r}_{2i-1})\phi_i(\mathbf{r}_{2i})
\big].
\end{align}
Moreover, for $i,j = 1,\ldots, N$ the permutation coefficients
$$
c_{\mathbf{r}_{i},\mathbf{r}_{j}}= (-1)^{i+j+1},
\quad
c_{\mathbf{r}_{j},\mathbf{r}_{i}}=(-1)^{s_{N-1,N}}(-1)^{i+j+1},
$$
account for the antisymmetry of \eqref{eq:TwoOrbital} and
the spin part is given by
$$
\lvert S,M\rangle=\otimes^{\frac{N}{2}}_{i=1}
\lvert s_{2i-1,2i},m_{2i-1,2i}\rangle.
$$
The effective Hamiltonian accounting for all possible configurations
of interactions
is built according to the expectation values of the initial Hamiltonian representing
all electron-electron and electron-nuclei interactions.
The effective state associated with \eqref{eq:TwoOrbital} is represented by
$\big\lvert\Psi^{\tau}_{S,M}\big\rangle$, where $\tau=(k,k')$.
The generic wave function describing the multiple bridged structure includes all
probabilities related with the spatial distribution of unpaired electrons.
Thus, we may compute
the eigenstates of the ensuing spin Hamiltonian in terms of an appropriate linear
combination of the effective states.
In the case of \eqref{eq:TwoOrbital}, this is given by the superposition
\begin{equation}\label{eq:AllPathsStatesTriplet}
\lvert s,m\rangle
=\sum_{\tau}c^\tau_{S}
\big\lvert\Psi^{\tau}_{S,M}\big\rangle,
\end{equation}
where
$\tau$ runs over the number of all existing electrons' configurations
that depend on all intermediate bridges and
$s=0,1$, $m=0,\pm1$ are the corresponding effective spin
and magnetic quantum numbers, that obey the conservation of angular momentum
$S\equiv s$, $M\equiv m$. Further, $c^\tau_{S}$ is the associated probability
coefficient that depends on $S$.
The expectation values of the canonical Hamiltonian are a part of the eigenstates
relevant to \eqref{eq:AllPathsStatesTriplet}.
Within the framework of the given example they are represented by the sum
\begin{equation}\label{eq:EffEnergies}
E_{s,m}=\sum_{\tau} \big|c^\tau_{S}\big|^2E^{\tau}_{S,M}.
\end{equation}
The elements in the last sum are functions of the Coulomb
$U_\tau, V_\tau$, hopping
$t_\tau$ and direct exchange integrals $D_\tau$ relevant to each intermediate bridge.
For example,
$$
E^\tau_{1,M}=V_\tau-D_\tau,
\qquad
M=0,\pm1,
$$
$$
E^\tau_{0,0}= D_\tau+\frac{U_\tau+V_\tau}{2}-
\sqrt{4t^2_\tau+\frac{\left(U_\tau-V_\tau\right)^2}{4}}.
$$
Within the effective spin space, determined by $s$ and $m$, one can account for only one
transition with energy $|\Delta E|=|E_{1,m}-E_{0,0}|$ due to the
$3$-fold degeneracy of the triplet level. However, for different
spatial distributions of the considered electrons the values of \eqref{eq:EffEnergies}
alter and accordingly the energy of the transition changes.
Such effect is not related neither to the magnetic anisotropy nor higher order multiple interactions.
Therefore, conventional bilinear spin Hamiltonians with only one exchange coupling
and additional interacting terms is not able to account for the variation in $\Delta E$.
In order to address these features we introduce Hamiltonian
\eqref{eq:AddHamiltonian} that depends upon the parameters described hereafter.
\subsection{Properties of the $\hat{\boldsymbol{\sigma}}$--operators}
For a single spin the square and the $z$ component of each operator
$\hat{\boldsymbol\sigma}$ are completely determined in the
basis of the total spin component $\hat{s}^z$, such that for all $i$ and
$\alpha \in \{x,y,z\}$
\begin{equation}\label{eq:Sigma_i}
\hat{\sigma}^\alpha_i \lvert \ldots ,s_i,m_i, \ldots \rangle
=
a^{s_i,m_i}_i \hat{s}^{\alpha_{\phantom{i}}}_i
\lvert \ldots ,s_i,m_i, \ldots \rangle,
\end{equation}
where $a^{s_i,m_i}_i \in \mathbb{R}$.
Furthermore, the $\sigma$ rising and lowering operators
obey the equations
\begin{equation}\label{eq:Sigma_LR_i}
\hat{\sigma}^{\pm}_i \lvert \ldots ,s_i,m_i, \ldots \rangle
=
a^{s_i,m_i}_i \hat{s}^{{\pm}_{\phantom{i}}}_i
\lvert \ldots ,s_i,m_i, \ldots \rangle.
\end{equation}
For all $i$, the square of $\sigma_i$ commutes
only with its $z$ component.
Its eigenvalues depend on $m_i$ and according to
\eqref{eq:Sigma_i} and \eqref{eq:Sigma_LR_i}
one can distinguish two cases:
(1) $m_i=\pm s_i$; (2) $-s_i<m_i<s_i$,
where $s_i\ne0$, with the respective eigenvalues
\begin{subequations}\label{eq:SigmaSquareEigenvalue_i}
\begin{equation}\label{eq:SigmaSquareEigenvalue_mi=si}
\left( a^{s_i,\pm s_i}_i\right)^2 s^2_i+a^{s_i,\pm s_i}_i
a^{s_i,\pm(s_i-1)}_i s_i,
\end{equation}
\begin{align}\label{eq:SigmaSquareEigenvalue_mi}
& \tfrac12 a^{s_i,m_i}_i \left[a^{s_i,m_i+1}_i +
a^{s_i,m_i-1}_i\right]s_i(s_i+1)
+\left( a^{s_i,m_i}_i\right)^2 m^2_i
\nonumber \\
& - \tfrac12 a^{s_i,m_i}_i m_i \left[a^{s_i,m_i+1}_i(m_i+1)
+ a^{s_i,m_i-1}_i (m_i-1)\right].
\end{align}
\end{subequations}
On the other hand when the spins of $i$th and $j$th magnetic ions are
coupled, with total spin operator
$\hat{\mathbf{s}}_{ij}=\hat{\mathbf{s}}_{i}+\hat{\mathbf{s}}_{j}$,
the relation \eqref{eq:Sigma_i} enters a more general and complex expression.
To explore the properties of the coupled spins one has to
work with the total $\sigma$-operator $\hat{\boldsymbol{\sigma}}_{ij}$.
Its $z$ component and square are completely determined in the basis of
the spin operator $\hat{\mathbf{s}}^2_{ij}$.
Similar to eq. \eqref{eq:Sigma_i} for all $i\ne j$ and
$\alpha \in \{x,y,z\}$, we have
\begin{equation}\label{eq:Sigma_ij}
\hat{\sigma}^\alpha_{ij} \lvert \ldots ,s_{ij},m_{ij}, \ldots \rangle
=
a^{s_{ij},m_{ij}}_{ij} \hat{s}^{\alpha_{\phantom{j}}}_{ij}
\lvert \ldots ,s_{ij},m_{ij}, \ldots \rangle,
\end{equation}
where $a^{s_{ij},m_{ij}}_{ij} \in \mathbb{R}$.
The corresponding rising and lowering operators obey
\begin{equation}\label{eq:Sigma_LR_ij}
\hat{\sigma}^{\pm}_{ij} \lvert \ldots ,s_{ij},m_{ij}, \ldots \rangle
=
a^{s_{ij},m_{ij}}_{ij} \hat{s}^{{\pm}_{\phantom{i}}}_{ij}
\lvert \ldots ,s_{ij},m_{ij}, \ldots \rangle.
\end{equation}
The eigenvalues of $\hat{\boldsymbol{\sigma}}^2_{ij}$ depend on $m_{ij}$.
Therefore having in mind the following two cases $m_{ij}=\pm s_{ij}$,
and $-s_{ij}<m_{ij}<s_{ij}$, where $s_{ij}\ne0$, the eigenvalues read
\begin{subequations}\label{eq:SigmaSquareEigenvalue_ij}
\begin{equation}\label{eq:SigmaSquareEigenvalue_mij=sij}
\left( a^{s_{ij},\pm s_{ij}}_{ij}\right)^2 s^2_{ij}+a^{s_{ij},
\pm s_{ij}}_{ij} a^{s_{ij},\pm(s_{ij}-1)}_{ij} s_{ij},
\end{equation}
\begin{align}\label{eq:SigmaSquareEigenvalue_mij}
& \tfrac12 a^{s_{ij},m_{ij}}_{ij} \left[a^{s_{ij},m_{ij}+1}_{ij} +
a^{s_{ij},m_{ij}-1}_{ij}\right]s_{ij}(s_{ij}+1)
\nonumber \\
& +\left( a^{s_{ij},m_{ij}}_{ij}\right)^2 m^2_{ij}
- \tfrac12 a^{s_{ij},m_{ij}}_{ij} m_{ij}
\nonumber \\
& \times\left[a^{s_{ij},m_{ij}+1}_{ij}(m_{ij}+1)
+ a^{s_{ij},m_{ij}-1}_{ij} (m_{ij}-1)\right].
\end{align}
\end{subequations}
The corresponding $\sigma$-operators share a single
coefficient and for $i \ne j$ and $\alpha \in \{x,y,z\}$,
we have
\begin{equation}\label{eq:Sigma_k}
\hat{\sigma}^{\alpha}_i \lvert \ldots ,s_{ij},m_{ij}, \ldots \rangle
=
a^{s_{ij},m_{ij}}_{ij} \hat{s}^{\alpha_{\phantom{j}}}_i
\lvert \ldots ,s_{ij},m_{ij}, \ldots \rangle.
\end{equation}
We further assume that
the $\sigma$-operators preserve the corresponding
spin magnetic moment and for a non coupled spin
obey the following constraints
\begin{subequations}\label{eq:Constraint_i}
\begin{equation}\label{eq:SigmaZ_i}
\hat{\sigma}^z_i \lvert \ldots ,s_i,m_i, \ldots \rangle
=
m_i \lvert \ldots ,s_i,m_i, \ldots \rangle,
\end{equation}
\begin{equation}\label{eq:SigmaSquare_i}
\hat{\boldsymbol{\sigma}}^2_i \lvert \ldots ,s_i,m_i, \ldots \rangle
=
s_i(s_i+1) \lvert \ldots ,s_i,m_i, \ldots \rangle.
\end{equation}
\end{subequations}
Similarly, when the $i$th and $j$th spins are
coupled, for all $i \ne j$ we have
\begin{subequations}\label{eq:Constraint_ij}
\begin{equation}\label{eq:SigmaZ_ij}
\hat{\sigma}^z_{ij}
\lvert\ldots,s_{ij},m_{ij},\ldots \rangle =
m_{ij} \lvert\ldots,s_{ij},m_{ij},\ldots \rangle,
\end{equation}
\begin{equation}\label{eq:SigmaSquare_ij}
\hat{\boldsymbol{\sigma}}^2_{ij}
\lvert\ldots,s_{ij},m_{ij},\ldots \rangle =
s_{ij}(s_{ij}+1)
\lvert\ldots,s_{ij},m_{ij},\ldots \rangle.
\end{equation}
\end{subequations}
Taking into account \eqref{eq:Constraint_i} together with
expressions \eqref{eq:SigmaSquareEigenvalue_mi} for all $m_i$ we have
$a^{s_i,m_i}_i=1$.
Further, according to constraints \eqref{eq:Constraint_ij} and
eqs. \eqref{eq:SigmaSquareEigenvalue_ij} we distinguish three cases:
\textbf{(1)} $s_{ij} \ne 0$, $m_{ij} \ne 0$:
Then, for all $m_{ij}$, $a^{s_{ij},m_{ij}}_{ij}=1.$
As a result the transformations of eigenvectors
via the $\sigma$-operator coincide with those defined by
its corresponding spin operator.
Therefore, all couplings will be constants and the Hamiltonian
\eqref{eq:AddHamiltonian} will capture the same features as its
Heisenberg parent.
\textbf{(2)} $s_{ij} \ne 0$ and $m_{ij}=0$:
The corresponding coefficient cannot be determined from Eq.
\eqref{eq:SigmaZ_ij} and from eqs. \eqref{eq:SigmaSquareEigenvalue_mij} and
\eqref{eq:SigmaSquare_ij} one obtains
\begin{equation}\label{eq:as0_ij}
a^{s_{ij},\pm1}_{ij}=a^{s_{ij},0}_{ij}=\pm1.
\end{equation}
We would like to point out that the ``minus''
sign is irrelevant to the case of two unpaired electrons.
\textbf{(3)} $s_{ij} = 0$: The associated parameter
remains unconstrained and there exist a
set of coefficients $c^{n}_{ij}\in\mathbb{R}$ $\forall n\in\mathbb{N}$,
such that
\begin{equation}\label{eq:a00_ij}
a^{0,0}_{ij} \in \{c^{n}_{ij}\}_{n\in\mathbb{N}}.
\end{equation}
The values of $c^{n}_{ij}$
depend on the number of unpaired valence electrons and
intermediate nonmagnetic ions of the respective exchange bridges.
Depending on the type of exchange these effective
coefficients are functions of the Coulomb, hopping and
exchange integrals.
Thereby, for a linear cluster with only one bonding anion between
magnetic cations and a unique electron's spatial distribution, one would obtain the limit
$|c^{n}_{ij}-c^{k}_{ij}| \to 0, \ \forall \ n \ne k$,
where $c^{n}_{ij}\to1$. Accordingly,
the changes in electron's distribution could be considered as negligible pointing
to sharpened peaks in the magnetic
spectrum.
On the other hand, the inequality $|c^{n}_{ij}-c^{k}_{ij}|>0$ for all
$n \ne k$, would have to be considered as a sign
for the presence of exchange paths of different energy,
\textit{i.e.} more than one favorable spatial distribution of unpaired electrons,
and therefore of increased
excitation width in energy.
As an example, if $i$th and $j$th magnetic centers are linked via more than one
exchange bridge of complex chemical structure,
then one may expect
that the exchange path is not unique.
In such case according to \eqref{eq:AllPathsStatesTriplet} and
\eqref{eq:EffEnergies} the transition energy
reads
$$
|\Delta E_n|=\sum_{\tau}
\left| \big|c^\tau_{n,1}\big|^2 E^{\tau}_{1,M}
-\big|c^\tau_{n,0}\big|^2 E^{\tau}_{0,0}\right|,
$$
where $n$ assigns a unique transition energy to a certain
number of favorable spatial distributions.
Hence, with Hamiltonian \eqref{eq:AddHamiltonian} and taking into account
\eqref{eq:a00_ij} we can express all existing transitions by
\begin{equation}\label{eq:TranEnergy_a1}
\Delta E_n=\tfrac12 J_{ij} (3c^{n}_{ij}\pm1),
\qquad
a^{1,0}_{ij}=\pm1.
\end{equation}
The set of values $\Delta E_n$ will correspond to a broadened
peak in the magnetic spectrum. Applying the relation $\Delta E_n=2J_{c^{n}_{ij}}$,
where $J_{c^{n}_{ij}}$ is the $n$th value of the Heisenberg type exchange coupling
from \eqref{eq:TranEnergy_a1}
we thus obtain
\begin{equation}\label{eq:cij_a1}
c^{n}_{ij}= \frac{4}{3}\frac{J_{c^{n}_{ij}}}{J_{ij}}\mp\frac13,
\qquad
a^{1,0}_{ij}=\pm 1.
\end{equation}
As we will see later this approach
allows one to explain in details the experimentally observed splitting and
broadening of magnetic spectra in the molecular magnet
Ni$_4$Mo$_{12}$, see e.g. FIG. \ref{fig:NiExp}.
\section{The tetramer N\lowercase{i}$_4$M\lowercase{o}$_{12}$}
The indistinguishable spin-one $\mathrm{Ni^{2+}}$ ions of the spin
cluster compound $\mathrm{Ni_4Mo_{12}}$, are arranged
on the vertices of a distorted tetrahedron, see
FIG. \ref{fig:Nickel}. For this molecule
the bonds Ni1-Ni2 and Ni3-Ni4
are slightly shorter about 0.03 \AA\ than the
other four \cite{furrer_magnetic_PRB_2010}. Notice that the
intermediate bridges contain ions of molybdenum, oxygen and hydrogen.
\begin{figure}[!ht]
\centering
\includegraphics[scale=0.8]{Ni4Mo12.pdf}
\caption{
Sketch of the structure of the molecular
nanomagnet Ni$_4$Mo$_{12}$. The inset represents a schematic view of
the arrangement of Ni ions (blue balls). The gray
lines represent the two shorter distances, while the red lines show the
effective spin-1 dimers. \label{fig:Nickel}}
\end{figure}
To perform a thorough analysis of the magnetic excitations of the
compound $\mathrm{Ni_4Mo_{12}}$ obtained by INS experiments (see e.g.
FIG. \ref{fig:NiExp}) reported
in Ref. \cite{nehrkorn_inelastic_2010,furrer_magnetic_PRB_2010} we
employ Hamiltonian \eqref{eq:AddHamiltonian}.
Notice that the symmetry of the magnetic cluster imply
$J_{ij}=J$ and further we assume that
the magnetic centers Ni1-Ni2 and Ni3-Ni4 are coupled,
which defines these bonds as the intersection of two different planes.
Therefore, we have the total spin eigenstates
$\lvert s_{12},s_{34},s,m \rangle$,
four $\hat{\boldsymbol{\sigma}}$ operators for each constituent magnetic ion and two
bond operators corresponding to both Ni1-Ni2 and Ni3-Ni4 spin pairs.
We point out that
$\hat{\boldsymbol{\sigma}}_1$ and $\hat{\boldsymbol{\sigma}}_2$
are related with the
Ni1-Ni2 pair that share the coefficients
$a^{s_{12},m_{12}}_{12}$ of the total bond
$\hat{\boldsymbol{\sigma}}_{12}$. The operators $\hat{\boldsymbol{\sigma}}_3$
and $\hat{\boldsymbol{\sigma}}_4$ are associated with the
coefficient $a^{s_{34},m_{34}}_{34}$ of
$\hat{\boldsymbol{\sigma}}_{34}$.
Consequently from \eqref{eq:AddHamiltonian} we obtain the Hamiltonian
\begin{align}\label{eq:NickelHamilton}
\hat{\mathcal{H}} = & \ J
\left(
\hat{\boldsymbol{\sigma}}_1\cdot\hat{\mathbf{s}}_2 +
\hat{\boldsymbol{\sigma}}_2\cdot\hat{\mathbf{s}}_1 +
\hat{\boldsymbol{\sigma}}_3\cdot\hat{\mathbf{s}}_4 +
\hat{\boldsymbol{\sigma}}_4\cdot\hat{\mathbf{s}}_3
\right)
\nonumber \\
& + J
\left(
\hat{\boldsymbol{\sigma}}_{12}\cdot\hat{\mathbf{s}}_{34} +
\hat{\boldsymbol{\sigma}}_{34}\cdot\hat{\mathbf{s}}_{12}
\right).
\end{align}
With the applied effective spin-one spins the tetramer
exhibits in total eighty one eigenstates without taking into account the
quadrupolar, octupolar and other eigenfunctions related with higher
symmetries.
The ground state of this nanomagnet is a singlet with possible eigenstates
$\{\lvert 0,0,0,0 \rangle,\lvert 1,1,0,0 \rangle, \lvert 2,2,0,0 \rangle\}$.
On the other hand, the selection rules imply that the ground state
excitations must be related with singlet-triplet transitions and
since the quantum numbers $s_{14}$ and $s_{23}$ cannot be
simultaneously varied, we
deduce that the ground state is,
related to the formation of two local triplets, i.e. $s_{14}=1$ and $s_{23}=1$.
The triplet
eigenstates are eighteen. Those, three in total,
characterized by the local quintets $s_{14}=2$ and $s_{23}=2$
are not adequate to the
established selection rules and nine are directly connected
to experimental spectra.
\begin{figure*}[ht!]
\centering
\includegraphics[scale=1]{Ni4Mo12_spec.pdf}
\caption{
Energy level structure and the corresponding transitions of
Ni$_4$Mo$_{12}$. The blue line and
arrows stands for the ground state energy and the ground state excitations, respectively.
The red arrow marks the excited transition and the corresponding initial
level is shown in red. The dashed lines represent the centers of the two bands.
All transitions are denoted with respect to the experimental data shown on
FIG. \ref{fig:NiExp} provided from Ref. \cite{nehrkorn_inelastic_2010}.
}
\label{fig:NickelSpectra}
\end{figure*}
\subsection{Energy levels}
According to the selected coupling scheme we denote the
eigenvalues of \eqref{eq:NickelHamilton} by
$E^{m}_{s_{12},s_{34},s}$.
The ground state is the singlet $\lvert 1,1,0,0 \rangle$. Therefore, using
\eqref{eq:Constraint_ij} we get $a^{1,m_{12}}_{12}=a^{1,m_{34}}_{34}=1$
and taking into account \eqref{eq:NickelHamilton} we obtain
$$
E^{0}_{1,1,0} = -8J.
$$
With the triplet eigenstates $\lvert 0,1,1,m\rangle$, $m_{34}\equiv m=0,\pm1$,
the spins of Ni1 and Ni2 ions are coupled in singlet,
the parameter $a^{0,0}_{12}$ remains unconstrained
and can be determined using INS experimental data.
For the corresponding energies we get
\begin{equation*}
E^{m}_{0,1,1}= -2Ja^{1,m}_{34}-4Ja^{0,0}_{12}, \quad m=0, \pm 1,
\end{equation*}
where according to \eqref{eq:as0_ij} we have $a^{1,0}_{34}=\pm1$.
Analyzing the Ni$_4$Mo$_{12}$ spectrum we further obtain
$a^{0,0}_{12}\in\{c^1_{12},c^2_{12}\}$.
When the singlet bond is Ni3-Ni4 the eigenstates are
$\lvert 1,0,1,m\rangle$, $m_{12}\equiv m=0,\pm1$,
the value of $a^{0,0}_{34}$ remains unconstrained
and according to \eqref{eq:NickelHamilton} we have
\begin{equation*}
E^{m}_{1,0,1}= -2Ja^{1,m}_{12}-4Ja^{0,0}_{34}, \quad m=0, \pm 1,
\end{equation*}
where $a^{1,0}_{12}=\pm1$. Without loss of generality we set $a^{0,0}_{34}=c_{34}$.
For all remaining triplets, i.e.
$\lvert 1,1,1,m\rangle$, $\lvert2,2,1,m\rangle$, $\lvert 2,1,1,m\rangle$
and $\lvert 1,2,1,m\rangle$, where $m=0,\pm 1$, the corresponding
coefficient are constrained, $a^{s_{12},m_{12}}_{12}=1$ and
$a^{s_{34},m_{23}}_{34}=1$. Thus, we obtain
$$
E^{m}_{1,1,1} = E^{m}_{2,2,1} =
E^{m}_{2,1,1} = E^{m}_{1,2,1} = -6J.
$$
The tetramer Ni$_4$Mo$_{12}$ exhibits also a singlet bond at the quintet level.
The energies associated with the Ni1-Ni2 singlet bond and eigenstates
$\lvert 0,2,2,m \rangle$, where $m_{34}\equiv m$ are
\begin{equation*}
E^{m}_{0,2,2} = 2Ja^{2,m}_{34}-4Ja^{0,0}_{12}, \quad m=0, \pm 1,\pm 2,
\end{equation*}
where, the coefficients are determined by $a^{2,0}_{34}=\pm1$ and
$a^{0,0}_{12}\in\{c^1_{12},c^2_{12}\}$.
Similarly, if the spins of third and fourth ions are in a singlet state,
where the corresponding eigenstates are
$\lvert 2,0,2,m \rangle$, then the Hamiltonian in \eqref{eq:NickelHamilton}
yield the following energy values
\begin{equation*}
E^{m}_{2,0,2} = 2Ja^{2,m}_{12}-4Ja^{0,0}_{34}, \quad m=0, \pm 1,\pm 2,
\end{equation*}
where $a^{2,0}_{12}=\pm1$.
With respect to the other twelve quintet eigenstates the coefficients
$a^{s_{12},m_{12}}_{12}=a^{s_{34},m_{34}}_{34}=1$ and
therefore,
$$
E^{m}_{2,2,2} = E^{m}_{1,1,2} =
E^{m}_{2,1,2} = E^{m}_{1,2,2} = -2J.
$$
For the remaining two levels we obtain
$a^{s_{12},m_{12}}_{12}=1$ and $a^{s_{34},m_{34}}_{34}=1$.
The energy sequence follows the Land\'e interval rule $E_{s+1} -
E_{s} = 2Js$, see e.g. FIG. \ref{fig:NickelSpectra}.
The septet level is twenty one fold degenerate and it is defined by the
eigenstates $\lvert 2,1,3,m \rangle, \lvert 1,2,3,m \rangle, \lvert
2,2,3,m \rangle$ with $m=0,\pm1,\pm2,\pm3$. All corresponding
energies have equal value
$$
E^{m}_{2,1,3} = E^{m}_{1,2,3} =
E^{m}_{2,2,3} = 4J.
$$
Applying the nonet state $\lvert 2,2,4,m \rangle$,
where $m=0,\ldots,\pm4$ we end up with
$E^{m}_{2,2,4} = 12J$.
The described energy level structure is illustrated on
FIG. \ref{fig:NickelSpectra}.
In what follows we find the following notations more convenient
\begin{equation*}
\begin{array}{lll}
E_0 = -8J, & E_1 = -2J-4Jc^1_{12}, \\
E_2 = -2J-4Jc^2_{12}, & E_3 = -2J-4Jc_{34}, \\
E_4=-6J, & E_5 = 2J-4Jc^1_{12}, \\
E_6 = 2J-4Jc^2_{12}, & E_7 = 2J-4Jc_{34}, \\
E_8=-2J, & E_9=4J, \\
E_{10}=12J. & {}
\end{array}
\end{equation*}
\subsection{Scattering Intensities}
The INS selection rules are $\Delta s = 0, \pm 1$, $\Delta
m = 0, \pm 1$ and $\Delta s_{12} = 0, \pm 1$, $\Delta s_{34} = 0, \pm
1$. Here the transitions $\Delta s_{12} \ne 0$
and $\Delta s_{34} \ne 0$ are not allowed simultaneously.
Within the applied coupling scheme we obtain
$S^{\alpha\beta}(\mathbf{q},\omega_{n'n})+S^{\beta\alpha}(\mathbf{q},\omega_{n'n})=0$,
$\forall \ n,n'$, $\alpha \ne\beta$ and $\sum_{\alpha} \Theta^{\alpha\alpha}=2$.
The energy of the first experimental magnetic excitation is
approximately $0.4$ meV. The corresponding peak is depicted on FIG. \ref{fig:NiExp},
see also~\cite{furrer_magnetic_PRB_2010}.
This excitation is related with the transition between the ground state and the
local singlet state $\lvert 0,1,1,\pm1 \rangle$.
The associated scattering functions are
\begin{equation}\label{eq:ScatteringNikel12States1}
S^{\alpha \alpha}(\mathbf{q},\omega_{10}) = \tfrac{4}{9}
[1-\cos(\mathbf{q}\cdot\mathbf{r}_{12})]p_0,
\ \ \
S^{zz}(\mathbf{q},\omega_{10}) = 0,
\end{equation}
where $p_n$ is the population factor and $\alpha = x,y$.
The magnetic excitation at $0.6$ meV shown on FIG. \ref{fig:NiExp}
is associated with the eigenstate $\lvert 1,0,1,\pm1 \rangle$
and the scattering functions
\begin{equation}\label{eq:ScatteringNikel34States1}
S^{\alpha \alpha}(\mathbf{q},\omega_{30}) =
\tfrac{4}{9} [1-\cos(\mathbf{q}\cdot\mathbf{r}_{34})]p_0,
\ \ \
S^{zz}(\mathbf{q},\omega_{30}) = 0,
\end{equation}
where $\alpha = x,y$.
The functions
\eqref{eq:ScatteringNikel12States1} differ from
\eqref{eq:ScatteringNikel34States1} due to the spatial orientations
of the spin bonds with
$\mathbf{r}_{12}\cdot\mathbf{r}_{34} = 0$.
For the same reason, we
deduce that the third cold peak at $1.7$ meV, see FIG. \ref{fig:NiExp},
is related with the transition between the ground state
and non magnetic triplet $\lvert 0,1,1,0 \rangle$.
For $\alpha = x,y$ the corresponding
scattering functions read
\begin{equation*}
S^{zz}(\mathbf{q},\omega_{50}) =
\tfrac{4}{9} [1-\cos(\mathbf{q}\cdot\mathbf{r}_{12}) ]p_0,
\quad
S^{\alpha \alpha}(\mathbf{q},\omega_{50}) = 0.
\end{equation*}
The excited magnetic transition at around $1.2$ meV shown by green and red items
on FIG. \ref{fig:NiExp} is nicely reproduced by the scattering functions
\begin{equation*}
S^{\alpha \alpha}(\mathbf{q},\omega_{64}) =
\tfrac{2}{3} [1-\cos(\mathbf{q}\cdot\mathbf{r}_{12})] p_4,
\quad
S^{zz}(\mathbf{q},\omega_{64}) = 0,
\end{equation*}
where $\alpha = x,y$. The initial state is
given by the triplet state $\lvert 1,1,1,\pm1 \rangle$ with two triplet bonds
and the final one appears to be $\lvert 0,1,1,0 \rangle$.
Hence if the neutron scatters from the Ni3-Ni4 dimer, then we have
$\mathbf{q}\cdot\mathbf{r}_{12} = 0$ and
$\mathbf{q}\cdot\mathbf{r}_{34} > 0$.
Nevertheless, with the coefficients $a^{s_{12},m_{12}}_{12}$ and
$a^{s_{34},m_{34}}_{34}$ one can uniquely identify the
two spin bonds and distinguish $I_{10}$ from $I_{30}$.
Moreover, one can distinguish the eigenvalues
of tetramer Hamiltonian corresponding to $m=0$ and $m\ne0$,
with $S^{zz}(\mathbf{q},\omega_{n'n}) = 0$ and
$S^{xx}(\mathbf{q},\omega_{n'n})=0$, $S^{yy}(\mathbf{q},\omega_{n'n})=0$,
respectively.
This affects directly the integrated intensities, such that choosing
$\mathbf{r}_{12} = (0,0,r^z)$ and $\mathbf{r}_{34} = (r^x,0,0)$ from
\eqref{eq:ScatteringIntensities} yields
\begin{equation*}\label{eq:NickelIntegIntensities}
\begin{array}{l}
\displaystyle
I_{10} \propto \gamma_{10}
\left[
1-\frac{\sin(\mathrm{q}\mathrm{r})}{\mathrm{q}\mathrm{r}}
\right] F^2(\mathrm{q}),
\\ [0.5cm]
\displaystyle
I_{30} \propto \gamma_{30}
\left[
1 - 6\frac{\sin(\mathrm{q}\mathrm{r})}{5(\mathrm{q}\mathrm{r})^3} - 3\frac{\sin(\mathrm{q}\mathrm{r})}{5\mathrm{q}\mathrm{r}} + 6\frac{\cos(\mathrm{q}\mathrm{r})}{5(\mathrm{q}\mathrm{r})^2}
\right] F^2(\mathrm{q}),
\\[0.5cm]
\displaystyle
I_{50} \propto \gamma_{50}
\left[
1-3\frac{\sin(\mathrm{q}\mathrm{r})}{(\mathrm{q}\mathrm{r})^3}+
3\frac{\cos(\mathrm{q}\mathrm{r})}{(\mathrm{q}\mathrm{r})^2}
\right] F^2(\mathrm{q}),
\\[0.5cm]
\displaystyle
I_{64} \propto \gamma_{64}
\left[
1-\frac{\sin(\mathrm{q}\mathrm{r})}{\mathrm{q}\mathrm{r}}
\right] F^2(\mathrm{q}),
\end{array}
\end{equation*}
where
$$
\gamma_{10} = \tfrac{8}{9}p_0, \ \gamma_{30} = \tfrac{20}{27}p_0, \
\gamma_{50} = \tfrac{8}{27}p_0, \ \gamma_{64} = \tfrac{4}{3}p_4,
$$
and $r=|\mathbf{r}_{12}|=|\mathbf{r}_{34}|$.
The integrated intensities as a function of temperature
are shown on FIG. \ref{fig:NiIntensity}. According to
Ref. \cite{furrer_magnetic_PRB_2010} the average distance between
Ni-Ni ions is $\mathrm{r}=6.68 \ \mathrm{\mathring{A}}$.
The magnitude of the scattering vector is fixed at $\mathrm{q}=1 \
\mathrm{\mathring{A}}^{-1}$ and the calculated form factor
for Ni$^{2+}$ di-cations is
$
F(\mathrm{q}) = 256(16+\mathrm{q}^2 \mathrm{r}^2_{\mathrm{o}})^{-2},
$
where $\mathrm{r}_{\mathrm{o}} = 0.529 \, \mathrm{\mathring{A}}$ is the Bohr radius.
The dependence of normalized intensities, $I_{n'n} \to
I_{n'n}/\gamma_{n'n}$, on the scattering vector is shown on FIG. \ref{fig:NiIntFFactor}.
\begin{figure}[th!]
\includegraphics[]{IN.pdf}
\caption{
Intensities as a function of the absolute temperature. $I_{10}$, $I_{30}$
and $I_{50}$ correspond to the ground state transitions with
energies 0.4 meV, 0.6 meV and 1.7 meV, respectively. The
intensity $I_{64}$ in the inset stands for the excited transition with
energy 1.15 meV.
The blue squares, the green circles and red triangles point to the
values of intensities in table~\ref{tab:NickelIntensThreeTemp}.
}
\label{fig:NiIntensity}
\end{figure}
\begin{figure}[th!]
\includegraphics[]{IQEXPNi.pdf}
\caption{
Normalized by $\gamma_{n'n}$ intensities as a function of the
scattering vector along with the experimental data of
Ref. \cite{furrer_magnetic_PRB_2010}. $I_{10}$, $I_{30}$ and $I_{50}$ correspond
to ground state transitions with energies 0.4 meV, 0.6 meV and
1.7 meV, respectively. The intensity $I_{64}$ stands for the
excited transition with energy 1.15 meV. The inset
shows the intensity $I_{64}$ that coincides
with the function $I_{10}$.
}
\label{fig:NiIntFFactor}
\end{figure}
\begin{table}[!ht]
\caption{
Calculated values of integrated intensities $I_{n'n}$ [arb. units] at
temperatures 2.4, 9.3 and 23 K, shown on
FIG. \ref{fig:NiIntensity} as blue squares, green circles and red triangles,
respectively.
}
\label{tab:NickelIntensThreeTemp}
\begin{center}
\begin{tabular}{lcccc}
\hline\noalign{\smallskip}
Transitions & I & II & III & IV \\
\noalign{\smallskip}\hline\noalign{\smallskip}
Intensities & $I_{10}$ & $I_{30}$ & $I_{50}$ & $I_{64}$ \\
\noalign{\smallskip}\hline\noalign{\smallskip}
2.4~[K] & 0.137(6) & 0.120(3) & 0.051(5) & 0.008(9) \\
9.3~[K] & 0.039(8) & 0.034(8) & 0.014(9) & 0.026(5) \\
23~~[K] & 0.019(5) & 0.017(1) & 0.007(3) & 0.021(1) \\
\noalign{\smallskip}\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[!h]
\caption{
Values of the coupling constants and the quantities
$c^1_{12}$, $c^2_{12}$ and $c_{34}$
for all magnetic excitations with energies given in
eq. \eqref{eq:TransitionEnergies}. The results are obtained
by taking into account the experimental data of
Ref. \cite{nehrkorn_inelastic_2010,furrer_magnetic_PRB_2010}.
}
\label{tab:Nickel}
\begin{center}
\begin{tabular}{lcccc}
\hline\noalign{\smallskip}
Transitions & I & II & III & IV \\
\noalign{\smallskip}\hline\noalign{\smallskip}
$E_{n'n}$[meV] & $E_{10}$ & $E_{30}$ & $E_{50}$ & $E_{64}$ \\
{} & 0.4 & 0.6 & 1.7 & 1.15 \\
\noalign{\smallskip}\hline\noalign{\smallskip}
$J_{\phantom{c^1_{12}}}$~[meV] & 0.325 & 0.325 & 0.325 & 0.325 \\
$J_{c^1_{12}}$~[meV] & 0.372 & $-$ & 0.372 & $-$ \\
$J_{c^2_{12}}$~[meV] & $-$ & $-$ & $-$ & 0.353 \\
$J_{c_{34}}$~[meV] & $-$ & 0.334 & $-$ & $-$ \\
$c^1_{12}$ & 1.1923 & $-$ & 1.1923 & $-$ \\
$c^2_{12}$ & $-$ & $-$ & $-$ & 1.1153 \\
$c_{34}$ & $-$ & 1.0384 & $-$ & $-$ \\
\noalign{\smallskip}\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Energy of the magnetic transitions}
The energy transition $E_{ij}$ between $i$th and $j$th levels,
corresponding to the calculated scattering intensities are
\begin{equation}\label{eq:TransitionEnergies}
\begin{array}{ll}
E_{10} = 6J-4Jc^1_{12},
&
E_{30} = 6J-4Jc_{34},
\\ [0.2cm]
E_{50} = 10J-4Jc^1_{12},
&
E_{64} = 8J-4Jc^2_{12}.
\end{array}
\end{equation}
From the last equations we can take advantage of one more constraint to
determine $J$, $E_{50}-E_{10} = 4J$. According to the
experimental data
\cite{nehrkorn_inelastic_2010,furrer_magnetic_PRB_2010} the ground
state magnetic excitations are grouped in two relatively broadened
peaks. The first peak is centered at about 0.5 meV and the second one at 1.7
meV.
Furthermore, the first peak is composed of two subbands with energies
$E_{10}=0.4$ meV and $E_{30}=0.6$ meV. The width of the second peak can be
explained by the presence of an energy band, where the transition
energies are restricted in the region 1.6 meV to 1.8 meV.
Therefore,
setting $E_{50} = 1.7$ meV we obtain $E_{50}-E_{10} = 1.3$ meV and $J
= 0.325$ meV. The computed energy
transitions are depicted on FIG. \ref{fig:NickelSpectra}. The
centers of both energy bands referring
to the value $c^{2}_{12} = 1.1153$ are shown by
dashed lines. The energies of all transitions and
the corresponding parameters are given in table~\ref{tab:Nickel}.
\section{Conclusion}\label{sec:conclusion}
We propose a formalism for exploring the physical properties
of molecular magnets with non trivial bridging structure.
The underlying concept lies on the hypothesis
that due the cluster symmetry, as well as its shape,
size and the chemical structure that surrounds
the magnetic ions, the spatial distribution of
valence electrons
is not unique leading to a variation of the relevant Coulomb, hopping
and direct exchange energies.
Studying the INS spectra of the compound $\mathrm{Ni_4Mo_{12}}$
with the Hamiltonian \eqref{eq:AddHamiltonian} we were
able to derive a detailed picture for the neutron scattering
intensities FIG. \ref{fig:NiIntensity} and FIG. \ref{fig:NiIntFFactor}.
Hamiltonian \eqref{eq:NickelHamilton} leads to
energy spectrum with two energy bands,
shown in FIG. \ref{fig:NickelSpectra}. These bands are related to
the fact that the tetramer cluster exhibits two distinguishable
with respect to the coefficients $a^{s_{12},m_{12}}_{12}$ and
$a^{s_{34},m_{34}}_{34}$ bonds.
We ascribe this feature to the
difference in the chemical environment around Ni1-Ni2 and Ni3-Ni4 couples
that give rise to distinct spatial distributions of the valence electrons.
This allowed a unique identification of the magnetic
excitations. Thereby, the obtained energy bands explain the width of second
ground state peaks centered at 1.7 meV and the splitting of the first
one centered at 0.5 meV.
In particular, for $s_{12} = 0$, $s_{34} = 0$ and $i=1,2$
we get $|c^i_{12}|>1$ and
$|c_{34}|>1$, respectively.
Besides, according to \eqref{eq:cij_a1} we have $J<J_{c^{i}_{12}}$ and $J<J_{c_{34}}$,
see table~\ref{tab:Nickel}.
These inequalities signals that the strength
of the exchange is amplified.
Furthermore, the inequality $J_{c_{34}}<J_{c^{i}_{12}}$
indicates that most probably
the density of electrons along Ni3-Ni4 bond is lower than that along the Ni1-Ni2.
To conclude we would like to anticipate that the results for the magnetization and
low-field magnetic field susceptibility of the tetramer $\mathrm{Ni_4Mo_{12}}$ are in
concert with the experimental measurements. These results will be the
subject of a separate paper.
\section*{Acknowledgment}
The authors are indebted to Prof. N.S. Tonchev and Prof. N. Ivanov
for the very helpful discussions and to Prof. J. Schnack for providing
us the experimental results on FIG. \ref{fig:NiExp}.
This work was supported by the Bulgarian National Science Fund under
contract DN/08/18.
|
train/arxiv
|
BkiUcbDxK6nrxjHzAxs6
| 5 | 1 |
\section{Introduction}
We consider continuous dynamical systems with a state $u(\mathbf{x}, t) \in \mathbb{R}$ that evolves over time $t \in \mathbb{R}_+$ and spatial locations $\mathbf{x} \in \Omega \subset \mathbb{R}^D$ of a bounded domain $\Omega$. We assume the system is governed by an unknown partial differential equation (PDE)
\begin{align} \label{eq:pde}
\dot{u}(\mathbf{x},t) := \frac{du(\mathbf{x},t)}{dt} &= F(\mathbf{x}, u, \nabla_\mathbf{x} u, \nabla_\mathbf{x}^2 u, \ldots),
\end{align}
where the temporal evolution $\dot{u}$ of the system depends on the current state $u$ and its spatial first and higher-order partial derivatives w.r.t. the coordinates $\mathbf{x}$. Such PDE models are the cornerstone of natural sciences, and are widely applicable to modelling of propagative systems, such as behavior of sound waves, fluid dynamics, heat dissipation, weather patterns, disease progression or cellular kinetics \citep{courant2008methods}. Our objective is to learn the differential $F$ from data.
There is a long history of manually deriving mechanistic PDE equations for specific systems \citep{cajori1928early}, such as the Navier-Stokes fluid dynamics or the Schrödinger's quantum equations, and approximating their solution forward in time numerically \citep{ames2014numerical}. These efforts are complemented by data-driven approaches to infer any unknown or latent coefficients in the otherwise known equations \citep{isakov2006inverse,berg2017neural,santo2019data}, or in partially known equations \citep{freund2019dpm,seo2019differentiable,seo2020physics}. A series of methods have studied neural proxies of known PDEs for solution acceleration \citep{lagaris1998artificial,raissi2017physics,weinan2018deep,sirignano2018dgm} or for uncertainty quantification \citep{khoo2017solving}.
\paragraph{Related work.} Recently the pioneering work of \citet{long2017pde} proposed a fully non-mechanistic method \mbox{PDE-Net}, where the governing equation $F$ is learned from system snapshot observations as a convolutional neural network (CNN) over the input domain discretised into a spatio-temporal grid. Further works have extended the approach with residual CNNs \citep{ruthotto2019deep}, symbolic neural networks \citep{long2019pde}, high-order autoregressive networks \citep{geneva2020modeling}, and feed-forward networks \citep{xu2019dl}. These models are fundamentally limited to discretizing the input domain with a sample-inefficient grid, while they also do not support continuous evolution over time, rendering them unable to handle temporally or spatially sparse or non-uniform observations commonly encountered in realistic applications.
Models such as \citep{battaglia2016interaction,chang2016compositional,sanchez2018graph} are related to the interaction networks where object's state evolves as a function of its neighboring objects, which forms dynamic relational graphs instead of grids. In contrast to the dense solution fields of PDEs, these models apply message-passing between small number of moving and interacting objects, which deviates from PDEs that are strictly differential functions.
In \citet{poli2019graph} graph neural ordinary differential equations (GNODE) were proposed as a framework for modeling continuous-time signals on graphs. The main limitations of this framework in application to learning PDEs are the lack of spatial information about physical node locations and lack of motivation for why this type of model could be suitable. Our work can be viewed as
connecting graph-based continuous-time models with data-driven learning of PDEs in spatial domain through a classical PDE solution technique.
\paragraph{Contributions.} In this paper we propose to learn free-form, continuous-time, a priori fully unknown PDE model $F$ from sparse data measured on arbitrary timepoints and locations of the coordinate domain $\Omega$ with graph neural networks (GNN). Our contributions are:
\begin{itemize}
\item We introduce continuous-time representation and learning of the dynamics of PDE-driven systems
\item We propose efficient graph representation of the domain structure using the method of lines with message passing neural networks
\item We achieve state-of-the-art learning performance on realistic PDE systems with irregular data, and our model is highly robust to data sparsity
\end{itemize}
Scripts and data for reproducing the experiments can be found in \href{https://github.com/yakovlev31/graphpdes_experiments}{this github repository}.
\begin{table}[ht]
\caption{Comparison of machine-learning based PDE learning methods.}
\begin{center}
\resizebox{\columnwidth}{!}{
\begin{tabular}{l cccc r}
\toprule
& Unknown PDE & Continuous & Free-form & Free-form & \\
Model & learning & time & spatial domain & initial/boundary conditions & Reference
\\
\midrule
PINN & \textcolor{red!60!black}{\ding{55}} & \textcolor{green!60!black}{\ding{51}} & \textcolor{red!60!black}{\ding{55}} & \textcolor{red!60!black}{\ding{55}} & \citet{raissi2017physics} \\
AR & \textcolor{green!60!black}{\ding{51}} & \textcolor{red!60!black}{\ding{55}} & \textcolor{red!60!black}{\ding{55}} & \textcolor{red!60!black}{\ding{55}} & \citet{geneva2020modeling} \\
PDE-net & \textcolor{green!60!black}{\ding{51}} & \textcolor{red!60!black}{\ding{55}} & \textcolor{red!60!black}{\ding{55}} & \textcolor{green!60!black}{\ding{51}} & \citet{long2017pde} \\
DPM & \textcolor{red!60!black}{\ding{55}} & \textcolor{green!60!black}{\ding{51}} & \textcolor{red!60!black}{\ding{55}} & \textcolor{green!60!black}{\ding{51}} & \citet{freund2019dpm} \\
DPGN & \textcolor{green!60!black}{\ding{51}} & \textcolor{red!60!black}{\ding{55}} & \textcolor{green!60!black}{\ding{51}} & \textcolor{green!60!black}{\ding{51}} & \citet{seo2019differentiable} \\
PA-DGN & \textcolor{red!60!black}{\ding{55}} & \textcolor{green!60!black}{\ding{51}} & \textcolor{green!60!black}{\ding{51}} & \textcolor{green!60!black}{\ding{51}} & \citet{seo2020physics} \\
\midrule
Ours & \textcolor{green!60!black}{\ding{51}} & \textcolor{green!60!black}{\ding{51}} & \textcolor{green!60!black}{\ding{51}} & \textcolor{green!60!black}{\ding{51}} & \\
\bottomrule
\end{tabular}
}
\end{center}
\label{tab:methods2}
\end{table}
\section{Methods}
In this Section we consider the problem of learning the unknown function $F$ from observations $(\mathbf{y}(t_0), \ldots, \mathbf{y}(t_M) ) \in \mathbb{R}^{N \times (M+1)}$ of the system's state $\mathbf{u}(t) = (u(\mathbf{x}_1,t),\ldots,u(\mathbf{x}_N,t))^T$ at $N$ arbitrary spatial locations $(\mathbf{x}_1, \dots, \mathbf{x}_N)$ and at $M+1$ time points $(t_0, \dots, t_M)$. We introduce efficient graph convolution neural networks surrogates operating over continuous-time to learn PDEs from sparse data. Note that while we consider arbitrarily sampled spatial locations and time points, we do not consider the case of partially observed vectors $\mathbf{y}(t_i)$ i.e. when data at some location is missing at some time point. Partially observed vectors, however, could be accounted by masking the nodes with missing observations when calculating the loss. The function $F$ is assumed to not depend on global values of the spatial coordinates i.e. we assume the system does not contain position-dependent fields (Section \ref{sec:pos_invar_gnn_diffs}).
We apply the method of lines (MOL) \citep{schiesser2012numerical} to numerically solve Equation $\ref{eq:pde}$.
The MOL consists of selecting $N$ nodes in $\Omega$ and discretizing spatial derivatives in $F$ at these nodes. We place the nodes to the observation locations $(\mathbf{x}_1, \dots, \mathbf{x}_N)$. The discretization leads to $F$ being approximated by $\hat{F}$ and produces the following system of ordinary differential equations (ODEs) whose solution asymptotically approximates the solution of Equation \ref{eq:pde}
\begin{align} \label{eq:odes}
\dot{\mathbf{u}}(t) = \begin{pmatrix} \dot{u}_1(t) \\ \vdots \\ \dot{u}_N(t) \end{pmatrix} = \begin{pmatrix} \frac{d u(\mathbf{x}_1, t)}{dt} \\ \vdots \\ \frac{d u(\mathbf{x}_N, t)}{dt} \end{pmatrix} &\approx \begin{pmatrix} \hat{F}(\mathbf{x}_1, \mathbf{x}_{\mathcal{N}(1)}, u_1, u_{\mathcal{N}(1)}) \\ \vdots \\ \hat{F}(\mathbf{x}_N, \mathbf{x}_{\mathcal{N}(N)}, u_N, u_{\mathcal{N}(N)}) \end{pmatrix} \in \mathbb{R}^N.
\end{align}
As the discretized $\hat{F}$ inherits its unknown nature from the true PDE function $F$, we approximate $\hat{F}$ by a learnable neural surrogate function.
The system's state at $\mathbf{x}_i$ is defined as $u_i$, while $\mathcal{N}(i)$ is a set of indices of neighboring nodes other than $i$ that are required to evaluate $\hat{F}$ at $\mathbf{x}_i$, and $\mathbf{x}_{\mathcal{N}(i)}$ with $u_{\mathcal{N}(i)}$ are positions and states of nodes $\mathcal{N}(i)$. This shows that the temporal derivative $\dot{u}_i$ of $u_i$ depends not only on the location and state at the node $i$, but also on locations and states of neighboring nodes, resulting in a locally coupled system of ODEs.
Each ODE in the system follows the solution at a fixed location $\mathbf{x}_i$. Numerous ODE solvers have been proposed (such as Euler and Runge-Kutta solvers) to solve the full system
\begin{align}\label{eq:ode}
\mathbf{u}(t) &= \mathbf{u}(0) + \int_0^t \dot{\mathbf{u}}(\tau) d\tau,
\end{align}
where $0 \le \tau \le t$ is a cumulative intermediate time variable. Solving equation \ref{eq:ode} forward in time scales linearly both with respect to the number of nodes $N$ and the number of evaluated time points $M$, while saturating the input space $\Omega$ requires a large number of nodes. In practice, PDEs are often applied for two- and three-dimensional spatial systems where the method is efficient.
\subsection{Position-invariant graph neural network differential}\label{sec:pos_invar_gnn_diffs}
After introducing Equation $\ref{eq:odes}$, we transition from learning $F$ to learning $\hat{F}$. The value of $\hat{F}$ at a node $i$ must depend only on the nodes $i$ and $\mathcal{N}(i)$. Furthermore, the number of arguments and their order in $\hat{F}$ is not known in advance and might be different for each node. This means that our model $\hat{F}$ must be able to work with an arbitrary number of arguments and must be invariant to permutations of their order. Graph neural networks (GNNs) \citep{wu2020comprehensive} satisfy these requirements. In a more restricted setting, where the number of neighbors and their order is known, (e.g. if the grid is uniform) other types of models such as multilayer perceptrons and convolutional neural networks can be used as well.
We consider a type of GNNs called message passing neural networks (MPNNs) \citep{DBLP:journals/corr/GilmerSRVD17} to represent $\hat{F}$ as
\begin{align}
\hat{F}_\theta( \mathbf{x}_{\mathcal{N}(i)} - \mathbf{x}_i, u_i, u_{\mathcal{N}(i)}),
\end{align}
where $\mathbf{x}_{\mathcal{N}(i)} - \mathbf{x}_i = \{\mathbf{x}_j - \mathbf{x}_i : j \in \mathcal{N}(i)\}$ and $\theta$ denote parameters of the MPNN.
\begin{wrapfigure}[13]{r}{0.3\textwidth}
\centering
\includegraphics[width=0.28\textwidth]{figures/delaunay_traingulation_example.png}
\caption{Delaunay triangulation for a set of points. Green and orange points are considered to be neighbors as they share the same edge.}
\label{fig:delaunay_traingulation_example}
\end{wrapfigure}
This formulation assumes the absence of position-dependent quantities in $\hat{F}$, but models based on this formulation are invariant to translations and rotations of $\Omega$, which makes generalization to systems with different node positions feasible, and prevents overfitting by memorizing position-specific dynamics.
We use MPNNs, which is a type of spatial-based GNNs, due to their flexibility and computational efficiency. The main alternative -- spectral-based GNNs -- have relatively poor scaling with the number of nodes and learn global, or domain-dependent, filters due to the need to perform eigenvalue decomposition of the Laplacian matrix.
\subsection{Message passing neural networks}
Let a graph $G = (V, E)$ contain nodes $V = \{\mathbf{x}_i\}_{i=1}^N$, defined by the measurement positions, and undirected edges $E = \{e_{ij}\}$, and assume each node and edge are associated with a node feature $\mathbf{v}_i$ and an edge feature $\mathbf{e}_{ij}$, respectively. We use
the node neighborhood $\mathcal{N}(i)$ to define edges. Neighbors for each node were selected by applying Delaunay triangulation to the measurement positions. Two nodes were considered to be neighbors if they lie on the same edge of at least one triangle (Figure \ref{fig:delaunay_traingulation_example}). Delaunay triangulation has such useful properties as maximizing the minimum angle within each triangle in the triangulation and containing the nearest neighbor of each node which helps to obtain a good quality discretization of $\Omega$.
In message passing graph neural networks we propagate a latent state for $K \geq 1$ graph layers, where each layer $k$ consists of first aggregating \emph{messages} $\mathbf{m}_i^{(k)}$ for each node $i$, and then updating the corresponding node \emph{states} $\mathbf{h}_i
^{(k)}$,
\begin{align}\label{eq:mpnn_update_formula}
\mathbf{m}_i^{(k+1)} &= \underset{{j \in \mathcal{N}(i)}}{\bigoplus} \phi^{(k)}\left(\mathbf{h}_i^{(k)}, \mathbf{h}_j^{(k)}, \mathbf{e}_{ij}\right), \\
\mathbf{h}_i^{(k+1)} &= \gamma^{(k)}\left(\mathbf{h}_i^{(k)}, \mathbf{m}_i^{(k+1)} \right),
\end{align}
where $\oplus$ denotes a permutation invariant aggregation function (e.g. sum, mean, max), and $\phi^{(k)}, \gamma^{(k)}$ are differentiable functions parameterized by deep neural networks. At any time $\tau$, we initialise the latent states $\mathbf{h}_i^{(0)} = \mathbf{v}_i = u_i(\tau)$ and node features to the current state $u_i(\tau)$ of the system.
We define edge features $\mathbf{e}_{ij} := \mathbf{x}_j - \mathbf{x}_i$ as location differences. Finally, we use the node states at the last graph layer of the MPNN to evaluate the PDE surrogate
\begin{align}
\frac{d\hat{u}(\mathbf{x}_i, t)}{dt}=\hat{F}_\theta( \mathbf{x}_{\mathcal{N}(i)} - \mathbf{x}_i, u_i, u_{\mathcal{N}(i)}) &= \mathbf{h}_i^{(K)},
\end{align}
which is used to solve Equation \ref{eq:ode} for the estimated states $\hat{\mathbf{u}}(t)=(\hat{u}(\mathbf{x}_1,t),\dots,\hat{u}(\mathbf{x}_N,t))$.
\subsection{Adjoint method for learning continuous-time MPNN surrogates}
Parameters of $\hat{F}_\theta$ are defined by $\theta$ which is the union of parameters of functions $\phi^{(k)}$, $\gamma^{(k)}$, $k=1,\dots,K$ in the MPNN. We fit $\theta$ by minimizing the mean squared error between the observed states $\left( \mathbf{y}(t_0), \ldots, \mathbf{y}(t_M) \right)$ and the estimated states $\left( \hat{\mathbf{u}}(t_0), \ldots, \hat{\mathbf{u}}(t_M) \right)$,
\begin{align}
\mathcal{L}(\theta) &= \int_{t_0}^{t_M} \ell(t, \hat{\mathbf{u}}) dt
= \int_{t_0}^{t_M} \frac{1}{M+1} \sum_{i=0}^M || \hat{\mathbf{u}}(t_i) - \mathbf{y}(t_i) ||_2^2 \delta(t - t_i) dt \\
&= \frac{1}{M+1}\sum_{i=1}^M || \hat{\mathbf{u}}(t_i) - \mathbf{y}(t_i) ||_2^2.
\end{align}
While discrete-time neural PDE models evaluate the system state only at measurement time points, more accurate continuous-time solution for the estimated state generally requires many more evaluations of the system state. If an adaptive solver is used to obtain the estimated states, the number of time steps performed by the solver might be significantly larger than $M$. The amount of memory required to evaluate the gradient of $\mathcal{L}(\theta)$ by backpropagation scales linearly with the number of solver time steps. This typically makes backpropagation infeasible due to large memory requirements. We use an alternative approach, which allows computing the gradient for memory cost, which is independent from the number of the solver time steps. The approach was presented in \citet{chen2018neural} for neural ODEs and is based on the adjoint method \citep{pontryagin2018mathematical}. The adjoint method consists of a single forward ODE pass \ref{eq:ode} until state $\hat{\mathbf{u}}(t_M)$ at the final time $t_M$, and subsequent backward ODE pass solving the gradients. The backward pass is performed by first solving the adjoint equation
\begin{align}
\dot{\boldsymbol\lambda}(t)^T &= \frac{\partial \ell}{\partial \hat{\mathbf{u}}(t)} - \boldsymbol\lambda(t)^T \frac{\partial \hat{F}}{\partial \hat{\mathbf{u}}(t)}.
\end{align}
for the adjoint variables $\boldsymbol\lambda$ from $t=t_M$ until $t=0$ with $\boldsymbol\lambda(t_M)=0$, and then computing
\begin{align}\label{eq:adj_grad}
\frac{d\mathcal{L}}{d\theta} &= -\int_0^T \boldsymbol\lambda(t)^T \frac{\partial \hat{F}}{\partial \theta} dt
\end{align}
to obtain the final gradient.
\section{Experiments}
We evaluate our model's performance in learning the dynamics of known physical systems. We compare to state-of-the-art competing methods, and begin by performing ablation studies to measure how our model's performance depends on measurement grid sizes, interval between observations, irregular sampling, amount of data and amount of noise.
\subsection{Convection-diffusion ablation studies}\label{sec:conv_diff_ablation_studies}
The convection-diffusion equation is a partial differential equation that can be used to model a variety of physical phenomena related to the transfer of particles, energy, and other physical quantities inside a physical system. The transfer occurs due to two processes: convection and diffusion. The convection-diffusion equation is defined as
\begin{equation}
\frac{\partial u(x,y,t)}{\partial t} = D\nabla^2u(x,y,t)-\mathbf{v}\cdot\nabla u(x,y,t),
\end{equation}
where $u$ is the concentration of some quantity of interest (full problem specification and setup are in Appendix \ref{appendix_convdiff_ablation}). Quality of the model's predictions was evaluated using the relative error between the observed states $\mathbf{y}(t_i)$ and the estimated states $\hat{\mathbf{u}}(t_i)$:
\begin{equation}
Err = \frac{\left\| \mathbf{y}(t_i) - \hat{\mathbf{u}}(t_i) \right\|}{\left\| \mathbf{y}(t_i) \right\|}.
\end{equation}
In all following experiments, unless otherwise stated, the training data contains 24 simulations on the time interval $[0, 0.2]\ \text{sec}$ and the test data contains 50 simulations on the time interval $[0, 0.6]\ \text{sec}$. The data is randomly downsampled from high fidelity simulations, thus all train and test simulations have different node positions while the number of nodes remains constant. Examples from the train and test sets are shown in Figure \ref{fig:iclr_convdiff_train_test}.
\paragraph{Different grid sizes.}
This experiment tests our model's capability to learn from data with different density of observation points.
The time step was set to $0.02\ \text{sec}$ resulting in 11 training time points per simulation. The number of observation points $\mathbf{x}_i$ (and consequently nodes in the GNN) was set to $3000$, $1500$ and $750$. The resulting grids are shown in the first column of Figure \ref{fig:comp_diff_grids}. Figure \ref{fig:rel_errs_different_grids_and_comp_diff_grids} shows relative test errors and models' predictions.
\begin{figure}[t]
\centering
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[width=0.8\linewidth]{figures/rel_errs_different_grids.png}
\caption{}
\label{fig:rel_errs_different_grids}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[width=1\linewidth]{figures/comp_diff_grids.png}
\caption{}
\label{fig:comp_diff_grids}
\end{subfigure}
\caption{a) Relative test errors for different grid sizes. b) Visualization of the true and learned system dynamics (grids are shown in the first column).}
\label{fig:rel_errs_different_grids_and_comp_diff_grids}
\end{figure}
The performance of the model decreases with the number of nodes in the grid. Nonetheless, even with the smallest grid, the model was able to learn a reasonably accurate approximation of the system's dynamics and generalize beyond the training time interval.
\paragraph{Different measurement time interval.}
As will be shown in the following experiments, models with a constant time step are sensitive to the length of the time interval between observations. While showing good performance when the time step is small, such models fail to generalize if the time step is increased. This experiment shows our model's ability to learn from data with relatively large time intervals between observations.
We used $11$, $4$ and $2$ evenly spaced time points for training. The number of nodes was set to $3000$. Figure \ref{fig:rel_errs_different_time_grids_and_comp_diff_time_grids} shows relative test errors and models' predictions.
\begin{figure}[t!]
\centering
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[width=0.8\linewidth]{figures/rel_errs_different_time_grids.png}
\caption{}
\label{fig:rel_errs_different_time_grids}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{figures/comp_diff_time_grids.png}
\caption{}
\label{fig:comp_diff_time_grids}
\end{subfigure}
\caption{a) Relative test errors for different time grids. b) Visualization of the true and learned system dynamics (grids are shown in the first column).}
\label{fig:rel_errs_different_time_grids_and_comp_diff_time_grids}
\end{figure}
The model is able to recover the continuous-time dynamics of the system even when trained with four time point per simulation. Increasing the frequency of observation does not significantly improve the performance. An example of a training simulation with four time points is shown in Figure \ref{fig:example_sim_with_4_time_points}.
\paragraph{Irregular time step.}
Observations used for training might not be recorded with a constant time step. This might cause trouble for models that are built with this assumption. This experiment tests our model's ability to learn from data observed at random points in time.
\begin{wrapfigure}[13]{r}{0.4\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{figures/rel_errs_noisy_noiseless_time_grids.png}
\caption{Relative test errors for regular and irregular time grids.}
\label{fig:rel_errs_noisy_noiseless_time_grids}
\end{wrapfigure}
The model is trained on two time grids. The first time grid has a constant time step $0.02\ \text{sec}$. The second grid is the same as the first one but with each time point perturbed by noise $\epsilon \sim \mathcal{N}(0, (\frac{0.02}{6})^2)$. This gives a time grid with an irregular time step. The time step for test data was set to $0.01\ \text{sec}$. The number of nodes was set to $3000$. Relative test errors are shown in Figure \ref{fig:rel_errs_noisy_noiseless_time_grids}. In both cases the model achieves similar performance. This demonstrates the continuous-time nature of our model as training and predictions are not restricted to evenly spaced time grids as with most other methods. None of the previous methods that learn free form (i.e., neural network parameterised) PDEs can be trained with data that is sampled irregularly over time.
\begin{wrapfigure}[13]{r}{0.4\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{figures/rel_errs_diff_num_of_sims.png}
\caption{Relative test errors for different amounts of training data.}
\label{fig:rel_errs_diff_num_of_sims}
\end{wrapfigure}
\paragraph{Different amount of data.}
In this experiment, the model is trained on 1, 5, 10 and 24 simulations. The test data contains 50 simulations. The time step was set to $0.01\ \text{sec}$. The number of nodes was set to $3000$. Relative test errors are shown in Figure \ref{fig:rel_errs_diff_num_of_sims}.
Performance of the model improves as the amount of training data increases. It should be noted that despite using more data, the relative error does not converge to zero.
\paragraph{Varying amount of additive noise.}
We apply additive noise $\epsilon \sim \mathcal{N}(0, \sigma^2)$ to training data with $\sigma$ set to 0.01, 0.02, and 0.04 while the largest magnitude of the observed states is 1. The time step was set to 0.01 sec. The number of nodes was set to 3000. Noise was added only to the training data. The relative test errors are shown in Figure \ref{fig:noisy_data_errs}. The model's performance decreases as $\sigma$ grows but even at $\sigma=0.04$ it remains quite high.
\newpage
\subsection{Benchmark method comparison}
\begin{wrapfigure}[14]{r}{0.4\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{figures/noisy_data_errs.png}
\caption{Relative test errors for different amounts of noise in the training data.}
\label{fig:noisy_data_errs}
\end{wrapfigure}
The proposed model was compared to two models presented in the literature: PDE-Net \citep{long2017pde} and DPGN \citep{DBLP:journals/corr/abs-1902-02950}. PDE-Net is based on a convolutional neural network and employs a constant time-stepping scheme resembling the Euler method. DPGN is based on a graph neural network and implements time-stepping as an evolution map in the latent space.
We used the PDE-Net implementation provided in \citet{long2017pde} except that we pass filter values through an MLP consisting of 2 hidden layers 60 neurons each and \texttt{tanh} nonlinearities which helps to improve stability and performance of the model. We use $5 \times 5$ and $3 \times 3$ filters without moment constraints and maximum PDE order set to $4$ and $2$ respectively.The number of $\delta t$-blocks was set to the number of time steps in the training data. Our implementation of DPGN followed that from \citet{DBLP:journals/corr/abs-1902-02950} with latent diffusivity $\alpha=0.001$. The number of parameters in all models was close to 20k.
\begin{wrapfigure}[15]{r}{0.4\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{figures/model_comp.png}
\caption{Mean relative errors of models trained with different time steps.}
\label{fig:model_comp}
\end{wrapfigure}
The training data contains 24 simulations on the time interval $[0, 0.2]\ \text{sec}$ with the following time steps: $0.01$, $0.02$ and $0.04\ \text{sec}$. The test data contains 50 simulations on the time interval $[0, 0.6]\ \text{sec}$ with the same time steps. The data was generated on a $50 \ \times \ 50$ regular grid as PDE-net cannot be applied to arbitrary spatial grids. Separate models were trained for each time step. The performance of the models was evaluated using the mean of relative test error averaged over time.
Mean relative test errors of the models are shown in Figure \ref{fig:model_comp}. The figure shows that performance of the discrete-time models is strongly dependent on the time step while performance of the continuous-time model remains at the same level. At the smallest timestep, PDE-Net with $5 \times 5$ filters outperforms other models due to having access to a larger neighborhood of nodes which allows the model to make more accurate predictions. However, larger filter size does not improve stability.
We note that some discrete-time models, e.g. DPGN, could be modified to incorporate the time step as their input. Comparison with this type of models would be redundant since Figure \ref{fig:model_comp} already demonstrates the best case performance for such models (when trained and tested with constant time step).
\begin{wrapfigure}[15]{r}{0.4\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{figures/iclr_ours_vs_gnode.png}
\caption{Mean relative test errors of models with and without relative node positions.}
\label{fig:iclr_ours_vs_gnode}
\end{wrapfigure}
\paragraph{Importance of relative positional information.}
We test our model with and without relative node positions that are encoded as the edge features in our MPNN on grids with a different number of nodes. Smaller number of nodes results in
higher distance variability between neighboring nodes (Figure \ref{fig:relative_node_dists_for_graphs_with_diff_num_of_nodes})
which should increase the dependence of the model accuracy on the relative spatial information. By removing spatial information from our model, we recover GNODE. The models were tested on the heat (Appendix \ref{app_heat_eq_exp}) and convection-diffusion equations. A full description of the experiment is in Appendix \ref{app:rel_pos_info_exp}. The results are shown in Figure \ref{fig:iclr_ours_vs_gnode}.
Surprisingly, GNODE shows good results on the purely diffusive heat equation. Nonetheless, the performance of GNODE noticeably differs from that of our model that includes the spatial information. Furthermore, the performance difference almost doubles as the number of nodes is decreased from 100\% to 50\%.
When applied to the convection-diffusion equation, GNODE fail to learn the dynamics irrespective of the number of nodes. This can be explained by the presence of the convective term which transports the field in a specific direction thus making positional information particularly important for accurately predicting changes in the field.
\subsection{Other dynamical systems}
The model was tested on two more dynamical systems in order to evaluate its ability to work with a wider range of problems. We selected the heat equation and the Burgers' equations for that purpose. The heat equation is one of the simplest PDEs while the Burgers' equations are more complex than the convection-diffusion equation due to the presence of nonlinear convective terms. The increase in the problems' difficulty allows to trace the change in the model's performance as we move from simpler to more complex dynamics while keeping the number of model parameters fixed.
\paragraph{Heat equation.}
The heat equation describes the behavior of diffusive systems. The equation is defined as $\frac{\partial u}{\partial t}=D\nabla^2u$, where $u$ is the temperature field (see Appendix \ref{app_heat_eq_exp} for details).
Figure \ref{fig:rel_errs_heat_and_comp_heat} shows relative errors and model predictions for a random test case.
\begin{figure}[t!]
\centering
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[width=0.8\linewidth]{figures/rel_errs_heat.png}
\caption{}
\label{fig:rel_errs_heat}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{figures/comp_heat.png}
\caption{}
\label{fig:comp_heat}
\end{subfigure}
\caption{a) Relative test errors for heat equation. b) True and learned system dynamics.}
\label{fig:rel_errs_heat_and_comp_heat}
\end{figure}
The heat equation describes simpler dynamics than the convection diffusion equation which allowed the model to achieve slightly smaller test errors.
\paragraph{Burgers' equations.}
The Burgers' equations is a system of two coupled nonlinear PDEs. It describes the behavior of dissipative systems with nonlinear propagation effects. The equations are defined in a vector form as $ \frac{\partial \mathbf{u}(x,y,t)}{\partial t} = D\nabla^2\mathbf{u}(x,y,t) - \mathbf{u}(x,y,t) \cdot \nabla \mathbf{u}(x,y,t) $, where $\mathbf{u}$ is the velocity vector field (see Appendix \ref{app_burgers_eq_exp} for details). For visualization and error measurement purposes, the velocity vector field is converted to a scalar field defined by the velocity magnitude at each node. Figure \ref{fig:rel_errs_burgers_and_comp_burgers} shows relative errors and model predictions for a random test case.
\begin{figure}[t!]
\centering
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[width=0.8\linewidth]{figures/rel_errs_burgers.png}
\caption{}
\label{fig:rel_errs_burgers}
\end{subfigure}
~
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{figures/comp_burgers.png}
\caption{}
\label{fig:comp_burgers}
\end{subfigure}
\caption{a) Relative test errors for Burgers' equations. b) True and learned system dynamics.}
\label{fig:rel_errs_burgers_and_comp_burgers}
\end{figure}
The Burgers' equations describe more complex dynamics than the previous two cases which is reflected in higher relative test errors. Visual comparison of the true and predicted states shows that the model was able to achieve sufficient accuracy at approximating the unknown dynamics.
\section{Conclusion}
We present a continuous-time model of dynamical systems whose behavior is governed by PDEs. The model accurately recovers the system's dynamics even when observation points are sparse and the data is recorded at irregular time intervals. Comparison with discrete-time models reveals the advantage of continuous-time models for datasets with larger time intervals between observations, which is typical for real-world applications where measurements can be either tedious or costly, or both. Discretization of the coordinate domain with the method of lines provides a general modeling framework in which arbitrary surrogate functions can be used for approximating $\hat{F}$. The continuous-time nature of the model enables the use of various time integrators ranging from the Euler method to highly accurate adaptive methods. This allows to optimize the choice of the surrogate function and time integration scheme depending on the structure of the data.
|
train/arxiv
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BkiUd1M4ubng80eumZg5
| 5 | 1 |
\section{Introduction}
\label{section: introduction}
In the U.S., if federal zero-emission vehicle sales targets are met, there could be more than 48 million electric vehicles (EVs) on the road in 2030 \cite{mckinsey}. In order to provide charge to this growing EV population, it is estimated that over 1.2 million public EV chargers need to be installed at on-the-go locations and at destinations where EVs are parked for long periods \cite{mckinsey}. Furthermore, the estimated cost for hardware, planning, and installation of this future public charging infrastructure exceeds \$35 billion (U.S.D.) \cite{mckinsey}.
Due to the increasing numbers of EVs, large investment cost of EV charging infrastructure, and the need to charge EVs through cheaper and cleaner energy resources, it is evident that smart charging strategies are required to schedule the power delivery to EVs to maximize the benefits of both the EVs and the public charging infrastructure \cite{ferguson2018optimal, turan_ITSC, moradipari_TTE}. In response, there has been much academic work in the area of EV smart charging at public facilities in recent years \cite{tucker2019online, 8917101, 8635950}; however, most of the effective and implementable solutions require user input, are often facility/infrastructure specific, or ignore critical infrastructure details for modeling purposes. Additionally, the fact that some smart charging frameworks require user input, specifically energy request amounts and departure time estimates, severely constrains their deployment potential. First, most charging infrastructure installed today does not have 2-way communication and would require a third-party user interface (e.g., a touchscreen or smartphone application) to enable user input. Second, the accuracy of user input information is often quite poor. The authors of \cite{lee2019acn, lee_mag} showed that user input data in their real-world smart charging system has 18.6\% and 26.9\% mean absolute error percentages for departure time and energy request estimates, respectively.
Furthermore, while the research area of smart charging has been thoroughly explored in theory, there are only a few frameworks that are actually deployable in the real-world. This is often due to simplifications of the charging infrastructure, assumptions on data availability from the users (i.e., assuming energy request and/or departure times are known a priori), or the lack of real-world data to test and validate the performance of new smart charging strategies. As such, detailed and repeatable case studies for effective and deployable smart charging algorithms could be valuable to the community.
\textit{Specific Challenges:} There are several key challenges to designing a deployable framework to schedule the charging profiles of numerous EVs. First and foremost, the algorithm must run in real-time without knowledge of the future EV arrivals. The algorithm must adapt its planned power schedules as more information is revealed (i.e., as more EVs arrive to the parking lot). Second, contrary to most smart charging algorithms presented in the literature, if the system is to run on the most common standard chargers under the J1772 standard, the algorithm must be able to function with limited information from each EV \cite{7586042}, \cite{6919255}. Specifically, when an EV plugs in, the algorithm does not get access to the EV's State of Charge (SoC) nor does it know the EV's future departure time (most level 2 chargers do not sense EV SoC nor do they request user input for future departure times). As such, our smart charging algorithm must predict how much energy an EV may consume as well as the EV's future departure time. Such challenges have been acknowledged in past papers including \cite{8585856, 8586075, 8585744, 8586132}. Third, all of the EV charging schedules within a parking lot are coupled due to the local transformer capacity constraint \cite{POWELL2020115352}. As such, the algorithm cannot over-allocate power at any given time; therefore, the algorithm should make use of a model of the future EV arrivals to avoid over allocating power due to unexpected arrivals.
\textit{Contribution:} In this paper, we present a deployable and customizable framework for real-time smart charging to be implemented at real large-scale charging facilities. Three of the main contributions of our framework are as follows:
\begin{enumerate}
\item The smart charging strategy is readily deployable and customizable for a wide-array of facilities, infrastructure, objectives, and constraints.
\item The online optimization framework can be easily modified to operate with or without user input for energy request amounts and/or departure time estimates which allows our framework to be implemented on standard chargers with 1-way communication or new chargers with 2-way communication.
\item The online optimization framework outperforms other real-time strategies (including first-come-first-serve, least-laxity-first, earliest-deadline-first, etc.) in multiple real-world test cases. We showcase our framework with charging session data obtained from collaboration with SLAC and Google campuses \cite{tucker2022real, 9281897} in the Bay Area even with poor accuracy on users' departure time predictions.
\end{enumerate}
\section{Problem Description}
\label{section: problem description}
In the following, we consider an EV charging facility running a real-time smart charging strategy. Specifically, we are examining a time horizon $t=1,\dots,T$ where the facility manager wants to schedule the energy delivery to the EVs that utilize the chargers within the facility. We denote the variable $e_i(t)$ as a $T$x1 vector for the amount of energy (in kWh) that is delivered at each time $t$ to EV $i$. It is through this variable that the smart-charging algorithm can determine when to deliver energy and how much to deliver. Furthermore, throughout the time horizon, we index the EV arrivals as $i=1,\dots,I$ with each EV having an arrival time $t_i^a$, departure time $t_i^d$, and energy demand/request amount $d_i$. We note that the departure time and energy request for each arrival $i$ is unknown unless the facility has communication infrastructure in place to solicit this information from the users. Additionally, we note that any real-time smart charging algorithm must be able to function without knowledge of the arrival sequence a priori. That is, a real-time smart charging algorithm must operate as information is revealed through arrivals and departures within the system. The online optimization framework we present is able to function in real-time with or without the user information for departure times and energy requests, an aspect that is critical for deploying a smart-charging framework to existing charging infrastructure.
\subsection{Objectives}
Due to the fact that we are designing a framework that functions with or without user input (i.e., a framework agnostic to users' energy requests and departure times), we must design an appropriate objective function that is fair to all users, does well in controlling the costs incurred by the facility, and outputs desirable load profiles for the facility.
Accordingly, in this section, we discuss several objectives that the manager of a workplace charging facility might want to optimize. These include maximizing the utility of the EV owners, minimizing Time-Of-Use (TOU) electricity cost, minimizing demand charges, promoting load flattening and equal energy sharing, or utilizing behind-the-meter renewable generation. Likely, a facility manager would want to consider multiple objectives simultaneously with varying importance for each objective. As such, the main objective function of our optimization framework is a summation of various utility functions multiplied by weights to signify their importance:
\begin{align}
\max_{\substack{e}} U(e) = \max_{\substack{e}} \sum_{f=1}^{F} w_f u_f(e).
\end{align}
Note that $f=1,\dots,F$ are the various utility functions included in the global objective. Additionally, $w_1,\dots,w_F$ are the weights used to tune the global objective and determine the relative importance of each utility function. Now we will discuss several utility functions that could be included by a charging facility operator (some of the following utility functions were adopted from \cite{lee_tsg1, lee_tsg2} which present a similar real-time smart charging algorithm and case studies; however, they make the assumption that user input for energy request amounts and departure times is always available).
\textit{EV Owner Utility:} The main purpose of many workplace charging facilities is to provide utility to the users of the system (e.g., employees or visitors that drive EVs). As such, one objective that is common to many smart charging systems is to maximize user utility based on how much energy their EV receives during their stay (this objective can be used if energy requests are unknown):
\begin{align}
\label{eq:OU}
u_{OU}(e) = \sum_{i}\log (\sum_{t}e_i(t)+1).
\end{align}
In \eqref{eq:OU}, we represent the utility of user $i$ as a logarithmic function dependant on how much energy is delivered to user $i$'s EV. The logarithmic utility term was chosen to model the diminishing returns effect in user utility for EVs receiving excessive amounts of energy (e.g., the first 20kWh charged is more valuable to the EV owner than the second 20kWh). We note that this objective does not depend on when the energy is delivered; therefore, if the facility operator wants to emphasize quickly charging the EVs, the following objective can be used.
\textit{Quick Charging:} Here, the facility manager's goal is to deliver as much energy as possible to the EVs (and does not require departure time or energy request information):
\begin{align}
\label{eq:qc}
u_{QC}(e) = \sum_{t}\frac{T-t+1}{T}\sum_{i}e_i(t).
\end{align}
In \eqref{eq:qc}, the utility for energy delivered at time $t$ is scaled by a term that decreases as time progresses. That is, energy delivered to EVs in the near future is more valuable than energy delivered later on, thus prioritizing quick charging.
\textit{Maximizing Profit (or Minimizing Cost):} In addition to maximizing user utility, many charging facilities aim to maximize profit or minimize their operational costs. In the following, let $q(t)$ be the price that a user has to pay the facility for 1 kWh of energy at time $t$ and $p(t)$ be the price of electricity from the time-of-use (TOU) rates from the local utility. Furthermore, let us denote $z(t)$ as the energy used by other loads that are not the EVs from behind the same meter (e.g., other buildings, lights, etc). As such, we can write the profit maximization function as follows:
\begin{align}
u_{PM}(e) = \sum_{t}q(t)\sum_{i} e_i(t) - \sum_{t}p(t)\Big(\sum_{i}e_i(t) + z(t)\Big).
\end{align}
If $z(t)$ is unknown or unmeasured, it can be set to zero and only the revenue and cost of the EVs will be considered. Furthermore, setting $q(t)=0,\forall t$ allows for the manager to only consider cost minimization.
\textit{Minimizing Demand Charges:} Alongside TOU energy costs, the facility manager needs to be conscious of the maximum demand each month. We denote the monthly demand charge as $\hat{p}$ \$/KW that gets charged based on the peak load each month:
\begin{align}
u_{DC}(e) = -\hat{p} \cdot \max_{\substack{t}}\big(\sum_{i}e_i(t)+z(t)\big).
\end{align}
This can be a difficult objective to minimize in real-time; therefore, we use an estimate of the peak demand from the previous month, denoted as $\hat{e}_{old}$ and charge the facility operator for any increase $\hat{e}_{inc}$ to the peak load estimate. As such, we adopt the following strategy to include demand charges in our real-time optimization formulation:
\begin{align}
&u_{DC}(e) = -\hat{p} \cdot \hat{e}_{inc}\\
&\nonumber \text{where }\\
&\hat{e}_{inc} = \max_{\substack{t}}\bigg\{\sum_{i}\big(e_i(t)+z(t)\big)-\hat{e}_{old},0\bigg\}.
\end{align}
\textit{Load Flattening:} Another desirable outcome for a facility manager implementing a smart charging algorithm is load flattening. This minimizes the variations in the load from the facility which can help the local utility and reduce the need for extra generation to account for sudden changes:
\begin{align}
u_{LF}(e) = -\sum_{t} \big( \sum_{i} e_i(t) + z(t) \big) ^2.
\end{align}
\textit{Equal Sharing:} Another useful objective is to promote equal sharing of the facility's resources among the plugged-in EVs. We write this utility function as follows:
\begin{align}
u_{ES}(e) = -\sum_{t, i} e_i(t)^2.
\end{align}
Additionally, the inclusion of this term in the global objective function ensures a unique optimal solution; specifically, if there are multiple energy delivery schedules that yield the same total utility, then the addition of the equal sharing objective will force the optimal solution to be the one that most evenly distributes energy among the plugged-in EVs.
\textit{Energy Demand: } Lastly, in the offline optimal and online solutions, if users' energy demands are known either from historical data or user input, then the facility tries to deliver a certain amount of energy to each EV denoted as the energy demand $d_i$ of EV $i$. In the offline case, the energy demand can be ensured with a constraint; however, in the online case, to ensure feasibility, sometimes it can be beneficial to include a penalty function for not fulfilling the energy demand. We can denote such a penalty term as follows:
\begin{align}
\label{eq:ED}
u_{ED}(e) = -\sum_{i}\bigg(|\sum_{t}e_i(t)-d_i|\bigg).
\end{align}
When the demand for all the EVs is met, then $\sum_{t}e_i(t)=d_i,\forall i$ and the penalty is 0.
\subsection{Constraints}
There are numerous constraints that limit how the smart charging algorithm can distribute energy to the users' EVs:
\begin{subequations}
\begin{align}
& \label{c: energy min max}0 \leq e_i(t) \leq e_{max}, \quad&&\forall t,i\\
& \label{c: energy outside}e_i(t) = 0, \quad&&\forall t\notin [t_i^a,t_i^d]\\
& \label{c: energy demand}\sum_{t} e_i(t) \leq d_i, \quad&&\forall i\\
& \label{c: energy trans}\sum_{i} e_i(t) \leq e_{trans}, \quad&&\forall t\\
&\label{c: demand increase}\hat{e}_{inc}\geq \sum_{i} e_i(t) - \hat{e}_{old}, \hspace{4pt}&&\forall t\\
&\label{c: demand increase 0}\hat{e}_{inc}\geq0.
\end{align}
\end{subequations}
The constraints are described as follows: \eqref{c: energy min max} constrains the minimum and maximum amount of energy that a charger can deliver to an EV in one time slot. Constraint \eqref{c: energy outside} ensures that an EV has to be plugged-in for it to receive energy. Constraint \eqref{c: energy demand} ensures that the optimization does not exceed the energy demand of each EV if the energy request amounts for the EVs are known. Constraint \eqref{c: energy trans} is a coupling constraint across all EVs that limits the total energy delivered at any time slot due to the infrastructure limits or the local transformer. Constraints \eqref{c: demand increase} and \eqref{c: demand increase 0} describe the calculation of the incremental increase to the maximum energy demand for demand charge calculation.
\subsection{Offline Optimization}
If the entire sequence of EV arrivals was known for a given time span (i.e., their arrival times, departure times, and energy demands), then one can formulate an offline optimization using the utility functions in \eqref{eq:OU}-\eqref{eq:ED} and constraints \eqref{c: energy min max}-\eqref{c: demand increase 0} to solve for the optimal smart charging strategy for a given time period:
\begin{subequations}
\begin{align}
&\nonumber\max_{\substack{e}} U(e)\\
&\nonumber\text{subject to:}\\
&\nonumber\eqref{c: energy min max}-\eqref{c: demand increase 0}
\end{align}
\end{subequations}
However, this is not the case in real-world charging facilities. Instead, information is revealed to the facility manager in an online fashion, and charging decisions must be made without perfect knowledge of the future. As such, in the following section we discuss an online solution akin to model-predictive-control (MPC) that solves the smart charging scheduling problem in real-time. Additionally, it readily handles challenges that come from inaccurate departure time information.
\subsection{Real-Time Smart Charging Algorithm (RTSC-A)}
In this section, we discuss the online optimization framework to solve the smart charging problem in real-time without knowledge of the future arrivals. The approach is akin to model predictive control (MPC) and solves a convex optimization problem at each time step. Specifically, for each time step $t=1,\dots,T$ the optimization solves for the energy output of each EVSE for the next $T-t$ periods. Then, the facility enacts the control strategy for the next time slot, and recomputes the next set of actions at $t=t+1$ (i.e., rolling horizon).
\textit{Departure time scenario generation:} As discussed in the introduction, users' departure time estimates tend to be quite inaccurate or they are unavailable. As such, we generate $n$ potential departure times that the user might leave at (either from historical data or user input, if available), thus allowing the Real-Time Smart Charging Algorithm (RTSC-A) to plan for multiple scenarios. Using these $N$ candidate departure times, we create a scenario in our optimization problem for each potential departure time and solve the optimization across all scenarios. As time progresses, if a potential departure time is no longer feasible (i.e., the potential departure time is the current time step and the EV has not yet departed), then that scenario is removed from the optimization via dynamic scenario weights (let $C_n$ be a weight coefficient for each scenario that is set to 0 if the scenario is no longer feasible). Furthermore let $x_{i,n}$ be the Tx1 binary vector that indicates when EV $i$ is available to charge in scenario $n$.
\textit{Certainty equivalent control for future EV arrivals:} We make use of our dataset to generate a model for an average day that consists of estimated arrival times, departure times, and energy requests for each day of the week. We then use these daily models in the real-time optimization to account for the unknown future EV arrivals. Specifically, at time $t$, let us assume that there are $J$ EVs in the certainty equivalent daily model that are expected to arrive in the future. Let $x_j$ be the Tx1 binary vector indicating when EV $j$ is available to charge. The decision variables that determine how much energy is delivered at a given time $t$ are Tx1 vectors labeled as $e_1,\dots,e_I$ for the actual EVs plugged in and $e_{I+1},\dots,e_{I+J}$ for the future EV arrivals from the model:
\begin{small}
\begin{subequations}
\begin{align}
\label{eqn:obj mpc}&\max_{\substack{e}}
\sum_{i=1}^I \hspace{-1pt}\sum_{n=1}^N \frac{1}{C_n} \Big[U(e_i,x_{i,n})\Big]
+\sum_{j=I+1}^{I+J} \Big[U(e_j,x_{j})\Big]
\\
\nonumber&\textrm{subject to:}
\\
&0 \leq e_k \leq e_{max}, \hspace{40.5pt} \forall k=1,\dots,I+J
\label{eqn:0 mpc}
\\
&e_i^{\;T} x_{i,n} \leq d_i,
\label{eqn:deliveredmin mpc actual}
\;\;\hspace{46pt}\forall i=1,\dots,I,
\\
&\nonumber\hspace{23.8ex}\forall n=1,\dots,N,
\\
&e_j^{\;T} x_{j} \geq d_j^{min},
\label{eqn:deliveredmin mpc model}
\hspace{46pt}\forall j=I+1,\dots,I+J,
\\
&\sum_{k=1}^{I+J} e_k(t) \leq e_{trans}, \;\;\hspace{20pt} \forall t=1,\dots,T,
\label{eqn:transformer}
\\
&\hat{e}_{inc}\geq \sum_{k=1}^{I+J} e_k(t) - \hat{e}_{old}, \hspace{7pt}\forall t=1,\dots,T.
\label{eqn: peak 1 }
\end{align}
\end{subequations}
\end{small}
The first term of the objective function \eqref{eqn:obj mpc} accounts for all $I$ EVs currently plugged in and their $N$ potential departure times each while the second term of the objective function accounts for all $J$ EVs in the future model. Constraint \eqref{eqn:0 mpc} ensures that the energy delivered is nonnegative and less than the EVSE max $e_{max}$. Constraint \eqref{eqn:deliveredmin mpc actual} ensures that the energy demand of EV $i$ is not exceeded (if energy demand is known), constraint \eqref{eqn:deliveredmin mpc model} ensures a minimum amount of energy is delivered to each EV in the future model, and constraint \eqref{eqn:transformer} ensures that the sum of all energy delivered by the EVSEs at each time $t$ does not exceed the transformer constraint $e_{trans}$. Constraint \eqref{eqn: peak 1 } keeps track of any increase to the current month's peak load for the demand charge. We note that $\hat{e}_{old}$ corresponds to the previous peak energy demand that has been observed during the month. The pseudocode for the daily algorithm can be viewed in Algorithm \ref{algorithm}.
\begin{algorithm}[]
\begin{small}
\caption{\textsc{Real-Time Smart Charging}}
\label{algorithm}
\begin{algorithmic}[1]
\FOR{ each day}
\STATE Update current parking lot state
\FOR{ each time interval $t$}
\IF{new departure from parking lot}
\STATE Update parking lot state
\ENDIF
\IF{new arrival to parking lot}
\STATE Generate $N$ potential departure times for new arrival
\STATE Update Parking lot state
\ENDIF
\STATE \textbf{Formulate optimization for time $t$:}
\FOR{each EV $i$ plugged in at time $t$}
\STATE Add EV $i$ to total objective function
\eqref{eqn:obj mpc}
\STATE Add EV $i$ to active constraints \eqref{eqn:0 mpc}-\eqref{eqn: peak 1 }
\ENDFOR
\FOR{each future EV $j$ in daily model $t_{model}>t$}
\STATE Add EV $j$ to total objective function \eqref{eqn:obj mpc}
\STATE Add EV $j$ to active constraints \eqref{eqn:0 mpc}-\eqref{eqn: peak 1 }
\ENDFOR
\STATE \textbf{Solve optimization \eqref{eqn:obj mpc}-\eqref{eqn: peak 1 } for time $t$}
\STATE Store planned energy schedule for each EV $i$
\STATE Set each EVSE's output power for the current time interval
\STATE Update peak load $\hat{e}_{old}$ for demand charge calculation (if a new peak load is observed)
\ENDFOR
\ENDFOR
\end{algorithmic}
\end{small}
\end{algorithm}
\section{Test Cases}
In the following section we compare the performance of our Real-Time Smart Charging Algorithm with various other scheduling strategies. These include the offline optimal (i.e., the optimal schedule solved offline with perfect knowledge of arrival times, departure times, and energy requests), uncontrolled (e.g., First-Come-First-Serve), Least-Laxity-First, and Earliest-Deadline-First. Furthermore, we vary the accuracy of the departure time information that is input to our Real-Time Smart Charging algorithm to showcase its ability to handle inaccurate departure time estimates and still yield good performance. Additionally, we vary the coupling constraint (infrastructure/transformer size) to showcase the performance of our algorithm in a constrained setting. The first test case considers a facility manager that wants to maximize user utility and only slightly cares about TOU electricity costs (i.e., this would be the case of a large company campus who wants to provide free and effective charging for employees). The second test case considers a facility manager that wants to maximize profit while delivering adequate energy to each customer (i.e., this would be the case of a for-profit third-party parking structure equipped with chargers and wants to minimize TOU electricity costs and demand charges).
\subsection{Test Case Specifics}
\label{section: case study}
We examine a two week period from June 17 - June 29 in 2019 at a Bay Area workplace from our Google EV dataset. The location has 57 level 2 EVSEs \textcolor{black}{with 50-100 EVs arriving each weekday} and is under PG\&E's E-19 rate structure.
First, the EV charging session data was filtered by weekday and then filtered again by arrival time. Namely, each charging session was put into one of 12 possible groups corresponding to 2-hour windows for the arrival times (e.g., an EV charging session that started at 9:48am would be stored in the 8:00am-10:00am group). Once this was done, daily arrival time histograms were generated and the average stay duration and average energy consumption were calculated for each of the 12 groups. The average arrivals per weekday, the average arrivals per 2 hour window, the groups' average stay durations, and the groups' average energy consumption were then used to create the algorithm's future model each day and to generate potential departure times for each EV arrival.
Last, we note that all of these simulations were done in Python with CVX and Mosek on a Laptop with an i7 processor and 16gb of RAM. Moreover, the optimization problem's complexity is not affected by the number of arriving EVs each day; rather, the problem size grows only as the number of chargers increases. Additionally, for implementation, (13a)-(13f) has to be solved every 15 minutes and for the 57 chargers in our case study, (13a)-(13f) was solved in less than a second. Thus, the algorithm is scalable and there is significant extra time for computation for a larger dataset (i.e., more chargers at the parking lot).
\subsection{Test Case 1: User Utility Maximization with TOU Rates}
\begin{figure*}[]
\centering
\includegraphics[width=0.8\textwidth]{EM_bars1.png}
\caption{Total energy delivered for the various cases including Least-Laxity-First and Earliest-Deadline-First (both with perfect departure time knowledge) with varying transformer capacities.}
\label{fig: EM_bars1}
\end{figure*}
\begin{figure}[]
\centering
\includegraphics[width=0.9\columnwidth]{EM_bars2.png}
\caption{Cost per KWh from TOU rates for the uncontrolled, offline optimal, and 4 MPC test cases}
\label{fig: EM_bars2}
\end{figure}
In this section, we consider the case of a facility manager that wants to maximize user utility and only slightly cares about TOU electricity costs (i.e., this would be the case of a large company campus who wants to provide free and effective charging for employees). The following objective function was used in the Real-Time Smart Charging Algorithm and the offline optimization:
\begin{align}
U_{1}(e) =
15 u_{OU}(e) + u_{PM}(e) + 10^{-9}\Big( u_{LF}(e) + u_{ES}(e)\Big)
\end{align}
In this objective function, the smart charging strategy heavily favors the EV owner's utility with a small emphasis on TOU electricity cost. Load flattening and energy sharing are also included to promote a desirable energy distribution amongst EVs and a flat demand for the utility.
In this test case, we vary the coupling constraint from the local transformer capacity from 160KW to 100KW. We note that in the uncontrolled dataset, the peak load during the two week period was 160.5KW.
Additionally, we vary the accuracy of the departure time estimates that are input to our RTSC-A. This is due to the fact that users often cannot predict their departure times with perfect accuracy (if the system solicits user information) or due to the fact that historical data does not accurately forecast departure times. As such, for our RTSC-A, for each EV charging session, we consider 10 scenarios. Each of the 10 scenarios makes use of a departure time sampled from a normal distribution centered on the real departure time. We vary the standard deviation of the normal distribution to model EV owners' inaccurate departure time estimates. In the following plots, we denote MPC0 as our RTSC-A with perfect departure time estimates (standard deviation equal to 0) and we denote MPC5 as our RTSC-A with departure time estimates sampled from a normal distribution with standard deviation equal to 5 time steps (1hr 15min). MPC0, MPC5, MPC10, MPC20 make use of normal distributions with standard deviation equal to 0, 5, 10, and 20 time steps, respectively. Furthermore, we note that in the case of MPC20, the standard deviation in the departure time estimate is 20x15min = 5 hours, which is a very inaccurate estimate.
Figure \ref{fig: EM_bars1} presents the total energy delivered to EVs for the various cases including Least-Laxity-First and Earliest-Deadline-First (both with perfect departure time knowledge). Figure \ref{fig: EM_bars2} presents the cost per KWh from TOU rates for the uncontrolled, offline optimal, and 4 MPC test cases.\\
\noindent\textit{Test Case 1 Key Results:}
\begin{enumerate}
\item As shown in Figure \ref{fig: EM_bars1}, all of the charging strategies were able to deliver ~900KWh of energy per day with a 160KW coupling constraint. However, as the coupling constraint becomes more restrictive (i.e., 140KW, 120KW, or 100KW), many of the charging strategies begin delivering less energy than the offline optimal.
\item Our RTSC Algorithm with inaccurate departure times consistently beats out EDF and LLF (which are given the exact departure times). This is likely due to the fact that the RTSC algorithm makes use of a future model for arrivals that have not yet shown up, while EDF and LLF are myopic strategies.
\item As shown in Figure \ref{fig: EM_bars2}, the RTSC algorithm purchases cheaper energy from TOU rates than the uncontrolled case. The uncontrolled case consistently purchases energy at over \$0.21 per KWh for all transformer capacities while the RTSC algorithm remains closer to the offline optimal cost per KWh, below \$0.195 per KWh.
\end{enumerate}
\subsection{Test Case 2: Profit Maximization with TOU Rates and Demand Charges}
\begin{figure}[]
\centering
\includegraphics[width=0.95\columnwidth]{PM_bars.png}
\caption{Total revenue, total electricity cost (from both TOU rates and demand charges), electricity cost per KWh, demand charges, and total profit for the uncontrolled, offline optimal, RTSCA with varying departure time accuracies, least-laxity-first, and earliest-deadline-first strategies.}
\label{fig: PM_bars}
\end{figure}
\begin{figure}[]
\centering
\includegraphics[width=0.8\columnwidth]{PM_loads.png}
\caption{Daily loads of the charging facility for various charging strategies.}
\label{fig: PM_loads}
\end{figure}
In this section, we consider a facility manager that wants to maximize profit while delivering adequate energy to each customer (i.e., this would be the case of a for-profit third-party parking structure equipped with chargers and wants to minimize TOU electricity costs and demand charges). The following objective function was used in the Real-Time Smart Charging Algorithm:
\begin{align}
&U_{2}(e) = \\
&\nonumber 10 \Big(u_{PM}(e) + u_{DC}(e)\Big)+ u_{OU}(e) + 10^{-9}\Big( u_{LF}(e) + u_{ES}(e)\Big)
\end{align}
In this objective function, the smart charging strategy prioritizes profit maximization and cost minimization from both TOU rates and demand charges. Load flattening and energy sharing are also included to promote a desirable energy distribution amongst EVs and a flat demand for the utility.
As in the previous test case, we vary the accuracy of the departure time estimates that are input to our RTSC-A. We vary the standard deviation of the normal distribution to model EV owners' inaccurate departure time estimates. Furthermore, in this test case, we do not vary the transformer constraint, it is set to 160KW for all tests. Additionally, we assume energy is sold to the EV users at \$0.30 per kWh.
Figure \ref{fig: PM_bars} presents results for total revenue, total electricity cost (from both TOU rates and demand charges), electricity cost per KWh, demand charges, and total profit for the uncontrolled, offline optimal, RTSC-A with varying departure time accuracies, least-laxity-first, and earliest-deadline-first strategies. Figure \ref{fig: PM_loads} presents the daily load profile of the charging facility for various charging strategies. The top plot includes a comparison of LLF and EDF with the uncontrolled and optimal load. The bottom plot shows the real-time smart charging algorithm's daily load compared to the offline optimal and uncontrolled.
{\textit{Test Case 2 Key Results:}
\begin{enumerate}
\item As shown in Figure \ref{fig: PM_bars}, total revenue, total electricity cost (from TOU rates and demand charges), and electricity cost per KWh, are similar across all the charging strategies, with the offline optimal performing the best and the RTSC-A with perfect departure time information performing the second best.
\item However, in plot 4 of Figure \ref{fig: PM_bars}, we see that the demand charges from the uncontrolled case, EDF, and LLF are significantly more than the offline optimal and RTSC-A.
\item Due to the offline solution's and RTSC-A's ability to optimize for demand charges, the daily net profit (plot 5 in Fig. \ref{fig: PM_bars}) for these strategies significantly outperforms the uncontrolled case as well as EDF and LLF. Even the profit of RTSC-A with bad departure time estimates (RTSC-A with stddev=20) outperforms LLF and EDF with perfect departure time estimates.
\item RTSC-A's average daily profit is $\sim$\$40, LLF's average daily profit is $\sim$\$23, EDF's average daily profit is $\sim$\$19, the offline optimal average daily profit is $\sim$\$52, and the uncontrolled average daily profit is $\sim$\$3.
\item As shown in the top plot of Figure \ref{fig: PM_loads}, EDF and LLF load profiles look similar to the uncontrolled profile and reach the coupling constraint of 160KW in the mid-morning. However, in the bottom plot of Figure \ref{fig: PM_loads}, the RTSC algorithm is able to flatten the daily profile to mimic the offline optimal, which is a perfectly flat profile, thus significantly reducing demand charges.
\end{enumerate}}
\section{Conclusion}
In this paper we presented a customizable online optimization framework for real-time EV smart charging to be readily implemented at real large-scale charging facilities. The smart charging strategy is readily deployable and customizable for a wide-array of facilities, infrastructure, objectives, and constraints. The online optimization framework can be easily modified to operate with or without user input for energy requests and/or departure time estimates. Our online optimization framework outperforms other real-time strategies (including first-come-first-serve, least-laxity-first, earliest-deadline-first, etc.) in multiple real-world test cases using real charging session data from SLAC and Google campuses in the Bay Area even with poor accuracy on users' departure time predictions.
\begin{footnotesize}
\textsc{Acknowledgment:} This work was funded by the California Energy Commission under grant EPC-17-020. SLAC National Accelerator Laboratory is operated for the US Department of Energy by Stanford University under Contract DE-AC02-76SF00515. Thank you to Rolf Schreiber from Google for providing data.
\end{footnotesize}
\linespread{0.95}
\bibliographystyle{IEEEtran}
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\section{Introduction}
A graph $G$ is $q$-Ramsey for $H$, denoted by $G\to (H)_q$, if every $q$-colouring of the edges of $G$ contains a monochromatic copy of $H$. Many interesting questions arise when we consider those graphs $G$ which are minimal with respect to $G\to (H)_q$. A graph $G$ is $q$-Ramsey-minimal for $H$ (or $q$-minimal for $H$) if $G\to (H)_q$ and $G'\nto(H)_q$ for every proper subgraph $G'\sne G$. We denote the family of such graphs by $\mathcal{M}_q(H)$. The fact that $\mathcal{M}_q(H)\neq\varnothing$ for every graph $H$ and every integer $q\ge 2$ is a consequence of Ramsey's theorem \cite{r1930}.
Burr, Erd\H{o}s, and Lov\'asz~\cite{bel1976} initiated the study of properties of graphs in ${\mathcal M}_2(K_k)$ in 1976, where as usual $K_k$ denotes the complete graph on $k$ vertices. Their seminal paper raised numerous questions on minimal Ramsey graphs that were addressed by various mathematicians in subsequent years~\cite{befs1981,bnr1984,l1994,bdks2001,rs2008}.
Various graph parameters have been studied for graphs in $\mathcal{M}_q(H)$, the most prominent being the Ramsey number $r_q(H)$ which is the smallest number of vertices of a graph in ${\mathcal M}_q(H)$.
When $H$ is the complete graph we also write $R_q(k)$ for $r_q(K_{k})$.
Estimating $R_q(k)$ or even $R_2(k)$ is one of the fundamental open problems in Ramsey theory. It is known that $2^{k/2+o(k)}\leq R_2(k)\leq 2^{2k-o(k)}$ where the best lower bound is due to Spencer~\cite{s1977} improving a result by Erd\H{o}s~\cite{e1947}, and the best known upper bound is due to Conlon~\cite{c2009}, improving earlier bounds by Erd\H{o}s and Szekeres~\cite{es1935}, R\"odl~\cite{gr1987}, and Thomason~\cite{t1988}.
Quite surprisingly, some other parameters could be determined precisely. Ne\v set\v ril and R\"odl~\cite{nr1976} showed, for example, that the smallest clique number of a graph in ${\mathcal M}_q(H)$ is exactly the clique number of $H$, extending earlier work by Folkman~\cite{f1970}. Furthermore, the smallest chromatic number and the smallest connectivity of a graph in ${\mathcal M}_q(H)$ are known for all $H$ and $q\ge 2$, see~\cite{bel1976} and~\cite{bnr1984}.
A parameter of ongoing interest is $s_q(H)$, the smallest minimum degree of a graph $G\in {\mathcal M}_q(H)$. The value of $s_2(H)$ is known for some graphs $H$, including cliques~\cite{bel1976}, complete bipartite graphs~\cite{fl2006}, trees and cycles~\cite{szz2010}, and complete graphs with a pendant edge~\cite{fglps2014}. The asymptotic behaviour of $s_q(K_k)$ was considered when $q\to\infty$ in~\cite{fglps2016,gw2017}, and when $k\to \infty$ in~\cite{hrs2018}.
In this paper we are interested in the interplay between ${\mathcal M}_q(H)$ and ${\mathcal M}_r(H')$ when $q\neq r$ or when $H$ and $H'$ are nonisomorphic.
Clearly, every graph $G$ that is a $q$-minimal graph for some graph $H$ is $r$-Ramsey for $H$, for all $2\le r\le q$, and thus contains an $r$-minimal graph as an induced subgraph. Our first contribution complements this observation in the sense that every $r$-minimal graph $G$ can be obtained this way from a $q$-minimal graph $G'$, as long as $H$ satisfies some connectivity conditions. Following standard notation we write $H\cong H'$ if $H$ and $H'$ are isomorphic.
\begin{theorem}\thlab{rodlscorollary}
Let $H$ be a $3$-connected graph or $H\cong K_3$ and let $q>r \geq 2$ be integers.
Then for every $F\in {\mathcal M}_r(H)$
there are infinitely many graphs $G\in {\mathcal M}_q(H)$ such that $F$ is an induced subgraph of $G$.
\end{theorem}
In fact, this result is an immediate consequence of the following more general statement.
\begin{theorem}\thlab{thm:containment}
Let $H$ be a $3$-connected graph or $H\cong K_3$, let $q\geq 2$ be an integer and
let $F$ be a graph which is not $q$-Ramsey for $H$.
Then there are infinitely many graphs $G\in {\mathcal M}_q(H)$ such that $F$ is an induced subgraph of $G$.
\end{theorem}
For the assertions of \thref{rodlscorollary,thm:containment} to hold it is clearly necessary that $H$ is {\em Ramsey infinite}, that is ${\mathcal M}_q(H)$ is infinite. Some graphs including, for example, star forests with an odd number of edges, are known not to be Ramsey infinite. Faudree~\cite{f1991} provided a full characterization of forests that are Ramsey infinite. Furthermore, it follows from~\cite[Corollary 4]{rr1995} by R\"odl and Ruci\'nski that $H$ is Ramsey infinite if $H$ contains a cycle. It may well be possible that the assertions of \thref{rodlscorollary,thm:containment} hold for all graphs $H$ that are Ramsey infinite.
The 2-colour version of \thref{thm:containment} was proved by Burr, Ne\v{s}et\v{r}il and R\"odl~\cite{bnr1984}, extending earlier work by Burr, Faudree and Schelp~\cite{BFS} who proved the statement for $q=2$ and when $H$ is a complete graph. Yet, it is this multi-colour version which implies \thref{rodlscorollary} as a corollary.
As in \cite{bnr1984} for $q=2$, \thref{thm:containment} also implies the existence of multicolour Ramsey-minimal graphs with arbitrarily large maximum degree, genus and chromatic number. Indeed, it is well-known that, for a fixed graph $H$ containing a cycle and for a fixed integer $k$, the uniform random graph $G(n,p)$ does not contain $H$ as a subgraph and has maximum degree, genus and chromatic number at least $k$ with probability tending to 1 as $n\to\infty$, for some $p=\Theta(1/n)$. Take $F$ in \thref{thm:containment} to be such a graph drawn from $G(n,p)$.
Another implication of \thref{thm:containment} that we find noteworthy is the following.
\begin{corollary}\thlab{wow}
Let $H$ be a $3$-connected graph or $H\cong K_3$ and let $q\geq 2$ be an integer.
Suppose that $\mathcal{M}_q(H)\se\mathcal{M}_q(H')$ for some arbitrary graph $H'$.
Then $\mathcal{M}_q(H)=\mathcal{M}_q(H')$.
\end{corollary}
We provide the short argument in Section~\ref{sec:containment}.
Another way to view \thref{wow} is that if both $H$ and $H'$ are 3-connected or isomorphic to $K_3$ then the two sets $\mathcal{M}_q(H)$ and $\mathcal{M}_q(H')$ are either equal or incomparable with respect to the subset relation, i.e.~the set $\{{\mathcal M}_q(H) : H \text{ is $3$-connected or } K_3\}$ forms an antichain with respect to the subset relation.
We find it instructive to note at this point that for such $H, H'$, in fact, $\mathcal{M}_q(H)=\mathcal{M}_q(H')$ is only possible if $H$ is isomorphic to $H'.$
\begin{theorem}\thlab{dennisObs}
Let $H$ and $H'$ be non-isomorphic graphs that are either 3-connected or isomorphic to $K_3$. Then $\mathcal{M}_q(H)\neq\mathcal{M}_q(H')$ for all $q\ge 2$.
\end{theorem}
It is now natural to ask which pairs of graphs $H$ and $H'$ do satisfy ${\mathcal M}_q(H)={\mathcal M}_q(H')$. For an integer $q\ge 2$ let us call two graphs $H$ and $H'$ {\em $q$-Ramsey equivalent} (or just {\em $q$-equivalent}) if ${\mathcal M}_q(H)={\mathcal M}_q(H')$.
The notion was introduced by Szab\'o, Zumstein and Z\"urcher~\cite{szz2010} in the case of two colours to capture the fact that $s_2(H)=s_2(H')$ for some graphs $H$ and $H'$ merely because ${\mathcal M}_2(H)={\mathcal M}_2(H')$.
We are particularly interested in the relationship between 2-colour equivalence and multi-colour equivalence, i.e.~what can we infer from known results for 2 colours to more colours?
To briefly survey which pairs of graphs are known to be 2-equivalent,
let $H+sH'$ denote the graph formed by the vertex disjoint union of a copy of $H$ and $s$ copies of $H'$, where we omit $s$ when $s=1$.
It is straight-forward to see that $K_k$ is 2-equivalent to $K_k+sK_1$ if and only if $s\le R(k)-k$, see e.g.~\cite{szz2010}. For $k\ge 4$, $K_k$ and $K_k+K_2$ are known to be $2$-equivalent. In fact, Szab\'o, Zumstein and Z\"urcher~\cite{szz2010} proved that for $2\le t\le k-2$ and $s< (R(k-t+1,k)-2(k-t))/2t$ the graphs $K_k$ and $K_k+sK_t$ are $2$-equivalent, where $R(k,\ell)$ denotes the smallest integer $n$ such that every red/blue-colouring of the edges of $K_n$ contains a red copy of $K_k$ or a blue copy of $K_{\ell}$.
For the case $t=k-1$, Bloom and the second author~\cite{bl2018} show that $K_k$ and $K_k+K_{k-1}$ are 2-equivalent for all $k\ge 4$.
(The requirement $k\ge 4$ is necessary in both~\cite{szz2010} and~\cite{bl2018}. Furthermore, the result in~\cite{szz2010} is optimal up to a factor of roughly 2, the result in~\cite{bl2018} is optimal in the sense that $K_k+K_{k-1}$ cannot be replaced by $K_k+2K_{k-1}$. We comment on these {\em non-equivalence} results further below.)
Axenovich, Rollin, and Ueckerdt~\cite{aru2015} provide a tool to lift these 2-equivalence results to $q$-equivalence.
\begin{theorem}[Theorem 10 in \cite{aru2015}] \thlab{aru}
If two graphs $H$ and $H'$ are $2$-equivalent and $H\se H'$ then $H$ and $H'$ are $q$-equivalent for every $q\ge 3$.
\end{theorem}
In particular, the pairs $K_k$ and $K_k+s K_t$ are $q$-equivalent for every $q\ge 3$ whenever they are 2-equivalent. It would be desirable to remove the condition $H\se H'$ from \thref{aru}.
In general, the following lifts $2$-equivalence (without the subgraph requirement) to $q$-equivalence for even $q$.
\begin{observation}\thlab{positiveLC}
Let $a,b,q,r$ be non-negative integers such that $q,r\ge 2$. If $H$ and $H'$ are $q$- and $r$-equivalent then they are $(aq+br)$-equivalent.
\end{observation}
Indeed, the result follows by induction on $a+b\geq 1$ with the case $a+b=1$ given by assumption. Without loss of generality suppose that $H$ and $H'$ have been shown to be $n$-equivalent, where $n=(a-1)q+br$. Now suppose $G$ is a graph such that $G\to(H)_{n+q}$. We claim that then $G\to(H')_{n+q}$ as well. Fix an $(n+q)$-colouring $c:E(G)\to[n+q]$ of the edges of $G$, where $[m]$ denotes the set $\{1,\ldots,m\}$, and consider the (uncoloured) subgraphs $G_1$ given by the $q$ colour classes $1,\ldots, q$ and $G_2$ given by the $n$ colour classes $q+1,\ldots, q+n$. Note that we must have $G_1\to(H)_q$ or $G_2\to(H)_n$ since we could otherwise recolour $G$ with $n+q$ colours without a monochromatic copy of $H$, a contradiction.
By equivalence in $n$ and $q$ colours we then have that $G_1\to(H')_q$ or $G_2\to(H')_n$ and hence the original colouring of $G$ admits a monochromatic copy of $H'$, so $G\to(H')_{n+q}$ as claimed. Similarly, every graph $G$ that is $(n+q)$-Ramsey for $H'$ needs to be $(n+q)$-Ramsey for $H$, which implies ${\mathcal M}_{n+q}(H)={\mathcal M}_{n+q}(H').$
\thref{positiveLC} implies in particular that if two graphs $H$ and $H'$ are 2- {\em and} 3-equivalent, then they are $q$-equivalent for every $q\ge 2$.
We wonder whether it is true that two graphs $H$ and $H'$ are 3-equivalent if they are 2-equivalent and whether this can be shown using ad-hoc methods.
So far, we have investigated what we can deduce for $q\ge 3$ colours when we know that $H$ and $H'$ {\em are} 2-equivalent. What can we deduce when $H$ and $H'$ are {\em not} 2-equivalent?
To examine this question let us return to the example of disjoint cliques from above.
It is easy to see that $K_6$ is 2-Ramsey for $K_3$, yet fails to be Ramsey for the triangle and a disjoint edge, see e.g.~\cite{szz2010}. This shows that $K_3$ and $K_3+K_2$ are not $2$-equivalent.
The following then implies that, in general, nothing can be deduced from non-2-equivalence.
\begin{theorem}\thlab{thm:3equiv}
The graphs $K_3+K_2$ and $K_3$ are $q$-equivalent for all $q\ge 3$.
\end{theorem}
In fact, there are infinitely many pairs of graphs that are not 2-equivalent, yet they are $q$-equivalent for some $q\ge 3$.
To see this let us first mention how the criterion in~\cite{szz2010} generalises to more than two colours.
For integers $q,k_1,\ldots, k_q\ge 2$ let $R(k_1,\ldots,k_q)$ denote the smallest integer $n$ such that any colouring of the edges of $K_n$ with colours $[q]$ contains a monochromatic copy of $K_{k_i}$ in colour $i$, for some $i\in [q]$. We write $R_q(k_1,k_2,\ldots,k_2)$ when $k_2=k_3=\ldots=k_q$.
\begin{theorem}\thlab{GeneralLowerBdd}
Let $k,t,q$ be integers such that $q\ge 2$ and $k>t\ge 2$. If $s<(R_q(k-t+1,k,\ldots,k)-q(k-t))/qt$ then $K_k$ and $K_k+s K_t$ are $q$-equivalent.
\end{theorem}
For $q=2$ and $t\le k-2$ this is Corollary~5.2~(ii) in~\cite{szz2010}, and the argument easily generalises to $q\ge 3$ colours. We provide the proof for completeness in Section~\ref{section4}.
For $q=2$, \thref{GeneralLowerBdd}
is known to be best possible up to a factor of roughly 2. Specifically, Fox, Grinshpun, Person, Szab\'o and the second author~\cite{fglps2014} show that for $k>t\ge 3$ the graphs $K_k$ and $K_k+s K_t$ are {\em not} $2$-equivalent if $s> (R(k-t+1,k)-1)/t$. This result implies the optimality of the equivalence of $K_k$ and $K_k+K_{k-1}$ in~\cite{bl2018} and the optimality up to a factor of roughly 2 in~\cite{szz2010} mentioned above.
The consequence of this non-equivalence result in~\cite{fglps2014} and \thref{GeneralLowerBdd} is that, for given $k>t\ge 3$, the graphs $K_k$ and $K_k+sK_t$ are not 2-equivalent, but they are $q$-equivalent for some large enough $q$, if we take $s$ such that $(R(k-t+1,k)-1)/t < s< (R_q(k-t+1,k,\ldots,k)-q(k-t))/qt$.
The previous discussion shows that in general we cannot deduce non-$q$-equivalence for $q\ge 3$ from non-2-equivalence. However, all of the examples above that witness this phenomenon have at least one of $H$, $H'$ being disconnected.
When both graphs $H$ and $H'$ are 3-connected or isomorphic to $K_3$ then $H$ and $H'$ are not $q$-equivalent for any $q\ge 2$, by \thref{dennisObs}. In fact, it remains an open question, first posed in~\cite{fglps2014}, whether there are two non-isomorphic connected graphs $H$ and $H'$ that are 2-equivalent.
A theorem by Ne\v set\v ril and R\"odl~\cite{nr1976} implies that any graph that is $q$-equivalent to the clique $K_k$, for some $q\ge 2$, needs to contain $K_k$ as a subgraph. Fox, Grinshpun, Person, Szab\'o and the second author~\cite{fglps2014} show that $K_k$ is not $2$-equivalent to $K_k\cdot K_2$, the graph on $k+1$ vertices formed by adding a pendant edge to $K_k$. We lift this result to any number of colours.
\begin{theorem}\thlab{thm:nonequiv}
For all $k, q\geq 3$, $K_k$ and $K_k\cdot K_2$ are not $q$-Ramsey equivalent.
\end{theorem}
Together with the result in~\cite{nr1976} this implies that, for all $q\ge 3$, $K_k$ is not $q$-equivalent to any connected graph other than $K_k$.
We wonder whether one can prove in general that if two graphs $H$ and $H'$ are connected and not 2-equivalent, then they are not $q$-equivalent for any $q\ge 3$. In our proof of \thref{thm:nonequiv} the graph $K_k$ cannot be replaced by, say, $K_k$ missing an edge.
The rest of the paper is organised as follows. In Section 2, we fix our notation and describe the method of signal senders. We also include the proof of \thref{dennisObs} there. In Section 3 we prove \thref{thm:containment,wow}. Section 4 contains the results related to Ramsey equivalence, that is we prove \thref{GeneralLowerBdd}, which we obtain as a corollary to a slightly more general result, as well as both \thref{thm:3equiv} and \thref{thm:nonequiv}. In the final section we discuss open problems.
\section{Preliminaries}\lab{sec:preliminaries}
{\bf Notation.}
For a graph $G=(V,E)$ we write $V(G)$ and $E(G)$ for its vertex set and edge set, respectively,
and we set $v(G)=|V(G)|$ and $e(G)=|E(G)|$. Throughout the paper we assume that $E(G)\se \binom{V(G)}{2}$ and that both $V$ and $E$ are finite.
A graph $F$ is called a {\em subgraph of a graph $G$}, denoted by $F\se G$,
if $V(F)\se V(G)$ and $E(F)\se E(G)$.
Let $G$, $F$, and $H$ be graphs such that $F\se G$ and $V(G)\cap V(H) =\emptyset$.
We write $G-F$ for the graph with vertex set $V(G)$ and edge set $E(G)\setminus E(F)$;
and $G+H$ for the graph formed by the vertex-disjoint union of $G$ and $H$, i.e.~the graph with vertex set $V(G)\cup V(H)$ and edge set $E(G)\cup E(H)$. When $F$ or $H$ consist of a single edge $e$ we also write $G-e$ and $G+e$, respectively.
For a subset $A\subseteq V(G)$ denote by $G[A]$ the {\em induced subgraph} on $A$, i.e.~the graph with vertex set $A$ and edge set consisting of all edges of $G$ with both endpoints in $A$.
A subgraph $F$ of $G$ is called an {\em induced subgraph} if $F = G[V(F)]$.
Given a path $P$ in a graph $G$, the {\em length of $P$} is the number of edges of $P$.
For two subsets $A,B\subseteq V(G)$, we write $\operatorname{dist}_G(A,B)$
for the distance between $A$ and $B$, i.e.~the length
of a shortest path in $G$ with one endpoint in $A$
and the other endpoint in $B$. Given a subgraph $F\subseteq G$,
we also write $\operatorname{dist}_G(A,F)$ for $\operatorname{dist}_G(A,V(F))$ and $\operatorname{dist}_G(A,e)$ if $F$ consists of a single edge $e$.
A {\em $q$-colouring of a graph $G$} is a function $c$ that assigns colours to edges, where the set $S$ of colours has size $q$ and, unless specified otherwise, we assume that $S=[q]=\{1,\ldots,q\}$.
We call a $q$-colouring {\em $H$-free} if there is no monochromatic copy of $H$.
\bigskip
{\bf Signal senders.}
For the proofs of \thref{thm:containment,dennisObs,thm:nonequiv}
we use the idea of signal sender graphs which was first introduced by Burr, Erd\H{o}s and Lov\'asz~\cite{bel1976}.
Let $H$ be a graph and $q\ge 2$ and $d\ge 0$ be integers.
A \emph{negative (positive) signal sender $S=S^-(q, H,d)$ ($S=S^+(q, H,d)$)}
is a graph $S$ containing distinguished edges $e,f\in E(S)$
such that
\begin{enumerate}
\item[(S1)] $S\nto (H)_q$;
\item[(S2)] in every $H$-free $q$-colouring of $E(S)$,
the edges $e$ and $f$ have different (the same) colours; and
\item[(S3)] $\operatorname{dist}_S(e,f)\ge d$.
\end{enumerate}
The edges $e$ and $f$ in the definition above are called
\emph{signal edges} of $S$.
The following was proved by R\"odl and Siggers~\cite{rs2008}, generalising earlier proofs by
Burr, Erd\H{o}s and Lov\'asz~\cite{bel1976} and by Burr, Ne\v{s}et\v{r}il and R\"odl~\cite{bnr1984}.
\begin{lemma}\thlab{lem:signal}
Let $H$ be 3-connected or $H=K_3$, and let $q,d\geq 2$ be integers. Then there exist
negative and positive signal senders $S^-(q, H, d)$ and $S^+(q, H, d)$.
\end{lemma}
In the proofs of \thref{thm:containment,dennisObs,thm:nonequiv}
we construct graphs using several signal senders.
Assume that $G$ is some graph and let $e_1,e_2\in E(G)$ be two disjoint edges.
We say that we {\it join $e_1$ and $e_2$ by a signal sender $S(q,H,d)$}
if we add a vertex disjoint copy $\tilde S$ of a signal sender $S(q,H,d)$ to $G$ and then identify the signal edges of $\tilde S$ with
$e_1$ and $e_2$, respectively.
\thref{dennisObs} is an easy consequence of the existence of signal senders, we prove it here to serve as a simple example of the method of signal senders.
\begin{proof}[Proof of \thref{dennisObs}]
Without loss of generality let $H\nse H'$.
Let $S = S^+(q, H', d)$ be a positive signal sender, where
$d = v(H)+1$. If $S \to (H)_q$, then we are done since $S \nto (H')_q$ by (S1). So we may
assume that there is an $H$-free colouring $\phi : E(S) \to [q]$.
Now construct a graph $G$ as follows. Fix a copy $\tilde H$ of $H'$ and an edge $e$ that is vertex-disjoint
from $\tilde H$. Then, for every $f \in E(\tilde H)$ join $e$ and $f$ by a copy of the signal sender $S$ so that $e$ is always identified with the same signal edge of $S$.
Then, $G \to (H')_q$. Indeed, for a $q$-colouring of $G$, there is a monochromatic copy of $H'$ in one of the copies of the signal sender $S$, or every edge in $\tilde H$ has the same colour as $e$, by (S2) and by construction of $G$. In either case, there is a monochromatic copy of $H'$.
Moreover, $G \nto (H)_q$. Consider the colouring of $E(G)$ defined by colouring each copy of $S$ using $\phi$. Note that any two copies of $S$ intersect in the edge $e$ only (and at most one vertex in $\tilde H$). Since $e$ is always identified with the same signal edge in $S$ this colouring is well-defined.
Now every copy of $H$ in $G$ is contained in a copy of the signal sender $S$
since $H\nse H'$, $H$ is 3-connected or $H\cong K_3$, and since $\operatorname{dist}_G(e,\tilde H) > v(H)$ by choice of $S$ and (S3).
However, $\phi$ is $H$-free (on each copy of $S$), so none of these copies of $H$ is monochromatic.
\end{proof}
\section{Proof of \thref{thm:containment}}
\lab{sec:containment}
In order to prove \thref{thm:containment} we first establish the existence of certain gadget graphs.
Let $F$ and $H$ be graphs and let $d,q\geq 2$ be integers.
Let $G$ be a graph containing both an induced subgraph $\tilde F$
that is isomorphic to $F$
and an edge $e$ that is vertex-disjoint from $\tilde F$.
$G$ is called an \emph{$(H, F, e,q,d)$-indicator}
if $\operatorname{dist}_G(\tilde F,e)\ge d$ and
the following hold for every $i,j\in [q]$:
\begin{enumerate}
\item[(I1)] There exists an $H$-free $q$-colouring of $G$
such that $\tilde F$ is monochromatic of colour $i$.
\item[(I2)] In every $H$-free $q$-colouring of $G$
in which $\tilde F$ is monochromatic of colour $i$, $e$ has colour~$i$.
\item[(I3)] If $f$ is any edge of $\tilde F$, then there exists an
$H$-free colouring of $G-f$ in which $\tilde F-f$ is monochromatic of colour~$i$
and in which $e$ has colour~$j$.
\end{enumerate}
Note that it would be enough to say that the subgraphs and edges in the Properties~(I2) and~(I3) above should have the same or different colours respectively, without mentioning explicit colours $i$ and $j$ (since we can
swap the colours by symmetry). Nevertheless, we find it more convenient
to state the properties in the above manner, so that we do not need
to repeat the argument of swapping colours over again.
The notion of indicators for $q=2$ was introduced by Burr, Faudree and Schelp~\cite{BFS}
who established their existence in the case when $H$ is a clique and $F\not\supseteq H$, but with $d$ not being specified; see Lemma 3 in \cite{BFS}.
We find the definition above to be a suitable generalisation for $q\ge 3$ to be able to prove existence while still being useful gadgets for the proof of \thref{thm:containment}.
By definition it is necessary that $F$ does not contain a copy of $H$ for an $(H,F,e,q,d)$-indicator to exist. Under the assumption that $H$ is suitably connected this turns out to be sufficient. We need one more ingredient though which allows us to combine indicators (and signal senders) by identifying certain edges without creating new copies of $H$.
We say that an $(H, F, e,q,d)$-indicator $G$ {\em has Property~$\cT$}
if there is a collection of subgraphs $\{T_f\se G \mid f\in E(\tilde F)\}$
such that
\begin{enumerate}
\item[(T1)] $V(T_f)\cap V(\tilde F)=f$ and $f\in E(T_f)$ for all $f\in E(\tilde F)$,
\item[(T2)] $V(G)= \bigcup_{f\in E(\tilde F)} V(T_f)$ and $E(G) = \bigcup_{f\in E(\tilde F)} E(T_f)$, and
\item[(T3)] for all distinct $f_1,f_2\in E(\tilde F)$ and all $v\in V(T_{f_1})\cap V(T_{f_2})$ it holds that
$v\in V(\tilde F)$ or $\operatorname{dist}_G(v,\tilde F)\geq d$,
\end{enumerate}
where $\tilde F$ is the fixed induced copy of $F$ in $G$.
\begin{lemma}\thlab{lem:indicator}
Let $H$ be 3-connected or $H=K_3$, let $F$ be a graph that does not contain a copy of $H$, let $e$ be an edge that is vertex-disjoint from $F$, and let $q,d\geq 2$ be integers.
Then there exists an $(H, F, e,q,d)$-indicator $G$ that has Property~$\cT$ .
\end{lemma}
Similar to the convention for signal senders we say that, for given graphs $F\se G$ and an edge $e\in E(G)$ that is vertex-disjoint from $F$, we {\it join $F$ and $e$ by an $(H,F,e,q,d)$-indicator} when we add a vertex-disjoint copy of an $(H,F,e',q,d)$-indicator $G'$ to $G$ and identify the copy of $F$ in $G'$ with $F\se G$ and identify the edge $e'$ in $G'$ with $e$ in $G$.
The proof of \thref{lem:indicator} proceeds by induction on $e(F)$. When $F$ is a matching of two edges, however, we need gadget graphs with a stronger property than $(I3)$. We prove their existence first.
\begin{lemma}\thlab{Case2}
Let $H, F, e, q, d$ be as in \thref{lem:indicator} and assume that $F=\{f_1,f_2\}$ is a matching. Then there exists an $(H, F, e,q,d)$-indicator $G_2$ with $\operatorname{dist}_{G_2}(f_1,f_2)\geq d$ that has Property~$\cT$, where instead of $(I3)$ we have that
\begin{enumerate}
\item[$(I3')$] for $\ell\in\{1,2\}$ there exists an $H$-free colouring of $G_2$
in which $f_\ell$ has colour $i$ and both, $e$ and $f_{3-\ell}$ have colour $j$.
\end{enumerate}
\end{lemma}
\begin{proof}
We construct $G_2$ as follows. Start with a copy $\tilde F$ of $F$ and an edge $e$ that is vertex disjoint from $F$. By a slight abuse of notation we refer to $f_1$ and $f_2$ for the copies of the two edges of $F$.
Let $\{e_1,e_2,\ldots,e_{q-1}\}$ be a matching of $q-1$ edges that are vertex-disjoint from $\tilde F$ and $e$. Let $H_1,H_2,\ldots,H_{q-1}$ be copies of $H$ that are vertex-disjoint from $f_1,f_2,e_1,e_2,\ldots,e_{q-1}$ and such that any two copies $H_i$ and $H_j$ intersect in one fixed edge which we identify with $e$.
Furthermore,
\begin{enumerate}
\item[(i)] join $f_1$ and $e_1$ by a negative signal sender $S_1=S^{-}(q,H,d)$
and for every $2\leq k\leq q-1$ join $f_2$ and $e_k$ by a negative signal sender
$S_k=S^{-}(q,H,d)$;
\item[(ii)] for every $1\leq k<\ell < q$ join $e_k$ and $e_{\ell}$
by a negative signal sender $S_{k,\ell}=S^{-}(q,H,d)$;
\item[(iii)] for every $1\leq k\leq q-1$ and every edge $g\in E(H_k-e)$ join $e_k$ and $g$ by a positive signal sender $S_{k,g}=S^{+}(q,H,d)$.
\end{enumerate}
Note that the existence of the signal senders in (i)-(iii) is given by Lemma~\ref{lem:signal}.
Call the resulting graph $G_2$; an illustration can be found in Figure~\ref{fig:Case2} for the case that $q=4$.
\begin{figure}
\begin{center}
\includegraphics[scale=0.75,page=1]{pictures}
\end{center}
\caption{Indicator for $q=4$ and $F=\{f_1,f_2\}$ being a matching.}
\label{fig:Case2}
\end{figure}
It should be clear that $\operatorname{dist}_{G_2}(e,\tilde F)\geq d$ and $\operatorname{dist}_{G_2}(f_1,f_2)\geq d$. Thus, it remains to prove that $G_2$ satisfies Properties~$(I1)$, $(I2)$, $(I3')$ and Property $\cT$. Without loss of generality we may assume that~$i=q$.
In the light of these properties, we first observe that every copy of $H$ in $G_2$
either is one of the subgraphs $H_k$ with $k\in [q-1]$
or is contained completely in one of the signal senders from (i)-(iii).
Indeed, let a copy $H'$ of $H$ be given and assume first that $H'$ contains at least
one vertex $v$ from a signal sender $S$ such that $v$ is not incident with one of the signal edges of $S$.
Due to the fact that $H$ is 3-connected or $H=K_3$ and the fact that signal edges always
have distance at least $d>v(H)$, it must hold that $H\subseteq S$.
Assume then that $H'$ does not contain such a vertex.
Then $H'$ must be contained in the union of all $H_k$ with $k\in [q-1]$.
As these subgraphs all intersect only in the edge $e$
and since $H$ is 3-connected or $H\cong K_3$, we must have $V(H')=V(H_k)$ for some $k\in [q-1]$.
For Property~$(I1)$, define a $q$-colouring of $G_2$ as follows.
Colour the edges of $\tilde F$ and $e$ with colour $q$, and for every $k\in [q-1]$ colour the edges of $H_k-e$ and $e_k$ with colour $k$. Moreover, colour every signal sender from (i)-(iii)
with an $H$-free $q$-colouring preserving the colours already chosen
for the signal edges. Note that this is possible by Properties~$(S1)$ and $(S2)$,
because the signal senders may only intersect in their signal edges and the colours above
have been chosen in such a way that the signal edges of negative/positive signal senders
receive different/identical colours. The resulting $q$-colouring of $G_2$ is
$H$-free as it is $H$-free on every signal sender and on every subgraph $H_k$ with $k\in [q-1]$.
For Property~$(I2)$, let $c:E(G_2)\rightarrow [q]$ be an $H$-free $q$-colouring of $G_2$
such that $\tilde F$ is monochromatic of colour $q$.
Then $c(e_1)\neq c(f_1)=q$ and $c(e_k)\neq c(f_2)=q$ for every $k\in [q-1]$, by Property~$(S2)$ for the negative signal senders in (i).
Similarly, by Property~$(S2)$ for the negative signal senders in (ii) we obtain that
$c(e_k)\neq c(e_{\ell})$ for every $1\leq k<\ell \leq q-1$.
Therefore, it must hold that $\{c(e_k):k\in [q-1]\}=[q-1]$.
Applying $(S2)$ for the positive signal senders in (iii) we finally deduce that
$H_k-e$ must be monochromatic in colour $c(e_k)$.
Therefore, in order to prevent any copy $H_k$ of $H$ from becoming monochromatic
we must have $c(e)\notin [q-1]$, i.e.~$c(e)=q$.
For Property~$(I3')$, let $f=f_{\ell}$, $\ell \in [2]$ be one of the two edges of $\tilde F$. We define a colouring $c:E(G_2)\rightarrow [q]$ as follows.
Set $c(f_{\ell})=c(e_{3-\ell})=q$, $c(e)=c(f_{3-\ell})=j$
and colour the edges $e_{\ell},e_3,e_4,\ldots,e_{q-1}$
with distinct colours from $[q-1]\setminus \{j\}$.
Colour the edges of $H_k-e$ with colour $c(e_k)$ for every $k\in [q-1]$. Finally, colour every signal sender from (i)-(iii) with an $H$-free colouring
preserving the colours already chosen for the signal edges.
Analogously to the verification of Property~$(I1)$ this is possible
and it results in an $H$-free $q$-colouring of~$G_2$. Property~$(I3')$ follows.
For Property $\cT$\ note that the choice $T_{f_1}=S_1$ and $T_{f_2}= G_2[V(G_2)\setminus V(S_1-e_1)]$ satisfies (T1)-(T3).
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:indicator}]
Without loss of generality we may assume that $d>v(H)$. We proceed by induction on $e(F)$.
If $e(F)=1$ then let $G=S^+(q,H,d)$ be a positive signal sender, which exists by \thref{lem:signal}, and identify its signal edges with $e$ and $f$, where $f$ is the unique edge of $F$. Then Properties~$(I1)$ and $(I2)$ hold by Properties~$(S1)$ and $(S2)$ for positive signal senders. Property~$(I3)$ follows since $F-f$ has no edges, and by $(S1)$ again, after possibly swapping colours.
Property $\cT$\ holds with $T_f=G$.
Suppose now that $e(F)\ge2$.
We construct $G$ as follows. Start with a copy $\tilde F$ of $F$ and an edge $e$ that is vertex disjoint from $\tilde F$. Let $e_1$ be an edge that is vertex-disjoint from $e$ and $\tilde F$, and let $f_1,f_2,\ldots,f_{e(F)}$ be
the edges of $F$ in any order.
For clarity of presentation, we assume that the edges of $\tilde F$ are labelled $f_1,f_2,\ldots,f_{e(F)}$ as well.
Let $G_1$ be an $(H, F-f_1,e_1,q,d)$-indicator that has Property~$\cT$\ as given by induction, and let $G_2$ be an $(H,\{f_1,e_1\},e,q,d)$-indicator that has Property~$\cT$\ as given by \thref{Case2}. Now join $\tilde F-f_1$ and $e_1$ by $G_1$ and join $f_1$ and $e_1$ by $G_2$.
An illustration can be found in Figure~\ref{figCase3}.
\begin{figure}[thb]
\begin{center}
\includegraphics[scale=0.7,page=2]{pictures}
\end{center}
\caption{Recursive construction of indicators.}
\label{figCase3}
\end{figure}
First observe that $\operatorname{dist}_G(e,\tilde F)\geq \min\{\operatorname{dist}_{G_2}(e,f_1),\operatorname{dist}_{G_2}(e,e_1)\}\ge d$ and $\operatorname{dist}_G(e_1,f_1)\geq d$. Furthermore, every copy of $H$ in $G$ must be either a subgraph of $G_1$ or of $G_2$.
To see this, let $H'$ be a copy of $H$ in $G$.
Assume first that $H'$ contains a vertex from $V(G_2)\setminus (e_1\cup f_1)$. Since $\operatorname{dist}_{G_2}(e_1,f_1)\geq d > v(H)$, $H'$ cannot use vertices from both $e_1$ and $f_1$, and thus,
we conclude that $H'\subseteq G_2$ since either $H\cong K_3$ or $H$ is 3-connected.
Assume then that $H'$ does not use vertices from $V(G_2)\setminus (e_1\cup f_1)$.
This implies that $V(H')\se V(G_1)$ so we are done unless $f_1$ is an edge of $H'$ (note that by definition, the vertices of $f_1$ are vertices of $\tilde F-f_1$ and hence of $G_1$).
Assume towards a contradiction that $f_1$ is an edge of $H'$.
Furthermore, we may assume that $H'$ contains a vertex
$v\in V(G_1)\setminus (V(\tilde F)\cup e_1)$ since $\tilde F$ does not contain a copy of $H$.
Let $T_{f_2},\ldots,T_{f_{e(F)}}$ be the subgraphs of $G_1$
given by Property $\cT$. Then $v\in V(T_g)$ for some $g\in E(F-f_1)$, by (T2).
Furthermore, $g$ is unique since otherwise
$\operatorname{dist}_{G}(v,f_1)\geq \min\{\operatorname{dist}_{G_1}(v,\tilde F),\operatorname{dist}_{G_2}(e_1,f_1)\}\geq d>v(H)$ by (T3),
a contradiction since $H$ is connected.
In fact, this shows that no vertex of $V(H')\setminus V(\tilde F)$ is contained in the intersection $V(T_{g_1})\cap V(T_{g_2})$ for distinct $g_1,g_2\in E(\tilde F-f_1)$.
Now, $g$ cannot be incident with both endpoints of $f_1$, and $V(T_g)\cap V(\tilde F) = g$ by (T1). When $H\cong K_3$ this already implies that $v$ together with the vertices of $f_1$ cannot form a copy of $H$.
When $H$ is 3-connected, we then find a vertex $w\notin V(T_g)$
which is incident with $f_1$ and
three internally vertex-disjoint $v$-$w$-paths in~$H'$.
Among these paths there is at least one path
that does not contain a vertex from $g$, call this path $P$.
Since $v\in V(T_g)$ and $w\notin V(T_g)$
there must be an edge $e=xy$ on $P$ such that
$x\in V(T_g)$ and $y\notin V(T_g)$. But now, $x\notin g = V(T_g)\cap V(\tilde F)$, and thus we conclude that $x\notin V(\tilde F)$
and $e=xy \notin E(\tilde F)$. Using (T2) and $y\notin V(T_g)$
it follows that $e\in E(T_{g'})$ for some $g'\in E(\tilde F - f_1 - g)$. But then $x$ is a vertex in $V(H')\setminus V(\tilde F)$
which is contained in the intersection $V(T_g)\cap V(T_{g'})$
for distinct $g,g'\in E(\tilde F - f_1)$. We already explained that such a vertex does not exist, a contradiction.
For Property~$(I1)$, let $c_1$ be an $H$-free $q$-colouring of $G_1$with $\tilde F-f_1$ and $e_1$ having colour $q$, as provided by Properties~$(I1)$ and $(I2)$ for $G_1$. Analogously, let $c_2$ be an $H$-free $q$-colouring of $G_2$ with $\{e_1,f_1\}$ and $e$ having colour $q$. The combination of both colourings together is an $H$-free $q$-colouring $c$ of $G$, as every copy of $H$ is contained either in $G_1$ or in $G_2$. Moreover, $\tilde F$ is monochromatic in colour $q$, as claimed.
For Property~$(I2)$, let $c$ be an $H$-free $q$-colouring of $G$ such that $\tilde F$ is monochromatic of colour $q$. Then $c(e_1)=q$ by Property~$(I2)$ of $G_1$. But then $\{e_1,f_1\}$ is monochromatic in colour $q$ which implies that
$c(e)=q$ by Property~$(I2)$ of $G_2$.
For Property~$(I3)$, let $f\in E(\tilde F)$. Assume first that $f=f_1$.
As in $(I1)$ there exists an $H$-free $q$-colouring $c_1$ of $G_1$
with $\tilde F-f_1$ and $e_1$ having colour $q$. Moreover,
using Property~$(I3)$ of $G_2$ we know that there is an $H$-free $q$-colouring $c_2$
of $G_2-f_1$ such that $c_2(e_1)=q$ and $c_2(e)=j$.
The combination of both colourings is a $q$-colouring as desired,
since every copy of $H$ is contained either in $G_1$ or in $G_2$.
Now, assume that $f\neq f_1$. By Property~$(I3)$ of $G_1$
there is an $H$-free $q$-colouring $c_1$ of $G_1-f$ such that
$\tilde F-\{f_1,f\}$ is monochromatic in colour $q$ and with $e_1$ having colour $j$.
By Property~$(I3')$ of $G_2$ there is an $H$-free $q$-colouring of $G_2$
such that $c_2(f_1)=q$ and $c_2(e)=c_2(e_1)=j$.
The combination of both colourings is a $q$-colouring as desired for Property~$(I3)$.
For Property~$\cT$, let $T_{f_2},\ldots,T_{f_{e(F)}}$ be the subgraphs for $f_2,\ldots,f_{e(F)}$
given by Property~$\cT$\ of $G_1\subseteq G$.
Moreover, set $T_{f_1}=G_2$. Then (T1)
holds for $G$, since (T1) holds for $G_1$
by induction and since $V(G_2)\cap V(\tilde F)=f_1$ and $f_1\in E(G_2)$.
Property~(T2) is given for $G$,
since $V(G)=V(G_1)\cup V(G_2)=\bigcup_{f\in \tilde F-f_1}V(T_f)\cup V(T_{f_1})$ by Property~(T2) for $G_1$; and since $E(G) = E(G_1)\cup E(G_2) = \bigcup_{g\in E(\tilde F)}E(T_g)$.
For (T3), let $v\in V(T_{f_i})\cap V(T_{f_j})$
for some $i\neq j$ where $v\notin V(\tilde F)$.
If $i=1$ or $j=1$, then
$v\in e_1$ and thus $\operatorname{dist}_G(v,\tilde F)\geq d$.
Otherwise, by (T3) for $G_1$ and
the construction of $G$, we conclude that
$
\operatorname{dist}_G(v,\tilde F)\geq \min\{\operatorname{dist}_{G_1}(v,\tilde F-f_1), \operatorname{dist}_{G_2}(e_1,f_1)\}\geq d
$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:containment}.]
Let $H$, $q$, and $F$ be as in the theorem statement. By assumption, there exists an $H$-free $q$-colouring of $F$. Let $F_1,\ldots,F_q$ denote its colour classes.
We construct a graph $G$ as follows.
Let $\{r_1,\ldots,r_q,e_1,\ldots,e_q,f_1,\ldots,f_q\}$ be a matching that
is vertex-disjoint from $F =F_1\cup\ldots\cup F_q$.
Now join these matching edges and the edges of $F$ by signal senders and indicators as follows. Set $d=v(H)+1$.
\begin{enumerate}
\item[(i)] For every $1\leq k<\ell\leq q$ join $r_k$ and $r_{\ell}$
by a negative signal sender $S_{k,\ell} =S^{-}(q,H,d)$;
\item[(ii)] for every $k\in [q]$ and every $g\in F_k$ join
$r_k$ and $g$ by a positive signal sender
$S_{k,g}=S^{+}(q,H,d)$;
\item[(iii)] for every $k\in [q]$ join $F_k$ and $e_k$
by an $(H,F_k,e_k,q,d)$-indicator $I_k$
that has Property~$\cT$ ;
\item[(iv)] for every $k\in [q]$ join $e_k$ and $f_k$ by a negative signal sender
$S_{k}^-=S^{-}(q,H,d)$;
\item[(v)] for every $k\in [q-1]$ join $f_k$ and $f_{k+1}$ by a positive signal sender
$S_{k}^+=S^{+}(q,H,d)$.
\end{enumerate}
The existence of the signal senders and indicators in (i)-(v) follows from \thref{lem:signal,lem:indicator}.
An illustration of the construction can be found in Figure~\ref{fig:containment}.
\begin{figure}
\begin{center}
\includegraphics[scale=0.75,page=4]{pictures}
\end{center}
\caption{Construction of $G$ for $q=3$.}
\label{fig:containment}
\end{figure}
Similar to the constructions for Lemma~\ref{lem:indicator},
we first show that every copy of $H$ in $G$ is a subgraph of either $F$ or one of these signal senders or one of these indicators. Let $H'$ be a copy of $H$ in $G$.
Assume first that there is a signal sender $S$ from (i), (ii), (iv) or (v), and a vertex $v\in V(H')\cap V(S)$ that is not incident to any of the signal edges of $S$. Then $H'\se S$, since the signal edges have distance at least $d$ in $S$ and since $H$ is $3$-connected or a triangle.
So we may assume that $V(H')\se V(F)\cup \bigcup_{k\in[q]}V(I_k)$.
If $V(H')\se V(F)$ then we are done. Thus, we may also assume that $H'$ contains a vertex $v$ from $V(I_k)\setminus V(F)$ for some $k\in [q]$. Let $\{T_g \mid g\in E(F_k)\}$ be the collection of subgraphs of $I_k$
given by Property~$\cT$\ of $I_k$.
By (T2)
we know that $v\in V(T_g)\setminus V(F)$ for some $g\in E(F_k)$.
Moreover, the only edges of $G$ containing $v$ are contained in $I_k$ or in $S_k^-$, by construction.
When $H'\cong K_3$ we immediately can deduce that
$V(H')\se V(I_k)$, since we already assumed
that $V(H')\cap (V(S_k^-)\setminus e_k) = \emptyset$.
Let $V(H')=\{v,x,y\}$ in this case. By Property~(T2)
we have $vx\in E(T_{g'})$ for some $g'\in E(F_k)$.
If $g\neq g'$ holds, then $v\in V(T_g)\cap V(T_{g'})$ and therefore
$\operatorname{dist}_{I_k}(v,F)\geq d = 4$ by Property~(T3),
which yields $H'\subseteq I_k$.
Hence, we may assume that $g=g'$ and $vx\in E(T_g)$,
analogously we may assume that $vy\in E(T_g)$.
If $xy\in E(I_k)$, then we are done.
So, we may also assume that $xy\in E(F)$.
Then by Property~(T1) we get
$\{x,y\}\subseteq V(T_g)\cap V(F) = g$
and thus $xy=g\in E(T_g)$, again implying that $H'\subseteq I_k$.
Consider next the case when $H'$ is $3$-connected.
We already know that we may assume that
$V(H')\se V(F)\cup \bigcup_{k\in[q]}V(I_k)$
and that there is a vertex $v$ from $V(I_k)\setminus V(F)$
for some $k\in [q]$. Again, by (T2) there is an edge $g\in E(F_k)$
with $v\in V(T_g)\setminus V(F)$. If $H'\subseteq T_g$ then we are done.
So, we may assume that $H'$ contains an edge
which does not belong to $E(T_g)$. By the 3-connectivity
of $H'$ we then find a $v$-$y$-path $P$ in $H'$
which does not use vertices from $g$ and such that
the edge $xy\in E(P')$ which is incident with the endpoint $y$ does not belong to $E(T_g)$. Let $P$ be a shortest such path. Then
$x\in V(T_g)$. Since $P$ does not use vertices of $g$
and since $V(T_g)\cap V(F)=g$ by (T1),
we have $x\notin V(F)$. In particular,
the only neighbours of $x$ in $V(F)$ need to belong to $g$,
and hence $y\notin V(F)$. It follows that $y\in V(I_k)\setminus V(F)$ and $xy\in E(I_k)$, since all the indicators intersect in $V(F)$ only. Hence, using (T1) and since $xy\notin E(T_g)$ by the choice of $P$, we find an edge $g'\in E(F_k-g)$ such that $xy\in E(T_{g'})$. But then $x\in V(T_g)\setminus V(F)$ is a vertex that lies in the intersection
$V(T_g)\cap V(T_{g'})$ for distinct $g,g'\in E(F_k)$.
Applying (T3) leads to $\operatorname{dist}_{I_k}(x,F)\geq d$
and hence $H'\subseteq I_k$.
We now prove that $(G-f)\nto (H)_q$ for every $f\in E(F)$.
Without loss of generality let $f\in F_q$.
We define a colouring $c:E(G-f)\to [q]$ as follows. Colour all edges of $F_q-f$ and $r_q$ with colour $q$, and for every $k\in [q-1]$ colour the edges of $F_k+\{e_k,r_k\}$ with colour $k$. Set $c(e_q)=1$ and $c(f_k)=q$ for every $k\in [q]$. Finally, colour every indicator from (iii) and every signal sender from (i), (ii), (iv) and (v) with an $H$-free $q$-colouring preserving the colours already chosen. For $I_q$ this is possible by Property~$(I3)$ and for all other indicators this is possible by Properties~$(I1)$ and $(I2)$. For the signal senders this is possible by Properties~$(S1)$ and $(S2)$, as the colours above have been chosen in such a way that the signal edges of negative/positive signal senders receive different/identical colours.
We claim that $c$ is $H$-free. Indeed, any copy of $H$ is contained as a subgraph either in $F$ or in one of the indicators or signal senders as we have shown above.
The colouring on each indicator and signal sender is $H$-free, and it is $H$-free on $F$ since each of $F_1,\ldots F_q$ receives a distinct colour, and each $F_i$ is $H$-free by assumption.
We next show that $G\to(H)_q$.
Assume that there exists an $H$-free $q$-colouring $c$.
By Property~$(S2)$ of the negative signal senders
in (i), we find that $c(r_k)\neq c(r_{\ell})$ for all $k,\ell\in[q]$ with $k\neq \ell$.
Without loss of generality let $c(r_k)=k$ for all $k\in [q]$.
By Property~$(S2)$ of the positive signal senders in (ii) it then follows
that $F_k$ needs to be monochromatic in colour $k$ for every $k\in [q]$.
Using Property~$(I2)$ of the indicators in (iii) we conclude that
$c(e_k)=k$ must hold for every $k\in [q]$,
and applying Property~$(S2)$ of the negative signal senders in (iv)
we then deduce $c(f_k)\neq k$ for every $k\in [q]$.
But then, using Property~$(S2)$ of the positive signal senders in (v),
we obtain $c(f_1)=c(f_k)\neq k$ for every $k\in [q]$, a contradiction.
Finally, let $G'\subset G$ be a subgraph of $G$ that is $q$-Ramsey-minimal for $H$.
Then $f\in E(G')$ for every $f\in E(F)$ since $(G-f)\nto(H)_q.$ Thus, $G'$ is a $q$-Ramsey-minimal graph for $H$ which contains $F$ as an induced subgraph.
In order to obtain infinitely many such $q$-Ramsey-minimal graphs set $G_0=G'$ and obtain further such $q$-Ramsey-minimal graphs $G_i$ iteratively as follows. Let $F_i$ be the disjoint union of $v(G_{i-1})$ copies of $F$. Since $F_i$ is not $q$-Ramsey for $H$, we can repeat the above argument and thus create a
$q$-Ramsey-minimal graph $G_i$ for $H$ which contains $F_i$ as an induced subgraph. Note that then $G_i$ also contains $F$ as an induced subgraph and $v(G_i)\geq v(F_i)>v(G_{i-1})$ holds.
\end{proof}
\begin{proof}[Proof of \thref{wow}]
Suppose that for two graphs $H$ and $H'$ we have
$\mathcal{M}_q(H)\se\mathcal{M}_q(H')$ and $\mathcal{M}_q(H')\nse\mathcal{M}_q(H)$.
Let $G \in \mathcal{M}_q(H')\sm\mathcal{M}_q(H).$
If $G$ is $q$-Ramsey for $H$ then for some subgraph $G'$ of $G$ we have that $G'\in {\mathcal M}_q(H)\se{\mathcal M}_q(H')$ by assumption.
If $G'=G$ this contradicts $G\not\in {\mathcal M}_q(H)$, and if $G'$ is a proper subgraph of $G$ then this contradicts $G\in {\mathcal M}_q(H')$ as $G$ is not minimal then.
On the other hand, if $G$ is not $q$-Ramsey for $H$ then there exists a graph $G'$ such that $G\se G'\in{\mathcal M}_q(H)\se{\mathcal M}_q(H')$, by \thref{thm:containment} and assumption. Since $G\in{\mathcal M}_q(H')$ by assumption it follows that $G=G'$, a contradiction to $G\not\in{\mathcal M}_q(H)$.
\end{proof}
\section{Ramsey equivalence results}\label{section4}
In this section we prove \thref{thm:3equiv,GeneralLowerBdd,thm:nonequiv}.
We start with the proof of \thref{GeneralLowerBdd} which is a
corollary of the following slightly more general statement. This multi-colour version is a straight-forward generalisation of the argument for 2 colours in \cite[Theorem 3.1]{szz2010}. \thref{GeneralLowerBdd}
follows by repeatedly applying this theorem to pairs $H_{i-1} = K_k+(i-2)K_t$ and $H_{i}=K_k+(i-1)K_t$ with $2\leq i\leq s +1$.
\begin{theorem}\thlab{Thm3Pt1FromTibor}
Let $q\ge 2$, let $a_1\ge a_2\ge\ldots \ge a_s\ge 1$ and define $H_i:=K_{a_1}+\cdots + K_{a_i}$ for $1\le i \le s$. If
$R_q(a_1-a_s+1,a_1,\ldots,a_1) >q(a_1+\ldots+a_{s-1}),$ then $H_s$ and $H_{s-1}$ are $q$-equivalent.
\end{theorem}
\begin{proof}
It is clear that every graph $G$ that is $q$-Ramsey for $H_{s}$ is also $q$-Ramsey for $H_{s-1}$. Now let $G$ be a graph that is $q$-Ramsey for $H_{s-1}$. We need to show that $G$ is $q$-Ramsey for $H_s$.
Suppose for a contradiction that $G\not\to (H_s)_q$ and let $c:E(G)\to[q]$ be a $q$-colouring of the edges of $G$ without a monochromatic copy of $H_s$. Without loss of generality, we may assume that there is a copy of $H_{s-1}$ in colour 1, and let $S_1$ be its vertex set.
Since $c$ has no copy of $H_s$ in colour 1 the colouring restricted to $V(G)\sm S_1$ has no copy of $K_{a_s}$ in colour 1.
Now, recursively for every colour $j=2,\ldots,q$, let $i_j$ be the largest index such that
$V(G)\sm (S_1\cup\ldots\cup S_{j-1})$ contains a monochromatic copy of $H_{i_j}$ in colour $j$ (where we take $H_0$ to be the empty graph), and let $S_j$ be its vertex set.
Since $c$ has no monochromatic copy of $H_s$ we have that $i_j<s$ for all $j\in[q]$.
Now $c$ restricted to $V(G)\sm (S_1\cup\ldots\cup S_{q})$ does not contain a monochromatic copy of $K_{a_1}$,
since by the maximality of $i_j$ there is no copy of
$H_{i_j+1}= H_{i_j} + K_{a_{i_j+1}}$ in colour $j$
in $V(G)\setminus (S_1\cup\ldots\cup S_{j-1})$
and since $a_{i_j+1}\leq a_1$ for every $j\in [q]$.
As in the proof of Theorem 3.1 in \cite{szz2010} we now recolour some edges of $G$.
We have that
\begin{align*}
|S_1\cup\ldots S_q|=|V(H_{s-1})|+|V(H_{i_2})|+\ldots+|V(H_{i_q})|
& \le q(a_1+\ldots+a_{s-1})\\
& < R_q(a_1-a_s+1,a_1,\ldots,a_1).
\end{align*}
Hence, by the definition of the Ramsey number we can recolour the edges inside $S_1\cup\ldots\cup S_q$ without a monochromatic copy of $K_{a_1-a_s+1}$ in colour 1 and without a monochromatic copy of $K_{a_1}$ in colour $j$, for all $2\le j\le q$.
All edges between $S_1\cup\ldots\cup S_q$ and $V\sm (S_1\cup\ldots\cup S_q)$ receive colour 1, and all remaining edges retain their original colour.
It is now easy to see that there is no monochromatic copy of $K_{a_1}$ which is a contradiction to $G\to(H_{s-1})_q$.
\end{proof}
It turns out that \thref{GeneralLowerBdd} already implies \thref{thm:3equiv} for $q\ge 4$. We need two more ingredients for the case~$q=3$.
\begin{observation}\thlab{obsii}
Let $G$ be a graph such that $G\to (K_3)_3$, and let $c$ be a 3-colouring of the edges of $G$. If there is a monochromatic copy of $K_3$ in every colour, then there is a monochromatic copy of $K_3+K_2$.
\end{observation}
\begin{proof}
We first note that $\chi(G)\ge R_3(3) = 17$, where $\chi(G)$ is the chromatic number of $G$, see, e.g., Theorem 1 in \cite{l1972}.
Let $V_0$ be the set of vertices belonging to the three monochromatic triangles, each of a different colour, which exist by assumption.
Then $G[V(G)\backslash V_0]$ contains an edge as otherwise $\chi(G)\leq\chi (G[V_0])+1\leq 10$. This edge then forms a monochromatic copy of $K_3+K_2$ along with one of the three monochromatic triangles.
\end{proof}
The next theorem was proved by Bodkin and Szab\'o (see \cite{b2015}, and \cite{bl2018} for a proof).
\begin{theorem}[Theorem 2 in \cite{bl2018}] \thlab{K6Distinguisher}
If $G\to (K_3)_2$ and $G\not\to (K_3+K_2)_2$ then $K_6\se G$.
\end{theorem}
\begin{proof}[Proof of \thref{thm:3equiv}]
For $k=3$ and $t=2$, \thref{GeneralLowerBdd} implies that $K_3$ and $K_3+K_2$ are $q$-equivalent if $R_q(2,3,\ldots,3) = R_{q-1}(3) > 3q$. This inequality follows for $q\ge 4$ easily by induction on $q$ with the induction start given by the fact $R_3(3)=17>4\cdot 3$.
It remains to prove that $K_3$ and $K_3+K_2$ are $3$-equivalent.
Clearly, any graph which is $3$-Ramsey for $K_3+K_2$ is also $3$-Ramsey for $K_3$.
Let now $G$ be a graph that is $3$-Ramsey for $K_3$ and let $c$ be a 3-colouring of $G$ using colours red, blue, and yellow. Let $R$, $B$, and $Y$ denote the subgraphs formed by the red, blue, and yellow edges, respectively. We need to show that we can find a copy of $K_3+K_2$ in one of $R$, $B$, or $Y$.
Suppose first that none of the subgraphs of $G$ formed by the union of any two of $R, B, Y$ is a $2$-Ramsey graph for $K_3$. Then the subgraph $R\cup B$ can be recoloured red-blue without monochromatic copies of $K_3$.
Hence there must exist a (yellow) copy of $K_3$ in $Y$, since $G\to (K_3)_3$.
Similarly we argue that there is also both a blue and a red copy of $K_3$ in $G$.
We are then done by \thref{obsii}.
Suppose now that without loss of generality $R\cup B$ is $2$-Ramsey for $K_3$.
Then by \thref{K6Distinguisher} either there is a copy of $K_3+K_2$ in $R$ or in $B$ (and we are done); or $K_6$ is a subgraph of $R\cup B$, say on vertex set $S$. Now we find either a red or a blue copy of $K_3+K_2$ in $S$; or both a red and a blue copy of $K_3$ on $S$.
We claim that $G$ contains a further (not necessarily monochromatic) copy of $K_3$ in $V(G)\sm S$. Suppose not. Then we recolour $G$ as follows.
Let $v\in S$ and colour the edges of $G[V(S)\setminus \{v\}]$ with red and blue without a monochromatic copy of $K_3$ (i.e.~a red and a blue $C_5$). Colour all edges incident to $v$ in $S$ yellow and colour all edges in $V(G)\sm S$ blue. Finally, colour all edges between $V(S)\setminus \{v\}$ and $V(G)\sm S$ yellow and all those between $v$ and $V(G)\sm S$ red. Unless there is a triangle in $V(G)\sm S$ this colouring does not contain a monochromatic copy of $K_3$, a contradiction to $G\to (K_3)_3$.
%
Let $T$ be this triangle in $G-S$. If any of the edges of $T$ is red or blue, then this edge forms a monochromatic copy of $K_3+K_2$ with one of the monochromatic triangles in $S$. Otherwise, all edges of $T$ are yellow, and we are done again by \thref{obsii}.
\end{proof}
We now turn to the proof of \thref{thm:nonequiv}. To show the non-equivalence of two graphs $H$ and $H'$ we need to construct a graph that is $q$-Ramsey for one of the graphs, say for $H$, and not $q$-Ramsey for $H'$. Recall that the signal senders in Section~\ref{sec:preliminaries} provide us with graphs that can enforce certain predefined colour patterns. We now introduce suitable colour patterns.
Following notation of~\cite{fglps2016}, we call a graph $F$ on $n$ vertices
$(n,r,k)$-critical if $K_{k+1}\not\se F$ and every subset $S\se V(F)$ of size $|S|\ge n/r$ satisfies $K_k\se F[S]$.
A sequence of pairwise edge-disjoint graphs $F_1,\ldots, F_r$ on the same vertex set $V$ is called a {\em colour pattern} on $V$.
\begin{lemma}[Lemmas 4.2 and 4.4 in \cite{fglps2016}]\thlab{nrkcritical}
Let $k\ge 2$, $r\ge 3$ be integers. Then there exists a colour pattern $F_1,\ldots,F_r$ on vertex set $[n]$, for some $n$, such that each $F_i$ is $(n, r, k)$-critical.
\end{lemma}
\begin{remark}
The results in \cite{fglps2016} include bounds on $n$ in terms of $r$, which is unnecessary for our purpose. Without these bounds, the lemma can actually be proved by a now standard application of the probabilistic method.
\end{remark}
Next we state a lemma which captures the effect of repeated application of the pigeonhole principle in a coloured bipartite graph. Its proof is a straight-forward generalisation of the proof of Lemma 2.6~(a) in~\cite{fglps2014}.
\begin{lemma}\thlab{focusing}
Let $G=(A\cup B, E)$ be a complete bipartite graph with a $q$-colouring $c:E\to[q]$ of its edges. Then there exists a subset $B'\se B$ with $|B'|\ge |B|/q^{|A|}$ such that for every vertex $a\in A$ the set of edges from $a$ to $B'$ is monochromatic.
\end{lemma}
We are ready to prove \thref{thm:nonequiv}.
\begin{proof}[Proof of \thref{thm:nonequiv}]
Fix $k\geq 3$.
The proof proceeds by induction on $q$.
For $q=2$, there exists a graph $G_2$ that satisfies $G_2\to K_k$ and $G_2\nto K_k\cdot K_2$, by~\cite{fglps2014}.
So assume that $q\ge 2$ and let $G_q$ be a graph such that is $G_q\to (K_k)_q$ and $G_q \nto (K_k\cdot K_2)_q$. We construct a graph $G_{q+1}$ with the properties $G_{q+1} \to (K_k)_{q+1}$ and $G_{q+1} \nto (K_k\cdot K_2)_{q+1}$.
Let $r=q^{|V(G_q)|+qk^2}+1$, and let
$F=F_1\cup\ldots\cup F_q$ be a colour pattern such that each $F_i$ is $(n,r,k-1)$-critical for some $n$. The existence of $F$ follows from \thref{nrkcritical}. (Note that we only use $q$ pairwise edge-disjoint graphs, where the lemma in fact provides $r$ such graphs $F_i$.)
We construct $G_{q+1}$ as follows.
Let $\tilde G_q$ be a copy of $G_q$, say on vertex set $V_0$.
Let $V_1,\ldots, V_{k-2}$ be pairwise vertex disjoint sets of size $n=|V(F)|$ that are disjoint from $V_0$. Let $\{e_1,\ldots,e_q\}$ be a matching of size $q$, (vertex-) disjoint from $V_0\cup\ldots\cup V_{k-2}$.
For each $1\le j \le k-2$ let $F^{(j)}=F_1^{(j)}\cup\ldots\cup F_q^{(j)}$ be a copy of $F$ on vertex set $V_j$.
Additionally, add all edges between $V_i$ and $V_j$ for all $0\le i<j\le k-2$.
Finally, we join edges by signal senders in the following way.
For all $1\le i < j\le q$, join $e_i$ and $e_j$ by a negative signal sender
$S^{-}=S^{-}(q+1,K_k,k)$.
And for all $1\le i \le q$ and every edge $e\in F_i^{(1)}\cup\ldots\cup F_i^{(k-2)}$ join $e$ and $e_i$ by a positive signal sender $S^{+}=S^{+}(q+1,K_k,k)$.
Both signal senders $S^{-}$ and $S^{+}$ exist by \thref{lem:signal}.
The resulting graph is $G_{q+1}$, an illustration can be found in Figure~\ref{fig:nonEquiv}.
\begin{figure}
\begin{center}
\includegraphics[width=0.7\textwidth]{nonEquiv}
\end{center}
\caption{An illustration of the graph $G_{q+1}$ when $q=3$ and $k=5$. Bold edges indicate complete bipartite graphs.}
\label{fig:nonEquiv}
\end{figure}
\begin{claim}
$G_{q+1}\nto (K_k\cdot K_2)_{q+1}$.
\end{claim}
\begin{proof}
Consider the following $(q+1)$-colouring of the edges of $G_{q+1}$.
By inductive hypothesis of $G_q$, there exists a $(K_k\cdot K_2)$-free
colouring $c_0:E(\tilde G)\to [q]$ of the edges in $V_0$.
For all $1\le i \le q$, colour the edges of $F_i^{(1)}\cup\ldots\cup F_i^{(k-2)}$ and the edge $e_i$ in colour $i$.
Colour all edges between any $V_i$ and $V_j$, $0\le i<j\le k-2$, with colour $q+1$.
Note that all pairs of edges that are joined by copies of $S^+$ have the same colour.
There exists a $K_k$-free $(q+1)$-colouring $c^+$ of $S^+$ by Property~$(S1)$,
and by Property~$(S2)$ both signal edges have the same colour in $c^+$. Extend the partial colouring of $G_{q+1}$ to every copy of $S^+$ using $c^+$ (possibly permuting the colours so that the colouring agrees on the already coloured signal edges). Similarly, any two edges that are joined by copies of $S^-$ received distinct colours (edge $e_i$ received colour $i$ for $i\in [q]$); and there exists a $K_k$-free $(q+1)$-colouring $c^-$ of $S^-$ by Property~$(S1)$ in which the two signal edges have distinct colours. Extend the partial colouring further to every copy of $S^-$ using $c^-$, again permuting colours when needed.
We claim that this gives a $(K_k\cdot K_2)$-free $(q+1)$-colouring of $G_{q+1}$.
First note that any copy of $K_k$ is either contained in $V_0\cup\ldots\cup V_{k-2}$, or is contained in one of the copies of a signal sender. This follows since the intersection of the vertex set of every copy of $S^+$ (or $S^-$) and $V_0\cup\ldots\cup V_{k-2}\cup \bigcup_{k\in [q]} e_k$ contains at most the two signal edges of the signal sender, and since the distance between those two edges is at least $k$ in $G_{q+1}$.
The colouring is $K_k$-free on every copy of a signal sender by the choice of the colourings $c^+$ and $c^-$. Next, note that the edges of colour $(q+1)$ in $V_0\cup\ldots\cup V_{k-2}$ form a (complete) $(k-1)$-partite graph as no edge inside $V_i$, $0\le i\le k-2$, has colour $q+1$. Thus there is no monochromatic copy of $K_k$ in colour $q+1$ in $G_{q+1}.$
Furthermore, for every $1\le i\le q$, the graph formed by edges of colour $i$ on vertex set $V_1\cup\ldots\cup V_{k-2}$ is isomorphic to the vertex-disjoint union of copies of $F_i$ which is $(n,r,k-1)$-critical and thus $K_k$-free.
It follows that the only monochromatic copies of $K_k$ are contained in $V_0$. The colouring on $V_0$ only uses the colours $[q]$, whereas all edges between $V_0$ and $V(G_{q+1})\sm V_0$ have colour $q+1$. Furthermore, the colouring on $V_0$ is $K_k\cdot K_2$-free, by inductive assumption. Therefore, if there is a monochromatic copy of $K_k$, then it must be contained in $V_0$, and then there is no pendant edge to that copy of the same colour.
\end{proof}
\begin{claim}
$G_{q+1}\to (K_k)_{q+1}$.
\end{claim}
\begin{proof}
Let $c:E(G_{q+1})\to[q+1]$ be a $(q+1)$-colouring and suppose that there is no monochromatic copy of $K_k$ in this colouring.
Then $c$ is $K_k$-free on every copy of $S^-$. Thus the two edges $e_i$ and $e_j$ receive different colours for all $1\le i<j\le q$, by Property~$(S2)$ of a negative signal sender. After permuting colours we may henceforth assume that the edge $e_i$ has colour~$i$ for $1\le i\le q$.
Furthermore, $c$ is $K_k$-free on every copy of $S^+$ which joins $e$ and $e_i$, for each~$i\in[q]$ and $e\in F_i^{(1)}\cup\ldots\cup F_i^{(k-2)}$. This implies that the graph
\begin{align}\lab{aux665}
& F_i^{(1)}\cup\ldots\cup F_i^{(k-2)} \text{ is monochromatic of colour $i$ for every }i\in [q],
\end{align}
by Property~$(S2)$ for positive signal senders.
We now apply \thref{focusing} to the bipartite graph between $V_0$ and $V_1$ and deduce that there is a set $V_1'\se V_1$ with $|V_1'|\ge |V_1|/q^{|V_0|}$ such that
for every vertex $v\in V_0$ the set of edges from $v$ to $V_1'$ is monochromatic.
Now, $|V_1|/q^{|V_0|}\ge |V_1|/r$ by choice of $r$. Hence, for every~$i\in[q]$ there is a monochromatic copy of $K_{k-1}$ in colour $i$ in $V_1'$, say on vertex set $W_1^{(i)}$, since $F_1^{(i)}$ is $(n,r,k-1)$-critical and monochromatic of colour $i$, by~\eqref{aux665}.
Let $W_1= \bigcup_{i\in [q]} W_1^{(i)}$ and note that $|W_1|\le qk$.
If there exists a vertex $v\in V_0$ such that all the edges from $v$ to $W_1\se V_1'$ have colour $i$ for some $i\in[q]$ then the vertices $W_1^{(i)}\cup\{v\}$ form a monochromatic copy of $K_k$ in colour $i$ and we are done. We may thus assume that all edges between $V_0$ and $W_1$ have colour $q+1$.
Iteratively assume that we have defined $W_1,\ldots, W_{\ell}$ for some $\ell=1,\ldots,k-3$, such that for every $i,j\in [\ell]$ with $i\ne j$ we have that $W_i\se V_i$ of size $|W_i|\le qk$, $W_i$ contains a monochromatic copy of $K_k$ in every colour $j\in[q]$, all edges between $V_0$ and $\bigcup_{i\in[\ell]} W_i$ have colour $q+1$, and all edges between $W_i$ and $W_j$ have colour $q+1$.
We then obtain $W_{\ell+1}$ in $V_{\ell+1}$ by repeating the argument above where $V_0$ is replaced by $V_0\cup W_1\cup \ldots \cup W_{\ell}$. Note that this set has size at most $|V_0|+qk^2$. Thus the subset $V_{\ell+1}'\se V_{\ell+1}$ that we obtain by application of \thref{focusing} has size at least $|V_{\ell+1}|/q^{|V_0|+qk^2}\ge |V_{\ell+1}|/r$ by choice of $r$. The rest of the argument is analogous.
Thus either we find a monochromatic copy of $K_k$ in one of the colours $1,\ldots,q$; or we obtain sets $W_1,\ldots, W_{k-2}$ that form a complete $(k-2)$-partite graph in colour $q+1$ and such that all edges between $V_0$ and
$\bigcup_{i\in[k-2]} W_i$ are present and have colour $q+1$. If any of the edges in $V_0$ has colour $q+1$, then this edge together with one vertex from each $W_i$, $i\in [k-2]$, forms a monochromatic copy of $K_k$ in colour $q+1$, and we are done again. Otherwise, no edge in $V_0$ has colour $q+1$. But the graph on $V_0$ is isomorphic to $G_q$ which means that in any $q$-colouring of the edges in $V_0$ there is a monochromatic copy of $K_k$ in at least one of the colours.
\end{proof}
This finishes the proof of \thref{thm:nonequiv}.
\end{proof}
\section{Concluding remarks}
\noindent
{\bf Minimal minimum degree of minimal Ramsey graphs.\\}
We have proved that $K_k$ and $K_k\cdot K_2$ are not $q$-equivalent for any $q\ge 3$. The proof proceeds by induction on $q$ with the base case given by the non-equivalence in two colours from~\cite{fglps2014}. The {\em $2$-distinguishing} graph $G_2$ constructed in~\cite{fglps2014} actually has a stronger property, namely that $G\nto (K_k\cdot K_2)_2$ and every $(K_k\cdot K_2)$-free colouring of $G_2$ has a {\em fixed} copy of $K_k$ being monochromatic.
This stronger property was used there to construct a graph $G'$ that is 2-minimal for $K_k\cdot K_2$ and that contains a vertex of degree $k-1$, i.e.~$s_2(K_k\cdot K_2)\le k-1$. The classical paper by Burr, Erd\H{o}s, and Lov\'asz contains the proof of $s_2(K_k) = (k-1)^2$, i.e.~adding a pendant edge to $K_k$ changes the behaviour of $s_2(\cdot)$ drastically.
\begin{problem}
Determine $s_q(K_k\cdot K_2)$ for $q\ge 3$.
Specifically, is it true that $s_q(K_k\cdot K_2) \le s_q(K_k)$, and if so, how small is the ratio $s_q(K_k\cdot K_2)/s_q(K_k)$?
\end{problem}
It is known that $s_q(K_k) = O(q^2(\ln q)^{8(k-1)^2})$ for $k\ge 4$ where the implicit constant is independent of $q$ \cite{fglps2016}. For fixed $k$, this bound is tight up to a factor that is polylogarithmic in $q$. Furthermore, $s_q(K_3) = \Theta(q^2\log q)$ \cite{gw2017}.
The construction of $G_2$ in~\cite{fglps2014} does not generalise in a straight-forward manner to more than 2 colours. The $q$-distinguishing graph $G_q$, $q\ge 3$, from the proof of \thref{thm:nonequiv} contains signal senders and thus does not have the stronger property of having a fixed copy of $K_k$ that is monochromatic in every $(K_k\cdot K_2)$-free $q$-colouring of $G_q$ as $G_2$. In particular, our graphs $G_q$ cannot be used (per se) for constructions showing upper bounds on $s_q(K_k\cdot K_2)$.
\medskip
\noindent
{\bf From $2$-(non)-equivalence to multicolour-(non)-equivalence.\\}
We have seen in the introduction that $2$-equivalence of $H$ and $H'$ implies $q$-equivalence for every even $q$. More generally, \thref{positiveLC} implies that two graphs are $q$-equivalent for every $q\ge 3$ if they are known to be $2$-equivalent {\em and} $3$-equivalent. We reiterate our question from the introduction here.
\begin{question}
Is it true that any two 2-equivalent graphs $H$ and $H'$ are also 3-equivalent?
\end{question}
Or are there two graphs $H$ and $H'$ that are, say, $100$-equivalent but not $101$-equivalent?
We have also said in the introduction that in general one cannot deduce that $H$ and $H'$ are not $q$-equivalent for $q\ge 3$ from the mere fact that they are not $2$-equivalent. All examples had $H$ or $H'$ being disconnected. Is this a coincidence?
\begin{question}
Let $H$ and $H'$ be both connected graphs that are $3$-equivalent. Is it true that they are 2-equivalent as well?
\end{question}
This question may have an affirmative answer for the trivial reason that there are no two connected non-isomorphic graphs $H$ and $H'$ that are $q$-equivalent for any $q\ge 2$. This question was first posed in~\cite{fglps2014} for two colours, and we extend it here to any number of colours.
\begin{question}
For given $q\ge 2$, are there two non-isomorphic connected graphs $H$ and $H'$ that are $q$-equivalent?
\end{question}
Since $K_k$ is not $q$-equivalent to any other connected graph
(see the discussion preceding \thref{thm:nonequiv})
and since any two 3-connected graphs are not $q$-equivalent for any $q\ge 2$ by \thref{dennisObs} it is generally believed that the answer to this question is no.
\medskip
\noindent
{\bf Adding a connected graph to a clique.\\}
We have seen that $K_k$ is Ramsey equivalent to $K_k +H$ where $H$ is a collection of vertex-disjoint cliques. What other graphs $H$ have that property? Here we concentrate on the 2-colour case to highlight how little is known. Of course, all the following questions have natural analogues in the multicolour setting.
We know that $K_k$ and $K_k+K_k$ are not Ramsey equivalent (since the clique on $R_2(k)$ vertices is a distinguisher) and that $K_k$ and $K_k+ K_{k-1}$ are Ramsey equivalent. The following three questions are, of course, related, we find each of them interesting.
\begin{question}
\begin{itemize}
\item
What is the largest value of $t=t(k)$ such that there is a connected graph $H$ on $t$ vertices so that $K_k$ and $K_k+H$ are Ramsey equivalent?
\item
What is the largest value of $t=t(k)$ such that $K_k$ and $K_k+S_t$ are Ramsey equivalent, where by $S_t$ we denote the star with $t$ vertices (in alignment with the previous question)?
\item
What is the largest value of $t=t(k)$ such that $K_k$ and $K_k+P_t$ are Ramsey equivalent, where by $P_t$ we denote the path with $t$ vertices?
\end{itemize}
\end{question}
The second question is from \cite{fglps2014}. Note that the equivalence of $K_k$ and $K_k+K_{k-1}$ implies that the answer to these questions is at least $k-1$. Moreover, it is easy to obtain an upper bound of roughly $R(k)$, i.e.~exponential in $k$. To the best of our knowledge nothing better is known.
Specifically, we wonder whether $K_k$ and $K_k+S_k$ are Ramsey-equivalent. If the answer is affirmative then this may shed light on whether $K_k +K_{k-1}\cdot K_2$ and $K_k$ are Ramsey equivalent. Slightly more ambitious is the following.
\begin{problem}
Are $K_k$ and $K_k+K_k^-$ Ramsey equivalent, where $K_k^-$ denotes the clique on $k$ vertices with one edge deleted?
\end{problem}
An affirmative answer would imply that $R(K_k^-)<R(K_k)$, an inequality conjectured to be true, but only known for $k\le 6$, see e.g.~\cite{BLS}.
\bigskip
\noindent
{\bf Acknowledgement.}
This research was started at Monash University,
during a research stay of the first and third author
who would like to express their gratitude for
hospitality and for providing a perfect working environment.
The third author would also like to thank Vojtech R\"odl for a helpful discussion.
\bibliographystyle{pagendsort}
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\section{Introduction}
\label{sec:introduction}
Most computer vision systems cannot take advantage of the abundance of Internet video content as training data. This is because current algorithms typically learn under strong supervision and annotating video content is expensive. Our goal is to remove the need for expensive manual annotations and instead reliably recover spatiotemporal correspondences between deformable objects under weak supervision. For instance, given a collection of animal documentary videos, can we automatically match pixels on a tiger in one video to those on a different tiger in another video (Figs.~\ref{fig:teaser} and \ref{fig:alignment})?
\begin{figure}
\begin{center}
\includegraphics[scale =0.42]{images/teasernew.pdf}
\end{center}
\caption{\small{
We recover point-to-point spatiotemporal correspondences across a collection
of unstructured videos of deformable objects.
Here, we display the recovered correspondences by mapping tigers from frames in two different
videos (top) onto each other (bottom).
Our method maps each part of a tiger in the first video to the corresponding part in the second
(\emph{e.g.}\xspace, head goes to head, front-right paw goes to front-right paw, etc.).
We use motion cues to find short video intervals where the foreground moves in a consistent manner.
This enables finding correspondences despite large variations in appearance (\emph{e.g.}\xspace, white and orange tigers).
}}
\vspace{-4pt}
\label{fig:teaser}
\end{figure}
Recovering point-to-point spatiotemporal correspondences across videos is powerful
because it enables to assemble a collection of \emph{aligned}
foreground masks from a collection of videos of the same object class (Fig.~\ref{fig:teaser}).
Accomplishing this task in the presence of significant object appearance variations is
particularly important in order to capture the richness of the visual concept (\emph{e.g.}\xspace, different coloring and textures of an animal).
Achieving this could replace the expensive manual annotations required by several
popular methods for learning visual concepts~\cite{dalal05cvpr,felzenszwalb10pami,
viola:nips05,
cinbis13iccv,wang13iccv,girshick14cvpr},
including methods that require annotations at the part level~\cite{Felzenszwalb03pictorialstructures,BourdevMalikICCV09,azizpour12eccv}.
Additionally, it can enable novel applications, such as replacing one instance of
an object with a suitable instance from a different video (like
the orange and the white tiger in Fig.~\ref{fig:teaser}).
Instances of the same class in different videos exhibit
large variations in appearance.
Hence, traditional methods for matching still images using local
appearance descriptors~\cite{barnes10eccv,liu08eccv,Hartley00,lowe04ijcv}
typically do not find reliable correspondences.
We do this more effectively by aligning short temporal intervals where the objects exhibit consistent motion patterns. We exploit the characteristic motion of an object
class (\emph{e.g.}\xspace, a tiger's prowl) to identify suitable correspondences,
and combine motion and edge features to align them with great accuracy.
We present a new technique to align two sequences of frames spatiotemporally using a set of Thin Plate Splines (TPS),
an expressive non-rigid mapping that has primarily been used
for registration~\cite{Chui03} and shape matching~\cite{ferrari10ijcv}
in still images.
We extend these ideas to video by fitting a TPS that varies smoothly in time to minimize the distance between edge points in corresponding frames from the two sequences.
We evaluate our method on a new set of ground-truth annotations:
19 landmarks (\emph{e.g.}\xspace, left eye,
front left knee, neck, etc.) for two classes (horses and tigers).
We annotated $\sim$100 video shots per class, for a total of
$\sim$35,000 annotated frames (25 minutes of video).
The tiger shots come from a dataset of high-quality nature
documentaries filmed by professionals~\cite{delpero15cvpr}. The horse shots are sourced
from the YouTube-Objects dataset~\cite{prest12cvpr}, which are primarily
low-resolution footage filmed by amateurs.
This enables quantitative analysis on a large scale in two different settings.
Experiments show that our method recovers around a 1000 pairs of correctly aligned sequences
from 100 real-world video shots of each class.
As the recovered alignment is between {\em sequences}, this amounts to having correspondences between 10,000 pairs of frames.
This significantly outperforms the traditional approach of matching SIFT keypoints~\cite{lowe04ijcv} and the popular SIFT Flow algorithm~\cite{liu08eccv}.
The contributions of our work are:
(1) a weakly supervised system that goes from a large collection of unstructured video of an object class to a tight network of spatiotemporal correspondences between object instances;
(2) a method for aligning sequences of frames with consistent motion using TPS;
(3) publicly releasing the ground-truth annotations above. To our knowledge, this is the largest benchmark for sequence alignment to date.
\section{System architecture}
\label{sec:overview}
Our method takes as input a large set of video shots containing instances of an object class (\emph{e.g.}\xspace tigers).
These input shots are neither temporally segmented nor pre-aligned in any way.
The output is a collection of pairwise correspondences between frames.
Each correspondence is both temporal, \emph{i.e.}\xspace we find
correspondences between frames in different shots,
and spatial, \emph{i.e.}\xspace we recover the transformation
mapping points between the two frames (Fig.~\ref{fig:teaser}, bottom).
\paragraph{Overview.}
Fig.~\ref{fig:overview} shows an overview of our system. The key
idea is to first identify pairs of frame sequences from two videos
that exhibit consistent foreground motion.
For this, we use~\cite{papazoglou13iccv} to
extract foreground masks from each shot using motion cues and~\cite{delpero15cvpr}
to cluster short intervals with similar foreground motion.
Within each cluster, we identify pairs of sequences of fixed length
$T=10$ containing similar foreground motion;
we term these \emph{consistent motion pairs} (CMPs).
By focusing on similar motion, CMPs provide reliable correspondences
even across object instances with very different appearance
(such as the white and orange tigers in Fig.~\ref{fig:teaser} or the
cub and adult in Fig.~\ref{fig:bboxvsransac}).
These are fed to the next stage, which
spatiotemporally aligns the two sequences in each CMP.
\vspace{-6pt}
\paragraph{Foreground masks.}
We use the fast video segmentation technique~\cite{papazoglou13iccv} to automatically segment the foregound object from the background. These foreground masks remove confusing features on the background and facilitate the alignment process.
\vspace{-6pt}
\paragraph{CMP extraction.}
Attempting to spatially align all possible pairs of sequences
would be prohibitively expensive (there are over a billion in
just 20 minutes of video). Clustering based on motion with~\cite{delpero15cvpr}
significantly limits the search space. We prune further by considering only the top
10 ranked pairs between two intervals $p$ and $q$ in the same cluster according to
the following metric (Fig.~\ref{fig:candidates}).
We describe each frame using a bag of words (BoW) over the
Trajectory Shape and Motion Boundary Histogram
descriptors~\cite{wang_ICCV_2013} of trajectories starting in that frame. Let
$d_{ij}$ be the histogram intersection between the BoWs for frame $i$ in $p$
and frame $j$ in $q$. The similarity between the $T$-frame sequence pair
starting at $i$ and $j$ is
\begin{equation}
\mathrm{s}\!\left([f_{i}^{p},\ldots,f_{i+T-1}^{p}] , [f_{j}^{q},\ldots,f_{j+T-1}^{q}]\right)=\sum_{t=0}^{T-1}d_{(i+t)(j+t)} \, .
\label{eq:candidates}
\end{equation}
This measure preserves the temporal order of the frames, whereas a BoW
aggregated over the whole sequences would not.
We found this scheme extracts CMPs that reliably show similar
foreground motion and form good candidates for spatial alignment.
\paragraph{Sequence alignment.}
We have explored a variety of approaches for sequence alignment and report on two representative methods here.
The first is a coarse rigid alignment generated by fitting a single homography
to foreground trajectory descriptors matched across the two sequences
(Sec.~\ref{sec:homog}).
The second approach fits a non-rigid TPS mapping to edge points extracted from the foreground regions of each frame. This TPS is allowed to deform smoothly through time through the sequence (Sec.~\ref{sec:tps}).
Our experiments confirm that the more flexible model outperforms the rigid alignment (Sec.~\ref{sec:evaluation}).
\begin{figure}
\begin{center}
\includegraphics[scale =0.3]{images/candidates.png}
\end{center}
\caption{\small{
Extracting CMPs from two intervals.
First, we approximate the pairwise distance between frames
as the histogram distance between their BoWs (which contains all
motion descriptors through the frame, sec.~\ref{sec:overview}). Then we keep as CMPs
the top scoring pairs of sequences of length $T$ with respect to
(\ref{eq:candidates}). For the intervals above,
the number of pairs of sequences to examine is $(n-T)\cdot (m-T)$.
}}
\label{fig:candidates}
\end{figure}
\begin{figure*}[t]
\begin{center}
\setlength{\tabcolsep}{0.8pt}
\begin{tabular}{c c c c c c}
\footnotesize{\textbf{foreground masks}} & \footnotesize{\textbf{trajectory matches}} & \footnotesize{\textbf{homography}} & \footnotesize{\textbf{TPS mapping}} & \footnotesize{\textbf{foreground edge points}} \\
\includegraphics[scale =0.25]{images/walkleft1.jpg} &
\includegraphics[scale =0.25]{images/walkleft2.jpg} &
\includegraphics[scale =0.25]{images/walkleft3.jpg} &
\includegraphics[scale =0.25]{images/walkleft5.jpg} &
\includegraphics[scale =0.25]{images/walkleft4.png} \\
\small{(a)} & \small{(b)} & \small{(c)} & \small{(d)} & \small{(e)} \\
\end{tabular}
\end{center}
\caption{\small{
Aligning sequences with similar foreground motion. We first estimate
a foreground mask (green) using motion segmentation (a). We then fit a
homography to matches between point trajectories (b, sec.~\ref{sec:homogtraj}).
In (c) we project the foreground pixels in the first sequence
(top) onto the second (bottom) with the recovered homography.
This global, coarse mapping is often not accurate (note the
misaligned legs and head).
We refine it by fitting Thin-Plate Splines (TPS) to
edge points extracted from the foreground (e, sec.~\ref{sec:tps}).
The TPS mapping is non-rigid and provides a more accurate alignment
for complex articulated objects (d).
}
}
\label{fig:alignment} \end{figure*}
\section{Rigid sequence alignment}
\label{sec:homog}
Traditionally, homographies are used to model the mapping between two
still images, and are estimated from a set
of noisy 2D point correspondences~\cite{Hartley00}.
We consider instead the problem of estimating a homography from
trajectories correspondences between two sequences (in a CMP).
Below we first review the standard approach for still images, and then present
our extensions.
\subsection{Homography between still images}
\label{sec:homogstill}
A 2D homography $H_{uv}$ is a $3\times3$ matrix that
can be determined from four or
more point correspondences $X_{u} \leftrightarrow X_{v}$ by solving
\begin{equation} X_u = H_{uv}X_v
\label{eq:fittinghomo}
\end{equation}
RANSAC~\cite{Fischler81} estimates a homography from a set
of putative correspondences $\mathcal{P}_{uv} = \{(x_u, y_u) \leftrightarrow
(x_v, y_v)\}$ that may include outliers. Traditionally, $\mathcal{P}_{uv}$
contains matches between local appearance descriptors, like SIFT~\cite{lowe04ijcv}.
At each iteration, a hypothesis is
generated by fitting a homography to four samples from
$\mathcal{P}_{uv}$; the computed homography with the smallest number of outliers
is kept.
\subsection{Homography between video sequences}
\label{sec:homogtraj}
In video sequences, we use point trajectories as units for matching, instead of SIFT keypoints.
We extract trajectories in each sequence and match them using a modified Trajectory Shape (TS) descriptor~\cite{wang_ICCV_2013} (Fig.~\ref{fig:modifiedtraj}).
We match each trajectory in the first sequence to its nearest neighbor
in the second with respect to Euclidean distance.
We use trajectories that are 10 frames long and only match those that start in the same frame in both sequences.
Each trajectory match provides 10 point correspondences (one per frame).
We consider two alternative ways to fit a homography to trajectory matches.
In the first, we treat the point correspondences generated by a single trajectory match independently during RANSAC. We call this strategy `Independent Matching' (IM).
In the second alternative, we sample four {\em trajectory} matches at each RANSAC iteration
instead of four point correspondences.
We solve~\eqref{eq:fittinghomo} using the $4T$ associated point
correspondences, in the least squares sense.
A trajectory match is considered an outlier
only if fewer than half of its point correspondences are outliers.
We call this strategy `Temporal Matching' (TM).
TM encourages geometric consistency over the duration of the CMP.
Instead, IM could overfit to point correspondences from just a few
frames. Our experiments show that TM is superior to IM.
\begin{figure}
\begin{center}
\includegraphics[scale = 0.78]{images/trajaugment.jpg}
\end{center}
\caption{\small{
Modifying the TS descriptor. The TS descriptor is the concatenation
of the 2D displacement vectors (green) of a trajectory across consecutive frames.
This descriptor works well when aggregated in unordered representations
like Bag-of-Words~\cite{wang_ICCV_2013}, but matches found between
individual trajectories are not very robust.
For example, the TS descriptors for the trajectories on the torso of a tiger walking are almost identical.
We make TS more discriminative by appending the vector (yellow) between the trajectory
and the center of mass of the foreground mask (green) in the frame where
the trajectory starts. We normalize this vector by the diagonal of the bounding box of the foreground mask
to preserve scale invariance.
}}
\vspace{-6pt}
\label{fig:modifiedtraj}
\end{figure}
\subsection{Using the foreground mask as a regularizer}
\label{sec:homogreg}
The homography estimated from trajectories
tends to be inaccurate when the input
matches do not cover the entire foreground (Fig.~\ref{fig:bboxvsransac}).
To address this issue, we note that the bounding boxes of the foreground masks~\cite{papazoglou13iccv} provide
a coarse, global mapping (Fig.~\ref{fig:bbox}).
Specifically, we consider the correspondences between the bounding box
corners, which we call `foreground matches' ($F_{u}$, $F_{v}$).
These are included in Eq.~\eqref{eq:fittinghomo} as additional
point correspondences (four per frame):
\begin{equation}
\min \|H_{uv}X_{v}- X_{u}\| + \|H_{uv}F_{v}- F_{u}\|~~~.
\label{eq:fittinghomomodified}
\end{equation}
This form of regularization makes our method much more stable (Fig.~\ref{fig:bboxvsransac}).
\begin{figure}
\begin{center}
\includegraphics[scale = 0.3]{images/bboxvsRansac.png}
\end{center}
\caption{\small{
Top: Trajectory matches (yellow) often cover
only part of the object (head and right leg here).
Here, the homography overfit to correspondences
on the head, providing an incorrect mapping for the legs (right).
Bottom: Adding correspondences from the foreground
bounding boxes provides a more stable mapping (right).
The correspondences in the bottom
row are also found automatically by our method (no manual intervention needed).
}}
\label{fig:bboxvsransac}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale = 0.21]{images/bbox_map.pdf}
\end{center}
\caption{\small{
Matching corners between the bounding boxes of the foreground mask provide
additional point correspondences between the two sequences. While these
correspondences are too coarse to provide a detailed spatial alignment
between the sequences, and are sensitive to errors in the foreground
segmentation (see Fig.~\ref{fig:tmvsfg}), they are useful as a regularizer
when combined with other point correspondences.
}}
\label{fig:bbox}
\end{figure}
\vspace{-10pt}
\section{Temporal TPS for sequence alignment}
\vspace{-4pt}
\label{sec:tps}
In this section we present a second approach to sequence alignment, based on time-varying thin plate splines (TTPS). Unlike the approach presented in the previous section, TTPS is a non-rigid mapping, which is more suitable for putting different object instances in correspondence.
We build on the popular TPS Robust Point Matching algorithm~\cite{Chui03}, originally developed to align sets of points between two still images (Sec.~\ref{sec:tpsrpm}).
We extend TPS-RPM to align two sequences of frames with a TPS that evolves smoothly over time (Sec.~\ref{sec:tpssequence}).
\subsection{TPS-RPM}
\label{sec:tpsrpm}
A TPS is a smooth, non-rigid mapping, $\mathrm{f}$, comprising an affine transformation $d$ and a non-rigid warp $w$. The mapping is a single closed-form function for the entire space,
with a smoothness term $\mathrm{L}(\mathrm{f})$ defined as the sum of the squares of the second
derivatives of $\mathrm{f}$ over the space~\cite{Chui03}.
Given two sets of points $\mathcal{U} = \{u_i\}$ and $\mathcal{V} = \{v_i\}$ in correspondence,
$\mathrm{f}$ can be estimated by minimizing
\begin{equation}
\label{eq:basictps}
\mathrm{E}(\mathrm{f}) = \sum_{i} || u_{i} - \mathrm{f}(v_{i})||^{2} + \lambda||\mathrm{L}(\mathrm{f})||.
\end{equation}
$\mathcal{U}$ and $\mathcal{V}$ are typically the position of detected image features (we use edge points, sec.~\ref{sec:tpssequence}).
As the point correspondences are typically not known beforehand, TPS-RPM jointly estimates $\mathrm{f}$
and a soft-assign correspondence matrix $M=\{m_{ij}\}$
by minimizing
\begin{equation}
\label{eq:tps-rpm}
\mathrm{E}(M, \mathrm{f}) = \sum_{i} \sum_{j} m_{ij}|| u_{i} - \mathrm{f}(v_{j})||^{2} + \lambda||\mathrm{L}(\mathrm{f})||.
\end{equation}
TPS-RPM alternates between updating $\mathrm{f}$ by keeping
$M$ fixed, and the converse. $M$ is continuous-valued, allowing
the algorithm to evolve through a continuous correspondence
space, rather than jumping around in the space
of binary matrices (hard correspondence).
It is updated by setting
$m_{ij}$ as a function of the distance between $u_{i}$
and $\mathrm{f}(v_{j})$~\cite{Chui03}.
The TPS is updated by fitting $\mathrm{f}$ between
$\mathcal{V}$ and the current estimates $\mathcal{Y}$ of the corresponding points,
computed from $\mathcal{U}$ and $M$.
TPS-RPM and optimizes \eqref{eq:tps-rpm} in a deterministic annealing framework,
which allows TPS-RPM to find a good solution even when starting from a relatively poor initialization. \vitto{need to mention 'robust to outliers', or 'decides to match not all the points'}
\subsection{Temporal TPS}
\label{sec:tpssequence}
Our goal is to find a series of mappings $\mathcal{F}=\{\mathrm{f}^{1},\ldots,\mathrm{f}^{T} \}$,
one at each frame in the input sequences. We enforce temporal smoothness by
constraining each mapping to use a set of point correspondences that is consistent over time.
Let $\mathcal{U}^t = \{u_i^t\}$ be a set of points for
frame $t$ in the first sequence (with $\mathcal{V}^t$ defined analogously). This set
contains both edge points extracted in
$t$ as well as edge points extracted in other frames of the sequence and
propagated to $t$ via optical flow (Fig.~\ref{fig:propagation}). Each
$\mathcal{U}^t$ stores points in the same order such that $u_i^t$ and
$u_i^{\tau}$ are related by flow propagation.
We solve for $\mathcal{F}$ by minimizing
\begin{equation}
\label{eq:tpsflow}
\mathrm{E}(\mathcal{M}, \mathcal{F}) = \sum_{t}\! \left(\sum_{i}\! \sum_{j} m_{ij}^t|| u_{i}^{t} - \mathrm{f}^{t}(v_{j}^{t})||^{2} + \lambda||\mathrm{L}(\mathrm{f}^{t})|| \!\right) .
\end{equation}
subject to the constraint that $m_{ij}^t = m_{ij}^{\tau}$. That is, if two points
are in correspondence in frame $t$, they must still be in correspondence after being
propagated to frame $\tau$.
\paragraph{Inference.}
Minimizing (\ref{eq:tpsflow}) is very challenging. In practice, we find an
approximate solution by first using TPS-RPM to fit a separate TPS $\mathrm{f}^t$ to
the edge points extracted at time $t$ only. This is initialized with the homography
found in Sec.~\ref{sec:homogreg}.
$\mathrm{f}^{t}$ fixes the correspondences, which we use to estimate $\mathrm{f}^{\tau}$ in all other
frames.
We repeat this process starting in each frame, generating a total of $T$ TTPS candidates and keep the highest scoring one according to (\ref{eq:tpsflow}).
Thanks to this efficient approximate inference, we can apply TTPS to align
thousands of CMPs.
\paragraph{Foreground edge points.}
We extract edges using~\cite{dollar13iccv}.
We remove clutter edges far from the object by multiplying the edge strength of each point with the Distance Transform (DT) of the image with respect to the foreground mask (\emph{i.e.}\xspace, the distance of each pixel to the closest point on the mask). We prune points scoring $\leq 0.2$.
This removes most background edges, and is robust to cases where the mask does not cover the complete object (Fig~\ref{fig:edges}).
To accelerate the TTPS fitting process, after pruning we subsample the edge points to at most 1,000 per image.
\begin{figure}[t]
\begin{center}
\setlength{\tabcolsep}{0.8pt}
\begin{tabular}{ccccc}
\footnotesize{\textbf{fg mask}} & \footnotesize{\textbf{all edges}} & \footnotesize{\textbf{fg edges}} & \footnotesize{\textbf{edges*DT}} \\
\includegraphics[scale =0.3]{images/a16.png} &
\includegraphics[scale =0.3]{images/b16.png} &
\includegraphics[scale =0.3]{images/c16.png} &
\includegraphics[scale =0.3]{images/d16.png} \\
\small{(a)} & \small{(b)} & \small{(c)} & \small{(d)} \\
\end{tabular}
\end{center}
\caption{\small{
Edge extraction. Using edges extracted from the entire image
confuses the TPS fitting due to background edge points (b).
Using only edges on the foreground mask (c) loses
useful edge points if the mask is inaccurate, \emph{e.g.}\xspace the missing legs in (a).
We instead weigh the edge strength (b) by the Distance
Transform (DT) with respect to the foreground mask. This is robust to errors
in the mask, while pruning most background edges (d).
}
}
\label{fig:edges} \end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale =0.32]{images/propagation.jpg}
\end{center}
\caption{\small{
Propagation using optical flow. In each sequence, we propagate
edge points extracted at time $t$ using optical flow,
independently in each sequence (dashed lines).
Our TTPS model (Sec.~\ref{sec:tpssequence}) enforces
that the correspondences between edge points at time $t$ (solid lines)
are consistent with their propagated version at time $t+1$.
}
}
\label{fig:propagation} \end{figure}
\section{Related work}
\paragraph{Still image alignment.} Most works on spatial alignment focus on
matching between images for a variety of applications such
as multi-view reconstruction~\cite{seitz2006comparison},
image stitching~\cite{Brown2007}, and object instance
recognition~\cite{ferrari06ijcv,lowe04ijcv}.
The traditional approach identifies candidate matches using a local appearance descriptor (\emph{e.g.}\xspace,
SIFT~\cite{lowe04ijcv}) with global geometric verification performed using RANSAC~\cite{Fischler81,chum08pami}
or semi-local consistency checks~\cite{Schmid96,ferrari06ijcv,jegou08eccv}.
PatchMatch~\cite{barnes10eccv} and SIFT Flow~\cite{liu08eccv} generalize this notion to match patches between
semantically similar scenes.
\paragraph{Sequence alignment.}
Our method differs from previous work on sequence
alignment~\cite{caspi00cvpr,caspi06ijcv,ukrainitz06eccv} in several ways.
First, we find correspondences between different scenes, rather than between
different views of the same scene~\cite{caspi00cvpr,caspi06ijcv}.
While the method in \cite{ukrainitz06eccv} is able to align actions across different scenes by directly maximizing local space-time correlations, it cannot handle the large intra-class appearance variations and diverse camera motions present in our videos.
As another key difference, all above approaches require temporally pre-segmented videos. Instead, we operate on
unsegmented videos and our method automatically finds which portions of each
can be spatiotemporally aligned.
Finally, these works have been evaluated only qualitatively
on 5-10 pairs of sequences, providing no quantitative analysis.
In the context of video action recognition,
there has been work on matching of spatiotemporal templates to
actor silhouettes~\cite{WeizmannActions,yilmaz05cvpr} or groupings
of supervoxels~\cite{ke07iccv}.
Our work is different because we map pixels from one unstructured video to another.
The method in~\cite{Jain2013} mines discriminative space-time patches and
matches them across videos. It focuses on rough alignment using sparse
matches (typically one patch per clip), while we seek a finer, non-rigid spatial
alignment.
Other works on sequence alignment focus on temporal rather than spatial alignment~\cite{rao03iccv} or
target a very specific application, like aligning presentation slides to videos of the corresponding
lecture~\cite{fan11tip}.
\begin{figure*}[t]
\begin{center}
\includegraphics[scale =0.085]{images/architecture3.jpg}
\end{center}
\caption{\small{
Overview of our method. The input is a collection of shots showing the
same class (1). Each shot (which can be of any length) is partitioned into shorter temporal
intervals of 10--200 frames (2), which are then clustered together (3) using
motion cues. (\cite{delpero15cvpr} shows that using intervals shorter than
the original shots finds more compact clusters.) The clusters effectively limit
the search space: we extract CMPs only from pairs of intervals
in the same cluster (4). For each pair, we extract CMPs from all
possible pairs of sequences of fixed length (10 frames). An example of a pair is shown
in the bottom right (sec.~\ref{sec:overview} and Fig.~\ref{fig:candidates}).
Last (5), we align the two sequences of each CMP (see sec.~\ref{sec:homog} and~\ref{sec:tps}).
}}
\label{fig:overview}
\end{figure*}
\paragraph{TPS alignment.}
TPS were developed as a general purpose smooth functional mapping
for supervised learning~\cite{wahba90tps}.
TPS have been used for non-rigid point matching between still images~\cite{Chui03},
and to match shape models to images~\cite{ferrari10ijcv}.
The computer graphics community recently proposed semi-automated
video morphing using TPS~\cite{liao14cgf}.
However, this method requires manual point correspondences as input,
and it matches image brightness directly.
\paragraph{Learning from videos.}
A few recent works exploit video as a source of training data
for object class detectors~\cite{prest12cvpr,Tang2013}.
However, their use of video is limited to segmenting objects from their background.
Ramanan et al.\xspace~\cite{ramanan06pami} build a simple 2D pictorial structure model of an
animal from one video. None of these methods find spatiotemporal correspondences between
different instances of a class.
\section{Evaluation}
\label{sec:evaluation}
We evaluate our method on $108$ shots of tigers
from a dataset of documentary nature footage~\cite{delpero15cvpr}
and $98$ shots of horses from YouTube-Objects~\cite{prest12cvpr},
for a total of 17,000 frames per class (roughly 25 minutes of video).
\subsection{Evaluation protocol}
\label{sec:protocol}
\paragraph{Landmark annotations.}
In each frame, we annotate the 2D location
of 19 landmarks on each tiger/horse\footnote{If multiple are visible, we annotate the
animal closest to the camera.} (such as eyes, knees, chin, Fig.~\ref{fig:landmarks}).
We do not annotate occluded landmarks. We will make these annotations publicly available.
Unlike coarser annotations, such as bounding boxes,
landmarks enable evaluating the alignment of objects with
non-rigid parts with greater accuracy.
\begin{figure}
\begin{center}
\includegraphics[scale = 0.24]{images/landmarks2.pdf}
\end{center}
\caption{\small{
Examples of annotated landmarks. A total of 19 points are marked when
visible in over 17,000 frames for two different classes (horses and
tigers). Our evaluation measure uses to landmarks to evaluate
the quality of a sequence alignments (sec.~\ref{sec:protocol}). }}
\label{fig:landmarks}
\end{figure}
\paragraph{Evaluation measure.}
We evaluate the mapping found between the two sequences in a CMP as follows.
For each frame, we map each landmark in the first sequence onto the second and compute the Euclidean distance
to its ground-truth location.
The evaluation measure is the average between this distance and the reverse (\emph{i.e.}\xspace, the distance for landmarks mapped from the second sequence into the first).
We normalize the error by the scale of the object,
defined as the maximum distance between any two landmarks in the frame.
The overall error for a pair of sequences is the average error of all visible landmarks over all frames.
\ricco{Previous version in comments.}
After visual inspection of many sampled alignments (Fig.~\ref{fig:reperror}), we found that $0.18$ was a reasonable threshold for separating acceptable alignments from those with noticeable errors.
We count an alignment as \emph{correct} if the error is below this threshold and if the Intersection over
Union (IOU) of the two sets of visible landmarks in the sequence is above $0.5$ (to avoid rewarding accidental alignments of a few landmarks, bottom row of fig.~\ref{fig:reperror}).
\begin{figure}
\begin{center}
\includegraphics[scale = 0.19]{images/reperror.jpg}
\end{center}
\caption{\small{
Evaluation measure. We use the ground-truth landmarks
to measure the alignment error of the mappings estimated by our method (sec.~\ref{sec:protocol}).
As the error increases, the quality of the alignment clearly degrades. Around $0.18$ the alignments contain
some slight mistakes (\emph{e.g.}\xspace, the slightly misaligned legs in the top right image),
but are typically acceptable. We consider a mapping incorrect also when the IOU of the
visible landmarks in the aligned pair is below $0.5$ (bottom row).
}}
\label{fig:reperror}
\end{figure}
\vspace{-4pt}
\subsection{Evaluating CMP extraction}
\vspace{-4pt}
\label{sec:evalcandidates}
First, we evaluate our method for CMP extraction
in isolation (sec.~\ref{sec:overview}). Given a CMP, we use the ground-truth
landmarks to fit a homography, and check if it is correct according to the evaluation measure above.
If so, it means that it is in principle possible to align it (we call it \emph{alignable}).
Our method returns roughly 3000 CMP on the tiger data, of which $51\%$ are alignable.
As a baseline, we consider extracting CMPs by uniformly sampling sequences from pairs of shots. In this case, the percentage of alignable CMPs drops to $19\%$.
Results are similar on the the horse dataset: our method delivers $49\%$ alignable CMPs, vs $26\%$ by the baseline.
\subsection{Evaluating spatial alignment}
\label{sec:evalalignment}
We now evaluate various methods for automatic sequence alignment.
For each method, we generate a precision-recall curve as follows.
Let $n$ be the total number of CMPs returned by the method; $c$ the number of correctly aligned CMPs; and $a$ the total number of alignable CMPs (sec.~\ref{sec:evalcandidates}).
Recall is $c/a$, and precision is $c/n$.
Different operating points on the precision-recall curve are obtained by varying the maximum percentage of outliers allowed when fitting a homography.
\begin{figure}
\begin{center}
\includegraphics[scale = 0.22]{images/tigerresultscrop.png}
\includegraphics[scale = 0.22]{images/horseresultscrop.png}
\end{center}
\caption{\small{
Evaluation of sequence alignment. We separately evaluate our method
on two classes, horses and tigers.
With no regularization, trajectory methods are superior to SIFT on both classes,
with TM performing better than IM.
Adding regularization using the foreground matches
improves the performance of both TM and SIFT (compare
the dashed to the solid curves).
TTPS clearly outperform all trajectory methods,
as well as SIFT Flow and the FG baseline (see text). }}
\label{fig:results}
\end{figure}
\paragraph{Comparison to other methods.}
We compare our method against SIFT Flow~\cite{liu08eccv}.
We use ~\cite{liu08eccv} to align each pair of frames from the
two sequences independently.
We restrict the algorithm to match only the bounding boxes of the foreground masks,
after rescaling them to be the same size (without these two steps,
performances significantly drop).
Further, we also compare to fitting a homography to SIFT matches found in the two sequences.
We use only keypoints on the foreground mask, and preserve
temporal order by matching only keypoints in corresponding frames.
We tested this method alone (SIFT), and by adding spatial regularization using
the foreground masks (SIFT + FG, sec.~\ref{sec:homogreg}).
Finally, we consider a simple baseline that fits a homography to the bounding box of the foreground masks alone (FG).
\begin{figure}
\begin{center}
\setlength{\tabcolsep}{0.8pt}
\begin{tabular}{c c c c}
\footnotesize{\textbf{Video 1}} & \footnotesize{\textbf{Video 2}} & \footnotesize{\textbf{Homography}} & \footnotesize{\textbf{TPS}} \\
\includegraphics[scale =0.15]{images/f8_182_219_1_4_pair1_6.png} &
\includegraphics[scale =0.15]{images/e8_182_219_1_4_pair1_6.png} &
\includegraphics[scale =0.15]{images/a8_182_219_1_4_pair1_6.png} &
\includegraphics[scale =0.15]{images/d8_182_219_1_4_pair1_6.png} \\
\includegraphics[scale =0.15]{images/a41.png} &
\includegraphics[scale =0.15]{images/a180.png} &
\includegraphics[scale =0.15]{images/b7_1054_2459_pair1_6.png} &
\includegraphics[scale =0.15]{images/c7_1054_2459_pair1_6.png} \\
\small{(a)} & \small{(b)} & \small{(c)} & \small{(d)} \\
\end{tabular}
\end{center}
\caption{\small{
TTPS (d) provide a more accurate alignment for complex articulated objects than
homographies (c).}
}
\label{fig:tpsvshomo} \end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale = 0.38]{images/TMvsFG2.pdf}
\includegraphics[scale = 0.38]{images/SIFTvsTM.pdf}
\end{center}
\caption{\small{
Top two rows: Estimating the homography from the foreground masks alone fails
when the bounding boxes are not tight around the objects (first-second columns).
Adding trajectories (TM+FG) is more accurate (fourth column, sec.~\ref{sec:homogtraj}).
Bottom two rows: the striped texture of tigers often confuses estimating
the homography from SIFT keypoint matches (third row).
On this class, using trajectories (TM)
often performs better.
}}
\label{fig:tmvsfg}
\end{figure}
\vspace{-6pt}
\paragraph{Analysis of rigid alignment.}
Both trajectory methods (TM, IM, sec.~\ref{sec:homogtraj}) are superior to SIFT on both classes, with TM performing better than IM (Fig.~\ref{fig:results}).
Adding spatial regularization with the foreground masks (+FG) improves the performance of both TM and SIFT.
SIFT performs poorly on tigers, since the striped texture confuses matching SIFT
keypoints (Fig.~\ref{fig:tmvsfg}, bottom).
Trajectory methods work somewhat better on tigers than horses due to the poorer quality of YouTube video (\emph{e.g.}\xspace low resolution, shaky camera, abrupt pans).
As a result of these factors, TM+FG clearly outperforms SIFT+FG on tigers, but it is somewhat worse on horses.
\paragraph{Analysis of TTPS.}
The time-varying TPS model (TTPS+FG, sec.~\ref{sec:tps}) significantly improves upon its initialization (TM+FG) on both classes.
On tigers, it is the best method overall, as its precision-recall is above all other curves for the entire range.
On horses, the SIFT+FG and TTPS+FG curves intersect. However, TTPS+FG achieves a higher Average Precision (\emph{i.e.}\xspace the area under the curve): 0.265 vs 0.235.
The SIFT Flow software~\cite{liu08eccv} does not produce scores comparable across CMPs, so we cannot produce a full precision-recall curve. At the level of recall of SIFT Flow, TTPS achieves +0.2 higher precision on tigers, and +0.3 on horses. We also note that TM and TM+FG are closely
related to the method for fitting homographies
to trajectories in~\cite{caspi06ijcv}. TM+FG augments~\cite{caspi06ijcv} in several ways
(automatic CMP extraction, modified TS desriptor, regularization with the foreground masks),
but is still inferior to TTPS+FG. \luca{I put caspi here instead
of at the end of the last paragraph.}
Last, TTPS also achieves a significantly higher precision than the FG baseline.
This shows that our method is robust to errors in the foreground masks.
In supplemental material we provide example head-to-head qualitative results, showing that TTPS alignents
typically look more accurate than the other methods (Fig.~\ref{fig:tpsvshomo}).
For the tiger class, out of all CPMs returned by TTPS (rightmost point on the curve),
$1,000$ of them are correctly aligned tiger (\emph{i.e.}\xspace $10,000$ frames). The precision at this point is $0.5$, i.e. half of the returned CMPs are correctly aligned.
For the horse class, TTPS returns 800 correctly aligned CMPs, with precision $0.35$.
\section{Discussion}
We present a method that automatically extracts dense spatiotemporal
correspondences from a collection of videos showing a particular object class.
Our pipeline consumes raw video, without the need for manual annotations or
temporal segmentation. Using motion as the primary signal for identifying
correspondences allows us to match sequences despite significant appearance
variation. Ultimately, the thin plate spline matching results in
temporally-stable, high-quality alignments for thousands of sequence pairs.
Our method is not limited to a particular class of object but instead applies
to any objects that exhibit consistency in behavior and thus exhibit the same
characteristic motion patterns across different observations. The
correspondences we find can be used to learn a
general model of the object class without requiring any human supervision beyond
video-level object class labels. Additionally, they can enable
novel applications, such as replacing an instance of an object
with an instance from a different video, or retrieving videos in a
collection that tightly match the motion of the object in a query video.
\paragraph{Acknowledgments.}
We are very grateful to Anestis Papazoglou for helping with the data collection,
and to Shumeet Baluja for his helpful comments. This work was partly funded by a Google Faculty Research Award,
and by ERC Starting Grant ``Visual Culture for Image Understanding''.
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\section{Introduction}
In this paper, we study the Cauchy problem for the Korteweg-de
Vries-Burgers (KdV-B) equation with fractional dissipation
\begin{eqnarray}\label{eq:kdvb}
u_t+u_{xxx}+\epsilon |\partial_x|^{2\alpha}u+(u^2)_x=0, \ \
u(0)=\phi,
\end{eqnarray}
where $0<\epsilon, \alpha \leq 1$, $u$ is a real-valued function of
$(x, t) \in \mathbb{R}\times \mathbb{R}_+$. Eq. \eqref{eq:kdvb} has
been derived as a model for the propagation of weakly nonlinear
dipersive long waves in some physical contexts when dissipative
effects occur (cf. \cite{OS}). The global well-posedness of
\eqref{eq:kdvb} and the generalized KdV-Burgers equation has been
studied by many authors (see \cite{MR,MR2} and the reference
therein).
In \cite{MR} Molinet and Ribaud studied Eq. \eqref{eq:kdvb} in the
case $\alpha=1$ and showed that \eqref{eq:kdvb} is globally
well-posed in $H^{s}\ (s>-1)$. The main tool used in \cite{MR} is an
$X^{s,b}$-type space which contains the dissipative structure. Their
result is sharp in the sense that the solution map of
\eqref{eq:kdvb} fails to be $C^2$ smooth at $t=0$ if $s<-1$. In
particular, one can't get lower regularity simply using fixed-point
machinery. Note that $s=-1$ is lower than the critical index
$s=-3/4$ for the KdV equation and also lower than the critical index
$s=-1/2$ for the dissipative Burgers equation. The case $0<\alpha
<1$ was left open and it was conjectured in \cite{MR} that one can
get that \eqref{eq:kdvb} is globally well-posed in $H^s\
(s>s_c=(\alpha-3)/2(2-\alpha))$ by using the same strategy as
$\alpha=1$.
In the first part of this paper, we will study the global well
posedness of Eq. \eqref{eq:kdvb} by following some ideas in
\cite{MR}\footnote{After the paper was finished, the authors were
noted that the same results in this part were also obtained by
St\'{e}phane Vento \cite{Vento} using the similar method.}. The main
issue reduces to a bilinear estimate
\begin{equation}\label{eq:bi1}
\norm{\partial_x(uv)}_{X^{-1/2+\delta, s, \alpha}}\leq
C\norm{u}_{X^{1/2, s, \alpha}}\norm{v}_{X^{1/2, s, \alpha}}.
\end{equation}
For the definition of $X^{b, s, \alpha}$, one can refer to
\eqref{eq:Xs} below. We will apply the $[k;Z]$-multiplier method in
\cite{Tao} to prove \eqref{eq:bi1}. We obtain a critical number
\begin{eqnarray}\label{eq:sa}
s_\alpha=\left \{
\begin{array}{ll}
-3/4, & 0<\alpha\leq1/2,\\
-3/(5-2\alpha),& 1/2<\alpha\leq 1.
\end{array}
\right.
\end{eqnarray}
It is worth to note that $s_\alpha$ is strictly bigger than the
conjectured number $s_c$ for $0<\alpha<1$. We prove that
\eqref{eq:bi1} holds if and only if $s>s_\alpha$. So, it seems
that $s>s_\alpha$ is an essential limitation of this method.
In the second part of this paper, we study the inviscid limit
behavior of \eqref{eq:kdvb} when $\epsilon$ goes to $0$. Formally,
if $\epsilon=0$ then \eqref{eq:kdvb} reduces to the KdV equation
\begin{eqnarray}\label{eq:kdv}
u_t+u_{xxx}+(u^2)_x=0, \ \ u(0)=\phi.
\end{eqnarray}
The local well posedness of Eq. \eqref{eq:kdv} in $L^2$ was
established by Bourgain \cite{Bour} and the $X^{b,s}$-theory was
discovered. This local solution is a global one by using the
conservation of $L^2$ norm. The optimal result on local
well-posedness in $H^s$ was obtained by Kenig, Ponce, Vega
\cite{KPV}, where they developed the sharp bilinear estimates and
obtained that \eqref{eq:kdv} is locally well-posed for $s>-3/4$. The
sharp result on global well-posedness in $H^s$ was obtained in
\cite{Tao2}, it was shown that \eqref{eq:kdv} is globally well-posed
in $H^s$ for $s>-3/4$, where a kind of modified energy method, so
called I-method, is introduced.
A natural question is whether the solution of \eqref{eq:kdvb}
converges to that of \eqref{eq:kdv} if $\epsilon$ goes to $0$. We
will prove that the global solution of \eqref{eq:kdvb} converges to
the solution of \eqref{eq:kdv} as $\epsilon\to 0$ in the natural
space $C([0,T],H^s)$ for $-3/4<s\leq 0$. To achieve this, we need to
control the solution uniformly in $\epsilon$, which is independent
of the properties of dissipative term. We prove a uniform global
well-posedness result using $l^1$-variant $X^{b,s}$-type space and
the I-method. Notice that \eqref{eq:kdvb} is invariant under the
following scaling for $0<\lambda\leq 1$
\begin{equation}
u(x,t)\rightarrow \lambda^2 u(\lambda x, \lambda^3 t),\
\phi(x)\rightarrow \lambda^2\phi(\lambda x), \ \epsilon\rightarrow
\lambda^{3-2\alpha}\epsilon.
\end{equation}
The equation \eqref{eq:kdvb} has less symmetries than the KdV
equation \eqref{eq:kdv} due to the dissipative term. Hence the
proofs for the pointwise estimate of the multipliers in our argument
are different from those in the KdV equation \cite{Tao2}. The basic
idea is the same, and to exploit dedicated cancelation to remove
the singularity in the denominator.
For the limit behavior, we need to study the difference equation
between \eqref{eq:kdvb} and \eqref{eq:kdv}. We first treat the
dissipative term as perturbation and then use the uniform Lipschitz
continuity property of the solution map. Similar idea can be found
in \cite{Wang} for the inviscid limit of the complex Ginzburg-Landau
equation. For $T>0$, we denote $S_T^{\epsilon}$, $S_T$ the solution
map of \eqref{eq:kdvb}, \eqref{eq:kdv} respectively. Now we state
our main results. The notations used in this paper can be found in
Section \ref{notation}.
\begin{theorem}\label{t11}
Assume $0<\epsilon, \alpha\leq 1$. Let $s_\alpha$ be given in
\eqref{eq:sa}. Let $\phi \in H^s({\mathbb R})$, $s>s_\alpha$. For any $T>0$,
there exists a unique solution $u_\epsilon$ of \eqref{eq:kdvb} in
\begin{equation}
Z_T=C([0,T],H^s)\cap X^{1/2,s,\alpha}_T.
\end{equation}
Moreover, the solution map $S_T^\epsilon: \phi\rightarrow u$ is
smooth from $H^s({\mathbb R})$ to $Z_T$ and $u$ belongs to
$C((0,\infty),H^{\infty}({\mathbb R}))$.
\end{theorem}
Notice that the critical regularity for the fractional Burgers
equation is $s=3/2 -2\alpha$ in the sense of scaling. Thus if
$1/2<\alpha\leq 1$ then $s_\alpha$ is lower than the critical
regularity for the KdV and also for the fractional Burgers equation.
In the proof we need to exploit the properties of the dissipative
term both in bilinear estimates and regularity for the solution.
Therefore, the results in Theorem \ref{t11} depend on $\epsilon>0$.
For the uniform well-posedness, we have the following,
\begin{theorem}\label{t12}
Assume $0<\alpha\leq 1$ and $-3/4<s\leq 0$. Let $\phi \in H^s({\mathbb R})$.
Then for any $T>0$, the solution map $S_T^\epsilon$ in Theorem
\ref{t11} satisfies for all $0<\epsilon \leq 1$
\begin{equation}
\norm{S_T^\epsilon \phi}_{F^s(T)}{\lesssim} C(T,\norm{u}_{H^s})
\end{equation}
where $F^s(T)\subset C([0,T];H^s)$ which will be defined later and
$C(\cdot,\cdot)$ is a continuous function with $C(\cdot,0)=0$, and
also satisfies that for all $0<\epsilon \leq 1$
\begin{eqnarray}
\norm{S_T^\epsilon (\phi_1)-S_T^\epsilon
(\phi_2)}_{C([0,T],H^s)}\leq
C(T,\norm{\phi_1}_{H^s},\norm{\phi_2}_{H^s})\norm{\phi_1-\phi_2}_{H^s}.
\end{eqnarray}
\end{theorem}
We also have the uniform persistence of regularity, following the
standard argument. The similar conclusions in Theorem \ref{t12} also
hold for the complex-valued equation \eqref{eq:kdvb} for a small
$T=T(\norm{u}_{H^s})>0$. Our final result is on the limit behavior.
\begin{theorem}\label{t13}
Assume $0<\alpha\leq 1$. Let $\phi \in H^s({\mathbb R})$, $-3/4<s\leq 0$. For
any $T>0$, then
\begin{equation}\label{eq:limitthm}
\lim_{\epsilon\rightarrow
0^+}\norm{S_T^\epsilon(\phi)-S_T(\phi)}_{C([0,T],H^s)}=0.
\end{equation}
\end{theorem}
\begin{remark} \rm
We are only concerned with the limit in the same regularity space.
There seems no convergence rate. This can be seen from the linear
solution,
\begin{equation}
\norm{e^{-t\partial_{x}^3-t\epsilon|\partial_x|^{2\alpha}}\phi-e^{-t\partial_{x}^3}\phi}_{C([0,T],H^s)}\rightarrow
0, \quad \mbox{as }\epsilon\rightarrow 0,
\end{equation}
but without any convergence rate. We believe that there is a
convergence rate if we assume the initial data has higher regularity
than the limit space. For example, we prove that
\begin{equation}
\norm{S_T^\epsilon(\phi_1)-S_T(\phi_2)}_{C([0,T],L^2)}{\lesssim}
\norm{\phi_1-\phi_2}_{L^2}+\epsilon^{1/2}C(T,\norm{\phi_1}_{H^1},\norm{\phi_2}_{L^2}).
\end{equation}
We only prove our results in the case $s\leq 0$ and our method also
works for $s>0$. For the complex valued equation \eqref{eq:kdvb},
the limit behavior \eqref{eq:limitthm} holds for a small
$T=T(\norm{\phi}_{H^s})>0$.
\end{remark}
The rest of the paper is organized as following. In Section 2 we
present some notations and Banach function spaces. The proof of
Theorem \ref{t11} is given in Section 3. We present uniform LWP in
Section 4 and prove Theorem \ref{t12} in Section 5. Theorem
\ref{t13} is proved in Section 6.
\section{Notation and Definitions} \label{notation}
For $x, y\in {\mathbb R}$, $x\sim y$ means that there exist $C_1, C_2 > 0$
such that $C_1|x|\leq |y| \leq C_2|x|$. For $f\in {\mathcal{S}}'$ we denote
by $\widehat{f}$ or ${\mathcal{F}} (f)$ the Fourier transform of $f$ for both
spatial and time variables,
\begin{eqnarray*}
\widehat{f}(\xi, \tau)=\int_{{\mathbb R}^2}e^{-ix \xi}e^{-it \tau}f(x,t)dxdt.
\end{eqnarray*}
We denote by ${\mathcal{F}}_x$ the the Fourier transform on spatial variable
and if there is no confusion, we still write ${\mathcal{F}}={\mathcal{F}}_x$. Let
$\mathbb{Z}$ and $\mathbb{N}$ be the sets of integers and natural
numbers, respectively. ${\mathbb Z}_+={\mathbb N} \cup \{0\}$. For $k\in {\mathbb Z}_+$ let
\[{I}_k=\{\xi: |\xi|\in [2^{k-1}, 2^{k+1}]\}, \ k\geq 1; \quad I_0=\{\xi: |\xi|\leq 2\}.\]
Let $\eta_0: {\mathbb R}\rightarrow [0, 1]$ denote an even smooth function
supported in $[-8/5, 8/5]$ and equal to $1$ in $[-5/4, 5/4]$. For
$k\in {\mathbb N}$ let $\eta_k(\xi)=\eta_0(\xi/2^k)-\eta_0(\xi/2^{k-1})$ and
$\eta_{\leq k}=\sum_{k'=0}^k\eta_{k'}$. For $k\in {\mathbb Z}$ let
$\chi_k(\xi)=\eta_0(\xi/2^k)-\eta_0(\xi/2^{k-1})$. Roughly speaking,
$\{\chi_k\}_{k\in \mathbb{Z}}$ is the homogeneous decomposition
function sequence and $\{\eta_k\}_{k\in \mathbb{Z}_+}$ is the
non-homogeneous decomposition function sequence to the frequency
space.
For $k\in {\mathbb Z}_+$ let $P_k$ denote the operator on $L^2({\mathbb R})$ defined
by
\[
\widehat{P_ku}(\xi)=\eta_k(\xi)\widehat{u}(\xi).
\]
By a slight abuse of notation we also define the operator $P_k$ on
$L^2({\mathbb R}\times {\mathbb R})$ by the formula ${\mathcal{F}}(P_ku)(\xi,
\tau)=\eta_k(\xi){\mathcal{F}} (u)(\xi, \tau)$. For $l\in {\mathbb Z}$ let
\[
P_{\leq l}=\sum_{k\leq l}P_k, \quad P_{\geq l}=\sum_{k\geq l}P_k.
\]
We define the Lebesgue spaces $L_T^qL_x^p$ and $L_x^pL_T^q$ by the
norms
\begin{equation}
\norm{f}_{L_T^qL_x^p}=\normo{\norm{f}_{L_x^p}}_{L_t^q([0,T])}, \quad
\norm{f}_{L_x^pL_T^q}=\normo{\norm{f}_{L_t^q([0,T])}}_{L_x^p}.
\end{equation}
We denote by $W_0$ the semigroup associated with Airy-equation
\[
{\mathcal{F}}_x(W_0(t)\phi)(\xi)=\exp[i\xi^3t]\widehat{\phi}(\xi), \ \forall
\ t\in {\mathbb R},\ \phi \in \mathcal {S}'.
\]
For $0<\epsilon\leq 1$ and $0< \alpha \leq 1$, we denote by
$W_\epsilon^\alpha$ the semigroup associated with the free evolution
of \eqref{eq:kdvb},
\[
{\mathcal{F}}_x(W_\epsilon^\alpha(t)\phi)(\xi)=\exp[-\epsilon
|\xi|^{2\alpha}t+i\xi^3t]\widehat{\phi}(\xi),\ \forall \ t\geq 0,\
\phi \in \mathcal {S}',
\]
and we extend $W_\epsilon^\alpha$ to a linear operator defined on
the whole real axis by setting
\[
{\mathcal{F}}_x(W_\epsilon^\alpha(t)\phi)(\xi)=\exp[-\epsilon
|\xi|^{2\alpha}|t|+i\xi^3t]\widehat{\phi}(\xi),\ \forall \ t\in {\mathbb R},
\ \phi \in \mathcal {S}'.
\]
To study the low regularity of \eqref{eq:kdvb}, Molinet and Ribaud
introduce the variant version of Bourgain's spaces with dissipation
\begin{equation}\label{eq:Xs}
\norm{u}_{X^{b,s,\alpha}}=\norm{\jb{i(\tau-\xi^3)+|\xi|^{2\alpha}}^b\jb{\xi}^s\widehat{u}}_{L^2({\mathbb R}^2)},
\end{equation}
where $\jb{\cdot}=(1+|\cdot|^2)^{1/2}$. The standard $X^{b,s}$ space
for \eqref{eq:kdv} used by Bourgain \cite{Bour} and Kenig, Ponce,
Vega \cite{KPV} is defined by
\begin{eqnarray*}
\norm{u}_{X^{b,s}}=\norm{\jb{\tau-\xi^3}^b\jb{\xi}^s\widehat{u}}_{L^2({\mathbb R}^2)}.
\end{eqnarray*}
The space $X^{1/2,s,\alpha}$ turns out to be very useful to capture
both dispersive and dissipative effect. From the technical level,
the dissipation will give bounds below for the modulations. These
bounds will weaken the frequency interaction for $\alpha> 1/2$, but
won't for $\alpha\leq 1/2$.
In order to study the uniform global wellposedness for
\eqref{eq:kdvb} and the limit behavior, we use an $l^1$ Besov-type
norm of $X^{b,s}$. For $k\in {\mathbb Z}_+$ we define the dyadic
$X^{b,s}$-type normed spaces $X_k=X_k({\mathbb R}^2)$,
\begin{eqnarray*}
X_k=\{f\in L^2({\mathbb R}^2): &&f(\xi,\tau) \mbox{ is supported in }
I_k\times{\mathbb R} \mbox{ and }\\&& \norm{f}_{X_k}=\sum_{j=0}^\infty
2^{j/2}\norm{\eta_j(\tau-\xi^3)\cdot f}_{L^2}\}.
\end{eqnarray*}
Structures of this kind of spaces were introduced, for instance, in
\cite{Tata}, \cite{IKT} and \cite{In-Ke} for the BO equation. From
the definition of $X_k$, we see that for any $l\in {\mathbb Z}_+$ and $f_k\in
X_k$ (see also \cite{IKT}),
\begin{equation}
\sum_{j=0}^\infty 2^{j/2}\norm{\eta_j(\tau-\xi^3)\int
|f_k(\xi,\tau')|2^{-l}(1+2^{-l}|\tau-\tau'|)^{-4}d\tau'}_{L^2}{\lesssim}
\norm{f_k}_{X_k}.
\end{equation}
Hence for any $l\in {\mathbb Z}_+$, $t_0\in {\mathbb R}$, $f_k\in X_k$, and $\gamma
\in {\mathcal{S}}({\mathbb R})$, then
\begin{equation}
\norm{{\mathcal{F}}[\gamma(2^l(t-t_0))\cdot {\mathcal{F}}^{-1}f_k]}_{X_k}{\lesssim}
\norm{f_k}_{X_k}.
\end{equation}
For $-3/4<s\leq 0$, we define the following spaces:
\begin{eqnarray}
&&F^{s}= \{u\in \mathcal {S}'({\mathbb R}^2):\ \norm{u}_{F^s}^2=\sum_{k \in
{\mathbb Z}_+}2^{2sk}\norm{\eta_k(\xi){\mathcal{F}}(u)}_{X_k}^2<\infty\},\\
&& N^{s}= \{u\in \mathcal {S}'({\mathbb R}^2): \norm{u}_{N^s}^2=\sum_{k\in
{\mathbb Z}_+}2^{2sk}\norm{(i+\tau-\xi^3)^{-1}\eta_k(\xi){\mathcal{F}}(u)}_{X_k}^2<\infty\}.
\end{eqnarray}
The space $F^s$ is between $X^{1/2,s}$ and $X^{1/2+,s}$. It can be
embedded into $C({\mathbb R};H^s)$ and into the Strichartz-type space, say
$L^p_tL^q_x$ as $X^{1/2+,s}$. On the other hand, it has the same
scaling in time as $X^{1/2,s}$, which is crucial in the uniform
linear estimate, See section 4. That is the main reason for us
applying $F^s$.
For $T\geq 0$, we define the time-localized spaces
$X_T^{b,s,\alpha}$, $X_T^{b,s}$, $F^{s}(T)$, and $N^{s}(T)$
\begin{eqnarray}
&& \norm{u}_{X_T^{b,s,\alpha}}=\inf_{w\in
X^{b,s,\alpha}}\{\norm{w}_{X^{b,s,\alpha}},\ w(t)=u(t) \mbox{ on }
[0, T]\};\nonumber \\
&&\norm{u}_{X_T^{b,s}}=\inf_{w\in X^{b,s}}\{\norm{w}_{X^{b,s}}, \ \
w(t)=u(t) \mbox{ on } [0, T]\};\nonumber\\
&&\norm{u}_{F^{s}(T)}=\inf_{w\in F^{s}}\{\norm{w}_{F^{s}}, \
w(t)=u(t)
\mbox{ on } [0, T]\};\nonumber\\
&&\norm{u}_{N^{s}(T)}=\inf_{w\in N^{s}}\{\norm{w}_{N^{s}}, \
w(t)=u(t) \mbox{ on } [0, T]\}.
\end{eqnarray}
As a conclusion of this section we prove that the norm on $F^s$
controls some space-time norm as the norm $X^{1/2+,s}$. If applying
to frequency dyadic localized function, we see that the norm $F^s$
is almost the same as the norm $X^{1/2+,s}$. Fortunately, in
application we usually encounter this case. See \cite{Tao3} for a
survey on $X^{s,b}$ space.
\begin{proposition}\label{p21}
Let $Y$ be a Banach space of functions on ${\mathbb R}\times {\mathbb R}$ with the
property that
\[\norm{e^{it\tau_0}e^{-t\partial_x^3}f}_Y{\lesssim} \ \norm{f}_{H^s({\mathbb R})}\]
holds for all $f\in H^s({\mathbb R})$ and $\tau_0\in {\mathbb R}$. Then we have the
embedding
\begin{equation}
\left(\sum_{k\in {\mathbb Z}_+}\norm{P_k u}_{Y}^2 \right)^{1/2}\lesssim
\norm{u}_{F^s}.
\end{equation}
\end{proposition}
\begin{proof}
In view of definition, it suffices to prove that if $k\in {\mathbb Z}_+$
\begin{equation}
\norm{P_ku}_Y \lesssim 2^{sk}\norm{\eta_k(\xi){\mathcal{F}}(u)}_{X_k}.
\end{equation}
Indeed, we have
\begin{align}
P_ku =&\int \eta_k(\xi){\mathcal{F}} u(\xi,\tau) e^{ix\xi}e^{it\tau}d\xi d\tau \nonumber\\
=&\sum_{j=0}^\infty\int \eta_j(\tau-\xi^3)\eta_k(\xi){\mathcal{F}}
u(\xi,\tau) e^{ix\xi}e^{it\tau}d\xi d\tau \nonumber\\
=&\sum_{j=0}^\infty\int \eta_j(\tau)e^{it\tau} \int \eta_k(\xi){\mathcal{F}}
u(\xi,\tau+\xi^3) e^{ix\xi}e^{it\xi^3}d\xi d\tau.
\end{align}
From the hypothesis on $Y$, we obtain
\begin{eqnarray}
\norm{P_k u}_Y &{\lesssim}& \sum_{j=0}^\infty \int \eta_j(\tau)
\normo{e^{it\tau} \int \eta_k(\xi){\mathcal{F}} u(\xi,\tau+\xi^3)
e^{ix\xi}e^{it\xi^3}d\xi}_{Y}d\tau \nonumber\\
&{\lesssim}& 2^{sk}\norm{\eta_k(\xi){\mathcal{F}}(u)}_{X_k},
\end{eqnarray}
which completes the proof of the proposition.
\end{proof}
\section{Global well-posedness for KdV-B equation}
In this section, we prove a global wellposedness result for the
KdV-Burgers equation by following the idea of Molinet and Ribaud
\cite{MR}. Using Duhamel's principle, we will mainly work on the
integral formulation of the KdV-Burgers equation
\begin{equation}\label{eq:intekdvb}
u(t)=W_\epsilon^\alpha(t)\phi_1-\half{1}\int_0^tW_\epsilon^\alpha(t-\tau)\partial_x(u^2(\tau))d\tau,
\ t\geq 0.
\end{equation}
We will apply a fixed point argument to solve the following
truncated version
\begin{equation}\label{eq:trunintekdvb}
u(t)=\psi(t)\left[W_\epsilon^\alpha(t)\phi_1-\half{\chi_{{\mathbb R}_+}(t)}\int_0^tW_\epsilon^\alpha(t-\tau)\partial_x(\psi_T^2(\tau)u^2(\tau))d\tau
\right],
\end{equation}
where $t\in {\mathbb R}$ and $\psi$ is a smooth time cutoff function
satisfying
\begin{equation}\label{eq:cutoff}
\psi \in C_0^\infty({\mathbb R}), \quad {\mbox{supp}}\ \psi \subset [-2, 2], \quad
\psi\equiv 1 \mbox{ on } [-1, 1],
\end{equation}
and $\psi_T(\cdot)=\psi(\cdot/T)$. Indeed, if $u$ solves
\eqref{eq:trunintekdvb} then $u$ is a solution of
\eqref{eq:intekdvb} on $[0, T]$, $T\leq 1$.
Theorem \ref{t11} can be proved by a slightly modified argument in
\cite{MR} combined with the following bilinear estimate. See also
\cite{Vento}.
\begin{proposition}\label{p31}
Let $s_\alpha$ be given by \eqref{eq:sa}. Let $s \in (s_\alpha, 0]$,
$0<\delta\ll 1$, then there exists $C_{s,\alpha}>0$ such that for
any $u, v \in {\mathcal{S}}$,
\begin{equation}\label{eq:bikdvb}
\norm{\partial_x(uv)}_{X^{-1/2+\delta, s, \alpha}}\leq
C_{s,\alpha}\norm{u}_{X^{1/2, s, \alpha}}\norm{v}_{X^{1/2, s,
\alpha}}.
\end{equation}
\end{proposition}
This type of estimate was systematically studied in \cite{Tao}, see
also \cite{KPV} for an elementary method. We will follow the idea in
\cite{Tao} to prove Proposition \ref{p31}. Let $Z$ be any abelian
additive group with an invariant measure $d\xi$. In particular,
$Z={\mathbb R}^2$ in this paper. For any $k\geq 2$, Let $\Gamma_k(Z)$ denote
the hyperplane in ${\mathbb R}^k$
\[\Gamma_k(Z):=\{(\xi_1,\ldots, \xi_k)\in Z^k: \xi_1+\ldots+\xi_k=0\}\]
endowed with the induced measure
\[\int_{\Gamma_k(Z)}f:=\int_{Z^{k-1}}f(\xi_1,\ldots, \xi_{k-1},-\xi_1-\ldots-\xi_{k-1})d\xi_1\ldots d\xi_{k-1}.\]
Note that this measure is symmetric with respect to permutation of
the co-ordinates.
A function $m: \Gamma_k(Z)\rightarrow {\mathbb C}$ is said to to be a
$[k;Z]-multiplier$, and we define the norm $\norm{m}_{[k;Z]}$ to be
the best constant such that the inequality
\begin{equation}
\left|\int_{\Gamma_k(Z)}m(\xi)\prod_{j=1}^k f_i(\xi_i)\right|\leq
\norm{m}_{[k;Z]}\prod_{j=1}^k \norm{f_i}_{L^2}
\end{equation}
holds for all test functions $f_i$ on $Z$.
By duality and Plancherel's equality, it is easy to see that for
\eqref{eq:bikdvb}, it suffices to prove
\begin{equation}\label{eq:bi}
\normo{\frac{|\xi_3|\jb{\xi_3}^s\jb{\xi_1}^{-s}\jb{\xi_2}^{-s}\jb{i(\tau_3-\xi_3)+|\xi_3|^{2\alpha}}^{-1/2+\delta}}{\jb{i(\tau_2-\xi_2)+|\xi_2|^{2\alpha}}^{1/2}\jb{i(\tau_1-\xi_1)+|\xi_1|^{2\alpha}}^{1/2}}}_{[3;{\mathbb R}^2]}{\lesssim}
1.
\end{equation}
By comparision principle (see \cite{Tao}), it suffices to prove that
\begin{eqnarray}
&&\sum_{N_1,N_2,N_3}\sum_{L_1,L_2,L_3}\sum_{H}\frac{N_3\jb{N_3}^s\jb{N_1}^{-s}\jb{N_2}^{-s}}{\jb{L_1+N_1^{2\alpha}}^{1/2}\jb{L_2+N_2^{2\alpha}}^{1/2}\jb{L_3+N_3^{2\alpha}}^{1/2-\delta}}\nonumber\\
&&\qquad
\qquad\norm{\chi_{N_1,N_2,N_3;H;L_1,L_2,L_3}}_{[3;{\mathbb R}^2]}{\lesssim} 1,
\end{eqnarray}
where $N_i,L_i, H$ are dyadic, $h(\xi)=\xi_1^3+\xi_2^3+\xi_3^3$ and
\begin{eqnarray}
&&\chi_{N_1,N_2,N_3;H;L_1,L_2,L_3}=\chi_{|\xi_1|\sim N_1,|\xi_2|\sim
N_2,|\xi_3|\sim N_3}\nonumber\\
&&\qquad \cdot \chi_{|h(\xi)|\sim H}\chi_{|\tau_1-\xi_1^3|\sim
L_1,|\tau_2-\xi_2^3|\sim L_2,|\tau_3-\xi_3^3|\sim L_3}.
\end{eqnarray}
The issues reduce to an estimate of
\begin{equation}\label{eq:char}
\norm{\chi_{N_1,N_2,N_3;H;L_1,L_2,L_3}}_{[3;{\mathbb R}^2]}
\end{equation}
and dyadic summation. Since
\[\xi_1+\xi_2+\xi_3=0,\quad |h(\xi)|=|\xi_1^3+\xi_2^3+\xi_3^3|\sim N_1N_2N_3,\]
and
\[\tau_1-\xi_1^3+\tau_2-\xi_2^3+\tau_3-\xi_3^3+h(\xi)=0,\]
then we have
\begin{eqnarray}\label{eq:dyadic1}
N_{max}&\sim& N_{med},\nonumber\\
L_{max}&\sim& \max(L_{med}, H),
\end{eqnarray}
where we define $N_{max}\geq N_{med}\geq N_{min}$ to be the maximum,
median, and minimum of $N_1,\ N_2,\ N_3$ respectively. Similarly
define $L_{max}\geq L_{med}\geq L_{min}$. It's known (see Section 4,
\cite{Tao}) that we may assume
\begin{equation}\label{eq:dyadic2}
N_{max}{\gtrsim} 1, \quad L_1,L_2,L_3{\gtrsim} 1.
\end{equation}
Therefore, from Schur's test (Lemma 3.11, \cite{Tao}) it suffices to
prove that
\begin{eqnarray}\label{eq:bicase1}
&&\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_1,L_2,L_3\geq 1}\frac{N_3\jb{N_3}^s\jb{N_1}^{-s}\jb{N_2}^{-s}}{\jb{L_1+N_1^{2\alpha}}^{1/2}\jb{L_2+N_2^{2\alpha}}^{1/2}\jb{L_3+N_3^{2\alpha}}^{1/2-\delta}}\nonumber\\
&&\qquad \qquad \times
\norm{\chi_{N_1,N_2,N_3;L_{max};L_1,L_2,L_3}}_{[3;{\mathbb R}^2]}
\end{eqnarray}
and
\begin{eqnarray}\label{eq:bicase2}
&&\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{max}\sim L_{med}}\sum_{H \leq L_{max}}\frac{N_3\jb{N_3}^s\jb{N_1}^{-s}\jb{N_2}^{-s}}{\jb{L_1+N_1^{2\alpha}}^{1/2}\jb{L_2+N_2^{2\alpha}}^{1/2}\jb{L_3+N_3^{2\alpha}}^{1/2-\delta}}\nonumber\\
&&\qquad \qquad \times
\norm{\chi_{N_1,N_2,N_3;H;L_1,L_2,L_3}}_{[3;{\mathbb R}^2]}
\end{eqnarray}
are both uniformly bounded for all $N{\gtrsim} 1$.
\begin{proposition}[Proposition 6.1, \cite{Tao}]\label{pchar}
Let dyadic numbers $H,N_1,N_2,N_3,L_1,L_2,L_3>0$ obey \eqref{eq:dyadic1}, \eqref{eq:dyadic2}.\\
(i) If $N_{max}\sim N_{min}$ and $L_{max}\sim H$, then we have
\begin{equation}\label{eq:chari}
\eqref{eq:char} {\lesssim} L_{min}^{1/2}N_{max}^{-1/4}L_{med}^{1/4}.
\end{equation}
(ii) If $N_2\sim N_3 \gg N_1$ and $H\sim L_1{\gtrsim} L_2,L_3$, then
\begin{equation}\label{eq:charii}
\eqref{eq:char} {\lesssim}
L_{min}^{1/2}N_{max}^{-1}\min(H,\frac{N_{max}}{N_{min}}L_{med})^{1/2}.
\end{equation}
Similarly for permutations.\\
(iii) In all other cases, we have
\begin{equation}\label{eq:chariii}
\eqref{eq:char} {\lesssim} L_{min}^{1/2}N_{max}^{-1}\min(H,L_{med})^{1/2}.
\end{equation}
\end{proposition}
In order to estimate the denominator in \eqref{eq:bicase1},
\eqref{eq:bicase2}, we will need the following proposition to reduce
some cases.
\begin{proposition}\label{pdeno}
Let $k\in {\mathbb N}$. Assume that $a_1, a_2,\ldots, a_k$ and
$b_1,b_2,\ldots, b_k$ are non-negative numbers, and $A_1\leq A_2\leq
\ldots \leq A_k$, $B_1\leq B_2\leq \ldots \leq B_k$ are rearrange of
$\{a_i\}$, $\{b_i\}$ respectively. Then
\begin{equation}\label{eq:edeno}
\prod_{i=1}^k(a_i+b_i)\geq \prod_{i=1}^k(A_i+B_i).
\end{equation}
\end{proposition}
\begin{proof}
We apply an induction on $k$. The case $k=1$ is obviously. For
$k=2$, we have
\begin{eqnarray*}
(a_1+b_1)(a_2+b_2)&=&a_1a_2+b_1b_2+a_1b_2+a_2b_1\\
&\geq& A_1B_1+A_2B_2+A_1B_2+A_2B_1=(A_1+B_1)(A_2+B_2).
\end{eqnarray*}
We assume the lemma holds for all $ q\in {\mathbb N},\ q\leq k-1$. Now we
prove for $k$. If $a_1=A_1$, $b_1=B_1$, then we apply induction
assumption for $k-1$ and get \eqref{eq:edeno}. Otherwise, we may
assume $a_1=A_1$, $b_2=B_1$. By induction assumption for $2$, then
$k-1$, we get
\begin{eqnarray}
\prod_{i=1}^k(a_i+b_i)&=&(a_1+b_1)(a_2+b_2)\prod_{i=3}^k(a_i+b_i)\nonumber\\
&\geq& (A_1+B_1)(a_2+b_1)\prod_{i=3}^k(a_i+b_i)\nonumber\\
&\geq&\prod_{i=1}^k(A_i+B_i),
\end{eqnarray}
which completes the proof of the proposition.
\end{proof}
\begin{proof}[Proof of Proposition \ref{p31}.]
We will prove the proposition using case-by-case analysis. We first
bound \eqref{eq:bicase2}. Since we have
\begin{equation}
N_3\jb{N_3}^s\jb{N_1}^{-s}\jb{N_2}^{-s}{\lesssim}
N\jb{N_{min}}^{-s}+N^{-2s}N_{min}\jb{N_{min}}^s
\end{equation}
and from (iii) of Proposition \ref{pchar}, we obtain
\begin{eqnarray}
\eqref{eq:bicase2}&{\lesssim}&\sum_{N_{max}\sim N_{med}\sim
N}\sum_{L_i,L_{max}\geq
H}\frac{(N\jb{N_{min}}^{-s}+N^{-2s}N_{min}\jb{N_{min}}^s)L_{min}^{1/2}N_{min}^{1/2}}{L_{max}^{1/2-\delta}L_{med}^{1/2-\delta}L_{min}^{1/2-\delta}}\nonumber
\\
&{\lesssim}&\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{max}\geq
H}(N\jb{N_{min}}^{-s}+N^{-2s}N_{min}\jb{N_{min}}^s)
L_{max}^{-1+3\delta}N_{min}^{1/2}\nonumber\\
&{\lesssim}&\sum_{N_{min}\leq N^{-2}}(N+N^{-2s}N_{min})
N_{min}^{1/2}\nonumber\\
&&+\sum_{N^{-2}\leq N_{min}\leq 1}(N+N^{-2s}N_{min})
N^{-2+6\delta}N_{min}^{-1/2+3\delta}\nonumber\\
&&+\sum_{N_{min}\geq 1}(NN_{min}^{-s}+N^{-2s}N_{min}^{1+s})
N^{-2+6\delta}N_{min}^{-1/2+3\delta}\nonumber\\
&{\lesssim}&1,
\end{eqnarray}
provided that $-1<s\leq 0$.
We next bound \eqref{eq:bicase1}, which is more complicated. We
first assume that \eqref{eq:chari} applies. Then we have
\begin{eqnarray}
\eqref{eq:bicase1}&{\lesssim}&\sum_{N_{max}\sim N_{min}\sim
N}\sum_{L_1,L_2,L_3\geq
1}\frac{N^{3/4-s}L_{min}^{1/2}L_{med}^{1/4}\jb{L_{min}+N^{2\alpha}}^{-1/2+\delta}}{\jb{L_{max}+N^{2\alpha}}^{1/2-\delta}\jb{L_{med}+N^{2\alpha}}^{1/2-\delta}}\nonumber\\
&{\lesssim}&\sum_{N_{max}\sim N_{min}\sim N}\sum_{L_{med}}\frac{N^{3/4-s}L_{med}^{1/4+\delta}}{N^{3/2-3\delta}\jb{L_{med}+N^{2\alpha}}^{1/2-\delta}}\nonumber\\
&{\lesssim}&N^{-\frac{3}{4}-\half \alpha-s+4\delta}{\lesssim} 1,
\end{eqnarray}
provided that $-\frac{3}{4}-\half \alpha<s\leq 0$.
If \eqref{eq:chariii} applies, from Proposition \ref{pdeno}, we
obtain
\begin{eqnarray}
\eqref{eq:bicase1}&{\lesssim}&\sum_{N_i}\sum_{L_i}\frac{(N\jb{N_{min}}^{-s}+N^{-2s}N_{min}\jb{N_{min}}^s)L_{min}^{1/2}N^{-1}L_{med}^{1/2}}{(L_{max}+N^{2\alpha})^{1/2-\delta}\jb{L_{med}+N^{2\alpha}}^{1/2-\delta}\jb{L_{min}+N_{min}^{2\alpha}}^{1/2-\delta}}\nonumber\\
&{\lesssim}&\sum_{N_i}\frac{(N\jb{N_{min}}^{-s}+N^{-2s}N_{min}\jb{N_{min}}^s)N^{-1+4\alpha\delta}}{(N^2N_{min}+N^{2\alpha})^{1/2-3\delta}}\nonumber\\
&{\lesssim}&\sum_{N_{min}\leq N^{2\alpha-2}}\frac{(N+N^{-2s}N_{min})N^{-1+4\alpha\delta}}{N^{\alpha-6\delta}}\nonumber\\
&&+\sum_{N^{2\alpha-2}\leq N_{min}\leq 1}\frac{(N+N^{-2s}N_{min})N^{-1+4\alpha\delta}}{N^{1-6\delta}N_{min}^{1/2-3\delta}}\nonumber\\
&&+\sum_{N_{min}\geq
1}\frac{(NN_{min}^{-s}+N^{-2s}N_{min}^{1+s})N^{-1+4\alpha\delta}}{N^{1-6\delta}N_{min}^{1/2-3\delta}}\nonumber\\
&{\lesssim}&N^{-\alpha+10\delta}+N^{-2s-3+\alpha+6\delta}+N^{-2s-2+6\delta}+N^{-s-3/2+7\delta}\nonumber\\
&{\lesssim}&1,
\end{eqnarray}
provided that $-1<s\leq 0$.
If \eqref{eq:charii} applies, we have three cases:
\begin{eqnarray}
&&N_2\sim N_3\gg N_1,\quad L_1{\gtrsim} L_2, L_3,\label{eq:bicase121}\\
&&N_1\sim N_3\gg N_2,\quad L_2{\gtrsim} L_1, L_3,\label{eq:bicase122}\\
&&N_1\sim N_2\gg N_3,\quad L_3{\gtrsim} L_1, L_2.\label{eq:bicase123}
\end{eqnarray}
If \eqref{eq:bicase121} holds, then we have
\begin{eqnarray}
\eqref{eq:bicase1}&{\lesssim}&\sum_{N_i}\sum_{L_i}\frac{N\jb{N_{min}}^{-s}L_{min}^{1/2}N^{-1}\min(H,\frac{N_{max}}{N_{min}}L_{med})^{1/2}}{N_{min}^{1/2}N\jb{L_{med}+N^{2\alpha}}^{1/2-\delta}\jb{L_{min}+N^{2\alpha}}^{1/2}}\nonumber\\
&{\lesssim}&\sum_{N_i}\sum_{L_{med}\geq
NN_{min}^2}\frac{N\jb{N_{min}}^{-s}\log(L_{med})N^{-1}N_{min}^{1/2}N}{N_{min}^{1/2}N\jb{L_{med}+N^{2\alpha}}^{1/2-\delta}}\nonumber\\
&&+\sum_{N_i}\sum_{L_{med}\leq
NN_{min}^2}\frac{N\jb{N_{min}}^{-s}\log(L_{med})L_{med}^{1/2}N^{-1}N_{min}^{-1/2}N^{1/2}}{N_{min}^{1/2}N\jb{L_{med}+N^{2\alpha}}^{1/2-\delta}}\nonumber\\
&=&A_1+A_2.
\end{eqnarray}
We first bound $A_1$.
\begin{eqnarray}
A_1&{\lesssim}&\sum_{N^{-2}\leq N_{min}\leq 1}\sum_{L_{med}\geq
NN_{min}^2}\frac{L_{med}^\delta}{\jb{L_{med}+N^{2\alpha}}^{1/2-\delta}}\nonumber\\
&&+\sum_{N_{min}\geq 1}\sum_{L_{med}\geq NN_{min}^2}\frac{L_{med}^\delta N_{min}^{-s}}{\jb{L_{med}}^{1/2-\delta}}\nonumber\\
&{\lesssim}& N^{-\alpha+7\delta}+\sum_{N_{min}\geq
1}N_{min}^{-s-1+4\delta}N^{-1/2+2\delta}{\lesssim} 1,
\end{eqnarray}
provided $-1<s\leq 0$.
For $A_2$, we have
\begin{eqnarray}
A_2&{\lesssim}&\sum_{N^{-1/2}\leq N_{min}\leq 1}\sum_{L_{med}\leq
NN_{min}^2}\frac{L_{med}^{\delta+1/2}N_{min}^{-1}N^{-1/2}}{\jb{L_{med}+N^{2\alpha}}^{1/2-\delta}}\nonumber\\
&&+\sum_{N_{min}\geq 1}\sum_{L_{med}\leq
NN_{min}^2}\frac{L_{med}^{\delta+1/2}N_{min}^{-1-s}N^{-1/2}}{\jb{L_{med}+N^{2\alpha}}^{1/2-\delta}}\nonumber\\
&{\lesssim}&\sum_{N^{-1/2}\leq N_{min}\leq
1}N^{2\delta-1/2}N_{min}^{4\delta-1}+\sum_{N_{min}\geq
1}N_{min}^{-1-s+4\delta}N^{-1/2+2\delta}\nonumber\\
&{\lesssim}&1,
\end{eqnarray}
provided $-1<s\leq 0$.
From symmetry, the case
\eqref{eq:bicase121} is identical to the case \eqref{eq:bicase122}.
Now we assume that \eqref{eq:bicase123} holds, and we obtain
\begin{eqnarray}
\eqref{eq:bicase1}&{\lesssim}&\sum_{N_i}\sum_{L_i}\frac{N^{-2s}\jb{N_{min}}^{s}N_{min}L_{min}^{1/2}N^{-1}\min(H,\frac{N_{max}}{N_{min}}L_{med})^{1/2}}{N_{min}^{1/2-\delta}N^{1-2\delta}\jb{L_{med}+N^{2\alpha}}^{1/2-\delta}\jb{L_{min}+N^{2\alpha}}^{1/2}}\nonumber\\
&{\lesssim}&\sum_{N_i}\sum_{L_{med}\geq
NN_{min}^2}\frac{N^{-2s}\jb{N_{min}}^{s}N_{min}\log(L_{med})N^{-1}N_{min}^{1/2}N}{N_{min}^{1/2-\delta}N^{1-2\delta}\jb{L_{med}+N^{2\alpha}}^{1/2-\delta}}\nonumber\\
&&+\sum_{N_i}\sum_{L_{med}\leq
NN_{min}^2}\frac{N^{-2s}\jb{N_{min}}^{s}N_{min}\log(L_{med})L_{med}^{1/2}N^{-1}N_{min}^{-1/2}N^{1/2}}{N_{min}^{1/2-\delta}N^{1-2\delta}\jb{L_{med}+N^{2\alpha}}^{1/2}}\nonumber\\
&=&B_1+B_2.
\end{eqnarray}
We first bound $B_1$.
\begin{eqnarray}
B_1&{\lesssim}&\sum_{N^{-2}\leq N_{min}\leq 1}\sum_{L_{med}\geq
NN_{min}^2}\frac{N^{-2s-1+2\delta}N_{min}^{1+\delta}L_{med}^\delta}{\jb{L_{med}+N^{2\alpha}}^{1/2-\delta}}\nonumber\\
&&+\sum_{N_{min}\geq 1}\sum_{L_{med}\geq
NN_{min}^2}\frac{N^{-2s-1+2\delta}N_{min}^{1+\delta+s}L_{med}^\delta}{\jb{L_{med}+N^{2\alpha}}^{1/2-\delta}}\nonumber\\
&{\lesssim}&\sum_{N^{-2}\leq N_{min}\leq 1}\frac{N^{-2s-1+2\delta}N_{min}^{1+\delta}}{\jb{NN_{min}^2+N^{2\alpha}}^{1/2-2\delta}}\nonumber\\
&&+\sum_{N_{min}\geq
1}\frac{N^{-2s-1+2\delta}N_{min}^{1+\delta+s}}{\jb{NN_{min}^2+N^{2\alpha}}^{1/2-2\delta}}.
\end{eqnarray}
We discuss it in the following two cases. If $1/2\leq \alpha\leq 1$,
then
\begin{eqnarray}
B_1&{\lesssim}&N^{-2s-1-\alpha+6\delta} +\sum_{N_{min}\geq
N^{\alpha-1/2}}N^{-2s-3/2+4\delta}N_{min}^{5\delta+s}\nonumber\\
&&+\sum_{1\leq N_{min}\leq
N^{\alpha-1/2}}N^{-2s-1-\alpha+6\delta}N_{min}^{1+\delta+s},
\end{eqnarray}
provided that $-\frac{3}{5-2\alpha}<s\leq 0$. If $0<\alpha \leq
1/2$, then
\begin{eqnarray}
B_1&{\lesssim}&\sum_{N^{\alpha-1/2}\leq N_{min}\leq
1}N^{-2s-3/2+4\delta}N_{min}^{5\delta}+\sum_{N_{min}\geq
1}N^{-2s-3/2+4\delta}N_{min}^{5\delta+s}\nonumber\\
&&+\sum_{N^{-2}\leq N_{min}\leq
N^{\alpha-1/2}}N^{-2s-1-\alpha+6\delta}N_{min}^{1+\delta}\nonumber\\
&{\lesssim}&1,
\end{eqnarray}
provided that $-3/4<s\leq 0$.
For $B_2$, we have
\begin{eqnarray}
B_2&{\lesssim}&\sum_{N^{-1/2}\leq N_{min}\leq 1}\sum_{L_{med}\leq
NN_{min}^2}\frac{N^{-2s-3/2+2\delta}N_{min}^\delta L_{med}^{1/2+\delta}}{\jb{L_{med}+N^{2\alpha}}^{1/2}}\nonumber\\
&&+\sum_{N_{min}\geq 1}\sum_{L_{med}\leq
NN_{min}^2}\frac{N^{-2s-3/2+2\delta}N_{min}^{\delta+s}
L_{med}^{1/2+\delta}}{\jb{L_{med}+N^{2\alpha}}^{1/2}}\nonumber.
\end{eqnarray}
and get
\begin{eqnarray}
B_2&{\lesssim}&\sum_{N^{-1/2}\leq N_{min}\leq
1}\frac{N^{-2s-1+3\delta}N_{min}^{1+3\delta}}{\jb{NN_{min}^2+N^{2\alpha}}^{1/2}}+\sum_{N_{min}\geq
1}\frac{N^{-2s-1+3\delta}N_{min}^{1+s+3\delta}}{\jb{NN_{min}^2+N^{2\alpha}}^{1/2}}\nonumber.
\end{eqnarray}
If $1/2\leq \alpha \leq 1$, then
\begin{eqnarray}
B_2&{\lesssim}&N^{-2s-1-\alpha+3\delta}+\sum_{N_{min}\geq
N^{\alpha-1/2}}N^{-2s-3/2+3\delta}N_{min}^{s+3\delta}\nonumber\\
&&+\sum_{1\leq N_{min}\leq
N^{\alpha-1/2}}N^{-2s-1-\alpha+3\delta}N_{min}^{1+s+3\delta}\nonumber\\
&{\lesssim}&1,
\end{eqnarray}
provided that $-\frac{3}{5-2\alpha}<s\leq 0$. If $0< \alpha \leq
1/2$, then
\begin{eqnarray}
B_2&{\lesssim}&\sum_{N^{-1/2}\leq N_{min}\leq
N^{\alpha-1/2}}N^{-2s-1-\alpha+3\delta}N_{min}^{1+3\delta}\nonumber\\
&&+\sum_{N^{\alpha-1/2}\leq N_{min}\leq
1}N^{-2s-3/2+3\delta}N_{min}^{3\delta}+\sum_{N_{min}\geq
1}N^{-2s-3/2+3\delta}N_{min}^{s+3\delta}\nonumber \\
&{\lesssim}&1,
\end{eqnarray}
provided that $-3/4<s\leq 0$. Therefore, we complete the proof of
Proposition \ref{p31}.
\end{proof}
\begin{proposition}
If $s\leq s_\alpha$, then for any $0<\delta\ll 1$, there doesn't
exist $C>0$ such that for any $u, v \in {\mathcal{S}}$,
\begin{equation}\label{eq:bikdvb}
\norm{\partial_x(uv)}_{X^{-1/2+\delta, s, \alpha}}\leq
C\norm{u}_{X^{1/2, s, \alpha}}\norm{v}_{X^{1/2, s, \alpha}}.
\end{equation}
\end{proposition}
\begin{proof}
From the proof of the Proposition \ref{p31}, we see that the
restriction on $s$ is caused by high-high interaction, and hence we
construct the worst case. The idea is due to C. Kenig, G. Ponce and
L. Vega \cite{KPV}. In view of definition, \eqref{eq:bikdvb} is
equivalent to
\begin{eqnarray}\label{eq:count}
&& \norm{\frac{\xi(1+|\xi|)^s}{(1+|\xi|^{2\alpha}+|\tau-\xi^3|)^{1/2-\delta}}\nonumber\\
&&\times \int
\frac{f(\xi_1,\tau_1)(1+|\xi_1|)^{-s}f(\xi-\xi_1,\tau-\tau_1)(1+|\xi-\xi_1|)^{-s}d\xi_1
d\tau_1}{\jb{|\xi_1|^{2\alpha}+|\tau_1-\xi_1^3|}^{1/2}\jb{|\xi-\xi_1|^{2\alpha}+|\tau-\tau_1-(\xi-\xi_1)^3|}^{1/2}}}_{L_{\xi,\tau}^2}\nonumber\\
&{\lesssim}& \norm{f}_{L_{\xi,\tau}^2}^2.
\end{eqnarray}
If $0<\alpha\leq 1/2$, fix $N\gg 1$, we set
\begin{eqnarray*}
f(\xi,\tau)=\chi_A(\xi,\tau)+\chi_{-A}(\xi,\tau),
\end{eqnarray*}
where
\begin{eqnarray*}
A=\{(\xi,\tau)\in {\mathbb R}^2|N\leq \xi \leq N+1, N\leq |\tau-\xi^3|\leq
2N\},
\end{eqnarray*}
and
\begin{eqnarray*}
-A=\{(\xi,\tau)\in {\mathbb R}^2| -(\xi,\tau)\in A\}.
\end{eqnarray*}
Clearly,
\begin{equation}
\norm{f}_{L_{\xi,\tau}^2}\sim N^{1/2}.
\end{equation}
On the other hand, $A$ contains a rectangle with $(N, N^3+N)$ as a
vertex, with dimension $N^{-1}\times N^2$ and longest side pointing
in the $(1, 3N^2)$ direction. Therefore,
\begin{equation}
|f*f(\xi,\tau)|{\gtrsim} N\chi_{R}(\xi,\tau),
\end{equation}
where R is a rectangle centered at the origin of dimensions
$N^{-1}\times N^2$ and longest side pointing in the $(1, 3N^2)$
direction. Taking the one-third rectangle away from origin, then we
have $|\xi|\sim 1$, and therefore \eqref{eq:count} implies that
\begin{equation}
N^{-1+2\delta}N^{-2s}N^{-1}NN^{-1/2}N{\lesssim} N,
\end{equation}
which implies that $s> -3/4$.
If $1/2\leq \alpha \leq 1$, then take
\begin{eqnarray*}
f(\xi,\tau)=\chi_B(\xi,\tau)+\chi_{-B}(\xi,\tau),
\end{eqnarray*}
where
\begin{equation}
B=\{(\xi,\tau)\in {\mathbb R}^2|N\leq \xi \leq N+N^{\alpha-1/2},
N^{2\alpha}\leq |\tau-\xi^3|\leq 2N^{2\alpha}\},
\end{equation}
and
\begin{eqnarray*}
-B=\{(\xi,\tau)\in {\mathbb R}^2| -(\xi,\tau)\in B\}.
\end{eqnarray*}
Clearly,
\begin{equation}
\norm{f}_{L_{\xi,\tau}^2}\sim N^{\half {3\alpha}-\frac{1}{4}}.
\end{equation}
On the other hand, $B$ contains a rectangle with $(N,
N^3+N^{2\alpha})$ as a vertex, with dimension $N^{2\alpha-2}\times
N^{\alpha+3/2}$ and longest side pointing in the $(1, 3N^2)$
direction. Therefore,
\begin{equation}
|f*f(\xi,\tau)|{\gtrsim} N^{3\alpha-1/2}\chi_{R}(\xi,\tau),
\end{equation}
where R is a rectangle centered at the origin of dimensions
$N^{2\alpha-2}\times N^{\alpha+3/2}$ and longest side pointing in
the $(1, 3N^2)$ direction. Taking the one-third rectangle away from
origin, then we have $|\xi|\sim N^{\alpha-1/2}$, and therefore
\eqref{eq:count} implies that
\begin{equation}
N^{(\alpha-1/2)(1+s)}N^{(\alpha+3/2)(-1/2+\delta)}N^{-2s}N^{-2\alpha}N^{3\alpha-1/2}N^{\alpha-1}N^{\alpha/2+3/4}{\lesssim}
N^{3\alpha-1/2},
\end{equation}
which implies that $s> -3/(5-2\alpha)$.
\end{proof}
\begin{remark}\label{r35} \rm
The constant in Proposition \ref{p31} depends on $\alpha$, which is
the main reason for gaining $\delta$-order derivative in time in the
bilinear estimates. In proving global well-posedness we also need to
exploit the smoothing effect of the dissipative term and then $L^2$
conservation law. Therefore, the result of Theorem \ref{t11} is
dependent of $\epsilon$.
\end{remark}
\section{Uniform LWP for KdV-B equation}
In this section we study the uniform local well posedness for the
KdV-Burgers equation. We will prove a time localized version of
Theorem \ref{t12} where $T=T(\norm{\phi}_{H^{s}})$ is small. In view
of Remark \ref{r35}, the space $X^{b,s}$ we used in the last section
is not proper in this situation. We will use the space $F^s$. Let us
recall that \eqref{eq:kdvb} is invariant in the following scaling
\begin{equation}\label{eq:scaling}
u(x,t)\rightarrow \lambda^2 u(\lambda x, \lambda^3 t),\
\phi(x)\rightarrow \lambda^2\phi(\lambda x), \ \epsilon\rightarrow
\lambda^{3-2\alpha}\epsilon, \ \ \forall \ 0<\lambda\leq 1.
\end{equation}
This invariance is very important in the proof of Theorem \ref{t12}
and also crucial for the uniform global-well posedness in the next
section. We first show that $F^s(T)\hookrightarrow C([0, T], H^{s})$
for $s\in {\mathbb R}$, $T\in (0, 1]$ in the following proposition.
\begin{proposition}\label{p41}
If $s\in {\mathbb R}$, $T\in (0, 1]$, and $u\in F^s(T)$, then
\begin{equation}
\sup_{t\in [0, T]}\norm{u(t)}_{H^s}{\lesssim} \norm{u}_{F^s(T)}.
\end{equation}
\end{proposition}
\begin{proof}
In view of definition, it suffices to show that for $k\in {\mathbb Z}_+$,
$t\in [0, 1]$,
\begin{equation}\label{eq:s42}
\norm{\eta_k(\xi){\mathcal{F}}_x{u}(t)}_{L^2}{\lesssim}
\norm{\eta_k(\xi){\mathcal{F}}{u}}_{X_k}.
\end{equation}
From the fact
\[
\eta_k(\xi){\mathcal{F}}_x{u}(t)= \sum_{j\in \mathbb{Z}_+}\int_{\mathbb R}
\eta_j(\tau-\xi^3)\eta_k(\xi){\mathcal{F}}(u)(\tau)e^{it\tau}d\tau,
\]
we easily see that \eqref{eq:s42} follows from the Minkowski's
inequality, Cauchy-Schwarz inequality and the definition of $X_k$.
\end{proof}
We prove an embedding property of the space $N^s$ in the next
proposition which can be viewed as a dual version of Proposition
\ref{p41}. This property is important in proving the limit behavior
in Section 6.
\begin{proposition}\label{p42}
If $s\in {\mathbb R}$ and $u\in L_t^2H_x^s$, then
\begin{equation}
\norm{u}_{N^s}{\lesssim} \norm{u}_{L_t^2H_x^s}.
\end{equation}
\begin{proof}
We may assume $s=0$. By definition it suffices to prove that for
$k\in {\mathbb Z}_+$,
\begin{equation}
\norm{(i+\tau-\xi^3)^{-1}\eta_k(\xi){\mathcal{F}}(u)}_{X_k}{\lesssim}
\norm{\eta_k(\xi){\mathcal{F}}(u)}_{L^2},
\end{equation}
which immediately follows from the definition of $X_k$.
\end{proof}
\end{proposition}
As in the last section we will mainly work on the correspondng
integral equation of eq. \eqref{eq:kdvb}. But for technical reason
we will mainly work on the following integral equation
\begin{equation}\label{eq:trunintekdvbuni}
u(t)=\psi(t)\left[W_\epsilon^\alpha(t)\phi_1-L \big(\partial_x
(\psi^2 u^2)\big)(x,t) \right],
\end{equation}
where $\psi$ is as in \eqref{eq:cutoff} and
\begin{eqnarray} \label{defL}
L(f)(x,t)=W_0(t)\int_{{\mathbb R}^2}e^{ix\xi}\frac{e^{it\tau'}-e^{-\epsilon
|t||\xi|^{2\alpha}}}{i\tau'+\epsilon
|\xi|^{2\alpha}}{\mathcal{F}}(W_0(-t)f)(\xi,\tau')d\xi d\tau'.
\end{eqnarray}
One easily sees that
\begin{eqnarray}
\chi_{{\mathbb R}_+}(t) \psi(t)L(f)(x,t)&=&\chi_{{\mathbb R}_+}(t)\psi(t)\int_0^t
W_\epsilon^\alpha(t-\tau) f(\tau)d\tau. \label{eq:nonhomo}
\end{eqnarray}
Indeed, taking $w=W_0(\cdot)f$, the right hand side of
\eqref{eq:nonhomo} can be rewritten as
\begin{eqnarray*}
&& W_0(t)\left[\chi_{{\mathbb R}_+}(t)\psi(t)\int_{{\mathbb R}^2}
e^{ix\xi}e^{-\epsilon t |\xi|^{2\alpha}}\widehat{w}(\xi,
\tau')\int_{0}^t e^{i\tau\tau'}e^{e^{\epsilon \tau
|\xi|^{2\alpha}}}d\tau d\xi
d\tau'\right]\\
&& =W_0(t)\left[\chi_{{\mathbb R}_+}(t)\psi(t)\int_{{\mathbb R}^2}
e^{ix\xi}\frac{e^{it\tau'}-e^{-\epsilon t
|\xi|^{2\alpha}}}{i\tau'+\epsilon|\xi|^{2\alpha}}\widehat{w}(\xi,
\tau') d\xi d\tau'\right].
\end{eqnarray*}
Thus, if $u$ solves \eqref{eq:trunintekdvbuni} then $u$ is a
solution of \eqref{eq:intekdvb} on $[0, 1]$. We first prove a
uniform estimate for the free solution.
\begin{proposition}\label{p43}
Let $s\in {\mathbb R}$. There exists $C>0$ such that for any $0\leq
\epsilon\leq 1$
\begin{equation}
\norm{\psi(t)W_\epsilon^\alpha(t)\phi}_{F^s}\leq C\norm{\phi}_{H^s},
\quad \forall \ \phi \in H^s({\mathbb R}).
\end{equation}
\end{proposition}
\begin{proof}
We only prove the case $0<\epsilon\leq 1$. By definition of $F^s$,
it suffices to prove that for $k\in {\mathbb Z}_+$
\begin{equation}
\norm{\eta_k(\xi){\mathcal{F}}(\psi(t)W_\epsilon^\alpha(t)\phi)}_{X_k}{\lesssim}
\norm{\eta_k(\xi)\widehat{\phi}(\xi)}_{L^2}. \label{eq:4.7}
\end{equation}
In view of the definition, if $k=0$, then by Taylor's expansion
\begin{eqnarray*}
&&\norm{\eta_0(\xi){\mathcal{F}}(\psi(t)W_\epsilon^\alpha(t)\phi)}_{X_0}\\
&{\lesssim}&
\sum_{j=0}^{\infty}2^{j/2}\normo{\eta_0(\xi)\widehat{\phi}(\xi){\mathcal{F}}_t\brk{\psi(t)\sum_{n\geq 0}\frac{(-1)^n\epsilon^n |\xi|^{2n\alpha}}{n!}|t|^n}(\tau)\eta_j(\tau)}_{L_{\xi,\tau}^2}\\
&{\lesssim}& \sum_{n\geq
0}\frac{4^n}{n!}\norm{\eta_0(\xi)\widehat{\phi}(\xi)}_{L^2}\norm{|t|^n\psi(t)}_{H^1}
{\lesssim}\norm{\eta_0(\xi)\widehat{\phi}(\xi)}_{L^2},
\end{eqnarray*}
which is the estimate \eqref{eq:4.7}, as desired. We now consider
the cases $k\geq 1$. We first observe that if $|\xi|\sim 2^k$, then
for any $ j\geq 0$,
\begin{equation}\label{eq:obser}
\norm{P_j(e^{-\epsilon |\xi|^{2\alpha}|t|})(t)}_{L^2}{\lesssim}
\norm{P_j(e^{-\epsilon 2^{2k\alpha}|t|})(t)}_{L^2},
\end{equation}
which follows from Plancherel's equality and the fact that
\[{\mathcal{F}}(e^{-|t|})(\tau)=C\rev{1+|\tau|^2}.\]
It follows from the definition that
\begin{eqnarray*}
\norm{\eta_k(\xi){\mathcal{F}}(\psi(t)W_\epsilon^\alpha(t)\phi)}_{X_k}&{\lesssim}&\sum_{j=0}^\infty
2^{j/2}
\normo{\eta_k(\xi)\widehat{\phi}(\xi)\eta_j(\tau){\mathcal{F}}_t\brk{\psi(t)e^{-\epsilon
|t||\xi|^{2\alpha}}}(\tau)}_{L_{\xi,\tau}^2}\\
&{\lesssim}&\sum_{j=0}^\infty 2^{j/2}
\normo{\eta_k(\xi)\widehat{\phi}(\xi)P_j\brk{\psi(t)e^{-\epsilon
|t||\xi|^{2\alpha}}}(t)}_{L_{\xi,t}^2}\\
&{\lesssim}&\sum_{j=0}^\infty 2^{j/2}
\normo{\eta_k(\xi)\widehat{\phi}(\xi)}_{L^2} \sup_{|\xi|\sim
2^k}\normo{P_j\brk{\psi(t)e^{-\epsilon
|t||\xi|^{2\alpha}}}(t)}_{L_t^2}.
\end{eqnarray*}
It suffices to show that for any $k\geq 1$,
\begin{eqnarray}\label{eq:p442}
\sum_{j=0}^\infty 2^{j/2} \sup_{|\xi|\sim 2^k}
\normo{P_j\brk{\psi(t)e^{-\epsilon
|t||\xi|^{2\alpha}}}(t)}_{L_t^2}{\lesssim} 1.
\end{eqnarray}
We may assume $j\geq 100$ in the summation. Using the para-product
decomposition, we have
\begin{align}
u_1u_2 = \sum^\infty_{r=0} [(P_{r+1} u_1) (P_{\le r+1}u_2)+ (P_{\le
r}u_1) (P_{r+1}u_2)], \label{eq:para-pr}
\end{align}
and
\begin{align}
P_j(u_1u_2) = P_j \Big(\sum_{r\ge j-10} [(P_{r+1} u_1) (P_{\le
r+1}u_2)+ (P_{\le r}u_1) (P_{r+1}u_2)]\Big):=P_j(I+II).
\label{eq:para-pr-2}
\end{align}
Now we take $u_1=\psi(t)$ and $u_2= e^{-\epsilon
|t||\xi|^{2\alpha}}$. It follows from Bernstein's estimate,
H\"older's inequality and \eqref{eq:obser} that
\begin{align}
\sum_{j \ge 100} 2^{j/2} \|P_j(II)\|_{L^\infty_\xi L^2_t} & \lesssim
\sum_{j \ge 100} 2^{j/2} \sum_{r\ge j-10} \| P_{r+1}
u_2\|_{L^{\infty}_\xi L^{2}_t}
\|P_{\le r+1}u_1 \|_{L^\infty_{\xi,t}} \nonumber\\
& \lesssim \sum_{j\ge 100} 2^{(j-r)/2} \sum_{r\ge j-10} 2^{r/2} \|
P_{r+1} u_2\|_{L^{\infty}_\xi L^{2}_t}
\nonumber\\
& \lesssim \sum_{r} 2^{r/2} \| P_{r+1} (e^{-\epsilon
|t|2^{2k\alpha}})\|_{L^{2}_t} {\lesssim} 1,
\end{align}
where we used the fact that $\dot{B}_{2,1}^{1/2}$ has a scaling
invariance and $e^{-|t|}\in\dot{B}_{2,1}^{1/2}$. the first term
$P_j(I)$ in \eqref{eq:para-pr-2} can be handled in an easier way.
Therefore, we complete the proof of the proposition.
\end{proof}
From the proof we see that $F^s$ norm has a same scale in time as
$B^{1/2}_{2,1}$ and $e^{-\epsilon C |t|}$. If applying $X^{1/2+,s}$
norm, one can not get a uniform estimate. Similarly for the
inhomogeneous linear operator we get
\begin{proposition}\label{p45}
Let $s\in {\mathbb R}$. There exists $C>0$ such that for all $v \in
{\mathcal{S}}({\mathbb R}^2)$ and $0\leq \epsilon\leq 1$,
\begin{equation}
\norm{\psi(t)L(v)}_{F^s}\leq C \norm{v}_{N^s}.
\end{equation}
\end{proposition}
\begin{proof}
The idea is essential due to Molinet and Ribaud \cite{MR}. See also
section 5 in \cite{In-Ke}. We only prove the case $0<\epsilon\leq
1$. In view of definition, it suffices to prove that if $k\in {\mathbb Z}_+$,
\begin{eqnarray}
\norm{\eta_k(\xi){\mathcal{F}}(\psi(t)L(v))}_{X_k} {\lesssim}
\norm{(i+\tau-\xi^3)^{-1}\eta_k(\xi){\mathcal{F}}(v)}_{X_k}.
\end{eqnarray}
We set \[w(\tau)=W_0(-\tau)v(\tau), \quad k_\xi(t)=\psi(t)\int_{\mathbb R}
\frac{e^{it\tau'}-e^{-\epsilon t
|\xi|^{2\alpha}}}{i\tau'+\epsilon|\xi|^{2\alpha}}\widehat{w}(\xi,
\tau') d\tau',\]
Therefore, by the definition, it suffices to prove
that
\begin{equation}\label{eq:p452}
\sum_{j=0}2^{j/2}\norm{\eta_k(\xi)\eta_j(\tau){\mathcal{F}}_t(k_\xi)(\tau)}_{L_{\xi,\tau}^2}
{\lesssim} \sum_{j=0}2^{-j/2}\norm{\eta_k(\xi)\eta_j(\tau)\widehat{w}(\xi,
\tau)}_{L_{\xi,\tau}^2}.
\end{equation}
We first write
\begin{eqnarray*}
k_\xi(t)&=&\psi(t)\int_{|\tau|\leq
1}\frac{e^{it\tau}-1}{i\tau+\epsilon|\xi|^{2\alpha}}\widehat{w}(\xi,\tau)d\tau+\psi(t)\int_{|\tau|\leq
1}\frac{1-e^{-\epsilon
|t||\xi|^{2\alpha}}}{i\tau+\epsilon|\xi|^{2\alpha}}\widehat{w}(\xi,\tau)d\tau\\
&&+\psi(t)\int_{|\tau|\geq
1}\frac{e^{it\tau}}{i\tau+\epsilon|\xi|^{2\alpha}}\widehat{w}(\xi,\tau)d\tau-\psi(t)\int_{|\tau|\geq
1}\frac{e^{-\epsilon
|t||\xi|^{2\alpha}}}{i\tau+\epsilon|\xi|^{2\alpha}}\widehat{w}(\xi,\tau)d\tau\\
&=&I+II+III-IV.
\end{eqnarray*}
We now estimate the contributions of $I-IV$. First, we consider the
contribution of $IV$.
\begin{eqnarray*}
\sum_{j=0}2^{j/2}\norm{\eta_k(\xi)P_j(IV)(t)}_{L_{\xi,t}^2}&\leq&
\sum_{j=0}2^{j/2}\sup_{\xi\in I_k}\norm{\eta_k(\xi)P_j(\psi(t)e^{-\epsilon|t||\xi|^{2\alpha}})(t)}_{L_{t}^2}\nonumber\\
&&\quad \cdot\int_{|\tau|\geq 1}\frac{\norm{|\eta_k(\xi)\widehat{w}(\xi,\tau)|}_{L_\xi^2}}{|\tau|}d\tau\\
&{\lesssim}&\sum_{j=0}2^{-j/2}\norm{\eta_k(\xi)\eta_j(\tau)\widehat{w}(\xi,
\tau)}_{L_{\xi,\tau}^2},
\end{eqnarray*}
where we use Taylor expansion for $k=0$ and \eqref{eq:p442} for
$k\geq 1$. Next, we consider the contribution of $III$. Setting
$g(\xi,\tau)=\frac{|\widehat{w}(\xi,\tau)|}{|i\tau+\epsilon|\xi|^{2\alpha}|}\chi_{|\tau|\geq
1}$ we have
\begin{eqnarray*}
\sum_{j=0}2^{j/2}\norm{\eta_k(\xi)P_j(III)(t)}_{L_{\xi,t}^2}&{\lesssim}&\sum_{j=0}2^{j/2}\norm{\eta_k(\xi)\eta_j(\tau)\widehat{\psi}*_\tau g(\xi,\tau)}_{L_{\xi,\tau}^2}\\
&{\lesssim}&\sum_{j\geq 1} 2^{j/2}\normo{
\frac{\eta_j(\tau')\norm{\eta_k(\xi)\widehat{w}(\xi,\tau')}_{L_\xi^2}}{|i\tau'|}\chi_{|\tau'|\geq
1}}_{L_{\tau'}^2}\nonumber\\
&{\lesssim}&\sum_{j=0}2^{-j/2}\norm{\eta_k(\xi)\eta_j(\tau)\widehat{w}(\xi,
\tau)}_{L_{\xi,\tau}^2},
\end{eqnarray*}
where we used the fact that $B_{2,1}^{1/2}$ is a multiplication
algebra and that ${\mathcal{F}}^{-1}(|\widehat{\psi}|)\in B_{2,1}^{1/2}$.
Thirdly, we consider the contribution of $II$. For
$\epsilon|\xi|^{2\alpha}\geq 1$, as for $IV$, we get
\begin{eqnarray*}
\sum_{j=0}2^{j/2}\norm{\eta_k(\xi)P_j(II)(t)}_{L_{\xi,t}^2}&{\lesssim}&
\sum_{j=0}2^{j/2}\sup_{\xi\in I_k}\norm{\eta_k(\xi)P_j(\psi(1-e^{-\epsilon|t||\xi|^{2\alpha}}))(t)}_{L_{t}^2}\\
&&\cdot
\int\frac{\norm{\widehat{w}(\xi,\tau)}_{L_\xi^2}}{\jb{\tau}}d\tau
\nonumber\\
&{\lesssim}&
\sum_{j=0}2^{-j/2}\norm{\eta_k(\xi)\eta_j(\tau)\widehat{w}(\xi,
\tau)}_{L_{\xi,\tau}^2}.
\end{eqnarray*}
For $\epsilon|\xi|^{2\alpha}\leq 1$, using Taylor's expansion, we
have
\begin{eqnarray*}
&&\sum_{j=0}2^{j/2}\norm{\eta_k(\xi)P_j(II)(t)}_{L_{\xi,t}^2}\\
&{\lesssim}& \sum_{n\geq
1}\sum_{j=0}2^{j/2}\normo{\eta_k(\xi)\int_{|\tau|\leq
1}\frac{\widehat{w}(\xi,\tau)}{i\tau+\epsilon|\xi|^{2\alpha}}d\tau
P_j(|t|^n\psi(t))\frac{\epsilon^n|\xi|^{2\alpha
n}}{n!}}_{L_{\xi,t}^2} \\
&{\lesssim}& \normo{\int_{|\tau|\leq
1}\frac{\epsilon|\xi|^{2\alpha}|\eta_k(\xi)\widehat{w}(\xi,\tau)|}{|i\tau+\epsilon|\xi|^{2\alpha}|}d\tau}_{L_\xi^2}{\lesssim}\sum_{j=0}2^{-j/2}\norm{\eta_k(\xi)\eta_j(\tau)\widehat{w}(\xi,
\tau)}_{L_{\xi,\tau}^2},
\end{eqnarray*}
where in the last inequality we used the fact
$\norm{|t|^n\psi(t)}_{B^{1/2}_{2,1}}\leq\norm{|t|^n\psi(t)}_{H^{1}}\leq
C 2^n$. Finally, we consider the contribution of $I$.
\begin{eqnarray*}
I=\psi(t)\int_{|\tau|\leq 1}\sum_{n\geq
1}\frac{(it\tau)^n}{n!(i\tau+\epsilon|\xi|^{2\alpha})}\widehat{w}(\tau)d\tau.
\end{eqnarray*}
Thus, we get
\begin{eqnarray*}
&&\sum_{j=0}2^{j/2}\norm{\eta_k(\xi)P_j(I)(t)}_{L_{\xi,t}^2}\\
&{\lesssim}&
\sum_{n\geq 1}\normo{\frac{t^n\psi(t)}{n!}}_{B^{1/2}_{2,1}}
\normo{\int_{|\tau|\leq1}\frac{|\tau|}{|i\tau+\epsilon|\xi|^{2\alpha}|}|\eta_k(\xi)\widehat{w}(\xi,\tau)|d\tau}_{L_\xi^2}\\
&{\lesssim}&\sum_{j=0}2^{-j/2}\norm{\eta_k(\xi)\eta_j(\tau)\widehat{w}(\xi,
\tau)}_{L_{\xi,\tau}^2}.
\end{eqnarray*}
Therefore, we complete the proof of the proposition.
\end{proof}
In order to apply the standard fixed-point machinery, we next turn
to a bilinear estimate in $F^s$. The proof is divided into several
cases. We will use the estimate for the characterization multiplier
in Proposition \ref{pchar}. The first case is $low\times
high\rightarrow high$ interaction.
\begin{proposition}\label{punif}
If $k\geq 10$, $|k-k_2|\leq 5$, then for any $u\in F^s,\ v \in F^s$
\begin{equation}
\norm{(i+\tau-\xi^3)^{-1}\eta_k(\xi)i\xi\widehat{P_0u}*\widehat{P_{k_2}v}}_{X_k}{\lesssim}
\norm{\widehat{P_0u}}_{X_0}\norm{\widehat{P_{k_2}v}}_{X_{k_2}}.
\end{equation}
\end{proposition}
\begin{proof}
For simplicity of notation we only prove the case that $k=k_2$,
since the other cases can be handled in the same way. From
definition of $X_k$, we get
\begin{eqnarray}
\norm{(i+\tau-\xi^3)^{-1}\eta_k(\xi)i\xi\widehat{P_0u}*\widehat{P_kv}}_{X_k}\les2^k\sum_{j,j_1,j_2\geq
0}2^{-j/2}\norm{1_{D_{k,j}}u_{0,j_1}*v_{k,j_2}}_2,
\end{eqnarray}
where
\[u_{0,j_1}=\eta_0(\xi)\eta_{j_1}(\tau-\xi^3)\widehat{u},\
v_{k,j_2}=\eta_k(\xi)\eta_{j_2}(\tau-\xi^3)\widehat{v}.\] Thus, in
view of definition it suffices to show that
\begin{equation}\label{eq:biunicase1}
\norm{1_{D_{k,j}}u_{0,j_1}*v_{k,j_2}}_2{\lesssim} 2^{-k}
2^{(j_1+j_2)/2}\norm{u_{0,j_1}}_2\norm{v_{k,j_2}}_2.
\end{equation}
By duality and
$\xi_1^3+\xi_2^3-(\xi_1+\xi_2)^3=-3\xi_1\xi_2(\xi_1+\xi_2)$,
\eqref{eq:biunicase1} is equivalent to
\begin{eqnarray}
&&\aabs{\int \int u(\xi_1,\tau_1)v(\xi_2,\tau_2)g(\xi_1+\xi_2,
\tau_1+\tau_2-3\xi_1\xi_2(\xi_1+\xi_2))d\xi_1d\xi_2d\tau_1d\tau_2}\nonumber\\
&&{\lesssim} \ 2^{-k} 2^{(j_1+j_2)/2}\norm{u}_2\norm{v}_2 \norm{g}_2
\end{eqnarray}
for any $u,v,g\in L^2$ supported in $I_0\times I_{j_1}$, $I_k\times
I_{j_2}$, $I_k\times I_{j}$ respectively. Therefore, it suffices to
show that
\begin{eqnarray}\label{eq:biunicase12}
&&\int_{|\xi_1|\leq 2} \int_{|\xi_2|\sim 2^k}
u(\xi_1)v(\xi_2)g(\xi_1+\xi_2,
-3\xi_1\xi_2(\xi_1+\xi_2))d\xi_1d\xi_2\nonumber\\
& & {\lesssim} \ 2^{-k}\norm{u}_2\norm{v}_2 \norm{g}_2
\end{eqnarray}
for any $u,v,g\in L^2$ supported in $I_0$, $I_k$, $I_k\times
\widetilde{I}_{j_{max}}$ respectively where
$j_{max}=\max(j,j_1,j_2)$ and $\widetilde{I}_{j_{max}}=\cup_{l=-3}^3
I_{j_{max}+l}$.
Indeed, by changing the coordinates $\mu_1=\xi_1,\
\mu_2=\xi_1+\xi_2$, the left-side of \eqref{eq:biunicase12} is
bounded by
\begin{equation}\label{eq:biunicase13}
\int_{|\mu_1|\leq 2} \int_{|\mu_2|\sim 2^k}
u(\mu_1)v(\mu_2-\mu_1)g(\mu_2,
-3\mu_1(\mu_2-\mu_1)\mu_2)d\mu_1d\mu_2.
\end{equation}
Since in the integration area
\begin{equation}
\left|\frac{\partial}{\partial_{\mu_1}}[-3\mu_1(\mu_2-\mu_1)\mu_2]\right|\sim
2^{2k},
\end{equation}
then by Cauchy-Schwarz inequality we get
\begin{eqnarray}
\eqref{eq:biunicase13}&{\lesssim}& \norm{u}_2\norm{v}_2 \norm{g(\mu_2,
-3\mu_1(\mu_2-\mu_1)\mu_2)}_{L_{|\mu_1|\leq 2, |\mu_2|\sim 2^k}^2}\nonumber\\
&{\lesssim}&2^{-k} \norm{u}_2 \norm{v}_2 \norm{g}_2,
\end{eqnarray}
which completes the proof.
\end{proof}
\begin{proposition}
If $k\geq 10$, $|k-k_2|\leq 5$ and $1\leq k_1\leq k-9$. Then for any
$u, v\in F^s$
\begin{equation}
\norm{(i+\tau-\xi^3)^{-1}\eta_k(\xi)i\xi\widehat{P_{k_1}u}*\widehat{P_{k_2}v}}_{X_k}{\lesssim}\
k^32^{-k/2}2^{-k_1}
\norm{\widehat{P_{k_1}u}}_{X_{k_1}}\norm{\widehat{P_{k_2}v}}_{X_{k_2}}.
\end{equation}
\end{proposition}
\begin{proof}
We only prove the case $k=k_2$. From the definition, we get
\begin{eqnarray}
\norm{(i+\tau-\xi^3)^{-1}\eta_k(\xi)i\xi\widehat{P_{k_1}u}*\widehat{P_kv}}_{X_k}\les2^k\sum_{j,j_1,j_2\geq
0}2^{-j/2}\norm{1_{D_{k,j}}u_{k_1,j_1}*v_{k,j_2}}_2,
\end{eqnarray}
where
\[u_{k_1,j_1}=\eta_{k_1}(\xi)\eta_{j_1}(\tau-\xi^3)\widehat{u},\
v_{k,j_2}=\eta_k(\xi)\eta_{j_2}(\tau-\xi^3)\widehat{v}.\] By
checking the support properties of the functions $u_{k_1,j_1},\
v_{k_2,j_2}$ and using the fact that
$|\xi_1^3+\xi_2^3-(\xi_1+\xi_2)^3|\sim 2^{2k+k_1}$, we get that
$1_{D_{k,j}}u_{k_1,j_1}*v_{k,j_2}\equiv 0$ unless $j_{max}\geq
2k+k_1-10$. Using \eqref{eq:charii}, we get
\begin{eqnarray}
&&2^k\sum_{j,j_1,j_2\geq 0}2^{-j/2}\norm{1_{D_{k,j}}u_{k_1,j_1}*v_{k,j_2}}_2\nonumber\\
&&{\lesssim} \ 2^k\sum_{j,j_1,j_2\geq 0}2^{-j/2}2^{j_{min}/2}2^{-k/2}2^{-k_1/2}2^{j_{med}/2}\norm{u_{k_1,j_1}}_2\norm{v_{k,j_2}}_2\nonumber\\
&&{\lesssim} \ 2^k\sum_{j_{max}\geq
2k+k_1-10}k^32^{-k/2}2^{-k_1/2}2^{-j_{max}/2}\norm{\widehat{P_{k_1}u}}_{X_{k_1}}\norm{\widehat{P_kv}}_{X_k}\nonumber\\
&&{\lesssim} \
k^32^{-k/2}2^{-k_1}\norm{\widehat{P_{k_1}u}}_{X_{k_1}}\norm{\widehat{P_kv}}_{X_k},
\end{eqnarray}
which completes the proof of the proposition.
\end{proof}
The second case is $high\times high \rightarrow low$. This case is
the worst and where the condition is imposed. This is easy to be
seen, since $s\leq 0$ and $\norm{u}_{F^s}, \norm{v}_{F^s}$ are
small for $u,v$ with very high frequency.
\begin{proposition}
If $k\geq 10$, $|k-k_2|\leq 5$, then for any $u,\ v \in F^s$
\begin{equation}\label{eq:biunicase2}
\norm{(i+\tau-\xi^3)^{-1}\eta_0(\xi)i\xi\widehat{P_{k}u}*\widehat{P_{k_2}v}}_{X_0}{\lesssim}
\ k^32^{-3k/2}
\norm{\widehat{P_{k}u}}_{X_{k}}\norm{\widehat{P_{k_2}v}}_{X_{k_2}}.
\end{equation}
\end{proposition}
\begin{proof}
As before we assume $k=k_2$. From the definition, we get
\begin{eqnarray}
\norm{(i+\tau-\xi^3)^{-1}\eta_0(\xi)i\xi\widehat{P_{k}u}*\widehat{P_kv}}_{X_0}{\lesssim}\sum_{k'=-\infty}^0
2^{k'}\sum_{j,j_1,j_2=0}2^{-j/2}\norm{1_{D_{k',j}}u_{k,j_1}*v_{k,j_2}}_2,
\end{eqnarray}
where
\begin{equation}\label{eq:ukj}
u_{k,j_1}=\eta_{k}(\xi)\eta_{j_1}(\tau-\xi^3)\widehat{u},\
v_{k,j_2}=\eta_k(\xi)\eta_{j_2}(\tau-\xi^3)\widehat{v}.
\end{equation}
We may
assume that $k'\geq -10k$ and $j,j_1,j_2\leq 10k$. Otherwise, from
the following simple estimate which follows from H\"older's
inequality and Young's inequality
\[\norm{1_{D_{k',j}}u_{k,j_1}*v_{k,j_2}}_2{\lesssim} 2^{j_{min}/2}2^{k'/2}\norm{u_{k,j_1}}_2\norm{v_{k,j_2}}_2\]
we immediately obtain \eqref{eq:biunicase2}. For the same reason as
in the proof of last proposition, we see that $j_{max}\geq
2k+k'-10$. Using \eqref{eq:charii}, we get
\begin{eqnarray}
&&\norm{(i+\tau-\xi^3)^{-1}\eta_0(\xi)i\xi\widehat{P_{k}u}*\widehat{P_kv}}_{X_0}\nonumber\\
&&{\lesssim} \ \sum_{k'=-10k}^0
2^{k'}\sum_{j,j_1,j_2\geq 0}2^{-j/2}\norm{1_{D_{k',j}}u_{k,j_1}*v_{k,j_2}}_2\nonumber\\
&&{\lesssim} \ \sum_{k'=-10k}^0\sum_{j,j_1,j_2\geq 0}2^{-j/2}
2^{k'}2^{j_{min}/2}2^{-k/2}2^{-k'/2}2^{j_{med}/2}\norm{u_{k,j_1}}_2\norm{v_{k,j_2}}_2\nonumber\\
&&{\lesssim} \ \sum_{k'=-10k}^0\sum_{j_{max}\geq
2k+k'}k^22^{-k/2}2^{k'/2}2^{-j_{max}/2}\norm{\widehat{P_{k}u}}_{X_{k}}\norm{\widehat{P_kv}}_{X_k}\nonumber\\
&&{\lesssim} \
k^32^{-3k/2}\norm{\widehat{P_{k}u}}_{X_{k}}\norm{\widehat{P_kv}}_{X_k}.
\end{eqnarray}
Therefore, we complete the proof of the proposition.
\end{proof}
\begin{proposition}
If $k\geq 10$,$|k-k_2|\leq 5$ and $1\leq k_1\leq k-9$, then for any
$u,\ v \in F^s$
\begin{equation}
\norm{(i+\tau-\xi^3)^{-1}\eta_{k_1}(\xi)i\xi\widehat{P_{k}u}*\widehat{P_{k_2}v}}_{X_{k_1}}{\lesssim}
\ k^32^{-3k/2}
\norm{\widehat{P_{k}u}}_{X_{k}}\norm{\widehat{P_{k_2}v}}_{X_{k_2}}.
\end{equation}
\end{proposition}
\begin{proof}
As before we assume $k=k_2$. From the definition of $X_{k_1}$, we
get
\begin{eqnarray}\label{eq:highhigh2}
\norm{(i+\tau-\xi^3)^{-1}\eta_{k_1}(\xi)i\xi\widehat{P_{k}u}*\widehat{P_kv}}_{X_{k_1}}{\lesssim}
\ 2^{k_1}\sum_{j,j_1,j_2\geq
0}2^{-j/2}\norm{1_{D_{k_1,j}}u_{k,j_1}*v_{k,j_2}}_2,
\end{eqnarray}
where $u_{k,j_1},v_{k,j_2}$ are as in \eqref{eq:ukj}. For the same
reason as before we have $j_{max}\geq 2k+k_1-10$ and we may assume
$j,j_1,j_2\leq 10k$. It follows from \eqref{eq:charii} that the
right-hand side of \eqref{eq:highhigh2} is bounded by
\begin{eqnarray*}
&&\sum_{j,j_1,j_2\geq 0}2^{-j/2}
2^{k_1}2^{j_{min}/2}2^{-k/2}2^{-{k_1}/2}2^{j_{med}/2}\norm{u_{k,j_1}}_2\norm{v_{k,j_2}}_2\nonumber\\
&&{\lesssim} \ \sum_{j_{max}\geq
2k+k_1}k^22^{-k/2}2^{{k_1}/2}2^{-j_{max}/2}\norm{\widehat{P_{k}u}}_{X_{k}}\norm{\widehat{P_kv}}_{X_k}{\lesssim}
\
k^32^{-3k/2}\norm{\widehat{P_{k}u}}_{X_{k}}\norm{\widehat{P_kv}}_{X_k}.
\end{eqnarray*}
Therefore we complete the proof of the proposition.
\end{proof}
\begin{proposition}
If $k\geq 10$, $|k-k_2|\leq 5$ and $k-9\leq k_1\leq k+10$, then for
any $u,\ v \in F^s$
\begin{equation}
\norm{(i+\tau-\xi^3)^{-1}\eta_{k_1}(\xi)i\xi\widehat{P_{k}u}*\widehat{P_{k_2}v}}_{X_{k_1}}{\lesssim}
\ k^32^{-3k/4}
\norm{\widehat{P_{k}u}}_{X_{k}}\norm{\widehat{P_{k_2}v}}_{X_{k_2}}.
\end{equation}
\end{proposition}
\begin{proof}
As before we assume $k=k_2$. From the definition of $X_{k_1}$, we
get
\begin{eqnarray}\label{eq:highhigh3}
\norm{(i+\tau-\xi^3)^{-1}\eta_{k_1}(\xi)i\xi\widehat{P_{k}u}*\widehat{P_kv}}_{X_{k_1}}{\lesssim}
\ 2^{k_1}\sum_{j,j_1,j_2\geq
0}2^{-j/2}\norm{1_{D_{k_1,j}}u_{k,j_1}*v_{k,j_2}}_2,
\end{eqnarray}
where $u_{k,j_1},v_{k,j_2}$ are as in \eqref{eq:ukj}. For the same
reason as before we have $j_{max}\geq 2k+k_1-10$ and we may assume
$j,j_1,j_2\leq 10k$. It follows from \eqref{eq:chari} that the
right-hand side of \eqref{eq:highhigh3} is bounded by
\begin{eqnarray*}
&&\sum_{j,j_1,j_2\geq 0}2^{-j/2}
2^{k_1}2^{j_{min}/2}2^{-k/4}2^{j_{med}/4}\norm{u_{k,j_1}}_2\norm{v_{k,j_2}}_2{\lesssim}
\
k^32^{-3k/4}\norm{\widehat{P_{k}u}}_{X_{k}}\norm{\widehat{P_kv}}_{X_k},
\end{eqnarray*}
which completes the proof of the proposition.
\end{proof}
The final case is $low\times low {\rightarrow} low$ interaction. Generally
speaking, this case is always easy to handle in many situations.
\begin{proposition}\label{punil}
If $0\leq k_1,k_2,k_3\leq 100$, then for any $u,\ v \in F^s$
\begin{equation}
\norm{(i+\tau-\xi^3)^{-1}\eta_{k_1}(\xi)i\xi\widehat{P_{k_2}u}*\widehat{P_{k_3}v}}_{X_{k_1}}{\lesssim}
\norm{\widehat{P_{k_2}u}}_{X_{k_2}}\norm{\widehat{P_{k_3}v}}_{X_{k_3}}.
\end{equation}
\end{proposition}
\begin{proof}
From the definition of $X_{k_1}$, we get that
\begin{eqnarray}\label{eq:highhigh3}
\norm{(i+\tau-\xi^3)^{-1}\eta_{k_1}(\xi)i\xi\widehat{P_{k_2}u}*\widehat{P_{k_3}v}}_{X_{k_1}}{\lesssim}
\ 2^{k_1}\sum_{j,j_1,j_2\geq
0}2^{-j/2}\norm{1_{D_{k_1,j}}u_{k_2,j_1}*v_{k_3,j_2}}_2,
\end{eqnarray}
where $u_{k_2,j_1},v_{k_3,j_2}$ are as in \eqref{eq:ukj}. By
checking the support properties of the function
$u_{k_2,j_1},v_{k_3,j_2}$, we get that
$1_{D_{k_1,j}}u_{k_2,j_1}*v_{k_3,j_2}\equiv 0$ unless
$|j_{max}-j_{med}|\leq 10$ or $j_{max}\leq 1000$ where $j_{max},
j_{med}$ are the maximum and median of $j,j_1,j_2$ respectively. It
follows immediately from Young's inequality that
\begin{equation}
\norm{1_{D_{k,j}} u_{k_1,j_1}*v_{k_2,j_2}}_{L_{\xi,\tau}^2}{\lesssim}
2^{k_i}2^{j_i}\norm{u_{k_1,j_1}}_2\norm{v_{k_2,j_2}}_2,\ i=1,2.
\end{equation}
From definition and summing in $j_i$, we complete the proof of the
proposition.
\end{proof}
With these propositions in hand, we are able to prove the bilinear
estimate. The idea is to decompose the bilinear product using
para-product, and then divide it into many cases according to the
interactions. Finally we use discrete Young's inequality.
\begin{proposition}\label{p412}
Fix any $s\in (-3/4, 0]$, $\forall s\leq \sigma \leq 0$, there
exists $C>0$ such that for any $u,v\in F^\sigma$,
\begin{equation}\label{eq:biuni}
\norm{\partial_x(uv)}_{N^\sigma}\leq
C(\norm{u}_{F^s}\norm{v}_{F^\sigma}+\norm{v}_{F^s}\norm{u}_{F^\sigma}).
\end{equation}
\end{proposition}
\begin{proof}
In view of definition, we get that
\begin{equation}
\norm{\partial_x(uv)}_{N^\sigma}^2=\sum_{k_3\in {\mathbb Z}_+}2^{2\sigma
k_3}\norm{(i+\tau-\xi^3)^{-1}\eta_{k_3}(\xi)i\xi\widehat{u}*\widehat{v}}_{X_{k_3}}^2.
\end{equation}
We decompose $\widehat{u}, \widehat{v}$ and get
\begin{eqnarray}\label{eq:biunipara}
\norm{(i+\tau-\xi^3)^{-1}\eta_{k_3}(\xi)i\xi\widehat{u}*\widehat{v}}_{X_{k_3}}{\lesssim}
\sum_{k_1,k_2\in {\mathbb Z}_+}
\norm{(i+\tau-\xi^3)^{-1}\eta_{k_3}(\xi)i\xi\widehat{P_{k_1}u}*\widehat{P_{k_2}v}}_{X_{k_3}}.
\end{eqnarray}
By checking the support properties we get that
$\eta_{k_3}(\xi)\widehat{P_{k_1}u}*\widehat{P_{k_2}v}\equiv 0$
unless $|k_{max}-k_{med}|\leq 5$ where $k_{max}, k_{med}$ are the
maximum and median of $k_1,k_2,k_3$ respectively. We may assume that
$k_1\leq k_2$ from symmetry. By dividing the summation into
$high\times high$, $high\times low$ four parts, we get that the
right-hand side of \eqref{eq:biunipara} is bounded by
\begin{eqnarray}
\big(\sum_{j=1}^4\sum_{k_1,k_2\in A_j}\big)
\norm{(i+\tau-\xi^3)^{-1}\eta_{k_3}(\xi)i\xi\widehat{P_{k_1}u}*\widehat{P_{k_2}v}}_{X_{k_3}},
\end{eqnarray}
where $A_j$, $j=1,2,3,4$ are defined by
\begin{eqnarray*}
&&A_1=\{k_2\geq 10, |k_2-k_3|\leq 5, k_1\leq k_2-10\};\\
&&A_2=\{k_2\geq 10, |k_2-k_3|\leq 5, k_2-9\leq k_1\leq k_2+10\};\\
&&A_3=\{k_2\geq 10, |k_2-k_1|\leq 5, k_3\leq k_1-10\};\\
&&A_4=\{k_1,k_2,k_3\leq 100\}.
\end{eqnarray*}
Therefore, \eqref{eq:biuni} from the Proposition
\ref{punif}-\ref{punil}, discrete Young's inequality and the
assumption that $s>-3/4$.
\end{proof}
We next show \eqref{eq:kdvb} is uniformly (on $0<\epsilon \leq 1$)
locally well-posed in $H^s$, $-3/4<s\leq 0$. The procedure is quite
standard. See \cite{KPV}, for instance. By the scaling
\eqref{eq:scaling}, we see that $u$ solves \eqref{eq:kdvb} if and
only if $u_\lambda (x,t)= \lambda^2 u(\lambda x, \lambda^3 t)$
solves
\begin{eqnarray}\label{eq:kdvb-m}
\partial_t u_\lambda + \partial^3_x u_{\lambda}+\epsilon \lambda^{3-2\alpha} |\partial_x|^{2\alpha}u_\lambda+\partial_x (u^2_\lambda)=0, \ \
u_\lambda (0)= \lambda^2\phi (\lambda \, \cdot).
\end{eqnarray}
Since $-3/4<s\leq 0$,
\begin{equation}
\norm{\lambda^2\phi(\lambda
x)}_{H^s}=O(\lambda^{3/2+s}\norm{\phi}_{H^s}) \quad \mbox{as }
\lambda\rightarrow 0,
\end{equation}
thus we can first restrict ourselves to considering \eqref{eq:kdvb}
with data $\phi$ satisfying
\begin{equation}
\norm{\phi}_{H^s}=r\ll 1.
\end{equation}
As in the last section, we will mainly work on the integral equation
\eqref{eq:trunintekdvbuni}. We define the operator
\begin{equation}
\Phi_{\phi}(u)=\psi(t)W_\epsilon^\alpha(t)\phi- \psi(t)L
\big(\partial_x (\psi^2 u^2)\big),
\end{equation}
where $L$ is defined by \eqref{defL}. We will prove that $\Phi_\phi
(\cdot)$ is a contraction mapping from
\begin{equation}
\mathcal{B}=\{w\in F^s:\ \norm{w}_{F^s}\leq 2cr\}
\end{equation}
into itself. From Propositions \ref{p42}, \ref{p43} and \ref{p45} we
get if $w\in \mathcal{B}$, then
\begin{eqnarray}
\norm{\Phi_\phi(w)}_{F^s}&\leq&
c\norm{\phi}_{H^s}+\norm{\partial_x(\psi(t)^2w^2(\cdot, t))}_{N^s}\nonumber\\
&\leq& cr+c\norm{w}_{F^s}^2\leq cr+c(2cr)^2\leq 2cr,
\end{eqnarray}
provided $r$ satisfies $4c^2r\leq 1/2$. Similarly, for $w, h\in
\mathcal{B}$
\begin{eqnarray}
\norm{\Phi_\phi(w)-\Phi_\phi(h)}_{F^s}
&\leq& c\normo{ L \partial_x(\psi^2(\tau)(u^2(\tau)-h^2(\tau)))}_{F^s}\nonumber\\
&\leq&c\norm{w+h}_{F^s}\norm{w-h}_{F^s}\nonumber\\
&\leq&4c^2r\norm{w-h}_{F^s}\leq \frac{1}{2}\norm{w-h}_{F^s}.
\end{eqnarray}
Thus $\Phi_\phi(\cdot)$ is a contraction. There exists a unique
$u\in \mathcal{B}$ such that
\begin{equation}
u=\psi(t)W_\epsilon^\alpha(t)\phi- \psi(t)L \big(\partial_x (\psi^2
u^2)\big).
\end{equation}
Hence $u$ solves the integral equation \eqref{eq:intekdvb} in the
time interval $[0,1]$.
We prove now that $u\in X^{1/2,s,\alpha}$. Indeed, from the slightly
modified argument as the proof for Proposition 2.1, 2.3 \cite{MR},
we can show that
\begin{eqnarray*}
&&\norm{\psi(t)W_\epsilon^\alpha(t)\phi}_{X^{1/2,s,\alpha}}{\lesssim}
\norm{\phi}_{H^s};\\
&&\norm{\psi(t)L(v)}_{X^{1/2,s,\alpha}}{\lesssim}
\norm{v}_{X^{-1/2,s,\alpha}}+\brk{\int\jb{\xi}^{2s}\big(\int
\frac{|\widehat{v}(\tau)|}{\jb{i\tau+\epsilon|\xi|^{2\alpha}}}d\tau\big)^2d\xi}^{1/2}{\lesssim}
\norm{v}_{N^s},
\end{eqnarray*}
which then imply $u\in X^{1/2,s,\alpha}$, as desired. For general
$\phi \in H^s$, by using the scaling \eqref{eq:scaling} and the
uniqueness in Theorem \ref{t11}, we immediately obtain that Theorem
\ref{t12} holds for a small $T=T(\norm{\phi}_{H^s})>0$.
\section{Uniform global well-posedness for KdV-B equation}
In this section we will extend the uniform local solution obtained
in the last section to a uniform global solution. The standard way
is to use conservation law. Let $u$ be a smooth solution of
\eqref{eq:kdvb}, multiply $u$ and integrate, then we get
\begin{equation}\label{eq:L2law}
\half{1}\norm{u(t)}_2^2+\epsilon \int_0^t\norm{\Lambda^\alpha
u(\tau)}_2^2d\tau=\half{1}\norm{\phi}_2^2.
\end{equation}
By a standard limit argument, \eqref{eq:L2law} holds for
$L^2$-strong solution. Thus if $\phi\in L^2$, then we get that
\eqref{eq:kdvb} is uniformly globally well-posed.
For $\phi\in H^s$ with $-3/4<s<0$, there is no such conservation
law. We will follow the idea in \cite{Tao2} (I-method) to extend the
solution. Let $m: {\mathbb R}^k \rightarrow {\mathbb C}$ be a function. We say $m$ is
symmetric if $m(\xi_1,\ldots, \xi_k)=m(\sigma(\xi_1,\ldots, \xi_k))$
for all $\sigma \in S_k$, the group of all permutations on $k$
objects. The symmetrization of $m$ is the function
\begin{equation}
[m]_{sym}(\xi_1,\xi_2,\ldots, \xi_k)=\rev{k!}\sum_{\sigma\in
S_k}m(\sigma(\xi_1,\xi_2,\ldots,\xi_k)).
\end{equation}
We define a $k-linear$ functional associated to the multiplier $m$
acting on $k$ functions $u_1,\ldots,u_k$,
\begin{equation}
\Lambda_k(m;u_1,\ldots,u_k)=\int_{\xi_1+\ldots+\xi_k=0}m(\xi_1,\ldots,\xi_k)\widehat{u_1}(\xi_1)\ldots
\widehat{u_k}(\xi_k).
\end{equation}
We will often apply $\Lambda_k$ to $k$ copies of the same function
$u$. $\Lambda_k(m;u,\ldots,u)$ may simply be written $\Lambda_k(m)$.
By the symmetry of the measure on hyperplane, we have
$\Lambda_k(m)=\Lambda_k([m]_{sym})$.
The following statement may be directly verified by using the KdV-B
equation \eqref{eq:kdvb}. Compared to the KdV equation, the KdV-B
equation has one more term caused by the dissipation.
\begin{proposition}
Suppose $u$ satisfies the KdV-B equation \eqref{eq:kdvb} and that
$m$ is a symmetric function. Then
\begin{equation}\label{eq:menergy}
\frac{d}{dt}\Lambda_k(m)=\Lambda_k(mh_k)-\epsilon
\Lambda_k(m\beta_{\alpha,k})-i\half k
\Lambda_{k+1}(m(\xi_1,\ldots,\xi_{k-1},\xi_k+\xi_{k+1})(\xi_k+\xi_{k+1})),
\end{equation}
where
\[h_k=i(\xi_1^3+\xi_2^3+\ldots+\xi_k^3), \quad \beta_{\alpha,k}=|\xi_1|^{2\alpha}+|\xi_2|^{2\alpha}+\ldots+|\xi_k|^{2\alpha}.\]
\end{proposition}
We follow the I-method \cite{Tao2} to define a set of modified
energies. Let $m:{\mathbb R}\rightarrow {\mathbb R}$ be an arbitrary even ${\mathbb R}$-valued
function and define the operator by
\begin{equation}
\widehat{If}(\xi)=m(\xi)\widehat{f}(\xi).
\end{equation}
We define the modified energy $E_I^2(t)$ by
\begin{equation}
E_I^2(t)=\norm{Iu(t)}_{L^2}^2.
\end{equation}
By Plancherel and the fact that $m$ and $u$ are ${\mathbb R}$-valued, and $m$
is even,
\[E_I^2(t)=\Lambda_2(m(\xi_1)m(\xi_2)).\]
Using \eqref{eq:menergy}, we have
\begin{eqnarray}
\frac{d}{dt}E_I^2(t)&=&\Lambda_2(m(\xi_1)m(\xi_2)h_2)-\epsilon\Lambda_2(m(\xi_1)m(\xi_2)\beta_{\alpha,2})\nonumber\\
&&-i\Lambda_3(m(\xi_1)m(\xi_2+\xi_3)(\xi_2+\xi_3)).
\end{eqnarray}
The first term vanishes. The second term is non-positive, hence
good. We symmetrize the third term to get
\begin{equation}
\frac{d}{dt}E_I^2(t)=-\epsilon\Lambda_2(m(\xi_1)m(\xi_2)\beta_{\alpha,2})+\Lambda_3(-i[m(\xi_1)m(\xi_2+\xi_3)(\xi_2+\xi_3)]_{sym}).
\end{equation}
Let us denote
\begin{equation}
M_3(\xi_1,\xi_2,\xi_3)=-i[m(\xi_1)m(\xi_2+\xi_3)(\xi_2+\xi_3)]_{sym}.
\end{equation}
Form the new modified energy
\[E_I^3(t)=E_I^2(t)+\Lambda_3(\sigma_3)\]
where the symmetric function $\sigma_3$ will be chosen momentarily
to achieve a cancellation. Applying \eqref{eq:menergy} gives
\begin{eqnarray}\label{eq:E3}
\frac{d}{dt}E_I^3(t)&=&-\epsilon\Lambda_2(m(\xi_1)m(\xi_2)\beta_{\alpha,2})+\Lambda_3(M_3)\nonumber\\
&&+\Lambda_3(\sigma_3h_3)-\epsilon
\Lambda_3(\sigma_3\beta_{\alpha,3})-\half 3
i\Lambda_4(\sigma_3(\xi_1,\xi_2,\xi_3+\xi_4)(\xi_3+\xi_4)).
\end{eqnarray}
Compared to the KdV case \cite{Tao2}, there is one more term to
cancel, so we choose
\begin{equation}
\sigma_3=-\frac{M_3}{h_3-\epsilon \beta_{\alpha, 3}}
\end{equation}
to force the three $\Lambda_3$ terms in \eqref{eq:E3} to cancel.
Hence if we denote
\begin{equation}
M_4(\xi_1,\xi_2,\xi_3,\xi_4)=-i\half
3[\sigma_3(\xi_1,\xi_2,\xi_3+\xi_4)(\xi_3+\xi_4)]_{sym}
\end{equation}
then
\begin{eqnarray}
\frac{d}{dt}E_I^3(t)=-\epsilon\Lambda_2(m(\xi_1)m(\xi_2)\beta_{\alpha,2})+\Lambda_4(M_4).
\end{eqnarray}
Similarly defining
\[E_I^4(t)=E_I^3(t)+\Lambda_4(\sigma_4)\]
with
\begin{equation}
\sigma_4=-\frac{M_4}{h_4-\epsilon \beta_{\alpha,4}},
\end{equation}
we obtain
\begin{eqnarray}
\frac{d}{dt}E_I^4(t)=-\epsilon\Lambda_2(m(\xi_1)m(\xi_2)\beta_{\alpha,2})+\Lambda_5(M_5)
\end{eqnarray}
where
\begin{equation}
M_5(\xi_1,\ldots,\xi_5)=-2i[\sigma_4(\xi_1,\xi_2,\xi_3,\xi_4+\xi_5)(\xi_4+\xi_5)]_{sym}.
\end{equation}
Now we give pointwise bounds for the multipliers. We will only be
interested in the value of the multiplier on the hyperplane
$\xi_1+\xi_2+\ldots+\xi_k=0$. There is a flexibility of choosing the
multiplier $m$. In application, we consider $m(\xi)$ is smooth,
monotone, and of the form
\begin{eqnarray}\label{eq:Imul}
m(\xi)=\left \{
\begin{array}{l}
1, \quad |\xi|<N,\\
N^{-s}|\xi|^s,\quad |\xi|>2N.
\end{array}
\right.
\end{eqnarray}
It is easy to see that if $m$ is of the form \eqref{eq:Imul}, then
$m^2$ satisfies
\begin{eqnarray}\label{eq:eImul}
&&m^2(\xi)\sim m^2(\xi') \mbox{ for } |\xi|\sim|\xi'|,\nonumber\\
&&(m^2)'(\xi)=O(\frac{m^2(\xi)}{|\xi|}),\nonumber\\
&&(m^2)''(\xi)=O(\frac{m^2(\xi)}{|\xi|^2}).
\end{eqnarray}
We will need two mean value formulas which follow immediately from
the fundamental theorem of calculus. If $|\eta|,|\lambda|\ll |\xi|$,
then we have
\begin{equation}\label{eq:mvt}
|a(\xi+\eta)-a(\xi)|{\lesssim} |\eta|\sup_{|\xi'|\sim |\xi|}|a'(\xi')|,
\end{equation}
and the double mean value formula that
\begin{equation}\label{eq:dmvt}
|a(\xi+\eta+\lambda)-a(\xi+\eta)-a(\xi+\lambda)+a(\xi)|{\lesssim}
|\eta||\lambda|\sup_{|\xi'|\sim |\xi|}|a''(\xi')|.
\end{equation}
\begin{proposition}
If $m$ is of the form \eqref{eq:Imul}, then for each dyadic
$\lambda\leq \mu$ there is an extension of $\sigma_3$ from the
diagonal set
\[\{(\xi_1,\xi_2,\xi_3)\in \Gamma_3({\mathbb R}), |\xi_1|\sim \lambda, |\xi_2|, |\xi_3|\sim \mu\}\]
to the full dyadic set
\[\{(\xi_1,\xi_2,\xi_3)\in {\mathbb R}^3, |\xi_1|\sim \lambda, |\xi_2|, |\xi_3|\sim \mu\}\]
which satisfies
\begin{equation}\label{eq:m3}
|\partial_1^{\beta_1}\partial_2^{\beta_2}\partial_3^{\beta_3}\sigma_3(\xi_1,\xi_2,\xi_3)|\leq
C m^2(\lambda)\mu^{-2}\lambda^{-\beta_1}\mu^{-\beta_2-\beta_3},
\end{equation}
where $C$ is independent of $\epsilon$.
\end{proposition}
\begin{proof}
Since on the hyperplane $\xi_1+\xi_2+\xi_3=0$,
\[h_3=i(\xi_1^3+\xi_2^3+\xi_3^3)=3i\xi_1\xi_2\xi_3\]
is with a size about $\lambda \mu^2$ and
\[M_3(\xi_1,\xi_2,\xi_3)=-i[m(\xi_1)m(\xi_2+\xi_3)(\xi_2+\xi_3)]_{sym}=i(m^2(\xi_1)\xi_1+m^2(\xi_2)\xi_2+m^2(\xi_3)\xi_3),\]
if $\lambda \sim \mu$, we extend $\sigma_3$ by setting
\begin{equation}
\sigma_3(\xi_1,\xi_2,\xi_3)=-\frac{i(m^2(\xi_1)\xi_1+m^2(\xi_2)\xi_2+m^2(\xi_3)\xi_3)}{3i\xi_1\xi_2\xi_3-\epsilon(|\xi_1|^{2\alpha}+|\xi_2|^{2\alpha}+|\xi_3|^{2\alpha})},
\end{equation}
and if $\lambda\ll \mu$, we extend $\sigma_3$ by setting
\begin{equation}
\sigma_3(\xi_1,\xi_2,\xi_3)=-\frac{i(m^2(\xi_1)\xi_1+m^2(\xi_2)\xi_2-m^2(\xi_1+\xi_2)(\xi_1+\xi_2))}{3i\xi_1\xi_2\xi_3-\epsilon(|\xi_1|^{2\alpha}+|\xi_2|^{2\alpha}+|\xi_3|^{2\alpha})}.
\end{equation}
From \eqref{eq:mvt} and \eqref{eq:eImul}, we see that \eqref{eq:m3} holds.
\end{proof}
We define on the hyperplane $\{(\xi_1,\xi_2,\xi_3)\in \Gamma_3({\mathbb R}),
|\xi_1|\approx \lambda, |\xi_2|, |\xi_3|\approx \mu\}$
\begin{equation}
\sigma_3^{-}(\xi_1,\xi_2,\xi_3)=-\frac{i(m^2(\xi_1)\xi_1+m^2(\xi_2)\xi_2+m^2(\xi_3)\xi_3)}{3i\xi_1\xi_2\xi_3+\epsilon(|\xi_1|^{2\alpha}+|\xi_2|^{2\alpha}+|\xi_3|^{2\alpha})},
\end{equation}
and extend it as for $\sigma_3$. Then \eqref{eq:m3} also holds for
$\sigma_3^{-}$, and on the hyperplane $\xi_1+\xi_2+\xi_3=0$ we get
\begin{equation}\label{eq:diffm3}
|\sigma_3(\xi_1,\xi_2,\xi_3)-\sigma_3^{-}(\xi_1,\xi_2,\xi_3)|{\lesssim}
\frac{\epsilon
|\xi|_{max}^{2\alpha}m^2(|\xi|_{min})|\xi|_{min}}{(\xi_1\xi_2\xi_3)^2+\epsilon^2|\xi|_{max}^{4\alpha}},
\end{equation}
where
\[|\xi|_{max}=\max(|\xi_1|,|\xi_2|,|\xi_3|), \quad |\xi|_{min}=\min(|\xi_1|,|\xi_2|,|\xi_3|).\]
Now we give the pointwise bounds for $\sigma_4$ which is key to
estimate the growth of $E^4_I(t)$. It has the same bound as in the
KdV case.
\begin{proposition}
Assume $m$ is of the form \eqref{eq:Imul}. In the region where
$|\xi_i|\sim N_i,|\xi_j+\xi_k|\sim N_{jk}$ for $N_i, N_{jk}$ dyadic,
\begin{equation}\label{eq:m4}
\frac{|M_4(\xi_1,\xi_2,\xi_3,\xi_4)|}{|h_4-\epsilon
\beta_{\alpha,4}|}{\lesssim}
\frac{m^2(\min(N_i,N_{jk}))}{(N+N_1)(N+N_2)(N+N_3)(N+N_4)}.
\end{equation}
\end{proposition}
\begin{proof}
From symmetry, we can assume that $N_1\geq N_2\geq N_3\geq N_4$.
Since $\xi_1+\xi_2+\xi_3+\xi_4=0$, then $N_1\sim N_2$. We can also
assume that $N_1\sim N_2 {\gtrsim} N$, otherwise $M_4$ vanishes, since
$m^2(\xi)=1$ if $|\xi|\leq N$. If $\max(N_{12},N_{13},N_{14})\ll
N_1$, then $\xi_3\approx-\xi_1,\ \xi_4\approx -\xi_1$, which
contradicts that $\xi_1+\xi_2+\xi_3+\xi_4=0$. Hence we get
$\max(N_{12},N_{13},N_{14})\sim N_1$. The right side of
\eqref{eq:m4} may be reexpressed as
\begin{equation}
\frac{m^2(\min(N_i,N_{jk}))}{{N_1}^2(N+N_3)(N+N_4)}.
\end{equation}
Since $\xi_1+\xi_2+\xi_3+\xi_4=0$, then
$h_4=\xi_1^3+\xi_2^3+\xi_3^3+\xi_4^3=3(\xi_1+\xi_2)(\xi_1+\xi_3)(\xi_1+\xi_4)$,
and we can write that
\begin{eqnarray}\label{eq:rm4}
CM_4(\xi_1,\xi_2,\xi_3,\xi_4)&=&[\sigma_3(\xi_1,\xi_2,\xi_3+\xi_4)(\xi_3+\xi_4)]_{sym}\nonumber\\
&=&\sigma_3(\xi_1,\xi_2,\xi_3+\xi_4)(\xi_3+\xi_4)+\sigma_3(\xi_1,\xi_3,\xi_2+\xi_4)(\xi_2+\xi_4)\nonumber\\
&&+\sigma_3(\xi_1,\xi_4,\xi_2+\xi_3)(\xi_2+\xi_3)+\sigma_3(\xi_2,\xi_3,\xi_1+\xi_4)(\xi_1+\xi_4)\nonumber\\
&&+\sigma_3(\xi_2,\xi_4,\xi_1+\xi_3)(\xi_1+\xi_3)+\sigma_3(\xi_3,\xi_4,\xi_1+\xi_2)(\xi_1+\xi_2)\nonumber\\
&=&[\sigma_3(\xi_1,\xi_2,\xi_3+\xi_4)-\sigma_3^{-}(-\xi_3,-\xi_4,\xi_3+\xi_4)](\xi_3+\xi_4)\nonumber\\
&&+[\sigma_3(\xi_1,\xi_3,\xi_2+\xi_4)-\sigma_3^{-}(-\xi_2,-\xi_4,\xi_2+\xi_4)](\xi_2+\xi_4)\nonumber\\
&&+[\sigma_3(\xi_1,\xi_4,\xi_2+\xi_3)-\sigma_3^{-}(-\xi_2,-\xi_3,\xi_2+\xi_3)](\xi_2+\xi_3)\nonumber\\
&=&I+II+III.
\end{eqnarray}
The bound \eqref{eq:m4} will follow from case by case analysis.
{\bf Case 1.} $|N_4|{\gtrsim} \half{N}$.
{\bf Case 1a.} $N_{12}, N_{13}, N_{14}{\gtrsim} N_1$.
For this case, we just use \eqref{eq:m3}, then we get
\begin{equation}
\frac{|M_4(\xi_1,\xi_2,\xi_3,\xi_4)|}{|h_4-\epsilon
\beta_{\alpha,4}|}{\lesssim}\frac{|M_4(\xi_1,\xi_2,\xi_3,\xi_4)|}{|h_4|}{\lesssim}
\frac{m^2(N_4)}{N_1N_2N_3N_4},
\end{equation}
which is acceptable.
{\bf Case 1b.} $N_{12}\ll N_1$, $N_{13}{\gtrsim} N_1$, $N_{14}{\gtrsim} N_1$.
Contribution of I. We just use \eqref{eq:m3}, then we get
\begin{equation}
\frac{|I|}{|h_4-\epsilon \beta_{\alpha,4}|}{\lesssim}\frac{|I|}{|h_4|}{\lesssim}
\frac{m^2(\min(N_4, N_{12}))}{N_1N_2N_3N_4},
\end{equation}
which is acceptable.
Contribution of II. We first write
\begin{eqnarray}
II&=&[\sigma_3(\xi_1,\xi_3,\xi_2+\xi_4)-\sigma_3^{-}(-\xi_2,-\xi_4,\xi_2+\xi_4)](\xi_2+\xi_4)\nonumber\\
&=&[\sigma_3(\xi_1,\xi_3,\xi_2+\xi_4)-\sigma_3^{-}(\xi_1,\xi_3,\xi_2+\xi_4)](\xi_2+\xi_4)\nonumber\\
&&+[\sigma_3^{-}(\xi_1,\xi_3,\xi_2+\xi_4)-\sigma_3^{-}(-\xi_2,-\xi_4,\xi_2+\xi_4)](\xi_2+\xi_4)\nonumber\\
&=&II_1+II_2.
\end{eqnarray}
Then from \eqref{eq:diffm3} we get
\begin{eqnarray}
\frac{II_1}{|h_4-\epsilon \beta_{\alpha,4}|}{\lesssim}
\frac{II_1}{|\epsilon
\beta_{\alpha,4}|}{\lesssim}\frac{m^2(N_4)}{N_1N_1N_1N_3}.
\end{eqnarray}
We now consider $II_2$. If $N_{12}{\gtrsim} N_3$, then using
\eqref{eq:mvt} and \eqref{eq:m3}, or else if $N_{12}\ll N_3$, then
using \eqref{eq:mvt} twice and \eqref{eq:m3}, then
\begin{eqnarray}
\frac{II_2}{|h_4-\epsilon \beta_{\alpha,4}|}{\lesssim}
\frac{II_2}{h_4}{\lesssim}\frac{m^2(N_4)}{N_1N_1N_1N_3}.
\end{eqnarray}
Contribution of III. This is identical to II.
{\bf Case 1c.} $N_{12}\ll N_1$, $N_{13}\ll N_1$, $N_{14}{\gtrsim} N_1$.
Since $N_{12}\ll N_1$, $N_{13}\ll N_1$, then $N_1\sim N_2\sim
N_3\sim N_4$.
Contribution of I. We first write
\begin{eqnarray}
I&=&[\sigma_3(\xi_1,\xi_2,\xi_3+\xi_4)-\sigma_3^{-}(\xi_1,\xi_2,\xi_3+\xi_4)](\xi_3+\xi_4)\nonumber\\
&&+[\sigma_3^{-}(\xi_1,\xi_2,\xi_3+\xi_4)-\sigma_3^{-}(-\xi_3,\xi_2,\xi_3+\xi_4)](\xi_3+\xi_4)\nonumber\\
&&+[\sigma_3^{-}(-\xi_3,\xi_2,\xi_3+\xi_4)-\sigma_3^{-}(-\xi_3,-\xi_4,\xi_3+\xi_4)](\xi_3+\xi_4)\nonumber\\
&=&I_1+I_2+I_3.
\end{eqnarray}
We use \eqref{eq:diffm3} for the first term and \eqref{eq:m3},
\eqref{eq:mvt} for the last two terms, then we get
\begin{eqnarray}
\frac{I}{|h_4-\epsilon \beta_{\alpha,4}|} {\lesssim} \frac{I_1}{|\epsilon
\beta_{\alpha,4}|}+\frac{I_2}{|h_4|}+\frac{I_3}{|h_4|}{\lesssim}\frac{m^2(N_{12})}{N_1^4}.
\end{eqnarray}
Contribution of II. This is identical to I.
Contribution of III. We first write
\begin{eqnarray}
III&=&[\sigma_3(\xi_1,\xi_4,\xi_2+\xi_3)-\sigma_3^{-}(-\xi_2,-\xi_3,\xi_2+\xi_3)](\xi_2+\xi_3)\nonumber\\
&=&[\sigma_3(\xi_1,\xi_4,\xi_2+\xi_3)-\sigma_3^{-}(\xi_1,\xi_4,\xi_2+\xi_3)](\xi_2+\xi_3)\nonumber\\
&&+1/2[\sigma_3^{-}(\xi_1,\xi_4,\xi_2+\xi_3)-\sigma_3^{-}(-\xi_2,-\xi_3,\xi_2+\xi_3)\nonumber\\
&&-\sigma_3^{-}(-\xi_3,-\xi_2,\xi_2+\xi_3)+\sigma_3^{-}(\xi_4,\xi_1,\xi_2+\xi_3)](\xi_2+\xi_3)\nonumber\\
&=&III_1+III_2.
\end{eqnarray}
We use \eqref{eq:diffm3} for the first term and \eqref{eq:dmvt} four
times for the second term, then we get
\begin{equation}
\frac{III}{|h_4-\epsilon \beta_{\alpha,4}|} {\lesssim}
\frac{III_1}{|\epsilon
\beta_{\alpha,4}|}+\frac{III_2}{|h_4|}{\lesssim}\frac{m^2(N_{1})}{N_1^4}.
\end{equation}
{\bf Case 1d.} $N_{12}\ll N_1$, $N_{13}{\gtrsim} N_1$, $N_{14}\ll N_1$.
This case is identical to Case 1c.
{\bf Case 2.} $N_4\ll N/2$.
In this case we have $m^2(\min(N_i,N_{jk}))=1$, and $N_{13}\sim
|\xi_1+\xi_3|=|\xi_2+\xi_4|\sim N_1$. We discuss this case in the
following two subcases.
{\bf Case 2a.} $N_1/4>N_{12}{\gtrsim} N/2$.
Since $N_4\ll N/2$ and $|\xi_3+\xi_4|=|\xi_1+\xi_2|{\gtrsim} N/2$, then
$N_3{\gtrsim} N/2$. From $|h_4|\sim N_{12}N_1^2$, then we bound the six
terms in \eqref{eq:rm4} respectively, and get
\begin{equation}
\frac{|M_4|}{|h_4-\epsilon\beta_{\alpha,4}|}{\lesssim}
\frac{|M_4|}{|h_4|}{\lesssim} \frac{1}{N_1^2N_3N},
\end{equation}
which is acceptable.
{\bf Case 2b.} $N_{12}\ll N/2$.
Since $N_{12}=N_{34}\ll N/2$ and $N_4\ll N/2$, then we must have
$N_3\ll N/2$, and $N_{13}\sim N_{14}\sim N_1$.
Contribution of I. Since $N_3, N_4, N_{34}\ll N/2$, then we have
$\sigma_3^{-}(-\xi_3,-\xi_4,\xi_3+\xi_4)=0$. Thus it follows from
\eqref{eq:m3} that
\begin{equation}
\frac{|I|}{|h_4-\epsilon\beta_{\alpha,4}|}{\lesssim}
\frac{|\sigma_3(\xi_1,\xi_2,\xi_3+\xi_4)|}{N_1^2}{\lesssim}
\frac{1}{N_1^4}.
\end{equation}
Contribution of II and III. We have two items of $N_3, N_4, N_{12}$
in the denominator, which will cause a problem. Thus we can't deal
with II and III separately, but we need to exploit the cancelation
between II and III. We rewrite
\begin{eqnarray}
II+III&=&[\sigma_3(\xi_1,\xi_3,\xi_2+\xi_4)-\sigma_3^{-}(-\xi_2,-\xi_4,\xi_2+\xi_4)](\xi_2+\xi_4)\nonumber\\
&&+[\sigma_3(\xi_1,\xi_4,\xi_2+\xi_3)-\sigma_3^{-}(-\xi_2,-\xi_3,\xi_2+\xi_3)](\xi_2+\xi_3)\nonumber\\
&=&[\sigma_3(\xi_1,\xi_3,\xi_2+\xi_4)-\sigma_3^{-}(-\xi_2,-\xi_4,\xi_2+\xi_4)]\xi_4\nonumber\\
&&+[\sigma_3(\xi_1,\xi_4,\xi_2+\xi_3)-\sigma_3^{-}(-\xi_2,-\xi_3,\xi_2+\xi_3)]\xi_3\nonumber\\
&&+[\sigma_3(\xi_1,\xi_3,\xi_2+\xi_4)-\sigma_3^{-}(-\xi_2,-\xi_4,\xi_2+\xi_4)\nonumber\\
&&\quad+\sigma_3(\xi_1,\xi_4,\xi_2+\xi_3)-\sigma_3^{-}(-\xi_2,-\xi_3,\xi_2+\xi_3)]\xi_2\nonumber\\
&=&J_1+J_2+J_3.
\end{eqnarray}
We first consider $J_1$. From
\begin{eqnarray}
\frac{|J_1|}{|h_4-\epsilon\beta_{\alpha,4}|}&\leq&
\frac{|[\sigma_3(\xi_1,\xi_3,\xi_2+\xi_4)-\sigma_3(-\xi_2,-\xi_4,\xi_2+\xi_4)]\xi_4|}{|h_4|}\nonumber\\
&&+\frac{|[\sigma_3(-\xi_2,-\xi_4,\xi_2+\xi_4)-\sigma_3^{-}(-\xi_2,-\xi_4,\xi_2+\xi_4)]\xi_4|}{|\epsilon\beta_{\alpha,4}|},
\end{eqnarray}
and \eqref{eq:diffm3} for the second term, \eqref{eq:mvt} if
$N_{12}\ll N_3$ (in this case, $N_3\sim N_4$), and \eqref{eq:m3} if
$N_{12}{\gtrsim} N_3$ for the first term, then we get
\begin{equation}
\frac{|J_1|}{|h_4-\epsilon\beta_{\alpha,4}|}{\lesssim} \frac{1}{N_1^4}.
\end{equation}
The term $J_2$ is identical to the term $J_1$. Now we consider
$J_3$. We first assume that $N_{12}{\gtrsim} N_3$. Then by the symmetry
of $\sigma_3$, we get
\begin{eqnarray}
J_3&=&[\sigma_3(\xi_1,\xi_3,\xi_2+\xi_4)-\sigma_3^{-}(-\xi_2,-\xi_4,\xi_2+\xi_4)\nonumber\\
&&\quad+\sigma_3(\xi_1,\xi_4,\xi_2+\xi_3)-\sigma_3^{-}(-\xi_2,-\xi_3,\xi_2+\xi_3)]\xi_2\nonumber\\
&=&[\sigma_3(\xi_1,\xi_3,\xi_2+\xi_4)-\sigma_3(-\xi_2-\xi_3,\xi_3,\xi_2)\nonumber\\
&&\quad+\sigma_3(\xi_1,\xi_4,\xi_2+\xi_3)-\sigma_3(-\xi_2-\xi_4,\xi_4,\xi_2)]\xi_2.
\end{eqnarray}
From \eqref{eq:mvt} and $N_{12}{\gtrsim} N_3$, we get
\begin{equation}
\frac{|J_3|}{|h_4-\epsilon\beta_{\alpha,4}|}{\lesssim}
\frac{|J_3|}{|h_4|}{\lesssim} \frac{1}{N_1^4}.
\end{equation}
If $N_{12}\ll N_3$, then $N_3\sim N_4$. We first write
\begin{eqnarray}
J_3&=&[\sigma_3(\xi_1,\xi_3,\xi_2+\xi_4)-\sigma_3^{-}(\xi_1,\xi_3,\xi_2+\xi_4)\nonumber\\
&&\quad+\sigma_3(-\xi_2,-\xi_3,\xi_2+\xi_3)-\sigma_3^{-}(-\xi_2,-\xi_3,\xi_2+\xi_3)]\xi_2\nonumber\\
&&+[\sigma_3^{-}(-\xi_2,\xi_3,\xi_2+\xi_4)-\sigma_3^{-}(-\xi_2,-\xi_4,\xi_2+\xi_4)\nonumber\\
&&\quad+\sigma_3(\xi_1,\xi_4,\xi_2+\xi_3)-\sigma_3(\xi_1,-\xi_3,\xi_2+\xi_3)]\xi_2\nonumber\\
&&+[\sigma_3^{-}(\xi_1,\xi_3,\xi_2+\xi_4)-\sigma_3^{-}(-\xi_2,\xi_3,\xi_2+\xi_4)\nonumber\\
&&\quad+\sigma_3(\xi_1,-\xi_3,\xi_2+\xi_3)-\sigma_3(-\xi_2,-\xi_3,\xi_2+\xi_3)]\xi_2\nonumber\\
&=&J_{31}+J_{32}+J_{33}.
\end{eqnarray}
It follows from \eqref{eq:mvt} that
\begin{equation}
\frac{|J_{33}|}{|h_4-\epsilon\beta_{\alpha,4}|}{\lesssim}
\frac{|J_{33}|}{|h_4|}{\lesssim} \frac{1}{N_1^4}.
\end{equation}
It remains to bound $J_{31}$ and $J_{32}$. First we consider
$J_{31}$. Since $m^2(\xi_3)=1$, we rewrite $J_{31}$ by
\begin{eqnarray}\label{eq:J31}
J_{31}&=&[\sigma_3(\xi_1,\xi_3,\xi_2+\xi_4)-\sigma_3^{-}(\xi_1,\xi_3,\xi_2+\xi_4)\nonumber\\
&&\quad+\sigma_3(-\xi_2,-\xi_3,\xi_2+\xi_3)-\sigma_3^{-}(-\xi_2,-\xi_3,\xi_2+\xi_3)]\xi_2\nonumber\\
&=&A(\xi_1,\xi_3,\xi_2+\xi_4)(m^2(\xi_1)\xi_1+\xi_3+m^2(\xi_2+\xi_4)(\xi_2+\xi_4))\xi_2\nonumber\\
&&+A(-\xi_2,-\xi_3,\xi_2+\xi_3)(-m^2(\xi_2)\xi_2-\xi_3+m^2(\xi_2+\xi_3)(\xi_2+\xi_3))\xi_2\nonumber\\
&=&[A(\xi_1,\xi_3,\xi_2+\xi_4)-A(-\xi_2,-\xi_3,\xi_2+\xi_3)]\xi_3\xi_2\nonumber\\
&&-[A(\xi_1,\xi_3,\xi_2+\xi_4)-A(-\xi_2,-\xi_3,\xi_2+\xi_3)]\xi_2\nonumber\\
&&\times[-m^2(\xi_2)\xi_2+m^2(\xi_2+\xi_3)(\xi_2+\xi_3)]\nonumber\\
&&+A(\xi_1,\xi_3,\xi_2+\xi_4)\xi_2\nonumber\\
&&\times[m^2(\xi_1)\xi_1+m^2(\xi_2+\xi_4)(\xi_2+\xi_4)-m^2(\xi_2)\xi_2+m^2(\xi_2+\xi_3)(\xi_2+\xi_3)]
\end{eqnarray}
where
\[A(\xi_1,\xi_2,\xi_3)=\frac{2\epsilon (|\xi_1|^{2\alpha}+|\xi_2|^{2\alpha}+|\xi_3|^{2\alpha})}{|\xi_1\xi_2\xi_3|^2+\epsilon^2(|\xi_1|^{2\alpha}+|\xi_2|^{2\alpha}+|\xi_3|^{2\alpha})^2}.\]
It's easy to see that $A(\xi_1,\xi_2,\xi_3)$ satisfies
\begin{equation}
|\partial_{\xi_i}A(\xi_1,\xi_2,\xi_3)|{\lesssim}
\frac{|A(\xi_1,\xi_2,\xi_3)|}{|\xi_i|},\quad i=1,2,3.
\end{equation}
For the first two terms in \eqref{eq:J31} we use \eqref{eq:mvt} by
writing
\begin{eqnarray*}
&&A(\xi_1,\xi_3,\xi_2+\xi_4)-A(-\xi_2,-\xi_3,\xi_2+\xi_3)\\
&=&A(\xi_1,\xi_3,\xi_2+\xi_4)-A(-\xi_2,\xi_3,\xi_2+\xi_4)\\
&&+A(-\xi_2,\xi_3,\xi_2+\xi_4)-A(-\xi_2,\xi_3,\xi_2+\xi_3).
\end{eqnarray*}
For the third term, we note that
\begin{eqnarray}
&&m^2(\xi_1)\xi_1+m^2(\xi_2+\xi_4)(\xi_2+\xi_4)-m^2(\xi_2)\xi_2+m^2(\xi_2+\xi_3)(\xi_2+\xi_3)\nonumber\\
&=&m^2(\xi_2+\xi_4)(\xi_2+\xi_4)-m^2(\xi_2)\xi_2\nonumber\\
&&-m^2(\xi_2+\xi_3+\xi_4)(\xi_2+\xi_3+\xi_4)+m^2(\xi_2+\xi_3)(\xi_2+\xi_3),
\end{eqnarray}
thus we can apply \eqref{eq:dmvt}. Therefore, we get
\begin{equation}
\frac{|J_{31}|}{|h_4-\epsilon\beta_{\alpha,4}|}{\lesssim}
\frac{|J_{31}|}{|\epsilon\beta_{\alpha,4}||}{\lesssim} \frac{1}{N_1^4}.
\end{equation}
Last we consider $J_{32}$. We denote
\begin{eqnarray}
B(\xi_1,\xi_2,\xi_3)&=&\frac{1}{i\xi_1\xi_2\xi_3-\epsilon(|\xi_1|^{2\alpha}+|\xi_2|^{2\alpha}+|\xi_3|^{2\alpha})}-\frac{1}{i\xi_1\xi_2\xi_3}\nonumber\\
&=&\frac{\epsilon(|\xi_1|^{2\alpha}+|\xi_2|^{2\alpha}+|\xi_3|^{2\alpha})}{[i\xi_1\xi_2\xi_3-\epsilon(|\xi_1|^{2\alpha}+|\xi_2|^{2\alpha}+|\xi_3|^{2\alpha})]i\xi_1\xi_2\xi_3}.
\end{eqnarray}
It's easy to see that $B(\xi_1,\xi_2,\xi_3)$ satisfies
\begin{equation}\label{eq:eB}
|\partial_{\xi_i}B(\xi_1,\xi_2,\xi_3)|{\lesssim}
\frac{|B(\xi_1,\xi_2,\xi_3)|}{|\xi_i|},\quad i=1,2,3.
\end{equation}
Let
\begin{equation}\label{eq:m3kdv}
\tilde{\sigma}_3(\xi_1,\xi_2,\xi_3)=\frac{M(\xi_1,\xi_2,\xi_3)}{i\xi_1\xi_2\xi_3},
\end{equation}
then we can rewrite $J_{32}$ by
\begin{eqnarray}\label{eq:J32}
J_{32}&=&[\sigma_3^{-}(-\xi_2,\xi_3,\xi_2+\xi_4)-\sigma_3^{-}(-\xi_2,-\xi_4,\xi_2+\xi_4)\nonumber\\
&&\quad+\sigma_3(\xi_1,\xi_4,\xi_2+\xi_3)-\sigma_3(\xi_1,-\xi_3,\xi_2+\xi_3)]\xi_2\nonumber\\
&=&B(-\xi_2,\xi_4,\xi_2+\xi_4)[-m^2(-\xi_2)\xi_2-\xi_4+m^2(\xi_2+\xi_4)(\xi_2+\xi_4)]\xi_2\nonumber\\
&&+B(\xi_1,\xi_4,\xi_2+\xi_3)[m^2(\xi_1)\xi_1+\xi_4+m^2(\xi_2+\xi_3)(\xi_2+\xi_3)]\xi_2\nonumber\\
&&-B(\xi_2,\xi_3,\xi_2+\xi_4)[-m^2(-\xi_2)\xi_2+\xi_3+m^2(\xi_2+\xi_4)(\xi_2+\xi_4)]\xi_2\nonumber\\
&&-B(\xi_1,-\xi_3,\xi_2+\xi_3)[m^2(\xi_1)\xi_1-\xi_3+m^2(\xi_2+\xi_3)(\xi_2+\xi_3)]\xi_2\nonumber\\
&&+[\tilde{\sigma}_3(-\xi_2,\xi_3,\xi_2+\xi_4)-\tilde{\sigma}_3(\xi_1,-\xi_3,\xi_2+\xi_3)\nonumber\\
&&\quad-\tilde{\sigma}_3(-\xi_2,-\xi_4,\xi_2+\xi_4)+\tilde{\sigma}_3(\xi_1,\xi_4,\xi_2+\xi_3)]\xi_2.
\end{eqnarray}
For the first four terms in \eqref{eq:J32}, we can bound them by the
same way as for $J_{31}$, using \eqref{eq:eB} and the symmetry of
$B$ that $B(\xi_1,-\xi_2,\xi_3)=B(-\xi_1,\xi_2,\xi_3)$. For the last
term, it follows from \eqref{eq:m3kdv} and $m^2(\xi_3)=m^2(\xi_4)=1$
that
\begin{eqnarray}
J_L&=&[\tilde{\sigma}_3(-\xi_2,\xi_3,\xi_2+\xi_4)-\tilde{\sigma}_3(\xi_1,-\xi_3,\xi_2+\xi_3)\nonumber\\
&&\quad-\tilde{\sigma}_3(-\xi_2,-\xi_4,\xi_2+\xi_4)+\tilde{\sigma}_3(\xi_1,\xi_4,\xi_2+\xi_3)]\xi_2\nonumber\\
&=&\frac{-m^2(\xi_2)\xi_2+\xi_3+m^2(\xi_2+\xi_4)(\xi_2+\xi_4)}{-\xi_2\xi_3(\xi_2+\xi_4)}\xi_2\nonumber\\
&&\quad
-\frac{-m^2(\xi_2)\xi_2-\xi_4+m^2(\xi_2+\xi_4)(\xi_2+\xi_4)}{\xi_2\xi_4(\xi_2+\xi_4)}\xi_2\nonumber\\
&&+\frac{m^2(\xi_1)\xi_1+\xi_4+m^2(\xi_2+\xi_3)(\xi_2+\xi_3)}{\xi_1\xi_4(\xi_2+\xi_3)}\xi_2\nonumber\\
&&\quad
-\frac{m^2(\xi_1)\xi_1-\xi_3+m^2(\xi_2+\xi_3)(\xi_2+\xi_3)}{-\xi_1\xi_3(\xi_2+\xi_3)}\xi_2.
\end{eqnarray}
Note that there is a cancelation. Therefore,
\begin{eqnarray}\label{eq:JL}
J_L&=&-\frac{\xi_3+\xi_4}{\xi_3\xi_4}\frac{-m^2(\xi_2)\xi_2+m^2(\xi_2+\xi_4)(\xi_2+\xi_4)}{\xi_2(\xi_2+\xi_4)}\xi_2\nonumber\\
&&+\frac{\xi_3+\xi_4}{\xi_3\xi_4}\frac{m^2(\xi_1)\xi_1+m^2(\xi_2+\xi_3)(\xi_2+\xi_3)}{\xi_1(\xi_2+\xi_3)}\xi_2.
\end{eqnarray}
We rewrite \eqref{eq:JL} by
\begin{eqnarray*}
&&-\frac{\xi_3+\xi_4}{\xi_3\xi_4}\frac{-m^2(\xi_2)\xi_2+m^2(\xi_2+\xi_4)(\xi_2+\xi_4)+m^2(\xi_1)\xi_1+m^2(\xi_2+\xi_3)(\xi_2+\xi_3)}{\xi_2(\xi_2+\xi_4)}\xi_2\\
&&+\frac{\xi_3+\xi_4}{\xi_3\xi_4}[m^2(\xi_1)\xi_1+m^2(\xi_2+\xi_3)(\xi_2+\xi_3)][\frac{1}{\xi_1(\xi_2+\xi_3)}+\frac{1}{\xi_2(\xi_2+\xi_4)}]\xi_2.
\end{eqnarray*}
Therefore, we use \eqref{eq:dmvt} for the first term, and
\eqref{eq:mvt} for the second term, and finally we conclude that
\begin{equation}
\frac{|J_L|}{|h_4-\epsilon\beta_{\alpha,4}|}{\lesssim}
\frac{|J_L|}{|h_4|}{\lesssim} \frac{1}{N_1^4},
\end{equation}
which completes the proof of the proposition.
\end{proof}
With the estimate of $\sigma_4$, we immediately get the estimate of
$M_5$. We have the same bound as in the KdV case.
\begin{proposition}
If $m$ is of the form \eqref{eq:Imul}, then
\begin{equation}
|M_5(\xi_1,\ldots,\xi_5)|{\lesssim}
\left[\frac{m^2(N_{*45})N_{45}}{(N+N_1)(N+N_2)(N+N_3)(N+N_{45})}\right]_{sym},
\end{equation}
where
\[N_{*45}=\min(N_1,N_2,N_3,N_{45},N_{12},N_{13},N_{23}).\]
\end{proposition}
So far we have showed that the multipliers $M_i$, $i=3,4,5$ have the
same bounds as for the KdV equation. We list now some propositions.
\begin{proposition}\label{pm5}
Let $w_i(x,t)$ be functions of space-time with Fourier support
$|\xi|\sim N_i$, $N_i$ dyadic. Then
\begin{equation}
\left|\int_0^\delta\int \prod_{i=1}^5w_i(x,t)dxdt\right|{\lesssim}
\prod_{j=1}^3\norm{w_j}_{F^{1/4}(\delta)}\norm{w_4}_{F^{-3/4}(\delta)}\norm{w_5}_{F^{-3/4}(\delta)}.
\end{equation}
\end{proposition}
\begin{proof}
It follows from the same argument as for the proof of Lemma 5.1 in
\cite{Tao2} with the Proposition \ref{p21}.
\end{proof}
\begin{proposition}
If the associated multiplier $m$ is of the form \eqref{eq:Imul} with
$s=-3/4+$, then
\begin{equation}
\left|\int_0^\delta \Lambda_5(M_5;u_1,\ldots,u_5)dt\right|{\lesssim}
N^{-\beta}\prod_{i=1}^5\norm{Iu_i}_{F^0(\delta)},
\end{equation}
where $\beta=3+\frac{3}{4}-$.
\end{proposition}
\begin{proof}
This proposition can be proved by following the proof of Lemma 5.2
in \cite{Tao2} and using proposition \ref{pm5}. We omit the details.
\end{proof}
\begin{proposition}
Let $I$ be defined with the multiplier $m$ of the form
\eqref{eq:Imul} and $s=-3/4$. Then
\begin{equation}
|E_I^4(t)-E_I^2(t)|{\lesssim} \norm{Iu(t)}_{L^2}^3+\norm{Iu(t)}_{L^2}^4.
\end{equation}
\begin{proof}
Since $E_I^4(t)=E_I^2(t)+\Lambda_3(\sigma_3)+\Lambda_4(\sigma_4)$
and the bound for $\sigma_3$, $\sigma_4$ are the same as in the KdV
case, this proposition follows immediately from Lemma 6.1 in
\cite{Tao2}.
\end{proof}
\end{proposition}
We state a variant local well-posedness result which follows from
slight argument in the last section. This is used to iterate the
solution in the I-method.
\begin{proposition}
If $s>-3/4$, then \eqref{eq:kdvb} is uniformly locally well-posed
for data $\phi$ satisfying $I\phi\in L^2({\mathbb R})$. Moreover, the
solution exists on a time interval $[0,\delta]$ with lifetime
\begin{equation}
\delta\sim \norm{I\phi}_{L^2}^{-\alpha},\ \alpha>0,
\end{equation}
and the solution satisfies the estimate
\begin{equation}
\norm{Iu}_{F^s(\delta)}{\lesssim} \norm{I\phi}_{L^2}.
\end{equation}
\end{proposition}
With these propositions and the scaling \eqref{eq:scaling}, we can
show Theorem \ref{t12} by using the same argument in \cite{Tao2}. We
omit the details.
\section{Limit Behavior}
In this section we prove our third result. It is well-known that
\eqref{eq:kdv} is completely integrable and has infinite
conservation laws, and as a corollary one obtains that let $v$ be a
smooth solution to \eqref{eq:kdv}, for any $k\in {\mathbb Z}_{+}$,
\begin{equation}
\sup_{t\in{\mathbb R}}\norm{v(t)}_{H^k}{\lesssim} \norm{v_0}_{H^k}.
\end{equation}
There are less symmetries for \eqref{eq:kdvb}. We can still expect
that the $H^k$ norm of the solution remains bounded for a finite
time $T>0$, since the dissipative term behaves well for $t>0$. We
already see that for $k=0$ from \eqref{eq:L2law}. Now we prove for
$k=1$ which will suffice for our purpose. We do not pursue for
$k\geq 2$.
Assume $u$ is a smooth solution to \eqref{eq:kdvb}. Let
$H[u]=\int_{\mathbb R} (u_x)^2-\frac{2}{3}u^3+u^2 dx$, then by the equation
\eqref{eq:kdvb} and partial integration
\begin{eqnarray*}
\frac{d}{dt}H[u]&=&\int_{\mathbb R} 2u_x \partial_x(u_t)-2u^2u_t+2uu_t dx\\
&=&\int_{\mathbb R} 2u_x(-u_{xxxx}-\epsilon |\partial_x|^{2\alpha}\partial_xu-(u^2)_{xx})dx\\
&&+\int_{\mathbb R} 2u^2(u_{xxx}+\epsilon |\partial_x|^{2\alpha}u+(u^2)_{x})dx+\int_{\mathbb R}-2\epsilon(\Lambda^\alpha u)^2 dx\\
&=&\int_{\mathbb R} -2\epsilon (\Lambda^{1+\alpha}u)^2+2\epsilon
u^2\Lambda^{2\alpha}u-2\epsilon(\Lambda^\alpha u)^2 dx\\
&\leq& -\epsilon\int_{\mathbb R}
(\Lambda^{2\alpha}u)^2+2u^2\Lambda^{2\alpha}u dx,
\end{eqnarray*}
where we denote $\Lambda=|\partial_x|$. Thus we have
\begin{equation}
\frac{d}{dt}H[u]+\frac{\epsilon}{2} \norm{\Lambda^{2\alpha}u}_2^2
{\lesssim} \norm{u}_4^4.
\end{equation}
Using Galiardo-Nirenberg inequality
\[
\norm{u}_3^3 {\lesssim} \norm{u}_2^{5/2}\norm{u_x}_2^{1/2}, \quad
\norm{u}_4^4 {\lesssim} \norm{u}_2^{3}\norm{u_x}_2
\]
and Cauchy-Schwarz inequality, we get
\begin{equation}\label{eq:H1law}
\sup_{[0,T]}\norm{u(t)}_{H^1}+\epsilon^{1/2}\brk{ \int_{0}^T
\norm{\Lambda^{2\alpha}u(\tau)}_2^2d\tau}^{1/2}\leq
C(T,\norm{\phi}_{H^1}), \quad \forall\ T>0.
\end{equation}
Assume $u_\epsilon$ is a $L^2$-strong solution to \eqref{eq:kdvb}
obtained in the last section and v is a $L^2$-strong solution to
\eqref{eq:kdv} in \cite{Tao2}, with initial data $\phi_1,\phi_2\in
L^2$ respectively. We still denote by $u_\epsilon, v$ the extension
of $u_\epsilon, v$. From the scaling \eqref{eq:scaling}, we may
assume first that $\norm{\phi_1}_{L^2},\norm{\phi_2}_{L^2}\ll 1$.
Let $w=u_\epsilon-v$, $\phi=\phi_1-\phi_2$, then $w$ solves
\begin{eqnarray}\label{eq:diff}
\left \{
\begin{array}{l}
w_t+w_{xxx}+\epsilon |\partial_x|^{2\alpha}u_\epsilon+(w(v+u_\epsilon))_x=0, t\in {\mathbb R}_{+}, x\in {\mathbb R},\\
v(0)=\phi.
\end{array}
\right.
\end{eqnarray}
We first view $\epsilon |\partial_x|^{2\alpha}u_\epsilon$ as a
perturbation to the difference equation of the KdV equation, and
consider the integral equation of \eqref{eq:diff}
\begin{equation}
w(x,t)=W_0(t)\phi-\int_0^tW_0(t-\tau)[\epsilon
|\partial_x|^{2\alpha}u_\epsilon+(w(v+u_\epsilon))_x]d\tau, \ t\geq
0.
\end{equation}
Then $w$ solves the following integral equation on $t\in [0,1]$,
\begin{eqnarray}
w(x,t)&=&\psi(t)[W_0(t)\phi-\int_0^tW_0(t-\tau)
\chi_{{\mathbb R}_+}(\tau)\psi(\tau)\epsilon
|\partial_x|^{2\alpha}u_\epsilon (\tau)d\tau\nonumber\\
&&\quad
-\int_0^tW_0(t-\tau)\partial_x(\psi^2(\tau)w(v+u_\epsilon))(\tau)d\tau
].
\end{eqnarray}
By Proposition \ref{p42} and Proposition
\ref{p43},\ref{p45},\ref{p412}, we get
\begin{eqnarray}
\norm{w}_{F^0}{\lesssim}
\norm{\phi}_{L^2}+\epsilon\norm{u_\epsilon}_{L^2_{[0,2]}\dot{H}_x^{2\alpha}}+\norm{w}_{F^0}(\norm{v}_{F^0}+\norm{u_\epsilon}_{F^0}).
\end{eqnarray}
Since from Theorem \ref{t12} we have
\[\norm{v}_{F^0}{\lesssim} \norm{\phi_2}_{L^2}\ll 1,\quad \norm{u_\epsilon}_{F^0}{\lesssim} \norm{\phi_1}_{L^2}\ll 1,\]
then we get that
\begin{equation}
\norm{w}_{F^0}{\lesssim}
\norm{\phi}_{L^2}+\epsilon\norm{u_\epsilon}_{L^2_{[0,2]}\dot{H}_x^{2\alpha}}.
\end{equation}
From Proposition \ref{p41} and \eqref{eq:H1law} we get
\begin{equation}
\norm{u_\epsilon-v}_{C([0,1], L^2)}{\lesssim}
\norm{\phi_1-\phi_2}_{L^2}+\epsilon^{1/2}C(\norm{\phi_1}_{H^1},\norm{\phi_2}_{L^2}).
\end{equation}
For general $\phi_1,\phi_2 \in L^2$, using the scaling
\eqref{eq:scaling}, then we immediately get that there exists
$T=T(\norm{\phi_1}_{L^2},\norm{\phi_2}_{L^2})>0$ such that
\begin{equation}\label{eq:limitL2}
\norm{u_\epsilon-v}_{C([0,T], L^2)}{\lesssim}
\norm{\phi_1-\phi_2}_{L^2}+\epsilon^{1/2}C(T,\norm{\phi_1}_{H^1},\norm{\phi_2}_{L^2}).
\end{equation}
Therefore, \eqref{eq:limitL2} automatically holds for any $T>0$, due
to \eqref{eq:L2law} and \eqref{eq:H1law}.
\begin{proof}[Proof of Theorem \ref{t13}]
For fixed $T>0$, we need to prove that $\forall\ \eta>0$, there
exists $\sigma>0$ such that if $0<\epsilon<\sigma$ then
\begin{equation}\label{eq:limitHs}
\norm{S_T^\epsilon(\varphi)-S_T(\varphi)}_{C([0,T];H^s)}<\eta.
\end{equation}
We denote $\varphi_K=P_{\leq K}\varphi$. Then we get
\begin{eqnarray}
&&\norm{S_T^\epsilon(\varphi)-S_T(\varphi)}_{C([0,T];H^s)}\nonumber\\
&\leq&\norm{S_T^\epsilon(\varphi)-S_T^\epsilon(\varphi_K)}_{C([0,T];H^s)}\nonumber\\
&&+\norm{S_T^\epsilon(\varphi_K)-S_T(\varphi_K)}_{C([0,T];H^s)}+\norm{S_T(\varphi_K)-S_T(\varphi)}_{C([0,T];H^s)}.
\end{eqnarray}
From Theorem \ref{t12} and \eqref{eq:limitL2}, we get
\begin{eqnarray}
\norm{S_T^\epsilon(\varphi)-S_T(\varphi)}_{C([0,T];H^s)}{\lesssim}
\norm{\varphi_K-\varphi}_{H^s}+\epsilon^{1/2}C(T,K,
\norm{\varphi}_{H^s}).
\end{eqnarray}
We first fix $K$ large enough, then let $\epsilon$ go to zero,
therefore \eqref{eq:limitHs} holds.
\end{proof}
\noindent{\bf Acknowledgment.} Part of the work was finished while
the first named author was visiting the Department of Mathematics
at the University of Chicago under the auspices of China Scholarship
Council. The authors are grateful to Professor Carlos E. Kenig for
his valuable suggestions. This work is supported in part by the
National Science Foundation of China, grant 10571004; and the 973
Project Foundation of China, grant 2006CB805902, and the Innovation
Group Foundation of NSFC, grant 10621061.
\footnotesize
|
train/arxiv
|
BkiUbo7xK6nrxpQczzR0
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\begin{document}
\begin{titlepage}
\begin{center}
{\Large \bf $OSp(1|4)$ supergravity and its noncommutative extension }
\\[3em]
{\large {\bf Leonardo Castellani} } \\ [2em] {\sl Dipartimento di Scienze e Innovazione Tecnologica
\\ INFN Gruppo collegato di Alessandria,\\Universit\`a del Piemonte Orientale,\\ Viale T. Michel 11, 15121 Alessandria, Italy
}\\ [4em]
\end{center}
\begin{abstract}
\vskip 0.2cm
We review the $OSp(1|4)$-invariant formulation of $N=1$, $D=4$ supergravity and present its noncommutative
extension, based on a $\star$-product originating from an abelian twist with deformation parameter $\theta$. After use of a geometric generalization of the Seiberg-Witten map, we obtain an extended (higher derivative) supergravity theory, invariant under usual $OSp(1|4)$ gauge transformations. Gauge fixing breaks the $OSp(1|4)$ symmetry to its Lorentz subgroup, and
yields a Lorentz invariant extended theory whose classical limit $\theta \rightarrow 0$ is the usual $N=1$, $D=4$ $AdS$ supergravity.
\end{abstract}
\vskip 8.5cm \noindent \hrule \vskip.2cm \noindent {\small
[email protected]}
\end{titlepage}
\newpage
\setcounter{page}{1}
\sect{Introduction}
We present a noncommutative (NC) extension of the $OSp(1|4)$-invariant action of $N=1$, $D=4$ anti-De Sitter supergravity , obtained by the use of a twisted $\star$-product, and a geometric generalization \cite{AC3} of the Seiberg-Witten map \cite{SW} for abelian twists. We thus find a higher derivative extension of $OSp(1|4)$ supergravity where the higher order couplings are dictated by the noncommutative structure of the original NC action. The resulting extended theory is geometric (diffeomorphic invariant) and gauge invariant under usual $OSp(1|4)$ gauge transformations.
Noncommutativity of spacetime coordinates
\begin{equation}
[x^\mu , x^\nu] = i \theta^{\mu\nu} \label{CR}
\end{equation}
is a recurrent theme in physics, being advocated already by Heisenberg in the hope that uncertainty relations between spacetime coordinates
could resolve UV divergences arising in quantum field theory \cite{Heis}.
This motivation still holds, in particular for nonrenormalizable theories of gravity where
finiteness is the only option for consistency.
The issue was explored initially by Snyder in \cite{Snyder}, and
since then noncommutative geometry has found applications in many
branches of physics, mainly in the last two decades. Some
comprehensive reviews can be found in references \cite{Connes},
\cite{Landi}, \cite{madorebook}, \cite{Castellani1}, \cite{revDN},\cite{revSz1}, \cite{book}.
Relations (\ref{CR}) provide a (kinematical) way to encode quantum properties directly in the texture of spacetime. Field theories on noncommuting spacetime can be reformulated as field theories on ordinary (commuting) spacetime, but with a deformed $\star$-product between fields. When the deformation originates from a twist, as in the present paper, the resulting $\star$-product is a twisted product, associative and noncommutative.
This product between fields generates infinitely many derivatives and introduces a
dimensionful noncommutativity parameter $\theta$. The
prototypical example of twisted product is the
Moyal-Groenewold product \cite{MoyalGroenewold} (historically
arising in phase-space after Weyl quantization \cite{Weyl}) :
\begin{eqnarray} & & f(x) \star g(x) \equiv \exp
\left( {i \over 2} \theta^{\mu\nu} {\partial \over \partial x^\mu}
{\partial \over \partial y^\nu} \right) f(x) g(y) |_{y \rightarrow x} \nonumber \\
& & = f(x) g(x) + {i\over 2} \theta^{\mu \nu}
\partial_\mu f \partial_\nu g + \cdots + {1 \over n!} {\left( i \over 2 \right)^n} \theta^{\mu_1\nu_1}
\cdots \theta^{\mu_n\nu_n} (\partial_{\mu_1} \cdots \partial_{\mu_n} f )(\partial_{\nu_1} \cdots \partial_{\nu_n} g )+ \cdots \nonumber\\
& & \label{starproduct}
\end{eqnarray}
\noindent with a constant $\theta$. Using this deformed product
one finds $x^\mu\star x^\nu- x^\nu\star x^\mu=i\theta^{\mu\nu}$, realizing the commutation relations (\ref{CR}).
A straightforward generalization is provided by the twisted $\star$-product, where the partial derivatives in (\ref{starproduct}) are replaced by a set of commuting
tangent vectors $X_A \equiv X_A^\mu \partial_\mu$. Dealing with (super)gravity theories, it is desirable to
extend the twisted $\star$-product to forms. This can be done simply by replacing
the tangent vectors $X_A$, acting on functions, with Lie derivatives along $X_A$, acting
on forms.
Replacing
products between fields with $\star$-products yields nonlocal actions (called twisted, or NC actions), containing
an infinite number of new interactions and higher derivative terms. In this way
twisted Yang-Mills theories in flat space have been constructed (see for ex. \cite{Madore,starYM,AJSW}), as well as twisted metric gravity \cite{Wessgroup,book}. Noncommutative $D=4$ vielbein gravity has been treated in \cite{Cham2002,Cham2003}, where
deformations of conformal gravity and complex vielbein gravity were considered, and in \cite{CZ}, where a $U(2,2)$ $\star$-gauge invariant NC action with constraints was proposed as a NC deformation of Einstein gravity. More recently twisted vielbein gravity and its couplings to fermions \cite{AC1}, gauge fields \cite{AC4} and scalars \cite{AC5} have been constructed, as well as a NC deformation of $D=4$, $N=1$ supergravity \cite{AC2}.
These twisted theories are invariant under deformations of the original symmetries. For example the NC action for gauge fields is :
\begin{equation}
S= {1\over 4 g^2} \int Tr(F_{\mu\nu} \star F^{\mu\nu})
\end{equation}
where
\begin{eqnarray}
& & F_{\mu\nu} = \partial_{\mu} A_\nu - \partial_{\nu} A_\mu - (A_\mu \star A_\nu-
A_\nu \star A_\mu)\\
& & A_\mu = A_\mu^I T_I, ~~~~ Tr(T^I T^J)=\delta^{IJ}
\end{eqnarray}
The noncommutative gauge
transformations:
\begin{eqnarray}
& & \delta_{\varepsilon} A_\mu=\partial_{\mu} \varepsilon
-(A_\mu \star \varepsilon - \varepsilon \star A_\mu)\\
& &\delta_{\varepsilon} F_{\mu\nu} = -(F_{\mu\nu}
\star \varepsilon - \varepsilon \star F_{\mu\nu} ) \end{eqnarray} leave the action invariant, because of the cyclicity of the trace, and
of the property
\begin{equation}
\int f \star g = \int g \star f
\end{equation}
(cyclicity of integral) holding up to boundary terms.
Noncommutativity apparently comes with a price, i.e. a proliferation of new degrees
of freedom. This can be understood by considering the $\star$-deformation of
the Yang-Mills field strength:
\begin{equation}
F_{\mu\nu}^{I}T_{I} = \partial_{\mu} A_{\nu}^{I}T_{I}-\partial_{\nu} A_{\mu}^{I}T_{I}-(A_{\mu}^{I} \star A_{\nu}^{J}-
A_{\nu}^{I} \star A_{\mu}^{J})T_{I}T_{J}
\end{equation}
Because of noncommutativity of the $\star$-product, anticommutators as well as commutators of
group generators appear in the right-hand side, and therefore the $T_{I}$ must be a basis
for the whole universal enveloping algebra of $G$. Thus $I$ runs in principle on the
infinite set of universal enveloping algebra elements (all symmetrized products of the original gauge generators)
and the number of independent $A^{I}_{\mu}$ field components increases to infinity. This proliferation
can be drastically reduced by choosing a specific representation for the generators $T_{I}$. For example if the gauge group is $SU(2)$ and we take its generators to be the in the defining $2 \times 2$ representation, these are just
the Pauli matrices, and a basis for the enveloping algebra only requires an additional matrix proportional
to the unit matrix.
We may get rid even of these additional degrees of freedom if we use the Seiberg-Witten map, which
allows to express all the fields appearing in the NC action (usually called the NC fields) in terms of series expansions in $\theta$ containing only the original fields of the undeformed theory, the so called classical fields. The map is engineered so that the classical gauge transformations on the classical fields induce the NC gauge transformations on the NC fields.
In the $SU(2)$ example, the map relates the four noncommutative fields to
the three classical $SU(2)$ gauge fields.
Substituting in the action the NC fields with their expressions in terms of the classical fields yields an infinite series in powers of $\theta$, whose 0-th order term is the classical action. This higher derivative action is {\it invariant under the
classical gauge variations}, since these by construction induce the NC symmetries of the NC action. Every higher order term in the $\theta$ expansion is actually separately invariant, because the classical symmetries do not involve $\theta$.
With this procedure the NC deformation of vielbein gravity, found in \cite{AC1}, has been re-expressed in \cite{AC3} in terms of the classical vielbein and spin connection, and its Lorentz invariant (and higher derivative) geometric action has been computed up to second order in the noncommutativity parameter \cite{ACD2}. The Seiberg-Witten (SW) map was also used in
\cite{CZ} to compute the first order correction of the deformed $U(2,2)$ gauge invariant and constrained theory, and
in \cite{Top1,Top2,Top3} for the Mac Dowell-Mansouri gauge theory of gravity.
We also mention the NC extension of $SO(2,3)$ $AdS$ gravity of ref. \cite{DRS}, which contains its expansion to order
$\theta^2$, and ref. \cite{DGEFV} where the SW map for pure gravity is examined at second order.
In the present paper we apply this method to $OSp(1|4)$ supergravity. For reviews on the $OSp(1|4)$ formulation of supergravity see for example \cite{CDF,PvN1,PvN2}. The classical theory contains
the vielbein $V^a$, the spin connection $\omega^{ab}$, the gravitino $\psi$ and nondynamical auxiliary fields (a scalar, a pseudoscalar, a vector and a spin 1/2 fermion) necessary to ensure the full off-shell invariance (and closure) under local $OSp(1|4)$ gauge transformations. The auxiliary fields satisfy
$OSp(1|4)$-invariant constraints. The $OSp(1|4)$ symmetry
can be exploited to reach a gauge (the soldering gauge) in which the auxiliary fields take constant values. This gauge choice breaks
the supergroup $OSp(1|4)$ to its Lorentz $SO(1,3)$ subgroup, and reproduces the Mac Dowell-Mansouri action \cite{MDM}, equivalent up to boundary terms to the action of usual $N=1$, $D=4$ anti De Sitter supergravity. For this action supersymmetry is not a {\it gauge} symmetry any more, since it gets broken along with the translations (the $SO(2,3)$ boosts). However supersymmetry is still ``alive" in the gauge fixed theory. This can be seen in two distinct ways:
i) by solving the supertorsion constraint and passing to second order formalism (expressing the spin connection in terms of the vielbein and the gravitino fields) \cite{FFvN};
ii) or, remaining in first order formalism, by an appropriate modification of the spin connection supersymmetry variation \cite{DZ}.
Note that the supersymmetry transformations leaving
the gauge fixed action invariant do not close off-shell (whereas the
$OSp(4|1)$ gauge variations close off-shell by construction).
After $\star$-deforming the product in the $OSp(1|4)$ supergravity action, and using the geometric Seiberg-Witten map , the resulting higher derivative theory contains the same fields as the classical theory, and is invariant under the same local $OSp(1|4)$ symmetries.
The reason we start from the $OSp(1|4)$ gauge-invariant theory resides in that all local symmetries (except general coordinate invariance) are contained in a gauge supergroup. The derivation of the Seiberg-Witten map in \cite{SW} is purely algebraic, and nothing changes in the derivation if groups are replaced by supergroups, connections by superconnections etc. In the present paper we apply the map to
$OSp(1|4)$ superconnections and supermatrix (adjoint) auxiliary fields, containing all the fields of $N=1$, $D=4$ supergravity. Thus we are guaranteed that supersymmetry (part of the $OSp(1|4)$ symmetry) survives in the extended theory.
By choosing the same gauge as in the classical theory (the gauge group $OSp(1|4)$ is the same), we obtain an extended theory containing only the vielbein, spin connection and gravitino fields, reducing in the commutative limit to $N=1$, $D=4$ $AdS$ supergravity.
The ``mother", non gauge-fixed extended theory is $OSp(1|4)$-invariant, and as such is a locally supersymmetric higher derivative theory. The price to pay for realizing this local gauge supersymmetry (closing off-shell) is the presence of constrained auxiliary fields.
The plan of the paper is as follows. In Section 2 we briefly review $OSp(1|4)$ supergravity. In Section 3 we recall its manifestly
$OSp(1|4)$-invariant action. The noncommutative deformation is presented in Section 4. Section 5 deals with the geometric Seiberg-Witten map, applied in Section 6 to obtain the extended $OSp(1|4)$ supergravity action to second order in $\theta$. Section 7 contains some conclusions.
\sect{Classical $OSp(1|4)$ supergravity}
\subsection{Geometric MacDowell - Mansouri action}
The Mac Dowell-Mansouri action \cite{MDM} for $N=1$, $D=4$ supergravity can be recast in an index-free form:
\begin{equation}
S = 2i \int Tr( R \wedge R \gamma_5 + 2 \Sigma \wedge \overline \Sigma \gamma_5) \label{action1}
\end{equation}
where the trace is taken on spinor indices, and the $2$-form curvatures $R$ (bosonic) and $\Sigma$ (fermionic) originate from the $1$-form $OSp(1|4)$ connection supermatrix:
\begin{equation}
{\bf \Om} \equiv
\left(
\begin{array}{cc}
\Omega & \psi \\
\bar \psi & 0 \\
\end{array}
\right), ~~~ \Omega \equiv {1 \over 4} \omega^{ab} \gamma_{ab} - {i \over 2} V^a \gamma_a
\label{Omdef}
\end{equation}
whose corresponding $OSp(1|4)$ curvature supermatrix is
\begin{equation}
{\bf R} = d {\bf \Om} - {\bf \Om} \wedge {\bf \Om}~
\equiv \left(
\begin{array}{cc}
R & \Sigma \\
\overline \Sigma & 0 \\
\end{array}
\right) \label{Rdef}
\end{equation}
Immediate matrix algebra yields\footnote{we omit wedge products between forms, and all index contractions involve the Minkowski metric $\eta_{ab}$}:
\begin{eqnarray}
& & R = {1 \over 4} R^{ab} \gamma_{ab} - {i \over 2} R^a \gamma_a \label{defR}\\
& & \Sigma = d \psi - {1 \over 4} \omega^{ab} \gamma_{ab} \psi + {i \over 2} V^a \gamma_a \psi \label{defSigma} \\
& & \overline \Sigma = d \bar \psi - {1 \over 4} \bar \psi \omega^{ab} \gamma_{ab} + {i \over 2} \bar \psi V^a \gamma_a
\end{eqnarray}
\noindent with
\begin{eqnarray}
& & R^{ab} \equiv d \omega^{ab} - \omega^{ac} \omega^{cb} + V^a V^b + {1 \over 2} \bar \psi \gamma^{ab} \psi \\
& & R^{a} \equiv d V^{a} - \omega^{ab} V^{b} - {i \over 2} \bar \psi \gamma^a \psi
\end{eqnarray}
\noindent We have also used the Fierz identity for $1$-form Majorana spinors:
\begin{equation}
\psi \bar \psi = {1 \over 4} ( \bar \psi \gamma^a \psi \gamma_a - {1 \over 2} \bar \psi \gamma^{ab} \psi \gamma_{ab} )
\end{equation}
\noindent (to prove it, just multiply both sides by $\gamma_c$ or $\gamma_{cd}$ and take the trace on spinor indices). The $1$-forms $V^a$, $\omega^{ab}$ and $\psi$ are respectively the vielbein, the spin connection and the gravitino field (a Majorana spinor, i.e. $\bar \psi = \psi^T C$, where $C$ is the charge conjugation matrix).
Carrying out the spinor trace in the action (\ref{action1}) yields the familiar MacDowell-Mansouri action:
\begin{equation}
S = 2 \int {1 \over 4} R^{ab} \wedge R^{cd} \varepsilon_{abcd} - 2i \overline \Sigma \wedge \gamma_5 \Sigma
\end{equation}
After inserting the curvature definitions the action takes the form
\begin{equation}
S = \int {\cal R}^{ab} V^c V^d \varepsilon_{abcd} + 4 \bar \rho \gamma_a \gamma_5 \psi V^a + {1 \over 2} (V^a V^b V^c V^d + 2 \bar \psi \gamma^{ab} \psi V^c V^d ) \epsilon_{abcd} \label{adsSG}
\end{equation}
with
\begin{equation}
{\cal R}^{ab} \equiv d \omega^{ab} - \omega^{ac} \omega^{cb} , ~~\rho \equiv d \psi - {1 \over 4} \omega^{ab} \gamma_{ab} \psi \equiv {\cal D} \psi
\label{RLordef}
\end{equation}
We have dropped the topological term ${\cal R}^{ab} {\cal R}^{cd} \epsilon_{abcd}$ (Euler form) and used
the gravitino Bianchi identity
\begin{equation}
{\cal D} \rho = - {1 \over 4} {\cal R}^{ab} \gamma_{ab}
\end{equation}
and the gamma matrix identity $2 \gamma_{ab} \gamma_5 = i \epsilon_{abcd} \gamma^{cd}$ to recognize that ${1 \over 2} {\cal R}^{ab} \bar \psi \gamma^{cd} \psi \epsilon_{abcd} - 4 i \bar \rho \gamma_5 \rho$
is a total derivative. Bianchi identities are easily obtained by taking the exterior derivative of the curvature definitions
in (\ref{Rdef}), or in (\ref{RLordef}). The action (\ref{adsSG}) describes $N=1$, $D=4$ anti-De Sitter supergravity,
the last term being the supersymmetric cosmological term. After rescaling the vielbein and the gravitino as
$V^a \rightarrow \lambda V^a$, $\psi \rightarrow \sqrt{\lambda} \psi$ and dividing the action by $\lambda^2$, the usual (Minkowski)
$N=1$, $D=4$ supergravity is retrieved by taking the limit $\lambda \rightarrow 0$. This corresponds to the
Inon\"u-Wigner contraction of $OSp(1|4)$ to the superPoincar\'e group.
The action (\ref{action1}) can be rewritten even more compactly using the $OSp(1|4)$ curvature supermatrix ${\bf R}$:
\begin{equation}
S = 4 \int {\rm STr} ( {\bf R} ({\bf 1} + {{\bf \Gamma}^2 \over 2}) {\bf R} {\bf \Gamma}) \label{action2}
\end{equation}
where STr is the supertrace and ${\bf \Gamma}$ is the following constant matrix:
\begin{equation}
{\bf \Gamma} \equiv \left(
\begin{array}{cc}
i \gamma_5 & 0 \\
0 & 0 \\
\end{array}
\right) \label{Gammabold}
\end{equation}
\noindent All boldface quantities are 5 $\times$ 5 supermatrices.
\subsection{$OSp(1|4)$ gauge variations}
The gauge transformation of the connection ${\bf \Om}$
\begin{equation}
\delta_{\boldsymbol {\epsilon}} {\bf \Om} = d \boldsymbol {\epsilon} - {\bf \Om} \boldsymbol {\epsilon} + \boldsymbol {\epsilon} {\bf \Om}
\end{equation}
where $\boldsymbol {\epsilon}$ is the $OSp(1|4)$ gauge parameter:
\begin{equation}
\boldsymbol {\epsilon} \equiv
\left(
\begin{array}{cc}
{1 \over 4} \varepsilon^{ab} \gamma_{ab} - {i \over 2} \varepsilon^a \gamma_a & \epsilon \\
{\bar \epsilon} & 0 \\
\end{array}
\right) \label{gaugeparam}
\end{equation}
becomes, on the component fields entering ${\bf \Om}$:
\begin{eqnarray}
& &
\delta \omega^{ab} = d \varepsilon^{ab} - \omega^{ac} \varepsilon^{cb} + \omega^{bc} \varepsilon^{ca}- \varepsilon^a V^b + \varepsilon^b V^a - {\bar \epsilon} \gamma^{ab} \psi \\
& & \delta V^a = d \varepsilon^a - \omega^{ab} \varepsilon^b + \varepsilon^{ab} V^b + i {\bar \epsilon} \gamma^a \psi \\
& & \delta \psi = d \epsilon - {1 \over 4} \omega^{ab} \gamma_{ab} \epsilon + {i \over 2} V^a \gamma_a \epsilon + {1 \over 4} \varepsilon^{ab} \gamma_{ab} \psi - {i \over 2} \varepsilon^a \gamma_a \psi \label{gaugetransf}
\end{eqnarray}
Similarly from the gauge variation of the curvature ${\bf R}$:
\begin{equation}
\delta_{\boldsymbol {\epsilon}} {\bf R} = - {\bf R} \boldsymbol {\epsilon} + \boldsymbol {\epsilon} {\bf R} \label{Rgauge}
\end{equation}
we find the gauge transformations of the curvature components:
\begin{eqnarray}
& & \delta R^{ab} = - R^{ac} \varepsilon^{cb} + R^{bc} \varepsilon^{ca}- \varepsilon^a R^b + \varepsilon^b R^a - {\bar \epsilon} \gamma^{ab} \Sigma \\
& & \delta R^a = - R^{ab} \varepsilon^b + \varepsilon^{ab} R^b + i {\bar \epsilon} \gamma^a \Sigma \\
& & \delta \Sigma = - {1 \over 4} R^{ab} \gamma_{ab} \epsilon + {i \over 2} R^a \gamma_a \epsilon + {1 \over 4} \varepsilon^{ab} \gamma_{ab} \Sigma - {i \over 2} \varepsilon^a \gamma_a \Sigma
\end{eqnarray}
As is well known, the action (\ref{action2}), although a bilinear in the $OSp(1|4)$ curvature, is {\it not} invariant under the
$OSp(1|4)$ gauge transformations. In fact it is not a Yang-Mills action (involving the exterior product of ${\bf R}$ with its Hodge dual), nor a topological action of the form $\int {\bf R} {\bf R}$: the constant supermatrix ${\bf \Gamma}$ ruins the $OSp(1|4)$ gauge invariance, and breaks it to its Lorentz subgroup. This can be seen easily by noting that the gauge parameter in (\ref{gaugeparam}) commutes with ${\bf \Gamma}$ only when restricted to Lorentz rotations ($\varepsilon^{a} = \epsilon =0$), so that Lorentz rotations indeed leave the action invariant since the supertrace is cyclic. On the other hand a gauge parameter supermatrix containing also translation and/or supersymmetry parameters does not commute with ${\bf \Gamma}$, and therefore the action is not invariant under $OSp(1|4)$ translations or supersymmetry transformations.
However supersymmetry is still there: to see it one needs to modify the $\omega^{ab}$ supersymmetry transformation.
\subsection{Supersymmetry}
The (non vanishing) variation of the action (\ref{action2}) under gauge supersymmetry can be computed rather quickly by using $\delta {\bf R} = [\boldsymbol {\epsilon} , {\bf R}]$ with $\boldsymbol {\epsilon}$ containing only the off-diagonal fermionic supersymmetry parameter $\epsilon$. The result is
\begin{equation}
\delta S = -4 \int R^a \bar \rho \gamma_a \gamma_5 \epsilon
\end{equation}
Now consider instead the variation of the action under an {\it arbitrary} variation of the spin connection $\omega^{ab}$, i.e. the variation that defines the $\omega^{ab}$ field equation. To compute it with a minimum of algebra, first vary (\ref{action2}) with respect to ${\bf \Om}$, and then set $\delta V^a = \delta \psi =0$ in $\delta {\bf \Om}$ as defined by (\ref{Omdef}). The result is
\begin{equation}
\delta S = 16 \int R^a V^b \delta \omega^{cd} \epsilon_{abcd}
\end{equation}
Requesting this variation to vanish for arbitrary $\delta\omega^{ab}$ yields the spin connection
field equation $R^a =0$.
Thus if we consider a supersymmetry variation of the action, where the variation of $\omega^{ab}$ is modified by an extra piece (in addition to its gauge variation):
\begin{equation}
\delta \omega^{ab} = \delta_{gauge} \omega^{ab} + \delta_{extra} \omega^{ab}
\end{equation}
the corresponding variation of the action (\ref{action2}) will be
\begin{equation}
\delta S = - 4 \int R^a ( \bar \rho \gamma_a \gamma_5 \epsilon - 2 \delta_{extra} \omega^{bc} V^d \epsilon_{abcd})
\end{equation}
This variation can be made to vanish in two distinct ways:
\vskip .4cm
\noindent 1) by enforcing the constraint $R^a = 0$, which is really equivalent to the field equation of $\omega^{ab}$. As is well known $R^a =0$ allows to express the spin connection $\omega^{ab}$ in terms of the vielbein and gravitino fields. Substituting back $\omega^{ab} (V, \psi)$ in the action leads to the supersymmetric action of AdS supergravity in second order formalism. In this formalism one never needs to vary the fields inside the ``package" $\omega^{ab} (V, \psi)$, since any variation of $S$ due to $\delta \omega^{ab}$ vanishes identically, being proportional to $R^a$ (then one works in the so-called ``1.5 order formalism").
\vskip .4cm
\noindent 2) by choosing $\delta_{extra} \omega^{ab}$ so that
\begin{equation}
\bar \rho \gamma_a \gamma_5 \epsilon - 2 \delta_{extra} \omega^{bc} V^d \epsilon_{abcd}=0
\end{equation}
This equation can be solved for $\delta_{extra} \omega^{ab}$ in the same way one solves $R^a =0$ for $\omega^{ab}$. The result is
\begin{equation}
\delta_{extra} \omega^{ab} = {1 \over 2} \epsilon^{abcd} (\bar \rho_{de} \gamma_c \gamma_5 \epsilon + \bar \rho_{ec} \gamma_d \gamma_5 \epsilon - \bar \rho_{cd} \gamma_e \gamma_5 \epsilon) V^e
\end{equation}
\noindent where $\bar \rho_{cd}$ are the components along the vielbein basis of the gravitino curvature, i.e. $\bar \rho \equiv \bar \rho_{cd} V^c V^d$.
Thus the first order action (\ref{action2}) is invariant under the supersymmetry transformations, given by eq.s (\ref{gaugetransf}) for the vielbein and the gravitino:
\begin{equation}
\delta V^a = -i {\bar \epsilon} \gamma^a \psi, ~~\delta \psi = d \epsilon - {1 \over 4} \omega^{ab} \gamma_{ab} \epsilon \equiv {\cal D} \epsilon
\end{equation}
and by the modified rule for $\omega^{ab}$:
\begin{equation}
\delta \omega^{ab} = \delta_{gauge} \omega^{ab}+ \delta_{extra} \omega^{ab}= - {\bar \epsilon} \gamma^{ab} \psi + {1 \over 2} \epsilon^{abcd} (\bar \rho_{de} \gamma_c \gamma_5 \epsilon + \bar \rho_{ec} \gamma_d \gamma_5 \epsilon - \bar \rho_{cd} \gamma_e \gamma_5 \epsilon) V^e
\end{equation}
More details can be found for ex. in \cite{PvN1,PvN2}.
\sect{The manifestly $OSp(1|4)$-invariant action}
Can we reformulate supergravity in an explicit $OSp(1|4)$-invariant way? The answer is yes \cite{Cham1,Cham2,DDFR,PV}, and generalizes the $SO(2,3)$ formulation of AdS gravity of ref.s \cite{West,Fre,StelleWest,DDFR}. Indeed looking at (\ref{action2}), we see that promoting the constant matrix ${\bf \Gamma}$ to a field supermatrix ${\bf \Phi}$
transforming under $OSp(1|4)$ as
\begin{equation}
\delta {\bf \Phi} = - {\bf \Phi} \boldsymbol {\epsilon} + \boldsymbol {\epsilon} {\bf \Phi}
\end{equation}
the action $S$ becomes:
\begin{equation}
S = \int {\rm STr} ( {\bf R} ({\bf 1} + {{\bf \Phi}^2 \over 2}) {\bf R} {\bf \Phi}) \label{action3}
\end{equation}
and is manifestly $OSp(1|4)$-invariant. By doing so, we are introducing new, auxiliary fields contained in ${\bf \Phi}$. We have to ensure, however, that a particular gauge choice exists such that ${\bf \Phi}$ reduces to the constant supermatrix ${\bf \Gamma}$: only if this gauge choice exists the theory is equivalent to the one described by (\ref{action2}). To satisfy this requirement we choose ${\bf \Phi}$ in the symmetric (traceless) 5 -dimensional representation of $OSp(1|4)$ \cite{Cham1}:
\begin{equation}
{\bf \Phi} (x) \equiv \left(
\begin{array}{cc}
{1 \over 4} \pi (x) + i \phi (x) \gamma_5 + \phi^a (x) \gamma_a \gamma_5 & \zeta (x) \\
-\bar \zeta (x) & \pi (x) \\
\end{array}
\right) \label{Phidef}
\end{equation}
Now translations and supersymmetries of $OSp(1|4)$ can be used to set $\phi^a$ and $\zeta$ to zero \cite{Cham1}.
Moreover, the $OSp(1|4)$-invariant constraint
\begin{equation}
{\bf \Phi} ^3 + {\bf \Phi} = 0 \label{constraint}
\end{equation}
enforces $\pi =0$ and $\phi = \pm 1$, reducing ${\bf \Phi}$ to the constant supermatrix $ \pm {\bf \Gamma}$ (ignoring the trivial solution ${\bf \Phi} = 0$. If we want to exclude it, we can instead impose the $OSp(1|4)$-invariant constraints $ {\rm STr}({\bf \Phi}^2) = 4 (const)^2$, ${\rm STr}({\bf \Phi}^3)=0$ see ref. \cite{Cham2}).
The simplest way to implement the constraint (\ref{constraint}) is to add a ($OSp(1|4)$-invariant) Lagrange multiplier term in the action:
\begin{equation}
S_{\boldsymbol {\lambda}} = \int {\rm STr} ( \boldsymbol {\lambda} {\bf \Phi} ({\bf \Phi}^2 + {\bf 1}) {\bf \Phi} ~D {\bf \Phi} D {\bf \Phi} D {\bf \Phi} D {\bf \Phi} )
\end{equation}
where the Lagrange multiplier $\boldsymbol {\lambda} (x)$ is proportional to the unit matrix, i.e. $\boldsymbol {\lambda} (x) = \lambda (x) {\bf 1}$, generalizing the analogous term in the $SO(2,3)$-invariant formulation of gravity (see for ex. \cite{StelleWest,Fre}).
Another interesting possibility is to give dynamics (cf. \cite{Cham1}) to the fields $\pi(x)$ and $\phi(x)$ with a potential admitting a stable minimum for the values $\pi =0$ and $\phi = const$. In this paper the constrained auxiliary fields
are considered as {\it background} fields, on the same footing of the background vector fields that define the
$\star$-product (see next Section). We do not introduce Higgs fields to break spontaneously the $OSp(1|4)$
invariance. The breaking of $OSp(1|4)$, and contact with $AdS$ $D=4$ supergravity, is made by explicit gauge fixing.
The $OSp(1|4)$ gauge invariant formulation of $N=1$, $D=4$ anti De Sitter supergravity is our starting point for a noncommutative supersymmetric extension.
\sect{Noncommutative $OSp(1|4)$ supergravity}
\subsection{The NC action}
The NC theory is obtained by a $\star$-deformation of the action in (\ref{action3}):
\begin{equation}
S = \int {\rm STr} ( {\bf R} \star ({\bf 1} + {{\bf \Phi} \star {\bf \Phi} \over 2}) \wedge_{\star} {\bf R} \star {\bf \Phi}) \label{action4}
\end{equation}
where the curvature $2$-form ${\bf R}$ is now:
\begin{equation}
{\bf R} = d {\bf \Om} - {\bf \Om} \wedge_\star {\bf \Om} \label{Rstar}
\end{equation}
and the $\star$-exterior product between forms is defined as
\begin{eqnarray}
& & \tau \we_\star \tau' \equiv \sum_{n=0}^\infty \left({i \over 2}\right)^n \theta^{A_1B_1} \cdots \theta^{A_nB_n}
(\ell_{A_1} \cdots \ell_{{A_n}} \tau) \wedge (\ell_{{B_1}} \cdots \ell_{{B_n}} \tau') \nonumber \\
& & ~~ = \tau \wedge \tau' + {i \over 2} \theta^{AB} (\ell_{A} \tau) \wedge (\ell_{B} \tau') + {1 \over 2!} {\left( i \over 2 \right)^2} \theta^{A_1B_1} \theta^{A_2B_2} (\ell_{{A_1}} \ell_{{A_2}} \tau) \wedge
(\ell_{{B_1}} \ell_{{B_2}} \tau') + \cdots \nonumber \\
\label{defwestar}
\end{eqnarray}
\noindent where $\ell_{A}$ are Lie derivatives along commuting
vector fields $X_A$. This noncommutative product is associative due to $[X_A , X_B]=0$. If the vector fields $X_A$ are chosen to coincide with the partial derivatives
$\partial_\mu$, and if $\tau$, $\tau'$ are $0$-forms, then $\tau\star\tau'$ reduces to the well-known Moyal-Groenewold product
\cite{MoyalGroenewold}.
The $\star$-gauge transformations of the NC fields are:
\begin{eqnarray}
& & \delta_{\boldsymbol {\epsilon}} {\bf \Om} = d \boldsymbol {\epsilon} - {\bf \Om} \star \boldsymbol {\epsilon} + \boldsymbol {\epsilon} \star {\bf \Om} \label{Omstargauge} \\
& & \delta_{\boldsymbol {\epsilon}} {\bf \Phi} = - {\bf \Phi} \star \boldsymbol {\epsilon} + \boldsymbol {\epsilon} \star {\bf \Phi} \label{Phistargauge}
\end{eqnarray}
Recalling the $\star$-gauge transformation of the curvature induced by (\ref{Omstargauge}):
\begin{equation}
\delta_{\boldsymbol {\epsilon}} {\bf R} = - {\bf R} \star \boldsymbol {\epsilon} + \boldsymbol {\epsilon} \star {\bf R} \label{Rstargauge}
\end{equation}
and the cyclicity of the supertrace and of the integral\footnote{twisted differential geometry is treated for ex. in \cite{book}; see the Appendix of \cite{AC2} for a summary.} , the action (\ref{action4}) is manifestly invariant under the $\star$-gauge symmetry.
Because of noncommutativity, the $\star$-symmetry group is enhanced to $U(1,3|1)$ so as to contain all enveloping algebra generators. Thus the NC $1$-form connection is given by
\begin{equation}
{\bf \Om} =
\left(
\begin{array}{cc}
\Omega & \psi \\
\bar \psi & w \\
\end{array}
\right), ~~~ \Omega \equiv {1 \over 4} \omega^{ab} \gamma_{ab} + i \omega I + \tilde \om \gamma_5 - {i \over 2} V^a \gamma_a
- {i \over 2} \widetilde{V}^a \gamma_a \gamma_5
\label{NCOmdef} \end{equation}
and correspondingly the gauge parameter supermatrix $\boldsymbol {\epsilon}$ becomes
\begin{equation}
\boldsymbol {\epsilon} =
\left(
\begin{array}{cc}
\varepsilon & \epsilon \\
{\bar \epsilon} & \eta \\
\end{array}
\right), ~~~ \epsilon \equiv {1 \over 4} \varepsilon^{ab} \gamma_{ab} + i \varepsilon I + \widetilde{\epsi} \gamma_5 - {i \over 2} \varepsilon^a \gamma_a
- {i \over 2} \widetilde{\epsi}^a \gamma_a \gamma_5
\label{NCepsidef} \end{equation}
containing all the gauge parameters of the superalgebra $U(1,3|1)$.
The curvature supermatrix ${\bf R}$,
\begin{equation}
{\bf R}
\equiv \left(
\begin{array}{cc}
R & \Sigma \\
\overline \Sigma & r \\
\end{array}
\right)
\end{equation}
\noindent defined in (\ref{Rstar}), is now given by
\begin{eqnarray}
& & R = d \Omega - \Omega \we_\star \Omega - \psi \we_\star \bar \psi \\
& & \Sigma = d \psi - \Omega \we_\star \psi - \psi \we_\star w \\
& & \overline \Sigma = d \bar \psi - \bar \psi \we_\star \Omega - w \we_\star \bar \psi \\
& & r = dw - \bar \psi \we_\star \psi - w \we_\star w
\end{eqnarray}
\noindent where $R$ has components along the complete Dirac basis.
As usual in NC theories, the algebra of gauge transformations
closes as follows:
\begin{equation}
[\delta_{\boldsymbol {\epsilon}_1}, \delta_{\boldsymbol {\epsilon}_2}] = \delta_{\boldsymbol {\epsilon}_1 \star \boldsymbol {\epsilon}_2 - \boldsymbol {\epsilon}_2 \star \boldsymbol {\epsilon}_1}
\end{equation}
Consistency with the $\star$-gauge transformations requires for the $0$-form ${\bf \Phi}$ a similar expansion:
\begin{equation}
{\bf \Phi} =
\left(
\begin{array}{cc}
\Phi & \zeta \\
-\bar \zeta & \pi \\
\end{array}
\right), ~~~ \Phi \equiv {i \over 4} \phi^{ab} \gamma_{ab} + {1\over 4} \pi I + i\phi \gamma_5 + \phi^a \gamma_a
+ {\widetilde \phi}^a \gamma_a \gamma_5
\label{NCPhidef} \end{equation}
The crucial difference between the two supermatrix fields ${\bf \Om}$ and ${\bf \Phi}$ (besides their different form degree) is their
commutative limit. We will see in Section 6 how the Seiberg-Witten map ensures that, in the $\theta \rightarrow 0$ limit,
${\bf \Om}$ contains only $C$-antisymmetric gamma matrices (cf. (\ref{Omdef})) and ${\bf \Phi}$ only $C$-symmetric gamma matrices
(cf. (\ref{Phidef})).
In analogy with the classical case we also require the $U(1,3|1)$-invariant constraint:
\begin{equation}
{\bf \Phi} \star {\bf \Phi} \star {\bf \Phi} + {\bf \Phi} = 0 \label{constraint2}
\end{equation}
reducing to (\ref{constraint}) for $\theta \rightarrow 0$. In alternative, we can require ${\rm STr} ({\bf \Phi} \star {\bf \Phi}) = 4 (const)^2$,
${\rm STr} ({\bf \Phi} \star {\bf \Phi} \star {\bf \Phi}) = 0$.
\subsection{Hermiticity conditions and reality of the NC action}
In the expansions (\ref{NCOmdef}) and (\ref{NCPhidef}) all fields are taken to be real. This
is equivalent to the relations
\begin{equation}
{\bf \Om}^\dagger = - {\bf \Gamma}_0 {\bf \Om} {\bf \Gamma}_0, ~~ {\bf \Phi}^\dagger = {\bf \Gamma}_0 {\bf \Phi} {\bf \Gamma}_0, ~~ {\bf \Gamma}_0 \equiv
\left(
\begin{array}{cc}
\gamma_0 & 0 \\
0 & -1 \\
\end{array}
\right)
\end{equation}
due to $\gamma_{ab}$ and $\gamma_5$ being $\gamma_0$ antihermitian (i.e. $\gamma_{ab}^\dagger = - \gamma_0 \gamma_{ab} \gamma_0$ etc), while $1$, $\gamma_a$ and $\gamma_a \gamma_5$ are $\gamma_0$ -hermitian. Noting that ${\bf \Gamma}_0^2 = {\bf 1}$, and that the ${\bf \Gamma}_0$-antihermiticity of ${\bf \Om}$ implies ${\bf \Gamma}_0$-antihermiticity of ${\bf R}$, one easily proves that the NC action is real.
\subsection{Charge conjugation invariance}
The NC action is also invariant under substitution of the fields by their charge conjugates
\begin{equation}
{\bf \Om}^c \equiv -{\bf C}^{-1} {\bf \Om}^T {\bf C} ~ \Rightarrow {\bf R}^c = - {\bf C}^{-1} {\bf R}^T {\bf C}, ~~~{\bf \Phi}^c \equiv {\bf C}^{-1} {\bf \Phi}^T {\bf C},~~~~
{\bf C} \equiv
\left(
\begin{array}{cc}
C & 0 \\
0 & 1 \\
\end{array}
\right)
\end{equation}
and simultaneously changing $\theta$ into $-\theta$ in the $\star$-products. Indeed
\begin{eqnarray}
& & S^c = \int {\rm Str}({\bf C}^{-1} {\bf R}^T {\bf C} ({\bf 1} + {1 \over 2} {\bf C}^{-1} {\bf \Phi}^T {\bf C} {\bf C}^{-1} {\bf \Phi}^T {\bf C}) {\bf C}^{-1} {\bf R}^T {\bf C} {\bf C}^{-1} {\bf \Phi}^T {\bf C})_{-\theta} \nonumber\\
& & ~~~~~~= \int {\rm Str}( {\bf R}^T ({\bf 1} + {1 \over 2} {\bf \Phi}^T {\bf \Phi}^T) {\bf R}^T {\bf \Phi}^T )_{-\theta}\nonumber \\
& & ~~~~~~= \int {\rm Str}( {\bf \Phi} {\bf R} ({\bf 1} + {1 \over 2} {\bf \Phi}\Phibold) {\bf R} )^T_{\theta} \nonumber\\
& & ~~~~~~= \int {\rm Str}({\bf R} ({\bf 1} + {1 \over 2} {\bf \Phi}\Phibold) {\bf R} {\bf \Phi} )^T_{\theta}=S
\end{eqnarray}
\noindent using ciclicity of the integral and of the supertrace, and invariance of the supertrace under matrix transposition. We have defined
$(ABC...)_{\theta}$ to be the $\star$-(exterior) product between the forms $A,B,C...$ and
$(ABC...)_{-\theta}$ to be the same product with opposite $\theta$. Note that for ex. $(AB)^T_{\theta} = \pm (B^T A^T)_{-\theta} $ for $A(x),B(x)$
matrix valued fields (the minus sign when $A$ and $B$ are both forms of odd degree), i.e. the transposition acts only on the matrix structure of $A$ and $B$. To interchange the ordering of $A$ and $B$ as functions of $x$ one needs $\theta \rightarrow -\theta$, since
$(f \star g)_\theta = (g \star f)_{-\theta}$, as follows from the definition (\ref{starproduct}).
\sect{The geometric Seiberg-Witten map}
The results of this Section hold for any gauge group. Here we denote by $\widehat{\Omega}$ the NC
gauge field, and by ${\widehat{\varepsilon}}$ the NC gauge parameter.
The Seiberg-Witten map relates $\widehat{\Omega}$
to the ordinary $\Omega$, and
${\widehat{\varepsilon}}$ to the ordinary $\varepsilon$ so as to
satisfy:
\begin{equation}
\widehat{\Omega} (\Omega) + {\widehat{\delta}}_{\widehat{\varepsilon}}
\widehat{\Omega} (\Omega) = \widehat{\Omega} (\Omega + \delta_\varepsilon \Omega)
\label{SWcondition}
\end{equation}
with
\begin{eqnarray}
& &
\delta_\varepsilon \Omega_\mu = \partial_\mu \varepsilon + \varepsilon \Omega_\mu - \Omega_\mu
\varepsilon~, \\
& &
{\widehat\delta} _{\widehat{\varepsilon}}{} \widehat{\Omega}_\mu = \partial_\mu {\widehat{\varepsilon}} + {\widehat{\varepsilon}} \star \widehat{\Omega}_\mu - \widehat{\Omega}_\mu \star
{\widehat{\varepsilon}}~.
\end{eqnarray}
\noindent In words: the dependence of the noncommutative gauge field on the ordinary gauge field is fixed
by requiring that ordinary gauge variations of $\Omega$ inside $\widehat{\Omega}(\Omega)$ produce the noncommutative
gauge variation of $\widehat{\Omega}$.
Similarly noncommutative ``matter fields" are related to the commutative
ones by requiring
\begin{equation}
\widehat{\Phi}(\Phi,\Omega) + {\widehat{\delta}}_{\widehat{\varepsilon}}
\widehat{\Phi} (\Phi,\Omega) = \widehat{\Phi} (\Phi + \delta_\varepsilon \Phi, \Omega+\delta_\varepsilon\Omega) ~.
\label{SWcondition1}
\end{equation}
The conditions (\ref{SWcondition}), (\ref{SWcondition1}) are satisfied if the following differential equations in the
noncommutativity parameter $\theta^{AB}$ hold \cite{SW, AC3}:
\begin{eqnarray}
& &{ \partial ~~\over \partial \theta^{AB}} \widehat{\Omega} = {i \over 4} \{ \widehat{\Omega}^{}_{[A}, \ell^{}_{B]} \widehat{\Omega} +\widehat{R}^{}_{B]} \}_\star ~,\label{diffeqOm} \\
& & { \partial ~~\over \partial \theta^{AB}} \widehat{\Phi} = {i \over 4} \{ \widehat{\Omega}_{[A}, {I\!\!L}^{}_{B]} \widehat{\Phi} \}_\star ~,\label{diffeqphi} \\
& & { \partial ~~\over \partial \theta^{AB}} {\widehat{\varepsilon}} = {i \over 4} \{ \widehat{\Omega}_{[A}, \ell^{}_{B]} {\widehat{\varepsilon}} \}_\star ~,\label{diffeqepsi}
\end{eqnarray}
\noindent where:\\
$\bullet$ $\widehat{\Omega}_A$, $\widehat{R}_A$ are defined as the contraction $i_A$ along the tangent
vector $X_A$ of the exterior forms $\widehat{\Omega}$, $\widehat{R}$, i.e. $\widehat{\Omega}_A\equiv i_A\widehat{\Omega}$,
$\widehat{R}_A \equiv i_A \widehat{R}$.\\[.2em]
$\bullet$ The bracket $[A\, B]$ denotes antisymmetrization of the indices $A$ and $B$
with weight 1, so that for example
$\widehat{\Omega}_{[A} \widehat{R}_{B]} =\frac{1}{2}(\widehat{\Omega}_A\widehat{R}_B-\widehat{\Omega}_B\widehat{R}_A)$.
The bracket $\{~,~\}_\star$ is the usual $\star$-anticommutator, for
example
$\{\Omega_A,R_B\}_\star=\Omega_A\star R_B+R_B\star\Omega_A$. \\[.2em]
$\bullet$ The second differential equation holds for fields transforming
in the adjoint representation. Notice that $\widehat{\Phi}$ can also be an
exterior form. The ``fat" Lie derivative ${I\!\!L}_B$ is defined by
${I\!\!L}_B\equiv \ell_B+L_B\,$ where $L_B$ is the covariant Lie derivative along the
tangent vector $X_B$; it acts on the field $\widehat{\Phi}$ as
$$L_B \widehat{\Phi} = \ell_B \widehat{\Phi} - [\widehat{\Omega}_B , \widehat{\Phi}]_\star\;,$$ with
$[\widehat{\Omega}_B , \widehat{\Phi}]_\star=\widehat{\Omega}_B \star\widehat{\Phi}- \widehat{\Phi}\star\widehat{\Omega}_B$.
In fact the covariant Lie derivative $L_B$ can be written in Cartan form:
\begin{equation}
L_B = i_B D + D i_B~,
\end{equation}
where $D$ is the covariant derivative.
\vskip .4cm
The differential equations (\ref{diffeqOm})-(\ref{diffeqepsi}) hold for any abelian twist defined by arbitrary commuting vector
fields $X_A$ \cite{AC3}. They reduce to the usual Seiberg-Witten
differential equations \cite{SW} in the case of a Moyal-Groenewold twist, i.e. when $X_A\to \partial_\mu$.
\vskip .4cm
We can solve these differential equations order by order in $\theta$
by expanding $\widehat{\Omega}$, ${\widehat{\varepsilon}}$ and $\widehat{\Phi}$ in power series of $\theta$
\begin{eqnarray}
& & \widehat{\Omega}=\Omega+ \Omega^1+\Omega^2\;\ldots+\Omega^n\ldots \\
& & {\widehat{\varepsilon}}= \varepsilon +\varepsilon^1 + \varepsilon^2 \;\ldots + \varepsilon^n\ldots \\
& & \widehat{\Phi}=\Phi+\Phi^1+\Phi^2\;\ldots +\Phi^n\ldots
\end{eqnarray}
where the fields $\Omega^n$, $\varepsilon^n$ and $\Phi^n$ are homogeneous polynomials in
$\theta$ of order $n$.
By multiplying the differential equations by $\theta^{AB}$ and using the
identities
$\theta^{AB}\frac{\partial}{\partial
\theta^{AB}}\Omega^{n+1}=(n+1)\Omega^{n+1}$ and similar for $\varepsilon^{n+1}$ and $\Phi^{n+1}$,
we obtain the recursive relations
\begin{eqnarray}
& & \Omega^{n+1} = \frac{i\,\theta^{AB}}{4(n+1)} \{\widehat{\Omega}_A, \ell_B \widehat{\Omega} +\widehat{R}_B\}_\star^{n} \label{Omn+1} ~,\\
& & \Phi^{n+1} =\frac{i\,\theta^{AB}}{4(n+1)} \{\widehat{\Omega}_A, {I\!\!L}_B \widehat{\Phi} \}_\star^{n} \label{phin+1}~, \\
& & \varepsilon^{\;n+1} =\frac{i\,\theta^{AB}}{4(n+1)} \{\widehat{\Omega}_A, \ell_B
{\widehat{\varepsilon}} \}_\star^{n} \label{epsin+1}~,
\end{eqnarray}
where for any field $P$ (also composite, as for ex. $\{\widehat{\Omega}_A,
{I\!\!L}_B \widehat{\Phi} \}_\star$),
$P^n$ denotes its component of
order $n$ in $\theta$. These recursion relations reduce to the ones found in ref. \cite{Ulker}
in the special case of a Moyal twist.
In the following we omit the hat
denoting noncommutative fields, the $\star$ and $\wedge_\star$
products, and simply write $\{~,~\}$, $[~,~]$ for
$\{~,~\}_\star$, $[~,~]_\star$.
If $P$ and $Q$ are forms in the adjoint representation of the gauge group (i.e. if $\delta_\varepsilon P = - P \varepsilon + \varepsilon P$ etc.) the following recursion relation for the product $PQ$ holds \cite{ACD2}:
\begin{equation}
(PQ)^{n+1}=\frac{i\,\theta^{AB}}{4(n+1)}\Big(\{\Omega_A,{I\!\!L}_B(P Q)\}
+2L_A P \, L_B Q
\Big)^{n}~\label{RRCF}.
\end{equation}
Some other useful identities are \cite{ACD2}:
\begin{eqnarray}
& & \theta^{AB} L_A L_B P = - {1 \over 2} \theta^{AB} \{R_{AB},P \} \\
& & \theta^{AB} {I\!\!L}_A \Omega_B = \theta^{AB} R_{AB} \\
& & \theta^{AB}\int Tr\Big( \{\Omega_A,I\!\!L_B(PQ)\} +2L_AP\,L_B
Q\,\Big)=\,\theta^{AB}\int Tr \Big(\{R_{AB}, P\}Q\Big)~
\end{eqnarray}
\noindent where $R_{AB} \equiv i_B i_A R $. Finally, using (\ref{RRCF}) one can find the recursion relation for the curvature:
\begin{equation}
R^{n+1}
=\frac{i\,\theta^{AB}}{4(n+1)}\Big(\{\Omega_A,{I\!\!L}_B R\} -
[R_A,R_B]\Big)^{n} \label{Rn+1}
\end{equation}
Some basic formulae of Cartan calculus, used in deriving the above identities, are listed in Appendix A.
\vskip .4cm
We list below the first order corrections to the classical
$OSp(1|4)$ fields and curvatures, obtained by using the general recursion formulas (\ref{Omn+1}), (\ref{phin+1}) and (\ref{Rn+1}) for $n=0$. On the right-hand sides all products are ordinary exterior products, and all fields are classical.
\subsection{$OSp(1|4)$ fields and curvatures at first order in $\theta$}
\noindent {\bf ${\bf \Om}$ connection components}
\begin{eqnarray}
& & \Omega^1={i \over 4}\theta^{AB} (\{ \Omega_A,\ell_B \Omega + R_B \} + \psi_A \ell_B \bar \psi
+ \ell_B \psi \bar \psi_A + \psi_A \overline \Sigma_B + \Sigma_B \bar \psi_A ) \\
& & \psi^1 = {i \over 4}\theta^{AB} (\Omega_A \ell_B \psi + \ell_B \Omega \psi_A
+ \Omega_A \Sigma_B +R_B \psi_A) \\
& & w^1 = {i \over 4}\theta^{AB} (\bar \psi_A \ell_B \psi + \ell_B \bar \psi \psi_A + \bar \psi_A \Sigma_B+ \overline \Sigma_B \psi_A)
\end{eqnarray}
\noindent {\bf ${\bf \Phi}$ field components}
\begin{eqnarray}
& & \Phi^1 = -{1 \over 4}\theta^{AB} (\{ \Omega_A , \gamma_5 \Omega_B - \Omega_B \gamma_5 \} +
\psi_A \bar \psi_B \gamma_5) \\
& & \zeta^1 = - {1 \over 4}\theta^{AB}(2 \Omega_A \gamma_5 \psi_B + \gamma_5 \Omega_B \psi_A) \\
& & \pi^1 = -{1 \over 2}\theta^{AB} (\bar \psi_a \gamma_5 \psi_B)
\end{eqnarray}
\noindent {\bf ${\bf R}$ curvature components}
\begin{eqnarray}
& & R^1 = {i \over 4}\theta^{AB} (\{\Omega_A , {I\!\!L}_B R \}
- [R_A,R_B]+ \{\Omega_A,-\psi_B \overline \Sigma + \sigma \bar \psi_B \}
\nonumber\\
& & ~~~~~~+ \psi_A {I\!\!L}_B \overline \Sigma + {I\!\!L}_B \Sigma \bar \psi_A - \psi_A \bar \psi_B R + R \psi_B \bar \psi_A - 2 \Sigma_A \overline \Sigma_B)\\
& & \Sigma^1 = {i \over 4}\theta^{AB} (\Omega_A ({I\!\!L}_B \Sigma +
R \psi_B) - \psi_A (\bar \psi_B \Sigma + \overline \Sigma \psi_B )+
(\Sigma \bar \psi_B - \psi_B \overline \Sigma) \psi_A \nonumber\\
& &~~~~~~ + {I\!\!L}_B R \psi_A - 2 R_A \Sigma_B) \\
& & r^1={i \over 4}\theta^{AB}(\bar \psi_A {I\!\!L}_B \Sigma + {I\!\!L}_B \overline \Sigma \psi_A + 2 \bar \psi_A R \psi_B - 2 \overline \Sigma_A \Sigma_B)
\end{eqnarray}
\sect{The extended $OSp(1|4)$ supergravity action}
We now discuss the $\theta$ expansion of the NC action (\ref{action4}) , where the NC supermatrix fields
${\bf \Om}$ and ${\bf \Phi}$ have been substituted by their SW expansion in terms of the classical fields.
\subsection{The action is even in $\theta$}
We first note that the SW map is such that:
\begin{eqnarray}
& & {\bf \Om}_\theta^c \equiv -{\bf C}^{-1} {\bf \Om}^T_\theta {\bf C} = {\bf \Om}_{-\theta}, ~~~ \Rightarrow {\bf R}_\theta^c = - {\bf C}^{-1} {\bf R}^T_\theta {\bf C} = {\bf R}_{-\theta} \label{CCrel1} \\
& & {\bf \Phi}_\theta^c \equiv {\bf C}^{-1} {\bf \Phi}^T_\theta {\bf C} = {\bf \Phi}_{-\theta}
\label{CCrel2} \end{eqnarray}
\noindent where the $\theta$ dependence is explicitly indicated as a subscript. The proof by induction, using (\ref{Omn+1}) and (\ref{phin+1}), is straightforward. Suppose that relations (\ref{CCrel1}) hold up to order
$\theta^n$. Then
\begin{eqnarray}
& &- {\bf C}^{-1} ({\bf \Om}^T)^{n+1}_\theta {\bf C} = \nonumber \\
& &~~~~~~ = {-i \theta^{AB} \over 4(n+1)} ({\bf C}^{-1} {\bf \Om}^T_\theta {\bf C} {\bf C}^{-1} (\ell_B {\bf \Om} + {\bf R})^T_\theta {\bf C} + {\bf C}^{-1}(\ell_B {\bf \Om} + {\bf R})^T_\theta {\bf C} {\bf C}^{-1}{\bf \Om}^T_\theta {\bf C} )^n
\nonumber \\
& & ~~~~~~ = {-i \theta^{AB} \over 4(n+1)} ({\bf \Om}_{-\theta} (\ell_B {\bf \Om} + {\bf R})_{-\theta} + (\ell_B {\bf \Om} + {\bf R})_{-\theta} {\bf \Om}_{-\theta})^n \nonumber \\
& & ~~~~~~ = {\bf \Om}_{-\theta}^{n+1}
\end{eqnarray}
Similarly one proves (\ref{CCrel2}). Exploiting now the invariance of the NC action $S$ under charge conjugation, proved in Section 4, and using the relations (\ref{CCrel1}) and (\ref{CCrel2}) one finally finds:
\begin{equation}
S_\theta = S^c_\theta = S_{-\theta}
\end{equation}
i.e. the NC action $S$ {\it is even } in $\theta$. Therefore the $\theta$-expansion of $S$ has the form:
\begin{equation}
S = S^0 + S^2 + S^4 + \cdots
\end{equation}
and the first nonvanishing correction to the classical action $S^0$ is at order $\theta^2$.
Note that the relations (\ref{CCrel1}) and (\ref{CCrel2}) imply the following conditions on the NC component fields:
\begin{eqnarray}
& & \omega^{ab}_\theta=\omega^{ab}_{-\theta},~~ V^{a}_\theta=V^{a}_{-\theta}, ~~C \bar \psi^T_\theta = \psi_{-\theta}\\
& & \omega_\theta=- \omega_{-\theta}, ~~ \tilde \om_\theta= - \tilde \om_{-\theta}, ~~ \widetilde{V}^{a}_\theta= - \widetilde{V}^{a}_{-\theta}
\end{eqnarray}
\noindent and
\begin{eqnarray}
& & \pi_\theta= \pi_{-\theta}, ~~ \phi_\theta= \phi_{-\theta}, ~~ {\widetilde \phi}^{a}_\theta= {\widetilde \phi}^{a}_{-\theta},~~C \bar \zeta^T_\theta = \zeta_{-\theta} \\
& & \phi^{ab}_\theta= - \phi^{ab}_{-\theta},~~ \phi^{a}_\theta= - \phi^{a}_{-\theta}
\end{eqnarray}
\noindent where the $\theta$ dependence of the NC fields is indicated with a subscript. Thus in the limit $\theta \rightarrow 0$
we see that only $\omega^{ab}$, $V^a$ and $\psi$ survive in ${\bf \Om}$, and only $\pi$, $\phi$, ${\widetilde \phi}^a$ and $\zeta$ survive
in ${\bf \Phi}$, in agreement with the classical fields in (\ref{Omdef}) and (\ref{Phidef}). Finally we recall that $C \bar \psi^T_\theta = \psi_{-\theta}$
(and similar for $\zeta$) is the noncommutative definition for a Majorana spinor \cite{AC1,AC2}, consistent with the NC
gauge transformations and reducing to the usual definition for $\theta=0$.
\subsection{The action at order $\theta^2$}
We can compute the $\theta^2$ correction with the help of the recursion relations (\ref{RRCF}) for composite fields, and the identities
at the end of section 5. The result reads:
\begin{equation}
S^2 = S^2_{RR\Phi} + S^2_{R \Phi \Phi R \Phi}
\end{equation}
\noindent with
\begin{eqnarray}
& & S^2_{RR\Phi} = - {1 \over 16} \theta^{AB} \theta^{CD}
\int {\rm STr} (R_{AB} R_{CD} RR\Phi + {1 \over 2} \{R_{CD},RR \} R_{AB} \Phi-2 R_{AC}R_{BD} \{RR,\Phi\} \nonumber \\ & & ~~+ \{R_{AB},L_CR\} L_D R \Phi + \{R_{AB}, \Phi \} L_C R L_D R \Phi + 2 \{R_{AC},L_D R \} [L_B R ,\Phi] \nonumber \\ & & ~~ - \{R_{CD},R_A R_B \} R \Phi - \{R_{CD} , R \}R_A R_B \Phi - R_{AB} \{R_C R_D , R \}\{ \Phi, R_{AB}\} \nonumber \\
& & ~~+ R_{AB} L_C (RR) L_D \Phi + RR L_C R_{AB} L_D \Phi - 2 L_A (R_C R_D) \{L_B R . \Phi \}+ 2 L_A R (L_C L_B R ) L_D \Phi \nonumber \\
& & ~~ - L_C R_A L_D R_B R \Phi - 2 R L_C (R_A R_B) L_D \Phi - 2 R
L_C R_A L_D R_B \Phi \nonumber \\
& & ~~ +2 i_A (R_C R_D) ( \{R_B,R \Phi \}+ [R_B, \Phi R]) + 2 R_A R_B L_C R L_D \Phi + 4 R_A R_B R_C R_D \Phi)
\end{eqnarray}
\begin{eqnarray}
& & S^2_{R \Phi \Phi R \Phi} = - {1 \over 16} \theta^{AB} \theta^{CD} \int {\rm STr} (( {1 \over 2} \{R_{CD}, \{R_{AB},R \Phi \Phi \} \} - \{ R_{CD} , \{ R_A R_B \Phi , \Phi \} \} \nonumber \\
& & ~ + \{ R_{CD}, L_A R \{ \Phi , L_B \Phi \} \} + \{ R_{CD} , R L_A \Phi L_B \Phi \} + \{ R_{CD} , \Phi L_A R L_B \Phi \} - \{ \{ R_{AC}, R_{BD} \}, R \Phi\Phi \} \nonumber \\
& & ~ + [L_C R_{AB} , L_D (R \Phi\Phi)]+ \{ R_{AB}, L_C R L_D(\Phi\Phi)- R_C R_D \Phi\Phi + R L_C \Phi L_D \Phi \} \nonumber \\
& & ~- [L_C (R_A R_B ) , L_D \Phi] - \{L_C R_A L_D R_B \Phi, \Phi \} + \{ [ i_A (R_C R_D ) , R_B \Phi] , \Phi \} + L_C L_A (R \Phi) L_D L_B \Phi \nonumber \\
& & ~ + ((L_CL_A R)L_D \Phi + \{R_{AC}, L_D R \} \Phi - L_A(R_C R_D\Phi))L_B \Phi + L_A (R \Phi )\{R_{BC}, L_D \Phi \} \nonumber \\
& & ~+( L_CRL_DL_A \Phi L_B \Phi + R \{R_{AC},L_D \Phi \} L_B \Phi + [L_C(L_ARL_B\Phi),L_D \Phi] \nonumber \\
& & ~ + \{ L_C L_A R L_D L_B \Phi + \{ R_{AC} , L_D R \} L_B \Phi - L_A (R_CR_D)L_B \Phi + L_AR \{R_{BC}, L_D \Phi \} , \Phi \} ) R \Phi \nonumber \\
& & ~+2 ({1 \over 2} \{R_{AB}, R\Phi\Phi \} - \{R_AR_B \Phi , \Phi \} + L_A(R \Phi) L_B \Phi+ \{L_AR LL_B \Phi, \Phi \})(L_CR L_D\Phi - R_C R_D \Phi) ) \nonumber \\
\end{eqnarray}
Here all products are ordinary exterior products between classical fields. These corrections to the classical $OSp(1|4)$ action are invariant under local (ordinary) $OSp(1|4)$ gauge variations, as is manifest since all quantities appearing in $S^2$ are gauge covariant, and transform in the adjoint (i.e. as commutators with the gauge parameter). The SW map is designed to ensure this invariance: to find explicitly gauge invariant corrections, order by order in $\theta$, is a powerful check on the computations.
To recover the usual $N=1$, $D=4$ AdS supergravity (without auxiliary fields) in the $\theta \rightarrow 0$ limit one still needs to break $OSp(1|4)$ to its Lorentz subroup. This is done exactly as in the classical case, by choosing the gauge where ${\bf \Phi}$ becomes the constant supermatrix ${\bf \Gamma}$ defined in (\ref{Gammabold})
(the constrained auxiliary fields take constant values). This gauge breaks translations and supersymmetry. We have seen how supersymmetry can be uncovered in the classical ($\theta=0$) gauge fixed action. The question whether a hidden supersymmetry is present also in the gauge fixed extended ($\theta \neq 0$) action is left to future investigations.
\sect{Conclusions}
The fascinating idea that (super)gravity has some kind of conformal phase, before breaking occurs and dimensionful constants emerge is rather old and the $OSp(4|1)$ actions we have been discussing are part of this idea.
The result we have presented here is a noncommutative extension of $OSp(4|1)$ supergravity, the novelty being on one
side a $D=4$ {\it supergravity} action $S$ invariant under local $\star$-supersymmetry (part of the supergroup noncommutative gauge symmetry), and on the other side explicit invariance of $S$ under diffeomorphisms, thanks to a geometrical formulation of abelian twists. Previous works have addressed noncommutative extensions of Mac Dowell-Mansouri gravity actions, but without treating their supersymmetric versions.
We have then used
a generalization of the Seiberg-Witten map (adapted to abelian twists and suitably ``geometrized"), obtaining a higher derivative $D=4$ supergravity, with constrained auxiliary fields, invariant under the {\it usual} gauge transformations of the whole supergroup $OSp(1|4)$. Recursion formulae for the SW higher order corrections have been applied to
compute the $\theta^2$ correction to the classical $OSp(1|4)$ action.
In short, noncommutativity has been used as a guide to construct an extended, locally supersymmetric higher derivative theory with the same symmetries of its classical $\theta \rightarrow 0$ limit.
\vskip .4cm\sk
\noindent {\bf Acknowledgements}
\vskip .4cm
\noindent We thank Paolo Aschieri for useful comments on the manuscript.
|
train/arxiv
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BkiUeEXxK0-nUKDhrHUY
| 5 | 1 |
\section{Introduction}
\label{sec:intro}
True muonium is the as-yet undiscovered $(\ensuremath{\mu^+\mu^-})$ bound state. Its
spectrum, with lifetimes in the range of ps to
ns~\cite{Brodsky:2009gx}, is well defined since the 2.2 $\mu$s
weak-decay lifetime of the muon is much longer. The levels and
transitions of true muonium are dominated by QED effects because its
purely leptonic nature relegates the influence of QCD to vacuum
polarization, where it contributes a small effect at
$\mathcal{O}(\alpha^5)$~\cite{Jentschura:1997tv,Jentschura:1997ma}.
Electroweak effects are suppressed further, and become relevant only
when all $\mathcal{O}(\alpha^7)$ terms are
considered~\cite{PhysRevD.91.073008}. The existing discrepancies in
muon physics (the muon anomalous magnetic moment
$(g-2)_\mu$~\cite{PhysRevD.73.072003}, the proton charge radius
$r_P$~\cite{Antognini:1900ns}, $B^+\rightarrow K^+\ensuremath{\ell^+\ell^-}$
decays~\cite{Aaij:2014ora}, $\bar{B}^0 \to
D^{*+}\ell^{-}\bar{\nu}_{\ell}$~\cite{Aaij:2015yra}) motivate a
serious investigation of true muonium, which has been shown to have
strong discriminating power among alternative resolutions to these
anomalies~\cite{TuckerSmith:2010ra,
PhysRevD.91.073008,Lamm:2015fia,Lamm:2015gka}. Using the methods
developed in this paper, nonperturbative corrections to bound states
from these new physics proposals could be investigated through the
inclusion of new matrix elements, allowing for more stringent
constraints than those obtained through conventional perturbative
studies.
The atom's non-observation to date is due to difficulties in producing
associated low-energy muon pairs, as well as its short lifetime. Many
proposed methods of production exist~\cite{Nemenov:1972ph,
Moffat:1975uw,Holvik:1986ty,Kozlov:1987ey,Ginzburg:1998df,
ArteagaRomero:2000yh,Brodsky:2009gx,Chen:2012ci,Banburski:2012tk,
Ellis:2015eea}. The Heavy Photon Search (HPS) experiment in 2016 will
begin a search for true muonium at a fixed
target~\cite{Celentano:2014wya,Banburski:2012tk}. Additionally, the
DImeson Relativistic Atom Complex (DIRAC) might observe the atom in an
upgraded run~\cite{Benelli:2012bw,dirac}. Given enough statistics,
DIRAC could obtain a value for the Lamb shift using methods developed
for ($\pi^+\pi^-$)~\cite{Nemenov:2001vp}. These experiments produce
relativistic true muonium: In general, the $\mu^+$ and $\mu^-$ are
produced relativistically, both with respect to the lab frame and each
other. Unfortunately, instant-form (conventional fixed-time) wave
functions are not functions of boost-invariant variables (because the
$\mu^+$ and $\mu^-$ rest frames are not the same); thus, production
and decay rates can be modified. To reduce this uncertainty, we
produce boost-invariant wave functions through light-front
techniques~\cite{Lamm:2013oga,Lamm:2016ong}.
To establish the context for the work presented here, a discussion of
the history of this and related problems is appropriate. Weinberg,
interested in the \textit{infinite-momentum frame} (in which a state's
momentum component $p_z\rightarrow\infty$), discovered in the case of
the $\phi^3$ theory that creating or annihilating particles from the
vacuum was forbidden~\cite{Weinberg:1966jm}. This observation
eventually led to the understanding that the vacuum of such a field
theory is trivial ({\it i.e.}, empty of ordinary particles), and that
Fock states with fixed particle content are well-defined. Instead of
taking the infinite-momentum limit of instant-form field theory, one
can obtain equivalent results by quantizing at fixed values of
light-front time $x^+ \equiv t + z$ (called {\it front
form})~\cite{Dirac:1949cp}. In front form, one is able to develop a
rigorous, closed-form Hamiltonian formalism~\cite{Brodsky:1997de}. In
this formalism, an analogue of the Schr\"{o}dinger equation exists,
since an infinite but denumerable set of coupled integral equations
for eigenstates of the Hamiltonian occurs. The front form admits a
perturbation theory, and its Feynman rules were derived by Kogut and
Soper~\cite{Kogut:1969xa}. Because of the inequivalent nature of
instant-form and front-form quantization, it has been a crucial, but
highly nontrivial, matter to show that the traditional instant-form
calculations give equivalent results to those from the front
form~\cite{Ligterink:1994tm,
Chang:1972xt,Yan:1973qf,Chang:1973qi,Bakker:2002ih,Bakker:2005mb,
Misra:2005dt,Bakker:2006pn,Patel:2010vd}.
Perturbative front-form methods have shown success in the study of
non-Abelian gauge theories. In QCD, these methods have been used to
obtain results for exclusive processes by Lepage and
Brodsky~\cite{Lepage:1980fj}, where equivalent instant-form
expressions did not exist. Analytical results using light-front
techniques have also reproduced the correct leading-order Lamb shift
and hyperfine splitting for QED bound
states~\cite{Jones:1996pz,Jones:1996vy, Jones:1997cb,Jones:1997iu}.
The Yukawa theory has been used to understand the differences between
instant-form and front-form approaches and how they can be
reconciled~\cite{Glazek:1992bs,
Bakker:2006pn,Bakker:2002ih,ManginBrinet:2001tc,ManginBrinet:2003nm}.
The complete Standard model has also been formulated in light-front
quantization~\cite{PhysRevD.66.045019}.
As mentioned, the existence of a closed-form Hamiltonian allows for a
Schr\"odinger-like equation that can be expressed in an
infinite-dimensional Fock space, which can be used to solve
nonperturbative field theory, and allows techniques from
nonrelativistic quantum mechanics to be applied to quantum field
theory. To make these problems tractable, the infinite set of coupled
equations must be truncated in a suitable way. In analogy to results
in instant form, these truncations can produce divergent results and
must be regularized to obtain sensible answers. The topic of how to
renormalize such a Hamiltonian was first considered
in~\cite{Glazek:1993rc}. One method proceeds by truncating the Fock
space to a finite number of states based on particle content. In this
truncation, renormalization is possible through Fock state
sector-dependent counterterms~\cite{Karmanov:2001te,
ManginBrinet:2001tc, Carbonell:2002xs,ManginBrinet:2003nm,
Karmanov:2008br,Karmanov:2010ih, Karmanov:2010zz,Karmanov:2012aj},
Pauli-Villars regulators~\cite{Chabysheva:2009vm,Chabysheva:2010vk,
Malyshev:2013eca,Hiller:2015bic,Chabysheva:2015vga}, or the use of
flow equations~\cite{Gubankova:1998wj,Gubankova1998,Gubankova:1999cx,
Gubankova:2000cia,Gubankova:2000ci}. While each method works in
principle, the practical difficulty of renormalizing nonperturbative
Hamiltonians remains daunting.
In order to solve these field theories numerically, the Fock states
are furthermore discretized in momentum Fourier modes on a lattice, a
method called Discretized Light Cone Quantization (DLCQ). This method
was pioneered by Pauli and Brodsky, working with a 1+1-dimensional
Yukawa theory~\cite{Pauli:1985ps}. The special feature of
super-renormalizability of field theories in 1+1 has been particularly
amenable to DLCQ, and these theories have been investigated in depth.
Sawicki used the method to solve scalar $\rm
QED_{1+1}$~\cite{Sawicki:1985uq,Sawicki:1985vs}, while Harindranath
and Vary investigated the structure of the vacuum and bound states of
$\phi^3_{1+1}$ and $\phi^4_{1+1}$ models~\cite{Harindranath:1987db,
Harindranath:1988zt,Harindranath:1988zs}. Pushing further,
Hornbostel {\it et al.\/} presented results for the meson and baryon
eigenstates of $\rm QCD_{1+1}$~\cite{Hornbostel:1988fb}, while Swenson
and Hiller studied more field-theoretical properties of the
light-front in the Wick-Cutkosky model~\cite{Swenson:1993ci}. The
Schwinger model, which admits analytical solutions in both instant
form and front form, was first studied by Eller {\it et al.\/} in
1986~\cite{Eller:1986nt}, and since has become an important test bed
for developing improvements that can then be used in other
theories~\cite{Heinzl:1991dw,
McCartor:1991gn,McCartor:1994im,McCartor:1996pe,Nakawaki:1999ee,
Strauss:2008zx,Strauss:2009zza,Strauss:2010zza}.
Since DLCQ produces both the wave functions and the energy levels,
Hiller was able to compute the $R$-ratio in $\rm
QED_{1+1}$~\cite{Hiller:1990xy}. In one spatial dimension, DLCQ has
also been applied to solving 't~Hooft's model of large-N
QCD~\cite{vandeSande:1996mc}, adjoint QCD~\cite{Trittmann:2000uj,
Trittmann:2001dk,Trittmann:2015oka}, and supersymmetric
models~\cite{Matsumura:1995kw,Antonuccio:1998kz,Antonuccio:1998jg,Antonuccio:1998tm,
Antonuccio:1998mq,Antonuccio:1998zp,Antonuccio:1998mw,Antonuccio:1998zu,
Antonuccio:1999jj,Haney:1999tk,Lunin:1999ib,Antonuccio:1999iz,Lunin:2000im,
Pinsky:2000rn,Hiller:2000nf,Hiller:2001mh,Hiller:2001tv,Hiller:2002cg,
Hiller:2002pj,Hiller:2003qe,Hiller:2003jd,Hiller:2004rb,Hiller:2005vf,
Hiller:2007sc,Trittmann:2009dw}. Although spontaneous symmetry
breaking is manifested in a distinctly different way in 1+1, it is
also possible to study using DLCQ~\cite{Pinsky:1993yi,Pinsky:1994si,
Bender:1992yd}. Finally, research has been undertaken using DLCQ to
test Maldecena's AdS/CFT conjecture in 1+1
theories~\cite{Antonuccio:1999iz,Hiller:2000nf, Hiller:2005vf}.
Extending DLCQ beyond 1+1 dimensions is complicated in two ways:
first, higher-dimensional theories require regularization and
renormalization, as discussed above. Second, the number of Fock
states grows so rapidly that tractable numerical calculations allow
only a small number of states to be included. Despite these
difficulties, DLCQ was applied first to positronium by Tang {\it et
al.}~\cite{Tang:1991rc}. In that work, the effective Hamiltonian
matrix equation was derived for a model including only the
$|\ensuremath{e^+e^-}\rangle$ and $|\ensuremath{e^+e^-\gamma}\rangle$ Fock states. Variational methods
were applied to this effective model and produced upper limits on the
triplet state. Attempts to apply DLCQ to QCD were undertaken at the
same time by Hollenberg~\cite{Hollenberg:1991fk}, but renormalization
and computational resources prevented much success. Further
developments in understanding the connection between light-front and
instant-form techniques were studied by Kalu\v{z}a and Pauli,
reproducing the expected results for the hyperfine splitting and Bohr
states in the limit of $\alpha\rightarrow0$~\cite{Kaluza:1991kx}.
Krautg\"artner {\it et al.}, implementing the \textit{Coulomb
counterterm} techniques developed by W\"olz~\cite{Wolz:1990zzz},
solved the effective matrix equation for
positronium~\cite{Krautgartner:1991xz}. They found that it was
possible to reproduce the correct Bohr spectrum, as well as the
leading relativistic hyperfine splitting, for both $\alpha_{\rm
QED}=1/137$ and $\alpha=0.3$, albeit with some cutoff dependence.
Concerned with the effect of \textit{zero modes\/} (nontrivial field
configurations in the Fock vacuum), Kalloniatis and Pauli undertook
numerical simulations based upon perturbative solutions to the
zero-mode constraint equations~\cite{Kalloniatis:1993jh}.
Krautg\"artner further developed these techniques and began to
analytically study the two-photon exchange interaction and its
relationship to the observed divergences in his
dissertation~\cite{Krautgartner:1992zzz}. W\"olz, in his
dissertation, applied DLCQ to QCD by including the
$|q\bar{q}gg\rangle$ Fock state~\cite{Wolz:1995zzz}. Numerical
limitations at the time prevented implementation of the counterterm
techniques being concurrently developed, so that a slow convergence in
the number of discretization points and a strong dependence on the
momentum cutoff precluded these results from suggesting any conclusive
statements. Synthesizing all these techniques, Trittmann computed the
first results for positronium with the inclusion of the annihilation
$\ensuremath{e^+e^-}\rightarrow\gamma$
channel~\cite{Trittmann:1997xz,Trittmann:1997tt,Trittmann:2000gk}.
Utilizing the good quantum number $J_z$, he was able to split the
problem into sectors and investigate the breaking of rotational
invariance inherent in light-front form in the effective equation.
Cutoff dependence and inadequate computational resources were the
major limits to Trittmann's work. With improved computing resources
and the introduction of a special counterterm to cancel a divergent
matrix element, DLCQ was applied by the current authors to two-flavor
QED to obtain bound states of positronium and true muonium
simultaneously~\cite{Lamm:2013oga}. This work built upon the prior
methods by incorporating a number of features of QED bound states in
front-form field theory.
Beyond DLCQ, other numerical methods have been developed for
light-front systems. Basis light-front quantization (BLFQ) follows
from discretizing the momenta into harmonic-oscillator modes in the
transverse direction instead of using Fourier modes. This method
aspires to decrease the number of basis states needed by more
accurately representing the functional behavior of the wave function.
BLFQ has shown initial success in solving bound-state problems in
QED~\cite{Vary:2009gt,Zhao:2011ct,Maris:2013qma,Zhao:2013cma,
Zhao:2013jia,Chakrabarti:2014cwa,Zhao:2014xaa} and
QCD~\cite{Li:2015zda}. Using Monte Carlo methods developed for
instant-form lattice gauge theory, transverse lattice theory has
investigated simple models of QCD in 3+1
dimensions~\cite{Pauli:1995dt,Dalley:1996ns,vandeSande:1996gh,
Dalley:1999ay}. Tube-based, collinear QCD and other
effective-Hamiltonian methods also exist~\cite{vandeSande:1995cg,
Brisudova:1996vw,Chakrabarti:2001yh}. In recent years, the AdS/QCD
conjecture has been extended to light-front field theory to produce
the low-energy meson and baryon spectra~\cite{deTeramond:2008ht,
deTeramond:2010we,Brodsky:2010ur,deTeramond:2010ge,Brodsky:2013ar,
Brodsky:2014yha,deTeramond:2014asa,Brodsky:2015oia,Dosch:2015bca,
Dosch:2015nwa}.
The limitations of Fock-state truncation in renormalization have also
prompted the study of other methods of truncation. Drawing upon the
techniques found in many-body physics,
coupled-cluster~\cite{Chabysheva:2011ed, Chabysheva:2012bf,
Elliott:2014fsa} and coherent-basis truncations~\cite{Misra:1993ps,
Misra:1995sd,Misra:2000ub,Misra:2005dt,More:2012ey,More:2013lua} have
shown promise in simpler systems.
This paper is organized as follows. In Sec.~\ref{sec:model} we
briefly review the model of true muonium studied here.
Section~\ref{sec:result} is devoted to presenting the numerical
results obtained for the energy levels and the decay constants, with
emphasis on the effect of the annihilation channel and of the presence
of multiple flavors on the states. We conclude in Sec.~\ref{sec:cons}
with some discussion of our results and possible directions for future
work.
\section{True Muonium model}
\label{sec:model}
We review here the major points of our model, which are described in
detail in a previous work~\cite{Lamm:2013oga}. In front form, the
eigenvalue equation for a bound state is given by:
\begin{align}
\label{eq:ham}
\bigg(M^2 - \sum_i&\frac{m^2_i+\bm{k}^{2}_{\perp i}}{x_i} \bigg)
\psi (x_i,\bm{k}_{\perp i};h_i)\nonumber\\ =\sum_{h_j}&\int_D
\mathrm{d}x'_{j} \mathrm{d}^2 \bm{k}'_{\perp j}
\langle x_i,\bm{k}_{\perp i};h_i \left| V_{\rm eff}
\right| x'_{j}, \bm{k}'_{\perp j};h_j \rangle\nonumber\\&\times \psi
(x'_{j},\bm{k}'_{\perp,j}; h_j),
\end{align}
where $M$ is the invariant mass of the state, $m$ indicates a mass
term, $i,j$ are component particle indices, $x$ and $\bm{k}_\perp$ are
the conventional longitudinal and transverse momentum light-front
coordinates, respectively, $h$ is shorthand for all intrinsic quantum
numbers of a state, and $V_{\rm eff}$ are interaction terms given by
the light-front Hamiltonian. The domain $D$ of Eq.~(\ref{eq:ham}) is
made well defined by the introduction of cutoff $\Lambda$, and we
choose~\cite{Lepage:1980fj}
\begin{equation}
\frac{ m^2 + \bm{k}_\perp^2 }{x(1-x)} \le \Lambda^2 + 4m^2 \, .
\end{equation}
Our model considers only the truncated Fock space of
$|\ell_i\bar{\ell}_i\rangle$, $|\ell_i\bar{\ell}_i\gamma\rangle$, and
$|\gamma\rangle$. The single-photon interaction allows for mixing
between flavors via the annihilation channel. The wave functions are
in the form of helicity states only for pure lepton states ({\it
e.g.}, $\left|\ensuremath{\mu^+\mu^-}\right>$). The $\left|\gamma\right>$ and
$\left|\ell_i\bar{\ell}_i\gamma\right>$ components are folded into
$V_{\rm eff}$ by means of the method of iterated
resolvents~\cite{Pauli:1997ns,Trittmann:1997xz}.
Discretization in ($x,\bm{k}_\perp$) space results in an asymmetric
matrix in the discretized form of Eq.~(\ref{eq:ham}), which
significantly increases the computational effort, so instead it is
numerically superior to use the polar coordinates utilized initially
by Karmanov~\cite{Karmanov:1980mc} to study a toy model of the
deuteron, and later by Sawicki~\cite{Sawicki:1985vs,Sawicki:1985uq} in
studying relativistic scalar-field bound states on the light front.
These coordinates are defined by
\begin{equation} \label{eq:xdefn}
x=\frac{1}{2}\left(1+\frac{\mu\cos \theta }
{\sqrt{m_i^2+\mu^2}}\right) \, ,
\end{equation}
\begin{equation} \label{eq:kperpdefn}
\bm{k}_\perp = \mu ( \sin \theta \cos \phi ,
\sin \theta \sin \phi , 0 ) \, .
\end{equation}
Using these variables, one may exchange $\phi$ for the discrete
quantum number $J_z$~\cite{Trittmann:1997xz} and compute using only
$\mu$, $\theta$. The new variable $\mu$ can be considered an
off-shell momentum, due to the relation
\begin{equation} \label{eq:mudef}
\frac{m_i^2+\bm{k}^2_\perp}{x(1-x)}=4(\mu^2+m_i^2) \, .
\end{equation}
Since these coordinates depend upon the fermion mass $m_i$, different
sets of $\mu$, $\theta$ values result from the same sets of $x$ and
$\bm{k}_\perp$ values in the multiple-flavor system.
It has been shown~\cite{Krautgartner:1991xz,Trittmann:1997xz,
Lamm:2013oga} that strong dependence in ${}^1S_0$ states on $\Lambda$
arises from the matrix element between antiparallel-helicity states
called $G_2$. In the limit of $k_\perp \equiv |\bm{k}_\perp|$ or
$k'_\perp \equiv |\bm{k}'_\perp|\to
\infty$, this interaction approaches
\begin{equation}
\label{eq:g2lim}
\lim_{\ensuremath{k_{\perp}}\rightarrow
\infty}G_2=-\frac{\alpha}{\pi}\frac{2}{x+x'-2xx'}\delta_{J_z,0} \, ,
\end{equation}
which, in the absence of the dependence of $|\psi_{\ensuremath{\ell^+\ell^-}}\rangle$ upon
$\ensuremath{k_{\perp}}$, would result in a $\delta$ function-like behavior in
configuration space. Reference~\cite{Krautgartner:1991xz} chose to
regularize this singularity by deleting the entire divergent term.
Instead, a numerically superior subtraction scheme is obtained by only
removing its limit as $k_\perp$ or $k'_\perp \to \infty$,
\begin{equation}
\label{eq:greg}
G_{2,\rm reg}=G_2+\bigg\{\frac{\alpha}{\pi}\frac{2}{x+x'-2xx'}
\delta_{J_z,0}\bigg\},
\end{equation}
which retains part of the term (including $x$ and $x'$ dependence).
This scheme removes the strongest $\Lambda$ dependence of ${}^1S_0$
states in both QED~\cite{Lamm:2013oga,Wiecki:2014ola} and
QCD~\cite{Li:2015zda} models. It is important to note that the $\ensuremath{\bm{k}_{\perp}}$
dependence of $|\psi_{\ensuremath{\ell^+\ell^-}}\rangle$ varies with $\alpha$, and
therefore it should be anticipated that the strength of this apparent
divergence should also depend upon $\alpha$. With this regularization
scheme, the model allows for taking the $\Lambda\rightarrow\infty$
limit, albeit with a regularization dependence determined by
mathematical, rather than purely physical, considerations.
Much of the previous work on QED with DLCQ has focused upon the
unphysically large value $\alpha = 0.3$. In this regime, QED
perturbative calculations can potentially become unreliable. We use
this strong coupling value of $\alpha$ to study flavor mixing. New to
this work, we investigate the approach to the physical
$\mathcal{O}(10^{-2})$ value of the QED coupling constant.
\section{Results}
\label{sec:result}
Previous work has given results sensitive to numerical artifacts,
limiting the reliability of the results that could be obtained. To
overcome some of these limitations, we have produced a new numerical
code, TMSWIFT (True Muonium Solver With Front-from Techniques), which
is available online~\cite{TMSWIFT}. This code uses the parallel
eigenvalue-solver package SLEPc~\cite{Hernandez:2005:SSF}, both to
increase the number of Fock states and to decrease the time of
calculation. TMSWIFT allows an arbitrary number of flavors, each
specified by a distinct mass $m_i$, cutoff $\Lambda$, and
discretization numbers $N_\mu$ and $N_\theta$ (although throughout
this work we will fix $N_\mu=N_\theta=N$). Different discretization
schemes are available in TMSWIFT for exploration of numerical errors
and efficiency. Our code also allows easy implementation of new
effective interactions ({\it e.g.}, from $|\gamma\gamma\rangle$
states). These improvements have also allowed us to investigate lower
values of $\alpha$, where the extrapolation to
$\Lambda,N\rightarrow\infty$ becomes more difficult. In order to
examine these limits, except for Subsec.~\ref{subsec:Multiple} which
explicitly studies multiple-flavor effects, we restrict ourselves to
the case of single-flavor true muonium.
In this section, we explore a number of properties of true muonium,
dedicating a subsection to each: the invariant squared mass $M^2_n$,
the ground-state hyperfine splitting, the singlet and triplet wave
functions, the decay constants, and multiflavor effects.
\subsection{Invariant Squared Mass}
With larger $N$ and improved regularization, we found it possible to
fit the energy levels, $M^2_n$, to Pad\'{e} approximants of second
order. To perform these fits, we first fit the $N$ dependence for
each value of $\Lambda$ for which simulations were computed:
\begin{equation}
\label{eq:mfit}
M^2(N,\Lambda)=\frac{M^2({\Lambda})+\frac{b}{N}+\frac{c}{N^2}}
{1+\frac{d}{N}+\frac{e}{N^2}}\, .
\end{equation}
Then, the final $N \to \infty$ and $\Lambda \to \infty$ results can be
obtained from a second fit to:
\begin{equation}
\label{eq:mfit2}
M^2({\Lambda})=\frac{M^2_\infty+\frac{f}{\Lambda}+\frac{g}{\Lambda^2}}
{1+\frac{h}{\Lambda}+\frac{i}{\Lambda^2}}\, .
\end{equation}
These functions are well defined separately in the
$N\rightarrow\infty$ and $\Lambda\rightarrow\infty$ limits, and
therefore one can extract the continuum- and cutoff-independent
values, $M^2_\infty$. While in principle the entire data set could be
simultaneously fit in $N$ and $\Lambda$, the large cancellations that
can occur between Pad\'{e} coefficients, and the large number of
parameters to fit in practice, make the process more difficult, and
initial conditions for the fit must be carefully chosen to avoid local
minima of the fits. Moreover, the two parameters have different
origins: $N$ is a numerical artifact, while $\Lambda$ is a theoretical
artifact. By fitting separately, these issues are largely avoided.
Results for the ground-state singlet and triplet states are tabulated
in Table~\ref{tab:esp}.
\begin{table*}[t]
\caption{Extrapolated results for the bound-state invariant
squared mass $M^2$ in units of $m_\mu^2$, and the decay constants
$f_V$, $f_P$ in units of $m_\mu$, for a range of $\alpha$ values. The
column labeled $C_{\rm HFS,LF}$ is the computed hyperfine coefficient
$C_{\rm HFS}$ from Eq.~(\ref{eq:ffchf}). The column labelled $C_{\rm
HFS,ET}$ is the instant-form prediction for $C_{\rm HFS}$ from
Eq.~(\ref{eq:ifchf}).}
\label{tab:esp}
\begin{center}
\input{table_ef.tex}
\end{center}
\end{table*}
\begin{figure}
\includegraphics[angle=-90,width=\linewidth]{N_dep}
\caption{\label{fig:ndep}Example of the dependence of $M^2$ upon $N$,
normalized to the continuum and infinite limit for $\alpha=0.2$,
$\Lambda = 5m_\mu \alpha$.}
\end{figure}
An example of the dependence of $M^2$ upon $N$ is shown in
Fig.~\ref{fig:ndep} for a fixed value of $\Lambda$ and $\alpha$. This
dependence is qualitatively the same for all values of $\alpha$ and
$\Lambda$. As can be seen, with increasing $N$, $M^2$ at first rises
to a peak and then decreases toward a continuum value. The location
of this peak is found to be proportional to $1/\alpha^2$ and linear in
$\Lambda$. It is therefore more difficult to numerically simulate
small $\alpha$ and large $\Lambda$, because any results that only
sample to the left of the peaks systematically overestimate $M^2$, by
not respecting that the functions decrease to the right of the peaks.
From Fig.~\ref{fig:ndep} it can also be seen that the triplet state
reaches its (smaller) maximum at a larger $N$. It is also empirically
found that, while the singlet state peaks at lower $N$, the variance
of $M^2$ is much larger. With this understanding of the space of $N$,
$\alpha$, and $\Lambda$, one can study the approach to the
perturbative regime of $\alpha$. The analytic, instant-form values of
$M$ to $\mathcal{O}(\alpha^4)$ are given for $1^1S_0$ and $1^3S_1$
by~\cite{bethe1957quantum}:
\begin{equation}
\label{eq:m2s}
M^2_{1^1S_0}=\left(2m-\frac{1}{4}m\alpha^2
-\frac{21}{64}m\alpha^4\right)^2,
\end{equation}
\begin{equation}
\label{eq:m2s2}
M^2_{1^3S_1}=\left(2m-\frac{1}{4}m\alpha^2
+\frac{49}{192}m\alpha^4\right)^2.
\end{equation}
Since $m=1$ in our units, to test these formulas, one can fit to
\begin{equation}
\label{eq:efit}
M^2(\alpha)=(N_0+N_2\alpha^2+N_4\alpha^4+N_4\alpha^5)^2.
\end{equation}
From the Fock space considered in our model, a perturbative
calculation should not have any higher-order contributions, but one
could anticipate a possible $\mathcal{O}(\alpha^5)$ term due to the
contributions of higher-order terms arising from our nonperturbative
procedure and regularization scheme. The results of the fit are found
in Table~\ref{tab:Ealpha}.
\begin{table*}[t]
\caption{Parameters of Eq.~(\ref{eq:efit}) for the singlet and triplet
states of true muonium, fit over two ranges of $\alpha$. The
$\mathcal{O}(\alpha^4)$ perturbative predictions are $N_0=2$,
$N_2=-\frac{1}{4}$, $N_{4,1^1S_0}=-\frac{21}{64}\approx-0.328$,
$N_{4,1^3S_1}=\frac{49}{192}\approx0.255$. The expected value of
$N_5$ is unknown, but anticipated to be small. Reported uncertainties
result solely from the fitting procedure.}
\label{tab:Ealpha}
\input{eafit.tex}
\end{table*}
Comparing the singlet-state results to Eqs.~(\ref{eq:m2s}), one sees
that TMSWIFT reproduces within uncertainty the $\mathcal{O}(\alpha^4)$
calculation over the entire range of $\alpha$. Extracting possible
higher-order coefficients would be possible by increasing $N$ beyond
what has been presented here. In contrast, for the triplet state,
only the terms up to $\alpha^2$ of Eqs.~(\ref{eq:m2s2}) are correctly
reproduced. The $\alpha^4$ coefficient reproduced the anticipated
sign, but it is larger than the result of the instant-form
calculation. Additionally, there is a large, unanticipated $\alpha^5$
coefficient. Such results are indicative of issues in the
annihilation channel, which affects only the triplet at this order.
\subsection{Hyperfine Splitting}
To study these effects further, one can check how accurately our
front-form model reproduces the expected instant-form results through
the hyperfine coefficient, which is defined as
\begin{equation}
\label{eq:ffchf}
C_{\rm HFS}\equiv\frac{E_{\rm{HFS}}}{m_\mu\alpha^4}=
\frac{\sqrt{M^2(1^3S_1)}-\sqrt{M^2(1^1S_0)}}{m_\mu\alpha^4}.
\end{equation}
If all Fock states were included in our model, then the full known
$\mathcal{O}(\alpha^7)$ instant-form prediction of $E_{\rm HFS}$ of
Ref.~\cite{PhysRevD.91.073008} could be compared to our results. But
because of our Fock-state truncations, there is a mismatch in the
higher-order contributions. Since we can only extract up to
$\mathcal{O}(\alpha^4)$, it is useful to compare to the leading-order
value of $C_{\rm HFS}=\frac{7}{12}$.
Our model would be expected to partially resum the relativistic
corrections from the single-photon exchange and annihilation diagrams.
Therefore, we present the values of $C_{\rm HFS}$ given by the exact
Dirac-Coulomb solutions~\cite{Breit:1930}:
\begin{align}
\label{eq:ifchf}
C_{\rm HFS}&=\frac{1}{m_\mu\alpha^4}\left(\frac{E_F}
{\sqrt{1-\alpha^2}[2\sqrt{1-\alpha^2}-1]}\right)\nonumber\\
&=\frac{7}{12}\left(1+\frac{3}{2}\alpha^2+\frac{17}{8}\alpha^4
+\mathcal{O}(\alpha^6)\right),
\end{align}
where $E_F=\frac{7}{12}m_\mu\alpha^4$ is the lowest-order hyperfine
splitting of true muonium. If higher precision could be attained,
these effects might be resolvable, but at the current levels they are
not yet visible.
Previous results for $C_{\rm HFS}$ at $\alpha=0.3$ without the
regularization term are found in Table 4.2 of
Ref.~\cite{Trittmann:1997xz}, and can be calculated from the results
found in Ref.~\cite{Gubankova:1999cx}. The $C_{\rm HFS}$ obtained in
these works appears to have a logarithmic singularity in the singlet
state, indicating that no $\Lambda\rightarrow\infty$ limit could be
taken. The severity of the divergence can be seen in
Ref.~\cite{Trittmann:1997xz}, where $C_{\rm HFS}$ rises from
$\approx0.313$ at $\Lambda=m_f$ to $\approx 1.27$ at $\Lambda=18m_f$.
In contrast, we find that for our regularization scheme, $C_{\rm HFS}$
is finite because the two energy levels are finite in the
$N\rightarrow\infty$ and $\Lambda\rightarrow\infty$ limits.
The numerical results in Table~\ref{tab:esp} are roughly consistent
over the entire range of $\alpha$, albeit with large uncertainty.
While the results are finite, we find that the central values are
systematically larger than the anticipated $\frac{7}{12}\approx0.58$,
being in the range $0.7$--$0.8$ except for $\alpha=0.3$, where
observables approach their asymptotic values more slowly due to
changes in the wave function large-$\bm{k}$ dependence, as
discussed in the next section.
Clearly, a disagreement is seen between the two instant-form
predictions and the results on the light front. Previously, several
authors~\cite{Tang:1991rc,Krautgartner:1991xz,Trittmann:1997xz} have
also pointed out that the correct value of $C_{\rm HFS}$ is best
obtained for $\Lambda\approx m\alpha$, and the results from TMSWIFT
support this point of view. Unfortunately, the divergences spoil this
agreement at larger $\Lambda$, necessitating renormalization. The
larger splitting in the infinite-$\Lambda$ limit can be understood
thusly: Although the regularization procedure developed allows for
extrapolation to $\Lambda\rightarrow\infty$, the $\Lambda$ dependences
of the singlet and triplet states are different, as was seen
in~\cite{Lamm:2013oga}, leading to an asymptotic HFS that, while
finite, is larger than the known result.
These results are in contrast to the situation in which the
annihilation-channel interaction is excluded. Choosing the
intermediate case of $\alpha=0.1$, we performed an exploratory search
with a smaller number of simulations. In this case, we found in the
continuum and infinite-$\Lambda$ limits that $C_{\rm HFS}=0.35(11)$,
in agreement with the anticipated value at leading order of $C_{\rm
HFS}\approx0.333$. A similar small study for $\alpha=0.3$ with
$J_z=1$ also found a value of $C_{\rm HFS}\approx0.75$, indicating
that both the dynamical and instantaneous annihilation interactions
are affected. This evidence further suggests that the
annihilation-channel interaction is the source of the discrepancies.
To understand why the annihilation channel gives trouble, it is useful
to recall how this term is included in instant form. In standard,
perturbative nonrelativistic calculations, these contributions in
coordinate space are represented as a contact term $\propto
\delta^{(3)}({\bf r})$; therefore, in momentum space these terms are
very sensitive to large momenta, and imposing a cutoff $\Lambda$
prevents these momenta from contributing. Furthermore, we have
already seen that obtaining numerical results for large $\Lambda$ is
complicated by the need to include much larger $N$ than is currently
possible. Put together, these facts indicate that regularization and
renormalization is a more complicated affair in the annihilation
channel.
\subsection{Wave Functions}
In order to understand the effect of the regularization term on the
effective interaction, we have studied the large-$\mu$ behavior of the
wave functions. The momentum-space wave function obtained from the
nonrelativistic Schr\"odinger equation is
\begin{equation}
\Psi(\bm{k})=\frac{\sqrt{8}}{\pi}\frac{1}{(1+\bm{k}^2)^2},
\end{equation}
where the instant-form 3-momentum carries units of Bohr momentum
$\frac{1}{2}m\alpha$. It is known that higher-order corrections to
the interaction lead to a modified power law, changing the
large-$\bm{k}$ power scaling from the nonrelativistic value of $-4$.
Since according to Eq.~(\ref{eq:mudef}) $\bm{k}$ is linear in $\mu$,
for our studies it suffices to compute the dependence upon $\alpha$ at
large $\mu$. The large-$\mu$ behavior is parametrized as
\begin{equation}
\label{eq:kfit}
\Psi(\mu)=a\mu^{-\kappa},
\end{equation}
where $\kappa=4$ is the result for the nonrelativistic Schr\"odinger
equation.
In~\cite{Krautgartner:1991xz}, it was found that for $\alpha_{\rm
QED}=1/137$, the large-$\mu$ behavior of the $\uparrow\downarrow$
singlet wave function is $\kappa=4.0$, in agreement with expectations,
and that for $\alpha=0.3$ the behavior is $\kappa=2.5$. We believe
this large reduction in $\kappa$ is related to the strong $\Lambda$
dependence found in~\cite{Krautgartner:1991xz,Trittmann:1997xz}. To
further understand the relation between regularization and $\kappa$,
we fit the large-$\mu$ tail of our wave functions to
Eq.~(\ref{eq:kfit}) with the results for a selected few values of
$\alpha$ shown in Fig.~\ref{fig:sing}. In all of these cases, we have
implemented our regularization subtraction scheme.
\begin{figure}
\includegraphics[width=\linewidth]{Singlet_largeK}
\caption{\label{fig:sing}Dependence of the $\uparrow\downarrow$
component of the $1^1S_0$ state upon $\mu$ for a fixed value of
$x=0.5$ for different values of $\alpha$. The points indicate the
numerical results, and the solid lines are the fits used to extract
$\kappa$.}
\end{figure}
The values of $\kappa$ for $\alpha=0.01,0.07$ appear to show only
small deviations from the nonrelativistic value, consistent
with~\cite{Krautgartner:1991xz}. In contrast, our value of
$\kappa=3.59$ for $\alpha=0.3$ is dramatically larger than found in
the unregulated results of~\cite{Krautgartner:1991xz}. Since the
large-$\mu$ tail decays much faster than
in~\cite{Krautgartner:1991xz}, the contribution of any potentially
divergent terms will be reduced, explaining why the results
of~\cite{Lamm:2013oga} showed such a dramatic improvement.
Using our entire set of $\alpha$ results, it is possible to study the
effect of varying $\alpha$ upon $\kappa$. Shown in
Fig.~\ref{fig:kappaalpha} are the extracted values of $\kappa$ for
both the dominant $\uparrow\downarrow$ component of the singlet state
and the subleading $\uparrow\uparrow$ component. We have also
obtained values of $\kappa$ for a smaller set of $\alpha$ without
using our regularization scheme.
\begin{figure}
\includegraphics[width=\linewidth]{kappa_alpha}
\caption{\label{fig:kappaalpha}$\kappa$ vs.\ $\alpha$ for the
$\uparrow\uparrow$ and $\uparrow\downarrow$ components of the
$1^1S_0$ state. Open (closed) symbols indicate results excluding
(including) the regularization term.}
\end{figure}
Because the regularization term is only needed for $G_2$, it makes
sense that only the $\uparrow\downarrow$ has a dramatic change in its
$\alpha$ dependence by the introduction of the regularization term,
whereas the $\uparrow\uparrow$ wave functions are mostly unaffected.
\subsection{Decay Constants}
In addition to the invariant masses, the decay constants offer an
interesting observable that can be extracted from the wave functions.
They also serve as an good test bed for understanding how the
properties of the wave function are affected by regularization and
renormalization. The decay constants in the vector $V$ and
pseudoscalar $P$ channels are defined by
\begin{align}
\langle0|\bar{\psi}\gamma^\mu\psi|V(p),\lambda\rangle&=
\epsilon^\mu_\lambda m_V f_V,\nonumber\\
\langle0|\bar{\psi}\gamma^\mu\gamma^5\psi|P(p)\rangle&=ip^\mu f_P,
\end{align}
where $\epsilon^\mu_\lambda(p)$ is the polarization vector for the
boson, and $\lambda=0,\pm1$. In front-form field theory, the decay
constants can be computed directly from the $+$ components of these
currents which, following Ref.~\cite{Branz:2010ub,Li:2015zda}, are
given for QED bound states by
\begin{widetext}
\begin{align}
&f_{V(P)}=\int \frac{\ensuremath{\mathrm{d}} x}{\sqrt{x(1-x)}}
\frac{\ensuremath{\mathrm{d}} ^2\bm{k}_\perp}{(2\pi)^3}\left[\psi^J_{J_z=0}
(\bm{k}_\perp,x,\uparrow\downarrow)\mp\psi^J_{J_z=0}
(\bm{k}_\perp,x,\downarrow\uparrow)\right],
\end{align}
\end{widetext}
where the vector (pseudoscalar) decay constant is given by the
difference (sum) of the two terms in the equation. Taking the
component wave functions from TMSWIFT calculations, it is possible to
obtain $f_V$ for the singlet state and $f_P$ for the triplet state as
a function of $\alpha$. Like the invariant masses, the decay
constants are found to be well fit to the functional form of
Eq.~(\ref{eq:mfit}), and therefore an infinite-cutoff values for them
can be obtained. These results can be found in Table~\ref{tab:esp}.
For the decay constants, one expects
$f_i\propto|\psi_i(0)|/\sqrt{M_i}$, which suggests a $\alpha^{3/2}$
power law at leading order. To check this prediction, a fit is
performed to the function
\begin{equation}
\label{eq:ffit}
f_i(\alpha)=N\alpha^\beta,
\end{equation}
and the results are exhibited in Table~\ref{tab:falpha}\@.
\begin{table}[ht]
\begin{center}
\caption[Fit Parameters of Eq.~(\ref{eq:ffit}) for the Vector Decay
Constant of the Singlet State and the Pseudoscalar Decay Constant of
the Triplet State for Two Ranges of $\alpha$. The Leading-Order
Perturbative Prediction Is $\beta=3/2$.]{Fit parameters of
Eq.~(\ref{eq:ffit}) for the vector decay constant $f_V$ of the
singlet state and the pseudoscalar decay constant $f_P$ of the
triplet state for two ranges of $\alpha$. $N$ has units of $m$.
The leading-order perturbative prediction is $\beta=3/2$.}
\label{tab:falpha}
\input{fafit.tex}
\end{center}
\end{table}
Similar to the invariant masses, the $f_V$ values for the singlet
state seem to reproduce the perturbative form to leading order very
well over for all values of $\alpha$. The agreement between $f_P$ for
the triplet state shows a poorer agreement, especially for large
$\alpha$, where the inclusion of the annihilation channel enables
higher-order corrections to the decay rate.
\subsection{Multiple-Flavor Effects} \label{subsec:Multiple}
True muonium is acutely sensitive to the effects of multiple flavors.
The large mass difference $m_\mu/m_e\approx207$ causes electronic loop
corrections to be the largest corrections to the spectrum of true
muonium. Additionally, the ratio $m_\tau/m_\mu\approx16$ is small
enough to produce appreciable effects on the system at
$\mathcal{O}(\alpha^5)$. While the vacuum polarization in the
exchange diagrams is neglected by our model, it is possible to study
these effects in the annihilation channel.
\begin{figure}[h]
\includegraphics[width=\linewidth]{m_mT_J0p}
\caption{$\Delta M^2$ corrections from a second flavor of leptons to
true muonium as a function of the second flavor's mass $m_e$.
Errors are estimated from the numerical fit alone.}
\label{fig:eloop}
\end{figure}
\begin{figure*}
\includegraphics[width=\linewidth]{wave_3c}
\caption{\label{fig:waves}The $1^3\!S_1^0$ probability density of the
$\uparrow\downarrow$ components of (left) $\ensuremath{e^+e^-}$, (center) $\ensuremath{\mu^+\mu^-}$, and
(right) $\ensuremath{\tau^+\tau^-}$, with $J_z=0$, as a function of $x$ and $k_\perp$, for
$\alpha=0.3$, $m_\mu/m_e=m_\tau/m_\mu=2$, $\Lambda_i=5\alpha m_i$,
and $N_\mu=37$, $N_\tau=31$, $N_e=71$.}
\end{figure*}
Previous results~\cite{Lamm:2013oga} found large, nonlinear $N$ and
$\Lambda_e$ dependence from the electronic contribution, even for the
unphysically large ratio of $m_\mu/m_e=2$. With TMSWIFT, we have been
able to further study this dependence. Numerical limitations prevent
the collection of a sufficiently large number of simulations to fit to
Pad\'{e}-approximants. Instead, we fix $\alpha = 0.3$, $N_\mu=21,
\Lambda_\mu=10$ ($\Lambda_i$ is given in units of $m_i \alpha$), and
then obtain estimates for $\Delta M^2$ (the shift of squared mass
eigenvalues due to the inclusion of additional lepton flavors) by
averaging over the ranges $N_e \in [27,35]$ and $\Lambda_e \in
[1,35]$.
We have been able to further reduce the uncertainty through two new
ideas. First, simulations were made using two different
discretization schemes, Gauss-Legendre and Curtis-Clenshaw. The use
of two discretization schemes for the same $N_e$ allows us to explore
the effects of discretization on the continuum electron states with
smaller $N$. Additionally, $f_P$ is a sensitive probe of the coupling
of electron continuum states to the bound state. Empirically, we find
that if the value of $f_P$ differs by more than 10\% from the
single-flavor case, the simulation has sampled the continuum in an
inaccurate way and can be excluded from the average.
Producing results for the physical value of the electron mass remains
difficult numerically because of the large separation of scales. Our
results for the corrections to true muonium from electronic loops in
the annihilation channel are shown in Fig.~\ref{fig:eloop}, compared
to the anticipated instant-form result, and the previous results
of~\cite{Lamm:2013oga}. One can see that TMSWIFT's parallel
implementation, while still numerically limited, can produce better
agreement with the instant form than found in~\cite{Lamm:2013oga},
with smaller uncertainty.
TMSWIFT has also been written to allow for an arbitrary number of
flavors. We present here results from a three-flavor true muonium
model, albeit with unphysical ratios $m_\mu/m_e=m_\tau/m_\mu=2$,
keeping $\alpha=0.3$. In Fig.~\ref{fig:waves} are shown the
probability densities of the $\uparrow\downarrow$ components of each
flavor for the triplet state. In Table~\ref{tab:prob} we present the
relative probability for each component in this case.
\begin{table}[ht]
\caption{Integrated probability for each flavor in the true muonium
$1^3S_1^0$ state. The parameters used are $\alpha=0.3$,
$m_\mu/m_e=m_\tau/m_\mu=2$, $\Lambda_i=5\alpha m_i$, and
$N_\mu=37,N_\tau=31,N_e=71$.}
\begin{tabular}{l|c}
\hline\hline
Flavor & $\int\mathrm{d}x \, \mathrm{d}^2 \! \bm{k}_\perp
P(x,{k}_\perp)$\\ \hline
$|\ensuremath{\mu^+\mu^-}\rangle$&$0.992$\\
$|\ensuremath{e^+e^-}\rangle$&$0.008$\\
$|\ensuremath{\tau^+\tau^-}\rangle$&$\approx 1.2\times10^{-5}$\\
\hline
\end{tabular}
\label{tab:prob}
\end{table}
\section{Discussion and Conclusion}
\label{sec:cons}
In this work we have presented results for the invariant mass and
decay constants of the true muonium system. For the first time, we
have gone beyond the case $\alpha=0.3$ and shown that the approach to
$\alpha_{\rm QED}$ is possible with sufficient numerical resources.
The purpose of this program is not to produce energy levels
competitive in the weak-field limit with perturbative calculations.
Instead, our goals in calculating at $\alpha_{\rm QED}$ are to produce
true muonium wave functions that can be used in relativistic
situations, and as to provide an independent check on our methods,
allowing one to be confident in the strong-field predictions.
Furthermore, using our previously developed regularization scheme, the
simultaneous limits of $N\rightarrow\infty$ and
$\Lambda\rightarrow\infty$ have been taken and stable results found.
These values have been compared to the instant-form perturbative
calculations, and reasonable agreement has been obtained. Finally,
initial studies have been undertaken to compute the fully
nonperturbative contribution to the bound state arising from
additional flavors, both lighter and more massive then the muon.
Improved agreement with instant-form predictions have been obtained
for a range of masses of a second flavor, and simulations of the
three-flavor model have been produced.
Currently, work is underway to include the $|\gamma\gamma\rangle$
state and the pair of states $|\ell\bar{\ell}\ell\bar{\ell}\rangle$
and $|\ell\bar{\ell}\ell'\bar{\ell}'\rangle$, which are required for
gauge invariance. These corrections are crucial for precision true
muonium predictions and are a necessary step for QCD bound states as
well.
Proper renormalization of the Hamiltonian is the remaining obstacle.
In order to make accurate predictions, the $\Lambda$ dependence found
in this work must be systematically removed, which involves not just
including new Fock sectors, but imposing gauge invariance at each
stage. A proper implementation of charge renormalization and the
running of the coupling $\alpha$ should address a large part of the
issue. A first step in this direction would focus upon implementing a
renormalized vacuum polarization into the effective interactions.
With a robust renormalization scheme, multiple values of $\Lambda$
would not be needed to take the $\Lambda\rightarrow\infty$, greatly
reducing the numerical effort to produce reliable results. With
TMSWIFT, Fock-space limitations have been greatly decreased. This
improvement allows for the implementation of explicit Fock-state
renormalization methods like Pauli-Villars
regulators~\cite{Chabysheva:2009vm,Chabysheva:2010vk,
Malyshev:2013eca} and sector-dependent
counterterms~\cite{Karmanov:2008br, Karmanov:2012aj}. Using the
exchange properties of leptons could further reduce the number of
basis states, similar to the methods used in
Ref.~\cite{Chabysheva:2014rra} for bosons. More time-intensive
renormalization schemes like the Hamiltonian-flow
method~\cite{Gubankova:1998wj,Gubankova:1999cx} also become viable
with a parallel implementation.
\begin{acknowledgments}
This work was supported by the National Science Foundation under
Grants PHY-1068286 and PHY-1403891. This work used the Extreme
Science and Engineering Discovery Environment (XSEDE), which is
supported by National Science Foundation Grant Number ACI-1053575, and
the ASU Physics Computing Cluster.
\end{acknowledgments}
|
train/arxiv
|
BkiUeZY4dbggz-0v0PdP
| 4 | 0.8 |
\section{INTRODUCTION}
The finite element method has achieved great success in many fields of science and technology since it was
first suggested in elasticity in the fifth decade of $ 20^{th}$ century.Today it has become a powerful tool for
solving partial differential equations \cite{brenner2008mathematical,ciarlet2002finite}.The key issue of the finite
element method is using a discrete solution on the finite element space,usually consisting of piecewise polynomials,to
approximate the exact solution on the given space according to a certain kind of variational principle.When the
finite element space is a subspace of the solution space, the method is called conforming.It is known that in this case,the
finite element solution converges to true solution provided the finite element space approximates the given space in some sense.
In general, for a $2m$ order elliptic boundary-value problem,the conforming finite element space is a $C^{m-1}$
subspace.It means that the shape function in this conforming finite element space is continuous together with its $m-1$ order derivatives.That is for a second-order problem, the shape function is continuous and for a fourth order problem, the shape functions and its derivatives are continuous.It is a rather strong restriction put on the shape functions in the latter case.
It is prove that to build up a conforming finite element space with $ C^{1}$-continuity for a two dimensional
fourth-order problem, like the plate bending problem in elasticity, at least a quintic polynomial with 18 parameters is required for a triangular element and a bi-cubic polynomial with 16 parameters for a rectangular element.It causes some computational difficulties because the dimension of the related finite element space is fairly large and its structure is rather complicated.
Therefore, it is desirable to relax directly the $ C^{m-1}$ continuity of the finite element space.It comes to the so called 'nonconforming' finite element method which had and still has a great impact on the development of finite element methods.However, it was found that some nonconforming element elements converge and some do not. The convergence behaviour sometimes depends on the mesh configuration.
Up to now there has been proposed a vast number of engineer devices based on different mechanical interpolations,like unconventional elements,energy-orthogonal elements with free formulation,quasi-conforming elements,generalised conforming elements and many others.The approximate spaces related to all these elements mentioned above are not included in the given solution space.Hence they are simply called 'nonstandard' finite elements.A unified mathematical treatment for analysis of these nonstandard finite elements has been proposed by various authors.\\
A brief description and analysis of some interesting and important conforming and nonconforming finite elements has been given below.\\
$\textit{$P^{1}$ nonconforming element}$ \cite{ciarlet2002finite,brenner2008mathematical}. This is a triangular element which is not $C^{0} $. The shape function is a linear polynomial with three nodal parameters at mid points of three edges of the triangle.This element converges for second order elliptic problem with optimal rate.\\
$\textit{Wilson-element} $\cite{taylor1976non,lesaint1976convergence,lesaint1980convergence}. This is a nonconforming rectangular element.The shape function is a quadratic polynomial with six parameters, four at vertices of the rectangle and two internal degrees of freedom, like the second order derivatives.This element converges for rectangular mesh, but does not converge for arbitrary quadrilaterals.It is interesting to mention that the behaviour of the Wilson element is better than the corresponding bilinear $Q_{1} $ conforming element as many engineering example have indicated.
$\textit{The rotated $ Q_{1} $ element}$ \cite{li1998nonconforming,ming2001nonconforming,rannacher1990simple}. $\ $This is a newly established non-conforming rectangular element.The shape function consists of four terms as[1,x,y,$x^{2}-y^{2} $].There are two versions of choosing
nodal parameters.The first one uses four function values at the mid-point of each edge of the rectangle.The second version uses four mean values of the shape function along edges.
Both versions are convergent for rectangular meshes.However,the first version is not convergent for arbitary quadrilaterals unless certain mess conditions are satisfied.
$\textit{Morley element} $ \cite{morley1968triangular} This is an old and simplest plate element.The shape function is a quadratic polynomial with six nodal parameters.They are three function values at vertices and three normal derivatives at mid-points of three edges.This element does not even belong to $C^{0} $ class,nevertheless it is convergent for the fourth-order problem\cite{lascaux1975some} . Surprisingly, it is recently proved that the Morley element is divergent for the second-order elliptic problem \cite{nilssen2001robust}.
In contrast, it is well known that there exists for long time
the $P^{2} $ conforming element for the second order problem.Its shape function is again a quadratic polynomial with six parameters as three function values at vertices and three function values at mid-points of edges.This quadratic $C^{0}$ element is convergent for the second order problem, but divergent for the fourth-order problem.
$\textit{Zienkiewicz incomplete cubic triangular element} $ \cite{bazeley1965triangular}. The shape function consists of incomplete cubic polynomials with specially chosen nine terms.The nine nodal parameters are three function values together with six first partial derivatives with respect to $x$ and $y$ at three vertices.This is a $ C^{0}$ element but not $C^{1}$ .It is proved that this element is convergent only for very special meshes, namely, all edges of triangles are parallel to the three given directions.It is a very interesting phenomenon that the Zienkiewicz
element using the cross-diagonal mesh actually tend to a limit,but it is not the true solution of the given problem,rather of another problem \cite{cai1986limitation} .
Another new nonconforming piecewise quadratic finite element on triangles has been discussed in \cite{fortin1983non}.This element satisfy patch test of Irons and Razzaqque \cite{irons1972experience}.This implies that on element interfaces,one should ensure the continuity of the approximation at the Gauss-Legendre quadrature points needed for the exact integration of third-degree polynomials. Optimal-error estimates and regularity properties for Dirichlet's problem has been studied by authers.\\
In this paper we will propose a new finite element which is a bridge between conforming and nonconforming finite element.This new element piecewise quadratic and quasi conforming.We have studied error estimation for Dirichlet's problem,and observed that this element does not give optimal convergence rate,which has been generally considered as a major drawback against the use of this element.Though the element is piecewise quadratic but it does not require two point continuity restriction on each interface of $ \tau^{h}$,which is needed for the above mentioned piecewise quadratic nonconforming element \cite{fortin1983non}.We shall show here that these elements are in fact very simple to use and they are nothing but combination of $P^{1} $ nonconforming element with incomplete $P^{2} $ - conforming element.
\section{Discretization}
\subsection{Continuous Problem}
We consider the following model problem \\
\begin{align}
-\Delta u & =f \quad \text{in} \ \Omega \quad \\
u & =u_{D} \quad \text{on} \ \partial\Omega \quad \nonumber \\ \nonumber
\end{align}
where $\Omega \subset \mathbb{R}^2$
\subsection{Notation}
Let $\tau_{h} $ be a conforming triangulation of $ \Omega$.The subscript h refer to the maximum element size $\text h= \text max_{k \in \tau_{h}} \text h_{k} $, where $h_{k}$ is the diameter of an element $K\in \tau_{h} $.$\varepsilon^{h} $ is the set of the edges in $ \tau_{h}$, $\textbf{n} $ is the unit outward normal along $ \partial K$ and the jump $[u] $ across an edge e is a vector defined as follows-\\
Let e be an interior edge shared by two triangles $K_{1}$ and $K_{2}$ in $ \tau^{h}$ , and $\omega_{j}=\omega \vert_{K_{j}} $ for j=1,2 . We define on e
\begin{center}
$[\omega]=\omega_{1} \textbf{n}_{1}+\omega_{2} \textbf{n}_{2} $
\end{center}
where $n_{j} $ is the unit normal of e pointing towards the outside of $K_{j} $ .If e is an edge on the boundary of $\Omega $,then we define on e
\begin{center}
$[\omega]=\omega \textbf{n} $
\end{center}
where $\textbf{n}$ is the unit outer normal of e pointing towards the outside of $\Omega $.
\subsection{Weak Formulation}
Find $u\in V=H^{1}_{0}(\Omega)$ such that
\begin{center}
$a(u,v)=F(v)\quad \forall \ v \in V=H^{1}_{0}(\Omega) $
\end{center}
where
\begin{align}
a(u,v) &=\int_{\Omega}\nabla u.\nabla v \\
F(v) &=\int_{\Omega}fv
\end{align}
$\Omega$ is convex polygon and $f\in L^{2}(\Omega) $ therefore $u\in H^{2}(\Omega) $ by elliptic regularity theory $\cite{s1989topics}.$
\begin{figure}
\caption{NC1-C2 Element}
\centering
\includegraphics[scale=0.6]{last2.eps}
\end{figure}
$\linebreak $
\subsection{NC1-C2 methods}
In order to define a nonconforming space,we introduce some further notation.\\
$V^{1}_{h}= \{ v \in L^{2}(\Omega) \ : v \vert_{K} \ \text {is linear}\ \forall \ K\in \tau_{h},v \ \text {is continuous at the mid points of the edges of} \ \tau_{h} \} $
The above space is basically $p_{1} $ -nonconforming space.\\
We define
$D_{K}^{2}= \text{span} \{ \phi_{1},\phi_{2},\phi_{3} \} $;where $\phi_{i}=\hat{\phi}_{i} \circ F^{-1}_{K} $ and $F_{K}$ is Affine mapping from $\hat{K}$ to K [Fig 2]. $\hat{\phi}_{1},\hat{\phi}_{2},\hat{\phi}_{3}$ are basis fn on reference triangle $\hat{K}$ corresponding to the vertices $\hat{b}_{1},\hat{b}_{2},\hat{b}_{3} $ respectively, which is defined by
\begin{align}
\hat{\phi_{1}} &=(-1+2x+2y)(-1+x+y) \nonumber \\
\hat{\phi_{2}} &=(2x-1)x \nonumber \\
\hat{\phi_{3}} &=(2y-1)y \nonumber
\end{align}
$\phi_{i}, \ 1\leq i \leq 3$ is continuous along edge on each element.In this paper we use the following finite element space-
\begin{center}
$ V_{h}:= V_{h}^{1} \oplus V_{h}^{2}, \quad V_{h}^{2}:= \{v_{h}\in L^{2}(\Omega) : v_{h}\vert_{K} \in D_{K}^{2} \} $
\end{center}
Finite element space $V_{h} $ consists of piecewise quadratic function which is discontinuous along edge of each triangle except at mid points of edges [Fig 1].\\
A typical element $\omega \in V_{h} $ is demonstrated below-
\begin{center}
$\omega=\omega^{1}+\omega^{2} \ \text{where}\ \omega^{1}\in V_{h}^{1}\ \text{and} \ \omega^{2}\in V_{h}^{2} $
\end{center}
since $\omega^{1}$ is discontinuous along edges except at the midpoints and $\omega^{2}$ is continuous along edge, $\omega$ is discontinuous along edges except at the mid points.\\
On interior edges jump of typical element is reduced to $P_{1}$ polynomial .Let e be an interior edge which is shared by two triangle $K^{1}$ and $ K^{2}$ . $\omega\vert_{K^{1}} $ and $\omega\vert K^{2} $ are restrictions of $\omega$ on $K^{1}$ and $K^{2}$ respectively.
\begin{equation}
\begin{split}
[\omega] & = \omega\vert K^{1} -\omega \vert K^{2} \\
& = ( \omega^{1}\vert K^{1}+\omega^{2}\vert K^{1} )-( \omega^{1}\vert K^{2}+ \omega^{2}\vert K^{2})\\
& = (\omega^{1}\vert K^{1}-\omega^{1}\vert K^{2} )+( \omega^{2}\vert K^{1}-\omega^{2}\vert K^{2})\\
& = (\omega^{1}\vert K^{1}-\omega^{1}\vert K^{2} )
\end{split}
\end{equation}
This space contains the space of continuous piecewise-quadratic and space of nonconforming piecewise-linear,since
\begin{center}
$V_{h}=V_{h}^{1}+P_{h}^{2}$
\end{center}
where $P_{h}^{2}$ is piecewise-quadratic conforming finite element space.
\subsection{Discretization}
The space $V_{h} $ is not continuous and hence it is no longer in $ H^{1}_{0}(\Omega)$.We must modify the variational form $a_{h}(.,.) $ in the discretized problem.We define the following bilinear form on $V_{h}+V_{\ast} $.where $V_{\ast} $ is subset of V and exact solution belongs to $V_{\ast} $.
\begin{align}
a_{h}(v,\omega) &=\sum_{K\in \tau_{h}}\int_{K}\nabla v.\nabla \omega \ dx \\
F_{h}(\omega)&=\sum_{K\in \tau_{h}} \int_{K}f\omega
\end{align}
\subsection{Consistency}
Let $u \in V_{\ast} \subset V $ such that u satisfies weak formulation,i,e
\begin{equation}
a(u,\omega)=F(\omega)\ \forall\ \omega \in V
\end{equation}
Then it is obvious that
\begin{equation}
a_{h}(u,\omega)=F_{h}(\omega) \ \forall \ \omega \in V_{h}
\end{equation}
Hence the discrete bilinear form is consistent.It also satisfies Galerkin's orthogonality condition since
\begin{equation}
a_{h}(u-u_{h},\omega)=0 \ \forall \ \omega \in V_{h}
\end{equation}
\subsection{Discrete stability} The discrete bilinear form $ a_{h}(.,.) $ enjoys discrete stability on $V_{h} $ if there is $C_{sta}>0 $ such that
\begin{equation}
c_{sta} \parallel v_{h} \parallel \leq \text{sup}_{\omega_{h}\in V_{h}\setminus \{0 \}} \frac{a_{h}(v_{h},\omega_{h})}{\parallel \omega_{h} \parallel}
\end{equation}
\begin{equation}
a_{h}(v_{h},\omega_{h})=\sum_{K\in \tau_{h}}\int_{K}\nabla v_{h}.\nabla \omega_{h}
\end{equation}
considering $v_{h}=\omega_{h}$ \\
\begin{equation}
\begin{split}
\parallel v_{h} \parallel^{2} & =\sum_{K\in \tau_{h}}\int_{K} \vert \nabla v_{h} \vert^{2} \\
& =a_{h}(v_{h},v_{h})
\end{split}
\end{equation}
Hence
\begin{equation}
\begin{split}
\parallel v_{h} \parallel & = \frac{a_{h}(v_{h},v_{h})}{\parallel v_{h}\parallel}\\
& \leq \text{sup}_{\omega_{h} \in V_{h}\setminus \{ 0\}}\frac{a_{h}(v_{h},\omega_{h})}{\parallel \omega_{h}\parallel}
\end{split}
\end{equation}
Hence $a_{h}(.,.)$ satisfies discrete stability condition.This implies the discrete bilinear form
\begin{equation}
a_{h}(u,\omega)=F_{h}(\omega) \ \forall \ u,\omega \in V_{h}
\end{equation}
is well posed i.e. the discrete bilinear form has unique solution.
\section{Apriori Error Estimation}
$\textbf{Lemma 1}$ $ \ $ Assume dim $V_{h} < \infty $.Let $ a_{h}(..)$ be a symmetric positive definite bilinear form on $ V+V_{h}$ which reduces to a(.,.) on V.Let $u \in V $ solve
\begin{equation}
a(u,v)=F(v) \ \forall \ v\in V \nonumber
\end{equation}
where $F\in V' \cap V_{h}' $.Let $u_{h} \in V_{h} $ solve
\begin{equation}
a_{h}(u_{h},v)=F(v) \ \forall \ v \in V \nonumber
\end{equation}
Then
\begin{equation}
\parallel u-u_{h} \parallel_{h} \leq \text{inf}_{v \in V_{h}} \parallel u-v \parallel _{h}+\text{sup}_{\omega \in V_{h}\setminus \{ 0\}} \frac{\vert a_{h}(u-u_{h},\omega)}{\parallel \omega \parallel_{h}}
\end{equation}
where $\parallel . \parallel_{h}=\sqrt{a_{h}(.,.)} $\\
proof:- Let $\tilde{u_{h}} \in V_{h} $ satisfies
\begin{equation}
a_{h}(\tilde{u_{h}},v)=a_{h}(u,v) \ \forall \ v \in V_{h} \nonumber
\end{equation}
which implies that
\begin{equation}
a_{h}(\tilde{u_{h}}-u,v) =0 \ \forall \ v\in V_{h} \nonumber
\end{equation}
\begin{equation}
\Rightarrow \ \parallel u-\tilde{u_{h}} \parallel_{h} =\text{inf}_{v \in V_{h}} \parallel u-v \parallel_{h} \nonumber
\end{equation}
Then
\begin{equation}
\begin{split}
\parallel u-u_{h}\parallel_{h} &\leq \parallel u- \tilde{u_{h}} \parallel_{h}+\parallel \tilde{u_{h}}-u_{h} \parallel_{h} \\
&\leq \parallel u-\tilde{u_{h}} \parallel_{h}+\text{sup}_{\omega \in V_{h}\setminus \{0 \}} \frac{\vert a_{h}(\tilde{u_{h}}-u_{h},\omega)\vert}{\parallel \omega \parallel_{h}} \nonumber
\end{split}
\end{equation}
$\linebreak $
$\textbf{Lemma 2} $ $\quad$ Let K be an arbitrary element of conforming triangulation $\tau_{h}$.Then the following inequality holds
\begin{equation}
\vert e \vert^{-1} \parallel \zeta \parallel_{L^{2}(e)}^{2} \leq \ C(h_{K}^{-2} \parallel \zeta \parallel_{L^{2}(K)}^{2}+ \vert \zeta \vert_{H^{1}(K)}^{2})\ \forall \ \zeta \in H^{1}(K)
\end{equation}
where $\vert e \vert $ denotes the length of edge $e\subset \partial K$, $h_{K}$ =diam$\ $K,and the positive constant depends only on the chunkiness parameter of K.\\
Proof: See the details in \cite{brenner2008mathematical}\\
$\textbf{Lemma 3} $ \quad Let all assumptions of Lemma 2 hold and $\omega $ be an arbitrary element of $ V_{h} $.Then
\begin{equation}
\vert e \vert \parallel [\omega] \parallel_{L^{2}(e)}^{2} \leq \ C \ \sum_{K \in \tau_{e}} h_{K}^{2} \vert \omega \vert_{H^{1}(K)}^{2}
\end{equation}
where $[\omega] $ denotes jump of $\omega $ along edge $e \in \varepsilon^{h}$.\\
Proof: Using lemma 2 we can write
\begin{align}
\vert e \vert^{-1} \parallel [\omega] \parallel_{L^{2}(e)}^{2} \ &\leq \ C \sum_{K \in \tau_{e}}(h_{K}^{-2} \parallel \omega \nonumber \parallel_{L^{2}(K)}^{2} +\vert \omega \vert_{H^{1}(K)}^{2} ) \\ \nonumber
\vert e \vert \parallel [\omega] \parallel_{L^{2}(e)}^{2} &\leq \ C \sum_{K \in \tau_{e}} (\vert e \vert^{2} h_{K}^{-2} \parallel \omega \parallel_{L^{2}(K)}^{2}+\vert e \vert^{2} \vert \omega \vert_{H^{1}(K)}^{2}) \nonumber
\end{align}
Where $\tau_{e} $ is the set of triangles in $ \tau_{h}$ containing e on their boundaries.
Again $[\omega]$ =0 at midpoint of each edge of K hence first part of of right-hand side will be vanished.Therefore we have
\begin{equation}
\begin{split}
\vert e \vert \parallel [\omega] \parallel_{L^{2}(e)}^{2} &\leq \ C \sum_{K \in \tau_{e}} \vert e \vert^{2} \vert \omega \vert_{H^{1}(K)}^{2}\\ \nonumber
& \leq C \sum_{K \in \tau_{e}} h_{K}^{2} \vert \omega \vert_{H^{1}(K)}^{2} \nonumber
\end{split}
\end{equation}
$ \linebreak $
$\textbf{Lemma 4}$ \quad Let B be a ball in $\Omega $ such that $\Omega $ is star-shaped with respect to B and such that its radius $ \rho > (\frac{1}{2}) \rho_{max}$. Let $Q^{m}u $ be the Taylor polynomial of order m of u averaged over B where $u \in W_{p}^{m}(\Omega) $ and $p\geq 1$.Then
\begin{equation}
\vert u-Q^{m}u \vert_{W^{k}_{p}(\Omega)} \leq C_{m,n,\gamma} d^{m-k} \vert u \vert_{W_{p}^{m}(\Omega)} \ k=0,1,\dot{...} ,m,
\end{equation}
where d=diam$(\Omega)$ and $ \rho_{max}$=sup $\{ $ $\rho:\Omega$ is star-shaped with respect to a ball of radius $\rho$ $\}$
Proof: See the details in \cite{brenner2008mathematical}\\
\subsection{}
The important ingredient in the error analysis is a bound on the approximation error $\parallel u-u_{I} \parallel $ where $u_{I} \in V_{h} $ is a suitable interpolation which agrees with u at mid point of each edges of $\varepsilon^{h}$ of exact solution u.The interpolation operator is defined at the element level. We just require the local approximation property
\begin{equation}
\vert u - u_{I} \vert_{H^{s}(K)} \leq C h^{p+1-s}_{K} \vert u \vert_{H^{p+1}(K)} \quad \forall \ K \in \tau_{h} \ ,s=0,1,2 \nonumber
\end{equation}
It will be useful to define it explicitly.It is defined in two steps.\\
\begin{figure}
\caption{Affine Mapping}
\centering
\includegraphics[scale=0.5]{figure50.eps}
\end{figure}
Let $\hat{K} $ be the reference triangle with vertices$ \ \hat{b_{1}},\hat{b_{2}},\hat{b_{3}} \ $ whose coordinates are $\ $ (0,0),(1,0),(0,1) respectively and $\hat{m_{i}} $ be the midpoint of the side joining i and i+1(modulo 3) vertices.\\
We define
\begin{equation}
\hat{I}^{1}(\hat{u})=\hat{u}(\hat{m}_{1}) \hat{\phi}_{4}+\hat{u}(\hat{m}_{2})\hat{\phi}_{5}+\hat{u}(\hat{m}_{3})\hat{\phi}_{6} \nonumber
\end{equation}
\begin{equation}
\hat{I}^{2}(\hat{u})=(\hat{u}(\hat{b}_{1})-\hat{I}^{1}(\hat{u})(\hat{b}_{1}))\hat{\phi}_{1}+(\hat{u}(\hat{b}_{2})-\hat{I}^{1}(\hat{u})(\hat{b}_{2}))\hat{\phi}_{2}+(\hat{u}(\hat{b}_{3})-\hat{I}^{1}(\hat{u})(\hat{b}_{3}))\hat{\phi}_{3} \nonumber
\end{equation}
Finally we define interpolation as
\begin{equation}
\begin{split}
\hat{I}(\hat{u}) & = \hat{I}^{1}(\hat{u})+ \hat{I}^{2}(\hat{u}) \\
& = \sum_{j=1}^{6} \hat{L}_{j}(\hat{u})\hat{\phi}_{j}
\end{split}
\end{equation}
where $\hat{L}_{i} $ for i=1,2,3,4,5,6 $\ $ continuous linear functional.\\
$\linebreak$
Now we will show that $P_{2}(\hat{K}) $ is unisolvent with respect to these functionals, i.e.for an arbitrary polynomial $\hat{p} \in P_{2}(\hat{K}) $ $ \ $ $ \hat{L}_{i}(\hat{p})=0$ implies $\hat{p}=0 $. \\
Since $\hat{p} \in P_{2}(\hat{K}) $ implies that $\hat{p} $ can be written as linear combination of basis of $P_{2}(\hat{K}) $ .Then
\begin{align}
\ \hat{p} \ &=\sum_{i=1}^{6} C_{i} \phi_{i}. \nonumber \\ \nonumber
C_{4} & =\hat{p}(\hat{m}_{1})=\hat{L}_{4}(\hat{p})=0\\ \nonumber
C_{5} & =\hat{p}(\hat{m}_{2})=\hat{L}_{5}(\hat{p})=0\\ \nonumber
C_{6} & =\hat{p}(\hat{m}_{3})=\hat{L}_{6}(\hat{p})=0\\ \nonumber
\Rightarrow \hat{p} &=C_{1}\hat{\phi}_{1}+C_{2}\hat{\phi}_{2}+C_{3}\hat{\phi}_{3}
\end{align}
again
\begin{align}
\ C_{1} &=\hat{p}(\hat{b}_{1})=\hat{L}_{1}(\hat{p})=0 \nonumber \\ \nonumber
C_{2} & =\hat{p}(\hat{m}_{2})=\hat{L}_{2}(\hat{p})=0\\ \nonumber
C_{3} & =\hat{p}(\hat{m}_{3})=\hat{L}_{3}(\hat{p})=0\\ \nonumber
\Rightarrow \ \hat{p} &=0
\end{align}
Similarly it can be shown that for arbitrary $ \hat{p}\in P_{2}(\hat{K})$,$\ $ $\hat{I}(\hat{p})=\hat{p} $ .
Let $(K,P_{2}(K),\Sigma)$ be an affine finite element of $(\hat{K},P_{2}(\hat{K}),\hat{\Sigma)}$ $\ $ where$\ $ $\Sigma=\{L_{1},L_{2},L_{3},L_{4},L_{5},L_{6}\} $ and $ \ $ $\hat{\Sigma}=\{\hat{L}_{1},\hat{L}_{2},\hat{L}_{3},\hat{L}_{4},\hat{L}_{5},\hat{L}_{6}\} $ .\\
Then for all $v \in H^{m+1}(K) $ we have
\begin{equation}
\parallel D^{s}(u-I_{h}u)\parallel_{L^{2}(K)} \leq C\ h_{K}^{m+1-s}\ \parallel D^{m+1} u \parallel_{L^{2}(K)}
\end{equation}
where $s \leq m+1 $ and m=0,1,2.
\subsection{}
We want to find out $\parallel u-u_{h}\parallel_{h}$
Using lemma 1 we can write
\begin{equation}
\parallel u-u_{h}\parallel_{h} \leq \text{inf}_{v \in V_{h}} \parallel u-v \parallel_{h}+\text{sup}_{\omega \in V_{h}\setminus \{ 0\}} \frac{a_{h}(u-u_{h},\omega)}{\parallel \omega \parallel_{h}} \nonumber
\end{equation}
\begin{equation}
\begin{split}
\text{inf}_{v \in V_{h}} \parallel u-v \parallel_{h} &\leq \parallel u-u_{I} \parallel_{h}\\
&\leq \ C\ h^{2}\ \vert u \vert_{H^{3}(\Omega)} \quad \text{by} \quad (20)
\end{split}
\end{equation}
Now we have to estimate
\begin{center}
$\text{sup}_{\omega \in V_{h}\setminus \{ 0\}} \frac{\vert a_{h}(u-u_{h},\omega)\vert}{\parallel \omega \parallel_{h}}$
\end{center}
\begin{equation}
\begin{split}
a_{h}(u-u_{h},\omega) & = \sum_{K \in \tau_{h}} \int_{K} \nabla u. \nabla \omega dx -\int_{\Omega}f \omega dx\\
&=\sum_{k\in \tau_{h}} [\int_{\partial k} \nabla u.\omega \textbf{n} ds - \int_{K} \Delta u \omega d\Omega]-\int_{K}f\omega d \Omega \\
&=\sum_{K \in \tau_{h}} \int_{\partial K } \nabla u.\omega \textbf{n} ds\\
&=\sum_{e \in \varepsilon^{h}} \int_{e} \nabla u.[\omega] ds
\end{split}
\end{equation}
Again we have the following estimate
\begin{equation}
\sum_{e\in \varepsilon^{h}} \int_{e} \nabla u.[\omega] ds=\sum_{e \in \varepsilon^{h}} \int_{e}(\nabla u.\textbf{n}_{e}-c_{e})\textbf{n}_{e}.[\omega]ds
\end{equation}
since $ \int_{e}[\omega]=0$ \quad by (4).
\begin{equation}
\begin{split}
\sum_{e \in \varepsilon^{h}} \int_{e} \nabla u.[\omega] ds &=\sum_{e \in \varepsilon ^{h}} \int_{e} (\nabla u.\textbf{n}_{e}-c_{e})\textbf{n}_{e}.[\omega]ds \\
& \leq \sum_{e \in \varepsilon^{h}} \vert e \vert^{-\frac{1}{2}} \parallel \nabla u. \textbf{n}_{e} -c_{e} \parallel_{L^{2}(e)} \vert e \vert^{\frac{1}{2}} \parallel [\omega] \parallel_{L^{2}(e)}\\
& \leq (\sum_{e \in \varepsilon^{h}} \vert e \vert^{-1} \parallel \nabla u.\textbf{n}_{e}-C_{e} \parallel^{2}_{L^{2}(e)} )^{\frac{1}{2}}(\sum_{e \in \varepsilon^{h}} \vert e \vert \parallel[\omega] \parallel^{2}_{L^{2}(e)})^{\frac{1}{2}}\\
& \leq C [\sum_{e \in \varepsilon^{h}} \text{min}_{K \in \tau_{e}}(h_{K}^{-2} \parallel \nabla u .\textbf{n}_{e}-C_{e} \parallel^{2}_{L^{2}(T)}+\vert u \vert ^{2}_{H^{2}(K)})]^{\frac{1}{2}} [\sum_{e \in \varepsilon^{h}} \sum_{K \in \tau_{e}} h_{K}^{2} \vert \omega \vert^{2}_{H^{1}(K)} ] ^{\frac{1}{2}} \ \text{by} \ (16,17) \\
& \leq C h \vert u \vert _{H^{2}(\Omega)} \parallel \omega \parallel_{h} \nonumber
\end{split}
\end{equation}
Hence we have
\begin{equation}
\vert a_{h}(u-u_{h},\omega)\vert \leq C\ h \ \vert u \vert_{H^{2}(\Omega)} \parallel \omega \parallel_{h} \nonumber
\end{equation}
\begin{equation}
\Rightarrow \quad \frac{\vert a_{h}(u-u_{h},\omega)\vert}{\parallel \omega \parallel_{h}} \leq C \ h \ \vert u \vert_{H^{2}(\Omega)} \nonumber
\end{equation}
\begin{equation}
\Rightarrow \quad \parallel u-u_{h} \parallel_{h} \leq C\ h \ \vert u \vert_{H^{2}(\Omega)}\quad \text{using \ (21)}
\end{equation}
\subsection{$ L^{2} $-Error Estimate}
Let $ \eta \in H^{2}(\Omega)\bigcap H^{1}_{0}(\Omega) $ satisfy
\begin{equation}
a(\eta,v)=\int_{\Omega} v (u-u_{h}) \ \forall \ v \in H^{1}_{0}(\Omega) \nonumber
\end{equation}
and $\eta_{h} \in V_{h} $ satisfy
\begin{equation}
a_{h}(\eta_{h},v)=\int_{\Omega} v \ (u-u_{h}) \ \forall \ v \in V_{h} \nonumber
\end{equation}
Again we have
\begin{equation}
\begin{split}
\parallel u-u_{h} \parallel^{2}_{L^{2}(\Omega)} &= \int_{\Omega}(u-u_{h})^{2} \\
&= \int_{\Omega}(u-u_{h})(u-u_{h}) \\
&=\int_{\Omega}u(u-u_{h})-\int_{\Omega}u_{h}(u-u_{h})\\
&=a(u,\eta)-a_{h}(u_{h},\eta_{h}) \\
&=a_{h}(u,\eta)-a_{h}(u_{h},\eta)+a_{h}(u_{h},\eta)-a_{h}(u_{h},\eta_{h})\\
&=a_{h}(u-u_{h},\eta)+a_{h}(u_{h},\eta-\eta_{h})\\
&=a_{h}(u-u_{h},\eta-\eta_{h})+a_{h}(u-u_{h},\eta_{h})+a_{h}(u_{h},\eta-\eta_{h})
\end{split}
\end{equation}
Using estimation (24) we can write
\begin{equation}
\begin{split}
a_{h}(u-u_{h},\eta-\eta_{h}) & \leq \parallel u-u_{h} \parallel_{h} \parallel \eta-\eta_{h} \parallel_{h} \quad (\text{using \ Cauchy-Schwarz }) \\
& \leq \ C \ h^{2} \ \vert u \vert_{H^{2}(\Omega)} \ \vert \eta \vert_{H^{2}(\Omega)}
\end{split}
\end{equation}
Where $\ $ C $\ $ is generic constant.
\begin{equation}
a_{h}(u-u_{h},\eta_{h})=a_{h}(u-u_{h},\eta_{h}-\eta_{I})+a_{h}(u-u_{h},\eta_{h}) \nonumber
\end{equation}
\begin{equation}
\begin{split}
a_{h}(u-u_{h},\eta_{h}-\eta_{I}) & \leq \parallel u-u_{h} \parallel_{h} \ \parallel \eta_{h} -\eta_{I} \parallel_{h} \\
& \leq \ C \ h^{2} \ \vert u \vert_{H^{2}(\Omega)} \ \vert \eta \vert_{H^{2}(\Omega)} \nonumber
\end{split}
\end{equation}
\begin{equation}
\begin{split}
\parallel \eta_{h}-\eta_{I}\parallel_{h} & = \parallel \eta_{h}-\eta+\eta-\eta_{I} \parallel_{h} \\
& \leq \parallel \eta_{h}-\eta \parallel_{h} +\parallel \eta - \eta_{I} \parallel_{h} \\
&\leq \ C_{1}\ h \ \vert \eta \vert_{H^{2}(\Omega)} + C_{2} \ h \ \vert \eta \vert_{H^{2}(\Omega)} \\
&=\ C \ h \ \vert \eta \vert_{H^{2}(\Omega)} \nonumber
\end{split}
\end{equation}
\begin{equation}
\begin{split}
a_{h}(u-u_{h},\eta_{I}) &= \sum_{e \in \varepsilon^{h}} \ \int_{e}\ \nabla u.[\eta_{I}] \ ds \quad \quad by \ (22) \\
& =\sum_{e \in \varepsilon^{h}} \ \int_{e}\ \nabla u.[\eta_{I}-\eta] \ ds \\
& =\sum_{e \in \varepsilon^{h}} \ \int_{e} \ (\nabla u.\textbf{n}_{e}-C_{e})\ \textbf{n}_{e}.[\eta_{I}-\eta]\ ds \nonumber
\end{split}
\end{equation}
Where $ \ C_{e} \ $ is arbitrary constant and we have used the following two identity
\begin{center}
$ [\eta]=0 \quad \& \quad \int_{e}[\eta_{I}]=0 $
\end{center}
\begin{equation}
\begin{split}
a_{h}(u-u_{h},\eta_{I}) & \leq \sum_{e \in \varepsilon^{h}} \vert e \vert^{-\frac{1}{2}} \parallel \nabla u.\textbf{n}_{e}-C_{e} \parallel_{L^{2}(e)} \vert e \vert^{\frac{1}{2}} \parallel \eta-\eta_{I} \parallel_{L^{2}(e)} \nonumber \\
& \leq C \ [\sum_{e \in \varepsilon^{h}} \text{min}_{K \in \tau_{e}}(h^{-2}_{K} \text{inf}_{C_{e} \in \mathbb{R}} \parallel \nabla u.\textbf{n}_{e} -C_{e} \parallel^{2}_{L^{2}(K)} + \vert u \vert^{2}_{H^{2}(K)} )]^{\frac{1}{2}} \\
& \times [\sum_{e \in \varepsilon^{h}} \vert e \vert^{2} \sum_{K \in \tau_{e}} (h^{-2}_{K} \parallel \eta-\eta_{I} \parallel^{2}_{L^{2}(K)} + \vert \eta-\eta_{I}\vert^{2}_{H^{1}(K)} )]^\frac{1}{2} \\
&\leq\ C \ \vert u \vert_{H^{2}(\Omega)} (h^{2} \ \vert \eta \vert_{H^{2}(\Omega)})\quad \text{using} \ (18,20)
\end{split}
\end{equation}
Therefore
\begin{equation}
a_{h}(u-u_{h},\eta_{h}) \leq \ C \ h^{2} \ \vert u \vert_{H^{2}(\Omega)} \ \vert \eta \vert_{H^{2}(\Omega)}
\end{equation}
Again we have
\begin{equation}
\begin{split}
a_{h}(u_{h},\eta-\eta_{h}) & = a_{h}(\eta-\eta_{h},u_{h}) \\
& =a_{h}(\eta-\eta_{h},u_{h}-u_{I})+a_{h}(\eta-\eta_{h},u_{I})
\end{split}
\end{equation}
\begin{equation}
a_{h}(\eta-\eta_{h},u_{h}-u_{I}) \ \leq \ \parallel \eta-\eta_{h} \parallel_{h} \parallel u_{h}-u_{I} \parallel_{h} \nonumber
\end{equation}
and
\begin{equation}
\begin{split}
\parallel u_{h}-u_{I}\parallel_{h} &= \parallel u_{h}-u+u-u_{I} \parallel_{h} \\
& \leq \ \parallel u-u_{h} \parallel_{h}+\parallel u-u_{I} \parallel_{h} \\
& \leq \ C \ h \vert u \vert_{H^{2}(\Omega)} \nonumber
\end{split}
\end{equation}
Therefore
\begin{equation}
a_{h}(\eta-\eta_{h},u_{h}-u_{I}) \leq \ C \ h^{2} \vert \eta \vert_{H^{2}(\Omega)} \ \vert u \vert_{H^{2}(\Omega)}
\end{equation}
We can write second term of (28) as
\begin{equation}
\begin{split}
a_{h}(\eta-\eta_{h},u_{I}) &= \sum_{e \in \varepsilon^{h}} \int_{e} \nabla \eta.[u_{I}] \\
&= \sum_{e \in \varepsilon^{h}} \int_{e} (\nabla \eta.\textbf{n}_{e}-C_{e})\ \textbf{n}_{e}.[u-u_{I}] \ ds \nonumber
\end{split}
\end{equation}
where we have used the following two identity
\begin{center}
$[u]=0 \quad \& \quad \int_{e}[u_{I}]= 0 $ ,e is an interior edge.
\end{center}
Therefore
\begin{equation}
\begin{split}
a_{h}(\eta-\eta_{h},u_{I}) &= \sum_{e \in \varepsilon^{h}} \ \int_{e} (\nabla \eta.\textbf{n}_{e}-C_{e})\ \textbf{n}_{e}.[u-u_{I}] \ ds \\
& \leq \sum_{e \in \varepsilon^{h}} \ \vert e \vert^{-\frac{1}{2}} \parallel \nabla \eta .\textbf{n}_{e}-C_{e} \parallel_{L^{2}(e)} \ \vert e \vert^{\frac{1}{2}} \parallel u-u_{I}\parallel_{L^{2}(e)} \\
& \leq \ C \ \vert \eta \vert_{H^{2}(\Omega)}(h^{2} \ \vert u \vert_{H^{2}(\Omega)} )
\end{split}
\end{equation}
Hence
\begin{equation}
\begin{split}
a_{h}(u_{h},\eta-\eta_{h}) &=a_{h}(\eta-\eta_{h},u_{h}-u_{I})+a_{h}(\eta-\eta_{h},u_{I}) \\
& \leq \ C \ h^{2} \ \vert \eta \vert_{H^{2}(\Omega)} \ \vert u \vert_{H^{2}(\Omega)}
\end{split}
\end{equation}
Using estimations (26),(27),(31) in (25) we can write
\begin{equation}
\parallel u-u_{h}\parallel^{2}_{L^{2}(\Omega)} \leq \ C \ h^{2} \ \vert \eta \vert_{H^{2}(\Omega)}\ \vert u \vert_{H^{2}(\Omega)}
\end{equation}
using elliptic regularity we can write
\begin{equation}
\parallel \eta \parallel_{H^{2}(\Omega)} \leq \ C \ \parallel u-u_{h} \parallel_{L^{2}(\Omega)}
\end{equation}
Therefore
\begin{equation}
\quad \parallel u-u_{h} \parallel_{L^{2}(\Omega)} \leq \ C \ h^{2} \ \vert u \vert_{H^{2}(\Omega)} \nonumber
\end{equation}
\section{Numerical Test}
In this section we perform grid convergence studies for the proposed 'NC1-C2' method.The new method was implemented using penalization technique.We estimate the experimental order of convergence by the formula
\begin{equation}
Eoc=log(\frac{E(2h)}{E(h)}) \nonumber
\end{equation}
where $E(h)=\parallel u-u_{h}\parallel $ is the error in the specified norm.The result indicate the same convergence behaviour for NC1-C2 method and and $p^{1}$-nonconforming method.We have glued second order conforming element with first order nonconforming element suitably and observed that first order nonconforming element dominated second order conforming element and we have obtained an EOC of 2.0 in the $L^{2}$ norm and $1.0 $ in the $H^{1} $ norm which is same as $p^{1}$ nonconforming method.EOC stands for experimental order of convergence.\\
\textbf{Stationary Diffusion Problem :}
We consider the Poisson equation
\begin{equation}
-\Delta u=f \quad \text{on} \quad \Omega=(0,1)\times(0,1)
\end{equation}
with homogeneous boundary conditions and the right hand side $\ f=2\pi^{2}sin(\pi x)sin(\pi y)$. The exact solution is given by $u(x,y)=sin(\pi x)sin(\pi y) $.\\
\section*{Acknowledgments}
Authors thank Prof. Alexei Lozinski(Université de Franche-Comté, Besancon) and Prof Frederic Hecht(LJLL-Université Pierre et Marie Curie)
for fruitful discussions on implementation of NC1-C2 method during the workshop CIMPA-2015 held at IIT BOMBAY, INDIA.
\bibliographystyle{unsrt}
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BkiUdULxK7IDM3wqf1-n
| 5 | 1 |
\section{Introduction}
Granular materials, including natural sand, silica spheres, and glass beads, have garnered increasing interest within the geophysics and geotechnical communities, as they enable the physical and geological modeling of various complex structures \cite{Sherlock,Krawczyk}. This is due to the wide range of shapes and sizes of their grains, as well as their intrinsic mechanical parameters.
In particular, glass microbeads (GBs) have recently been employed in laboratory-scale physical models to study the propagation of seismic waves in unconsolidated \cite{Jacob,Bodet,Bergamo} and porous \cite{Pasquet} heterogeneous media. Small-scale physical modelling, combined with laser measurement techniques, is often used to address theoretical and methodological issues related to seismic exploration, especially when experimental tests or inversion techniques are needed \cite{Campman,Bodet3,Bodet4,Dewangan,Bretaudeau}.\\
Under gravity loads, the elastic properties of granular materials can be defined according to the Hertzian theory, which describes the contact forces between the beads \cite{Gassmann}. The theory predicts that, near the free surface, the longitudinal $v_P$ and shear $v_S$ wave velocity profiles exhibit a power-law dependence on the compacting pressure $p=\rho gz$ along the depth $z$, where $\rho$ is the medium density and $g$ the gravitational constant. Consequently, the wave velocity profiles of a granular material can be expressed as $v_{P,S} = \gamma_{P,S} (\rho g z)^{\alpha_{P,S}}$, with $\gamma_{P,S}$ designating a coefficient that is mainly influenced by the medium elastic properties and porosity and $\alpha_{P,S}$ denoting the power-law exponent; estimated to equal 1/6, when uniform beads randomly arranged in a close-packed structure are considered. Whether this coefficient is correct when describing actual granular materials is still debated \cite{Makse}. Recent experiments suggest that imperfections in the geometry or dimension of the beads might affect the value of $\alpha_{P,S}$. Regardless of the physical origins of such a variation in the power-law coefficients, it is worth noting the consequences this has on the acoustic wave propagation. In particular, the increase in material stiffness with depth causes the upward bending of the acoustic waves (also known as the ``mirage" effect \cite{Liu}) toward the free surface \cite{Gusev2}. Thus, the depth-increasing stiffness profile of the medium combined with the presence of a mechanically free surface leads to the formation of guided surface acoustic modes (GSAMs) channeled between the free surface and the increasingly rigid material \cite{Gusev,Bonneau}. Among these GSAMs are vertically polarized P-SV waves, composed of coupled longitudinal (P) and shear vertical (SV) modes \cite{Aleshin}. Experimental dispersion analysis of these P-SV acoustic modes propagating along the surface of granular media has been conducted in existing literature, with the purpose of estimating the elastic power-law coefficients \cite{Jacob,Bodet2,Bodet}. In addition, experimentally determined dispersion relations have revealed unusual low propagation velocities, ascribed to very low pressures of the near-surface layers \cite{Jacob}.\\
While the dynamics of the P-SV waves has been extensively investigated, the attenuation of their amplitude due to geometric spreading and material damping remains uncharted. The geometric spreading is mainly controlled by the source type (e.g., point or line source) and the elastic profile of the medium \cite{Foti}, whereas the material (or internal) damping is linked to the energy dissipated within the skeletal frame of the medium \cite{Rix}. Both geometric spreading and material damping strongly influence the seismic site response. Determination of attenuation characteristics of shallow unconsolidated soil layers is fundamental to the study of several problems in geophysics and geotechnical engineering, such as site amplification or ground borne vibrations \cite{Field}. The objective of this work is to determine the P-SV wave geometric spreading in order to gain insights into the attenuation mechanisms of vertically polarized waves traveling at the surface of unconsolidated substrates with power-law rigidity profiles at the seismic scale. In addition, frequency-dependent P-SV wave attenuation coefficients are determined to characterize the material damping of the employed granular medium.\\
In geophysics, surface wave methods (SWMs) are among the most established in situ test methods for the evaluation of geometric and material attenuation properties of near-surface soils \cite{Rix,Foti}. The SWMs include the spectral slope \cite{Jongmans,Stewart,Gibbs}, waveform inversion \cite{Askan}, half-power bandwidth \cite{Badsar}, circle fit \cite{Verachtert} techniques, and the regression of the amplitude of particle motion as a function of the distance from the source, among others.\\
In this work, the particle motion amplitude regression technique is used to determine the attenuation properties of a granular layer of GBs. The amplitude decay of the displacement associated with the P-SV surface modes is analysed as the wavefront propagates away from the source. We replicate the experimental setup designed in Refs. \cite{Jacob,Bodet,Bergamo}, which consists of a wooden box filled with an unconsolidated granular material made of silica microbeads. A mechanical point source and a laser-Doppler vibrometer are employed to record a small-scale seismic line at the surface. After recalling the dispersion properties of vertically polarized surface modes detected in the experiment, we analyse the effects of geometric spreading and material damping on the multimode P-SV wave displacement. According to the GSAM theory described in Refs. \cite{Aleshin,Jacob}, the P-SV acoustic modes are modeled as linear modes and the granular medium is treated as continuous and elastic, featuring a depth-increasing rigidity gradient. In virtue of the linearity of these modes, the geometric spreading is calculated by adopting an approach originally developed to quantify the spatial attenuation law of seismic Rayleigh-like waves in inhomogeneous stratified media \cite{Foti}. Next, by performing a regression analysis on the particle displacements, as calculated from the experimental velocity data, we determine the frequency-dependent attenuation coefficients, which account for the intrinsic energy dissipation induced by material damping.
\section{Sample preparation and experimental setup}
The experimental setup, illustrated in Fig.~\ref{ExpSetup}, consists of a wooden box of dimensions 2000 ${\times}$ 1500 ${\times}$ 1000 mm filled with an unconsolidated granular material, 150 ${\mu} $m-diameter silica microbeads. To minimize the amplitude of the reflected signals, a paperboard layer is arranged at the bottom of the wooden box. This setup has been originally designed to study the propagation of the GSAMs in granular media \cite{Jacob,Bodet,Bergamo}, while similar ones have been built to analyse their interaction with arrays of mechanical sub-wavelength oscillators \cite{Zaccherini1,Zaccherini2}.\\
The deposition process of the silica particles is designed to ensure a good homogeneity of the medium compaction and, at the same time, estimate its bulk density. Following the procedure adopted in Ref. \cite{Bodet}, the glass beads (approximately 3800 ${kg}$) are gently poured into the wooden box, step by step, in packages of 25 ${kg}$. The thickness of the obtained granular layer is measured to estimate the medium bulk density $\rho$, which is found to be 1600 ${kg/m^{3}}$.
\begin{figure}
\includegraphics[width=0.5\textwidth]{Fig1new}
\caption{The experimental setup including a wooden box filled with granular material, a laser-Doppler vibrometer, and an electromagnetic shaker driven by a waveform generator, which produces a Ricker pulse centered at 500 Hz. The laser vibrometer, mounted on a scanning stage, records the particle vertical velocity along the 800 mm-long middle line of the box (green line), with constant steps of $\Delta x=6.6$ mm.}
\label{ExpSetup}
\end{figure}
\noindent
We generate vertically polarized surface waves through a mechanical point excitation at the surface of the medium, situated at a distance of 375 mm from the box edge, and we record the particle vertical velocity field using a Laser-Doppler vibrometer (Polytec, OFV-500). The mechanical point excitation is realized by means of an 8 mm-diameter metal rod in contact with the surface forming an angle of 20$^{\circ}$ with the normal vector to the surface of the granular medium. The rod is attached to an electromagnetic shaker (Tira, Vibration Test Systems N1000 to N2700), driven through a waveform generator (Agilent, 33220), which produces the excitation input signal: a Ricker wavelet centered at 500 Hz. The vibrometer, mounted on a scanning stage moving along a two-dimensional plane parallel to the surface, records the particle vertical velocity along the 800 mm-long middle line of the box, with constant steps of $\Delta x=6.6$ mm (see the green line of Fig.~\ref{ExpSetup}). The chosen step allows for acquisition of 9 points for the shortest wavelength. Following the data acquisition procedure described in Refs. \cite{Jacob,Bodet}, for each point, we average 32 signals of 0.5 s duration at a sampling rate of 500 kHz. Following the procedure detailed in Ref. \cite{Jacob}, all measurements are carried out with small amplitude excitations to ensure that the applied acoustic pulses do not affect the state of the sample. In particular, in the far-field, the wave displacement amplitude is kept below 2 $\mu$m, considerably below the size of the grains (150 $\mu$m). In addition, to limit the effect of non-linearities between the metal rod and the granular material, the maximum vertical velocity close to the source is kept below 3 $\times 10^{-4}$ m/s (in accordance with the procedure described in Ref. \cite{Jacob}, where the maximum vertical velocity was kept below 5 $\times 10^{-4}$).
\section*{Experimental and numerical dispersion analysis}
\subsection{Experimental data and dispersion analysis}
Figure~\ref{DispRel}(a) illustrates the seismogram obtained from the data collected along the surface green line of Fig.~\ref{ExpSetup}. Several propagation modes arise due to constructive interference phenomena, but mainly two wavetrains dominate the seismogram: a quasi-compressional P wave and a packet of sagittal P-SV waves that propagate with lower velocities. The displacement field associated with these vertically polarized acoustic modes can be described via the following depth-dependent Helmholtz equations:
\begin{equation}
({v_S}^2(z)u'_x)'+\Big((\omega/k)^2-{v_P}^2(z)\Big)u_x =
i({v_P}^2(z)-2{v_S}^2(z)u'_z+({v_S}^2(z)u_z)')
\label{eq:Helmholtz1}
\end{equation}
\begin{equation}
({v_P}^2(z)u'_z)'+\Big((\omega/k)^2-{v_S}^2(z)\Big)u_z = i\Big({v_S}^2(z)u'_x+\big(({v_P}^2(z)-2{v_S}^2(z))u_x\big)'\Big)
\label{eq:Helmholtz2}
\end{equation}
where $u_x$ and $u_z$ denote the horizontal and the vertical displacement component, respectively, $\omega$ represents the angular frequency, $k$ the wavenumber, $i$ is the imaginary unit, and the prime encodes the derivative with respect to the vertical ($z$) axis.
\begin{figure*}
\includegraphics[width=1\textwidth]{Fig2new}
\caption{Dispersion Analysis. (a) Seismogram of a Ricker pulse propagating in the granular medium. (b) Experimental and numerical dispersion relations of P-SV acoustic surface modes propagating at the surface of an unconsolidated granular medium. (b) A drawing of the 3D granular unit cell developed in COMSOL MULTIPHYSICS®.}
\label{DispRel}
\end{figure*}
\noindent
Figure~\ref{DispRel}(b) shows the frequency-wavenumber spectrum derived after applying the two dimensional (2D) discrete Fourier transform (DFT) to the zero-padded time waveforms. The latter exhibits the frequency content of the acoustic waves detected in the seismogram. The low frequency (0-600 Hz) components of the signal are associated with the two lowest-order P-SV waves (labeled as "P-SV$_1$" and "P-SV$_2$"). It is worth noting that the P-SV$_2$ mode, associated with long-wavelength surface waves, propagates faster than the P-SV$_1$ mode, characterized by shorter wavelengths. Since most of the strain energy associated with surface wave motion is confined within a depth of about one wavelength $\lambda$ from the free boundary \cite{Achenbach}, the P-SV$_2$ mode penetrates deeper into the interior of a medium and, further influenced by the underlying stiffer layers, propagates faster than the P-SV$_1$ mode.\\
The high frequency (600-800 Hz) components of the signal are, instead, associated with the purely compressive (P) vertical wave traveling with higher velocities. These quasi-compressional fast modes can be easily detected in experiments, when the excitation has a significant vertical component of surface displacement \cite{Jacob,Bodet}, as it was the case in our experiment. The displacement field associated with these P modes can be described via Eqs. (\ref{eq:Helmholtz1}-\ref{eq:Helmholtz2}) by neglecting shear rigidity ($v_s$=0) \cite{Jacob}.
\subsection{Numerical dispersion analysis via Finite Element approach}
We confine our analysis to the P-SV acoustic waves and we numerically calculate their dispersion curves by modelling a portion of the granular medium with a Bloch wave - Finite Element (FE) approach, using the software COMSOL Multiphysics®. It has been shown that this numerical procedure allows for properly computing the GSAM dispersion relations \cite{Palermo,Zaccherini2}. We consider a 3D FE model of the unit cell, a column of granular material 1000 mm-high and 30 $\times$ 30 mm wide, as depicted in Fig.~\ref{DispRel}(b). Since the medium is homogeneous along the wave-propagation direction (i.e., along the $x$-axis), the width of the granular column ($D=30$ mm) can be arbitrarily chosen to derive the dispersive properties in the wavenumber range of interest (i.e., from 0 to 100 rad/m, as shown in the experimental $f$-$k$ spectrum illustrated in Fig.~\ref{DispRel}(b)). Since the maximum wavenumber within the range of interest is $k_{max}=100$ rad/m, it follows that $D = \pi/k_{max} \cong 30$ mm.\\
According to the GSAM theory described in Ref. \cite{Aleshin}, the medium is modeled as a linear elastic continuum, assuming constant density and depth-dependent longitudinal and shear velocities profiles: $\displaystyle v_P,_S=\gamma_P,_S(\rho gz)^{\alpha_P,_S}$ with $\gamma_P=14.4$, $\gamma_S=6.42$, and $\alpha_P,_S = 0.31$. We adopted the same elastic power-law coefficients ($\gamma_P,_S$, $\alpha_P,_S$) assumed in Ref. \cite{Palermo} to model the same material. Wave reflections from the bottom of the model are prevented via insertion of absorbing conditions and periodic Bloch boundary conditions are applied to the sides. We perform eigenfrequency analyses by sweeping the wave vector along the Irreducible Brillouin Contour, i.e., by considering the wavenumber spanning in the range of $0 - k_{max}$. The numerical dispersion curves, depicted as solid lines in Fig.~\ref{DispRel}(a), well match the experimental energy distribution of the two P-SV acoustic surface modes.
\section{Wave attenuation mechanisms}
\subsection{Amplitude attenuation of P-SV modes in granular media}
The amplitude of the particle motion associated with the vertically polarized surface waves, generated by a time-harmonic point load, which is applied at the free surface of an unconsolidated granular medium, attenuates not only due to the energy dissipated within the skeletal frame of the medium (i.e., material damping), but also due to the geometric spreading. Consequently, in the far-field, the vertical displacement amplitude induced at the surface by a harmonic point vertical source $F_z \cdot e^{i \omega t}$ in granular media can be expressed as follows \cite{Lai,Rix,Foti}:
\begin{equation}
\left|U_z(r,{\omega})\right|= F_z\cdot \Upsilon_{P-SV}(r,{\omega},z=0)\cdot e^{-\alpha(\omega) r}
\label{eq:fitting}
\end{equation}
where $\left|U_z(r,{\omega})\right|$ represents the spectral particle displacement amplitude at the surface, $F_z$ is the amplitude of the excitation, $\Upsilon_{P-SV}(r,{\omega},z=0)$ denotes the function describing the surface geometric spreading of the multimode P-SV waves, and $\alpha(\omega)$, the so-called attenuation coefficient or attenuation constant, is the parameter accounting for the material damping.\\
The geometric spreading function defines how the wave energy spreads into the inhomogeneous medium; in particular, how the P-SV displacement amplitude, for a given frequency, modifies as the wavefront travels away from the source, due to constructive wave interference phenomena occurring between the interfaces of in-depth layers with increasing rigidity. The geometric spreading is mainly controlled by the source type (e.g., point or line source) and the rigidity profile of the medium \cite{Foti}.\\
Instead, the frequency-dependent attenuation constant $\alpha(\omega)$ defines the exponential decay of the wave amplitude caused by the energy dissipated within the soil skeletal frame \cite{Rix}. In geophysics, $\alpha(\omega)$ is a widely used parameter to describe material damping and corresponds to the imaginary part of the complex wavenumber $k*=Re(k*)+Im(k*)=k-i\alpha(\omega)$. Theoretically, at each frequency, there exist as many attenuation coefficients as the total number of modes $M$. However, Eq. \ref{eq:fitting} considers a unique apparent attenuation coefficient that may represent the combination of the several propagating modes \cite{Foti}.\\
The theoretical model represented by Eq. \ref{eq:fitting} independently accounts for the geometric and material attenuation contributions. Since the geometric attenuation depends on the phase velocity, which is, usually, not independent of the material attenuation as a result of material dispersion, we recognize that the geometric attenuation may not be independent from the material one. However, since the numerical dispersion curves, calculated in absence of material damping-induced dissipation, well match the experimental ones, we consider that neglecting this dependence does not significantly affect our analysis.
\subsection{Geometric attenuation law in granular media}
The geometric spreading function of the GSAMs is computed using the explicit solution of the geometric attenuation law of Rayleigh-like waves in layered media, i.e., vertically heterogeneous media composed of multiple homogeneous linear elastic layers overlaying a homogeneous half-space. Indeed, the P-SV surface acoustic modes display a polarization in the vertical plane similar to that of the Rayleigh wave and the granular substrate featuring the power-law velocity profile can be considered as a stratified medium composed of infinitesimally thin layers. In addition, the dynamics of the first sagittal mode $P-SV_1$, which carries most of the energy in our experiment, resembles that of the Rayleigh wave in homogeneous media, tending to it in the presence of strong particle adhesion \cite{Aleshin}.
\subsubsection{Geometric spreading functions computed via modal superposition}
In lossless, vertically layered media, the far field wave motion induced by a time-harmonic, vertical point load applied at the free surface results via the superposition of several propagation modes each traveling, for a given frequency, at a different phase velocity. These multiple modes originate from interference phenomena arising among waves undergoing multiple reflections and refractions at the layer interfaces. The geometric attenuation law $\Upsilon(r,{\omega},z)$ resulting from the superposition of $M$ different Rayleigh-like modes can be expressed as follows \cite{Lai}:
\begin{equation}
\Upsilon(r,{\omega},z)=\frac{1}{4\sqrt{2\pi r}} \bigg\{\sum_{i=1}^M \sum_{j=1}^M \frac{r_z(k_i,\omega,z)r_z(k_j,\omega,z)r_z(k_i,\omega,z=0)r_z(k_j,\omega,z=0)\cos[r(k_i-k_j)]}{\sqrt{k_ik_j}[V_iU_iI_i][V_jU_jI_j]} \bigg\}^{0.5}
\label{eq:Spreading}
\end{equation}
where $r_z(k_j,\omega,z)$ and $r_x(k_j,\omega,z)$ are the vertical and the horizontal eigenfunctions associated with the $j$-th mode of propagation characterized by a wavenumber $k_j$. $V_j$ and $U_j$ are the phase and group velocity evaluated at $\omega$-$k_j$, whereas $I_j$ represents the Rayleigh-like energy integral \cite{Aki} evaluated at $\omega$-$k_j$ and defined as:
\begin{equation}
I_j(k_j,\omega,z)=\frac{1}{2} \int_{0}^{\infty}\rho(z)[{r_x^2(k_j,\omega,z)}+{r_z^2(k_j,\omega,z)}] dz
\label{eq:energy}
\end{equation}
\noindent
The total number of modes $M$ required to properly derive the surface geometric spreading function depends on the nature of the elastic profile of the soil, i.e., how the shear modulus varies with depth, and on the excitation frequency. Here, the spatial attenuation law is calculated considering the first two lowest-order P-SV modes, which dominate the experimental frequency-wavenumber spectrum illustrated in Fig.~\ref{DispRel}(a), hence, $M=2$. Since the main frequency content of these modes extends from 300 to 550 Hz, we compute the geometric spreading function at the surface ($z=0$), for values of $r$ ranging from 0 to 1 m, and for six equispaced frequency values within the aforementioned frequency range (i.e., at 300, 350, 400, 450, 500, 550 Hz). All input parameters including wavenumbers, eigenfunctions, phase and group velocities, and Rayleigh-like energy integral are obtained numerically from the eigenfrequency study performed on the granular unit cell. Figure~\ref{ModeShape} illustrates the vertical ($z$) and horizontal ($x$) displacement profiles of the two P-SV modes associated with the six aforementioned frequency values.
\begin{figure*}
\includegraphics[width=1\textwidth]{Fig3}
\caption{Numerical mode shapes. Vertical ($r_{z,P-SV_1}$, $r_{z,P-SV_2}$) and horizontal ($r_{x,P-SV_1}$, $r_{x,P-SV_2}$) displacement component of the first and second lowest-order P-SV acoustic surface modes, respectively.}
\label{ModeShape}
\end{figure*}
\noindent
The maximum displacement occurs in the near-surface layers, consistently for any surface mode and the wave energy, localized at the surface, decays rapidly with depth. As expected, the mode penetration depth, which is typically roughly equal to one wavelength $\lambda$, increases for decreasing frequency values. Furthermore, as disclosed in Refs. \cite{Gusev,Aleshin}, the number of "phases" in the displacement profile, as a function of the depth, corresponds to the order of modes.\\
The computed geometric spreading functions $\Upsilon(r,{\omega},z=0)$ describing the spatial attenuation of the P-SV modes at the surface are depicted as solid black lines in Fig.~\ref{GeomSpread}.
\begin{figure*}
\includegraphics[width=1\textwidth]{Fig5new}
\caption{Geometric spreading functions computed at the surface ($z=0$) at 300, 350, 400, 450, 500, and 550 Hz, using the explicit solution of the geometric attenuation law of Rayleigh-like waves in stratified media (solid black lines) and performing numerical time-domain simulations (grey dots). The solid red lines represent the geometric spreading functions $\Upsilon_{P-SV_1}$ calculated considering only the contribution of the fundamental P-SV$_1$ mode.}
\label{GeomSpread}
\end{figure*}
\noindent
The interference among the acoustic modes produces pronounced oscillations in the spreading functions that, as revealed in Ref. \cite{Foti}, multiply for increasing frequencies. These oscillations are not present in the simple classical attenuation law of the Rayleigh wave in homogeneous media proportional to $1/\sqrt{r}$ or in the case of single-mode wave propagation (see the solid red lines in Fig.~\ref{GeomSpread} depicting the spreading functions, labelled here $\Upsilon_{P-SV_1}$, calculated considering only the contribution of the fundamental P-SV$_1$ mode). Indeed, if we take into account only the contribution of the P-SV$_1$ mode, then $M=1$, $k_i=k_j=k_1$, $V_i=V_j=V_1$, $U_i=U_j=U_1$, $I_i=I_j=I_1$, and the geometric spreading function reduces to the relation $\Upsilon_{P-SV_1}= E_z / \sqrt{r} $, with $E_z= r_z(z,\omega) r_z(0,\omega) / 4 V_1 U_1 I_1 \sqrt{2\pi k}$.
\subsubsection{Geometric spreading functions computed by means of 3D numerical simulations}
Next, we compare the surface geometric spreading functions calculated via modal superposition with the wave displacement amplitude decay computed by means of three-dimensional (3D) time-transient numerical simulations performed using SPECFEM3D \cite{Komatitsch}, a spectral element-based software able to solve large-scale time-dependent elastodynamic problems. In particular, we consider a 1000-mm-deep granular half-space, 2000 $\times$ 1500 mm wide, modeled as a linear elastic continuum, defined by the previously assumed power-law rigidity profile, according to the GSAM theory described in Ref. \cite{Aleshin} (see Fig.~\ref{model}).
\begin{figure*}
\includegraphics[width=1\textwidth]{Fig4new}
\caption{Numerical time domain simulation. The computational domain characterized by the granular substrate excited by a vertical, time-harmonic point load applied at the free surface.}
\label{model}
\end{figure*}
\noindent
A mesh of hexahedral elements is generated using the software TRELIS 16.5 through PYTHON scripting. Since the velocity profiles are discretized accordingly to the discretization of the computational domain, we choose the element size equal to 5 mm to accurately model the exponential increase of $v_s$ and $v_s$, especially in the near-surface layers. The inset on the right depicts the adopted discretization of the compressional and shear velocity profiles. Furthermore, the GSAM theory establishes that, in absence of adhesion, the compressional and shear velocities vanish at the free surface. However, the employed computational software (SPECFEM3D) requires non-vanishing velocity values in each of the mesh points. Therefore, we impose at the free surface velocity values calculated for a depth $z$ equal to 2 mm.\\
A vertical, time-harmonic point load applied at the free surface and located at a distance of 300 mm from the left edge, generates vertically polarized modes. To avoid reflections from the boundaries, we apply absorbing conditions to the bottom and side edges.\\
The spatial attenuation laws of the P-SV modes obtained by means of 3D numerical simulations are depicted as dotted grey lines in Fig.~\ref{GeomSpread}. These are derived by computing the vertical component of the wave displacement at the surface, along the 1-m long line illustrated in blue in Fig.~\ref{model}. The wave displacement amplitude decays computed by means of 3D numerical simulations (dotted grey lines) well match the geometric spreading functions calculated via modal superposition (solid black lines). This confirms that the analytical technique is able to correctly describe the P-SV wave displacement amplitude as well as capture its oscillations at different frequencies.
\subsection{Material damping-induced dissipation}
To determine the frequency-dependent attenuation coefficients, we consider the data collected in the laboratory experiment. In particular, we compute the vertical displacement time histories from the velocity data recorded at the surface along the green line illustrated in Fig.~\ref{ExpSetup}. The numerical integration is performed using a simple trapezoidal rule, after employing a linear high-pass filter in order to avoid signal drifts. For each acquisition point along the surface line, we calculate the surface spectral displacement amplitude $\left|U_z(r,{\omega})\right|$ associated with each of the six frequency values previously considered\footnote{In the experiment, the vertical motion is realized by means of a rod exciting the granular medium with an angle of 20$^\circ$ from the normal to the free surface. As a result, the excitation features a small ($F_x=F_z \cdot \tan (20^\circ)$) horizontal component along the $x$-axis. Therefore, we verify that the presence of $F_x$ does not significantly affect our analysis. In particular, we perform 3D numerical simulations with and without the presence of $F_x$ using SPECFEM3D and we numerically compute the spectral coherence between the time waveforms obtained from the two simulations.}. The results are depicted as grey dots in Fig.~\ref{expdata}.
\begin{figure*}
\includegraphics[width=1\textwidth]{Fig6_updated}
\caption{Spectral P-SV surface wave displacement. Experimental P-SV wave spectral displacement (grey dots) and theoretical displacement functions (solid black lines) resulting from the non-linear regression analysis at 300, 350, 400, 450, 500, and 550 Hz. With yellow dots we depict the displacement data discarded in the regression analysis because located in the near field. The upper and lower bounds of the confidence intervals define the boundaries of 95$\%$ confidence region, which is marked as a red shaded area.}
\label{expdata}
\end{figure*}
\noindent
Then, according to Eq. \ref{eq:fitting}, we perform a non-linear regression analysis on the experimental data $\left|U_z(r,{\omega})\right|$ to derive the material attenuation coefficients $\alpha_{mat}(\omega)$. We discard from the analysis the acquisition points lying at a distance less than $\lambda_{P-SV}/2$ from the source, with $\lambda_{P-SV}$ representing the wavelength of the P-SV waves (see the yellow points in Fig.~\ref{expdata}). Indeed, numerical studies of surface wave propagation in vertically inhomogeneous soil profiles \cite{Holzlohner,Vrettos,Tokimatsu} have shown that, in normally dispersive media, i.e., media with a mechanical impedance $\rho v_s$ that is constant or smoothly increases with depth, the near-field (body wave) effects become negligible and, consequently, the wavefield is dominated by surface waves when the distance from the source exceeds about $\lambda/2$.
Aside from the material attenuation constant $\alpha_{mat}(\omega)$, we also consider the excitation magnitude $F_z$ as an unknown parameter, in order to alleviate uncertainties due to coupling between the source and the medium \cite{Foti}. The regression analysis yields a series of material attenuation coefficients, shown in Fig.~\ref{attcurve} and collected in Table~\ref{symbols}.
\begin{table}
\caption{Frequency-dependent material attenuation coefficients obtained from the regression analysis that was carried out on the experimental displacement data. The latter are computed from the velocity data that has been recorded along the surface green line, illustrated in Fig.~\ref{ExpSetup}.}
\label{symbols}
\begin{tabular}{@{}lllllll}
$f$ (Hz) & 300 & 350 & 400 & 450 & 500 & 550\\
\hline
$\alpha_{mat}$ (1/m) & 0.370 & 0.413 & 0.510 & 0.669 & 0.746 & 0.887\\
\hline
\end{tabular}
\end{table}
\begin{figure}
\includegraphics[width=0.5\textwidth]{Fig7_updated}
\caption{P-SV wave attenuation curve. Attenuation curve of P-SV surface acoustic waves propagating at the surface of a granular medium composed of 150 ${\mu} $m-diameter silica microbeads. The obtained attenuation values are depicted as black dots together with their confidence intervals, while the blue line represents the trend of shear attenuation data found in literature for 105-210 ${\mu} $m-diameter glass beads.}
\label{attcurve}
\end{figure}
\noindent
As expected, the material attenuation coefficient $\alpha_{mat}(\omega)$ increases with frequency. The theoretical fitted functions, represented as solid black lines in Fig.~\ref{expdata}, well match the experimental displacement data, thoroughly capturing the oscillations arising as a result of the mode superposition. However, there exist additional oscillations that are not reproduced by the theoretical model, but are visible in the experimental data, in particular at 300 and 400 Hz. These oscillations may be attributed to the influence of the pure compressional wave. Indeed, by observing the experimental dispersion curve (Fig.~\ref{DispRel}(a)), we note that part of the energy of the P wave is localized at low frequencies, particularly for the values of 300 and 400 Hz, thus justifying this mismatch.\\
The 95$\%$ confidence intervals for the coefficients $\alpha_{mat}$ of the nonlinear regression model, computed using the bootstrap method, are depicted in black in Fig.~\ref{attcurve}. The upper and lower bounds of the confidence intervals define the boundaries of 95$\%$ confidence region, which is marked as a red shaded area in Fig.~\ref{expdata}.\\
To validate the findings, our results are compared to attenuation values found in literature for unconsolidated granular materials. Experimental studies \cite{Bell} report shear wave attenuation measurements in 105-210 ${\mu} $m-diameter dry glass beads for frequencies ranging from 500 Hz to 20 kHz. Although the P-SV modes do not propagate within the material bulk as shear waves, but in the near-surface layers, similar attenuation coefficients may be expected owing to their predominant shear component. The reported data, which was subsequently validated via the grain-shearing theory, described in Ref. \cite{Buckingham}, indicates a linear frequency dependence (represented by the blue dashed line in Fig.~\ref{attcurve}), which proves consistent with the attenuation coefficients $\alpha_{mat}$ obtained from the regression analysis. In addition, the linear frequency dependence of the attenuation coefficients, predicted by the grain-shearing theory, indicates that the results of this work can be extended to lower frequencies.\\
Finally, note that laboratory and field tests on cohesionless sandy soils \cite{Seed} suggest values of damping ratios consistent with those obtained for our granular medium.
\section{Conclusions}
In conclusion, we experimentally and numerically analyse the dynamics and the attenuation of vertically polarized acoustic waves propagating at the surface of an unconsolidated granular medium, which features a power-law stiffness profile. Dispersion analysis reveals that the surface wavefield is dominated by the two lowest-order P-SV modes, whose frequency content mainly extends from 300 to 550 Hz. The existence, for a given frequency, of multiple vertically polarized acoustic modes, each characterized by its own vertical and horizontal displacement profile along the depth, is due to constructive interference phenomena occurring in the inhomogeneous granular material.\\
Subsequently, we analyse the particle displacement amplitude decay, investigating the two main attenuation mechanisms that contribute to wave energy dissipation during dynamic excitation, namely the geometric spreading and the material damping. First, we calculate the GSAM geometric spreading function by adopting an approach originally developed to quantify the spatial attenuation law of seismic Rayleigh-like waves in inhomogeneous stratified media. In virtue of the linearity of these surface acoustic modes, the medium is modeled as perfectly continuous and elastic, despite its granular nature. As an effect of the mode superposition, the obtained geometric spreading functions exhibit pronounced oscillations that multiply for increasing frequencies. Then, frequency-dependent attenuation coefficients are determined by means of non-linear regression analysis, which was carried out on experimental particle displacement data, to assess the intrinsic energy dissipation induced by material damping. The outcomes of the nonlinear regression reveal attenuation coefficient values increasing with the frequency. The latter proves consistent with attenuation values found in literature for dry glass beads with similar grain size and damping ratios of cohesionless sandy soils.\\
In addition, the pronounced amplitude oscillations of the experimental particle displacement are well captured by the theoretical spatial attenuation law of seismic Rayleigh-like waves in inhomogeneous stratified media. Firstly, this demonstrates that the continuous medium approximation can be applied not only to estimate the elastic properties of the material or investigate the GSAM dispersive properties (as the previous studies show), but further to determine the wave geometric spreading caused by the medium inhomogeneity and the attenuation coefficients accounting for the energy dissipated between the grains. Furthermore, it confirms that techniques that are commonly used at the geophysics scale prove successful also when applied to laboratory-scaled physical models.\\
Future studies should investigate how the attenuation coefficient value is affected upon increase of the adhesion between grains. This can be achieved by gradually injecting various types of pore fluid into the granular layer in a controlled manner \cite{Bodet}. In the presence of strong adhesion, the medium is expected to become almost homogeneous near the surface. The lowest-order mode should transform into the Rayleigh mode, while the further surface modes delocalize. As a result, the GSAM geometric spreading should gradually tend to the classical attenuation law of the Rayleigh wave (the amplitude oscillations should disappear) and the attenuation coefficient value increase.
\begin{acknowledgments}
This work was partially supported by the ETH Research Grant (49 17-1) to E.C.
\end{acknowledgments}
\begin{dataavailability}
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
\end{dataavailability}
|
train/arxiv
|
BkiUa2PxK6-gDz87NwgJ
| 5 | 1 |
\section{Do we expect IGM to be turbulent?}
A fluid of viscosity $\nu$ becomes turbulent when the rate of viscous dissipation, which is $\sim \nu/L^2$ at the energy injection scale $L$, is much smaller than the energy transfer rate $\sim V_L/L$, where $V_L$ is the velocity dispersion at the scale $L$. The ratio of the two rates is the Reynolds number $Re=V_LL/\nu$. In general, when $Re$ is larger than $Re_{crit}\sim
10-100$ the system becomes turbulent. Chaotic structures develop gradually as $Re$ increases, and those with $Re\sim10^3$ are appreciably less chaotic than those with $Re\sim10^8$.
It is widely accepted that medium in clusters should be turbulent.
Mergers of galactic subclusters may be one of the major energy injection
mechanism (see Sarazin 2002, Brunetti 2003 and references therein). The details
of the injection and the energy transfer are poorly understood.
A crude picture includes the energy injection scale of 100-500 kpc and
the injection velocity of the order of $10^3$ km/s.
A difficulty that one faces trying to understand turbulence is clusters
is that the diffusivity is very different along and perpendicular to
magnetic field. This is related to the fact that the mean free path of
ions is substantial. For {\it non-magnetized} intracluster gas the
$Re$ number is only marginally larger than $Re_{crit}$ and
therefore formally is only just sufficient for initiating the turbulence.
At the same time, even small magnetization makes the Reynolds number for
the motions perpendicular to magnetic field very large (e.g. more than
$10^{10}$). This poses the question of what a reasonable choice of
the Reynolds number is for intracluster medium.
Strangely enough we do not have a good answer to this basic question. There
are several processes that definitely influence the effective diffusivity
of intracluster plasma. First of all, compressions of collisionless
plasma should result
in various instabilities that would decrease the mean free paths of protons.
In addition, bending of magnetic field lines does not allow
mean free path of protons to be larger than the scale at which the turbulent
velocity is equal to the Alfven velocity (henceforth, the $l_A$ scale).
Therefore the effective Reynolds number at the injection scale will be
$\sim 10^3$ for the field of 1 $\mu$G field, which is the field
that fills according to Enslin et al. (2005) 90\% of the intracluster volume.
Initially the turbulence is superAlfvenic and hydrodynamic motions easily bend
magnetic field lines. Such turbulence is analogous to hydrodynamic one until
the scales of the order of $l_A\equiv L (V_A/V_L)^3$ are reached.
Assuming $V_L=10^3$ km/s and $L=100$ kpc we get 30~pc, if we adopt, following Ensslin et al (2005)
that 1 $\mu$G field fills 90\% of the ingracluster volume. The
turbulence gets mangetohydrodynamic (MHD) for scales $l<l_A$ as magnetic fields control fluid
motions. However, in both ``hydrodynamic'' and MHD regimes
plasma effects continue to be important as they drain and redistribute energy
of compressible motions (see Schekochihin et al. 2005).
\section{What do we know about MHD turbulence?}
There have long been understanding that the MHD turbulence
is anisotropic (e.g. Shebalin et al.~1983). Substantial progress has been achieved
by Goldreich \& Sridhar (1995; hereafter GS95), who made an
ingenious prediction regarding relative motions parallel and
perpendicular to magnetic field {\bf B} for incompressible
MHD turbulence.
An important observation that leads to understanding of the GS95
scaling is that magnetic field
cannot prevent mixing motions
of magnetic field lines if the motions
are perpendicular to the magnetic field. Those motions will cause, however,
waves that will propagate along magnetic field lines.
If that is the case,
the time scale of the wave-like motions along the field,
i.e. $\sim l_{\|}/V_A$,
($l_{\|}$ is the characteristic size of the perturbation along
the magnetic field and
$V_A=B/\sqrt{4 \pi \rho}$ is
the local Alfven speed) will be equal to the hydrodynamic time-scale,
$l_{\perp}/v_l$,
where $l_{\perp}$ is the characteristic size of the perturbation
perpendicular to the magnetic field.
The mixing motions are
hydrodynamic-like.
They obey Kolmogorov scaling,
$v_l\propto l_{\perp}^{1/3}$, because incompressible turbulence is assumed.
Combining the two relations above
we can get the GS95 anisotropy, $l_{\|}\propto l_{\perp}^{2/3}$
(or $k_{\|}\propto k_{\perp}^{2/3}$ in terms of wave-numbers).
If we interpret $l_{\|}$ as the eddy size in the direction of the
local
magnetic field.
and $l_{\perp}$ as that in the perpendicular directions,
the relation implies that smaller eddies are more elongated.
The latter is natural as it the energy in hydrodynamic motions
decreases with the decrease of the scale. As the result it gets more
and more difficult for feeble hydrodynamic motions to bend magnetic
field lines.
GS95 predictions have been confirmed
numerically (Cho \& Vishniac 2000; Maron \& Goldreich 2001;
Cho, Lazarian \& Vishniac 2002, 2003);
they are in good agreement with observed and inferred astrophysical spectra
(see Cho \& Lazarian 2005). What happens in a compressible MHD? Does any part
of GS95 model survives? The answer depends on the mode coupling.
According to closure calculations (Bertoglio,
Bataille, \& Marion 2001; see also Zank \& Matthaeus 1993),
the energy in compressible modes in {\it hydrodynamic} turbulence scales
as $\sim M_s^2$ if $M_s<1$.
Cho \& Lazarian (henceforth CL03) conjectured that this relation can be extended to MHD turbulence
if, instead of $M_s^2$, we use
$\sim (\delta V)_{A}^2/(a^2+V_A^2)$.
(Hereinafter, we define $V_A\equiv B_0/\sqrt{4\pi\rho}$, where
$B_0$ is the mean magnetic field strength.)
However, since the Alfven modes
are anisotropic,
this formula may require an additional factor.
The compressible modes are generated inside the so-called
Goldreich-Sridhar cone, which takes up $\sim (\delta V)_A/ V_A$ of
the wave vector space. The ratio of compressible to Alfvenic energy
inside this cone is the ratio given above.
If the generated fast modes become
isotropic (see below), the diffusion or, ``isotropization'' of the
fast wave energy in the wave vector space increase their energy by
a factor of $\sim V_A/(\delta V)_A$. This results in
\begin{equation}
\frac{\delta E_{comp}}{\delta E_{Alf}}\approx \frac{\delta V_A V_A}{V_A^2+a^2},
\label{eq_high2}
\end{equation}
which suggests that the drain of energy from
Alfvenic modes is marginal along the cascade.
Our considerations above about the mode coupling can guide us
in the discussion below. Indeed,
if Alfven cascade evolves on its own, it is natural to assume that
slow modes exhibit the GS95 scaling.
Indeed, slow modes in gas
pressure dominated environment (high $\beta$ plasmas) are
similar to the pseudo-Alfven modes in incompressible regime
(see GS95; Lithwick \& Goldreich 2001). The latter modes do follow
the GS95 scaling.
In magnetic pressure dominated environments or low $\beta$ plasmas,
slow modes are density perturbations propagating with the
sound speed $a$ parallel to the mean magnetic field.
Those perturbations are essentially
static for $a\ll V_A$.
Therefore Alfvenic turbulence is expected to mix density
perturbations as if they were passive scalar. This also induces the
GS95 spectrum.
The fast waves in low $\beta$ regime propagate at $V_A$ irrespectively
of the magnetic field direction.
In high $\beta$ regime, the properties of fast modes are similar,
but the propagation speed is the sound speed $a$.
Thus the mixing motions induced by Alfven waves should marginally
affect the fast wave
cascade. It is expected to
be analogous to the acoustic wave cascade and hence be isotropic.
\section{How does turbulence scatter and accelerate cosmic rays?}
The propagation of cosmic rays (CRs) is affected by their interaction
with magnetic field. This field is turbulent and therefore, the resonant
interaction of cosmic rays with MHD turbulence has been discussed
by many authors as the principal mechanism to scatter and isotropize
cosmic rays. Although cosmic ray diffusion can
happen while cosmic rays follow wandering magnetic fields (Jokipii 1966), the acceleration of
cosmic rays requires efficient scattering.
For instance, scattering of cosmic rays back into the shock is a
vital component of the first order Fermi acceleration.
While most investigations are restricted to Alfv\'{e}n modes propagating
along an external magnetic field (the so-called slab model of Alfv\'{e}nic
turbulence) (see Schlickeiser 2002), obliquely propagating MHD modes have been included in
Fisk et al. (1974) and later studies (see Pryadko \& Petrosian 1999). A more complex models
were obtained by combining the
results of the Reduced MHD with parallel slab-like
modes . Models that better correspond to the current understanding of MHD turbulence
(see above) have been considered lately. Chandran (2000, henceforth C00) considered resonant
scattering and acceleration by incompressible MHD turbulence. Resonant scattering and
acceleration by the compressible MHD turbulence was considered in Yan \& Lazarian (2002, 2004,
henceforth YL02, YL04).
There the following result was obtained for the diffusion coefficients that governed by
Alfvenic scattering
\begin{eqnarray}
\left[\begin{array}{c}
D_{\mu\mu}\\
D_{pp}\end{array}\right]&=&\frac{v^{2.5}\mu^{5.5}}{\Omega^{1.5}L^{2.5}(1-\mu^2)^{0.5}}\Gamma[6.5,k_{max}^{-\frac{2}{3}}k_{\parallel,res}L^{\frac{1}{3}}]\nonumber\\
& &\left[\begin{array}{c}
1\\
m^{2}V_{A}^{2}\end{array}\right],\label{ana}
\end{eqnarray}
where $\Gamma[a,z]$ is the incomplete gamma function. This result was obtained using
a tensor of magnetic fluctuations that was obtained in the CLV02 study. It provides
scattering and acceleration rates orders of
magnitude larger than those
in C00 for the most of energies considered. However, if energy is injected through random driving
at large
scale $L\gg k_{\parallel,res}^{-1}$
the scattering frequency,
\begin{equation}
\nu=2D_{\mu\mu}/(1-\mu^{2}),\label{nu}
\end{equation}
are still much smaller
than the estimates for isotropic and slab model.
What does
scatter cosmic rays? Our work in YL02 identified fast modes as the principal agent responsible
for CR scattering and acceleration. A study in YL04 showed that this is true in spite of the fact
that fast modes are subjected to much more dissipation compared to the Alfven modes. In the
next section we shall discussed another possibility proposed in Lazarian \& Beresnyak (2006), namely, instabilities related to cosmic rays that inject Alfven modes at resonant scales (see \S 4). This
possibility may provide a physical justification for the earlier calculations in Brunetti et al. (2004).
The work above was done in relation to galactic cosmic rays. However, the problem of MHD
turbulence with CR is a general one. Therefore in a recent paper Cassano \&
Brunetti (2005) considered fast modes as the principal component of CR acceleration in galaxy
clusters (see also Brunetti 2004). A further work in this direction is in Brunetti \& Lazarian (2006).
Resonant acceleration (we include Transit-Time-Damping acceleration to this category) is not the
only process by which magnetized turbulence accelerates CR. For instance, the
acceleration of cosmic rays by the large scale compressible motions was described in the literature
rather long time ago (see Ptuskin 1988). In this regime slow and fast diffusion limit exist. The slow
limit corresponds to the rate of particle diffusion out of compressible eddies, which is slower than the
evolution rate of the eddies. On the contrary, in the fast diffusion limit particles leave the eddies before they turnover. Large scale compressions associated with
anisotropic slow modes were used to accelerate Solar Flare CR in Chandran (2003) who considered
slow diffusion regime.
Fast diffusion regime was used in Chandran \& Maron (2004ab) in application to CR acceleration in
galaxy clusters. A comprehensive study of these processes has been done in Cho \& Lazarian (2006).
We found, first of all, in slow diffusion limit the resonance scattering, that is a part and parcel of
the slow diffusion process, is the dominant acceleration process. Then, we identified fast modes as the principal cause of non-resonant acceleration. In addition, we showed that weak turbulence (Galtier
et al. 2005) may be
important for cosmic ray acceleration in the fast diffusion limit.
While the importance of fast modes for the scattering and the acceleration of CR is unquestionable,
plasma and CR instabilities may make the actual turbulence in galaxy clusters more involved compared to a simple picture presented above. For instance, below we consider a robust CR instability that should modify the properties of MHD turbulence.
\section{How can cosmic rays modify turbulence?}
Studies of the backreaction of the energetic particles on the
turbulence are usually limited by the damping of turbulence on
energetic particles (see Brunetti \& Blasi 2005,
Ptuskin et al. 2005, Petrosian, Yan \& Lazarian 2006). However,
this is not the only effect of CR. For instance, Lazarian \& Beresnyak
(2006) have considered a transfer of turbulent energy to small (but
not dissipation!) scales
that is mediated by CR. As CR present a collisionless fluid,
compressions CR through the compression of magnetic field
preserve the adiabatic invariant $p^2_\perp/B$, where $p_\perp$ is the CR momentum
perpendicular to magnetic field. CR with the anisotropic
distribution of momenta, i.e. with nonzero
$A=(p_\perp-p_\|)/p_\|$ are subjected to an instability, which growth rate can be estimated as
\begin{equation}
\Gamma_i(k_{\|}) = \omega_{pi} \frac{n_{CR}(p>m\Omega/k_{\|})}{n}AQ,
\label{stream}
\end{equation}
where $\Omega=eB/mc$ is a cyclotron frequency, $m$ -- proton mass, $n$
is the density of plasma, $\omega_{pi}$ is the ion plasma frequency,
refering to $n$ and $n_{CR}(p>m\Omega/k)$ is the number density of
cosmic rays with momentum larger than minimal resonant momentum for a
wavevector value of $k$. $Q$ is a dimensionless numerical factor, depending
only on cosmic ray power-law index $\alpha$.
\begin{figure}
\includegraphics[width=.5\textwidth]{f1.eps}
\caption{Energy density of compressive modes and Alfv\'enic slab-type
waves, induced by CRs, in galaxy clusters. The energy is transferred from
the mean free path scale to the CR Larmor radius scale. If mean free path
falls below compressive motions cutoff, spectrum of slab waves becomes
steeper (from Lazarian \& Beresnyak, 2006).}
\end{figure}
The outcome of this instability is the direct transfer of the energy from mean free path of
a CR, which is determined by magnetic scattering of the CR, to the CR gyroradius. The magnetic
perturbations at the gyroradius scale are the most efficient in CR scattering and this decreases
the mean free path $\lambda$ and therefore $A\sim v_{\lambda}/V_A$.
At large scales the instability is limited
by damping arising from the ambient Alfvenic turbulence ($r_{p, crit}$ in Fig.~1), while at small
scales steepening limits the amplitude the perturbations. All in all, the interaction of CR with MHD
turbulence results in an additional slab-like small-scale component of Alfvenic perturbations that
interacts with CR much more efficiently than the Alfvenic mode being excited by the energy injection
scale.
\section{Can magnetic reconnection accelerate CR?}
Magnetic reconnection is a process that we expect to happen routinely
in magnetized
turbulent fluid. This even more true for superAlfvenic turbulence
when fluid motions bend magnetic fields easily. Indeed, magnetic
reconnection should happen whenever magnetic field with non-parallel
direction interact.
Several schemes of magnetic reconnection are known (see Fig.~2).
The most famous one is
the Sweet-Parker (Parker 1957, Sweet 1958)
reconnection, which naturally occurs when magnetic fields
get into contact over long current sheets. This is the most natural arangement
of magnetic field to get into within the turbulent fluid as random moving
flux tubes collide and push their way through one another. The scheme provide
ridiculously low rates of reconnection to be of astrophysical importance,
however. Indeed, the reconnection rate scales as $V_A Rm^{-1/2}$, where
$Rm=V_A L/\eta_{mag}$ is the Lunquist number, which for realistic values
of magnetic diffusivity $\eta_{mag}$ is so humangous that any processing
of magnetic energy via reconnection gets absolutely negligible for any
astrophysical system (e.g. stars, interstellar medium) not to speak about
intracluster medium with its much larger scales.
The Petcheck (1964) scheme emerged as an answer to the evidence
of fast reconnection that cannot be explained within the Sweet-Parker
model. Within this scheme the reconnection is concentrated along
thin filaments from which oppositely directed magnetic field lines diverge.
It is clear
at this moment that it cannot be sustained at large $R_L$
for smooth resistivities.
Whether or not anomalous effects, e.g. those related to Hall term,
can provide reconnection at rates comparable with the Alfv\'en speed
is hotly debated\footnote{ For instance, the necessary condition
for the anomalous effects to be important, e.g. for
that the electron mean free path is less than the current sheet thickness (see
Trintchouk et al. 2003) is difficult to satisfy for the ISM, where
the Sweet-Parker current sheet thickness is typically {\it much} larger
than the ion Larmor radius.} (see Biskamp, Schwarz \& Drake 1997,
Shay et al. 1998, Shay \& Drake 1998,
Bhattacharjee, Ma \& Wang 2001). We feel, however, that
the issue of
satisfying boundary outflow conditions is the most controversial element
in applying Petscheck scheme to astrophysical conditions. If very special
global geometry or magnetic fluxes, e.g. convex magnetized regions,
is required to enable reconnection, then
for a generic astrophysical case the reconnection is slow (see Fig. 3).
\begin{figure}
\includegraphics[width=.45\textwidth]{ethan.eps}
\caption{ {\it Upper plot}:
Sweet-Parker model of reconnection. The outflow
is limited by a thin slot $\Delta$, which is determined by Ohmic
diffusivity. The other scale is an astrophysical scale $L\gg \Delta$.
{\it Middle plot}: Turbulent reconnection model that accounts for the
stochasticity
of magnetic field lines. The outflow is limited by the diffusion of
magnetic field lines, which depends on field line stochasticity.
{\it Low plot}: An individual small scale reconnection region. The
reconnection over small patches of magnetic field determines the local
reconnection rate. The global reconnection rate is substantially larger
as many independent patches come together.}
\end{figure}
The turbulent reconnection model
proposed in Lazarian \& Vishniac (1999, henceforth LV99) deals with
magnetic field configurations with flat long current sheets.
However, these sheets consist of small sheets related to the interaction
of individual turbulent elements of magnetic flux. The outflow within
the model is limited not by the thickness of the current sheet, but by
magnetic field wondering. As the result the rate of reconnection is
proportional to the turbulence intensity. The model can explain flaring
associated with reconnection as well as high rates of reconnection that are
required by both observations and theory (e.g. dynamo theory). Calculations
in Lazarian, Vishniac \& Cho (2004) confirm main points of the LV99 model
(e.g. the rate of field line wondering)
and extend it to reconnection in partially ionized gas.
\begin{figure}
\includegraphics[width=0.45\textwidth]{petscheck.eps}
\caption{ {\it Upper plot}:
Petscheck reconnection scheme has a magnetic diffusion region (rectangular
area at the tip of the magnetic field line bending) for which both
the longitudinal and transversal dimensions are determined
by the Ohmic diffusivity.
To enable $V_R\sim V_A$ the model requires field line opening over the whole astrophysical scale
involved, which is difficult to satisfy in practice. {\it Lower plot}:
External forcing, e.g. the forcing present in the ISM, is likely
to close the opening required by the Petscheck model. In this case
the global outflow constraint is not satisfied and the resulting
reconnection speed is $V_R'\ll V_A$. As the result the Petscheck reconnection
cannot operate steadily and therefore cannot deal with the amount of flux
that, for instance, an astrophysical dynamo would require to reconnect.}
\end{figure}
There are several ways how magnetic
reconnection can accelerate cosmic rays. It is well known
that electric fields in the current sheet can do the job. For Sweet-Parker
reconnection this may be an important process in those exceptional
instances when Sweet-Parker reconnection is fast in
astrophysical settings. For Petscheck reconnection only an insubstantial
part of magnetic energy is being released within the reconnection
zone, while bulk of the energy is being released in shocks that
support X-point. Therefore one would expect the shock acceleration
of cosmic rays to accompany Petscheck reconnection.
\begin{figure}
\includegraphics[width=.35\textwidth]{acceler.eps}
\caption{Cosmic rays spiral about a reconnected magnetic
field line and bounce back at points A and B. The reconnected
regions move towards each other with the reconnection velocity
$V_R$. The advection of cosmic rays entrained on magnetic field
lines happens at the outflow velocity, which is in most cases
of the order of $V_A$. Bouncing at points A and B happens
because either of streaming instability or turbulence in the
reconnection region.}
\end{figure}
Similarly to the Petscheck scheme
the turbulent reconnection process assumes that only
small segments of magnetic field lines enter the reconnection zone
and are subjected to Ohmic annihilation. Thus only small fraction of
magnetic energy, proportional to ${R}_L^{-2/5}$ (LV99), is
released in the current sheets. The rest of the energy is
released in the form of non-linear Alfv\'en
waves that are generated as reconnected magnetic field lines straighten
up. Such waves are likely to cause second order Fermi acceleration.
This idea was briefly discussed in Lazarian et al. (2001) in relation
to particle acceleration during the gamma-ray burst events. In addition,
large amplitude Alfv\'enic motions in low $\beta$, i.e. magnetically
dominated, plasmas are likely to induce shocks (see Beresnyak, Lazarian
\& Cho 2005), which can also cause particle acceleration.
However, the most interesting process is the first-order Fermi acceleration
that is intrinsic to the turbulent reconnection. To understand it
consider a particle entrained on a reconnected
magnetic field line (see Fig.~4). This particle
may bounce back and force between magnetic mirrors formed by oppositely
directed magnetic fluxes moving towards each other with the velocity
$V_R$. Each of such bouncing will increase the energy of a particle
in a way consistent with the requirements of the first-order Fermi
process. The interesting property of this mechanism that potentially
can be used to test observationally the idea is that the
resulting spectrum is different from those arising from shocks.
Gouveia Dal Pino \& Lazarian (2003) used particle acceleration
within turbulent reconnection regions to explain
the synchrotron power-law spectrum arising from the flares of the
microquasar GRS 1915+105. Note, that the mechanism acts in the
Sweet-Parker scheme as well as in the scheme of turbulent reconnection.
However, in the former the rates of reconnection and therefore the
efficiency of acceleration are marginal in most cases.
\section{How can we test the turbulence model?}
There are several ways that information about turbulence in intracluster
medium can be obtained from observations. For instance, magnetic power
spectrum has been obtained using Faraday rotation (see Ensslin 2004, Ensslin, Vogt
\& Pfromer 2005). The issues of detectibility of velocity turbulence
have been discussed in Sunyaev, Norman \& Bryan (2003). Recent advances
in the techniques of studies of turbulence via velocity fluctuations (see review by
Lazarian 2004 and references therein) enable us to get spectra of turbulent fluctuations.
The techniques described there can potentially separate compressible and incompressible
motions.
For instance, a Velocity Coordinate Spectrum (VCS) technique can be used to
extract turbulence statistics from the Doppler-shifted spectra even when the
measurements are made over a limited number of directions (even one) or
the object is not resolved (Lazarian \& Pogosyan 2006). Such measurements
would already be possible, if not for the failure of the high velocity resolution
instrument on board of the ASTRO-E2. An example of such
a study that can be available with the forthcoming X-ray telescope
Constellation X is presented in Fig.~5.
\begin{figure}
\includegraphics[width=.45\textwidth]{spect2.eps}
\includegraphics[width=.45\textwidth]{spect.eps}
\caption{An example of the velocity line profile (upper plot) and velocity spectrum (lower plot) for underlying
Kolmogorov turbulence that can be produced with observations using Constellation X forthcoming
X-ray facilty with 1 hour exposure (Chepurnov \& Lazarian 2006).}
\end{figure}
\section{What are the prospects of the field?}
The detailed description of the CR acceleration and scattering is a challenging
problem that requires coordinated efforts of both theorists and observers.
It is clear that at the moment we do not have an adequate description of
MHD turbulence in the collisionless fluid and this is an impediment for the
progress. To get such a description one needs to account for different
instabilities that affect magnetized collisionless intercluster
plasmas. This, together with the change of effective $Re$ fluid number as
magnetic field and instabilities evolve, make the description of turbulence
quite challenging. Together with turbulence driving
the above processes determine the efficiency of the CR interaction with
magnetized turbulence. We know by now that if the energy is injected at
large scales, the interaction of CR of low energies with Alfvenic part of
MHD cascade is suppressed. This, however, may not be true if some part of
Alfven modes is generated by small scale plasma instabilities. In particular,
some of these instabilities may be due to CR themselves. In addition,
reconnection within chaotic fields may accelerate CR directly.
Therefore, with more observational data, clusters of galaxies
Nevertheless, with more observational data, clusters of galaxies may serve
as a good testing ground for studies of the fundamental processes involving CR.
\acknowledgements
AL acknowledges
the support from the NSF Center for Magnetic Self-Organization in
Laboratory and Astrophysical Plasmas.
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\section{Introduction}
Recurrence relations of the form
\[
P_{n+1}(z) = \sigma_{n+1} (z -v_{n+1}) P_{n}(z) - u_{n+1}(z-a_{n})(z-b_{n}) P_{n-1}(z), \quad n \geq 1,
\]
with initial conditions $P_0(z) = 1$ and $P_1(z) = \sigma_1(z - v_1)$, have been studied by Ismail and Masson \cite{Ismail-Masson-JAT1995}. In \cite{Ismail-Masson-JAT1995} these recurrence relations were referred to as those associated with continued fractions of $R_{II}$ type. One of the interesting results shown in \cite{Ismail-Masson-JAT1995} is that given such a recurrence relation, if $u_{n+1} \neq 0$, $P_{n}(a_{k}) \neq 0$ and $P_{n}(b_{k}) \neq 0$, for all $n,k$, then associated with this recurrence relation there exists a linear functional $\mathcal{L}$ such that the rational functions $\mathcal{S}_n(z) = P_n(z) \prod_{j =1}^{n}[(z-a_j)^{-1}(z-b_j)^{-1}]$, $n \geq 1$,
satisfy the orthogonality
\begin{equation} \label{Eq-IsmailMasson-Orthogonality}
\mathcal{L}[z^{j}\mathcal{S}_n(z)] = 0 \quad \mbox{for} \quad 0 \leq j < n.
\end{equation}
The importance of these recurrence relations were further highlighted in the work of Zhedanov \cite{Zhedanov-JAT1999}, where the author shows that they are connected to generalized eigenvalue problems involving two tri-diagonal matrices. Just as in \cite{Zhedanov-JAT1999} we will refer to these recurrence relations as $R_{II}$ type recurrence relations.
Our objective in the present manuscript is to consider the special $R_{II}$ type recurrence relation
\begin{equation} \label{Eq-Special-R2Type-RR}
P_{n+1}(x) = (x - c_{n+1}) P_{n}(x) - d_{n+1} (x^2 + 1) P_{n-1}(x), \quad n \geq 1,
\end{equation}
with $P_0(x) = 1$ and $P_{1}(x) = x - c_1$, where $\{c_n\}_{n \geq 1}$ is a real sequence and $\{d_{n+1}\}_{n \geq 1}$ is a positive chain sequence.
We denote by $\{\ell_{n+1}\}_{n \geq 0}$ the minimal parameter sequence of the positive chain sequence $\{d_{n+1}\}_{n \geq 1}$. Thus, $(1-\ell_n)\ell_{n+1} = d_{n+1}$, $n \geq 1$, with $\ell_1 = 0$ and $0 < \ell_n < 1$ for $n \geq 2$.
It is known that a positive chain sequence can have multiple (infinitely many) parameter sequences or just a single parameter sequence. By Wall's criteria the positive chain sequence $\{d_{n+1}\}_{n \geq 1}$ has only a single parameter sequence if and only if the series
\[
\sum_{n=1}^{\infty} \prod_{j=1}^{n} \frac{\ell_{j+1}}{(1-\ell_{j+1})}
\]
is divergent. When the positive chain sequence $\{d_{n+1}\}_{n \geq 1}$ has multiple parameter sequences one could talk about its maximal parameter sequence $\{M_{n+1}\}_{n \geq 0}$ such that
\[0 < g_{n} < M_{n} < 1 \quad \mbox{and} \quad (1-g_{n})g_{n+1} = (1-M_{n})M_{n+1} = d_{n+1}, \quad \mbox{for} \quad n \geq 1,
\]
where $\{g_{n+1}\}_{n \geq 0}$ is any other parameter sequence of $\{d_{n+1}\}_{n \geq 1}$ different from the minimal and maximal parameter sequences. For all the information concerning positive chain sequences that we use in this manuscript we refer to Chihara \cite{Chihara-book}.
Let $u_0(x) = P_0(x)$,
\begin{equation} \label{Eq-Rationals-1}
u_n(x) = \frac{(-1)^{n}}{(x-i)^{n} \prod_{j=1}^{n}\sqrt{d_{j+1}}} P_n(x) , \quad n \geq 1,
\end{equation}
and $\mathbf{u}_{n}(x) = [u_0(x), u_1(x), \ldots, u_{n-1}(x)]^{T}$, where $\{P_{n}\}_{n \geq 0}$ and $\{d_{n+1}\}_{n \geq 1}$ are as in \eqref{Eq-Special-R2Type-RR}. Then the following theorem can be stated.
\begin{theo} \label{Thm-GenEigenProblem} For any $n \geq 2$,
\begin{equation} \label{Eq-Matrix-equality}
\mathbf{A}_{n} \mathbf{u}_{n}(x) = x\, \mathbf{B}_{n} \mathbf{u}_{n}(x)+ \sqrt{d_{n+1}}(x-i) u_{n}(x)\,\mathbf{e}_{n},
\end{equation}
where the $n \times n$ tridiagonal matrices $\mathbf{A}_{n}$, $\mathbf{B}_{n}$ are respectively
\[
\left[
\begin{array}{cccccc}
c_1 & \! i\sqrt{d_2} & 0 & \!\!\! \cdots & 0 & 0 \\[1ex]
\!\!\! -i\sqrt{d_2} & c_2 & \!\! i\sqrt{d_3} & \!\!\! \cdots & 0 & 0 \\[1ex]
0 & \!\!\!\!\! -i\sqrt{d_3} & c_3 & \!\!\! \cdots & 0 & 0 \\[1ex]
\vdots & \vdots & \vdots & & \vdots & \vdots \\[1ex]
0 & 0 & 0 & \!\!\! \cdots & c_{n-1} & \!\! i\sqrt{d_{n}} \\[1ex]
0 & 0 & 0 & \!\!\! \cdots & \!\!\!\!\! -i\sqrt{d_{n}} & c_{n}
\end{array} \! \right],
\quad
\left[
\begin{array}{cccccc}
1 & \!\sqrt{d_2} & 0 & \cdots & 0 & 0 \\[1ex]
\!\sqrt{d_2} & 1 & \!\sqrt{d_3} & \cdots & 0 & 0 \\[1ex]
0 & \!\sqrt{d_3} & 1 & \cdots & 0 & 0 \\[1ex]
\vdots & \vdots & \vdots & & \vdots & \vdots \\[1ex]
0 & 0 & 0 & \cdots & 1 & \!\sqrt{d_{n}} \\[1ex]
0 & 0 & 0 & \cdots & \!\sqrt{d_{n}} & 1 \\[1ex]
\end{array} \right],
\] \\[0ex]
and $\mathbf{e}_n$ is the $n^{th}$ column of the $n \times n$ identity matrix. Moreover, the zeros of $P_{n}$ ($n \geq 2$) are the eigenvalues of the generalized eigenvalue problem
\begin{equation} \label{Eq-Gen-Eigenproblem}
\mathbf{A}_{n} \mathbf{u}_{n}(x)= x\, \mathbf{B}_{n} \mathbf{u}_{n}(x).
\end{equation}
\end{theo}
\vspace{1ex}
The generalized eigenvalue problem given by Theorem \ref{Thm-GenEigenProblem} forms the basis for our study in this manuscript.
Given a nontrivial positive measure $\mu$ on the unit circle $\mathbb{T}: = \{\zeta=e^{i\theta}\!\!: \, 0 \leq \theta \leq 2\pi \}$ let $\{\Phi_{n}\}_{n \geq 0}$ be the associated sequence of monic orthogonal polynomials. The Verblunsky coefficients $\alpha_{n} = -\overline{\Phi_{n+1}(0)}$, $n \geq 0$, which satisfy $|\alpha_{n}| < 1$, are known to play an important role in the theory of orthogonal polynomials on the unit circle. For basic information concerning the theory of orthogonal polynomials on the unit circle we refer to \cite{Ismail-Book, Simon-Book-p1, Szego-Book}.
A monic para orthogonal polynomial on the unit circle of degree $n$ is given by $z\Phi_{n-1}(z) \linebreak - \tau \Phi^{\ast}_{n-1}(z)$, where $|\tau| =1$ and $\Phi^{\ast}_{n}(z) = z^n \overline{\Phi_{n}(1/\overline{z})}$. The name para-orthogonal polynomials first appeared in Jones, Nj\aa stad and Thron \cite{JoNjTh-1989}, where it was also shown that the zeros of these polynomials are simple and are of modulus equal to one. For some of the contributions that bring interesting properties of the zeros of these polynomials we cite \cite{CMV-EJA2002}, \cite{Golinskk-AMH2002}, \cite{Simon-JMAA2007} and \cite{Wong-JAT2007}. One of the well known applications of para-orthogonal polynomials is with regards to quadrature formulas on the unit circle and, in \cite{BracLiRanga-2005} and \cite{DarNjaVAssh-RMJ2003}, such quadrature formulas have been used to study the frequency analysis problem.
The sequence of polynomials $\{P_{n}\}_{n \geq 0}$ given by the $R_{II}$ type recurrence \eqref{Eq-Special-R2Type-RR} will be shown to be related to a certain sequence $\{z\Phi_{n-1}(z) - \tau_{n} \Phi^{\ast}_{n-1}(z)\}_{n \geq 0}$ of para-orthogonal polynomials on the unit circle. Thus, the generalized eigenvalue problem given by Theorem \ref{Thm-GenEigenProblem} provide us with another approach to study the properties of measures and orthogonal polynomials on the unit circle.
With respect to the orthogonality \eqref{Eq-IsmailMasson-Orthogonality} associated with the $R_{II}$ type recurrence \eqref{Eq-Special-R2Type-RR} the following can be stated.
\begin{theo} \label{Thm-Special-R2Type-MF} Consider the $R_{II}$ type recurrence \eqref{Eq-Special-R2Type-RR}, where we assume $\{d_{n+1}\}_{n \geq 1}$ to be a positive chain sequence having multiple parameter sequences. Let $\{M_{n+1}\}_{n \geq 0}$ be the maximal parameter sequence of $\{d_{n+1}\}_{n \geq 1}$ and let $\nu$ be the probability measure on the unit circle such that its Verblunsky coefficients are
\[
\beta_{n-1} = \frac{1}{\tau_{n-1}}\,\frac{1-2M_{n} - i c_n}{1 - i c_n}, \quad n \geq 1,
\]
where $\tau_{0}= 1$ and $\tau_{n} = \tau_{n-1}(1-ic_n)/(1+ic_n)$, $n \geq 1$.
If $d\varphi(x) = - d\nu((x+i)/(x-i))$ then
\[
\int_{-\infty}^{\infty} x^{j} \, \frac{P_{n}(x)}{(x^2+1)^{n}} \, d \varphi(x) = \gamma_n \delta_{n,j}, \quad j=0,1, \ldots, n,
\]
where $\gamma_{0} = \int_{-\infty}^{\infty} d\varphi(x) = \int_{\mathbb{T}} d \nu(\zeta) = 1$ and $\gamma_{n} = (1- M_{n})\gamma_{n-1}$, $n \geq 1$.
\end{theo}
\vspace{1ex}
Apart from the results that we have presented in this manuscript, a further important advantage that we could point out is the availability of nice numerical techniques for the generation of eigenvalues of our generalized eigenvalue problem. Subsequently, from these eigenvalues one could easily determine the zeros of the respective para-orthogonal polynomials.
The manuscript is organized as follows. The proof of Theorem \ref{Thm-GenEigenProblem} and some properties of the zeros of the polynomials $P_n$ are given in Section \ref{Sec-InitialResults}. In Section \ref{Sec-EVP-to-Orthogonality}, using techniques from linear algebra, we obtain the relation between the polynomials $P_n$ and orthogonal polynomials on the unit circle. In Section \ref{Sec-OPUC-to-POPUC2} we show how the results obtained in Section \ref{Sec-EVP-to-Orthogonality} are related to those established in \cite{BracRangaSwami-2016} and use this to obtain information concerning the orthogonality satisfied by the polynomials $P_{n}$. In particular, in this section we also give the proof of Theorem \ref{Thm-Special-R2Type-MF}. Finally, in Section \ref{Sec-Examples} examples are given to justify the results.
\setcounter{equation}{0}
\section{Polynomials from the special $R_{II}$ recurrence } \label{Sec-InitialResults}
In this section we will consider some properties of the polynomials $P_{n}$, $n \geq 1$, given by the $R_{II}$ recurrence relation \eqref{Eq-Special-R2Type-RR}.
\begin{theo} \label{Thm-LeadingCoefficient}
The polynomial $P_n$ is of exact degree $n$ with positive leading coefficient. Precisely, if we denote by $\mathfrak{p}_n$ the leading coefficient of $P_n(z)$ then $\mathfrak{p}_0 =1$, $\mathfrak{p}_1 = 1$ and
\[
0 < (1-\ell_n) = \frac{\mathfrak{p}_{n}}{\mathfrak{p}_{n-1}} < 1, \quad n \geq 2.
\]
Here, $\{\ell_{n+1}\}_{n \geq 0}$ is the minimal parameter sequence of the positive chain sequence $\{ d_{n+1}\}_{n \geq 1}$.
\end{theo}
\begin{proof}[\bf Proof]
Clearly, $\mathfrak{p}_0 =1$ and $\mathfrak{p}_1 = 1$ and one can directly verify from \eqref{Eq-Special-R2Type-RR} that $\mathfrak{p}_2 = 1 - d_2 = 1 - \ell_2 > 0$. Hence, the theorem holds for $n =2$.
Assume that $\mathfrak{p}_{n-1} > 0$, $\mathfrak{p}_{n} > 0$ and $1 - \ell_{n} = \mathfrak{p}_{n}/\mathfrak{p}_{n-1}$. Observe that \eqref{Eq-Special-R2Type-RR} also leads to
%
\[
\frac{x^2+1}{(x-c_{n+1})(x-c_{n})} d_{n+1} = \frac{P_n(x)}{(x-c_n)P_{n-1}(x)}\Big(1 - \frac{P_{n+1}(x)}{(x-c_{n+1})P_{n}(x)}\Big), \quad n \geq 1.
\]
Hence, by letting $x \to + \infty$ in the above equation we have
\[
d_{n+1} = (1-\ell_{n})\, \lim_{x \to \infty} \Big(1 - \frac{P_{n+1}(x)}{(x-c_{n+1})P_{n}(x)}\Big).
\]
Since $\{\ell_{n+1}\}_{n \geq 0}$ is a parameter sequence of the positive chain sequence $\{ d_{n+1}\}_{n \geq 1}$, we thus have
\[
0\ < \ \lim_{x \to \infty} \Big(1 - \frac{P_{n+1}(x)}{(x-c_{n+1})P_{n}(x)}\Big) = \ell_{n+1} \ < \ 1.
\]
Hence, $\mathfrak{p}_{n+1} \neq 0$ and $\mathfrak{p}_{n+1}/\mathfrak{p}_{n} = 1 - \ell_{n+1} > 0$. The theorem follows by induction.
\end{proof}
Now we give the proof of Theorem \ref{Thm-GenEigenProblem}.
\begin{proof}[\bf Proof of Theorem \ref{Thm-GenEigenProblem}] From \eqref{Eq-Special-R2Type-RR} we have for any $n \geq 2$,
\[
\begin{array}l
(x-i)\sqrt{d_2}\, u_{1} + (x-c_{1})u_{0} = 0 , \\[1ex]
(x-i)\sqrt{d_{k+1}}\, u_{k} + (x-c_{k})u_{k-1} + (x+i)\sqrt{d_{k}}\, u_{k-2} = 0, \quad k=2,3, \ldots, n,
\end{array}
\]
where $u_{j}$ refers to $u_{j}(x)$. This we can write in the equivalent form
\[
\begin{array}{rl}
c_{1}\, u_{0} + i \sqrt{d_{2}}\, u_{1} &\!\! = x\, [u_{0} + \sqrt{d_{2}}\, u_{1}], \\[1ex]
-i \sqrt{d_{k}}\,u_{k-2} + c_{k}\, u_{k-1} + i \sqrt{d_{k+1}}\, u_{k} &\!\! = x\, [ \sqrt{d_{k}}\, u_{k-2} + u_{k-1} + \sqrt{d_{k+1}}\, u_{k} ],
\end{array}
\]
for $k = 2, 3, \ldots, n$, which is exactly the system of equations given by \eqref{Eq-Matrix-equality}.
Observe that since $P_{n}(i) = \prod_{j=1}^{n}(i-c_{j})$, the point $i$ is not a zero of $P_{n}(x)$. Moreover, the $n$ zeros of the polynomial $P_{n}(x)$ are the only zeros of $(x-i)u_{n}(x)$. Hence, we obtain from \eqref{Eq-Matrix-equality} that the zeros of $P_{n}$ are the eigenvalues of the generalized eigenvalue problem \eqref{Eq-Gen-Eigenproblem}.
\end{proof}
The $n \times n$ matrices $\mathbf{A}_{n}$ and $\mathbf{B}_{n}$ in Theorem \ref{Thm-GenEigenProblem} are Hermitian and, since $\{d_{n+1}\}_{n \geq 1}$ is a positive chain sequence, the matrix $\mathbf{B}_{n}$ is also positive definite. We recall from \cite[p.\,75]{Wall-Book} the positive definiteness of $\mathbf{B}_n$ can be easily verified from the identity
\[
\mathbf{v}_{n}^{H} \mathbf{B}_n \mathbf{v}_{n} = \sum_{j=1}^{n-1}\big|\sqrt{1-\ell_{j}}\, v_{j} - \sqrt{\ell_{j+1}} \,v_{j+1}\big|^2 \ + (1-\ell_{n})\, |v_{n}|^2,
\]
where $\mathbf{v}_{n} = [v_1, v_2, \ldots, v_{n}]^{T}$. Another way (see \cite{Wilkinson-book}) to confirm the positive definiteness of $\mathbf{B}_n$ is its Cholesky decomposition $\mathbf{L}_n\mathbf{L}_n^{T}$ given in Section \ref{Sec-EVP-to-Orthogonality}.
Since $\mathbf{B}_n$ is positive definite, from \eqref{Eq-Gen-Eigenproblem} we have
\[
x_j^{(n)} = \frac{\mathbf{u}_{n}(x_j^{(n)})^{H}\mathbf{A}_{n}\, \mathbf{u}_{n}(x_j^{(n)}) }{ \mathbf{u}_{n}(x_j^{(n)})^{H}\mathbf{B}_{n} \, \mathbf{u}_{n}(x_j^{(n)} )}, \quad j=1,2,\ldots,n,
\]
where $x_{j}^{(n)}$ are the eigenvalues of the generalized eigenvalue problem \eqref{Eq-Gen-Eigenproblem}. Thus, one can verify that the eigenvalues of the generalized eigen-system \eqref{Eq-Gen-Eigenproblem} (i.e., the zeros of $P_n$) are all real (see, for example, \cite[Chap.\,5]{BDDRV-2000}). The next theorem shows that we can say more about the zeros of the polynomials $P_n$.
\begin{theo} \label{Thm-InterlacingZeros}
The zeros $x_j^{(n)}$, $j =1,2, \ldots, n$ of $P_n$ are real and simple. Assuming the ordering $x_{j}^{(n)} < x_{j-1}^{(n)}$, $1 \leq j \leq n$, for the zeros, we also have the interlacing property
\[
x_{n+1}^{(n+1)} < x_{n}^{(n)} < x_{n}^{(n+1)} < \cdots < x_{2}^{(n+1)} < x_{1}^{(n)} < x_{1}^{(n+1)},
\]
for $n \geq 1$.
\end{theo}
\begin{proof}[\bf Proof]
Let us consider the Wronskians
\begin{equation} \label{Eq-Wronskians}
\mathcal{G}_n(x) = P_{n}^{\prime}(x) P_{n-1}(x) - P_{n-1}^{\prime}(x) P_{n}(x), \quad n \geq 1.
\end{equation}
From the recurrence \eqref{Eq-Special-R2Type-RR} we find
\begin{equation} \label{Eq-WronskianRelation}
\mathcal{G}_{n+1}(x) = P_n(x) [ P_n(x) - 2 d_{n+1} x P_{n-1}(x)] + d_{n+1} [x^2 + 1] \mathcal{G}_{n}(x), \quad n \geq 1.
\end{equation}
The only zero of $P_1$ is $x_1^{(1)} = c_1$, which is real. Since the leading coefficient of $P_1$ is positive, we then have from \eqref{Eq-Wronskians} that $\mathcal{G}_{1}(x_1^{(1)}) = P_{1}^{\prime}(x_{1}^{(1)})P_{0}(x_{1}^{(1)}) > 0$. Hence, from \eqref{Eq-WronskianRelation}
\[
-P_{1}^{\prime}(x_{1}^{(1)}) P_{2}(x_1^{(1)}) = \mathcal{G}_{2}(x_{1}^{(1)}) = d_{2} [ (x_1^{(1)})^2+ 1]\,\mathcal{G}_{1}(x_1^{(1)}) > 0.
\]
Thus, with the leading coefficient of $P_2$ positive we conclude that the zeros $x_{1}^{(2)}$ and $x_{2}^{(2)}$ of $P_2$ can be ordered such that $x_{2}^{(2)} < x_{1}^{(1)} < x_{1}^{(2)}$.
Now suppose that the zeros of $P_n$ and $P_{n-1}$ interlace as stated in the theorem. Then since the leading coefficients of $P_n$ and $P_{n-1}$ are positive we have from \eqref{Eq-Wronskians} that
\[
\mathcal{G}_{n}(x_j^{(n)}) = P_{n}^{\prime}(x_j^{(n)}) P_{n-1}(x_j^{(n)}) > 0 , \quad j = 1,2, \ldots n.
\]
Consequently, from \eqref{Eq-WronskianRelation},
\[
- P_{n}^{\prime}(x_j^{(n)}) P_{n+1}(x_j^{(n)}) = \mathcal{G}_{n+1}(x_j^{(n)}) = d_{n+1} [ (x_{j}^{(n)})^2 + 1]\, \mathcal{G}_{n}(x_j^{(n)}) > 0,
\]
for $ j = 1,2, \ldots n$. This leads to the required interlacing of the zeros of $P_n$ and $P_{n+1}$. Hence, we conclude the proof of the theorem by induction.
\end{proof}
Now if the matrix $\mathbf{A}_n$ is also positive definite (negative definite) then the zeros of $P_n$ are all positive (negative).
\begin{theo}
A sufficient condition for all the zeros of $P_n$ ($n \geq 2$) to be positive is that
\[
c_1 > \sqrt{d_2}, \quad c_{j} > \sqrt{d_j} + \sqrt{d_{j+1}}, \ j =2 , 3, \ldots, n-1 \quad \mbox{and} \quad c_{n} > \sqrt{d_{n}}.
\]
Likewise, a sufficient condition for all the zeros of $P_{n}$ to be negative is that
\[
c_1 < -\sqrt{d_2}, \quad c_{j} < -\sqrt{d_j} - \sqrt{d_{j+1}}, \ j =2 , 3, \ldots, n-1 \quad \mbox{and} \quad c_{n} < -\sqrt{d_{n}}.
\]
\end{theo}
\begin{proof}[\bf Proof]
We prove this by verifying that the Hermitian matrix $\mathbf{A}_n$ has all its eigenvalues positive (negative). So the proof is a simple application of the Gershgorin Theorem.
\end{proof}
\setcounter{equation}{0}
\section{Orthogonality from the generalized eigenvalue problem} \label{Sec-EVP-to-Orthogonality}
As part of the proof of Theorem \ref{Thm-InterlacingZeros} we also find that $\mathcal{G}_{n}(x_{j}^{(n)}) > 0$ and $\mathcal{G}_{n+1}(x_{j}^{(n)}) > 0$, for $j =1 ,2 , \ldots, n$ and $n \geq 1$. The following theorem shows that $\mathcal{G}_{n}(x) > 0$ for all real $x$.
\begin{theo} \label{Thm-Positive-Wronskian}
With $x$ and $y$ real, let
\[
G_{n}(x,y) = \frac{P_{n}(x) P_{n-1}(y) - P_{n-1}(x) P_{n}(y)}{x-y}, \quad n \geq 1.
\]
Then $G_{1}(x,y) = 1$ and
\[
\begin{array}l
\displaystyle \frac{G_{n}(x,y)}{(x-i)^{n-1}(y+i)^{n-1} d_2d_3 \cdots d_{n}} = \mathbf{u}_n(y)^{H} \mathbf{B}_{n} \mathbf{u}_{n}(x),
\end{array}
\]
for $n \geq 2$. Moreover, for the Wronskians $\mathcal{G}_{n}(x)$ given by \eqref{Eq-Wronskians} there hold
\[
\begin{array}{l}
\displaystyle \frac{\mathcal{G}_{n}(x)}{(x^2+1)^{n-1} d_2 d_3 \cdots d_{n}} = \mathbf{u}_n(x)^{H} \mathbf{B}_{n} \mathbf{u}_n(x) > 0,
\end{array}
\]
for $n \geq 2$.
\end{theo}
\begin{proof}[\bf Proof]
Post-multiplication of \eqref{Eq-Matrix-equality} by the conjugate transpose of $\mathbf{u}_{n}(y)$ gives
\begin{equation} \label{Eq-traceApp1}
\mathbf{A}_{n} \mathbf{u}_{n}(x)\, \mathbf{u}_{n}(y)^{H} = x\, \mathbf{B}_{n} \mathbf{u}_{n}(x)\, \mathbf{u}_{n}(y)^{H}+ \sqrt{d_{n+1}}(x-i) u_{n}(x)\,\mathbf{e}_{n}\, \mathbf{u}_{n}(y)^{H}.
\end{equation}
Likewise, pre-multiplication of the conjugate transpose of \eqref{Eq-Matrix-equality} given in terms of $y$ by $\mathbf{u}_{n}(x)$ gives
\begin{equation} \label{Eq-traceApp2}
\mathbf{u}_{n}(x)\,\mathbf{u}_{n}(y)^{H}\mathbf{A}_{n} = y\, \mathbf{u}_{n}(x)\,\mathbf{u}_{n}(y)^{H} \mathbf{B}_{n} + \sqrt{d_{n+1}}(y+i) \overline{u_{n}(y)}\, \mathbf{u}_{n}(x)\,\mathbf{e}_{n}^{T}.
\end{equation}
It is known that given any two matrices $\mathbf{M}$ and $\mathbf{N}$ if the products $\mathbf{M}\mathbf{N}$ and $\mathbf{N}\mathbf{M}$ exist then $Tr(\mathbf{M}\mathbf{N}) = Tr(\mathbf{N}\mathbf{M})$. Hence, by observing the equality in the traces of the matrices on the left hand sides of \eqref{Eq-traceApp1} and \eqref{Eq-traceApp2}, we have
\[
\begin{array}l
\displaystyle -\sqrt{d_{n+1}} \left[(x-i) u_{n}(x) \overline{u_{n-1}(y)}-(y+i)\overline{u_{n}(y)} u_{n-1}(x)\right]\\[2ex]
\qquad \qquad \displaystyle = (x-y) \Big[\sum_{j=0}^{n-1} \overline{u_{j}(y)} u_{j}(x) + \sum_{j=1}^{n-1} \sqrt{d_{j+1}}\big[\, \overline{u_{j-1}(y)} u_{j}(x)+ \overline{u_{j}(y)}u_{j-1}(x) \big]\Big].
\end{array}
\]
We can easily verify that the expression on the above right hand side is equal to \linebreak $(x-y)\mathbf{u}_n(y)^{H} \mathbf{B}_{n} \mathbf{u}_{n}(x)$ and the term on left hand side is equal to
\[ \begin{array}{l}
\displaystyle \frac{(x-y) G_{n}(x,y)}{(x-i)^{n-1}(y+i)^{n-1} d_2d_3 \cdots d_{n}}.
\end{array}
\]
This proves the first part of the theorem.
The latter part of the theorem follows from $\lim_{y \to x} G_{n}(x,y) = \mathcal{G}_{n}(x)$ and the positive definiteness of the matrix $\mathbf{B}_{n}$.
\end{proof}
It is not difficult to verify that the Cholesky decomposition $\mathbf{L}_{n} \mathbf{L}_{n}^{T}$ of the real positive definite matrix $\mathbf{B}_{n}$ is such that
\[
\mathbf{L}_{n} = \left[
\begin{array}{cccccc}
\sqrt{1-\ell_{1}} & 0 & \cdots & 0 & 0 \\[1ex]
\sqrt{\ell_2} & \!\!\!\sqrt{1-\ell_{2}} & \cdots & 0 & 0 \\[1ex]
\vdots & \vdots & & \vdots & \vdots \\[1ex]
0 & 0 & \cdots & \!\!\!\sqrt{1-\ell_{n-1}} & 0 \\[1ex]
0 & 0 & \cdots & \sqrt{\ell_{n}} & \!\!\!\sqrt{1-\ell_{n}} \\[1ex]
\end{array} \right],
\]
where $\{\ell_{n+1}\}_{n \geq 0}$ is the minimal parameter sequence of $\{ d_{n+1}\}_{n \geq 1}$.
Now consider the ordinary eigenvalue problem
\[
\mathbf{L}_{n}^{-1} \mathbf{A}_{n} (\mathbf{L}_{n}^{-1})^{T} \mathbf{L}_{n}^{T} \mathbf{u}_{n}(x) = x \mathbf{L}_{n}^{T} \mathbf{u}_{n}(x),
\]
which is equivalent to \eqref{Eq-Gen-Eigenproblem}. The eigenvectors are $\mathbf{L}_{n}^{T} \mathbf{u}_{n}(x_k^{(n)})$, $k=1,2, \ldots, n$ and the matrix involved $\mathbf{L}_{n}^{-1} \mathbf{A}_{n} (\mathbf{L}_{n}^{-1})^{T}$ is Hermitian. Hence, we must have that the eigenvectors $\mathbf{L}_{n}^{T} \mathbf{u}_{n}(x_j^{(n)})$ and $\mathbf{L}_{n}^{T} \mathbf{u}_{n}(x_k^{(n)})$ are orthogonal to each other when $j \neq k$. We can also verify this from Theorem \ref{Thm-Positive-Wronskian} by taking $x = x_j^{(n)}$ and $y = x_k^{(n)}$. That is, from Theorem \ref{Thm-Positive-Wronskian},
\[
[\mathbf{L}_{n}^{T}\mathbf{u}_{n}(x_{k}^{(n)})]^{H}\, [\mathbf{L}_{n}^{T}\mathbf{u}_{n}(x_{j}^{(n)})] = \mathbf{u}_n(x_{k}^{(n)})^{H} \mathbf{B}_{n} \mathbf{u}_{n}(x_{j}^{(n)}) = \frac{\mathcal{G}_{n}(x_{k}^{(n)})}{[(x_{k}^{(n)})^2+1]^{n-1} d_2 d_3 \cdots d_{n}} \, \delta_{j,k}.
\]
Thus, we consider the matrix
\[
\mathbf{Y}_{n} = \big[\mathbf{y}_{1}^{(n)}, \mathbf{y}_{2}^{(n)}, \ldots , \mathbf{y}_{n}^{(n)}\big] ,
\]
where
\(
\mathbf{y}_r^{(n)} = \sqrt{\lambda_{r}^{(n)}} \, \mathbf{L}_{n}^{T} \mathbf{u}_{n}(x_r^{(n)})
\)
and
\begin{equation} \label{Eq-Lambda-rn}
\lambda_{r}^{(n)} = \frac{[(x_r^{(n)})^2+1]^{n-1} d_2 d_3 \cdots d_{n}}{\mathcal{G}_{n}(x_r^{(n)})}.
\end{equation}
Clearly $\mathbf{Y}_{n}^{H} \mathbf{Y}_{n} = \mathbf{I}_{n}$ and thus, $\mathbf{Y}_{n}^{H} = \mathbf{Y}_{n}^{-1}$.
Now if we define the new rational functions $\hat{u}_{k}(x)$ by
\[
\begin{array}{ll}
\hat{u}_{k}(x) &= \sqrt{\ell_{k+1}}\,u_{k}(x) + \sqrt{1-\ell_{k}}\,u_{k-1}(x), \\[2ex]
& = \frac{(-1)^{k} \sqrt{\ell_{k+1}}}{(x-i)^k \prod_{j=1}^{k}\sqrt{d_{j+1}}}\big[P_{k}(x) - (1-\ell_{k})(x-i) P_{k-1}(x)\big],
\end{array} \quad k \geq 1,
\]
then from $ \mathbf{Y}_{n} \mathbf{Y}_{n}^{H} = \mathbf{I}_{n}$ we have
\begin{equation} \label{Eq-Discrete-Orthogonality-1}
\begin{array}l
\displaystyle \sum_{r=1}^{n} \lambda_{r}^{(n)}\, \overline{ \hat{u}_{m}(x_{r}^{(n)})}\, \hat{u}_{k}(x_{r}^{(n)}) = \delta_{m,k},
\end{array}
\end{equation}
for $m,k = 1,2, \ldots, n$.
The transformation
\begin{equation} \label{Eq-Transformation}
\zeta = \zeta(x) = \frac{x+i}{x-i},
\end{equation}
maps the real line $(-\infty, \infty)$ on to the (cut) unit circle $\mathbb{T} = \{ \zeta = e^{i\theta}, 0 < \theta < 2 \pi\}$. The inverse of this transformation is $x(\zeta) = i(\zeta+1)/(\zeta-1)$.
With this transformation let $\{R_{n}\}_{n \geq 0}$ be given by $R_{0}(\zeta) = u_{0}(x) = 1$ and
\begin{equation} \label{Eq-P-to-R}
R_{k}(\zeta) = \frac{2^{k} P_{k}(x)}{(x-i)^{k}} = u_{k}(x)\, (-2)^{k}\prod_{j=1}^{k}\sqrt{d_{j+1}} , \quad k \geq 1.
\end{equation}
Clearly, $R_{k}(\zeta)$ is a polynomial in $\zeta$ of exact degree $k$. To be precise, from the recurrence relation \eqref{Eq-Special-R2Type-RR} we find $R_{1}(\zeta) = (1+ic_{1})\zeta + (1-i c_{1})$ and
\begin{equation} \label{Eq-TTRR-R}
R_{k+1}(\zeta) = [(1+ic_{k+1})\zeta + (1-ic_{k+1})] R_{k}(\zeta) - 4 d_{k+1} \zeta R_{k-1}(\zeta), \quad k \geq 1.
\end{equation}
Hence, the leading coefficient of $R_{k}$ is $\prod_{j=1}^{k} (1+ic_{j})$. We also have
\[
\hat{u}_{k}(x) = \frac{(-1)^{k} \sqrt{\ell_{k+1}}}{2^{k} \prod_{j=1}^{k}\sqrt{d_{j+1}}}\, \hat{R}_{k}(\zeta), \quad k \geq 1,
\]
where $\hat{R}_{k}(z) = R_{k}(z)- 2(1-\ell_{k}) R_{k-1}(z)$. Hence, from \eqref{Eq-Discrete-Orthogonality-1}
\begin{equation} \label{Eq-Discrete-Orthogonality-2}
\begin{array}{l}
\displaystyle \sum_{r=1}^{n} \lambda_{r}^{(n)}\, \overline{\hat{R}_{m}(\zeta_{r}^{(n)})}\, \hat{R}_{k}(\zeta_{r}^{(n)})
= \frac{2^{2k} \prod_{j=1}^{k}d_{j+1}}{\ell_{k+1}}\, \delta_{m,k},
\end{array}
\end{equation}
for $m,k = 1,2, \ldots, n$, where
\[
\zeta_{r}^{(n)} = \frac{x_{r}^{(n)}+i}{x_{r}^{(n)}-i}, \quad r =1,2, \ldots, n.
\]
Moreover, since $\lim_{x \to \infty} \hat{u}_{k}(x) = 0$, we also have $\hat{R}_{k}(1)= 0$. Hence, we introduce the polynomials
\begin{equation} \label{Eq-OPUC-Relation1}
\Phi_{k-1}(z) = \frac{1}{\prod_{j=1}^{k} (1+ic_{j})}\frac{\hat{R}_{k}(z)}{z-1}, \quad k \geq 1.
\end{equation}
Note that $\Phi_{k}$ is a monic polynomial of degree $k$. With the transformation \eqref{Eq-Transformation} one can also write
\begin{equation} \label{Eq-OPUC-Relation2}
\Phi_{k-1}(\zeta) = \frac{-i\,2^{k-1}}{\prod_{j=1}^{k}(1+ic_{j})}\frac{1}{(x-i)^{k-1}}\big[P_{k}(x) - (1-\ell_{k}) (x-i) P_{k-1} \big], \quad k \geq 1.
\end{equation}
\begin{theo} \label{Thm-Orthogonality1-UC}
Given the $R_{II}$ type recurrence \eqref{Eq-Special-R2Type-RR} let $\mu$ be the positive measure on the unit circle such that its Verblunsky coefficients are
\begin{equation} \label{Eq-Verblunsky-Characterization}
\alpha_{n-1} = - \frac{1}{\tau_{n}}\,\frac{1-2\ell_{n+1} - i c_{n+1}}{1 - i c_{n+1}}, \quad n \geq 1,
\end{equation}
where $\tau_{0}= 1$ and $\tau_{n} = \tau_{n-1}(1-ic_n)/(1+ic_n)$, $n \geq 1$. The polynomials $\{\Phi_n\}_{n \geq 0}$ given by \eqref{Eq-OPUC-Relation2} are the monic orthogonal polynomials on the unit circle that satisfy
\[
\int_{\mathbb{T}} \overline{\Phi_{m}(\zeta)}\,\Phi_{k}(\zeta)\, d \mu(\zeta) = \frac{(1+c_{1}^2)\,2^{2k} \prod_{j=1}^{k+1}d_{j+1}}{\ell_{k+2} \prod_{j=1}^{k+1} (1+c_{j}^2)}\, \delta_{m,k} .
\]
Moreover, if $\{d_{n+1}\}_{n \geq 1}$ is a positive chain sequence with multiple parameter sequences then the measure $\mu$ is such that $\int_{\mathbb{T}} |\zeta-1|^{-2} d \mu(\zeta)$ exists.
\end{theo}
\begin{remark}
The measure $\mu$ characterized by the Verblunsky coefficients in \eqref{Eq-Verblunsky-Characterization} may or may not be such that the integral $\int_{\mathbb{T}} |\zeta-1|^{-2} d \mu(\zeta)$ exists. However, this is definitely the case if $\{d_{n+1}\}_{n \geq 1}$ is a positive chain sequence with multiple parameter sequences.
\end{remark}
\begin{proof}[\bf Proof of Theorem \ref{Thm-Orthogonality1-UC}]
Observe that $z-1 = \hat{R}_{1}(z)/(1+ic_1)$. Hence, from \eqref{Eq-Discrete-Orthogonality-2},
\[
\sum_{r=1}^{n} \lambda_{r}^{(n)} |\zeta_{r}^{(n)} - 1|^2 = \frac{1}{1+c_1^2} \sum_{r=1}^{n} \lambda_{r}^{(n)} |\hat{R}_{1}(\zeta_{r}^{(n)})|^2 = \frac{4}{1+c_1^2}.
\]
Thus, we rewrite \eqref{Eq-Discrete-Orthogonality-2} as
\[
\begin{array}{l}
\displaystyle \sum_{r=1}^{n} \hat{\lambda}_{r}^{(n)}\, \overline{\Phi_{m}(\zeta_{r}^{(n)})}\,\Phi_{k}(\zeta_{r}^{(n)})
= \frac{(1+c_1^2)\,2^{2k} \prod_{j=1}^{k+1}d_{j+1}}{\ell_{k+2} \prod_{j=1}^{k+1} (1+c_{j}^2)}\, \delta_{m,k},
\end{array}
\]
for $m, k = 0,1, \ldots,n-1$, where
\[
\hat{\lambda}_{r}^{(n)} = \frac{1}{4} (1+c_1^2)\,|\zeta_{r}^{(n)} - 1|^2\lambda_{r}^{(n)} > 0, \quad r=1,2,\ldots, n
\]
and $\sum_{r=1}^{n} \hat{\lambda}_{r}^{(n)} = 1$. Therefore, if we introduce the series of probability measures
\[
\Lambda_{n}(\zeta) = \sum_{r=1}^{n} \hat{\lambda}_{r}^{(n)}\, \delta_{\zeta_{r}^{(n)}}
\]
where $\delta_{w}$ is the Dirac measure concentrated at the point $w$, then
\[
\int_{\mathbb{T}} \overline{\Phi_{m}(\zeta)}\,\Phi_{k}(\zeta)\, d \Lambda_{n}(\zeta) = \frac{(1+c_1^2)\, 2^{2k} \prod_{j=1}^{k+1}d_{j+1}}{\ell_{k+2} \prod_{j=1}^{k+1} (1+c_{j}^2)}\, \delta_{m,k},
\]
for $m, k = 0,1, \ldots,n-1$ and $n \geq 2$. It is also not difficult to verify that by expressing the monomial $\zeta^{k}$ as a linear combination of the set of monic polynomials $\{\Phi_{j}\}_{j=0}^{k}$ that
\[
\int_{\mathbb{T}}{\zeta^k} d \Lambda_{n}(\zeta) = \int_{\mathbb{T}}{\zeta^k} d \Lambda_{n+m}(\zeta) \quad \mbox{and} \quad \int_{\mathbb{T}}{\overline{\zeta}^k} d \Lambda_{n}(\zeta) = \int_{\mathbb{T}}{\overline{\zeta}^k} d \Lambda_{n+m}(\zeta),
\]
for $n > k$ and $m \geq 1$. Thus, we could conclude that the sequence $\{\Lambda_{n}\}$ has a {\em subsequence} that converges to a probability measure $\mu$ (see, for example, \cite{Billingsley-Book}) and that
\[
\int_{\mathbb{T}}{\zeta^k} d \Lambda_{n}(\zeta) = \int_{\mathbb{T}}{\zeta^k} d \mu(\zeta) \quad \mbox{and} \quad \int_{\mathbb{T}}{\overline{\zeta}^k} d \Lambda_{n}(\zeta) = \int_{\mathbb{T}}{\overline{\zeta}^k} d \mu(\zeta),
\]
for $n > k$. Thus, the orthogonality relation given in the theorem holds. The Verblunsky coefficients are then obtain from $-\overline{\Phi_n(0)}$.
Now to prove the latter part of the theorem, note that
\[
\begin{array}{l}
\displaystyle \int_{\mathbb{T}}\frac{1}{|\zeta-1|^2} d \Lambda_{n}(\zeta) = \sum_{r=1}^{n} \hat{\lambda}_{r}^{(n)} \frac{1}{|\zeta_{r}^{(n)}-1|^2} = \frac{1}{4} (1+c_1^2) \sum_{r=1}^{n} \lambda_{r}^{(n)} , \quad n \geq 2.
\end{array}
\]
Since one can verify that the set of polynomials $\{\hat{R}_{1}, \hat{R}_{2}, \ldots, \hat{R}_{n-1}, R_{n-1}\}$ form a basis in $\mathbb{P}_{n-1}$, by considering the linear combination
\[
1 = b_1 \hat{R}_{1}(z) + b_2 \hat{R}_{2}(z) + \ldots + b_{n-1} \hat{R}_{n-1}(z) - 2b_{n} (1-\ell_n) R_{n-1}(z),
\]
we find
\begin{equation} \label{Eq-Cobination-for-1}
b_k= \frac{-1}{2^{k}(1-\ell_1)\cdots (1-\ell_{k})}, \quad k=1,2, \ldots, n.
\end{equation}
Hence, by observing
\[
1 = \sum_{k=1}^{n-1} b_{k} \hat{R}_{k}(\zeta_{r}^{(n)}) \ - 2b_{n}(1-\ell_{n})R_{n-1}(\zeta_{r}^{(n)}) = \sum_{k=1}^{n} b_{k} \hat{R}_{k}(\zeta_{r}^{(n)}) = \sum_{k=1}^{n} b_{k} \overline{\hat{R}_{k}(\zeta_{r}^{(n)})},
\]
we can write
\[
\begin{array}{l}
\displaystyle \int_{\mathbb{T}}\frac{1}{|\zeta-1|^2} d \Lambda_{n}(\zeta) = \frac{1}{4} (1+c_1^2) \sum_{r=1}^{n} \lambda_{r}^{(n)} \Big[ \sum_{k=1}^{n} b_{k} \overline{\hat{R}_{k}(\zeta_{r}^{(n)})}\sum_{k=1}^{n} b_{k} \hat{R}_{k}(\zeta_{r}^{(n)}) \Big], \quad n \geq 2.
\end{array}
\]
Thus, by using the orthogonality \eqref{Eq-Discrete-Orthogonality-2} and then the values for $b_{k}$ in \eqref{Eq-Cobination-for-1},
\[
\begin{array}{ll}
\displaystyle \int_{\mathbb{T}}\frac{1}{|\zeta-1|^2} d \Lambda_{n}(\zeta) & \displaystyle = \frac{1}{4} (1+c_1^2) \sum_{k=1}^{n}b_{k}^2 \sum_{r=1}^{n} \lambda_{r}^{(n)}|\hat{R}_{k}(\zeta_{r}^{(n)})|^2 \\[2.5ex]
&\displaystyle = \frac{1}{4} (1+c_1^2) \Big[1 + \sum_{k=2}^{n} \prod_{j=2}^{k}\frac{\ell_{j}}{(1-\ell_{j})} \Big], \quad n \geq 2.
\end{array}
\]
By Wall's criteria for the maximal parameter sequence of a positive chain sequence (see \cite[p.101]{Chihara-book}), the series $\sum_{k=2}^{\infty} \prod_{j=2}^{k}\frac{\ell_{j}}{(1-\ell_{j})}$ converges if $\{ d_{n+1}\}_{n \geq 1}$ is a multiple parameter positive chain sequence and diverges if $\{ d_{n+1}\}_{n \geq 1}$ is a single parameter positive chain sequence. Thus, if $\{ d_{n+1}\}_{n \geq 1}$ is a multiple parameter positive chain sequence then
\[
\displaystyle \int_{\mathbb{T}}\frac{1}{|\zeta-1|^2} d \mu(\zeta) = \frac{1}{4} (1+c_1^2) \Big[1 + \sum_{k=2}^{\infty} \prod_{j=2}^{k}\frac{\ell_{j}}{(1-\ell_{j})} \Big]
\]
is finite, which concludes the second part of the theorem.
\end{proof}
\begin{remark}
The initial part of Theorem \ref{Thm-Orthogonality1-UC} is a kind of Favard type theorem on the unit circle, where a measure $\mu$ is derived from the pair of sequences $[\{c_n\}_{n \geq 1}, \{d_{n+1}\}_{n \geq 1}]$. See an alternative proof of this in \cite{BracRangaSwami-2016}. The classical Favard theorem on the unit circle is where one starts with the Verblunsky coefficients to derive the measure $\mu$. See \cite{ENZG1} for a simple proof the classical Favard theorem on the unit circle.
\end{remark}
We now state the above theorem in terms of the rational functions $\hat{u}_{n}(x) = \linebreak \sqrt{\ell_{n+1}}\, u_{n}(x) + \sqrt{1-\ell_{n}}\, u_{n-1}(x)$, $ n \geq 1$.
\begin{coro} Let the positive measure $\mu$ be as in Theorem \ref{Thm-Orthogonality1-UC} and let $\psi$, defined on the real line, be given by $d\psi(x) = -d\mu((x+i)/(x-i))$. Then the following hold.
\[
\int_{-\infty}^{\infty} \hat{u}_{m}(x)\, \hat{u}_{k}(x)\, (x^2+1) \, d \psi(x) = (1+c_1^2) \delta_{m,k},
\]
for $m, n = 1, 2, \ldots$.
\end{coro}
\begin{remark}
The sequence of polynomials $\{R_n\}_{n \geq 1}$ are in fact a sequence of para-orthogonal polynomials associated with the sequence of orthogonal polynomials on the unit circle $\{\Phi_{n}\}_{n \geq 0}$.
\end{remark}
\begin{remark}
According to the results shown in \cite{Ismail-Masson-JAT1995}, there is a moment functional $\mathcal{L}$ such that $\mathcal{L}[x^{k}P_n(x)] = 0$ for $0 \leq k < n$. What can we say about moment functional $\mathcal{L}$?
\end{remark}
The above remarks will become clearer with results presented in the next section.
\setcounter{equation}{0}
\section{Para-orthogonal polynomials to $R_{II}$ type recurrence } \label{Sec-OPUC-to-POPUC2}
From \eqref{Eq-Verblunsky-Characterization} it is easily verified that
\begin{equation} \label{Eq-Tau-Recurrence-0}
\tau_1 = \frac{1-ic_1}{1+ic_1} \quad \mbox{and} \quad \tau_{n+1} = \frac{\tau_n + \overline{\alpha}_{n-1}}{1+ \tau_{n}\alpha_{n-1}}, \ \ n \geq 1.
\end{equation}
Moreover, we can also verify the following formulas for $\{c_{n}\}_{n \geq 1}$ and $\{\ell_{n+1}\}_{n \geq 0}$ in terms of the sequences $\{\tau_{n}\}_{n \geq 1}$ and $\{\alpha_{n}\}_{n \geq 0}$.
\begin{equation*}\label{***}
\begin{array}{ll}
\displaystyle c_{1} = i\frac{\tau_1 - 1}{\tau_1 + 1}, & \displaystyle c_{n+1} = \frac{ \mathcal{I}m(\tau_{n}\alpha_{n-1})} {1 + \mathcal{R}e(\tau_{n}\alpha_{n-1})} = \frac{ \mathcal{I}m(\tau_{n+1}\alpha_{n-1})} {1 - \mathcal{R}e(\tau_{n+1}\alpha_{n-1})}, \quad n \geq 1 \\[3ex]
& \displaystyle \ell_{n+1} = \frac{1}{2}\, \frac{|1 + \tau_{n} \alpha_{n-1}|^2}{1 + \mathcal{R}e( \tau_{n} \alpha_{n-1})} = \frac{1}{2}\, \frac{1 - |\tau_{n+1} \alpha_{n-1}|^2}{1 - \mathcal{R}e( \tau_{n+1} \alpha_{n-1})}, \quad n \geq 1.
\end{array}
\]
To verify the two different expressions for $c_{n+1}$ and $\ell_{n+1}$ we require
\[
(1- \tau_{n+1}\alpha_{n-1})(1 + \tau_{n}\alpha_{n-1}) = 1 - |\alpha_{n-1}|^2, \quad n \geq 1,
\]
which is a consequence of the recurrence formula in \eqref{Eq-Tau-Recurrence-0}.
Hence, given the pair of sequences $\{c_{n}\}_{n \geq 1}$ and $\{d_{n+1}\}_{n \geq 1}$, the probability measure $\mu$ on the unit circle given by Theorem \ref{Thm-Orthogonality1-UC} is unique. That is, the measure $\mu$ for which the polynomials $\Phi_{n}$ given by \eqref{Eq-OPUC-Relation1} are orthogonal, is unique.
However, the reciprocal of the above observation is not true. There are infinitely many pairs of sequences $\{c_{n}\}_{n \geq 1}$ and $\{d_{n+1}\}_{n \geq 1}$ produce the same measure $\mu$ and the same orthogonal polynomials $\Phi_{n}$. This follows from the following results established in \cite{BracRangaSwami-2016}.
With any $\tau$ chosen such that $|\tau| = 1$, $\tau \neq -1$, let the sequence of numbers $\{\tau_n\}_{n=0}^{\infty}$ be given by
\begin{equation*} \label{Eq-TauRecurence1}
\tau_{1} = \tau \quad \mbox{and} \quad \tau_{n+1} = \frac{\tau_{n} + \overline{\alpha}_{n-1}}{1 + \tau_{n} \alpha_{n-1}}, \quad n \geq 1,
\end{equation*}
where $\alpha_{n-1} = - \overline{\Phi_n(0)}$, $n \geq 1$, are the Verblunsky coefficients. Then the sequence $\{R_n\}$ of para-orthogonal polynomials on the unit circle given by
\[
R_n(z) \frac{1+\tau_1}{2} \prod_{k=1}^{n-1} \frac{ 1 + \Re(\tau_{k}\alpha_{k-1})}{1 + \tau_{k} \alpha_{k-1}} = z\Phi_{n-1}(z) + \tau_{n}\Phi_{n-1}^{\ast}(z), \quad n \geq 1,
\]
satisfy the three term recurrence formula \eqref{Eq-TTRR-R}, where the real sequence $\{c_n\}$ and the positive chain sequence $\{d_{n+1}\}_{n=1}^{\infty} $ are such that
\begin{equation*}\label{Eq-CoeffsTTRR-11}
c_{1} = i\frac{\tau_1 - 1}{\tau_1 + 1}, \quad c_{n+1} = \frac{ \Im(\tau_{n}\alpha_{n-1})} {1 + \Re(\tau_{n}\alpha_{n-1})} \quad \mbox{and} \quad d_{n+1} = (1-\ell_{n})\ell_{n+1}, \quad n \geq 1,
\end{equation*}
where $\{\ell_{n+1}\}_{n\geq 0}$ is the minimal parameter sequence of $\{d_{n+1}\}_{n \geq 1}$ given by
\[
\ell_1 = 0 \quad \mbox{and} \quad \ell_{n+1} = \frac{1}{2} \frac{|1 + \tau_{n} \alpha_{n-1}|^2}{1 + \Re( \tau_{n} \alpha_{n-1})}, \quad n \geq 1.
\]
Moreover, the measure $\mu$ is such that the integral $\int_{\mathbb{T}} |\zeta-1|^{-2}d\mu(\zeta)$ exists if and only if there exists a $\tau$ $(|\tau| = 1, \tau \neq -1)$ such that the corresponding positive chain sequence $\{d_{n+1}\}_{n \geq 1}$ is not a single parameter positive chain sequence.
\begin{remark}
The above results are precisely those given in \cite[Thm.\,4.1]{BracRangaSwami-2016}, where the sequence $\{\rho_{n}\}_{n \geq 0}$ that appear in \cite[Thm.\,4.1]{BracRangaSwami-2016} is such that $\rho_{n} = - \tau_{n+1}$, $n \geq 0$.
\end{remark}
The following Theorem, also given in \cite{BracRangaSwami-2016}, is a restatement of the above results under the additional assumption that the measure $\mu$ belongs to the class of measures for which the principal value integral
\begin{equation} \label{Eq-PVI-UC}
I(\mu) = \Xint-_\mathcal{\mathbb{T}} \frac{\zeta}{\zeta-1} d \mu(\zeta) = \lim_{\epsilon \to 0} \int_{\epsilon}^{2\pi - \epsilon} \frac{e^{i\theta}}{e^{i\theta}-1} d\mu(e^{i \theta}),
\end{equation}
exists. Observe that this class of measures also includes measures for which the integral $\int_{\mathbb{T}} |\zeta-1|^{-2}d\mu(\zeta)$ does not exist. The simplest example of such measures is the Lebesque measure $d\mu(\zeta) = (2\pi i \zeta)^{-1} d\zeta$, where $I = 1/2$ but $\int_{\mathbb{T}} |\zeta-1|^{-2}(2\pi i \zeta)^{-1} d\zeta$ does not exist.
\begin{theo}[\cite{BracRangaSwami-2016}] \label{Thm-BRS2016-2}
Let $\mu$ be a nontrivial probability measure on the unit circle such that $I = I(\mu)$ exists and let $\{\Phi_n\}_{n=0}^{\infty}$ be the sequence of monic orthogonal polynomials on the unit circle with respect to $\mu$. With any $s$ be such that $-\infty < s < \infty$, let the sequence $\{\tau_{n}(s)\}_{n=1}^{\infty}$ be given by
\[
\displaystyle \tau_{1}(s)= \frac{I+is}{\overline{I} - is}\quad \mbox{and} \quad \tau_{n+1}(s) = \frac{\tau_{n}(s) + \overline{\alpha}_{n-1}}{1 + \tau_{n}(s) \alpha_{n-1}}, \quad n \geq 1.
\]
Then for the sequence $\{R_n(s;z)\}$ of para orthogonal polynomials, in $z$, given by
\[
\quad R_n(s;z) \frac{1+\tau_1(s)}{2} \prod_{k=1}^{n-1} \frac{ 1 + \Re[\tau_{k}(s)\alpha_{k-1}]}{1 + \tau_{k}(s) \alpha_{k-1}} = z\Phi_{n-1}(z) + \tau_{n}(s)\Phi_{n-1}^{\ast}(z), \quad n \geq 1,
\]
the following three term recurrence formula holds.
\begin{equation*}
R_{n+1}(s;z) = \big[(1+ic_{n+1}(s))z + (1-ic_{n+1}(s))\big] R_{n}(s;z) - 4\,d_{n+1}(s) z R_{n-1}(s;z),
\end{equation*}
with $R_{0}(s;z) = 1$ and $R_{1}(s;z) = (1+ic_{1}(s))z + (1-ic_{1}(s))$, where the real sequences $\{c_n(s)\}$ and $\{d_{n+1}(s)\}$ are such that
\begin{equation*}\label{Eq-CoeffsTTRR-12}
\begin{array}l
\displaystyle c_{1}(s) = i\frac{\tau_1(s) - 1}{\tau_1(s) + 1} = -2[s + \Im(I)] \quad \mbox{and} \\ [2ex]
\displaystyle c_{n+1}(s)= \frac{ \Im[\tau_{n}(s)\alpha_{n-1}]} {1 + \Re[\tau_{n}(s)\alpha_{n-1}]}, \ \ d_{n+1}(s) = [1-\ell_{n}(s)]\ell_{n+1}(s), \ \ n \geq 1.
\end{array}
\end{equation*}
Here, $\{d_{n+1} (s)\}_{n=1}^{\infty}$ is a positive chain sequence and $\{\ell_{n+1}(s)\}_{n=0}^{\infty}$ is its minimal parameter sequence, which is given by
\[
\ell_{1}(s) = 0 \quad \mbox{and} \quad \ell_{n+1}(s) = \frac{1}{2}\, \frac{|1 + \tau_{n}(s) \alpha_{n-1}|^2}{1 + \Re\big( \tau_{n}(s) \alpha_{n-1}\big)}, \quad n \geq 1.
\]
With respect to the measure $\mu$, the polynomials $R_n(s;.)$ also satisfy the orthogonality property $\mathcal{N}^{(s)}[z^{-n+k}R_n(s;z)] = 0$, $k=0,1,\ldots, n-1$,
with respect to the moment fuctional $\mathcal{N}^{(s)}$ given by
\begin{equation} \label{Eq-TheoMain4}
\mathcal{N}^{(s)}[f(z)] = \Xint-_{\mathbb{T}} f(\zeta) \frac{\zeta}{\zeta-1} d\mu (\zeta) + i\,s\, f(1).
\end{equation}
\end{theo}
\vspace{1.5ex}
The orthogonal polynomials $\{\Phi_n(z)\}_{n=0}^{\infty}$ can be recovered from $\{R_n(s;z)\}_{n=0}^{\infty}$ by
\[
(z-1)\Phi_n(z)\prod_{k=1}^{n+1}\big(1+ic_{k}(s)\big) = R_{n+1}(s;z) - 2\big(1-\ell_{n+1}(s)\big)R_{n}(s;z), \quad n \geq 0,
\]
which we can identify with the relation \eqref{Eq-OPUC-Relation1}.
Observe that $I$ in Theorem \ref{Thm-BRS2016-2} is such that $I + \overline{I} = \mu_0 = 1$. Hence, $\mathcal{R}e(I) = 1/2$ and $\tau_{1}(s) \neq -1$ if $-\infty < s < \infty$.
It was also shown in \cite{BracRangaSwami-2016} that the sequence $\{d_{n+1} (s)\}_{n=1}^{\infty}$ given in Theorem \ref{Thm-BRS2016-2} is always a single parameter positive chain sequence if $s \neq 0$. However, $\{d_{n+1} (0)\}_{n=1}^{\infty}$ is a single parameter positive chain sequence if and only if the integral $\int_{\mathbb{T}} |\zeta-1|^{-2}d\mu(\zeta)$ does not exists. \\[-1ex]
\noindent {\bf Somes remarks with respect to the measure obtained in Theorem \ref{Thm-Orthogonality1-UC}}.
\begin{itemize}
\item There are infinitely many pairs $[\{c_{n}\}_{n \geq 1}, \{d_{n+1}\}_{n \geq 1}]$, where the $\{c_{n}\}_{n \geq 1}$ is a real sequence and $\{d_{n+1}\}_{n \geq 1}$ is a positive chain sequence, considered as in Theorem \ref{Thm-Orthogonality1-UC} produce the same measure $\mu$. However, as we have shown above, there is a one to one correspondence between any pair $[\{c_{n}\}_{n \geq 1}, \{d_{n+1}\}_{n \geq 1}]$ and the pair $[\{\alpha_{n}\}_{n \geq 0}, \tau]$, where $\{\alpha_{n}\}_{n \geq 0}$ is the sequence of Verblunsky coefficients associated with the measure $\mu$ and $\tau = (1-ic_1)/(1+ic_1)$.
\item If the resulting measure $\mu$ is such that the principal value integral $I(\mu)$ defined by \eqref{Eq-PVI-UC} exists, then each pair $[\{c_{n}\}_{n \geq 1}, \{d_{n+1}\}_{n \geq 1}]$ that produces $\mu$ can be identified to be equal to a pair $[\{c_{n}(s)\}_{n \geq 1}, \{d_{n+1}(s)\}_{n \geq 1}]$ given by Theorem \ref{Thm-BRS2016-2}, where $s = -\Im(I(\mu)) - c_1/2$.
\item In a pair $[\{c_{n}\}_{n \geq 1}, \{d_{n+1}\}_{n \geq 1}]$ that equals to $[\{c_{n}(s)\}_{n \geq 1}, \{d_{n+1}(s)\}_{n \geq 1}]$, where $s \neq 0$, the positive chain sequence $\{d_{n+1}\}_{n \geq 1}$ can not be a multiple parameter positive chain sequence. However, in the pair equals to $[\{c_{n}(0)\}_{n \geq 1}, \{d_{n+1}(0)\}_{n \geq 1}]$ the sequence $\{d_{n+1}\}_{n \geq 1}$ is a multiple parameter positive chain sequence only if $\mu$ is such that integral $\int_{\mathbb{T}}|\zeta-1|^{-1} d \mu(\zeta)$ exists.
\item Observe that if the integral $\int_{\mathbb{T}}|\zeta-1|^{-1} d \mu(\zeta)$ exists then the principal value integral $I(\mu)$ also exists as a normal integral.
\end{itemize}
Hence, if the measure $\mu$ obtained from Theorem \ref{Thm-Orthogonality1-UC} is such that the principal value integral $I = I(\mu)$ exists then from Theorem \ref{Thm-BRS2016-2} we have that the polynomials $R_{n}$ given by \eqref{Eq-TTRR-R} satisfy
\begin{equation} \label{Eq-Lorthogonality-2}
\Xint-_{\mathbb{T}} \zeta^{-n+k}R_{n}(\zeta) \frac{\zeta}{\zeta-1} d\mu (\zeta) - i\,[\Im(I)+c_{1}/2]\, R_{n}(1) = 0, \quad 0 \leq k \leq n-1.
\end{equation}
Now by considering the transformation $\zeta = (x+i)/(x-i)$ we have
\[
I(\mu) = \Xint-_\mathcal{\mathbb{T}} \frac{\zeta}{\zeta-1} d \mu(\zeta)
= \lim_{y \to \infty} \int_{-y}^{y} \frac{1}{2}(1 - i x) d \psi(x),
\]
where
\begin{equation} \label{Eq-RL-Measure}
d \psi(x) = - d\mu\big((x+i)/(x-i)\big).
\end{equation}
Hence, by considering the principal value integral on the real line defined by
\begin{equation} \label{Eq-PVI-RealLine}
\Xint-_{-\infty}^{\infty} f(x) d\psi(x) =
\lim_{y \to \infty} \int_{-y}^{y} f(x) d \psi(x),
\end{equation}
if $I(\mu)$ exists then
\[
\Xint-_{-\infty}^{\infty} x d \psi(x) = -2 \Im(I(\mu)).
\]
Consequently, from \eqref{Eq-Lorthogonality-2},
\begin{equation} \label{Eq-Ortogonality-P-3}
\Xint-_{-\infty}^{\infty} r_{k}^{(n)}(x) \frac{P_{n}(x)}{(x^2+1)^{n}} \,(x^2+1) d \psi(x) + \big[c_{1} - \Xint-_{-\infty}^{\infty} x\, d\psi(x)\big] \mathfrak{p}_{n} = 0, \quad 0 \leq k \leq n-1.
\end{equation}
where $r_{k}^{(n)}(x) = (x+i)^{k} (x-i)^{n-1-k}$ and $\mathfrak{p}_n = (1-\ell_{1}) \cdots(1-\ell_{n})$ is the leading coefficient of $P_{n}$.
\begin{theo} \label{Thm-PV-Orthonality-M} Given the $R_{II}$ type recurrence, let $\mu$ be the probability measure on the unit circle obtained as in Theorem \ref{Thm-Orthogonality1-UC}. If the measure $\mu$ is such that the principal value integral $ I(\mu)$ defined by \eqref{Eq-PVI-UC} exists then with $\psi(x)$ given by \eqref{Eq-RL-Measure} and with the principal value integral on the real line defined by \eqref{Eq-PVI-RealLine},
\[
\Xint-_{-\infty}^{\infty} x^{k} \, \frac{P_{n}(x)}{(x^2+1)^{n}} \, (x^2+1) d \psi(x) = - \Big[c_{1} - \Xint-_{-\infty}^{\infty} x\, d\psi(x)\Big] \mathfrak{p}_{n}\, \delta_{n-1,k}, \quad 0 \leq k \leq n-1,
\]
for $n \geq 1$.
\end{theo}
\begin{proof}
Since the set of polynomials $\{r_j^{(n)}\}_{j=0}^{n-1}$ is linearly independent in $\mathbb{P}_{n-1}$ we can write
\[
x^{k} = \sum_{j=0}^{n-1}q_{k,j}^{(n)}\,r_j^{(n)}(x), \quad 0 \leq k \leq n-1.
\]
%
Hence, from \eqref{Eq-Ortogonality-P-3}
%
\[
\Xint-_{-\infty}^{\infty} x^{k} \, \frac{P_{n}(x)}{(x^2+1)^{n}} \, (x^2+1) d \psi(x) + \Big[c_{1} - \Xint-_{-\infty}^{\infty} x\, d\psi(x)\Big] \mathfrak{p}_{n}\, \sum_{j=1}^{n-1}q_{k,j}^{(n)} = 0 , \quad 0 \leq k \leq n-1.
\]
Now considering the limit of $x^{-n+1} \sum_{j=0}^{n-1}q_{k,j}^{(n)}r_j^{(n)}(x)$ as $x \to \infty$ we find that $\sum_{j=0}^{n-1}q_{k,j}^{(n)} = 0$ for $k=0,1, \ldots, n-2$ and $\sum_{j=0}^{n-1}q_{n-1,j}^{(n)} = 1$, which gives the required result in the theorem.
\end{proof}
We are now able to consider the proof of Theorem \ref{Thm-Special-R2Type-MF}.
\begin{proof}[\bf Proof of Theorem \ref{Thm-Special-R2Type-MF}]
Since the positive chain sequence $\{d_{n+1}\}_{n \geq 1}$ has multiple parameter sequences, from Theorem \ref{Thm-Orthogonality1-UC} the associated measure is such that the integral $\int_{\mathbb{T}} |\zeta-1|^{-2} d \mu(\zeta)$ exists. Moreover, from the remarks that follow Theorem \ref{Thm-BRS2016-2} the polynomials $R_{n}$ given by \eqref{Eq-TTRR-R} satisfy $R_n(z) = R_n(0; z)$, $n \geq 0$ and
\begin{equation} \label{Eq-Ortho3-Rn}
\int_{\mathbb{T}} \zeta^{-n+k}R_{n}(\zeta) (1-\zeta) d \nu(\zeta) = \mathcal{N}^{(0)}[z^{-n+k}R_n(0;z)] = 0, \quad 0 \leq k \leq n-1,
\end{equation}
where the probability measure on the unit circle $\nu$ is given by
\[
\mathop{\rm const} \, d \nu(\zeta) = \frac{\zeta}{(\zeta-1)(1-\zeta)} d \mu(\zeta) = \frac{1}{|\zeta-1|^{2}} d \mu(\zeta).
\]
Now from results obtained in \cite{Costa-Felix-Ranga-JAT2013} and from some further observations made later (see for example \cite{BracRangaSwami-2016}), if $\{\beta_{n}\}_{n \geq 0}$ is the sequence of Verblunsky coefficients with respect to the measure $\nu$ then for the coefficients of the recurrence relation \eqref{Eq-TTRR-R} of $R_{n}$ we also have
\begin{equation}\label{Eq-CoeffsTTRR-2}
\begin{array}{l}
\displaystyle c_{n} = \frac{-\mathcal{I}m \big(\tau_{n-1}\beta_{n-1}\big)} {1-\mathcal{R}e\big(\tau_{n-1}\beta_{n-1}\big)} \quad \mbox{and} \quad d_{n+1} = \big(1-g_{n}\big)g_{n+1}, \quad n \geq 1
\end{array}
\end{equation}
and
\[
\frac{1}{\prod_{j=1}^{n}(1-g_{j})}R_n(z) = K_n(z, 1),
\]
where $K_n(z, w)$ represents the Christoffel-Darboux kernels with respect to the probability measure $\nu$. Here, $\tau_0 =1$,
\[
g_{n} = \frac{1}{2} \frac{\big|1 - \tau_{n-1} \beta_{n-1}\big|^2}{\big[1 - \mathcal{R}e\big(\tau_{n-1}\beta_{n-1}\big)\big]} \quad \mbox{and} \quad \tau_{n} = \frac{\tau_{n-1} - \overline{\beta}_{n-1}}{1 - \tau_{n-1} \beta_{n-1}}, \quad n \geq 1.
\]
Since the measure $\nu$ does not have a pure point at $z=1$, the above parameter sequence $\{g_{n+1}\}_{n \geq 0}$ of the positive chain sequence $\{d_{n+1}\}_{n \geq 1}$ is also its maximal parameter sequence $\{M_{n+1}\}_{n \geq 0}$.
The following reciprocal formulas can also be easily verified.
\[
\beta_{n-1} = \frac{1}{\tau_{n-1}}\,\frac{1-2M_{n} - i c_n}{1 - i c_n} \quad \mbox{and} \quad \tau_{n} = \tau_{n-1}\frac{1-ic_n}{1+ic_n}
\]
for $n \geq 1$, with $\tau_{0}= 1$.
Now we consider the transformation $\zeta = (x+i)/(x-i)$. Then we have from \eqref{Eq-P-to-R} and \eqref{Eq-Ortho3-Rn}
\[
\int_{-\infty}^{\infty} x^{k} \, \frac{P_{n}(x)}{(x^2+1)^{n}} \, d \varphi(x) = 0, \quad k=0,1, \ldots, n-1.
\]
where $d \varphi(x) = - d \nu((x+i)/(x-i))$.
To obtain the value of $\gamma_{n} = \int_{-\infty}^{\infty} \frac{x^{n}P_{n}(x)}{(x^2+1)^{n}} \, d \varphi(x)$, first we observe from the $R_{II}$ type recurrence \eqref{Eq-Special-R2Type-RR},
\[
(x^2+1) \frac{P_{n+1}(x)}{(x^2+1)^{n+1}} = (x - c_{n+1}) \frac{P_{n}(x)}{(x^2+1)^{n}} - d_{n+1} \frac{P_{n-1}(x)}{(x^2+1)^{n-1}}, \quad n \geq 1.
\]
Hence, $ \gamma_{n+1} = \gamma_{n} - d_{n+1} \gamma_{n-1}$, which is equivalent to
\[
d_{n+1} = \frac{\gamma_{n}}{\gamma_{n-1}}\big[1 - \frac{\gamma_{n+1}}{\gamma_{n}} \big], \quad n \geq 1.
\]
Clarely, $\gamma_{0} = \int_{-\infty}^{\infty} d \varphi(x) = \int_{\mathbb{T}} d \nu(\zeta) = 1$. Thus, all we have to show is $\gamma_{1} = (1-M_1)$.
With $\int_{-\infty}^{\infty} \frac{P_{1}(x)}{(x^2+1)} \, d \varphi(x) = 0$ we can write
\[
\gamma_{1} = \int_{-\infty}^{\infty} (x+c_1) \frac{P_{1}(x)}{x^2+1} d \varphi(x) = 1 - (1+c_1^{2}) \int_{-\infty}^{\infty} \frac{1}{x^2+1} d \varphi(x).
\]
However,
\[
\int_{-\infty}^{\infty} \frac{1}{x^2+1} d \varphi(x) = \frac{1}{4}\int_{\mathbb{T}} \zeta^{-1}(\zeta-1)^2 d \nu(\zeta) = \frac{1}{2} [1 - \Re(\tilde{\alpha}_0)].
\]
Hence, together with the result for $c_1$ in \eqref{Eq-CoeffsTTRR-2} we find $\gamma_{1} = (1-M_1)$.
\end{proof}
\setcounter{equation}{0}
\section{Examples } \label{Sec-Examples}
We justify the statements made in Theorems \ref{Thm-Special-R2Type-MF} and \ref{Thm-Orthogonality1-UC}, and also the results given in Section \ref{Sec-OPUC-to-POPUC2} with the following examples. \\[-0.5ex]
\noindent{\bf Example 1}. We consider the $R_{II}$ type recurrence \eqref{Eq-Special-R2Type-RR} with $c_n = 0$, $n \geq 1$, $d_{2} = 1/2$ and $d_{n+1} = 1/4$, $n \geq 2$. The sequence $\{d_{n+1}\}_{n\geq 1}$ is a single parameter positive chain sequence with its minimal parameter sequence $\{\ell_{n+1}\}_{n \geq 0} = \{0, 1/2, 1/2, 1/2, \ldots\}$.
From the theory of difference equations it is easily verified that
\[
P_{n}(x) = \big(\frac{x-i}{2}\big)^n + \big(\frac{x+i}{2}\big)^n, \quad n \geq 1.
\]
Hence, from \eqref{Eq-Rationals-1} we have
\[
u_{0}(x) = 1, \quad \mbox{and} \quad u_{n}(x) = \frac{(-1)^n}{\sqrt{2}}\big[1 + \frac{(x+i)^n}{(x-i)^n}\big], \ \ n \geq 1.
\]
From this we successively find from the results given in Section \ref{Sec-EVP-to-Orthogonality} that
\[
\hat{u}_{n}(x) = \frac{(-1)^n}{2} \Big[\frac{(x+i)^{n}}{(x-i)^{n}} - \frac{(x+i)^{n-1}}{(x-i)^{n-1}} \Big], \ \ \ n \geq 1,
\]
\[
R_{0}(z) = 1, \quad R_{n}(z) = z^n +1, \ \ n \geq 1,
\]
\[
\hat{R}_{n}(z) = z^{n} - z^{n-1}, \ \ \ n \geq 1
\]
and
\[
\Phi_{n}(z) = z^{n}, \quad n \geq 0.
\]
Thus, clearly the probability measure $\mu$ that follows from Theorem \ref{Thm-Orthogonality1-UC} is the Lebesgue measure. That is, $d \mu(\zeta) = \frac{1}{2\pi i\zeta} d \zeta$ and all the associated Verblunsky coefficients are equal to $0$.
For the Lebesgue measure the integral $\int_{\mathbb{T}}|\zeta-1|^{-2} d \mu(\zeta)$ does not exist. However, we observe that the principal value integral $I(\mu)$ exists. To be precise, we have
\[
I(\mu) = \frac{1}{2\pi} \lim_{\epsilon \to 0} \Big[ \int_{0+\epsilon}^{\pi} \frac{e^{i\theta}}{e^{i\theta}-1} d \theta + \int_{0+\epsilon}^{\pi} \frac{e^{-i\theta}}{e^{-i\theta}-1} d \theta\Big] = \frac{1}{2\pi} \lim_{\epsilon \to 0} \int_{0+\epsilon}^{\pi} d\theta,
\]
from which $I(\mu) = 1/2$. Hence, from Theorem \ref{Thm-BRS2016-2} (see also \cite{BracRangaSwami-2016})
\[
\tau_{n+1}(s) = \frac{1 +i 2s}{1-i2s} , \quad n \geq 0
\]
and for the polynomials
\[
\quad R_n(s;z) \frac{1+\tau_1(s)}{2} \prod_{k=1}^{n-1} \frac{ 1 + \Re[\tau_{k}(s)\alpha_{k-1}]}{1 + \tau_{k}(s) \alpha_{k-1}} = z\Phi_{n-1}(z) + \tau_{n}(s)\Phi_{n-1}^{\ast}(z), \quad n \geq 1,
\]
there hold $R_{1}(s;z) = (1+ic_{1}(s))\zeta + (1-i c_{1}(s))$ and
\[
R_{k+1}(s;z) = [(1+ic_{k+1}(s))z + (1-ic_{k+1}(s))] R_{k}(z) - 4 d_{k+1}(s) z R_{k-1}(z), \quad k \geq 1,
\]
where
\begin{equation*} \label{Eq-Example1}
c_1(s) = -2s, \quad c_{n+1}(s) = 0, \ n \geq 1, \qquad d_2(s) = d_2 = \frac{1}{2}, \quad d_{n+2}(s) = d_{n+1} = \frac{1}{4}, \ n \geq 1.
\end{equation*}
Thus, we conclude that with a choice of $s$ such that $-\infty < s < \infty$ the real sequence $\{c_n(s)\}_{n \geq 1} = \{-2s, 0, 0, 0, \ldots\}$ and and positive chain sequence $\{d_{n+1}\}_{n \geq 1} = \linebreak \{1/2, 1/4, 1/4, 1/4, \ldots\}$ used together with Theorem \ref{Thm-Orthogonality1-UC} leads to the same measure which is the Lebesgue measure.
If the sequence of polynomials $\{P_{n}(s;x)\}$ are given by $P_{0}(s; x) = 1$, $P_{1}(s; x) = x - c_1(s)$ and
\[
P_{n+1}(s; x) = [x - c_{n+1}(s)] P_{n}(s;x) - d_{n+1} (x^2 + 1) P_{n-1}(s;x), \quad n \geq 1.
\]
then from Theorem \ref{Thm-PV-Orthonality-M} we have
\[
d \psi(x) = \frac{1}{\pi} \frac{1}{x^2+1} dx, \qquad \Xint-_{-\infty}^{\infty} x\, d \psi(x)= \lim_{y \to \infty} \Big[\frac{1}{2\pi}\ln (x^2+1)\Big]_{-y}^{y} = 0
\]
and
\[
\frac{1}{\pi} \Xint-_{-\infty}^{\infty} x^{k} \, \frac{P_{n}(s;x)}{(x^2+1)^{n}} \, d x = - c_{1}(s) \mathfrak{p}_{n} \, \delta_{n-1, k}, \quad 0 \leq k \leq n-1,
\] \\[0ex]
where $\mathfrak{p}_n = (1-\ell_{1}) \cdots(1-\ell_{n}) = 1/2^{n-1}$.
\vspace{4ex}
\noindent {\bf Example 2}. With $s$ such that $-\infty < s < \infty$, we consider the $R_{II}$ type recurrence \eqref{Eq-Special-R2Type-RR}, where $c_n = c_n(s)$ and $d_{n+1} = d_{n+1}(s) = [1- \ell_{n}(s)] \ell_{n+1}(s)$, for $n \geq 1$, are given by
$c_{1}(s) = \kappa - 2s$, $\ell_{1}(s) =0$,
\begin{equation*} \label{Eq-c-psi1-2}
\begin{array}{ll}
\displaystyle c_{4m+2}(s) &\!\! = \displaystyle \frac{-\kappa}{1+4m\kappa}\,
\frac{(1-\kappa)^2[1+(4m+1)\kappa] + 4s\kappa(1-\kappa) - 4s^2 [1+(4m-1)\kappa]}
{(1-\kappa)^2[1+(4m+1)\kappa]+4s^2 [1+(4m-1)\kappa]},
\\[3ex]
\displaystyle c_{4m+3}(s) &\!\! = \displaystyle \frac{-4s\kappa}{1+(4m+1)\kappa}\,
\frac{(1-\kappa)[1+(4m+1)\kappa] + 2 s \kappa }
{(1-\kappa)^2[1+(4m+1)\kappa] + 4s\kappa(1-\kappa) + 4s^2[1+(4m+1)\kappa]},
\\[3ex]
\displaystyle c_{4m+4}(s) &\!\! = \displaystyle \frac{\kappa}{1+(4m+2)\kappa}\,
\frac{(1-\kappa)^2[1+(4m+1)\kappa] + 4s\kappa(1-\kappa) - 4s^2 [1+(4m+3)\kappa]}
{(1-\kappa)^2[1+(4m+1)\kappa]+4s^2 [1+(4m+3)\kappa]},
\\[3ex]
\displaystyle c_{4m+5}(s) &\!\! = \displaystyle \frac{-2\kappa (1-\kappa)}{1+(4m+3)\kappa}\,
\frac{(1-\kappa) \kappa - 2 s [1+(4m+3)\kappa]}
{(1-\kappa)^2[1+(4m+3)\kappa] - 4s\kappa(1-\kappa) + 4s^2[1+(4m+3)\kappa]},
\end{array}
\end{equation*}
\[
\begin{array}{l}
\ell_{4m+2}(s) = \displaystyle \frac{1+(4m-1)\kappa}{2[1+(4m)\kappa]^2} \times \\[2ex]
\displaystyle \quad \frac{(1-\kappa)^2[1+(4m+1)\kappa]^2 +4s\kappa(1-\kappa)[1+(4m+1)\kappa] +4s^2\big[\kappa^2 + [1+(4m)\kappa]^2\big]}
{(1-\kappa)^2[1+(4m+1)\kappa] + 4s^2[1+(4m-1)\kappa]},
\end{array}
\]
\[
\begin{array}{l}
\ell_{4m+3}(s) = \displaystyle \frac{1+(4m)\kappa}{2[1+(4m+1)\kappa]^2} \times \\[2ex]
\displaystyle \quad \frac{(1-\kappa)^2[1+(4m+1)\kappa]^2 +4s\kappa(1-\kappa)[1+(4m+1)\kappa] +4s^2\big[\kappa^2 + [1+(4m+2)\kappa]^2\big]}
{(1-\kappa)^2[1+(4m+1)\kappa] + 4s\kappa (1-\kappa) + 4s^2[1+(4m+1)\kappa]},
\end{array}
\]
\[
\begin{array}{l}
\ell_{4m+4}(s) = \displaystyle \frac{1+(4m+1)\kappa}{2[1+(4m+2)\kappa]^2} \times \\[2ex]
\displaystyle \quad \frac{(1-\kappa)^2\big[\kappa^2 + [1+(4m+2)\kappa]^2\big] -4s\kappa(1-\kappa)[1+(4m+3)\kappa] + 4s^2[1+(4m+3)\kappa]^2}
{(1-\kappa)^2[1+(4m+1)\kappa] + 4s^2[1+(4m+3)\kappa]} ,
\end{array}
\]
and
\[
\begin{array}{l}
\ell_{4m+5}(s) = \displaystyle \frac{1+(4m+2)\kappa}{2[1+(4m+3)\kappa]^2} \times \\[2ex]
\displaystyle \quad \frac{(1-\kappa)^2\big[\kappa^2 + [1+(4m+4)\kappa]^2\big] -4s\kappa(1-\kappa)[1+(4m+3)\kappa] +4s^2[1+(4m+3)\kappa]^2}
{(1-\kappa)^2[1+(4m+3)\kappa] - 4s\kappa (1-\kappa) + 4s^2[1+(4m+3)\kappa]} ,
\end{array}
\]
for $m \geq 0$. Here, we take $\kappa$ to be such that $0 < \kappa < 1$. We also denote by $P_{n}(s;x)$ the polynomials obtain from \eqref{Eq-Special-R2Type-RR}.
Independent of the value of $s$, the measure that follows from Theorem \ref{Thm-Orthogonality1-UC} is (see \cite{BracSilvaSR-AMC2015})
\[
d\mu(\zeta) = (1 - \kappa) \frac{1}{2\pi i \zeta} d \zeta + \kappa\, \delta_{i}.
\]
The above probability measure $\mu$ is a modification of the Lebesgue measure so that it has an isolated mass (pure point) of size $\kappa$ at the point $z = i$.
For the principal value integral
\[
I(\mu) = \Xint-_\mathcal{\mathbb{T}} \frac{\zeta}{\zeta-1} d \mu(\zeta) = \frac{1-\kappa}{2\pi} \lim_{\epsilon \to 0} \Big[ \int_{0+\epsilon}^{2\pi-\epsilon} \frac{e^{i\theta}}{e^{i\theta}-1} d \theta\Big] + \frac{\kappa\, i}{i - 1},
\]
we obtain $I(\mu) = (1 - i \kappa)/2$. Hence, by using the results given by Theorem \ref{Thm-PV-Orthonality-M} we have
\[
d \psi(x) = \frac{1}{\pi} \frac{1-\kappa}{x^2+1} dx + \kappa\, \delta_{1}, \qquad \Xint-_{-\infty}^{\infty} x\, d \psi(x)= \kappa
\]
and
\[
\frac{1}{\pi} \Xint-_{-\infty}^{\infty} x^{k} \, \frac{P_{n}(s;x)}{(x^2+1)^{n}} \, (x^2+1) d \psi(x) = - [c_{1}(s) - \kappa] \mathfrak{p}_{n}(s) \, \delta_{n-1, k}, \quad 0 \leq k \leq n-1,
\]
where $\mathfrak{p}_n(s) = (1-\ell_{1}(s)) \cdots(1-\ell_{n}(s))$. Here, $\delta_{1}$ represents the Dirac measure concentrated at the point $x=1$.
\vspace{4ex}
\noindent {\bf Example 3}. We now consider the $R_{II}$ type recurrence \eqref{Eq-Special-R2Type-RR} with
\begin{equation} \label{Eq-Example3}
c_n = 0 \quad \mbox{and} \quad d_{n+1} = 1/4, \quad n \geq 1.
\end{equation}
The sequence $\{d_{n+1}\}_{n\geq 1}$ is not a single parameter positive chain sequence. Its minimal parameter sequence is $\{\ell_{n+1}\}_{n \geq 0}$, where $\ell_{n+1} = n/(2(n+1))$, $n \geq 0$ and its maximal parameter sequence $\{M_{n+1}\}_{n \geq 0}$ is such that $M_{n+1} = 1/2$, $n \geq 0$.
Again from the theory of difference equations it is easily verified that
\[
P_{n}(x) = i \big(\frac{x-i}{2}\big)^{n+1} - i \big(\frac{x+i}{2}\big)^{n+1}, \quad n \geq 0.
\]
Hence, from \eqref{Eq-Rationals-1} we have
\[
u_{n}(x) = (-1)^n\frac{1}{2}i(x-i)\big[1 - \frac{(x+i)^{n+1}}{(x-i)^{n+1}}\big], \ \ n \geq 0.
\]
From this we successively find from the results given in Section \ref{Sec-EVP-to-Orthogonality} that
\[
\hat{u}_{n}(x) = (-1)^n\frac{\sqrt{\ell_{n+1}}}{2} i (x-i) \Big[1 - \frac{n+1}{n}- \frac{(x+i)^{n+1}}{(x-i)^{n+1}} + \frac{n+1}{n} \frac{(x+i)^{n}}{(x-i)^{n}} \Big], \ \ \ n \geq 1,
\]
\[
R_{n}(z) = \frac{z^{n+1} - 1}{z - 1}, \ \ n \geq 0,
\]
\[
\hat{R}_{n}(z) = \frac{1}{n(z-1)}\big[n (z^{n+1}-1) - (n+1) (z^{n} -1 ) \big], \ \ \ n \geq 1
\]
and
\[
\Phi_{n-1}(z) = \frac{1}{n(z-1)^2}\big[n (z^{n+1}-1) - (n+1) (z^{n} -1 ) \big], \ \ \ n \geq 1.
\]
The corresponding probability measure $\mu$ is found to be
\[
d \mu(\zeta) = \frac{(1-\zeta)(\zeta -1)}{4\pi i \zeta^2} d \zeta.
\]
Now we derive the polynomials $R_n(s;z)$ and their orthogonality property given by Theorem \ref{Thm-BRS2016-2}. Again, many information regarding the orthogonal polynomials on the unit circle associated with this measure are well known.
For example, the associated Verblunsky coefficients are
\[
\alpha_{n-1} = - \frac{1}{n+1}, \quad n \geq 1.
\]
Clearly the integral $\int_{\mathbb{T}}|\zeta - 1|^{-2} d \mu(\zeta)$ exists and that one can also easily verify that
\[
I(\mu) = \int_{\mathcal{C}} \frac{\zeta}{\zeta - 1} d \mu(\zeta) = \frac{1}{2}.
\]
We also easily verify by induction that the sequence $\{\tau_{n}(s)\}$ generated by
\[
\displaystyle \tau_{1}(s)= \frac{I+is}{\overline{I} -is}\quad \mbox{and} \quad \tau_{n+1}(s) = \frac{\tau_{n}(s) + \overline{\alpha_{n-1}}}{1 + \tau_{n}(s) \alpha_{n-1}}, \quad n \geq 1,
\]
can be explicitly given by (see \cite{BracRangaSwami-2016})
\[
\tau_{n+1}(s) = \frac{1 + i (n+1)(n+2)s}{1 - i (n+1)(n+2)s}, \quad n\geq 0.
\]
With this we obtain
\[
c_n(s) = - \frac{2ns}{1 + (n^2-1)(n+1)^2 s^2} \quad \mbox{and} \quad d_{n+1}(s) = [1-\ell_{n}(s)]\ell_{n+1}(s),
\]
for $n \geq 1$, where
\[
\ell_{n+1}(s) = \frac{n}{2(n+1)} \frac{1 + (n+1)^2 (n+2)^2 s^2}{1 + n (n+1)^2 (n+2)s^2}, \quad n \geq 0.
\]
Since,
\[
1 - \ell_{n+1}(s) = \frac{n+2}{2(n+1)} \frac{1 + n^2 (n+1)^2 s^2}{1 + n (n+1)^2 (n+2)s^2}, \quad n \geq 1,
\]
we have
\[
\sum_{n =1}^{\infty} \prod_{k=1}^{n} \frac{\ell_{k+1}(s)}{1 - \ell_{k+1}(s)} = \frac{2}{1 + 4s^2}\sum_{n =1}^{\infty} \Big[\frac{1}{(n+1)(n+2)} + (n+1)(n+2) s^2 \Big].
\]
The above series converges if $s = 0$ and diverges otherwise. Hence, by Wall's criteria the positive chain sequence $\{d_{n+1}(0)\}$ is a multiple parameter positive chain sequence and $\{d_{n+1}(s)\}$ for $s \neq 0$ is a single parameter positive chain sequence.
Now if the sequence of polynomials $\{P_{n}(s;x)\}$ are given by $P_{0}(s; x) = 1$, $P_{1}(s; x) = x - c_1(s)$ and
\[
P_{n+1}(s; x) = [x - c_{n+1}(s)] P_{n}(s;x) - d_{n+1}(s) (x^2 + 1) P_{n-1}(s;x), \quad n \geq 1.
\]
then from Theorem \ref{Thm-PV-Orthonality-M}
\[
d \psi(x) = \frac{2}{\pi} \frac{1}{(x^2+1)^2} dx, \qquad \int_{-\infty}^{\infty} x\, d \psi(x) = 0
\]
and
\[
\frac{2}{\pi} \int_{-\infty}^{\infty} x^{k} \, \frac{P_{n}(s;x)}{(x^2+1)^{n}} \, \frac{1}{x^2+1} d x \ = - \ c_{1}(s) \mathfrak{p}_{n}(s)\, \delta_{n-1,k}, \quad 0 \leq k \leq n-1,
\] \\[0ex]
where $\mathfrak{p}_n(s) = (1-\ell_{1}(s)) \cdots(1-\ell_{n}(s))$.
Since $\{d_{n+1}(0)\}_{n \geq 1} = \{1/4\}_{n \geq 1}$ is a multiple parameter positive chain sequence, from Theorem \ref{Thm-Special-R2Type-MF} we have for the polynomials $\{P_n\} = \{P_n(0;.)\}$ given by \eqref{Eq-Special-R2Type-RR} and \eqref{Eq-Example3}
\[
\frac{2}{\pi} \int_{-\infty}^{\infty} x^{k} \, \frac{P_{n}(x)}{(x^2+1)^{n}} \, \frac{1}{x^2+1} d x = \frac{1}{2^n} \delta_{n,k}, \quad k=0,1, \ldots, n.
\]
\vspace{4ex}
\noindent {\bf Example 4}. Here we consider the $R_{II}$ type recurrence \eqref{Eq-Special-R2Type-RR} with $\{c_n\}_{n\geq 1}$ and $\{d_{n+1}\}_{n\geq 1}$ given by
\begin{equation} \label{Eq-Example4}
\begin{array}l
\displaystyle c_n = \frac{\eta}{\lambda+n}, \quad d_{n+1} = \frac{1}{4} \frac{n(2\lambda+n+1)}{(\lambda+n)(\lambda+n+1)}, \ \ n\geq 1,
\end{array}
\end{equation}
where \ $\lambda, \eta \in \mathbb{R}$ and $\lambda > -1$.
Observe that for the minimal parameter sequence of the positive chain sequence $\{d_{n+1}\}_{n\geq 1}$ we have
\begin{equation*} \label{Eq-Special-ParamSeq}
\quad \ell_{n+1} = \frac{n}{2(\lambda+n+1)}, \quad \ n \geq 0.
\end{equation*}
The sequence $\{d_{n+1}\}_{n=1}^{\infty}$ is a single parameter chain sequence for $-1/2 \geq \lambda > -1$ and it is a multiple parameter sequence otherwise. When $\lambda > -1/2$ the maximal parameter sequence $\{M_{n+1}\}_{n=0}^{\infty}$ of $\{d_{n+1}\}_{n=1}^{\infty}$ is such that
\[
M_{n+1} = \frac{1}{2}\frac{2\lambda +n +1}{\lambda+n+1}, \quad n \geq 0.
\]
The polynomials $R_n$ obtained from the above sequences $\{c_n\}$ and $\{d_{n+1}\}$, together with the recurrence formula \eqref{Eq-TTRR-R} have already been studied in \cite{BracRangaSwami-2016}, and we have
\begin{equation*} \label{Eq-HypergeometricRn}
R_n(z) = \frac{(2\lambda+2)_n}{(\lambda+1)_n} \, _2F_1(-n,b+1;\,b+\overline{b}+2;\,1-z), \quad n \geq 1,
\end{equation*}
where $b = \lambda + i \eta$. Here, $ _2F_1(-n,b;\,c;\,z)$ represents a Gaussian hypergeometric polynomial of degree $n$ in $z$. More about the properties of such polynomials and about general Gaussian hypergeometric functions $ _2F_1(a,b;\,c;\,z)$, we cite \cite{Andrews-Book}.
From \eqref{Eq-P-to-R} we then have
\[
P_{n}(x) = \frac{(2\lambda+2)_n}{(\lambda+1)_n} \frac{(x-i)^n}{2^n}\, _2F_1(-n,b+1;\,b+\overline{b}+2;\,-2i/(x-i)), \quad n \geq 1.
\]
The nontrivial probability measure $\mu$ that follows from Theorem \ref{Thm-Orthogonality1-UC} is found to be
\[
d \mu(e^{i\theta})= \frac{2^{b+\overline{b}+2}|\Gamma(b+2)|^2 }{2\pi\, \Gamma(b+\overline{b}+3)} e^{(\pi-\theta)\Im(b)} [\sin^{2}(\theta/2)]^{\Re(b)+1} d\theta . \qquad
\]
The associated monic orthogonal polynomials on the unit circle and Verblunsky coefficients are (see \cite{SR-PAMS2010})
\[
\begin{array}l
\displaystyle \Phi_n(z) = \frac{(b+\overline{b}+3)_n}{(b+2)_n} \, _2F_1(-n,b+2;\,b+\overline{b}+3;\,1-z), \\[3ex]
\displaystyle \alpha_{n-1} = - \frac{(b+1)_{n}}{(\overline{b}+2)_{n}},
\end{array} n \geq 1.
\]
When $\lambda > -1/2$ the sequence $\{d_{n+1}\}_{n\geq 1}$ is a multiple parameter positive chain sequence and its maximal parameter sequence $\{M_{n+1}\}_{n\geq 0}$ is such that
\[
M_{n+1} = \frac{1}{2}\, \frac{2\lambda+n +1}{\lambda+n+1}, \quad n \geq 0.
\]
Hence, for $\lambda > -1/2$ we can consider the probability measure
\[
d \nu(e^{i\theta}) = \mathop{\rm const} \frac{1}{|\zeta-1|^2} d \mu(e^{i\theta}) = \frac{2^{b+\overline{b}}|\Gamma(b+1)|^2 }{2\pi\, \Gamma(b+\overline{b}+1)} e^{(\pi-\theta)\Im(b)} [\sin^{2}(\theta/2)]^{\Re(b)} d\theta,
\]
and from Theorem \ref{Thm-Special-R2Type-MF} the polynomials $P_{n}$ given by \eqref{Eq-Special-R2Type-RR} and \eqref{Eq-Example4} satisfy the orthogonality
\[
\int_{-\infty}^{\infty} x^{j} \, \frac{P_{n}(x)}{(x^2+1)^{n}} \, d \varphi(x) = \gamma_n \delta_{n,k}, \quad j=0,1, \ldots, n,
\]
where
\[
d \varphi(x) = - d \nu\big(\frac{x+i}{x-i}\big) = \frac{2^{b+\overline{b}+1}|\Gamma(b+1)|^2 }{2\pi\, \Gamma(b+\overline{b}+1)}\, e^{[\pi -2\,arccot(x)]\eta} (x^2 + 1)^{-\lambda-1} dx,
\]
$\gamma_{0} = \int_{-\infty}^{\infty} d\varphi(x) = \int_{\mathbb{T}} d \nu(\zeta) = 1$ and $\gamma_{n} = (1- M_{n})\gamma_{n-1}$, $n \geq 1$.
\vspace{2ex}
|
train/arxiv
|
BkiUdl7xaJJQnMsb0DKf
| 5 | 1 |
\section{Introduction}
Since it was suggested by 't~Hooft \cite{tH76} and Mandelstam \cite{Ma76} that monopole condensation in nonabelian gauge theories may be an explanation for the confinement mechanism, many studies have been devoted to the magnetic properties of these theories \cite{KT98,Co98,En98,DGT82,BLS81}.
In the present study, we probe the vacuum structure of pure SU(2) gauge theory by inserting a static monopole-antimonopole pair into the vacuum and measuring its free energy at different separations and different temperatures. While in the classical theory there is a Coulomb potential between SU(2) monopoles, we expect the quantum theory to show a Yukawa-potential if there is a monopole condensate, and a Coulomb-potential, if quantum fluctuations do not produce a magnetically screening object.
\section{Simulation method}
The method we use for inserting a static SU(2) monopole pair on the lattice was put forward by 't~Hooft and others \cite{tH78,MP79,MP80,Ya79,UWG80,SS81} and essentially amounts to introducing a magnetic flux tube by multiplying the couplings with an element of the center of the gauge group ($\beta\rightarrow -\beta$ for SU(2)) on a string of plaquettes in every timeslice (schematically shown in fig.\ref{chain}).
\begin{figure}
\begin{center}
\epsfig{file=monpic.eps,width=7cm}
\end{center}
\caption{\label{chain}Monopoles sit at the end of a magnetic flux tube}
\end{figure}
This tube carries a flux of $\Phi=\pi\sqrt{\beta}$, so at each end of the flux tube is a elementary cube with net magnetic outflux $\Phi=\pm\pi\sqrt{\beta}$. But since for a compact lattice theory magnetic flux is only defined modulo $2\pi\sqrt{\beta}$, both monopoles are indistinguishable and coincide with their own antimonopole.
In order to measure the free energy of a monopole pair, we split the SU(2) action into two parts with different couplings
\begin{equation}
\label{act}
S(\beta,{\beta^\prime})=\frac{1}{2}\left(\beta\sum_{P\not\in M}\mbox{Tr}(U_P)+{\beta^\prime}\sum_{P\in M}\mbox{Tr}(U_P)\right)
\end{equation}
where $M$ is the set of all the plaquettes in the monopole string. In the case where ${\beta^\prime}=\beta$, (\ref{act}) is the action of a plain SU(2) theory and for ${\beta^\prime}=-\beta$ it is the action of SU(2) with the static monopole pair. So the free energy difference upon a monopole pair insertion is
\begin{equation}
\label{freen}
\Delta F=-T\ln\frac{Z(\beta,-\beta)}{Z(\beta,\beta)}
\end{equation}
where $T=1/N_t a$ is the temperature of the system and $Z(\beta,{\beta^\prime})$ is the partition function
\begin{equation}
Z(\beta,{\beta^\prime})=\sum_{\mathcal C} e^{-S(\beta,{\beta^\prime})}
\end{equation}
We can calculate $\Delta F$ using the Ferrenberg-Swendsen multihistogram method \cite{FS} on histograms of the flux tube energy
\begin{equation}
E^{\prime}=\frac{1}{2}\langle\sum_{P\in M}\mbox{Tr}(U_P)\rangle_{\beta^\prime}
\end{equation}
for a set of ${\beta^\prime}$'s between $-\beta$ and $\beta$ (where the separation between successive ${\beta^\prime}$'s has to be small enough, that neighboring histograms have a substantial overlap).
Our simulations were performed with couplings between $\beta=2.476$ and $\beta=2.82$ on systems of volumes, ranging from $N_x\times N_y\times N_z=16^2\times 32$ to $N_x\times N_y\times N_z=32^2\times 64$. The time extent in our simulation varied between $N_t=2$ and $N_t=16$. For each system, we measured the free energy at monopole separations from $a$ to $6a$.
We used a combined overrelaxation and 3-hit Metropolis update algorithm. For every simulation, we started at ${\beta^\prime}=\beta$ with $5000$ thermalization updates followed by $200$ to $800$ independent measurements of $E^{\prime}$, separated by $50$ updates. We then decreased ${\beta^\prime}$ in $11$ to $61$ steps to ${\beta^\prime}=-\beta$ and performed $500$ thermalization updates followed again by the measurement updates. Here, a single $E^{\prime}$ measurement consisted of an average over $384$ configurations of the plaquettes on the monopole string on a fixed background configuration.
\section{Results}
\begin{figure}
\epsfig{file=fmass.eps,width=7cm}
\caption{\label{fmass}Free energy of the monopole-antimonopole pair vs.~distance in lattice units at $\beta=2.6$ for different values of $N_t$. The lines represent fits to a Yukawa potential.}
\end{figure}
In fig. \ref{fmass} we plot the free energy vs.~monopole pair separation for a $N_x\times N_y\times N_z=20^2\times 40$ system at $\beta=2.6$ and different temperatures. The lines are fits to a Yukawa-potential
\begin{equation}
F(r)=F_0-c\frac{e^{-mr}}{r}
\end{equation}
\begin{table}
\begin{tabular}{r|l|l|l|l}
$N_t$ & $T/\sqrt{\sigma}$ & $m/\sqrt{\sigma}$ & $ma$ & Q\\
\hline
2 & 3.676 & 17.3(4.1) & 2.29(55) & 0.41 \\
4 & 1.838 & 5.43(59) & 0.720(78) & 0.03 \\
6 & 1.225 & 4.13(41) & 0.548(54) & 0.02 \\
16 & 0.460 & 3.24(42) & 0.430(56) & 0.96
\end{tabular}
\caption{\label{masstab}Screening masses for $\beta=2.6$ in units of the string tension at zero temperature $\sqrt{\sigma}$ and in lattice units. For comparison, the temperature of the system is given. Q is the fit quality.}
\end{table}
\begin{table}
\begin{tabular}{r|l|l|l}
$\beta$ & $m_g a$ & $ma$ & $m_ra$ \\
\hline
2.5 & 0.66(2) & 0.46(8) & 0.64(20) \\
2.6 & 0.51(3) & 0.43(6) & 0.53(12)
\end{tabular}
\caption{\label{gb}Comparison of the screening masses ($m,m_r$) in the confined phase to the lightest glueball mass ($m_g$, taken from \cite{E98}). $m_r$ is the screening mass from a restricted fit ignoring the data point at monopole separation $a$.}
\end{table}
Table \ref{masstab} shows the screening masses obtained from this fit. A Coulombic behavior ($m=0$) is clearly ruled out. The quality of a Coulomb fit is $Q\approx 10^{-7}$ for $N_t=16$ and even worse ($Q<10^{-15}$) for all other cases.
At low temperature, we can compare our screening masses to the known mass of the lightest glueball state (table \ref{gb}). The masses are roughly in agreement, especially when one ignores the data point at separation $a$, which is most affected by discretization errors.
\begin{figure}
\epsfig{file=hit.eps,width=7cm}
\caption{\label{fhit} Plot of the screening mass vs.~temperature. The last data point for $\beta=2.6$ is from a simulation with $N_t=2$ and should be taken with caution. The object acquires a thermal mass and there is no indication for the deconfinement transition in this observable. The dashed line indicates the critical temperature (taken from \cite{FHK92}).}
\end{figure}
In fig. \ref{fhit} we plot the screening masses vs.~temperature. There clearly is an increase of the screening mass with temperature, but we can see no signal of the phase transition.
\section{Conclusion}
We have studied the free energy of a monopole pair in pure SU(2) gauge theory at finite temperature. We find, that in both phases it exhibits a screened behavior. At low temperature, the screening mass is roughly in agreement with the mass of the lightest glueball state. At high temperature, we observe an increase in the screening mass with no apparent discontinuity at the phase transition.
|
train/arxiv
|
BkiUeGY5qX_Bxod3mRkY
| 4 | 0.8 |
\section{Introduction} \label{sec:intro}
The incompressible Navier-Stokes equations on the Euclidean space $\mathbb{R}^d$,
with $d \in \mathbb{Z}^+$, are the key governing equations of viscous fluid flows
with a divergence-free fluid velocity vector field occupying all the
$\mathbb{R}^d$ space, which are given by
\begin{equation*} \label{Eq:NS}
\tag{NS}
\left\{
\begin{array}{l}
\partial_t {\textbf{u}} = \Delta {\textbf{u}} - ({\textbf{u}} \cdot \nabla)
{\textbf{u}}- \nabla p +\nabla \cdot \mathbb{F}, \\ \\
\nabla \cdot {\textbf{u}} =0,
\end{array}
\right.
\end{equation*}
where ${\textbf{u}} := (u_1, \ldots ,u_d)$ is the fluid velocity, $p$ is the fluid
pressure, $\mathbb{F}:= (F_{i,j})_{1\le i,j\le d}$ is the forcing tensor,
$\displaystyle \nabla \cdot \mathbb{F}:= \left (\sum_{i=1}^d \partial_i F_{i,1},
\ldots, \sum_{i=1}^d \partial_i F_{i,d} \right)$, and
$\displaystyle \nabla \cdot \textbf{u}:= \sum_{i=1}^d \partial_i u_i$.
We now introduce several notations, preliminary results and definitions
that will be helpful in the sequel. For all
$y := (y_1, \ldots, y_d) \in \mathbb{R}^d$, we write
$|y|:= \sqrt{|y_1|^2 + \cdots + |y_d|^2}$. For any $d \in \mathbb{Z}^+$,
and for every $\gamma > 0$, we denote by $w_{\gamma}:=w_{d, \gamma}$,
the weight function, provided by
\begin{align*}
w_{\gamma} \colon \mathbb{R}^d &\longrightarrow \mathbb{R} \\
x &\longmapsto w_\gamma (x):= \frac{1}{(1+|x|)^{\gamma}}.
\end{align*}
For all $d \in \mathbb{Z}^+$, for every $\delta > 0$, and for each
$1\le p <+\infty$, we use the notation $L^p_{w_\delta}(\mathbb{R}^d)$
to represent the weighted $L^p$-space given by
$L^p(\mathbb{R}^d, w_\delta (x) \, dx)$. Moreover, for all
$d \in \mathbb{Z}^+$, and for any $j \in \{1, \ldots, d \}$,
$\mathcal{R}_j:=\mathcal{R}_{d, j}$ denotes the $j$-th \emph{Riesz transform},
given by
\begin{equation*}
\mathcal{R}_j := \frac{\partial_j}{\sqrt{-\Delta}}.
\end{equation*}
Let ${\textbf{b}}$ and ${\textbf{u}}$ be two vector fields on
$\mathbb{R}^d$. The \textit{tensor product},
${\textbf{b}} \otimes {\textbf{u}}$, of ${\textbf{b}}$ and
${\textbf{u}}$, is defined as the $d \times d$-matrix, given,
for all $i, j \in \{1, \ldots, d\}$, by
$({\textbf{b}} \otimes {\textbf{u}})_{i, j} := b_i u_j$. Thus,
if $\nabla \cdot {\textbf{b}} = 0$, we have that
$({\textbf{b}} \cdot \nabla) {\textbf{u}}
= \nabla \cdot ({\textbf{b}}\otimes {\textbf{u}})$.
Recently, P.G. Fern\'andez-Dalgo and P.G. Lemarié-Rieusset
(\hspace{1sp}\cite{PF_PG2}) gave a general characterization for the pressure
term in the incompressible Navier-Stokes equations (\ref{Eq:NS}) on the whole
Euclidean space $\mathbb{R}^d$, with $d \in \{2,3\}$. In \cite{PF_PG2}, the
authors consider velocities $\textbf{u} = (u_1,...,u_d)$ belonging to
$L^2 ((0,T), L^2_{w_\gamma}(\mathbb{R}^{d}))$, with $\gamma \in \{ d , d+1 \}$.
In addition, if $\textbf{u} \in L^2 ((0,T), L^2_{w_d}(\mathbb{R}^d))$, and
$\mathbb{F} (t,x) := \left(F_{i,j}(t,x)\right)_{1\leq i,j\leq d}$
belongs to $L^1\left((0,T), L^1_{w_{d}}(\mathbb{R}^d)\right)$,
it was also shown in \cite{PF_PG2} that $p$ is given, up to constants, by
\begin{equation*}
p=\sum_{i,j=1}^d(\varphi \, \partial_i \partial_j G_d) * (u_iu_j-F_{i,j}) +
\sum_{i,j=1}^d((1-\varphi) \, \partial_i \partial_j G_d) * (u_iu_j-F_{i,j}),
\end{equation*}
where $\varphi \in C_0^{\infty}(\mathbb{R}^d) :=
\{ \psi:\mathbb{R}^d \to \mathbb{C}; \; \psi \in C^{\infty}(\mathbb{R}^d)
\text{ and } \psi \text{ has compact support} \}$
is a real-valued function such that $\varphi = 1$ on
a neighborhood of $0$, $*$ stands for the \textit{convolution product},
and $G_d$ is the \textit{fundamental solution of the Laplace equation} on
$\mathbb{R}^d$, i.e., $-\Delta G_d=\delta$. So, for all
$x \in \mathbb{R}^d \smallsetminus \{0\}$, with $d \in \{2, 3\}$,
\begin{equation*}
G_2(x) = \frac 1{2\pi}\ln \left(\frac 1{\vert x\vert}\right)
\quad \text{ and } \quad G_3(x) = \frac 1{4\pi\vert x\vert}.
\end{equation*}
Furthermore, in the more favorable case, when
$\textbf{u} \in L^\infty ((0,T), L^2_{w_\gamma}(\mathbb{R}^3))$,
$\nabla {\textbf{u}} \in L^2((0,T), L^2_{w_\gamma}(\mathbb{R}^3))$, and
$\mathbb{F} \in L^2((0,T), L^2 _{w_{\gamma} }(\mathbb{R}^3))$,
with $0 < \gamma < \frac{5}{2}$, the authors in \cite{PF_PG} showed
that $p$ is determined, up to constants, by the more simple formula
\begin{equation*}
p = \sum_{i,j=1}^3 \mathcal{R} _i \mathcal{R} _j (u_i u_j - F_{i,j}).
\end{equation*}
The following definition concerns the \textit{Muckenhoupt class} of
weights.
\begin{Definition}
Let $d \in \mathbb{Z}^+$, and $1<p<+\infty$. A nonnegative real-valued
measurable function, $w :\mathbb{R}^d \to [0, + \infty)$, belongs to the
\emph{Muckenhoupt} $\mathcal{A}_p(\mathbb{R}^d)$ \emph{class} if it is
locally integrable and it satisfies the \emph{reverse H\"older inequality},
with conjugate exponents $p$ and $q:= \frac{p}{p-1}$, given by
\begin{equation} \label{Eq:muckenhoupt}
\sup_{x\in\mathbb{R}^d, R>0}
\left(\frac 1{\vert B(x,R)\vert} \int_{B(x,R)}
\! \! \! \! \! w(y)\, dy \right)^{\frac 1 p}
\cdot
\left( \frac 1{\vert B(x,R)\vert} \int_{B(x,R)}
\! \! \! \! \! w(y)^{-\frac 1{p-1}} dy
\right)^{1-\frac 1 p} \!\!<+\infty.
\end{equation}
\end{Definition}
The next result is a $d$-dimensional generalization of Lemma 1 in
\cite{PF_PG}, where it was assumed that $d=3$.
\begin{Lemma}[Muckenhoupt weights] \label{Lemma:Muck}
Let $d \in \mathbb{Z}^+$, $0<\delta<d$, and $1<p<+\infty$. Then,
$w_\delta$ belongs to the Muckenhoupt $\mathcal{A}_p(\mathbb{R}^d)$
class.
\end{Lemma}
\begin{proof}
We suppose that $d \in \mathbb{Z}^+$, $0<\delta<d$, and $1<p<+\infty$.
Let $x\in \mathbb{R}^d$. Let $y \in \mathbb{R}^d$ be such that
$\vert y-x\vert<R$. \\
{\bf{(i)}} First, we assume that $R\in(0, 1]$. Then,
$1+\vert y\vert \le 1 + \vert x-y \vert + \vert x \vert
< 1 + R + \vert x \vert \le 2 + \vert x \vert \le 2(1 + \vert x \vert)$.
Proceeding similarly as in the last expression, we get
$1+ \vert x \vert < 2(1 + \vert y \vert)$. Hence,
\begin{equation} \label{Eq:Rsmall}
\frac 1 2 (1+\vert x\vert) < 1+\vert y\vert < 2 (1+\vert x\vert).
\end{equation}
It follows from (\ref{Eq:Rsmall}) that
\begin{align*}
& \left(\frac 1{\vert B(x,R)\vert}
\int_{B(x,R)} w_{\delta}(y)\, dy\right)^{\frac 1 p}
\cdot \left( \frac 1{\vert B(x,R)\vert}
\int_{B(x,R)} w_{\delta}(y)^{-\frac 1{p-1}} \, dy \right)^{1-\frac 1 p} \\
\le & \left( \frac{2^{\delta}}{(1+ \vert x \vert)^{\delta}} \right)^{\frac{1}{p}}
\cdot \left( 2^{\frac{\delta}{p-1}} \cdot
(1+\vert x \vert)^{\frac{\delta}{p-1}} \right)^{\frac{p-1}{p}}
= 4^{\frac{\delta}{p}}.
\end{align*}
\noindent
{\bf{(ii)}} We now pretend that $R > 1$.
\begin{itemize}
\item We first consider the case when
$\vert x \vert > 10R$. Then,
$1+\vert y \vert \le 1 + \vert x-y \vert + \vert x \vert
< 1 + R + \vert x \vert < 1 + \left( \frac{1}{10} + 1 \right) \vert x \vert
\le \frac{11}{10}(1 + \vert x \vert)$. Moreover,
$1+ \vert x \vert \le 1 + \vert x-y \vert + \vert y \vert
< 1 + R + \vert y \vert < 1 + \frac{1}{10} \vert x \vert + \vert y \vert$.
Thus, $\frac{9}{10} \left( 1 + \vert x \vert \right)
< 1 + \frac{9}{10} \vert x\vert < 1 + \vert y \vert$. So,
\begin{equation} \label{Eq:Rbig-xbig}
\frac{9}{10} \left( 1 + \vert x \vert \right) < 1 + \vert y \vert
< \frac{11}{10} \, (1 + \vert x \vert).
\end{equation}
By using (\ref{Eq:Rbig-xbig}), we get
\begin{align*}
& \left(\frac 1{\vert B(x,R)\vert}
\int_{B(x,R)} w_{\delta}(y)\, dy\right)^{\frac 1 p}
\cdot \left( \frac 1{\vert B(x,R)\vert}
\int_{B(x,R)} w_{\delta}(y)^{-\frac 1{p-1}} \, dy \right)^{1-\frac 1 p} \\
\le & \left[ \left(\frac{10}{9} \right)^{\delta}
\cdot \frac{1}{(1+ \vert x \vert)^{\delta}} \right]^{\frac{1}{p}}
\cdot \left[ \left( \frac{11}{10} \right)^{\frac{\delta}{p-1}}
\cdot (1+\vert x \vert)^{\frac{\delta}{p-1}} \right]^{\frac{p-1}{p}}
= \left( \frac{11}{9} \right)^{\frac{\delta}{p}}.
\end{align*}
\item Finally, we presume that $\vert x\vert \le 10 R$. Since
$|B(0,R)| = |B(x,R)|$, and $B(x,R ) \subset B(0,11R)$, we obtain that
\begin{align*}
& \left(\frac 1{\vert B(x,R)\vert}
\int_{B(x,R)} w_{\delta}(y)\, dy\right)^{\frac 1 p}
\cdot \left( \frac 1{\vert B(x,R)\vert}
\int_{B(x,R)} w_{\delta}(y)^{-\frac 1{p-1}} \, dy \right)^{1-\frac 1 p} \\
\le & \left(\frac {1}{\vert B(0,R)\vert}
\int_{B(0,11 R)} w_\delta (y)\, dy\right)^{\frac 1 p}
\cdot \left( \frac {1}{\vert B(0,R)\vert}
\int_{B(0,11 R)} w_\delta (y)^{-\frac 1{p-1}} \, dy\right)^{1-\frac 1 p} \\
= & \left(\frac 1 { R^d} \int_0^{11\, R} \frac{1}{(1+r)^\delta}
\cdot r^{d-1} \, dr\right)^{\frac 1 p}
\cdot \left(\frac{1}{R^d} \int_0^{11R} (1+r)^{\frac{\delta}{p-1}}
\cdot r^{d-1} \, dr\right)^{1-\frac{1}{p}} \\
\le & \; c_{\delta, p} \left(\frac 1 {R^d} \int_0^{11R}
\frac{1}{r^\delta} \cdot r^{d-1} \, dr \right)^{\frac{1}{p}} \\
& \cdot \left[\left(\frac{1}{R^d} \int_0^{11R} r^{d-1} \,dr \right)^{1-\frac{1}{p}}
+ \left( \frac 1 {R^d} \int_0^{11R}
r^{\frac{\delta}{p-1}+d-1} \, dr\right)^{1-\frac{1}{p}} \right]
\end{align*}
\begin{align*}
= & \; c_{\delta, p} \cdot \left( \frac{11^{d-\delta}}
{R^{\delta} \cdot (d-\delta)} \right)^{\frac{1}{p}} \cdot
\left[ \left( \frac{11^d}{d} \right)^{1-\frac{1}{p}}
+ \left( \frac{11^{\frac{\delta}{p-1}+d} \cdot R^{\frac{\delta}{p-1}}}
{\frac{\delta}{p-1} +d} \right)^{1-\frac{1}{p}} \right] \\
= & \; c_{\delta, p} \cdot \frac{11^d}{(d-\delta)^{\frac{1}{p}}} \cdot
\left( \frac{(11R)^{-\frac{\delta}{p}}}{d^{1-\frac{1}{p}}}
+ \frac{1}{\left( \frac{\delta}{p-1} + d \right)^{1-\frac{1}{p}}} \right) \\
\le & \; c_{\delta, p} \cdot \frac{11^d}{(d-\delta)^{\frac{1}{p}}} \cdot
\left( \frac{1}{d^{1-\frac{1}{p}}}
+ \frac{1}{\left( \frac{\delta}{p-1} + d \right)^{1-\frac{1}{p}}} \right).
\end{align*}
\end{itemize}
From {\bf{(i)}} and {\bf{(ii)}}, the lemma follows.
\end{proof}
The following result will be used in the proof of Theorem~\ref{Theo:pr}.
\begin{Corollary} \label{Cor:operonb}
Let $d \in \mathbb{Z}^+$, $0<\delta<d$, and $1<p<+\infty$. Then, the
following assertions hold.
\begin{itemize}
\item For all $j \in \{1, \ldots, d \}$, the $j$-th \emph{Riesz transform},
$\mathcal{R}_j := \frac{\partial_j}{\sqrt{-\Delta}}$, is a bounded linear
operator on $L^p_{w_\delta}(\mathbb{R}^d)$, i.e., there exists a constant
$C:=C_{d, \delta, p} > 0$ such that for all $f \in L^p_{w_\delta}(\mathbb{R}^d)$,
\begin{equation*}
\|\mathcal{R}_j f\|_{L^p_{w_\delta}(\mathbb{R}^d)}
\le \, C \, \|f\|_{L^p_{w_\delta}(\mathbb{R}^d)}.
\end{equation*}
\item The \emph{Hardy-Littlewood maximal function operator} is a
bounded nonlinear operator on $L^p_{w_\delta}(\mathbb{R}^d)$, i.e.,
there is a constant $K:=K_{d, \delta, p} > 0$ such that for all
$g \in L^p_{w_\delta}(\mathbb{R}^d)$,
\begin{equation*}
\|\mathcal{M} g\|_{L^p_{w_\delta}(\mathbb{R}^d)}
\le \, K \, \|g\|_{L^p_{w_\delta}(\mathbb{R}^d)}.
\end{equation*}
\end{itemize}
\end{Corollary}
\begin{proof}
These are properties of the Muckenhoupt $\mathcal{A}_p(\mathbb{R}^d)$ class
(we refer to \cite{Gr09}).
\end{proof}
The main result of this paper and its proof are presented in the next
section. It is an improvement of Corollary 2 in \cite{PF_PG}, where
it was given a characterization of the pressure term, $\nabla p$, in the
incompressible 3D Navier-Stokes equations (\ref{Eq:NS}) on the whole
Euclidean space $\mathbb{R}^3$. More specifically, Theorem~\ref{Theo:pr}
below considers the dimension of the Euclidean space $d\in \{2,3\}$,
while Corollary 2 in \cite{PF_PG} refers only to the case of dimension 3.
Moreover, in the hypotheses of Theorem~\ref{Theo:pr}, for $d=3$,
we suppose that $\gamma \in (0, 3)$, where $\gamma$ is the parameter of
the weight function, $w_\gamma$, whereas Corollary 2 in \cite{PF_PG}
concerns the case $\gamma \in (0,5/2)$. It deserves to remark that
the hypotheses of Theorem~\ref{Theo:pr} imply the assumptions of
Theorem 1 in \cite{PF_PG2}; however, Theorem~\ref{Theo:pr} allows us
to manipulate the pressure term in the Navier-Stokes equations
more easily, by using the Riesz transforms, than employing the techniques
of Theorem 1 in \cite{PF_PG2}. Furthermore, we notice that in
a recent work, Z. Bradshaw and T.P. Tsai (\hspace{1sp}\cite{BTpr}) study
local expansions for the pressure term in the Navier-Stokes equations. To
end this section, we observe that some related results can also be found
in \cite{BTK}, \cite{CW18, PF_OJ}, and
\cite{JS14, KS07, LR99, LR02, LR16, Le34, VF, Wo17}.
\section{Determination of the pressure term in the incompressible
Navier-Stokes equations on the full Euclidean space}
The primary goal of this section is Theorem~\ref{Theo:pr} below. It can be
used to obtain a priori controls for solutions of the 2D incompressible
Navier-Stokes equations in weighted $L^2$-spaces, and to study existence,
uniqueness, and regularity of these solutions. We note that A. Basson
(\hspace{1sp}\cite{Ba06}) proved uniqueness of the solutions of the 2D
incompressible Navier-Stokes equations, for uniformly locally square
integrable initial data. It was also proposed in \cite{Ba06} the open
problem concerning the uniqueness of solutions of the 2D incompressible
Navier-Stokes equations, for initial data ${\textbf{u}}_0$ belonging to
the space $B_2(\mathbb{R}^2)$, where
\begin{equation*}
\| {\textbf{u}}_0 \|_{B_2(\mathbb{R}^2)} ^2 :=
\sup_{R \ge 1} \frac{1}{R^2} \int_{|y|\le R}
|{\textbf{u}}_0 (y)|^{2} \, dy < +\infty.
\end{equation*}
It deserves to remark that our estimates for the pressure term, given in
(\ref{Eq:2}) and (\ref{Eq:4}) below, may be used to study the problem regarding
the uniqueness of solutions of the 2D incompressible Navier-Stokes equations,
when the initial data belongs to the weighted $L^2$-space,
$L^2_{w_\gamma}(\mathbb{R}^2)$, with $0<\gamma<2$. With respect to the
aforementioned spaces, we observe that
(see Section 7 in \cite{PF_PG2})
\begin{equation*}
L^2_{w_{\gamma}}(\mathbb{R}^2) \subset B_{2}(\mathbb{R}^2)
\subset L^p_{w_{\delta}}(\mathbb{R}^2),
\end{equation*}
for $0 < \gamma \leq 2 < \delta$. Now, we go to the major theorem of this
manuscript.
\begin{Theorem} \label{Theo:pr}
Let $d \in \{ 2 , 3\}$, $0<\gamma<d$, and $0<T<+\infty$. We suppose
that $\mathbb{F}:=\left(F_{i,j}\right)_{1\le i,j\le d}
\in L^2((0,T), L^2_{w_{\gamma}}(\mathbb{R}^d))$. Let $\textup{\textbf{u}}$ be a
solution of the problem
\begin{equation} \label{Eq:NSpr}
\left\{
\begin{array}{l}
\partial_t \textup{\textbf{u}} = \Delta \textup{\textbf{u}}
- (\textup{\textbf{u}} \cdot \nabla) \textup{\textbf{u}}
- \nabla q + \nabla \cdot \mathbb{F}, \\ \\
\nabla \cdot \textup{\textbf{u}} = 0,
\end{array}
\right.
\end{equation}
such that $\textup{\textbf{u}} \in L^\infty((0,T), L^2_{w_\gamma}(\mathbb{R}^d))$,
and $\nabla \textup{\textbf{u}} \in L^2((0,T), L^2_{w_\gamma}(\mathbb{R}^d))$.
We also assume that the pressure, $q$, belongs to the space of distributions
$\mathcal{D}'((0,T) \times \mathbb{R}^d)$. Then, the gradient of the
pressure is given by the formula
\begin{equation*}
\nabla q = \nabla \left(\sum_{i,j=1}^d \mathcal{R}_i
\mathcal{R}_j(u_i u_j - F_{i,j}) \right).
\end{equation*}
If we, moreover, suppose that
$r \in \left(1, \min \{ \frac{d}{d-1}, \frac{d}{\gamma} \}\right)$,
and $a \in \mathbb{R}$ satisfies $\frac{2}{a} + \frac{d}{r} = d$, we get
\begin{equation*}
\sum_{i,j=1}^d \mathcal{R}_i \mathcal{R}_j (u_i u_j)
\in L^{a}((0,T),L^{r}_{w_{ r \gamma}}(\mathbb{R}^d)),
\end{equation*}
and
\begin{equation*}
\sum_{i,j=1}^d \mathcal{R}_i \mathcal{R}_j F_{i,j}
\in L^{2}((0,T),L^{2}_{w_\gamma}(\mathbb{R}^d)).
\end{equation*}
\end{Theorem}
\begin{proof}
We let
\begin{equation*}
p := \sum_{i,j=1}^d \mathcal{R}_i \mathcal{R}_j (u_i u_j - F_{i,j}).
\end{equation*}
We will show that $\nabla q - \nabla p = 0$, where $q$ is the pressure term
in the incompressible Navier-Stokes equations on $\mathbb{R}^d$ given in
(\ref{Eq:NSpr}). We begin by demonstrating that $p$ is well-defined. Let
$ 1 < r < \min \{ \frac{d}{d-1}, \frac{d}{\gamma} \}$. We take
$a \in \mathbb{R}$ such that $\frac{2}{a} + \frac{d}{r} = d$,
and $b \in \mathbb{R}$ such that $\frac{1}{r} = \frac{1}{2} + \frac{1}{b}$.
Thus, $\frac{2}{a} = d - \frac{d}{r} = \frac{d}{2} - \frac{d}{b}$, and
$0 < d - \frac{d}{r} < 1$. \\
As $\sqrt{w_\gamma} \, {\textbf{u}} \in L^\infty((0,T), L^2(\mathbb{R}^d))$,
and $\sqrt{w_\gamma} \, \nabla {\textbf{u}} \in L^2((0,T), L^2(\mathbb{R}^d))$,
by using the Gagliardo-Nirenberg interpolation inequality, we obtain that
$\sqrt{w_\gamma} \, {\textbf{u}} \in L^a((0,T), L^b(\mathbb{R}^d))$. In fact,
for almost all $t \in (0, T)$, we get
\begin{align*}
\| \sqrt{w_\gamma} \, {\textbf{u}}(t) \|_{L^b(\mathbb{R}^d)}
& \le C_{d, b} \; \| \nabla (\sqrt{w_\gamma} \, {\textbf{u}}(t)) \|_{L^2
(\mathbb{R}^d)}^{\frac{d}{2}-\frac{d}{b}} \; \cdot \;
\| \sqrt{w_\gamma} \, {\textbf{u}}(t)
\|_{L^2 (\mathbb{R}^d)}^{1-(\frac{d}{2}-\frac{d}{b})} \\
& \le \gamma \; C_{d, b}
\left(\| \sqrt{w_\gamma} \, {\textbf{u}}(t) \|_{L^2(\mathbb{R}^d)}
+ \| \sqrt{w_\gamma} \, \nabla {\textbf{u}}(t) \|_{L^2(\mathbb{R}^d)}
\right)^{\frac{d}{2}-\frac{d}{b}} \\
& \;\;\;\; \cdot \, \| \sqrt{w_\gamma} \, {\textbf{u}}(t)
\|_{L^2 (\mathbb{R}^d)}^{1-(\frac{d}{2}-\frac{d}{b})}.
\end{align*}
As $\frac{2}{a} = \frac{d}{2} - \frac{d}{b}$, and integrating with respect
to time, we see that
\begin{align} \label{Eq:0a}
\int_0^T \| \sqrt{w_\gamma} \, {\textbf{u}}(s) \|_{L^b(\mathbb{R}^d)}^a ds
& \le \gamma \; C_{d, b} \;
\| \sqrt{w_\gamma} \, {\textbf{u}} \|_{L^\infty((0,T), L^2(\mathbb{R}^d)) }^{a-2}
\nonumber \\
& \;\;\;\; \cdot \int_0^T ( \| \sqrt{w_\gamma} \, {\textbf{u}}(s) \|_{L^2(\mathbb{R}^d)}
+ \| \sqrt{w_\gamma} \, \nabla {\textbf{u}}(s) \|_{L^2(\mathbb{R}^d)})^2 ds \\
& < +\infty. \nonumber
\end{align}
Since $\sqrt{w_\gamma} \, {\textbf{u}} \in L^a((0,T), L^b(\mathbb{R}^d))$, and
using H\"older's inequality with indices $\frac{2}{r}$ and $\frac{b}{r}$,
we obtain that for all $i,j \in \{1, \ldots ,d\}$,
$w_\gamma \, u_i u_j \in L^a((0,T), L^r(\mathbb{R}^d))$, and
\begin{align} \label{Eq:0b}
\| u_i u_j \|_{L^{a}\left((0,T), L^{r}_{w_{r \gamma}}(\mathbb{R}^d)\right)}
& = \| \, \| \sqrt{w_\gamma} \, u_i (\cdot) \, \sqrt{w_\gamma} \,
u_j (\cdot) \|_{L^{r}(\mathbb{R}^d))} \, \|_{L^{a}(0,T) }
\nonumber \\
& \le \| \, \| \sqrt{w_\gamma} \, u_i (\cdot) \|_{L^{2}(\mathbb{R}^d))} \, \cdot \,
\| \sqrt{w_\gamma} \, u_j (\cdot) \|_{L^{b}(\mathbb{R}^d))} \, \|_{L^{a}(0,T) }
\nonumber \\
& \le \| \sqrt{w_\gamma} \, u_i \|_{L^{\infty} \left((0,T), L^{2}(\mathbb{R}^d) \right)}
\, \cdot \,
\| \sqrt{w_\gamma} \, u_j \|_{L^{a} \left((0,T), L^{b}(\mathbb{R}^d) \right)}.
\end{align}
\noindent
By the continuity of the Riesz transforms on $L^r_{w_{r \gamma}}(\mathbb{R}^d)$
(see Corollary~\ref{Cor:operonb} above), and employing the fact that
$0 < r \, \gamma < d$, we get
\begin{equation} \label{Eq:1}
\sum_{i,j=1}^d \mathcal{R}_i \mathcal{R}_j (u_i u_j)
\in L^{a}((0,T), L^{r}_{w_{r \gamma}}(\mathbb{R}^d)),
\end{equation}
and, more precisely, the following estimate holds
\begin{align} \label{Eq:2}
& \Big{\|} \, \sum_{i,j=1}^d \mathcal{R}_i \mathcal{R}_j (u_i u_j)
\, \Big{\|}_{L^{a}((0,T), L^{r}_{w_{r \gamma}}(\mathbb{R}^d))}
\nonumber \\
& \le C_{d, \gamma, r}
\sum_{i,j=1}^d \| u_i u_j \|_{L^{a}((0,T), L^{r}_{w_{r \gamma}}(\mathbb{R}^d))}
\nonumber \\
& \le C_{d, \gamma, r} \sum_{i,j=1}^d
\|\sqrt{w_\gamma} \, u_i \|_{L^{\infty} \left((0,T), L^{2}(\mathbb{R}^d) \right)}
\, \cdot \,
\| \sqrt{w_\gamma} \, u_j \|_{L^{a} \left((0,T), L^{b}(\mathbb{R}^d) \right)}
\nonumber \\
& \le \gamma^{\frac{1}{a}} \; {\tilde{C}}_{d, \gamma, r} \;
\| {\textbf{u}} \|_{L^{\infty}((0,T), L^{2}_{w_\gamma}(\mathbb{R}^d))}^{1+\frac{a-2}{a}}
\nonumber \\
& \;\;\;\; \cdot
\left( \int_0^T ( \| {\textbf{u}}(s) \|_{L^2_{w_\gamma}(\mathbb{R}^d)}
+ \| \nabla {\textbf{u}}(s) \|_{L^2_{w_\gamma}(\mathbb{R}^d)})^2 ds
\right)^{\frac{1}{a}},
\end{align}
where we have used (\ref{Eq:0b}) and (\ref{Eq:0a}). Using again the continuity
of the Riesz transforms, but this time on the space $L^2_{w_{\gamma}}(\mathbb{R}^d)$,
we obtain that
\begin{equation} \label{Eq:3}
\sum_{i,j=1}^d \mathcal{R}_i \mathcal{R}_j F_{i, j}
\in L^{2}((0,T), L^{2}_{w_\gamma}(\mathbb{R}^d)),
\end{equation}
and, moreover,
\begin{equation} \label{Eq:4}
\Big{\|}\sum_{i,j=1}^d \mathcal{R}_i \mathcal{R}_j F_{i,j} \, \,
\Big{\|}_{L^{2}((0,T), L^{2}_{w_\gamma}(\mathbb{R}^d))}
\le C_{d, \gamma} \sum_{i,j=1}^d
\| F_{i,j} \|_{L^{2}((0,T), L^{2}_{w_\gamma}(\mathbb{R}^d))}.
\end{equation}
It follows from (\ref{Eq:1})-(\ref{Eq:4}) that $p$ is well-defined, and
\begin{equation*}
p:= \sum_{i,j=1}^d \mathcal{R}_i \mathcal{R}_j (u_i u_j)
- \sum_{i,j=1}^d \mathcal{R}_i \mathcal{R}_j F_{i,j}
\in L^a((0,T), L^{r}_{w_{s\gamma}}(\mathbb{R}^d))
+ L^2 ((0,T), L^2_{w_\gamma}(\mathbb{R}^d )).
\end{equation*}
We now consider the expression $\nabla q - \nabla p$. By taking the
divergence of both members of the first equation of (\ref{Eq:NSpr}),
we find that $\Delta (q-p)=0$. In fact, as $\nabla \cdot {\textbf{u}} = 0$,
we have that
$\nabla \cdot \partial_t {\textbf{u}} = \nabla \cdot \Delta {\textbf{u}} = 0$,
and
\begin{align} \label{Eq:Lapl}
\Delta q = \nabla \cdot \nabla q \nonumber
& = -\nabla \cdot (({\textbf{u}} \cdot \nabla) {\textbf{u}}
- \nabla \cdot \mathbb{F}) \nonumber \\
& = -\nabla \cdot (\nabla \cdot ({\textbf{u}} \otimes {\textbf{u}})
- \nabla \cdot \mathbb{F}) \nonumber \\
& = -\sum_{i,j=1}^d \partial_i \partial_j
(u_i u_j - F_{i,j}) \nonumber \\
& = \Delta \sum_{i,j=1}^d \frac{\partial_i}{\sqrt{-\Delta}}
\frac{\partial_j}{\sqrt{-\Delta}} (u_i u_j - F_{i,j}) \nonumber \\
& = \Delta p.
\end{align}
Let $\varepsilon_0 \in \left(0, \frac{T}{2}\right)$. Let
$\alpha \in C_0^{\infty}(\mathbb{R})$ be a nonnegative real-valued function
such that for all $t \in \{s \in \mathbb{R};|s| \ge \varepsilon_0 \}$,
$\alpha (t)= 0$, and $\int_{\mathbb{R}} \alpha(s) \, ds= 1$. Moreover, let
$\beta \in C_0^{\infty}(\mathbb{R}^d)$ be a nonnegative real-valued
function such that $\int_{\mathbb{R}^d}\beta(x) \, dx = 1$. We use the
symbol $\alpha\otimes\beta$ to indicate the real-valued function given by
\begin{align*}
\alpha\otimes\beta \colon \mathbb{R} \times \mathbb{R}^d
&\longrightarrow \mathbb{R} \\
(s, x) &\longmapsto (\alpha\otimes\beta) (s, x):= \alpha(s) \, \beta(x).
\end{align*}
Also, for all $(s, x) \in \mathbb{R} \times \mathbb{R}^d$, we write
$(\alpha \otimes \Delta \beta)(s, x) := \alpha(s) \, \Delta \beta(x)
\in \mathbb{R}$, and $(\alpha \otimes \nabla \beta)(s, x) := (\alpha(s) \,
\partial_1 \beta (x), \ldots, \alpha(s) \, \partial_d \beta (x)) \in \mathbb{R}^d$.
Then, the \textit{mollification},
$A_{\alpha, \beta, t}$, of $\nabla q - \nabla p$ given by
\begin{equation*}
A_{\alpha,\beta,t}(x) := ((\alpha \otimes \beta)
* \nabla q - (\alpha \otimes \beta) * \nabla p) (t,x)
\end{equation*}
is well-defined on $(\varepsilon_0, T-\varepsilon_0) \times \mathbb{R}^d$.
Furthermore, for all $t \in (\varepsilon_0, T-\varepsilon_0)$,
\begin{align}\label{Eq:convol}
A_{\alpha,\beta,t}
& =((-\partial_t \, \alpha \otimes \beta + \alpha \otimes \Delta\beta)
* {\textbf{u}})(t,\cdot )
+ ((\alpha \otimes \nabla \beta)
* (- {\textbf{u}} \otimes {\textbf{u}} +\mathbb{F}))(t,\cdot )
\nonumber \\
& \;\;\;\; -((\alpha \otimes \nabla \beta) * p) (t,\cdot),
\end{align}
where for every $s \in (\varepsilon_0, T-\varepsilon_0)$, and for almost
all $x \in \mathbb{R}^d$, $((\alpha \otimes \nabla \beta) *
(-{\textbf{u}} \otimes {\textbf{u}} +\mathbb{F}))(s,x)$ denotes the
vector whose i-th coordinate is given by $\displaystyle \sum_{j=1}^d
((-u_i u_j + F_{i,j}) * (\alpha \otimes \partial_i \beta)(s,x)$, for
each $i \in \{1, \ldots, d\}$. For almost all
$t \in (\varepsilon_0, T-\varepsilon_0)$, we will prove
that $A_{\alpha,\beta,t}$ is necessarily zero, by demonstrating that it is a
harmonic tempered distribution, and therefore it is a polynomial, and finally
by showing that $A_{\alpha,\beta,t}$ belongs to a space which does not contain
nontrivial polynomials. Since $\gamma \in (0, d)$, and
\begin{equation*}
\lim_{r \to \, 1^+} \frac{d(2-r)}{2r} + \frac{\gamma}{2}
= \frac{d}{2} + \frac{\gamma}{2} < d,
\end{equation*}
there exist real numbers $\sigma$ and $\eta$ such that
\begin{equation} \label{Eq:5}
1 < \sigma < \min \Big{\{} \frac{d}{d-1}, \frac{d}{\gamma} \Big{\}},
\end{equation}
and
\begin{equation} \label{Eq:6}
\max \Big{\{} \gamma , \, \frac{d(2-\sigma)}{2\sigma}
+\frac{\gamma}{2} \Big{\}} < \, \eta < \, \frac{d}{\sigma} \, < \, d.
\end{equation}
For almost every $t \in (0,T)$, by using (\ref{Eq:1}), (\ref{Eq:3})
and (\ref{Eq:5}), we see that
\begin{equation*}
p(t) \, \in \, L^{\sigma}_{w_{\sigma\gamma}}(\mathbb{R}^d)
+ L^2_{w_\gamma}(\mathbb{R}^d),
\end{equation*}
and, of course, by employing (\ref{Eq:0b}) and (\ref{Eq:5}), we get
\begin{equation*}
(-{\textbf{u}} \otimes {\textbf{u}} +\mathbb{F})(t)
\, \in \, L^{\sigma}_{w_{\sigma \gamma}}(\mathbb{R}^d)
+ L^2_{w_\gamma}(\mathbb{R}^d).
\end{equation*}
Since for all function $\varphi \in \mathcal{C}_0^{\infty}(\mathbb{R}^d)$, and for
every locally integrable function $f:\mathbb{R}^d \to \mathbb{C}$,
$\vert f*\varphi\vert\le C_\varphi \, \mathcal{M} f$, it follows that
the convolution with a function belonging to
$\mathcal{C}_0^{\infty}(\mathbb{R}^d)$ is a bounded linear operator defined on
$L^2_{w_\gamma}(\mathbb{R}^d)$, and also on
$L^{\sigma}_{w_{\sigma \gamma}}(\mathbb{R}^d)$.
Thus, from (\ref{Eq:convol}), for almost every
$t \in (\varepsilon_0, T-\varepsilon_0)$, we obtain that
\begin{equation*}
A_{\alpha,\beta,t} \in L^{\sigma}_{w_{\sigma \gamma}}(\mathbb{R}^d)
+ L^2_{w_\gamma}(\mathbb{R}^d),
\end{equation*}
and therefore $A_{\alpha,\beta,t}$ is a tempered distribution. Moreover, for
all $t \in (\varepsilon_0, T-\varepsilon_0)$, $A_{\alpha,\beta,t}$ is a harmonic
polynomial. In fact, let $s \in (\varepsilon_0, T-\varepsilon_0)$. Then,
\begin{equation*}
\Delta A_{\alpha, \beta, s}=(\alpha \otimes \beta)*
\nabla (\Delta (q-p))(s, \cdot)=0,
\end{equation*}
where in the last expression we have used (\ref{Eq:Lapl}). In addition, by
using H\"older's inequality, we obtain that
\begin{equation*}
L^{\sigma}_{w_{\sigma \gamma}}(\mathbb{R}^d) +L^2_{w_\gamma}(\mathbb{R}^d)
\subset L^{\sigma}_{w_{\sigma \eta}}(\mathbb{R}^d).
\end{equation*}
In fact, for all
$g \in L^{\sigma}_{w_{\sigma \gamma}}(\mathbb{R}^d) + L^2_{w_\gamma}(\mathbb{R}^d)$,
we see that
\begin{align*}
\int |g(x)|^{\sigma} w_{\sigma \eta}(x) \, dx
& \le \left(\int (|g(x)|^{\sigma}
w_{\frac{\sigma \gamma}{2}}(x))^{\frac{2}{\sigma}} \, dx
\right)^{\frac{\sigma}{2}} \\
&\;\;\;\; \cdot \left( \int (1+|x|)^
{-(\sigma \eta \, - \, \frac{\sigma \gamma}{2})(\frac{2}{2-\sigma})}
\, dx \right)^{1-\frac{\sigma}{2}} \\
& < + \infty,
\end{align*}
where in the last expression we have used the fact that
$\displaystyle \left(\eta - \frac{\gamma}{2}\right) \cdot
\left(\frac{2\sigma}{2-\sigma}\right) > d$, which follows from (\ref{Eq:6}).
By using (\ref{Eq:6}), we see that $\sigma \eta < d$. Then, the space
$L^{\sigma}_{w_{\sigma \eta}}(\mathbb{R}^d)$ does not contain nontrivial
polynomials. Hence, for almost all $t \in (\varepsilon_0, T-\varepsilon_0)$,
$A_{\alpha,\beta,t}= 0$. For all $\varepsilon > 0$, for every
$t \in \mathbb{R}$, and for each $x \in \mathbb{R}^d$, we now let
\begin{equation*}
\alpha_\varepsilon (t) := \frac {1}{\varepsilon}
\alpha \left( \frac{t}{\varepsilon} \right),
\;\;\; \text{ and } \;\;\;
\beta_\varepsilon (x) := \frac {1}{\varepsilon^d}
\beta \left( \frac {x}\varepsilon \right).
\end{equation*}
Since for any $\varepsilon > 0$, $\alpha_\varepsilon \otimes \beta_\varepsilon
\in \mathcal{C}_0^{\infty}(\mathbb{R} \times \mathbb{R}^d)$ is a
nonnegative real-valued function satisfying
\begin{equation*}
\int_{\mathbb{R} \times \mathbb{R}^d}
(\alpha_\varepsilon \otimes \beta_\varepsilon)(s, x) \, ds \, dx = 1,
\end{equation*}
we conclude that for almost all $t \in (0, T)$,
\begin{equation*}
\nabla (q-p)(t,\cdot ) = \lim_{\varepsilon \to \, 0}
A_{\alpha_\varepsilon, \beta_\varepsilon, t} = 0.
\end{equation*}
\end{proof}
|
train/arxiv
|
BkiUauzxaJJQnMsbx-Cy
| 5 | 1 |
\subsection*{Introduction}
\label{intro}
A peak in Einstein's endeavor to extract as much information as possible about
the nature of radiation from the Planck distribution is his paper
``On the Quantum Theory of Radiation'' of 1916 \cite{Ein}.\footnote{The paper was first published in: Physikalische Gesellschaft Z\"{u}rich. Mitteilungen 18 (1916): 47-62. The same paper was received 3 March and published 15 March 1917 in: Physikalische Zeitschrift 18 (1917): 121-128.}
In the first part he gives a derivation
of Planck's formula which has become part of many textbooks on
quantum theory.
This part of the paper is now considered as the theoretical foundation of the laser,
that was technically realized almost half a century later.
In the second part of his fundamental paper, Einstein discusses the
exchange of momentum between atoms and radiation by making
use of the theory of Brownian motion. Using a truly beautiful argument,
which will be presented below, he shows that in every elementary process
of radiation, and in particular in spontaneous emission, an amount $h\nu/c$
of momentum is emitted in a random direction and that the atomic system
suffers a corresponding recoil in the opposite direction.
This recoil was
first experimentally confirmed in 1933 by showing that a long and
narrow beam of excited sodium atoms widens up after spontaneous
emissions have taken place \cite{Fri}.
Einstein's paper \cite{Ein} ends with the following remarkable statement
concerning the role of ``chance'' in his description of the
radiation processes by statistical laws, to which Pauli \cite{Pau}
drew special attention:
\begin{quote}
\textit{``The weakness of the theory lies, on the one hand, in the
fact that it does not bring us any closer to a merger with the
undulatory theory, and, on the other hand, in the fact that it
leaves the time and direction of elementary processes to `chance';
in spite of this I harbor full confidence in the trustworthiness
of the path entered upon.''}
\end{quote}
\subsection*{Derivation of the Planck distribution}
\label{sec:1}
In his novel derivation of the Planck distribution, Einstein
added the hitherto unknown process of
induced emission\footnote{Einstein's derivation shows that without
assuming a non-zero probability for induced emission one would
necessarily arrive at Wien's instead of Planck's radiation law.}, next
to the familiar processes of spontaneous emission and induced
absorption. For each pair of energy levels he described the
statistical laws for these processes by three coefficients
(the famous $A$- and $B$-coefficients) and established two
relations between these coefficients on the basis of his earlier
correspondence argument in the classical Rayleigh-Jeans limit
and Wien's displacement law. In addition, the latter implies
that the energy difference $\varepsilon_n-\varepsilon_m$ between
two internal energy states of the atoms in equilibrium with thermal
radiation has to satisfy Bohr's frequency condition:
$\varepsilon_n-\varepsilon_m=h\nu_{nm}$.
Einstein was very pleased by this derivation, about
which he wrote on August 11, 1916 to Besso: ``An amazingly simple
derivation of Planck's formula, I should like to say \emph{the}
derivation'' (CPAE, Vol.\,8, Doc.\,250).
In Dirac's 1927 radiation
theory these results follow -- without any correspondence
arguments -- from first principles.
\subsection*{Brownian motion of an atom in the thermodynamic radiation field}
\label{sec:2}
In the second part, Einstein regarded in his paper as more important, he investigates the exchange of momentum between atoms and radiation by studying the Brownian motion of molecules in the thermodynamic radiation field. A molecule experiences thereby two types of forces:
$\bullet$ A systematic drag force $Rv$, where $v$ is the velocity of the molecule, that leads in a small time interval $(t,t+\tau)$ to the momentum change $Rv\tau$.
$\bullet$ An irregular change of momentum $\Delta$ in the time $\tau$, due to fluctuations
of the radiation pressure.
In thermal equilibrium the following equation holds:
\[\langle(Mv-Rv\tau+\Delta)^2\rangle=\langle(Mv)^2\rangle. \]
Assuming that $\langle v\cdot\Delta\rangle=0$ we obtain
the important relation, repeatedly used by Einstein in his earlier work on Brownian motion,
\begin{equation}
\langle\Delta^2\rangle=2R\langle M v^2\rangle\tau=2RkT\tau.
\label{Eq:1}
\end{equation}
\paragraph{Calculation of the friction $R$.}
The computation of $R$ is not simple. For interested readers we present the details of Einstein's instructive calculation, thereby including some flesh of his profound investigation. (Others may prefer to jump directly to the simple result, given in equation (\ref{Eq:4}) below.)
The moving atom (molecule) sees an anisotropic radiation field. The friction force is the result of absorption and induced emission processes\footnote{The spontaneous emission has in the average no preferred direction, and therefore does in the average not transmit momentum to the atom.}. Let $K$ be the rest system of the radiation field, with respect to which the radiation intensity is isotropic. With $K'$ we denote the rest system of the atom. Let $S$ denote the partition sum
\[ S:=g_ne^{-E_n/kT}+g_me^{-E_m/kT}+\cdot\cdot\cdot~\]
for the energies $E_n$ and degeneracies $g_n$ of the atom. The fraction of time in the state $n$ is equal to $g_ne^{-E_n/kT}/S$, and similarly for the state $m$. Therefore, the number of absorptions $n\rightarrow m$ per unit time
into the solid angle $d\Omega'$ is equal to
\[\frac{1}{S}g_ne^{-E_n/kT}B^n_m\rho'_{\nu_0}\frac{d\Omega'}{4\pi},
~~~\nu_0=\frac{E_m-E_n}{h}\,,\]
where $\rho_{\nu}$ denotes the energy density of the radiation as a function of frequency $\nu$.
Correspondingly, for induced emission $m\rightarrow n$ the number is
\[\frac{1}{S}g_me^{-E_m/kT}B^m_n\rho'_{\nu_0}\frac{d\Omega'}{4\pi}.\]
With Einstein's relation $g_mB^m_n=g_nB^n_m$ from the first part of the paper, the momentum transfer per
unit time in the $x$-direction is
\begin{equation*}
-Rv=\frac{h\nu_0}{c}\frac{1}{S}g_nB^n_m\left(e^{-E_n/kT}-e^{-E_m/kT}\right)
\times\int\rho'_{\nu_0}(\theta',\varphi')\cos\theta'\frac{d\Omega'}{4\pi}\,.
\label{Eq:2}
\end{equation*}
\paragraph{Calculation of $\rho_{\nu_0}(\theta',\varphi')$.}
With elementary calculations one can prove the following Lorentz invariance
\[\frac{\rho'_{\nu'}}{\nu'^3}=\frac{\rho_\nu}{\nu^3}\,.\]
(This follows more directly from the fact that $\rho(\nu)$ is proportional to $\nu^3$ times the Lorentz invariant distribution function $f$.) Using also the Doppler shift to first order in $v/c$:
$\nu=\nu'\left(1+\frac{v}{c}\cos\theta'\right)$. This gives
\[
\rho'_{\nu'} \simeq \left(1+\frac{v}{c}\cos\theta'\right)^{-3}\rho_{[1+\frac{v}{c}\cos\theta']\nu'}~,\]
thus
\begin{eqnarray*}
\rho'_{\nu_0}&\simeq&\left[\rho_{\nu_0}+\left(\frac{\partial\rho_\nu}{\partial\nu}\right)_{\nu_0}
\nu_0\frac{v}{c}\cos\theta'\right]\left(1-3\frac{v}{c}\cos\theta'\right)\\
&\simeq&\rho_{\nu_0}+\frac{v}{c}\cos\theta'\left[\nu_0\left(\frac{\partial\rho_\nu}{\partial\nu}\right)_{\nu_0}
-3\rho_{\nu_0}\right]\,.
\end{eqnarray*}
To first order in $v/c$ we obtain
\begin{equation*}
\int\rho'_{\nu_0}\cos\theta'\frac{d\Omega'}{4\pi}=-\frac{v}{c}\left[\rho_{\nu_0}-\frac{1}{3}
\nu_0\left(\frac{\partial\rho_\nu}{\partial\nu}\right)_{\nu_0}\right]
=\frac{v}{c}\left\{\frac{\nu^4}{3}\frac{\partial}{\partial\nu}\left(\frac{\rho_\nu}{\nu^3}\right)\right\}_{\nu=\nu_0}.
\end{equation*}
Writing $\nu$ instead of $\nu_0$, we arrive at
\begin{equation*}
R=\frac{h\nu}{c^2}\frac{1}{S}g_nB^n_me^{-E_n/kT}\left(1-e^{-h\nu/kT}\right)
\times\left[-\frac{\nu^4}{3}\frac{\partial}{\partial\nu}\left(\frac{\rho_\nu}{\nu^3}\right)\right].
\end{equation*}
For the Planck distribution we obtain
\begin{equation}
R=\frac{1}{3}\left(\frac{h\nu}{c}\right)^2\frac{1}{kT}\frac{g_n}{S}e^{-E_n/kT}B^n_m\rho_\nu\,.
\label{Eq:3}
\end{equation}
Note that $(g_n/S)\exp (-E_n/kT)B^n_m\rho_\nu$ is the number of
absorptions per unit time. Since this is equal to 1/2 times the number $Z$ of
elementary processes (induced and spontaneous emissions plus absorptions)
per unit time, we find for the friction force the simple result:
\begin{equation}
R=\frac{1}{3}\left(\frac{h\nu}{c}\right)^2\frac{1}{2kT}Z.
\label{Eq:4}
\end{equation}
\paragraph{Interpretation}
Using this in (\ref{Eq:1}) we arrive at the crucial relation
\begin{equation}
\langle\Delta^2\rangle=\left(\frac{h\nu}{c}\right)^2\langle\cos^2\vartheta\rangle_{S^2} Z\tau \,.
\label{Eq:5}
\end{equation}
From classical electrodynamics (and experience) we know that to a pencil of light with energy $\varepsilon$ belongs a momentum $\varepsilon/c$ in the direction of the light beam. From this Einstein concluded that the atom receives in an absorption process the momentum $(E_m-E_n)/c$ in the direction of the infalling radiation, and in an induced emission process the same momentum in the opposite direction. The crucial relation (\ref{Eq:5}) then implies that \emph{spontaneous} emission must also be directed in such a way that for every elementary process of radiation an amount $h\nu/c$ of momentum is emitted in a random direction, and that the atomic system suffers a corresponding recoil in the opposite direction. ``There is no radiation in spherical waves''\footnote{``Ausstrahlung in Kugelwellen gibt es nicht''.}, Einstein adds.
Einstein regarded this conclusion as the main result of his paper. In a letter to Michele Besso he states:
\begin{quote}
``\textit{With this, the existence of light-quanta is practically assured.}''\footnote{``Damit sind die Lichtquanten so gut wie gesichert'' (6. September 1916).}
\end{quote}
Two years later he added:
\begin{quote}
``\textit{I do not doubt anymore the \emph{reality} of radiation
quanta, although I still stand quite alone in this conviction.}''
\end{quote}
We conclude with the following interesting remark of Einstein to O. Stern:
\begin{quote}
``\textit{Ich habe hundertmal mehr \"{u}ber Quantenprobleme
nachgedacht, als \"{u}ber die allgemeine Relativit\"{a}tstheorie.}''
\end{quote}
|
train/arxiv
|
BkiUdgA4uBhizKycjBnd
| 5 | 1 |
\section{Multi-armed Bandits}\label{sec:multi-armed-bandit}
The stochastic $K$-armed bandit problem~\citep{bellman1956problem,lattimore2018bandit} involves a learner sequentially choosing among $K$ different arms. At time $t$, the algorithm \emph{draws an arm} $a_t \in \Act$ and the arms \emph{generate outcomes} $\bx_t = [x_{t,j}]_{j=1}^K \in \real^K$. Then the learner \emph{obtains reward} $r_t = x_{t,a_t}$, without observing the other arms.
In the \emph{stochastic} setting, the learner is acting within an environment $\model$, with each action generating outcomes with distribution $f_a$ and means $\mu_a \defn \expect(f_a)$, and optimal expected reward $\mu^* \defn \max_a \mu_a$ so that the reward distribution for each arm is $\prob_\model(X_t = r_{t,a}) = f_a$ for all $t$.
The learner's policy $\pol$ for selecting actions is generally a stochastic mapping $\pol : \Hist \to \dist(\Act)$. Here $\Hist$ is the \emph{observed history}, i.e. the sequence of actions taken and rewards obtained by the learner.
\ifdefined \workshop
\
\else
The objective is to \emph{maximise the expected cumulative reward},
\begin{align}
S(\pol, \model, T) \defn
\sum_{t=1}^T \mathbb{E}_{\model \pol} [r_t]
= \sum_{a=1}^K \mathbb{E}_{\model \pol} \left[N_a(T)\right]\mu_a,
\end{align}
where $N_a(T)$ denotes the number of time arm $a$ is pulled till time step $T$ and $\mu_a$ is the expected reward of the arm $a$.
\fi
The quality of a learning algorithm $\pol$ is best summarised by its \emph{(expected cumulative) regret}, i.e. its loss in total reward relative to an oracle that knows $\model$:
\begin{equation}
\reg(\pol, \model, T) \defn
\sum_{a=1}^K \mathbb{E}_{\model \pol}\left[N_a(T)\right](\mu^* - \mu_a).\vspace*{-.5em}
\end{equation}
$N_a(T)$ denotes the number of time arm $a$ is pulled till time step $T$.
This is the cost incurred by the algorithm due to the incomplete information, as it has to play the suboptimal arms to gain information about the suboptimal arms. This process decreases the uncertainty in decision making and facilitates maximisation of the expected cumulative reward.
Lower bounds illustrate the inherent hardness of bandit problems.
\citet{lai1985asymptotically} proved that any consistent bandit policy $\pol$ must incur at least logarithmic growth ($\Omega(\log T)$) in expected cumulative regret. This lower bound is problem-dependent as it has a multiplicative factor dependent on the given environment $\model$.
\citet{vogel1960asymptotic} proved an environment-independent lower bound of $\Omega(\sqrt{KT})$.
This also called a \emph{problem-independent minimax lower bound} as the minimax regret is as $\reg_{\mathrm{minimax}}(T) \defn \min_{\pol}\max_{\model}\reg(\pol,\model,T)$.
A similar lower bound of $\Omega(\sqrt{KT})$ is established for the \emph{Bayesian regret} under any prior \citep{lattimore2018bandit}.
A detailed description of the existing lower bounds is provided in Appendix~\ref{sec:lbounds}.
\iffalse
One specific way to implement differential privacy is to ensure that the input to the algorithm is already a differentially private version of the original data. This notion of privacy is called \emph{local differential privacy}~\citep{duchi2013local}, and it allows the algorithm to be agnostic about privacy. For this reason, this notion is presently adapted by Apple and Google for their large-scale systems.
\fi
\subsubsection*{\bibname}}
\ShortHeadings{Privacy in Multi-armed Bandits}{Basu, Dimitrakakis and Tossou}
\firstpageno{1}
\begin{document}
\title{Privacy in Multi-armed Bandits:\\ Fundamental Definitions and Lower Bounds}
\author{Debabrota Basu, Christos Dimitrakakis, \and Aristide Y. Tossou\\
\addr{Chalmers University of Technology, Sweden}}
\maketitle
\begin{abstract}
Based on differential privacy (DP) framework, we introduce and unify privacy definitions for the multi-armed bandit algorithms. We represent the framework with a unified graphical model and use it to connect privacy definitions. We derive and contrast lower bounds on the regret of bandit algorithms satisfying these definitions. We leverage a unified proving technique to achieve all the lower bounds. We show that for all of them, the learner's regret is increased by a multiplicative factor dependent on the privacy level $\epsilon$. We observe that the dependency is weaker when we do not require local differential privacy for the rewards.
\end{abstract}
\input{writeup.tex}
\section*{Acknowledgement}
This work was partially supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation.
\bibliographystyle{abbrvnat}
\section{Proofs of Section~\ref{sec:dp_mab} (Differential Privacy in Bandits: Definitions)}\label{sec:proofs_def}
\begin{lemmarep}
Iff a bandit algorithm $\pol$ is DP with respect to the outcome sequence $\bx$ then it is DP with respect to the reward sequence $r$. However, in the general partial monitoring setting the equivalence does not hold.
\label{rem:equivalence}
\end{lemmarep}
\begin{proof
The first claim is direct:
\begin{align*}
\pol(a_1, \ldots, a_T \mid \bx_1, \ldots, \bx_T)
&= \prod_{t=1}^T \pol(a_t \mid a_1, \ldots, a_{t-1}, \bx_1, \ldots, \bx_{t-1})\\
&= \prod_{t=1}^T \pol(a_t \mid a_1, \ldots, a_{t-1}, x_{1,a_1}, \ldots, x_{t-1,a_{t-1}})\\
&= \prod_{t=1}^T \pol(a_t \mid a_1, \ldots, a_{t-1}, r_1, \ldots, r_{t-1})\\
&= \pol(a_1, \ldots, a_T \mid r_1, \ldots, r_T)
\end{align*}
The second follows from the fact that in the general setting the algorithm may observe the complete outcome $\bx_t$ even if the reward it obtains only depends on part of $r_t$.
\end{proof}
\begin{lemmarep}
If a policy $\pol$ satisfies differential privacy (Definition~\ref{def:DP_mab}) with privacy level $\epsilon$, $\pol$ will also satisfy instantaneous privacy (Definition~\ref{def:IDP_mab}) with privacy level $2\epsilon$. Conversely, if a policy satisfies $\epsilon$ instantaneous privacy, it only achieves $t \epsilon$ differential privacy after $t$ steps.
\end{lemmarep}
\begin{proof
For simplicity, we write $a^t = \{a_1, \ldots, a_t\}$ and $\br^t = \{\br_1, \ldots, \br_t\}$ for reward and action sequences respectively.
If $\pol$ is $2 \epsilon$-instantaneous private then (by definition) the following ratio must be bounded from above by $e^{2 \epsilon}$ for any two neighbouring reward sequences $\br^{t}, \hat{\br}^{t}$ :
\begin{align*}
\frac{\pol(a_t \mid a^{t-1}, \br^{t-1})}{\pol(a_t \mid a^{t-1}, \hat{\br}^{t-1})}
&\underset{(a)}{=}
\frac{\pol(a^t \mid \br^{t-1})}{\pol(a^t \mid \hat{\br}^{t-1})}
\frac{\pol(a^{t-1} \mid \hat{\br}^{t-1})}{\pol(a^{t-1} \mid \br^{t-1})}\\
&\underset{(b)}{=}
\frac{\pol(a^t \mid \br^{t-1})}{\pol(a^t \mid \hat{\br}^{t-1})}
\frac{\pol(a^{t-1} \mid \hat{\br}^{t-2})}{\pol(a^{t-1} \mid \br^{t-2})}\\
&\underset{(c)}{\leq}
e^{2 \epsilon}.
\end{align*}
The equality in step (a) is obtained through the definition of conditional probability.
Step (b) is resulted by independence of actions on current rewards.
The final inequality is through assumption of $\epsilon$-differential privacy and that $\br, \hat{\br}$ are neighbours.
The converse follows from composition of differential privacy~\citep[Corollary 3.15]{dwork2014algorithmic}.
\end{proof}
\section{Proofs of Section~\ref{sec:theory} (Regret Lower Bounds for Private Bandits)}\label{sec:proofs_lb}
First, let us remind ourselves of the chain rule of KL divergence for two probability measures $P,Q$ on a product space $\CX^T$ for a given $T$:
\begin{align*}
\kldiv{P}{Q}
&\defn
\int\limits_{\CX^T} \ln \frac{\dd P(x^T)}{\dd Q(x^T)} \dd P(x^T)\\
&=
\int\limits_{\CX^T} \ln \frac{\dd [P(x_T \mid x^{T-1}) P(x^{T-1})]}{\dd [Q(x_T \mid x^{x-1}) Q(x^{T-1})]} \dd [P(x_T \mid x^{T-1}) P(x^{T-1})]
\\
&=
\int\limits_{\CX^T} \left[\ln \frac{\dd P(x_T \mid x^{T-1})}{\dd Q(x_T \mid x^{x-1})} + \ln \frac{\dd P(x^{T-1})}{\dd Q(x^{T-1})}\right] \dd [P(x_T \mid x^{T-1}) P(x^{T-1})]
\\
&=
\int\limits_{\CX^T} \ln \frac{\dd P(x_T \mid x^{T-1})}{\dd Q(x_T \mid x^{x-1})} \dd P(x^{T}) + \int\limits_{\CX^{T-1}} \ln \frac{\dd P(x^{T-1})}{\dd Q(x^{T-1})} \dd P(x^{T-1})
\\
&=
\sum_{t=1}^T \int\limits_{\CX^t} \ln \frac{\dd P(x_t \mid x^{t-1})}{\dd Q(x_t \mid x^{t-1})} \dd P(x^{t}) \\
&=
\sum_{t=1}^T
\expect_{P(x^{t-1})}\left[\kldiv{P(x_t \mid x^{t-1})}{Q(x_t \mid x^{t-1})}\right].
\end{align*}
Here, $x_t$ denotes the reward at time $t$ and $x^{t}$ denotes the sequence of rewards obtained from the beginning to time $t$, i.e. $\lbrace x_1, \ldots, x_t\rbrace$. Now, the conditional KL-divergence is defined here as
\[
\kldiv{P(x\mid y)}{Q(x\mid y)} \defn \int_{\CX \times \CY} \ln \frac{\dd P(x\mid y)}{\dd Q(x\mid y)} \dd P(x, y).
\]
Thus, we get the chain rule of KL-divergence
\begin{align*}
\begin{split}
\kldiv{P}{Q}
&\defn
\int\limits_{x^T \in \CX^T} \ln \frac{P(x^T)}{Q(x^T)} \dd P(x^T)\\
&=
\sum_{t=1}^T \kldiv{P(x_t \mid x^{t-1})}{Q(x_t \mid x^{t-1})}.
\end{split}
\end{align*}
\iffalse
\cdcomment{But I don't see how we can write the KL decompositionm in terms of individual possible actions. Let's take just one part of the decomposition:
\begin{align*}
\kldiv{P(A_t | H_t)}{Q(A_t | H_t)}
&= \sum_{a} P(A_t =a | H_t) \ln P(A_t = a| H_t)/Q(A_t =a | H_t)
\\
&\neq \sum_{a} \kldiv{P(A_t=a | H_t)}{Q(A_t=a | H_t)} ??
\end{align*}
}\dbcomment{True. the decomposition is wrong. Edited.}
\fi
\begin{lemmarep}[KL-divergence Decomposition]
Given a bandit algorithm $\pol$, two environments $\model_1$ and $\model_2$, and a probability measure $\prob_{\pol\model}$ satisfying Equation~\ref{eqn:indep},
\begin{align}\label{eqn:kldecomp}
\begin{split}
\kldiv{\prob^T_{\pol{\model_1}}}{\prob^T_{\pol{\model_2}}} &= \sum_{t=1}^T \mathbb{E}_{\pol\model_1}\left[\kldiv{\pol(A_t|\mathcal{H}_t, \model_1)}{\pol(A_t|\mathcal{H}_t, \model_2)}\right] + \sum_{a=1}^K \mathbb{E}_{\pol\model_1}\left[N_a(T)\right]\kldiv{f_a \in \model_1}{f_a \in \model_2}.
\end{split}
\end{align}
\iffalse
\cdcomment{Isn't this just the following? Why are you breaking it up into actions? I think you can do that for the rhs term, but the bit with the action on the lhs of the dependence, it doesn't work. See my detailed derivation of the chain rule} \dbcomment{True This decomposition $\sum_{a=1}^K \sum_{t=1}^{T} \prob^t_{\pol{\model_1}}[A_t=a] \kldiv{\pol(a|\mathcal{H}_t, \model_1)}{\pol(a|\mathcal{H}_t, \model_2)}$ is wrong. Edited.}
\fi
\end{lemmarep}
\begin{proof}
\begin{align*}
&\kldiv{\prob^T_{\pol{\model_1}}}{\prob^T_{\pol{\model_2}}}\\
\underset{(a)}{=} &\sum_{t=1}^{T} \expect_{\prob^T_{\pol{\model_1}}} [ \kldiv{\pol(A_t|\mathcal{H}_t, \model_1)}{\pol(A_t|\mathcal{H}_t, \model_2)} + \kldiv{f(X_t|A_t,\mathcal{H}_t, \model_1)}{f(X_t|A_t,\mathcal{H}_t, \model_2)} ]\\
\underset{(b)}{=} &\sum_{t=1}^T \expect_{\prob^T_{\pol{\model_1}}}\left[\kldiv{\pol(A_t|\mathcal{H}_t, \model_1)}{\pol(A_t|\mathcal{H}_t, \model_2)}\right] + \sum_{t=1}^{T} \expect_{\prob^T_{\pol{\model_1}}}\left[ \sum_{a=1}^K \mathds{1}_{A_t=a} \kldiv{f_a(X_t) \in \model_1}{f_a(X_t) \in \model_2}\right]\\
\underset{(c)}{=} &\sum_{t=1}^T \expect_{\prob^T_{\pol{\model_1}}}\left[\kldiv{\pol(A_t|\mathcal{H}_t, \model_1)}{\pol(A_t|\mathcal{H}_t, \model_2)}\right] + \sum_{a=1}^K \left[\sum_{t=1}^{T} \expect_{\prob^T_{\pol{\model_1}}}[\mathds{1}_{A_t=a}] \kldiv{f_a(X_t) \in \model_1}{f_a(X_t) \in \model_2}\right]\\
\underset{(d)}{=} &\sum_{t=1}^T \expect_{\prob^T_{\pol{\model_1}}}\left[\kldiv{\pol(A_t|\mathcal{H}_t, \model_1)}{\pol(A_t|\mathcal{H}_t, \model_2)}\right] + \sum_{a=1}^K \expect_{\prob^T_{\pol{\model_1}}}\left[N_a(T)\right]\kldiv{f_a \in \model_1}{f_a \in \model_2}.
\end{align*}
The equality in step (a) is followed by the chain rule of KL-divergence and Equation~\ref{eqn:indep}.
Step (b) is obtained by conditioning.
Step (c) is a consequence of the linearity of expectation.
The equality in step (d) comes from the fact that expectation of an indicator function of an event returns its probability of occurrence.
\end{proof}
This style of KL-divergence decomposition appeared in proofs of~\citep{auer2002nonstochastic,garivier2018explore,lattimore2018bandit}. We adopt the proof in our context and notations with enough generality to proof the differentially private versions of it later.
\ifdefined \workshop
\subsection{Lower Bounds for Local Differential Privacy (LDP)}
\else
\subsection{Proofs for Local Differential Privacy (LDP)}
\fi
\ifdefined \workshop
\paragraph{Sketch of Results and Discussion.}
At first, we derive an upper bound on the KL-divergence decomposition for locally private reward generation. This upper bound would allow us to lower bound the sum of regret in two environments such that the best policy in one is the worst in another.
\begin{lemmarep}[Local Private KL-divergence Decomposition
If the reward generation process is $\epsilon$-local differentially private for both the environments $\model_1$ and $\model_2$,\vspace{-.5em}
\begin{align*}
\kldiv{\prob^T_{{\model_1}\pol}}{\prob^T_{{\model_2}\pol}} &\leq~2\min\lbrace4,e^{2\epsilon}\rbrace (e^{\epsilon}-1)^2\\ &\sum_{a=1}^K \mathbb{E}_{\pol \model_1}\left[N_a(T)\right]\kldiv{f_a \in \model_1}{f_a \in \model_2}.
\end{align*}
\end{lemmarep}
Now, as we derive this lower bound on sum of regrets in these two environments. We focus on specifying the environments.
Specifically, we choose such suboptimality gaps among rewards such that we can get a lower bound on the best regret in the worst environment.
\begin{theoremrep}[Local Private Minimax Regret Bound
Given an $\epsilon$-locally private reward generation mechanism with $\epsilon \in \real$, and a time horizon $T \geq g(K,\epsilon)$, then for any environment with finite variance, any algorithm $\pol$ satisfies\vspace{-1em}
\begin{equation}
\reg_{\mathrm{minimax}}(T) \geq \frac{c}{\min\lbrace 2,e^{\epsilon}\rbrace(e^{\epsilon}-1)}\sqrt{(K-1)T}.
\end{equation}\vspace*{-1.5em}
\end{theoremrep}
For small $\epsilon$, $e^{\epsilon}-1\approx \epsilon$. Thus, for small $\epsilon$, the minimax regret bound for local privacy worsens by a multiplicative factor $\frac{1}{\epsilon{e^{\epsilon}}}$. If the $\epsilon = 0$ which means the rewards obtained are completely randomised, the arms would not be separable any more and would lead to unbounded minimax regret.
Following this, we establish a lower bound for Bayesian regret of local private bandits.
In the Bayesian setup, the bandit algorithm assumes a prior distribution $Q_0$ over the possible environments $\nu \in \mathcal{E}$. As the algorithm $\pi$ plays further and observe corresponding rewards at each time $t$, it updates the prior over the possible environments to a posterior distribution $Q_t$. This framework is adopted for efficient algorithms like Thompson sampling~\citep{thompson1933} and Gittins indices~\citep{gittins1989multi}. In the Bayesian setting, we define the Bayesian regret as $\reg_{\mathrm{Bayes}}(\pol, T,Q) \defn \int_{\model^T} \reg(\pol, \model, T)dQ(\model)$.
\iffalse
Bayesian regret is a weaker measure than minimax regret as Bayesian regret is the average regret over believed environments whereas minimax regret is the worst-case regret. \cdcomment{I don't think this is true. If the possible priors have full support over all singular model distributions then the Bayesian regret is higher the classical minimax regret:
\[
\max_P f(P) = \max_P \int_\Theta f(\theta) dP(\theta) \geq \max_\theta f(\theta),
\]
as for every $\theta$ there is one $P$ such that $P(\theta) = 1$.
}\dbcomment{For me this choice of prior is like setting Bayesian minimax regret. I see the last statement was loose. Should remove it from the text but shall preserve the comment for my note.}
Bayesian minimax regret~\cite{lattimore2018bandit} is defined as the worst possible Bayesian regret for any prior: $\reg^*_{\mathrm{Bayes}}(T) \defn \min_{\pol}\max_{Q}\reg_{\mathrm{Bayes}}(\pol, T,Q)$.
\fi
\begin{corollaryrep}[Local Private Bayesian Minimax Regret Bound
Given an $\epsilon$-locally private reward generation mechanism with $\epsilon \in \real$, and a finite time horizon $T \geq g(K,\epsilon)$, then for any environment with bounded rewards $\br \in [0,1]^K$, any algorithm $\pol$ satisfies\vspace{-.5em}
\begin{equation}
\reg^*_{\mathrm{Bayes}}(T) \geq \frac{c}{\min\lbrace 2,e^{\epsilon}\rbrace(e^{\epsilon}-1)}\sqrt{(K-1)T}.
\end{equation}\vspace*{-1.5em}
\end{corollaryrep}
Both minimax and Bayesian minimax regret bounds are problem independent. They represent the worst-case regret for any environment and any prior over environments respectively. Someone may want to design algorithms that is optimal for a given environment $\nu$ and the minimax and Bayesian minimax bounds are too pessimistic for them. Thus, researchers study the problem-dependent lower bounds of regret involving environment dependent quantities. \cite{lai1985asymptotically} proved that a bandit algorithm achieves $\Omega(\log T)$ problem-dependent lower bound. We prove that for $\epsilon$-local privacy this lower bound worsens by a multiplicative factor $\frac{1}{e^{2\epsilon}(e^{\epsilon}-1)^2}$.
\begin{theoremrep}[Problem-dependent Local-Private Regret Bound
For any asymptotically consistent bandit algorithm $\pol$, an environment $\model$ with optimal reward distribution $f^*$, and an $\epsilon$-locally private reward generation mechanism, the expected cumulative regret\vspace*{-.5em}
\begin{equation*}\hspace{-1em}
\liminf_{T \rightarrow \infty} \frac{\reg(\pol, \model, T)}{\log T} \geq \sum_{a \neq a^*} \frac{\Delta_a}{2\min\lbrace4,e^{2\epsilon}\rbrace(e^{\epsilon}-1)^2 \kldiv{f_a}{f^*}}.
\end{equation*}\vspace*{-1em}
\end{theoremrep}
\paragraph{Proof Details.}
\else
\
\fi
\begin{lemmarep}[Locally Private KL-divergence Decomposition]
If the reward generation process is $\epsilon$-local differentially private for both the environments $\model_1$ and $\model_2$
\begin{align*
\begin{split}
&\kldiv{\prob^T_{{\model_1}\pol}}{\prob^T_{{\model_2}\pol}} \leq 2\min\lbrace4,e^{2\epsilon}\rbrace (e^{\epsilon}-1)^2 \sum_{a=1}^K \mathbb{E}_{\pol \model_1}\left[N_a(T)\right]\kldiv{f_a \in \model_1}{f_a \in \model_2}.
\end{split}
\end{align*}
\end{lemmarep
\begin{proof}
In case of local privacy, the reward observed by the algorithm $\pi$ is obtained at time $t$, $\CX_t$, from a privatised version of generated rewards $\BZ_t$. Thus, $x_t = z_{t,a}$, where $a$ refers to the action selected at time $t$. We denote the distribution over the privatised generated reward of arm $a$ as $g_a(z)$. $g_a(z)$ is obtained by imposing a local privacy mechanism $\mech$ on the original reward distribution $f_a(z)$.
We note that the KL-divergence decomposition of Lemma~\ref{thm:kldecomp} is obtained on the probability space over observed histories. Since the observed rewards are now coming from $g_a(z)$ rather than $f_a(x)$, we begin our derivations of local-private KL-divergence decomposition by substituting $g_a(z)$ in Equation~\ref{eqn:kldecomp}. For brevity, we denote $g_a^1$ and $g_a^2$ to represent the privatised reward distributions for arm $a$ in environments $\model_1$ and $\model_2$ respectively. Thus,
\begin{align*
\kldiv{\prob^T_{\pol{\model_1}}}{\prob^T_{\pol{\model_2}}}
\underset{(a)}{=} &\sum_{t=1}^T \mathbb{E}_{\pol\model_1}\left[\kldiv{\pol(A_t|\mathcal{H}_t, \model_1)}{\pol(A_t|\mathcal{H}_t, \model_2)}\right] + \sum_{a=1}^K \mathbb{E}_{\pol\model_1}\left[N_a(T)\right]\kldiv{g_a^1(Z)}{g_a^2(Z)}\\
\underset{(b)}{=} & \sum_{a=1}^K \mathbb{E}_{\pol\model_1}\left[N_a(T)\right]\kldiv{g_a^1(Z)}{g_a^2(Z)}\\
\underset{(c)}{\leq} & \sum_{a=1}^K \mathbb{E}_{\pol\model_1}\left[N_a(T)\right]\left[\kldiv{g_a^1(Z)}{g_a^2(Z)}+\kldiv{g_a^2(Z)}{g_a^1(Z)}\right]\\
\underset{(d)}{\leq} &\min\lbrace4,e^{2\epsilon}\rbrace (e^{\epsilon}-1)^2 \sum_{a=1}^K \mathbb{E}_{\pol \model_1}\left[N_a(T)\right]\lp{f_a^1(X)}{f_a^2(X)}{TV}^2\\
\underset{(e)}{\leq} &2\min\lbrace4,e^{2\epsilon}\rbrace (e^{\epsilon}-1)^2 \sum_{a=1}^K \mathbb{E}_{\pol \model_1}\left[N_a(T)\right]\kldiv{f_a^1(X)}{f_a^2(X)}
\end{align*}
Step (a) is from the generic KL-divergence decomposition in Lemma~\ref{thm:kldecomp}.
Step (b) is due to the fact that given the same history, $\pol(A_t|\mathcal{H}_t, \model_1)=\pol(A_t|\mathcal{H}_t, \model_2)$ as they do not vary with the model and depends only on the internal randomisation of the algorithm $\pi$.
The inequality in step (c) is derived from non-negativity of KL-divergence~\citep{cover2012elements} and the fact that for two non-negative numbers $a$ and $b$, $a \leq a+b$.
The inequality in the step (d) is obtained from Theorem 1 in~\citep{duchi2013local}.
The inequality in step (e) is obtained by applying Pinsker's inequality~\citep{cover2012elements}. Pinsker's inequality states that for any two distributions $P$ and $Q$, square of their total variance distance is upper bounded by $2$ times their KL-divergence: $\lp{P}{Q}{TV}^2 \leq 2\kldiv{P}{Q}$.
\end{proof}
\begin{theoremrep}[Locally Private Minimax Regret Bound]
Given an $\epsilon$-locally private reward generation mechanism with $\epsilon \in \real$, and a time horizon $T \geq g(K,\epsilon)$, then for any environment with finite variance, any algorithm $\pol$ satisfie
\begin{equation}
\reg_{\mathrm{minimax}}(T) \geq \frac{c}{\min\lbrace 2,e^{\epsilon}\rbrace(e^{\epsilon}-1)}\sqrt{(K-1)T}.
\end{equation}
\end{theoremrep}
\begin{proof}\ \\%[Proof Sketch]
\textbf{Step 1:} Fix two environments $\model_1$ and $\model_2$ such that drawing arm $1$ in $\model_1$ for more than $T/2$ times is good but doing the same is bad for $\model_2$.
We define $\model_1$ to be a set of $K$-reward distribution with mean reward $\lbrace \Delta, 0, \ldots, 0\rbrace$ and finite Fisher information $I$. Similarly, we define $\model_2$ to be to be a set of $K$-reward distribution with mean reward $\lbrace \Delta, \ldots, 0, 2\Delta\rbrace$ and finite Fisher information $I$. Drawing arm $1$ is the optimal choice in $\model_1$ whereas drawing arm $K$ is the optimal choice in $\model_2$.
\textbf{Step 2:} We get the lower bounds of expected cumulative regret for a policy $\pol$, and the environments $\model_1$ and $\model_2$ as follows:
\begin{align*}
\reg(\pol, \model_1, T) &\geq \prob_{\pol\model_1}\left(N_1(T) \leq T/2\right) \frac{T\Delta}{2},\\
\reg(\pol, \model_2, T) &> \prob_{\pol\model_2}\left(N_1(T) > T/2\right) \frac{T\Delta}{2}.
\end{align*}
Let us denote the event $N_1(T) \leq T/2$ as $E$.
Hence, we get
\begin{align}
\reg(\pol, \model_1, T) + \reg(\pol, \model_2, T) &\geq \frac{T\Delta}{2} \left(\prob_{\pol\model_1}(E) + \prob_{\pol\model_2}(E^C)\right) \notag \\
&\geq \frac{T\Delta}{4} \exp(-\kldiv{\prob_{\pol\model_1}}{\prob_{\pol\model_2}}).\label{eqn:regret_to_KL}
\end{align}
We obtain the last inequality from the Lemma 2.1 in~\citep{bretagnolle1979estimation}. This is called Bretagnolle-Huber inequality or probabilistic Pinsker's inequality~\citep{auer2002nonstochastic,lattimore2019information} and used for several proofs of bandit algorithms. This states that for any two distributions $P$ and $Q$ defined on the same measurable space and an event $E$, $P(E)+Q(E^C) \geq \exp(-\kldiv{P}{Q})$.
\textbf{Step 3:} From Lemma~\ref{thm:kldecomp_local}, we get
\begin{align}
\kldiv{\prob_{\pol\model_1}}{\prob_{\pol\model_2}}
&= \expect_{\pol\model_1}[N_K(T)]\kldiv{\mech(f_K(0,I))}{\mech(f_K(2\Delta,I))} \notag\\
&\underset{(a)}{\leq} 2\min\lbrace4,e^{2\epsilon}\rbrace (e^{\epsilon}-1)^2 \mathbb{E}_{\pol\model_1}\left[N_K(T)\right]\kldiv{f_K(0,I)}{f_K(2\Delta,I)} \notag\\
&\leq 2\min\lbrace4,e^{2\epsilon}\rbrace (e^{\epsilon}-1)^2 \mathbb{E}_{\pol\model_1}\left[N_K(T)\right] (2\Delta^2) \notag\\
&\leq 4\min\lbrace4,e^{2\epsilon}\rbrace (e^{\epsilon}-1)^2 \frac{T\Delta^2}{K-1}. \label{eqn:kl_local}
\end{align}
Step (a) is a consequence of Lemma~\ref{thm:kldecomp_local}.
From Equations~\eqref{eqn:regret_to_KL} and~\eqref{eqn:kl_local}, we obtain the regret bound to be
\begin{align*}
\max\lbrace \reg(\pol, \model_1, T), \reg(\pol, \model_2, T) \rbrace
&\geq \frac{1}{2}\left(\reg(\pol, \model_1, T) + \reg(\pol, \model_2, T)\right)\\
&\geq \frac{T\Delta}{4}\exp\left[-4\min\lbrace4,e^{2\epsilon}\rbrace (e^{\epsilon}-1)^2 \frac{T\Delta^2}{K-1}\right].
\end{align*}
\textbf{Step 4:} Let us choose the optimality gap $\Delta$ to be $\sqrt{\frac{(K-1)}{\min\lbrace 4,e^{2\epsilon}\rbrace (e^{\epsilon}-1)^2 T}} \leq \frac{1}{2}$. This holds for any for $T \geq \frac{(K-1)}{\min\lbrace 4,e^{2\epsilon}\rbrace (e^{\epsilon}-1)^2} \defn g(K,\epsilon)$. Hence, by using the results of Step 3, we obtain:
\begin{equation*}
\reg_{\mathrm{minimax}}(T) \geq \frac{1}{4e^{4}} \sqrt{\frac{(K-1)T}{\min\lbrace 4,e^{2\epsilon}\rbrace (e^{\epsilon}-1)^2}} = \dfrac{\sqrt{(K-1)T}}{4e^{4}\min\lbrace 2, e^{\epsilon}\rbrace (e^{\epsilon}-1)}.
\end{equation*}
For $0 < c\leq \frac{1}{4e^{4}}$, we obtain the theorem statement.
\end{proof}
\begin{corollaryrep}[Locally Private Bayesian Minimax Regret Bound]
Given an $\epsilon$-locally private reward generation mechanism with $\epsilon \in \real$, and a finite time horizon $T \geq g(K,\epsilon)$, then for any environment with bounded rewards $\br \in [0,1]^K$, any algorithm $\pol$ satisfies\vspace{-.5em}
\begin{equation}
\reg^*_{\mathrm{Bayes}}(T) \geq \frac{c}{\min\lbrace 2,e^{\epsilon}\rbrace(e^{\epsilon}-1)}\sqrt{(K-1)T}.
\end{equation}
\end{corollaryrep}
\begin{proof}
Let us denote the space of all plausible priors for a given bandit problem to be $\mathcal{L} \defn \lbrace \mu \rbrace$.
\cite{lattimore2019information} proves in their recent paper that if $\mathcal{L}$ is the space of all finitely supported probability measures on $[0,1]^{KT}$, the Bayesian regret and the minimax regret would be the same.
\begin{fact}[Theorem~1 in~\citep{lattimore2019information}]\label{thm:lattimore}
Let $\mathcal{L}$ be the space of all finitely supported probability measures on $\mathcal{R}^T$, where $\mathcal{R} \defn [0,1]^K$. Then
\begin{equation*}
\reg_{\mathrm{minimax}}(T) = \reg^*_{\mathrm{Bayes}}(T).
\end{equation*}
\end{fact}
Since variance of bounded random variable in $[0,1]$ is less than $\frac{1}{4}$, the bounded reward assumption satisfies the finite variance requirement of Theorem~\ref{thm:lb_local}.
Thus, the results of Theorem~\ref{thm:lb_local} and Fact~\ref{thm:lattimore} prove the statement of Corollary~\ref{cor:lp_bayes} for bounded rewards.
\end{proof}
\begin{theorem}[Problem-dependent Local-Private Regret Bound (Theorem~\ref{thm:lb_log_local} in text)
For any asymptotically consistent bandit algorithm $\pol$, an environment $\model$ with optimal reward distribution $f^*$, and an $\epsilon$-locally private reward generation mechanism, the expected cumulative regret
\begin{equation*
\liminf_{T \rightarrow \infty} \frac{\reg(\pol, \model, T)}{\log T} \geq \sum_{a \neq a^*} \frac{\Delta_a}{2\min\lbrace4,e^{2\epsilon}\rbrace(e^{\epsilon}-1)^2 \kldiv{f_a}{f^*}}.
\end{equation*}
\end{theorem}
\begin{remark}
Step 1 and Step 2 of this proof are similar in essence as that of Theorem~\ref{thm:lb_local}. Step 3 differs from the fact that rather than substituting the KL-divergence and the expected number of draws using the problem-independent terms, we keep the problem dependent terms. This leads to a problem-dependent bound in Step 4.
\end{remark}
\begin{proof}\ \\%[Proof Sketch]
\textbf{Step 1:} Fix two environments $\model_1$ and $\model_2$ such that $\model_1$ contains $K$ reward distributions $\lbrace f_1, \ldots, f_K\rbrace$ and $\model_2$ contains $K-1$ same reward distributions while the reward distribution $i$-th arm $f_i$ is replaced by $f'_i$ such that $\kldiv{f_i}{f'_i} \leq \kldiv{f_i}{f^*} + \delta$ for some $\delta > 0$ and $\mu(f'_i) > \mu(f^*)$. Here, $f^*$ represents the privatised reward distribution obtained for the optimal arm $a^*$.
After imposing the $\epsilon$-local private mechanism, we obtain privatised reward distribution $\lbrace g_1, \ldots, g_i, \ldots, g_K\rbrace$ and $\lbrace g_1, \ldots, g'_i, \ldots, g_K\rbrace$.
Let us denote the expected privatised rewards corresponding to the distributions as $\lbrace \mu_1, \ldots, \mu_K\rbrace$ and $\mu'_i$.
\textbf{Step 2:} We get the lower bounds of expected cumulative regret for a policy $\pol$, and the environments $\model_1$ and $\model_2$ as follows:
\begin{align*}
\reg(\pol, \model_1, T) &\geq \prob_{\pol\model_1}\left(N_1(T) \leq T/2\right) \frac{T}{2} (\mu_i - \mu^*),\\
\reg(\pol, \model_2, T) &> \prob_{\pol\model_2}\left(N_1(T) > T/2\right) \frac{T}{2} (\mu'_i - \mu^*).
\end{align*}
Let us denote the event $N_1(T) \leq T/2$ as $E$.
Hence, we get
\begin{align*}
\reg(\pol, \model_1, T) + \reg(\pol, \model_2, T)
&\geq \frac{T}{2} \left(\prob_{\pol\model_1}(E) (\mu_i - \mu^*) + \prob_{\pol\model_2}(E^C)(\mu'_i - \mu^*)\right)\\
&\geq \frac{T}{2} \left(\prob_{\pol\model_1}(E) + \prob_{\pol\model_2}(E^C)\right)\min \lbrace (\mu_i - \mu^*), (\mu'_i - \mu^*)\rbrace\\
&\underset{(a)}{\geq} \frac{T}{4} \exp(-\kldiv{\prob_{\pol\model_1}}{\prob_{\pol\model_2}})\min \lbrace (\mu_i - \mu^*), (\mu'_i - \mu^*)\rbrace.
\end{align*}
We obtain the inequality (a) from the Lemma 2.1 in~\citep{bretagnolle1979estimation} as mentioned in the proof of Theorem~\ref{thm:lb_local}.
\textbf{Step 3:} From Lemma~\ref{thm:kldecomp_local}, we get
\begin{align*}
\kldiv{\prob_{\pol\model_1}}{\prob_{\pol\model_2}}
&= \expect_{\pol\model_1}[N_i(T)]\kldiv{g_i}{g'_i}\\
&\underset{(b)}{\leq} 2\min\lbrace4,e^{2\epsilon}\rbrace (e^{\epsilon}-1)^2 \mathbb{E}_{\pol\model_1}\left[N_i(T)\right]\kldiv{f_i}{f'_i}\\
&\leq 2\min\lbrace4,e^{2\epsilon}\rbrace (e^{\epsilon}-1)^2 \mathbb{E}_{\pol\model_1}\left[N_i(T)\right] (\kldiv{f_i}{f^*} + \delta)
\end{align*}
Step (b) is obtained from Lemma~\ref{thm:kldecomp_local}.
Hence, we get the regret bound to be
\begin{align*}
&\reg(\pol, \model_1, T) + \reg(\pol, \model_2, T)\\
\geq &\frac{T}{4}\min \lbrace (\mu_i - \mu^*), (\mu'_i - \mu^*)\rbrace\exp\left[-2\min\lbrace4,e^{2\epsilon}\rbrace (e^{\epsilon}-1)^2 \mathbb{E}_{\pol\model_1}\left[N_i(T)\right] (\kldiv{f_i}{f^*} + \delta)\right].
\end{align*}
Taking logarithm on both sides and simplifying, we get
\begin{align*}
\quad\frac{\expect_{\pol\model_1}\left[N_i(T)\right]}{\log T}
&\geq \frac{1}{2 \min\lbrace4,e^{2\epsilon}\rbrace (e^{\epsilon}-1)^2 (\kldiv{f_i}{f^*} + \delta)}\frac{\log\left(\frac{T \min \lbrace (\mu_i - \mu^*), (\mu'_i - \mu^*)\rbrace}{4(\reg(\pol, \model_1, T) +\reg(\pol, \model_2, T))}\right)}{\log T}\\
&\geq \dfrac{1+\frac{\log\left(0.25 \min \lbrace (\mu_i - \mu^*), (\mu'_i - \mu^*)\rbrace\right)}{\log T}-\frac{\log\left(\reg(\pol, \model_1, T) + \reg(\pol, \model_2, T)\right)}{\log T}}{2 \min\lbrace4,e^{2\epsilon}\rbrace (e^{\epsilon}-1)^2 (\kldiv{f_i}{f^*} + \delta)}.
\end{align*}
\textbf{Step 4:} We obtain the asymptotic lower bound by taking limit inferior on both sides
\begin{align*}
&\quad\liminf_{T \rightarrow \infty} \frac{\expect_{\pol\model_1}\left[N_i(T)\right]}{\log T}\\
&\geq \liminf_{T \rightarrow \infty} ~~(2 \min\lbrace4,e^{2\epsilon}\rbrace (e^{\epsilon}-1)^2 (\kldiv{f_i}{f^*} + \delta))^{-1}\\
&\qquad\qquad~~[1+\frac{\log\left(0.25 \min \lbrace (\mu_i - \mu^*), (\mu'_i - \mu^*)\rbrace\right)}{\log T} -\frac{\log\left(\reg(\pol, \model_1, T) + \reg(\pol, \model_2, T)\right)}{\log T}]\\
&= (2 \min\lbrace 4,e^{2\epsilon}\rbrace (e^{\epsilon}-1)^2 (\kldiv{f_i}{f^*} + \delta))^{-1}\\
&\qquad[1+ \liminf\limits_{T \rightarrow \infty} \frac{\log\left(0.25 \min \lbrace (\mu_i - \mu^*), (\mu'_i - \mu^*)\rbrace\right)}{\log T} - \limsup\limits_{T \rightarrow \infty} \frac{\log\left(\reg(\pol, \model_1, T) + \reg(\pol, \model_2, T)\right)}{\log T}]\\
&\underset{(b)}{\geq} \dfrac{1 - p}{2 \min\lbrace 4,e^{2\epsilon}\rbrace (e^{\epsilon}-1)^2 (\kldiv{f_i}{f^*} + \delta)}\\
&\underset{(c)}{\geq} \dfrac{1}{2 \min\lbrace 4,e^{2\epsilon}\rbrace (e^{\epsilon}-1)^2 \kldiv{f_i}{f^*}}.
\end{align*}
We obtain inequality (b) because:
1. The first limit contains a constant in the numerator. Thus the limit goes to $0$ as $T \rightarrow \infty$.
2. In order to obtain the other limit. We use the asymptotic consistency property of $\pi$. Since $\pi$ is assumed to be asymptotically consistent, $\reg(\pol, \model_1, T) + \reg(\pol, \model_2, T) \leq C T^p$ for some constant $p \in (0,1)$ and large enough time horizon $T$. Thus, $\limsup\limits_{T \rightarrow \infty} \frac{\log\left(\reg(\pol, \model_1, T) + \reg(\pol, \model_2, T)\right)}{\log T} \leq \limsup\limits_{T \rightarrow \infty} \frac{p \log T + \log C}{\log T} = p$.
We obtain the inequality (c) in two steps: first, $1\geq p > 0$ and second, we take the limit as $\delta \rightarrow 0$.
\textbf{Step 5:} Using the definition of regret and the resulting inequality of Step 4, we obtain
\begin{align*}
\liminf_{T \rightarrow \infty} \frac{\reg(\pol, \model_1, T)}{\log T}
&= \liminf_{T \rightarrow \infty} \sum_{a\neq a^*} \frac{\expect_{\pol\model_1}\left[N_a(T)\right] (\mu_a - \mu^*)}{\log T}\\
&\geq \sum_{a\neq a^*} \dfrac{(\mu_a - \mu^*)}{2 \min\lbrace 4,e^{2\epsilon}\rbrace (e^{\epsilon}-1)^2 \kldiv{f_a}{f^*}}\\
&= \sum_{a\neq a^*} \dfrac{\Delta_a}{2 \min\lbrace 4,e^{2\epsilon}\rbrace (e^{\epsilon}-1)^2 \kldiv{f_a}{f^*}}.
\end{align*}
\end{proof}
\textbf{Summary:} \textit{These results establish that the lower bounds for both minimax and Bayesian minimax regret degrade by a multiplicative factor $\dfrac{1}{\min\lbrace 2,e^{\epsilon}\rbrace (e^{\epsilon}-1)}$ whereas the problem-dependent lower bound degrades by a multiplicative factor $\dfrac{1}{2\min\lbrace 4,e^{2\epsilon}\rbrace (e^{\epsilon}-1)^2}$ in an $\epsilon$-LDP bandit problem.}
\ifdefined \workshop
\
\else
\subsection{Proofs for Instantaneous Differential Privacy (IDP)}
\iffalse
\paragraph{Sketch of Results and Discussion.}
We proceed in the same framework as the local DP proofs.
We begin with proving an upper bound on KL-divergence for instantaneous DP.
\begin{lemmarep}[Instantaneous DP KL-divergence Decomposition
For a private bandit algorithm $\pol$ satisfying $l(T) \leq \expect_{\pol\model}[N_a(T)]$ for any arm $a$,
and two environments $\model_1$ and $\model_2$
\begin{align*}
\begin{split}
\kldiv{\prob^T_{\pol{\model_1}}}{\prob^T_{\pol{\model_2}}} &\leq 2\epsilon(e^{2\epsilon}-1)\frac{1-2e^{-\frac{T}{l(T)}}}{1 - e^{-\frac{T}{l(T)}}}\\
&+ \sum_{a=1}^K \mathbb{E}_{\model_1}\left[N_a(T)\right](\kldiv{f_a \in \model_1}{f_a \in \model_2}).
\end{split}
\end{align*}
\end{lemmarep}
\begin{theoremrep}[Instantaneous DP Minimax Regret Bound
Given a finite privacy level $\epsilon \leq a/2$, and a time horizon $T \geq h(K,\epsilon)$, then for any environment with finite variance, any algorithm $\pol$ that is $\epsilon$-DP satisfie
\begin{equation}
\reg_{\mathrm{minimax}}(T) \geq c(a)
\sqrt{\frac{(K-1)T}{2\epsilon(e^{2\epsilon}-1)}}.
\end{equation}\vspace*{-1.5em}
\end{theoremrep}
This implies that for small $\epsilon$, the minimax regret bound for $\epsilon$-DP worsens by a multiplicative factor $\frac{1}{\epsilon}$. Thus, the lower bound of minimax regret for DP bandits is better than LDP bandits by factors $e^{\epsilon/2}$ and $\sqrt{\frac{2}{\epsilon(e^{\epsilon} +1)}}$ for small and large $\epsilon$'s respectively.
\begin{corollaryrep}[Instantaneous DP Bayesian Minimax Regret Bound
Given a finite privacy level $\epsilon \in \real$, and a finite time horizon $T \geq h(K,\epsilon)$, then for any environment with finite variance and bounded reward $\br \in [0,1]^K$, any algorithm $\pol$ that is $\epsilon$-DP satisfies\vspace*{-.5em}
\begin{equation}
\reg^*_{\mathrm{Bayes}}(T) \geq c(\epsilon)
\sqrt{\frac{(K-1)T}{2\epsilon(e^{2\epsilon}-1)}}.
\end{equation}\vspace*{-2em}
\end{corollaryrep}
\paragraph{Proof Details.}
\fi
\begin{lemmarep}[Instantaneous DP KL-divergence Decomposition]
For an $\epsilon$-IDP bandit algorithm $\pol$ satisfying $l(T) \leq \expect_{\pol\model}[N_a(T)]$ for any arm $a$,
and two environments $\model_1$ and $\model_2$
\begin{align*}
\begin{split}
\kldiv{\prob^T_{\pol{\model_1}}}{\prob^T_{\pol{\model_2}}} &\leq 2\epsilon(e^{2\epsilon}-1)\frac{1-2e^{-\frac{T}{l(T)}}}{1 - e^{-\frac{T}{l(T)}}}+ \sum_{a=1}^K \mathbb{E}_{\pol \model_1}\left[N_a(T)\right](\kldiv{f_a \in \model_1}{f_a \in \model_2}).
\end{split}
\end{align*}
\end{lemmarep}
\begin{proof
The second term in the generic KL-divergence decomposition (Lemma~\ref{thm:kldecomp}) remains the same.
The first term is bounded as follows:
\begin{align*}
\sum_{t=1}^T \kldiv{\pol(A_t|\mathcal{H}^t, \model_1)}{\pol(A_t|\mathcal{H}^t, \model_2)}
&= \sum_{t=1}^T \abs{\kldiv{\pol(A_t|\mathcal{H}^t, \model_1)}{\pol(A_t|\mathcal{H}^t, \model_2)}}\\
&\leq \sum_{t=1}^T\max_{A_t,\mathcal{H}_t}\abs{\pol(A_t|\mathcal{H}^t, \model_1)}\abs{\log\frac{\pol(A_t|\mathcal{H}^t, \model_1)}{\pol(A_t|\mathcal{H}^t, \model_2)}}\\
&\leq 2\epsilon(e^{2\epsilon}-1) \sum_{t=1}^T \max_{A_t,\mathcal{H}_t} \pol(A_t|\mathcal{H}^t, \model_1)\\
&\leq 2\epsilon(e^{2\epsilon}-1) \max_{A_t,\mathcal{H}_t} \sum_{t=1}^T \pol(A_t|\mathcal{H}^t, \model_1)\\
&\leq 2\epsilon(e^{2\epsilon}-1)\frac{1-2e^{-\frac{T}{l(T)}}}{1 - e^{-\frac{T}{l(T)}}}
\end{align*}
for $\expect_{\pol \model_1}[N_a(T)] \geq l(T)$ for all arms $a$.
\end{proof}
\begin{theoremrep}[Instantaneous DP Minimax Regret Bound]
Given a finite privacy level $\epsilon \leq a/2$, and a time horizon $T \geq h(K,\epsilon)$, then for any environment with finite variance, any algorithm $\pol$ that is $\epsilon$-IDP satisfies
\begin{equation*}
\reg_{\mathrm{minimax}}(T) \geq c(a)
\sqrt{\frac{(K-1)T}{2\epsilon(e^{2\epsilon}-1)}}.
\end{equation*}
\end{theoremrep}
\begin{proof}\ \\%[Proof Sketch]
Repeat the \textbf{Steps 1} and \textbf{2} described in the proof sketch of Theorem~\ref{thm:lb_local}.
\textbf{Step 3:} Use Lemma~\ref{thm:kldecomp_global} for KL-divergence decomposition under differential privacy to obtain
\begin{align*}
\kldiv{\prob_{\pol\model_1}}{\prob_{\pol\model_2}} \leq 2\epsilon(e^{2\epsilon}-1) + \frac{2T\Delta^2}{K-1}.
\end{align*}
Hence, we get the regret bound to be
\begin{align*}
\max\lbrace \reg_{\model_1}(\pol, T), \reg_{\model_2}(\pol, T) \rbrace
&\geq \frac{1}{2}\left(\reg_{\model_1}(\pol, T) + \reg_{\model_2}(\pol, T)\right)\\
&\geq \frac{T\Delta}{4}\exp\left[-2\epsilon(e^{2\epsilon}-1) - \frac{2T\Delta^2}{K-1}\right].
\end{align*}
\textbf{Step 4:} Let us choose the optimality gap $\Delta$ to be $$\Delta = \sqrt{\frac{(K-1)C(\epsilon)}{4\epsilon(e^{2\epsilon}-1) T}} \leq \frac{1}{2}.$$
Here, we choose $$C(\epsilon) = - 2\epsilon(e^{2\epsilon}-1) W\left(-\frac{e^{-\delta(a)+2\epsilon(e^{2\epsilon}-1)}}{2\epsilon(e^{2\epsilon}-1)}\right),$$ where $W$ is the Lambert function or the product-log function and $\delta(a)$ is a function of $a$ such that $2\epsilon(e^{2\epsilon}-1) \leq \delta(a)$ for $\epsilon \leq a$. In the given range of $\epsilon$, $C(\epsilon) \leq 1$.
$\Delta$ being less than $\frac{1}{2}$ holds for any for $T \geq \frac{2(K-1)C(\epsilon)}{4\epsilon(e^{2\epsilon}-1)} \defn h(K,\epsilon)$. Hence, by combining the results of Step 3 and the upper bound on privacy level $\epsilon \leq a$ and $T \geq h(K,\epsilon)$, we obtain:
\begin{equation}
\reg_{minimax}(T) \geq \frac{e^{\delta(a)}}{4\sqrt{2}} \sqrt{\frac{(K-1)T}{\min\lbrace 2,e^{\epsilon}\rbrace (e^{2\epsilon}-1)}}.
\end{equation}
\end{proof}
\begin{corollaryrep}[Instantaneously Private Bayesian Minimax Regret Bound]
Given a finite privacy level $\epsilon \in \real$, and a finite time horizon $T$, then for any environment with bounded reward $\br \in [0,1]^K$, any algorithm $\pol$ that is $\epsilon$-IDP satisfies
\begin{equation*}
\reg^*_{\mathrm{Bayes}}(T) \geq c(\epsilon)
\sqrt{\frac{(K-1)T}{2\epsilon(e^{2\epsilon}-1)}}.
\end{equation*}
\end{corollaryrep}
\begin{proof}
Similar to the proof of Corollary~\ref{cor:lp_bayes}, the results of Theorem~\ref{thm:lb_global} and Fact~\ref{thm:lattimore} prove the statement of Corollary~\ref{cor:lb_bayes_global}.
\end{proof}
\fi
\ifdefined \workshop
\subsection{Lower Bounds for Differential Privacy (DP)}
\else
\subsection{Proofs for Differential Privacy (DP)}
\fi
There are two possibilities under DP which basically takes same set of $T$ actions $a^T$ for two neighbouring set of generated rewards $\bx^T$ and $\bx'^T$ with a difference of $e^{\epsilon}$.
In the first case, the one differently generated reward at step $k$, $x'_{k, a}$ does not come from the chosen arm i.e. $a_k \neq a$. In this case, the bound becomes trivial and derivation goes back to the classical version.
The second case is more interesting one. In this case, the arm $a$ is chosen at step $k$ and the differently generated reward $x'_{k, a}$ becomes the observed reward $r'_k$.
We derive all the following bounds under this second scenario.
\begin{lemma}\label{lemma:hist_dp}
If a policy $\pi$ satisfy $\epsilon$-DP for an environment $\model$ and $\ln \sup_{a, x_a,x'_a} \frac{\prob_{\model}(x_a)}{\prob_{\model}(x'_a)} \leq L \Delta$, we get
\begin{equation}
\prob_{\model \pol}(\Hist_T) \leq e^{\epsilon + L\Delta} \prob_{\model \pol}(\Hist'_T).
\end{equation}
Here, $L$ is the Lipschitz constant corresponding to the reward distributions in the environment and $\Delta \triangleq \max \rho(x_a, x'_a)$.
\end{lemma}
\begin{proof}
History till horizon $T$ is a collection of actions and observed rewards $\Hist_T = (a^T, r^T) \triangleq \lbrace a_1, \ldots, a_T, r_1, \ldots, r_k, \ldots, r_T\rbrace = \lbrace a_1, \ldots, a_T, x_{1, a_1}, \ldots, x_{k, a_k}, \ldots, x_{T, a_T}\rbrace$. Thus, we can define a neighbouring history for DP definition as $\Hist_T = (a^T, r'^T) \triangleq \lbrace a_1, \ldots, a_T, r_1, \ldots, r'_k, \ldots, r_T\rbrace$.
$S$ and $S'$ denote sets of all $\bx^T$ and $\bx'^T$s for which we get $r^T$ and $r'^T$ as the rewards sequence given a sequence of actions $a^T$.
We observe that $S \cup \{\bx_k(a_k) = r'_k\} = S' \cup \{\bx_k(a_k) = r_k\} = \real^{KT}$.
So, the sets of generated rewards that lead to two neighbouring reward sequences comes from two subsets of $\real^{KT}$ with two specific punctures at $r'_k$ and $r_k$ respectively.
With this underlying structure, we now derive the relation between the densities induced by two neighbouring histories.
\begin{align*}
\prob_{\model \pol}(\Hist_T)
&= \int_S \prob_{\pol}(\Hist_T| \bx^T) d\prob_{\model}(\bx^T) \\
&= \int_S \prob_{\pol}(a^T, r^T| \bx^T) d\prob_{\model}(\bx^T) \\
&= \int_S \prob_{\pol}(a^T | \bx^T) \prob_{\pol}(r^T | a^T, \bx^T) d\prob_{\model}(\bx^T) \\
&\underset{(a)}{=} \int_{S} \prob_{\pol}(a^T | \bx^T ) \prob(r'^T | a^T, \bx^T + \Delta_k) d\prob_{\model}(\bx^T) e^{L\Delta_k}\\
&\underset{(b)}{\leq} \int_{S} e^{\epsilon} \prob_{\pol}(a^T | \bx^T + \Delta ) \prob(r'^T | a^T, \bx^T + \Delta_k) d\prob_{\model}(\bx^T) \\
&\underset{(b)}{\leq} e^{\epsilon} \int_{S} \prob_{\pol}(a^T | \bx^T + \Delta_k ) \prob(r'^T | a^T, \bx^T + \Delta_k)~e^{L\abs{\Delta_k}} d\prob_{\model}(\bx^T + \Delta_k) \\
&\underset{(d)}{=} e^{\epsilon+L\abs{\Delta_k}} \int_{S'} \prob_{\pol}(a^T | \bx'^T) \prob(r'^T | a^T, \bx'^T) d\prob_{\model}(\bx'^T)\\
&\leq e^{\epsilon+L\Delta} \prob_{\model \pol} (\Hist'_T)
\end{align*}
Here, $\Delta_k = r_k - r'_k \neq 0$ and $\Delta = \max |r_k - r'_k|$.
\iffalse
\cdcomment{If $\hat{\bx}^T$ is constant:
\begin{align}
P(H)
&= \int_S \prob_{\pol}(a^T | \bx^T) \prob_{\pol}(r^T | a^T, \bx^T) d\prob_{\model}(\bx^T) \\
& \leq \int_S e^{\epsilon |\bx^T - \hat{\bx}^T|} \prob_{\pol}(a^T | \hat{\bx}^T) \prob_{\pol}(r^T | a^T, \bx^T) d\prob_{\model}(\bx^T) \\
&= \prob_{\pol}(a^T | \hat{\bx}^T) \int_S e^{\epsilon |\bx^T - \hat{\bx}^T|} \prob_{\pol}(r^T | a^T, \bx^T) d\prob_{\model}(\bx^T)
\end{align}
If $\hat{\bx}^T = f(\bx)^T$:
\begin{align}
P(H)
&= \int_S \prob_{\pol}(a^T | \bx^T) \prob_{\pol}(r^T | a^T, \bx^T) d\prob_{\model}(\bx^T) \\
& \leq \int_S e^{\epsilon |\bx^T - \hat{\bx}^T|} \prob_{\pol}(a^T | \hat{\bx}^T) \prob_{\pol}(r^T | a^T, \bx^T) d\prob_{\model}(\bx^T)
\end{align}
and we just don't take the term out of the integral.
}\dbcomment{This discussion leads to the change of variable above.}
\fi
Step (a) is obtained from the fact that the observed rewards $r_t$ and $r'_t$ are deterministic given the corresponding actions and generated rewards.
Inequality in step (b) comes from the definition of $\epsilon$-DP i.e. $\prob_{\pol}(a^T | \bx^T) \leq e^{\epsilon} \prob_{\pol}(a^T | \bx'^T) = e^{\epsilon} \prob_{\pol}(a^T | \bx^T + \Delta_k)$.
The second inequality in step (c) is obtained from the Lipschitzness assumption on the reward distributions of the given environment. Specifically, $\sup_{a} \frac{\prob_{\model}(x_a)}{\prob_{\model}(x'_a)} \leq e^{L |x_a - x'_a|}$, where L is the Lipschitz constant for the corresponding reward distributions.
Finally, step (d) is change of variable in the integral. This shift is possible if and only if the joint probability density on generated reward $\bx$ is smooth and thus non-zero for a given environment with $K$ reward distributions. Thus, both $r_k$ and $r'_k$ has well-defined double sided limit and their corresponding measures under this density function $\prob_{\model}$ are zero\footnote{Note that $\prob_\model \triangleq \prod_{a=1}^K f_a$.}.
\iffalse
Thus, it won't be satisfied by any reward generation distribution such as delta function or Bernoulli that has non-smooth densities or can have zero density for a subset of its support.
\fi
Hence, we get that $\prob_{\model \pol}(\Hist_T) \leq e^{\epsilon + L\Delta} \prob_{\model \pol}(\Hist'_T)$.
\iffalse
\begin{align*}
\frac{\prob_{\model \pol}(\Hist_T)}{\prob_{\model \pol}(\Hist'_T)}
&= \frac{\prob_{\model \pol}(a^T \mid r^{T})}{\prob_{\model \pol}(a^T \mid r'^{T})}\frac{\prob_{\model \pol}(r^{T})}{\prob_{\model \pol}(r'^{T})}\\
&=\frac{\prob_\pol(a^T \mid \bx^{T} \in S)}{\prob_\pol(a^T \mid \bx'^{T} \in S')}\frac{\prob_\pol(r^{T})}{\prob_\pol(r'^{T})}\\
&\underset{(a)}{\leq} e^\epsilon \frac{\prob_{\model \pol}(r^{T})}{\prob_{\model \pol}(r'^{T})}\\
&= e^\epsilon \frac{\int_S P(r^T | \bx^T) P(\bx^T)
}{\int_{S'} P(r'^T | \bx'^T) P(\bx'^T)}\\
&{=} e^\epsilon \frac{\prob_{\model}(x_{k, a_k} = r_k)}{\prob_{\model}(x_{k, a_k} = r'_k)}\\
&\underset{(c)}{\leq} e^{\epsilon + L\Delta_{x}}
\end{align*}
$P(r^T) = \int P(r^T | x^T) P(x^T)
P(r^T | x^T) = \prod_t P(r_t | a_t, x_t) P(a_t | r^{t-1}, a^{t-1})$
The inequality in Step (a) is obtained from Definition~\ref{def:DP_mab}.
\dbcomment{Removed step (b) or similar argument}Step (b) is obtained through a proof analogous to that of post-processing inequality~\citep[Proposition 2.1]{dwork2014algorithmic}.
In order to be specific, we can define a mapping $\phi$ from generated reward to observed reward \cdcomment{But policies are stochastic. Shouldn't you write that there exists at least one $x,x'$ pair for which, given the current actions, you obtain these $r$? So that means
\[
P(r^T) = \int P(r^T, a^T | x^T) dP(x^T)
\]
and
\[
P(r^T, a^T | x^T) = \prod_t P(r_t | a_t, x_t) P(a_t | r^{t-1}, a^{t-1}, x^{t-1})
\]
giving
\[
P(r^T) = \int \prod_t P(r_t | a_t, x_t) P(a_t | r^{t-1}, a^{t-1}, x^{t-1}) dP(x^T)...
\]
You also need this bit:
\[
P(r) = \int P(r | x) P(x) \leq \int P(r | x) P(x') L \Delta
= \int r(x, a) P(a | x) P(x') L \Delta
?=? \int r(x', a) P(a | x') P(x') L \Delta
\]
This is too complicated maybe, but you need to tease out the step where $r(x,a) = r$ and $r(x',a) = r'$.
The definition is just about $\Pr(a^T \mid x^T)$ for differential privacy. Why are you going back to rewards? Why not just talk about $\Pr(a^T \mid x^T)$ directly?
} for a given policy $r_t = \bx_t \mathds{1}_{a_t = a| \pol} \triangleq \phi(\bx_t| \pol)$. Now, we define $T \triangleq \{\bx \in \mathcal{X}: \phi(\bx) \in S\}$ \cdcomment{You are not using this set $T$ anywhere..}for any subset $S \subseteq R$. Then, $\prob_{\model \pol}(r_t) = \prob_{\model}(\phi(\bx_t| \pol)) = \prob_{\model}(\bx_t)$.
Step (c) is obtained from the assumption that the $\sup_a \frac{\prob_{\model}(x_a)}{\prob_{\model}(x'_a)} \leq L \rho(x_a, x_a') \leq L \Delta_x$.
Here, $L$ is the Lipschitz constant of the reward distributions and $\Delta_x$ is the sensitivity of the generated rewards.
Similarly,
\begin{align*}
\frac{\prob_{\model \pol}(\Hist_T)}{\prob_{\model \pol}(\Hist'_T)} &\geq e^{-\epsilon - L\Delta_x}
\end{align*}
Hence, we get that $\prob_{\model \pol}(\Hist_T) \leq \epsilon + L\Delta_x \prob_{\model \pol}(\Hist'_T)$.
\fi
\end{proof}
This assumption on Lipschitz continuity of the reward distribution is used in several occasions in literature.
It does not hold if there are deterministic distributions, like Dirac delta on a single reward value, but for most of the distributions, such as exponential family.
For exponential family, if the sufficient statistics is smooth enough with Lipschitz constant less than or equal to $\frac{L}{\eta}$ for some natural parametrisation $\eta$, we naturally satisfy this assumption.
Thus, this assumption allows us to deal with a large family of reward distributions used in practice, including Gaussian, Beta, Exponential, Bernoulli etc., while allowing the continuation of analysis.
\begin{lemmarep}[Differentially Private KL-divergence Decomposition]\label{thm:kldecomp_DP}
If the policy is $\epsilon$-local differentially private for both the environments $\model_1$ and $\model_2$, we get for two neighbouring histories $\Hist_T$ and $\Hist'_T$
\begin{align*}
\kldiv{\prob_{\pol{\model_1}}(\Hist_T)}{\prob_{\pol{\model_2}}(\Hist_T)} &\leq e^{\epsilon + c} ( 2(\epsilon + c)
+ \kldiv{\prob_{\pol{\model_1}}(\Hist'_T)}{\prob_{\pol{\model_2}}(\Hist'_T)} ).
\end{align*}
Here, $c=L\Delta$ where $L$ is the Lipschitz constant corresponding to the reward distributions of the environments and $\Delta$ is the difference in generated rewards.
\end{lemmarep}
\begin{proof}
The idea is bounding
\[\frac{\prob_\pol(\Hist|\model_1)}{\prob_\pol(\Hist|\model_2)} = \frac{\prob_\pol(\Hist|\model_1)}{\prob_\pol(\Hist'|\model_1)} \frac{\prob_\pol(\Hist'|\model_1)}{\prob_\pol(\Hist'|\model_2)}
\frac{\prob_\pol(\Hist'|\model_2)}{\prob_\pol(\Hist|\model_2)} \leq e^{2(\epsilon + c)} \frac{\prob_\pol(\Hist'|\model_1)}{\prob_\pol(\Hist'|\model_2)}
\]
This will give us the bound on KL:
\begin{align*}
\kldiv{\prob_{\pol{\model_1}}(\Hist_T)}{\prob_{\pol{\model_2}}(\Hist_T)} &= \int d\prob_{\pol{\model_1}}(\Hist_T) \ln \left( \frac{\prob_{\pol{\model_1}}(\Hist_T)}{\prob_{\pol{\model_2}}(\Hist_T)} \right)\\
&= \int d\prob_{\pol{\model_1}}(\Hist_T)\ln \frac{\prob_{\pol{\model_1}}(\Hist_T)}{\prob_{\pol{\model_2}}(\Hist_T)} \\
&\leq \int \ln e^{2\epsilon + 2c}\frac{\prob_{\pol{\model_1}}(\Hist'_T)}{\prob_{\pol{\model_2}}(\Hist'_T)} d\prob_{\pol{\model_1}}(\Hist_T)\\
&= 2(\epsilon+c) + \int \ln \frac{\prob_{\pol{\model_1}}(\Hist'_T)}{\prob_{\pol{\model_2}}(\Hist'_T)} d\prob_{\pol{\model_1}}(\Hist'_T) \frac{d\prob_{\pol{\model_1}}(\Hist_T)}{d\prob_{\pol{\model_1}}(\Hist'_T)}\\
&\leq 2(\epsilon + c) + e^{2(\epsilon + c)} \kldiv{\prob_{\pol{\model_1}}(\Hist'_T)}{\prob_{\pol{\model_2}}(\Hist'_T)}
\end{align*}
\end{proof}
\begin{theoremrep}[DP Minimax Regret Bound
Given a finite privacy level $\epsilon > 0$, and a time horizon $T \geq h(K,\epsilon)$, then for any environment with finite variance, any algorithm $\pol$ that is $\epsilon$-DP satisfies
\begin{align*}
\reg_{\mathrm{minimax}}(T) &\geq \frac{1}{8 e^{3(\epsilon+c)}} \sqrt{\frac{(K-1)T \ln(\epsilon+1) }{\epsilon^{(1+\frac{1}{\epsilon})}(\epsilon^2+1)^{\frac{1}{\epsilon}}}}
\end{align*}
Here, $c=L\Delta$ where $L$ is the Lipschitz constant corresponding to the reward distributions of the environments and $\Delta$ is the difference in generated rewards.
\end{theoremrep}
\begin{proof}\ \\%[Proof Sketch]
Repeat the \textbf{Steps 1} and \textbf{2} described in the proof sketch of Theorem~\ref{thm:lb_local}.
\textbf{Step 3: }Use Lemma~\ref{thm:kldecomp_DP} for KL-divergence decomposition under differential privacy to obtain
\begin{align*}
\kldiv{\prob_{\pol{\model_1}}(\Hist_T)}{\prob_{\pol{\model_2}}(\Hist_T)} &\leq 2(\epsilon + c) + e^{2(\epsilon + c)} \kldiv{\prob_{\pol{\model_1}}(\Hist'_T)}{\prob_{\pol{\model_2}}(\Hist'_T)}\\
&\leq 2(\epsilon + c) + e^{2(\epsilon + c)}\frac{2T\Delta^2}{K-1}
\end{align*}
Hence, we get the regret bound to be
\begin{align*}
\max\lbrace \reg_{\model_1}(\pol, T), \reg_{\model_2}(\pol, T) \rbrace
&\geq \frac{1}{2}\left(\reg_{\model_1}(\pol, T) + \reg_{\model_2}(\pol, T)\right)\\
&\geq \frac{T\Delta}{4}\exp\left[-2(\epsilon + c) - e^{2(\epsilon + c)} \frac{2T\Delta^2}{K-1}\right].
\end{align*}
\textbf{Step 4:} Let us choose the optimality gap $\Delta$ to be $$\Delta = \sqrt{\frac{(K-1) \ln(\epsilon^2+1) }{4T\epsilon e^{2(\epsilon + c)}} } \leq \frac{1}{2}.$$
$\Delta$ being less than $\frac{1}{2}$ holds for any for $T \geq \frac{(K-1) \ln(\epsilon^2+1) }{\epsilon e^{2(\epsilon + c)}} \defn h(K,\epsilon)$.
Hence, by substituting this value of suboptimality gap in the result of Step 3, we obtain for $T \geq h(K,\epsilon)$:
\begin{align*}
\reg_{\mathrm{minimax}}(T) &\geq \frac{1}{8 e^{\epsilon+c}} \sqrt{\frac{(K-1)T \ln(\epsilon^2+1) }{\epsilon}} \exp\left[ -2(\epsilon + c ) - \frac{\ln(\epsilon^2+1) }{2\epsilon} \right]\\
&\geq \frac{1}{8 e^{3(\epsilon+c)}} \sqrt{\frac{(K-1)T \ln(\epsilon+1) }{\epsilon^{(1+\frac{1}{\epsilon})}(\epsilon^2+1)^{\frac{1}{\epsilon}}}}
\end{align*}
\end{proof}
\begin{corollaryrep}[DP Bayesian Minimax Regret Bound
Given a finite privacy level $\epsilon \in \real$, and a finite time horizon $T \geq h(K, \epsilon)$, then for any environment with bounded reward $\br \in [0,1]^K$, any algorithm $\pol$ that is $\epsilon$-IDP satisfies
\begin{align*}
\reg_{\mathrm{minimax}}(T) &\geq \frac{1}{8 e^{3(\epsilon+c)}} \sqrt{\frac{(K-1)T \ln(\epsilon+1) }{\epsilon^{(1+\frac{1}{\epsilon})}(\epsilon^2+1)^{\frac{1}{\epsilon}}}}
\end{align*}
Here, $c=L\Delta$ where $L$ is the Lipschitz constant corresponding to the reward distributions of the environments and $\Delta$ is the difference in generated rewards.
\end{corollaryrep}
\begin{proof}
Similar to the proof of Corollary~\ref{cor:lp_bayes}, the results of Theorem~\ref{thm:lb_dp} and Fact~\ref{thm:lattimore} prove the statement of Corollary~\ref{cor:lb_bayes_dp}.
\end{proof}
\textbf{Summary:} \textit{These results establish that the lower bounds for both minimax and Bayesian minimax regret degrade by a multiplicative factor $\sqrt{\dfrac{\ln(\epsilon^2+1) }{e^{6\epsilon}\epsilon^{(1+\frac{1}{\epsilon})}(\epsilon+B)^{\frac{1}{\epsilon}}}}$ for an $\epsilon$-DP bandit algorithm.
The limit of the lower bound goes to infinity as $\epsilon \rightarrow 0$ i.e. the policy becomes completely random. In such a situation, the regret for bandits gets capped by the order of $T$. As $\epsilon$ increases, the privacy dependent factor in lower bound also decreases monotonically.}
\iffalse
\textbf{Sketch for Problem Independent Bounds.}
Now, proceeding with Step 1 and Step 2 of the problem-independent lower bound proof, we get:
\begin{align*}
\kldiv{\prob_{\pol{\model_1}}(\Hist_T)}{\prob_{\pol{\model_2}}(\Hist_T)} &\leq 2(\epsilon + c) e^{\epsilon + c} + e^{\epsilon + c} \kldiv{\prob_{\pol{\model_1}}(\Hist'_T)}{\prob_{\pol{\model_2}}(\Hist'_T)}\\
&\leq 2(\epsilon + c) + e^{2(\epsilon + c)}\frac{2T\Delta^2}{K-1}
\end{align*}
Hence, we get
\begin{align*}
\max\lbrace \reg_{\model_1}(\pol, T), \reg_{\model_2}(\pol, T) \rbrace
&\geq \frac{1}{2}\left(\reg_{\model_1}(\pol, T) + \reg_{\model_2}(\pol, T)\right)\\
&\geq \frac{T\Delta}{4}\exp\left[-2(\epsilon + c) - e^{2(\epsilon + c)}\frac{2T\Delta^2}{K-1}\right].
\end{align*}
\dbcomment{Now rest is algebra of finding a good $\Delta$.}
The simplest choice is:
$\Delta^2 = \frac{(K-1) \ln(\epsilon^2+1) }{4T\epsilon e^{2(\epsilon + c)}} \leq \frac{1}{4}.$ Thus, $T \geq \frac{(K-1) \ln(\epsilon^2+1) }{\epsilon e^{2(\epsilon + c)}}.$
We get,
\begin{align*}
\reg_{\mathrm{minimax}}(T) &\geq \frac{1}{8 e^{\epsilon+c}} \sqrt{\frac{(K-1)T \ln(\epsilon^2+1) }{\epsilon}} \exp\left[ -2(\epsilon + c ) - \frac{\ln(\epsilon^2+1) }{2\epsilon} \right]\\
&\geq \frac{1}{8 e^{3(\epsilon+c)}} \sqrt{\frac{(K-1)T \ln(\epsilon^2+1) }{\epsilon}} \exp\left[- \frac{\ln(\epsilon^2+1)}{2\epsilon} \right]\\
&\geq \frac{1}{8 e^{3(\epsilon+c)}} \sqrt{\frac{(K-1)T \ln(\epsilon+1/\epsilon) }{\epsilon^{(1+\frac{1}{\epsilon})}(\epsilon^2+1)^{\frac{1}{\epsilon}}}}
\end{align*}
\fi
\iffalse
\ifdefined \workshop
\
\else
\newpage
\begin{lemma}
If a policy $\pi$ satisfy $\epsilon$-DP for an environment $\model$, we get
\begin{equation}
e^{-\epsilon} \frac{\prob_{\model \pol}(r^{T})}{\prob_{\model \pol}(r'^{T})} \leq \frac{\prob_{\model \pol}(\Hist_T)}{\prob_{\model \pol}(\Hist'_T)} \leq e^{\epsilon} \frac{\prob_{\model \pol}(r^{T})}{\prob_{\model \pol}(r'^{T})}.
\end{equation}
\end{lemma}
\begin{proof}
\begin{align*}
\frac{\prob_{\model \pol}(\Hist_T)}{\prob_{\model \pol}(\Hist'_T)}
&= \frac{\prob_{\model \pol}(a^T \mid r^{T})}{\prob_{\model \pol}(a^T \mid r'^{T})}\frac{\prob_{\model \pol}(r^{T})}{\prob_{\model \pol}(r'^{T})}\\
&=\frac{\prob_\pol(a^T \mid \bx^{T})}{\prob_\pol(a^T \mid \bx'^{T})}\frac{\prob_\pol(r^{T})}{\prob_\pol(r'^{T})}\\
&\underset{(a)}{\leq} e^\epsilon \frac{\prob_{\model \pol}(r^{T})}{\prob_{\model \pol}(r'^{T})}
\end{align*}
The inequality in Step (a) is obtained from Definition~\ref{def:DP_mab}.
Similarly,
\begin{align*}
\frac{\prob_{\model \pol}(\Hist_T)}{\prob_{\model \pol}(\Hist'_T)} &\underset{(a)}{\geq} e^{-\epsilon} \frac{\prob_{\model \pol}(r^{T})}{\prob_{\model \pol}(r'^{T})}
\end{align*}
\end{proof}
\begin{corollary}
For a policy following $\epsilon$-DP,
$$\kldiv{\prob_{\pol{\model_1}}(\Hist_T)}{\prob_{\pol{\model_2}}(\Hist_T)} \leq e^{\epsilon} \left( 2(\epsilon) + \kldiv{\prob_{\pol{\model_1}}(\Hist'_T)}{\prob_{\pol{\model_2}}(\Hist'_T)} \right)$$
\end{corollary}
\begin{proof}
The idea is bounding
\[\frac{\prob_\pol(\Hist|\model_1)}{\prob_\pol(\Hist|\model_2)} = \frac{\prob_\pol(\Hist|\model_1)}{\prob_\pol(\Hist'|\model_1)} \frac{\prob_\pol(\Hist'|\model_1)}{\prob_\pol(\Hist'|\model_2)}
\frac{\prob_\pol(\Hist'|\model_2)}{\prob_\pol(\Hist|\model_2)} \leq e^{2\epsilon} \frac{\prob_{\model_1 \pol}(r^{T})}{\prob_{\model_1 \pol}(r'^{T})} \frac{\prob_{\model_2 \pol}(r'^{T})}{\prob_{\model_2 \pol}(r^{T})} \frac{\prob_\pol(\Hist'|\model_1)}{\prob_\pol(\Hist'|\model_2)} \overset{??}{=} e^{2\epsilon}\frac{\prob_\pol(\Hist'|\model_1)}{\prob_\pol(\Hist'|\model_2)}
\]
\dbcomment{Na, we need to bound ratio of reward probabilities.}
This will give us the bound on KL:
\begin{align*}
\kldiv{\prob_{\pol{\model_1}}(\Hist_T)}{\prob_{\pol{\model_2}}(\Hist_T)} &= \int d\prob_{\pol{\model_1}}(\Hist_T) \ln \left( \frac{\prob_{\pol{\model_1}}(\Hist_T)}{\prob_{\pol{\model_2}}(\Hist_T)} \right)\\
&= \int \frac{d\prob_{\pol{\model_1}}(\Hist_T)}{d\prob_{\pol{\model_1}}(\Hist'_T)} d\prob_{\pol{\model_1}}(\Hist'_T)\ln \frac{\prob_{\pol{\model_1}}(\Hist_T)}{\prob_{\pol{\model_2}}(\Hist_T)} \\
&\leq e^{\epsilon} \int \ln e^{2\epsilon}\frac{\prob_{\pol{\model_1}}(\Hist'_T)}{\prob_{\pol{\model_2}}(\Hist'_T)} d\prob_{\pol{\model_1}}(\Hist'_T)\\
&= 2\epsilon e^{2\epsilon} + e^{2\epsilon} \kldiv{\prob_{\pol{\model_1}}(\Hist'_T)}{\prob_{\pol{\model_2}}(\Hist'_T)}
\end{align*}
\end{proof}
\fi
\fi
%
%
%
%
%
\section{Existing Lower Bounds for Non-private and Private Bandits}\label{sec:lbounds
\textbf{Problem-independent Non-private Lower Bounds:} Minimax regret is the worst case regret that a bandit algorithm can incur if the environment is unknown. Thus, it is often referred as the problem-independent regret.
\cite{vogel1960asymptotic} performed the first minimax analysis of two-armed Bernoulli bandits. \cite{auer2002nonstochastic} generalised it to $K$-arm Bernoulli distributions.
\cite{gerchinovitz2016refined} provided a novel technique to establish high probability regret lower bounds for adversarial bandits with bounded reward.
For any bandit algorithm, the minimax regret is lower bounded by
$\mathrm{Reg}_{\mathrm{minimax}}(T) \geq c \sqrt{(K-1)T}$.
A bandit algorithm $\pol$ is called \emph{minimax optimal} if its minimax regret is upper bounded by $C\sqrt{(K-1)T}$.
In the Bayesian setup, the bandit algorithm assumes a prior distribution $Q_0$ over the possible environments $\nu \in \mathcal{E}$. \cite{lattimore2018bandit} proved that for any bandit algorithm $\pi$ there exists a prior distribution $Q_0$ that the Bayesian regret $\mathrm{Reg}_{\mathrm{Bayes}}(T) \geq C \sqrt{KT}$. This indicates that the minimax regret and the Bayesian regret lower bounds are identical for non-private bandits. This also holds for private bandits. \cite{lattimore2019information} provides the reasoning behind this connection using a modified minimax theorem.
\textbf{Existing Lower Bounds for Differentially Private Bandits.}
\cite{mishra2015nearly} proposed differentially private variants of UCB and Thompson sampling algorithms. \cite{tossou2016algorithms} improved the differentially private variant of UCB algorithm to obtain regret upper bound of $\Omega(\sum_{a\neq a^*}\frac{\Delta_a}{\kldiv{f_a}{f^*}}\log T + \frac{1}{\epsilon})$ for instantaneous privacy with time varying privacy loss. Our results show that the stricter sequential privacy definition that a constant privacy loss can be achieved with only an additive term on the regret cannot be true.
\cite{shariff2018differentially} proves a finite-time problem-dependent lower bound for contextual bandits. It indicates that the finite-time lower bound of regret is $\Omega(\log T)$ like the non-private lower bounds but with a modified multiplicative factor $(\sum_{a\neq a^*}\frac{\Delta_a}{\kldiv{f_a}{f^*}}+ \frac{K}{\epsilon})$. Though their definition of privacy is a bit ambiguous as it can be reduced to either of Definition~\ref{def:DP_mab} and~\ref{def:IDP_mab}. Among them, the first being unsuitable for continual observation defeats the purpose for bandits. \emph{We try to clarify at this point with the pan, instantaneous, and sequential privacy definitions.}
\cite{gajane2017corrupt} uses an analogous local privacy definition. They proved a finite-time problem-dependent lower bound of regret for locally private multi-armed bandits where the local privacy is induced by the corrupt bandit mechanism. In this case also, the $\Omega(\log T)$ regret bound of non-private bandits is maintaines with a modified multiplicative factor $\sum_{a\neq a^*}\frac{\Delta_a}{\kldiv{g_a}{g^*}}$. Here, $g_a$ and $g^*$ are the corrupt versions of the reward distributions $f_a$ and $f^*$ ensuring $\epsilon$-local privacy. Theorem~\ref{thm:lb_log_local} shows that the algorithms they have proposed, namely kl-UCB-CF and TS-CF, are suboptimal by at least by a factor $(1+e^{-\epsilon})^2$. \emph{This opens up the problem of designing optimal local-private bandit algorithms.}
We are not aware of any problem-independent minimax lower bounds for (locally and standard) differentially private bandits, before this paper.
\cite{tossou2017achieving} proposed a differentially private variant of EXP3 algorithm that achieves privacy for adversarial bandits with regret upper bound $O(\frac{\sqrt{T}\log T}{\epsilon})$. \emph{The lower bound of Theorem~\ref{thm:lb_global} shows that designing an optimal private algorithm for adversarial bandits is an open problem.}
\section{Introduction}
\emph{Differential Privacy} (DP) is a rigorous and highly successful definition of algorithmic privacy introduced by~\citet{dwork2006calibrating}. Given a definition of neighbourhood for inputs to an algorithm, it defines a notion of privacy that ensures that the algorithm's output renders neighbouring inputs indistinguishable.
Informally, if an algorithm is $\epsilon$-DP, then the amount of information inferred by an adversary about the algorithm's input is bounded by $\epsilon$.
\begin{definition}[$\epsilon$-Differential Privacy]\label{def:GDP}
A privacy-preserving mechanism $\mech$ composed with a randomised algorithm $\pol : \mathcal{D} \to \mathcal{A}$ is $\epsilon$-differentially private if for all
inputs $x, x' \in \mathcal{D}$ with $\lp{x}{x'}{H}= 1$:
$$\pol(A \mid x) \leq \pol(A \mid x') e^\epsilon, \qquad A \subset \mathcal{A}.$$
\end{definition}
Our notation suggests a view of the algorithm $\pol$ as a conditional distribution. Typically, the inputs to the algorithm are structured so that $x = (x_1, \ldots, x_n)$ corresponds to the data of $n$ individuals, with $x_i$ being the data of the $i$-th individual. $\lp{x}{x'}{H} = 1$ if $x_i \neq x'_i$ for an $i \in [n]$.
However, in many online learning settings, the algorithm might not see all of the data before making a decision, and in partial monitoring problems, where the algorithm might be unable to see all of the data before making a decision. In these cases, Definition~\ref{def:GDP} is not trivial to apply. Thus, different works analysing the privacy of algorithms in these domains adopt subtly different privacy definitions. Our aim is to fix precise definitions and to provide lower bounds on learning while satisfying privacy under each definition.
We consider the \emph{multi-armed bandit} setting~\citep{bellman1956problem,lattimore2018bandit} in particular. This involves an algorithm $\pol$ sequentially choosing among $K$ different arms to maximise expected cumulative reward. At time $t$, the algorithm \emph{draws an arm} $a_t \in \Act$ and the arms \emph{generate outcomes} $\bx_t = [x_{t,j}]_{j=1}^K \in \real^K$. Then the learner \emph{obtains a reward} $r_t = x_{t,a_t}$ which is equal to the outcome of the selected arm\footnote{More generally, we can consider a reward function $r_t = u(\bx_t, a_t)$.} without observing the other arms. As a motivating example, consider the case of patients arriving sequentially and the doctor assigning different treatments $a_t$. The doctor only sees the effect of the selected treatment, and there is a need to maintain patient's confidentiality.
The natural way to conceive differential privacy is to perceive the algorithm as a Turing machine with input tape $\bx = \bx_1, \bx_2, \ldots$, output tape $a = a_1, a_2, \ldots$, and a random input tape $\omega = \omega_1, \omega_2$ in order to admit stochastic decision making\footnote{We often denote a sequence $[s_1, \ldots, s_t, \ldots, s_T]$ as $s^T$ and $[s_1, \ldots, s'_t, \ldots, s_T]$ as $s'^T$.}. A bandit algorithm $\pol$ will satisfy differential privacy if, for any decision horizon $T$ and privacy level $\epsilon > 0$
\begin{align*}
\pol(a_1, \ldots, a_t \mid \bx_1, \ldots, \bx_k, \ldots, \bx_T)
\leq &\pol(a_1, \ldots, a_t \mid \bx_1, \ldots, \bx'_k, \ldots, \bx_T) e^\epsilon.
\end{align*}
One may reasonably ask why should we not consider a bandit algorithm differentially private only with respect to the observed rewards, i.e.
\begin{align*}
\pol(a_1, \ldots, a_t \mid r_1, \ldots, r_k, \ldots, r_T)
\leq &\pol(a_1, \ldots, a_t \mid r_1, \ldots, r'_k, \ldots, r_T) e^\epsilon.
\end{align*}
As we show later, the two definitions are equivalent in the bandit setting. However, the input to the algorithm is the outcomes generated by the arms, rather than the observed rewards, and since we cannot know a priori whether an arm will be chosen at a particular time. Similarly, we cannot know which sections of the input tape a Turing machine will see. Thus, the generated outcomes are a more suitable definition for differential privacy.
This brings us to the question: what are the mechanisms to use to guarantee privacy? One popular approach for achieving DP is the \emph{local privacy}~\cite{duchi2013local}. This creates a differentially private version $\by$ of the input $\bx$, such that $\prob(\by \mid \bx) \leq \prob(\by \mid \bx') e^\epsilon$.
The private rewards $\by$ is then used by the algorithm $\pi$ to make decisions.
In our paper, we examine the efficiency loss i.e. regret relative to algorithms both for the local and non-local mechanisms.
\paragraph{Our Contribution.}
We define, discuss, and unify different notions of differential privacy (Section~\ref{sec:dp_mab}). In
particular, we examine the effect of considering different notions of private input, observable output and neighbourhoods on the regret of multi-armed bandits that are operating under a differential privacy
constraint.
We also discuss mechanisms to achieve differential privacy: both local and non-local (Section~\ref{sec:dp_mech}). We illustrate the differences between those definitions using graphical models.
We provide a unified framework to prove minimax lower bounds on the regret of both differentially private multi-armed bandits (Section~\ref{sec:theory}).
This is based on a generalised KL-divergence decomposition lemma adapted for local and standard differential privacy definitions. Though the literature consists of problem-dependent regret bounds, these are the first minimax and Bayesian regret bounds for both differentially private bandits. We show that both in general and when differential privacy is achieved using a local mechanism, the regret scales as a multiplicative factor of $\epsilon$. As expected, local privacy mechanisms have a slightly worse performance.
The regret lower bounds that we prove and the corresponding privacy definitions are summarised in Table~\ref{tab:contributions}.
In Section~\ref{sec:multi-armed-bandit}, we provide the required technicalities of multi-armed bandit problems. In Section~\ref{sec:lbounds}, we elaborate that the proposed lower bounds pose several open problems of designing optimal bandit algorithms that satisfy different notions of privacy.
Appendix~\ref{sec:proofs_def} and ~\ref{sec:proofs_lb} contain detailed proofs.
\input{results_table}
\input{bandits_basics}
\section{Differential Privacy in Bandits: Definitions}
\label{sec:dp_mab}
In the classical differential privacy setting, we would like the output distribution of the algorithm to be similar for two neighbouring inputs. However, while it is clear what we should take for the output of the bandit algorithm (the actions), it is unclear whether we should consider the generated outcomes or the observed rewards as the input to the algorithm. In this paper, we will focus on the former definition, because it also implies the latter, and it is also applicable to other settings, like partial monitoring.
\begin{definition}[Differentially Private Bandits] \label{def:DP_mab}
An algorithm $\pol$ is $\epsilon$-DP if, $\forall t \leq T$:
\begin{align*}
&\prob_\pol(a^T \mid \bx_1, \ldots, \bx_t, \ldots, \bx_T) \leq
\prob_\pol(a^T \mid \bx_1, \ldots, \bx'_t, \ldots, \bx_T) e^\epsilon,
\end{align*}
for every sequence of actions $a^T \triangleq [a_1, \ldots, a_T]$ and neighbouring reward sequences $\bx^T$, $\bx'^T$ with $\lp{\bx^T}{\bx'^T}{H}= 1$.
\end{definition}
Let us consider the example of patients contributing their data to a clinical trial or a medical diagnostic experiment. $x_{t,i}$ would correspond to what the physician would see if they were to treat patient $t$ with treatment $i$. Since we cannot know \emph{a priori} which treatment the doctor would choose, we must protect every possible outcome generated by the patients. More concretely, consider actions that are diagnostic tests and where $x_{t,i}$ denotes the absence or presence of a disease marker. Then potentially revealing individual disease markers clearly violates privacy.
In bandits, we only ever see the $x_{t,i}$ chosen. Thus, we can get away with a weaker privacy definition, as shown in Lemma~\ref{rem:equivalence}. However, because of the conceptual difficulty in referring to the probability of actions given a sequence of rewards that would have only been obtained had those actions been taken, we use the outcomes to define differential privacy. This also allows us to apply this definition to more general sequential decision making problems.
\begin{lemma}
If and only if a bandit algorithm $\pol$ is $\epsilon$-DP with respect to the outcome sequence $\bx$ then it is $\epsilon$-DP with respect to the reward sequence $r$. However, in the general partial monitoring setting the equivalence does not hold.
\label{rem:equivalence}
\end{lemma}
However, a number of authors~\citep{mishra2015nearly,tossou2016algorithms,shariff2018differentially} have suggested differential privacy definitions for bandits of the following form, which we will refer to as \emph{instantaneous} differential privacy.
\begin{definition}[Instantaneous DP Bandits]\label{def:IDP_mab}
An algorithm $\pol$ is $\epsilon$-Instantaneous DP if, $\forall t \leq T$ and $\{a_t\}, \{r_t\}, \{r_t'\}$,
\begin{align*}
&\prob_\pol(a_T \mid a^{T-1}, r_1, \ldots, r_t, \ldots, r_{T-1})
\leq
\prob_\pol(a_T \mid a^{T-1}, r_1, \ldots, r'_t, \ldots, r_{T-1})
e^\epsilon.
\end{align*}
\end{definition}
Through Remark~\ref{rem:equivalence}, this definition is equivalent to
\begin{align*}
&\prob_\pol(a_T \mid a_1, \ldots, a_{T-1}, \bx_1, \ldots, \bx_t, \ldots, \bx_{T-1})
\leq
\prob_\pol(a_T \mid a_1, \ldots, a_{T-1}, \bx_1, \ldots, \bx'_t, \ldots, \bx_{T-1})e^\epsilon
\end{align*}
Consequently, it is easy to show that instantaneous differential privacy implies differential privacy and vice-versa.
\begin{lemma}
If a policy $\pol$ satisfies $\epsilon$-DP (Definition~\ref{def:DP_mab}) with privacy level $\epsilon$, $\pol$ will also satisfy $\epsilon$-instantaneous DP (Definition~\ref{def:IDP_mab}) with privacy level $2\epsilon$. Conversely, if a policy satisfies $\epsilon$-instantaneous DP, it achieves $t \epsilon$ DP after $t$ steps.
\label{lem:seq-ins-priv}
\end{lemma}
\section{Mechanisms for Achieving Differential Privacy}\label{sec:dp_mech}
One method to achieve differential privacy is to rely on differentially private inputs to the algorithm. This is called the \emph{Local Differential Privacy} model~\citep[c.f.][]{duchi2013local}. Local DP allows the algorithm to be agnostic about privacy. For this reason, this notion is presently adapted by Apple and Google for their large-scale systems.
If the bandit algorithm only observes a private version of the reward sequence, the algorithm's output is differentially private with respect to the \emph{generated outcome sequence} $\lbrace \bx_1, \ldots, \bx_T \rbrace = \lbrace \bx_i\rbrace_{i=1}^T$. In particular, let the input to the algorithm at time $t$ be $\bz_t$, so that the complete input sequence is $\bz_1, \ldots, \bz_t$. Then if a pre-processing mechanism $\mech$ generates differentially private inputs, i.e.
\begin{align*}
&\prob_\mech(\bz_1, \ldots, \bz_T \mid r_1, \ldots, r_t, \ldots, r_T)
\leq
\prob_\mech(\bz_1, \ldots, \bz_T \mid r_1, \ldots, r_t', \ldots, r_T) e^\epsilon
\end{align*}
then the resulting algorithm $\pol(a_1, \ldots, a_T \mid \bz_1, \ldots \bz_T)$ is differentially private with respect to $\bx$ (and so $r$) through post-processing.
\begin{definition}[$\epsilon$-Local Differentially Private Bandits]\label{def:LDP_mab}
A bandit algorithm $\pol$ is locally differentially private if its inputs are generated through an $\epsilon$-DP mechanism $\mech$.
\end{definition}
Here, we are not specifying how the input should be related to the rewards. If the input consists of a private reward sequence $\hat{r}$, the setting of Definition~\ref{def:LDP_mab} is analogous to the local privacy definition in the corrupted bandit setting of~\citet{gajane2017corrupt}.
If we only maintain differentially private statistics about each arm, i.e. we retain a differentially private mean for the reward of each arm, and use i to select new arms, then we end up with local differential privacy in the way it was used in~\cite{mishra2015nearly}.
\input{bayesnet_bandit}
\paragraph{Unified Graphical Model Representation of Privacy for Bandits.} Figure~\ref{fig:pgm_dp_mab} provides a unified graphical model perspective of non-private, private (locally and not) multi-armed bandits. The shaded nodes represent the observed variable. The clear nodes represent the hidden variables. The dashed-rectangle covers the input quantities with respect to which the privacy has to be maintained. All of these representations treat the generated rewards as input and all possible action sequences as output with local and sequential privacy-mechanisms acting at two different levels ensuring local and sequential privacies respectively. This representation allows us to define a new notion of privacy for bandits, namely \emph{environment privacy}.
\paragraph{Environment Privacy for Bandits.} For a bandit with a stationary environment $\model$, the reward generation mechanism can be represented as a distribution with an environment-dependent parameter $\model$. The user may consider this \emph{environment parameter to be the input} and \emph{the generated histories} $\gen_T$, i.e. the sequence of generated rewards $\lbrace \br_i\rbrace_{i=1}^T \in \real^{KT}$ and actions taken $\lbrace a_i\rbrace_{i=1}^T \in [K]^T$ by environment parameter $\model$ and the policy $\pol$ as \emph{the output}.
\begin{definition}[$\epsilon$-Environment Privacy for Bandits] \label{def:EDP_mab}
A privacy-preserving mechanism $\mathcal{M}$ preserves $\epsilon$-environment privacy if for all generated histories $\gen_T \defn \lbrace (a_i, \br_i)\rbrace_{i=1}^T \in ([K]\times \real^K)^T$, and environment parameters $\model,\model' \in \real^d$,\vspace{-.5em}
$$\log\left({\left |\frac{\prob_{\pol\model}\left(\lbrace (a_i, \br_i) \rbrace_{i=1}^T \mid \model\right)}{\prob_{\pol\model'}\left(\lbrace (a_i, \br_i)\rbrace_{i=1}^T \mid \model'\right)}\right|}\right) \leq \epsilon\rho(\model,\model'),$$
where $\rho$ is a distance metric defined in the space of $\model$.
\end{definition}
\paragraph{An Example of Data Privacy in Bandits.} In order to understand how each definition affects privacy, let us consider the example of web advertising for a specific individual. At time $t$, the individual is presented with some set of advertisements $a_t$. These advertisements generate potential responses $\bx_t$. Out of these generated responses of the user, we only see the clicked response $r_t$. We use a bandit algorithm $\pol$ in order to perform adaptive web advertising. If $\pol$ is $\epsilon$-differentially private (Definition~\ref{def:DP_mab}) with respect to $\{\br_t\}$, an adversary cannot distinguish similar responses between individuals. If we are locally $\epsilon$-DP (Definition~\ref{def:LDP_mab}), even the authority of the algorithm would not be able to distinguish between $\{\br_t\}$. Thus indistinguishability of individuals can be achieved with respect to both the adversary and the authority of the algorithm.
Finally, if we are $\epsilon$-envrionment private with respect to $\model$ (Definition~\ref{def:EDP_mab}), an adversary cannot infer the inherent preferences of the individuals.
For all of our definitions, the privacy loss is bounded by a constant privacy level $\epsilon$ independent of the length of interactions.
\section{Regret Lower Bounds for Private Bandits}\label{sec:theory
\paragraph{Trade-off between Privacy and Performance.} Now, the main question is how much additional regret that we would suffer if we want to satisfy DP. Although a number of DP bandit algorithms with upper bounds on the regret have been proposed, a clear lower bound on the regret that \emph{any} differentially private policy must accrue is lacking. In literature, DP for bandits has been achieved in two ways: (a) through local DP on the (statistics of) rewards, and (b) through instantaneous DP. The first method has the drawback of potentially adding too much noise to the reward signal, while the second method requires that the instantaneous differential privacy level is carefully controlled to achieve a constant (or sublinear) amount of privacy loss.
\paragraph{Regret Lower bound and Intrinsic Hardness of a Bandit Problem.} \textit{Lower bounds on the performance measure of a problem provides us insight about the intrinsic hardness of the problem and sets a target for optimal algorithm design.} In this section, we prove minimax and Bayesian minimax lower bounds for local, instantaneous and general DP bandits. We also prove a problem-dependent lower bound for local DP. Results are listed in Table~\ref{tab:thm}.
\begin{table*}[t!]
\centering
\begin{tabular}{c|c|c|c|c}
\hline
Privacy Type & Definition & Minimax & Bayesian Minimax & Problem-dependent \\
& for Bandits & Regret & Regret & Regret \\
\hline
DP & Definition~\ref{def:DP_mab} & Theorem~\ref{thm:lb_dp} & Corollary~\ref{cor:lb_bayes_dp} & -- \\
Instantaneous DP & Definition~\ref{def:IDP_mab} & Theorem~\ref{thm:lb_global} & Corollary~\ref{cor:lb_bayes_global} & -- \\
Local DP & Definition~\ref{def:LDP_mab} & Theorem~\ref{thm:lb_local} & Corollary~\ref{cor:lp_bayes} & Theorem~\ref{thm:lb_log_local} \\
\hline
\end{tabular}
\caption{Regret lower bound results different privacy definitions/mechanisms of bandits.}
\end{table*}
\paragraph{KL-divergence Decomposition in Sequential Setting.} In order to prove the lower bounds, we adopt the general canonical bandit model~\citep{lattimore2018bandit} that is general enough not to impose additional constraints on the bandit algorithm and the environment.
A privacy-preserving bandit algorithm $\pol$ and an environment $\model$ interacts up to a given time horizon $T$ to produce \emph{observed history} $\Hist_T \defn \lbrace (A_i, X_i)\rbrace_{i=1}^T$.
Thus, an observed history $\mathcal{H}_T$ is a random variable sampled from the measurable space $\left(([K]\times \real)^T, \mathcal{B}([K]\times \real)^T\right)$ and a probability measure $\prob_{\pol\model}$. Here, $\mathcal{B}([K]\times \real)^T$ is the Borel set on $([K]\times \real)^T$. $\prob_{\pol\model}$ is the probability measure induced by the algorithm $\pol$ and environment $\model$ such that
\begin{enumerate}[leftmargin=*]
\item[1.] the probability of choosing an action $A_t = a$ at time $t$ is dictated only by the algorithm $\pol(a|\Hist_{t-1})$,
\item[2.] the distribution of reward $X_t$ is $f_{A_t}$ and is conditionally independent of the previous observed history $\Hist_{t-1}$.
\end{enumerate
Hence, we get for any observed history $\Hist_T$,
\begin{align}\label{eqn:indep}
\begin{split}
\prob^T_{\pol{\model}} \defn \prob_{\pol\model}(\Hist_T) = \prod_{t=1}^T \pol(A_t|\Hist_{t-1})f_{A_t}(X_t).
\end{split
\end{align}
This canonical bandit framework allows us to state Lemma~\ref{thm:kldecomp} on KL-divergence decomposition. This decomposition allows us to separately treat the effect of policy and reward generation. It further leads to Lemma~\ref{thm:kldecomp_local},~\ref{thm:kldecomp_global} and~\ref{thm:kldecomp_DP} for local DP, instantaneous DP, and DP respectively.
\begin{lemma}[KL-divergence Decomposition]\label{thm:kldecomp}
Given a bandit algorithm $\pol$, two environments $\model_1$ and $\model_2$, and a probability measure $\prob_{\pol\model}$ satisfying Equation~\ref{eqn:indep},\vspace{-.5em}
\begin{align*}
\kldiv{\prob^T_{\pol{\model_1}}}{\prob^T_{\pol{\model_2}}} &= \sum_{t=1}^{T} \kldiv{\pol(A_t|\mathcal{H}_t, \model_1)}{\pol(A_t|\mathcal{H}_t, \model_2)}\\ &+ \sum_{a=1}^K \mathbb{E}_{\model_1}\left[N_a(T)\right]\kldiv{f_a \in \model_1}{f_a \in \model_2}.
\end{align*
\end{lemma}
For non-private and locally-private algorithms, the first term vanishes and the rest remains. The corresponding equality for non-private bandit algorithms was first proposed in~\citep{garivier2018explore}. The non-private decomposition of~\citep{garivier2018explore} is also used in~\citep{gajane2017corrupt} to derive a regret lower bound for locally private corrupt bandits.
In this paper, we derive two novel upper bounds of the KL-divergence for local DP and DP bandits.
Bounding KL-divergence is a classic technique in minimax risk analysis methods like Le Cam, Fano and Assaoud~\citep{Yu1997}.
Here, we propose three sequential and private versions of that for local, instantaneous, and general DP (Lemma~\ref{thm:kldecomp_local},~\ref{thm:kldecomp_global}, and~\ref{thm:kldecomp_DP}).
\begin{lemma}[Local Private KL-divergence Decomposition]\label{thm:kldecomp_local}
If the reward generation process is $\epsilon$-local differentially private for both the environments $\model_1$ and $\model_2$
\begin{align*}
\kldiv{\prob^T_{{\model_1}\pol}}{\prob^T_{{\model_2}\pol}} \leq~&2\min\lbrace4,e^{2\epsilon}\rbrace (e^{\epsilon}-1)^2 \sum_{a=1}^K \mathbb{E}_{\model_1}\left[N_a(T)\right]\kldiv{f_a \in \model_1}{f_a \in \model_2}.
\end{align*}
\end{lemma
\begin{proof}[Proof Sketch]
The first term of Lemma~\ref{thm:kldecomp} vanishes for local privacy. The second term yields this upper bound by applying Equation 9 in~\citep{duchi2013local} and Pinsker's inequality consecutively.
\end{proof}
\begin{lemma}[Instantaneously Private KL-divergence Decomposition]\label{thm:kldecomp_global}
For a sequentially private bandit algorithm $\pol$ satisfying $l(T) \leq \expect_{\pol\model}[N_a(T)]$ for any arm $a$,
and two environments $\model_1$ and $\model_2$
\begin{align*}
\begin{split}
\kldiv{\prob^T_{\pol{\model_1}}}{\prob^T_{\pol{\model_2}}} &\leq 2\epsilon(e^{2\epsilon}-1)\frac{1-2e^{-\frac{T}{l(T)}}}{1 - e^{-\frac{T}{l(T)}}} 1+ \sum_{a=1}^K \mathbb{E}_{\model_1}\left[N_a(T)\right](\kldiv{f_a \in \model_1}{f_a \in \model_2}).
\end{split}
\end{align*}
\end{lemma}
\begin{lemma}[Differentially Private KL-divergence Decomposition]\label{thm:kldecomp_DP}
If the policy is $\epsilon$-local differentially private for both the environments $\model_1$ and $\model_2$, we get for two neighbouring histories $\Hist_T$ and $\Hist'_T$ and a constant $c>0$
\vspace{-.5em}
\begin{align*}
\kldiv{\prob_{\pol{\model_1}}(\Hist_T)}{\prob_{\pol{\model_2}}(\Hist_T)} &\leq 2(\epsilon + c) + e^{2(\epsilon + c)} \kldiv{\prob_{\pol{\model_1}}(\Hist'_T)}{\prob_{\pol{\model_2}}(\Hist'_T)} ).
\end{align*}
\end{lemma}
\begin{proof}[Proof Sketch]
We use Lemma~\ref{lemma:hist_dp} to show that the ratio of two histories would be bounded by $e^{\epsilon + c}$ for a given environment. Here, $c= L \Delta$ such that the reward distributions in the environment $\model$ satisfy: $\ln \sup_{a, x_a,x'_a} \frac{\prob_{\model}(x_a)}{\prob_{\model}(x'_a)} \leq L \Delta$. This allows us to proof this lemma.
\end{proof}
\subsection{Minimax Regret
Now, we leverage these upper bounds on the KL-divergences to derive the minimax regret bounds for these two cases.
We know from a probabilistic Pinsker's inequality~\citep[Lemma 2.1]{bretagnolle1979estimation} that the probability of an event $E$ and its complement $E^C$ for two distributions $P$ and $Q$ defined on same event space is lower bounded by $\exp(-\kldiv{P}{Q})$.
Thus, the KL-divergence upper bound allows us to lower bound the sum of regret in two environments such that the best policy in one environment is the worst in another.
Now, as we derive this lower bound on sum of regrets in these two environments. We focus on specifying the environments.
In the end, we choose such suboptimality gaps among rewards such that we can get a lower bound on the best regret in the worst environment.
\begin{theorem}[Local Private Minimax Regret Bound]\label{thm:lb_local}
Given an $\epsilon$-locally private reward generation mechanism with $\epsilon \in \real$, and a time horizon $T \geq g(K,\epsilon)$, then for any environment with finite variance, any algorithm $\pol$ satisfie
\begin{equation}
\reg_{\mathrm{minimax}}(T) \geq \frac{c}{\min\lbrace 2,e^{\epsilon}\rbrace(e^{\epsilon}-1)}\sqrt{(K-1)T}.
\end{equation
\end{theorem}
For small $\epsilon$, $e^{\epsilon}-1\approx \epsilon$. Thus, for small $\epsilon$, the minimax regret bound for local privacy worsens by a multiplicative factor $\frac{1}{\epsilon{e^{\epsilon}}}$. If the $\epsilon = 0$ which means the rewards obtained are completely randomised, the arms would not be separable any more and would lead to unbounded minimax regret. For bandits, the regret gets capped by $\Theta(T)$ if the expected rewards of the arms are finite.
\begin{theorem}[Instantaneously Private Minimax Regret Bound]\label{thm:lb_global}
Given a finite privacy level $\epsilon \leq a/2$, and a time horizon $T \geq h(K,\epsilon)$, then for any environment with finite variance, any algorithm $\pol$ that is $\epsilon$-instantaneous DP satisfie
\begin{equation}
\reg_{\mathrm{minimax}}(T) \geq c(a)
\sqrt{\frac{(K-1)T}{2\epsilon(e^{2\epsilon}-1)}}.
\end{equation
\end{theorem}
This implies that for small $\epsilon$, the minimax regret bound for instantaneous DP worsens by a multiplicative factor $\frac{1}{\epsilon}$. Thus, the lower bound of minimax regret for instantaneously private bandit is better than the locally private bandit by factors $e^{\epsilon/2}$ and $\sqrt{\frac{2}{\epsilon(e^{\epsilon} +1)}}$ for small and large $\epsilon$'s respectively.
\begin{theorem}[DP Minimax Regret Bound]\label{thm:lb_dp}
Given a finite privacy level $\epsilon > 0$, and a time horizon $T \geq h(K,\epsilon)$, then for any environment with finite variance, any algorithm $\pol$ that is $\epsilon$-DP satisfies
\begin{align*}
\reg_{\mathrm{minimax}}(T) &\geq \frac{1}{8 e^{3(\epsilon+c)}} \sqrt{\frac{(K-1)T \ln(\epsilon+1) }{\epsilon^{(1+\frac{1}{\epsilon})}(\epsilon^2+1)^{\frac{1}{\epsilon}}}}
\end{align*}
Here, $c=L\Delta$ where $L$ is the Lipschitz constant corresponding to the reward distributions of the environments and $\Delta$ is the difference in generated rewards.
\end{theorem}
The minimax regret lower bound degrade by a multiplicative factor $\sqrt{\dfrac{\ln(\epsilon^2+1) }{e^{6\epsilon}\epsilon^{(1+\frac{1}{\epsilon})}(\epsilon+B)^{\frac{1}{\epsilon}}}}$ for an $\epsilon$-DP bandit algorithm.
The limit of the lower bound goes to infinity as $\epsilon \rightarrow 0$ i.e. the policy becomes completely random. In such a situation, the regret for bandits gets capped by the order of $T$. As $\epsilon$ increases, the privacy dependent factor in lower bound also decreases monotonically.
\subsection{Bayesian Minimax Regret}
In the Bayesian setup, the bandit algorithm assumes a prior distribution $Q_0$ over the possible environments $\nu \in \mathcal{E}$. As the algorithm $\pi$ plays further and observe corresponding rewards at each time $t$, it updates the prior over the possible environments to a posterior distribution $Q_t$. This framework is adopted for efficient algorithms like Thompson sampling~\citep{thompson1933} and Gittins indices~\citep{gittins1989multi}. In the Bayesian setting, we define the Bayesian regret as $$\reg_{\mathrm{Bayes}}(\pol, T,Q) \defn \int_{\model^T} \reg(\pol, \model, T)dQ(\model).$$
In the Bayesian setting, we define the Bayesian minimax regret as the worst possible regret for any prior
$$\reg^*_{\mathrm{Bayes}}(T) \defn \min_{\pol} \max_Q \int_{\model^T} \reg(\pol, \model, T)dQ(\model)=\min_{\pol}\max_{Q}\reg_{\mathrm{Bayes}}(\pol, T,Q).$$
We use a recent result on equivalence of minimax and Bayesian minimax regrets for bounded rewards to proceed~\cite{lattimore2019information}.
\begin{corollary}[Local Private Bayesian Minimax Regret Bound]\label{cor:lp_bayes}
Given an $\epsilon$-locally private reward generation mechanism with $\epsilon \in \real$, and a finite time horizon $T \geq g(K,\epsilon)$, then for any environment with bounded rewards $\br \in [0,1]^K$, any algorithm $\pol$ satisfies\vspace{-.5em}
\begin{equation}
\reg^*_{\mathrm{Bayes}}(T) \geq \frac{c}{\min\lbrace 2,e^{\epsilon}\rbrace(e^{\epsilon}-1)}\sqrt{(K-1)T}.
\end{equation}
\end{corollary}
\begin{corollary}[Instantaneous DP Bayesian Minimax Regret Bound]\label{cor:lb_bayes_global}
Given a finite privacy level $\epsilon \in \real$, and a finite time horizon $T \geq h(K,\epsilon)$, then for any environment with finite variance and bounded reward $\br \in [0,1]^K$, any algorithm $\pol$ that is $\epsilon$-instantaneous DP satisfies\vspace*{-.5em}
\begin{equation}
\reg^*_{\mathrm{Bayes}}(T) \geq c(\epsilon)
\sqrt{\frac{(K-1)T}{2\epsilon(e^{2\epsilon}-1)}}.
\end{equation}
\end{corollary}
\begin{corollary}[DP Bayesian Minimax Regret Bound]\label{cor:lb_bayes_dp}
Given a finite privacy level $\epsilon \in \real$, and a finite time horizon $T \geq h(K, \epsilon)$, then for any environment with bounded reward $\br \in [0,1]^K$, any algorithm $\pol$ that is $\epsilon$-IDP satisfies
\begin{align*}
\reg_{\mathrm{minimax}}(T) &\geq \frac{1}{8 e^{3(\epsilon+c)}} \sqrt{\frac{(K-1)T \ln(\epsilon+1) }{\epsilon^{(1+\frac{1}{\epsilon})}(\epsilon^2+1)^{\frac{1}{\epsilon}}}}
\end{align*}
Here, $c=L\Delta$ where $L$ is the Lipschitz constant corresponding to the reward distributions of the environments and $\Delta$ is the difference in generated rewards.
\end{corollary}
\subsection{Problem-dependent Regret}
Both minimax and Bayesian minimax regret bounds are problem independent. They represent the worst-case regret for any environment and any prior over environments respectively. Someone may want to design algorithms that is optimal for a given environment $\nu$ and the minimax and Bayesian minimax bounds are too pessimistic for them. Thus, researchers study the problem-dependent lower bounds of regret involving environment dependent quantities. \cite{lai1985asymptotically} proved that a bandit algorithm achieves $\Omega(\log T)$ problem-dependent lower bound.
We prove that for $\epsilon$-local privacy this lower bound worsens by a multiplicative factor $\frac{1}{e^{2\epsilon}(e^{\epsilon}-1)^2}$. It is an open problem for instantaneous and general DP.
\begin{theorem}[Problem-dependent Local-Private Regret Bound]\label{thm:lb_log_local}
For any asymptotically consistent bandit algorithm $\pol$, an environment $\model$ with optimal reward distribution $f^*$, and an $\epsilon$-locally private reward generation mechanism, the expected cumulative regret\vspace*{-.5em}
\begin{equation*}\hspace{-1em}
\liminf_{T \rightarrow \infty} \frac{\reg(\pol, \model, T)}{\log T} \geq \sum_{a \neq a^*} \frac{\Delta_a}{2\min\lbrace4,e^{2\epsilon}\rbrace(e^{\epsilon}-1)^2 \kldiv{f_a}{f^*}}.
\end{equation*}\vspace*{-1em}
\end{theorem}
\subsection{Discussion}
This shows that for both local and non-local DP in bandits the regret lower bound changes by a multiplicative factor dependent on the privacy level $\epsilon$. The factors are such that the regret goes to infinity if $\epsilon \rightarrow 0$ i.e. as the privacy mechanism gets completely random. In that case, the regret get capped by $T$. The bounds also show that the lower bound for local privacy is worse than that the general case, as expected. It also shows instantaneous DP is worse than the general DP if we maintain a constant $\epsilon$ at all steps $t$. This shows approaching the optimal policy from randomised rewards using the local privacy model is slower than to using a policy that is inherently DP. In Section~\ref{sec:lbounds}, we show our lower bounds falsify the conjecture of an additive factor in the lower bound by~\cite{tossou2016algorithms} and point out existing gaps in optimal algorithm design for private bandits.
\input{related_work}
\section{Discussion and Future Work}\label{sec:conc
We provide a unifying set of definitions for differentially private bandit algorithms that distinguishes between local and instantaneous DP used in the literature, as well as for general DP.
We proved corresponding minimax and Bayesian minimax lower bounds for both local and general DP bandit algorithms.
These bounds also pose design of optimal local and sequential private bandit algorithms as open problems since the existing algorithms are suboptimal.
We are now working on deriving the problem-dependent lower bound for DP.
This is based on a generalised KL-divergence decomposition lemma adapted for local and standard differential privacy definitions. Though the literature consists of problem-dependent regret bounds, these are the first minimax and Bayesian regret bounds for both differentially private bandits. We show that both in general and when differential privacy is achieved using a local mechanism, the regret scales as a multiplicative factor of $\epsilon$.
In future, researchers can utilise these lower bounds to design optimal private bandit algorithms for real-life applications, such as recommender systems and clinical trials. We are now working towards the two remaining problem dependent bounds for instantaneous and general DP to complete this problem.
\iffalse
Our definitions of sequential differential privacy provide a framework to look into differentially private bandit algorithms.
This also resolves the eclectic definitions available in the literature and discusses their applicability.
The KL-divergence decomposition based method provides a unified proving mechanism for lower bounds for bandit.
This allows us to propose the minimax lower bounds for both locally and standard sequentially private multi-armed bandits which was absent in literature.
These bounds also pose design of optimal local and sequential private bandit algorithms as open problems since the existing algorithms are suboptimal.
We are now working on deriving the problem-dependent lower bounds for sequentially and environment private bandits and problem-independent bounds for environment private bandits.
In future, researchers can utilise these bounds to design optimal private bandit algorithms for real-life applications, such as recommender systems and web advertising.
\fi
|
train/arxiv
|
BkiUdwQ5qg5A59Bh1au4
| 5 | 1 |
\section{Introduction} Martingale inequalities have played an important role in applications of probability to problems in analysis. Many of these applications rest on the celebrated Burkholder-Davis-Gundy inequalities (commonly referred to these days as the {\it ``BDG inequalities"}) which compare the $L^p$-norms of martingales to the $L^p$-norms of their quadratic variations. For a survey describing some of the early applications of these inequalities to several longstanding problems in analysis, we refer the reader to \cite{BanDav} and the many references therein. A predecessor to the BDG inequalities which also has had a huge number of applications in analysis is Burkholder's 1966 inequality which proves the $L^p$ boundedness of martingale transforms \cite{Bur1}. In 1984 Burkholder extended the martingale transform inequality in various directions (including to the setting of Banach spaces with the unconditional martingale difference sequence property, the so called {\it ``UMD spaces"}) and obtained the sharp constants \cite{Bur2}. In recent years the sharp martingale transform inequalities have been widely applied to prove $L^p$-boundedness of Fourier multipliers on $\mathrm{I\! R\!}^n$, and in particular to the study of multipliers arising from the Riesz transforms, the Beurling-Ahlfors operator, operators of Laplace transform-type, and related singular integrals. We refer the reader to \cite{Ban1} for references to this now very large literature and to some of the applications of these techniques to other problems of interest. These include connections to a longstanding conjecture of T. Iwaniec on the $L^p$-norm of Beurling-Ahlfors operator and a celebrated open problem of Morrey concerning rank-one convex and quasiconvex functions \cite[\S5]{Ban1}. In \cite{BanBieBog} and \cite{BanBog} sharp martingale transform inequalities were used to study a new class of Fourier multipliers which arise by certain natural transformations of L\'evy martingales associated to {\it L\'{e}vy processes}, i.e. the most general class of stochastic processes enjoying stationary and independent increments (of which Brownian motion is a special case.) A novel aspect of these new multipliers is that they are associated to processes whose paths contain jump discontinuities and whose infinitesimal generators are non-local operators. A distinctive feature of Fourier multiplier inequalities obtained from sharp martingale transform inequalities is that in many instances the inequalities inherit the sharpness of the martingale transform inequalities. This happens, for example, in the important case of second order Riesz transforms and other multipliers with certain homogeneity properties \cite{BanOse}, including some of the classical Marcinkiewicz-type multipliers \cite[p.110]{Ste2}.
The purpose of this paper is to show that the construction of the operators in \cite{BanBieBog} and \cite{BanBog} can be carried out in the general setting of Lie groups, leading to a new collection of linear operators which are bounded on the $L^p$-spaces of the group with bounds that are optimal with respect to $p$ and which generalize in a natural way the classical Riesz transforms and the imaginary powers of the Laplacian. The results in \cite{BanBog} have recently seen applications to $L^p$ regularity for solutions to non-local elliptic problems in Euclidean space; see \cite{DonKim1, DonKim2}. In the same way, we expect that the results of this paper will be of interest to those working in the analysis of Lie groups and its applications.
The paper is organized as follows. In \S2.1, we recall the sharp Burkholder inequalities for martingales under the assumption of subordination. While the discrete martingale versions serve as motivation, the results in this paper rest on versions of Burkholder's inequalities for continuous time martingales. In \S2.2, we review some of the basic facts about L\'{e}vy processes on Lie groups and recall the fundamental theorem of Hunt which describes the structure of their infinitesimal generators. In \S3, we define the martingale transforms and prove that they are bounded operators on $L^p$, for $1<p<\infty$ (Theorem \ref{mart-trasforms-bound}). The martingale transforms lead naturally to operators defined on the Lie group with $L^p$ bounds which are exactly the same as those for martingale transforms (Corollary \ref{proj-transform-T} and Theorem \ref{projections1}). In fact, the operators on the Lie group are limits of conditional expectations of the martingale transforms and for this reasons it is natural to call them ``projections of martingale transforms". (For the latter point of view, we refer the reader to \cite{BanMen} and \cite[\S3.10]{Ban1} where this is done in the setting of $\mathrm{I\! R\!}^n$.) In \S4 we specialize our construction to compact Lie groups where via the (non-commutative) Fourier transform we show that in various cases, including second order Riesz transforms and operators of Laplace transform-type, our operators are Fourier multipliers. We show that a large class of multipliers can be obtained from central L\'{e}vy processes, i.e. those that have conjugate invariant laws. We obtain a new L\'{e}vy-Khintchine type formula for such processes on compact, connected semi-simple Lie groups which was anticipated in work of Liao \cite{Liao1, Liao}. This formula plays an important role in the construction of multipliers but may also be of independent interest. In \S5 we revisit, as a special case of the results for general Lie groups, the setting of $\mathrm{I\! R\!}^n$ and derive the results of these two papers, rather directly, and without the technicalities of \cite{BanBog} and \cite{BanBieBog} concerning L\'evy systems. But even more, our construction yields a larger class of operators than those studied in \cite{BanBog} and \cite{BanBieBog} (Corollary \ref{projection-Rn}). When further specializing the transformations, these operators consequently yield a wider class of Fourier multipliers \eqref{multiplier-3} than those in \cite{BanBog} and \cite{BanBieBog}. Indeed the results of these papers are given by the multipliers in \eqref{multiplier-4} which are a sub-class of those presented in \eqref{multiplier-3}.
\vspace{5pt}
{\bf Notation.} If $G$ is a Lie group, ${\mathcal{B}}(G)$ is the Borel $\sigma$-algebra of $G, C_{0}(G)$ is the Banach space of real-valued continuous functions on $G$ that vanish at infinity (equipped with the usual supremum norm), and $C_{c}^{\infty}(G)$ is the dense linear subspace of $C_{0}(G)$ comprising infinitely differentiable functions of compact support. In any metric space $(M,d)$, the open ball of radius $r > 0$ centered on $p \in M$ is denoted by $B_{r}(p)$. If $a, b \in \mathbb{R}$, we define $a \vee b: = \max\{a,b\}$ and $a \wedge b:= \min\{a, b\}$. If $T > 0$ and $f:[0,T] \rightarrow \mathbb{R}$ has a left limit at $0 < s \leq T$ we write $f(s-):=\lim_{u \uparrow s}f(u)$ and we define the jump in $f$ at $s$ by $\Delta f(s) = f(s) - f(s-)$. Functions that are right continuous on $[0, T)$ and have left limits at every point in $(0,T]$ are said to be c\`adl\`ag~(from the French ``continue \`{a} droite et limit\'{e} \`{a} gauche''.) For such functions the set $\{s \in (0, T); \Delta f(s) \neq 0\}$ is at most countable. Consequently we obtain identities such as $\int_{0}^{T}|f(s-)|^{p}ds = \int_{0}^{T}|f(s)|^{p}ds$ (for $1 \leq p < \infty$) which we will use freely within this paper. $M_n(\mathrm{I\! R\!})$ denotes the space of all $n\times n$ matrices with real entries while $M_n(\mathbb{C})$ denotes the space of all $n \times n$ matrices with complex entries. If $(M, \mathcal{M}, \mu)$ is a $\sigma$-finite measure space, the norm of $f \in L^{\infty}(M, \mathcal{M}, \mu; M_{n}(\mathbb{R}))$ is $||f||:= \mbox{ess.sup}_{x \in M}\sup\{|f(x)v|;v\in \mathrm{I\! R\!}^n, |v|\leq 1\}$. The norm in $L^{\infty}(M, \mathcal{M}, \mu; M_{n}(\mathbb{C}))$ is defined similarly.
\section{Preliminaries}
\subsection{Sharp martingale inequalities} We begin by recalling the celebrated sharp martingale transform inequalities of Burkholder.
Let $f=\{f_n, n\geq 0\}$ be a martingale (defined on some probability space) with difference sequence $d=\{d_k, k\geq 0\}$, where $d_k=f_k-f_{k-1}$ for $k\geq 1$ and $d_0=f_0$. Given a predictable sequence of random variables $\{v_k, k\geq 0\}$ with $v_k\in [-1,1]$ a.s. for all $k$, the martingale difference sequence $\{v_kd_k, k\geq 0\}$ generates a new martingale called the {\it martingale transform} of $f$ and denoted here by $g$. We set $\|f\|_p=\sup_{n\geq 0}\|f_n\|_p$. In \cite{Bur1} Burkholder proved that for all $1<p<\infty$, $\|g\|_p\leq C_p\|f\|_p$ for some constant $C_p$ depending only on $p$. In \cite{Bur2} he sharpened this result by proving that under these assumptions
\begin{equation}\label{bur2}
\|g\|_p\leq (p^*-1)\|f\|_p,
\end{equation}
for all $1<p<\infty$, where $p^{*}:=\max\left\{p,\frac{p}{p-1}; 1 < p < \infty\right\}$ and that the constant $p^*-1$ is best possible. In the sequel the constant $p^*-1$ will appear often.
Martingale inequalities of these type have a long history both in analysis and probability and we refer the reader to \cite{Ban1} for some of this literature and applications.
By considering dyadic martingales, inequality \eqref{bur2} contains the classical inequality of Marcinkiewicz \cite{Mar} and Paley \cite{Pal} for Paley-Walsh martingales with the optimal constant. That is, let $\{h_k, k\geq 0\}$ be the Haar system in the Lebesgue unit interval $[0, 1)$ so that $h_0=[0, 1), h_1=[0, \,1/2)-[1/2, \,1), h_3=[0, \,1/4)-[1/4, \,1/2), h_4=[1/2, \,3/4)-(3/4,\, 1), \dots$, where the same notation is used for an interval as for its indicator function. Then for any sequence $\{a_k, {k\geq 0}\}$ of real numbers and any sequence $\{\e_k, {k\geq 0}\}$ of signs,
\begin{equation}\label{Paleyreal}
\Big\|\sum_{k=0}^{\infty} \varepsilon_k a_k h_k\Big\|_p\leq (p^*-1)\Big\|\sum_{k=0}^{\infty} a_k h_k\Big\|_p, \qquad
\end{equation}
for $1<p<\infty$. The constant $p^*-1$ is best possible here as well.
An obvious question that arises from these sharp results is: what happens to the best constant when the predictable sequence is non-symmetric in the sense that it takes values in $[0, 1]$ rather than in $[-1, 1]$?
In \cite{Cho}, K.P. Choi used the techniques of Burkholder to identify the best constant in this case as well. While Choi's constant is not as explicit (and simple) as the $p^*-1$ constant of Burkholder, one does have considerable information about it. More precisely, Choi's result states that if $v=\{v_k, k\geq 0\}$ with $v_k\in [0,1]$ a.s. for all $k$, then
\begin{equation}\label{choi1}
\|g\|_p\leq c_p\|f\|_p,
\end{equation}
for all $1<p<\infty$ with the best constant $c_p$ satisfying
$$
c_p=\frac{p}{2}+ \frac{1}{2}\log\left(\frac{1+e^{-2}}{2}\right) +\frac{\alpha_2}{p}+\cdots
$$
where
$$\alpha_2=\left[\log\left(\frac{1+e^{-2}}{2}\right)\right]^2+\frac{1}{2}\log\left(\frac{1+e^{-2}}{2}\right)-2\left(\frac{e^{-2}}{1+e^{-2}}\right)^{2}. $$
Motivated by the inequalities of Burkholder and Choi, the following definition was introduced in \cite{BanOse}.
\begin{definition}\label{defC}
Let $-\infty<b<B<\infty$ and $1<p<\infty$ be given and fixed. We define $C_{p,b,B}$ as the least positive number $C$ such that for any real-valued martingale $f$ and for any transform $g$ of $f$ by a predictable sequence $v=\{v_k, k\geq 0\}$ with values almost surely in $[b,B]$, we have
\begin{equation}\label{martin}
||g||_p\leq C||f||_p.
\end{equation}
\end{definition}
For any $0<a<\infty$, we see that $C_{p,-a,a}=a(p^*-1)$ by Burkholder's inequality \eqref{bur2} and
$C_{p,0,a}=a\,c_p$ by Choi's inequality \eqref{choi1}. An easy computation gives that for $b, B$ as above,
\begin{equation}
\max\left\{\left(\frac{B-b}{2}\right)(p^*-1), \, \max\{|B|, |b|\}\right\} \leq C_{p,b,B}\leq \max\{B, |b|\}(p^*-1).
\end{equation}
However, the lack of any general ``scaling" or ``translation" properties of $C_{p, b, B}$, outside the cases $[-a, a]$ and $[0, a]$, makes it very difficult to compute the constant. For example, what is the value of $C_{p, 1, 2}$?
For the applications in this paper we require versions of the above inequalities for martingales indexed by continuous time. Suppose that $(\Omega,\mathcal{F},\mathbb{P})$ is a complete probability space, equipped with a right continuous filtration $(\mathcal{F}_t)_{t\geq 0}$ of sub-$\sigma$-algebras of $\mathcal{F}$. We assume that $\F_0$ contains all the events of probability $0$. Let $X=\{X_t, t\geq 0\}$ and $Y=\{Y_t, t\geq 0\}$ be adapted real valued martingales which have right-continuous paths with left-limits, i.e., {c\`adl\`ag} martingales. We will denote by $[X,Y]$ the quadratic co-variation process of $X$ and $Y$. When $X=Y$ we will simply write $[X]$ for $[X,X]$. We refer the reader to Dellacherie and Meyer \cite{DelMey} for details on the construction and properties of $[X, Y]$ and $[X]$.
Following \cite{BanWan} and \cite{Wan}, we say that $Y$ is differentially subordinate to $X$ if $|Y_0| \leq |X_0|$ and $([X]_t-[Y]_t)_{t\geq 0}$ is nondecreasing and nonnegative as a function of $t$. We have the following extension of the Burkholder inequalities proved in \cite{BanWan} for martingales with continuous paths (the so called ``Brownian martingales") and in \cite{Wan} in the general case. More precisely,
if $Y$ is differentially subordinate to $X$, then
\begin{equation}\label{burkholder}
\|Y_T\|_p\leq (p^*-1)\|X_T\|_p,
\end{equation}
for all $1<p<\infty$ and all $T>0$. The constant $p^*-1$ is best possible.
\begin{remark}\label{hilbert-valued}
We note here that Theorem \eqref{burkholder} holds in the exact same form if the martingales take values in a real or complex Hilbert space; see \cite{BanWan, Wan}. This is important for martingale transforms of complex valued functions such as the multipliers of Laplace transform-type given below. That is, while we define martingale transforms with functions $A$ that take values in the set of all $n\times n$ matrices with real entries, $M_{n}(\mathbb{R})$, we could replace them with functions with values in $M_{n}(\mathbb{C})$. A similar statement applies to the functions $\psi$ which we assume take values in $\mathrm{I\! R\!}$. These changes to complex valued matrices $A$ and functions $\psi$ would not affect the bound $(p^*-1)$ given in the results that we obtain in this paper.
\end{remark}
We end this section by recalling a result for non-symmetric subordination proved in \cite{BanOse} which is the replacement for continuous time martingales of Choi's result. This will be used below as well.
Suppose $-\infty<b<B<\infty$ and $X_t$, $Y_t$ are two real valued martingales with right-continuous paths and left-limits. Suppose further that $|Y_0|\leq |X_0|$ and that
\begin{equation}\label{banoseestiamte}
\left[\frac{B-b}{2}X,\frac{B-b}{2}X\right]_t-\left[Y-\frac{b+B}{2}X,Y-\frac{b+B}{2}X\right]_t\geq 0
\end{equation}
and is nondecreasing for all $t\geq 0$. (We call this property ``non-symmetric differential subordination".) Then
\begin{equation}\label{non-symm}
||Y_T||_p\leq C_{p,b,B}||X_T||_p, \quad 1<p<\infty,
\end{equation}
for all $T>0$. The constant $C_{p,b,B}$ is best possible.
Note that when $B=a>0$ and $b=-a$, we have the case of \eqref{burkholder}.
\subsection{L\'evy processes on Lie groups}
Let $(\Omega, {\mathcal{F}}, P)$
be a probability space and let $G$ be a Lie group of dimension $n$ with neutral element $e$ and Lie algebra $\mathfrak{g}$. If $\phi: = (\phi(t), t \geq 0)$ is a stochastic process taking values in $G$ then the {\it right increment} of $\phi$ between $s$ and $t$ (where $s < t$) is the random variable $\phi(s)^{-1}\phi(t)$. We say that $\phi$ is a (left) {\it L\'{e}vy process} on $G$ if $\phi(0) = e$ (a.s.), $\phi$ has stationary and independent right increments and $\phi$ is stochastically continuous in that $\lim_{t \rightarrow 0} P(\phi(t) \in A) = 0$ for all $A \in \mathcal{B}(G)$ for which $e \notin \overline{A}$.
Let $p_{t}(A):=P(\phi(t) \in A)$ for $A \in {\mathcal{B}}(G), t \geq 0$ so that $p_{t}$ is the law of $\phi(t)$. Then $(p_{t}, t \geq 0)$ is a weakly continuous convolution semigroup of probability measures on $G$ for which $p_{0} = \delta_{e}$. Conversely given any such semigroup $(p_{t}, t \geq 0)$, we can always construct a L\'{e}vy process $(\phi_{t}, t \geq 0)$ on the space of all paths from $[0, \infty)$ to $G$ by using the celebrated Kolmogorov existence theorem.
For each $t \geq 0, f \in C_{0}(G), \sigma \in G$, define $P_{t}f(\sigma) = \int_{G}f(\sigma \tau)p_{t}(d\tau)$. Then $(P_{t}, t \geq 0)$ is a (positivity preserving) $C_{0}$-contraction semigroup which commutes with left translations. That is, $L_{\sigma} P_{t} = P_{t} L_{\sigma}$ for all $t \geq 0, \sigma \in G$, where $L_{\sigma}f(\tau) = f(\sigma^{-1}\tau)$. In fact $(P_{t}, t \geq 0)$ is also a (almost everywhere positivity preserving) $C_{0}$-contraction semigroup on $L^{p}(G)$ for all $1 \leq p < \infty$ where $G$ is equipped with a {\it right-invariant} Haar measure (see e.g. \cite{Kun, App3}.)
We continue to work in $C_{0}(G)$ and let ${\mathcal{L}}$ denote the infinitesimal generator of $(P_{t}, t \geq 0)$. So ${\mathcal{L}}$ is a densely defined closed linear operator. We denote its domain by $\mbox{Dom}({\mathcal{L}})$. We next give a precise description of ${\mathcal{L}}$ that is due to Hunt \cite{Hunt} (see also \S3.1 in \cite{Liao}.)
Let $(X_{j}, 1 \leq j \leq n)$ be a fixed basis of $\mathfrak{g}$ and define a dense linear manifold $C^{2}(G)$ in $C_{0}(G)$ by
$C^{2}(G): = \{ f \in C_{0}(G); X_{i}f
\in C_{0}(G)~ \mbox{and}~ X_{i}X_{j}f \in C_{0}(G) ~\mbox{for
all}~ 1 \leq i,j \leq n\}$. There exist functions ${x_{i} \in
C_{c}^{\infty}(G), 1 \leq i \leq n}$ so that $x_{i}(e) = 0,
X_{i}x_{j}(e) = \delta_{ij}$ and $(x_{1}, \ldots , x_{n})$ are canonical co-ordinates in a neighbourhood of $e$.
A measure $\nu$ defined on ${\mathcal{B}}(G)$ is called a {\it
L\'{e}vy measure} whenever $\nu(\{e\}) = 0$,
\begin{equation}\label{levy-Lie}
\int_{U}\left(\sum_{i=1}^{n}x_{i}(\tau)^{2}\right)\nu(d\tau) < \infty\,\,\, \mbox{and}\,\,\, \nu(G-U) < \infty,
\end{equation}
for any Borel neighbourhood $U$ of $e$.
For the rest of the paper we will use the Einstein summation convention whereby we sum over repeated upper and lower indices.
\begin{theorem}[Hunt] \label{Hunt} With the notation introduced above we have
\begin{enumerate}
\item $C^{2}(G) \subseteq \mbox{Dom}({\mathcal{L}})$.
\bigskip
\item For each
$\sigma \in G, f \in C^{2}(G)$,
\begin{eqnarray} \label{hu}
{\mathcal{L}}f(\sigma) & = & b^{i}X_{i}f(\sigma) +
a^{ij}X_{i}X_{j}f(\sigma) \nonumber \\
& + & \int_{G}(f(\sigma \tau) - f(\sigma) -
x^{i}(\tau)X_{i}f(\sigma))\nu(d\tau),
\end{eqnarray}
where $b = (b^{1}, \ldots b^{n}) \in {\mathbb{R}}^{n}, a = (a^{ij})$ is a
non-negative-definite, symmetric $n \times n$ real-valued matrix
and $\nu$ is a L\'{e}vy measure on $G$.
\end{enumerate}
Conversely, any linear operator with a representation as above
is the restriction to $C^{2}(G)$ of the generator
corresponding to a unique convolution semigroup of probability
measures.
\end{theorem}
We call $(b, a, \nu)$ the characteristics of the L\'{e}vy process $\phi$.
From now on we will equip ${\mathcal{F}}$ with a filtration $({\mathcal{F}}_{t}, t \geq 0)$ of sub-$\sigma$-algebras and we will always assume that a given L\'{e}vy process $\phi$ is adapted to this filtration. That is, $\phi(t)$ is ${\mathcal{F}}_{t}$-measurable for each $t \geq 0$. We say that a L\'{e}vy process $\phi$ is c\`adl\`ag~if there exists $\overline{\Omega} \in {\mathcal{F}}$ with $P(\overline{\Omega}) = 1$ such that the mappings $t \rightarrow \phi(t)(\omega)$ are right continuous with left limits existing for all $\omega \in \overline{\Omega}$.
Clearly any c\`adl\`ag~L\'{e}vy process $\phi$ is a Markov process with respect to its own filtration
and so for each $f \in C_{2}(G), t \geq 0$,
$$ f(\phi(t)) - f(e) - \int_{0}^{t}{\mathcal{L}}f(\phi(s))ds$$
is a martingale
which we denote by $M_{f}(t)$ and which can be written as
\begin{equation}\label{decomposition}
M_{f}(t)= M_{f}^{c}(t) + M_{f}^{d}(t)
\end{equation}
where $M_{f}^{d}(t)$ and $M_{f}^{c}(t)$ are the discontinuous and continuous
parts, respectively, see \cite{DelMey}. In \cite{AK} these martingales were found to be
stochastic integrals against a Poisson random measure on $G$
and a Brownian motion in $\mathfrak{g}$, respectively so that $\phi$ is in
fact the unique solution of the following stochastic differential equation
\begin{eqnarray} \label{apku}
f(\phi(t)) & = & f(e) + \int_{0}^{t}X_{i}f(\phi(s-))dB_{a}^{i}(s)
+ \int_{0}^{t}{\mathcal{L}}f(\phi(s-))ds + \nonumber\\
& & + \int_{0}^{t+}\int_{G}(f(\phi(s-)\sigma)
- f(\phi(s-))\tilde{N}(ds,d\sigma)
\end{eqnarray}
for each $f \in C_{2}(G), t \geq 0$. Here $B_{a} = (B_{a}(t), t \geq 0)$ is an
$n$-dimensional Brownian motion of mean zero and covariance matrix given by
Cov$(B_{a}^{i}(t)B_{a}^{j}(t)) = 2ta^{ij}$ for $t \geq 0, 1 \leq i,j \leq n$, and ${\tilde N}$
is the compensator defined for each $t\geq 0, E \in {\mathcal{B}}(G)$ by
$$ {\tilde N}(t,E) = N(t,E) - t \nu(E),$$
where $N$ is a Poisson random measure on ${\mathrm{I\! R\!}}^{+} \times G$ with intensity
measure Leb$\times \nu$ which is independent of $B$. In fact,
$$ N(t, A) = \#\{0 \leq s \leq t, \Delta \phi(s) \in A\}$$
for each $t \geq 0, A \in {\mathcal{B}}(G)$ where $\Delta \phi(s) = \phi(s-)^{-1}\phi(s)$ denotes the jump of the process at time $s > 0$.
Fix $T > 0$ and let $F: G \times G \rightarrow \mathbb{R}$ be continuous and $\xi \in L^{\infty}(\mathbb{R}^{+} \times G, \mbox{Leb} \times \nu)$ be such that the mapping $s \rightarrow \xi(s, \tau)$ is left continuous for all $\tau \in G$ and assume that $\int_{0}^{T}\int_{G}\mathbb{E}(|F(\phi(s), \tau)|^{2})\nu(d \tau)ds < \infty$. We note that (see e.g. Lemma 4.2.2 in \cite{App1}, p.221) It\^{o}'s isometry in the framework of Poisson random measures yields,
\begin{eqnarray} \label{Itois}
\mathbb{E}\left(\left(\int_{0}^{T}F(\phi(s-), \tau)\xi(s, \tau)\tilde{N}(ds,d\tau)\right)^{2}\right) & = &
\int_{0}^{T}\int_{G}\mathbb{E}(|F(\phi(s), \tau)|^{2})|\xi(s, \tau)|^{2}\nu(d \tau)ds \nonumber \\
& \leq & ||\xi||^{2}\int_{0}^{T}\int_{G}\mathbb{E}(|F(\phi(s), \tau)|^{2})\nu(d \tau)ds,
\end{eqnarray}
In the sequel we will find it convenient to work with the standard Brownian motion $B = (B(t), t \geq 0)$ in $\mathbb{R}^{n}$ with covariance Cov$(B^{i}(t)B^{j}(t)) = t\delta^{ij}$. To implement this we choose an $n \times n$ matrix $\Lambda$ such that $\Lambda \Lambda^{T} = 2 a$ and define $Y_{i} \in \mathfrak{g}$ by $Y_{i} = \Lambda_{i}^{j}X_{j}$ for $1 \leq i \leq n$. Then the integral with respect to Brownian motion in (\ref{apku}) may be rewritten
\begin{eqnarray*}
\int_{0}^{t}X_{i}f(\phi(s-))dB_{a}^{i}(s) & = & \int_{0}^{t}Y_{i}f(\phi(s-))dB^{i}(s)\\
& = & \int_{0}^{t}\nabla_{Y}f(\phi(s-)) \cdot dB(s),\end{eqnarray*}
where $\nabla_{Y}: = (Y_{1}, \ldots, Y_{n})$ and $\cdot$ is the usual inner product in $\mathbb{R}^{n}$.
So far we have always assumed that the L\'{e}vy process $\phi$ starts at $e$ with probability one. In the sequel we will want to change the starting point to arbitrary $\rho \in G$ and we can achieve this by defining the c\`adl\`ag~Markov process $\phi^{(\rho)}(t) = \rho \phi(t)$ for each $t \geq 0$. Note that the process $\phi^{(\rho)}:=(\phi^{(\rho)}(t), t \geq 0)$ retains stationary and independent right increments. It follows easily from (\ref{apku}) that (with probability one)
\begin{eqnarray} \label{apku1}
f(\phi^{(\rho)}(t)) & = & f(\rho) + \int_{0}^{t}\nabla_{Y}f(\phi^{(\rho)}(s-)) \cdot dB(s)
+ \int_{0}^{t}{\mathcal{L}}f(\phi^{(\rho)}(s-))ds + \nonumber\\
& & + \int_{0}^{t+}\int_{G}(f(\phi^{(\rho)}(s-)\sigma)
- f(\phi^{(\rho)}(s-))\tilde{N}(ds,d\sigma),
\end{eqnarray}
for each $f \in C_{2}(G), t \geq 0$.
\section{Martingale transforms and their projections} In this section we will construct a family of operators which act on $L^p(G)$. These operators will arise as ``projections" of martingale transforms and will be bounded on $L^p(G)$ for all $1<p<\infty$. To begin we fix $T > 0$ and choose $f \in C_{c}^{\infty}(G), \sigma \in G$. Then for all $0 \leq t \leq T$, we define
\begin{equation}\label{key}
M_{f}^{(T)}(\sigma, t): = (P_{T-t}f)(\sigma \phi(t))=(P_{T-t}f)(\phi^{\sigma}({t}))
\end{equation}
Later on we will also require the notation $M_{f}^{(T)}(\sigma):= M_{f}^{(T)}(\sigma, T)=f(\sigma\phi(T))$.
It follows from (\ref{apku1}) that
\begin{eqnarray}\label{mart1}
M_{f}^{(T)}(\sigma, t)&=&(P_{T}f)(\sigma)+ \int_{0}^{t}\nabla_{Y}(P_{T-s}f)(\phi^{(\sigma)}(s-)) \cdot dB(s)\nonumber\\
& +& \int_{0}^{t+}\int_{G}[(P_{T-s}f)(\phi^{(\sigma)}(s-)\tau)
- (P_{T-s}f)(\phi^{(\sigma)}(s-))]\tilde{N}(ds,d\tau),
\end{eqnarray}
and hence $M_{f,\sigma}^{(T)}:=(M_{f}^{(T)}(\sigma, t), 0 \leq t \leq T)$ is an $L^{2}$-martingale starting at $(P_{T}f)(\sigma)$. We fix a right-invariant Haar measure $m_{R}$ on $G$ and note that inside integrals we will always write $m_{R}(dg) = dg$.
\begin{definition}
Let $A \in L^{\infty}(\mathbb{R}^{+} \times G, \mbox{Leb} \times m_{R};M_{n}(\mathbb{R}))$ and $\psi \in L^{\infty}(\mathbb{R}^{+} \times G \times G, \mbox{Leb} \times m_{R} \times \nu)$. We assume that $(A, \psi)$ are {\it regular} in that the mappings $(s,\rho) \rightarrow A(s, \rho)$ and $(s, \rho) \rightarrow \psi(s, \rho, \tau)$ (for all $\tau \in G$) are continuous and that
$||A|| \vee ||\psi|| \leq 1$.
For each $f \in C_{c}^{\infty}(G), \sigma \in G, t \geq 0$ we set
\begin{eqnarray}\label{mart2}
M_{f}^{(T;A,\psi)}(\sigma, t) & = & \int_{0}^{t}A(T-s, \phi^{(\sigma)}(s-))\nabla_{Y}(P_{T-s}f)(\phi^{(\sigma)}(s-)) \cdot dB(s)\nonumber\\ & + & \int_{0}^{t+}\int_{G}\left\{(P_{T-s}f)(\phi^{(\sigma)}(s-)\tau)
- (P_{T-s}f)(\phi^{(\sigma)}(s-)\right\}\\
&\times&\left\{\psi(T-s, \phi^{(\sigma)}(s-), \tau)\right\} \tilde{N}(ds,d\tau)\nonumber.
\end{eqnarray}
This gives the new martingale
$$M_{f, \sigma}^{(T; A,\psi)}:=(M_{f}^{(T;A,\psi)}(\sigma, t), 0 \leq t \leq T)$$
which we shall call the martingale transform of $M_{f, \sigma}^{(T)}$ by $(A, \psi)$.
\end{definition}
Computing the quadratic variations of both $(M_{f}^{(T)}(\sigma, t)$ and its transform $M_{f}^{(T;A,\psi)}(\sigma, t)$ (see e.g. equation (4.16) in \cite{App1} p.257) we find that
\begin{eqnarray}\label{qv1}
[(M_{f}^{(T)}(\sigma, \cdot)]_t & = & \int_{0}^{t}|\nabla_{Y}(P_{T-s}f)(\phi^{(\sigma)}(s-))|^{2}ds \nonumber \\ & + &
\int_{0}^{t+}\int_{G}[(P_{T-s}f)(\phi^{(\sigma)}(s-)\tau)
- (P_{T-s}f)(\phi^{(\sigma)}(s-))]^{2}N(ds,d\tau)
\end{eqnarray}
while
\begin{eqnarray}\label{qv2}
[(M_{f}^{(T;A, \psi)}(\sigma, \cdot)]_t & = & \int_{0}^{t}|A(T-s, \phi^{(\sigma)}(s-))\nabla_{Y}(P_{T-s}f)(\phi^{(\sigma)}(s-))|^{2}ds \nonumber \\ & + &
\int_{0}^{t+}\int_{G}\left\{[(P_{T-s}f)(\phi^{(\sigma)}(s-)\tau)
- (P_{T-s}f)(\phi^{(\sigma)}(s-))]^{2}\right\}\nonumber\\&\times&\left\{[\psi(T-s, \phi^{(\sigma)}(s-), \tau)]^{2}\right\}N(ds,d\tau).
\end{eqnarray}
From these formulas, the fact that $0=|M_{f}^{(T;A,\psi)}(\sigma, 0)|\leq |M_{f}^{(T)}(\sigma, 0)|$, and our assumption that $||A|| \vee ||\psi|| \leq 1$, it follows that $M_{f, \sigma}^{(T; A,\psi)}$ is differentially subordinate to $M_{f,\sigma}^{(T)}$. Now, for $1<p<\infty$, set
$$||X||_{p} = \left(\int_{G}\mathbb{E}(|X(\sigma)|^{p})d\sigma)\right)^{\frac{1}{p}},$$
for $X \in L^{p}(\Omega \times G)$. Applying Burkholder's result in the form given by the inequality \eqref{burkholder}, we obtain
\begin{equation}\label{fundineq}
||M_{f}^{(T;A,\psi)}||_{p} \leq (p^{*}-1)||M_{f}^{(T)}||_{p}.
\end{equation}
But using Fubini's theorem and the right invariance of the Haar measure we see that
\begin{eqnarray}\label{ineq1}
||M_{f}^{(T)}||_{p}^p&=&||f(\cdot \phi(T)||_{p}^{p}\nonumber\\
& = & \int_{G}\int_{G}|f(\sigma \tau)|^{p}p_{T}(d\tau)d\sigma\\
& = & \int_{G}|f(\sigma)|^{p}d\sigma = ||f||_{p}^{p}\nonumber.
\end{eqnarray}
We summarize the above calculations in the following
\begin{theorem}\label{mart-trasforms-bound} Let $A \in L^{\infty}(\mathbb{R}^{+} \times G, \mbox{Leb} \times m_{R};M_{n}(\mathbb{R}))$ and $\psi \in L^{\infty}(\mathbb{R}^{+} \times G \times G, \mbox{Leb} \times m_{R} \times \nu)$ with $(A,\psi)$ regular and
$||A|| \vee ||\psi|| \leq 1$. For any $0<T<\infty$, the map $f\to M_{f}^{(T;A,\psi)}$ defines a linear operator from $L^p(G)\to L^{p}(\Omega \times G)$, $1<p<\infty$, with
\begin{equation}\label{fundineq2}
\|M_{f}^{(T;A,\psi)}\|_{p} \leq (p^{*}-1)\|f\|_{p},
\end{equation}
for $1<p<\infty$. In particular, the bound is independent of $T$.
\end{theorem}
Now let $q = \frac{p}{p-1}$ and for given $g \in C_{c}^{\infty}(G)$, we define a linear functional $\Lambda_{g}^{T;A,\psi}$ on $C_{c}^{\infty}(G)$ by the prescription
\begin{equation}\label{lf}
\Lambda_{g}^{T;A,\psi}(f) = \int_{G}\mathbb{E}(M_{f}^{(T;A,\psi)}(\sigma)M_{g}^{(T)}(\sigma))d\sigma
\end{equation}
Using H\"{o}lder's inequality, inequality (\ref{fundineq}) and equality (\ref{ineq1}), we obtain
\begin{eqnarray}\label{ineq2}
|\Lambda_{g}^{T;A,\psi}(f)| & \leq & ||M_{f}^{(T;A,\psi)}||_{p}||M_{g}^{(T)}||_{q}\nonumber\\
& \leq & (p^{*} -1)||M_{f}^{(T)}||_{p}||M_{g}^{(T)}||_{q}\nonumber\\
& \leq & (p^{*} -1)||f||_{p}||g||_{q}.
\end{eqnarray}
Hence $\Lambda_{g}^{T;A,\psi}$ extends to a bounded linear functional on $L^{p}(G)$ and by duality, there exists a bounded linear operator $S_{A,\psi}^{T}$ on $L^{p}(G)$ for which
$$ \Lambda_{g}^{T;A,\psi}(f) = \int_{G}S_{A,\psi}^{T}f(\sigma)g(\sigma)d\sigma, $$
for all $f \in L^{p}(G), g \in L^{q}(G)$ and with $||S_{A,\psi}^{T}||_{p} \leq (p^{*}-1)$.
We summarize this in the following
\begin{corollary}\label{proj-transform-T} Let $A \in L^{\infty}(\mathbb{R}^{+} \times G, \mbox{Leb} \times m_{R};M_{n}(\mathbb{R}))$ and $\psi \in L^{\infty}(\mathbb{R}^{+} \times G \times G, \mbox{Leb} \times m_{R} \times \nu)$ with $(A,\psi)$ regular and
$||A|| \vee ||\psi|| \leq 1$. For any $0<T<\infty$, the map $f\to S_{A,\psi}^{T}f$ defines a linear operator from $L^p(G)\to L^{p}(G)$, $1<p<\infty$, with
\begin{equation}\label{fundineq3}
\|S_{A,\psi}^{T}f\|_{p} \leq (p^{*}-1)\|f\|_{p},
\end{equation}
for $1<p<\infty$. In particular, the bound is independent of $T$.
\end{corollary}
Next we probe the structure of (\ref{lf}) using It\^{o}'s isometry for stochastic integrals driven by both Brownian motion and a Poisson random measure (see \eqref{Itois} for the latter) with the aim of letting $T\to\infty$ in $S_{A,\psi}^{T}$. Again, using Fubini's theorem and right invariance of Haar measure as in the derivation of (\ref{ineq1}), we obtain
\begin{eqnarray}\label{PL}
\Lambda_{g}^{T;A,\psi}(f)&=& \int_{G}\int_{0}^{T}\mathbb{E}\{A(T-s, \phi^{(\sigma)}(s-))\nabla_{Y}(P_{T-s}f)(\phi^{(\sigma)}(s-))\nonumber\\
&\times&\cdot \nabla_{Y}(P_{T-s}g)(\phi^{(\sigma)}(s-))\}dsd\sigma \nonumber \\
& + & \int_{G}\int_{G}\int_{0}^{T}\mathbb{E}\{[(P_{T-s}f)(\phi^{(\sigma)}(s-)\tau)
- (P_{T-s}f)(\phi^{(\sigma)}(s-))] \nonumber \\ & \times & [(P_{T-s}g)(\phi^{(\sigma)}(s-)\tau)
- (P_{T-s}g)(\phi^{(\sigma)}(s-))]\nonumber \\
&\times&\psi(T-s, \phi^{(\sigma)}(s-),\tau)\}ds\nu(d\tau)d\sigma \nonumber \\
& = & \int_{0}^{T}\int_{G} A(T-s, \sigma)\nabla_{Y}(P_{T-s}f)(\sigma)\cdot \nabla_{Y}(P_{T-s}g)(\sigma)d\sigma ds \nonumber \\
& +& \int_{0}^{T}\int_{G}\int_{G}[(P_{T-s}f)(\sigma\tau)
- (P_{T-s}f)(\sigma)][(P_{T-s}g)(\sigma \tau)
- (P_{T-s}g)(\sigma)]\nonumber\\
&\times& \psi(T-s, \sigma,\tau)\nu(d\tau)d\sigma ds \nonumber \\
& = & \int_{0}^{T}\int_{G} A(s, \sigma)\nabla_{Y}(P_{s}f)(\sigma)\cdot \nabla_{Y}(P_{s}g)(\sigma)d\sigma ds\nonumber\\
& + & \int_{0}^{T}\int_{G}\int_{G}[(P_{s}f)(\sigma\tau)
- (P_{s}f)(\sigma)][(P_{s}g)(\sigma \tau)
- (P_{s}g)(\sigma)]\nonumber \\
&\times&
\psi(s,\sigma,\tau)\nu(d\tau)d\sigma ds
\end{eqnarray}
Choosing
$$ A(s, \sigma) = \displaystyle\frac{\nabla_{Y}(P_{s}f)(\sigma) \otimes \nabla_{Y}(P_{s}g)(\sigma)}{|\nabla_{Y}(P_{s}f)(\sigma)|\,|\nabla_{Y}(P_{s}g)(\sigma)|},$$
for $s \geq 0, \sigma \in G$ and observing that $\|A\|=1$, inequality \eqref{ineq2} (with $\psi\equiv 0$) gives that
\begin{equation}\label{LP1}
\int_{0}^{T}\int_{G} |\nabla_{Y}(P_{s}f)(\sigma)|\,|\nabla_{Y}(P_{s}g)(\sigma)|d\sigma ds\leq (p^{*} -1)||f||_{p}||g||_{q}.
\end{equation}
Thus the first integral in \eqref{PL} converges absolutely as $T \rightarrow \infty$ and
\begin{equation}\label{LP2}
\int_{0}^{\infty}\int_{G} |\nabla_{Y}(P_{s}f)(\sigma)|\,|\nabla_{Y}(P_{s}g)(\sigma)|d\sigma ds\leq (p^{*} -1)||f||_{p}||g||_{q}.
\end{equation}
We now consider the second integral in \eqref{PL}. This time we take $A\equiv 0$ and
$$
\psi(s, \sigma, \tau) = \mbox{sign}((P_{s}f)(\sigma \tau) - P_{s}f(\sigma))((P_{s}g)(\sigma \tau) - P_{s}g(\sigma)))
$$
and obtain that
\begin{equation*}\label{secondterm}
\int_{0}^{T}\int_{G}\int_{G}|(P_{s}f)(\sigma \tau) - P_{s}f(\sigma))| |(P_{s}g)(\sigma \tau) - P_{s}g(\sigma))|\nu(d\tau)d\sigma ds
\leq (p^{*} -1)||f||_{p}||g||_{q},
\end{equation*}
with the right hand side independent of $T$ and hence we can also let $T\to\infty$ in the second integral of \eqref{PL}.
We summarize the above result in the following
\begin{theorem}\label{projections1} Let $A \in L^{\infty}(\mathbb{R}^{+} \times G,\mbox{Leb} \times m_{R};M_{n}(\mathbb{R}))$ and $\psi \in L^{\infty}(\mathbb{R}^{+} \times G \times G, \mbox{Leb} \times m_{R} \times \nu)$ with $(A, \psi)$ regular and
$||A|| \vee ||\psi|| \leq 1$. There exists a bounded linear operator $S_{A, \psi}$ on $L^{p}(G)$, $1<p<\infty$, for which
\begin{eqnarray} \label{identity}
\int_{G}S_{A,\psi}f(\sigma)g(\sigma)d\sigma & = & \int_{0}^{\infty}\int_{G} A(s, \sigma)\nabla_{Y}(P_{s}f)(\sigma)\cdot \nabla_{Y}(P_{s}g)(\sigma)d\sigma ds \nonumber \\
& + & \int_{0}^{\infty}\int_{G}\int_{G}[(P_{s}f)(\sigma\tau)
- (P_{s}f)(\sigma)][(P_{s}g)(\sigma \tau)
- (P_{s}g)(\sigma)]\\
&\times& \psi(s,\sigma,\tau)\nu(d\tau)d\sigma ds,\nonumber
\end{eqnarray}
for al $f, g \in C_{c}^{\infty}(G)$. Furthermore, for all $f\in L^p(G)$ and $g\in L^q(G)$, $\frac{1}{p}+\frac{1}{q}=1$,
\begin{equation}\label{bound1}
\Big|\int_{G}S_{A,\psi}f(\sigma)g(\sigma)d\sigma \Big| \leq (p^*-1)\|f\|_p\,\|g\|_q
\end{equation}
and
\begin{equation}\label{bound2}
\|S_{A,\psi}f\|_p\leq (p^*-1)\|f\|_p.
\end{equation}
\end{theorem}
Suppose $A \in L^{\infty}(\mathbb{R}^{+} \times G,\mbox{Leb} \times m_{R};M_{n}(\mathbb{R}))$ is symmetric with the property that for all $\xi\in \mathbb{R}^n$,
\begin{equation}\label{non-symm-A}
b|\xi|^2\leq A(s, \sigma)\xi\cdot \xi \leq B|\xi|^2
\end{equation}
for all $(s, \sigma)\in \mathbb{R}^{+} \times G$, $-\infty<b<B<\infty$. Then by a simple computation (see \cite{BanOse}, \S4) the martingale $Y=M_{f, \sigma}^{(T; A, 0)}$
is subordinate to the martingale $X=M_{f, \sigma}^{(T)}$ in the sense of inequality \eqref{banoseestiamte}. From the construction of our operators $S_{A, \psi}$ we have (in this case the assumption $||A|| \leq 1$ is no longer needed):
\begin{theorem}\label{non-symm-thm} Let
$A \in L^{\infty}(\mathbb{R}^{+} \times G, \mbox{Leb} \times m_{R};M_{n}(\mathbb{R}))$ be continuous, symmetric and satisfying \eqref{non-symm-A}. Then for all $1<p<\infty$ and $f\in L^p(G)$,
\begin{equation}\label{nonsymmetric1}
\|S_{A, 0}f\|_p\leq C_{p,b,B}\|f\|_p,
\end{equation}
where $C_{p, b, B}$ is the constant in the inequality \eqref{non-symm}
\end{theorem}
As we shall see below (see \S5), the inequalities \eqref{bound2} and \eqref{nonsymmetric1} are sharp as they include the bounds for Riesz transforms in $\mathrm{I\! R\!}^n$ and other homogeneous multipliers (see \cite{BanOse}).
\section{Compact groups: Fourier multipliers}
In this action we are interested in cases of a compact Lie group where the operators $S_{A,\psi}$ are Fourier multipliers. Thus from now on we will assume that $G$ is a compact Lie group. Then every (left or right) Haar measure on $G$ is bi-invariant and also finite. In the sequel we will always assume that Haar measure is normalized to have total mass one. Let $\widehat{G}$ be the {\it unitary dual} of $G$, i.e. the set of all equivalence classes (modulo unitary equivalence) of irreducible representations of $G$. In the sequel we will often identify a class in $\widehat{G}$ with a representative element. The set $\widehat{G}$ is countable and each $\pi \in \widehat{G}$ is finite dimensional, so that that there exists a finite dimensional complex Hilbert space $V_{\pi}$, having dimension $d_{\pi}$ such that for each $\sigma \in G, \pi(\sigma)$ is a unitary matrix. We define the co-ordinate functions $\pi_{ij}(\sigma) = \pi(\sigma)_{ij}$ for $\sigma \in G, 1 \leq i,j \leq d_{\pi}$. We have $\pi_{ij} \in C^{\infty}(G)$ and the celebrated Peter-Weyl theorem tells us that $\{\sqrt{d_{\pi}}\pi_{ij}; 1 \leq i,j \leq d_{\pi}, \pi \in \widehat{G}\}$ is a complete orthonormal basis for $L^{2}(G, \mathbb{C})$. For each $f \in L^{2}(G, \mathbb{C})$, we define its {\it non-commutative Fourier transform} to be the matrix $\widehat{f}(\pi)$ defined by
$$ \widehat{f}(\pi)_{ij} = \int_{G}f(\sigma)\pi_{ij}(\sigma^{-1})d\sigma,$$ for each $1 \leq i,j \leq d_{\pi}$.
We will have need of the following version of the {\it Plancherel theorem} (which can be found in e.g. \cite{Far}, Theorem 6.4.2, p.110)
\begin{equation} \label{Planch}
\int_{G}f(\sigma)\overline{g(\sigma)}d\sigma = \sum_{\pi \in \widehat{G}}d_{\pi}\mbox{tr}(\widehat{f}(\pi)\widehat{g}(\pi)^{*}),
\end{equation}
for $f, g\in L^2(G, \mathbb{C})$. In particular if $T$ is a bounded linear operator on $L^{2}(G, \mathbb{C})$ we have
\begin{equation} \label{Planch1}
\int_{G}Tf(\sigma)\overline{g(\sigma)}d\sigma = \sum_{\pi \in \widehat{G}}d_{\pi}\mbox{tr}(\widehat{Tf}(\pi)\widehat{g}(\pi)^{*}).
\end{equation}
We say that the operator $T$ is a {\it Fourier multiplier} if for each $\pi \in \widehat{G}$ there exists a $d_{\pi} \times d_{\pi}$ complex matrix $m_{T}(\pi)$ so that
\begin{equation} \label{FM}
\widehat{Tf}(\pi) = m_{T}(\pi)\widehat{f}(\pi).
\end{equation}
We call the matrices $(m_{T}(\pi), \pi \in \widehat{G})$ the {\it symbol} of the operator $T$.\footnote{See \cite{RT} for a monograph treatment of pseudo differential operators and their symbols on compact Lie groups and \cite{RW} for a study of Fourier multipliers from this perspective. For probabilistic developments in the spirit of the present work, see \cite{App, App2}.}
Given $\pi \in \widehat{G}$ we obtain the {\it derived representation} $d\pi$ of the Lie algebra $\mathfrak{g}$ from the identity
$$ \pi(\exp(X)) = e^{d\pi(X)}, $$
for each $X \in \mathfrak{g}$. Then $d\pi(X)$ is a skew-hermitian matrix acting in $V_{\pi}$. We now equip $\mathfrak{g}$ with an Ad-invariant metric (which induces a bi-invariant Riemannian metric on $G$.) From now on $\{X_{1}, \dots, X_{n}\}$ will be an orthonormal basis for $\mathfrak{g}$ with respect to the given metric. We define the {\it Casimir operator} $\Omega_{\pi}$ by $\Omega_{\pi}: = \sum_{i=1}^{n}d\pi(X_{i})^{2}$. Then it can be shown (see e.g. \cite{Far} Corollary 6.7.2, p.122) that $\Omega_{\pi} = -\kappa_{\pi}I_{\pi}$ where $I_{\pi}$ is the identity matrix acting on $V_{\pi}$ and $\kappa_{\pi} \geq 0$ (with $\kappa_{\pi}=0$ if and only if $\pi$ is the trivial representation).
The Laplace-Beltrami operator $\Delta = \sum_{i=1}^{n}X_{i}^{2}$ is an essentially self-adjoint operator in $L^{2}(G, \mathbb{C})$ with domain $C(G, \mathbb{C})$ having discrete spectrum with
\begin{equation} \label{spec}
\Delta \pi_{ij} = - \kappa_{\pi}\pi_{ij},
\end{equation}
for all $\pi \in \widehat{G}, 1 \leq i,j \leq d_{\pi}$.
\subsection{Brownian motion}
Brownian motion (with twice the usual auto-covariance) on a Lie group $G$ is the L\'{e}vy process having characteristics $(b, I, 0)$. It is well known (see e.g. \cite{Elw}, \S6) that it can be obtained as the unique solution of the Stratonovich stochastic differential equation
\begin{equation} \label{sde}
d\phi(t) = \sqrt{2}X^{i}(\phi(t)) \circ dB_{i}(t),
\end{equation} with initial condition $\phi(0) = e$ (a.s.) In this case $(P_{t}, t \geq 0)$ is the heat semigroup with generator ${\mathcal{L}} = \Delta$.
The following is well-known but we include a short proof for the reader's convenience:
\begin{prop} \label{ftheat} For all $t \geq 0, \pi \in \widehat{G}, f \in L^{2}(G, \mathbb{C})$
$$ \widehat{P_{t}f}(\pi) = e^{-t\kappa_{\pi}}I_{\pi}.$$
\end{prop}
{\it Proof.} It follows from (\ref{spec}) that for all $1 \leq i,j \leq d_{\pi}$,
$$P_{t}\pi_{ij} = e^{-t\kappa_{\pi}}\pi_{ij}.$$
But then
\begin{eqnarray*} \widehat{P_{t}f}(\pi)_{ij} & = & \int_{G}P_{t}f(g)\pi(g^{-1})_{ij}dg\\
& = & \int_{G}P_{t}f(g)\overline{\pi(g)_{ji}}dg\\
& = & \int_{G}f(g)\overline{P_{t}\pi(g)_{ji}}dg\\
& = & e^{-t\kappa_{\pi}}\int_{G}f(g)\pi(g^{-1})_{ij}dg \\
& = & e^{-t\kappa_{\pi}}\widehat{f}(\pi)_{ij}, \end{eqnarray*}
as required. $\hfill \Box$
\subsection{Second order Riesz transforms}
The {\it first order Riesz transform} in the direction $X \in \mathfrak{g}$ is the operator $R_{X} = X(-\Delta)^{-\frac{1}{2}}$. It is shown in \cite{Arc} (using the martingale inequalities from \cite{BanWan}) that if $G$ is endowed with a bi-invariant Riemannian metric and $|X| = 1$ then $R_{X}$ is a bounded operator on $L^{p}(G)$ and $||R_{X}||_{p} \leq B_{p}$ for all $1 < p < \infty$, where $B_{p}: = \cot\left(\frac{\pi}{2p^{*}}\right)$ is the ``Pichorides constant'' (i.e. the $L^{p}$ norm of the Hilbert transform on the real line.) We write $R_{j}: = R_{X_{j}}$, for $1 \leq j \leq n$, so that $R_{i}R_{j}=X_{i}X_{j}{\Delta}^{-1}$. Let $C$ be a $n \times n$ matrix with $||C|| \leq 1$. We define the {\it second order Riesz transform} (see also \cite{GMS, BanBau}) $R_{C}^{(2)}$ to be
$$ R_{C}^{(2)}: = \sum_{i, j = 1}^{n}C_{ji}R_{i}R_{j} = \sum_{i, j = 1}^{n}C_{ji}X_{i}X_{j}\,{\Delta}^{-1}. $$
We will show that $R_{C}^{(2)}$ is precisely an operator of the form $S_{A, \psi}$ where $\psi = 0$. First we show that $R_{C}^{(2)}$ is a Fourier multiplier. Indeed this follows by using (\ref{Planch1}) and computing for $C^{\infty}(G)$
\begin{eqnarray*} \int_{G}R_{C}^{(2)}f(\sigma) g(\sigma)d\sigma & = & \sum_{i, j = 1}C_{ji}\sum_{\pi \in \widehat{G}}d_{\pi}\mbox{tr}(\widehat{R_{i}R_{j}f}(\pi)\widehat{g}(\pi)^{*})\\
& = & - \sum_{i, j = 1}C_{ji}\sum_{\pi \in \widehat{G}}d_{\pi}\frac{1}{\kappa_{\pi}}\mbox{tr}(d\pi(X_{i})d\pi(X_{j})\widehat{f}(\pi)\widehat{g}(\pi)^{*})\\
& = & \sum_{\pi \in \widehat{G}}d_{\pi}\mbox{tr}(\Omega_{C}(\pi)\widehat{f}(\pi)\widehat{g}(\pi)^{*}), \end{eqnarray*}
where for all $\pi \in \widehat{G}$,
$$ \Omega_{C}(\pi): = -\frac{1}{\kappa_{\pi}}\sum_{i, j = 1}^{n}C_{ji}d\pi(X_{i})d\pi(X_{j}). $$
Hence we conclude that $R_{C}^{(2)}$ is a Fourier multiplier with symbol $\Omega_{C}(\cdot)$.
We connect to the work of the previous section by taking $\psi = 0$ and $\phi$ to be a Brownian motion. We write $S_{A} = S_{A,0}$ and we take $A$ to be a constant matrix.
Using (\ref{identity}), (\ref{Planch1}) and Proposition \ref{ftheat}, we obtain, for $f, g \in C^{\infty}(G)$,
\begin{eqnarray*} \int_{G}S_{A}f(\sigma)g(\sigma)d\sigma & = & 2\int_{0}^{\infty}\int_{G} A \nabla_{X}(P_{s}f)(\sigma)\cdot \nabla_{X}(P_{s}g)(\sigma)d\sigma ds\\
& = & 2\sum_{i,j=1}^{n}A_{ij}\int_{0}^{\infty}\int_{G} X_{i}(P_{s}f)(\sigma)X_{j}(P_{s}g)(\sigma)d\sigma ds\\
& = & 2\sum_{i,j=1}^{n}A_{ij}\int_{0}^{\infty}\sum_{\pi \in \widehat{G}}d_{\pi}e^{-2s\kappa_{\pi}}\mbox{tr}(d\pi(X_{i})\widehat{f}(\pi)\widehat{g}(\pi)^{*}(-d\pi(X_{j}))ds\\
& = & -\sum_{i,j=1}^{n}A_{ij}\sum_{\pi \in \widehat{G}}d_{\pi}\frac{1}{\kappa_{\pi}}\mbox{tr}(d\pi(X_{j})d\pi(X_{i})\widehat{f}(\pi)\widehat{g}(\pi)^{*})\\
& = & \sum_{\pi \in \widehat{G}}d_{\pi}\mbox{tr}(\Omega_{A}(\pi)\widehat{f}(\pi)\widehat{g}(\pi)^{*}), \end{eqnarray*}
and so we deduce that $R_{C}^{(2)} = S_{C}$.
The following is a corollary of the above discussion and Theorem \ref{projections1}, inequality \eqref{bound1}.
\begin{corollary} Consider the second order Riesz transform defined by
\begin{equation}\label{secondriesz1}
R_{C}^{(2)}: = \sum_{i, j = 1}^{n}C_{ji}R_{i}R_{j} = \sum_{i, j = 1}^{n}C_{ji}X_{i}X_{j}{\Delta}^{-1}
\end{equation}
with symbol
\begin{equation}\label{secondriesz2}
\Omega_{C}(\pi): = -\frac{1}{\kappa_{\pi}}\sum_{i, j = 1}^{n}C_{ji}d\pi(X_{i})d\pi(X_{j}).
\end{equation}
Then for $1<p<\infty$, $f\in L^p(G)$,
\begin{equation}
\|R_{C}^{(2)}f\|_p\leq (p^*-1)\|f\|_p.
\end{equation}
\end{corollary}
The next corollary follows from Theorem \ref{non-symm-thm}. We record it here in the form stated in \cite{BanOse} in the case of $\mathbb{R}^n$ (see also \cite{BanBau} for compact Lie groups). Note that we here drop the condition $||C|| \leq 1$ as in Theorem \ref{non-symm-thm}.
\begin{corollary}\label{Riesz1}
Let $C$ be an $n\times n$ symmetric matrix with eigenvalues $\lambda_1\leq \lambda_2\leq \ldots \leq \lambda_n$.
Then for $1<p<\infty$, $f\in L^p(G)$,
\begin{equation}\label{generalriesz1}
\|R_{C}^{(2)}f\|_p\leq C_{p,\,\lambda_1,\lambda_d}\|f\|_p,
\end{equation}
where $C_{p,\,\lambda_1,\lambda_d}$ is the constant in \eqref{martin}. In particular, if $J\subsetneq \{1,2,\,\ldots,n\}$, then
\begin{equation}\label{squareriesz2}
\left|\left|\sum_{j\in J} R_j^2 f\right|\right|_p\leq C_{p,0,1}\|f\|_p=c_p\|f\|_p, \quad 1<p<\infty,
\end{equation}
where $c_p$ is the Choi constant in \eqref{choi1}.
\end{corollary}
We remark that in the case of $\mathrm{I\! R\!}^n$ both bounds above are sharp, see \cite{BanOse} and \cite{GMS}.
\subsection{Multipliers of Laplace transform-type}
We continue to work with Brownian motion (as in the previous subsection) and this time we take $A(s):=A(s,\cdot)$ to be a time-dependent matrix-valued function. Then (\ref{identity}), integration by parts and (\ref{Planch1}) yields for all $f, g \in C^{\infty}(G)$,
\begin{eqnarray*} \int_{G}S_{A}f(\sigma)g(\sigma)d\sigma & = &2 \int_{0}^{\infty}\int_{G} A(s) \nabla_{X}(P_{s}f)(\sigma)\cdot \nabla_{X}(P_{s}g)(\sigma)d\sigma ds,\\
& = &-2 \int_{0}^{\infty}\int_{G}A(s)\Delta P_{2s}f(\sigma)g(\sigma)d\sigma ds.\\
& = & 2\int_{0}^{\infty}A(s)\sum_{\pi \in \widehat{G}}d_{\pi}\kappa_{\pi}e^{-2s \kappa_{\pi}}\mbox{tr}(\widehat{f}(\pi)\widehat{g}(\pi))ds. \end{eqnarray*}
Hence for all $\pi \in \widehat{G}$ we have
\begin{equation} \label{LT}
\widehat{S_{A}f}(\pi) = \left(\int_{0}^{\infty}2\kappa_{\pi}e^{-2s \kappa_{\pi}}A(s)ds\right)\widehat{f}(\pi)
\end{equation}
and $S_{A}$ is an operator of {\it Laplace transform-type} (see \cite{Stein} p.58, \cite{Ban1}, \S3.11). It is clearly a Fourier multiplier in the sense of (\ref{FM}). A special case of particular interest is obtained by taking $A(s) = \displaystyle\frac{(2s)^{-i\gamma}}{\Gamma(1-i\gamma)}{I}$, where $\gamma \in \mathbb{R}$. In this case an easy computation shows that the symbol is $m_{S_{A}}(\pi) = \kappa_{\pi}^{-i\gamma}$ and so $S_{A} = (-\Delta)^{i\gamma}$ is an imaginary power of the Laplacian (see also \cite{SW}) and we have the following corollary of Theorem \ref{projections1}, inequality \eqref{bound1}. (Note that for this case we need to apply the Burkholder inequality for complex valued martingales using the results in \cite{BanWan} or \cite{Wan}, as indicated in Remark \ref{hilbert-valued}).
\begin{corollary} For $1<p<\infty$, $f\in L^p(G)$, we have
\begin{equation}
\|(-\Delta)^{i\gamma}f\|_{p} \leq \frac{p^{*}-1}{|\Gamma(1- i\gamma)|}\|f\|_{p}.
\end{equation}
\end{corollary}
This bound was first proved in $\mathrm{I\! R\!}^n$ in \cite[Eq. (7.3)]{Hyt}. For further discussion on this result in $\mathrm{I\! R\!}^n$ and comparison to previous known bounds, see \cite{Ban1}.
\subsection{Subordinate Brownian motion}
In this section we will consider an operator of the form $S_{\psi}:=S_{0, \psi}$ which is built from the non-local part of the L\'{e}vy process. To this end, let $(T(t), t \geq 0)$ be a subordinator, i.e. an almost surely non-decreasing real-valued L\'{e}vy process (so $T(t)$ takes values in $\mathbb{R}^{+}$ with probability one for all $t \geq 0$) . Then we have (see e.g. \cite{SSV}) that $\mathbb{E}(e^{-uT(t)}) = e^{-t h(u)}$ for all $u > 0, t \geq 0$ where $h:(0, \infty) \rightarrow \mathbb{R}^{+}$ is a Bernstein function for which $\lim_{u \rightarrow 0}h(u) = 0$. Hence $h$ must take the form
\begin{equation} \label{bern}
h(u) = c u + \int_{(0, \infty)}(1 - e^{-uy})\lambda(dy)
\end{equation}
where $c \geq 0$ and $\lambda$ is a Borel measure on $(0, \infty)$ satisfying $\int_{(0, \infty)}(1 \wedge y)\lambda(dy) < \infty$. We denote the law of $T(t)$ by $\rho_{t}$, for each $t \geq 0$. It is well-known that (see e.g. \cite{AG, App}) that if $\phi$ is a L\'{e}vy process in $G$ and $(T(t), t \geq 0)$ is an independent subordinator with Bernstein function $h$ then the process $\phi_{T} = (\phi_{T}(t), t \geq 0)$ is again a L\'{e}vy process in $G$ where $\phi_{T}(t):= \phi(T(t))$ for all $t \geq 0$. Let $(P_{t}, t \geq 0)$ be the usual semigroup of convolution operators on $C(G)$ that is associated to the process $\phi$ and $(P^{h}_{t}, t \geq 0)$ be the corresponding semigroup of convolution operators associated to the process $\phi_{T}$. Then it is well-known (and easily verified) that
\begin{equation} \label{subsemi}
P^{h}_{t}f(\sigma) = \int_{(0, \infty)} P_{s}f(\sigma)\rho_{t}(ds)
\end{equation}
for all $t \geq 0, f \in C(G), \sigma \in G$.
In the sequel we will always take $\phi$ to be Brownian motion on $G$ as given by \eqref{sde}.
The following is implicit in \cite{App} but we give a proof for completeness.
\begin{prop} \label{subsemiFT} If $(P_{t}, t \geq 0)$ is the heat semigroup on $G$ then for all $f \in C(G), \pi \in \widehat{G}, t \geq 0$,
$$\widehat{P^{h}_{t}f}(\pi) = e^{-t h(\kappa_{\pi})}\widehat{f}(\pi).$$
\end{prop}
{\it Proof.} Using Fubini's theorem we have from (\ref{subsemi}) and Proposition \ref{ftheat}
\begin{eqnarray*} \widehat{P^{h}_{t}f}(\pi) & = & \int_{G}\int_{(0, \infty)} P_{s}f(\sigma)\rho_{t}(ds)\pi(\sigma^{-1})d\sigma \\
& = & \int_{(0,\infty)}\left(\int_{G}P_{s}f(\sigma)\pi(\sigma^{-1})d\sigma\right)\rho_{t}(ds)\\
& = & \int_{(0,\infty)}e^{-s\kappa_{\pi}}\rho_{t}(ds)\widehat{f}(\pi)\\
& = & e^{-t h(\kappa_{\pi})}\widehat{f}(\pi).~~~~~~~~\hfill \Box \end{eqnarray*}
We now take $A=0$ in (\ref{identity}) and take the L\'evy process to be of the form $\phi_{T}$ as just described. For simplicity we also assume that $\psi$ only depends on the jumps of the process and so we write $\psi(\tau): = \psi(\cdot, \cdot, \tau)$ for each $\tau \in G$. As before, we assume that $\psi$ is regular and that $\|\psi\|\leq 1$. In this case we have
\begin{eqnarray} \label{notime1}
\int_{G}S_{\psi}f(\sigma)g(\sigma)d\sigma &=&
\int_{0}^{\infty}\int_{G}\int_{G}[(P^{h}_{s}f)(\sigma\tau)
- (P^{h}_{s}f)(\sigma)]\\
&\times& [(P^{h}_{s}g)(\sigma \tau)
- (P^{h}_{s}g)(\sigma)]\psi(\tau)\nu(d\tau)d\sigma ds\nonumber\end{eqnarray}
Now using (\ref{Planch1}) and Proposition \ref{subsemiFT} we obtain
\begin{eqnarray*} \int_{G}S_{\psi}f(\sigma)g(\sigma)d\sigma
&=& \int_{0}^{\infty}\int_{G}\sum_{\pi \in \widehat{G}}d_{\pi}e^{-2s h(\kappa_{\pi})}\mbox{tr}[(\pi(\tau) - I_{\pi})\widehat{f}(\pi)\widehat{g}(\pi)(\pi(\tau)^{*} - I_{\pi})]\\
&\times&\psi(\tau)\mu(d\tau)ds\\
& = & \int_{G}\sum_{\pi \in \widehat{G}}d_{\pi}\frac{1}{2 h(\kappa_{\pi})}\mbox{tr}[(2I_{\pi} - \pi(\tau) - \pi(\tau)^{*})\widehat{f}(\pi)\widehat{g}(\pi)]\psi(\tau)\nu(d\tau) \end{eqnarray*}
and so $S_{\psi}$ is a Fourier multiplier with
\begin{equation}\label{subordination-muilt}
m_{S_{\psi}}(\pi) = \frac{1}{2 h(\kappa_{\pi})}\int_{G}(2I_{\pi} - \pi(\tau) - \pi(\tau)^{*})\psi(\tau)\nu(d\tau),
\end{equation}
for $\pi \in \widehat{G}$. From Theorem \ref{projections1} we obtain
\begin{corollary} Let $S_{\psi}$ be the operator with Fourier multiplier given by \eqref{subordination-muilt}. \begin{enumerate}
\item If $\|\psi\|\leq 1$, then
\begin{equation}\label{subordination-1}
\|S_{\psi}f\|_p\leq (p^*-1)\|f\|_p,
\end{equation}
for all $1<p<\infty$, $f\in L^p(G)$.
\item If $\psi:G\to [b, B]$, with $-\infty<b<B <\infty$, then
\begin{equation}\label{subordination-2}
\|S_{\psi}f\|_p\leq C_{p, b, B}\|f\|_p,
\end{equation}
for all $1<p<\infty$, $f\in L^p(G)$, where $C_{p, b, B}$ is the constant in \eqref{martin}.
\end{enumerate}
\end{corollary}
To compare with results in \cite{BanBog} (see e.g. equation (32) therein), suppose that $G$ is connected as well as compact. Then $\exp: \mathfrak{g} \rightarrow G$ is onto and so for every $\sigma \in G, \sigma = \exp(X)$ for some $X \in \mathfrak{g}$. Then $\pi(\sigma) = e^{i(-id\pi(X))}$ where we note that the matrix $-id\pi(X)$ is hermitian. Then we can write $2I_{\pi} - \pi(\sigma) - \pi(\sigma)^{*} = 2(I_{\pi}-\cos(id\pi(X))$. Let $\tilde{\nu}: = \nu \circ \exp$ so $\tilde {\nu}$ is a Borel measure on $\mathfrak{g}$ and define $\tilde{\psi}:\mathfrak{g} \rightarrow \mathbb{R}$ by $\tilde{\psi}:=\psi\circ\exp$. Then in this case we obtain
$$m_{S_{\psi}}(\pi) = \frac{1}{h(\kappa_{\pi})}\int_{\mathfrak{g}}(I_{\pi}-\cosh(d\pi(X)))\tilde{\psi}(X)\tilde{\nu}(dX).$$
\subsection{Multipliers associated to central L\'{e}vy processes}
We say that a L\'{e}vy process $(\phi(t), t \geq 0)$ is {\it central} (or {\it conjugate invariant}) if its law $p_{t}$ is a central measure for all $t \geq 0$, i.e. for all $A \in {\mathcal{B}}(G), \sigma \in G$
$$ p_{t}(\sigma A \sigma^{-1}) = p_{t}(A).$$ Such processes have been investigated in \cite{Liao1} and \cite{App}. It follows that the L\'{e}vy measure $\nu$ is also a central measure (see \cite{Liao1} p.1567.)
Recall that if $\mu$ is a Borel probability measure on a compact Lie group $G$ then its Fourier transform (or characteristic function\footnote{Note that in this paper we adopt the analysts convention when defining the Fourier transform of a function and the probabilists convention for that of a measure. We emphasize that no mathematical inconsistencies arise from these choices.}) is defined by $\widehat{\mu}(\pi):= \int_{G}\pi(\sigma)\mu(d\sigma)$ for all $\pi \in \widehat{G}$. Moreover the matrices $\{\widehat{\mu}(\pi), \pi \in \widehat{G}\}$ uniquely determine $\mu$ (see e.g. Theorem 1 in \cite{KI}.) Returning to L\'{e}vy processes, it is easy to verify that for all $t \geq 0, \pi \in \widehat{G}, 1 \leq i,j \leq d_{\pi}$ we have
\begin{equation} \label{semFT}
\widehat{p_{t}}(\pi)_{ij} = P_{t}\pi_{ij}(e).
\end{equation}
Now suppose that the L\'{e}vy process $(\phi(t), t \geq 0)$ is central. By a slight generalization of the argument of Proposition 2.1 in \cite{App} we deduce that for all $\pi \in \widehat{G}$, there exists $\alpha_{\pi} \in \mathbb{C}$ with $\Re(\alpha_{\pi}) \leq 0$ so that
\begin{equation} \label{centFT}
\widehat{p_{t}}(\pi) = e^{t \alpha_{\pi}}I_{\pi}.
\end{equation}
It follows easily from this that for all $t \geq 0, \pi \in \widehat{G}, f \in C(G)$
\begin{equation} \label{centFTsemi}
\widehat{P_{t}f}(\pi) = e^{t \alpha_{\pi}}\widehat{f}(\pi)
\end{equation}
We now apply \eqref{identity}. For simplicity we again consider the case where $A$ is a constant matrix and $\psi=\psi(\tau)$ depends only on the jumps of the L\'{e}vy process. In this case we can apply Plancherel's theorem \eqref{Planch1} using very similar arguments to those applied in sections 4.1 and 4.4 to deduce that $S_{A, \psi}$ is Fourier multiplier. Indeed for all $f \in C^{\infty}(G), \pi \in \widehat{G}$ we have
$$ \widehat{S_{A, \psi}f}(\pi) = m_{A,\psi}(\pi){\widehat f}(\pi),$$ where
\begin{eqnarray*} m_{A,\psi}(\pi) & = & \frac{1}{\Re(\alpha_{\pi})}\sum_{i,j=1}^{n}A_{ji}d\pi(X_{i})d\pi(X_{j})\\
& - & \frac{1}{2\Re(\alpha_{\pi})}\int_{G}(2I_{\pi} - \pi(\tau) - \pi(\tau)^{*})\psi(\tau)\nu(d\tau).
\end{eqnarray*}
Two of our previously considered examples are in fact special cases of this formula. Indeed for the second order Riesz transform we have $\psi = 0$ and $\alpha_{\pi} = -\kappa_{\pi}$ while for subordinated Brownian motion, $A = 0$ and $\alpha_{\pi} = -h(\kappa_{\pi})$.
Under further constraints, we can find an explicit general formula for $\alpha_{\pi}$. The following is essentially contained in Theorem 4 (c) p.1567 in \cite{Liao1} under some stronger assumptions. It gives a L\'{e}vy-Khintchine type formula for the Fourier transform of the law of a central L\'{e}vy process. We will need the group character $\chi_{\pi}(\cdot):= \mbox{tr}(\pi(\cdot))$ and the normalized character $\varrho_{\pi}(\cdot):=\frac{1}{d_{\pi}}\chi_{\pi}$ for each $\pi \in \widehat{G}$.
\begin{theorem}
Assume that $(\phi(t), t \geq 0)$ is a central L\'{e}vy process on a compact, connected semi-simple Lie group $G$. Then for all $\pi \in \widehat{G}$,
$$ \alpha_{\pi} = -c \kappa_{\pi} + \int_{G}(\varrho_{\pi}(\tau) - 1)\nu(d\tau),$$
where $c \geq 0$.
\end{theorem}
{\it Proof.} First assume that the L\'{e}vy measure $\nu$ has a finite first moment, i.e. $\int_{G}|x_{i}(\sigma)|\nu(d\sigma) < \infty$ for all $1 \leq i \leq n$. It follows from Propositions 4.4 and 4.5 in \cite{Liao} p.99 that for all $f \in C^{2}(G), \sigma \in G$,
$$ {\mathcal{L}}f(\sigma) = c \Delta f(\sigma) + \int_{G}(f(\sigma\tau)- f(\sigma))\nu(d\tau).$$
For each $\pi \in \widehat{G}$, define the matrix ${\mathcal{L}}(\pi)$ by ${\mathcal{L}}(\pi)_{ij}:=({\mathcal{L}}\pi_{ij})(e)$ for $1 \leq i,j \leq d_{\pi}$. Then
$${\mathcal{L}}(\pi) = - c \kappa_{\pi}I_{\pi} + \int_{G}(\pi(\tau) - I_{\pi})\nu(d\tau),$$ and hence by \eqref{semFT},
$ \widehat{p_{t}}(\pi) = e^{t {\mathcal{L}}(\pi)}$ for all $t \geq 0$. Comparing this identity with \eqref{centFT}, we deduce that
$$ \alpha_{\pi}I_{\pi} = -c \kappa_{\pi}I_{\pi} + \int_{G}(\pi(\tau) - I_{\pi})\nu(d\tau),$$
and we obtain the required result on taking the trace (in $V_{\pi}$) of both sides of this last equation.
Now we turn to the general case and take $(\phi(t), t \geq 0)$ to be an arbitrary central L\'{e}vy process with characteristics $(b, a, \nu)$ and infinitesimal generator ${\mathcal{L}}$. Again by Proposition 4.4 in \cite{Liao}, we must have $a = cI$ for some $c \geq 0$ and $b=0$ by the argument in \cite{Liao}, Proposition 4.5. So the process has characteristics $(0, cI, \nu)$. Let $(U_{m}, m \in \mathbb{N})$ be a sequence of Borel sets in $G$ so that $U_{m} \uparrow G$ as $m \rightarrow \infty$ and $\int_{U_{m}}|x_{i}(\sigma)|\nu(d\sigma) < \infty$ for all $1 \leq i \leq n$. To see that such sets can always be constructed, recall that $G$ is equipped with a bi-invariant metric for which the mappings $\tau \rightarrow \sigma \tau \sigma^{-1}$ (where $\tau \in G$) are isometries for all $\sigma \in G$. So we may take e.g. $U_{m} = B_{\frac{1}{m}}(e)^{c}$. Now define the central L\'{e}vy measure $\nu_{m}$ on $G$ by $\nu_{m}(A): = \nu(A \cap U_{m})$ for all $A \in \mathcal{B}(G),m \in \mathbb{N}$. For each $m \in \mathbb{N}$, let $(\phi_{m}(t), t \geq 0)$ be a central L\'{e}vy process with characteristics $(0, cI, \nu_{m})$ and infinitesimal generator ${\mathcal{L}}_{m}$. Then it is easily verified that for all $f \in C^{2}(G)$,
$$ \lim_{m \rightarrow \infty}{\mathcal{L}}_{m}f = {\mathcal{L}}f.$$
If $p_{t}^{(m)}$ is the law of $\phi_{m}(t)$ then for all $\pi \in \widehat{G}$,
$$\widehat{p_{t}^{(m)}}(\pi) = \exp{\left\{ -c \kappa_{\pi} + \int_{U_{m}}(\varrho_{\pi}(\tau) - 1)\nu(d\tau)\right\}},$$ but from the construction in the first part of the proof, we have $\lim_{m \rightarrow \infty}\widehat{p_{t}^{(m)}}(\pi) = \widehat{p_{t}}(\pi)$ and the result follows. $\hfill \Box$
\section{$\mathrm{I\! R\!}^n$, revisited}
In this section we apply the results of \S3 to Euclidean space $\mathrm{I\! R\!}^n$ and compute the Fourier multipliers for the operators $S_{A,\psi}$ explicitly under some assumptions on $A$ and $\psi$. When $A$ is constant and $\psi$ is just a function of the jumps this was already done in \cite{BanBieBog} and \cite{BanBog}. Nevertheless, the formulas derived in \S3 elucidate these results further and provide a more uniform approach to these L\'evy-Fourier multipliers.
We start by recalling various standard notations and facts about Fourier transforms and L\'evy processes in $\mathrm{I\! R\!}^n$. We use the following normalization for the Fourier transform $\widehat{f}$ of $f \in {\mathcal{S}}(\mathbb{R}^{n})$ where ${\mathcal{S}}(\mathbb{R}^{n})$ is the usual Schwartz space of rapidly decreasing functions on $\mathbb{R}^{n}$. For all $\xi \in \mathbb{R}^{n}$,
\begin{equation}\label{Fo}
\widehat{f}(\xi)=\int_{\mathrm{I\! R\!}^n} e^{2\pi i\xi\cdot x}f(x)dx
\end{equation}
and for all $x \in \mathbb{R}^{n}$,
\begin{equation}\label{Foin}
f(x)=\int_{\mathrm{I\! R\!}^n} e^{-2\pi ix\cdot\xi}\widehat{f}(\xi)d\xi
\end{equation}
so that Plancherel's identity takes the form
\begin{equation}\label{plancherel}
\int_{\mathrm{I\! R\!}^n} f(x)g(x)dx=\int_{\mathrm{I\! R\!}^n}\widehat{f}(\xi)\overline{\widehat{g}(\xi)} d\xi,
\end{equation}
for $f, g \in {\mathcal{S}}(\mathbb{R}^{n})$.
With this normalization, if $\nabla=\left(\frac{\partial}{\partial x_1}, \dots, \frac{\partial}{\partial x_n}\right)=\left(\partial_1, \dots, \partial_n\right)$ denotes the standard gradient in $\mathrm{I\! R\!}^n$, then
\begin{equation} \label{nabla}
\widehat{\partial_j f}(\xi)=(-2\pi i) \xi_j \hat{f}(\xi) \,\,\,\,\text{and}\,\,\,\, \widehat{\nabla f}(\xi)=(-2\pi i)\xi \hat{f}(\xi).
\end{equation}
Recall that a Borel measure $\nu$
on $\mathrm{I\! R\!}^n$ with $ \nu(\{0\})=0$ and
\begin{equation}\label{levymu1}
\int_{\mathrm{I\! R\!}^n} \frac{|y|^2}{1+|y|^2} \,\nu(dy)<\infty
\end{equation}
is called a L\'evy measure. (Note that the definition of the L\'evy measure in $\mathrm{I\! R\!}^n$ coincides with that given by \eqref{levy-Lie} in \S2.2 for general Lie groups.)
We denote a L\'evy process in $\mathrm{I\! R\!}^n$ by $(X(t), t\geq 0)$. The celebrated L\'evy-Khintchine formula
\cite{App1}
guarantees the existence of a triple
$\left( b, a, \nu \right)$
such that the characteristic function of the process is given by
$\mathbb{E}\left[\, e^{i \xi \cdot X(t)}\,\right] = e^{t
\rho(\xi)},$
where
\begin{equation}\label{levykin}
\rho(\xi)= i b\cdot \xi - a\xi\cdot \xi +
\int_{\mathrm{I\! R\!}^n}\left[\, e^{i\,\xi \cdot y}- 1 - i (\xi\cdot y) \,\bb{I}_{B_{1}(0)} (y) \,\right]
\nu(dy).
\end{equation}
Here, $b=(b_1, \dots, b_n) \in \mathrm{I\! R\!}^n$, $a=(a_{ij})$ is a non-negative $n \times n$ symmetric matrix,
$\bb{I}_{B_1(0)}$
is the indicator function of the unit ball $B_{1}(0)\subset\mathrm{I\! R\!}^n$ centered at the origin and $\nu$
is a L\'evy measure on $\mathrm{I\! R\!}^n$. We remark that the L\'evy-Khintchine formula may be deduced as a corollary of Hunt's theorem (Theorem \ref{Hunt}) in the case where the group $G$ is $\mathbb{R}^{n}$ (see \cite{Hunt}, \S6, pp.281-3.)
We decompose $\rho$ into its real and imaginary parts so that
\begin{equation}\label{real-Imag}
\rho(\xi)=\Re{\rho}(\xi)+i\Im{\rho}(\xi),
\end{equation}
where
\begin{equation}\label{real}
\Re{\rho}(\xi)=- a\xi\cdot \xi +
\int_{\mathrm{I\! R\!}^n}\left[\, \cos{(\xi \cdot y)}- 1\,\right]
\nu(dy)
\end{equation}
and
\begin{equation}\label{imag}
\Im{\rho}(\xi)=b\cdot \xi+ \int_{\mathrm{I\! R\!}^n}\left[\, \sin{(\xi \cdot y)}- (\xi\cdot y) \,\bb{I}_{B_{1}(0)} (y) \,\right]\,
\nu(dy).
\end{equation}
We note that the convergence of the integrals in \eqref{real} and \eqref{imag} follows immediately from \eqref{levymu1}. Also, observe that $\Re{\rho}(-\xi)=\Re{\rho}(\xi)$ and that $\Re{\rho}(\xi)\leq 0$.
With this notation, the semigroup of the L\'evy process $(X(t), t\geq 0)$ acting on $f\in {\mathcal{S}}(\mathrm{I\! R\!}^n)$ is given as a pseudo-differential operator by
\begin{eqnarray}\label{semigroup}
P_tf(x)=\mathbb{E}[f(X(t)+x)] &=& \mathbb{E}\left(\int_{\mathrm{I\! R\!}^n}e^{-2\pi i(X(t)+x)\cdot\xi}\,\widehat{f}(\xi)d\xi\right)\nonumber\\
&=&\int_{\mathrm{I\! R\!}^n}e^{t\rho(-2\pi\xi)}\,e^{-2\pi ix\cdot\xi}\widehat{f}(\xi)d\xi.
\end{eqnarray}
Since ${\mathcal{S}}(\mathbb{R}^{n}) \subseteq \mbox{Dom}({\mathcal{L}})$ we also obtain the pseudo-differential operator representation
\begin{equation}
{\mathcal{L}}f(x)=\int_{\mathrm{I\! R\!}^n} \rho(-2\pi\xi) e^{-2\pi ix\cdot\xi}\,\widehat{f}(\xi)d\xi.
\end{equation}
Thus
\begin{equation}\label{fouriersemigroup}
\widehat{P_tf}(\xi)=e^{t\rho(-2\pi\xi)}\,\widehat{f}(\xi).
\end{equation}
Also, using \eqref{nabla} we can write $\mathcal{L}$ as
\begin{eqnarray*}
\mathcal{L}f(x)= b^i\partial_i f(x)&+& a^{ij}\partial_i\partial_j f(x)\\
&+& \int\Big[f(x+y)-f(x)-y\cdot\nabla f(x)\bb{I}_{B_{1}(0)} (y)\Big] \nu(dy).
\end{eqnarray*}
A detailed account of these results may be found in \cite{App1} \S3.2.2, pp.163-9.
With $\Lambda$ such that $\Lambda \Lambda^{T} = 2a$, we set, as in \S3, $\nabla_{Y}=\Lambda\nabla$. With this notation we have the following corollary of Theorem \ref{projections1}.
\begin{corollary}\label{projection-Rn}
Suppose $A \in L^{\infty}(\mathbb{R}^{+} \times \mathrm{I\! R\!}^n, \mbox{Leb} \times m_{R};M_{n}(\mathbb{R}))$ and $\psi \in L^{\infty}(\mathbb{R}^{+} \times \mathrm{I\! R\!}^n \times \mathrm{I\! R\!}^n; \mathrm{I\! R\!})$ with $(A, \psi)$ regular and $||A|| \vee ||\psi|| \leq 1$. There exists a bounded linear operator $S_{A, \psi}$ on $L^{p}(\mathrm{I\! R\!}^n)$ where $1<p<\infty$ for which
\begin{eqnarray} \label{identity-Rn}
\int_{\mathrm{I\! R\!}^n}S_{A,\psi}f(x)g(x)dx& = & \int_{0}^{\infty}\int_{\mathrm{I\! R\!}^n} A(s, x)\nabla_{Y}(P_{s}f)(x)\cdot \nabla_{Y}(P_{s}g)(x)dx ds \nonumber \\
& + & \int_{0}^{\infty}\int_{\mathrm{I\! R\!}^n}\int_{\mathrm{I\! R\!}^n}[(P_{s}f)(x+y)
- (P_{s}f)(x)][(P_{s}g)(x+y)
- (P_{s}g)(x)]\\
&\times& \psi(s,x, y)\nu(dy)dx ds,\nonumber
\end{eqnarray}
for al $f, g \in C_{c}^{\infty}(G)$.
Furthermore, for all $f\in L^p(\mathrm{I\! R\!}^n)$ and $g\in L^q(\mathrm{I\! R\!}^n)$, $\frac{1}{p}+\frac{1}{q}=1$,
\begin{equation}\label{bound1-Rn}
\Big|\int_{\mathrm{I\! R\!}^n}S_{A,\psi}f(x)g(x)dx \Big| \leq (p^*-1)\|f\|_p\,\|g\|_q
\end{equation}
and
\begin{equation}\label{bound2-Rn}
\|S_{A,\psi}f\|_p\leq (p^*-1)\|f\|_p.
\end{equation}
\end{corollary}
Let us now suppose that $A=A(s)$ is only a function of $s$ (time) and that $\psi(s, y)$ is only a function of $s$ (time) and $y$ (jumps). Set
\begin{equation}
I=\int_{0}^{\infty}\int_{\mathrm{I\! R\!}^n} A(s)\nabla_{Y}(P_{s}f)(x)\cdot \nabla_{Y}(P_{s}g)(x)dx ds
\end{equation}
and
\begin{equation}
II= \int_{0}^{\infty}\int_{\mathrm{I\! R\!}^n}\int_{\mathrm{I\! R\!}^n}[(P_{s}f)(x+y)
(P_{s}f)(x)][(P_{s}g)(x+y)
- (P_{s}g)(x)]\psi(s, y)\nu(dy)dx ds.
\end{equation}
Applying Plancherel's identity, \eqref{plancherel}, we find that
\begin{eqnarray}\label{I}
I &=&\int_{\mathrm{I\! R\!}^n}\left\{4\pi^2\int_0^{\infty} [A(s)\Lambda\xi\cdot \Lambda\xi]e^{2s\Re{\rho(2\pi\xi)}}ds\right\} \hat{f}(\xi)\overline{\hat{g}}(\xi)d\xi\\
&=&\int_{\mathrm{I\! R\!}^n} m_1(\xi)\hat{f}(\xi)\overline{\hat{g}}(\xi)d\xi,\nonumber
\end{eqnarray}
where
\begin{equation}\label{mult-1}
m_1(\xi)=4\pi^2\int_0^{\infty} [A(s)\Lambda\xi\cdot \Lambda\xi]e^{2s\Re{\rho(2\pi\xi)}}ds.
\end{equation}
In the same way,
\begin{eqnarray}\label{II}
II&=&\int_{\mathrm{I\! R\!}^n}\left\{\int_{\mathrm{I\! R\!}^n}\int_0^{\infty}e^{2s\Re{\rho}(2\pi\xi)}|e^{-2\pi\,i\xi\cdot y}-1|^2 \psi(s, y)ds d\nu(y) \right\} \hat{f}(\xi)\overline{\hat{g}}(\xi)d\xi\nonumber\\
&=&\int_{\mathrm{I\! R\!}^n}\left\{2\int_{\mathrm{I\! R\!}^n}\int_0^{\infty}e^{2s\Re{\rho}(2\pi\xi)}\left(1-\cos(2\pi\xi\cdot y)\right)\psi(s, y)ds \nu(dy) \right\} \hat{f}(\xi)\overline{\hat{g}}(\xi)d\xi\\
&=&\int_{\mathrm{I\! R\!}^n} m_2(\xi)\hat{f}(\xi)\overline{\hat{g}}(\xi)d\xi,\nonumber
\end{eqnarray}
where
\begin{equation}\label{multiplier-2}
m_2(\xi)=2\int_{\mathrm{I\! R\!}^n}\int_0^{\infty}e^{2s\Re{\rho}(2\pi\xi)}\left(1-\cos(2\pi\xi\cdot y)\right)\psi(s, y)ds d\nu(y).
\end{equation}
Thus under the assumption that $A=A(s)$ and $\psi = \psi(s,y)$ we conclude that the operator $S_{A,\psi}$ is a Fourier multiplier with
\begin{equation}
\widehat{S_{A,\psi}f}(\xi) =m(\xi) \widehat{f}(\xi),
\end{equation}
where
\begin{eqnarray}\label{multiplier-3}
m(\xi)&=&4\pi^2\int_0^{\infty} [A(s)\Lambda\xi\cdot \Lambda\xi]e^{2s\Re{\rho(2\pi\xi)}}ds\nonumber\\
&+&2\int_{\mathrm{I\! R\!}^n}\int_0^{\infty}e^{2s\Re{\rho}(2\pi\xi)}\left(1-\cos(2\pi\xi\cdot y)\right)\psi(s, y)ds \nu(dy)
\end{eqnarray}
Recalling from \eqref{real} that
$$
-2\Re{\rho}(2\pi \xi)= 8\pi^2a\xi\cdot \xi +
2\int_{\mathrm{I\! R\!}^n}\left(1- \cos{(2\pi\xi \cdot y)}\right) \nu(dy)
$$
and using the fact that $2a\xi\cdot \xi=\Lambda\xi\cdot \Lambda\xi$, we see that under the assumption $||A|| \vee ||\psi|| \leq 1$, $m\in L^{\infty}(\mathrm{I\! R\!}^n)$ and $\|m\|_{\infty}\leq 1$. Furthermore, if the matrix $A$ has constant entries and the function $\psi$ only depends on $y$, then a simple computation gives that
\begin{eqnarray}\label{multiplier-4}
m(\xi)=\frac{4\pi^2 A\Lambda\xi\cdot \Lambda\xi +
2\int_{\mathrm{I\! R\!}^n}\left(1- \cos{(2\pi\xi \cdot y)}\right)\psi(y) \nu(dy)}{8\pi^2a\xi\cdot \xi +
2\int_{\mathrm{I\! R\!}^n}\left(1- \cos{(2\pi\xi \cdot y)}\right) \nu(dy)}.
\end{eqnarray}
Note that $m(\xi)=\widetilde{m}(2\pi\xi)$,
where
$$
\widetilde{m}(\xi)=\frac{\frac{1}{2}\left(\Lambda^TA\Lambda\right)\xi\cdot \xi +
\int_{\mathrm{I\! R\!}^n}\left(1- \cos{(\xi \cdot y)}\right)\psi(y) \nu(dy)}{a\xi\cdot \xi +
\int_{\mathrm{I\! R\!}^n}\left(1- \cos{(\xi \cdot y)}\right) \nu(dy)}.
$$
From the Remark \ref{hilbert-valued} we know that the above multipliers have the same bounds if the matrix has complex entires and the function $\psi$ takes values in the complex plane. We summarize these results in the following
\begin{theorem}\label{multipliersRn}
Let $A\in M_{n}(\mathbb{C})$ and $\psi:\mathrm{I\! R\!}^n\to\mathbb{C}$ be such that $\|A\|\vee \|\psi\|\leq 1$. Suppose $a\in M_{n}(\mathrm{I\! R\!})$ is symmetric and non-negative definite. Let $\Lambda\in M_{n}(\mathrm{I\! R\!})$ be such that $\Lambda\Lambda^T=2a$. Then the $L^{\infty}$ function
\begin{equation}\label{multiplier-5}
{m}(\xi)=\frac{\frac{1}{2}\left(\Lambda^TA\Lambda\right)\xi\cdot \xi +
\int_{\mathrm{I\! R\!}^n}\left(1- \cos{(\xi \cdot y)}\right)\psi(y) \nu(dy)}{a\xi\cdot \xi +
\int_{\mathrm{I\! R\!}^n}\left(1- \cos{(\xi \cdot y)}\right) \nu(dy)},
\end{equation}
where $\nu$ is a L\'evy measure, defines an $L^p$-multiplier operator $S_{A,\psi}$ with \
\begin{equation}\label{bound2Rn}
\|S_{A,\psi}f\|_p \leq (p^*-1)\|f\|_p,
\end{equation}
for all $f\in L^p(\mathrm{I\! R\!}^n)$, $1<p<\infty$. The inequality is sharp.
\end{theorem}
This formula gives the multipliers studied in \cite{BanBieBog} and \cite{BanBog}. For several concrete examples of classical multipliers that arise from this formula, see \cite{Ban1} and \cite{BanBieBog}. In particular, if we take $\nu=0$, $a=I$ and we write $A=(A_{jk})$, then
$$
m(\xi)=\frac{\sum_{j, k=1}^n A_{jk}\xi_j\xi_k}{|\xi|^2}.
$$
With the right choice of $A$, this gives the multiplier $m(\xi)=\frac{\xi_j^2-\xi_k^2}{|\xi|^2}$. This corresponds to the second order Riesz transforms $R_k^2-R_j^2$ which by \cite{GMS} (see also \cite{BanOse}) has norm $(p^*-1)$. Hence the bound is sharp.
|
train/arxiv
|
BkiUbgPxK0wg09lKAujH
| 5 | 1 |
\section{Introduction}
In today's financial world, most markets rely on a {\em limit order
book} (LOB) to match buyers and sellers. High frequency traders and
market makers rely on real-time access to the LOB in order to
implement their trading algorithms.
Researchers, traders and regulators are interested in the dynamics of
LOB for their own individual reasons.
How stocks trade on the NYSE is discussed by Schwartz (1993)
\cite{NYSE1} and Hasbrouck, Sofianos and Sosebee (1993) \cite{NYSE2}.
A common feature of the dynamics of each LOB is clustering of events.
Each time that an event occurs, the likelihood increases that another
event will occur in the near figure. Processes with this feature are
called {\em self-exciting}, and the lengths of consecutive durations
(times between events) tend to be similar to each other. Two main
types of models for
self-exciting processes have been developed over the years.
One type consists of {\em intensity models}, and the other consists of
{\em duration models}. Intensity models focus on the conditional intensity
function, which gives the instantaneous conditional probability of an event
at each time given the history of the
process. Bauwens and Hautsch (2006) \cite{ABC5} give a good survey of
current intensity models. The most common intensity
model is the Hawkes process, introduced by Hawkes (1971)
\cite{HAWKES}, in which the conditional intensity function is modeled
as a linear combination of the effects of all of the past events. The
effect of each past event is modeled as a decaying function
of the time elapsed since that event. Zhao's thesis
(2010) \cite{ABC6} proposed another intensity model, in which the
conditional intensity function depends on the number of events in the
most recent
past time window of a fixed length. Duration models were
popularized by Engle and Russel (1997, 1998) \cite{ACD} who introduced the
{\em autoregressive conditional duration} (ACD) model. The ACD model
specifies that the conditional mean of the next duration has an
autoregressive structure
so as to capture the self-exciting effect. There is rich literature
extending the ACD model to other duration models. Pacurar (2006)
\cite{ACDsurvey} has given a comprehensive survey on the development
of ACD models. In particular, Bauwens and Veredas (2004) \cite{SCD}
proposed a stochastic conditional duration (SCD) model, in which they
introduce a stochastic noise in the autoregressive formula for the
expectation of duration.
Also, the log ACD model \cite{logACD}, the threshold ACD model
\cite{TACD}, the Markov Switching ACD model \cite{MSACD}, the
stochastic volatility duration (SVD) model \cite{SVD}, and the fractionally
integrated ACD model \cite{FIACD} are all designed to generalize and
improve the original ACD model in various ways.
Furthermore, Russell (1999) \cite{ACI} has developed an intensity model
based on the ACD which is called the autoregressive conditional
intensity (ACI) model.
Although models in the ACD family successfully capture the
self-exciting features of the duration processes, Bauwens, Giot,
Grammig and Veredas (2000) \cite{COMPARE} have shown that none of the
parametric ACD models can pass a model evaluation criterion that was
proposed by Diebold, Gunther and Tay (1998) \cite{denFORCAST} and is based on
the probability integral transform theorem. The criterion will be
described in Section~\ref{subsec:diagnostics}. In addition, all of the
models incorporate an intraday trend. Typically, a spline is fit to
the durations as a preprocessing step,
and then the durations are divided by the fitted
spline to remove the trend. In order to predict durations
on one day given what one learns from the previous day, it is useful
to have a model for how the intraday trend varies from day to day.
In this article, we develop a semiparametric duration model for the dynamics of
trade flow. We combine nonparametric
conditional density estimation, parametric time series models and an
online learning of the intraday trend. Since trades occur at
irregular times throughout the day, the trade duration process is
typically characterized as a marked point process, where the trades
are target events and the other associated features, including price,
spread, trade side, and other features of the LOB are marks.
The main contributions of this article lie in a new and more precise
model of the dynamics of the trade duration process as well as a new
way to deal with the intraday trend for the purposes of prediction.
The rest of the paper is organized as follows. In
Section~\ref{sec:acd}, we review the ACD model and some of its variants.
Section~\ref{sec:hsdm}, introduces our semiparametric framework for
modeling the duration process, including our estimation procedures.
Section~\ref{sec:expres} presents experimental results
on LOB data from the NYSE and compares our
framework with four models from the ACD family.
Section~\ref{sec:disc} summarizes our results and puts them into perspective.
\section{Review of ACD models}\label{sec:acd}
In this section, we briefly review the ACD model along with some of its
variants and analyze their limitations. These limitations serve as
the inspiration for our semiparametric model.
Let $X_i$ denote the elapsed time (duration) between two consecutive
events at times $time_{i-1}$ and $time_i$, i.e. $X_i = time_i - time_{i-1}$, with
$time_0$ being the time at which observation begins.
An ACD model attempts to capture the time dependence in the duration
process by modeling the conditional expectation of the next duration
given the past, i.e. $E(X_i | \mathcal{F}_{i-1})$, where $\mathcal{F}_{i-1}$
denotes the information available up to time $time_{i-1}$. A common ACD model
is:
\begin{eqnarray}
\label{eqn:acd1}
X_i& =& \Psi_i \epsilon_i,\\
\label{eqn:acd2}
\Psi_i &=& \omega + \alpha X_{i-1} + \beta \Psi_{i-1},
\end{eqnarray}
where $\{\epsilon_i\}_{i\geq1}$ is a process of IID positive random
variables with mean 1, $\omega>0$, $\alpha>0$ and $\beta>0$ are
parameters with $\alpha+\beta<1$ (to allow the $\Psi_i$ to have a
common mean.) Thus, $E(X_i | \mathcal{F}_{i-1}) = \Psi_i$.
The particular model specified above is called ACD(1,1) because of the
introduction of one lag for both $X$ and $\Psi$ in (\ref{eqn:acd2}).
The distribution of $\epsilon_i$ is assumed to be from a parametric
family with a long tail. Common choices include, Gamma, Weibull and
Burr families.
\subsection{Additional Explanatory Variables in ACD}
In a market microstructure data set, such as our NYSE data set, events
are usually associated with some additional explanatory variables,
such as volume, spread, price and so forth. These variables can be
characterized as marks in the point process and they can have an
impact on the intensity of the process. In the original ACD model and its
variants, the effects of additional explanatory variables are
incorporated by modifying the autoregressive formula (\ref{eqn:acd2}) to
\begin{equation}\label{eqn:AEV}
\Psi_i = \omega + \alpha X_{i-1} + \beta \Psi_{i-1} + \delta^T u_i,
\end{equation}
where, $u_i$ is a vector of additional explanatory variables and
$\delta$ is a vector of coefficients. Such a specification
indicates that the additional variables affect the distribution of
durations by a scale change.
\subsection{ACD Variants}
Researchers have created variations of the ACD model of two
main types. One type of variation modifies the autoregressive formula
(\ref{eqn:acd2}). The Log-ACD model by
Bauwens and Giot (2000) \cite{logACD} replaces (\ref{eqn:acd2}) by
\[\log(\Psi_i) = \omega + \alpha\log(X_{i-1}) + \beta\log(\Psi_{i-1}),\]
which, unlike (\ref{eqn:acd2}), requires no additional restrictions in
order to guarantee that $\Psi_i>0$. The Stochastic
Duration model \cite{SCD} introduces a random noise in (\ref{eqn:acd2})
to allow $\Psi_i$ to be a non-deterministic function of the past, as
in
\[\Psi_i = \omega + \alpha X_{i-1} + \beta \Psi_{i-1} + u_i,\]
where $u_i$ is IID Gaussian noise. The Fractional Integrated ACD model
\cite{FIACD} introduced a differencing in order to capture the long
memory of the duration sequence. This model will be described in more
detail in Section~\ref{sec:bench} because we use it as a benchmark for
comparison to our model. The Threshold ACD \cite{TACD},
\[ \Psi_i = \left\{
\begin{array}{lr}
\omega_1 + \alpha_1 X_{i-1} + \beta_1 \Psi_{i-1}, &
\mbox{if \ $0 < X_{i-1} \leq r_1$,} \\
\omega_2 + \alpha_2 X_{i-1} + \beta_2 \Psi_{i-1}, &
\mbox{if $r_1 < X_{i-1} \leq r_2$,} \\
\omega_3 + \alpha_3 X_{i-1} + \beta_3 \Psi_{i-1},
&\mbox{if $r_2 < X_{i-1} < \infty$.}\end{array}\right.,\]
allows different dependence in different
regimes.
The second type of variation is to allow more general
distributions for $\epsilon$ in (\ref{eqn:acd1}). Traditionally, $\epsilon$
is assumed to have a long-tailed distribution with mean 1. A semiparametric
version of the ACD model \cite{FIACD} estimates the parameters in
(\ref{eqn:acd2}) by quasi-maximum likelihood \cite{QML} and then
estimates the distribution of the residual $\epsilon$
nonparametrically. When we compare our model with
ACD variants, we will include both the parametric ACD with exponential
distribution and the semiparametric ACD using kernel density estimation to
estimate the distribution of $\epsilon$ nonparametrically.
\subsection{Limitations of ACD Models}\label{sec:limit}
In parametric ACD models and their variants, there is a strong parametric
assumption on the distribution of the $\epsilon$ process. Gamma and
Weibull distributions are the most common choices, and both of these
include exponential distributions as special cases. However, as we
will show in Section~\ref{sec:hsdm}, the trade durations do not
admit such an
ideal parametric distribution. Furthermore, the empirical distribution
of log-durations appears to be bimodal, which undermines the
performance of any model that relies on a parametric family of
distributions for log-durations. In
nonparametric versions of the ACD model, Gaussian kernel
density estimation also performs poorly because of the long tail of
the distribution of the residual $\epsilon$.
Another important restriction on ACD models and their variants is that
the time dependency is incorporated only in the expectation (which
happens to be the same as the scale because of the form of the model) of the
duration distribution. However, as we will show in our experimental
results in Section~\ref{sec:expres},
the previous trade duration affects the ensuing trade duration in
a more general way. In particular, it changes both the locations and
the relative sizes of the two modes of the bimodal conditional
distribution of durations. Therefore, modeling the time
dependency solely in terms of the duration mean/scale cannot capture the
more general effect of previous durations. Although some of the extensions
are designed to overcome this shortcoming, they are still not
flexible enough to capture the effects on the modes of the distribution.
\section{ Hierarchical Semi-parametric Duration Model
(HSDM)} \label{sec:hsdm}
In this section, we propose a semiparametric model for the
conditional distribution of durations. The model also allows
estimation of a conditional intensity function. Suppose that we
observe a series
of occurrence times $\{time_i\}_{i\geq0}$ from a point process. By taking the
differences of successive occurrence times, we get a series of
durations $\{\Delta time_i\}_{i\geq1}$, where $\Delta time_i=time_i-time_{i-1}$.
Suppose that the conditional density
function (given the past) for $\Delta time_i$ is $p_i(t)$ with CDF
$P_i(t)$. From the viewpoint of point processes, the conditional
intensity function $\lambda$ is defined as:
\[\lambda(t | \mathcal{F}_t) = \lim_{\Delta t\rightarrow 0}
\frac{1}{\Delta t} \Pr(\mbox{One\ event\ occurs\ in\ the\ time\
interval} \ [t, t+\Delta t] | \mathcal{F}_t)\]
In a duration-based point process, the estimated intensity $\hat
\lambda_t$ is essentially the hazard function, which can be obtained
by $\hat haz_i(\cdot) = \hat p_i(\cdot) / (1 - \hat P_i(\cdot) )$.
Since the duration has a long tail, we transform to the log scale in
the rest of the article and use $\{T_i\}_{i\geq1}$ to denote the logarithms of
the durations. The logarithms of durations are critical in that
the original durations have a distribution with a long tail, which
makes Gaussian kernel density estimation perform poorly. Instead,
kernel density estimation on the log scale is
equivalent to kernel density estimation with an asymmetric
bandwidth that increases the farther one
gets into the tail of the original scale. This ensures that the asymmetric
long-tailed distribution is captured well. Throughout the paper, the
analysis of HSDM focuses on logarithms of durations.
We let $f_i(\cdot)$ denote the density of $T_i$ given the past, and
$F_i(\cdot)$ denotes its
CDF. Naturally, $p_i(\cdot)$ and $P_i(\cdot)$ can be derived directly from
$f_i(\cdot)$ and $F_i(\cdot)$ and vice-versa. Therefore, our goal is
equivalent to estimating the density function
$f_i(\cdot)$ of $T_i $ given $\{T_j\}_{j=1}^{i-1}$.
In what follows, we present the model, a corresponding estimation method, and a
prediction algorithm for future events.
\subsection{Model} \label{sec:DGP}
In this section, we describe a model for general point processes. In
Section~\ref{subsec:estimation},
we give the specific version that we use for trade duration processes
along with the steps needed to fit the model.
In general, the duration sequence comes from a multi-layer
hierarchical model as shown in Figure~\ref{fig:DGP1},
\begin{figure}[htb]
\begin{center}
\begin{subfigure}{0.4\textwidth}
\caption{}
\includegraphics[width = \textwidth, height = 3.0in ]{DGP1.png}
\label{fig:DGP1}
\end{subfigure}
~
\begin{subfigure}{0.4\textwidth}
\caption{}
\includegraphics[width = \textwidth, height = 3.0in]{DGP2.png}
\label{fig:DGP2}
\end{subfigure}
\end{center}
\footnotesize
\caption{(a) General form of the model, and (b) specific version for trade
durations.}
\label{fig:DGP}
\end{figure}
with Figure~\ref{fig:DGP2} showing more specific information for the
model that we use for trade durations. The process that we used for
choosing the components of the model is described in
Section~\ref{sec:modelbuild}.
\begin{enumerate}
\item Underlying the process is a latent process $\{p_i\}_{i\geq1}$
that captures the long-memory dependence. The latent process is
modeled by a parametric time series model, e.g. autoregressive
moving average (ARMA) or autoregressive fractionally integrated
moving average (ARFIMA). If one needs to incorporate
additional explanatory variables, one can
augment the time series model with a regression component. Descriptions of
ARFIMA models and the augmentation that we use for regression are is given in
Appendix~\ref{sec:arfima}.
\item A general time trend can be modeled so that the distribution of
duration depends on both clock time and event time. Let
$R(\cdot|time_{i-1})$ be a one-to-one, clock-time dependent, function of a
real variable, where $time_{i-1}$ stands for the clock time at which event $i-1$
occurs. Applying the transformation produces the transformed process
\[p^T_i = R(p_i|time_{i-1}).\]
\item Finally, the log-duration $T_i$ is a general past-dependent
transformation of $p^T_i$, $T_i=H_i(p^T_i)$.
\end{enumerate}
Combining the above levels of the hierarchy, the distribution of
$\{T_i\}_{i\geq1}$ can be written in terms of the distribution of the
latent process $\{p_i\}_{i\geq1}$. Let $G_i(\cdot)$ denote the
conditional distribution function of $p_i$ given the past. Then, the
conditional distribution of $T_i$ given the past has CDF
\begin{equation}\label{eq:tcdf}
F_i^*(t)=\Pr(T_i\leq t)=\Pr(p_i^T\leq
H_i^{-1}(t))=G_i\left(R^{-1}[H_i^{-1}(t)|time_{i-1}]\right).
\end{equation}
\subsection{Estimation Method} \label{subsec:estimation}
Our proposed semiparametric estimation method proceeds by reversing the
steps in the data generating process described above.
\begin{enumerate}
\item First, express the general transformation $H_i$ as
$H_i(\cdot)=F_i^{-1}(\Phi(\cdot))$, where $F_i(\cdot)$ is a general
cumulative distribution function (CDF) that depends on the past and
$\Phi$ is the standard normal CDF. For trade durations, we
find a nonparametric kernel estimator of
$H_i$ as follows. Compute a conditional density estimator
$\hat f_i (\cdot)$ for the log-durations $T_i$ given the previous
log-duration $T_{i-1}$, and form the
corresponding CDF, $\hat F_i (\cdot)=\hat F(\cdot|T_{i-1})$. \label{step1}
Calculate the {\em generalized residuals} $c_i = \hat F_i (T_i)$
and the {\em transformed generalized residuals} $\hat{p}^T_i =
\Phi^{-1}(c_i)$.
\item\label{step2} Model the clock-time dependent trend. For trade
durations, we find that an intraday trend is both useful and
meaningful. The specific form we use is
\[R(p_i|time_{i-1})=p_i\tau_{\rm sd}(time_{i-1}) + \tau_{\rm mean}(time_{i-1}),\]
where $\tau_{\rm mean}$ and $\tau_{\rm sd}$ are respectively
functions of clock time that model changes in the mean and
standard deviation of
$\{\hat{p}^T_i\}_{i\geq1}$.
Estimate the trends as $\hat\tau_{\rm mean}(time_{i-1})$ and $\hat\tau_{\rm
sd}(time_{i-1})$, and then
calculate the detrended sequence
\[\hat{p}_i = \hat{R}^{-1}[\hat{p}^T_i|time_{i-1}]=\frac{\hat{p}^T_i-
\hat{\tau}_{\rm mean}(time_{i-1})}{\hat{\tau}_{\rm
sd}(time_{i-1})}.\]
Specifically, we let both $\tau_{\rm mean}$ and $\tau^2_{\rm sd}$ be
quadratic functions of clock time as described in more detail below.
\item Fit a parametric time series model to the detrended transformed
generalized residuals: $TSmodel(\hat{p}_i |\{\hat{p}_j\}_{j=1}^{i-1})$.
The fitted time series model will predict that each $\hat p_i$ given the
past has a normal distribution with mean $\hat\mu_i$ and standard
deviation $\hat\sigma_i$.
If additional explanatory variables are needed, an
appropriate modification is done at this
step. Section~\ref{sec:arfima} gives more details. \label{step3}
\end{enumerate}
After the fit, transform back to the log-duration scale.
The fitted value corresponding to the $i$th log-duration is
\[\hat{T}_i=\hat
F_i^{-1}\left[\Phi\left(\frac{\hat{p}_i-\hat{\mu}_i}{\hat{\sigma}_i}
\right)\right].\]
In step~\ref{step1}, $\hat{F}_i$ can be any
past-dependent CDF. For trade durations, we try to capture
a general form of the most important dependence, specifically the dependence
on the previous log-duration, $T_{i-1}$. So, we use a nonparametric
conditional density estimator for the density of $T_i$ given
$T_{i-1}$, and convert the density into its corresponding CDF.
If $\hat{F}_i$ were indeed the conditional CDF of
$T_i$ given the past, then the generalized residuals
$\{c_i\}_{i\geq1}$ would be independent
uniform random variables on the interval $(0,1)$, and
$\hat{p}^T_i=\Phi^{-1}(c_i)$ would be independent standard normal
random variables. Of course, empirical evidence with trade durations
suggests that the distribution of $\{\hat{p}^T_i\}_{i\geq1}$ is much
more complicated, having both time-varying mean and time-varying
standard deviation, not to mention long memory.
In step~\ref{step2}, we detrend the $\hat{p}^T_i$. Both the mean and
variance appear to be large in the middle of the day and smaller at
the start and end of the day. So, we fit a quadratic trend for each:
\[\tau_{\rm mean}(t)=at^2+bt+c,\quad\tau^2_{\rm sd}(t)=dt^2+et+f.\]
Our model says that the
\[p_i=\frac{p_i^T-\tau_{\rm mean}(time_{i-1})}{\tau_{\rm sd}(time_{i-1})},\]
given the past, are a normally distributed process with time-series
structure. We fit the trend parameters $\eta = (a,b,c,d,e,f)$ using
quasi-maximum likelihood. The log-quasi-likelihood function that we use is
\begin{equation}\label{eq:thetall}
-\sum_{i=2}^n\left[\log\tau_{\rm sd}(time_{i-1})+\frac{[\hat{p}_i^T-
\tau_{\rm mean}(time_{i-1})]^2}{2\tau_{\rm sd}(time_{i-1})^2}\right],
\end{equation}
where $n$ is the number of durations in the day. The function in
(\ref{eq:thetall}) would be the likelihood function if the
$\hat{p}_i$ were independent rather than following a time-series model.
In step~\ref{step3}, we fit an appropriate time series model to the
$\{\hat{p}_i\}_{i\geq1}$ sequence assuming that the noise terms are
normally distributed. With trade duration data, we fit an ARFIMA
model with orders chosen by BIC. Each time series model then says that
the distribution of $p_i$ is the normal distribution with a fitted
mean $\hat \mu_i$ and fitted variance $\hat \sigma_i^2$ determined by
the specific model.
Instead of maximizing the quasi-log-likelihood (\ref{eq:thetall}), we
could attempt to find the joint MLE of $\eta$
and the parameters of the ARFIMA model. The log-likelihood for both sets of
parameters is not (\ref{eq:thetall}), but rather
\begin{equation}\label{eq:thetall2}
-\sum_{i=2}^n\left[\log[ \tau_{\rm sd}(time_{i-1}) \sigma_i
]+\frac{[\hat{p}_i^T-
\tau_{\rm mean}(time_{i-1}) - \tau_{sd}(time_{i-1})
\mu_i]^2}{2\tau_{\rm sd}(time_{i-1})^2 \sigma_i^2}\right],
\end{equation}
where $\mu_i$ and $\sigma_i$ are functions of the ARFIMA parameters
that specify the mean and standard deviation of $p_i$ given the
past. Note that (\ref{eq:thetall}) is the
special case of (\ref{eq:thetall2}) when $\mu_i=0$ and $\sigma_i=1$.
Starting with $\mu_i=0$ and $\sigma_i=1$, steps~\ref{step2}
and~\ref{step3}
should be iteratively repeated, using (\ref{eq:thetall2}) in
step~\ref{step2}instead of (\ref{eq:thetall}), until the parameter estimates
converge. Specifically, after the first round of steps~\ref{step2}
and~\ref{step3}, set $\mu_i$ and $\sigma_i$ respectively to the estimated
mean and standard
deviation of $p_i$ given the past as fit by the ARFIMA model in
step~\ref{step3}. For later iterations, use the estimated $\mu_i$ and $\hat
\sigma_i$ to refit the trend parameters by maximizing the
log-likelihood in equation \ref{eq:thetall2}. With the new estimated
trend parameters, refit the ARFIMA parameters and alternated until the
estimated parameters converge. We compared this procedure to the
quasi-likelihood maximization described above and found that the
ARFIMA parameter estimates change negligibly (usually less than 1\%).
The trend parameters sometimes change as much as 15\%, but the
changes do not translate into noticeable changes in predictions. (We
give more evidence of this last claim in
Section~\ref{subsec:prediction}.) The
empirical results that we report in Section~\ref{sec:expres} use the
quasi-likelihood maximization.
\subsection{Prediction} \label{subsec:prediction}
When we wish to predict log-durations on a new day (which we will call {\em
test data}), we start with the fitted
model based on the previous day's data (which we will call {\em
training data}). We carry forward the conditional CDF $\hat
F(\cdot|T_{i-1})$ and coefficients from the ARFIMA process that were
estimated from the training data. For the the intraday trend, we assume
that a new coefficient vector $\eta = (a,b,c,d,e,f)$ is needed each day.
We start by using the estimated $\hat
\eta=(a_0,b_0,c_0,d_0,e_0,f_0)$ from the training data. In order to
perform updates to $\hat\eta$ in real
time as events occur, a fast update is needed for $\hat{\eta}$.
We propose the following penalized least squares estimation (LSE) method.
Whenever a new pair of $(time_{j-1}, p^T_j)$
arrives from the test data, we update our estimated $\hat
\eta$ as follows. Choose $(\hat a, \hat b, \hat c)$ to minimize
\[\sum_{i=1}^j (p^T_i - a \times time_{i-1}^2 - b \times time_{i-1} -c )^2 +
\lambda(a-a_0)^2 +\lambda(b-b_0)^2 + \lambda(c-c_0)^2,\]
and set
\begin{equation}\label{eq:tauhat1}
\hat{\tau}_{\rm mean}(time_{i-1})=\hat a\times
time_{i-1}^2+\hat b \times time_{i-1}+\hat c.
\end{equation}
Then choose $(\hat d,
\hat e, \hat f)$ to minimize
\[\sum_{i=1}^j [ (p^T_i-\hat \tau_{\rm mean}(time_{i-1}))^2 - d
\times time_{i-1}^2 - e \times time_{i-1} -f ]^2 + \lambda(d-d_0)^2 +
\lambda(e-e_0)^2 + \lambda(f-f_0)^2,\]
and set
\begin{equation}\label{eq:tauhat2}
\hat{\tau}_{\rm sd}(time_{i-1})=\sqrt{\hat d\times
time_{i-1}^2+\hat e \times time_{i-1}+\hat f.}
\end{equation}
Empirical results depend little on the value of $\lambda$ for
$\lambda$ in a large interval. We use $\lambda= 10$ in our calculations.
A more time-consuming, but perhaps more principled, method of updating
the trend parameters would be to compute the posterior mode (PM) after
each trade event. We could use the negatives of the penalizations in
the LSE method log-priors and maximize the sum of those log-priors and
the log-likelihood (\ref{eq:thetall2}).
The quality of the LSE approximation, compared to PM is illustrated in
Figures~\ref{fig:PMvsLSE} and~\ref{fig:individualLL_LSEvsPM}.
We see that the two update methods produce intraday trends that
are very similar with predictions of comparable quality. Because LSE
works many times faster
than PM, we use LSE for prediction in the remainder of the paper.
\begin{figure}[htb]
\begin{center}
\includegraphics[width = 1.0\textwidth, height = 3.0in]{PMvsLSE.png}
\caption{Estimated intraday trends in both mean and standard
deviation for one stock on one day. The black line gives the
estimates based on LSE, and the red
line gives the estimates based on PM.}\label{fig:PMvsLSE}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width = 0.7\textwidth, height = 3.0in]{individualLL_LSEvsPM.png}
\caption{Histogram of differences of individual prediction
log-likelihoods (one for each predicted duration) between LSE and PM for JPM
on one day. The mean and median are both close to 0, and the average
of the individual prediction log-likelihoods is $-2.15$, so the
percentage differences are also small. The other stocks and days have
similar patterns. }\label{fig:individualLL_LSEvsPM}
\end{center}
\end{figure}
In Section~\ref{sec:expres}, we evaluate our model
fit and compare it to the fits of other models. We fit all models
using training data and then base the evaluations and
comparisons on test data. We use the prediction
log-likelihood and goodness-of-fit tests based on generalized residuals.
\subsection{Predictive Distribution and Log-Likelihood}\label{sec:preddist}
In this section, we show how to use estimates from the training data
along with the continuously updated intraday trend described above to
compute the predictive distribution of test data.
The fitted conditional CDF of the next $T_i$ in the test data given
the past can be constructed using
(\ref{eq:tcdf}). The time series model says that our estimate of the
conditional CDF $G_i$ is $\hat
G_i(u)=\Phi([u-\hat\mu_i]/\hat\sigma_i)$, where $\hat\mu_i$ and
$\hat\sigma_i$ are based on the estimated ARFIMA parameters from the
training data along with the past durations in the test data.
Our estimate of $H_i$ from the training data is $\hat H_i=\hat
F_i^{-1}(\Phi)$, so the estimated conditional CDF of $T_i$ is
\begin{equation}\label{eq:estcdf}
\hat{F}_i^*(t)=\Phi\left(\frac{1}{\hat\sigma_i}\left[
\frac{\Phi^{-1}(\hat F_i(t))-\hat\tau_{\rm mean}(time_{i-1})}
{\hat\tau_{\rm sd}(time_{i-1})}-\hat{\mu}_i\right]\right),
\end{equation}
where $\hat{\tau}_{\rm mean}$ and $\hat{\tau}_{\rm sd}$ come from
(\ref{eq:tauhat1}) and (\ref{eq:tauhat2}) respectively, and are
recomputed each time that a new event occurs.
In order to compute the prediction log-likelihood of the test data, we
need the density corresponding to the CDF in (\ref{eq:estcdf}). This
is obtained by standard calculus operations as
\begin{equation}\label{eq:estdens}
\hat{f}_i^*(t)=\frac{1}{\hat{\tau}_{\rm sd}(time_{i-1})}
\phi_{\hat\mu_i,\hat\sigma_i}\left(\frac{\Phi^{-1}(\hat F_i(t))
-\hat\tau_{\rm mean}(time_{i-1})}{\hat\tau_{\rm sd}(time_{i-1})}\right)
\times\frac{\hat{f}_i(t)}{\phi_{0,1}(\Phi^{-1}(\hat{F}_i(t)))},
\end{equation}
where, $\phi_{m,s}(\cdot)$ denotes the density of the normal distribution
with mean $m$ and standard deviation $s$, and $\hat{f}_i$ is the
density that corresponds to $\hat{F}_i$.
The predictionlog-likelihood for the test data $\{T_i\}_{i=1}^n$ is
$\sum_{i=1}^n\log (\hat f_i^*(T_i))$.
The form (\ref{eq:estdens})
has a convenient interpretation for comparing our model to some
submodels. For example, if we ignore the intraday trend, then we just
set $\hat{\tau}_{\rm mean}(\cdot)\equiv0$ and $\hat{\tau}_{\rm sd}(\cdot)\equiv1$.
If we wish to ignore the ARFIMA modeling, we just set $\hat\mu_i=0$
and $\hat\sigma_i=1$.
The {\em final generalized residual} corresponding to $T_i$ from the
test data is computed by substituting $T_i$ for $t$ in (\ref{eq:estcdf}):
\begin{equation}\label{eq:fgr}
c^*_i = \hat F_i^*(T_i) =\Phi\left(\frac{1}{\hat\sigma_i}\left[
\frac{\Phi^{-1}(\hat F_i(T_i))-\hat\tau_{\rm mean}(time_{i-1})}
{\hat\tau_{\rm sd}(time_{i-1})}-\hat{\mu}_i\right]\right),
\end{equation}
If the model fits well, then the $c^*_i$ should look like a sample of
independent uniform random variables on the interval $(0,1)$.
We will present goodness-of-fit tests
results based on $\{c^*_i\}_{i=1}^n$ in Section~\ref{sec:expres}.
\comment{This log-likelihood can be
written as the sum of three terms that correspond to the three steps
in our estimation procedure and the three levels of the hierarchical model.
To make the formulas cleaner, define
\begin{eqnarray*}
\hat{p}^T_i&=&\Phi^{-1}(\hat F_i(T_i)),\\
\hat{p}_i&=&=\frac{\hat{p}_i^T-\hat{\tau}_{\rm mean}(time_{i-1})}
{\tau_{\rm sd}(time_{i-1})},\\
\hat{e}_i&=&\frac{\hat{p}_i-\hat\mu_i}{\hat\sigma_i},
\end{eqnarray*}
so that $c^*_i=\Phi(\hat{e}_i)$. Then
\begin{eqnarray}\nonumber
\sum_{i=1}^n\log (\hat f_i^*(T_i))&=&\sum_{i=1}^n\log(\hat{f}_i(T_i))
+\sum_{i=1}^n\log\left(\frac{\phi_{\hat{\tau}_{\rm mean}(time_{i-1}),
\hat{\tau}_{\rm sd}(time_{i-1})}(\hat p_i)}{\phi_{0,1}(\hat p_i^T)}\right)\\
&&+\sum_{i=1}^n\log\left(\frac{\phi_{\hat\mu_i,\hat\sigma_i}(\hat p_i)}
{\phi_{0,1}(\hat p_i)}\right). \label{eq:splitll}
\end{eqnarray}
The first sum on the right-hand side of (\ref{eq:splitll}) is the
prediction log-likelihood that we would get if we dropped the first
two stages of our hierarchical model from Section~\ref{sec:DGP} and
implemented only step~\ref{step1} of the estimation procedure in
Section~\ref{subsec:estimation} (nonparametric conditional
distribution given previous duration). The second sum on the right-hand
side is the additional prediction log-likelihood that comes from
including the second stage of the hierarchical model (intraday trend)
and step~\ref{step2} of the estimation procedure. The third sum
is the incremental contribution from the first stage of the model
(parametric time series model) and step~\ref{step3} of the estimation
procedure.}
\subsection{Additional Explanatory Variables} \label{subsec:bpi}
In a marked point process, marks are observed along with the target
events. These marks, or additional explanatory variables, may have an
impact on the distribution of log-duration. In the ACD model and its
variants, the
marks' information is incorporated in the autoregressive formula
(\ref{eqn:acd2}). Analogously, in our model,
it is natural to incorporate the additional variables' effects in the
parametric time-series model (step~\ref{step3}).
A straightforward way to incorporate additional variables into an
ARFIMA model is to extend the
ARMA model with regression \cite{book}. We call the extension {\em ARFIMA
with regression}.
Various methods for estimating ARFIMA models have been reviewed in
\cite{ARFIMAsurvey}.
In the trade duration example in Section~\ref{sec:expres}, we use ARFIMA with
regression in order to incorporate an additional explanatory variable
into our model.
\section{Experimental Results}\label{sec:expres}
In this section, the HSDM framework is applied to the trade flows from
the limit order books of four stocks (IBM, JC Penny, JP Morgan, and Exxon Mobil)
on the New York Stock Exchange (NYSE) for selected dates between
6 July 2010 and 29 July 2010, 18 consecutive trading days. In
Section~\ref{sec:modelbuild}, the first eight days are used to build
the model and discover patterns. In Section~\ref{subsec:diagnostics},
we use the remaining days to validate the model and to make
comparisons between HSDM and benchmark models. Each day is used as
training data to estimate parameters and the following day is used as
test data for prediction and goodness-of-fit tests. This pattern of
training data followed by test data is used during both model building
and validation.
Since the focus of this paper is on the dynamics of trade flow, the
limit order book data are preprocessed, and a list of triples
$(time_i, T_i, BPI_i)$ is produced, where $time_i$ is the clock time,
$T_i$ is log-duration, and $BPI_i$ (book pressure
imbalance) is our additional explanatory variable.
Table~\ref{table:triple} gives a sample of three consecutive such triples.
\begin{table}[hbt]
\centering
\caption{A sample of trade flow data. Clock time is in milliseconds
since midnight, and log-duration is in log-milliseconds since
previous trade.}\label{table:triple}
\begin{tabular}{l ccc }
\hline
clock time & log-duration & book pressure imbalance \\
\hline
43026177 & 3.91 & -0.588 \\
43026179 & 6.82 & -0.134 \\
43026180 & 6.10 & -1.946 \\
\hline
\end{tabular}
\end{table}
We observe such a triple whenever a trade occurs.
Book pressure imbalance is defined as the
logarithm of the ratio of the number of sell-side shares at the ask
price to number of buy-side shares at the bid price. In general, it
measures the imbalance of demand between the buy and sell sides of
the market. BPI varies from time to time and actually changes much
more frequently than trades occur. In this paper, we keep track of
the book pressure imbalance only when a trade happens. Moreover, since
the focus in this article is the duration between trades, it is the
degree of imbalance rather than the direction of imbalance that
matters. Thus, we use the absolute value of book pressure imbalance,
$|BPI|$, as our additional explanatory variable.
In Section~\ref{sec:modelbuild}, we present empirical evidence that
motivates each of the stages in the hierarchy of the HSDM framework.
Section~\ref{sec:bench} describes the benchmark models to which we
compare HSDM in Section~\ref{subsec:diagnostics}.
\subsection{Model Building}\label{sec:modelbuild}
Here we show why we chose the particular stages in the hierarchy of
the HSDM model of Section~\ref{sec:hsdm}. The choices are based on a
period of model building data ranging from 6 July 2010 to 16 July
2010. It is widely accepted that
duration processes in market microstructure data are self-exciting
(especially intertrade durations). Figure~\ref{fig:selfexciting}
\begin{figure}[hbt]
\begin{center}
\includegraphics[width = 0.8\textwidth, height =
4in]{consecutiveDurations.pdf}
\caption{Current log-duration (vertical axis) vs. previous
log-duration (horizontal axis) for one stock on one day. Two key
features are noted by the circled regions. The solid circles indicate
the self-exciting nature by showing the clustering of pairs with
both durations being short and with both durations being long. The
dashed circles indicate the bimodal nature by showing secondary
modes at both long and short durations regardless of the length of
the other duration.}\label{fig:selfexciting}
\end{center}
\end{figure}
shows a scatter plot of the pairs of consecutive
log-durations for one stock on one day. The plot shows that (i) long
durations tend to be
followed by long durations, while short durations tend to be followed
by short duration (the self-exciting property) and (ii) the marginal
distribution of log-duration and the conditional distribution of
log-duration given the previous log-duration are bimodal. Since we
expect the most recent durations to carry the most information,
step~\ref{step1} of the
estimation procedure employs nonparametric kernel
conditional density estimation as described by Hall, Racine and Li
(2004) \cite{ABC7}, conditioning on the previous log-duration
$T_{i-1}$. A technical issue arises due to the discreteness
of durations, as they are measured to the nearest millisecond. We explain
this issue in more detail in Appendix~\ref{sec:noise} along with
how we deal with
it. In particular, we explain why it makes sense
to base the estimation on the logarithms of the durations.
The estimated conditional CDF of $T_i$ given $T_{i-1}$ is denoted by
$\hat F_i(\cdot)$. Figure~\ref{fig:npcdens}
\begin{figure}[htb]
\begin{center}
\includegraphics[width = 1.0\textwidth, height = 5in]{figureNPCDENS.png}
\caption{Conditional density of $T_i$ given $T_{i-1}$ for 12 different
values of $T_{i-1}\in\{1,\ldots,12\}$. These range from
$2.7\times10^{-3}$ seconds to 2.7 minutes.}\label{fig:npcdens}
\end{center}
\end{figure}
shows the conditional densities
that were estimated conditional on
different values of
the previous log-duration
$T_{i-1}$ for one day of one stock. The
conditional densities capture the self-exciting feature of trades.
Note how the conditional density is highest near zero when the
previous duration is short, but when the previous duration
is long, the
density is highest at larger values. No matter what the previous
duration $T_{i-1}$ is, the
distribution of the current duration $T_i$ has two local modes. And as
the previous
duration increases, both the height and the location of the second
mode increase. The bimodal characteristic partly explains why
parametric conditional duration models with a unimodal residual
distribution cannot capture the dynamics of trade flow very well.
Nonparametric conditional density estimation in step~\ref{step1}
captures not only some short memory information but also the bimodal
nature of the conditional distribution.
In step~\ref{step2} of the estimation, we introduce intraday trends
for both the mean and variance of the $\{p_i^T\}_{i\geq1}$ sequence.
Figure~\ref{fig:intradayTrend}
\begin{figure}[htb]
\begin{center}
\includegraphics[width = 1.0\textwidth, height = 3in]{intradayPattern.png}
\caption{Fitted intraday trends on eight different days (one color for each
day) for both the mean and variance of $p_i^T$ for a single stock (JPM).
The trends were fitted as moving average trends with window length equal 2000 trades fit to the $\hat{p}_i^T$.}
\label{fig:intradayTrend}
\end{center}
\end{figure}
shows empirical evidence for those trends for eight different days and one
stock. The mean
trends in Figure~\ref{fig:intradayTrend} are computed as moving
averages $m_i$ of the $\hat{p}^T_i$ sequences throughout each day. The
trends for variance were computed as moving averages of
$(\hat{p}^T_i-m_i)^2$ throughout the day.
The shapes suggest that an intraday trend is present in the data and
that a quadratic shape might provide a good fit. The mean and the
variance of $p_i^T$ have similar intraday patterns, but they change
slightly from day to day. This apparent change motivates our online
estimation of the parameters described in Section~\ref{subsec:prediction}.
After detrending the $\hat{p}_i^T$ in step~\ref{step2}, we compute their
autocorrelation function and partial autocorrelation function, which appear
in Figure~\ref{fig:acf} for a typical trading day of JPM. These plots
are typical of the patterns that we see across all of the four stocks
and all days.
\begin{figure}[htb]
\begin{center}
\includegraphics[width = 1.0\textwidth, height = 2.5in]{ACFsingle.pdf}
\caption{Sample ACF and PACF of transformed generalized residuals
after detrending. The lag-zero autocorrelation of 1 is truncated in
the ACF plot.}
\label{fig:acf}
\end{center}
\end{figure}
The patterns in these plots
suggest the presence of long memory. The
negative lag-one autocorrelation suggests that the conditional density
estimation may be overfitting the lag-one dependence.
In step~\ref{step3} of the estimation, we begin with a long-memory
time series model having no exogenous variables.
Appendix~\ref{sec:arfima} provides some detail about both
the time series model that we use (ARFIMA) and how we incorporate
regression into that model. The particular model that we choose is
ARFIMA(0,$d$,1), which has the form
\begin{equation}
\label{eqn:ARFIMA01}
(1 - B)^d P_i = (1 - \theta B)Z_i, \ \ where \ \ Z_i \sim N(0, \sigma^2).
\end{equation}
With our data sets, the differencing parameter $d$ is typically
estimated to be around
0.1 (with standard error around $0.01$), suggesting significant long memory.
We chose this from the family of ARFIMA($p,d,q$) models by
minimizing the BIC score among the potential choices of
$p\in\{0,1,2,3\}$ and $q\in\{0,1,2,3\}$.
\subsection{The Benchmark Models}\label{sec:bench}
In this section, we implement four common members of the ACD family of
models as benchmarks. These include exponential ACD, semiparametric
ACD, exponential FIACD, and semiparametric FIACD.
For each of these models, intraday patterns are estimated directly from
the durations by fitting a cubic spline with knots at each full hour
of clock-time. After fitting the spline, the $i$th duration is divided by
the fitted spline value at $time_{i-1}$. The resulting ratios are used
as the input data for the ACD model and its variants as Jaisk did in
\cite{FIACD}. For
predicting test data, we use the estimated intraday pattern based on the
previous day's (training) data.
All of the benchmark models start with the formula
\[X_i = \Psi_i \epsilon_i,\]
where $X_i$ is a duration (divided by the intraday trend), $\Psi_i$ is
the conditional mean of $X_i$, and $\epsilon_i$ are IID from a long-tailed
distribution with mean 1. The exponential ACD and exponential FIACD,
assume that $\epsilon_i$ has the standard exponential distribution
while the
semiparametric versions allow $\epsilon_i$ to have a general density
$f(\cdot)$ that is fit by kernel density estimation. For the ACD and exponential
ACD models, the conditional mean of $X_i$ evolves as
\[\Psi_i = \omega + \alpha X_{i-1} + \beta \Psi_{i-1}.\]
For the fractionally integrated versions, the conditional mean evolves
as
\[ (1-\beta) \Psi_i = w + [1-\beta - (1-\alpha-\beta)(1-B)^d] X_{i-1}.\]
After fitting these models we compare them all to HSDM
in terms of prediction log-likelihood and various model diagnostics as
described in Section~\ref{subsec:diagnostics}.
\subsection{Model Comparisons} \label{subsec:diagnostics}
In this section, we compare the five models based
on prediction log-likelihood and a number of diagnostic tests using
the validation data from 19 July 2010 through 29 July 2010.
Diebold, Gunther and Tay (1998) \cite{denFORCAST} (henceforth DGT)
proposed a method of evaluating prediction models based on the
probability integral transform. If the predictive
distribution of the $i$th observation $X_i$ given the past has the CDF
$F_i(\cdot)$, then the sequence of values $\{F_i(X_i)\}_{i\geq1}$ forms an
IID sample of uniform random variables on the interval (0,1).
Of course, we don't know $F_i$, but each model provides a fitted
$\hat{F}_i$ for each $i$. We can then see to what extent the sequence
of final generalized residuals,
$\{\hat{F}_i(X_i)\}_{i\geq1}$, looks like a sample of IID uniform
random variables on the interval (0,1).
The empirical distribution of the sequence should look like a
uniform distribution, and the sequence should not exhibit any autocorrelation.
For the HSDM model, the final generalized residuals, $c_i^*$ come from
(\ref{eq:fgr}). Each of the other models has a
corresponding final generalized residual to be tested, i.e. $c_i^* =
\hat F(X_i/\hat{\Psi}_i)$, where $\hat F(\cdot)$ is the estimated CDF of
$\epsilon$ for each model. In the exponential ACD and exponential
FIACD models, $\hat F(\cdot)$
is the CDF of the exponential distribution with mean 1, while in the
semiparametric ACD and semiparametric FIACD, $\hat F(\cdot)$ is an
estimate based on kernel density estimation using the fitted
$\epsilon_i$ values as suggested by \cite{FIACD}. In
Appendix~\ref{app:tail} we give more detail on the kernel density
estimation. In particular, we explain why it makes sense to base the
estimation on the $\log(\epsilon_i)$ values.
Bauwens, Giot, Grammig and Veredas (2000)
\cite{COMPARE} compared some of the most popular conditional duration models,
including ACD, log ACD, threshold ACD, SCD and SVD, by means of the DGT
method. Although they found that these models generally work well on the
price duration process and the volume duration process, none of them work
on the trade duration process.
There is one important distinction between the
diagnostics that we compute and those computed in most
other papers on self-exciting point process models. As in other
papers, we first fit models to training data. The difference is that
we compute the final generalized residuals by using the fitted
models to predict test data. Most papers compute their diagnostics
from the final generalized residuals obtained by predicting the same
training data that were used to fit the
models. There are two main reasons for using test data to perform the
diagnostics rather than using training data. First, it is well-known
that virtually all
statistical models fit better to the data from which they were
estimated than to new data that were not used for their estimation.
It is good statistical practice to evaluate the fit of every model on
test data, if such data are available. Second, we are comparing a
number of models that are semiparametric along with some that do not
form a nested
sequence. Traditional likelihood-ratio tests are useful for comparing
nested parametric models in order to see whether the additional
parameters provide significant improvement or merely overfit. With
semiparametric models, the theory of likelihood-ratio tests is still
being developed. In order to minimize the chance of overfitting with
semiparametric models, it is good practice to evaluate them with test
data that were not used in the fitting. (This procedure is good
practice even with parametric models.) If a model overfits the
training data, it will make noisy predictions with test data. In
comparing two or more models, comparing their predictions based on
test data is the safest way to avoid choosing a model that was overfit.
\subsubsection{Prediction Log-Likelihood}
In this section, we compare the five models based on prediction
log-likelihood for test data. This is essentially a comparison based
on how high is each model's predictive density at the observed test data.
The larger the prediction log-likelihood, the better the prediction
is. Figure~\ref{fig:ACDvsHSDM}
\begin{figure}[htb]
\begin{center}
\includegraphics[width = 1.0\textwidth, height = 3.5in]{ACDvsHSDM.png}
\caption{Comparison between HSDM model and benchmark models in terms
of negative prediction log-likelihood. Smaller is better.}
\label{fig:ACDvsHSDM}
\end{center}
\end{figure}
shows the negative prediction
log-likelihood for JPM on the eight consecutive test days in the
validation data for all five models. The HSDM model (black bars) is
consistently better than all of the benchmark models. And within the
four benchmark models, semi-parametric models are better than
exponential models. ACD models and FIACD models exhibit similar
performance. The other three stocks (IBM, XOM, JCP) have similar patterns.
the HSDM and benchmark model perform on each individual
observation is of great interest.
Figure~\ref{fig:HistIndividualLL}
\begin{figure}[htb]
\begin{center}
\includegraphics[width = 0.6\textwidth, height = 2.8in]{HistIndividualLL.png}
\caption{Histogram of differences between individual log-likelihoods,
HSDM minus the best benchmark model (semiparametric-FIACD) on a
typical day of JPM. The red vertical line is the mean ($\approx
0.122$). The median is 0.124, and the percentage of differences that
are positive is 62.4\%.}
\label{fig:HistIndividualLL}
\end{center}
\end{figure}
compares the performance of HSDM and the best benchmark model,
semiparametric FIACD, on a typical day of
JPM. The plot
displays the histogram of differences of individual prediction
log-likelihood between HSDM and semiparametric FIACD. The positive
part of the histogram
corresponds to those observations on which HSDM has a better
prediction than semiparametric FIACD, while the negative part corresponds to
observations on which HSDM performs worse than semiparametric FIACD. The mean of
the difference is around 0.122 (denoted by the red vertical
line), the median is around 0.124, and the percentage of positive
differences is 62.4\%.
All of the other days and stocks have
similar patterns.
\subsubsection{Uniform Test}\label{sec:utest}
In this section, we compare each set of final generalized residuals to
the uniform distribution on the interval (0,1) by means of the
Kolmogorov-Smirnov (KS) test.
The left subfigure in Figure~\ref{fig:qqplot}
\begin{figure}[hbt]
\begin{center}
\includegraphics[width = 1.0\textwidth, height =
2.8in]{pvalueQQplots.png}
\caption{Q-Q plots of $p$-values from KS test and Ljung-Box test on HSDM and
benchmark models.}\label{fig:qqplot}
\end{center}
\end{figure}
shows the Q-Q plot of 32
$p$-values from KS tests for each of the five models (8 test days for
each of 4 stocks). If the final
generalized residuals were really sampled from
their predictive distributions,
then the $p$-values should be uniformly
distributed on the interval $(0,1)$ and the the Q-Q plot should be
around the 45-degree line. If the final generalized residuals come
from different distributions, the $p$-values should be stochastically
smaller than uniform on $(0,1)$.
Since each model estimates a predicitive distribution from the observed
data (including training data), we cannot expect the $p$-values to be
uniformly distributed. When the estimated distributions come from a
finite-dimensional parametric family, there are modifications
available to the KS test so that the $p$-values have uniform
distribution asymptotically. When the predictive distributions are estimated
nonparametrically or semiparameterically, the appropriate
modifications have not yet been determined. Nevertheless, the KS test
statistics (or equivalently their $p$-values) still give a means for
comparing models based on how close to uniform the final generalized
residuals appear to be. From the comparison in
Figure~\ref{fig:qqplot}, it is clear the benchmark models have
(empirically) stochastically smaller $p$-values than HSDM.
The HSDM $p$-values are still stochastically smaller than the
uniform distribution, but they are much larger than those for the
benchmark models.
The reason that the $p$-values from the exponential models are all so small
is that the data come from a distribution with a much fatter tail than
that of the
exponential distribution. As a matter of fact, no popular parametric model
can perform satisfactorily because the empirical log-duration has a
bimodal shape. The semiparametric models perform relatively better
but still have very small $p$-values.
\subsubsection{Autocorrelation Test}\label{sec:actest}
In this section, we assess the degree of autocorrelation in the final
generalized residuals. We used the Ljung-Box test with lags of 5, 10,
and 15. The test is conducted on each of the 32 pairs of stock/test
day for each model.
The right subfigure in Figure~\ref{fig:qqplot} shows Q-Q plots of 32
$p$-values for the Ljung-Box test with lag 10 for all five models. The results
of lags 5 and 15 are similar. If there were no autocorrelations, the
$p$-vaules should be uniformly distributed on the interval $(0,1)$.
If there are autocorrelations, the $p$-values should be stochastically smaller.
The four benchmark models have $p$-values that are stochastically much
smaller than those of HSDM.
of the Box test. Although the qqplot of HSDM shows that the p values
are not from a standard uniform distribution, it is significantly
better than all of the four benchmark models, which almost have all 0
$p$-values.
The reason that the $p-$values are so low for the benchmark models is
that they don't capture the
information of the most recent duration very well. The HSDM model captures
this information non parametrically, which helps eliminate lag-one
autocorrelation from the final generalized residuals. In summary, the HSDM
model captures the time dependency significantly better than the
benchmark models.
\subsection{Additional Explanatory Variables} \label{subsec:AEV}
As mentioned earlier, the effects of
exogenous variables can be incorporated in step~\ref{step3} of the HSDM
framework by extending the parametric time series model with
regression. As an example, we took into account book pressure
imbalance (BPI) (defined in
Section~\ref{sec:expres}) as additional explanatory variables. Other
variables, such as spread and volume, may be considered
similarly. Figure~\ref{fig:BPIeffect}
\begin{figure}[htb]
\begin{center}
\includegraphics[width = 3.5in, height = 3.0in]{PvsBPI.png}
\caption{Effects of BPI on detrended P. Detrended P vs absolute value of BPI on a typical day of JPM. A significant downward trend is noticeable. }
\label{fig:BPIeffect}
\end{center}
\end{figure}
plots $\hat{p}_i$ (detrended
$\hat{p}^T_i$) against $|{\rm BPI}_i|$ on a typical day of JPM. The red curve
is a fitted smoothing spline, while the green line is the fitted
linear regression.
Apparently, as the absolute value of BPI increases, which means that
there is more imbalance between buy side and sell side, $\hat{p}_i$
becomes smaller, leading to a shorter duration as expected. And the
effect is close to a linear relationship.
We choose how many lags of BPI to include in the
regression by BIC. It turns out that the number of lags varies by
stock. For example, the inclusion of two lags of BPI improves the
prediction on JPM and XOM significantly, while there is
no noticeable improvement with IBM and JCP. There is no reason that
the same exogenous variables should be included in the models for all
stocks, hence each stock will be modeled either with or without
inclusion of BPI as we determine during the model building stage.
Equation (\ref{eqn:arfimaR}) is the ARFIMA with regression model for
those stocks that use two lags of BPI.
\begin{equation}
\label{eqn:arfimaR}
(1-B)^d ( P_i - b_0 - b_1 |BPI_{i-1}| - b_2 |BPI_{i-2}| ) = (1 -
\theta B) Z_i, \ \ Z_i \sim N(0, \sigma^2)
\end{equation}
Figure~\ref{fig:llimprovementBPI}
\begin{figure}[htb]
\begin{center}
\includegraphics[width = 4.5in, height = 2.5in]{llimprovementBPI.png}
\caption{Increase in prediction log-likelihood by incorporation
of $|BPI|$ for JPM as a fraction of the increase of HSDM (without
BPI) over semiparametric-FIACD, i.e., equation (\ref{eq:pllratio}).}
\label{fig:llimprovementBPI}
\end{center}
\end{figure}
shows the increase in prediction
log-likelihood from incorporating BPI as in \ref{eqn:arfimaR} as a
fraction of the amount by which HSDM (without $|BPI|$) improves over
the best benchmark
model (semiparametric-FIACD) for each of 8 days for JPM. That is, the
plot shows
\begin{equation}\label{eq:pllratio}
\frac{ll_{\rm HSDM\ with\ BPI} - ll_{\rm HSDM\ without\ BPI}}{ll_{\rm HSDM\ without\
BPI} - ll_{\rm semiparametric-FIACD}}.
\end{equation}
The consistent large positive ratios across 8
validation days' data indicate that the incorporation of BPI as in
equation \ref{eqn:arfimaR} further improves the model significantly on
JPM.
The model constructed in this section is merely an illustration of how
one might incorporate an exogenous variable into HSDM,
hence, we did not include BPI in the benchmark models for comparison.
\section{Discussion and Conclusion}\label{sec:disc}
We proposed a semiparametric framework for estimating the joint
distribution of a marked point process. In particular, we applied
our framework to trade duration processes. Using validation data that were
not used to fit the models, the DGT evaluation methods (Diebold,
Gunther and Tay, 1998 \cite{denFORCAST}),
show that our model does consistently better than a number of benchmark models
that are variants of the widely-used ACD model. The evaluation methods include
Kolmogorov-Smirnov tests for uniformity of final generalized residuals
and Ljung-Box tests for lack of autocorrelation.
Bauwens, Giot, Grammig and Veredas (2000)
\cite{COMPARE} claimed that the parametric ACD model and all of its
parametric variants
fail to pass both the Kolmogorov-Smirnov test and the Ljung-Box
test. This paper also shows that even semiparametric ACD models and
their variants perform poorly in the sense of DGT evaluation
methods, while HSDM shows a consistent
improvement over the benchmark models. In addition, the HSDM model has a
consistently better performance than the ACD model and its variants in
the sense of
prediction log-likelihood on validation data. Therefore, our framework has
great potential
for modeling the distributions of duration processes, especially
the trade duration process.
The framework has two important features. First, it recognizes that
both the shapes of distributions and the
time dependency must be modeled. Nonparametric conditional density estimation
captures the shapes of distributions along with the most recent time
dependence. Parametric time series models capture the longer-term time
dependency. Nonparametric estimation of the most recent time
dependency gives the model greater flexibility. Second, our
estimation procedure adaptively fits the
intraday trend so as to capture changes that occur from day to day.
The two features described above help to explain why the HSDM
procedure outperforms the existing ACD family on trade duration
processes. Some of these features could be incorporated into ACD
models and their fitting. Such incorporation will be the focus of
future work. However, every model that is based on equation
(\ref{eqn:acd1}) will continue to suffer from some of the limitations
mentioned in Section~\ref{sec:limit}.
|
train/arxiv
|
BkiUduzxaKPQokFdcGJE
| 5 | 1 |
\section{Introduction}
An important problem in analytic number theory is to understand the
distribution of values of $L$-functions on the central line $\Re(s) =
\frac{1}{2}$. Selberg \cite{selberg_zeta} famously proved that as $t$ varies in
large intervals $t \in [T, 2T]$, the real and imaginary parts of the
logarithm of Riemann's zeta function become distributed like independent
Gaussian random variables. Since that work, there have been several
efforts to extend the result to a more general setting. A few years
later, Selberg himself \cite{selberg_dirichlet} proved that for a
fixed value of $t$ the imaginary
part of $\log L(\frac{1}{2} +it; \chi)$ becomes normally distributed
as $\chi$ varies among Dirichlet characters to a large prime modulus
$q$. More recently, Bombieri and Hejhal
\cite{bombieri_hejhal} have shown that Selberg's result for zeta is
true for the values $\{L(\frac{1}{2} + it)\}_{t \in [T, 2T]}$ of a
quite general $L$-function, under mild assumptions about the zeros of
the function, and Wenzhi Luo \cite{luo} has verified this condition for the
$L$-function associated to any fixed modular form for $SL_2(\mathbb{Z})$.
Following the ground-breaking work of Katz and Sarnak \cite{katz_sarnak}, we
now understand the central values $L(\frac{1}{2} + it)$ of an $L$-function as
belonging in a family with a symmetry type governed by one
of the classical compact groups. The cases considered thus far, of a
fixed $L$-function with argument high in the critical strip, and of
central values of Dirichlet $L$-functions with varying character of
fixed conductor, arise as unitary families; on the basis of
calculations from random matrix theory, Keating and Snaith \cite{keating_snaith} have
proposed Selberg-type conjectures for the logarithms of central values of
$L$-functions from families of orthogonal and symplectic symmetry
type, as well. These conjectures appear far from reach,
however, because they involve only the real part of the logarithm of
$L$-functions at the fixed point $s = \frac{1}{2}$; even the best
known analytic methods have thus far only succeeded in proving that a positive
proportion of $L$-functions in a family are non-zero at a single
point, and even in the few special cases where the central value is
known to be positive, the real part of the logarithm is highly
sensitive to the 'low-lying' zeros, near $\frac{1}{2}$, which cannot
presently be controlled. Nonetheless, it is the purpose of this paper
to consider what partial results can be established theoretically in two such cases.
Let $S_k,\; k \equiv 0 \bmod{2}$ be the space of weight $k$
modular cusp forms
for ${\mathrm{SL}}_2(\mathbb{Z}) \backslash \mathbb{H}$ and let $H_k$ be it's basis of
$\sim \frac{k}{12}$ simultaneous eigenvectors of the Hecke
operators, normalized to have first Fourier coefficient equal to 1.
Let $f \in H_k$ have Fourier expansion
\[f(z) = \sum_{n = 1}^\infty n^{\frac{k-1}{2}}\lambda_f(n)e(nz); \]
the $L$-function $L(s,f)$ associated to $f$ is then
\begin{equation}\label{euler_product} L(s,f) = \sum_{n = 1}^\infty
\frac{\lambda_f(n)}{n^s} = \prod_{p}\left(1 -
\frac{\lambda_f(p)}{p^s} + \frac{1}{p^{2s}}\right)^{-1}.\end{equation}
This has completed $L$-function \[\Lambda(s,f) =
(2\pi)^{-s}
\Gamma(s + \frac{k-1}{2}) L(s,f),\] which satisfies the self-dual
functional equation \[\Lambda(s,f) = i^k\Lambda(1-s,f).\] When $k
\equiv 2\bmod 4$ this means that the central value $L(\frac{1}{2}; f)
= 0$, so for $k \equiv 0 \bmod 4$ we consider the family of values
$\{L(\frac{1}{2},f)\}_{f \in H_k},$ which is expected to have
orthogonal symmetry type. These central values have a certain extra
significance because Kohnen and Zagier \cite{kohnen_zagier} proved the striking formula
\[L(\frac{1}{2},f) = \frac{\pi^k}{(k-1)!} \frac{\langle f,
f\rangle}{\langle g, g\rangle} \] relating the central value
$L(\frac{1}{2},f)$ to the ratio of the Petersson norms of $f$ and a
half-integral weight form $g$ that lifts to $f$ under the Shimura
correspondance. A particular consequence is that $L(\frac{1}{2},f) >
0$; this is essentially the only family in which positivity of the
central value is known.
As a second example we let $d > 0$ be a
fundamental discriminant with associated quadratic character
$\chi_{d}(n) = \left(\frac{d}{n}\right)$ of conductor $d$. The
corresponding Dirichlet $L$-function is \[L(s, \chi_{d}) = \sum_{n =
1}^\infty \frac{\chi_{d}(n)}{n^s} = \prod_p \left(1 -
\frac{\chi_{d}(p)}{p^s}\right)^{-1},\qquad \qquad \Re(s) > 1\] with
completed $L$-function \[\Lambda(s, \chi_{d}) =
\left(\frac{8d}{\pi}\right)^{s/2} \Gamma(\frac{s}{2}) L(s,
\chi_{8d}).\] This also satisfies the self-dual functional equation
\[\Lambda(s, \chi_d) = \Lambda(1-s, \chi_{d})\] and conjecturally
$L(\frac{1}{2}, \chi_{8d}) > 0$, but this
is not known. For convenience we consider the family of central values
$\{L(\frac{1}{2}, \chi_{8d})\}_{d \in s(D)}$ where $s(D)$ denotes the
set of squarefree and odd $d$, $\frac{D}{2} < d \leq D$; this is
expected to be a family exhibiting symplectic symmetry.
We have two primary results. The first result proves,
unconditionally, 'one-half' of the Keating-Snaith conjectures.
\begin{cor}\label{upper_bound}
Let $k \equiv 0 \bmod{4}$. As $k \to \infty$ we have
\[{\mathbb{P}}\left[f \in H_k: \frac{1}{\sqrt{\log \log k}}\left(\log
L(\frac{1}{2},f)+\frac{1}{2}\log \log k\right) > A\right] \leq
\frac{1}{\sqrt{2\pi}}\int_A^\infty e^{-\frac{x^2}{2}} dx + o_A(1).\]
In particular, for any fixed $\epsilon >0$, $L(\frac{1}{2}, f) < (\log
k)^{-1/2 + \epsilon}$ with probability $1- o_\epsilon(1)$.
Also, as $D \to \infty$,
\[{\mathbb{P}}\left[d \in s(D): \frac{1}{\sqrt{\log \log D}}\left(\log
|L(\frac{1}{2},\chi_{8d})|-\frac{1}{2}\log \log D\right) > A\right] \leq
\frac{1}{\sqrt{2\pi}}\int_A^\infty e^{-\frac{x^2}{2}} dx + o_A(1).\]
\end{cor}
\noindent In \cite{sound_moments}, Soundararajan made the basic
observation that, on the Riemann Hypothesis, while zeros near $\frac{1}{2}+it$
can greatly alter the
value of $\log |\zeta(\frac{1}{2}+it)|$, they always decrease its
value as compared with that of $\log |\zeta(\frac{1}{2} + \sigma +
it)|$ at points off the critical line. Our proof of Corollary
\ref{upper_bound} is based upon an unconditional version of this fact,
together with the following slightly technical result.
\begin{theorem}\label{right_of_half}
Let $\sigma = \sigma(k)$ be a function of $k$, tending to 0 as $k \to
\infty$ in such a way that $\sigma \log k \to \infty$ but $\frac{\sigma \log
k}{\sqrt{\log \log k}}\to 0$. Also, for $f \in H_k$ put \[A(f) =
\frac{1}{\sqrt{\log \log k}}\left(\log |L(\frac{1}{2} + \sigma, f)| +
\frac{1}{2} \log \log k\right).\] Then
\[\frac{1}{|H_k|} \sum_{f \in H_k} \delta_{A(f)} \to N(0, 1), \qquad k
\to \infty.\] Here
$\delta_x$ is the point mass at $x$,
$N(0,1)$ is the standard normal distribution, and the convergence is
in the sense of distributions.
Similarly, let $\sigma = \sigma(D)$ be a function of $D$, tending to 0
as $D \to \infty$ in such a way that $\sigma \log D \to \infty$ but
$\frac{\sigma \log D}{\sqrt{\log \log D}} \to 0$. For $d \in s(D)$,
put \[A(d) = \frac{1}{\sqrt{\log \log D}} \left(\log |L(\frac{1}{2} +
\sigma, \chi_{8d})| - \frac{1}{2}\log \log D\right).\] Then
\[\frac{1}{|s(D)|} \sum_{d \in s(D)} \delta_{A(d)} \to N(0,1), \qquad
D \to \infty.\]
\end{theorem}
\noindent This Theorem is proven using Selberg's method in
\cite{selberg_dirichlet}; in particular it makes use of
'zero-density' estimates putting almost all of the low-lying zeros of
the corresponding $L$-functions very near the half-line. In the case
of $L(s, \chi_{8d})$, such a result is essentially available from the
work of Conrey and Soundararajan in \cite{conrey_sound}. For the case
of $L(s,f)$, this is a concurrent result of the author in
\cite{hough_zero_density}.
For our second main result we assume some weak conjectural information
about the low-lying zeros in the families $\{L(s,f)\}_{f \in H_k}$,
and $\{L(s, \chi_{8d})\}_{d \in s(D)}$ in order to deduce the full
Keating-Snaith conjectures for these families. Given $f \in H_k$ and
$s$ near $\frac{1}{2}$, $L(s,f)$ has conductor $\asymp k^2$, and
therefore for $1 \ll T = k^{o(1)}$ the number of zeros of $L(s,f)$ up
to height $T$ grows as $\frac{T}{\pi}\log k$. Thus, based upon purely
density considerations, we might expect that for most $f \in H_k$,
$\gamma_{min}(f) \gg \frac{1}{\log k}$, where \[\gamma_{min}(f) =
\min_{\rho =\frac{1}{2} + \beta + i\gamma} |\gamma|\] is the height of the
lowest
non-trivial zero of $L(s,f)$. Similarly, for $d \in s(D)$ and $s$
near $\frac{1}{2}$, $L(s,
\chi_{8d})$ has conductor $\asymp D$, and therefore we might typically
expect that $\gamma_{min}(d) \gg \frac{1}{\log D}$. We formalize this
heuristic in the following hypothesis.
\begin{hypothesis}[Low-lying Zero Hypothesis]\label{low_zero_hypothesis}
Assume $y = y(k) \to \infty$ with $k$. Then
\[{\mathbb{P}}\left[ f \in H_k: \gamma_{min}(f) < \frac{\pi}{y \log k}\right]=o(1),
\qquad k \to \infty.\] Similarly, if $y = y(D) \to \infty$ with $D$
then
\[{\mathbb{P}}\left[ d \in s(D): \gamma_{min}(d) < \frac{2\pi}{y \log
D}\right] = o(1), \qquad D \to \infty.\]
\end{hypothesis}
\noindent In fact, stronger and more detailed statements about the low-lying
zeros in these two families are expected to be true. Specifically,
Iwaniec, Luo and Sarnak \cite{iwaniec_luo_sarnak} have conjectured
that in essentially any natural family of $L$-functions of conductor $C$, the
one-level density of
zeros at a scale of $\frac{2\pi}{\log C}$ depends asymptotically only
on the symmetry type of the family. For our two families of
$L$-functions, their 'Zero Density Conjecture' takes the following shape.
\begin{conjecture}[Zero Density Conjecture]
Let $\phi(x)$ be a Schwarz class function on $\mathbb{R}$ with Fourier transform
having compact support. Define the densities \[W(SO_{\text{even}})(x)dx =
\left(1 +
\frac{\sin 2\pi x}{2\pi x}\right)dx, \qquad W(Sp)(x)dx = \left(1 - \frac{\sin
2\pi x}{2\pi x}\right) dx\] and write the non-trivial zeros of
$L(s)$ as $\rho = \frac{1}{2} + i\gamma$, with $\gamma$ possibly complex if
the Riemann Hypothesis for $L(s)$ is false. Then
\[\lim_{\substack{k \to \infty \\ k\equiv 0 \bmod 4}} \frac{1}{|H_k|}{\sum_{f
\in H_k}}
\sum_{\substack{\Lambda(\rho,f)=0\\ \rho = \frac{1}{2} + i\gamma}}
\phi\left(\frac{\gamma \log k}{\pi}\right) = \int_{-\infty}^\infty \phi(x)
W(SO_{\text{even}})(x)dx\]
and
\[\lim_{D \to \infty} \frac{1}{|s(D)|} \sum_{d \in s(D)}
\sum_{\substack{\Lambda(\rho, \chi_{8d}) = 0\\ \rho = \frac{1}{2} +
i\gamma}}\phi\left(\frac{\gamma \log D}{2\pi}\right) =
\int_{-\infty}^\infty \phi(x)W(Sp)(x)dx.\]
\end{conjecture}
\noindent It is a straightforward exercise to prove that our Low-lying
Zero Hypothesis is implied by the Zero Density Conjecture together
with the Riemann Hypothesis for the corresponding family of
$L$-functions.
We now state our second main result.
\begin{theorem}\label{conditional}
Suppose the Low-lying Zero Hypothesis holds in the family
$\{L(s,f)\}_{f \in H_k}$. For $f \in H_k$ put \[B(f) =
\frac{1}{\sqrt{\log \log k}}\left(\log L(\frac{1}{2},f) +
\frac{1}{2}\log \log k\right).\] Then, as distributions
\[ \frac{1}{|H_k|} \sum_{f \in H_k} \delta_{B(f)} \to N(0,1), \qquad k
\to \infty.\]
Similarly, assume the Low-lying Zero Hypothesis in the family $\{L(s,
\chi_{8d}\}_{d \in s(D)}$ and for $d \in s(D)$ put \[B(d) =
\frac{1}{\sqrt{\log \log D}}\left(\log |L(\frac{1}{2}, \chi_{8d})| -
\frac{1}{2}\log \log D\right).\] Then, in the sense of distributions,
\[ \frac{1}{|s(D)|} \sum_{d \in s(D)} \delta_{B(d)} \to N(0,1), \qquad
D \to \infty.\]
In particular, either of these results is true if both the Riemann
Hypothesis and the Zero Density Hypothesis is true for the
corresponding family of $L$-functions.
\end{theorem}
\section{Background}\label{lemma_section}
In this section we collect together standard facts regarding our two families
of $L$-functions, as well as the part of Selberg's work that we need for our
arguments.
\subsection{$L$-function coefficients, and orthogonality}
For $f\in H_k$, the Fourier coefficients of $f$ satisfy the
Hecke relations
\begin{equation}\label{hecke_relations} \lambda_f(m)\lambda_f(n) =
\sum_{d|(m,n)} \lambda_f(\frac{mn}{d^2}).\end{equation}
A specific consequence of this fact is that for distinct primes $p_1,
..., p_r$ we have
\begin{equation}\label{prime_product}\lambda(p_1)^{e_1}\lambda(p_2)^{e_2}\cdots \lambda(p_r)^{e_r} =
\sum_{0 \leq j_1 \leq \lfloor\frac{e_1}{2}\rfloor} \cdots \sum_{0 \leq
j_r \leq \lfloor\frac{e_r}{2}\rfloor}c(\mathbf{e},
\mathbf{j})\lambda_f(p_1^{e_1 - 2j_1}
\cdots p_r^{e_r - 2 j_r})\end{equation} for some positive coefficients
$c(\mathbf{e}, \mathbf{j})$.
\begin{lemma}
We have $c(\mathbf{2}, \mathbf{*}) = 1$ where $\mathbf{2}$ is the string
consisting entirely of 2's and $\mathbf{*}$ is any string containing 0's and
1's. Also, for general $\mathbf{e}, \mathbf{j}$, $c(\mathbf{e}, \mathbf{j})
\leq 2^{e_1 + ... + e_r}.$
\end{lemma}
\noindent Recall, also, Deligne's
bound $|\lambda_f(n)| \leq d(n)$.
We use the following basic orthogonality relation on $H_k$.
\begin{lemma}\label{H_k_orthogonality}
Let $0 < \delta < 2$. There exists $\gamma = \gamma(\delta)>0$ such
that if $m < k^{2-\delta}$ then
\[\frac{1}{|H_k|}\sum_{f\in H_k} \lambda_f(m) = \frac{\delta_{m =
\square}}{\sqrt{m}} + O(k^{-\gamma}).\]
\end{lemma}
\begin{proof}
Actually, this is a combination of two different estimates. Using the
Petersson Trace Formula, Rudnick and Soundararajan
(\cite{rudnick_sound_examples}, Lemma 2.1) prove that for $mn < \frac{k^2}{10000}$,
\[\sum_{f \in H_k} \frac{2\pi^2}{k-1}L(1, {\mathrm{sym}}^2 f)^{-1} \lambda_f(m)\lambda_f(n) =
\delta_{m=n} + O(e^{-k}).\] Here $w_f = \frac{2\pi^2}{k-1}L(1,{\mathrm{sym}}^2 f)^{-1}$ is
the so-called 'harmonic weight' of $f$, and $L(s, {\mathrm{sym}}^2 f)$ is the symmetric
square $L$-function attached to $L(s,f)$ with coefficients given by
\[L(s,{\mathrm{sym}}^2 f) = \sum_{n = 1}^\infty \frac{\rho_f(n)}{n^s} =
\zeta(2s)\sum_{n =1}^\infty \frac{\lambda_f(n^2)}{n^s}, \qquad \Re(s)
> 1.\]
A now-standard method of Kowalski-Michel (\cite{kowalski_michel},
Proposition 2) allows the removal of the harmonic weight by truncating
the Dirichlet series for $L(1, {\mathrm{sym}}^2 f)$; with $x = k^{1 - \delta/2}$
and recalling $|H_k| \sim \frac{k-1}{12}$, their method gives
\begin{align*}\frac{1}{|H_k|} \sum_{f \in H_k} \lambda_f(m) &= \frac{1}{|H_k|}
\sum_{f \in H_k} w_f \frac{k-1}{2\pi^2} L(1, {\mathrm{sym}}^2 f)^{-1}
\lambda_f(m) \\&= \frac{1}{\zeta(2)} \sum_{f \in H_k} w_f \lambda_f(m)
\sum_{\ell^2 d < x} \frac{\lambda_f(d^2)}{\ell^2 d} + O(k^{-\gamma}).\end{align*}
Substituting the bound of Rudnick and Soundararajan, one deduces the lemma.
\end{proof}
For the real characters $\chi_{8d}$, our basic orthogonality relation is the
following.
\begin{lemma}
Let $n < D^{2-\delta}$. Then there is $\gamma = \gamma(\delta)>0$ such that
\[ \sum_{d \in s(D)} \left(\frac{8d}{n}\right) = \delta_{n = \square}
\prod_{\substack{p|n \\ \text{odd}}} \left(\frac{p}{p+1}\right) +
O(D^{-\gamma}).\]
\end{lemma}
\begin{proof}
Note that $\mu(2d)^2$ is exactly the indicator function for odd,
squarefree $d$. Rudnick and Soundararajan (\cite{rudnick_sound_examples} Lemma
3.1) prove, for any $z > 3$, that if $n$ is a perfect square then
\[\sum_{d \leq z} \mu(2d)^2 \left(\frac{8d}{n}\right) = \frac{z}{\zeta(2)}
\prod_{p|2n}\left(\frac{p}{p+1}\right) + O(z^{\frac{1}{2} + \epsilon}
n^\epsilon)\]
and if $n$ is not a square then
\[\sum_{d \leq z} \mu(2d)^2 \left(\frac{8d}{n}\right) = \frac{z}{\zeta(2)}
\prod_{p|2n}\left(\frac{p}{p+1}\right) + O(z^{\frac{1}{2} }n^{\frac{1}{4}}
\log(2n).\]
The result follows on taking successively $z = D/2, D$.
\end{proof}
\subsection{Selberg's work: two expressions for the logarithm}
Writing the Euler product of $L(s,f)$ as
\[
L(s,f) = \prod_p \left(1 - \frac{\lambda_f(p)}{p^s} +
\frac{1}{p^{2s}}\right)^{-1} = \prod_{p} \left(1 -
\frac{\alpha_p}{p^s}\right)^{-1}\left(1-
\frac{\overline{\alpha_p}}{p^s}\right)^{-1}, \qquad \qquad \Re(s) > 1
\]
we have that for $m = 1, 2, ...$
\[ \lambda_f(p^m) = \alpha_p^m + \alpha_p^{m-2} + ... + \alpha_p^{-m+2} +
\alpha_p^{-m} \] where for each $p$, $\alpha_p$
is a complex number of modulus 1 solving
$\alpha_p + \overline{\alpha_p} = \lambda_f(p)$. Logarithmically differentiating
$L(s,f)$ term-by-term we obtain
\begin{equation}\label{def_Lambda_f} -\frac{L'}{L}(s,f) = \sum_{n = 1}^\infty
\frac{\Lambda_f(n)}{n^s} = \sum_{m = 1}^\infty \sum_p \frac{(\alpha_p^m +
\overline{\alpha}_p^m)\log p}{p^{ms}}, \qquad \qquad \Re(s)
> 1.\end{equation} In particular, $\Lambda_f(n)$ is supported on prime powers,
and is given explicitly by
\begin{equation}\label{Lambda_f_value} \Lambda_f(p^m) = (\lambda_f(p^m) -
\lambda_f(p^{m-2})) \log p, \qquad \qquad m \geq 1,\end{equation} with the
convention that $\lambda_f(p^{-1}) = 0$.
Similarly we have
\[L(s, \chi_{8d}) = \prod_p \left(1 - \frac{\chi_{8d}(p)}{p^s}\right)^{-1},
\qquad \Re(s) > 1\] and logarithmically differentiating this leads to
\begin{equation}\label{def_Lambda_8d} -\frac{L'}{L}(s, \chi_{8d}) = \sum_n
\frac{\Lambda_{8d}(n)}{n^s}, \qquad \Re(s) > 1 \end{equation} with
$\Lambda_{8d}$ supported on primes powers
and \begin{equation}\label{Lambda_8d_value}\Lambda_{8d}(p^n) =
\left(\frac{8d}{p^n}\right) \log p. \end{equation}
In a standard way one may write down an expression for $-\frac{L'}{L}(s)$
similar to (\ref{def_Lambda_f}) and (\ref{def_Lambda_8d}) when $\frac{1}{2} <
\Re(s) \leq 1$, although in this case the zeros of $L(s)$ enter into
the formula. The following lemma is the analog of \cite{selberg_zeta}, Lemma
10 with $L(s)$ replacing the Riemann zeta function.
\begin{lemma} Let $*$ stand in for either $8d$ or $f$, so that $L(s,*)$ is
either $L(s, \chi_{8d})$ for some $d \in s(D)$ or $L(s, f)$ for some $f \in
H_k$.
Let $x > 1$ be a parameter and define
\[ \Lambda_{x,*}(n) = \Lambda_*(n) a_x(n); \qquad \qquad a_x(n) =
\left\{\begin{array}{lll} 1,&& 1 \leq n
\leq x\\ \frac{\log^2\frac{x^3}{n} - 2 \log^2 \frac{x^2}{n}}{2
\log^2 x}, && x \leq n \leq x^2\\\frac{\log^2
\frac{x^3}{n}}{2 \log^2 x}, && x^2 \leq n \leq x^3 \end{array}\right..
\]
For $s$ not coinciding with a trivial or non-trivial zero of $L(s,*)$ we have
\begin{align}\label{euler_derivative}-\frac{L'}{L}(\frac{1}{2} + s,*) = \sum_{n
\leq x^3}
\frac{\Lambda_{x,*}(n)}{n^{\frac{1}{2} + s}} -& \frac{1}{\log^2
x}
\sum_{\substack{\rho:\; \Lambda(\rho,*) = 0\\ \text{non-trivial}}}\frac{x^{\rho
- \frac{1}{2} - s}(1 - x^{\rho - \frac{1}{2} -
s})^2} {(\frac{1}{2} +s - \rho)^3} \\ \notag &- \frac{1}{\log^2 x}
\sum_{\substack{q:\; L(-q,*) = 0,\; \Lambda(-q, *) \neq 0\\ \text{trivial}}}
\frac{x^{-q-\frac{1}{2}-s}(1 - x^{ - q-\frac{1}{2} -
s})^2}{(\frac{1}{2}+ q + s)^3}.\end{align}
\end{lemma}
\begin{proof}
The sum $\sum_{n \leq x^3} \frac{\Lambda_{x,*}(n) }{n^{\frac{1}{2}
+
s}}$ is the result of expanding $\frac{L'}{L}(z,*)$ in it's Dirichlet series
in
\[\frac{1}{2\pi i \log^2 x}\int_{(3)} \frac{x^{z - \frac{1}{2} -s}(1 - x^{z
- \frac{1}{2} - s})^2}{(z- \frac{1}{2} - s)^3}\frac{L'}{L}(z,*)dz\]
and integrating term-by-term.
The remainder of the expression is obtained by shifting the $z$-contour
leftward and evaluating residues.
\end{proof}
Put \begin{equation}\label{def_archimedian_factor} \gamma_f(s) =
(2\pi)^{-s}\Gamma(s + \frac{k-1}{2}), \qquad \qquad \gamma_{8d}(s) =
\left(\frac{8d}{\pi}\right)^{-s/2}\Gamma(\frac{s}{2})\end{equation}
so that we may write in a unified way
\[\Lambda(s, *) = \gamma_*(s) L(s, *)\] for the completed $L$-function
corresponding to either $L(s,f)$ or $L(s, \chi_{8d})$. The completed
$L$-function is entire of order 1 and hence has a Hadamard product running
over it's zeros,
\begin{equation}\label{hadamard_product}\Lambda(s,*) = e^{A +
Bs}\prod_{\rho: \; \Lambda(\rho, *) = 0}\left(1 -
\frac{s}{\rho}\right)e^{\frac{s}{\rho}}.\end{equation}
Logarithmically differentiating $\Lambda(s,*)$ one proves the following
lemma.
\begin{lemma}\label{hadamard}
For real $\sigma > 0$ we have
\begin{equation}\label{hadamard_derivative}
-\frac{L'}{L}(\frac{1}{2} + \sigma,*) =
\frac{\gamma_*'}{\gamma_*}(\frac{1}{2} + \sigma) -
{\sum_{\rho}}'\frac{1}{\frac{1}{2} + \sigma - \rho}.
\end{equation}
Furthermore, the sum over zeros is given by
\begin{equation}\label{zero_sum}
\frac{1}{2}\sum_{\substack{\rho= \frac{1}{2} + \beta + i \gamma}}
\left(\frac{1}{ \sigma - \beta + i\gamma} + \frac{1}{
\sigma - \beta - i\gamma}\right) = \sum_{\substack{\rho=
\frac{1}{2} + \beta + i \gamma}} \frac{ \sigma - \beta}{(
\sigma - \beta)^2 + \gamma^2}.
\end{equation}
\end{lemma}
One of Selberg's major achievements in \cite{selberg_zeta} was that he gave an
efficient way to compute
$\log\zeta(\frac{1}{2} + it)$ as a short sum over primes; by balancing the
expression for $-\frac{\zeta'}{\zeta}(s)$ coming from
the Hadamard product as in (\ref{hadamard_derivative}) against the expression
from the Euler product (\ref{euler_derivative}), he was able to bound the
contribution of the zeros in (\ref{euler_derivative}). To do so, Selberg
introduced a perturbation $\sigma_{x,t}$ depending on the
location of the zeros of $\zeta$ near height $t$, and evaluated $\log
\zeta(\frac{1}{2} + \sigma_{x,t} + it)$ in place of $\log \zeta(\frac{1}{2} +
it)$.
For $\log |L(\frac{1}{2},*)|$ the analog of
$\sigma_{x,t}$
is
\begin{equation}\label{def_sigma}\sigma_{x,*} = 2\max_{\rho\in
{\mathcal{G}}_{x,*}}\left(\beta, \frac{2}{\log x}\right); \qquad {\mathcal{G}}_* =
\left\{\rho = \frac{1}{2} + \beta + i\gamma : |\gamma|
\leq \frac{x^{3|\beta|}}{\log x},\; |\beta|
\geq \frac{2}{\log x}\right\}.\end{equation}
Selberg's argument for $\log \zeta(\frac{1}{2} + \sigma_{x,t} + it)$ carries
over with trivial modifications to bound the zero sum of $L(s,*)$ at
$s = \frac{1}{2} + \sigma_{x,*}$ and thus to the evaluation of
$\log L(\frac{1}{2} +
\sigma_{x,*},*);$ the result is the
following lemma.
\begin{lemma}\label{log_L_off}
Let $C = k^2$ for $L(s,f)$ or $C = 8d$ for $L(s, \chi_{8d})$ be the conductor
of the $L$-function near $s = \frac{1}{2}$. We have
\begin{equation}\label{zero_sum_bound}
\sum_{\substack{\rho = \frac{1}{2} + \beta + i\gamma\\ \Lambda(\rho,*) = 0}}
\frac{\sigma_{x,*}}{(\sigma_{x,*} - \beta)^2 + \gamma^2} = O\left(\left|\sum_{n
< x^3} \frac{\Lambda_{x,*}(n)}{n^{\frac{1}{2} + \sigma_{x,*}}} \right|\right) +
O(\log C)
\end{equation}
and
\begin{equation}\label{log_prime_sum}\log L(\frac{1}{2} + \sigma_{x,*},*) =
\sum_{n \leq x^3} \frac{\Lambda_{x,*}(n)}{n^{\frac{1}{2} +
\sigma_{x,*}} \log n} + O\left(\frac{1}{\log x} \left|\sum_{n \leq x^3}
\frac{\Lambda_{x,*}(n) }{n^{\frac{1}{2} +
\sigma_{x,*}}}\right|\right)
+ O\left(\frac{\log C}{\log x}\right).\end{equation}
\end{lemma}
\begin{proof}
See \cite{selberg_zeta} pp 22-26.
\end{proof}
In order to proceed further with Selberg's approach we need an understanding
of the perturbation $\sigma_{x,*}$, that is, we need input regarding the
distribution of zeros of $L(s,*)$ near the central point $s =
\frac{1}{2}$ as either $f$ varies in $H_k$ or $d$ varies in $s(D)$. Our basic
analytic ingredient is
the following.
\begin{theorem}\label{zero_density}
For a sufficiently small $\delta > 0$ there exists $\theta = \theta(\delta)$
such that, uniformly in $ \frac{2}{\log k}< \sigma < \frac{1}{2}$ and
$\frac{10}{\log k}< T <
k^{2\delta},$
\[ N(\sigma, T, k) \stackrel{\text{def}}{=} \frac{1}{|H_k|}{\sum_{f \in H_k}}
\#\left\{L(\frac{1}{2} + \beta + i \gamma, f) = 0: \sigma < \beta, \; |\gamma|<
T\right\} = O(T k^{ -2 \theta \sigma } \log k),\] and also, uniformly in
$\frac{4}{\log D} < \sigma < \frac{1}{2}$ and $\frac{10}{\log D} < T <
D^\delta$,
\[ N(\sigma, T, D) \stackrel{\text{def}}{=} \frac{1}{|s(D)|}{\sum_{d \in
s(D)}}
\#\left\{L(\frac{1}{2} + \beta + i \gamma, \chi_{8d}) = 0: \sigma < \beta, \;
|\gamma|<
T\right\} = O(T D^{ - \theta \sigma } \log D).\]
\end{theorem}
\begin{proof}
The part of the theorem regarding $f \in H_k$ is Theorem 1.1 in
\cite{hough_zero_density}. For $d \in s(D)$, this result is proved in
\cite{conrey_sound} for the family $\{L(s, \chi_{-8d})\}_{d \in s(D)}$, in the
most difficult range $|T| \ll \frac{1}{\log D}$. The changes needed to adapt
this to positive fundamental discriminants are trivial, and it is
straightforward to extend their result to $|T| \gg D^{\delta}$, for instance,
along the same lines as in \cite{hough_zero_density}. Alternatively, the
reader may take the second statement as a black box.
\end{proof}
As a consequence we derive the following essential lemma.
\begin{lemma}\label{zero_prob}
Let $\mathcal{F}$ be one of the two families of $L$-functions, either
$\mathcal{F} = \{L(s,f)\}_{f \in H_k}$ or $\mathcal{F} = \{L(s, \chi_{8d})\}_{d
\in s(D)}$. Denote ${\mathbb{P}}$ the uniform probability on $\mathcal{F}$. Let $C =
k^2$ or $C = D$ be the respective conductor of the family.
For $x = x(C)$ growing with $C$ in such a way that $ \frac{\log x}{\log C}
\to 0$ as $C \to \infty$ we have
\[ {\mathbb{P}}\left[ \exists \; \rho = \frac{1}{2} + \beta + i\gamma:
\Lambda(\rho,*) = 0,\; \beta > \frac{4}{\log x}, \; |\gamma| \leq
\frac{x^{3\beta}}{\log x}\right] = o(1)\] as $C \to \infty$.
\end{lemma}
\begin{proof}
Assume that $C$ is sufficiently large so that $\log x \leq \frac{1}{25} \log
C$. We have
\begin{align*}
{\mathbb{P}}\left[ \exists \; \rho :
\; \beta > \frac{4}{\log x}, \; |\gamma| \leq
\frac{x^{3\beta}}{\log x}\right] & \leq {\mathbb{P}}\left[\bigcup_{j =
4}^{\lceil \frac{\log x}{2}\rceil} \left\{ \exists \; \rho :
\; \beta > \frac{j}{\log x}, \; |\gamma| \leq
\frac{e^{3(j+1)}}{\log x}\right\}\right]
\\ & \leq \sum_{j = 4}^{\lceil \frac{\log x}{2}\rceil}
{\mathbb{P}}\left[\exists \; \rho :
\; \beta > \frac{j}{\log x}, \; |\gamma| \leq
\frac{e^{3(j+1)}}{\log x}\right].
\end{align*}
By applying Theorem \ref{zero_density}, the last sum is bounded by
\begin{align*} \ll \sum_{j = 4}^{\lceil \frac{\log x}{2}\rceil}
\frac{e^{3(j+1)}}{\log x} C^{-\frac{\theta j}{\log x}} \log C& \leq \frac{\log
C}{\log x} \sum_{j = 4}^{\lceil \frac{\log x}{2}\rceil}
e^{\left(-\theta \frac{\log
C}{\log x} + 3\right) j} \ll \frac{\log C}{\log x} e^{\frac{-\theta \log C}{
\log x}}
\end{align*}
and this tends to zero as $C \to \infty$.
\end{proof}
\subsection{Convergence in the sense of distributions}
Before turning to the main argument, we record, for repeated later use, the
following simple fact concerning convergence in the sense of distributions.
Suppose we have a sequence
of finite sets $\{R_n\}$; for each $n$ let there be two functions $f,
\tilde{f}:R_n \to \mathbb{R}$, so that we obtain two
sequences of probability measures $\{\mu_n\}$, $\{\tilde{\mu}_n\}$ on $\mathbb{R}$,
\[\mu_n = \frac{1}{|R_n|}\sum_{s \in R_n} \delta_{f(s)}, \qquad \qquad
\tilde{\mu}_n =
\frac{1}{|R_n|}\sum_{s \in R_n} \delta_{\tilde{f}(s)}.\]
\begin{lemma}\label{conv_dist_lemma}
Let $\mu$ be a finite (Borel)
measure on $\mathbb{R}$. Each of the following three conditions is sufficient to
guarantee the simultaneous convergence in distribution
\[ \mu_n {\; \stackrel{d}{\longrightarrow} \;} \mu \qquad \Leftrightarrow \qquad \tilde{\mu}_n
{\; \stackrel{d}{\longrightarrow} \;} \mu\]
of $\mu_n$ and $\tilde{\mu}_n$ to $\mu$.
\begin{align*}
\tag{i} \frac{1}{|R_n|}\sum_{\substack{s \in R_n \\ f(s) \neq \tilde{f}(s)}} 1
=
o(1), \qquad\qquad &n \to \infty \\
\tag{ii} \sup_{s \in R_n} |f(s) - \tilde{f}(s)| = o(1), \qquad\qquad &n
\to \infty\\
\tag{iii} \frac{1}{|R_n|}\sum_{s \in R_n} |f(s) - \tilde{f}(s)|^2 =
o(1),\qquad\qquad & n
\to \infty.
\end{align*}
\end{lemma}
\section{The distribution of the prime sum}
We first show that short prime sums $(x = C^{o(1)})$
\[\{\sum_{n \leq x}
\frac{\Lambda_f(n)}{\sqrt{n}}\}_{f \in H_k}, \qquad \{\sum_{n \leq x}
\frac{\Lambda_{8d}(n)}{\sqrt{n}}\}_{d \in S(d)}\] converge to the appropriate
Gaussian distributions as the the conductor $C \to \infty$. The main work will
then be in comparing $\log |L(\frac{1}{2}, *)|$ to these sums.
\begin{prp}\label{prime_sum}
Let $C = k^2$ for the family $\mathcal{F} = \{L(s,f)\}_{f \in H_k}$ and $C
= D$ for the family $\{L(s, \chi_{8d})\}_{d \in s(D)}$. Assume $x = x(C)$ grows
with $C$ in such a way that $\frac{\log x}{\log C}
\to 0$ as $C \to \infty$, but $\log \log x = \log \log C + o(\sqrt{\log
\log C})$. Define, for $f \in H_k$, \[P(f) = \frac{1}{\sqrt{\log \log k}}
\left( \sum_{n \leq x}
\frac{\Lambda_f(n)}{n^{\frac{1}{2}} \log n} + \frac{1}{2} \log \log
k\right),\] and for $d \in s(D)$,
\[P(d) = \frac{1}{\sqrt{\log \log D}}\left(\sum_{n \leq x}
\frac{\Lambda_{8d}(n)}{n^{\frac{1}{2}}} - \frac{1}{2} \log \log D\right).\] We
have
\begin{equation}\label{conv_of_prime_sum}\frac{1}{|\mathcal{F}|}\sum_{* \in
\mathcal{F}} \delta_{P(*)} {\; \stackrel{d}{\longrightarrow} \;}
N(0,1), \qquad C \to \infty.\end{equation}
Also, for each $C$ let $\{b_n(C)\}_{n = 1}^\infty$ be a sequence of real
numbers, bounded independently of $C$. Then
\begin{equation}\label{mean_error} \frac{1}{|\mathcal{F}|}\sum_{* \in
\mathcal{F}}\left| \sum_{n <
x^3} \frac{\Lambda_*(n) b_n}{n^{\frac{1}{2}}}\right|^{2} = O(\log^2 x), \qquad
\qquad C \to \infty. \end{equation}
\end{prp}
\begin{proof}
We show the proof for the family $\mathcal{F} = \{L(s,f)\}_{f \in H_k}$; the
argument for real characters is essentially the same, with the caveat that the
positive mean of $L(\frac{1}{2}, \chi_{8d})$ comes from the fact that
$\left(\frac{8d}{p^2}\right) = 1$ if $p \nmid 8d$.
For (\ref{conv_of_prime_sum}), let $f \in H_k$ and write
\[P(f) = \frac{1}{\sqrt{\log \log k}}\left[ \sum_{m = 1}^\infty \frac{1}{m}
\sum_{p < x^{\frac{1}{m}}} \frac{\Lambda_f(p^m)}{p^{\frac{m}{2}} \log p} -
\frac{1}{2} \log \log k \right].\] In view of the evaluation of the
coefficients $\Lambda_f(n)$ given in (\ref{Lambda_f_value}) we have
\begin{align*} P(f) &= \frac{1}{\sqrt{\log \log k}}\sum_{p <
x}\frac{\lambda_f(p)}{p^{ \frac{1}{2}}} + \frac{1}{2\sqrt{\log \log k}} \sum_{p
< \sqrt{x}} \frac{\lambda_f(p^2)-1}{p} + \frac{1}{2} \sqrt{\log \log k} +
o(1)\\ & = \frac{1}{\sqrt{\log \log k}}\sum_{p <
x}\frac{\lambda_f(p)}{p^{ \frac{1}{2}}} + \frac{1}{2\sqrt{\log \log k}} \sum_{p
< \sqrt{x}} \frac{\lambda_f(p^2)}{p} + o(1), \end{align*} by Merten's
theorem for $\sum \frac{1}{p}$. Now we may assume that $k$ is sufficiently
large so that $x^2 < k^{2-\delta}$. Then
\begin{align*} \frac{1}{|H_k|}\sum_{f \in H_k} \left[\sum_{p <
\sqrt{x}}\frac{\lambda_f(p^2)}{p}\right]^2 &= \sum_{p_1, p_2 < \sqrt{x}}
\frac{1}{p_1p_2} \frac{1}{|H_k|}\sum_{f \in H_k} \lambda_f(p_1)
\lambda_f(p_2) \\ &=
\sum_{p < \sqrt{x}} \frac{1}{p^2} + O(k^{-\gamma}\log^2 x) = O(1)
\end{align*}
by applying Lemma \ref{H_k_orthogonality}. Thus by Lemma
\ref{conv_dist_lemma} it suffices to prove
\begin{equation}\label{perturbed_convergence} \frac{1}{|H_k|}\sum_{f \in H_k}
\delta_{\tilde{P}(f)} {\; \stackrel{d}{\longrightarrow} \;} N(0,1); \qquad \qquad \tilde{P}(f) =
\frac{1}{\sqrt{\log \log k}} \sum_{p < x}
\frac{\lambda_f(p)}{p^{\frac{1}{2}}}.\end{equation} This we do by the method of
moments.
Let $m$ be fixed and assume
now that $k$ is sufficiently large so that $x^{2m} <
k^{2-\delta}$, $x^m < k^{\frac{\gamma}{2}}$. We have
\begin{align*} \frac{1}{|H_k|}\sum_{f \in H_k}\left| \sum_{p < x}
\frac{\lambda_f(p)}{p^{\frac{1}{2}}} \right|^{2m} &= \sum_{p_1, ..., p_{2m} <x}
\frac{1}{\sqrt{p_1\cdots p_{2m}}} \frac{1}{|H_k|} \sum_{f \in H_k}
\lambda_f(p_1)\cdots \lambda_f(p_2m). \end{align*}
When some $p_i$ appears an odd number of times in the list, we see from the
expression (\ref{prime_product}) that $\lambda_f(p_1)\cdots\lambda_f(p_r)$ can
be written as a linear combination of $O_m(1)$ terms $\lambda_f(n_i)$, for
which none of the $n_i$ are squares. Thus by Lemma \ref{H_k_orthogonality} the
contribution of all such terms is $\ll_m k^{-\frac{\gamma}{2}}$.
Among terms containing each $p_i$ an even number of times, those containing
some $p_i$ at least 4 times contribute $\ll_m (\log \log x)^{m-2}$, which is an
error term. We are left to consider terms containing each prime exactly twice.
These contribute
\[O_m(k^{-\gamma} (\log \log k)^m) +\sum_{\substack{p_1, ..., p_m < x\\
\text{distinct}}}
\frac{1}{p_1 \cdots p_m}\sum_{d|p_1\cdots p_m} \frac{1}{d} = \frac{(2m)!}{2^m
m!} (\log \log x)^m (1 + o_m(1)).\] The claimed convergence in
(\ref{perturbed_convergence}) thus
follows from the fact that the Gaussian distribution is determined by its
moments.
To prove (\ref{mean_error}), assume $x^6 < \min(k^{2-\delta}, k^\gamma)$ and
split the
primes, prime squares, and higher powers as
\[\left| \sum_{n < x^3} \frac{\Lambda_f(n) b_n}{n^{\frac{1}{2}}}\right|^2
\leq 3 \left[\left|\sum_{p < x^3}\frac{\lambda_f(p)
b_p \log p}{p^{\frac{1}{2}}}\right|^2 + \left|\sum_{p < x^{\frac{3}{2}}}
\frac{O(\log p)}{p}\right|^2 + O(1) \right]\]
Thus
\begin{align*}\frac{1}{|H_k|}\sum_{f \in H_k} \left| \sum_{n < x^3}
\frac{\Lambda_f(n)
b_n}{n^{\frac{1}{2}}}\right|^2 &\leq \frac{3}{|H_k|}\sum_{f \in
H_k}\left|\sum_{p <
x^3}\frac{\lambda_f(p)
b_p \log p}{p^{\frac{1}{2}}}\right|^2 + O(\log^2 x)
\\
& \leq 3\sum_{p_1, p_2 \leq x^3} \frac{b_{p_1}b_{p_2}\log p_1 \log
p_2}{\sqrt{p_1 p_2}} \frac{1}{|H_k|}\sum_{f \in H_k}
\lambda_f(p_1)\lambda_f(p_2) +
O(\log^2 x)\\
& \leq \sum_{p \leq x^3} \frac{O(\log^2 p)}{p} + O(k^{-\gamma/2}) + O(\log^2
x) \\&= O(\log^2 x).
\end{align*}
\end{proof}
\section{Proof of main results}
Throughout this section we let $\mathcal{F}$ be a family of $L$-functions,
either $\mathcal{F} =
\{L(s,f\}_{f \in H_k}$ or $\mathcal{F} = \{L(s, \chi_{8d}\}_{d \in s(D)}$, and
we let $C = k^2$ or $C = D$ for the conductor in the family. We also let $*$
stand in for the typical element in the family.
\begin{proof}[Proof of Theorem \ref{right_of_half}]
Choose $x=x(C)$
by $\frac{4}{\log x} = \sigma$ and observe that $\frac{\log
C}{\log x} \to \infty$ while $\frac{\log C}{\log x \sqrt{\log \log C}} \to 0$
as $C \to \infty$. In particular, $\log \log x = \log \log C + O(\log_3
C)$, fulfilling the conditions of Proposition \ref{prime_sum},
so that $\frac{1}{|\mathcal{F}|} \sum_{* \in \mathcal{F}} \delta_{P(*)} {\; \stackrel{d}{\longrightarrow} \;}
N(0,1)$ as $C \to \infty$.
Recall that we set \[A(f) = \frac{1}{\sqrt{\log \log k}} \left(
\log |L(\frac{1}{2} + \frac{4}{\log
x},f)|+ \frac{1}{2} \log \log
k\right)\] and \[ A(d) = \frac{1}{\sqrt{\log \log D}} \left( \log
|L(\frac{1}{2} + \frac{4}{\log x}, \chi_{8d})| - \frac{1}{2} \log \log
D\right).\]
By Lemma \ref{zero_prob} there is a set $E \subset \mathcal{F}$ of measure
$o(1)$ such that outside $E$,
$\sigma_{x,*} = \frac{4}{\log x}$. Thus by (\ref{log_prime_sum}) we have
\[\log L(\frac{1}{2} + \frac{4}{\log x},*) = \sum_{n < x^3}
\frac{\Lambda_{x,*}(n)}{n^{\frac{1}{2} + \frac{4}{\log x}} \log n} +
O\left(\frac{1}{\log x} \left|\sum_{n \leq x^3}
\frac{\Lambda_{x,*}(n) }{n^{\frac{1}{2} +
\frac{4}{\log x}}}\right|\right)
+ O\left(\frac{\log C}{\log x}\right)\]
except for on a set of measure $o(1)$. Note that the second
error term contributes $o(1)$ to $A(*)$. Now
\[\sum_{n < x^3}
\frac{\Lambda_{x, *}(n)}{n^{\frac{1}{2} + \frac{4}{\log x}} \log n} = \sum_{n <
x} \frac{\Lambda_*(n)}{n^{\frac{1}{2}}\log n} + \sum_{n <
x} \frac{\Lambda_*(n)}{n^{\frac{1}{2}}\log n}\left(n^{
\frac{-4}{\log x}} - 1\right) + \sum_{x \leq n < x^3}
\frac{\Lambda_{x,*}(n)}{n^{\frac{1}{2} + \frac{4}{\log x}} \log n}.\]
Applying (\ref{mean_error}) of Proposition \ref{prime_sum} successively with
corresponding choices of $b_n$, we find that
\begin{align*}
\frac{1}{|\mathcal{F}|}\sum_{* \in \mathcal{F}} \left[\frac{1}{\log x}
\sum_{n \leq x^3}
\frac{\Lambda_{x,*}(n) }{n^{\frac{1}{2} +
\frac{4}{\log x}}} \right]^2 = O(1), & \qquad b_n = a_x(n)\\
\frac{1}{|\mathcal{F}|}\sum_{* \in \mathcal{F}} \left[\sum_{n <
x} \frac{\Lambda_*(n)}{n^{\frac{1}{2}}\log n}\left(n^{
\frac{-4}{\log x}} - 1\right) \right]^2 = O(1), & \qquad b_n = \left\{
\begin{array}{lll} \frac{\log
x}{\log n}\left(n^{\frac{-4}{\log x}} - 1\right), & &n \leq x
\\ 0,&& x \leq n < x^3\end{array} \right. \\
\frac{1}{|\mathcal{F}|}\sum_{* \in \mathcal{F}} \left[ \sum_{x \leq n < x^3}
\frac{\Lambda_{x,*}(n)}{n^{\frac{1}{2} + \frac{4}{\log x}} \log n}\right]^2 =
O(1), & \qquad b_n = \left\{
\begin{array}{lll}0, && n < x\\ a_x(n) \frac{\log x}{\log n}, && x \leq n <
x^3 \end{array} \right..
\end{align*}
Thus by Lemma \ref{conv_dist_lemma}, \[\frac{1}{|\mathcal{F
}|} \sum_{* \in \mathcal{F}} \delta_{P(*)} {\; \stackrel{d}{\longrightarrow} \;} N(0,1) \qquad
\Rightarrow \qquad \frac{1}{|\mathcal{F}|} \sum_{* \in \mathcal{F}}
\delta_{A(*)} {\; \stackrel{d}{\longrightarrow} \;} N(0,1).\]
\end{proof}
We will deduce Corollary \ref{upper_bound} from Theorem \ref{right_of_half} by
applying the following Proposition, which is an analog of the upper bound in the
Proposition of \cite{sound_moments},
in the case when RH for the $L$-function is not assumed.
\begin{prp}\label{our_upper_bound}
Continue to let $\mathcal{F} = \{L(s,f)\}_{f \in H_k}$ or $\mathcal{F} = \{L(s,
\chi_{8d})_{d \in s(D)}$, and let $C$ be the conductor of the $L$-functions in
the family. For $\sigma_{x,*}$ as defined in
(\ref{def_sigma}) we have
\[ \log |L(\frac{1}{2},*)| \leq \log L(\frac{1}{2} + \sigma_{x,*},*) +
\sigma_{x,*}\log C + O(1).\]
\end{prp}
\begin{proof}
\begin{align} \notag\log |L(\frac{1}{2},*)|-&\log L(\frac{1}{2} +
\sigma_{x,*},*) = \int_{0}^{ \sigma_{x,*}} -
\frac{L'}{L}(\frac{1}{2} + \sigma,*)d\sigma
\\&\label{upper_bound_sum} = \log{\frac{\gamma_*(\frac{1}{2} +
\sigma_{x,*})}{\gamma_*(\frac{1}{2})}
} - \sum_{\rho = \frac{1}{2} + \beta + i \gamma: \Lambda(\rho,*) = 0}
\int_0^{\sigma_{x,*}} \frac{ \sigma - \beta}{( \sigma
-\beta)^2 + \gamma^2} d\sigma\\
\notag& \leq \sigma_{x,*} \log C + O(1) - \frac{1}{2}\sum_{\rho =
\frac{1}{2} + \beta + i \gamma: \Lambda(\rho, *)=0}
\log\left(\frac{( \sigma_{x,*} - \beta)^2 + \gamma^2}{ \beta^2 +
\gamma^2}\right)
\end{align}
from the formulas (\ref{hadamard_derivative}) and
(\ref{zero_sum}).\footnote{Note that by choice of $\sigma_{x,*}$, $L(s,*)$ has
no zero on the
real axis for $s = \frac{1}{2} + \sigma$ and $\sigma > \sigma_{x,*}.$ In the
case that $L(s,f)$ has a zero between $\frac{1}{2}$ and $\frac{1}{2} +
\sigma_{x,*}$ the logarithm of $L(s,*)$ at $\frac{1}{2}$ is defined by
continuous
variation along a path that makes a small semicircle in the upper half plane
around the zero, and the line integrals above also follow this path.} For
those
$\rho$ for which $|\beta | \leq \frac{\sigma_{x,*}}{2}
$ the logarithm is plainly positive. Among the remaining $\rho$, pair $\rho =
\frac{1}{2} + \beta + i\gamma$ and $\rho' = \frac{1}{2}-\beta + i\gamma$, with,
say, $\beta >
0$. Note that by (\ref{def_sigma}), $\rho, \rho' \not \in {\mathcal{G}}_{x,f}$ so that
$|\gamma| > \frac{x^{3\beta }}{\log x}.$
The combined contribution of $\rho$ and $\rho'$ to the sum in
(\ref{upper_bound_sum}) is
\begin{align} \notag \log &\left[\frac{( \sigma_{x,*} - \beta)^2 +
\gamma^2}{ \beta^2 + \gamma^2} \cdot \frac{(\sigma_{x,*} + \beta)^2 + \gamma^2}{
\beta^2 +
\gamma^2}\right]
\\& \notag \qquad= \log \left[\left( 1 + \frac{(\sigma_{x,*} -
2\beta)\sigma_{x,*}}{ \beta^2 + \gamma^2}\right) \left(1 +
\frac{(\sigma_{x,*} + 2 \beta)\sigma_{x,*}}{\beta^2
+\gamma^2}\right)\right]
\\& \label{paired_sum}\qquad = \log\left[1 + \frac{\sigma_{x,*}^2}{ \beta^2 +
\gamma^2}\left( 2 + \frac{\sigma_{x,*}^2 - 4\beta^2}{ \beta^2 +
\gamma^2}\right)\right]
\end{align}
Now observe that
\[ \gamma > \frac{x^{3\beta}}{\log x} \geq 3\beta \] so that
\begin{equation}\label{bracket_bound}2 \geq 2 + \frac{\sigma_{x,*}^2 - 4
\beta^2}{ \beta^2
+ \gamma^2} \geq 2 -\frac{4\beta^2}{10\beta^2}
\geq \frac{8}{5}\end{equation}
Hence
\[ \text{expr. }(\ref{paired_sum}) \geq \log\left[1 + \frac{8}{5}
\frac{\sigma_{x,*}^2}{\beta^2 + \gamma^2}\right] > 0.\] It follows that the
zeros contribute a
negative amount to (\ref{upper_bound_sum}), which proves the Proposition.
\end{proof}
\begin{proof}[Deduction of Corollary \ref{upper_bound}]
Take $x = x(C)$ growing such that $\frac{\log C}{\log x} \to \infty$ but
$\frac{\log C}{\log x \sqrt{\log \log C}} \to 0$ as $C \to \infty$, as in
the proof of Theorem \ref{right_of_half}. Then as before we have $\sigma_{x,*}
= \frac{4}{\log x}$ except for on a set of measure $o(1)$. It
follows from Proposition \ref{our_upper_bound} that
\[\frac{\log |L(\frac{1}{2},*)|}{\sqrt{\log \log C}} \leq \frac{\log
L(\frac{1}{2}+
\frac{4}{\log x},*)}{\sqrt{\log
\log C}} + o(1)\]
except on a set of measure $o(1)$, and the Corollary now follows from the
convergence in distribution of the right hand side proved in Theorem
\ref{right_of_half}.
\end{proof}
We now prove Theorem \ref{conditional} by bounding the negative contribution of
the zeros in Proposition \ref{our_upper_bound} by invoking the Low-lying Zero
Hypothesis.
\begin{proof}[Proof of Theorem \ref{conditional}]
Let $x = x(C)$ and $y = y(C)$ be parameters growing with $C$, satisfying the
conditions
\begin{enumerate}
\item $\frac{\log C}{\log x} \to \infty $
\item $\frac{\sqrt{\log \log C}}{\log y} \to \infty$, but
$\frac{\log
C}{y \log x} \to 0$
\item $\frac{\log C \log y}{\log x \sqrt{\log \log C}} \to 0$
\end{enumerate}
as $C \to \infty$. For instance, these are simultaneously satisfied with $\log
x = \log C (\log \log C)^{-\frac{1}{4}} $, $y = \log \log C$.
Since $\frac{\log C}{\log x} \to \infty$,
$\sigma_{x,*} = \frac{4}{\log x}$ except on a set of measure $o(1)$ in
$\mathcal{F}$. Thus, invoking the Low-lying Zero Hypothesis, there is a set
of $\mathcal{F}^*$ of measure $1-o(1)$ in $\mathcal{F}$ on which
$\sigma_{x,*} = \frac{4}{\log x}$ and for all $L(s,*) \in \mathcal{F}^*$, all
zeros $\rho = \frac{1}{2} + \beta + i\gamma$ of
$\Lambda(s,*)$ satisfy \[|\gamma| > \frac{1}{y \log x}.\] Restricting to
$\mathcal{F^*}$, by (\ref{upper_bound_sum})
we have
\[\log |L(\frac{1}{2} + \frac{4}{\log x},*)| - \log |L(\frac{1}{2},*)| =
O(\frac{\log C}{\log x}) + \frac{1}{2}\sum_{\rho =
\frac{1}{2} + \beta + i \gamma:\Lambda(\rho,*) = 0}
\log\left(\frac{( \frac{4}{\log x} - \beta)^2 + \gamma^2}{ \beta^2 +
\gamma^2}\right).\]
In view of $\frac{\log C}{\log x} = o(\sqrt{\log \log C})$ it suffices to prove
the bound
\begin{equation*}\tag{\dag}\label{bound} \frac{1}{|\mathcal{F}|} \sum_{* \in
\mathcal{F}^*} \left|\sum_{\rho =
\frac{1}{2} + \beta + i \gamma:\Lambda(\rho,*) = 0}
\log\left(\frac{( \frac{4}{\log x} - \beta)^2 + \gamma^2}{ \beta^2 +
\gamma^2}\right)\right|^2 = o(\log \log C)\end{equation*} in order to deduce
the theorem from Lemma \ref{conv_dist_lemma} together with
the normal approximation of $\log L(\frac{1}{2} + \frac{4}{\log x}; *)$ proved
in Theorem \ref{right_of_half}.
In the sum over zeros of (\ref{bound}), for $\rho$
with $|\beta| < \frac{2}{\log x}$ we bound the summand in absolute value by
\begin{align*}
\left|\log\left(\frac{( \sigma_{x,*} - \beta)^2 + \gamma^2}{ \beta^2 +
\gamma^2}\right)\right| &= \left|\log \left(1- \frac{(\sigma_{x,*} -
\beta)^2 - \beta^2}{(\sigma_{x,*} - \beta)^2 + \gamma^2}\right)\right|
\\& \leq \left| \log \left(1 - \frac{\sigma_{x,*}^2}{\sigma_{x,*}^2
+ \gamma^2}\right) \right| \\&= \left| \log \left(1 - \frac{1}{1
+ (\frac{\gamma \log x}{4})^2}\right) \right| \ll
\frac{\log y}{1 + (\frac{\gamma\log x}{4})^2},
\end{align*}
by using $|\gamma| \log x \gg \frac{1}{y}.$
Since $|\beta| \leq \frac{2}{\log x}$, the last quantity is bounded by
\begin{equation}\label{small_beta_bound} \ll \frac{\log y}{\log^2 x}
\frac{1}{(\frac{4}{\log x} - \beta)^2 +\gamma^2}.\end{equation}
For $|\beta| > \frac{2}{\log x} = \frac{\sigma_{x,*}}{2}$ we pair the
contributions of $\rho$ and $\rho' = \rho-2\beta$; by (\ref{paired_sum}) this
combined contribution is
\[\log\left[1 + \frac{\sigma_{x,*}^2}{ \beta^2 +
\gamma^2}\left( 2 + \frac{\sigma_{x,*}^2 - 4\beta^2}{ \beta^2 +
\gamma^2}\right)\right].\]
Since $|\beta| > \frac{\sigma_{x,*}}{2}$ we have $\rho, \rho' \not \in
\mathcal{G}_{x,f}$, and therefore $|\gamma| \geq \frac{x^{3\beta}}{\log x}
\geq 3 \beta$. By (\ref{bracket_bound}),
\[2 \geq 2 + \frac{\sigma_{x,*}^2 - 4\beta^2}{ \beta^2 +
\gamma^2} \geq \frac{8}{5},\] and therefore the combined contribution is
bounded in absolute value by
\begin{equation}\label{large_beta_bound}\ll \frac{\sigma_{x,*}^2}{\beta^2 +
\gamma^2} \ll \frac{1}{\log^2 x} \frac{1}{(\frac{4}{\log x} - \beta)^2 +
\gamma^2}.\end{equation}
Combining (\ref{small_beta_bound}) and (\ref{large_beta_bound}),
\begin{align*}\left|\sum_{\rho =
\frac{1}{2} + \beta + i \gamma}
\log\left(\frac{( \frac{4}{\log x} - \beta)^2 + \gamma^2}{ \beta^2 +
\gamma^2}\right) \right| &= O\left( \frac{\log y}{\log^2 x} \sum_\rho
\frac{1}{(\frac{4}{\log x} - \beta)^2 + \gamma^2}\right) \\&= O\left(\frac{\log
y}{\log x}\left| \sum_{n \leq x^3} \frac{\Lambda_{x,*}(n)}{n^{\frac{1}{2}+
\frac{4}{\log x}}}\right|\right) + O\left(\frac{\log y \log C}{\log
x}\right)\end{align*}
by (\ref{zero_sum_bound}) of Lemma \ref{log_L_off}.
Since
\[\frac{\log y \log C}{\log x } = o(\sqrt{\log \log C}),\] the bound
(\ref{bound}) now follows from
\[\frac{1}{|\mathcal{F}|}{\sum_{* \in \mathcal{F}^*}} \left| \frac{\log
y}{\log x }\sum_{n
\leq x^3} \frac{\Lambda_{x,*}(n)}{n^{\frac{1}{2}+
\frac{4}{\log x}}}\right|^2 = O\left(\log^2 y \right) =
o(\log \log C)\] by (\ref{mean_error}) of Proposition \ref{prime_sum}.
\end{proof}
\bibliographystyle{plain}
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| 5 | 1 |
\section{Introduction}
\label{sec:intro}
In his 1969 paper, James Miller envisioned a world where technology enables people to vote from their homes~\citep{miller1969program}. With the rise of participatory democracies, the formation of many overlapping online communities, and the increasing use of polls by companies and service providers, this vision is turning into reality.
New online voting apps provide an easy way for people to report and aggregate their preferences, from simple direct polls (such as those used by Facebook and Doodle), through encrypted large-scale applications (e.g. \url{electionbuddy.com}), to sophisticated tools that use AI to guide group selection, such as \url{robovote.org}.
As a result, each of us is prompted to vote in various formats multiple times a day: we vote for our union members and approve their decisions, on meeting times, and even on the temperature in our office.\footnote{\url{http://design-milk.com/comfy-app-} \url{brings-group-voting-workplace-thermostat}.
Is direct democracy coming back? Can it replace representative democracy and parliaments? As it turns out, many online voting instances and polls have low participation rates~\citep{christian2005voting,jonsson2011user}, presumably since most people consider them insignificant, low-priority, or simply a burden.
The actual decisions in many of these polls are often taken by a small group of dedicated and active voters, with little or no involvement from most people who could have voted. The outcome in such cases may be completely unrepresentative for the entire population, e.g. if the motivation of the active voters depends on their position or other factors. Even if the set of active voters is selected at random and is thus representative in expectation, there may be too few voters for a reliable outcome. For example, Mueller et al.~\citep{mueller1972representative} argue that to function well, such a ``random democracy'' would require over 1000 representatives.
\medskip
\emph{Proxy voting} lets voters who are unable or uninterested to vote themselves transfer their voting rights to another person---a proxy. Proxy voting is common in politics and in corporates~\citep{riddick1985riddick}, and plays an important role in existing and planned systems for e-voting and participatory democracies~\citep{petrik2009participation}. Yet there is only a handful of theoretical models dealing with proxy voting, and our understanding of its effects are limited (see Discussion).
In this paper, we model voters' positions as points in a metric space aggregated by some function $\mathbf{g}$ (specifically, Median, Mean, or Majority). For example, a voter's position may be her preferred pension policy in the union's negotiation with management (say, how much to save on a scale of 0 to 10). The optimal policy is an aggregate over the preferences of all employees. Since actively participating in union's meeting costs time and effort, we consider a subset of \emph{active voters} selected from the population (either at random or by strategic self-selection), and ask whether the accuracy can be improved by allowing inactive voters to use a proxy at no cost. Following Tullock~\citep{tullock1967proportional}, we weigh the few active voters (who are used as proxies) according to their number of followers, and assume that inactive voters select the ``nearest'' active voter as a proxy. For example, a person who is unable to attend the next union meeting could use an online app to select a colleague with similar preferences as her proxy, thereby increasing his weight and influencing the outcome in her direction.
The intuition for why proxy voting should increase accuracy is straight-forward: opinions that are more ``central'' or ``representative'' would attract followers and gain weight, whereas the weight of ``outliers'' that distort the outcome will be demoted. However as we will see, this reasoning does not always work in practice
Thus it is important to understand the conditions in which proxy voting is expected to improve accuracy, especially when voters behave strategically.
\vspace{-1mm}\subsection{Contribution and Structure}
We dedicate one section to each common mechanism, and show via theorems and empirical results that proxy voting usually has a significant positive effect on accuracy, and hence welfare. For the Median mechanism on a line (Section~\ref{sec:median}), proxy voting may only increase the accuracy, often substantially. For the Mean mechanism on a line (Section~\ref{sec:mean}), we show improvement in expectation if active voters are sampled from the population at random. The last domain contains multiple independent binary issues, where a Majority vote is applied to each issue (Section~\ref{sec:binary}). Here we show that proxy voting essentially leads to a ``dictatorship of the best expert,'' which increases accuracy when the sample is small and/or when voters have high disagreements. Interestingly, results on real preference data are even more positive, and we analyze the reasons in the text.
We further characterize equilibria outcomes when voters strategically choose whether to become active (i.e., use as proxies), and show that most of our results extend this strategic setting. Results are summarized in Table~\ref{tab:results}.
\section{Preliminaries}
$\mathcal X$ is the \emph{space}, or set of possible voter's preferences, or types. In this paper $\mathcal X\subseteq \mathbb R^k$ for some $k\geq 1$ dimensions, thus each type can be thought of as a position in space. We use the $\ell_\rho$ distance metric on $\mathcal X$.
In particular, we will consider two spaces: an unknown interval $\mathcal X=[a,b]$ for some $a,b\in {\mathbb R} \cup \{\pm \infty\}$, and multiple binary issues ${\mathcal X} = \{0,1\}^k$. Note that this means that all $\ell_\rho$ norms coincide (not true e.g. for ${\mathcal X}=\mathbb R^2$).
We assume an infinite population of voters, that is given by a distribution $f$ over ${\mathcal X}$. We say that $f$ over the interval $[a,b]$ is \emph{symmetric} if there is a point $c$ s.t. $f(c-x)=f(c+x)$ for all $x$. We say that $f$ over the interval $[a,b]$ is \emph{[weakly] single-peaked} if there is a point $c\in {\mathcal X}$ s.t. $f$ is [weakly] increasing in $[\min a,z]$ and [weakly] decreasing in $[z,b]$. $f$ is \emph{single-dipped} if the function $-f$ is single-peaked. For example, (truncated) Normal distributions are single-peaked, and Uniform distributions are weakly single peaked. We denote the cumulative distribution function corresponding to $f$ by $F(X)= Pr_{z\sim f}(z<x)$.
\vspace{-3mm}\paragraph{Mechanisms}
A \emph{mechanism} $\mathbf{g}:\mathcal X^n \rightarrow \mathcal X$ (also called a voting rule) is a function that maps any profile (set of positions) to a winning position.
Two particular mechanisms we will consider for the interval setting are the \emph{Mean mechanism}, $\mathbf{mn}(S) = \frac{1}{|S|}\sum_{s_i\in S}s_i$, and the \emph{Median mechanism}, $\mathbf{md}(S) = \min\{s_i\in S \text{ s.t. } |\{j:s_j\leq s_i\}| \geq |\{j:s_j> s_i\}|$ (see Fig.~\ref{fig:line}).
For the binary issues we will focus on a simple \emph{Majority mechanism} that aggregates each issue independently according to the majority of votes. That is, $(\mathbf{mj}(S))^{(j)} = 1$ if $|\{i:s_i^{(j)}=1\}| > |\{i:s_i^{(j)}=0\}|$ and $0$ otherwise, where $s^{(j)}$ is the $j$'th entry of position vector $s$. In all mechanisms we break ties lexicographically towards the lower outcome.
All of our three mechanisms naturally extend to such infinite populations, as the Median, Mean, and Majority of $f$ (in their respective domains) are well defined. The mechanisms also extend to weighted finite populations. E.g. for $n$ agents with positions $S$ and weights $\vec w=\{w_1,\ldots,w_n\}$, the weighted mean is defined as $\mathbf{mn}(S,\vec w) \equiv \frac{1}{\sum_{i
\leq n} w_j}\sum_{i \leq n}w_i s_i$, and similarly for the Median and Majority.
\medskip
In our model, a finite subset $N$ of $n$ \emph{agents} are selected out of the whole population, and only these agents can vote. We follow \citep{mueller1972representative} in assuming that positions $S_N=\{s_1,\ldots,s_n\}$ are sampled i.i.d. from $f$. We can think of these as voters who happen to be available at the time of voting, or voters for which this voting is important enough to consider participation.
In our basic setup, the unavailable voters abstain, while \emph{all agents vote}. The result is $\mathbf{g}(S_N)$. Yet two problems may prevent us from getting a good outcome. First, $N$ may be too small for $\mathbf{g}(S_{N})$, the decision made by the agents, to be a good estimation of $\mathbf{g}(f)$, the true preference of the population. Second, even selected agents may decide not to vote due to various reasons, and such strategic participation may bias the outcome. We will then have a set of active agents $M \subseteq N$, and the outcome $\mathbf{g}(S_M)$ may be very far from both $\mathbf{g}(S_N)$ and $\mathbf{g}(f)$, depending on the equilibrium outcome of the induce game (later described in more detail).
\vspace{-3mm}\paragraph{Proxies and weights}
Our main focus in this paper is characterizing the regime in which \emph{voting by proxy} is beneficial. In this setup each inactive voter specifies one of the active agents as a proxy to vote on her behalf. Given a set $M$ of active agents, the decisions of inactive voters are specified by a mapping $J_M:{\mathcal X} \rightarrow M$, where $J_M(x)\in M$ is the proxy of any voter located at $x\in {\mathcal X}$. We label the Proxy setup as $P$, in contrast to the Basic setup denoted as $B$. We highlight that all voters select a proxy, whether they are part of $N$ or not.
Without further constraints, we will assume that the proxy of a voter at $x$ is always its nearest active agent, i.e. the agent whose position (or preferences) are most similar to $x$. Thus for every set $M$, we get a partition (a Voronoi tessellation) of ${\mathcal X}$ and can compute the \emph{weight} of each active agent $j$ by integrating $f$ over the corresponding cell. Formally, $J_M(x) = \argmin_{j\in M}{\|x-s_j\|}$ and $w_j = \int_{x\in \mathcal X: J_M(x)= j}f(x)dx$.
The outcome of each mechanism $\mathbf{g}$ for agents $N$ is then defined as $\mathbf{g}^B(S_N) = \mathbf{g}(S_N)$ in the Basic scenario, and $\mathbf{g}^P(S_N) = \mathbf{g}(S_N,\vec w_N)$ in the Proxy scenario, where $\vec w_N$ is computed according to $f$ as above (see Fig.~\ref{fig:line}). The distribution $f$ should be inferred from the context.
\begin{figure}[t]
\input{line_fig}
\protect\caption{\label{fig:line}
The top figure shows the preferences of 4 agents on an interval, as well as the outcomes of the median and mean mechanisms. In the middle figure we see the weight of each agent under proxy selection, assuming $f$ is a uniform distribution on the whole interval, as well as the modified outcomes. The bottom figure shows the outcome under proxy voting if agent~2 becomes inactive, and $M=\{1,3,4\}$. The dotted line marks $\mathbf{mn}(f)=\mathbf{md}(f)=5$. \vspace{-5mm}}
\end{figure}
\vspace{-3mm}\paragraph{Equilibrium under strategic participation}
In our strategic scenarios the agents $N$ are players in a complete information game, whose (ordinal) utility exactly matches their preferences as voters. I.e., they prefer an outcome that is as close as possible to their own position. Each agent has two actions: active and inactive. In addition, a voter who is otherwise indifferent between the two possible outcomes (i.e. he is not pivotal) will prefer to remain inactive, a behavior known as \emph{lazy-bias}~\citep{elkind2015equilibria}. We refer to these strategic/lazy-bias scenarios by adding ${+}L$ to either $B$ or $P$. Agents may not misreport their position.
When there are no proxies (scenario $B{+}L$) this strategic decision is very simple, since each agent has a single vote which may or may not be pivotal (and when it is pivotal it always helps the agent). On the other hand, if voting by proxy is allowed (scenario $P{+}L$), any change in the set of active agents changes the proxy selection and thus the weights of all remaining agents.
Recall that $\vec w_{\vec M}$ denotes the weights we get under proxy selection with active set $M$.
Then for all $i\notin M$, agent~$i$ prefers to join set $M$ iff $\left\|\vec g(S_{M \cup \{i\}},\vec w_{M \cup \{i\}}) - s_i \right\| < \left\|\vec g(S_M,\vec w_{M}) - s_i \right\|$.
For example, if agent~2 in Fig.~\ref{fig:line} (bottom) becomes inactive, we get no change in the Median outcome $\mathbf{md}(S_{M},\vec w_M)$, and thus agent~2 prefers to become inactive (it is also possible that an agent strictly loses when becoming active).
A pure Nash equilibrium, or equilibrium for short, is a subset $M\subseteq N$ s.t. no agent in $M$ prefers to be inactive, and no agent in $N\setminus M$ prefers to be active. While it is possible that there are multiple equilibria (or none at all), this will turn out not to be a problem in most cases we consider. \rmr{ok for median and majority, what about mean?}
We thus define $\mathbf{g}^{B{+}L}(S_N) = \mathbf{g}(S_M)$ and $\mathbf{g}^{P{+}L}(S_N) = \mathbf{g}(S_M,\vec w_M)$, where $M\subseteq N$ is the set of active agents in equilibrium.
\medskip
To recap, an \emph{instance} is defined by a population distribution $f$, a scenario $Q\in\{\! B,P,B+L,P+L\!\}$, a mechanism $\mathbf{g}\!\in\!\{\!\mathbf{md},\mathbf{mn},\mathbf{mj}\!\}$ and a sample size $n$.
We sample a finite profile of $n$ agents i.i.d. from $f$, whose locations are $S_N$. Then, according to the scenario, either all of $N$ are active, or we get a subset $M$ of active agents. The votes of all active agents are aggregated according to $\mathbf{g}$, with or without being weighted by $\vec w_{\vec M}$, the number of their inactive followers. Finally, the outcome of mechanism $\mathbf{g}^Q(S_N)$ depends on a subset of these parameters, according to the scenario $Q$.
\vspace{-3mm}\paragraph{Evaluation}
We do not consider here the reasons for using one mechanism over another, and simply assume that $\mathbf{g}(f)$ reflects the best possible outcome to the society or to the designer.
We want to measure how close is $\mathbf{g}^Q(S_N)$ to the \emph{optimal outcome} $\mathbf{g}(f)$. We define the \emph{error} as the distance between $\mathbf{g}^Q(S_N)$ and $\mathbf{g}(f)$, i.e., $\|\mathbf{g}^Q(S_N)-\mathbf{g}(f)\|$. \rmr{It actually does not matter for these 3 domains which $\ell_p$ norm is used}
The \emph{loss} of a mechanism $\mathbf{g}$ is calculated according to its expected error---the expected squared distance from the optimum---over all samples of $m$ available voters.
\labeq{loss}
{\mathcal{L}^Q(n) =\mathbb{E}_{S_N\sim f^n}\left[ \|\mathbf{g}^Q(S_N)-\mathbf{g}(f)\|^2\right],}
where the mechanism $\mathbf{g}$ and the distribution $f$ can be inferred from the context, and the expectation is over all subsets of $n$ positions sampled i.i.d. from distribution $f$ (sometimes omitted from the subscript).
We note that the loss is the sum of two components~\citep{wackerly2007mathematical}: the (squared) bias $\mathbb{E}[\mathbf{g}^Q(S_N) - \mathbf{g}(f)]^2$ and the variance $\mathbb{V}[\mathbf{g}^Q(S_N)]$.
A mechanism $\mathbf{g}$ is \emph{unbiased} for $(Q,f)$ if $\mathbb{E}[\mathbf{g}^Q(S_N)] = \mathbf{g}(f).$
For example in the Basic scenario, mechanisms $\mathbf{mn}$ and $\mathbf{mj}$ are unbiased for $(B,f)$ regardless of $f$, and $\mathbf{md}$ is unbiased for $(B,f)$ if $f$ is symmetric, but not for other (skewed) distributions.
Our primary goal is to characterize the conditions under which proxy voting improves the outcome, i.e. $\mathcal{L}^{P[+L]}(n) < \mathcal{L}^{B[+L]}(n)$.
\section{Median Voting on an Interval}
\label{sec:median}
The Median mechanism is popular for two primary reasons. First, it finds the point that minimizes the sum of distances to all reported positions, i.e. $\mathbf{md}(S) \in \argmin_{x\in {\mathcal X}}\sum_{s_i\in S}|\! s_i-x\! |$. Second, in strategic settings where agents might misreport their positions, it is known that the Median mechanism is group strategyproof~\citep{moulin1980strategy}, meaning that no subset of agents can gain by misreporting.
\vspace{-1mm}\subsection{Random participation}
Suppose all $n$ agents sampled from $f$ are active.
Let $j^*\in N$ be the proxy closest to $x^*=\mathbf{md}(f)$, and $s^*=s_{j^*}$
\begin{lemma}\label{lemma:median_optimal} $\mathbf{md}^{P}(S_N)= s^*$ for any distribution $f$.
\end{lemma}
\begin{proof}
Recall that $\mathbf{md}^{P}(S_N)=\mathbf{md}(S_N,\vec w)$ where $w_j$ is the weight of voters using $j\in N$ as a proxy.
All voters $x\geq \mathbf{md}(f)$ are mapped to one of the proxies $j^*,j^*+1,\ldots,n$, thus $\sum_{j=j^*}^n \geq 1/2$ and $\mathbf{md}(S_M,\vec w)\geq s^*$. Similarly, all voters $x\leq \mathbf{md}(f)$ are mapped to one of the proxies $1,2,\ldots,j^*$, thus $\sum_{j=1}^{j^*} \geq 1/2$ and $\mathbf{md}(S_N,\vec w)\leq s^*$. Thus $\mathbf{md}(S_N,\vec w)=s^*$.
\end{proof}
Thus $\mathbf{md}^{P}(S_N)$ always returns the proxy closest to $x^*=\mathbf{md}(f)$, whereas $\mathbf{md}^{B}(S_N)$ returns some $j\in N$, meaning that the error is \emph{never higher} with proxy voting. I.e., $|\mathbf{md}^P(S)-x^*| \leq |\mathbf{md}^B(S)-x^*|$ for any $S$. In particular, the loss (=expected error) is weakly better.
\begin{corollary}For the Median mechanism, $\mathcal{L}^{P}(n) \leq \mathcal{L}^{B}(n)$ for any distribution $f$ and sample size $n$.
\end{corollary}
\begin{proof}
$$\mathcal{L}^{P}(n)= E[(\mathbf{md}^{P}(S_N)-x^*)^2] \leq E[(\mathbf{md}^{B}(S_N)-x^*)^2] = \mathcal{L}^{B}(n).$$
\rmr{ removed since for asymmetric distributions it does not equal the variance
$$\mathcal{L}^{P}(n) = \mathbb{V}_{S_N\sim f^n}[\mathbf{md}(S_N,\vec w)] = \mathbb{V}[s_k] $$ \\
$$ = E[(s_k-x^*)^2] \leq E[(\mathbf{md}(S_M)-x^*)^2] = \mathbb{V}[\mathbf{md}(S_M)] = \mathcal{L}^{VA}(\mathbf{md},n,f).$$
}
\end{proof}
Note that for symmetric distributions, both of $\mathbf{md}^{B}(S_N)=\mathbf{md}(S_N)$ and $\mathbf{md}^{P}(S_N)=\mathbf{md}(S_N,\vec w_N)$ are unbiased from symmetry arguments. Therefore to compute or bound the loss we just need to compute the variance of $\mathbf{md}^{Q}(S_N)$.
For the unweighted median, this problem was solved by Laplace (see \citep{stigler1973studies} for details):
Let $x^*=\mathbf{md}(f)$ be the median of symmetric distribution $f$ s.t. $f(x^*)>0$.\footnote{We will assume in this section that $f(x)>0$ in some environment of $x^*$, which is a very weak assumption.} The variance of $\mathbf{md}(S_N)$ is given by (approximately) $\frac{1}{4n f(x^*)^2}=\Theta(1/n)$. Since for any distribution the loss (or MSE) is lower-bounded by the variance, we get that for the Median mechanism, $\mathcal{L}^B(n)=\Omega(\frac{1}{n})$.
We argue that the loss decreases \emph{quadratically faster} with the number of agents once proxy voting is allowed.
\begin{conjecture} \label{conj:median}
For the Median mechanism, $\mathcal{L}^P(n) = O(\frac{1}{n^2})$ for any distribution $f$.
\end{conjecture}
The rest of this section is dedicated to supporting this conjecture. In particular, we prove it for symmetric distributions, and show empirically that it holds for other distributions as well. Further, for Uniform and single-peaked distributions, we can upper-bound the constant in the expression.
\begin{theorem} For the Median mechanism, $\mathcal{L}^{P}(n) = O(\frac{1}{n^2})$, for any symmetric distribution $f$.\label{th:median_bound}
\end{theorem}
\begin{proof}
W.l.o.g. we can assume $x^*=0$, and that the support of $f$ is the interval $[-1,1]$.
What is the expected distance between $s^*$ and $\mathbf{md}(f)=x^*=0$? We can translate each proxy $x_i$ to $y_i = |x_i-x^*|=|x_i|$. Note that $y_i$ come from some distribution $f'$ on $[0,1]$. By our assumption, $f(x)$ is strictly positive in some $\epsilon$ environment of $x^*=0$, i.e. $f(x)>\alpha$ for all $x\in [-\epsilon,\epsilon]$, for some $\alpha,\epsilon>0$. We thus have that $f'(z)>\alpha$ for all $z\leq \epsilon$. Then for the cumulative distribution $F'(z)$, we have that $F'(z)>z\alpha$ for all $z\leq \epsilon$, and $F'(z)> \epsilon\alpha$ for all $z>\epsilon$.
Recall that by Lemma~\ref{lemma:median_optimal}, the error is exactly $|s^*-x^*|=|s^*|$.
The random variable $s^*=\min y_i$ is the minimum of $n$ variables sampled i.i.d. from $f'[0,1]$. The distribution of the minimum is well known and in particular for all $z\in[0,1]$,
$$Pr(s^* > z)
= (Pr_{Z\sim f'[0,1]}(Z>z))^n = (1-F'(z))^n.$$
For $z=\epsilon$, we get $Pr(s^* > \epsilon)=(1-F'(\epsilon))^n \leq (1-\epsilon\alpha)^n$.
There is some $n_\epsilon$ s.t. for all $n>n_\epsilon$, $\Pr(s^*>\epsilon) < \frac{1}{n^3}$ since the left terms drops exponentially fast. Thus assume $n>n_\epsilon$. Let $T_n= \floor{2n \cdot \epsilon}$
\begin{align*}
\mathcal{L} &^{P}(n)
= VAR_f[s^*]
= E_f[(s^*)^2] \leq \sum_{t=1}^{2n} Pr(s^*\in [\frac{t-1}{2n},\frac{t}{2n}])(\frac{t}{2n})^2 \tag{bound by steps}\\
& = \sum_{t=1}^{T_n} Pr(s^*\in [\frac{t-1}{2n},\frac{t}{2n}])(\frac{t}{2n})^2 +\sum_{t=T_n+1}^{2n} Pr(s^*\in [\frac{t-1}{2n},\frac{t}{2n}])(\frac{t}{2n})^2 \\
& \leq \sum_{t=1}^{T_n} Pr(s^*\in [\frac{t-1}{2n},\frac{t}{2n}]) (\frac{t}{2n})^2 + \sum_{t=T_n+1}^{2n} Pr(s^*\in [\frac{t-1}{2n},\frac{t}{2n}]) \\
& \leq \sum_{t=1}^{T_n} Pr(s^*\in [\frac{t-1}{2n},\frac{t}{2n}]) (\frac{t}{2n})^2 + Pr(s^* > \epsilon) \\
& \leq \sum_{t=1}^{T_n} Pr(s^*\in [\frac{t-1}{2n},\frac{t}{2n}]) (\frac{t}{2n})^2 + \frac{1}{n^3} \\
&= \sum_{t=1}^{T_n} (Pr(s^* > \frac{t-1}{2n})-Pr(s^* > \frac{t}{2n}))(\frac{t}{2n})^2 + \frac{1}{n^3} \\
&\leq \sum_{t=1}^{T_n} Pr(s^* > \frac{t-1}{2n})(\frac{t}{2n})^2 + \frac{1}{n^3}\\
&\leq \frac{1}{4n^2} \sum_{t=1}^{T_n}Pr(s^* > \frac{t-1}{2n})t^2 + \frac{1}{n^3}
\leq \frac{1}{4n^2} \sum_{t=1}^{T_n}(1-F'(\frac{t-1}{2n}))^n t^2 + \frac{1}{n^3}\\
&\leq \frac{1}{4n^2} \sum_{t=1}^{T_n}(1-\alpha \frac{t-1}{2n})^n t^2 + \frac{1}{n^3}
\leq \frac{1}{4n^2} \sum_{t=1}^{T_n}e^{-\alpha \frac{t-1}{2}} t^2 + \frac{1}{n^3}\\
& = \frac{1}{4n^2} \sum_{t=1}^{T_n}e^{-\Theta(t)} t^2 + \frac{1}{n^3}
< \frac{1}{4n^2} C + \frac{1}{n^3} = O(\frac{1}{n^2}),
\end{align*}
where $C$ is some constant.
\end{proof}
Further, for Uniform $f=U[-1,1]$ we can derive a tighter bound on the sum of the series and show $\mathcal{L}^{P}(n) < \frac{4}{n^2}$. In fact for any single-peaked $f$ on $[-1,1]$ we have $\mathcal{L}^{P}(n) < \frac{7}{n^2}$.
\if 0
\begin{proof}[Proof sketch for Uniform] Suppose $f=U[-1,1]$.
We translate each position $s_i\in S_N$ to $y_i = |s_i-x^*|=|s_i|$ (as $x^*=0$). Note that $(y_i)_{i\in N}$ are uniformly distributed on $[0,1]$, and thus the random variable $s^*$ is the minimum of $n$ variables sampled i.i.d. from $U[0,1]$.
For all $z\in[0,1]$,
\labeq{min_dist}
{
Pr(s^* > z) = Pr(\forall j\in N, s_j>z)
= (1-z)^n.}
Note that $\mathcal{L} ^{P}(n) = \mathbb{V}_{S_N\sim f^n}[\mathbf{md}^P(S_N)]
= \mathbb{V}[s^*]
= \mathbb{E}[(s^*)^2]$.
Let $c\in \mathbb N$ a constant to be determined later.
We now bound the variance by steps:
$$\mathcal{L} ^{P}(n)=E[(s^*)^2] \leq \sum_{t=1}^{cn} Pr\(s^*\in \left[\frac{t-1}{cn},\frac{t}{cn}\right]\)\(\frac{t}{cn}\)^2,$$
and continue
\begin{align*}
\mathcal{L} ^{P}&(n) = \sum_{t=1}^{cn} (Pr(s^* > \frac{t-1}{cn})-Pr(s^* > \frac{t}{cn}))(\frac{t}{cn})^2 \\
&\leq \frac{1}{c^2n^2} \sum_{t=1}^{cn}((1-\frac{t-1}{cn})^n-(1-\frac{t}{cn})^n)t^2 \tag{By Eq.~\eqref{eq:min_dist}}\\
&\leq \frac{1}{c^2n^2} \sum_{t=1}^{\infty}(e^{-\frac{t-1}{c}}-e^{-\frac{t+1}{c}})t^2 \stackrel{c\rightarrow \infty}{\rightarrow}\frac{1}{c^2n^2} 4c^2,
\end{align*}
thus
$\mathcal{L}^{P}(n) < \frac{1}{c^2n^2} 4c^2 =\frac{4}{n^2}$.
\end{proof}
\fi
We simulated the effect of proxy voting on the Median mechanism in Figure~\ref{fig:loss_m}.
We can see that the (log of the) loss for each distribution closely resembles $\log(\frac{c}{n^2}) = c' - 2\log(n)$, where the constant $c'$ depends on the distribution. This also holds for the asymmetric distributions, which supports our Conjecture~\ref{conj:median}. In particular, this means that the loss under proxy voting drops much faster than the loss in the Basic scenario, which is roughly $\frac{1}{n}$.
\begin{figure}[t]
\centering
\includegraphics[scale=0.42]{loss_median.pdf}
\protect\caption{\label{fig:loss_m}The top figure shows $\mathcal{L}^Q(n)$ (in log scale) as a function of $n$ for various distributions. SP/SD/symm stands for Single-peaked / Single-dipped/ Symmetric distributions. Each point is based on 1000 samples of size $n$.
\vspace{-4mm}
}
%
\end{figure}
%
\vspace{-1mm}\subsection{Strategic participation}
We show that when participation is strategic the outcome of proxy voting is not affected, whereas the unweighted sample median becomes unboundedly worse.
Suppose all voters in $N$ are indexed in increasing order by their location, so that $\argmin S_N=1$.
\begin{proposition}
In the Basic scenario, for any distribution $f$ and any set of agents $N$, there is a unique equilibrium of $\mathbf{md}^{B+L}$ where $M=\{1\}$ (i.e., the lowest agent). Further, the game is weakly acyclic, i.e. there is a sequence of best replies from any initial state to this equilibrium.
\end{proposition}
Clearly this means that $\mathcal{L}^{B+L}(n)<\mathcal{L}^{B}(n)$, and only gets worse as we increase the sample size $n$.
The intuition is that due to tie-breaking, either all agents below current median, or all agents above it, are non-pivotal.
\cut{
\begin{proof}
Note first that if $M=\{1\}$ then the single active agent is pivotal by definition. Any other agent $i>1$ is non-pivotal since $\mathbf{md}(\{s_1,s_i\})=s_1$ by our tie-breaking assumption, thus $M=\{1\}$ is an equilibrium.
Consider any subset of active agents $M\subseteq N$ s.t. $|M|>1$.
If $m$ is even, then all agents above the median $\mathbf{g}(M)$ are non-pivotal. If $m$ is odd then all agents below the median are non-pivotal. Thus there is at least one agent in $M$ who prefers to become inactive. This continues until $|M|=1$.
Finally, if $M=\{i\}$ for some $i>1$, we have the following sequence of best-replies: any agent $j<i$ is pivotal, and in particular $j=1$. Thus agent~$1$ will become active. Now agent~$i$ is no longer pivotal so becomes inactive.
\end{proof}
}
On the other hand, while lazy bias decreases participation in the Proxy scenario, this does not increase the loss.
\begin{theorem}
In the Proxy scenario, for any distribution $f$ and any set of agents $N$, there is a unique equilibrium of $\mathbf{md}^{P+L}$ where $M=\{j^*\}$ (the agent closest to $x^*$). Further, the game is weakly acyclic, i.e. there is a sequence of best replies from any initial state to this equilibrium.
\end{theorem}
In particular, $\mathcal{L}^{P+L}(n)=\mathcal{L}^{P}(n)$ for any distribution $f$.
\begin{proof} If $j^*\notin M$ is inactive, then for $M\cup \{j^*\}$ the outcome becomes $s_{j^*}$ rather than $s_k$ (where $k=J_M(x^*)$), which $j^*$ prefers. If $j^*$ is active, and $j\neq j^*$ quits, then all votes above $s_{j^*}$ are still mapped to $j^*$ or higher (and similarly for votes below $s_{j^*}$). Thus the outcome remains the same which means $j$ is not pivotal.
\end{proof}
\section{Mean Voting on an Interval}
The Mean mechanism is perhaps the simplest and most common way to aggregate positions. For positions $S_N$ on the interval the outcome is $\mathbf{mn}(S_N)=\frac{1}{n}\sum_{i\in N}s_i$, which is known to minimize the sum of \emph{square distances} to all agents.
\label{sec:mean}
\vspace{-1mm}\subsection{Random Participation}
Assume that $f$ is a symmetric distribution, so that $\mathbf{mn}^Q(S_N)$ is unbiased under all scenarios.
When we apply the Mean mechanism, the loss in the basic scenario is simply the sample variance.
\begin{proposition}
Let $f$ be a symmetric, weakly single-peaked distribution, and suppose $|N|=2$. Then, for any $S_N$, $\|\mathbf{mn}^P(S_N)-x^*\| \leq \|\mathbf{mn}^B(S_N)-x^*\| $. That is, for any pair of agents the proxy-weighted mean is weakly better than the unweighted mean.
\end{proposition}
\rmr{We need this proposition in the strategic section to conclude $\mathcal{L}^{P+L} \leq \mathcal{L}^{P} < \mathcal{L}^{B} = \mathcal{L}^{B+L}$.}
\begin{proof}
Suppose w.l.o.g. that the support of $f$ is $[-1,1]$, that $f$ is symmetric around $x^*=0$, that $s_1<s_2$, and that $x=\mathbf{mn}(S_N) = \frac{s_1+s_2}{2} \geq 0$.
Then for the basic (unweighted) scenario,
$$\|\mathbf{mn}^B(S_N)-x^*\| =|\mathbf{mn}^B(S_N)|=|\mathbf{mn}(S_N)|= |x|=x.$$
Since $f(\cdot)$ in single-peaked, the CDF $F(\cdot)$ is convex in $[-1,0]$ and concave in $[0,1]$, thus for all $z\geq 0$, $F(z)\geq\frac{z+1}{2}$. In particular $F(x) \geq \frac{x+1}{2}$.
In the proxy (weighted) scenario, agent~1 gets all voters below point $x$, i.e. $w_1=F(x)$, whereas $w_2=1-F(x)$. Thus
\begin{small}
\begin{align*}
&\mathbf{mn}(S_N,\vec w) = w_1 s_1 + w_2 s_2 = F(\hat x) s_1 + (1-F(\hat x))s_2\\
&= s_2+F(\hat x)(s_1-s_2) \leq s_2 + \frac{\hat x+1}{2}(s_1-s_2) \tag{as $s_1-s_2<0$}\\
&= \frac{s_1+s_2}{2} + \hat x\frac{s_1-s_2}{2} = \hat x+ \hat x\frac{s_1-s_2}{2} \\
&= \hat x(1+\frac{s_1-s_2}{2})\in [-\hat x,\hat x].~~\Rightarrow \tag{since $-1 \leq \frac{s_1-s_2}{2} < 0$}\\
&\|\mathbf{mn}^P(S_N)-x^*\| = |\mathbf{mn}(S_N,\vec w)| \leq |\hat x| = \|\mathbf{mn}^B(S_N)-x^*\|,
\end{align*}
\end{small}
as required.
\end{proof}
\cut{
\begin{proof}[Proof sketch]
Suppose w.l.o.g. that the support of $f$ is $[-1,1]$, that $f$ is symmetric around $x^*=0$, that $s_1<s_2$, and that $\hat x=\mathbf{mn}(S_N) = \frac{s_1+s_2}{2} \geq 0$.
Then for the basic (unweighted) scenario,
$\|\mathbf{mn}^B(S_N)-x^*\| =|\mathbf{mn}^B(S_N)|=|\mathbf{mn}(S_N)|= |\hat x|=\hat x$.
Since $f$ in single-peaked, $F$ is convex in $[-1,0]$ and concave in $[0,1]$, thus for all $z\geq 0$, $F(z)\geq\frac{z+1}{2}$. In particular $F(\hat x) \geq \frac{\hat x+1}{2}$.
In the proxy (weighted) scenario, agent~1 gets all voters below point $\hat x$, i.e. $w_1=F(\hat x)$, whereas $w_2=1-F(\hat x)$. We can compute the weights and show that $\mathbf{mn}(S_N,\vec w) =(1+\frac{s_1-s_2}{2})\in [-x,x]$.
This means that $\|\mathbf{mn}^P(S_N)-x^*\| = |\mathbf{mn}(S_N,\vec w)| \leq |\hat x|$, i.e. weakly better than $\mathbf{mn}^B(S_N)$.
\end{proof}
\rmr{can shorten to a proof sketch}
For larger sets of agents this is not true in general. Even for the Uniform distribution there are examples with more agents where proxy voting leads to a less accurate outcome:
\begin{center}
\begin{tikzpicture}[scale=0.8,transform shape]
\def0.08{0.05}
\draw[thick] (0,0) -- (10,0);
\draw[fill] (2.5,0.1) circle [radius=0.08];
\draw[fill] (2.5,0.25) circle [radius=0.08];
\draw[fill] (10,0.1) circle [radius=0.08];
\node [above] at (0,0.2) {$0$};
\node [above] at (10,0.2) {$1$};
\node [above] at (4.5,0.0) {$\mathbf{mn}(S_N)=\frac12$};
\node at (5,0.05) {$*$};
\node [below] at (6.5,0.05) {$\mathbf{mn}(S_N,\vec w_N)=\frac{17}{32}$};
\node at ( 5.3125,-0.08) {$*$};
\end{tikzpicture}
\end{center}
Consider 3 agents on ${\mathcal X}=[0,1]$, located at $S_N=\{\frac{1}{4},\frac{1}{4},1\} $. For a Uniform distribution $f$, the optimal outcome is $ x^* = \mathbf{mn}(f)=\frac{1}{2}$. In the Basic scenario, $\mathbf{mn}^B(S_N)= \frac{1}{2}$ while with proxies,\\
$\mathbf{mn}^P(S_N) = \mathbf{mn}^P(S_N,\vec w_N=\{ \frac58,0,\frac38 \}) = \frac{1}{4}\frac58+1\frac38=\frac{17}{32} $.
The question is under which
distributions $f$ the loss is improved \emph{on average} by weighing the samples. We show analytically that this holds for uniform distributions and provide similar simulation results for other distributions.
\vspace{-3mm}\paragraph{Uniform distribution}
Consider the uniform distribution over
the interval $[-1,1]$ (w.l.o.g., as we can always rescale).
In the Basic scenario, we know from \citep{arnold1992first} that $\mathcal{L}^{B}(n)=\mathbb{V}[\mathbf{mn}(S_{N})]=\frac{1}{3n}$.
The next proposition indicates that the loss under proxy voting decreases quadratically faster than without proxies (as with the median mechanism).
\begin{proposition} \label{th:w_mean_var}
For the Mean mechanism, when $f=U[-1,1]$,\\
$\mathcal{L}^P(S_N) =\frac{8}{n^{2}}(1+O(\frac{1}{n}))$.
\end{proposition}
\begin{proof}[Proof]
We first note that the weighted mean $\mathbf{mn}\left(S_{N},\mathbf{w}_N\right)$,
is an unbiased estimator of the distribution mean from symmetry argument, and therefore $\mathcal{L}^P(S_N) = \mathbb{V}[\mathbf{mn}(S_{N},\vec{w}_N)]$.
We now turn to evaluate this term. $\mathbb{V}[\mathbf{mn}(S_{N})]=\mathbb{E}\left[\left(\mathbf{mn}(S_{N})\right)^{2}\right]$.
\[ \mathbf{mn}\left(S_{N},\mathbf{w}_N\right)=\frac{1}{n}\sum_{j=1}^{n}w_{j}s_{j} \]
Here $w_{j}$ is the number of voters that elect representative
$j$ as their proxy. Since the number of vote $n$ is large, $w_{\alpha}$
is the corresponding share of the probability distribution,
In the Uniform distribution $U[-1,1]$ we can compute the weights:
\begin{align*}
w_{j}&=F\left(\frac{s_{j+1}+s_{j}}{2}\right)-F\left(\frac{s_{j-1}+s_{j}}{2}\right)\\
&=\frac12 (\frac{s_{j+1}+s_{j}}{2}-(-1)) + \frac12(\frac{s_{j-1}+s_{j}}{2}-(-1))=
\frac{1}{4}\left(s_{j+1}-s_{j-1}\right)
\end{align*}
where we set $s_{0}=-2-s_{1}$, $s_{n+1}=2-s_{n}$ for convenience.
Therefore $\mathbf{mn}\left(S_{N},\mathbf{w}_N\right)$ can be written as
\labeq{w_avg}
{\sum_{i=1}^{n}w_{j}s_{j} = \frac{1}{4} \sum s_{j}\left(s_{j+1}-s_{j-1}\right)
= \frac{s_{n}+s_{1}}{2}+\frac{s_{1}^{2}-s_{n}^{2}}{4}
}
by telescopic cancellation. Here $s_{n}$ and $s_{1}$ are the two
extremes representatives. Now, since the joint distribution of $(s_{1},s_{n})$
is explicitly known~\citep{arnold1992first},
\[
\Pr\left(s_{1}=x,s_{n}=y\right)=n\cdot\left(n-1\right)\cdot\left(\frac{1}{2}\right)^{2}\cdot\left(\frac{y-x}{2}\right)^{n-2}
\]
it is possible to evaluate it precisely by integration. We get \rmr{I divided Eli's result by 4. hope it's ok}
\begin{align*}
& \mathbb{E}\left[\mathbf{mn}\left(S_{N},\mathbf{w}\right)^{2}\right] \\
&=\frac{2((n-5)n+14)n\cdot\left(n-1\right)}{\prod_{t=1}^6(n+t)}\\
&= \frac{2}{n^{2}}\left(1+O(n^{-1})\right). \\
&= \frac{2}{n^{2}}\left(1+O(\frac{1}{n})\right) .
\end{align*}
We should note that the estimator $\frac{s_1+s_n}{2}$ is known to minimize the MSE for the uniform distribution. It is interesting that the estimator obtained by proxy voting is so similar.
\rmr{The bound in the proposition is obtained by rescaling}
\end{proof}
\cut{
Recall that $\frac{s_1+s_n}{2}$ is the maximum likelihood estimator of $\mathbb{E}[f]=\mathbf{mn}(f)$ for the uniform distribution, and
$\mathbb{V}\left[s_{1}+s_{n}\right] =\frac{2}{n^{2}}\left(1+O(\frac{1}{n})\right)$, which means that $\mathcal{L^P}/\mathcal{L^B}\rightarrow 4$.}
While proxy voting may have adverse effect on the mean in specific samples, our proof shows that \emph{on average}, proxy voting leads to a substantial gain under the Uniform distribution. Other common distributions displayed the same effect. Fig.~\ref{fig:loss_mn} shows proxy voting leads to a substantial improvement over the unweighted mean of active voters for various distributions.
\begin{figure}[t]
\centering
\includegraphics[scale=0.42]{loss_mean.pdf}
\protect\caption{\label{fig:loss_mn}The ratio of $\mathcal{L}^B(n)$ and $\mathcal{L}^P(n)$ (in log scale) as a function of $n$
\vspace{-4mm}}
\end{figure}
\vspace{-1mm}\subsection{Strategic participation}
In the basic (non-proxy) scenario, it is easy to see that every voter is always pivotal with any active set unless $s_i=\mathbf{mn}(M)$. Thus in every equilibrium $M\subseteq N$, $\mathbf{mn}(S_M)=\mathbf{mn}(S_N)$, and for any distribution $f$, and $\mathcal{L}^{B+L}(n)=\mathcal{L}^B(n)$.
In the proxy setting things get more involved. The following lemma analyzes the best response of agents in cases where the voter's population is monotonic is some region.
\begin{lemma}
\label{lem:mean-strategic}
%
(A) It is a dominant strategy for both $\argmin_i\left\{ S_{N}\right\} $ and $\argmax_i\left\{ S_{N}\right\}$ to be active;
(B) Consider three agents, $s_{1}<s_{2}<s_{3}$ s.t. $\{1,3\}\subseteq M$. Suppose $f$ is strictly decreasing in $[s_1,s_3]$. Agent~2 prefers to be active if $\mathbf{mn}\left(S_{M \cup \{s_2\}}\right)<s_{2}$, and prefers to be inactive if $\mathbf{mn}\left(S_{M \setminus \{s_2\}}\right)\geq s_{2}$. The reverse condition applies for increasing $f$. If $f$ is constant, agent~2 always prefer to be inactive.
\end{lemma}
\begin{proof}
(A) is obvious.
For (B), consider a set of active agents $M^{-}$ such that $\{s_{1},s_{3}\}\subseteq M^{-}$
and $s_{2}\notin M^{-}$. Define $M^{+}=M^{-}\cup\{s_{2}\}$.
The population decision boundary
points are the intermediate points between the different agents $\alpha=\left(s_{1}+s_{2}\right)/2$, $\gamma=\left(s_{2}+s_{3}\right)/2$ and $\beta=\left(s_{1}+s_{3}\right)/2$. The result of the decision mechanism is denoted as $g_{M^{-}}$,
or correspondingly, $g_{M^{+}}$. We note that $\mathbf{mn}\left(S_{M^{+}}\right)-\mathbf{mn}\left(S_{M^{-}}\right)$ equals\\ $\left(s_{2}-s_{1}\right)\int_{\alpha}^{\beta} f(x)d(x)+\left(s_{2}-s_{3}\right)\int_{\beta}^{\gamma} f(x)d(x)
$.
By the intermediate value theorem, there exists a point $x_{1}\in[\alpha,\beta]$
such that
$\int_{\alpha}^{\beta}f(x)d(x) = \left(\beta-\alpha \right)f(x_{1})
= \frac{(s_{3}-s_{2})f(x_{1})}{2}$.
Similarly, there is $x_{2}\in[\beta,\gamma]$ such that \\
$
\int_{\beta}^{\gamma}f(x)d(x)=\frac{(s_{2}-s_{1})f(x_{2})}{2}
$.
Therefore,
\begin{small}
\begin{equation}
\mathbf{mn}(\! S_{M^{+}}\!)-\mathbf{mn}(\! S_{M^{-}}\! )=\frac{\left(s_{2}\!-\!s_{1}\right)\left(s_{3}\!-\!s_{2}\right)\left(f(x_{1})\!-\!f(x_{2})\!\right)}{2}\label{eq:condition}
\end{equation}
\end{small}
This expression is positive if $f\left(x_{1}\right)>f\left(x_{2}\right)$.
If $f$ is monotonic decreasing in $[s_{1},s_{3}]$ this holds, while
if $f$ is monotonic increasing we have $\mathbf{mn}\left(S_{M^{+}}\right)<\mathbf{mn}\left(S_{M^{-}}\right).$
\if 0
A) We prove for $s_{n}$. For every set $M$ , $\mathbf{mn}\left(S_{M^{-}}\right)\leq\mathbf{mn}\left(S_{M^{+}}\right)\leq s_{n}$,
where equality holds only if $M^{-}=\emptyset$ or there exists $s_{i}$
such that $s_{i}\in M^{-}$ and $s_{i}=s_{n}$. In the former case
it is beneficial for agent~$n$ to become active and the latter
case happens with zero probability for continuous probability distribution.
Therefore, it is beneficial for agent~$n$ to become active, as
\[
\left\Vert \mathbf{mn}\left(S_{M^{+}}\right)-s_{n}\right\Vert >\left\Vert \mathbf{mn}\left(S_{M^{-}}\right)-s_{n}\right\Vert
\]
\fi
If $\mathbf{mn}\left(S_{M^{-}}\right)<s_{2}$ and $f$ is increasing, it is not beneficial for $s_{2}$ to become active. Likewise, if $\mathbf{mn}\left(S_{M^{-}}\right)>s_{2}$
and $f$ is monotonic decreasing $s_{2}$ will not be active.
Finally, if $f$ is constant, then $f(x_1)=f(x_2)$ and agent $s_2$ does not affect the result and will be inactive.
\end{proof}
Before considering general probability distributions, we apply the
previous lemma for the particular case of the uniform distribution.
We show that even when the voters are strategic, the result equilibrium
is the optimal configuration.
\begin{proposition}\label{prop:uni_mean_eq}
In the Proxy scenario, for the Uniform distribution and any set of agents $N$, there is a unique equilibrium of $\mathbf{mn}^{P+L}$ where $M=\{\argmin S_N,\argmax S_N\}$ (i.e., the two extreme agents). Further, the game is weakly acyclic, i.e. there is a sequence of best replies from any initial state to this equilibrium.
\end{proposition}
\begin{proof}
Lemma~\ref{lem:mean-strategic}(A) says it is beneficial
that the two most extreme agents to be active. Due to Part~(B), all other agents will quit.
\end{proof}
Our last result for uniform distributions shows that strategic behavior, despite lowering the number of active agents, leads to a more accurate outcome than in the non-strategic case. In fact, it can be shown that \emph{no other estimator} outperforms $\mathbf{mn}^{P+L}$ for the Uniform distribution.
\begin{corollary}
For the Mean mechanism, for any sample $S_N$, under the unique equilibrium of $\mathbf{mn}^{P+L}$ for Uniform $f$,
$\mathbf{mn}^{P+L}(S_N) = \mathbf{mn}^P(S_N)$. In particular, $\mathcal{L}^{P+L}(n)= \mathcal{L}^{P}(n)$.
\end{corollary}
\cut{
\begin{proof}
w.l.o.g. $f=U[-1,1]$.
For any sample $S_N$, let $S_M=\{s_1,s_n\}$ contain the two extreme samples. Let $\vec w_M= (w^*_1,w^*_n)$ denote the weights of these samples under proxy voting, when there are no other agents.
We have that
\begin{align*}
\mathbf{mn}^{P+L}(S_N)&= \mathbf{mn}(S_M,\vec w_M)= \frac12 (s_1 w^*_1 + s_n w^*_n) \\
&= \frac{1}{2}\ensuremath{\left(s_{1}\ensuremath{\left(\frac{s_{1}+s_{n}}{2}+1\right)}+s_{n}\ensuremath{\left(1-\frac{s_{1}+s_{n}}{2}\right)}\right)}
= \frac{s_{1}+s_{n}}{2}+\ensuremath{\frac{s_{1}^{2}-s_{n}^{2}}{4}}\\
&= \mathbf{mn}(S_N,\vec w_N) = \mathbf{mn}^{P}(S_N), \tag{by Eq.~\eqref{eq:w_avg}}
\end{align*}
as required.
\end{proof}
}
\begin{figure}
\begin{centering}
\includegraphics[width=1\columnwidth]{singlePeak.png}\protect\caption{\label{fig:single-peak-and-dip}a) An equitable partition of a a single
peaked distribution. b) A feasible equilibrium in a single dip scenario.
Note that in both cases the distribution is not necessarily symmetric.}
\par\end{centering}
\end{figure}
\medskip
We now turn to analyze more general distributions, and we first focus on the single peak case. Denote the
peak location as $x$, the smallest agent in $[x,\infty]$ as $s_{+}$
and the largest agent in $[-\infty,x]$ as $s_{-}$. Set $\alpha=\left(s_{+}+s_{-}\right)/2$ as the intermediate point between $s_{+}$ and $s_{-}$. Assume, w.l.o.g, that $f\left(s_{-}\right)\geq f\left(s_{+}\right)$. We call a given set of agents $M$ an \emph{equitable partition} if $\mathbf{mn}^P(S_M)\in[s_-,s_+]$ and $f(A)\geq f(s_{+})$ (Fig. \ref{fig:single-peak-and-dip}(a)).
\begin{proposition}
Consider a single peaked distribution $f$ and a profile $S_{N}$. If $S_{N}$ is an equitable partition, then there is an equilibrium of $\mathbf{mn}^{P+L}$ where all agents
are active $M=N$. In particular, the error is the same as in $\mathbf{mn}^{P}$.
\end{proposition}
\begin{proof}
Consider the set $M=N$. Following Eq.~\ref{eq:condition},
$s_{+}$ will not quit from the active set if
$\left(s_{2}-s_{1}\right)\left(s_{3}-s_{2}\right)\left(f(x_{1})-f(x_{2})\right)>0$.
We shall now show that $f(x_{1})>f(s_{+})$. Assume $A<x$. As $f(A)\geq f(s_{+})$,
for every $y\in[A,x]$ we have $f(y)>f(A)\geq f(s_{+})$ as $f$ is
increasing in $[A,x]$. Likewise, $f(y)>f(s_{+})$ as $f$ is decreasing
in $[x,s_{+}]$ therefore $f(x_{1})$, the mean value of $f$ in $[A,s_{+}]$
satisfies $f(x_{1})>f(s_{+})$. Now, $f(s_{+})>f(x_{2})$ as $f$
is monotonic decreasing in $[x,\infty]$. Therefore, the former expression
is positive, and $s_{+}$ will stay in the equilibrium set.
Now, Lemma \ref{lem:mean-strategic}(A) shows that the most extreme
agents $s_{1},s_{n}$ will be active, while Lemma\ref{lem:mean-strategic}
(B) shows every agent between $[s_{+},s_{n}]$ and $[s_{1},s_{-}]$
are also active. Namely, all the agents are active.
\end{proof}
This shows that proxy voting may achieve maximal participation in a single peak setup. Next, we address the single dip setting.
\begin{proposition}
Consider a single dipped distribution where the dip location is $x$. Consider
any equilibrium $M\subseteq N$, and assume w.l.o.g that
$\mathbf{mn}\left(S_{M}\right)\leq x$. Then, $M$ contains at
most two agents in $[\min\left(S_{N}\right),\mathbf{mn}\left(S_{M}\right)]$
and at most two agents in $[\mathbf{mn}\left(S_{M}\right),\max\left(S_{N}\right)]$.\end{proposition}
\cut{
\begin{proof}
Lemma \ref{lem:mean-strategic}(A) shows that the two most extreme
agents $s_{1},s_{n}$ are always active. Denote the dip
location as $x$. Consider some active agents set $M$, and assume $\mathbf{mn}\left(S_{M}\right)\leq x$.
Lemma \ref{lem:mean-strategic}(B) shows that there can not be more
than two agent in $[x,s_{n}]$ and $[s_{1},\mathbf{mn}\left(S_{M}\right)]$.
Consider an equilibrium set that contain active agents in $\mathcal{A=}[\mathbf{mn}\left(S_{M}\right),x]$.
Denote the maximal active agent in $\mathcal{A}$ as $y$. Then Lemma \ref{lem:mean-strategic}(B) indicates that all agents in $[\mathbf{mn}\left(S_{M}\right),y]$
are active, while there is only one active agent in $[s_{1},\mathbf{mn}\left(S_{M}\right)]$,
which is $s_{1}.$
If there are no active agent in $\mathcal{A}$, then Lemma \ref{lem:mean-strategic}(B) show that are at most two agents in $[s_{1},\mathbf{mn}\left(S_{M}\right)]$
and in $[s,s_{n}]$.
\end{proof}
}
We see the possible emergence of four active agents, or parties, at
the center-right, center-left, extreme right and extreme left. If
the distribution is heavily skewed, we expect some parties
to emerge between the dip location and the decision rule, balancing
the result.
\section{Binary Issues}
\label{sec:binary}
In this section $\mathcal X = \{0,1\}^k$ and $\mathbf{mj}(S)$ outputs a binary vector according to the majority on each issue.
In the most general case, $f$ can be an arbitrary distribution over $\{0,1\}^k$. However, we assume that issues are conditionally independent in the following way: first a number $P$ is drawn from a distribution $h$ over $[0,1]$, and then the position on each issue is `1' w.p. $P$. That is, the position of a voter on all issues is $(s^{(1)},\ldots,s^{(k)})$, where $s^{(j)}$ are random variables sampled i.i.d from a Bernoulli distribution $Ber(P)$, and $P$ is a random variable sampled from $h$. Since $h$ induces $f$ we sometimes use them interchangeably.
\vspace{-3mm}\paragraph{Evaluation}
W.l.o.g. denote the majority opinion on each issue as 0, meaning that $x^* = \mathbf{mj}(f) = (0,0,\ldots,0)$. The expected rate of `1' opinions is $\mu\equiv \mathbb{E}_{P\sim h}[P]< 0.5$.
One interpretation of this model is that $\vec{0}$ is the ground truth, and $P_i$ is the probability that agent $i$ is wrong at any issue. Under this interpretation $1{-}\mu$ is the signal strength that the population has on the truth. In the lack of ground truth, the majority opinion is considered optimal. Here $P_i$ is the probability that agent~$i$ disagrees with the majority at each issue.
The error of a given outcome $z\in \{0,1\}^k$ is then $\|z-x^*\|=\sum_{j=1}^k z^{(j)}$ (coincides with the Hamming distance between $z$ and $x^*$). The loss is the expected error over samples as before.
We argue that when society has \emph{limited information} (small sample size $n$ and high mistakes probability $\mu$), then scenario $P$ does better than scenario $B$, i.e. $\mathcal{L}^{P}(n) < \mathcal{L}^{B}(n)$.
\vspace{-1mm}\subsection{Random participation}
Suppose that each agent is wrong w.p. \emph{exactly} $\mu<0.5$, i.e. $t_i\sim Ber(\mu)$ is the opinion of agent~$i$ on a particular issue. Then the probability that the majority is wrong on this issue is $\Pr(\sum_i t_i > \frac{n}{2})$, as stated by the \emph{Condorcet Jury Theorem}. In our case, $t_i \sim Ber(P_i)$, where $P_i$ differs among agents, and this case of independent heterogeneous variables was covered in \citep{grofman1983thirteen}, which showed:
\labeq{grofman}
{
\Pr(\sum_i t_i > \frac{n}{2}) = \Pr(Z_{\mu,n} > \frac{n}{2}),
}
where $Z_{\mu,n}\sim Bin(\mu,n)$ and $\mu=\frac{1}{n}\sum_i P_i$. Since the loss is additive along issues, $\mathcal{L}^B(n) = k\Pr(Z_{\mu,n} > \frac{n}{2})$.
We now turn to analyze the Proxy scenario.
Assume w.l.o.g. that $P_1,P_2,\ldots,P_n$ are sorted in increasing order.
As $k$ increases, $P_i$ provides a good prediction of how many 1's and 0's will be in $s_i$. This enables us to predict how inactive agents will select their proxies: an agent with parameter $P_i<0.5$ will almost always select agent~$1$ and an agent with $P_i>0.5$ will select agent~$n$ w.h.p.
\begin{lemma} \label{lemma:dictator} For every position $z<0.5$, $\Pr(\exists j\in N \text{ s.t. } \|s_j-z\| < \|s_{1}-z\| ) < n\cdot e^{-bk}$. for some constant $b$. The same holds for $z>0.5$ and $s_n$.
\end{lemma}
\begin{proof}
Note that $P_{1} < P_j$ for all $j>1$. In addition, we denote a topic disagreement indicator $I_{i,j}^{(l)} =\ind{s_i^{(l)} \ne s_j^{(l)}}$. For each agent $i$ with $P_i <0.5, \forall j > 1$,
\begin{align*}
&\Pr(\|s_i-s_j\| < \|s_i-s_{[1]}\|) = \Pr(\sum_{l=1}^{k}I_{i,j}^{(l)} < \sum_{l=1}^{k}I_{i,1}^{(l)} ) \\
&= \Pr(\sum_{l=1}^{k}I_{i,j}^{(l)} - \sum_{l=1}^{k}I_{i,1}^{(l)} < 0 ) \\
\end{align*}
define
\begin{align*}
&q_1 = P_i(1-P_{1})+(1-P_i)P_{1}\\
&q_2 = P_i(1-P_j)+(1-P_i)P_j
\end{align*}
Since $P_i<0.5, P_{1} > P_j \Rightarrow q_1 > q_2$
\begin{align*}
&X_1 = \sum_{l=1}^{k}I_{i,1}^{(l)} \sim Binomial(k,q_1) \\
&X_2 = \sum_{l=1}^{k}I_{i,j}^{(l)} \sim Binomial(k,q_2) \\
&\Pr(\sum_{l=1}^{k}I_{i,1}^{(l)}< \sum_{l=1}^{k}I_{i,j}^{(l)} ) = \Pr(X_1 - X_2 <0 )\\
\end{align*}
Since $k \to \infty $ and $q_1,q_2$ are constants, a normal approximation to binomial distribution will be sufficiently accurate for our purpose.
\begin{align*}
&X_1 \approx Z_1 \sim N(kq_1,kq_1(1-q_1)) \\
&X_2 \approx Z_2 \sim N(kq_2,kq_2(1-q_2)) \\
& (Z_1 - Z_2) \sim N(k(q_1-q_2),k(q_1(1-q_1)+q_2(1-q_2))) \\
& \Pr(X_1 -X_2 < 0 ) \approx \Pr(Z_1 -Z_2 < 0 ) \\
&= \Phi(\dfrac{0 - k(q_1-q_2)}{\sqrt{k(q_1(1-q_1)+q_2(1-q_2))}}) \\
&=\Phi(\dfrac{\sqrt{k}(q_1-q_2)}{q_1(1-q_1)+q_2(1-q_2)})= \Phi(-a\cdot \sqrt k) \\% \xrightarrow[k \to \infty]{}\Phi(-\infty) = 0 $$
\end{align*}
for some positive constant $a$. Note that for $x<-1$, $\Phi(x) < O(e^{-\frac{x^2}{2}})$, thus $\Pr(X_1>X_2) < e^{-bk}$ for some constant $b>0$. By the union bound, $\Pr\left(\exists j\in M \text{ s.t. } \|s_j-z\| < \|s_{1}-z\| \right) \leq (m-1)Pr(X_1>X_2) < me^{-bk}$.
\end{proof}
This means that when there are many issues, all voters with $z<0.5$ will cast their votes to agent~1, thus $w_1=\Pr_{z\sim h}(z<0.5), w_n=\Pr_{z\sim h}(z>0.5)$. Hence one of the agents $\{1,n\}$ \textbf{is effectively a dictator}, depending on whether the median of $h$ is below or above $0.5$. From now on we will assume that agent~1 is the dictator, as this occurs with high probability as $k\rightarrow \infty$ under most distributions with $\mu<0.5$. Thus (for sufficiently large $k$),
\labeq{dictator} {
\lVert \mathbf{mj}^P(S_N)-x^* \rVert = k\min_{i \in N}{(P_i)}=kP_1 .}
To recap, under scenario $B$ the majority mechanism is equivalent to unweighted majority of a size $n$ committee, while under scenario $P$, the mechanism is equivalent to a dictatorship of the best expert (i.e., the most conformist agent).
Given a particular distribution $h$, we can calculate $\mathcal{L}^P(n)$ analytically or numerically. E.g. when $h=U[0,a]$ (note $a=2\mu)$,
$$\mathcal{L}^{P}(n)= k\mathbb{E}_{P^n\sim U(0,a)^n}[\min_{i \in N}{P_i}] = \frac{ka}{n+1}= \frac{2\mu k}{n+1}.$$
\begin{figure}[t]
\centering
\includegraphics[scale=0.32 ]{loss_uni_p_vs_b.pdf}
\includegraphics[scale=0.32 ]{loss_normal_p_vs_b.pdf}
\protect\caption{The loss $\mathcal{L}^Q(n)$ (in log scale), for distributions $h=U[0,2\mu=0.66]$ (left); $h=N(\mu=0.33,\sigma=0.3)$ (right).
\label{fig:binary_theory}\vspace{-4mm}}
\end{figure}
We can infer from Eqs.~\eqref{eq:grofman} and \eqref{eq:dictator} that the proxy voting is beneficial in cases where the best expert out-performs the majority decision on average. Specifically, when the sample is small and/or the signal of agents is weak ($\mu$ is close to $0.5$). See Fig.~\ref{fig:binary_theory}.
\vspace{-1mm}\subsection{Strategic participation}
In general there may be multiple equilibria that are difficult to characterize, and whose outcomes $\mathbf{mj}(S_M)$ may be very different from $\vec x^*$.
However we can show that for a sufficiently high $k$, there is (w.h.p) only one equilibrium outcome in each of the mechanisms $\mathbf{mj}^{B+L},\mathbf{mj}^{P+L}$.
Intuitively, the reason is as follows. For every agent $i\in N$ there is w.h.p an issue for which she is pivotal, and thus the only equilibrium in scenario $B{+}L$ will be $M=N$ (w.h.p).
In scenario $P{+}L$, the entire weight is distributed between the \emph{active agents} with the lowest and highest $P_i$. This means that the best agent is always pivotal and thus active. Regardless of which other agents become active, we get that $\mathbf{mj}^{P+L}(S_N) = \mathbf{mj}(S_M,\vec w_M) = s_1 = \mathbf{mj}^{P}(S_N)$.
The probability that any other equilibrium exists and affects the loss goes to zero.
\paragraph{Basic setting}
For any $M\subseteq N$, denote by $Y_M$ the event that set $M$ is an equilibrium in the game $\mathbf{mj}^{B+L}(S_N)$. We bound the probability that $N$ is not the unique equilibrium.
\begin{lemma}
$\Pr(\neg Y_N \vee (\exists M\subsetneq N, Y_M)) < e^{2n-\frac{k}{2^{n}}}$. Note that for $k \gg n \cdot 2^{n+1}$ the bound tends to $0$.
\end{lemma}
\begin{proof}
For a binary vector $\vec q\in \{0,1\}^n$, we denote by $Z_{\vec q}$ the event that for some issue $j\leq k$, $q_i=s_i^{(j)}$ for all $i\in N$. We also denote $Z^* = \bigcup_{\vec q\in \{0,1\}^n}Z_{\vec q}$.
We first argue that $Z^*$ entails both $Y_N$ and $\neg Y_M$ for any $M\subsetneq N$.
Consider first the set $N$, and voter $i\in N$. If $n$ is odd consider some vector $\vec q$ where $q_i=1$ and all other voters split evenly between $0$ and $1$. Since $Z^*$ holds, there is an issue $j$ s.t. $q_{i'}=s_{i'}^{(j)}$ for all $i'\in N$. We get that $\mathbf{mj}(N)^{(j)}=1$ but $\mathbf{mj}(N \setminus \{i\})^{(j)}=0$, i.e. $i$ is pivotal and will thus not quit. If $n$ is even we proceed in a similar way except $q_i=0$ and all of $N$ split evenly between $0$ and $1$.
For any smaller set $M$, consider some $i\in N\setminus M$, where $|M|=m$. If $m$ is \emph{even} we consider a vector $\vec q$ where $q_i=1$ and and all voters \emph{in $M$} split evenly between $0$ and $1$. We get that there is an issue $j$ where $\mathbf{mj}(M)^{(j)}=0$ but $\mathbf{mj}(N \cup \{i\})^{(j)}=1$, i.e. $i$ is pivotal and will join ($M$ is not stable).
If $m$ is odd we proceed in a similar way except $q_i=0$ and all of $M \cup \{i\}$ split evenly between $0$ and $1$.
\medskip
It is left to bound $Pr(\neg Z^*)$.
Indeed, for any $\vec q$ and $j\leq k$, the probability that $\vec q= s^{(j)}$ is exactly $2^{-n}$, and thus
\begin{align*}
Pr(\neg Z^*)& \leq \sum_{\vec q}Pr(\neg Z_{\vec q}) = \sum_{\vec q \in \{0,1\}^n}\prod_{j\leq k}Pr(s^{(j)} \neq Z_{\vec q})\\
&= \sum_{\vec q \in \{0,1\}^n}\prod_{j\leq k}(1-2^{-n}) = \sum_{\vec q \in \{0,1\}^n}(1-2^{-n})^k = 2^n (1-2^{-n})^k \\
&\leq 2^n e^{-k/ 2^{n}} < e^{2n-\frac{k}{2^{n}}}.
\end{align*}
\end{proof}
Any other equilibrium occurs with negligible probability, and has a bounded effect on the loss.
\begin{corollary}
As $k\rightarrow \infty$, the probability that $N$ is the unique equilibrium of $\mathbf{mj}^{B+L}(S_N)$ tends to $1$.
In particular, $|\mathcal{L}^{B+L}(n)- \mathcal{L}^B(n)| \stackrel{k \rightarrow \infty}{\rightarrow} 0$.
\end{corollary}
\rmr{The bound on $k$ is very lax, but this is the simplest proof}
\paragraph{Proxy voting}
From Lemma~\ref{lemma:dictator}, we know that for every set $M\subseteq N$, the most extreme voter $j=1$ gets the votes of all inactive voters with $P_i<0.5$, and in particular is pivotal (w.h.p., as $k$ is large enough). Thus voter~1 is active in any equilibrium, and is in fact a dictator as in the non-strategic scenario.
Finally, since we assume that the median of $h$ is less than $0.5$, $j=1$ is a dictator. As no other voter in $N$ is pivotal on any issue, they all become inactive. Thus under the same assumptions of Lemma~\ref{lemma:dictator}:
\begin{corollary}
As $k\rightarrow 0$, the probability that $M=\{1\}$ is the unique equilibrium of $\mathbf{mj}^{S}(S_N)$ tends to $1$.
In particular, $|\mathcal{L}^{P+L}(n)- \mathcal{L}^P(n)| \stackrel{k \rightarrow \infty}{\rightarrow} 0$.
\end{corollary}
\rmr{here we need a much smaller $k$ than in the B+L scenario}
\vspace{-1mm}\subsection{Empirical Evaluation}
We evaluate proxy voting on real data to avoid two unrealistic assumptions in our theoretical model: that the number of issues $k$ is very large, and that $i$'s votes on all issues are i.i.d.
We examine several data sets from PrefLib~\citep{mattei2013preflib}:\rmr{do not remove citation} The first few datasets are Approval ballots of French presidential 2002 elections over 16 candidates in several regions (ED-26). We treat each candidate is an ``issue'' and each voter can either agree with the issue (approve this candidate) or disagree. $P_i$ is the fraction of issues on which voter~$i$ disagrees with the majority.
We also considered two datasets of ordinal preferences: sushi preferences (ED-14) and AGH course selection (AD-9).
The translation to a binary matrix is by checking for each pair of alternatives $(\alpha,\beta)$ whether $\alpha$ is preferred over $\beta$. This leaves us with 45 and 36 binary issues in the sushi and AGH datasets, respectively.\footnote{Note that Hamming distance between agents' positions equals the Kendal-Tau distance between their ordinal preferences.}
A subset of $k=15$ issues were sampled at each iteration in order to get results that are more robust (we thus get a ``sushi distribution'' and ``AGH distribution'' instead of a single dataset).
\medskip
We first consider the weight distribution among agents (Fig.~\ref{fig:binary_sushi}). The weight of agents is decreasing in $P_i$, meaning that agents with higher agreement with the majority opinion gets more followers, with the best agents getting a significantly higher weight. This is related to the theoretical result that the best expert get $>0.5$ weight, but is much less extreme. Also there is no weight concentration on the worst agent (this can be explained by the `Anna Karenina principle',\footnote{``Happy families are all alike; every unhappy family is unhappy in its own way''~\citep{tolstoy}.} as each bad agent errs on different issues). In other words, allowing proxies does not result in a dictatorship of the best active agent, but in meritocracy of the better active agents.
This leads us to expect better performance than the theoretical prediction when comparing Proxy voting to the Basic setting. Indeed, Fig.~\ref{fig:binary_sushi} (right) and Fig.~\ref{fig:binary_all} show that in all datasets $\mathcal{L}^P(n) < \mathcal{L}^B(n)$ except for very small samples in the French election datasets. This gap increases quickly with the sample size.
\begin{figure}[t]
\centering
\includegraphics[scale=0.32 ]{Sushi_BinaryCube_weight_per_rank_m51.pdf}
~~~
\includegraphics[scale=0.35 ]{Sushi_BinaryCube_Loss_m_5_500.pdf}
\protect\caption{On the left, the average weight of each agent, in increasing order of $P_i$ (best agent on the left). On the right, the loss with (red) and without (blue) proxies. Results for the other datasets were similar.
\label{fig:binary_sushi
}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[scale=0.45 ]{loss_all_binary.pdf}
\protect\caption{The ratio of $\mathcal{L}^B(n)$ and $\mathcal{L}^P(n)$ (in log scale) for all datasets. Each point based on 1000 samples.
\label{fig:binary_all}\vspace{-4mm}}
\end{figure}
\section{Discussion and Related Work}
\label{sec:discussion}
\def\vphantom{$2^{2^2}$}{\vphantom{$2^{2^2}$}}
\begin{table*}[t]
\begin{center}
\begin{tabular}{c||c|c|c}
\hline
Rule: & Median & Mean & Majority ($k\rightarrow \infty$) \\
& & & with many issues \\
\hline
\hline
Proxy better & Yes & $f$ SP + symmetric & No \\
for any $S_N$& & $n=2$ & \\
\hline
$\mathcal{L}^P<\mathcal{L}^B$ \vphantom{$2^{2^2}$} & Always & $f$ Uniform & depends on the best \\
\cline{1-3}
$\mathcal{L}^P\ll \mathcal{L}^B$ \vphantom{$2^{2^2}$} & $f$ symmetric (* most $f$) & $f$ Uniform (* most $f$ ) & agent (* real data)\\
\hline
\hline
unique equilibrium & always & $f$ Uniform & always \\
of $\mathbf{g}^{P+L}$\vphantom{$2^{2^2}$} & & & \\
\hline
$\mathcal{L}^{B+L} \geq \mathcal{L}^B$ \vphantom{$2^{2^2}$} & always & always & always \\
\hline
$\mathcal{L}^{P+L} \leq \mathcal{L}^P$ \vphantom{$2^{2^2}$} & always & $f$ Uniform (* some SP $f$) & always \\
\hline
\end{tabular}
\end{center}
\protect\caption{A summary of our results. The first three lines show the effect of proxy voting when all agents are active. Results marked by (*) are obtained by simulations. The bottom lines summarize the effect of strategic voting with lazy bias. \label{tab:results} }
\end{table*}
Our results, summarized in Table~\ref{tab:results}, provide a strong support for proxy voting when agents' positions are placed on a line, especially when the Median mechanism is in use. In contrast, when positions are (binary) multi-dimensional, proxy voting might concentrate too much power in the hands of a single proxy, and increase the error. However we also showed that on actual data this rarely happens and analyzed the reasons.
These findings corroborate our hypothesis that proxy voting can improve representation across several domains. We are looking forward to study the effect of proxy voting in other domains, including common voting functions that use voters' rankings.
\rmr{connection to learning and boosting?}
Proxy voting, and our model in particular, are tightly related to the \emph{proportional representation problem}, dealing with how to select representatives from a large population. A recent paper by Skowron~\citep{skowron2015we} considers the selection of representatives who then use voting to decide on issues that affect the society. In our case, selection is random as suggested in \citep{mueller1972representativ
}, and representatives are weighted proportionally to the number of voters that pick them as proxies, as originally suggested by Tullock~\citep{tullock1967proportional}. It is interesting to note that political systems where public representatives are selected at random (``sortition'') have been applied in practice~\citep{dowlen2015political}. Our results suggest that such systems could be improved by weighting the representatives after their selection. Setting the weight \emph{proportionally} to the number of followers seems natural, but it is an open question whether there are even better ways to set these weights.
Closest to out work is a model by Green-Armytage~\citep{green2015direct}, where voters select proxies and use the Median rule to decide on each of several continuous issues. Decisions are evaluated based on their square distance from the ``optimal'' one. However even if the entire population votes, the outcome may be suboptimal, as Green-Armytage assumes people perceive their own position (as well as others' position) with some error. He then focuses on how various options for delegating one's vote may contribute to reducing her \emph{expressive loss}, i.e. the distance from her true opinion to her ballot.
In contrast, expressive losses do not play a roll in our model, where the sources of inaccuracy are small samples and/or strategic behavior.
Alger~\citep{alger2006voting} considers a model with a fixed set of political representatives on an interval (as in our model), but focuses mainly on the ideological considerations of the voters and the political implications rather than on mathematical analysis. Our very positive results on the use of proxies in the Median mechanism support Alger's conclusions, albeit under a somewhat different model of voters incentives. Alger also points out that proxy voting significantly reduces the amount of communication involved in collecting ballots on many issues.
Other models allow chains of voters who use each other as proxies~\citep{green2005direct,degrave2014resolving}, or social influence that effectively increases the weight of some voters~\citep{alon2015robust}.
Indeed, we believe that a realistic model of proxy voting would have to take into account such topological and social factors in addition to statistics and incentives. E.g., \citep{alon2015robust} shows the benefits of a bounded degree, which in our model may allow a way to bound excessive weights.
Social networks may also be a good way to capture correlations in voters' preferences~\citep{procaccia2015ranked}, and can thus be used to extend our results beyond independent voters.
\vspace{-3mm}\paragraph{Strategic behavior}
We showed that most of our results hold when participation is strategic. What if voters (either active or inactive) could mis-report their position? Note that inactive voters have no reason to lie under the Median and the Majority mechanisms, due to standard strategyproofness properties. However active agents may be able to affect the outcome by changing the partition of followers.
\rmr{Another variation is to consider variable costs for proxy voting, which may depend on the location of the voter, her distance from her proxy etc.}
We can also consider more nuanced strategic behavior, for example where an agent also cares about her number of followers regardless of the outcome. More generally, strategic considerations under proxy voting combine challenges from strategic voting with those of strategic candidacy~\citep{hotelling29,dutta2001strategic}, and would require a careful review of the assumptions of each model.
Other open questions include the effect of proxy voting on \emph{diversity}, \emph{fairness}, and \emph{participation}. It is argued that diverse representatives often reach better outcomes~\citep{marcolino2013multi}, and fairness attracts much attention in the analysis of voting and other multiagent systems~\citep{bogomolnaia2005collective,walsh2007representing,dickerson2014computational}. The effect on participation and engagement may also be quite involved, since allowing voters to use a proxy may increase the participation level of some who would otherwise not be represented, but on the other hand may lower the incentive to vote actively, thereby reducing overall engagement of the society.
Finally, the future of proxy voting depends on the development and penetration of novel online voting tools and social apps, such as those mentioned in the Introduction. We hope that sharing of data and insights will promote research on the topic, and set new challenges for mechanism design.
\rmr{In our paper at least lazy bias indeed leads to reduced participation under proxy voting, but without increasing the error. In more complex models this may no longer hold.}
\input{Proxy_voting_Arxiv.bbl}
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train/arxiv
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BkiUejLxK7kjXIK5s2t2
| 5 | 1 |
\section{Introduction to Cold Atom Dissipative Transport}
Cold atom samples differ from electronic systems in
significant ways that offer new opportunities to investigate
transport phenomena \cite{Cheng2,Stadler2012,Brantut2012,Brantut2013,Stoof}. Cold atom systems are highly
versatile: many of them have tunable interactions, can be confined in a single potential well or in
lattices, as well as in different dimensions. Furthermore, at nanoKelvin temperatures, the dynamics
of atoms is very slow (milliseconds), in contrast to the fast dynamics (picoseconds) of electrons in materials. This allows a
detailed scrutiny of atomic motion. One powerful tool in recent cold atom experiments is \emph{in situ} imaging of atoms, which reveals
high space-time resolution images of atomic distributions in snapshots \cite{Gemelke2009,Bakr2009,Sherson2010}. In the condensed matter analogy,
this technique is equivalent to following every electron with a femtosecond temporal resolution.
In further pursuit of understanding both these dynamics and the analogy between
electrons and cold atoms, in this paper
we apply the theory of dissipative
transport (developed for electrons) to ultracold trapped atoms.
In the last section of the paper we address recent experimental data \cite{Cheng2}.
Dissipative transport
is to be contrasted with ballistic transport where there are no relevant scattering
processes to limit the transport lifetime. Thus, ballistic transport is observable
when the mean free path is long compared to the relevant dimensions of the system.
We stress that the analogy between electrons and cold atoms
requires careful consideration.
Universal transport laws developed for electrons need to be revisited when applied to bosonic atoms.
In addition, cold atom samples are trapped in a conservative potential and isolated in vacuum. The lack of
thermal and particle reservoirs means total particle and total energy are usually conserved quantities.
A statistical description based on a grand-canonical ensemble may apply only to local observables \cite{landau}.
Furthermore, atoms are neutral and the analogue of electrical current will be the particle
flow or mass flow driven, not by electric field, but by a chemical potential gradient $\nabla \mu$ or temperature gradient $\nabla T$ .
Interestingly, unlike neutral liquid Helium, cold gas superfluids allow the imposition of a non-zero $\nabla \mu$
and thus are rather uniquely amenable to the transport studies we present.
Finally, to employ dissipative, as distinguished from
ballistic transport theory, the samples should be in
the hydrodynamic regime with coherence length much
shorter than the sample and relaxation time less than the
measurement time resolution. When these criteria are satisfied
the atoms can reach a local equilibrium associated with coarse
graining the system over a proper length and time scale.
Thermodynamic
quantities such as temperature and chemical potential can therefore be defined locally, and their gradients constitute the thermodynamic forces.
This approximation, which we will call the
local equilibrium approximation (LEA), limits the resolution of both temporal and spatial measurements. It should not be confused with
the local density approximation that describes inhomogeneously trapped
atoms in equilibrium.
Since cold atoms are typically confined in conservative potentials, they are usually free from impurities and background ionic lattices, including their phononic excitations. The fact that atoms are in a clean environment, on the one hand, may simplify many-body calculations; this removes some complexity encountered in electron transport in materials. On the other hand, there
must necessarily be
a relaxation mechanism in order to achieve a steady state and apply
the fundamentals of
transport theory.
In this paper, we discuss the transport behavior of ultracold gases in the normal phase under the assumption of
the LEA, the applicability of which is described in more detail
below. We characterize transport in
two limits where theory is relatively straightforward
and well established. Thus we consider non-interacting
particles as well as (generally interacting) particles
but near superfluid condensation, emphasizing the similarities and differences between Fermi and Bose systems. When the interactions are essentially
negligible, a normal degenerate Fermi gas
in the dissipative transport regime
has universal thermoelectric properties. By contrast universality
is absent for the bosonic counterpart. As the system approaches the superfluid critical temperature, the transport tends to be dominated by condensate (superconducting and superfluid) fluctuations in which regime, the Fermi and Bose gases are more similar. It is in this latter regime where there are now (Bose gas) experiments \cite{Cheng2}. Importantly these appear to be reasonably consistent with the theory presented here.
\subsection{Transport Equations }
We define the transport coefficients in terms of the linear response of the particle current $\vect{J}_p$ and the heat current $\vect{J}_Q$ to
temperature gradient $\vect{\nabla} T$ and and chemical potential gradient $\vect\nabla \mu$ as
\begin{eqnarray}
\mathbf{J}_p & =& - L_{11} \nabla \mu ~~- L_{12} \nabla T \\
\mathbf{J}_Q &=& - L_{21} \nabla \mu ~~- L_{22} \nabla T,
\label{eq:1}
\end{eqnarray}
\noindent where $\nabla \mu$ is analogous to the electric field for a charged system. It is convenient
to introduce dimensionless ratios to write this equation in the form (We work with units such that
$\hbar=k_B=1$.)
\begin{equation} \left(\begin{array}{c} \vect{J}_p \\ \vect{J}_Q \\
\end{array}\right)=-\sigma
\left(\begin{array}{cc} 1 & \mathcal{S} \\ \mathcal{P}
& \,\,\,\,\, T\mathcal{L}+\mathcal{S}\mathcal{P} \\ \end{array} \right)
\left(\begin{array}{c} \vect{\nabla}\mu\\ \vect{\nabla}T \\ \end{array}\right),
\label{transportmatrix}
\end{equation}
\noindent where the conductivity $\sigma \equiv L_{11}$, the Seebeck coefficient (or thermopower) $\mathcal{S}=\mathcal{P}/T\equiv L_{12}/L_{11}$ and the Lorenz
number (which is given by ratio of thermal to mass conductivity, divided
by temperature) $\mathcal{L} \equiv(L_{22} L_{11}-L_{12} L_{21})/T L_{11}^2$. Note that $L_{12}$ and $L_{21}$ satisfy the
Onsager relation $L_{12}=L_{21}/T$.
\subsection{Kubo versus Boltzmann Approaches}
There are two rather complementary approaches to addressing
dissipative transport in quantum systems. One can adopt
the approach of linear response theory and apply the Kubo formula, or alternatively one can employ the Boltzmann transport equation which is based on
kinetic theory. Each has its strengths in accomodating different
physical aspects of the system at hand and the choice for
which to apply is governed by
the underlying goals.
The Kubo formalism is more suitable for imposing conservation principles and sum rules, but it does not generally build in the specifics of the dissipation
mechanism. By contrast, the Boltzmann transport equation focuses on the details of the collision processes leading to dissipation, but is not
as well suited to impose or to verify conservation principles.
For non-interacting gases treated in the Kubo formalism, the results are
essentially identical to those based on Boltzmann transport equation
at the level of simple relaxation time approximations.
In this paper, we show that Kubo based approaches are well suited to addressing transport of both non-interacting as well as
interacting atomic gases in the near condensation regime.
Since it is likely
that dissipation in the ultracold gases
is linked to the details of the experimental set up, we will introduce dissipation via a phenomenological parameterization.
The Kubo approach in this form is also suitable for delivering one of the messages of this paper: that in bosonic systems (despite strong evidence for universality in many contexts), transport is highly non-universal and depends on the details of the dissipation.
The philosophy behind our phenomenological approach to dissipation is rather similar to that articulated by Kadanoff and Martin
who in a series of papers emphasized the importance of the Kubo-based correlation functions
and their symmetries \cite{KadanoffMartin2}. In related work on superfluids \cite{KadanoffMartin}, they argued for the suitability of introducing a phenomenological parameterization of the lifetimes associated with transport.
In this paper we parameterize the relaxation time.
We introduce the quantity
$\tau(\epsilon)$ where $\epsilon$ effectively represents the energy; presumably at
low temperatures $\epsilon\to 0$.
This is
treated phenomenologically as $\tau(\epsilon)=\tau_0\epsilon^{\eta/2}$.
While $\eta$ appears to be arbitrary, in this paper we consider the two different values
chosen because they have been addressed in the solid state literature:
$\eta=-2$ and $\eta=0$.
The latter is associated with impurity scattering models \cite{Alexandrov}, while
the former is more naturally associated with strictly bosonic
transport as, for example in superconducting fluctuation theories of transport
\cite{VarlamovLarkin,Tinkham}.
Then, reflecting the odd energy or frequency dependene of the boson self energy,
to leading order (in small $\epsilon$) this constrains $\tau(\epsilon)$ and thus $\eta = -2$.
\subsection{Role of Dissipation}
Equilibration is necessary but not sufficient for establishing steady state transport. A relaxation mechanism is needed for equilibration. Such a
mechanism is also central to
many non-equilibrium studies, involving for example,
interaction quenches in the ultracold gases. This was
demonstrated \cite{ourquenchpaper} by our group and in recent experimental
studies \cite{Hung2011,Makotyn2014}.
It is, however, important to stress that
mass transport, for example, as in the particle conductivity, cannot exclusively depend on inter-particle collisions as total momentum is conserved in the presence of Galilean invariance. A well behaved transport requires that there be a source of momentum relaxation.
In cold atom experiments dissipation depends on the experimental design, and will arise in a variety of ways. It will be present in thin channel transport
and in experiments with optical lattices or speckle potentials. In addition, there are systematic \cite{Thomas} and intrinsic \cite{Dalibard} fluctuations of the optical potentials that contribute to relaxation. The presence of a harmonic trap alone, which breaks Galilean invariance, is insufficient to avoid an unphysical infinite mass conductivity if there were otherwise no source of relaxation \cite{Wu2014}.
A recent experiment by one of us \cite{Cheng2} has measured the
thermoelectric transport coefficients using three-body recombination loss and heating. These processes lead to locally measurable mass and energy flow or currents. This is in contrast to condensed matter experiments where the temperature and chemical potential (or scalar EM potential) are externally applied. With simple models one can quantify the associated $\nabla \mu$ and $\nabla T$ and thereby deduce transport coefficient ratios.
This paper aims to address transport more generally
applying to a variety of different experiments, not just to the three-body
recombination loss mechanism. However, it should be stressed that, for this
particular transport scheme, the
LEA will be demonstrated to be appropriate. Most importantly,
along with particle and energy loss, momentum relaxation is present,
through the loss of particles.
\subsection{Local Equilibrium Approximation}
Indeed, a central assumption of this paper is the applicability of
the local equilibrium approximation.
In dissipative transport theory developed for electrons in solids it
is assumed that in the presence of external fields or perturbations,
a steady state can be achieved.
This is possible when the atoms
are in the hydrodynamic regime. Here the coherence length $l_c$ is significantly
less than the system size $l_s$ and thermal relaxation time $\tau$ is much shorter
than the system lifetime $\tau_s$.
One can consider the local
chemical potential gradients $\nabla\mu(x; t)$ and temperature gradients $\nabla T(x; t)$
which are essential quantities in transport theory.
That the dynamics of an atomic sample can be
completely described by local thermodynamic variables relies on the rapid
and short-ranged scattering by an external potential, which leads to
relaxation and serves to establish a steady state.
Note that $\nabla\mu(x; t)$ and $\nabla T(x; t)$
can only be defined by coarse graining local variables over a proper length scale $l_m$ and time scale $\tau_m$. This
leads to the constraint $l_c<l_m\ll l_s$ and $\tau<\tau_m\ll\tau_s$ where the subscript $m$ represents the experimental or measurement
variables. When these inequalities hold one is in the regime of
validity of the Local Equilibrium Approximation. This
LEA is a stronger condition than a hydrodynamic approximation since it assumes a history-independent local and short-time thermodynamic equilibrium.
The LEA, which we assume to be valid throughout this study, will break down for atoms in the collisionless regime or in a smooth trapping potential, where we have $l_c\gg l_s$ and $\tau_m\gg\tau_s$, respectively.
In support of our use of the LEA is the observation that the density profiles are well
described by an equilibrium equation of state
\cite{Gemelke2009}. Also important is the fact that measurements of the coherence
times and correlation lengths (which are discussed in Section V) are consistent
with the constraints, as outlined above.
\section{Transport for non-interacting bosons and fermions}
The transport properties of weakly interacting, normal Bose and Fermi gases are strikingly
different. In a degenerate Fermi gas, it is well known that the phase space contributions to transport are confined to a narrow energy range around the Fermi energy. By contrast, for Bose gases, there is no such constraint. As a consequence, the magnitude of transport coefficients $L_{ij}$ for bosons tends to be much larger and much more sensitive to the detailed assumptions about the nature of the dissipation.
For a Hamiltonian with only one-body terms, we are able to compare the transport properties
of bosons and fermions using an exact expression based on the Kubo formula first derived for fermions in Ref.~\onlinecite{Chester1961} but readily generalized to bosons,
\begin{equation}
L_{ij}=T^{1-j}\int_0^\infty {\rm d}\epsilon\, (\epsilon-\mu)^{i+j-2}\frac{2\epsilon}{m d} \rho(\epsilon) \tau(\epsilon) b_\pm^{(1)},
\label{eq:Chester}
\end{equation}
\noindent where $i,j$=1 or 2, $b_\pm^{(1)}\equiv-\partial_{\omega} b_\pm(\omega)$, with $b_\pm(\omega)=(z_\pm^{-1}e^{\omega/T}\pm 1)^{-1}$ the Fermi/Bose distribution, and the fugacity $z_\pm$ is defined as $z_+=e^{\mu/T}$ for fermions, and $z_-=e^{(\mu-\mu_c)/T}$ for bosons below critical chemical potential $\mu<\mu_c$.
We introduce $\rho(\epsilon)=(m/2\pi)^{d/2}\epsilon^{d/2-1}/\Gamma(d/2)$ the density of states for free particles of mass $m$ in $d$ dimensions (we use units such that $\hbar=k_B=1$). In order to describe a bosonic system at positive chemical potential, one has to include the (non-dissipative) inter-boson interactions, so that
the chemical potential term is given by $\mu-\mu_c$. See for instance
\cite{Capogrosso-Sansone2010} for an estimate of $\mu_c$ in weak coupling.
The relaxation time introduced earlier
appears as
$\tau(\epsilon)$ in
Eq.~\eqref{eq:Chester}.
This parameter plays the role of the relaxation time in Boltzmann theories of transport \footnote{In Ref. \onlinecite{Chester1961}, the transport coefficients are proved to be of the form $L_{ij}=T^{1-j}\int_0^\infty {\rm d}\epsilon\,(\epsilon-\mu)^{i+j-2}G(\epsilon) b_+^{(1)}$, where $G(\epsilon)$ is an unknown function dependent of the details of the Hamiltonian. We can always rewrite $G(\epsilon)=\frac{2\epsilon}{m d} \rho(\epsilon) \tau(\epsilon)$, where $\tau(\epsilon)$ is now unknown. }.
The transport coefficients for non-interacting fermions/bosons can be derived from Eq.~(\ref{eq:Chester}) as
\begin{equation}
\begin{split}
\sigma&= \frac{\Gamma(\zeta+1)\tau(T)}{\Gamma(d/2+1)m\lambda_{\rm dB}^{d}} |{\rm Li}_{\zeta}(\mp z_\pm)|\\
\mathcal{S}&= (\zeta+1)\frac{{\rm Li}_{\zeta+1}(\mp z_\pm)}{{\rm Li}_{\zeta}(\mp z_\pm)}-\ln z_\pm\\
\mathcal{L}&= (\zeta+1)(\zeta+2)\frac{{\rm Li}_{\zeta+2}(\mp z_\pm)}{{\rm Li}_{\zeta}(\mp z_\pm)}-(\mathcal{S}+\ln z_\pm\big)^2 ,
\label{eq:SSL}
\end{split}
\end{equation}
where $\zeta=(d+\eta)/2$, $\lambda_{\rm dB}=\sqrt{\frac{2\pi}{mT}}$ is the de Brogile length, and
\begin{equation}
{\rm Li}_{\alpha}(\mp x)=\frac{\mp 1}{\Gamma(\alpha)}\int_0^\infty {\rm d}t \frac{t^{\alpha-1}}{x^{-1}e^{t}\pm 1}
\end{equation}
are the polylogarithm functions.
For degenerate fermions $z_+\gg1$, the factor $b_+^{(1)}$ is strongly peaked at the Fermi energy. Thus the energy dependence of $\tau$ is not relevant and the Lorenz number in this regime is a universal number $\mathcal{L}_f = \pi^2 /3$ (independent of $\tau$ and $\eta$), known as the
Wiedermann-Franz law. By contrast, for bosons all energies $\epsilon$ contribute to the integrals, so that $\mathcal{L}$ depends on the details of the dissipation and has a non-universal behavior.
\begin{figure}[t]
\includegraphics[width=2.6in,clip,angle=-0]
{comp_coef.eps}
\caption{
Lorenz number $\mathcal{L}$ computed from Eq. \eqref{eq:SSL} in $d=2$ with $\eta=-2$ for bosons (solid red lower line) and fermions (short-dashed
green) and with $\eta=0$ for bosons (solid blue upper line) and fermions (dashed magenta line), as function of $\ln(z_\pm)$. The dotted black
line is the Wiedermann-Franz law $\mathcal{L}=\pi^2/3$ which is reached for fermions in the degeneracy limit $z_+\gg1$. }
\label{fig_coeff}
\end{figure}
Figure \ref{fig_coeff} presents a comparison of the behavior of fermionic and bosonic transport for one particular transport coefficient ratio
(the Lorenz number) $\mathcal{L}$, for $d=2$ as a function of the log of the fugacity. The figure illustrates how different values of the dissipation
exponents ($\eta=-2$ and $\eta=0$) influence the transport. One sees that for very negative $\ln(z)$, the transport is independent of statistics
and also sensitive to the details of the dissipation. Each component of $\mathcal{L}$ is larger for the bosonic case, but this effect is
not apparent when plotted as a ratio. One sees that as the fermions cross over to the large positive fugacities, the Lorenz number approaches the universal Wiedermann-Franz law.
While the universality in the fermionic case is evident, this is clearly not the case for bosons. For Bose systems, we observe that even in the quantum
regime the behavior depends on the dissipation mechanism, implying a non-universality of the transport.
\section{Scale invariance of two-dimensional bosonic transport }
Although, experimentally it has not been
tested in Ref.
\onlinecite{Cheng2}, here we
argue that an scale invariance
should be observed in transport in a two-dimensional Bose gas \footnote{In two-dimensional gas, scale invariance is only approximate due to logarithmic corrections to the scaling.}.
This scaling
arises from physics similar to that found
in the thermodynamics of two-dimensional dilute Bose gases \cite{Hung2011,Yefsah2011}. This scale invariance reflects the
$\mu=T=0$
quantum phase
transition (QPT) between the vacuum and superfluid phase \cite{SachdevBook}.
This zero-temperature phase transition must not be confused with the finite-temperature phase transition between the normal and superfluid phase discussed in the rest of the paper.
The presence of this QCP implies that a
thermodynamic property such as the pressure is a universal function of the
form
\begin{equation}
P(\mu,T)=\frac{T^{2}}{m}\mathfrak{P}\Big(\frac{\mu}{T},\tilde g(T)\Big).
\end{equation}
Here $\mathfrak{P}(x,y)$ is a universal function (independent of the microscopic details of the system) and $\tilde g(x)$ is a renormalized interaction depending logarithmically on $x$, which
for weakly-interacting bosons can be taken as constant, which we assume in the following. (If
$\tilde g(T)$ varies significantly, the scale invariance is then lost.)
Under these circumstances
there is scale invariance, so that all thermodynamic
functions in the dilute two-dimensional Bose gas depend only on $\mu/T$ once
the interaction strength has been fixed. The scaling of the pressure is valid in the critical regime close to the QCP, that is $\mu m a^2, Tma^2\ll 1$ \cite{Rancon2012a} ($a$ is the s-wave scattering length). For dilute gases,
this corresponds to the whole $T$-$\mu$ plane in the range of $\mu$ and $T$
relevant for the experiments.
From these results at equilibrium, we can infer that due to this same QCP,
the transport coefficients also obey scaling relations. Following Ref. \cite{Fisher1990}, one shows that close to a critical point, the conductivity scale as
\begin{equation}
\sigma(\mu,T,g)=s^{2-d}\sigma\Big(\mu s^{1/\nu},T s^{z},\tilde g(s)\Big),
\end{equation}
when lengths are rescaled by a factor $1/s$, where the correlation length exponent $\nu$ and the dynamical exponent $z$ are given by $1/\nu=z=2$ at the QCP. (See Ref.~\onlinecite{Rancon2012a} for
a detailed discussion of the universal thermodynamics of Bose gases and its relation to the QCP.)
Choosing $s=T^{-1/z}$, one obtains
\begin{equation}
\sigma(\mu,T)=T^{(d-2)/z}\mathfrak{S}_d\Big(\frac{\mu}{T},\tilde g(T)\Big),
\end{equation}
where $\mathfrak{S}_d(x,y)=\sigma(x,1,y)$ is a universal function.
This reasoning is easily generalized to the other transport coefficients
\begin{equation}
\begin{split}
L_{12}(\mu,T)&=T^{(d-2)/z}\mathfrak{A}_d\Big(\frac{\mu}{T},\tilde g(T)\Big),\\
L_{22}(\mu,T)&=T^{(d+z-2)/z}\mathfrak{K}_d\Big(\frac{\mu}{T},\tilde g(T)\Big),
\end{split}
\end{equation}
and explicitly in dimension two
\begin{equation}
L_{ij}(\mu,T)=T^{i-1}\mathcal{L}_{ij}\Big(\frac{\mu}{T},\tilde g\Big),
\end{equation}
with $\mathfrak{A}_d$, $\mathfrak{K}_d$ and $\mathcal{L}_{ij}$ universal functions. Note that the scaling arguments are in principle valid only for the singular part of the transport coefficient, and one could expect the presence of a regular part, which would not scale accordingly. However, since all four transport coefficients vanish for all $\mu\leq 0$ at $T=0$ (the system is empty), we can infer that the regular parts are identically zero in the vicinity of the QCP (i.e. for $\mu m a^2, Tma^2\ll 1$).
The normal-superfluid phase transition at finite $T$
(as distinguished from the above vacuum-superfluid zero-temperature phase transition) is
characterized by a singularity of the pressure, which translates into a
singularity of $\mathfrak{P}(x,y)$ at a given $x_c(y)$, where $\mathfrak{P}(x,y)$ is
universal. One
finds that the critical chemical potential (at a
given temperature) is given by $\mu_c(T)=T \mathfrak{M}\big(\tilde g\big)$,
where $\mathfrak{M}(x)$ is also universal.
Building on the universality of $\mu_c(T)/T$, we arrive at
\begin{equation}
L_{ij}(\mu,T)=T^{i-1}\mathfrak{L}_{ij}\Big(\frac{\mu-\mu_c}{T},\tilde g,\eta\Big).
\label{eq_scal_trans}
\end{equation}
Here, we have shown explicitly the dependence of the transport coefficients on the dissipation mechanism through the dissipative coefficient $\eta$ introduced earlier. In particular, this scaling holds far from the normal-superfluid critical regime \footnote{In the critical regime of the normal-superfluid transition, the transport coefficients become scaling functions of $\delta\tilde\mu$ which depend on the finite-temperature fixed point ($O(2)$ Wilson-Fisher in $d=3$ and Berezinskii-Kosterlitz-Thouless in $d=2$), while still respecting the scaling forms given in Eq. \eqref{eq_scal_trans}.}.
The dependence of the scaling functions on $\eta$, in turn, implies that even though for a \emph{given kind} of dissipation (defined by $\eta$) the transport coefficients of a dilute Bose gas are universal (\emph{i.e.} described by a function $\mathfrak{L}_{ij}$) in the whole $\mu$-$T$ plane, it will be different for different kinds of dissipation mechanisms (\emph{i.e.} different power laws), thus defining different universality classes \footnote{Note that we work close to equilibrium (the transport coefficient are compute from linear response theory), and thus the scaling functions $\mathfrak{L}_{ij}$ can be computed at equilibrium for given dissipation mechanisms. }.
It follows directly from Eq.~\eqref{eq_scal_trans} that the Seebeck coefficient (or thermopower) $\mathcal{S}$ and the
Lorenz ratio $\mathcal{L}$ are also scale invariant.
\section{Fluctuation transport in Bose and Fermi systems}
\subsection{Fluctuation theory}
In Section II we discussed transport in non-interacting normal gases.
In this context we have seen that there is a dissipative constant parameterized by $\eta$
which is generally unconstrained. Moreover $\eta$ played a central role in determining the transport coefficients of Bose gases,
leading to highly non-universal behavior.
We turn now to Bose and Fermi systems which are normal but near condensation (Bose-Einstein condensation for bosons and the superconducting transition for fermions). This focus stems from recent
Bose gas experiments \cite{Cheng2}.
For the Fermi case, interactions are essential
and one cannot apply the theory of Section II. For the Bose case, the near-condensation regime provides
constraints on the character of dissipation. In this regime we can appeal to theories of fluctuation dynamics
(such as time dependent Landau Ginsburg theory (TDGL)) to constrain the
time dependences and hence frequency dependences of the dissipation. This, in turn, serves to constrain $\eta$,
as
these dynamical theories contain linear time derivatives leading to lifetimes which vary linearly with
frequency or energy.
Relating to the discussion in Section II, interestingly
one arrives at the same
limiting behavior of the Bose transport
coefficients using either
the fluctuation scheme
or Eq.~(\ref{eq:Chester}) with the equivalent $\eta=-2$.
Our focus on normal state transport in the ultracold gases near their condensation temperatures
stems both from recent Bose gas experiments \cite{Cheng2} and also
from the observation that Bose and Fermi superfluids have rather similar properties
in this regime. Both are dominated by a bosonic condensate fluctuation
contribution. In this context \textit{bosonic} transport contributions in superconductors (in the
narrow critical regime) have been
of great interest to the condensed matter community over many decades \cite{VarlamovLarkin,Tinkham}.
Here
one attributes the transport coefficients to fermionic pairs (``composite" bosons) in low
but non-zero momentum states.
These pre-formed pairs yield a greatly enhanced transport in the normal
state, as compared with the behavior deduced from weakly interacting fermion theories such as those in
the previous section.
More recently
there has been a focus on the normal state of high temperature superconductors where it appears
that condensate fluctuation transport may set in at much higher temperatures $T^*$
well above the transition at $T_c$. This reflects the so-called ``pseudogap"
physics \cite{ourreview} and this thought to be related to pre-formed pairs. We \cite{JS2,ourreview}
and others have argued that this pseudogap should be relevant to Fermi gases at unitarity,
where pairing is strong, but the system is still fermionic.
Thus, if indeed, there is a pseudogap at unitarity, the behavior of transport in
the Fermi gases may serve to reveal it in future experiments.
Transport properties associated with condensate fluctuations are based on two
assumptions. (i) The fluctuations (often called ``Gaussian fluctuations")
represent relatively independent bosons.
At most
one introduces Hartree correlations, and (ii)
The dynamics of the
fluctuations incorporates linear time derivatives. Higher order
derivatives are less important due to critical slowing down
effects. This,
importantly, has implications for the frequency dependence of the dissipation
which ultimately affects the behavior of transport.
With these assumptions, the
transport calculations are straightforward and one can evaluate
\cite{ourquenchpaper,Tan2004}
the various transport coefficients following Gaussian fluctuation models
\cite{VarlamovLarkin,Dorsey2,Huse}.
Entering are two factors of the bosonic spectral function, called $A\equiv A({\mathbf k},\omega)$:
\begin{align}
\hspace{-1mm}L_{ij}&=\frac{T^{1-j}}{2m^2}\int
\frac{\dif^dk}{(2\pi)^d}\frac{\dif\omega}{2\pi}
\omega^{i+j-2} \frac{k^2}{d}\big[A(\mathbf{k},\omega)\big]^2 b_-^{(1)}(\omega).
\label{eq:sigma}
\end{align}
\begin{figure}[t]
\includegraphics[width=1.6in,clip,angle=-90]
{trans_coeff.eps}
\caption{Behavior of the transport coefficients $\tilde L_{11}$, $\tilde L_{12}$ and $\tilde L_{22}$ as functions of $(\mu-\mu_c)/T$ for $\Gamma=0.1$ in the Ohmic model ($n=1$) in $d=2$. Their behavior for $(\mu-\mu_c)\to0$ is given by Eq. \eqref{eq_scal}. }
\label{fig2_coeff}
\end{figure}
Here, $b^{(1)}\equiv-\frac{\partial}{\partial\omega} b(\omega)$, with $b(\omega)=(e^{\omega/T}-1)^{-1}$ the Bose distribution.
We characterize the spectral function in terms of the bosonic propagator
\begin{equation}
\Dmat({\mathbf k},\omega) \equiv \Bigl(\omega-\GT({\mathbf k},\omega)-\frac{{\mathbf k}^2}{2m}+\mu-\mu_c
+\frac{i}{2}\FT({\mathbf k},\omega)\Bigr)^{-1}.
\label{eq:7}
\end{equation}
so that
\begin{equation}\label{eq:A}
A ({\mathbf k},\omega)=\frac{\FT({\mathbf k},\omega)}{\left(\omega-\frac{{\mathbf k}^2}{2m}+\mu-\mu_c-\GT({\mathbf k},\omega)\right)^2
+\frac{1}{4}\left(\FT({\mathbf k},\omega)\right)^2},
\end{equation}
where the critical chemical potential $\mu_c$ is a phenomenological parameter, and
we take $\GT(0,0)=0$.
These constraints are convenient, rather than reflective of any deep physics.
We view the effective Hamiltonian (including self energy and dissipation)
as a Hartree-like theory
\cite{Capogrosso-Sansone2010}
where by definition
$\GT(0,0)=0$ and in this way the superfluid transition occurs at a finite
(positive) chemical potential.
A
crucial feature of bosons is that $\Sigma_2$
changes sign at $\omega = 0$, so that the
spectral function $A$ has the sign of $\omega$.
This is satisfied in
the usual TDGL
dynamics, where at low $\omega$ one has
\begin{equation}
\Sigma_2 \propto \omega^{n}~~~\rm{with}~~n=1,
\end{equation}
corresponding to Ohmic dissipation.
It should be pointed out,
the apparent absence of vertex correction terms
in Eq.~(\ref{eq:sigma}) has been validated in the TDGL literature
(where the bosons only experience Hartree interactions)
and more directly in Ref. \onlinecite{Tan2004} it can be seen
to be consistent with the current conservation constraint
which provides the framework for including vertex terms
in a Kubo formalism.
For the purposes of completeness we briefly revisit the transport nomenclature
found in
superconducting fluctuation
theories,
\cite{Maki,VarlamovLarkin,Tinkham},
where the
fluctuation propagator is found to be proportional to
\begin{equation}
D^{fluc}(\mathbf{q},i \omega) \propto \frac{1} { [(-1 + i \lambda ) (i \omega) +
\mathcal{D}q^2 +
(8/\pi) (T-T_c)]}
\label{eq:15}
\end{equation}
Note that Eq.~(\ref{eq:15}) can be seen to be equivalent to
Eq.~(\ref{eq:7}), when the dissipation is Ohmic.
Here $\mathcal{D}$ is the so-called diffusion coefficient and
$\lambda = [2 \pi T / g N(0)][N'(0)/N(0)]$
depends on the density of states at the Fermi energy $N(0)$
and its derivative.
By way of further comparison, we note
that in solid state physics one usually describes the approach of the transition by $T-T_c$, whereas $\mu-\mu_c$ is the variable associated with
cold atom experiments, although these two parametrizations are equivalent.
\subsection{Divergences in Transport}
As $\delta\tilde\mu\equiv (\mu -\mu_c)/T$ goes to zero (but away from the
strictly critical regime) we can deduce the transport coefficients
for the two-dimensional case from the integrals in
Eqs.~(\ref{eq:sigma}) and their counterparts.
The literature
is based on the Ohmic case $n=1$ which defines the nature of the divergences
in the $L_{ij}$.
These can be summarized
\cite{Maki,VarlamovLarkin,Tinkham}
in terms of proportionality relations
\begin{equation}
\begin{split}
\tilde L_{11} &\propto \frac{1}{|\delta\tilde\mu|},\\
\tilde L_{12} &\propto - \ln |\delta\tilde\mu|,\\
\tilde L_{22} &\propto {\rm const.}+|\delta\tilde\mu| \ln |\delta\tilde\mu|.
\end{split}
\label{eq_scal}
\end{equation}
From
Eqs. \eqref{eq_scal} we deduce that the Seebeck coefficient (thermopower) and
Lorenz number, which involve ratios of these coefficients, behave as
\begin{equation}
\begin{split}
\mathcal{S} &\propto -|\delta\tilde\mu| \ln |\delta\tilde\mu|, \\
\mathcal{L} &\propto |\delta\tilde\mu|.
\end{split}
\end{equation}
Importantly,
the thermopower changes sign at condensation and the Lorenz number tends to zero linearly.
Recall that $\mathcal{L}$ must be greater or equal to zero for thermodynamic stability \footnote{In the case $\eta=0$, as used in Ref. \onlinecite{Alexandrov} based on a Boltzmann bosonic approach designed for high-$T_c$ superconductors, one finds $\tilde \sigma\propto \ln|\delta\tilde\mu|$ which is to be contrasted with the Ohmic case where the divergences at condensation are more evident. As a consequence $\mathcal{L} \propto 1/\ln |\delta\tilde\mu|$ and the thermopower $\mathcal{S}$ will similarly vanish logarithmically.}.
In Figure \ref{fig2_coeff} we plot the transport coefficients
for $d=2$ as obtained from the fluctuation theory (Eq. \eqref{eq:sigma})
as a function of scaled chemical potential,
for a typical $\Gamma=0.1$ \cite{ourquenchpaper}.
We have checked that the value of $\Gamma$ does not change qualitatively the
transport coefficients or power laws for $|\delta\tilde\mu|\lesssim 1$.
This figure shows that the thermal conductivity (related to $L_{22}$) and
the mass conductivity (related to $L_{11}$) as well as the off-diagonal conductivity
satisfying the Onsager relation
($L_{12}=L_{21}/T$) all diverge at the transition. It should be clear, however, that
this divergence is strongest for the mass conductivity.
Important here is that, because of the divergence of $\sigma = L_{11}$,
as $\mu$ goes to $\mu_c$,
$\mathcal{L}$ (and $\mathcal{S}$) vanish at $T_c$.
\section{Results: comparison with experiment near condensation }
We turn now to a comparison of theory and the experiments of Ref.~\onlinecite{Cheng2}. In Ref. \onlinecite{Cheng2}, a Bose gas was confined in an optical trap with peak density $n=5\times10^{13}$cm$^{-3}$, a large scattering length $a=22$~nm, and the temperature $T= 35$~nK. The trap
was highly oblate with very strong confinement in the $z-$direction and weak confinement in the two radial directions. The sample is thus in the quasi-2D regime. For these parameters we are able to address the validity of the LEA. In the normal gas regime, the scattering rate determines the thermalization time scale $\tau_c=\alpha/(4\pi n a^2 v) = 2$~ms where $\alpha=2.7$ is a constant \cite{Monroe1993}, and the coherence length scale is given by the mean-free path $l_c=3.5\mu$m. An optical potential corrugation of up to 3~nK leads to a measured relaxation time of 20~ms of collective excitation. Both these length and time scales are small compared to the the sample size $l_s=60\mu$m$
\gg l_c$ and sample lifetime of $\tau_s=1$s$\gg \tau_c$, which suggests that the LEA is valid over a coarse grained spatial scale of $3.5\mu$m and
time scale $20$~ms.
Our comparison between theory and experiment is plotted in
Figs. \ref{fig_L}. Here we show the thermopower (upper panel) and the
Lorenz number (lower panel) for a two-dimensional Bose gas, obtained from Eq.~(\ref{eq:sigma}).
In the simplest terms the data suggests that the thermopower changes sign at
or near the transition. Experimentally, the Lorenz number approaches zero at or
near $T_c$. The error bars are sufficiently large and one
should not infer that $\cal L$ becomes negative on the other side of the transition,
as this is unphysical.
We stress that our theory is applicable only to the normal phase. Indeed, one does
not have many reliable theoretical tools for treating bosons (in any capacity, not
just transport) near, but below the transition.
\begin{figure}[!h]
\includegraphics[width=2.1in,clip]
{trans_exp.eps}
\caption{Comparison with the experimental data of Ref.\onlinecite{Cheng2} for the thermopower and Lorenz number as functions of $(\mu-\mu_c)/T$ for $\Gamma=0.1$ for the Ohmic model ($n=1$) in $d=2$, see Eq.~\eqref{eq:sigma}.}
\label{fig_L}
\end{figure}
In the normal phase, the figure shows that the theory and experiment of
Ref.~\onlinecite{Cheng2} coincide rather well for
the thermopower $\mathcal{S}$, particularly near the condensation
regime. The Lorenz number is off by about
a factor of 2. However, given the simplicity of the model, a
disparity of order unity is not unreasonable.
It should be stressed that the behavior of $\mathcal{S}$ and $\mathcal{L}$ are very
strongly reflective of the divergence of the mass conductivity at the phase
transition. The fact that both are found in the data to vanish at or near the transition in the
experiments serves to help validate this experimental methodology
\cite{Cheng2}.
\section{Conclusions }
Many communities from nuclear physics, astrophysics and condensed matter are interested in
transport in the cold gases. The shear viscosity has been a rather recent focus \cite{Ourviscosity,NJOP,ThomasJLTP08,Qviscosity,
ThomasViscosity} and,
as in this paper, there is an emerging interest in
thermoelectric properties \cite{Cheng2,Brantut2012}.
There will undoubtedly be many more such experimental transport studies with time.
The theory in this paper is directed towards such experiments. We have emphasized that
the mechanisms for momentum dissipation are very sensitive to the experimental cold atom set up.
This has led us to adopt a Kubo formula approach which treats
dissipation at a non-specific phenomenological level.
An important consequence of our studies
is the observation of quantum critical scaling and
of the non-universality of (bosonic) transport, despite universality
in thermodynamics. This is
in contrast to Fermi liquid or gas systems where
the processes contributing to transport are localized around the Fermi energy, and one finds,
for example a Wiedermann-Franz law for the Lorenz ratio.
Another major finding of this work is the observation that bosonic transport in
the normal state near condensation appears to be semi-quantitatively consistent with observations
and theory in condensed matter studies of (bosonic) superconducting fluctuations
\cite{VarlamovLarkin,Tinkham}.
In this way
our paper establishes a connection between transport properties in condensed matter and in normal-phase cold atom systems. This connection has other implications for future experiments.
Because
ultracold gases are clean and well controlled, they may help elucidate transport in (fermionic) high $T_c$ systems
\cite{ourreview} where bosonic degrees of freedom (through pre-formed pairs, or vortices or bipolarons, \emph{
etc.}) are believed to dominate transport.
In particular,
high temperature
superconductors in condensed matter
appear to have an anomalously large critical regime where these
fluctuation effects are observed which is associated with the famous ``pseudogap".
In the cold Fermi gases near unitarity there is some debate about whether such a pseudogap exists or not.
This would suggest that future transport studies of fermionic superfluids may help
elucidate this issue.
In contrast to the behavior of degenerate Fermi gases, we reiterate the strong sensitivity of Bose gases to the dissipation mechanism. For bosons, different dissipation mechanisms imply different scaling laws, possibly defining different universality classes. For the future, the universality found in the thermodynamics of dilute Bose gases, either loaded or not in a lattice, \cite{Yefsah2011,Hung2011,Zhang2012,Rancon2012a}, will also need to be addressed in the context of transport. This would allow a direct verification of our prediction of the breakdown of universality in transport.
\vskip1mm
This work is supported by NSF-MRSEC Grant
0820054; CC acknowledges support from NSF PHY-1206095 and ARO-MURI 63834-PH-MUR. We thank
Chih-Chun Chien and Yan He for helpful conversations. CC thanks Grenier Charles and Antoine Georges for useful discussion.
\bibliographystyle{apsrev}
|
train/arxiv
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BkiUd2A5qoTBG_qrq9mA
| 5 | 1 |
\section{Introduction}
Quantum field theory has perplexed physicists
with the appearance of infinities on radiative corrections
of physical quantities,
but the problem has offered useful hints for a fundamental theory.
In well-behaved theories such as the quantum electrodynamics (QED)
and the standard model (SM),
infinities are removed by the regularization
and the renormalization procedure.
There, however, exists a non-renormalizable theory
such as the quantum version of Einstein gravity,
and it suggests the idea
that {\it an underlying theory must own a finiteness}.
Typical examples
are superstring theories (SSTs)~\cite{SST}
and finite field theories (FFTs)~\cite{FFT}.
The finiteness comes from
the world-sheet modular invariance in SSTs
and the vanishing of $\beta$-functions in FFTs.
Both have a feature that a fundamental energy scale $\varLambda$
exists, but its origin is different from each other.
In SSTs, $\varLambda$ is the string scale defined by the string tension.
The world-sheet modular invariance implies
that radiative corrections from the contributions below $\varLambda$
are equivalent to those above $\varLambda$
and are given by the integration of an independent region
called $\lq\lq$the fundamental region''.
There, $\varLambda$ plays the role of cut-off parameter
or the ultra-violet (UV) divergences are identified with
unphysical infra-red (IR) ones.
In FFTs, the theory becomes scale invariant with the vanishing of
$\beta$-functions at $\varLambda$.
Then, physical parameters do not run beyond the scale,
and the concept of scale becomes vague.
Models have a high calculability and predictability,
because the reduction of coupling constants is
realized and the particle contents are tightly restricted,
combined with the grand unification~\cite{FUT}.
Both theories are powerful candidates for the physics at $\varLambda$.
However, any evidences for new physics
beyond the SM, e.g., supersymmetry (SUSY),
compositeness and extra dimensions, have not been discovered.
Hence, it would be meaningful to pursue other possibilities.
In this paper, we use the secret of finiteness in SSTs
as a guide for constructing a new type of FFTs,
and propose a framework of
theories based on a hidden duality
between an UV and an IR region.
Physical quantities do not receive radiative corrections
at the fixed point of the duality transformation,
and this feature is compatible with models
possessing a specific fermionic symmetry.
Theories can be tested indirectly by relations among parameters,
reflecting underlying symmetries,
in case that the system has a large symmetry at a fundamental level.
The contents of this paper are as follows.
We present novel FFTs based on
a duality relating world lines in Sect. II.
We explain that the finiteness can be
assisted by an exotic symmetry concerning abnormal particles
and theories can be tested indirectly in Sect. III.
Section IV is devoted to conclusions and discussions.
\section{Finiteness based on world-line duality}
We take the standpoint that quantum field theory
is an effective description of an unknown underlying theory,
and FFTs can be constructed by bringing in features of
the fundamental theory.
First, we list the assumptions relating features
of the ultimate theory.\footnote{
Our idea is inspired by the world-sheet modular invariance
in closed string theories,
and the proposal of solving the gauge hierarchy problem
and the cosmological constant problem by Dienes~\cite{Dienes}.
}\\
(a) There is an energy scale $\varLambda$ that associated with
a property of fundamental objects such as the string scale.
The fundamental objects possess various states that
identified with elementary particles.
The ground states are regarded
as massless particles, and some of them acquire small masses
$m_k (\ll \varLambda)$ through a low-energy dynamics.
The excited states are particles with masses of $O(\varLambda)$.\\
(b) There is a duality between the physics
at a higher-energy scale ($\mu \gtrsim \varLambda$)
and that at a lower-energy scale ($\mu \lesssim \varLambda$).
Physical quantities are invariant under the duality transformation,
and are estimated as finite values using one of the energy regions.\\
(c) A remnant of the duality is hidden in quantities of
the low-energy physics involved with $\varLambda$,
e.g., radiative corrections on parameters~\cite{duality}.
Finite corrections can be incorporated in
our formulation with a slight modification of
the duality transformation,
in the presence of low-energy parameters such as $m_k$.
To illustrate our idea, let us consider quantum corrections
on a quantity $A$ at the one-loop level given by,
\begin{eqnarray}
&~& \delta A(q) = \int_{0}^{1} ds \int_0^{\infty} dt
\int_{0}^{\infty} \frac{d^4p}{(2\pi)^4}~f(p, q, g_i, m_k, s, t)
\nonumber\\
&~& ~~~~~~~~~~~
= \int_{0}^{1} ds \int_0^{\infty} dt~h(q, g_i, m_k, s, t),
\label{delta-a}
\end{eqnarray}
where $s$ is an integration variable,
$t$ is the integration variable called a $\lq\lq$proper time'',
$p$ is an Euclidean momentum of a particle running in a loop,
$q$ is an Euclidean momentum of a particle for an external line,
$g_i$ are coupling constants,
and $h(q, g_i, m_k, s, t)$ is a function of $q$, $g_i$, $m_k$, $s$ and $t$.
In case that $\delta A(q)$ diverges, the infinities come from
$t = 0$ (corresponding the UV divergences)
and/or $t = \infty$ (corresponding the IR ones),
it can be regularized as
\begin{eqnarray}
\delta A(q)_{\rm reg} = \int_{0}^{1} ds
\int_{1/\tilde{\varLambda}^2}^{1/\tilde{\mu}^2} dt~h(q, g_i, m_k, s, t),
\label{delta-a-reg}
\end{eqnarray}
where $\tilde{\varLambda} = \tilde{\varLambda}(\varLambda, s)$.
In most cases, $\tilde{\varLambda} = x(s) \varLambda$
and $\tilde{\mu} = y(s) q^2 + y_k(s) m_k^2$
where $x(s)$, $y(s)$ and $y_k(s)$ are functions of $s$.
Although these functions can vanish at some values of $s$,
we often substitute $\tilde{\varLambda}$ and $\tilde{\mu}$
for $\varLambda$ and $q$, respectively.
For instance, $\tilde{\varLambda}$ and $\tilde{\mu}$
are given by $\tilde{\varLambda}^2 = s \varLambda^2$
and $\tilde{\mu}^2 = s(1-s) q^2 + s \mu_{\gamma}^2 + (1-s) m_{\rm e}^2$
for the self-energy of electron in QED~\cite{duality}.
Here, $\mu_{\gamma}$ is a fictitious photon mass
for a regularization of IR divergences
and $m_{\rm e}$ is the electron mass.
We require that $\delta A(q)_{\rm reg}$ should be invariant
under the following duality transformation on the internal world-line,
\begin{eqnarray}
t \to \frac{1}{\tilde{\varLambda}^4 t}.
\label{t-transf}
\end{eqnarray}
Under the transformation (\ref{t-transf}),
$\delta A(q)_{\rm reg}$ transforms as
\begin{eqnarray}
\delta A(q)_{\rm reg} \to
\int_{0}^{1} ds
\int_{\tilde{\mu}^2/\tilde{\varLambda}^4}^{1/\tilde{\varLambda}^2}
dt~\frac{h(q, g_i, m_k, s, 1/(\tilde{\varLambda}^4 t))}{\tilde{\varLambda}^4 t^2}.
\label{delta-a-reg-t-transf}
\end{eqnarray}
From the equality of the right-hand side of
(\ref{delta-a-reg}) and that of (\ref{delta-a-reg-t-transf}),
the form of $h(q, g_i, m_k, s, t)$ is fixed as
$h(q, g_i, m_k, s, t)=c_{-1}(q, g_i, m_k, s)/t$
and then $\delta A(q)_{\rm reg}$ is determined as
\begin{eqnarray}
\delta A(q)_{\rm reg} = \int_0^1 ds~
c_{-1}(q, g_i, m_k, s) \ln \frac{\tilde{\varLambda}^2}{\tilde{\mu}^2}.
\label{delta-a-reg-det}
\end{eqnarray}
The $\delta A(q)_{\rm reg}$ contains
a $\varLambda$-dependent logarithmic part and finite corrections.\footnote{
It is not clear whether all corrections including higher loops
are obtained in the duality invariant form.
If not, such corrections can be regarded as a tiny violation of
the duality.
}
We find that quantum corrections
corresponding quadratic divergences are removed,
and hence the naturalness problem in the SM
(the quadratic divergence problem relating the Higgs boson mass)
can be solved~\cite{duality}.
So far, we study corrections at the lower energy region
($q^2 < \varLambda^2$) by considering
only contributions from the ground states
(corresponding massless particles at $\varLambda$).
Beyond $\varLambda$, threshold corrections
due to excited states (corresponding massive particles of $O(\varLambda)$)
appear, and following two problems occur.\\
(i) To examine whether physical quantities
are formulated in a manner consistent with the duality,
{\it for all energy regions}.\\
(ii) To examine whether an effective field theory
have a high calculability and predictability,
{\it without knowing full spectrum of excited states}.
The threshold corrections might be introduced,
in the form that $h$ contains $\varLambda$ such that
\begin{eqnarray}
h(q, g_i, m_k, s, t)=
\frac{\tilde{c}_{-1}(q, g_i, m_k, s, t+1/(\tilde{\varLambda}^4 t))}{t}.
\label{h-Lambda}
\end{eqnarray}
However, we have no method to determine
the form of $\tilde{c}_{-1}$ as a function of $t$,
in the framework of effective field theory, and hence
we pursue an alternative.
Here, we give a bold conjecture that
{\it there are no threshold corrections due to excited states}.
If it holds, the second problem can be solved,
as will be explained in the next section.
For the first problem,
let us make an adventurous attempt
that the duality can be also applied to the external lines.
We require that $\delta A(q)_{\rm reg}$ should be invariant
under the duality transformation
$q \to q'$ that corresponds to
\begin{eqnarray}
\tilde{\mu}^2 \to
\tilde{\mu}'^2 = \frac{\tilde{\varLambda}^4}{\tilde{\mu}^2}.
\label{mu-transf}
\end{eqnarray}
The invariance is written as
\begin{eqnarray}
\delta A(q)_{\rm reg} = \delta A(q')_{\rm reg}.
\label{delta-a-reg-duality}
\end{eqnarray}
From (\ref{delta-a-reg-det}), (\ref{mu-transf}) and (\ref{delta-a-reg-duality}),
we obtain the expressions,
\begin{eqnarray}
\delta A(q)_{\rm reg}
= \int_{0}^{1} ds \int_{1/\tilde{\varLambda}^2}^{1/\tilde{\mu}^2} dt~
h(q, g_i, m_k, s, t)~~~~
({\rm for}~~\tilde{\mu} \le \tilde{\varLambda})
\label{delta-a-reg-mu}
\end{eqnarray}
and
\begin{eqnarray}
\delta A(q')_{\rm reg}
= \int_{0}^{1} ds \int_{1/\tilde{\mu}'^2}^{1/\tilde{\varLambda}^2} dt~
h(q', g_i, m_k, s, t)~~~~
({\rm for}~~\tilde{\mu}' \ge \tilde{\varLambda}).
\label{delta-a-reg-mu'}
\end{eqnarray}
Under the transformation (\ref{t-transf}),
$\delta A(q)_{\rm reg}$ transforms into
$\delta A(q')_{\rm reg}$ and vice versa,
and they take a same value with $h(q, g_i, m_k, s, t)=c_{-1}(q, g_i, m_k, s)/t$.
The equality (\ref{delta-a-reg-duality}) comes from
the fact that the distance between
$1/\tilde{\varLambda}^2$ and $1/\tilde{\mu}^2$ equals to
that between $1/\tilde{\mu}'^2 (=\tilde{\mu}^2/\tilde{\varLambda}^4)$
and $1/\tilde{\varLambda}^2$
at the logarithmic scale.
We find that $\tilde{\varLambda}$ is the fixed point
under the transformation (\ref{mu-transf}).
The relation (\ref{delta-a-reg-mu'}) suggests a strange feature
that the value obtained by integrating out the degrees
from $\tilde{\varLambda}$ to $\tilde{\mu}' (> \tilde{\varLambda})$
is not the value at $\tilde{\varLambda}$
but that at $\tilde{\mu}'$ (or $q'$),
and the quantity does not receive radiative corrections
at $\tilde{\varLambda}$, i.e.,
$\delta A(q)_{\rm reg}|_{\tilde{\mu} =\tilde{\varLambda}} = 0$.
We give a speculation of such an opposite running of physical quantities
beyond $\tilde{\varLambda}$.
If the role of energy and momentum is exchanged
in the region beyond $\tilde{\varLambda}$,
as is the case that the role of time and space is exchanged inside a black hole,
$p^2$ and $m_k^2$ in $f(p, q, g_i, m_k, s, t)$ can change its sign.
Eventually, it can induce the exchange of integration region.
Then, $\varLambda$ might be the Planck scale $M_{\rm Pl}$.\footnote{
As another work to show the importance of the trans-Planckian physics,
Volovik gave the observation that the sub-Planckian and
trans-Planckian contributions to the vacuum energy
are canceled by the thermodynamical argument~\cite{Volovik}.
}
Our procedure is regarded as not a mere regularization
but a recipe to obtain finite physical values,
because $\tilde{\varLambda}$ is (big but) finite and
infinities are taken away by the symmetry relating integration variables,
like closed string theories.
It is also regarded as the operation to pick out duality invariant parts.
In case that $h(q, g_i, m_k, s, t)$ does not contain $\varLambda$,
it is simply denoted by
\begin{eqnarray}
&~& \delta A(q)_{\rm reg}
= {\rm Du}\left[\int_{0}^{1} ds \int_0^{\infty} dt~h(q, g_i, m_k, s, t)\right]
\nonumber \\
&~& ~~~~~~~~~~~~~~~~~
= {\rm Du}\left[\int_{0}^{1} ds \int_0^{\infty} dt~
\sum_n c_n(q, g_i, m_k, s) t^n\right]
\nonumber \\
&~& ~~~~~~~~~~~~~~~~~
= c_{-1}(q, g_i, m_k, s) \ln \frac{\tilde{\varLambda}^2}{\tilde{\mu}^2},
\label{delta-a-Du}
\end{eqnarray}
where ${\rm Du}[*]$ represents the operation,
and $h(q, g_i, m_k, s, t)$ is expanded in a series of $t$.
\section{Fermionic symmetry}
\subsection{Calculability}
Let us construct a theory with a high calculability
\footnote{
In this paper, a $\lq\lq$calculability'' means that
physical quantities can be calculated
in terms of free parameters by a theory,
and a $\lq\lq$predictability'' means that
some features beyond the theory,
such as relations among parameters, can be predicted.
},
based on the feature
that any threshold corrections do not appear around $\varLambda$.
We assume that fundamental objects have a specific fermionic symmetry
and most states become unphysical by a counterpart of
the quartet mechanism~\cite{K&O1,K&O2}
in a system with the BRST symmetry.
Concretely, all excited states form the quartets such as
($\varphi_a, c_a, \overline{c}_a, \overline{\varphi}_a$),
and they transform as
\begin{eqnarray}
\bm{\delta}_{\rm f} \varphi_a = (\pm) c_a,~~
\bm{\delta}_{\rm f} c_a = 0,~~
\bm{\delta}_{\rm f} \overline{c}_a = (\pm) \overline{\varphi}_a,~~
\bm{\delta}_{\rm f} \overline{\varphi}_a = 0,
\label{delta-f}
\end{eqnarray}
where $(\pm)$ represents $+1$ or $-1$.
The transformation is generated
by a fermionic conserved charge $Q_{\rm f}$
with the nilpotency, i.e., ${Q_{\rm f}}^2 = 0$.
If we impose suitable subsidiary conditions
containing the following one on states
in order to select physical states,
\begin{eqnarray}
Q_{\rm f} |{\rm phys}\rangle =0,
\label{phys}
\end{eqnarray}
all $Q_{\rm f}$-quartets (or two sets of $Q_{\rm f}$-doublets)
become unphysical.
Hereafter, we denote the set of $Q_{\rm f}$-quartets
as $(\{\varphi_{\rm q}\}, \{c_{\rm q}\})$
and refer to particles belonging in $\{c_{\rm q}\}$ as ghosts.
As a possible candidate of fermionic symmetry,
symmetries between ordinary particles
and their ghost counterparts
obeying opposite statistics have been proposed~\cite{YK1,YK2,YK3,YK4}.
In this case, $\overline{c}_a$ and $\overline{\varphi}_a$
are the hermitian conjugates of $c_a$ and $\varphi_a$, respectively.
The algebraic relations among relevant conserved charges are given by
\begin{eqnarray}
{Q_{\rm F}}^2 = 0, ~~{Q_{\rm F}^{\dagger}}^2 = 0,~~
\{Q_{\rm F}, Q_{\rm F}^{\dagger}\} = N_{\rm D},
\label{QF-rels}
\end{eqnarray}
where $Q_{\rm F}$ and $Q_{\rm F}^{\dagger}$ are
a fermionic charge and its hermitian conjugate,
respectively,
and $N_{\rm D}$ is an $U(1)$ charge.
The physical state conditions are imposed as
\begin{eqnarray}
Q_{\rm F} |{\rm phys}\rangle =0,~~
Q_{\rm F}^{\dagger} |{\rm phys}\rangle =0,~~
N_{\rm D} |{\rm phys}\rangle =0.
\label{phys-QF}
\end{eqnarray}
In case that parts of massless states are $Q_{\rm f}$-singlets
(whose set is denoted by $\{\varphi_{\rm s}\}$)
and are physical,
the system can be, in general, described by
\begin{eqnarray}
\mathcal{L}_{\rm total} = \mathcal{L}_{\rm s}(\{\varphi_{\rm s}\})
+ \mathcal{L}_{\rm q}(\{\varphi_{\rm q}\}, \{c_{\rm q}\})
+ \mathcal{L}_{\rm mix}(\{\varphi_{\rm s}\},
\{\varphi_{\rm q}\}, \{c_{\rm q}\})
= \mathcal{L}_{\rm s}(\{\varphi_{\rm s}\}) + \bm{\delta}_{\rm f} \mathcal{R},
\label{L}
\end{eqnarray}
where $\mathcal{L}_{\rm s}$, $\mathcal{L}_{\rm q}$
and $\mathcal{L}_{\rm mix}$
are the Lagrangian density for $Q_{\rm f}$-singlets, $Q_{\rm f}$-quartets
and interactions between $Q_{\rm f}$-singlets and $Q_{\rm f}$-quartets.
Under suitable subsidiary conditions
including $Q_{\rm f} |{\rm phys}\rangle =0$ on states,
all $Q_{\rm f}$-quartets become unphysical
and would not give any physical effects on $Q_{\rm f}$-singlets,
that is, $Q_{\rm f}$-singlets do not receive any radiative corrections
from $Q_{\rm f}$-quartets.
Hence, the theory is free from
the gauge hierarchy problem~\cite{YK1}.\footnote{
Other type of fermionic symmetry called $\lq\lq$misaligned
supersymmetry'' has been proposed to solve the gauge hierarchy
problem and to realize the finiteness,
in the absence of space-time SUSY~\cite{Dienes,Dienes2,DM&M}.
}
Let us discuss the calculability in our formulation.
If all massive modes with masses of $O(\varLambda)$
are $Q_{\rm f}$-quartets and unphysical,
there are no threshold corrections at $\varLambda$
and all physical quantities can be calculable
using $\mathcal{L}_{\rm s}(\{\varphi_{\rm s}\})$ alone.
That is, if values of coupling constants $g_i$ and masses $m_k$
are determined by precision measurements,
we can obtain values of physical quantities accurately.
Note that $g_i$ and $m_k$ are free parameters
and their values are not determined theoretically,
in the framework of a low-energy effective theory.
Ordinarily, non-renormalizable interactions are
generated as a result that heavy particles are integrated out.
In our system, heavy particles with masses of $O(\varLambda)$
appear by pairs in the interaction terms
because of the $Q_{\rm f}$ invariance.
In the process with ordinary physical particles alone
in the external lines,
heavy particles appear in loops
and the sum of contributions can be canceled out
by the fermionic symmetry.
That is, non-renormalizable interactions
are not induced due to the excited states at $O(\varLambda)$.
Then, there is a possibility that
$\mathcal{L}_{\rm s}(\{\varphi_{\rm s}\})$
is renormalizable at $\varLambda$,
neglecting the effect of gravity.
In this case,
we have an interesting expectation
that {\it the system of visible fields
is described by the Lagrangian density
containing renormalizable terms alone}.
Because visible fields come from the massless states at $\varLambda$,
the effective Lagrangian density must have symmetries
such as gauge symmetry, chiral symmetry
and/or conformal symmetry
to guarantee the masslessness.
After integrating out extra particles in $\{\varphi_{\rm s}\}$
other than the SM ones,
we arrive at the system described by
\begin{eqnarray}
\mathcal{L}_{\rm SM} + \varDelta\mathcal{L}_{\rm SM},
\label{L-SM}
\end{eqnarray}
where $\mathcal{L}_{\rm SM}$
stands for the renormalizable Lagrangian density for the SM particles
and and $\varDelta\mathcal{L}_{\rm SM}$
is the non-renormalizable one generated by
contributions from heavy particles in $\{\varphi_{\rm s}\}$
beyond the weak scale.
A dark matter and massive right-handed neutrinos
are included as candidates of extra ones.
The exploration of $\varDelta\mathcal{L}_{\rm SM}$
is important to probe the physics (at the terascale) beyond the SM
and to determine $\mathcal{L}_{\rm s}(\{\varphi_{\rm s}\})$
indirectly.
The system above $\varLambda$ can be also described
by $\mathcal{L}_{\rm total}$
(essentially $\mathcal{L}_{\rm s}(\{\varphi_{\rm s}\})$),
and has a grounding that the duality introduced in the previous section
holds for physical quantities.
\subsection{Predictability}
Next, we discuss the predictability in our formulation
based on $\mathcal{L}_{\rm total}$.
It is needed to specify a structure of system at $\varLambda$.
Let us take a reasonable conjecture that
{\it fundamental objects have a large symmetry intrinsically
and the symmetry is realized in an unbroken phase at $\varLambda$}.
Concretely, the system has the symmetry
whose transformation group is $G_{\rm U}$ in the unbroken phase.
By some mechanism, $G_{\rm U}$ is broken down to the subgroup $G$
and the system is described by $\mathcal{L}_{\rm total}$,
where $\{\varphi_{\rm s}\}$ are multiplets of $G$.
Here, we consider two scenarios
that $\mathcal{L}_{\rm total}$ is derived after the reduction of symmetry,
and find a prediction based on them.\\
($\alpha$) First one is that all particles belong to
the members of $Q_{\rm f}$-quartets
and are multiplets of $G_{\rm U}$, in the unbroken phase.
They are denoted by $(\{\varphi_{\rm q}^{\rm U}\}, \{c_{\rm q}^{\rm U}\})$.
The system is described by
\begin{eqnarray}
\mathcal{L}^{(\alpha)}
= \mathcal{L}_{\rm U}^{(\alpha)}(\{\varphi_{\rm q}^{\rm U}\})
+ \mathcal{L}_{\rm gh}^{(\alpha)}(\{c_{\rm q}^{\rm U}\})
+ \mathcal{L}_{\rm int}^{(\alpha)}(\{\varphi_{\rm q}^{\rm U}\}, \{c_{\rm q}^{\rm U}\})
= \bm{\delta}_{\rm f} \mathcal{R}_{\rm U}^{(\alpha)},
\label{L-alpha}
\end{eqnarray}
where $\mathcal{L}_{\rm U}^{(\alpha)}$, $\mathcal{L}_{\rm gh}^{(\alpha)}$
and $\mathcal{L}_{\rm int}^{(\alpha)}$
are the Lagrangian densities for ordinary particles
$\{\varphi_{\rm q}^{\rm U}\}$,
the ghost counterparts $\{c_{\rm q}^{\rm U}\}$,
and interactions between ordinary particles and ghosts.
The multiplets of $G_{\rm U}$ are decomposed into those of $G$ such that
\begin{eqnarray}
\{\varphi_{\rm q}^{\rm U}\}
\Rightarrow \{\varphi_{\rm q}\}_0 + \{\varphi_{\rm q}\}_1,~~
\{c_{\rm q}^{\rm U}\}
\Rightarrow \{c_{\rm q}\}_0 + \{c_{\rm q}\}_1.
\label{G-decomp}
\end{eqnarray}
If some ghosts $\{c_{\rm q}\}_0$ disappear,
ordinary particles $\{\varphi_{\rm q}\}_0$
turn out to be $Q_{\rm f}$-singlets
and the reduction of symmetry occurs.\footnote{
It has been reported that,
in a system with complex scalar fields on a higher-dimensional space-time,
some physical modes are released from
unphysical $Q_{\rm F}$-doublets
and the reduction of a large symmetry occurs
through the orbifold breaking mechanism~\cite{YK3}.
}
Then, the system is described by
\begin{eqnarray}
{\mathcal{L}'}^{(\alpha)}
= \mathcal{L}_{\rm U}^{(\alpha)}(\{\varphi_{\rm s}\}, \{\varphi_{\rm q}\})
+ {\mathcal{L}'}_{\rm gh}^{(\alpha)}(\{c_{\rm q}\})
+ {\mathcal{L}'}_{\rm int}^{(\alpha)}(\{\varphi_{\rm s}\}, \{\varphi_{\rm q}\}, \{c_{\rm q}\}),
\label{L'-alpha}
\end{eqnarray}
where $\{\varphi_{\rm q}\}_0$, $\{\varphi_{\rm q}\}_1$
and $\{c_{\rm q}\}_1$
are denoted by $\{\varphi_{\rm s}\}$, $\{\varphi_{\rm q}\}$
and $\{c_{\rm q}\}$.
In this scenario, ${\mathcal{L}'}^{(\alpha)}$ corresponds to
$\mathcal{L}_{\rm total}$ at $\varLambda$.\\
($\beta$) Second one is that some massless ordinary particles
belong to members of $Q_{\rm f}$-singlets $\{\varphi_{\rm s}^{\rm U}\}$
and others are $Q_{\rm f}$-quartets
$(\{{\varphi'}_{\rm q}^{\rm U}\}, \{{c'}_{\rm q}^{\rm U}\})$.
All particles form multiplets of $G_{\rm U}$,
in the unbroken phase.
The system is described by
\begin{eqnarray}
&~& \mathcal{L}^{(\beta)}
= \mathcal{L}_{\rm U}^{(\beta)}(\{\varphi_{\rm s}^{\rm U}\})
+ \mathcal{L}_{\rm q}^{(\beta)}(\{{\varphi'}_{\rm q}^{\rm U}\}, \{{c'}_{\rm q}^{\rm U}\})
+ \mathcal{L}_{\rm int}^{(\beta)}(\{\varphi_{\rm s}^{\rm U}\},
\{{\varphi'}_{\rm q}^{\rm U}\}, \{{c'}_{\rm q}^{\rm U}\})
\nonumber \\
&~& ~~~~~~~~~
= \mathcal{L}_{\rm U}^{(\beta)}(\{\varphi_{\rm s}^{\rm U}\})
+ \bm{\delta}_{\rm f} \mathcal{R}_{\rm U}^{(\beta)},
\label{L-beta}
\end{eqnarray}
where $\mathcal{L}_{\rm U}^{(\beta)}$, $\mathcal{L}_{\rm q}^{(\beta)}$
and $\mathcal{L}_{\rm int}^{(\beta)}$
are the Lagrangian densities for $\{\varphi_{\rm s}^{\rm U}\}$,
$Q_{\rm f}$-quartets,
and interactions among them.
The $Q_{\rm f}$-singlets are decomposed into those of $G$ such that
\begin{eqnarray}
\{\varphi_{\rm s}^{\rm U}\}
\Rightarrow \{\varphi_{\rm s}\}_0 + \{\varphi_{\rm s}\}_1.
\label{G-decomp-beta}
\end{eqnarray}
If some ghosts $\{c_{\rm s}\}_1$ appear and
they form $Q_{\rm f}$-quartets in company with $\{\varphi_{\rm s}\}_1$,
the reduction of symmetry occurs.
Then, the system is described by
\begin{eqnarray}
&~& {\mathcal{L}'}^{(\beta)}
= \mathcal{L}_{\rm U}^{(\beta)}(\{\varphi_{\rm s}\}, \{\varphi_{\rm q}\}_1)
+ {\mathcal{L}'}_{\rm gh}^{(\beta)}(\{c_{\rm q}\}_1)
+ \mathcal{L}_{\rm q}^{(\beta)}(\{{\varphi'}_{\rm q}^{\rm U}\}, \{{c'}_{\rm q}^{\rm U}\})
\nonumber \\
&~& ~~~~~~~~~~~~~
+ {\mathcal{L}'}_{\rm int}^{(\beta)}(\{\varphi_{\rm s}\}, \{\varphi_{\rm q}\}_1,
\{c_{\rm q}\}_1, \{{\varphi'}_{\rm q}^{\rm U}\}, \{{c'}_{\rm q}^{\rm U}\}),
\label{L'-beta}
\end{eqnarray}
where $\{\varphi_{\rm s}\}_0$, $\{\varphi_{\rm s}\}_1$
and $\{c_{\rm s}\}_1$
are denoted by $\{\varphi_{\rm s}\}$,
$\{\varphi_{\rm q}\}_1$ and $\{c_{\rm q}\}_1$,
and $\{{\varphi'}_{\rm q}^{\rm U}\}$ and $\{{c'}_{\rm q}^{\rm U}\}$
are also used
in place of their decompositions under $G$, for simplicity.
In this scenario, ${\mathcal{L}'}^{(\beta)}$ corresponds to
$\mathcal{L}_{\rm total}$ at $\varLambda$.\footnote{
There is a possibility that $\mathcal{L}_{\rm total}$ is also derived
after eliminating some parts ($\{\varphi_{\rm s}\}_1$)
of $Q_{\rm f}$-singlets,
starting from $\mathcal{L}^{(\beta)}$.
}
Unless extra contributions appear on the symmetry reduction,
${\mathcal{L}'}^{(\alpha)}$ and ${\mathcal{L}'}^{(\beta)}$
must fit closely with
$\mathcal{L}_{\rm total}$ at $\varLambda$ from the matching condition.
Because both
$\mathcal{L}_{\rm U}^{(\alpha)}(\{\varphi_{\rm s}\}, \{\varphi_{\rm q}\})$
and $\mathcal{L}_{\rm U}^{(\beta)}(\{\varphi_{\rm s}\},
\{\varphi_{\rm q}\}_1)$
respect the large symmetry $G_{\rm U}$,
we expect that specific relations among some parameters in
$\mathcal{L}_{\rm s}(\{\varphi_{\rm s}\})$
hold at the tree level such that
\begin{eqnarray}
\left. g_1 = g_2 = \cdots = g_l \right|_{\varLambda},
\label{g-rel}
\end{eqnarray}
reflecting the underlying symmetry.
Note that the symmetry of the system enhances
from $G$ to $G_{\rm U}$
in the limit that all ghost fields go to zero,
because other terms than
${\mathcal{L}}_{\rm U}^{(\alpha)}$
and ${\mathcal{L}}_{\rm U}^{(\beta)}$ contain ghosts.
The relations (\ref{g-rel}) hold at the quantum level
(up to some gravitational effects),
because parameters do not receive threshold corrections
at $\varLambda$ in our theory.
Then, (\ref{g-rel}) can become fingerprints or indirect proof
of the underlying symmetry and the existence of unphysical fields,
if gravitational effects are small enough or predictable.
We can test them, using the measured values for $g_k$
and the renormalization group equations,
based on $\mathcal{L}_{\rm s}(\{\varphi_{\rm s}\})$.
Note that (\ref{g-rel}) are not understood from $\mathcal{L}_{\rm s}$ alone,
because the symmetry of $\mathcal{L}_{\rm s}$
is not $G_{\rm U}$ but $G$,
as seen from the fact that
$\{\varphi_{\rm s}\}$ form not multiplets of $G_{\rm U}$
but those of $G$.
If fundamental objects have a large gauge symmetry to unify
the SM gauge group $SU(3)_{\rm C} \times SU(2)_{\rm L} \times U(1)_{\rm Y}$,
we predict the unification of gauge coupling constants~\cite{YK1}.
This prediction suggests the existence of extra particles
other than the SM ones beyond the weak scale.\footnote{
For instance, the unification of gauge coupling constants at $M_{\rm Pl}$
has been studied under the assumption
that extra particles appear at the terascale
as remnants of hypermultiplets~\cite{YK5}.
}
\section{Conclusions and discussions}
We have used the secret of finiteness in SSTs
as a guide for constructing a new type of FFTs,
and proposed a framework of
theories based on a hidden duality
between an UV and an IR region.
Concretely, quantum corrections are
invariant under the duality transformation
$t \to 1/(\tilde{\varLambda}^4 t)$,
where $t$ is a proper time for a particle running in the loop
and $\tilde{\varLambda}$ is a fundamental scale $\varLambda$
up to some factor.
From the requirement that radiative corrections should be also
invariant under the duality transformation
for external world lines,
we have arrived at the feature that
physical quantities cannot receive any radiative corrections
at $\tilde{\varLambda}$, neglecting the gravitational effects.
It is compatible with models possessing a specific fermionic symmetry.
The low-energy physics $(q^2 \ll \tilde{\varLambda}^2)$ can be described by
a renormalizable effective field theory
whose constituents are $Q_{\rm f}$-singlets,
and physical quantities can be calculated precisely
using the low-energy theory alone.
Theories can be tested indirectly by relations among parameters,
reflecting underlying symmetries, in case that the system
has a large symmetry at the ultimate level.
Here, let us summarize features of fundamental theory, in the broken phase.
The theory is defined just at the fundamental scale.
Values of fundamental parameters are given there
as initial conditions.
The sector with ordinary particles alone has the large symmetry
$G_{\rm U}$,
that can originate from a characteristics of fundamental objects.
Ordinary particles are decomposed into multiplets of the subgroup $G$.
Some submultiplets are observed,
and others form $Q_{\rm f}$-quartets
and become unphysical in the presence of ghosts.
The reduction of symmetry occurs from $G_{\rm U}$ to $G$,
as a result that ghosts are not multiplets of $G_{\rm U}$
but those of $G$.
The symmetry of the system enhances
in the limit that all ghost fields go to zero.
There are serious problems related to ghost fields.
First, in most cases, ghosts require the non-local interactions~\cite{YK1},
and this fact might suggest that
fundamental objects are not point particles but
extended objects, and
a formulation using extended objects should be required
to describe interactions containing ghosts consistently.
Second, the origin of incomplete multiplets for ghosts is not clear.
We have introduced two scenarios.
One is that some ghosts are reduced
for the system that
all members are $Q_{\rm f}$-quartets as a beginning.
The other is that ghosts emerge for the system that
some ordinary particles exist as $Q_{\rm f}$-singlets.
In this case, ghosts might appear as solitonic states.
In any case,
both scenarios seem to be unrealistic or surreal,
because a drastic change for the degrees of freedom
is needed to realize them.
Though we recognize that quantum field theory is an extremely useful tool
to describe the lower-energy dynamics than $\varLambda$,
there is a possibility that it has limitations, i.e.,
every physical phenomenon at $\varLambda$
cannot be described completely in the framework of field theory
with finite numbers of fields.
It is less unnatural that the number of particle species changes
for the system with infinite kinds of fields
than for that with finite kinds of ones.
Hence, it would be interesting to study the reduction
or emergence of ghosts,
using a theory of extended objects.
\section*{Acknowledgments}
The author thanks Prof. G. Zoupanos
for valuable discussions and useful information
of finite field theories.
This work was supported in part by scientific grants
from the Ministry of Education, Culture,
Sports, Science and Technology under Grant No.~22540272.
|
train/arxiv
|
BkiUbR7xK6EuNBRhGa5L
| 5 | 1 |
\section{Technical Details for Section \ref{sec:complexity}} \label{sec:complexity_appendix}
\subsection{Rigidity-from-system is uncomputable: Technical details}
Here we provide an illustration in Figure~\ref{fig:turing_machine} of an embedding of our construction at three different points.
\begin{figure}[htp]
\centering
\subfloat{%
\label{fig:flexible_turing_machine}%
\includegraphics[width=0.4\textwidth]{images/flexible_turing_machine}
}%
\subfloat{%
\label{fig:rigid_turing_machine}%
\includegraphics[width=0.4\textwidth]{images/rigid_turing_machine}
}%
\quad
\caption{(a) As the Turing machine is being simulated, the assembly is grown with flexible bonds, allowing every row to be flexible relative to one another and the entire assembly to fold back and forth like an accordion. (b) If the Turing machine ever halts, a ``backbone'' of tiles connected through rigid bonds is grown on the side of the assembly, making the entire structure rigid.}
\label{fig:turing_machine}
\end{figure}
First, we consider a general structure of a commonly used type of aTAM tile assembly system for simulating the behavior of Turing machines. A zig-zag aTAM system is one which grows in a strict row-by-row ordering. More specifically, the first row grows either left-to-right or right-to-left, completely, at which point the second row begins growth in the opposite direction. When it completes, the third grows, again in reversed direction, and so on. Given a Turing machine $M$, $M$ can be simulated by a temperature-$2$ zig-zag aTAM system, $\mathcal{P}$ say, such that if $M$ halts, a final ``halting'' tile attaches to the westernmost column in the northernmost row of the terminal assembly of $\mathcal{P}$. One can also show that $M$ can be simulated by a zig-zag system which produces a single terminal assembly such that the westernmost tiles of this assembly (possibly including the halting tile) are colinear. Moreover, such an aTAM system gives rise to an FTAM system, $\calT$ say, where the tile types of $\calT$ are identical to the tile types of the aTAM system and all glues are rigid.
To show Theorem~\ref{thm:Rigidity}, we consider the FTAM system $\calT'$ that is obtained from $\calT$ by (1) modifying the glues on north and south edges so that they are flexible, and (2) adding appropriate glues to tile types and tiles to the tile set of $\calT$ so that if $M$ halts, tiles of these types initially bind to the west edge of the halting tile (via an added strength-$2$ glue) of the terminal assembly of $\calT'$, and then cooperatively bind one at a time along the west edges of the westernmost tiles of this terminal assembly (via strength-$1$ glues added to these west edges and north/south glues of the tiles of the additional tile types) to form a single tile wide column of tiles each of which is bound to a westernmost tile in the terminal assembly of $\calT'$. Moreover, the north and south glues of the tile types that bind to form the column of westernmost tiles are rigid. We call such a column of tiles a ``backbone''. Then, as the east/west glues of tiles belonging to any row of tiles in the terminal assembly of $\calT'$ are rigid, and since a backbone of tiles self-assembles iff $M$ halts, we see that the terminal assembly of $\calT'$ is rigid iff $M$ halts. In other words, we have a system such that for any terminal assembly $\alpha$, $\alpha$ is rigid iff $M$ halts. See Figure~\ref{fig:turing_machine} for an intuitive description of self-assembly in $\calT'$ for a simulation of a machine that halts. This suffices to show Theorem~\ref{thm:Rigidity}.
%
\subsection{Rigidity-from-assembly is co-NP-complete: Technical details} \label{sec:assembly-rigidity-append}
Here we provide technical details for the proof of Lemma~\ref{lem:rigidity-from-assembly-NP-hard}.
To prove Lemma~\ref{lem:rigidity-from-assembly-NP-hard}, we will reduce 3SAT to the complement of the rigidity-from-assembly problem. For our reduction, we introduce a new construction that we will subsequently refer to as the \emph{3SAT machine}. This is a computable assembly that is made up of four gadgets that connect together and will be able to reconfigure if and only if the corresponding 3SAT formula has a satisfying assignment.
\begin{figure}[htp]
\centering
\includegraphics[width=0.70\textwidth]{images/simulater_trivial.png}
\caption{The trivial state that the assembly can exist in, as shown in the simulator}
\label{fig:trivial_in_simulator}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[width=0.80\textwidth]{images/simulater_satisfied_side.png}
\caption{The satisfied state, as shown in the simulator from the $-Y$ perspective}
\label{fig:satisfied_in_simulator_side}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[width=1.00\textwidth]{images/simulater_satisfied_top.png}
\caption{The satisfied state, as shown in the simulator from the $+Z$ perspective}
\label{fig:satisfied_in_simulator_top}
\end{figure}
\begin{figure}[htp]
\centering
\subfloat{%
\label{fig:trivial_state}%
\includegraphics[width=0.2\textwidth]{images/trivial_state.jpg}
}
\enskip
\subfloat{%
\label{fig:satisfying_state}%
\includegraphics[width=0.35\textwidth]{images/satisfying_state.jpg}
}
\subfloat{%
\label{fig:state_compass}%
\includegraphics[width=0.18\textwidth]{images/state_compass.jpg}
}
\enskip
\subfloat{%
\label{fig:invalid_state}%
\includegraphics[width=0.22\textwidth]{images/invalid_state.jpg}
}
\enskip
\caption{Mock-up of (a) the initial state, (b) the alternate state, (c) the axes, and (d) the invalid state.}
\label{fig:machine}
\end{figure}
Before we get into the details, we define terms and prove lemmas that we will use as tools in the overall proof. We define \emph{entanglement} to be the cause-effect relationship of connections within one set of faces in which the inversion of one connection causes the inversion of all other connections. In other words, a set of faces that have been entangled form a \emph{rigid component}, i.e. a subassembly that cannot reconfigure without fully inverting into a chiral configuration.
We define a \emph{traditional 4-sided loop}, as a cycle of faces in which each face is bound on opposite ends to two other faces. In other words, both pairs of faces that are opposite of each other will exist in planes in the same orientation $\{XY, YZ, ZX\}$. Notice that a traditional 4-sided loop will always be a rigid component. The four flexible bonds between the four faces that make up the loop will always have to be either a $\{U,U,U,U\}$ configuration or a $\{D,D,D,D\}$ configuration.
\begin{claim}\label{clm:one_piece_not_inverted}
Given an assembly $\alpha$ and a configuration $c_\alpha$, for any rigid component $comp_{rigid}$ in $\alpha$, if $\alpha$ has another valid, non-trivial configuration $c_\alpha'$ that it can exist in, then there must also exist another valid, non-trivial configuration $c_\alpha''$, where $c_\alpha'$ may or may not equal $c_\alpha''$, in which the rigid component $comp_{rigid}$ is not inverted.
\end{claim}
\begin{proof}
Assume that in $c_\alpha'$, $comp_{rigid}$ is not inverted. Then $c_\alpha' = c_\alpha''$. However, if we assume that in $c_\alpha'$, $comp_{rigid}$ is inverted, then $c_\alpha' = inverse(c_\alpha'')$.
\end{proof}
\begin{claim}\label{clm:one_piece_rigid}
As a corollary of Claim~\ref{clm:one_piece_not_inverted}, for any rigid component $comp_{rigid}$ in $\alpha$, if $\alpha$ in $c_\alpha$ does not have another valid configuration with $comp_{rigid}$ in the same orientation, $\alpha$ itself is a rigid component.
\end{claim}
These claims give us a powerful tool in proving that pieces of an assembly are rigid components, since it allows us to assume one smaller rigid component will not reorient and then examine if any other pieces can reorient in relation to it.
\begin{claim}\label{clm:two_piece_entanglement}
If two rigid components $comp_{rigid}^1$ and $comp_{rigid}^2$ share two faces $p_1$ and $p_2$ that are not coplanar, then $comp_{rigid}^1$ and $comp_{rigid}^2$ can be entangled into one rigid component.
\end{claim}
\begin{proof}
Proof by contradiction. Assume $comp_{rigid}^1$ can reorient relative to $comp_{rigid}^2$. Since $comp_{rigid}^1$ is rigid in and of itself, the only way it can reorient is to invert. Now, assume the inversion happens over $p_1$. In this case, $p_2$ must now be flipped over the plane that $p_1$ exists in. Since $p_2$ is also part of $comp_{rigid}^2$ and is in a new location, $comp_{rigid}^2$ must also have reoriented to avoid breaking bonds with $p_2$, the only way to do which would be inverting. Now, in the case that the inversion happens over another face, the plane that contains that face cannot contain both $p_1$ and $p_2$ (since they are not coplanar by the claim) and therefore at least one of the shared faces ($p_1$ or $p_2$) will be in a new location. This means that $comp_{rigid}^2$ still had to reorient. Since both of these cases lead to contradictions, $comp_{rigid}^1$ and $comp_{rigid}^2$ must be part of the same rigid component.
\end{proof}
This provides us another powerful tool, showing that two shared non-coplanar faces is all that is required to entangle two rigid components into one.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{images/sat_machine_layout.png}
\caption{The full layout of the 3SAT machine. Each box highlights a section of the assembly that is duplicated with respect to the size of the corresponding 3SAT formula.}
\label{fig:machine_layout}
\end{figure}
Now that we have some tools to use in our proof, we give a proof overview. Figure~\ref{fig:machine_layout} shows the machine assembly broken down into individual components in the form of a face graph. The highlighted pieces indicated the components that are duplicated as the corresponding 3SAT problem has more clauses and more variables. The first step in the proof will be to take some of these pieces and entangle them together into the major rigid components of our construction. These include the evaluation space, variable constraint gadgets, trival assignment hat, and satisfying assignment hat. Figure~\ref{fig:sat_machine_rigid_components} shows how these rigid components will be made, while Figure~\ref{fig:sat_machine_reduced} shows what the reduced layout will look like when considering these larger pieces. Finally, we will show that the pieces interact in such a way that the rigidity of the machine gives a valid reduction from the 3SAT problem.
\begin{figure}[htp]
\centering
\subfloat[][The layout of the machine with all faces included. The faces that are in the same box will be entangled together into rigid components.]{%
\label{fig:sat_machine_rigid_components}%
\includegraphics[width=\textwidth]{images/sat_machine_rigid_components.png}
}
\quad
\subfloat[][The layout when considering the rigid components as one piece. The remaining flexible sections are the only ones that can reorient and only in the case that the corresponding 3SAT instance has a satisfying assignment.]{%
\label{fig:sat_machine_reduced}%
\includegraphics[width=0.75\textwidth]{images/sat_machine_reduced.png}
}
\caption{The before and after diagrams with respect to the entanglement of rigid components.}
\label{fig:machine_reduction}
\end{figure}
\subsubsection{Evaluation Space (ES)} \label{evaluationSpace}
The evaluation space, shown in Figure~\ref{fig:evaluation_space}, is a frame that facilitates interactions between the variable constraint gadgets, trivial assignment hat, and satisfying assignment hat. The frame consists of three \emph{evaluation gadgets} per row (which we colloquially refer to as ``bumps''), with as many rows as clauses in the corresponding 3SAT instance. The evaluation gadget consist of three tiles that form a bump over or under the open holes in the evaluation space. The three tiles that compose the evaluation gadget consist of two tiles in the $ZX$ plane and a tile in the $XY$ plane connecting them. An evaluation gadget forced up (Figure~\ref{fig:variable_false}) represents a literal evaluated to \emph{False}, while an evaluation gadget forced down (Figure~\ref{fig:variable_true}) represents a literal evaluated to \emph{True}. In the situation where the corresponding 3SAT instance does not have a satisfying assignment, the evaluation gadgets will collide with tiles from the satisfying assignment hat, preventing any configuration other than the initial. Every variable in the 3SAT instance has a corresponding evaluation gadget unless that variable has only positive literals or negative literals. If this is the case, we exclude that particular evaluation gadget since all instances of that variable can be assumed to be \emph{True}, so it is unnecessary for that evaluation gadget to ever collide with the satisfying assignment hat.
Attached to the evaluation space on opposite sides are the satisfying assignment hat and the trivial assignment hat. These pieces are attached to the evaluation space in such a way that exactly one of the them will be pressed against the evaluation space in any possible configuration. The satisfying assignment hat can only reorient down onto the evaluation space (as shown in Figures \ref{fig:satisfying_state}, \ref{fig:satisfied_in_simulator_side}, and \ref{fig:satisfied_in_simulator_top}) if and only if there is a solution to the corresponding 3SAT instance; otherwise, the configuration where the trivial assignment hat is pressed against the evaluation space (as shown in Figures \ref{fig:trivial_state} and \ref{fig:trivial_in_simulator}) will be the only valid configuration that the assembly can exist in.
\begin{figure}[htp]
\centering
\subfloat[]{%
\label{fig:variable_false}%
\includegraphics[width=0.33\textwidth]{images/variable_true.png}
}%
\subfloat[]{%
\label{fig:variable_true}%
\includegraphics[width=0.33\textwidth]{images/variable_false.png}
}%
\subfloat[]{%
\label{fig:formula}%
\includegraphics[width=0.33\textwidth]{images/formula.png}
}
\caption{The setup of the evaluation space. The tiles that make up the evaluation gadgets are included only in (a) and (b) to show the difference between a \emph{False} variable and a \emph{True} variable, respectively. Three individual evaluation gadgets together make up a clause, and multiple clauses together make up a formula, shown in (c).}
\label{fig:evaluation_space}
\end{figure}
\subsubsection{Variable Constraint Gadgets (VCG's)} \label{variableConstraints}
The variable constraint gadgets sit below the evaluation space (in the $-Z$ direction) and interact with the evaluation gadgets to ensure that each instance of a variable, even the negated instances, agree with each other. Only variables that have at least one positive literal and at least one negative literal will have a variable constraint gadget associated with that specific variable. Since other variables with all positive or all negative instances have no evaluation gadgets, they also do not have variable constraint gadgets.
The variable constraint gadgets are designed such that all eligible variables are assigned a unique level at a different $Z$ value below the evaluation space. Each positive / negative instance of a single variable has two parallel strings of tiles called \emph{chimneys} that connect the $XY$ tile of all evaluation gadgets that correspond to that positive / negative variable to the \emph{variable constraint level} (VCL) of that positive / negative variable. For example, all evaluation gadgets for $x_1$ literals have chimneys that extend to the same variable constraint level at a specified $Z$ value, as well as all evaluation gadgets for $\overline{x_1}$ literals.
The variable constraint levels, shown in Figure~\ref{fig:variable_constraint_gadget}, connect all instances of a variable. There are two levels per variable, one for all positive literals, i.e. $x_1$, and one for all negative literals, i.e. $\overline{x_1}$. The two levels are then connected by a \emph{bridge}, which is attached to the end of both \emph{crossbars}. The bridge is simply a domino that connects the two levels in the $ZX$ plane. Its purpose is to ensure that the levels exist in $XY$ planes that are two units in the $Z$ direction apart. Since the levels are connected to the chimneys and to the respective evaluation gadgets, we will show that this ensures that the evaluation gadgets for positive and negative literals of the same variable always disagree.
\begin{figure}[htp]
\centering
\subfloat{
\label{fig:variable_contstraint_level}%
\includegraphics[width=0.8\textwidth]{images/variable_contstraint_level.jpg}
}
\subfloat{
\label{fig:vcl_compass}
\includegraphics[width=0.2\textwidth]{images/vcl_compass.jpg}}
\caption{The layout of a pair of variable constraint levels (VCL's). Each variable constraint gadget (VCG) has a ``positive'' level (blue) and a ``negative'' level (red). Each evaluation gadget is connected to its corresponding VCL by a vertical string of tiles (the chimneys) that descends in the $Z$ direction to the VCL designated for that variable. Once in that plane, the tiles within each VCL extend down the strips to the crossbars. The crossbars of two opposite levels are connected by a two tile long bridge.}
\label{fig:variable_constraint_gadget}
\end{figure}
\begin{claim} \label{clm:vcl_rigid}
The variable constraint gadgets can only reorient by changing truth value or being inverted.
\end{claim}
\begin{proof}
To recap, for any variable $x$ with $p \ge 1$ instances of the positive literal and $n \ge 1$ instances of the negated literal, the variable constraint gadget is made up two levels (one for $x$ and one for $\overline{x}$), $p$ pairs of chimneys to one level, connected to the evaluation space collectively by $p$ evaluation gadgets, $n$ pairs chimneys to the other level, connected to the evaluation space collectively by $n$ evaluation gadgets, and one bridge between the two levels. Evaluation gadgets are made up of three pieces, one in the $XY$ plane and two in the $ZX$ plane.
First, notice that an $XY$ piece of an evaluation gadget, a pair of chimneys, and the level are connected to make a 4 sided loop. Take anyone of these loops and assume it is rigid by Claim~\ref{clm:one_piece_rigid}. Obviously, the orientation of every other chimney loop that exists between the evaluation space and that specific level must also remain unchanged in order for them to connect to the level. Now, we have a rigid component of one level, all the chimneys attached to it, and $XY$ pieces of their respective evaluation gadgets. This rigid component can move up or down by two units due to reorientation of the $ZX$ pieces of the evaluation gadgets. We apply the same argument to level, chimneys, and evaluation gadgets corresponding to $\overline{x}$.
Now, our variable constraint gadget for $x$ consists of two entangled rigid components (which we will now call $l$ and $r$), two sets of $ZX$ pieces from the evaluation gadgets (one connected to $l$ and one connected and $r$), and the bridge. We already know that a reorientation can happen that moves either $l$ or $r$ up two units in the $Z$ direction, the other down two units in the $Z$ direction, and causes a transformation of all $ZX$ pieces from the evaluation gadgets and the bridge. However, this is the only reorientation that can happen. To show this, notice that any pair of $ZX$ pieces for the evaluation gadgets must agree to be connected to the evaluation space through both $U$ bonds or both $D$ bonds in order to have a gap of one between them for the $XY$ piece to fit into. Since the rigid component $l$ or $r$ has already been shown to be rigid from one $ZX$ evaluation piece to another, the orientation of this pair of $ZX$ evaluation pieces will force the orientation of all other pair of evaluation pieces also connected to the $l$ or $r$ piece. The same argument goes with the $ZX$ evaluation pieces connected to the other $l$ or $r$ rigid component. Notice that the $l$ and $r$ component can't be on the same level, otherwise the two ends connected by the bridge would be incident on each other, leaving no room for the bridge itself. Therefore, one rigid component (either $l$ or $r$) must have each of its $ZX$ evaluation pieces be up, while the other rigid component must have each of its $ZX$ evaluation pieces be down. Therefore, the only ambiguity in the configuration of the variable constraint gadget (other than the trivial ambiguity of inversion) is determining which rigid component between $l$ and $r$ is up and which rigid component is down.
Because all variable constraint gadgets have the same structure, this argument holds for each variable constraint gadget corresponding to each variables in the formula.
\end{proof}
\subsubsection{Trivial Assignment Hat (TAH)} \label{trivialAssignment}
The purpose of the trivial assignment hat is to prevent the reorientation of any piece in the assembly as long as it is pressed against the evaluation space. It does this in two ways. First, it has what we labeled as ``Force Bumps'' that force the orientation of all evaluation gadgets and variable constraint gadgets, as shown in Figure~\ref{fig:tah_inkscape}. These are designed such that all negative literals in the evaluation space have a corresponding Force Bump on the trivial assignment hat and all positive literals do not. This effectively assigns a value of \emph{False} to all variables in the formula, hence the name ``trivial assignment hat'', forcing the evaluation gadgets and variable constraint levels of all negative variables down and thereby forcing the evaluation gadgets and variable constraint levels of all positive variables up.
The second way the trivial assignment hat disables reorientation is by having an additional piece that blocks otherwise-free pieces in the satisfying assignment hat from moving. This piece can be seen in Figure~\ref{fig:tah_simulator}. It will later be proven that this is sufficient to show that the machine is rigid in the case when no satisfying assignment exists to the corresponding 3SAT problem.
\begin{figure}[htp]
\subfloat{%
\label{fig:tah_inkscape}
\includegraphics[width=0.5\textwidth]{images/trivial_assignment_hat.png}
}%
\subfloat{%
\label{fig:tah_simulator}
\includegraphics[width=0.5\textwidth]{images/tah_real.png}
}
\caption{The layout of the trivial assignment hat. The left image just shows the Main Bottom piece along with the Force Bumps. The right is an image take from FTAM visualizer. The bumps are aligned with variables in the evaluation space. From here, the gadget would be flipped over and pressed onto the top of the evaluation space to force the orientation of any variables that were free to move. The additional tiles at the bottom left of the right image are used to block the satisfying assignment checkers whenever the assembly is in the trivial state.}
\label{fig:trivial_assignment_hat}
\end{figure}
\begin{claim}
The trivial assignment hat can't reorient without being inverted.
\end{claim}
\begin{proof}
To clarify some of the terminology of this section refer to Figure~\ref{fig:tah_labeled}, Figure~\ref{fig:machine_layout}, or Figure~\ref{fig:machine_reduction}.
Here we will show that most faces in the trivial assignment hat are part of a traditional four sided loop utilizing the ``Left Side'' piece and one of the two ``Right Side'' pieces, i.e. every loop will look like: Left Side, piece one, Right Side, piece two, Left Side. The planar sections that will be substituted into ``piece one'' and ``piece two'' are as follows: (Base Bottom, Main Top), (Base Front, Base Back), (Main Bottom, Main Top), (Main Bottom, Blocker Top), (Connection Front, Connection Back), (Blocker Front, Blocker Back), (Blocker Bottom, Blocker Top). By Claim~\ref{clm:two_piece_entanglement}, since each loop contains Left Side and one of the two Right Side's, we are left with two rigid components (corresponding to the split up of the Right Side's). However, again by Claim~\ref{clm:two_piece_entanglement}, both rigid components share Left Side, Main Top, and Main Bottom, none of which are coplanar, and can therefore be entangled into one rigid component. The only remaining planar sections in the trivial assignment hat are the Force Bumps, each of which forms a traditional four sided loop with the Main Bottom. Since each of these loops is rigid, the piece that is opposite of the Main Bottom must be in the same orientation as Main Bottom but in a plane that is one unit away in either normal direction. Since the Main Top is in the same orientation and in the plane that is one unit above the Main Bottom, no Force Bump can invert because it would collide with the Main Top. Therefore, each Force Bump can be entangled into the rigid component, leaving us with one rigid trivial assignment hat.
\end{proof}
\begin{figure}[htp]
\centering
\subfloat{%
\label{fig:tah_labeled}
\includegraphics[width=\textwidth]{images/TAH_labeled.png}
}
\quad
\subfloat{%
\label{fig:sah_labeled}
\includegraphics[width=\textwidth]{images/SAH_labeled.png}
}
\quad
\caption{Labeled images showing the names of each face in the two assignment hats.}
\label{fig:hat_labels}
\end{figure}
\subsubsection{Satisfying Assignment Hat (SAH)} \label{satisfyingAssignment}
The satisfying assignment hat comes down and presses up against the evaluation space to make an additional valid configuration for the whole machine if and only if there is a solution to the inputted 3SAT problem. The satisfying assignment hat works by using a \emph{checker} for each clause in the formula. The checker is encased in a structure, shown in Figure~\ref{fig:SAH_side_view}, that allows it to be in three different orientations, shown in Figure~\ref{fig:SAH_top_view}. These different orientations each project a blocking tile, shown in yellow, to a different location to block one of the three variables in the clause. The checkers can only be in three different orientations due to the structure the checker is encased in. In Figure~\ref{fig:SAH_side_view}, we can see this illustrated by the rigid outer structure and the partly flexible checker inside the structure. Marked in yellow, the flexible points in the checker allow for the checker to only ``fold into'' one of the available, raised parts of the structure. If the checker were to fold into a non-raised portion of the structure, the tiles would simply conflict with the outer structure, shown in red in Figure~\ref{fig:SAH_side_view}. If the satisfying assignment hat is able to press against the evaluation space, that means the checker for each clause has found at least one of the variables to be \emph{True} in each clause, allowing the checker to be in the orientation that projects the blocking tile over the evaluation gadget that corresponds to that variable.
\begin{figure}[htp]
\centering
\includegraphics[width=\textwidth]{images/SAH_side_view.png}
\caption{The structure that allows the checker piece to move to one of three different locations. The bottom right orientation is invalid because the checker piece conflicts with the outer structure.}
\label{fig:SAH_side_view}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[width=0.5\textwidth]{images/SAH_top_view.png}
\caption{A top view of a satisfying assignment hat with the three checkers occupying the three different possible positions.}
\label{fig:SAH_top_view}
\end{figure}
\begin{claim}
The outer structure of the satisfying assignment hat can't reorient without being inverted.
\end{claim}
\begin{proof}
To clarify some of the terminology of this section refer to Figure~\ref{fig:sah_labeled}, Figure~\ref{fig:machine_layout}, or Figure~\ref{fig:machine_reduction}.
First, notice the traditional 4-sided loop between the Tunnel Bottom and Tunnel Top utilizing the Tunnel Top Side Support's. Similarly, the Tunnel Bottom and Connection 2 piece form a 4-sided loop. We can entangle these two loops because, if one were to invert without the other, the three Tunnel Slack~2 pieces would have to collide with the Tunnel Bottom piece in order to connect the two. Since the Connection 2 piece and the Tunnel Top are part of the same rigid component, we can also entangle the three Tunnel Slack 2 pieces. Now, notice that the Tunnel Bottom, Tunnel Back, and any Left Starter or Right Starter piece form a vertex. This means that, in relation to the Tunnel Bottom, the Tunnel Back must either be ``Up'' or ``Down'' but cannot be ``Straight''. However, if it were to be ``Down'' (thereby inverted relative to the rigid component), its connection to the Tunnel Back would be $4$ down in the $Z$ dimension and $4$ to the left in the $X$ dimension to the connection between the Tunnel Slack Front 1 and the Connection 2. Since this Manhattan distance is $8$ and the combined length of the four piece that need to make up this distance (Connection 1, Tunnel Slack Back 1, Tunnel Slack Top 1, and Tunnel Slack Front~1) is also $8$, the pieces would have to form a shortest path from one connection to the other. However, since every shortest path would intersect with either the Tunnel Bottom or Tunnel Back, it cannot make this distance. Therefore, the Tunnel Back (and Left Starters, Right Starters, and First Sectionals) can be entangled into the rigid component. Now, the Manhattan distance that needs to be made up by the 4 pieces is just $4$. Since the respective lengths of the 4 pieces in the order previously given are $1$, $2$, $3$, $2$, the only arrangement that would give a total displacement of $4$ would for the $2$ length pieces (Tunnel Slack Back 1 and Tunnel Slack Front 1) to cancel out and for the other two pieces (Connection~1 and Tunnel Slack Top 1) to make up the distance. This leaves us with two options, the back and front slack folded away from the Tunnel Bottom or towards the Tunnel Bottom. As mentioned earlier, if the slacks fold towards the Tunnel Bottom, the Tunnel Slack Top 1 would collide with the Tunnel Bottom. Therefore, there is only one configuration of these pieces with respect to the rigid component, meaning they can be entangled, leaving us with one big rigid component.
\end{proof}
\subsubsection{3SAT Checker Machine} \label{3SAT}
There is one last piece of the assembly we must introduce, which we refer to as the \emph{rope}. This is a $15 \times 1$ face of tiles that connects the trivial and satisfying assignment hats. It connects to the Tunnel Top on the SAH and the Main Top on the TAH. It's purpose is to prevent the invalid configuration from Figure~\ref{fig:invalid_state} where neither hat is pressed against the evaluation space.
Now that we have introduced all the pieces of the assembly, we need to focus on how the remaining flexible bonds can move relative to one another. We group the remaining flexible bonds into three groups: the main loop bonds, the checker bonds, and the variable constraint bonds. The main loop group consists of four bonds: the ES to the SAH, the SAH to the rope, the rope to the TAH, and the TAH to the ES. Notice that these bonds form a loop. By looking at Figure~\ref{fig:main_loop}, you will notice this loop can only take two configurations, ``UDSS'' and ``SSDU'' (with respect to ES-SAH, SAH-rope, rope-TAH, TAH-ES, with the normal of the rope pointed in the $+Z$ dimension and the normal of the ES pointed in the $-Z$ dimension). We refer to these two configurations as the trivial state and satisfied state, respectively.
\begin{figure}[htp]
\centering
\includegraphics[width=\textwidth]{images/main_loop.png}
\caption{A view of the main loop of the assembly from the negative $Y$ direction. The arrows represent the displacement between the bonds from the evaluation space to each hat and the bonds from each hat to the rope. The orientation labels are for the bonds between the evaluation space and each hat. The illustration is meant to show that the only positions that the two hats can be in such that their bonds to the rope are in the same $Y$ plane and exactly 15 units apart in the $X$ plane are the two labeled positions for the rope.}
\label{fig:main_loop}
\end{figure}
The checker group consists of 16 bonds for each checker gadget in the SAH. These 16 bonds can be viewed as two subgroups of 8 which must be the same configuration from the set of ``UDDUUDDU'', ``UDDUSSSS'', and ``SSSSSSSS'' (with respect to the sequences of bonds starting at the First Sectional piece and ending with the Third Sectional and Blocker piece, with the normal of these two pieces pointed at the Tunnel Top). The sequences for the two subgroups of any checker gadget must match since both sequences of bonds end with the Third Sectional and Blocker piece, which connects with both previous pieces at the same $X$ coordinate. As opposed to a formal proof, we direct the reader to Figure~\ref{fig:SAH_side_view} to see the reasoning that these are the only three possible configurations. Note that the invalid configuration shown in the illustration would be ``SSSSUDDU''.
The variable constraint group consists of all the bonds between the ES and the $ZX$ evaluation gadget pieces, all the bonds between the $ZX$ evaluation gadget pieces and the $XY$ evaluation gadget pieces, and all the bonds between the VCL's and the bridges. As we saw in Claim~\ref{clm:vcl_rigid}, this entire group of bonds can only have as many possible configurations as there are possible assignments to the variables in the corresponding boolean formula, i.e. $2^n$ where $n$ is the number of variable in the formula. An intuitive explanation of this is that each variable has a variable constraint gadget, each of which can only take two states, one where the variable is \emph{False} and one where the variable is \emph{True}.
Finally, we show what can happen when the main loop is in the trivial state and satisfied state.
\begin{claim}
Neither the trivial nor the satisfying assignment hat can invert without the other.
\end{claim}
\begin{proof}
Notice that the displacement vectors in Figure~\ref{fig:main_loop} between the bond where either hat connects to the ES and the bond where the same hat connects to the rope is some rotation of the vector $(2,3)$ in the $ZX$ plane (it is the reverse of the apparent vector in the Figure, since the illustration is from the $-Y$ perspective instead of the $+Y$ perspective). If one hat were to invert without the other, the displacement vector of the inverted hat would change to some rotation of the vector $(3,2)$. Then, for the two displacement vectors to have the same $Z$ value (to have the ends of the rope be in the same $Z$ plane) the $X$ values must be the negation of each other. This leaves the possible distance between the two bonds at the ends of the rope to be $9$, $11$, $19$, or $21$. These numbers were computed by taking the length of the evaluation space, $15$, and adding or subtracting both of the possible $X$ coordinates of the displacement vectors, $2$ or $3$, since the displacement vectors must point in opposite $X$ directions. Since none of these possible distances are $15$, the actual distance of the rope, then none of the possible configurations resulting from an inversion of one hat without the other are valid.
\end{proof}
\begin{claim} \label{clm:trivial_rigid}
If the main loop is in the trivial state, no other bond can reconfigure.
\end{claim}
\begin{proof}
First, we also prove that the variable constraint gadgets can't invert without the SAH and TAH. This is easy to see from Figures~\ref{fig:simulater_machine} and \ref{fig:machine}. Since the TAH lays across the entire ES, if the variable constrain gadgets were to invert without the other main components, the chimneys would collide with the Main Bottom and Main Top pieces in the TAH. Therefore, the variable constraint gadgets can't reorient without both of the other main components reorienting as well.
Now, we show the remaining flexible bonds (the checker group and the variable constraint group) can't reorient either. In the trivial state, the checkers in the SAH are blocked by the extension of the TAH, which prevents the bonds from taking any configuration other than ``UDDUUDDU''. The bonds in the variable constraint group are also forced into a configuration by the TAH, since the Force Bumps on the TAH force all evaluation gadgets for positive literals to be popped down, thereby forcing their respective VCL's down, thereby forcing their opposing VCL's up, and thereby forcing all evaluation gadgets for negative literals to be popped up. Therefore, if the main loop bonds are fixed in the trivial state, the entire assembly is rigid.
\end{proof}
\begin{claim} \label{clm:satisfied_flexible}
If the main loop is in the satisfied state, the other free flexible bonds can only configure in a way such that no tiles in the assembly are overlapping if and only if there is a satisfying assignment to the corresponding 3SAT formula.
\end{claim}
\begin{proof}
First, we address the inversion of variable constraint gadgets. Since the TAH is no longer pressed against the ES, the variable constraint gadgets could invert across the $XY$ evaluation gadget pieces to which that specific variable constraint gadget is bound. However, this inversion would not affect the functionality of the variable constraint gadget, since it would still cause all positive literals of one variable to be one truth value and all negative literals of that variable to be the other truth value. Therefore, we can ignore any potential inversion of these gadgets in this proof. For elegance, the Tunnel Top piece of the SAH could be extended such that it blocks these inversions.
For the forward direction of the bijection, we prove that, ``If the assembly configures such that no tiles overlap, there is a satisfying assignment''. We know that, in the satisfied state, the Third Sectional and Blocker piece in each checker gadget of the SAH must occupy the tile location that is one unit in the $Z$ dimension above one of the three evaluation gadgets in the clause to which the checker gadget corresponds. Therefore, if no tiles overlap, one of the literals in that clause must have evaluated to \emph{True} so that it is popped down in the $Z$ dimension, allowing the checker gadget to occupy the space above. If this evaluation gadget represents a positive literal, we can start building the satisfying assignment by setting that variable to \emph{True}. If the evaluation gadget represents a negative literal, we set the variable to \emph{False}. We do this for each checker gadget. We know that we will never have to reassign a variable a new value, since the variable constraint gadgets ensure that every literal of the same type agrees with each other. After we go through each checker gadget, we have a variable assignment that satisfies every clause in the boolean formula. Any variable that hasn't been assigned can be given either truth value.
For the reverse direction, we prove that, ``If there is a satisfying assignment, the assembly can configure such that no tiles overlap''. Here, we can take every variable in the satisfying assignment, and if it is assigned the value \emph{True}, we raise the VCL that corresponds to the negative literals of that variable (and pop up the connected evaluation gadgets) and lower the VCL that corresponds to the positive literals of that variable (and pop down the connected evaluation gadgets). If the variable is assigned the value \emph{False}, we do the opposite. Now, because we know this is a satisfying assignment, at least one evaluation gadget in each clause must be popped down, allowing each checker gadget to have at least one configuration that doesn't overlap with an evaluation gadget, giving us a valid configuration for the whole assembly.
\end{proof}
To prove Lemma~\ref{lem:rigidity-from-assembly-NP-hard}, we use the previous claims as follows. Take the 3SAT machine in the trivial state. By Claim~\ref{clm:satisfied_flexible}, if the corresponding 3SAT formula has a satisfying assignment, then the machine has at least one additional configuration in the satisfied state it can reconfigure into. If the corresponding 3SAT formula does not have a satisfying assignment, there are no valid configurations in the satisfied state. By Claim~\ref{clm:trivial_rigid}, there is only one configuration in the trivial state. Therefore, determining if the assembly has multiple valid configurations, the complement of rigidity-from-assembly, is polynomial time reducible from 3SAT and therefore NP-hard.
\subsection{Terminality-from-assembly is co-NP-complete: Technical details} \label{sec:assembly-terminality-append}
Here we provide technical details for the proof of Lemma~\ref{lem:terminality-from-assembly-NP-hard}.
\begin{proof}
We will again reduce this problem to 3SAT. Take the 3SAT machine used to prove Lemma \ref{lem:rigidity-from-assembly-NP-hard}. This assembly has a bond between a piece called the ``Rope'' and the TAH that is in a ``Straight'' orientation when the machine is in trivial state and in a ``Down'' orientation in the satisfied state. For both tiles that make up this bond, add a unique flexible glue that is on the side 90 degrees clockwise from the original bond on one tile and 90 degrees counterclockwise from the original bond on the other tile. This way, both unique flexible glues are pointed in the same dimension. In the trivial state, these glues are adjacent to two different tile locations that are adjacent themselves. However, if the assembly can reconfigure into the satisfied state, these glues become adjacent to a mutual tile location. Therefore, by adding a tile type to the system with the complements of both glues on adjacent sides such that it could bind in this location, we create a situation in which the assembly is not terminal if and only if the corresponding 3SAT problem has a satisfying solution. Therefore, the complement of terminality-from-assembly is NP-hard.
\end{proof}
\section{Complexity of FTAM Properties} \label{sec:complexity}
\vspace{-5pt}
In this section we consider the problem of deciding if a system produces rigid assemblies, the problem of deciding for a given assembly, if the assembly is rigid, and the problem of deciding for a given assembly, if the assembly is terminal. We first consider the problem of deciding if a system produces rigid assemblies.
\vspace{-10pt}
\subsection{Determining if a system produces a rigid assembly is uncomputable}
\vspace{-5pt}
We first show that, given an arbitrary FTAM system, determining if it produces a rigid terminal assembly is undecidable.
\begin{problem}[Rigidity-from-system]
Given an FTAM system $\mathcal{T}$, does there exist assembly $\alpha \in \termasm{\mathcal{T}}$ such that $\alpha$ is rigid?
\end{problem}
\begin{theorem} \label{thm:Rigidity}
Rigidity-from-system is undecidable.
\end{theorem}
\vspace{-5pt}
For any given Turing machine $M$, we show that $M$ can be simulated by an FTAM system that produces a single terminal rigid assembly iff $M$ halts. The full proof of Theorem~\ref{thm:Rigidity} is given in
Section~\ref{sec:complexity_appendix}.
\vspace{-10pt}
\subsection{Determining the rigidity of an assembly is co-NP-complete}\label{sec:assembly-rigidity}
\vspace{-5pt}
Now, we look at the complexity of determining the rigidity of a given assembly. %
\begin{problem}[Rigidity-from-assembly]
Given an FTAM system $\mathcal{T}$ and assembly $\alpha \in \prodasm{\mathcal{T}}$, is $\alpha$ rigid?
\end{problem}
\begin{theorem}\label{thm:rigidity-from-assembly}
Rigidity-from-assembly is co-NP-complete.
\end{theorem}
To prove Theorem~\ref{thm:rigidity-from-assembly}, we prove the following two lemmas.
\begin{lemma}\label{lem:rigidity-from-assembly-co-NP}
The complement of rigidity-from-assembly is in NP.
\end{lemma}
\begin{proof}
To illustrate this, we take an instance of the problem that contains the FTAM system $\mathcal{T}$ and assembly $\alpha \in \prodasm{\mathcal{T}}$. Our certificate in this instance will be configurations $c_\alpha$ and $c_\alpha'$. Since a configuration is simply a mapping from every flexible bond in $\alpha$ to an orientation, each configuration requires $O(|\alpha|)$ space, and thus the cerficate is polynomial in the size of $\alpha$. To determine if the certificate is valid, and thus if $\alpha$ is flexible (and therefore not rigid), we first check that $c_\alpha$ and $c_\alpha'$ are valid encodings of a configurations, meaning they each map every flexible bond in $\alpha$ to an orientation from \{``Up'', ``Down'', ``Straight''\}. Then we must ensure that $c_\alpha'$ is different than $c_\alpha$. Both of these can be done in linear time with respect to the number of flexible bonds in the assembly. Next, we compute embeddings of $\alpha$ from $c_\alpha$ and $c_\alpha'$, taking linear time in the number of tiles in the assembly. While computing the embeddings, we simply check that no tile is assigned a placement already taken by another tile, that no bonds overlap the same space, and that every tile is adjacent to the tiles it is connected to in $\alpha$ such that their glues line up correctly. Computing the embeddings and checking these conditions takes linear time with respect to the number of tiles in the assembly. If all of these conditions are met, then both $c_\alpha$ and $c_\alpha'$ are valid configurations of $\alpha$, and therefore $\alpha$ is not rigid. Since the certificate has polynomial size in relation to $\alpha$ and can be verified in polynomial time to show that $\alpha$ is not rigid, the problem of determining if $\alpha$ is rigid is in co-NP.
\end{proof}
\vspace{-10pt}
\begin{lemma}\label{lem:rigidity-from-assembly-NP-hard}
The complement of rigidity-from-assembly is NP-hard.
\end{lemma}
\vspace{-5pt}
We prove Lemma~\ref{lem:rigidity-from-assembly-NP-hard} by a 3SAT reduction. In particular, we give an FTAM system, $\mathcal{T}$ say, and show how to encode a 3SAT formula as a producible assembly of $\mathcal{T}$ in a configuration, $c$ say, such that there exists a configuration $c'$ of $\alpha$ that is distinct from $c$ iff the 3SAT formula is satisfiable.
(See Section~\ref{sec:complexity} for details.)
Finally, Theorem~\ref{thm:rigidity-from-assembly} is proven by Lemmas~\ref{lem:rigidity-from-assembly-co-NP} and \ref{lem:rigidity-from-assembly-NP-hard}.
\vspace{-15pt}
\begin{figure}[htp]
\centering
\subfloat{%
\includegraphics[width=0.240\textwidth]{images/simulater_trivial.png}}
\enskip
\subfloat{%
\includegraphics[width=0.290\textwidth]{images/simulater_satisfied_side.png}}
\enskip
\subfloat{%
\includegraphics[width=0.425\textwidth]{images/simulater_satisfied_top.png}}
\enskip
\caption{Example assembly representing a 3SAT instance, visualized in FTAM simulator, used in the proof of Lemma~\ref{lem:rigidity-from-assembly-NP-hard}.}
\label{fig:simulater_machine}
\end{figure}
\vspace{-10pt}
\vspace{-20pt}
\subsection{Determining the terminality of an assembly is co-NP-complete}\label{sec:assembly-terminality}
In addition to rigidity, terminality is another useful-to-know property of assemblies. Using much of the same logic from the previous result, we can prove a similar result regarding the terminality of arbitrary assemblies.%
\begin{problem}[Terminality-from-assembly]
Given an FTAM system $\mathcal{T}$ and assembly $\alpha \in \prodasm{\mathcal{T}}$, is $\alpha$ terminal?
\end{problem}
\begin{theorem}\label{thm:terminality-from-assembly}
Terminality-From-Assembly is co-NP-complete.
\end{theorem}
\vspace{-5pt}
To prove Theorem~\ref{thm:terminality-from-assembly}, we prove the following two lemmas.
\begin{lemma}\label{lem:terminality-from-assembly-co-NP}
The complement of terminality-from-assembly is in NP.
\end{lemma}
\begin{proof}
For an instance of the problem, we are given an FTAM system $\mathcal{T} = (T,\sigma,\tau)$ and assembly $\alpha$. Our certificate in this case includes a configuration $c_\alpha$ for the assembly $\alpha$ and a frontier location $f$. Similar to in the proof of Lemma~\ref{lem:rigidity-from-assembly-co-NP}, (and since the encoding of $f$ requires space $\le |\alpha|)$ we know that the certificate is polynomial in size to $\alpha$. Also, we can check the validity of configuration $c_\alpha$ in polynomial time. Now, we simply need to verify the frontier location $f$ (a) isn't already occupied by a tile and (b) is adjacent to tiles in $\alpha$ while it's in configuration $c_\alpha$ such that the adjacent glues allow the tile specified by $f$ to bind to $\alpha$ with bonds collectively $\ge \tau$ strength, which can be done in time $O(|T|) + O(|\alpha|)$. Since the certificate has polynomial size in relation to $\alpha$ and can be verified in polynomial time to show that $\alpha$ is not terminal, the problem of determining if $\alpha$ is terminal is in co-NP.
\end{proof}
Now, we will also show that the complement of terminality is NP-hard. A slight augmentation to the 3SAT machine assembly can be used to achieve this.
\begin{lemma}\label{lem:terminality-from-assembly-NP-hard}
The complement of terminality-from-assembly is NP-hard.
\end{lemma}
\vspace{-5pt}
To prove Lemma~\ref{lem:terminality-from-assembly-NP-hard}, we use almost identical techniques as for the proof of Lemma~\ref{lem:rigidity-from-assembly-NP-hard}, with a slight modification to the 3SAT machine so that, if and only if the 3SAT instance is satisfiable, then there will be a valid configuration of the assembly which represents the satisfying assignment, and in that configuration - and no other valid configuration - there will be a frontier location, which means that the assembly is not terminal.
Theorem~\ref{thm:terminality-from-assembly} is proven by Lemmas~\ref{lem:terminality-from-assembly-co-NP} and \ref{lem:terminality-from-assembly-NP-hard}.
\vspace{-10pt}
\section{Technical Details for Section \ref{sec:definitions}}\label{sec:definitions-append}
\begin{algorithm}[H]
\caption{A procedure to perform one step of the self-assembly process of FTAM system $\mathcal{T}$}
\label{alg:assembly-step}
\begin{algorithmic}[1]
\Procedure{\texttt{ASSEMBLY-STEP}}{$\alpha$, $\mathcal{T}$} \Comment{Takes an assembly $\alpha$ and FTAM system $\mathcal{T}$}
\If {$|\partial^{\mathcal{T}} \alpha| = 0$}
\State \textbf{return} $\alpha$ \Comment{No frontier locations remain, $\alpha$ is terminal}
\Else
\State Uniformly at random select $(t, B) \in \hat{\partial}^{\mathcal{T}} \alpha$ \Comment{Select a frontier location}
\State Attach a tile of type $t$ with bonds to tiles in $B$, $\alpha \to_1^{\mathcal{T}} \alpha'$ \Comment{Add a tile}
\State Uniformly at random select $c_\alpha' \in \mathcal{C}_{max}(\alpha')$ \Comment{Find new-bond-maximizing configuration}
\State Form all bonds possible in $c_\alpha'$ to yield $\alpha''$ \Comment{Form those bonds}
\State \textbf{return} $\alpha''$ \Comment{Return the new assembly}
\EndIf
\EndProcedure
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[H]
\caption{A procedure to perform the self-assembly process of FTAM system $\mathcal{T}$}
\label{alg:full-assembly}
\begin{algorithmic}[1]
\Procedure{\texttt{FULL-ASSEMBLY}}{$\alpha$, $\mathcal{T}$} \Comment{Takes an assembly $\alpha$ and FTAM system $\mathcal{T}$}
\State $\alpha' = \texttt{ASSEMBLY-STEP}(\alpha, \mathcal{T}$)
\If {$\alpha == \alpha'$}
\State \textbf{return} $\alpha'$
\Else
\State \textbf{return} $\texttt{FULL-ASSEMBLY}(\alpha', \mathcal{T})$
\EndIf
\EndProcedure
\end{algorithmic}
\end{algorithm}
\section{Definition of the FTAM} \label{sec:definitions}
\vspace{-5pt}
In this section we present definitions related to the Flexible Tile Assembly Model.
A \emph{tile type} $t$ in the FTAM is defined as a 2D unit square that can be translated, rotated, and reflected throughout 3-dimensional space, but can only occupy a location such that its corners are positioned on four adjacent, coplanar points in $\mathbb{Z}^3$. Each tile type $t$ has four sides $i \in \{N, E, S, W\}$, which we refer to as $t_i$. Let $\Sigma$ be an alphabet of labels and $\Bar{\Sigma} = \{a^* | a \in \Sigma\}$ be the alphabet of \emph{complementary labels}, then each side of each tile has a \emph{glue} that consists of a \emph{label} $label(t_i) \in \Sigma \cup \Bar{\Sigma} \cup \epsilon$ (where $\epsilon$ is the unique \emph{empty} label for the \emph{null glue}), a non-negative integer \emph{strength} $str(t_i)$, and a boolean valued \emph{flexibility} $flx(t_i)$. (See Figure~\ref{fig:flexible-positions} for a depiction of the positions allowable by a flexible glue.)
A \emph{tile} is an instance of a tile type. A \emph{placement} of a tile $p = (l,n,o)$ consists of a location $l \in \mathbb{Z}^3$, a \emph{normal} vector $n$ which starts at the center of the tile and points perpendicular to the plane in which the tile lies (i.e. $n \in \{+x,-x,+y,-y,+z,-z\}$\footnote{We refer to the vectors $\{(1,0,0),(-1,0,0),(0,1,0),(0,-1,0),(0,0,1),(0,0,-1)\})$ by the shorthand notation $\{+x,-x,+y,-y,+z,-z\}$ throughout this paper.}), and an \emph{orientation} $o$ which is a vector lying in the same plane as the tile which starts at the center of the tile and points to the $N$ side of the tile (i.e. $o \in \{+x,-x,+y,-y,+z,-z\}$). Note that by convention, to avoid duplicate location specifiers for a given tile, we restrict a location $l$ to refer to only the $3$ possible tile locations with corners at $l$ and which extend in positive directions from $l$ along one of the planes (i.e. tiles are located by their vertices with the smallest valued coordinates). For any given $l$, there can only be a max of one tile with $n \in \{+x,-x\}$, one tile with $n \in \{+y,-y\}$, and one tile with $n \in \{+z,-z\}$, as to avoid overlapping tiles.
Let $p = (l,n,o)$ and $p' = (l',n',o')$ be placements of tiles $t$ and $t'$, respectively, such that $p$ and $p'$ are non-overlapping\footnote{Non-overlapping placements refer to different tile locations. Formally, two tile placements are non-overlapping if (1) $l != l'$ or (2) $n != n'$ and $n!=\texttt{inverse}(n')$.} and for some $i,j \in \{N,E,S,W\}$, sides $t_i$ and $t_j'$ are adjacent (i.e. touching). We say that $p$ and $p'$, have \emph{compatible} normal vectors if and only if either (1) $n = n'$, (2) $n$ and $n'$ intersect, or (3) $\texttt{inverse}(n)$ and $\texttt{inverse}(n')$ intersect, where the $\texttt{inverse}$ function simply negates the signs of the non-zero components of a vector. (See Figures~\ref{fig:compatible-normals} and \ref{fig:incompatible-normals}.) We will refer to these three orientations as ``Straight'', ``Up'', and ``Down'', respectively. Furthermore, if (1) $label(t_i)$ is complementary to $label(t_j')$, (2) $str(t_i) = str(t_j')$, (3) $flx(t_i) = False$ and $flx(t_j') = False$, and (4) $n$ and $n'$ are in a ``Straight'' orientation, then the glues on $t_i$ and $t_j'$ can \emph{bind} with strength value $str(t_i)$ to form a \emph{rigid bond}. Similarly, if (1) $label(t_i)$ is complementary to $label(t_j')$, (2) $str(t_i) = str(t_j')$, (3) $flx(t_i) = True$ and $flx(t_j') = True$, and (4) $n$ and $n'$ are compatible, then the glues on $t_i$ and $t_j'$ can \emph{bind} with strength value $str(t_i)$ to form a \emph{flexible bond}. \footnote{Note that any glue can only bind to a single other glue. Also, we do not allow two pairs of coplanar tiles to bind through the same space (i.e. the two partial surfaces created by two pairs of bounded coplanar tiles are not allowed to intersect). Therefore, $4$ glues from $4$ different tiles that are all adjacent to each other can all form bonds only if they form two flexible bonds in non-straight orientations.}
\vspace{-15pt}
\begin{figure}[htp]
\centering
\subfloat[][Compatible normal vectors.]{
\label{fig:compatible-normals}
\includegraphics[width=1.8in]{images/compatible-normals.png}
}
\quad\quad
\subfloat[][Incompatible normal vectors.]{
\label{fig:incompatible-normals}
\includegraphics[width=1.8in]{images/incompatible-normals.png}
}
\caption{Possible normal vectors of pairs of tiles. Those in (a) are compatible and allow a bond to form between complementary glues in the orientations ``Up'', ``Down'', and ``Straight'', respectively. Those in (b) are not compatible.}
\label{fig:normals}
\end{figure} \hfill
We define an \emph{assembly} $\alpha$ as a graph whose nodes, denoted $V(\alpha)$, are tiles and whose edges, denoted $E(\alpha)$, represent bound complementary glues between adjacent edges of two tiles. An edge between sides $i$ and $j$ of tiles $t$ and $t'$, respectively, is represented by the tuple $(t_i,t_j')$, which specifies which sides of $t$ and $t'$ the bond is between. Whether it is flexible is denoted by $flx(t_i)$ and its strength is denoted by $str(t_i)$ (since those values must be equal for $t_i$ and $t_j'$).
We define a \emph{face} to be a set of coplanar tiles that are all bound together through rigid bonds. Additionally, we define a \emph{face graph} to be a graph minor of the assembly graph where every maximal subgraph in which every node can be reached from every other node using a path of rigid tiles is replaced by a single node in the face graph. Two nodes in the face graph that correspond to two groups of nodes in the assembly graph have an edge if and only if there is at least one flexible bond between any single node in the first group of the assembly graph and any single node in the second group of the assembly graph.
An FTAM system is a triple $\mathcal{T} = (T,\sigma,\tau)$ where $T$ is a finite set of tile types (i.e. tile set), $\sigma$ is an initial \emph{seed} assembly, and $\tau \in \mathbb{Z}^+$ is a positive integer called \emph{temperature} which specifies the minimum binding threshold for tiles. An assembly is $\tau$-stable if and only if every cut of edges of $\alpha$ which separates $\alpha$ into two or more components must cut edges whose strengths sum to $\ge \tau$. We will only consider assemblies which are $\tau$-stable (for a given $\tau$), and we use the term assembly to refer to a $\tau$-stable assembly.
Given an assembly $\alpha$, a \emph{configuration} $c_\alpha$ is a mapping from every flexible bond in $\alpha$ to an orientation from \{``Up'', ``Down'', ``Straight''\}. An \emph{embedding} $e_\alpha$ is a mapping from each tile in $\alpha$ to a placement. Given an assembly and a configuration, we can obtain an embedding by choosing any single initial tile and assigning it a placement and computing the placement of each additional tile according to how it is bonded with tiles that are already placed. Note that, given tiles to which it is bound, their placements, and an orientation, there is only one tile location at which each additional tile can be placed. We say a configuration $c_{\alpha}$ is \emph{valid} if and only if an embedding obtained from the configuration (1) does not place more than one tile at any tile location, (2) doesn't bond tiles through the same space, and (3) does not have contradicting bond loops. To elaborate on (2), while 4 glues can all be adjacent at one point, we allow them to bind in pairs in ``Up'' or ``Down'' orientations but do not allow both pairs to bind across the gap in ``Straight'' orientations. To elaborate on (3), contradicting bond loops occur when placing a series of tiles that are all bound in a loop causes the last tile to be placed at a location that is not adjacent to the first tile, therefore making the loop unable to close. Examples of configurations that follow and contradict (3) are given in Figure \ref{fig:validation}. Note that two embeddings that use different initial tiles and initial placements but the same configuration will be equivalent up to rotation and translation.
\begin{figure}
\centering
\includegraphics[width=0.65\textwidth]{images/configurationValidation.png}
\caption{Here we see an assembly, a valid configuration, and an invalid configuration. In the third image, because of the orientations of bonds 1, 2, and 4, bond 3 is between two tiles that are not connected, making the configuration invalid.}
\label{fig:validation}
\end{figure}
Let $\alpha$ be an assembly and $c_\alpha$ and $c_\alpha'$ be valid configurations of $\alpha$. If for every flexible bond $b \in \alpha$ either $c_\alpha(b) = Up$ and $c_\alpha'(b) = Down$, $c_\alpha(b) = Down$ and $c_\alpha'(b) = Up$, or $c_\alpha(b) = Straight$ and $c_\alpha'(b) = Straight$, we say that $c_\alpha$ is the \emph{chiral} configuration of $c_\alpha'$ and vice versa. Note that the embeddings achieved from $c_\alpha$ and $c_\alpha'$ are reflections of each other. We refer to the special reconfiguration of an assembly to its chiral as \emph{inversion}.
Given an assembly $\alpha$ and two different embeddings $e_\alpha$ and $e_\alpha'$, we say that $e_\alpha$ and $e_\alpha'$ are \emph{equivalent}, written $e_\alpha \equiv e_\alpha'$, if one can be rotated and/or translated into the other. If two embeddings are equivalent, this means they were computed from the same configuration, although possibly using a different placement for the initial tile.
We define the set of all valid configurations of $\alpha$ as $\mathcal{C}(\alpha)$. We say that an assembly $\alpha$ is \emph{rigid} if (1) $|\mathcal{C}(\alpha)| = 1$, or (2) $|\mathcal{C}(\alpha)| = 2$ and the two valid configurations are chiral versions of each other. Conversely, if $\alpha$ is not rigid, we say that it is \emph{flexible}.
The \emph{frontier} of a configuration $c_\alpha$, denoted $\partial^{\mathcal{T}} c_\alpha$, is the set composed of all pairs $(t,B)$ where $t \in T$ is a tile type from tile set $T$ and $B$ is a set of up to 4 tile/glue pairs such that an embedding of $c_\alpha$ would place each tile adjacent to one location such that a tile of type $t$ could bind to each glue for a collective strength greater than or equal to the temperature parameter $\tau$. Given an assembly $\alpha$ and a set of valid configurations $\mathcal{C}(\alpha)$, we define the multiset of frontier locations of assembly $\alpha$ across all valid configurations to be $\hat{\partial}^{\mathcal{T}}\alpha = \bigcup_{c_\alpha \in \mathcal{C}(\alpha)} \partial^{\mathcal{T}} c_\alpha$, i.e. $\hat{\partial}^{\mathcal{T}}\alpha$ is the multiset resulting from the union of the sets of frontier locations of all valid configurations of $\alpha$.
Given assembly $\alpha$ and valid configuration $c_\alpha$, $\#(c_\alpha)$ is the maximum number of new bonds which can be formed across adjacent tile edges in an embedding of $\alpha$ which are not already bound in $\alpha$ (i.e. these are tile edges which have been put into placements allowing bonding in configuration $c_\alpha$ but whose bonds are not included in $\alpha$). We then define $\mathcal{C}_{max}(\alpha) = \{ c_\alpha | c_\alpha \in \mathcal{C}(\alpha)$ and $\forall c_\alpha' \in \mathcal{C}(\alpha)$, $\#(c_\alpha) \ge \#(c_\alpha')\}$. Namely, $\mathcal{C}_{max}(\alpha)$ is the set of valid configurations of $\alpha$ in which the maximum number of bonds can be immediately formed.
Given an assembly $\alpha$ in FTAM system $\mathcal{T}$, a single step of the assembly process intuitively proceeds by first randomly selecting a frontier location from among all frontier locations over all valid configurations of $\alpha$. Then, a tile is attached at that location to form a new assembly $\alpha'$. Next, over all valid configurations of $\alpha'$, a configuration is randomly selected in which the maximum number of additional new bonds can be formed (i.e. the addition of the new tile may allow for additional bonds to form in alternate configurations, and a configuration which maximizes these is chosen), and all possible new bonds are formed in that configuration, yielding assembly $\alpha''$.
Assuming that $\alpha$ was not \emph{terminal} and thus $\alpha'' \neq \alpha$, we denote the single-tile addition as $\alpha \to_1^{\mathcal{T}} \alpha''$. To denote an arbitrary number of assembly steps, we use $\alpha \to_*^{\mathcal{T}} \alpha''$. For an FTAM system $\mathcal{T} = (T,\sigma,\tau)$, assembly begins from $\sigma$ and proceeds by adding a single tile at a time until the assembly is terminal (possibly in the limit).
(See Section~\ref{sec:definitions-append}
for pseudocode of the assembly algorithms.)
For any $\alpha'$ such that $\sigma \to_*^{\mathcal{T}} \alpha'$, we say that $\alpha'$ is a \emph{producible} assembly and we denote the set of producible assemblies as $\prodasm{\mathcal{T}}$. We denote the set of terminal assemblies as $\termasm{\mathcal{T}}$.
Note that in this section we have provided what is intended to be an intuitively simple version of the FTAM in which the full spectrum of all possible configurations of an assembly are virtually explored at each step, and only those which maximize the number of bonds formed at every step are selected. Logically, this provides a model in which assemblies reconfigure into globally optimal configurations, in terms of bond formation, between each addition of a new tile. Clearly, depending on the size of an assembly and the degrees of freedom of various components afforded by flexible bonds, such optimal reconfiguration could conceivably be precluded by faster rates of tile attachments. Various parameters which seek to balance the amount of configuration-space exploration versus tile attachment rates have been developed to study more kinetically realistic dynamics, but are beyond the scope of this paper.
\section{Introduction}
Proteins are a fantastically diverse set of biomolecules, with structures and functions that can vary wildly from each other, such as fibrous proteins (like collagen), enzymatic proteins (like catalase), and transport proteins (like hemoglobin). Truly amazing is the fact that such diversity arises solely from the linear combination of only $20$ amino acid building blocks. It is the specific sequence of amino acids, interacting with each other as they are combined, which causes each chain to fold in a specific way and each protein to assume its particular three-dimensional structure, and this in turn dictates its structural and functional properties. Inspired by the prowess of nature to build molecules with such precision and heterogeneity, scientists have studied the mechanisms of protein folding - to realize that the dynamics are so complex that predicting a protein's shape given its amino acid sequence is considered to be intractable \cite{fraenkel1993complexity, crescenzi1998complexity}, and engineers have begun to develop artificial systems which fold self-assembling molecules into complex structures~\cite{RotOrigami05, RothOrigami, OrigamiSeed, OrigamiTiles, dill1995} - but with results that to date still lack the diversity of biology.
In order to help progress understanding of the dynamics of systems which self-assemble out of folding components, and to provide a framework for studying such systems, in this paper we introduce the \emph{Flexible Tile Assembly Model} (FTAM). The FTAM is intended to be a simplified mathematical model of self-assembling systems utilizing components which are able to dynamically reconfigure their relative 3-dimensional locations via folding and unfolding of flexible bonds between components. It is based on the abstract Tile Assembly Model \cite{Winf98}, and as such the fundamental components are 2-dimensional square \emph{tiles} which bind to each other via \emph{glues} on their edges. In contrast to the aTAM, in the FTAM each glue type can be specified to either form \emph{rigid} bonds (which force two adjacent tiles bound by such a glue to remain fixed in co-planar positions) or \emph{flexible} bonds (which allow two adjacent tiles bound by such a glue to possibly alternate between being in any of three relative orientations, as shown in Figure~\ref{fig:flexible-positions}). Because the FTAM is meant to be a test bed for flexible, reconfigurable self-assembling systems, we present a version of the model which makes many simplifying assumptions about allowable positions of tiles and dynamics of the self-assembly process, but which also differs greatly from previously studied self-assembling systems which allow reconfigurability \cite{jRTAM, JonoskaSignals1, JonoskaSignals2, jSignals3D,jSignals,FlexibleCompModel,JonoskaFlexible,CommonUnfoldings, aichholzer2017folding}, other computational studies of folding such as \cite{CommonUnfoldings,aichholzer2017folding}, and algorithmic studies focused on constructing more simple 3D structures such as \cite{KaoFolding}.
\vspace{-15pt}
\begin{figure}[htp]
\centering
\includegraphics[width=2.0in]{images/flexible-positions.png}
\caption{The three relative positions possible for two tiles bound via a flexible glue.}
\label{fig:flexible-positions}
\end{figure}
\vspace{-10pt}
In Section~\ref{sec:definitions}, we formally introduce the FTAM and provide definitions and algorithms describing its dynamics. In Section~\ref{sec:robustness}, we show how to control flexibility in the model to build 3D shapes. In Section~\ref{sec:utilizing_flexibility}, we present a pair of constructions which demonstrate the potential utility of reconfigurability of assemblies in the FTAM. In the first construction, an FTAM system $\calT$ is given which produces a single terminal assembly that may be in many different configuration. In addition, a set $S$ of $n$ distinct types of tiles are given such that for each subset of $S$, adding this subset of tiles of $S$ to the types of tiles for $\calT$ gives a system with an assembly sequence that starts from the single terminal assembly of $\calT$ and yields a rigid terminal assembly (i.e., an assembly to which no tiles may bind and which, at a high-level, is in a configuration which cannot be folded via flexible glues to give another distinct configuration). Moreover, the resulting rigid assembly is distinct for each choice of subset of $S$. The second construction given in Section~\ref{sec:robustness} demonstrates how a reconfigurable initial assembly can be transformed into either a volume-maximizing hollow cube or a small, tightly compressed brick by selecting between and adding one of two small subsets of tile types. These two constructions demonstrate how algorithmic self-assembling systems could be designed which efficiently (in terms of ``input'' specified by tile type additions) make drastic changes to their surface structures and volumes. These constructions show that FTAM systems can be designed which utilize reconfigurability. In Section~\ref{sec:complexity}, we show that this utility comes at a cost in terms of the computational complexity of determining some important properties of arbitrary FTAM systems. In particular, we show that, given an arbitrary FTAM system, the problem of determining whether it produces an assembly which cannot be reconfigured (via folding along tile edges bonded by flexible glues) is undecidable. Moreover, we show that, given an assembly, it is co-NP-complete to determine whether the assembly is rigid, i.e. has exactly one valid configuration. Our final result modifies the previous to show that the problem of deciding if a given assembly for an FTAM system is terminal is also co-NP-complete. This is especially interesting since, in the aTAM, there is a simple polynomial time algorithm to determine if a given assembly is terminal.
\section{Technical Details for Section \ref{sec:robustness}} \label{sec:robustness_appendix}
\subsection{Filler tiles}
First, we discuss the ``filler tiles'' that are used to fill in edge frames. On perfectly square faces, this can trivially be done, with filler tiles allowing to attach as the assembly grows. However, in cases where the face has a concave corner, a rectangular decomposition of the face with each rectangle being assigned a unique filler tile would prevent the filler tiles from overgrowing their bounds.
\subsection{Vertices and perspectives} \label{sec:verts_and_perts}
We again provide the enumeration of all $2\times2\times2$ polycubes for reference.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{images/possible_outline_vertiecs.png}
\caption{All possible polycubes that can fit inside of a $2\times2\times2$ space, and furthermore, all possible vertex types that could exist on a polycube.}
\label{fig:possible_outline_vertices_appendix}
\end{figure}
We categorize the types of vertices into two groups, \emph{simple} and \emph{complex}. In the enumeration in Figure \ref{fig:possible_outline_vertices}, the polycubes in the blue squares actually don't have a vertex in the center. The vertices in the polycubes in red (1,2,5,9) have three edges and three faces incident on the center point, creating what we refer to as a \emph{simple} vertex. Of these 4 vertices, 1 and 9 are the same vertex type, which we will refer to as a \emph{convex} vertex, and 2 and 5 are the same vertex type, which we will refer to as a \emph{concave} vertex. The vertices in the polycubes in brown (3,4,6,7) have more than three edges and more than three faces incident on the center point, creating what we refer to as a \emph{complex} vertex. All of these complex vertices are unique, and we will refer to them by their number. The polycube in yellow (8) is a special case in which there are more than three edges and more than three faces incident on the center point, but the polycube is arranged in a way that the center point can be thought of as two different simple convex vertices, one for each location that is missing a cube.
A vertex can be \emph{symmetric}, meaning all edges have the same perspective, \emph{semi-symmetric}, meaning some edges have the same perspective, or \emph{asymmetric}, meaning no edges have the same perspective. Semi-symmetric and asymmetric vertices have to differentiate between the different perspectives the vertex can exist in. The simple convex vertex and vertex 3 are both symmetric, meaning they have only one perspective each. Vertex 4 and 7 are semi-symmetric, with vertex 4 having 2 perspectives (even though it has 4 edges) and vertex 7 having 3 perspectives (even though it has 6 edges). The simple concave vertex and vertex 6 and asymmetric, with the simple concave vertex having 3 perspectives (and 3 edges) and the vertex 6 having 5 perspectives (and 5 edges). An example of different perspectives can be seen in the difference between Figure \ref{fig:concave_vertex} and Figure \ref{fig:combined_vertex}. All together, there are 15 different perspective, meaning we need 15 different tiling protocols to handle all situations.
We will now look at the loop lengths and bond sequences of different vertices. For bond sequences, we will use the notation $(b,b, ... ,b)$ where $b \in \{R,F\}$ and $R$ stands for rigid and $F$ stands for flexible. The first bond in the sequence will represent the edge assembling up to the vertex in the assembly process, and will therefore always be flexible. The other bonds in the sequence will represent the other bonds within the loop of tiles incident on the vertex, following the ordering set by the loop in either direction, without loss of generality. The set of all bond sequences for a single vertex is a non-repeating cyclic permutation group, minus the elements that begin with a rigid bond instead of a flexible bond.
We start with the simple vertices. First is the simple convex vertex. It has a loop length of 3 tiles and a bond sequence $(F,F,F)$. It is symmetric, meaning we don't have to differentiate between the edges. Therefore, to make a simple convex vertex, an edge will just initiate a protocol where the last two tiles of the edge end in flexible glues and another tile that matches both glues attaches to complete the loop. The next vertex is the simple concave vertex. It has a loop length of 5 and can have a bond sequence of either $(F,R,R,F,F)$, $(F,F,R,R,F)$, or $(F,F,F,R,R)$. Using each bond orientation will result in a different perspective, meaning all 3 perspectives can be deterministically assembled. Therefore, attaching the loop of tiles with the first bond sequence will yield the vertex in \ref{fig:combined_vertex}, the second bond sequence will yield the vertex in \ref{fig:concave_vertex}, and the third sequence will yield the mirror opposite of the vertex in \ref{fig:combined_vertex}.
The most unique complex vertex is vertex 6. It has a loop length of 7 and can have a bond sequence of either $(F,R,F,R,F,F,F)$, $(F,F,R,F,R,F,F)$, $(F,F,F,R,F,R,F)$, $(F,F,F,F,R,F,R)$, or $(F,R,F,F,F,R,F)$. As with the simple concave vertex, each bond sequences results in a different perspective, allowing the system to differentiate between the 5 different perspectives of vertex 6. Similarly, vertex 4 also has its own unique combinations. It has a loop length of 6 and can have a bond sequence of $(F,R,F,F,R,F)$ or $(F,F,R,F,F,R)$. The two bond orientations correspond to the two perspectives of vertex 4, allowing both to be deterministically assembled.
Vertex 3 and 7, however, share a combination of a loop length and bond sequence. Both have a loop length of 6 and a bond sequence of $(F,F,F,F,F,F)$. Because of this, attaching a loop of 6 tiles using all flexible bonds at the end of an edge can result in either vertex. In addition, since flexible bonds can ``mimic'' rigid bonds using a ``Straight'' orientation, the loop of tiles can even configure into vertex 4. Note that the reverse is mitigated by the fact vertex 4 has rigid bonds and no bonds in vertex 3 or 7 are ``Straight''. All together, it can end up in the one perspective from vertex 3, one of the two perspectives from vertex 4, or one of the three perspective from vertex 7. Given all these possibilities, vertex 3 and 7 cannot be deterministically assembled.
\begin{figure}[htp]
\centering
\includegraphics[width=\textwidth]{images/complex_vertex_loops.png}
\caption{The loops of tiles that form each complex vertex.}
\label{fig:complex_vetex_loops}
\end{figure}
\subsection{Edge growth protocol} \label{sec:edge_protocol}
When attempting to deterministically assemble outlines of polycubes in the FTAM, edges in the edge frame can be grown using the following protocol. Side 1 and side 2 refer to the two columns of tiles on the two faces that make up the edge. The hinge refers to the series of flexible bonds between tiles on side~1 and tiles on side~2. \begin{enumerate}
\item An exposed rigid double strength glue on side 1 of the edge will attach a new tile $t$ on side 1,
\item A flexible glue on tile $t$ on side 1 and an exposed rigid glue on side 2 will cooperate to attach a new tile $t'$ on side 2 of the edge,
\item A rigid double strength glue on tile $t'$ on side 2 of the edge will attach a new tile $t''$ on side 2, and
\item A flexible glue on tile $t''$ on side 2 and a rigid glue on tile $t$ on side 1 will cooperate to attach a new tile $t'''$ on side 1 of the edge
\end{enumerate} An edge can grow indefinitely by repeating this process using unique glues to grow up to a certain length. Notice that each new tile attaches using at least one rigid bond, meaning that, additional flexibility cannot be added to the edge past the flexibility of the hinge. Furthermore, there will only ever be one frontier location (per configuration, if multiple, but these are the same tiles and bonds) on the edge at any assembly step, leaving no room for non-determinism.
\subsection{Combating multiple edge frames} \label{sec:combating_multiple_edge_frames}
The two methods of combating multiple edge frames mentioned in the main section are blocking and symmetry. Blocking refers to when the faces surrounded by one edge frame would collide with the faces of another if the chiralities of the edge frames disagreed. This is actually the case in the example given in Figure \ref{fig:dumbbell_shape}. In these situations, even if the additional edge frames are configured to the wrong chirality during the assembly process, eventually the tiles with the potential to collide will be added to the assembly and force the correct chirality of both edge frames with respect to each other. In Figure \ref{fig:dumbbell_path}, you can see on the right how the inversion of the additional edge frame would cause the collision of tiles in the assembly. In this example, the yellow tile on the left of Figure \ref{fig:dumbbell_path} would force the correct relative chirality.
One other aspect of a shape with multiple edge frames that may reduce the number of possible configurations it can exist in is, like with full shapes, symmetry of the edge frames. To utilize the same example, imagine if the shape in Figure \ref{fig:dumbbell_shape} did not contain the yellow tile on left end or the symmetric tile on the right end. In this case, the ends would be free to reconfigure into the wrong chirality. However, this doesn't result in 4 different shapes that the mismatching chirality of 2 additional edge frames should produce. Instead, inversion of the left end of the assembly and not the right yields the same shape as inversion of the right end of the assembly and not the left, resulting in only 3 different possible shapes.
\begin{figure}[ht]
\centering
\subfloat[][The full dumbbell shape, a narrow connection piece and two end pieces]{%
\label{fig:dumbbell_shape}%
\includegraphics[width=0.7\textwidth]{images/dumbbell_shape}
}%
\quad
\subfloat[][An outline of the edges in the shape, showing 3 different edge frames]{%
\label{fig:dumbbell_edges}%
\includegraphics[width=0.7\textwidth]{images/dumbbell_edges}
}%
\quad
\subfloat[][The rights shows how an inversion of an end piece would cause a collision. The left shows the tile that needs to be placed to force the correct orientation of the end pieces (the yellow tile).]{%
\label{fig:dumbbell_path}%
\includegraphics[width=0.7\textwidth]{images/dumbbell_path}
}%
\quad
\caption{Example of a shape with multiple edge frames}
\label{fig:dumbbell}
\end{figure}
\section{Controlling Flexibility to Build Structures} \label{sec:robustness}
Our goal in this section is to deterministically assemble certain shapes in the FTAM at temperature two. We define a \emph{shape} to be a collection of connected tile locations. A shape is invariant through translation and rotation. Rather than go through an endless case-by-case analysis of all possible shapes, we focus on collections of 2D tile locations that form the outlines of three-dimensional shapes. We refer to these 3D shapes as \emph{polycubes} and the set of 2D tile locations on their outer surface as an \emph{outline}. We say that an FTAM system $\mathcal{T} = (T,\sigma,\tau)$ \emph{deterministically assembles} a shape $s$ if the embedding of all configurations $\mathcal{C}_\alpha$ of all terminal assemblies $\termasm{\mathcal{T}}$ of the system $T$ have shape $s$.
Due to the definition of the model, the most prominent additional challenge that is present in FTAM systems over traditional 2D aTAM systems is controlling the orientation of different faces in the assembly relative to one another as the assembly process is occurring. In which case, the approach that we use to demonstrate shape building in the FTAM is to make an \emph{edge frame} for each polycube using unique tile types and filling in each face. We define an edge frame to be the collection of the outer-most tiles of each face in the outline of a polycube. For now, we will make the assumption that every edge of the shape is connected and will address this later in the section. We claim that studying edge frames is sufficient for unveiling the power of the FTAM to orient new faces in the assembly process since, intuitively, the cooperation of other tiles on the edges of adjacent faces doesn't provide additional help in correctly orienting those faces over just the tiles at the vertex. This intuition stems from the idea that the faces of a shape incident on a vertex interact on the same axes that the individual tiles incident on a vertex do.
One big deciding factor about whether the outline of a specific polycube can be made in the FTAM comes down to the types of vertices in that polycube. Because of this, we continue our analysis by breaking down the types of vertices that can exist on a polycube. Every type of vertex possible on a polycube can be enumerated by enumerating all polycubes that can fit inside a $2\times2\times2$ space that are distinct up to rotation and reflection. You can see the outcome of this enumeration in Figure \ref{fig:possible_outline_vertices}. In each polycube, the vertex type is illustrated at the center point of the $2\times2\times2$ space. The illustration has labels to later reference each vertex type.
\begin{figure}[htp]
\centering
\includegraphics[width=1.0\textwidth]{images/possible_outline_vertiecs.png}
\caption{All possible polycubes that can fit inside of a $2\times2\times2$ space, and furthermore, all possible vertex types that could exist on a polycube.}
\label{fig:possible_outline_vertices}
\end{figure}
This yields 6 distinct vertex types. Vertices 1, 8, and 9 are all the same, which we refer to as a \emph{convex} vertex. Vertices 3 and 5 are the same, which we refer to as a \emph{concave} vertex. Vertex 3, 4, 6, and 7 are all distinct and we refer to them collectively as the \emph{complex} vertices. In addition to the vertex type, the system must also be able to deterministically assemble the vertex from the correct perspective. A \emph{perspective} is the relative direction that the new edges that form with the vertex are pointing with respect to the tiles of the original edge that first grows up to the vertex in the assembly process. Each vertex can have any number of perspectives from 1 (\emph{symmetric} vertex) to the number of edges (\emph{asymmetric} vertex), inclusive.
Collectively, there are 15 unique perspectives among the 6 distinct vertices. Each perspective requires its own tiling protocol to get the vertex to configure correctly. We construct these protocols using (a) the number of tiles that are incident on the vertex that are bound in a loop (which we refer to as the \emph{loop length}) and (b) the sequence of flexibility values in the bonds of the loop (which we refer to as the \emph{bond sequence}). If a perspective has a unique protocol, then attaching a loop of tiles using the protocol will result in the only possible configuration available for the loop of tiles being the correct perspective. Of the 15 perspectives previously mentioned, 11 perspectives among 4 of the vertices will have their own unique protocol and will therefore be deterministic. The other 4 perspectives among 2 vertices will all share one protocol and will therefore not be deterministic. These 2 vertices are types 3 and 7, which we will subsequently refer to as \emph{reconfigurable} vertices. For a full enumeration and discussion of vertices, perspectives, and tiling protocols, see
Section \ref{sec:verts_and_perts}.
\begin{figure}[htp]
\centering
\subfloat{%
\label{fig:vertex_control_setup}%
\includegraphics[width=0.24\textwidth]{images/vertex_control_setup}
}
\subfloat{%
\label{fig:convex_vertex}%
\includegraphics[width=0.16\textwidth]{images/convex_vertex}
}
\subfloat{%
\label{fig:concave_vertex}%
\includegraphics[width=0.30\textwidth]{images/concave_vertex}
}
\subfloat{%
\label{fig:combined_vertex}%
\includegraphics[width=0.20\textwidth]{images/combined_vertex}
}%
\quad
\caption{(a) original edge, (b) convex vertex, (c) concave vertex, from one unique perspective, (d) concave vertex, from another perspective}
\label{fig:types_of_vertices}
\end{figure}
\paragraph{Assembly Process.}
Now, we consider the assembly process. Let's assume we start with a seed that is just the three tiles in a simple convex vertex. Notice that as the assembly process starts, the seed vertex and the edges that are growing out from it can invert as a whole but cannot otherwise reconfigure (since that would require removing a bond from the assembly). (For assembling an edge, we outline a trivial protocol in
Section~\ref{sec:edge_protocol}
.) Each time the assembly grows up to a vertex, it will attach the loop of tiles that make this new vertex. As long as the new vertex is not a reconfigurable vertex, it will be forced to take a configuration that agrees with configuration of the seed vertex. By this, we mean that, if the seed vertex were to invert at this point, the edge connecting the two vertices would invert, and the new vertex would therefore be forced to invert. This cause-effect relationship is true for any vertices (excluding reconfigurable vertices) connected by an edge, which means that, if any bond in the partial assembly were to reorient, the whole partial assembly must invert, i.e. inversion is the only possible reconfiguration. An example of an edge frame started from a potential seed is shown in Figure \ref{fig:edge_frame_example}.
\begin{figure}[htp]
\centering
\includegraphics[width=\textwidth]{images/edge_frame_example.png}
\caption{An assembling edge frame starting from a potential seed. Each edge grows up to a vertex and initializes other edges until the whole frame has filled out.}
\label{fig:edge_frame_example}
\end{figure}
We now prove a claim that assembling in the correct configuration or the chiral configuration is identical (since both configurations have the same frontier) and will therefore yield the same shape.
\begin{claim}
Every frontier location $f$ in an assembly $\alpha$ for a given configuration $c_\alpha$ has a corresponding frontier location $f'$ in $\alpha$ in the chiral configuration $c_\alpha'$, such that attaching $f$ to $\alpha$ in $c_\alpha$ produces the same assembly but in the chiral configuration of attaching $f'$ to $\alpha$ in $c_\alpha'$.
\end{claim}
\begin{proof}
Notice that a frontier location in the FTAM is dependent on 12 neighboring tile locations, an ``Up'', ``Straight'', and ``Down'' location for each of the 4 sides of the tile. Also remember chiral configurations of an assembly $\alpha$ produce embeddings of $\alpha$ that are the reflections of each other. Now, take any frontier location $f$ in $c_\alpha$. By reflecting an embedding of $c_\alpha$ over the plane that $f$ exists in, the 12 tile locations that make $f$ into a frontier location will still be neighboring $f$, with the ``Up'' and ``Down'' neighboring locations switching places and also reflecting, thereby keeping the same glues incident on the location of $f$. Since all the same glues are incident on the tile location, this location, which we will call $f'$, is also a frontier location in $c_\alpha'$ with the same tile type as in $c_\alpha$, even if $c_\alpha'$ includes some translation or rotation. Since the frontier locations are on the plane of symmetry that we used to get the chiral configurations, adding the tile to the assembly in either configuration will produce two configurations that are also chiral configurations of each other.
\end{proof}
Once the assembly process has finished, the terminal assembly could also flip between the correct shape in its chiral. When there is at least one plane of symmetry in the shape, then reconfiguration in the assembly process actually will not prevent the system from being deterministic. This is because the chiral of a symmetric polycube is itself. Therefore, although the system will technically make two different terminal assemblies, one can be rotated into the other, meaning that the two different terminal assemblies have the same shape by definition, making the system deterministic.
\paragraph{Multiple Edge Frames.}
Up to this point, we have assumed all the edges in a polycube are connected. However, this is not always the case. For example, anytime two pieces of a shape are connected by a set of coplanar tiles (i.e. when the face graph has a cut vertex). Shapes like this are a problem because they require multiple edge frames to build, and similar to the chirality of asymmetric shapes, additional edge frames can also have chiral reconfigurations. Therefore, disagreeing chiralities of the edge frames can configure the terminal assembly of a system into a shape that is neither the intended shape nor its chiral. In general, each additional edge frame doubles the number of configurations that the terminal assembly can exist in, only one of which (or two, if symmetric) is the desired shape. There are some exceptions to this (as discussed in
Section~\ref{sec:combating_multiple_edge_frames})
such as blocking and symmetry.
\paragraph{Summary.}
Combining the results of this section, we get the following theorem.
\begin{theorem}
A temperature two FTAM system can deterministically assemble the outline of any polycube that meets the following conditions:
\begin{enumerate}
\item the polycube is symmetric,
\item there are no reconfigurable vertices in the polycube, and
\item the edges of the polycube are all connected
\end{enumerate}
\end{theorem}
\section{Technical Details for Section \ref{sec:utilizing_flexibility}} \label{sec:utilizing_flexibility_appendix}
\subsection{Staged Functional Surface: Maximizing the number of reconfigurations}
For this section, we simply provide another visual to illustrate how secondary stages could be designed to make certain functional surfaces. This visual is provided in Figure \ref{fig:staged_functional_surface_example}, while Figure \ref{fig:staged_functional_surface_appendix} was included again for the reader's reference.
\begin{figure}[htp]
\centering
\includegraphics[width=0.9\textwidth]{images/staged_functional_surface.png}
\caption{An example of reconfigurable shape that can be used in a staged environment to display a functional surface only after additional tiles are added to the system.}
\label{fig:staged_functional_surface_appendix}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[width=\textwidth]{images/staged_functional_surface_example.png}
\caption{An example of a system that could be used to match pixels with tiles. Here, the tiles are shown that are used to make the image on the right of Figure \ref{fig:staged_functional_surface_appendix}. The tiles in the bottom row on the right can be added to pin the unused pixel location down into the surface of the assembly.}
\label{fig:staged_functional_surface_example}
\end{figure}
\subsection{Compressing/expanding structures}\label{sec:compress-expand=append}
In this section we provide technical details about the construction in Section~\ref{sec:compress-expand}.
Let $\mathcal{T}$ be a staged FTAM system which is capable of self-assembling assembly $\alpha$ which can be thought of as a set of $6$ ``sheets'' that are each approximately $n \times n$ squares of tiles (which can be seen laid out flat in Figure~\ref{fig:compression-animation}f and in more detail in Figure~\ref{fig:unfolded-labels} where $\alpha$ contains each $n \times n$ square minus the outer perimeter strips which are colored grey). The ordering of growth of the squares follows the path of the arrow shown in Figure~\ref{fig:unfolded-labels}, and the squares are built using standard aTAM techniques to efficiently build $n \times n$ squares using $O(\log(n))$ tile types \cite{RotWin00}.
For the second stage, if the expanded cube is desired, tile types are added which attach in counterclockwise order along the perimeter to form one additional row of tiles and fill out the squares to size $n \times n$ (starting from the topmost left corner as shown in Figure~\ref{fig:unfolded-labels}) so that the glues of the outside perimeter on the sides labeled in Figure~\ref{fig:unfolded-labels} are flexible strength-1 glues with the labels shown. These perimeter tiles also have rigid glues between each other (except at the boundaries between squares) which will provide the rigid frame of the cube (while the other glues between tiles in the interior of the squares are flexible). A constant number of unique tile types are sufficient to tile these perimeter locations in this second stage and thus form the rigid cube (shown in Figure~\ref{fig:compression-animation}a).
If, instead, the compressed structure is desired, for the second stage the tiles added to form the perimeter create an alternating pattern of glues and grow $2 \times 2$ \emph{tabs} as shown in Figure~\ref{fig:wrinkle-labels}, which forces them to fold into the wrinkled pattern shown in Figure~\ref{fig:compressed}. The tabs form ``caps'' which stably bind the structure. Note that for clarity, in Figure~\ref{fig:wrinkle-labels}, the labeled locations are given unique labels to make it easier to see which locations will bind, but due to the dynamics of the FTAM, as a square self-assembles, the perimeter glues will force it to fold into the desired pattern even if a constant-sized set of glue types is repeatedly used. This is because the ordering of perimeter growth is fixed (by cooperative growth) and the fact that the tabs are $2 \times 2$ squares allows enforcement of the growth ordering so that each tab is completed before perimeter growth proceeds beyond it. Since the squares fold into approximately $\sqrt{n}$ length wrinkled layers, the constant-sized pattern must repeat each $O(\sqrt{n})$ distance, which requires $O(\log(\sqrt{n}))$ unique tile types which can be embedded as part of the counter tile types which form the original squares (and thus the tile complexity remains at $O(\log(n))$, causing marker glues to be positioned with correct periodicity in case the tile types for forming the compressed structure are added. Otherwise, the tile types which force formation of the expanded cube still attach to those glues, but they don't propagate this pattern to the perimeter. Therefore, only a constant number of unique tile types are required for the second stage to form this configuration as well.
\begin{figure}[htp]
\centering
\includegraphics[width=3.0in]{images/unfolded-labels.png}
\caption{Label types for glues on each side.}
\label{fig:unfolded-labels}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[width=\textwidth]{images/wrinkle-labels.png}
\caption{Schematic depiction of a portion of the perimeter of a square where additional tiles attach, forming tabs, and glue labels are exposed which are needed to make a portion of a sheet fold into two ``wrinkled'' layers. The green line shows the boundary between two separate wrinkled layers. The left side will fold to make a wrinkled layered from left-to-right, and the right side will fold underneath to form a second wrinkled layer from right-to-left, also binding to the layer above it to form a stable structure.}
\label{fig:wrinkle-labels}
\end{figure}
\begin{figure}[H]
\hfill
\includegraphics[width=1.0in]{images/compressed-cropped.png}
\hfill
\includegraphics[width=1.0in]{images/compressed-capped.png}
\hfill
\,
\caption{(Left) The compressed configuration of the tiles which can also form a cube (shown without end~``caps''). (Right) A portion of the compressed configuration with end ``caps'', which fold from the tabs, included.}\label{fig:compressed}
\end{figure}
\section{Utilizing Flexibility} \label{sec:utilizing_flexibility}
As discussed previously, reconfigurability may be able to provide assembly systems with interesting properties that enable diverse applications. For example, changing geometry on the surface of a synthetic structure may allow it to interact with varying other structures in a system, or contracting/expanding volumes may impact how well it can diffuse through narrow channels. With a simple extension to the base FTAM model which allows an initial terminal assembly to form, and then at a later stage the addition of a new set of tile types allows the assembly to reconfigure, an assembly's final shape can be locked in based on these additional tiles. As previously mentioned, we extend the FTAM here to allow such staged assembly as the simplest mechanism for leveraging this type of reconfigurability, but note that alternative mechanisms could also work, such as glue activation and deactivation \cite{Signals}.
\subsection{Staged Functional Surface: Maximizing the number of reconfigurations}
For our first demonstration of a construction utilizing flexibility as a tool, we present a construction which maximizes the number of rigid configurations which a flexible assembly (formed during a first stage of assembly) can be locked into, based on the number of new tile types added during a second stage of assembly. Figure \ref{fig:staged_functional_surface} gives a high-level schematic of a simple example of such a system. (Note that we omit full details of each tile type as these components can all be easily constructed using standard aTAM techniques and techniques from Section~\ref{sec:robustness}.) It shows the inner-makings of an initial structure that can later be modified by adding new tiles types into solution. We refer to this structure as a \emph{film}. The film works by allowing the tiles in the very top layer to move freely. By adding select subsets of tile types during the second stage, prescribed tiles can be pinned up from the surface or pinned to the bottom layer of the assembly. Pinning up works by using the second layer of the film (from the bottom) to block the incoming tiles from folding down into the assembly, thereby forcing them to fold up. Pinning down works by connecting the top layer to the bottom layer of the film, forcing the tiles to fold down. The bumps formed from pinning up, also called \emph{pixels}, can be arranged into a specified geometry, or \emph{image}.
The setup of this system is shown in Figure \ref{fig:staged_functional_surface_example}.
The eight tiles on top are used to pin up the pixels in the image. The other tiles specified on the bottom can be used to pin down the rest of the free pieces in the assembly if this is required instead.
For this system, if the side lengths of the film are $n$, note that there are $O(n^2)$ potential pixel locations, meaning that there are a maximum of $O(2^{n^2})$ possible pixel configurations (i.e. each can be either up or down in any given configuration). To transform the flexible film into a rigid configuration with a particular set of pixels projecting upward, it is necessary to add tiles of $O(n^2)$ tile types corresponding to the up or down orientations, which is optimal as each tile type is encoded by a constant number of bits and $\log(O(2^{n^2})) = O(n^2)$ bits are necessary to uniquely identify each of the $O(2^{n^2})$ configurations. Note that although these reconfigurations are relatively trivial, the differences in the sizes of the reconfigurable sections can be arbitrarily large without requiring more unique tile types to be added in the second stage. This construction displays a maximum number of resulting rigid configurations from an optimal number of additional tile types in the second stage.
\vspace{-10pt}
\begin{figure}[htp]
\centering
\includegraphics[width=0.9\textwidth]{images/staged_functional_surface.png}
\caption{An example of reconfigurable shape that can be used in a staged environment to display a functional surface only after additional tiles are added.}
\label{fig:staged_functional_surface}
\end{figure}
\vspace{-25pt}
\subsection{Compressing/expanding structures}\label{sec:compress-expand}
We now demonstrate a construction that is able to take advantage of the flexibility of bonds in the FTAM to allow a base assembly to lock into an expansive, rigid but hollow configuration given the addition of one subset of tile types in the second stage, or to instead lock into a compressed, compact and dense configuration given the addition of a different subset of tile types.
\begin{figure}[htp]
\centering
\includegraphics[width=\textwidth]{images/compression.png}
\caption{Series of images giving a schematic depiction of a transformation from a hollow $n \times n \times n$ cube into a compressed, approximately $n \times \sqrt{n} \times \sqrt{n}$ configuration. \vspace{10pt}}
\label{fig:compression-animation}
\end{figure}
Figure~\ref{fig:compression-animation}(f) shows an assembly of six approximately $n \times n$ squares attached together and in a flattened ``sheet'' configuration. Such an assembly can be efficiently self-assembled using $O(\log(n))$ tile types in the first stage. In the second stage, one of two sets of a constant number of additional tile types could be added so that either (1) the sheet folds into a hollow, volume-maximizing cube of dimensions $n \times n \times n$ (i.e. volume $n^3$). A schematic representation of the transformation can be seen going backward from Figure~\ref{fig:compression-animation}(f) to Figure~\ref{fig:compression-animation}(a), or (2) the sheet folds into a compressed, compact ``brick'' of dimensions $O(n) \times O(\sqrt{n}) \times O(\sqrt{n})$ (i.e. volume $n^2$). A schematic representation of the transformation can be seen going forward from Figure~\ref{fig:compression-animation}(f) to Figure~\ref{fig:compression-animation}(k).
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\section{\label{sec:level1}Introduction
We needed a versatile and robust system for measuring magnetic susceptibilities over a wide temperature range ({\color{black} 1.7 - 100 K) with high sensitivity.} A number of tunnel diode oscillators have been built for limited purposes, either with high sensitivity (better than 100 ppm) over a limited range of temperatures,
\cite{Degrift1981,Drigo2010}
or with low sensitivity ($\geq 1000$ ppm) but capable of functioning over a broad range of external parameters (temperature and pressure).
\cite{Clover1970,Srikanth1999,Agosta2000,Mielke2009,Komatsu2004}
The design reported here
maintains the tunnel diode at a fixed low temperature (2 - 4K) while the sample is located in a separate resonant coil of Helmholtz design that can be varied in temperature {\color{black} up to} 100 K. The Helmholtz design allows the placement of electrodes across the sample (typically mm in dimension) providing static electric fields (up to $10^5$ V/m). AC electric fields applied at low frequency
can be used to detect magnetoelectric effects synchronously with the electric excitation. This feature is important as it has been shown that the magnetoelectric effects can have a strong frequency dependence.
\cite{Xia2017}
\section{Design Considerations}
Figure 1 provides a schematic representation of the system. The tunnel diode and its biasing circuitry is located on a small copper clad circuit board (\color{black}TD \color{black} block in Fig.\,1) at the lowest end of the interior cryostat and maintained at a fixed low temperature
{\color{black} using a hard pressed thermal contact via a spring loaded clip to bath B containing} superfluid helium. A compression force of the order of 100 N can provide a cooling power of a few tens of mW at helium temperatures.\cite{GKWhite} The superfuid helium bath B is supplied by a flow limiting impedance (Z $\approx 10^{10}$ cm$^{-3}$) that is connected to a normal (4.2 K) helium bath, and has a maximum cooling capacity at T $\approx 2$ K of $\sim$ 100 mW. The volume flow ${\dot V}_4 = \Delta P /(\eta Z)$ where $\eta$ is the viscosity of the helium (in Poise) and $\Delta P \approx ~ 10^5 $ Pa is the pressure drop across Z.\cite{DeLong1971} The filter F screens out small particles of frozen air that might be suspended in the liquid helium and could block the impedance Z. {\color{black} The entire assembly can be inserted in simple wide neck liquid helium storage dewar and thereby keeping the helium consumption to a low value.}
\begin{figure}[ht!]
\includegraphics[width=12pc]{Fig1}
\caption{\label{schematic}
Schematic representation of the low temperature apparatus (not to scale). The tunnel diode is held at constant temperature by a spring contact (red) at the bottom of the inner container which is the wall of the pumped liquid helium bath B. The sample S is located between Helmholtz windings L which allow for two electrodes E that apply an electric field across the sample. The thermal board TB is maintained at a fixed temperature by means of a feedback heater, the thermal connection to the TB block being very weak. The outer dimension was 2.25 in and the entire assembly could be inserted into a simple wide neck liquid helium storage dewar for ease of operation.
}
\end{figure}
Maintaining the tunnel diode (General Electric type BD4)\cite{Lowry} at a fixed temperature maximizes the stability of the frequency and amplitude of the oscillations and allows one to carefully select the operating point for which the frequency variations are a minimum with respect to bias voltage as {\color{black} noted} by Clover and Wolf\cite{Clover1970}). The {\color{black} sample cell was located at the center of a quasi-Helmholtz radiofrequency coil L that was thermally anchored} to an independent thermal board whose temperature could be carefully regulated by a local heater. The sample S was contained in a small {\color{black} Kel-F} container and thermally bonded to the thermal board with N-grease.{\color{black}\cite{Apiezon}} The inductance L was resonated with a silver-dipped mica capacitor C located on the TD block and, depending on choice of the coil L could be operated between 2 and 120 MHz. \color{black}The sample cell block was shielded by a copper cylinder that minimizes external noise and provides {\color{black} additional} thermal isolation for the sample. \color{black}
The fixed temperature tunnel diode block is separated from the sample cell block by a 10 cm length of semi-rigid cryogenic coaxial cable with stainless-steel outer conductor and silver plated beryllium-copper inner conductor\cite{Coaxitube} that provides adequate thermal isolation to maintain the sample board at a temperature between {\color{black} 1.7 K and 100 K} (with a precision of better than 1 \%) by using a regulated feedback circuit\color{black}, while the tunnel diode block is held at liquid helium temperatures \color{black}. The \color{black}sample cell and tunnel diode block \color{black} temperatures were measured {\color{black} using calibrated} Cernox thermometers.{\color{black}\cite{Lakeshore}} A small amount of $^4$He exchange gas was added to the sample space to provide additional cooling.
As shown in figure 2, the {\color{black} open} Helmholtz coil design allowed one to introduce small electrodes to place \color{black} a static \color{black} electric field (up to 100 kV.m$^{-1}$) across the sample to detect magnetoelectric effects. In addition to a static field a small AC field can be applied and used in conjuction with FM detection for high resolution, ($\sim$ hundreds of ppb), detection of magnetoelectric effects.
\begin{figure}[ht]
\includegraphics[width=10pc]{Fig2}
\caption{\label{Cell} Picture of {\color{black} Kel-F} sample cell showing the Helmholtz coil pair (5 mm diameter) and the electrodes for generating an electric field across the sample (typically 1 mm in lateral dimension. The sample is thermalized by applying Apiezon N-grease which makes contact with the thermal board.
}
\end{figure}
\section{Performance}
{\color{black} In order to test the sensitivity we used a commercial sample of MgB${_2}$ \cite{Sigma} that had a known magnetic susceptibility change at the superconducting transition temperature of 40.05$\pm$ 0.10 K (in good agreement with the values for the critical temperature reported in the literature.\cite{Buzea_2001} The fractional change in oscillator frequency
\begin{equation}
\frac{\Delta f}{f} =-\frac{1}{2}\eta \chi^{'} +O(Q^{-2}\chi^{"})
\end{equation}
where $\eta$ is the sample filling factor and $\chi^{'}$ and $\chi^{"}$ are the real and imaginary components of the sample susceptibility, respectively.
is in good agreement with the estimated value of {\color{black} $\eta \sim 4$\%.
The noise level in the low resolution channel was 12ppm and in the high resolution AC channel was of the order of 0.01 ppm, comparable to the best performance reported by Degrift\cite{Degrift1975} for a low current diode BD6 at a fixed temperature.} Similar sensitivity was reported by Van Riet and Van Gerver who used a low temperature tunnel diode circuit to observe NMR in a micromolar sample at 4K.\cite{Riet1982}
\begin{figure}[h]
\hbox{\hspace{+0.6em}\includegraphics[width=25pc]{Fig3}}
\caption{\label{MgB2} Frequency response {\color{black} measured for the magnetic susceptibility} of high-temperature superconductor MgB$_2$.{\color{black} The fractional change in frequency corresponds to a filling factor of 3.8\%.} }
\end{figure}
\section{\label{sec:level3}Conclusion}
We have demonstrated the use of a novel cryogenic design for a tunnel diode oscillator for measuring the magnetic susceptibility of small samples over a wide temperature range (1.7-100 K). High sensitivity and high stability were achieved by thermally separating the sample cell from the tunnel diode with the latter held at fixed helium temperatures. The compact geometry allows on to insert the complete system in a simple liquid helium dewar keeping the boil off to less than 0.2 liquid liters per hour at the highest sample temperatures.
\begin{acknowledgments}
We would like to thank the University of Florida Cryogenics Facility and the Machine and Electronic shops of the Physics Department for their assistance with this project. This work was supported as part of the Center for Molecular Magnetic Quantum Materials, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award \# DE-SC0019330.
\end{acknowledgments}
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\section{Introduction}
The possibility of a quantum speedup for finding the solution of classical optimization problems is tantalizing, as a quantum advantage for this class of problems would provide a wealth of new applications for quantum computing. The goal of many optimization problems can be formulated as finding an $n$-bit string $x_{\mathrm{opt}}$ that minimizes a given cost function $f(x)$, which can be interpreted as the energy of a classical
Ising spin system whose ground state is $x_{\mathrm{opt}}$. Finding the ground state of such systems can be hard if, e.g., the system is strongly frustrated, resulting in a complex energy landscape that cannot be efficiently explored with any known algorithm due to the presence of many local minima \cite{Nishimori-book}.
This can occur, e.g., in classical simulated annealing (SA) \cite{kirkpatrick_optimization_1983},
when the system's state is trapped in a local minimum.
Thermal hopping and quantum tunneling provide two starkly different mechanisms for solving optimization problems, and finding optimization problems that favor the latter continues to be an open theoretical question \cite{EPJ-ST:2015,Heim:2014jf}.
It is often stated that quantum annealing
(QA) \cite{PhysRevB.39.11828,finnila_quantum_1994,kadowaki_quantum_1998,farhi_quantum_2001,RevModPhys.80.1061} uses tunneling
instead of thermal excitations to escape from local
minima, which can be advantageous in systems
with tall but thin barriers that are
easier to tunnel through than to thermally climb
over \cite{Heim:2014jf,RevModPhys.80.1061,Suzuki-book}.
It is with this potential tunneling-induced advantage over classical annealing that QA and the quantum adiabatic algorithm \cite{farhi_quantum_2000} were proposed. Our goal in this work is to address the question of the role played by tunneling in providing a quantum speedup, and to elucidate it by studying a number of illustrative examples. We shall demonstrate that the role of tunneling is significantly more subtle than what might be expected on the basis of the ``tall and thin barrier'' picture.
In order to make progress on this question, the potential with respect to which tunneling occurs must be clearly specified. Tunneling is defined with respect to a semi-classical potential which delineates classically allowed and forbidden regions. In QA, one typically initializes the system in the known ground state of a simple Hamiltonian and evolves the system towards a Hamiltonian representing the final cost function.
We shall argue that when one takes a natural semi-classical limit, the semi-classical potential does not become the final cost-function. Instead one obtains a potential appearing in the action of the spin-coherent path-integral representation of the quantum dynamics. This potential, which here we call the spin-coherent potential, has been used profitably before~\cite{Farhi-spike-problem,FarhiAQC:02,Schaller:2007uq,Boixo:2014yu}. We provide comprehensive evidence that multi-spin tunneling can be understood with respect to this spin-coherent potential.
We analyze the spin-coherent potential for several examples from a well-known class of problems known as perturbed Hamming weight oracle (PHWO) problems. These are problems for which instances can be generated where QA either has an advantage over classical random search algorithms with local updates, such as SA \cite{Farhi-spike-problem,Reichardt:2004}, or has no advantage \cite{vanDam:01,Reichardt:2004}. Moreover, because PHWO problems exhibit qubit permutation symmetry, their quantum evolutions are easily classically simulatable, and furthermore, their spin-coherent potential is one-dimensional. Tunneling becomes clear and explicit for these problems when using the spin-coherent potential.
We focus on a particular PHWO problem that has a plateau in the final cost function (henceforth,``the Fixed Plateau"). This problem offers a counter-example to two commonly held views: (1) QA has an advantage, due to tunneling, over SA only on problems where the barrier in the final cost function is tall and thin; (2) tunneling is necessary for a quantum speedup in QA. We refute the first statement by showing that for the Fixed Plateau, which is a short and wide cost function, QA significantly outperforms SA by using tunneling. Indeed, we find numerically that adiabatic QA (AQA) needs a time of $\mathcal{O}(n^{0.5})$ to find the ground state, where $n$ is the number of spins or qubits. Moreover, using the spin-coherent potential, we observe the presence of tunneling during the quantum anneal. On the other hand, we prove that single-spin update SA takes a time of $\mathcal{O}(n^\mathrm{plateau\ width})$. Thus, we have essentially an arbitrary polynomial tunneling speedup of QA over SA on a cost-function that is not tall and thin. We remark that the result about SA's performance is also a rigorous proof of a result due to Reichardt~\cite{Reichardt:2004} that classical local search algorithms will fail on a certain class of PHWO problems and is of independent interest.
We refute the second statement by showing that, for the Fixed Plateau, it is actually optimal to run QA diabatically (henceforth, DQA for diabatic quantum annealing). The system leaves the ground state, only to return through a sequence of diabatic transitions associated with avoided-level crossings. In this regime, the runtime for QA is $\mathcal{O}(1)$. Moreover, in this regime, we do not observe any of the standard signatures of tunneling. We show that this feature --- that the optimal evolution time $t_f$ for QA is far from being adiabatic --- is present in a few other PHWO problems and that this optimal evolution involves no multi-qubit tunneling.
Given that the optimal evolution involves no tunneling, we are inspired to investigate a classical algorithm, spin vector dynamics (SVD), which can be interpreted as a semi-classical limit of the quantum evolution with a product-state approximation. We observe that SVD evolves in an almost identical manner to DQA, and is able to recover the speedup seen by DQA. Thus, in these problems, we show that what may be suspected to be a highly quantum-coherent process---diabatic transitions---can be mimicked by a quantum-inspired classical algorithm.
The structure of this paper is as follows. In Sec.~\ref{sec:PHWOintro}, we list the PHWO problems we study. In Sec.~\ref{sec:SCtunnel}, we use these problems to present evidence that tunneling can be understood with respect to the spin-coherent potential. In Sec.~\ref{sec:FixedPlateau}, we focus on the Fixed Plateau PHWO problem, and exhaustively analyze the performance of various algorithms for this problem. In particular we numerically characterize AQA (Sec.~\ref{subsec:adDyn}), provide a rigorous proof of SA's performance (Sec.~\ref{subsec:SArandom}), and numerically analyze DQA (Sec.~\ref{subsec:diabaticQA}), SVD (Sec.~\ref{subsec:SVDplat}), and a quantum Monte Carlo algorithm (Sec.~\ref{subsec:SQAplat}). We conclude in Sec.~\ref{sec:discuss} by discussing the implications of our work and possible directions for future work. Additional background information and technical details can be found in the Appendix.
\section{Perturbed Hamming weight optimization problems and the examples studied}\label{sec:PHWOintro}
The cost function of a PHWO problem is defined as,
\begin{equation}
\label{eq:pertham}
f(x) = \begin{cases} \abs{x} + p(\abs{x}) & l<\abs{x}<u,\\
\abs{x} & \text{elsewhere} \end{cases} \ ,
\end{equation}
where $\abs{x}$ denotes the Hamming weight of the bit string $x\in\{0,1\}^n$. For SA, this is the cost-function. For QA, this will be the final Hamiltonian. More precisely, we define QA as the closed-system quantum evolution governed by the time-dependent Hamiltonian,
\begin{equation}
\label{eqt:QuantumH}
H(s) =\frac{1}{2}\left(1-s \right) \sum_i \left(\openone - \sigma_i^x \right)+ s \sum_{x} f(x) \ket{x} \bra{x} \ ,
\end{equation}
where we have chosen the standard transverse field ``driver'' Hamiltonian $H(0)$ that assumes no prior knowledge of the form of $f(x)$, and a linear interpolating schedule, with $s\equiv t/t_f$ being the dimensionless time parameter. The initial state is the ground state of $H(0)$.
Below, we list several of PHWO examples that we study in greater detail. We refer to the case with $p=0$ as the \textbf{Plain Hamming Weight} problem.
\begin{enumerate}
\item \textbf{Fixed Plateau:}
\begin{equation}
\label{eqt:plateau}
f(x) = \begin{cases}
u -1, & l<\abs{x}<u,\\
\abs{x}, & \text{otherwise}
\end{cases} \ .
\end{equation}
Clearly, this forms a plateau in Hamming weight space. We take $u,l=\mathcal{O}(1)$. Since the location of the plateau does not change with $n$, we refer to it as ``fixed." An instance of this cost function with $l=3$ and $u=8$ is illustrated in Fig.~\ref{fig:Plateau}. By numerical diagonalization we find that QA has a constant gap for this cost-function.
\begin{figure}[t]
\centering
\includegraphics[width=0.65\columnwidth]{Figure01}
\caption{$l=3,u=8$}
\label{fig:Plateau}
\end{figure}
\item \textbf{Reichardt:}
\begin{equation}
\label{eqt:Reichardt}
f(x) = \begin{cases}
\abs{x}+h(n), & l(n)<\abs{x}<u(n)\\
\abs{x} & \text{otherwise}
\end{cases} \ ,
\end{equation}
with $h\frac{u-l}{\sqrt{l}}= o(1)$. For this case, Reichardt~\cite{Reichardt:2004} proved a constant lower bound on the minimum spectral gap during the quantum anneal. In Appendix~\ref{app:review} we provide a pedagogical review of this proof and fill in some details not explicitly provided in the original proof.
\item \textbf{Moving Plateau:}
\begin{equation}
\label{eqt:movingplateau}
f(x) = \begin{cases}
u -1, & l(n)<\abs{x}<u(n)\\
\abs{x} & \text{otherwise}
\end{cases} \ ,
\end{equation}
with $l(n) = n/4$, and $u(n)=\mathcal{O}(1)$. This is termed ``moving" since the location of the plateau changes with $n$. Note that this is a special case from the Reichardt class.
\item \textbf{Grover:}
\begin{equation}
\label{eqt:Grover}
f(x) = \begin{cases}
n, & \abs{x}\geq 1,\\
0, & \abs{x}=0.
\end{cases} \
\end{equation}
This is a minor modification of the standard Grover problem: the marked state is the all-zeros string with energy $0$, and the energy of all the other states is $n$. Scaling the energy by $n$ keeps the maximum energy of all the PHWO problems we consider comparable.
\item \textbf{Spike:}
\begin{equation}
\label{eqt:spike}
f(x) = \begin{cases}
n, & \abs{x}=n/4,\\
\abs{x}, & \text{otherwise}
\end{cases} \ .
\end{equation}
This was studied by Farhi \textit{et al}. in~\cite{Farhi-spike-problem}, where it was argued that the quantum minimum gap scales as $\mathcal{O}(n^{-1/2})$ and that SA will take exponential time to find the ground state. However, we show below (Fig.~\ref{fig:SpikeMovingVanDam_TTSopt}) that SVD is more efficient than QA for this problem.
\item \textbf{Precipice:}
\begin{equation}
\label{eqt:precipice}
f(x) = \begin{cases}
-1, & \abs{x}=n,\\
\abs{x}, & \text{otherwise}
\end{cases} \ .
\end{equation}
This was studied by van Dam \textit{et al}. in~\cite{vanDam:01}, where it was proved that the quantum minimum gap for this problem scales as $\mathcal{O}(2^{-n/2})$.
\item \textbf{$\alpha$-Rectangle:}
\begin{equation}
\label{eqt:alpharect}
f(x) = \begin{cases}
\abs{x} + n^\alpha, & \frac{n}{4} - \frac{1}{2} c n^\alpha <\abs{x}< \frac{n}{4} + \frac{1}{2} c n^\alpha ,\\
\abs{x}, & \text{otherwise}
\end{cases} \ .
\end{equation}
We call this an $\alpha$-Rectangle because the width of the perturbation ($c n^\alpha$) is $c$ times the height. This was studied in~\cite{Brady:2015rc}, where evidence for the following conjecture for the scaling of the quantum minimum gap $g_{\mathrm{min}}$ was presented,
\begin{equation} \label{eqt:VanDam}
g_\mathrm{min} = \begin{cases}
\mathrm{constant}, & \alpha<\frac{1}{4}, \\
1 / \mathrm{poly}(n), & \frac{1}{4}<\alpha<\frac{1}{3}, \\
1 / \exp(n), & \alpha>\frac{1}{3}.
\end{cases}
\end{equation}
Note that $\alpha<1/4$ is a member of the Reichardt class, and thus the constant lower-bound on the minimum gap is a theorem, and not a conjecture. We restrict ourselves to the case of $c=1$.
\end{enumerate}
We remark that all the problems listed above are representative members of a large family of problems: if the input bit-string to any of the above problems is transformed by an $\mathrm{XOR}$ mask, then all of our analysis below will hold. For QA, the $\mathrm{XOR}$ mask can be represented as a unitary transformation: $\bigotimes_{i=1}^n (\sigma_i^x)^{a_i}$, with $a \in \{0,1\}^n$ being the mask string. As this unitary commutes with the QA Hamiltonian at all times, none of our subsequent analysis is affected. Similar arguments go through for SA and all the other algorithms that we consider.
We note that PHWO problems are strictly toy problems since these problems are typically represented by highly non-local Hamiltonians (see Appendix~\ref{app:locality}) and thus are not physically implementable, in the same sense that the adiabatic Grover search problem is unphysical \cite{Roland:2002ul,RPL:10}. Nevertheless, these problems provide us with important insights into the mechanisms behind a quantum speed-up, or lack thereof.
\section{The semi-classical potential and tunneling }\label{sec:SCtunnel}
In order to study tunneling, we need a potential arising from a semi-classical limit, which defines classically allowed and forbidden regions. One approach to writing a semi-classical potential for quantum Hamiltonians is to use the spin-coherent path-integral formalism \cite{klauder1979path}. This semi-classical potential has been used profitably in various QA studies, e.g., Refs.~\cite{Farhi-spike-problem,Schaller:2007uq,FarhiAQC:02,Boixo:2014yu}, and we extend its applications here. For the quantum evolution, since the initial state [the ground state of $H(0)$] is symmetric under permutations of qubits and the unitary dynamics preserves this symmetry (it is a symmetry of $H(s)$ for all $s$), we can consistently restrict ourselves to spin-1/2 symmetric coherent states $\ket{\theta,\phi} $:
\begin{equation}
\ket{\theta,\phi} = \bigotimes_{i=1}^n \left[ \cos\left( \frac{\theta}{2} \right) \ket{0}_i + \sin \left( \frac{\theta}{2} \right) e^{i \varphi} \ket{1}_i \right] \ .
\end{equation}
The spin-coherent potential is then given by:
\begin{equation} \label{eq:vsc}
V_{{\mathrm{SC}}}(\theta,\phi,s) = \bra{\theta,\phi} H(s) \ket{\theta,\phi} \ .
\end{equation}
We show that for all the examples defined above except the Reichardt class (we address this below), this potential captures important features of the quantum Hamiltonian [Eq.~\eqref{eqt:QuantumH}] and reveals the presence of tunneling. Specifically:
\begin{enumerate}
\item {\emph{The spin-coherent potential displays a degenerate double-well almost exactly at the point of the minimum gap.}} In Fig.~\ref{fig:Veff} we plot, for the Fixed Plateau the potential near the minimum gap. The potential transitions from having a single minimum on the right to a single minimum on the left. In between, it becomes degenerate and displays a degenerate double well. Since the instantaneous ground state corresponds to the position of the global minimum, which exhibits a discontinuity, the degeneracy point is where tunneling should be most helpful. In Fig.~\ref{fig:MinGapsWellLocs}, we show that the location of the minimum gap of the quantum evolution is very close to the location of the degenerate double-well in the spin-coherent potential.
\item {\emph{The ground state predicted by the spin-coherent potential is a good approximation to the quantum ground state except near the degeneracy point.}} As expected from a potential that arises in a semi-classical limit, the ground state predicted by the spin-coherent potential (i.e., the spin-coherent state corresponding to the instantaneous global minimum in $V_{\mathrm{SC}}$) agrees well with the quantum ground state, except where tunneling is important. In particular, delocalization when the spin-coherent potential is a degenerate double-well (or is close to being one) should imply that approximating the ground state with a wavefunction localized in one of the wells fails. Indeed, we find this to be the case. We illustrate this for the Fixed Plateau in Fig.~\ref{fig:GSdistanceSC-QA}; similar results hold for the other examples we have considered.
\item {\emph{There is a sharp change in the ground state of the adiabatic quantum evolution at the degeneracy point.}} Tunneling should be accompanied by a sharp change in the properties of the ground state at the degeneracy point as the state state shifts from being localized in one well to the other. We quantify this change by calculating the expectation value of the Hamming weight operator, defined as $\mathrm{HW} = \frac{1}{2} \sum_{i=1}^n \left( \openone - \sigma^z_i \right)$.
We expect a discontinuity in the spin-coherent ground state expectation value $\wich{\textrm{HW}}$, because the spin-coherent ground state changes discontinuously at the degeneracy point. We find that there is a nearly identical change in the quantum ground state expectation value $\wich{\textrm{HW}}$, for all of the examples listed above. This is illustrated explicitly for the Fixed Plateau in Fig.~\ref{fig:GSGibbs}. In Fig.~\ref{fig:HWdropsWellLocs}, we show that there is close and increasing agreement (as a function of $n$) between the position of the sudden drop in $\wich{\textrm{HW}}$ and the position of the degeneracy point, for all of the problems considered.
\item {\emph{The scaling of the barrier height in the spin-coherent potential is positively correlated with the scaling of the minimum gap of the quantum Hamiltonian.}} In Fig.~\ref{fig:heightsandgaps}, we see that as the barrier height increases, the inverse of the quantum minimum gap also increases.
\end{enumerate}
Note that the Reichardt class is absent from the discussion above. The reason is that for these problems, the barrier in the spin-coherent potential is very small, which makes its numerical detection difficult. Fortunately, we can make some analytical claims about this class of problems. By adapting Reichardt's proof (reviewed in Appendix~\ref{app:review}) that these problems have a constant minimum gap, we are able to prove that the barrier height in the spin-coherent potential for this class vanishes as $n \to \infty$. Therefore, for these easy-for-AQA problems, there is a vanishing barrier in the spin-coherent potential. More precisely, we can show, for any perturbed Hamming weight problem,
\begin{align}
V_{{\mathrm{SC}}}^{\mathrm{pert}} - V_{{\mathrm{SC}}}^{\mathrm{unpert}} &= s \sum_{l<k<u} f(k) \binom{n}{k} p(\theta)^k (1-p(\theta))^{n-k} \notag \\
&= \mathcal{O}\left(h \frac{u-l}{\sqrt{l}}\right)
\end{align}
where the unperturbed case refers to $h(n)=0$ in Eq.~\eqref{eqt:Reichardt}.
Recall that $h \frac{u-l}{\sqrt{l}}$ = $o(1)$ for the Reichardt class. Thus asymptotically, the spin-coherent potential for this class approaches the spin-coherent potential of the unperturbed Hamming weight problem. It is easy to check that the latter has a single minimum throughout the evolution, and hence no barriers.
\begin{figure*}[htbp]
\subfigure[]{\includegraphics[width=0.32\textwidth]{Figure02a} \label{fig:Veff}}
\subfigure[]{\includegraphics[width=0.32\textwidth]{Figure02b} \label{fig:GSdistanceSC-QA}}
\subfigure[]{\includegraphics[width=0.32\textwidth]{Figure02c} \label{fig:GSGibbs} }
\caption{Results for the Fixed Plateau problem with $l=0$ and $n=512$. (a) The semi-classical potential with $u=6$ exhibits a double-well degeneracy at the position $s \approx 0.89$ (solid), but is non-degenerate before and after this point (dotted and dashed), thus leading to a discontinuity in the position of its global minimum. The same is observed for other the PHWO problems we studied (not shown). (b) The trace-norm distance between the quantum ground state (obtained by numerical diagonalization) and the spin-coherent state corresponding to the instantaneous global minimum in $V_{\mathrm{SC}}$, as a function of $t/t_f$. The peak corresponds to the location of the tunneling event, at which point the semi-classical approximation breaks down. (c) $\wich{\mathrm{HW}}$ in the instantaneous quantum ground state state (GS) and the instantaneous quantum ground state as predicted by the semi-classical potential (SC GS), as a function of $t/t_f$. The sharp drop in the GS and SC GS curves is due to a tunneling event wherein $\sim u$ qubits are flipped which occurs at the degeneracy point observed in the spin-coherent potential.}
\end{figure*}
\begin{figure*}[t]
\subfigure[]{\includegraphics[width=0.4\textwidth]{Figure03a} \label{fig:MinGapsWellLocs}}
\subfigure[]{\includegraphics[width=0.4\textwidth]{Figure03b} \label{fig:HWdropsWellLocs}}
\caption{(a) The difference in the position of the minimum gap from exact diagonalization and the position of the double-well degeneracy (as seen in Fig.~\ref{fig:Veff} for the Fixed Plateau) from the semi-classical potential, as a function of $n$ for the {Fixed Plateau}, the {Spike}, the {$0.5$-Rectangle}, the {$0.33$-Rectangle}, the {Precipice}, and {Grover} (log-log scale). (b) The difference in the position of the sudden drop in Hamming weight [as seen in Fig.~\ref{fig:GSGibbs}] and the position of the double-well degeneracy from the semi-classical potential [as seen in Fig.~\ref{fig:Veff}], as a function of the number of qubits $n$ for the {Fixed Plateau}, the {Spike}, the {$0.5$-Rectangle}, the {$0.33$-Rectangle}, the {Precipice}, and {Grover} (log-log scale). }
\end{figure*}
\begin{figure*}[t]
\subfigure[]{\includegraphics[width=0.4\textwidth]{Figure04a} \label{fig:BarrierHeights}}
\subfigure[]{\includegraphics[width=0.4\textwidth]{Figure04b} \label{fig:MinGaps}}
\caption{(a) The height of the barrier between the two wells at the degeneracy point of the spin-coherent potential [as seen in Fig.~\ref{fig:Veff}], as a function of $n$ for the {Fixed Plateau}, the {Spike}, the {$0.5$-Rectangle}, the {$0.33$-Rectangle}, the {Precipice}, and {Grover} (log-log scale). (b) The inverse of the minimum gap as a function of $n$ for the {Fixed Plateau}, the {Spike}, the {$0.5$-Rectangle}, the {$0.33$-Rectangle}, the {Precipice}, and {Grover} (log-log scale). The important thing to note is that the \emph{order} of scaling is preserved in both plots. That is, steeper the scaling of the barrier height, steeper the scaling of the inverse minimum gap.}
\label{fig:heightsandgaps}
\end{figure*}
Taken together, these observations indicate that the spin-coherent potential (not the cost function alone) is the appropriate potential with respect to which tunneling is to be understood for these problems.
\section{Fixed Plateau: Performance of algorithms} \label{sec:FixedPlateau}
Having motivated the spin-coherent potential for understanding tunneling, we now exhaustively analyze the {Fixed Plateau}. We choose this problem because it forces us to confront some intuitions about the performance of certain algorithms. Considering the final cost function, the Fixed Plateau has neither local minima nor a barrier going from large to small $|x|$: it just has a long, flat section before the ground state at $|x|=0$. This might suggest that it is easy for an algorithm such as SA, and is not a candidate for a quantum speedup. Moreover, given the absence of a barrier, one might suspect that the quantum evolution would not even involve multi-qubit tunneling.
We dispel both of these intuitions and summarize our findings first. In the previous section, we already provided evidence that tunneling is unambiguously present for this problem. The spin-coherent potential involves energy barriers, despite their absence in the final cost function, and the adiabatic quantum evolution is forced to tunnel in order to follow the ground state. By a simulation of the Schr\"odinger equation, we find that AQA needs a time of $\mathcal{O}(n^{0.5})$ in order to reach a given success probability (see Sec.~\ref{subsec:adDyn}). Therefore, the adiabatic algorithm, via tunneling, is able to solve this problem efficiently.
Turning to SA, an algorithm which performs a local stochastic search on the final cost function, we prove that simulated annealing with single spin-updates will take time $\mathcal{O}(n^{u-l-1}) = \mathcal{O}(n^\mathrm{plateau\ width})$ to find the ground state (see Sec.~\ref{subsec:SArandom}). This result is due to the fact that a random walker on the plateau has no preferred direction and becomes trapped there. More precisely, the probability of a leftward transition while on the plateau is proportional to the probability of flipping one of a constant number of bits (given by the Hamming weight) out of $n$, which scales as $\sim 1/n$ if $l,u = \mathcal{O}(1)$. And since the walker needs to make as many consecutive leftward transitions as the width of the plateau in order to fall off the plateau, the time taken for this to happen scales as $\mathcal{O}(n^\mathrm{plateau\ width})$. Consequently, we obtain a polynomial speedup of AQA over SA that can be made as large as desired. Therefore, using the Fixed Plateau, we are able to demonstrate that a quantum speedup over SA is possible via tunneling in the adiabatic regime.
However, is the adiabatic evolution optimal? In order to find the optimal evolution time, we employ the optimal time to solution (TTS$_\mathrm{opt}$), a metric that is commonly used in benchmarking studies \cite{speedup} (also see Appendix~\ref{app:TTS}). It is defined as the minimum total time such that the ground state is observed at least once with desired probability $p_d$:
\begin{equation}
\label{eqt:TTSopt}
\text{TTS}_\mathrm{opt} = \min_{t_f > 0} \left\{ t_f \frac{\ln( 1- p_d)}{\ln \left(1-p_{\mathrm{GS}}(t_f) \right)} \right\} \ ,
\end{equation}
where $t_f$ is the duration (in QA) or the number of single spin updates (in SA) of a single run of the algorithm, and $p_{\mathrm{GS}}(t_f)$ is the probability of finding the ground state in a single such run. The use of TTS$_\mathrm{opt}$ allows for the possibility that multiple short runs of the evolution, each lasting an optimal annealing time $({t_f})_{\textrm{opt}}$, result in a better scaling than a single long (adiabatic) run with an unoptimized $t_f$. The quantum evolution that gives the optimal annealing time relative to this cost function is actually DQA, with an asymptotic scaling of $\mathcal{O}(1)$. Importantly, this diabatic evolution does not contain any of the signatures of tunneling discussed in the previous section. Therefore, for the Fixed Plateau, tunneling does not give rise to the optimal quantum performance.
Motivated by the fact that the optimal quantum evolution involves no multi-qubit tunneling, we consider spin-vector dynamics \cite{Smolin} (see, also Refs.~\cite{Albash:2014if,owerre2015macroscopic}), a model that evolves according to the spin-coherent potential in Eq.~\eqref{eq:vsc}. SVD can be derived as the saddle-point approximation to the path integral formulation of QA in the spin-coherent basis \cite{owerre2015macroscopic}. The SVD equations are equivalent to the Ehrenfest equations for the magnetization under the assumption that the density matrix is a product state, i.e., $\rho = \otimes_{i=1}^n \rho_{i}$, where $\rho_i$ denotes the state of the $i$th qubit. This algorithm is useful since it is derived under the assumption of continuity of the angles $(\theta, \phi)$, so tunneling, which here would amount to a discrete jump in the angles, is absent.
We also consider a quantum Monte Carlo based algorithm, often called simulated quantum annealing (SQA) \cite{sqa1,Santoro}. We show that SQA has a scaling that is better than SA's. Indeed, this is consistent with the fact that SQA thermalizes not just relative to the final cost function, but also during the evolution.
We provide further details of our implementations of each of these algorithms in Appendix \ref{app:Methods}. We now turn to each of the algorithms individually and detail their performance for the Fixed Plateau problem.
\subsection{Adiabatic dynamics} \label{subsec:adDyn}
In order to study the scaling of adiabatic dynamics, we consider the minimum time $\tau_{0}$ required to reach the ground state with some probability $p_0$, where we choose $p_0$ to ensure that we are exploring a regime close to adiabaticity for QA. We call this benchmark metric the ``threshold criterion,'' and set $p_0=0.9$. As seen in Fig.~\ref{fig:ThC}, we observe a scaling for AQA that is approximately $\sim n^{0.5}$. As is to be expected given that the tunneling for the Fixed Plateau problem is controlled by the width of the plateau, which is constant (does not scale with $n$), we find that $\tau_{0}$ scales in the same way for the Fixed Plateau and the Plain Hamming Weight problems (see Appendix~\ref{app:review}). This suggests that the dominant contribution to the scaling at large $n$ is not the time associated with tunneling but rather the time associated with the Plain Hamming Weight problem.
As also seen in Fig.~\ref{fig:ThC}, we find that the textbook adiabatic criterion \cite{Messiah:book} given by
\begin{equation}
t_f \gtrsim \max_{s \in [0,1]} \frac{ |\bra{\varepsilon_0(s) } \partial_s H(s) \ket{\varepsilon_1(s)}|}{\text{Gap}(s)^2} \ ,
\label{eqt:AdC}
\end{equation}
serves as an excellent proxy for the scaling of AQA %
\footnote{We note that while this adiabatic criterion matches the numerical scaling we observe for the quantum evolution, it is well known to be neither exact nor general; see, e.g., Refs.~\cite{Jansen:07,Amin:09,lidar:102106}.}.
The scaling of AQA is matched by the scaling of the numerator of the adiabatic condition, which is explained by the fact that we find a constant minimum gap for the case $l,u = \mathcal{O}(1)$. This numerator turns out to be well approximated in our case by the matrix element of $H(s)$ between the ground and first excited states, leading to $t_f\sim n^{0.5}$ in the adiabatic limit. Note that calculating this matrix element can easily be done for arbitrarily large systems, and is hence much easier to check directly than the scaling of AQA.
\begin{figure}[t]
\includegraphics[width=0.8 \columnwidth]{Figure05}
\caption{Performance of different algorithms for the Fixed Plateau problem with $l=0$ and $u =6$. Shown is a log-log plot of the scaling of the time to reach a threshold success probability of $0.9$, as a function of system size $n$ for AQA, SQA ($\beta = 30$, $N_{\tau} = 64$) and SA ($\beta_f = 20$). The time for SQA and SA is measured in single-spin updates (for SQA this is $N_{\tau}$ times the number of sweeps times the number of spins, whereas for SA this is the number of sweeps times the number of spins), where both are operated in `solver' mode as described in Appendix~\ref{app:Methods}. Also shown is the scaling of the numerator of the adiabatic condition as defined in Eq.~\eqref{eqt:AdC}. The scaling for AQA and the adiabatic condition extracted by a fit using $n \gtrsim 10^2$ is approximately $n^{0.44}$. However, the true asymptotic scaling is likely to be $\sim n^{0.5}$ since the scaling for the Fixed Plateau problem is clearly lower-bounded by the Plain Hamming Weight problem, for which we have verified $\tau_0 \sim n^{0.5}$ (see Appendix~\ref{app:review}), and we expect the effect of the plateau to become negligible in the large $n$ limit. SQA scales more favorably ($\sim n^{1.5}$) than SA ($\sim n^{5}$). We have checked that the scaling of SQA does not change even if we double the number of Trotter slices $N_{\tau}$ and keep the temperature $1/\beta$ fixed.}
\label{fig:ThC}
\end{figure}
\subsection{Simulated annealing using random spin selection} \label{subsec:SArandom}
We consider a version of SA with random spin-selection as the rule that generates candidates for Metropolis updates. Our main motivation is to understand the behavior of a local, stochastic search algorithm which has access only to the final cost function. We note that our analysis below is general for any Plateau problem, and is not limited to the {Fixed Plateau} or the {Moving Plateau}.
If we pick a bit-string at random, then for large $n$ we will start with very high probability at a bit-string with Hamming weight close to $n/2$. The plateau may be to the left or to the right of $n/2$; if the plateau is to the right, then the random walker is unlikely to encounter it and can quickly descend to the ground state.
Thus, the more interesting case is when the random walker arrives at the plateau from the right. We proceed to analyze these two cases separately.
\subsubsection{Walker starts to the right of the plateau}
In this case, how much time would it take, typically, for the walker to fall off the left edge? It is intuitively clear that traversing the plateau will be the dominant contribution to the time taken to reach the ground state, as after that the random walker can easily walk down the potential. We show below (for the walker that starts to the left of the plateau) that this time can be at most $\mathcal{O}(n^2)$ if $\beta = \Omega(\log n)$.
To evaluate the time to fall off the plateau, note that the perturbation is applied on strings of Hamming weight $l+1,l+2,\dots,u-1$, so the width of the plateau is $w \equiv u-l-1$. Consider a random walk on a line of $w+1$ nodes labelled $0,1,\dots w$. Node $i$ represents the set of bit strings with Hamming weight $l+i$, with $0\leq i \leq w$. We may assume that the random walker starts at node $w$.
Only nearest-neighbor moves are allowed and the walk terminates if the walker reaches node $0$.
Our analysis will provide a lower bound on the actual time to fall off the left edge, because in the actual PHWO problem one can also go back up the slope on the right, and in addition we disallow transitions from strings of Hamming weight $l$ to $l+1$. This is justified because the Metropolis rule exponentially (in $\beta$) suppresses these transitions.
The transition probabilities $p_{i \to j}$ for this problem can be written as a $(w+1) \times (w+1)$ row-stochastic matrix $p_{ij} = p_{i \to j}$. Here $p$ is a tridiagonal matrix with zeroes on the diagonal, except at $p_{00}$ and $p_{ww}$. First consider $1\leq i \leq w-1$. If the walker is at node $i$, then the transition to node $i+1$ (which has Hamming weight $l+i+1$) occurs with probability $\frac{n-(l+i)}{n}$ (the chance that the bit picked had the value $0$). Similarly, for $1\leq i \leq w$, the Hamming weight will decrease to $l+i-1$ with probability $\frac{l+i}{n}$ (the chance that the bit picked had the value $1$). Combining this with the fact that a walker at node $0$ stays put, we can write:
\bes
\begin{align}
b_i &\equiv p_{i\to i}=\begin{cases} 1 \text{ if } i=0 \\
0 \text{ if } 1\leq i \leq (w-1) \\
1-\frac{l+w}{n} \text{ if } i=w
\end{cases},\\
c_i &\equiv p_{i-1\to i} = \begin{cases} 0 \text{ if } i=1\\ 1-\frac{l+i-1}{n} \text{ if } i=2,\dots,w \end{cases}, \\
a_i &\equiv p_{i\to i-1} = \frac{l+i}{n} \text{ if } i=1,2,\dots,w.
\end{align}
\ees
Let $X(t)$ be the position of the random walker at time-step $t$. The random variable measuring the number of steps the random walker starting from node $r$ would need to take to reach node $s$ for the first time is
\begin{equation}
\tau_{r,s} \equiv \text{min} \{ t>0: X(t)=s,X(t-1)\neq s | X(0) =r\}\ .
\end{equation}
The quantity we are after is $\mathbb{E}\tau_{w,0}$, the expectation value of the random variable $\tau_{w,0}$, i.e., the mean time taken by the random walker to fall off the plateau. Since only nearest neighbor moves are allowed we have
\begin{equation} \label{eq:timetofalloff}
\mathbb{E}\tau_{w,0} =\sum_{r=1}^w \mathbb{E}\tau_{r,r-1}\ .
\end{equation}
Stefanov~\cite{stefanov1995mean} (see also Ref.~\cite{krafft1993mean}) has shown that
\begin{equation} \label{eq:stefanovformula}
\mathbb{E}\tau_{r,r-1} = \frac{1}{a_r} \left( 1 + \sum_{s=r+1}^{w} \prod_{t=r+1}^{s} \frac{c_t}{a_t} \right),
\end{equation}
where $c_{w+1}\equiv 0$. Evaluating the sum term by term, we obtain:
\bes
\label{eq:62}
\begin{align}
\mathbb{E}\tau_{w,w-1} &= \frac{n}{l+w}, \\
\vdots \nonumber \\
\mathbb{E}\tau_{w-k,w-k-1} &= \frac{n}{l+w-k} \left[1+ \frac{n-(l+w-k)}{l+w-(k-1)}+\dots \right. \nonumber \\
& +\frac{n-(l+w-k)}{l+w-(k-1)}\times\cdots \nonumber \\
& \left. \times\frac{n-(l+w-2)}{l+w-1} \times \frac{n-(l+w-1)}{l+w}\right].
\end{align}
\ees
Now consider the following cases:
\begin{enumerate}
\item \emph{Fixed Plateau}, $l,u = \mathcal{O}(1)$: Here, using the fact that $k=\mathcal{O}(w)=\mathcal{O}(1)$, we conclude that $\mathbb{E}\tau_{w-k,w-k-1} = \mathcal{O}(n^{k+1})$. Since the leading order term is $\mathbb{E}\tau_{w-(w-1),w-w}=\mathbb{E}\tau_{1,0}$, the time to fall off the plateau is $\mathcal{O}(n^w) = \mathcal{O}(n^{u-l-1}).$ This result about SA's performance is confirmed numerically in Fig.~\ref{fig:ThC}.
\item In order for Reichardt's bound (see Appendix~\ref{app:review}) to give a constant lower-bound to the quantum problem, we need $u=l+o(l^{1/4})$. Since at most we can have $l = \mathcal{O}(n)$, we can conclude $\mathbb{E}\tau_{w-k,w-k-1} = \mathcal{O}\left(\frac{n}{l}\right)^{k+1}$. Therefore, the time to fall-off becomes $\mathbb{E}\tau_{w,0} = \mathcal{O}\left(w(\frac{n}{l})^w\right)$.
\begin{itemize}
\item \emph{Moving Plateau}: If $l = \Theta(n)$ and $w = \mathcal{O}(1)$, we can see that $\mathbb{E}\tau_{w,0} = \mathcal{O}(1)$, which is a constant time scaling.
\item \emph{Moving Plateau with changing width}: If $l = \Theta(n)$ and $w = \mathcal{O}(n^a)$, where $0<a<1/4$, then $\mathbb{E}\tau_{w,0} = \mathcal{O}(n^a \mathcal{O}(1)^{n^a})$, which is super-polynomial.
\item \emph{Most general plateau in the Reichardt class}: More generally, if $l = \mathcal{O}(n^b)$, with $b \leq 1$ and $w = \mathcal{O}(n^a)$, where $0\leq a < b/4$, then we get the scaling $\mathbb{E}\tau_{w,0} = \mathcal{O}(n^{a}\mathcal{O}(n^{1-b})^{n^a})$
\end{itemize}
\end{enumerate}
\subsubsection{Walker starts to the left of the plateau}
\label{subsubsec:SAHamWt}
Note that this case is equivalent to the unperturbed Hamming weight problem, which is a straightforward gradient descent problem. We may therefore consider a simple fixed temperature version of SA (i.e., the standard Metropolis algorithm). We will show that the performance of SA on this problem provides an upper bound of $\mathcal{O}(n^2)$ on the time for a random walker to arrive at the plateau, and on the time for a random-walker to reach the ground state after descending from the plateau. Moreover, our analysis provides a lower bound of $\mathcal{O}(n\log n)$ on the efficiency of such algorithms.
For this problem, the transition probabilities are:
\bes
\begin{align}
c_i &\equiv p_{i-1\to i} = \frac{n-i+1}{n} e^{-\beta} \ , \\
a_i &\equiv p_{i\to i-1} = \frac{i}{n} \ ,
\end{align}
\ees
with $i=1,2,\dots,n$ denoting strings of Hamming weight $i$, and $\beta$ is the inverse temperature. Using the Stefanov formula~\eqref{eq:stefanovformula}, we can write (after much simplification):
\begin{equation}
\mathbb{E}\tau_{n-k,n-k-1} = \frac{n}{n-k} \binom{n}{k}^{-1} \sum_{l=0}^{k} e^{-l \beta} \binom{n}{k-l} \ .
\end{equation}
We will bound
\begin{equation} \label{eqt:hamwttime}
\mathbb{E}\tau_{n,0} = \sum_{k=0}^{n-1} \frac{n}{n-k} \binom{n}{k}^{-1} \sum_{l=0}^{k} e^{-l \beta} \binom{n}{k-l} \ ,
\end{equation}
the expected time to reach the all-zeros string starting from the all-ones string. This is the worst-case scenario as we are assuming that we are starting from the string farthest from the all-zeros string. Note again that if we start from a random spin configuration, then with overwhelming probability we will pick a string with Hamming weight close to $n/2$. Thus, most probably, $\mathbb{E}\tau_{n/2,0}$ will be the time to hit the ground state.
We first show that $\beta = \mathcal{O}(1)$ will lead to an exponential time to hit the ground state, irrespective of the walker's starting string.
Toward that end,
\bes
\begin{align}
\mathbb{E}\tau_{1,0} &= \mathbb{E}\tau_{n-(n-1),n-n} \\
&=\sum_{l=0}^{n-1} e^{-l \beta} \binom{n}{n-1-l} \\
&=e^\beta\left[(e^{-\beta}+1)^n-1\right],
\end{align}
\ees
which is clearly exponential in $n$ if $\beta= \mathcal{O}(1)$.
Next, let $\beta(n) = \log n$, i.e., we decrease the temperature logarithmically in system size. In this case,
\begin{align}
\mathbb{E}\tau_{1,0} = n \left[\left(1 + \frac{1}{n}\right)^n-1\right] \leq n (e - 1) = \mathcal{O}(n)\ .
\end{align}
Now it is intuitively clear that $\mathbb{E}\tau_{1,0} > \mathbb{E}\tau_{r,r-1}$ for all $r>1$, which implies that $n \mathbb{E}\tau_{1,0} \geq \mathbb{E}\tau_{n,0} $. Thus, if $\beta = \log n$, then $\mathbb{E}\tau_{n,0} = \mathcal{O}(n^2)$ at worst.
To obtain a lower-bound on the performance of the algorithm, we take $\beta \to \infty$. Thus, for each $k$ in Eq.~\eqref{eqt:hamwttime}, only the $l=0$ term will survive. Hence,
\begin{align}
\lim_{\beta \to \infty} \mathbb{E}\tau_{n,0} &= \sum_{k=0}^{n-1} \frac{n}{n-k} = n \sum_{i=1}^n \frac{1}{i} \approx n (\log n + \gamma)\ ,
\end{align}
for large $n$, with $\gamma$ being the Euler-Mascheroni constant. The scaling here is $\mathcal{O}(n \log n)$. This is the best possible performance for single-spin update SA with random spin-selection on the plain Hamming weight problem. Therefore, if $\beta = \Omega(\log n)$, the scaling will be between $\mathcal{O}(n \log n)$ and $\mathcal{O}(n^2)$. Of course, this cost needs to be added to the time taken for the walker starting to the right of the plateau.
Two clarifications are in order regarding the comparison between our theoretical bound on SA's performance and the associated numerical simulations we have presented. First, while Fig.~\ref{fig:ThC} displays the time to cross a threshold probability, our theoretical bound of $\mathcal{O}(n^{u-l-1})$ is on the expected time for the random walker to hit the ground state [Eq.~\eqref{eq:timetofalloff}]. However, we found that both metrics show identical scaling. Second, while the SA data in Fig.~\ref{fig:ThC} was generated using sequential spin updates, the theoretical bound assumes random spin updates (see Appendix~\ref{app:SA} for more details on the update schemes). However, we found that the asymptotic scaling for both cases is nearly identical in the long-time regime, and thus have plotted only the former.
\subsection{Optimal QA via Diabatic Transitions} \label{subsec:diabaticQA}
\begin{figure*}[t]
\centering
\subfigure[]{\includegraphics[width=0.32\textwidth]{Figure06a} \label{fig:QA_EnergyOverlap}}
\subfigure[]{\includegraphics[width=0.32\textwidth]{Figure06b} \label{fig:TTSScaling}}
\subfigure[]{\includegraphics[width=0.32\textwidth]{Figure06c} \label{fig:QA_O3_AveHW}}
\caption{Diabatic QA \textit{vs} SA and SVD for the {Fixed Plateau} problem with $l=0$. (a) Population $P_i$ in the $i$th energy eigenstate along the diabatic QA evolution at the optimal TTS for $n=512$ and $u=6$. Excited states are quickly populated at the expense of the ground state. By $t/t_f = 0.5$ the entire population is outside the lowest $9$ eigenstates. In the second half of the evolution the energy eigenstates are repopulated in order. This kind of dynamics occurs due to a lining-up of avoided level crossings as seen in Fig.~\ref{fig:Cascade}. (b) Scaling of the optimal TTS with $n$ for $u=6$, with an optimized number of single-spin updates for SA, and equal $({t_f})_\textrm{opt}$ for DQA and SVD. SA scales as $\mathcal{O}(n)$, a consequence of performing sequential single-spin updates. DQA and SVD both approach $\mathcal{O}(1)$ scaling as $n$ increases. Here we set $p_d=0.7$ in Eq.~\eqref{eqt:TTSopt}, in order to be able to observe the saturation of SVD's TTS to the point where a single run suffices, i.e., TTS$_\textrm{opt} = ({t_f})_\textrm{opt}$. The conclusion is unchanged if we increase $p_d$: this moves the saturation point to larger $n$ for both SVD and DQA, and we have checked that SVD always saturates before DQA.
Inset: scaling as a function of $u$ for $n=1008$. SVD is again seen to exhibit the best scaling, while for this value of $n$ the scaling of DQA and SA is similar (DQA's scaling with $n$ improves faster than SA's as a function of $n$, at constant $u$). (c) $\wich{\mathrm{HW}}$ of the QA wavefunction and the SVD state (defined as the product of identical spin-coherent states) for $n=512$ and $u = 6$. The behavior of the two is identical up to $t/t_f \approx 0.8$, when they begin to differ significantly, but neither displays any of the sharp changes observed in Fig.~\ref{fig:GSGibbs} for the instantaneous ground state. Inset: the trace-norm distance between the DQA and SVD states, showing that they remain almost indistinguishable until $t/t_f \approx 0.8$.}
\end{figure*}
\begin{figure}[t]
\centering
\includegraphics[width=0.8 \columnwidth]{Figure07}
\caption{The eigenenergy spectrum along the evolution for the {Fixed Plateau} with $n=512$, $l=0$, and $u=6$. Note the sequence of avoided level crossings that unmistakably line up in the spectrum to reach the ground state. This is the pathway through which DQA is able to achieve a speedup over AQA.}
\label{fig:Cascade}
\end{figure}
Having established that for the Fixed Plateau AQA enjoys a quantum speedup over local search algorithms such as SA via tunneling, we are motivated to ask: Is tunneling \emph{necessary} to achieve a quantum speedup on these problems? In order to answer this question, we demonstrate using the optimal TTS criterion defined in Eq.~\eqref{eqt:TTSopt} that the optimal annealing time for QA is far from adiabatic. Instead, as shown in Fig.~\ref{fig:QA_EnergyOverlap}, the optimal TTS for QA is such that the system leaves the instantaneous ground state for most of the evolution and only returns to the ground state towards the end. The cascade down to the ground state is mediated by a sequence of avoided energy level-crossings as seen in Fig.~\ref{fig:Cascade}. We consider this a diabatic form of QA (DQA) and call this mechanism through which DQA achieves a speedup a \emph{diabatic cascade}.
As $n$ increases for fixed $u$, repopulation of the ground state improves for fixed $({t_f})_{\textrm{opt}}$, hence causing TTS$_\mathrm{opt}$ to decrease with $n$, as seen Fig.~\ref{fig:TTSScaling}, until it saturates to a constant at the lowest possible value, corresponding to a single run at $({t_f})_{\textrm{opt}}$. At this point the problem is solved in constant time $({t_f})_{\textrm{opt}}$, compared to the $\sim\mathcal{O}(n^{0.5})$ scaling of the adiabatic regime. Moreover, as shown in Fig.~\ref{fig:QA_O3_AveHW}, there are no sharp changes in $\wich{\mathrm{HW}}$, suggesting that the non-adiabatic dynamics do not entail multi-qubit tunneling events, unlike the adiabatic case. Thus, this establishes that we may have speedups in QA that do not involve multi-qubit tunneling.
One may worry that for this diabatic evolution to be successful, the optimal annealing time may need to be very finely tuned. We address this concern in Appendix~\ref{app:tfprecision}, where we show that if $\epsilon$ is the precision desired in $p_{\mathrm{GS}}$, we need only have a precision of $\mathrm{polylog}(1/\epsilon)$ in setting $t_f$, which means that the diabatic speedup is robust.
Figure~\ref{fig:SpikeMovingVanDam_TTSopt} shows that the speedup of DQA and SVD over AQA exists for three other PHWO problems: the {Moving Plateau}, the {Spike}, and the {0.5-Rectangle} problems. Importantly, DQA and SVD have an exponential speedup over AQA for the 0.5-Rectangle problem. We do not observe a diabatic speedup for the {Precipice} or {Grover} problems.
\begin{figure*}[t]
\subfigure[\ Spike]{\includegraphics[width=0.6\columnwidth]{Figure08a}}
\subfigure[\ Moving Plateau]{\includegraphics[width=0.6\columnwidth]{Figure08b}}
\subfigure[\ 0.5-Rectangle]{\includegraphics[width=0.6\columnwidth]{Figure08c}}
\subfigure[\ Spike]{\includegraphics[width=0.6\columnwidth]{Figure08d}}
\subfigure[\ Moving Plateau]{\includegraphics[width=0.6\columnwidth]{Figure08e}}
\subfigure[\ 0.5-Rectangle]{\includegraphics[width=0.6\columnwidth]{Figure08f}}
\caption{(a-c) The optimal TTS for the {Spike}, {Moving Plateau}, and {0.5-Rectangle} problems respectively. Inset for (a) and (c): the optimal TTS for small problem sizes, where we observe SVD at first scaling poorly. However, as $n$ grows, this difficulty vanishes and it quickly outperforms DQA. (d-f) Population $P_i$ in the $i$-th energy eigenstate along the diabatic QA evolution at the optimal TTS. We observe similar diabatic transitions for these problem (shown are the cases with $n=512$ and $t_f = 9.85$ for the Spike, $n=512$ and $t_f = 10$ for the Moving Plateau, and $n=529$ and $t_f = 9.8$ for the 0.5-Rectangle) as we observed for the Fixed Plateau [Fig.~\ref{fig:QA_EnergyOverlap}].}
\label{fig:SpikeMovingVanDam_TTSopt}
\end{figure*}
\subsection{Spin Vector Dynamics} \label{subsec:SVDplat}
Given the absence of tunneling in the time-optimal quantum evolution, we are motivated to consider the behavior of Spin-Vector Dynamics (SVD), which arise in a semi-classical limit (see Appendix~\ref{app:methods_SVD} for an overview of this algorithm). As we show in Fig.~\ref{fig:TTSScaling}, the scaling of SVD's optimal TTS also saturates to a constant time, i.e., $({t_f})_{\textrm{opt}}$. Moreover, it reaches this value earlier (as a function of problem size $n$) than DQA, thus outperforming DQA for small problem sizes, while for large enough $n$ both achieve $\mathcal{O}(1)$ scaling. As seen in the inset, SVD's advantage persists as a function of $u$ at constant $n$.
The dynamics of DQA are well approximated by SVD until close to the end of the evolution, as shown in Fig.~\ref{fig:QA_O3_AveHW}: the trace-norm distance between the instantaneous states of DQA and SVD is almost zero until $t/t_f \approx 0.8$, after which the states start to diverge. This suggests that SVD is able to replicate the DQA dynamics up to this point, and only deviates because it is more successful at repopulating the ground state than DQA.
In Fig.~\ref{fig:SpikeMovingVanDam_TTSopt}, we show that SVD's speedup over AQA is replicated for the {Spike}, {Moving Plateau}, and 0.5-Rectangle problems as well. Remarkably, while the 0.5-Rectangle problem has an exponentially small gap [see Eq.~\eqref{eqt:VanDam} and Fig.~\ref{fig:MinGaps}], SVD and DQA both achieve $\mathcal{O}(1)$ scaling, and hence the diabatic cascades provides an exponential speedup relative to AQA.
It is important to note that SVD is ineffective if one desires to simulate the adiabatic evolution. In the absence of unitary dynamics (which allow for tunneling) or thermal activation (to thermally hop over the barrier), SVD gets trapped behind the barrier that forms in the semi-classical potential separating the two degenerate minima [see Fig.~\ref{fig:Veff}] and is unable to reach the new global minimum. In this sense, SVD does not enjoy the guarantee provided by the quantum adiabatic theorem for the unitary evolution~\cite{Jansen:07,Amin:09,lidar:102106}, that for sufficiently long $t_f$ dictated by the adiabatic condition, the ground state can be reached with any desired probability.
Likewise, it is important to keep in mind the distinction between a classical algorithm being able to match, or sometimes outperform, a quantum algorithm (as SVD does here), and the classical algorithm approximating the evolution or instantiating the physics of the quantum algorithm (as SVD fails to do here). Indeed, in both the diabatic and adiabatic regimes, SVD provides a poor approximation to the instantaneous quantum state. For example, in the diabatic regime, it is clear from Fig.~\ref{fig:QA_O3_AveHW} that the trace-norm distance between the instantaneous SVD state and the instantaneous quantum state starts to increase significantly for $s\gtrsim 0.8$. In the same spirit, consider the instantaneous semi-classical ground state, i.e., the spin-coherent state evaluated at the minimum of the spin-coherent potential, which may be suspected to provide a good approximation to the instantaneous quantum ground state, but does not as shown in Fig.~\ref{fig:GSdistanceSC-QA}. Thus the unentangled semi-classical ground state also fails to provide a good approximation to the quantum ground state.
\subsection{Simulated Quantum Annealing}\label{subsec:SQAplat}
Simulated Quantum Annealing (SQA) is a quantum Monte Carlo algorithm performed along the annealing schedule (see Appendix~\ref{app:methods_SQA} for further details). It is often used as a benchmark against which QA is compared (though see Ref.~\cite{Heim:2014jf} for caveats). SQA scales better than SA for the Fixed Plateau problem using the threshold criterion (see Fig.~\ref{fig:ThC}). In order to understand why SQA enjoys an advantage over SA using this benchmark metric, it is useful to study the behavior of the state of SQA along the annealing schedule. We show the behavior of $\langle \mathrm{HW} \rangle$ for SQA in Fig.~\ref{fig:AveHWSQAA}, where we observe that SQA at the optimal number of sweeps (the case of $1500$ sweeps shown in Fig.~\ref{fig:AveHWSQAA}) does \emph{not} follow the instantaneous ground state. Instead it reaches the threshold success probability by thermally relaxing to the ground state \emph{after} the minimum gap point (and tunneling event) of the quantum Hamiltonian. Therefore, SQA's advantage over SA stems from the fact that it thermalizes in a different energy landscape than SA.
We also contrast the behavior of SQA and AQA using the threshold criterion. While SQA is able to follow the instantaneous ground state for a sufficiently large number of sweeps and thus mimic the tunneling of AQA (see Fig.~\ref{fig:AveHWSQAA}), this is not the optimal way for it to reach the threshold criterion. For a fixed threshold success probability, the process of thermal relaxation after the minimum gap point uses fewer sweeps (and hence is more efficient) than following the instantaneous ground state closely throughout the anneal \footnote{This may be an artifact of our implementation of SQA, whereby we only include cluster updates along the imaginary-time direction and not along the spatial (problem) direction. An implementation with space-like cluster updates may allow SQA via its thermal relaxation to mimic the tunneling of AQA more efficiently. Whether this is the case will be addressed in future work.}. This is in contrast to AQA, where tunneling is the only means for it to reach a high success probability and nevertheless is more efficient than SQA, as seen in Fig.~\ref{fig:ThC}.
\begin{figure}[t]
\centering
\includegraphics[width=0.8 \columnwidth]{Figure09}
\caption{The expectation value of the Hamming weight operator for the quantum ground state, SQA, and AQA for the {Fixed Plateau} problem with $n=512$, $l = 0$, and $u = 6$ and annealing time chosen so as to reach a success probability of $0.9$. The expectation value for SQA ($\beta = 30$, $N_{\tau} = 64$) at a given $t/t_f$ is calculated by averaging over the Hamming weight of the $N_\tau$ imaginary time states at that time and over $10^5$ independent trials. The case of $1500$ sweeps is the minimum number of sweeps required for SQA (in `annealer' mode) to reach the threshold ground state probability of $p_0 =0.9$, and similarly for the annealing time value of $t_f = 4931.16$ for AQA. While AQA is able to approximately follow the quantum ground state (i.e., the evolution is very close to being adiabatic), the optimal SQA evolution (i.e., that requires the fewest sweeps) for achieving the threshold criterion involves \emph{not} following the ground state at the minimum gap point and instead thermally relaxing towards the ground state after this point. As shown using the higher $N_{\textrm{sw}}$ values, only after increasing the number of sweeps by more than two orders of magnitude does SQA follow the instantaneous ground state closely.}
\label{fig:AveHWSQAA}
\end{figure}
We note that SQA's threshold criterion advantage over SA does not carry over to the optimal TTS criterion. In fact, we find that using the optimal TTS criterion, SQA scales as $O(n^{1.5})$, while SA scales as $O(n)$, as seen in Fig.~\ref{fig:TTSScaling}. The reason for the latter scaling is that the optimal number of sweeps for SA is $1$, simply because there is a small but non-zero probability that in the first sweep all the $1$s are flipped to $0$s.
\section{Discussion} \label{sec:discuss}
\begin{figure}[t]
\centering
\includegraphics[width=0.32\textwidth]{Figure10}
\caption{The optimal TTS for the potential given in Eq.~\eqref{eqt:jarret}. QA outperforms SVD over the range of problem sizes we were able to check. The reason can be seen in the inset, which displays the ground state probability for SVD and QA for different annealing times $t_f$, with $n = 512$. The optimal annealing time for SVD occurs at the first peak in its ground state probability ($t_f \approx 8.98$), whereas the optimal annealing time for QA occurs at the much larger \emph{second} peak in its ground state probability ($t_f \approx 10.91$).}
\label{fig:ConvexPGS}
\end{figure}
It is often assumed that the shape of the final cost-function determines how hard it is for QA to solve the problem (in fact, this was partly the motivation for the {Spike} problem in Ref.~\cite{Farhi-spike-problem}), and that potentials with tall and thin barriers should be advantageous for AQA, since this is where tunneling dominates over thermal hopping (e.g., \cite[p.215]{Heim:2014jf}, \cite[p.1062]{RevModPhys.80.1061}, \cite[p.226]{Suzuki-book}). It is then assumed that problems where the final potential has this feature are those for which there should be a quantum speedup. We have given several counterexamples to such claims, and shown that tunneling is not necessary to achieve the optimal TTS. Instead, the optimal trajectory may use diabatic transitions to first scatter completely out of the ground state and return via a sequence of avoided level crossings.
That diabatic transitions can help speed up quantum algorithms has also been noted and advantageously exploited in Refs.~\cite{Somma:2012kx, crosson2014different, Hen2014,Steiger:2015fk}. Moreover, we have shown that the instantaneous semi-classical potential provides important insight into the role of tunneling, while the final cost function can be rather misleading in this regard.
While both adiabatic and diabatic QA outperform SA for the Fixed Plateau problem, the faster quantum diabatic algorithm is not better than the classical SVD algorithm for this problem. The PHWO problems due to Reichardt~\cite{Reichardt:2004}, which includes problems very similar to the Fixed Plateau, have widely been considered an example where tunneling provides a quantum advantage; we have shown that this holds if one limits the comparison to SA, but that there is in fact no quantum speedup in the problem when one compares the quantum diabatic evolution (which outperforms adiabatic quantum annealing) to SVD.
These results of the diabatic optimal evolution extend beyond the plateau problems:
even the {Spike} problem studied in Ref.~\cite{Farhi-spike-problem}---which is in some sense the antithesis of the plateau problem since it features a sharp spike at a single Hamming weight---also exhibits the diabatic-beats-adiabatic phenomenon, indicating that tunneling is not required to efficiently solve the problem. Thus diabatic evolution, especially via diabatic cascades, is an important and relatively unexplored mechanism in quantum optimization that is different from tunneling. The fact that we observe a speedup relative to AQA for several problems, especially an exponential speedup for the 0.5-Rectangle, motivates the search for algorithms exploiting this mechanism and may yield fruitful results. However, we also already know that diabatic cascades are not generic. E.g., we have checked that this mechanism is absent in the {Grover} and {Precipice} problems, even though the Grover problem is equivalent to a `giant' plateau problem.
In summary, our work provides a counterargument to the widely made claims that tunneling should be understood with respect to the final cost function, that speedups due to tunneling require tall and thin barriers; and that tunneling is needed for a quantum speedup in optimization problems. Which features of Hamiltonians of optimization problems favor diabatic or adiabatic algorithms remains an open question, as is the understanding of tunneling for non-permutation-symmetric problems.
We finish on a positive note for QA. We have given several examples where SVD outperforms QA, e.g., the Spike problem~\cite{Farhi-spike-problem}. However, we make no claim that SVD will always have an advantage over QA. A simple and instructive example comes from the class of cost functions that are convex in Hamming weight space, which have a constant minimum gap \cite{jarret2014fundamental}:
\begin{equation}
\label{eqt:jarret}
f(x) = \begin{cases} 2, & |x| =0 \\
\abs{x}, & \text{otherwise} \end{cases} \ .
\end{equation}
We have observed similar diabatic transitions for this problem as for the Fixed Plateau (not shown), but find that DQA outperforms SVD, as shown in Fig.~\ref{fig:ConvexPGS}. This results because the optimal TTS for QA occurs at a slightly higher optimal annealing time, i.e., there is an advantage to evolving somewhat more slowly, though still far from adiabatically. Thus, this provides an example of a ``limited'' quantum speedup \cite{speedup}.
\acknowledgments{
Special thanks to Ben Reichardt for insightful conversations and for suggesting the plateau problem, and to Bill Kaminsky for inspiring talks~\cite{Kaminsky:2014,Kaminsky:USC-talk-2014}. We also thank Itay Hen, Joshua Job, Iman Marvian, Milad Marvian, and Rolando Somma for useful comments. The computing resources were provided by the USC Center for High Performance Computing and Communications and by the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. This work was supported under ARO grant number W911NF-12-1-0523 and ARO MURI Grant No. W911NF-11-1-0268.}
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\section{Introduction}
Automatic speech recognition (ASR) systems are being increasingly deployed in real-life situations (e.g. home assistants like Amazon Echo, Google Home, etc.), where the acoustic environment may present distortions such as background noise and reverberation. To tackle these problems, the field of noise-robust ASR has organized various challenges over the past several years, such as the REVERB~\cite{kinoshita2016summary} or the CHiME~\cite{barker2013pascal,barker2015third} challenges.
In this paper, we propose a new approach for noise-aware training of deep neural network (DNN) based acoustic models in hybrid ASR systems. We leverage a speech activity detection (SAD) model to identify the speech frames and the silence frames in the utterance. Our ``noise vector'', then, is obtained by concatenating the means of the speech frames and the silence frames, respectively. We conjecture that such a noise estimate is well-informed by the distortions occurring throughout the utterance, including additive and multiplicative noises. The use of non-speech frames to estimate the noise in an utterance has also motivated previous work on noise-robust ASR~\cite{seltzer2013investigation} and speech denoising~\cite{Xu2020ListeningTS}. Furthermore, in our estimation strategy, this additional information is acquired at no extra computational cost, since we make use of a GMM-HMM model that is already trained for generating alignments for training the DNN acoustic model.
In online speech recognition, we are only provided a partial input sequence, and the model is required to output streaming transcriptions. In such a setting, waiting until the whole utterance has been seen in order to compute the means of the speech and silence frames is not feasible, thus requiring online estimates of the noise vector. Towards this objective, we propose to use maximum likelihood (ML) estimates obtained from a probabilistic model of the means of speech and silence. This makes our noise-adaptive training scheme compatible with streaming ASR systems.
\section{Related Work}
\label{sec:related}
In the past, several methods have been proposed to make ASR systems robust to environmental noise~\cite{li2014overview}, either through feature normalization (like spectral subtraction~\cite{boll1979suppression} and Weiner filtering~\cite{loizou2013speech}) or through model adaptation, such as MLLR-like schemes~\cite{leggetter1995maximum}. Recently, with the ubiquitous adoption of neural networks in hybrid as well as end-to-end ASR, data-driven supervised approaches have emerged as potential alternatives to traditional unsupervised methods~\cite{zhang2018deep}. These include front-end enhancement techniques like masking~\cite{srinivasan2006binary} or back-end methods like noise-aware and multi-condition training~\cite{seltzer2013investigation}.
For robust streaming ASR, several online front-end processing methods have been proposed, such as time-frequency masks and steering vectors~\cite{higuchi2016robust} or online-enabled Generalized Eigenvalue (GEV) beamformers~\cite{kitza2016robust}. In this work, we do not perform any front-end processing of the acoustic signals; instead, we obtain an informed estimate of the noise from the observed chunks of speech and non-speech frames, and use this estimate to adapt the acoustic model.
Several researchers have proposed the use of utterance-level embeddings for noise/environment adaptation of acoustic models. Saon et al.~\cite{saon2013speaker} used i-vectors, proposed originally for speaker verification~\cite{Dehak2011FrontEndFA}. An ``online'' version of these i-vectors, estimated every 10 frames, is supported in the Kaldi~\cite{povey2011kaldi} ASR toolkit. Seltzer et al.~\cite{seltzer2013investigation} used noise-aware training similar to our approach, but their noise vectors were estimated by averaging the first and last 10 frames in the utterance, whereas we estimate noise by leveraging a SAD module. In \cite{Kim2016EnvironmentalNE}, the authors trained a network to predict the environment label, and used the embedding from the bottleneck-layer to inform the acoustic model. Similarly, \cite{Feng2017AnEF} trained a linear discriminant analysis (LDA) model using environment labels to project i-vectors into a space that more accurately represents noise variability in the utterance, calling them e-vectors. They also used a bottleneck DNN similar to~\cite{Kim2016EnvironmentalNE} for estimating e-vectors, with the exception that they used i-vectors as input, instead of the acoustic features. We will look at some of these approaches in more detail in Section~\ref{sec:baselines}.
Since we use the means of acoustic features, comparisons may also be drawn to the popular cepstral mean normalization (CMN)~\cite{liu-etal-1993-efficient} approach. However, while CMN just subtracts the cepstral mean, our method allows the DNN to learn non-linear transforms of the noise vector.
\section{Noise Vectors}
\label{sec:noise}
We make the acoustic model noise-aware by appending additional information with the input feature sequence in the form of ``noise vectors.'' Let $\xvec_i=(\xvec_{i1},\ldots,\xvec_{it},\ldots,\xvec_{iT_i})$, $\xvec_{it}\in \mathbb{R}^d$, be the input feature frames corresponding to the utterance $i$. Consider an oracle SAD function, $f:\mathbb{R}^d \rightarrow \{\text{speech},\text{sil}\}, f(\xvec_{it}) = s_{it}$, which classifies a frame as speech or silence. Let $\svec_i=(s_{i1},\ldots,s_{it},\ldots,s_{iT_i})$ be the corresponding SAD predictions for the input sequence. We compute the means of speech and silence frames, $\muvec_{\rms_i}$ and $\muvec_{\rmn_i}$, as
\begin{equation}
\label{eqn:noise_vec}
\muvec_{\rms_i} = \frac{\sum_{t \in \text{speech}} \xvec_{it}} {\sum_{t \in \text{speech}} 1},~~\text{and}~~
\muvec_{\rmn_i} =\frac{\sum_{t \in \text{sil}} \xvec_{it}} {\sum_{t \in \text{sil}} 1}.
\end{equation}
In our hybrid ASR system, the acoustic model is trained using a sequence discriminative criterion~\cite{vesely2013sequence} such as lattice-free MMI~\cite{povey2016purely}, on alignments generated using a GMM-HMM model with the training transcriptions. Since we know which phones map to silence phones, we use the same alignments to obtain the SAD label sequence $\svec_i$. An overview of our noise-aware training scheme is shown in Fig.~\ref{fig:overview}. We append the noise vectors to the input features through a linear transformation (called a ``control layer'' in literature~\cite{Rownicka2019EmbeddingsFD})\footnote{This is conceptually similar to an LDA feature transformation~\cite{Somervuo2003FeatureTA}.}.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figs/noise_vector.pdf}
\caption{Noise-aware training/inference pipeline with a TDNN-based acoustic model. Dotted lines denote paths that are only required during training.}
\label{fig:overview}
\end{figure}
At inference time, we run a first-pass decoding with the same GMM-HMM model to obtain the decoding lattice, which is then used to obtain the SAD labels for estimating the noise vectors. Since we only need approximate representations of speech and noise frames, the classification need not be perfect, and so this strategy suffices. Next, we describe an extension of this estimation technique to the online decoding scenario using maximum likelihood estimate.
\subsubsection*{Maximum likelihood (ML) estimate}
The ML estimate of the noise vectors is similar to equation~\eqref{eqn:noise_vec}, with $\svec_i$ being the partial sequence observed until time $t$. If no speech or silence frames are observed, the corresponding components are set to 0.
In Section~\ref{sec:mle}, we will show that this estimate of the noise vectors quickly converges to the offline estimate in most settings.\footnote{In addition to this maximum likelihood estimate of the noise vectors, we can also derive a \emph{maximum a posteriori} (MAP) estimate which ties together the characteristics of both the speech and silence classes. The derivation is given in Appendix~\ref{appendix:map}. In our preliminary experiments, however, this model did not show any improvements over the simple ML estimate.}
\section{Experimental setup}
\subsection{Datasets and model}
We performed experiments on two different datasets: AMI and Aurora-4. AMI~\cite{carletta2005ami} consists of 100 hours of recorded meetings, and speech from close-talk, single distant microphone (SDM), and array microphones is provided. We used the SDM recordings for our experiments, which is ideal for single-channel far-field evaluation. Aurora-4~\cite{yeung2004improved} is a 14-hour close-talk speech corpus based on Wall Street Journal (WSJ0) data with artificially added noise. We evaluated on the \texttt{eval92} and \texttt{0166} subsets. All our experiments were conducted using the Kaldi speech recognition toolkit~\cite{povey2011kaldi} using neural networks trained with lattice-free MMI~\cite{povey2016purely}. The acoustic model for Aurora-4 was based on factored TDNN~\cite{povey2018semi}, while the one for AMI was TDNN-based. The hyperparameters for each setup are shown in Table~\ref{tab:am_details}. For the Aurora-4 setup, we used a pruned 3-gram language model (LM) trained on WSJ transcriptions using the IRSTLM toolkit~\cite{federico2008irstlm}. For AMI, we trained a pruned 3-gram LM with KN-smoothing on a combination of AMI and Fisher~\cite{cieri2004fisher} data using the SRILM toolkit~\cite{stolcke2002srilm}.
\begin{table}[t]
\centering
\caption{Hyperparameters for the acoustic model training for our experimental setups.}
\label{tab:am_details}
\begin{adjustbox}{max width=\linewidth}
\begin{tabular}{@{}llcccc@{}}
\toprule
\textbf{Setup} & \textbf{Model} & \textbf{\# layers} & \textbf{Dim} & \textbf{Bottleneck} & \textbf{\# epochs} \\
\midrule
AMI & TDNN & 9 & 450 & - & 9 \\
Aurora-4 & TDNN-F & 12 & 1024 & 128 & 10 \\
\bottomrule
\end{tabular}
\end{adjustbox}
\end{table}
\subsection{Baselines}
\label{sec:baselines}
As described earlier in Section~\ref{sec:related}, several embedding-based methods have been proposed for noise-aware training of acoustic models. Here, we perform comparisons with 4 of these methods, described below.
\begin{enumerate}[wide, labelwidth=!, labelindent=0pt]
\item \textbf{i-vector}: i-vectors were first introduced in \cite{Dehak2011FrontEndFA} for speaker verification, and later used for speaker adaptation of acoustic models in \cite{saon2013speaker}. They are generative embeddings which model both speaker and channel variability in utterances. We used two variants in our experiments: \textit{offline}, which were estimated over the whole utterance, and \textit{online}, which were estimated every 10 frames. For both these variants, we extracted 100-dim i-vectors using extractors trained on the training data.
\item \textbf{NAT-vector}: Seltzer et al.~\cite{seltzer2013investigation} proposed ``noise vectors'' for noise-aware training, by averaging the first and last 10 frames of the utterance. We refer to these vectors as NAT-vectors, to avoid confusion with our proposed noise vectors. Since we use 40-dim MFCC features for training our models, these vectors are also 40-dimensional.
\item \textbf{e-vector}: Since i-vectors capture both speaker and channel variability, \cite{Feng2017AnEF} proposed a technique for extracting the ``environment'' variability by transforming the i-vectors using environment labels, either using linear discriminant analysis (LDA), or bottleneck neural networks (NNs), and called them e-vectors. Here, we use the LDA variant of e-vectors for our comparison. For Aurora-4, we trained the e-vectors on 14 noise label types from the official development set. Since AMI does not have annotated environment labels, we trained the LDA on CHiME-4 data~\cite{barker2015third}, which consists of 4 different noise types. For both the datasets, we transformed the 100-dim i-vectors to 50-dim e-vectors.
\item \textbf{Bottleneck NN}: Since the above baselines estimate environment embeddings at the utterance level, Kim et al.~\cite{Kim2016EnvironmentalNE} argued that they cannot effectively capture non-stationary noise. Instead, they trained bottleneck NNs to classify noise types with frame-level acoustic features provided as input. Frame-level noise embeddings were then extracted from the bottleneck layer and used for noise-aware training. For our experiments, we trained a 5-layer feed-forward neural network with 1024 hidden units (similar to \cite{Kim2016EnvironmentalNE}), and a 80-dim bottleneck was placed at the 4th layer. Noise labels used as training targets for the bottleneck NNs were obtained similar to the e-vector LDA training.
\end{enumerate}
In addition to these embedding methods from literature, we report results using (i) CMN, and (ii) embeddings obtained by averaging the entire utterance, which we call \texttt{utt-mean}. This ablation is performed to demonstrate the importance of obtaining separate means for speech and silence frames. Our implementation for these baselines and our proposed method will be made publicly available at: {\footnotesize\href{https://github.com/desh2608/kaldi-noise-vectors}{\texttt{https://github.com/desh2608/kaldi-noise-vectors}}}.
\section{Results and Discussion}
\subsection{Comparison with baselines}
\begin{table}[t]
\centering
\caption{WER results for our proposed noise vectors (offline and MLE) compared with other embedding-based techniques from literature. Methods marked with $^\dagger$ require environment labels for training embedding extractor.}
\label{tab:results}
\begin{adjustbox}{width=\linewidth}
\begin{tabular}{@{}lcccccc@{}}
\toprule
\multirow{2}{*}{\textbf{Method}} & \phantom{a} & \multicolumn{2}{c}{\textbf{AMI}} & \phantom{a} & \multicolumn{2}{c}{\textbf{Aurora-4}} \\
\cmidrule(l{2pt}r{2pt}){3-4} \cmidrule(l{2pt}r{2pt}){6-7}
&& \textbf{Dev} & \textbf{Eval} && \textbf{Eval-92} & \textbf{0166} \\
\midrule
Base model && 38.8 & 42.6 && 7.94 & 8.12 \\
CMN && 38.5 & 42.8 && 7.79 & 7.94 \\
utt-mean && 36.8 & 40.8 && 7.77 & 7.78 \\
\midrule
i-vector (offline)~\cite{saon2013speaker} && 35.2 & 39.4 && 8.24 & 8.53 \\
i-vector (online) && 35.6 & 40.0 && 8.51 & 8.78 \\
NAT-vector~\cite{seltzer2013investigation} && 38.1 & 42.0 && 8.17 & 8.38 \\
e-vector (LDA)$^{\dagger}$~\cite{Feng2017AnEF} && 37.6 & 41.5 && 7.63 & 7.73 \\
Bottleneck NN$^{\dagger}$~\cite{Kim2016EnvironmentalNE} && 38.7 & 42.4 && 7.77 & 7.98 \\
\midrule
Noise vector (offline) && \textbf{34.9} & \textbf{38.8} && \textbf{7.37} & \textbf{7.61} \\
Noise vector (MLE) && 35.4 & 39.2 && 7.72 & 7.93 \\
\bottomrule
\end{tabular}
\end{adjustbox}
\end{table}
\begin{table}[t]
\centering
\caption{Insertion (ins), deletion (del), and substitution (sub) error comparison for noise vectors compared with the base model, for AMI (dev) and Aurora-4 (eval92).}
\label{tab:breakdown}
\begin{adjustbox}{width=0.9\linewidth}
\begin{tabular}{@{}llcccc@{}}
\toprule
\textbf{Dataset} & \textbf{System} & \textbf{Ins} & \textbf{Del} & \textbf{Sub} & \textbf{WER} \\
\midrule
\multirow{2}{*}{AMI (dev)} & TDNN & 4.7 & 12.6 & 21.5 & 38.8 \\
& + noise vector & 3.2 & 11.5 & 20.3 & 34.9 \\
\midrule
\multirow{2}{*}{\shortstack[l]{Aurora-4\\(eval92)}} & TDNN-F & 0.50 & 2.57 & 4.87 & 7.94 \\
& + noise vector & 0.58 & 2.12 & 4.67 & 7.37 \\
\bottomrule
\end{tabular}
\end{adjustbox}
\end{table}
\begin{figure*}[t]
\begin{subfigure}{\linewidth}
\centering
\cutimage{figs/AMI_ES2011a_SDM_FEE043_0084557_0084784.pdf}{0.65}{0.65}
\end{subfigure}\hfill
\vspace{-1em}
\begin{subfigure}{\linewidth}
\centering
\cutimage{figs/AMI_ES2011a_SDM_FEE041_0081408_0081591.pdf}{0.65}{0.65}
\end{subfigure}\hfill
\vspace{-1em}
\caption{Frame-level phone alignments obtained from the Baseline TDNN and noise vector models for the sentences: {\normalfont \texttt{I MEAN IT JUST SEEMS LIKE YEAH}} (top) and {\normalfont \texttt{TILL THE MEETING OH RIGHT}} (bottom) in the AMI Dev set. The shaded region shows where the baseline model makes insertion and deletion errors, respectively. We have only shown part of the spectograms for each utterance.}
\label{fig:analysis}
\end{figure*}
Table~\ref{tab:results} shows the WER results for the proposed method and the baselines on AMI and Aurora-4. It can be seen from the table that adding offline noise vectors to the input features provided WER improvements of 10\% and 7.1\% on AMI and Aurora-4, respectively. Furthermore, these vectors consistently outperformed the \texttt{utt-mean} baseline, indicating that the speech/silence classification is important for the model. MLE-based vectors are also effective, although they perform slightly worse than the utterance-level vectors, as expected. We conjecture that this may be because of incorrect estimates of speech and silence in the beginning of the utterance.
Among the baseline embedding-based systems, i-vectors (both offline and online) showed significant gains for AMI, but degraded results on Aurora-4. This may be because Aurora-4 utterances contain long silences, which can corrupt i-vector estimation. NAT vectors also followed the same trend, indicating that noise estimation from only the first and last few frames leads to inaccuracies. On the other hand, e-vectors and bottleneck NNs improved WERs over the base models on both datasets. Our proposed method provided the best performance across the board, demonstrating its robustness. In the next section, we will analyze the performance of our noise vectors further using some ablation experiments.
\begin{table}[t]
\centering
\caption{Effect of speech and non-speech frames in the noise vector.}
\label{tab:speech_sil}
\begin{adjustbox}{width=0.85\linewidth}
\begin{tabular}{@{}cccc@{}}
\toprule
\textbf{Speech mean} & \textbf{Non-speech mean} & \textbf{AMI Dev} & \textbf{AMI Eval} \\
\midrule
\xmark & \xmark & 38.8 & 42.6 \\
\xmark & \cmark & 38.1 & 41.9 \\
\cmark & \xmark & 36.7 & 40.8 \\
\cmark & \cmark & 34.9 & 38.8 \\
\bottomrule
\end{tabular}
\end{adjustbox}
\end{table}
\subsection{Where does the improvement come from?}
\textbf{WER breakdown by error type.} Table~\ref{tab:breakdown} shows the insertion, deletion, and substitution error rates for the proposed method. On the AMI dataset, which contains late reverberations and interfering speakers, our noise vectors reduce insertion errors substantially, suggesting that utterance-level characteristics help in speaker selection. In Fig.~\ref{fig:analysis} (top), we see that the baseline model inserted extraneous phonemes between \texttt{LIKE} and \texttt{YEAH}, as a result of cross-talk from the interfering speaker, but the noise-aware model correctly identified the frames as silence. Since Aurora-4 does not contain far-field or overlapping speech, noise vectors do not improve insertion errors on that setting. On both datasets, noise vectors consistently reduced deletion and insertion errors. Qualitative analysis (such as that in Fig.~\ref{fig:analysis} (bottom)) showed that this was a result of better recognition on noisy frames, where the base model did not predict any phones, or predicted incorrect phones.
\textbf{Effect of speech and non-speech frames.} In Table~\ref{tab:results}, comparison with the \texttt{utt-mean} baseline showed that averaging speech and non-speech frames separately was important for the effectiveness of our noise vectors. To measure the individual contributions of these components, we conducted experiments on the AMI dataset by selecting each of the components alternatively (Table~\ref{tab:speech_sil}). We found that speech frames are more important than non-speech frames, but their combination results in the best performance. The relatively smaller WER improvement using non-speech frames is also consistent with the NAT vector results in Table~\ref{tab:results}.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figs/ivec_vs_nvec.pdf}
\vspace{-1em}
\caption{Effect of combining i-vectors and noise vectors on the AMI dataset. Noise vectors outperform i-vectors, but they are complementary.}
\label{fig:ivec_vs_nvec}
\end{figure}
\textbf{Combination with i-vectors.} In Fig.~\ref{fig:ivec_vs_nvec}, we show the effect of combining i-vectors and noise vectors on the AMI dataset. We found that the WER gains using these embedding methods are complementary, and their combination achieved a relative improvement of approx. 12.8\% compared with the baseline TDNN model. This is likely because i-vectors and noise vectors capture speaker and environment characteristics in the utterance, respectively.
\textbf{Combination with multi-condition training.} We also compared our method with multi-condition (MC) training on Aurora-4, where we applied speed and volume perturbation, and added simulated reverberation and point-source noises to the training data, resulting in 6x the original training set. MC training outperformed noise vectors; however, their combination achieved the best performance, reducing the WER by approx. 21.6\% relative over the baseline TDNN-F model.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figs/nvec_vs_mc.pdf}
\vspace{-1em}
\caption{Effect of combining noise vectors with multi-condition (MC) training on the Aurora-4 dataset. MC training outperforms noise vectors, but they are complementary.}
\label{fig:nvec_vs_mc}
\end{figure}
\subsection{Offline vs. MLE noise vectors}
\label{sec:mle}
Fig.~\ref{fig:online_analysis} shows 4 (out of 80) coefficients from the offline and MLE noise vectors for an utterance selected at random from the Aurora-4 \texttt{test\_0166} set. We see that the online estimates for speech coefficients (15 and 35) quickly approached the utterance-level values. For the noise coefficients (55 and 75), the utterance-level averages for the online estimates were higher because of fewer silence frames in the middle of the utterance, but the overall estimate over the whole utterance was close to the offline vector. Similar observations were made across both datasets.
\section{Conclusions}
We proposed a simple method that leverages a SAD to perform noise-aware training of acoustic models in hybrid ASR systems by computing the means of speech and silence frames. Our method, which we call ``noise vectors,'' provided significant improvements in WER in diverse conditions such as noise and far-field reverberant speech, as demonstrated through experiments conducted on Aurora-4 and AMI. On both conditions, our method outperformed strong baselines that use i-vectors, e-vectors trained with LDA, or bottleneck neural networks.
Through error analysis, we hypothesized that the improvements are a results of better performance of our model on frames containing high background noise or interference in the form of overlapping speech. Finally, we extended our method to the case of streaming ASR by showing that similar WER gains can be obtained using maximum likelihood estimates of the noise vectors.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figs/aurora4_442c0206b.pdf}
\caption{Online noise estimates using MLE and MAP for a randomly selected utterance in the Aurora-4 {\normalfont\texttt{test\_0166}} set. The top and bottom row coefficients correspond to means of speech and silence frames, respectively. For each coefficient, the solid red and green lines are the ``offline'' and MLE estimates, respectively. The dashed green line represents the mean of the MLE noise vector over the whole utterance.}
\label{fig:online_analysis}
\end{figure}
\bibliographystyle{IEEEtran}
{\small
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BkiUcQQ5jDKDx8o5rdGz
| 5 | 1 |
\section{Introduction}
The study of the viscosity of strongly interacting quantum fluids has brought together very different areas of
physics -- black holes and string theory, quark-gluon plasmas, quantum fluids and cold atoms -- which, at first sight,
appear to have little in common~\cite{Son-review,Schafer09}.
This extraordinary development originated with the work of Son, Starinets and coworkers~\cite{Son-review,Policastro01,Kovtun05}
who calculated the shear viscosity in a strongly interacting quantum field theory,
the ${\cal{N}}=4$ supersymmetric Yang-Mills (SYM) theory,
and conjectured a lower bound
\begin{equation}
{\eta}/{s} \ge {\hbar}/{(4\pi k_B)}
\label{bound}
\end{equation}
for the ratio of the shear viscosity $\eta$ to the entropy density $s$ of \emph{any} system.
These results were obtained using the AdS/CFT formalism where certain strongly coupled
field theories can be mapped onto weakly coupled gravity theories.
Although a number of counterexamples have since been proposed~\cite{Cohen07,Brigante08,Kats09,Buchel09}, there are no known experimental violations of the bound given by Eq.~(\ref{bound}). Remarkably, two very different experimental systems come close to saturating the bound:
the quark-gluon plasma at Brookhaven's Relativistic Heavy Ion Collider~\cite{RHIC,Teaney01}, and
ultracold atomic Fermi gases~\cite{Turlapov08,Gelman05}
close to a Feshbach scattering resonance, where the $s$-wave scattering length becomes infinite~\cite{Trentoreview}.
This is the strongly interacting unitary regime that lies at the center of the BCS-BEC crossover. These two systems are amongst the hottest and coldest systems every realized in a laboratory.
\medskip
In this paper we focus on \emph{non-relativistic} quantum fluids, with particular emphasis on \emph{strongly
interacting Fermi gases}. These are systems for which the most controlled experiments should be possible.
A ``perfect fluid'' with the minimum shear viscosity is necessarily in a quantum regime, since the bound involves $\hbar$.
In addition, it must also be in a strongly interacting regime where \emph{well-defined quasiparticle excitations do not exist}.
If the system had sharp quasiparticles, then their mean scattering rate $\tau^{-1}$ would be much less than the average energy
per particle $\epsilon_0$, so that $\hbar/\tau \ll \epsilon_0$. We can then use Boltzmann's kinetic theory approach to obtain
$\eta \sim n \epsilon_0 \tau$, where $n$ is the number density.
Using $s \sim nk_B$, we find a large $\eta/s \sim \epsilon_0 \tau / k_B \gg {\hbar}/{k_B}$.
Thus, in order to find perfect fluids that come close to saturating the lower bound given by Eq.~(\ref{bound}), one must
look at strongly interacting quantum fluids where the quasiparticle approximation fails.
In this paper we use Kubo formulas for the frequency-dependent
spectral functions for shear viscosity $\eta(\omega)$ and
bulk (or second) viscosity $\zeta(\omega)$, and derive several exact, non-perturbative results without making
weak coupling or quasiparticle approximations. Our main results are:
\medskip
$\bullet$ We establish a microscopic connection between the shear viscosity $\eta$ and the normal fluid density $\rho_n$
and show that a non-zero $\rho_n$ is a necessary condition for a non-vanishing $\eta$.
\medskip
$\bullet$ We derive sum rules for $\eta(\omega)$ and $\zeta(\omega)$ of any Bose or Fermi system with
an arbitrary two-body interaction; see Eqs.~(\ref{etasumruleiso}) and (\ref{zetasumruleiso}).
\medskip
$\bullet$ For a dilute two-component Fermi gas, we find the shear viscosity sum rule
\begin{eqnarray}
\frac{1}{\pi}\int^{\infty}_{0}\!\!d\omega \left[\eta(\omega) -\frac{C}{10\pi\sqrt{m\omega}}\right]=
\frac{\varepsilon}{3}-\frac{ C}{10\pi m a},
\label{etasumrule0finite}
\end{eqnarray}
valid for arbitrary temperature and $1/(k_F a)$, where $a$ is the $s$-wave scattering length.
Here, $\varepsilon$ is the energy density and $C$ is the \textit{contact}~\cite{Tan08}. A central quantity in many of our results,
$C = k_F^4 {\cal C}[1/(k_F a), T/\epsilon_F]$ can be defined via the large-$k$ tail of the momentum distribution
$n_{{\bf k}} \simeq C/k^4$ for $k \gg k_F$, and characterizes the short-distance properties of the many-body state.
\medskip
$\bullet$ For the bulk viscosity, we obtain the sum rule
\begin{eqnarray}
\frac{1}{\pi}\int^{\infty}_{0}d\omega\; \zeta(\omega)=\frac{1}{72 \pi m a^2} \left( \frac{\partial C}{\partial a^{-1}} \right)_s,
\label{zetasumrule0}
\end{eqnarray}
where the derivative is at fixed entropy per particle $s \equiv S/N$. [Different from Eq.~(\ref{bound}), in the remainder of this paper we use $s$ to denote this quantity rather than the entropy density.]
Below the superfluid transition, the bulk viscosity that enters Eq.~(\ref{zetasumrule0}) is $\zeta_2$,
associated with the damping of in-phase motions of the superfluid and normal components. The positivity
of $\zeta(\omega)$ implies that $(\partial C / \partial a^{-1})_s \geq 0$.
\medskip
$\bullet$ At unitarity, the bulk viscosity spectral function vanishes at \emph{all} frequencies and \emph{all} temperatures. Quite generally, $\zeta(\omega) \geq 0$,
but the sum rule in Eq.~(\ref{zetasumrule0}) vanishes for $|a|=\infty$ and thus $\zeta(\omega) = 0$ for the unitary Fermi gas.
This generalizes the result~\cite{Son07} that the static bulk viscosity $\zeta(0)$ vanishes at unitarity.
\medskip
$\bullet$ It follows from the previous result that, at unitarity, the shear viscosity spectral function $\eta(\omega)$ can be
related to density-density correlations as
\begin{eqnarray}
\eta(\omega)
= \lim_{q\to 0}\frac{3\omega^3}{4 q^4}\;\mathrm{Im}\chi_{\rho\rho}({\bf q},\omega) \ \ \ \ \ (|a| = \infty).
\end{eqnarray}
Thus, $\eta(\omega)$ for the unitary Fermi gas
can be measured spectroscopically using, for instance, two-photon Bragg spectroscopy.
\medskip
$\bullet$ We show from our sum rules that various dynamic response functions for Fermi gases
have high-frequency tails characterized by odd-integer power laws,
whose magnitudes are controlled by the contact $C$. The tail $C/\sqrt{\omega}$ of $\eta(\omega)$ is evident from
Eq.~(\ref{etasumrule0finite}). Using this we find that the dynamic structure factor has a tail of the form~\cite{Son10}
$S({{\bf q}},\omega) \sim C q^4/\omega^{7/2}$ for
$q \to 0$ and $\omega \to \infty$, which is shown to be a generic feature of short range physics.
\bigskip
In the remainder of this Section we describe how the rest of the paper is organized.
In Section~\ref{Kubosec}, we begin with a careful derivation of Kubo formulas for the spectral functions
$\eta(\omega)$ and $\zeta(\omega)$ in terms of current-current correlation functions,
Eqs.~(\ref{KuboTR}) and (\ref{KuboLR}), and, equivalently, in terms of the stress-stress correlator, Eq.~(\ref{KuboPiFull}).
In Section~\ref{rhon-section} we recall some elementary facts about the shear viscosity of a fluid and why it
is analogous to the resistivity, and not the conductivity, of a metal. We then establish a connection between the viscosity $\eta$ and
the normal fluid density $\rho_n$ using microscopic response functions.
After establishing the positivity of $\eta(\omega)$ and of $\zeta(\omega)$ in Section~\ref{posdefsec},
we derive sum rules for these quantities in Section~\ref{sumrulesec}. The most general sum rules for
the shear and bulk viscosities of any Bose or Fermi system with an arbitrary isotropic interaction potential $V(p)$,
and valid for all temperatures, are given in Eqs.~(\ref{etasumruleiso}) and (\ref{zetasumruleiso}).
In Section~\ref{diluteviscositysec}, we specialize to the dilute Fermi gas, with interparticle spacing
$k_F^{-1}$ and $s$-wave scattering length $a$ both much larger than the characteristic range $r_0$ of the potential.
We obtain the $\zeta$ sum rule in Eq.~(\ref{zetasumrule0}), which is finite in the zero range limit.
The $\eta$ sum rule, however, has an ultraviolet divergence; see Eq.~(\ref{etasumrule2}).
We identify, in Section~\ref{highfrequencytailsec}, the $C/\sqrt{\omega}$ high-frequency tail of the shear viscosity spectral function,
and derive the sum rule given by Eq.~(\ref{etasumrule0finite}), which is manifestly finite for $r_0 \to 0$.
The sum rules given by Eqs.~(\ref{etasumrule0finite}) and (\ref{zetasumrule0}) are valid in both the normal and superfluid phases,
with $\zeta$ replaced by $\zeta_2$ in the latter state.
In Section~\ref{Crossoversec} we show from the $\zeta$ sum rule that, at unitarity, $\zeta(\omega)$ vanishes at all frequencies
and all temperatures.
We also discuss the $1/(k_F a)$-dependence of the $\eta$ and $\zeta$ sum rules across the BCS-BEC crossover, using
available quantum Monte Carlo data for the energy density at $T=0$.
We discuss the connection between viscosity and density-density correlations in Section~\ref{SqomegaSec} and find
two interesting results. First, we show how a density probe such as two-photon Bragg spectroscopy can in principle be used to measure the shear
viscosity spectral function $\eta(\omega)$ at unitarity.
Second, we identify the high-frequency $\omega^{-7/2}$ tail in the dynamic structure factor $S({{\bf q}},\omega)$.
In Section~\ref{comparisonsec}, we briefly compare the sum rules that we have derived for non-relativistic
quantum fluids with those obtained in relativistic quantum field theories. Finally in Section~\ref{conclusionssec}
we conclude with open questions.
There are five Appendices which contain technical details of derivations or review certain results which are used at
various places in the paper. In Appendix~\ref{KuboPi0}, we briefly discuss an alternate stress tensor operator often used to calculate the shear viscosity. Some results from dissipative two-fluid hydrodynamics, which we use in our paper, are reviewed in Appendix~\ref{hydrosec}.
We review in Appendix \ref{ContactAppendix} results related to the contact that are used at several places in the paper,
and also give a detailed derivation of certain equations that involve the contact.
In Appendix~\ref{Pressuresec}, we derive a microscopic expression for the pressure. Finally, in
Appendix \ref{universalthermosec} we give details of the derivation of the $\zeta$ sum rule which make use of the scaling form
of thermodynamic functions across the BCS-BEC crossover.
\section{Kubo formula for viscosity}
\label{Kubosec}
We begin by deriving Kubo formulas for the bulk and shear viscosity. Although the results of this Section
are, for the most part, ``well known'', we could not find a complete derivation at any one place in the literature.
In particular, there are several subtle points not dealt with adequately
elsewhere, not least the definition of the stress tensor operator
$\widehat{\Pi}_{\alpha\beta}$ for non-relativistic systems.
To introduce notation, we start with the Euler equation
\begin{eqnarray}
m\partial_t j_{\alpha}({\bf r},t)= -\partial_{\beta}\Pi_{\alpha\beta}({\bf r},t),
\label{euler}
\end{eqnarray}
where $m$ is the mass of the particles, $j_{\alpha}$ is the (number) current and
$\Pi_{\alpha\beta}$ is the momentum flux density tensor, which we call the \emph{stress tensor}, for short.
Here, $\alpha$ and $\beta$ take on values $x,y,z$ (and there is no difference between upper and lower indices
in our non-relativistic formulation).
In general, the stress tensor is given by~\cite{LLFM}
\begin{eqnarray}
\Pi_{\alpha\beta} = P\delta_{\alpha\beta} + \rho u_{\alpha}u_{\beta} -\sigma'_{\alpha\beta},
\label{stress}
\end{eqnarray}
where $P$ is the pressure, $\rho$ the mass density and ${\bm u}$ the velocity.
The viscous term $\sigma'_{\alpha\beta}$ is given by
\begin{eqnarray}
\sigma'_{\alpha\beta}= \eta\left[\partial_{\beta} u_{\alpha} \!+\! \partial_{\alpha}u_{\beta} \!-\! \frac{2}{3}\delta_{\alpha\beta}({\bm \nabla}\cdot{\bm u})\right] \!+\! \zeta\delta_{\alpha\beta}({\bm \nabla}\cdot{\bm u}),
\label{stresstensor}
\end{eqnarray}
where $\eta$ is the shear viscosity and $\zeta$ the bulk viscosity.
The generalization of Eq.~(\ref{stresstensor}) to the superfluid state is well known~\cite{LLFM} and
involves additional bulk viscosities.
At the end of Section~\ref{currentcorrelatorsec}, we show that the Kubo formula we derive for $\zeta$
describes the bulk viscosity $\zeta_2$ in the superfluid phase.
Our goal is to obtain Kubo formulas for frequency-dependent generalizations of
the long-wavelength viscosities, $\eta$ and $\zeta$, in terms of equilibrium correlation functions
of the many-body system.
The Kubo formulas for viscosities are often written in terms of the stress-stress
correlators; see, e.g., Sec.~90 of Ref.~\cite{LLStatPhysII}.
However, the form of the stress tensor (or momentum flux density) \emph{operator}
$\widehat{\Pi}_{\alpha\beta}$ is not obvious, and many different, complicated
expressions \cite{Forster} which are presumably equivalent can be found in the literature.
Part of the problem is to write down an operator expression
for the pressure $P$ in terms of particle positions and momenta. In high-energy physics, a simple way to calculate the stress-energy tensor $\widehat{\Pi}_{\alpha\beta}$ is to vary the action with respect to the metric in curved space-time. We prefer, however, to describe non-relativistic fluids without going to curved space-time.
To begin with, in~\ref{currentcorrelatorsec}, we adopt an approach that permits us to get around the
complexities of defining the stress operator $\widehat{\Pi}$. We
consider the linear response of a fluid to an externally imposed velocity field
and derive Kubo formulas for the bulk and shear viscosities in terms of \emph{current-current correlators}.
The results of this subsection are the same as those of Kadanoff and Martin~\cite{Kadanoff63}.
In~\ref{Kubocomparisonsec}, we use an operator form of Euler's equation to make the connection between bulk and shear
viscosities and \emph{stress-stress correlators}.
In Appendix~\ref{KuboPi0}, we derive an alternative form of the stress correlator, which works
only for the shear viscosity in the zero-frequency limit, but is often used in calculations.
\subsection{Current correlators}
\label{currentcorrelatorsec}
We calculate within linear response theory~\cite{NozieresPines1,Baymbook} the current flow in a fluid
subjected to an external velocity field ${\bm u}({\bf r},t) = {\bm u}({\bf r}) e^{-i\omega t} e^{0^+t}$ which is
turned on adiabatically.
Our goal is to relate the imaginary part of this response function
to viscosity through the dissipative part of the stress tensor.
The response of a fluid to the ``moving walls'' of its container is a standard concept in the theory of
superfluidity~\cite{Baymbook}. Here, we generalize this analysis to a non-uniform and time-varying
external perturbation ${\bm u}({\bf r},t)$, taking the long wavelength limit at the end.
We write the Hamiltonian of the system $\hat{H}$ plus external perturbation $\hat{H}'$ as~\cite{wallnote}
\begin{eqnarray}
\hat{H}_{\rm total} &=& \frac{1}{2m}\sum_{i=1}^N \int d{\bf r}\;
\left[\hat{{\bf p}}_i - m{\bm u}({\bf r},t)\delta({\bf r}-\hat{{\bf r}}_i)\right]^2 + \hat{V}\nonumber\\
=\hat{H} \! &-& \!\frac{1}{2}\sum_{i=1}^N \int d{\bf r} {\bm u}({\bf r},t)\cdot
\left\{ \hat{{\bf p}}_i,\delta({\bf r}\!-\!\hat{{\bf r}}_i) \right\}
+ {\cal{O}}[{\bm u}^2],\label{Hshift}\end{eqnarray}
where $\hat{{\bf p}}_i$ and $\hat{{\bf r}}_i$ are the momentum and position operators, respectively, for the $i$th particle,
$m$ is the mass, and $\hat{V}$ is the potential energy operator.
The anticommutator $\{ \hat{A},\hat{B} \} = \hat{A}\hat{B} + \hat{B}\hat{A}$ is used to
symmetrize products.
We thus see that to linear order in ${\bm u}$, the external perturbation is
\begin{eqnarray}
\hat{H}'(t) = - m\int d{\bf r} e^{-i\omega t} e^{0^+t} {\bm u}({\bf r})\cdot\hat{\bm{j}}({\bf r},t),
\label{Hpert}
\end{eqnarray}
where
$\hat{\bm{j}} = \sum_{i=1}^N \left\{ \hat{{\bf p}}_i , \delta({\bf r}\!-\!\hat{{\bf r}}_i) \right\}/2m$
is the current density operator.
Linear response theory gives the result~\cite{NozieresPines1,Baymbook}
\begin{eqnarray}
\lefteqn{\langle \hat{j}_{\alpha}({\bf r},t)\rangle = m \int d{\bf r}'\!\int^{\infty}_{-\infty}\!\!\! dt'e^{0^+t'}e^{-i\omega t'}\times}&&\nonumber\\&&\;\;\;\;\;\;\;\;\;\;\;\; \chi^{\alpha\beta}_{J}({\bf r}-{\bf r}',t-t')u_{\beta}({\bf r}').\label{lr1}
\end{eqnarray}
Here and below, we use the standard convention of summing over repeated indices.
The retarded current correlation function
$\chi^{\alpha\beta}_J$ is obtained by using
$\hat{A}=\hat{j}_{\alpha}$ and $\hat{B}=\hat{j}_{\beta}$ in Eq.~(\ref{chiAB}) below.
For later use, we provide a general definition for the \emph{retarded} response function,
or correlator, for operators $\hat{A}$ and $\hat{B}$:
\begin{eqnarray}
\lefteqn{\chi_{A,B}({\bf r}-{\bf r}',t-t') \equiv}&&\nonumber\\&& i\Theta(t-t')\langle[\hat{A}({\bf r},t),\hat{B}^{\dagger}({\bf r}',t')]\rangle.
\label{chiAB} \end{eqnarray}
Here, $\langle \hat{Q}\rangle = {\rm Tr} [\hat{Q} \exp(-\hat{H}/T)]/{\cal{Z}}$ is
the thermal expectation value at temperature $T$ and
${\cal{Z}} = {\rm Tr}[\exp(-\hat{H}/T)]$ is the partition function.
The step-function $\Theta(t-t')$ enforces causality.
We will use the convention of unit volume $\Omega = 1$ and set $\hbar = k_B = 1$,
unless explicitly stated otherwise.
We find the spectral representation for $\chi_{A,B}$ using the exact eigenstates and eigenvalues of the fully interacting
many-body Hamiltonian $\hat{H}|a\rangle = E_a|a\rangle$, and Fourier transform the result to obtain
\begin{eqnarray}
\lefteqn{\chi_{A,B}({\bf q},\omega) = \frac{1}{{\cal{Z}}}\sum_{a,b}e^{-E_a/T}\times}&&\nonumber
\\&&\left[
\frac{\langle a|\hat{B}^{\dagger}_{{\bf q}}|b\rangle\langle b|\hat{A}_{{\bf q}}|a\rangle}{\omega + E_{ba} + i0^+}
- \frac{\langle a|\hat{A}_{{\bf q}}|b\rangle\langle b|\hat{B}^{\dagger}_{{\bf q}}|a\rangle}{\omega - E_{ba} + i0^+}
\right],
\label{FourierchiAB}
\end{eqnarray}
where $E_{ba}\equiv E_b-E_a$.
The quantity of central interest to us in this paper is the imaginary part of $\chi$, given by
\begin{eqnarray}
\lefteqn{\mathrm{Im}\chi_{A,B}({\bf q},\omega) = \pi(1-e^{-\omega/T})}&&\nonumber\\&&
\times\frac{1}{\cal{Z}}\sum_{a,b}e^{-\beta E_a}\langle a|\hat{A}_{{\bf q}}|b\rangle\langle b|\hat{B}^{\dagger}_{{\bf q}}|a\rangle
\delta(\omega -E_{ba}).
\label{ImchiAB}
\end{eqnarray}
Returning to the problem of interest, we find that the induced current,
obtained by Fourier transforming Eq.~(\ref{lr1}), is
\begin{eqnarray}
\langle \hat{j}^{\alpha}({\bf q},\omega)\rangle = m \chi^{\alpha\beta}_{J}({\bf q},\omega)u_{\beta}.
\label{lr}
\end{eqnarray}
$\chi^{\alpha\beta}_J$ is given by Eq.~(\ref{FourierchiAB}) with $\hat{A}_{{\bf q}}=\hat{j}^{\alpha}_{{\bf q}}$ and
$\hat{B}^{\dagger}_{{\bf q}} = \hat{j}^{\beta}_{-{\bf q}}$, where
\begin{eqnarray}
\hat{j}^{\alpha}_{{\bf q}} = \frac{1}{2m}\sum_{{\bf k}\sigma} (2k_{\alpha}+q_{\alpha}) \hat{c}^{\dagger}_{{\bf k}\sigma}\hat{c}_{{\bf k}+{\bf q}\sigma}
\label{jdef}\end{eqnarray}
is the current operator with $\sigma$ denoting the different internal states of interest (e.g., spin).
Next, we need to relate Eq.~(\ref{lr}) to viscosity,
using ``constitutive relations'' between the current and transport coefficients.
For this we use Eqs.~(\ref{stress}) and (\ref{stresstensor}) substituted into Eq.~(\ref{euler}),
where the symbols $j_{\alpha}$ and $\Pi_{\alpha\beta}$,
\emph{without the hats used for operators}, are understood to
denote expectation values. In the long-wavelength limit, the contributions to the stress tensor coming from viscous terms dominate
over contributions from pressure fluctuations, while the convective term $\partial_{\beta}u_{\alpha}u_{\beta}$ is beyond linear order in velocity. We thus get
$m\partial_t j^{\alpha}=\zeta\partial_{\alpha}({\bm \nabla}\cdot{\bm u}) + \eta\left[\nabla^2 u_{\alpha}+ \partial_{\alpha}({\bm \nabla}\cdot{\bm u})/3\right]$.
Fourier transforming and comparing with Eq.~(\ref{lr}), we obtain
\begin{eqnarray}
\zeta q_{\alpha}q_{\beta}u_{\beta} + \eta\left(\!q^2u_{\alpha}\! +\! \frac{1}{3}q_{\alpha}q_{\beta}u_{\beta}\!\right)\! =\!- i\omega m^2 \chi^{\alpha\beta}_{J}({\bf q},\omega)u_{\beta}.\nonumber\\
\label{Kubo1}
\end{eqnarray}
We decompose the current correlation function into its
longitudinal ($\chi_L$) and transverse ($\chi_T$) components:
\begin{eqnarray}
\chi^{\alpha\beta}_{J} = \chi_L\frac{q_{\alpha}q_{\beta}}{q^2} + \chi_T\left(\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^2}\right)
\label{chiTL}
\end{eqnarray}
By taking appropriate $q \to 0$ limits \cite{limits} of Eq.~(\ref{Kubo1}) we find
\begin{eqnarray}
\eta(\omega) = \lim_{q\to 0}{(-i\omega)}m^2 \chi_T({\bf q},\omega)/{q^2}
\label{KuboT}
\end{eqnarray}
and
\begin{eqnarray}
\zeta(\omega) + {4\eta(\omega)}/{3} = \lim_{q\to 0}{(-i\omega)}m^2 \chi_L({\bf q},\omega)/{q^2}.
\label{KuboL}\end{eqnarray}
These expressions define the \emph{complex} shear and bulk viscosities.
We will be interested in the properties and sum rules of
the \emph{spectral functions}:
\begin{eqnarray}
\mathrm{Re}\; \eta(\omega) = \lim_{q\to 0}{m^2 \omega}\mathrm{Im}\chi_T({\bf q},\omega)/{q^2}
\label{KuboTR}
\end{eqnarray}
and
\begin{eqnarray}
\mathrm{Re}\; \zeta(\omega) + {4\mathrm{Re} \;\eta(\omega)}/{3} = \lim_{q\to 0}
{m^2 \omega}\mathrm{Im}\chi_L({\bf q},\omega)/{q^2}.
\label{KuboLR}
\end{eqnarray}
The static viscosities $\eta$ and $\zeta$ introduced in Eq.~(\ref{stresstensor})
are $\eta \equiv \mathrm{Re}\; \eta(\omega=0)$ and $\zeta \equiv \mathrm{Re}\; \zeta(\omega=0)$.
In closing this subsection, we note that the Kubo formulas for the viscosity derived here and below
are valid in both the normal and superfluid phases, provided we recognize that the bulk viscosity
in the superfluid state refers to $\zeta_2$, which describes damping associated with an in-phase motion of the superfluid and normal fluid components~\cite{LLFM}. To understand this in more detail, we recall Landau's two-fluid hydrodynamics~\cite{LLFM} for the
superfluid state. In this theory, three bulk viscosities, $\zeta_1$, $\zeta_2$, and $\zeta_3$, are required
to describe the dissipation associated with different types of relative motions of the superfluid and normal components.
The \emph{longitudinal} response does not distinguish between the superfluid and normal components \cite{Baymbook} and
thus forces the superfluid and normal fluid velocities to be equal: ${\bf v}_s = {\bf v}_n = {\bm u}$. When both components
flow with the same velocity, the two-fluid hydrodynamic stress tensor [see Eq.~(140.5) in Ref.~\cite{LLFM}] reduces to the expression in
Eq.~(\ref{stresstensor}), with $\zeta$ replaced by $\zeta_2$, the bulk viscosity associated with the damping of the
in-phase motions of the superfluid and normal fluid components.
One can also show by direct application of Eq.~(\ref{KuboLR2}) to the two-fluid
hydrodynamic density response function in Eq.~(\ref{chirhorho}) that the left-hand side of Eq.~(\ref{KuboLR2}) is $\zeta_2+4\eta/3$ in the
low-frequency two-fluid hydrodynamic regime.
\subsection{Stress correlators}
\label{Kubocomparisonsec}
We next derive Kubo formulas equivalent to those derived above but expressed in terms of
the correlators of a suitably defined stress tensor operator $\widehat{\Pi}_{\alpha\beta}$.
These are useful to make connections with the literature~\cite{Kovtun05,Bruun05,Peshier05}.
We will also use these results in connection with the positivity of the bulk viscosity
spectral function and its vanishing for the unitary Fermi gas.
The $\widehat{\Pi}_{\alpha\beta}$ operator must satisfy
\begin{eqnarray}
i m [\hat{j}_{\alpha},\hat{H}] = \partial_{\beta}\widehat{\Pi}_{\alpha\beta},
\label{euler-op}
\end{eqnarray}
which is the operator version of the Euler equation, Eq.~(\ref{euler}), and is simply a statement of momentum conservation.
We go to Fourier space and relate matrix elements of the current operator to those of the stress tensor
by sandwiching Eq.~(\ref{euler-op}) between exact many-body eigenstates.
Using the spectral representation in Eq.~(\ref{FourierchiAB}) we can then relate the current correlator
$\chi^{\alpha\beta}_{J}({\bf q},\omega)$ to the stress correlator
$\chi^{\alpha\beta,\mu\nu}_{\Pi}({\bf q},\omega)$. The latter is defined by choosing
$\hat{A}=\hat{\Pi}^{\alpha\beta}({\bf q})$ and $\hat{B}=\hat{\Pi}^{\mu\nu}(-{\bf q})$ in Eq.~(\ref{FourierchiAB}).
For simplicity we calculate only $\chi^{xx}_{J}$, which will suffice for our purposes.
The final result, after some simple algebra, is
\begin{eqnarray}
m^2\omega^2\chi^{xx}_{J}({\bf q},\omega) &=&q_{\alpha}q_{\beta}\chi^{x\alpha,x\beta}_{\Pi}({\bf q},\omega) -
mq_{\alpha}\langle[\widehat{\Pi}^{x\alpha}_{{\bf q}},\hat{j}^{x}_{-{\bf q}}]\rangle.\nonumber\\ \!\!\!\!\!\!\!
\label{Correlationrelation}
\end{eqnarray}
Note that $\widehat{\Pi}^\prime_{\alpha\beta} = \widehat{\Pi}_{\alpha\beta} + \hat{\Lambda}_{\alpha\beta}$,
with any symmetric tensor $\hat{\Lambda}$ satisfying $\partial_{\beta}\hat{\Lambda}_{\alpha\beta} = 0$, will
also be a solution to the Euler equation, Eq.~(\ref{euler-op}).
This non-uniqueness in the definition of $\widehat{\Pi}$ does not affect our final results
for the viscosity, related to $\chi^{xx}_{J}$, since a symmetric $\hat{\Lambda}$
with $q_{\beta}\hat{\Lambda}_{\alpha\beta} = 0$ makes no contribution to
Eq.~(\ref{Correlationrelation}).
Using the decomposition given by Eq.~(\ref{chiTL}), and taking the appropriate limits, we find
\begin{eqnarray}
m^2\omega^2\lim_{q\to 0}\frac{\chi_T}{q^2} = \lim_{q\to 0}
\left[ \chi^{xy,xy}_{\Pi} - \frac{m}{q} \langle[\widehat{\Pi}^{xy}_{{\bf q}},\hat{j}^{x}_{-{\bf q}}]\rangle \right],
\label{TPi}
\end{eqnarray}
where we have taken $q_x$ and $q_z$ to zero before $q_y$, and
\begin{eqnarray} m^2\omega^2\lim_{q\to 0}\frac{\chi_L}{q^2} = \lim_{q\to 0} \left[ \chi^{xx,xx}_{\Pi}- \frac{m}{q} \langle[\widehat{\Pi}^{xx}_{{\bf q}},\hat{j}^{x}_{-{\bf q}}]\rangle \right],
\label{LPi}
\end{eqnarray}
where we have taken $q_y$ and $q_z$ to zero before $q_x$. We note that the commutators on the right hand sides
of Eqs.~(\ref{TPi}) and (\ref{LPi}) only affect the real parts of $\chi_T$ and $\chi_L$ and not the
spectral functions of interest, shown in the next two equations.
Using the Kubo formulas given by Eqs.~(\ref{KuboTR}) and (\ref{KuboLR})
that were derived above, we find
\begin{eqnarray}
\mathrm{Re}\; \eta(\omega) = \lim_{q\to 0}{\mathrm{Im}\chi^{xy,xy}_{\Pi}({\bf q},\omega)}/{\omega}
\label{KuboPiT}
\end{eqnarray}
and
\begin{eqnarray}
\mathrm{Re}\; \zeta(\omega) + {4\mathrm{Re} \;\eta(\omega)}/{3}
= \lim_{q\to 0}{\mathrm{Im}\chi^{xx,xx}_{\Pi}({\bf q},\omega)}/{\omega}.
\label{KuboPiL}
\end{eqnarray}
In an isotropic system, in the $q \to 0$ limit, the only fourth rank tensor allowed by
symmetry is of the form $A \delta_{\alpha\beta} \delta_{\mu\nu}
+ B\left( \delta_{\alpha\mu} \delta_{\beta\nu} + \delta_{\alpha\nu} \delta_{\beta\mu} \right)$.
We can thus combine Eqs.~(\ref{KuboPiT}) and (\ref{KuboPiL}) to write
\begin{eqnarray}
\left[\mathrm{Re}\; \zeta - \frac{2}{3} \mathrm{Re}\; \eta \right] \delta_{\alpha\beta} \delta_{\mu\nu}
&+& \mathrm{Re}\; \eta \left( \delta_{\alpha\mu} \delta_{\beta\nu} + \delta_{\alpha\nu} \delta_{\beta\mu} \right)
\nonumber
\\
&=& \lim_{q\to 0}\frac{\mathrm{Im}\chi^{\alpha\beta,\mu\nu}_{\Pi}({\bf q},\omega)}{\omega}.
\label{KuboPiFull}
\end{eqnarray}
A very useful formula for the bulk viscosity follows from Eq.~(\ref{KuboPiFull})
by looking at its $(xx,yy)$ component and combining it with the $(xx,xx)$ component in Eq.~(\ref{KuboPiL}).
Using the summation convention, we thus obtain
\begin{eqnarray}
\mathrm{Re}\; \zeta(\omega) = \lim_{q\to 0}\frac{\mathrm{Im}\chi^{\alpha\alpha,\beta\beta}_{\Pi}({\bf q},\omega)}{9\;\omega}.
\label{KuboPiZeta}
\end{eqnarray}
We emphasize again that the Kubo formulas for the bulk and shear viscosities expressed in terms
of the stress-stress correlation function are equivalent to
those expressed in terms of current-current correlations, Eqs.~(\ref{KuboTR}) and (\ref{KuboLR}).
The two sets of equations are simply related by the exact conservation law, Eq.~(\ref{euler-op}).
Above, we focused on the \emph{dissipative} parts of the response, i.e., the \emph{real} parts of the viscosities.
Comparing Eqs.~(\ref{KuboT}) and (\ref{KuboL}) with Eqs.~(\ref{TPi}) and (\ref{LPi}), we
see that the imaginary part of $\eta$ and the imaginary part of $\left(4\eta/3 + \zeta\right)$ are \textit{not}
given by $\lim_{\omega\to 0}\lim_{q\to 0}\mathrm{Re}\chi^{xy,xy}_{\Pi}/\omega$ and $\lim_{\omega\to 0}\lim_{q\to 0}\mathrm{Re}\chi^{xx,xx}_{\Pi}/\omega$, respectively. $\mathrm{Im}\; \eta$ and $\mathrm{Im}\; \zeta$,
when written in terms of stress correlators, also involve the frequency-independent, equal-time commutator terms in Eqs.~(\ref{TPi}) and (\ref{LPi}).
This point seems to be missed in treatments that start out with the stress correlator formalism.
The imaginary parts of the transport coefficients
are most simply expressed in terms of the current correlation functions, Eqs.~(\ref{KuboT}) and (\ref{KuboL}).
In the $\omega \to 0$ limit, the validity of this assertion can be seen quite independently from
hydrodynamics (see Appendix~\ref{hydrosec}). Allowing $\eta$ to be complex in the hydrodynamic expression for the transverse current correlation function in Eq.~(\ref{chiT}), for instance, one can readily confirm that the imaginary part of the shear viscosity is indeed given by Eq.~(\ref{KuboT}).
\section{Shear viscosity and normal fluid density}
\label{rhon-section}
In this Section we discuss the relation between the static shear viscosity $\mathrm{Re}\; \eta(\omega=0)$
and the normal fluid density $\rho_n$, both of which can be written in terms of the \emph{transverse} current-current
correlation function. This allows us to prove that a non-zero
normal fluid density $\rho_n$ is a necessary condition for a non-vanishing shear viscosity $\eta$.
This is, perhaps, not entirely unexpected on physical grounds, but we are unaware of
a microscopic proof, valid for all Galilean invariant Bose or Fermi quantum fluids,
that does not rely on a quasiparticle approximation.
Before turning to the calculation, it may be useful to review some elementary facts
about the shear viscosity $\eta$. Given that there is a Kubo formula for $\eta(\omega)$
in terms of the current-current correlation function, Eq.~(\ref{KuboTR}), and that in kinetic theory
$\eta$ is proportional to the mean free path, it may seem natural to assume
that the shear viscosity of a fluid is the analog of metallic conductivity.
This, however, is completely misleading. The shear viscosity is, in fact, the analog of the \emph{resistivity}.
This is clear, e.g., from the classical formula of Poiseuille
for the flow rate $Q = {\pi R^4 \Delta P}/(8 \eta L)$, with a pressure difference $\Delta P$ across
a cylindrical pipe of radius $R$ and length $L$.
We will see below that zero viscosity in a superfluid is the analog of zero resistance in
a superconductor.
We begin by rewriting the Kubo formula for the shear viscosity, given by Eq.~(\ref{KuboTR}), using the spectral representation in Eq.~(\ref{ImchiAB}):
\begin{eqnarray}
\mathrm{Re}\eta(\omega) = \lim_{(T)} \frac{\pi m^2}{\cal{Z}}
\sum_{a,b}\left[e^{-\beta E_a}\;-\;e^{-\beta E_b}\right] E_{ba}
\nonumber\\
\times\frac{\vert \langle b|\hat{j}^x_{{\bf q}}|a\rangle \vert^2}{q^2}
\delta(\omega -E_{ba}),
\label{etaomega}
\end{eqnarray}
Here and below, the ``transverse limit'', denoted by $ \lim_{(T)}$, means that for $\chi_J^{xx}$
we first set $q_x = 0$ and then take the limit $q_y \to 0$.
The normal fluid density $\rho_n$ characterizes the response of a fluid to
moving walls and determines the moment of inertia of a cylinder containing the fluid; see, e.g.,
the detailed discussion in Refs.~\cite{Baymbook,NozieresPines2}. It is defined in terms of the
real part of the static transverse current correlator:
\begin{eqnarray}
\rho_n = \lim_{q \to 0} m^2 \mathrm{Re}\chi_T({\bf q},\omega=0).
\label{rhon1}
\end{eqnarray}
Using the spectral representation in Eq.~(\ref{FourierchiAB}) for $\chi_J^{xx}$, we
can rewrite this result as
\begin{eqnarray}
\rho_n = \lim_{(T)} \frac{m^2}{\cal{Z}}
\sum_{a,b}\frac{\left[e^{-\beta E_a}\;-\;e^{-\beta E_b}\right]}{E_{ba}}
{\vert \langle b|\hat{j}^x_{{\bf q}}|a\rangle \vert^2}.
\label{rhon2}
\end{eqnarray}
Our goal now is to understand the connection between the shear viscosity
$\eta$, which is obtained by taking the $\lim_{\omega \to 0}\lim_{q \to 0}$ of $\mathrm{Im}\chi_T$
in Eq.~(\ref{etaomega}), and the normal fluid density $\rho_n$, which is
the $\lim_{q \to 0}\lim_{\omega \to 0}$ of $\mathrm{Re}\chi_T$ in Eq.~(\ref{etaomega}).
In lattice models of superconductors, it has been suggested \cite{Scalapino} that the order of the
$q$ and $\omega$ limits can be safely interchanged for the \emph{transverse} current correlator,
because all ``transverse'' excitations are gapped (unlike longitudinal excitations such as
phonons in charge-neutral systems). However, this argument is \emph{not} valid for the systems
of interest to us. This can be seen, e.g., from the hydrodynamic form of $\chi_T$ in Eq.~(\ref{chiT})
which has a ``diffusion pole'' that makes the order of limits quite different.
To prove the result stated at the beginning of this Section, we will show that
$\rho_n = 0$ implies $\eta = 0$. The starting condition $\rho_n = 0$ makes sense
only at $T=0$, since at any finite temperature there will necessarily be some thermal excitations.
Furthermore, the vanishing of the normal fluid density
\begin{eqnarray}
\rho_n = \lim_{(T)} 2 m^2
\sum_{b}\frac{\vert \langle b|\hat{j}^x_{{\bf q}}|0\rangle \vert^2}{E_{b0}}
\label{rhon3}
\end{eqnarray}
at $T=0$ implies that each term in the sum $\sum_b$ over states vanishes.
This means that, for each state $|b\rangle $, if the excitation energy varies as $\lim_{(T)}E_{b0} \sim q^{\alpha_b}$, with
$\alpha_b \geq 0$, then the matrix element of the current operator vanishes even faster:
$\lim_{(T)}\vert \langle b|\hat{j}^x_{{\bf q}}|0\rangle \vert \sim q^{\alpha_b + \beta_b}$
with $\beta_b > 0$. Note that we are not making any assumptions about the nature of the
spectrum since both gapless ($\alpha_b > 0$) and gapped ($\alpha_b = 0$) excitations are permitted.
In either case, the matrix element of $\hat{j}^x_{{\bf q}}$ vanishes, since the $q \to 0$ limit
of $\hat{j}^x_{{\bf q}}$ is the total momentum, which commutes with the Hamiltonian in a Galilean invariant system.
It is only in such a system that $\rho_n$ vanishes at $T=0$~\cite{NozieresPines2,Paramekanti}.
Now that we have constrained the matrix elements for any form of the excitation spectrum given $\rho_n = 0$,
we now ask how these constraints impact the shear viscosity. We look separately at the contribution from
gapless and gapped states to Eq.~(\ref{etaomega}), which at $T=0$ can be written as
\begin{eqnarray}
\eta(\omega) = \lim_{(T)} {\pi m^2}
\sum_{b} E_{b0}\frac{\vert \langle b|\hat{j}^x_{{\bf q}}|0\rangle \vert^2}{q^2}
\delta(\omega -E_{b0}).
\label{etaomega2}
\end{eqnarray}
Each gapless state $b$, with $\alpha_b > 0$, will contribute a term
$\lim_{(T)} q^{2\alpha_b + \beta_b - 2}\delta(\omega - A_b q^{\alpha_b})$,
which gives a vanishing contribution \cite{eta-zero} in the limit $q \to 0$ for all $\omega > 0$.
Finally taking the $\omega \to 0$ limit, we find that the contribution of the gapless states
to $\eta$ vanishes.
Next, consider the gapped states with $\alpha_b = 0$, so that
$\lim_{(T)}E_{b0} \equiv \Delta_b > 0$. Their contribution to Eq.~(\ref{etaomega2})
yields an expression of the form
$\eta(\omega) = \lim_{(T)}
\sum_{b}^\prime C_b q^{\beta_b - 2} \delta(\omega -\Delta_{b})$,
where the prime indicates a sum over all gapped states. This result contributes
to both the $\eta$ sum rule and the high-frequency tail that we will
derive later in the paper. The important point here is that
for $0< \omega < \min_b^\prime\left\{\Delta_b\right\}$, i.e., below the minimum gap
of all excitations, $\eta(\omega) = 0$.
Thus, we conclude that
the vanishing of the normal fluid density implies that the static limit of the shear viscosity
vanishes as well: $\eta = 0$. This means that the Galilean invariant ground state of a superfluid
has zero shear viscosity~\cite{zeroviscosity}. This is similar to the zero d.c. resistivity of a charged superconductor,
as already mentioned at beginning of this Section.
There is, however, an important difference in that the vanishing resistivity persists all the way
up to the transition temperature $T_c$. Even though there are normal fluid excitations in a
superconductor, the infinite conductivity of the condensate ``shorts out'' the normal fluid in
a superconductor. In marked contrast, in a neutral superfluid,
$\eta$ vanishes only at $T=0$. For $0 < T < T_c$, even though a condensate exists,
the normal fluid excitations give rise to a non-zero shear viscosity.
\section{Positivity of spectral functions}
\label{posdefsec}
We simplify notation and write from now on
\begin{eqnarray}
\eta(\omega) \equiv \mathrm{Re}\; \eta(\omega)
\; \; \mathrm{and} \; \;
\zeta(\omega) \equiv \mathrm{Re}\; \zeta(\omega),
\label{simplifyReal}
\end{eqnarray}
unless explicitly stated otherwise. This should cause no confusion since we will not be dealing
with the corresponding imaginary parts.
Before deriving sum rules for
$\eta(\omega)$ and $\zeta(\omega)$ in Section~\ref{sumrulesec},
it is important to discuss here their positivity properties.
Every time we say `positive' we actually mean
`non-negative', a term we find awkward for repeated use.
We will show that
\begin{eqnarray}
\eta(\omega)\ge 0\;\; \mathrm{and}\;\; \zeta(\omega) \ge 0
\;\;\;\; \forall \omega.
\label{Work}
\end{eqnarray}
The simplest approach is to make explicit use of the spectral representation. We will see that this
is sufficient to prove the positivity of $\eta(\omega)$, but \emph{not} that of $\zeta(\omega)$.
To prove the latter, we will calculate the power absorbed by the
fluid from an external velocity perturbation with $\nabla \cdot {\bm u} \neq 0$.
Let us begin with Eqs.~(\ref{KuboTR}) and (\ref{KuboLR}) and
use the spectral representation given by Eq.~(\ref{ImchiAB}) with $\hat{A}_{{\bf q}}=\hat{j}^x_{{\bf q}}$
and $\hat{B}^{\dagger}_{{\bf q}} = \hat{j}^x_{-{\bf q}}$. The transverse and longitudinal components
are obtained, as usual, by taking suitable $q \to 0$ limits~\cite{limits}.
Using $|\langle n|\hat{j}^x_{-{\bf q}}|m\rangle|^2 \geq 0$ and $\omega[1-\exp(-\beta\omega)]\geq 0$ for all $\omega$,
we see that both
$\omega\mathrm{Im}\chi_T({\bf q},\omega)$ and $\omega\mathrm{Im}\chi_L({\bf q},\omega)$ are positive.
Thus we obtain
\begin{eqnarray}
\eta(\omega)\ge 0\;\; \mathrm{and}\;\; \zeta(\omega) +4\eta(\omega)/3\ge 0
\;\;\;\; \forall \omega.
\label{Work2}
\end{eqnarray}
The inequality for $\zeta(\omega)$ is much weaker than what we wish to prove.
One reason to expect that a stronger result should exist for $\zeta(\omega)$ is that
it is known from hydrodynamics (see Sec.~49 of Landau and Lifshitz~\cite{LLFM}) that
the \emph{static} bulk viscosity $\zeta(0)$ must be positive.
To generalize this to all frequencies, we exploit the idea that
the time-averaged power absorbed by the system from an external perturbation
is necessarily positive.
The rate at which the external velocity perturbation given by Eq.~(\ref{Hpert}) does work on the fluid is given
by
\begin{eqnarray}
\frac{d W}{dt} = i\omega m\int d{\bf r} e^{-i\omega t} e^{0^+t}{\bm u}({\bf r})\cdot\langle \hat{\bm{j}}({\bf r},t)\rangle. \label{Power}
\end{eqnarray}
Following Ref.~\cite{ChaikinLubensky}, one finds that the time average of the
power absorbed by the fluid is
\begin{eqnarray} \overline{\frac{dW}{dt}} = \frac{m^2}{2} \sum_{\bf q}
u_{\alpha}(-{\bf q})\left[\omega\mathrm{Im}\chi^{\alpha\beta}_J({\bf q},\omega)\right]u_{\beta}({\bf q})>0.
\label{Power2}\end{eqnarray}
$\overline{dW}/dt > 0$ follows from the fact that energy can only be dissipated
for \emph{any} choice of the external velocity field.
This implies that the real, symmetric matrix $\omega\mathrm{Im}\chi^{\alpha\beta}({\bf q},\omega)$ must be positive definite,
which is equivalent to the positivity of its eigenvalues.
Using Eq.~(\ref{chiTL}), we see that these eigenvalues are precisely
$\omega \mathrm{Im}\chi_L({\bf q},\omega)$ and $\omega \mathrm{Im}\chi_T({\bf q},\omega)$,
so that we simply rederive Eq.~(\ref{Work2}), and do not obtain $\zeta(\omega) \ge 0$.
To constrain $\zeta(\omega)$, without any $\eta(\omega)$ contribution, we must
look at an external velocity field ${\bm u}({\bf r},t) = {\bm u}({\bf r})e^{-i\omega t}$ with
${\bm u}({\bf r}) = a{\bf r}$, where $a = \left(\nabla \cdot {\bm u}\right)/3$ is
spatially uniform.
To analyze the effect of such a perturbation, we first need to rewrite Eq.~(\ref{Power2})
in terms of the stress correlator so that $\partial_\alpha u_\beta$ is directly involved. Second,
${\bm u}({\bf r}) = a{\bf r}$ is not Fourier transformable, so we must work in ${\bf r}$-space, rather than
${\bf q}$-space used elsewhere in the paper.
We use the same derivation that led from the operator Euler equation given by Eq.~(\ref{euler-op}) to Eq.~(\ref{Correlationrelation}),
to get
\begin{eqnarray}
m^2\omega^2\mathrm{Im}\chi^{\alpha\beta}_{J}({\bf q},\omega) = q_{\mu}q_{\nu}\mathrm{Im}\chi^{\alpha\mu,\beta\nu}_{\Pi}({\bf q},\omega).
\label{ImCorrelationrelation}
\end{eqnarray}
Using this in Eq.~(\ref{Power2}) and rewriting the resulting expression in real space, we get
\begin{eqnarray}
\lefteqn{\overline{\frac{dW}{dt}} = \frac{1}{2}\int \!d{\bf r} \!\int \!d{\bf r}' \times}&&\nonumber\\&&\partial_{\alpha}u_{\mu}({\bf r})\left[\frac{\mathrm{Im}\chi^{\alpha\mu,\beta\nu}_{\Pi}({\bf r}-{\bf r}',\omega)}{\omega}\right]\!\partial_{\beta}u_{\nu}({\bf r}'),
\label{Power3}
\end{eqnarray}
which must hold for arbitrary velocity fields ${\bm u}({\bf r})$.
To isolate the contribution of the bulk viscosity, we choose the velocity field ${\bm u} = a{\bf r}$,
for which the shear term (in square brackets)
in the viscous stress tensor, Eq.~(\ref{stresstensor}), vanishes.
Using $\partial_\alpha u_\beta = a \delta_{\alpha\beta}$ in Eq.~(\ref{Power3}) we get
$\mathrm{Im}\chi^{\alpha\alpha,\beta\beta}_{\Pi}({\bf q} \to 0,\omega)/\omega \ge 0$. From the result
given by Eq.~(\ref{KuboPiZeta}) for the bulk viscosity, it immediately follows that $\zeta(\omega)\ge 0$ for all $\omega$.
\section{Sum rules}
\label{sumrulesec}
We now derive sum rules for the shear and bulk viscosities,
$\int^{\infty}_0 d\omega \eta(\omega)$ and $\int^{\infty}_0 d\omega \zeta(\omega)$.
We will first show that
\begin{eqnarray}
\lefteqn{\frac{1}{\pi}\int^{\infty}_0 \!\!d\omega
\lim_{q\to 0}\!\frac{\omega}{q^2}\mathrm{Im}\chi^{xx}_{J}({\bf q},\omega)=}&&\nonumber
\\&&\!\!\!\!\!\!\!\!\lim_{q\to 0}
\frac{\langle[\hat{j}^{x}_{-{\bf q}},[\hat{H},\hat{j}^{x}_{{\bf q}}]]\rangle}{2 q^2}
+\lim_{\omega\to 0}\lim_{q\to 0}\frac{\omega^2}{2q^2}\mathrm{Re}\chi^{xx}_{J}({\bf q},\omega).
\label{noncommutator}
\end{eqnarray}
Then we will simplify the two terms on the right hand side of Eq.~(\ref{noncommutator}):
the first term by explicit evaluation of the commutators, and the second by appealing to
hydrodynamics.
To see what is involved in deriving Eq.~(\ref{noncommutator}), let us first be na\"ive and
ignore the $q \to 0$ limit.
Evaluating the integral on the left hand side by using the
spectral representation in Eq.~(\ref{ImchiAB}) for $\mathrm{Im}\chi^{xx}_{J}$,
we only obtain the first commutator term on the right.
But taking the $q \to 0$ limit after doing the $\omega$-integration
is \emph{not} the same as interchanging the order of these operations!
In order to do it correctly ($q \to 0$ limit before the $\omega$-integration),
we exploit the Kramers-Kronig (K-K) relations to evaluate the
integral in Eq.~(\ref{noncommutator}).
The only subtle point in this approach is that we need to ensure that the
analytic functions which we K-K transform
decay sufficiently rapidly for $\omega \to \infty$.
Using the expression in Eq.~(\ref{FourierchiAB}), it is straightforward to expand
the current correlator in powers of $\omega^{-1}$ for large frequencies.
One finds~\cite{Pitaevskiibook},
\begin{eqnarray}
\lim_{\omega\to \infty}\chi^{xx}_J({\bf q},\omega) &=& \frac{\langle[\hat{j}^{x}_{-{\bf q}},\hat{j}^{x}_{{\bf q}}]\rangle}{\omega} - \frac{\langle[\hat{j}^{x}_{-{\bf q}},[\hat{H},\hat{j}^{x}_{{\bf q}}]]\rangle}{\omega^2} + \ldots.
\nonumber\\
\label{chiJasymptote}
\end{eqnarray}
The $\omega^{-1}$ term vanishes since $\langle[\hat{j}^{x}_{-{\bf q}},\hat{j}^{x}_{{\bf q}}]\rangle = -(2q_{x}/m^2)\sum_{{\bf k}\sigma}n_{{\bf k}\sigma} k_{x} = 0$ in a uniform system.
We further note that this expansion is strictly valid only for a smooth potential~\cite{divergence}, a point
which we will elaborate on in later Sections.
Let us define a function $F(\omega)$ as
\begin{eqnarray} F(\omega) \equiv \lim_{q\to 0}\frac{\omega^2}{q^2}\left[\chi^{xx}_J({\bf q},\omega) + \frac{\langle[\hat{j}^{x}_{-{\bf q}},[\hat{H},\hat{j}^{x}_{{\bf q}}]]\rangle}{\omega^2}\right],
\label{F} \end{eqnarray}
where the $q \to 0$ limit is defined appropriately~\cite{limits} for the longitudinal and
transverse cases.
From Eq.~(\ref{chiJasymptote}), we see that $\lim_{\omega\to\infty}F(\omega)$ vanishes
at least as fast as $\omega^{-1}$ and we can K-K transform it. We thus obtain
\begin{eqnarray}
\lim_{\omega\to 0}\mathrm{Re} F(\omega) &=& \frac{{\cal{P}}}{\pi}\int^{\infty}_{-\infty}d\omega'\frac{\mathrm{Im}F(\omega')}{\omega'}\\
&=&\frac{2}{\pi}\int^{\infty}_{0}d\omega'
\lim_{q\to 0}\frac{\omega'}{q^2}\mathrm{Im}\chi^{xx}_J({\bf q},\omega').\nonumber
\label{noncommutator2}
\end{eqnarray}
where we have used the fact that $\omega \mathrm{Im}\chi^{xx}_J({\bf q},\omega)$ is an even function
of $\omega$.
Using Eq.~(\ref{F}) on the left-hand side of this expression immediately leads to
the result, Eq.~(\ref{noncommutator}), quoted above.
As mentioned earlier, $\lim_{\omega\to 0}\lim_{q\to 0}(\omega^2/2q^2)\mathrm{Re}\chi^{xx}_J$
in Eq.~(\ref{noncommutator}) arises from the noncommutativity of the $\omega\to 0$ and $q\to 0$ limits.
Since this term involves the zero-frequency, long-wavelength limit where hydrodynamics is applicable, we can use hydrodynamic expressions for the current correlation function to evaluate it.
In Appendix~\ref{hydrosec}, we review such expressions and show that for any simple hydrodynamic liquid, one has
\begin{eqnarray}
\lim_{\omega\to 0}\lim_{q\to 0}\frac{m^2\omega^2}{2q^2}\mathrm{Re}\chi_T({\bf q},\omega) &=& 0,\nonumber\\
\label{difference2}\\
\lim_{\omega\to 0}\lim_{q\to 0}\frac{m^2\omega^2}{2q^2}\mathrm{Re}\chi_L({\bf q},\omega) &=& - \frac{\rho c_s^2}{2},\nonumber
\end{eqnarray}
where the adiabatic sound speed is $c_s \equiv (\partial P/\partial\rho)^{1/2}$ at fixed $s=S/N$.
Equation~(\ref{difference2}) is valid for both normal fluids and superfluids
(within two-fluid hydrodynamics).
Combining Eqs.~(\ref{KuboTR}), (\ref{KuboLR}), (\ref{noncommutator}), and (\ref{difference2}), we find the following sum rules:
\begin{eqnarray} \frac{1}{\pi}\int^{\infty}_0 \!\!d\omega
\eta(\omega)=\lim_{q\to 0}\frac{m^2\langle[\hat{j}^{x}_{-{\bf q}},[\hat{H},\hat{j}^{x}_{{\bf q}}]]\rangle_T}{2 q^2},\label{etasumrule}\end{eqnarray}
\begin{eqnarray} \frac{1}{\pi}\!\int^{\infty}_0 \!\!\!d\omega\!\!
\left[\!\zeta(\omega)\!+\!\frac{4\eta(\omega)}{3}\right] \!=\! \lim_{q\to 0}\frac{m^2\langle[\hat{j}^{x}_{-{\bf q}},[\hat{H},\hat{j}^{x}_{{\bf q}}]]\rangle_L}{2 q^2}\! -\! \frac{\rho c_s^2}{2}. \nonumber\\ \label{zetasumrule}\end{eqnarray}
Here, $\langle\cdots\rangle_{T(L)}$ denotes the $q \to 0$ limit appropriate to the transverse (longitudinal) case~\cite{limits}.
The last remaining step in our derivation is
to evaluate the commutators in Eqs.~(\ref{etasumrule}) and (\ref{zetasumrule}).
We consider a system of fermions or bosons described by the Hamiltonian
\begin{eqnarray}
\hat{H}\! &=& \hat{K} + \hat{V}
\\
&=& \! \sum_{{\bf k}\sigma}\varepsilon_{{\bf k}}\hat{c}^{\dagger}_{{\bf k}\sigma}\hat{c}_{{\bf k}\sigma}\! +\!\frac{1}{2}\!\sum_{\substack{{\bf k}\bk'{\bf p}\\ \sigma\sigma'}}\!V(p)\hat{c}^{\dagger}_{{\bf k}+{\bf p}\sigma}\hat{c}^{\dagger}_{{\bf k}'\!-{\bf p}\sigma'}\hat{c}_{{\bf k}'\!\sigma'}\hat{c}_{{\bf k}\sigma}.\nonumber
\label{H}\end{eqnarray}
For a single-component Bose system, $\sigma=\sigma'$ assumes one value;
for fermions, $\sigma = \uparrow,\downarrow$ can take one of two ``spin'' values.
It is straightforward, but tedious, to evaluate the commutator in Eq.~(\ref{noncommutator}) for this Hamiltonian.
One finds, for both fermions and bosons,
\begin{eqnarray}
\lefteqn{\frac{m^2}{2}\langle[\hat{j}^{x}_{-{\bf q}},[\hat{H},\hat{j}^{x}_{{\bf q}}]]\rangle=}&&
\label{Jsumrule}
\\
&&\frac{\langle \hat{K} \rangle}{3}\left(2q^2_{x} + q^2\right) + n\frac{ q^2q^2_{x}}{8m}-
\frac{1}{2} \langle\!\langle 2V(p)p^2_{x}
\nonumber
\\
&-&V\!(|{\bf p}-{\bf q}|)(p_{x}\!-\!q_{x})^2
- V\!(|{\bf p}+{\bf q}|)(p_{x}\!+\!q_{x})^2 \rangle\!\rangle.
\nonumber
\end{eqnarray}
Here, $\langle \hat{K} \rangle = \sum_{{\bf k}\sigma}\varepsilon_{{\bf k}} n_{{\bf k}\sigma}$ is the kinetic energy density,
and we have introduced the shorthand notation
\begin{eqnarray}
\left\langle\!\left\langle {\cal Q} \right\rangle\!\right\rangle \equiv \frac{1}{2}\sum_{\substack{{\bf k}\bk'{\bf p}\\ \sigma\sigma'}}
{\cal Q} \langle \hat{c}^{\dagger}_{{\bf k}+{\bf p}\sigma}\hat{c}^{\dagger}_{{\bf k}'-{\bf p}\sigma'}
\hat{c}_{{\bf k}'\sigma'}\hat{c}_{{\bf k}\sigma}\rangle.
\label{intaverage}
\end{eqnarray}
Related expressions specific to Bose liquids are given in Ref.~\cite{Dalfovo92}. We also note in passing that the longitudinal component of Eq.~(\ref{Jsumrule}) is related by Eq.~(\ref{Nconserv}) to the so-called ``$\langle \omega^3\rangle $ sum rule" discussed for electronic systems~\cite{Puff65}.
The right-hand side of Eq.~(\ref{Jsumrule}) varies as $q^2$ as $q\to 0$,
which cancels the $1/q^2$ in Eqs.~(\ref{etasumrule}) and (\ref{zetasumrule}).
Evaluating the transverse and longitudinal limits of Eq.~(\ref{Jsumrule}), one finds the following viscosity sum rules:
\begin{eqnarray}
\frac{1}{\pi}\int^{\infty}_{0}\!\!d\omega \eta(\omega)= \frac{\varepsilon}{3}-\frac{\langle \hat{V} \rangle}{3} + \frac{2\overline{V}'}{15}
+ \frac{\overline{V}''}{30}
\label{etasumruleiso}
\end{eqnarray}
and
\begin{eqnarray}
\frac{1}{\pi}\int^{\infty}_{0}\!\!d\omega \zeta(\omega) =
\frac{5\varepsilon}{9}+\frac{4\langle \hat{V} \rangle}{9} +
\frac{5\overline{V}'}{9}+
\frac{\overline{V}''}{18}-\frac{\rho c_s^2}{2}.
\label{zetasumruleiso}
\end{eqnarray}
Here,
$\varepsilon=\langle \hat{K} \rangle+\langle \hat{V} \rangle$ is the total energy density,
$\langle \hat{V} \rangle$ the potential energy density, and the terms
$\overline{V}'$ and $\overline{V}''$ are defined using Eq.~(\ref{intaverage}) as
\begin{eqnarray}
\overline{V}' \equiv \left\langle\!\left\langle p\left({\partial V}/{\partial p}\right) \right\rangle\!\right\rangle \ \
{\rm and} \ \
\overline{V}'' \equiv \left\langle\!\left\langle p^2\left({\partial^2V}/{\partial p^2}\right) \right\rangle\!\right\rangle.
\label{Vpp}
\end{eqnarray}
These sum rules are valid at all temperatures (i.e., in the superfluid as well as normal phase) for any Bose or Fermi system with an arbitrary,
spin-independent, isotropic interaction potential $V(p)$. We emphasize that these are exact results obtained without making
any quasiparticle approximations.
In the next Section (Sec.~\ref{diluteviscositysec}), we simplify these sum rules
for the case of a two-component Fermi gas with short range interactions,
which is of relevance to experiments on ultracold atomic Fermi gases with Feshbach scattering resonances.
Before closing this Section, let us briefly discuss viscosity sum rules using the stress correlator representation.
For the shear viscosity spectral function, the sum rule
\begin{eqnarray}
\int^{\infty}_0 d\omega \eta(\omega) &=& \frac{1}{\pi}\int^{\infty}_0 d\omega \lim_{q\to 0}
\frac{\mathrm{Im}\chi^{xy,xy}_{\Pi}({\bf q},\omega)}{\omega} \nonumber\\ &=&
\lim_{\omega\to 0}\lim_{q\to 0}\frac{1}{2}\mathrm{Re}\chi^{xy,xy}_{\Pi}({\bf q},\omega)
\label{etasumrulePi}\end{eqnarray}
follows trivially from the Kramers-Kronig relation.
To show that this is the same as Eq.~(\ref{etasumrule}), we
use Eq.~(\ref{TPi}) in the second line of Eq.~(\ref{etasumrulePi}).
One can rewrite the commutator in Eq.~(\ref{TPi}) using the Fourier transform of
Eq.~(\ref{euler-op}), and set $\omega^2 \mathrm{Re}\chi_T/q^2$ to zero
using the hydrodynamic result, Eq.~(\ref{difference2}), to obtain Eq.~(\ref{etasumrule}).
\section{Dilute two-component Fermi gas}
\label{diluteviscositysec}
We now specialize to the case of a two-component Fermi gas in the dilute limit, where the effective range $r_0$ of the
potential (van der Waals at ``long" distances, with $r_0\sim 100 a_0$) is much smaller than the $s$-wave scattering length $a$ and the mean interparticle spacing $k^{-1}_F$. (In typical experiments, $k^{-1}_F\sim 1\mu m$ and $500 a_0\lesssim |a| \lesssim \infty$.)
In the zero range limit $r_0\to 0$, all physical observables are
universal ($r_0$-independent) functions of the
energy scale $\epsilon_F$ (or length scale $k^{-1}_F$) and the dimensionless parameters
$T/\epsilon_F$ (temperature) and $1/(k_F a)$ (interaction).
We will show that for Fermi gases, the results given by
Eqs.~(\ref{etasumruleiso}) and (\ref{zetasumruleiso}) of the previous Section,
reduce to the simple expressions given by Eqs.~(\ref{etasumrule0finite}) and (\ref{zetasumrule0})
in the Introduction.
Our main task is to calculate the terms $\overline{V}'$ and $\overline{V}''$, involving \textit{gradients}
of the interaction potential, defined in Eq.~(\ref{Vpp}).
We use the real-space approach developed by Zhang and Leggett~\cite{Zhang08}, which is a simple
way to derive results first obtained by Tan~\cite{Tan08,Braaten08}.
Using the two-body density matrix
\begin{eqnarray}
\lefteqn{{\cal F}({\bf r}) = }&&\nonumber\\&&\!\!\!\!\!\int\!\! d^3{\bf R} \Big\langle \hat{\psi}^{\dagger}_{\uparrow}({\bf R}\!+\!\frac{{\bf r}}{2})
\hat{\psi}^{\dagger}_{\downarrow}({\bf R}\!-\!\frac{{\bf r}}{2})
\hat{\psi}_{\downarrow}({\bf R}\!-\!\frac{{\bf r}}{2})\hat{\psi}_{\uparrow}({\bf R}\!+\!\frac{{\bf r}}{2})\Big\rangle
\nonumber
\\
\label{rho2def}
\end{eqnarray}
we rewrite $\overline{V}'$ and $\overline{V}''$ in real space as
\begin{eqnarray}
\overline{V}' = \int d^3{\bf r} \; r \frac{\partial V(r)}{\partial r} {\cal F}({\bf r})
\label{moment1}
\end{eqnarray}
and
\begin{eqnarray}
\overline{V}''
= \int d^3{\bf r}\; r^2 \frac{\partial^2 V(r)}{\partial r^2} {\cal F}({\bf r}).
\label{moment2}
\end{eqnarray}
Since $V(r)$ is short-ranged, these expressions are only sensitive to the short-distance ($r_0\lesssim r \ll k^{-1}_F$) structure of the two-body density matrix. (The non-universal contribution from distances smaller than $r_0$ is assumed to be small.) For a two-component dilute Fermi gas, at these short distances, the two-body density matrix is~\cite{Zhang08}
\begin{eqnarray}
{\cal F}({\bf r}) = \frac{C}{16\pi^2}\left(\frac{1}{r}-\frac{1}{a}\right)^2.
\label{rho2}
\end{eqnarray}
Here, $C$ is the contact~\cite{Tan08,Zhang08,Braaten08} mentioned in the Introduction.
In Appendix \ref{ContactAppendix} we remind the reader how the contact $C$ governs both the short-distance
behavior of the two-body density matrix in Eq.~(\ref{rho2}), and the
large-$k$ tail of the momentum distribution function
$\lim_{k\to \infty} n_{{\bf k}\sigma} = {C}/{k^4}$.
Using integration by parts, we transform gradients of the potential $V(r)$
in Eqs.~(\ref{moment1}) and (\ref{moment2}) into gradients of the
two-body density matrix, Eq.~(\ref{rho2}). We thus find
\begin{eqnarray}
\overline{V}' = \frac{C}{4\pi}\int dr V(r)\left(-1 + {4r}/{a}\right)\label{moment1b}
\end{eqnarray}
and
\begin{eqnarray} \overline{V}'' =\frac{C}{2\pi}\int dr V(r)\left(1 - {6r}/{a}\right).\label{moment2b}
\end{eqnarray}
All that remains is to evaluate the two integrals
$X_n = C\int dr V(r) (r/a)^n/4\pi$ with $n=0,1$ in the limit
where the range of the potential $r_0 \to 0$. The Tan relations
are precisely what we need to evaluate such (possibly divergent) integrals.
The details of this analysis are described in Appendix \ref{ContactAppendix}.
We use the potential energy density~\cite{Tan08,Braaten08}
\begin{eqnarray}
\langle \hat{V} \rangle = - \frac{C \Lambda}{2\pi^2 m} + \frac{ C}{4\pi m a},
\label{eint}
\end{eqnarray}
where $\Lambda \equiv 1/r_0$ is the ultraviolet cutoff ,
and the pressure
\begin{eqnarray}
P = 2\varepsilon/3 + {C}/(12\pi m a)
\label{tanP}
\end{eqnarray}
to determine $X_0$ and $X_1$.
In deriving these results, we also use an expression for the pressure
$P$ in terms of $\varepsilon$, $\langle \hat{V} \rangle$ and $\overline{V}'$
which is derived in Appendix~\ref{Pressuresec} using the Feynman-Hellmann theorem.
Our final results for $\overline{V}'$ and $\overline{V}''$,
derived in Appendix \ref{ContactAppendix}, are
\begin{eqnarray}
\overline{V}' =
-\langle \hat{V} \rangle -2\varepsilon + 3P = C\Lambda/{2\pi^2 m}
\label{identity1}
\end{eqnarray}
and
\begin{eqnarray} \overline{V}''
= 2\langle \hat{V}\rangle + 8\varepsilon - 12P = - \frac{C \Lambda}{\pi^2 m} - \frac{ C}{2\pi ma}
\label{identity2}
\end{eqnarray}
Using these results in the general sum rules given by Eqs.~(\ref{etasumrule}) and (\ref{zetasumrule}), we
obtain the $\eta$ and $\zeta$ sum rules for the two-component dilute
Fermi gas which are valid for \emph{all} values of $1/(k_F a)$ throughout the BCS-BEC crossover,
so long as $a,k^{-1}_F\gg r_0$, and at all temperatures, both in the superfluid and normal phases, so long as $T\ll 1/mr^2_0$.
For the shear viscosity, we find
\begin{eqnarray}
\int^{\Lambda^2/m}_{0}d\omega\; \eta(\omega)/{\pi} &=& {\varepsilon}/{3} - {2\langle \hat{V} \rangle}/{5}
\nonumber
\\
&=& \frac{\varepsilon}{3} - \frac{C}{10 \pi m a} + \frac{C \Lambda}{5\pi^2 m},
\label{etasumrule2}
\end{eqnarray}
where we have imposed the energy cutoff $\Lambda^2/m = 1/mr_0^2$~\cite{uv-cutoff}.
In the zero-range limit as $\Lambda = 1/r_0 \to \infty$, the right hand side diverges.
(Strictly speaking, every physical potential has a small non-zero effective range $r_0$,
which leads to a well-defined, finite results, but one that is ``non-universal'' in that it depends
on short distance physics.) We will see in the following Section, Sec.~\ref{highfrequencytailsec},
how to make sense of this divergence and find a modified sum rule that remains finite as $r_0\to 0$.
For the bulk viscosity we find
\begin{eqnarray}
\int^{\infty}_{0}d\omega\; \zeta(\omega)/{\pi} &=& P - \varepsilon/9 - {\rho c_s^2}/{2}
\\
&=& \frac{5\varepsilon}{9} + \frac{C}{12 \pi m a} - \frac{\rho c_s^2}{2}.
\label{zetasumrule2}
\end{eqnarray}
Below the superfluid transition, the bulk viscosity $\zeta$ that enters Eq.~(\ref{zetasumrule0}) is the bulk viscosity $\zeta_2$,
as explained earlier.
We can rewrite the right hand side of this sum rule in a useful way using
simple facts about the scaling form of thermodynamic functions across the entire BCS-BEC crossover,
as described in detail in Appendix~\ref{universalthermosec}. The final result is
\begin{eqnarray}
\int^{\infty}_{0}d\omega\; \zeta(\omega)/{\pi} =
\frac{1}{72 \pi m a^2} \left( \frac{\partial C}{\partial a^{-1}} \right)_s,
\label{zetasumrule2A}
\end{eqnarray}
where the derivative is taken at constant entropy per particle $s = S/N$.
The positivity of the sum rule, given that of its integrand, implies that the contact is a
monotonically increasing function of $1/(k_Fa)$ through the BCS-BEC crossover.
We will discuss this further in Section~\ref{Crossoversec}.
\section{High-frequency tails}
\label{highfrequencytailsec}
In this Section we derive a modified shear viscosity sum rule that is manifestly finite
in the $\Lambda = 1/r_0 \to \infty$ limit. This is obtained by relating the
linear (in $\Lambda$) divergence in the sum rule, Eq.~(\ref{etasumrule2}), to a high-frequency tail
in $\eta(\omega)\sim 1/\sqrt{\omega}$, and then ``subtracting out'' the contribution of this tail.
We use ``high frequency'' or $\omega \to \infty$, to mean
$\epsilon_F \ll \omega \lesssim 1/mr_0^2$.
We also argue that a high-frequency tail of the form
$\omega^{-n/2}$, with odd integer $n$, in a variety of spectral functions
is a generic feature of short-range physics. As discussed below, it shows up in many contexts,
even outside dilute quantum gases.
We can rewrite the $\eta$ sum rule in Eq.~(\ref{etasumrule2}) as
\begin{eqnarray}
\frac{1}{\pi}\int^{\infty}_{0}\!\!d\omega \left[\eta(\omega) -\frac{C\Theta(\omega - \Omega_0)}{10\pi\sqrt{m\omega}}\right]
\nonumber \\
= \frac{\varepsilon}{3} - \frac{ C}{10\pi m a} + \frac{C}{5\pi^2}\sqrt{\frac{\Omega_0}{m}},
\label{etasumrule3}
\end{eqnarray}
where $\Omega_0$ is an arbitrary energy scale. If we choose $\Omega_0$ to be $\Lambda^2/m$
we recover Eq.~(\ref{etasumrule2}). But for any finite $\Omega_0$,
subtracting out the $\omega^{-1/2}$ tail makes the integral ultraviolet
convergent and we can take the cutoff $\Lambda$ to infinity.
If we choose $\Omega_0 = 0$, we obtain the
finite sum rule
\begin{eqnarray}
\frac{1}{\pi}\int^{\infty}_{0}\!\!d\omega \left[\eta(\omega) -\frac{C}{10\pi\sqrt{m\omega}}\right]
= \frac{\varepsilon}{3} - \frac{ C}{10\pi m a}.
\label{etasumrule3A}
\end{eqnarray}
The price we pay for using this finite, $r_0$-independent result
(in the $r_0 \to 0$ limit) is that we sacrifice the positivity
of the integrand. At sufficiently small $\omega$, we must necessarily have
$\eta(\omega) < C/(10\pi\sqrt{m\omega})$ since $\eta(0)$ is finite.
One can, in principle, exploit the freedom in Eq.~(\ref{etasumrule3}) and
choose $\Omega_0$ to be large enough so
that the integrand is always positive, however.
The finiteness of the right hand side of Eq.~(\ref{etasumrule3A})
implies that the integrand on the left must vanish at least as fast as $\omega^{-3/2}$ for the integral
to converge at large $\omega$.
Thus the asymptotic behavior of the spectral function
$\eta(\omega)$ is of the form
\begin{eqnarray}
\eta(\omega \to \infty) \simeq \frac{C}{10\pi\sqrt{m\omega}}.
\label{etahighomega}
\end{eqnarray}
We note that a high-frequency tail in the imaginary part of a
retarded correlation function which goes like
$\omega^{-n/2}$, with positive integer $n$, is a general feature of short-range two-body physics.
Suppose that for some operator $\hat{A}$, the corresponding $n$-th moment sum rule has the form
\begin{eqnarray}
\frac{1}{\pi}\int^{\infty}_0 d\omega \omega^n \mathrm{Im}\chi_{A,A}(\omega) = \alpha \langle V\rangle + \cdots,\label{sumrulediv}
\end{eqnarray}
where we only show the divergent term explicitly;
the ellipses denote regular terms. $\alpha$ is some combination of parameters and is not, in general, dimensionless.
In addition to the current correlation function ($n=1$), diverging sum rules of the form given by Eq.~(\ref{sumrulediv}) arise for the radio frequency (RF) spectral function ($n=1$)~\cite{RFsumrule}, and, as we show below, the density response function ($n=3$). Using the same reasoning as above, a divergence of the form given by Eq.~(\ref{sumrulediv})
implies a high-frequency tail.
For a dilute two-component Fermi gas with $a\gg r_0$,
the high-frequency tail is given by
\begin{eqnarray}
\mathrm{Im}\chi_{A,A}(\omega \to \infty) \simeq \frac{\alpha C}{4\pi m^{1/2}}\frac{1}{\omega^{n+1/2}}.
\end{eqnarray}
As seen from the above arguments, an $\omega^{-3/2}$ tail arises in the
radio-frequency spectroscopy response function $I(\omega)$~\cite{Schneider09,Strinati09} for Fermi gases.
Another interesting example is the $\omega^{-7/2}$ tail in the density response of a dilute Fermi gas which we derive
in Section~\ref{SqomegaSec}.
There, we also point out that an identical asymptotic behavior is found for the dense Bose liquid $^4$He, which further
emphasizes the generality of the short-distance physics in all quantum fluids.
\section{Sum rules through the BCS-BEC crossover}
\label{Crossoversec}
In this Section we consider the bulk and shear viscosity sum rules through the BCS-BEC crossover, going from the weakly attractive BCS limit
($a$ small and negative) with large Cooper pairs to the BEC limit ($a$ small and positive) with weakly interacting, tightly bound molecules.
The crossover can be traversed by changing $x = 1/(k_F a)$ from $x = -\infty$ (BCS limit) to $x = +\infty$ (BEC limit).
In experiments, the scattering length $a$ is varied by tuning a magnetic field about a Feshbach resonance. Precisely at resonance, $x=0$, the
scattering length diverges and the Fermi gas is in a very strongly interacting ``unitary regime" where the pair size is of the order of the interparticle
spacing.
To actually compute the viscosity sum rules given by Eqs.~(\ref{zetasumrule2A}) and (\ref{etasumrule3A})
for arbitrary coupling $x = 1/(k_F a)$ and temperature $T$,
we need to know the energy density $\varepsilon = n \epsilon_F {\cal{E}}(x,T/\epsilon_F)$, from which we can determine the contact $C$ as described below.
In general, we will need to use quantum Monte Carlo (QMC) data for the energy density to evaluate the sum rules.
However, as shown below, we are able to analytically constrain the bulk viscosity spectral function at unitarity.
\begin{figure}
\begin{center}
\epsfig{file=bulksumrule.eps, angle=0,width=0.47\textwidth}
\caption{(Color
online) The value of the bulk viscosity sum rule given by the right-hand side of Eq.~(\ref{zetasumrule2A}) at $T=0$ in units of $n\epsilon_F$ through the BCS-BEC crossover. }
\label{bulksumrulefig}
\end{center}
\end{figure}
We see from Eq.~(\ref{zetasumrule2A}) that the bulk viscosity sum rule vanishes at unitarity:
\begin{eqnarray}
\frac{1}{\pi}\int^{\infty}_{0}d\omega\; \zeta(\omega)= 0 \ \ \ \ (|a| = \infty).
\label{ZetaSumUnitarity}
\end{eqnarray}
We are using here the fact that $(\partial C / \partial a^{-1})_s$ is finite (i.e., non-infinite) at
$x = 1/(k_F a) = 0$ at all temperatures. One can also see this using
elementary arguments that do not involve the contact. From ``universal thermodynamics"~\cite{Ho04}, the only energy scales at unitarity are $\epsilon_F$ and the temperature, and
we can directly show that $P - \varepsilon/9- \rho c_s^2/2=0$ (see Appendix~\ref{universalthermosec}).
The vanishing sum rule, Eq.~(\ref{ZetaSumUnitarity}), together with the positivity condition
$\zeta(\omega) \geq 0$ derived in Section~\ref{posdefsec}, implies
\begin{eqnarray}
\zeta(\omega) = 0 \ \ \ \forall \omega \ \ \ (|a| = \infty).
\label{ZetaUnitarity}
\end{eqnarray}
That the \emph{static} bulk viscosity $\zeta(0)$ vanishes is a well-known
consequence~\cite{Son07} of scale or conformal invariance at unitarity~\cite{Castin04}. Our result generalizes this to arbitrary frequencies.
As discussed below in Section~\ref{SqomegaSec}, our result actually has important implications for measuring the
frequency dependent shear viscosity of a unitary Fermi gas using a density probe such as two-photon Bragg scattering.
Another general consequence of $\zeta(\omega) \geq 0$ is that its sum rule must be positive for
all $x = 1/(k_F a)$ and $T$. Equation~(\ref{zetasumrule2A}) then implies that
\begin{eqnarray}
\left(\frac{\partial C}{\partial a^{-1}}\right)_{{s}}\geq 0\;\;\;\;\forall a,
\end{eqnarray}
so that the contact must be a monotonically increasing function of $1/(k_Fa)$ through the BCS-BEC crossover
at fixed entropy per particle. We can understand this inequality intuitively as follows: the contact $C$, which
is related to the probability of finding two particles of opposite spin close to each other, can only increase with
increasing attraction $a^{-1}$.
In Fig.~\ref{bulksumrulefig} we show the bulk viscosity sum rule in Eq.~(\ref{zetasumrule2A}) at $T=0$
calculated using QMC data~\cite{Astrakharchik04} for the energy density $\varepsilon$.
The contact $C$ is obtained from $\varepsilon$ using Tan's ``adiabatic relation"~\cite{Tan08}
\begin{eqnarray}
\left(\partial \varepsilon / \partial a^{-1} \right)_{s} = - C/(4\pi m),
\label{tanA}
\end{eqnarray}
where the derivative is taken at fixed entropy per particle $s\equiv S/N$.
We fitted the QMC data and took numerical derivatives with respect to $a^{-1}$.
Since the $\zeta$ sum rule involves the second derivative of QMC data
for the energy density, the results may not be very accurate far from unitarity in either direction.
\begin{figure}
\begin{center}
\epsfig{file=shearsumrule.eps, angle=0,width=0.47\textwidth}
\caption{(Color online) The value of the finite shear viscosity sum rule, with the contribution from the high-frequency tail in Eq.~(\ref{etahighomega}) subtracted out, given by the right-hand side of Eq.~(\ref{etasumrule3A}) at $T=0$ in units of $n\epsilon_F$ through the BCS-BEC crossover.}
\label{shearsumrulefig}
\end{center}
\end{figure}
Both the vanishing of the $\zeta$ sum rule at $x= 1/(k_F a) =0$ and its positivity away from
unitarity are apparent in Fig.~\ref{bulksumrulefig}. This is due to the
$1/a^2$ dependence of the sum rule in the vicinity of unitarity.
We emphasize the nontriviality of the result given by Eq.~(\ref{zetasumrule2A}) in the unitarity region.
In the form first derived in Eq.~(\ref{zetasumrule2}), the right hand side is
$(P - \varepsilon/9- \rho c_s^2/2)$. Each term in this expression has both constant and order $x$
contributions, which must all cancel to give a final result which goes like $x^2$ at small $x$.
In the BCS limit, the $\zeta$ sum rule vanishes as $2n\epsilon_F/(27\pi|x|$) since $C\to 4\pi^2n^2a^2_s$~\cite{Tan08}. In the BEC limit, the energy density is dominated by the negative molecular binding energy, $\varepsilon\approx nE_b/2$, with $E_b = -1/(ma^2)$. Thus, $C\to 4\pi n/a$ and the sum rule grows as $n|E_b|/18$.
Next, in Fig.~\ref{shearsumrulefig}, we plot the shear viscosity sum rule given by Eq.~(\ref{etasumrule3A}) at $T = 0$
again using the QMC data of Ref.~\cite{Astrakharchik04}.
Because of the $1/\sqrt{\omega}$ subtraction extending all the way down to $\omega = 0$, the $\eta$ sum rule
in Eq.~(\ref{etasumrule3A}) is \emph{not} constrained to be positive. Using the above analytic result for the contact in the BCS limit, one finds that the $\eta$ sum rule asymptotes to $0.2 n\epsilon_F$
in the BCS limit.
At unitarity, $|a| = \infty$ and the $\eta$ sum rule is $\varepsilon/3 \simeq 0.4 \times (3 n\epsilon_F/5) \times (1/3) = 0.08 n\epsilon_F$.
On the BEC side of the resonance the sum rule changes sign, tending to $(17/30)nE_b$ in the BEC limit.
\section{Dynamic Structure Factor}
\label{SqomegaSec}
We now discuss the connection between viscosity and the density-density correlator or dynamic structure factor.
This analysis leads to two interesting results for the two-component Fermi gas. First, we predict that
a density probe such as two-photon Bragg spectroscopy~\cite{Iacuppo} can in principle be used to measure the
frequency dependent $\eta(\omega)$ at unitarity:
\begin{eqnarray}
\eta(\omega)
= \lim_{q\to 0}\frac{3\omega^3}{4 q^4}\;\mathrm{Im}\chi_{\rho\rho}({\bf q},\omega) \ \ \ \ \ (|a| = \infty).
\label{bragg}
\end{eqnarray}
Second, we derive the high-frequency tail~\cite{Son10}
\begin{eqnarray}
\lim_{\omega\to\infty}\lim_{q\to 0}S({\bf q},\omega) = \frac{2q^4C}{15 \pi^2 m^{1/2}}\frac{1}{\omega^{7/2}},
\label{sqomegatail}
\end{eqnarray}
a result that is valid for all $1/(k_F a)$ and all temperatures. As discussed below, such non-analytic
tails are also known in other strongly interacting quantum fluids like $^4$He.
We start with the operator form of the continuity equation
\begin{eqnarray}
i [\hat{\rho},\hat{H}] = m \partial_{\alpha}\hat{j}_{\alpha},
\label{continuity-op}
\end{eqnarray}
where $\hat{\rho} = m\hat{n}$ is the mass density operator, and take its matrix elements
between exact many-body eigenstates. Using the spectral representation, Eq.~(\ref{FourierchiAB}),
we relate the density correlator $\chi_{\rho\rho}$ to the the \emph{longitudinal} current correlator [see Eq.~(\ref{Nconserv})].
The latter is related to the viscosity as shown in Eq.~(\ref{KuboLR}),
namely $\zeta(\omega) + 4 \eta(\omega)/3 = \lim_{q\to 0} {m^2 \omega}\mathrm{Im}\chi_L({\bf q},\omega)/{q^2}$.
We thus obtain
\begin{eqnarray}
\zeta(\omega) + {4\eta(\omega)}/{3} = \lim_{q\to 0}{\omega^3}\mathrm{Im}\chi_{\rho\rho}({\bf q},\omega)/{q^4}.
\label{KuboLR2}
\end{eqnarray}
We discuss two situations where the contribution of $\zeta(\omega)$ vanishes and we can obtain
interesting results connecting $\eta(\omega)$ and density correlations.
First, we focus on the unitary Fermi gas where $\zeta(\omega)$ vanishes at all $\omega$ (as shown in
Section \ref{Crossoversec}) and Eq.~(\ref{KuboLR2}) simplifies to Eq.~(\ref{bragg}).
Thus, the frequency-dependent shear viscosity $\eta(\omega)$ in a unitary Fermi gas can in principle be measured
using an experiment like Bragg scattering, which directly probes $\mathrm{Im}\chi_{\rho\rho}$.
Second, let us look at the high-frequency regime $\epsilon_F \ll \omega \lesssim 1/(mr_0^2)$.
The $\zeta$ sum rule in Eq.~(\ref{zetasumrule2}) is convergent in the $r_0\to 0$ limit,
and thus $\zeta(\omega)$ must decay faster than $1/\omega$, while $\eta(\omega) \sim 1/\sqrt{\omega}$
(see Section \ref{highfrequencytailsec}). Thus, as $\omega \to \infty$, the bulk viscosity $\zeta(\omega)$
is much smaller than the shear viscosity $\eta(\omega)$ for all $1/(k_F a)$ and all $T/\epsilon_F$.
Using Eq.~(\ref{KuboLR2}) we thus find
\begin{eqnarray}
\eta(\omega \to \infty) &\simeq& \lim_{\omega\to\infty}\lim_{q\to 0}\frac{3\omega^3}{4q^4}\mathrm{Im}\chi_{\rho\rho}({\bf q},\omega)
\nonumber\\ &=&
\lim_{\omega\to\infty}\lim_{q\to 0}\frac{3\pi\omega^3}{4q^4}S({\bf q},\omega),
\label{etahighomegaSqw}
\end{eqnarray}
The dynamic structure factor $S({\bf q},\omega)$ is related to $\mathrm{Im}\chi_{\rho\rho}$ via the
fluctuation-dissipation theorem
\begin{eqnarray}
S({\bf q},\omega) = \frac{\mathrm{Im}\chi_{\rho\rho}({\bf q},\omega)}{\pi[1-\exp(-\beta\omega)]}.
\end{eqnarray}
Our final result, given by Eq.~(\ref{sqomegatail}), for the high-frequency tail of $S({\bf q},\omega)$ is obtained
by using the high-frequency tail, Eq.~(\ref{etahighomega}), of $\eta(\omega)$ in Eq.~(\ref{etahighomegaSqw}).
The high-frequency $\omega^{-7/2}$ tail of the dynamic structure factor result is a universal feature of short-range two-body interactions.
Remarkably, such a tail was first noticed in deep inelastic neutron scattering studies of superfluid $^4$He~\cite{Wong77}
and was subsequently understood in terms of hard-sphere gases~\cite{Kirkpatrick84}.
The high-frequency neutron scattering experiments probe the
short distance properties of the two-body pair distribution function.
[In dilute Fermi gases, this is directly related to the contact $C$; see Eq.~(\ref{rho2})].
It may seem surprising that such anomalous high-frequency tails arise even in dense systems like $^4$He.
Recall that this behavior should be visible in a frequency range $n^{2/3}/m < \omega < 1/mr^2_0$
in which the interaction ``looks" short-range. Even in $^4$He, where $nr^3_0\lesssim 1$, such a frequency range can be found using deep inelastic neutron scattering, although the range is obviously much smaller than in dilute gases with $nr^3_0\ll 1$.
\section{Comparison with sum rules for Relativistic Field Theories}
\label{comparisonsec}
There has been a considerable effort in the high-energy literature to understand the
properties of viscosity spectral functions and their sum rules; see, e.g., Refs.~\cite{Teaney06,Romatschke09,Moore08}).
In addition to understanding the transport coefficients within the AdS/CFT framework,
this work seems to be motivated in part by an interest in reliably extracting transport coefficients
of the quark-gluon plasma from lattice QCD calculations of Euclidean correlation functions.
We briefly discuss here some similarities and differences between the results for
relativistic quantum field theories and those derived in this paper for
non-relativistic Fermi gases: Eqs.~(\ref{zetasumrule2A}) and (\ref{etasumrule3A}).
There exist a number of Boltzmann calculations of the viscosity spectral functions in
weak coupling QCD~\cite{Teaney06,Schafer09}.
For the shear viscosity, the authors of Ref.~\cite{Schafer09} find the shear viscosity sum rule
\begin{eqnarray}
\frac{1}{\pi}\int^{\omega_c}_0d\omega \eta(\omega) = \frac{\varepsilon+P}{5},
\label{QCDshearsumrule}
\end{eqnarray}
where $g^4T\ll \omega_c \ll g^2T$ is a cutoff that removes a diverging contribution from a high-frequency tail.
For the ${\cal{N}}=4$ supersymmetric Yang-Mills theory (SYM), Romatschke and Son~\cite{Romatschke09}
derived the following shear viscosity sum rule:
\begin{eqnarray}
\frac{1}{\pi}\int^{\infty}_0 d\omega [\eta(\omega)-\eta_{T=0}(\omega)] = \frac{\varepsilon}{5}.
\label{SUSY}
\end{eqnarray}
Here, a diverging vacuum contribution from a $T$-independent high-frequency tail has been subtracted out.
We note that our $\eta$ sum rule in Eq.~(\ref{etasumrule3A}), though similar in structure, has one key difference.
The high-frequency tail for the Fermi gas is in general $T$-\emph{dependent}, because its coefficient is
set by the contact $C = k_F^4 {\cal C}[1/(k_F a), T/\epsilon_F]$.
A non-perturbative calculation of the bulk viscosity sum rule in ${\cal{N}}=4$ supersymmteric Yang-Mills theory and pure Yang-Mills theory (QCD with no quarks) has been given recently by Romatschke and Son~\cite{Romatschke09}:
\begin{eqnarray}
\lefteqn{\frac{1}{\pi}\int^{\infty}_0 d\omega [\zeta(\omega)-\zeta_{T=0}(\omega)] =}&&\nonumber\\&&(3\varepsilon+P)(1-3c^2)- 4(\varepsilon-3P),
\label{QCDbulksumrule}
\end{eqnarray}
where $c\equiv \sqrt{\partial P/\partial \varepsilon}$ is the sound speed in relativistic hydrodynamics (with the speed of light equal to unity)~\cite{LLFM}. There are some differences and one very
interesting similarity with our $\zeta$ sum rule in Eq.~(\ref{zetasumrule2A}). In contrast to the Fermi gas spectral function $\zeta(\omega)$, there is a need
to subtract out a divergent tail in Eq.~\ref{QCDbulksumrule} and this tail appears to be $T$-independent.
The interesting similarity is that in the ``conformal limit" $P = \varepsilon/3$, the right hand
side of Eq.~(\ref{QCDbulksumrule}) vanishes, analogous to the unitary Fermi gas.
\section{Conclusions}
\label{conclusionssec}
In this paper we have derived various exact, non-perturbative results for the shear and bulk viscosities
of non-relativistic quantum fluids, focusing on the strongly interacting Fermi gas. Our main results were already
summarized in the Introduction. To conclude, we discuss some open questions and how our results relate to them.
Most calculations~\cite{Bruun05,Schafer07} of the viscosity in strongly interacting Fermi gases
have so far been restricted to solving Boltzmann equations or using diagrammatic perturbation theory, in essence making a
quasiparticle approximation. Such results are valid in the high and low temperature regimes, but
not in the most interesting regime near and above $T_c$ where a quasiparticle approximation is questionable
and the shear viscosity is known to be the smallest. It was recognized some time back~\cite{Randeria-reviews}
that there is a breakdown of Fermi liquid theory in the normal (i.e., non-superfluid) state of the
strongly interacting regime of the BCS-BEC crossover. It was shown that precursor pairing correlations
lead to a pseudogap \cite{Randeria-reviews}, which is a strong suppression of low-energy spectral weight
in various response functions. It is likely that no sharp quasiparticle excitations exist in this regime
near unitarity and just above $T_c$, but controlled calculations of dynamic quantities are very difficult.
Quantum Monte Carlo methods have played an important role in determining the equilibrium thermodynamic properties
of the unitary Fermi gas. However, results for transport coefficients are much less common, since they require
analytic continuation of imaginary time (Euclidean) data to the real axis~\cite{Aarts02}.
The sum rules we derive could serve as useful constraints on similar calculations for
strongly interacting Fermi gases.
From an experimental point of view, the (static) shear viscosity for strongly interacting Fermi gases
has been estimated from studies of the damping of collective oscillations~\cite{Turlapov08}. We have shown
above that, at unitarity, the full frequency dependence of the shear viscosity spectral function
$\eta(\omega)$ can be obtained from two-photon Bragg spectroscopy. While it would be a challenging experiment (the density response being very small for small-$q$), this would give extremely important insights
into the strongly interacting Fermi gas, analogous to optical conductivity measurements of solids.
Finally, we return to the conjectured bound~\cite{Son-review} on the shear viscosity, Eq.~(\ref{bound}).
Proving or disproving the existence of a bound~\cite{bound} for non-relativistic quantum fluids like the strongly interacting Fermi gas
remains a challenging open problem. We hope that the spectral functions and sum rules derived here
constitute a step in this direction, just as they have for other well known inequalities
in quantum many-body physics.
\begin{acknowledgments}
ET would like to thank Shizhong Zhang, Georg Bruun, Joaqu\'in Drut, Vijay Shenoy, and Jason Ho for stimulating discussions.
MR would like to thank the participants at the International Conference on Recent Progress in Many Body Theories
(RPMBT15) last summer for spurring his interest in this problem. We thank Eric Braaten, Dam Son, and Sandip Trivedi for comments on the manuscript and Stefano Giorgini for sharing with us the Monte-Carlo data of Ref.~\cite{Astrakharchik04}.
We gratefully acknowledge support from NSF-DMR 0706203, ARO W911NF-08-1-0338, and NSF-DMR 0907366.
\end{acknowledgments}
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\def{Integrability of moduli and regularity of Denjoy counterexamples}{{Integrability of moduli and regularity of Denjoy counterexamples}}
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\begin{document}
\title{Integrability of moduli and regularity of Denjoy counterexamples}
\date{\today}
\keywords{Exceptional circle diffeomorphism, moduli of continuity, length spectrum}
\subjclass[2010]{Primary: 37E10; Secondary: 37C05, 37C15, }
\author[S. Kim]{Sang-hyun Kim}
\address{School of Mathematics, Korea Institute for Advanced Study (KIAS), Seoul, 02455, Korea}
\email{[email protected]}
\urladdr{http://cayley.kr}
\author[T. Koberda]{Thomas Koberda}
\address{Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA}
\email{[email protected]}
\urladdr{http://faculty.virginia.edu/Koberda}
\begin{abstract}
We study the regularity of exceptional actions of groups by $C^{1,\alpha}$ diffeomorphisms on the circle, i.e. ones
which admit exceptional minimal sets, and whose elements have first derivatives that are continuous with
concave modulus of continuity $\alpha$.
Let $G$ be a finitely generated group admitting a $C^{1,\alpha}$
action $\rho$ with a free orbit on the circle, and such that the logarithms of derivatives of group elements
are uniformly bounded at some point of the circle.
We prove that if $G$ has spherical growth bounded by $c n^{d-1}$ and if the function $1/\alpha^d$ is integrable near zero,
then under some mild technical assumptions on $\alpha$, there is a sequence of exceptional $C^{1,\alpha}$ actions of $G$
which converge to $\rho$ in the $C^1$ topology. As a consequence for a single diffeomorphism, we obtain that
if the function $1/\alpha$ is integrable near zero, then there exists a $C^{1,\alpha}$ exceptional
diffeomorphism of the circle. This corollary accounts for all previously known moduli of continuity for derivatives of
exceptional diffeomorphisms.
We also obtain a partial converse to our main result.
For finitely generated free abelian groups, the existence of an exceptional
action, together with some natural hypotheses on the derivatives of group elements, puts integrability restrictions on the modulus
$\alpha$. These results are
related to a long-standing question of D. McDuff concerning the length spectrum of exceptional $C^1$ diffeomorphisms of the circle.
\end{abstract}
\maketitle
\section{Introduction}
Let $f\in\Homeo^+(S^1)$ be an orientation preserving homeomorphism of the circle without any periodic points. It is well-known that in this
case $f$ has an irrational rotation number $\theta$. Here, we make the identification $S^1=\bR/\bZ$, and we have the rotation number
\[\mathrm{rot}(f)=\lim_{n\to\infty}\frac{F^n(x)-x}{n}\pmod 1\in\bR/\bZ,\] where $F$ is any lift of $f$ to $\bR$ and $x\in\bR$ is arbitrary. It is well-known
that $\mathrm{rot}(f)$ is independent of $x$ and of the choice of a lift.
In this introduction, we shall always assume that $f\in\Homeo^+(S^1)$ has irrational rotation number $\theta$
unless otherwise noted. A standard
fact going back to Poincar\'e
asserts that if $f$ has a dense orbit, then $f$ is topologically conjugate to an irrational rotation by $\theta$.
If $f$ does not have a dense orbit then
it must have a \emph{wandering interval}, which is to say a nonempty interval $J$ such that $f^n(J)\cap J=\varnothing$ for $n\neq 0$.
A classical result of Denjoy asserts that if $f$ is twice differentiable (or if in fact the logarithm of the
derivative of $f$ has bounded variation, equivalently just the derivative of $f$ has bounded variation),
then $f$ is topologically conjugate to a rotation by $\theta$. In lower levels of regularity, this fact
ceases to hold. It is easy to produce continuous examples which are not topologically conjugate to a rotation, and Denjoy showed that one
can construct differentiable examples for every irrational rotation number $\theta$.
Examples of this type were also know to Bohl~\cite{Bohl1916}.
Such examples will be called \emph{exceptional} diffeomorphisms, since they have a so-called
exceptional minimal set, which in this case will be homeomorphic to a Cantor set (see Theorem 2.1.1 of~\cite{Navas2011}).
In general, an \emph{exceptional group action} is one which admits an exceptional minimal set.
In what follows, we will consider not just single diffeomorphisms, but also non-cyclic groups acting with exceptional minimal sets on the circle,
and the regularity which can be required of such actions.
\subsection{Main results}
In this paper, we consider the problem of determining which moduli of continuity can be imposed on the derivatives of elements
in exceptional diffeomorphism groups of the circle.
A \emph{(concave) modulus of continuity} (or, \emph{concave modulus}) is a homeomorphism
\[\alpha\colon[0,\infty)\to[0,\infty)\]
that is concave as a map.
We say a function $g\colon S^1\to\bR$ is \emph{$\alpha$--continuous} if it satisfies \[\sup_{x\ne y} \frac{|g(x)-g(y)|}{\alpha(|x-y|)}<\infty.\] Here, we interpret $|x-y|$ by identifying $S^1$ with $\bR/\bZ$
and computing this difference modulo $1$.
The value of this supremum
is sometimes called the $\alpha$--norm of $g$, and is written $[g]_{\alpha}$.
Conversely, every continuous map $g$ on $S^1$ is $\alpha$--continuous for some concave modulus $\alpha$; see~\cite{CKK2019} for instance.
A $C^1$--diffeomorphism $f\co S^1\to S^1$ is said to be $C^{1,\alpha}$ if $f'$ is $\alpha$--continuous. One commonly denotes by $\Diff_+^{1,\alpha}(S^1)$ the set of orientation--preserving $C^{1,\alpha}$ diffeomorphisms of $S^1$.
It follows from the concavity of $\alpha$ that $\Diff_+^{1,\alpha}(S^1)$ is a group.
Note that if $\alpha(x)=x$, then $\alpha$--continuity is just Lipschitz continuity. More generally, if $\alpha(x)=x^{\tau}$ for some $0<\tau\leq 1$
then $\alpha$--continuity is $\tau$--H\"older continuity. For every $0<\tau< 1$, it is known that there exist $C^{1,\tau}$ exceptional
diffeomorphisms of the circle for arbitrary $\theta$ (see~\cite{Herman1979}, also Theorem 3.1.2 of~\cite{Navas2011}). In fact,
Herman~\cite{Herman1979} proved that for arbitrary $\theta$ there exist $C^{1,\alpha}$ exceptional diffeomorphisms of the circle for
\[\alpha(x)=x(\log{1/x})^{1+\epsilon}\] for all $\epsilon>0$, which implies the corresponding conclusion for H\"older moduli. Moreover,
he proved that such diffeomorphisms can be chosen arbitrarily $C^1$--close to a rotation by $\theta$, and with
uniformly bounded $C^{1,\alpha}$--norms.
On the other hand, he showed that if $\theta$ satisfies a certain strong Diophantine hypothesis then for
\[\alpha(x)=x\left(\log\log\log{1/x}\right)^{1-\epsilon},\]
every $f\in\Diff_+^{1,\alpha}(S^1)\cap\rot^{-1}(\theta)$ is topologically conjugate to a rotation~\cite[X.4.4]{Herman1979}.
It is a well-known open problem to determine whether or not there exist $C^{1,\alpha}$ exceptional diffeomorphisms
for \[\alpha(x)=x\log{1/x}.\]
Before stating our results, we introduce some notation and terminology.
For a single diffeomorphism $f$, we consider a maximal wandering set of open intervals for $f$ in the circle and write them
as $\{J_i\}_{i\in\bZ}$. These intervals are characterized by
the property that $f(J_i)=J_{i+1}$, and that any interval properly containing $J_i$ for any index
$i$ must meet the exceptional minimal set of $f$. We will write $\ell_i$ for the length of the interval $J_i$, and we will
oftentimes refer to the collection
$\{J_i\}_{i\in\bZ}$ as a \emph{maximal wandering set}. Observe that a maximal wandering set may not be unique. An exceptional
diffeomorphism of the circle will necessarily have a wandering interval.
The discussion in this paragraph also generalizes to arbitrary group actions.
An exceptional action $\rho$ necessarily admits a wandering interval,
which in this case is an interval $J$ such that every pair of intervals in $\{g J\}_{g\in G}$ are either equal or disjoint.
\begin{thm}\label{t:c1al-intro}
Let $d\in\bN$, and let $\alpha$ be a concave modulus satisfying
\[
\int_0^1 1/\alpha(x)^ddx<\infty.\]
If $d>1$, we further assume \[\sup_{0<y<1} \alpha\!\left( y^{d+1}/\alpha(y)^d\right)/y<\infty.\]
Suppose that a finitely generated group $G$ admits an action
$\rho\co G\to \Diff_+^{1,\alpha}(S^1)$ with a free orbit $\rho(G).0$ such that \[\sup_{g\in G} \abs*{\log g'(0)}<\infty.\]
Suppose furthermore that the spherical growth function of $G$ is at most $c n^{d-1}$ for some $c>0$.
Then $\rho$ admits a sequence of exceptional actions
\[\{\rho_k\co G\to\Diff_+^{1,\alpha}(S^1)\}\]
such that $\rho_k\to \rho$ in the $C^1$--topology. Moreover, we may require that
\[\sup_k\{[\rho_k(g)]_{\alpha}\}<\infty\] for all $g\in G$.
\end{thm}
In the statement of Theorem~\ref{t:c1al-intro}, the spherical growth function
of a finitely generated group denotes the number of elements of length exactly $n$ with respect to some
generating set. We remark that finite $G$--orbits are allowed under the hypotheses of the theorem.
For a single diffeomorphism (i.e. $G=\bZ$), we have the following immediate consequence of Theorem~\ref{t:c1al-intro}:
\begin{thm}\label{t:int-alpha}
Let $\theta$ be arbitrary, and suppose that we have \[\int_0^1\frac{1}{\alpha(x)}\,dx<\infty.\]
Then there exists a $C^{1,\alpha}$ exceptional diffeomorphism $f$ of $S^1$ with rotation number $\theta$. Moreover, we may arrange for $f$ to be arbitrarily $C^{1}$--close to a rotation by $\theta$.
\end{thm}
Theorem~\ref{t:int-alpha} recovers all previously known possible moduli of continuity for derivatives of
exceptional diffeomorphisms of the circle.
In particular, Theorem~\ref{t:int-alpha} recovers the fact
that there are $C^{1,\tau}$ exceptional diffeomorphisms for every irrational rotation number for
all $0<\tau<1$, as well as Herman's corresponding result for $\alpha(x)=x(\log{1/x})^{1+\epsilon}$. Moreover, we can immediately assert the existence
of exceptional $C^{1,\alpha}$ diffeomorphisms for moduli which were previously unrecorded in the literature, such as
\[\alpha(x)=x(\log{1/x})\left(\log\log{1/x}\right)^{1+\epsilon}.\]
The next result is an attempt to control the integrability properties of the modulus, given the existence of an exceptional action. In the
statement, a semi-conjugacy is a monotone, surjective, degree one map which
intertwines two actions of a group; see Section~\ref{sec:blowup}
below.
\begin{thm}\label{t:al-inv-intro}
Let $\alpha$ be a concave modulus, and let
\[\rho\co \bZ^d\to \Diff_+^{1,\alpha}(S^1)\]
be an exceptional action.
Suppose that $\rho$ is semi-conjugate to a faithful action
\[\bar\rho\co\bZ^d\to\Diff_+^1(S^1)\]
by a semi-conjugacy map $H$ such that $\bar\rho$ is $C^1$--conjugate into $\SO(2,\bR)$.
Assume for each generator $s$ of $G$ and $x$ in the exceptional minimal set of $\rho$ we have that
\[ \rho(s)'(x) = \bar\rho(s)'\circ H(x).\]
Then we have that \[\int_0^1 \alpha^{-1}(t)/t^{d+1}\;dt<\infty.\]
\end{thm}
The next result is a partial converse to Theorem~\ref{t:int-alpha}, which follows easily from Theorem~\ref{t:al-inv-intro} in the case of
a single diffeomorphism.
\begin{thm}\label{t:alpha-inv}
Suppose $f$ is an exceptional $C^{1,\alpha}$ diffeomorphism, and suppose that $\ell_{i+1}/\ell_i\to 1$ as $i\to\infty$. Then there is a positive
constant $A>0$ such that \[\frac{A}{i}\leq \alpha(\ell_i)\] for all $i>0$.
\end{thm}
Theorem~\ref{t:alpha-inv} may be compared with Lemma 3.1.3 of~\cite{Navas2011}.
We immediately obtain the following consequence of Theorem~\ref{t:alpha-inv}, by a straightforward change of variables.
\begin{cor}\label{cor:int-alpha-inv}
Suppose $f$ is an exceptional $C^{1,\alpha}$ diffeomorphism, and suppose that $\ell_{i+1}/\ell_i\to 1$ as $i\to\infty$. Then
\[\int_0^1\frac{\alpha^{-1}(t)}{t^2}\, dt<\infty.\]
\end{cor}
Notice that since $\alpha$ is a homeomorphism of the non-negative reals, we may make sense of the notation $\alpha^{-1}$ as a function.
Theorem~\ref{t:alpha-inv} recovers the fact that if $\ell_{i+1}/\ell_i\to 1$ as $i\to\infty$, then $f'$ cannot be $\alpha$--continuous for
$\alpha(x)=x\log{1/x}$ (see Exercise 4.1.26 and the examples in section 4.1.4 of~\cite{Navas2011}).
The final main result is as follows, and provides a partial converse to Theorem~\ref{t:al-inv-intro}.
\begin{thm}\label{t:int-comparison-intro}
Let $d$ be a positive integer, and let $\alpha$ be a concave modulus such that
\[\int_0^1\frac{\alpha^{-1}(t)}{t^{d+1}}\, dt<\infty,\] and such that \[\sup_{t> 0}\frac{\alpha(t)}{t\alpha'(t)}<\infty.\]
If $\rho\co \bZ^d\to\Diff_+^{1,\alpha}(S^1)$ is a faithful representation that is $C^1$--conjugate into the rotation group $\SO(2,\bR)$, then there exists a sequence
\[\rho_k\co \bZ^d\to\Diff_+^{1,\alpha}(S^1)\]
of exceptional actions
which converges to $\rho$ in the $C^1$--topology.\end{thm}
\begin{rem}\label{rem:alpha-diff}
Here, the assumption that $\alpha$ is differentiable does not result in any loss of generality~\cite{Medvedev2001}. We also note that the
supremum in Theorem~\ref{t:int-comparison-intro} is bounded below by $1$ as follows from the standard concavity estimate
$t\alpha'(t)\leq\alpha(t)$.\end{rem}
The specialization of Theorem~\ref{t:int-comparison-intro} to a single diffeomorphism is as follows.
\begin{thm}\label{t:int-alpha-conv}
Let $\theta\in\bR\setminus\bQ$ be arbitrary. Suppose that \[\int_0^1\frac{\alpha^{-1}(t)}{t^2}\, dt<\infty,\] and that \[\sup_{t> 0}\frac{\alpha(t)}{t\alpha'(t)}<\infty.\]
Then there exists a $C^{1,\alpha}$ exceptional diffeomorphism $f$ with rotation number $\theta$, and such that $\ell_{i+1}/\ell_i\to 1$
as $i\to\infty$.
\end{thm}
Theorems~\ref{t:int-comparison-intro} and~\ref{t:int-alpha-conv} are close being converses to Theorems~\ref{t:c1al-intro}
and~\ref{t:int-alpha}, since the supremum hypotheses on $\alpha$ imply that $1/\alpha^d$ and $1/\alpha$ are integrable near zero,
respectively.
\subsection{Denjoy counterexamples beyond moduli of continuity}
Identifying the precise conditions under which a even a single
homeomorphism $f\in\Homeo^+(S^1)$ with irrational rotation number is necessarily
topologically conjugate
to an irrational rotation is tantalizing. Some regularity of $f^{-1}$ is necessary, as was demonstrated by Hall~\cite{Hall81}. Even a
hypothetical characterization
of moduli of continuity for which there exist exceptional diffeomorphisms does not appear to be the end of the discussion.
For instance, Sullivan proved that if the logarithm of the derivative of $f$ satisfies the Zygmund condition (also called the ``big"
Zygmund condition), then $f$ is topologically conjugate
to an irrational rotation~\cite{Sullivan1992}.
Note that the Zygmund condition for $\log f'$ implies that $f'$ is $\alpha$--continuous with $\alpha(x) = x\log 1/x$.
More generally, Hu and Sullivan~\cite{HuSu1997} show that if the derivative of $f$ has bounded quadratic variation and
bounded Zygmund variation, then $f$ is topologically conjugate to an irrational rotation. In the same paper, they show that if the
logarithm of the cross ratio distortion of $f$ has bounded variation, then $f$ cannot have any wandering intervals and hence must by
topologically conjugate to an irrational rotation.
None of these conditions on $f$ or $f'$ seem to be
expressible in terms of moduli of continuity for $f'$.
\subsection{Remarks on the structure of the paper}
Throughout this paper, we strive for generality insofar as it is possible. However,
we have taken steps to make the exposition easier to follow for
readers who are not interested in general group actions but rather in single diffeomorphisms.
Section~\ref{sec:blowup} discusses generalities about finitely generated group actions on the circle, blowups of such actions,
and constructions of exceptional diffeomorphism actions by finitely generated groups. At first, general group actions are
discussed, though later the subexponential growth hypothesis comes into play, culminating in Corollary~\ref{c:stab0} and
the rather technical Theorem~\ref{t:c1al}.
Section~\ref{s:bounding} establishes Theorem~\ref{t:alpha-inv} and Theorem~\ref{t:int-alpha-conv}, which further clarify the relationship
between the lengths of wandering intervals and the possible moduli of continuity of exceptional diffeomorphisms of the circle.
The content of Section~\ref{s:bounding} applies to single diffeomorphisms and not to general group actions.
Finally, Appendix~\ref{s:appendix} proves Theorem~\ref{t:int-alpha} directly, without the added layers of
technical difficulty coming from the relaxed
hypotheses of Theorem~\ref{t:c1al}. The content of the appendix again applies to single diffeomorphisms as opposed to general group
actions, and may be used by the reader both as a self-contained discussion and
as a guide to understand the intuition behind the proof of Theorem~\ref{t:c1al}.
\section{Blowing-up finitely generated groups}\label{sec:blowup}
Throughout this section, we assume that
\[
\rho\co G\to\Homeo^+(S^1)\]
is an action of a finitely generated group $G$.
We often suppress $\rho$ in expressions involving group actions; that is, when the meaning is clear, we will simply write
\[
gx=g(x) = \rho(g)(x), \quad
g'(x) =\rho(g)'(x).\]
The reader may imagine that $\rho$ is injective for the ease of reading.
Another action $\tilde \rho$ of $G$ is said to be \emph{semi-conjugate} to $\rho$ if there exists a
monotone surjective (hence continuous) degree--one map $H$ of $S^1$ such that
\[ H\circ \tilde\rho(g) = \rho(g)\circ H\]
for all $g\in G$.
We sometimes call $H$ a \emph{semi-conjugacy map} from $\tilde\rho$ to $\rho$.
The semi-conjugacy is not an equivalence relation; however, one may define an equivalence
relation on the set of representations $\Hom(G,\Homeo_+(S^1))$
by declaring that $\rho_0$ and $\rho_1$ are \emph{semi-conjugate} if they have a ``common blow-up''
in a certain natural sense~\cite[Definition 2.3 and Theorem 2.2]{KKM2019}; see also~\cite{Ghys1999, CD2003IM, BFH2014}.
It is essentially due to Poincar\'e that every countable group action is semi-conjugate to a minimal one, and also that a
semi-conjugacy preserves the rotation numbers.
If $H$ is a homeomorphism, then the actions $\tilde\rho$ and $\rho$ are conjugate; otherwise, we call $\tilde\rho$
a \emph{nontrivial blow-up} of $\rho$ at the $\rho(G)$--equivariant set
\[
\{ x\in S^1\mid H^{-1}(x)\text{ is not a singleton }\}.\]
If $\rho$ does not admit a finite orbit, then every nontrivial blow-up of $\rho$ is exceptional.
For a single minimal diffeomorphism of the circle, there is a standard technique of producing
nearby exceptional diffeomorphisms in the regularity $C^0$ due to Denjoy~\cite{Denjoy1932},
and also in $C^1$ and $C^{1,\alpha}$ due to Herman~\cite{Herman1979}; see
also~\cite{Tsuboi1995, DKN2007} for free abelian groups.
In this section, we will describe a common framework to produce exceptional actions of
finitely generated groups from a given one with various regularities.
This will generalize the aforementioned result of Herman. The reader is referred to the appendix for a discussion of Herman's construction
in the case of a single diffeomorphism.
\subsection{Exceptional $C^0$--actions}
The following $C^0$--blow-up process is well-known~\cite{Denjoy1932,Herman1979}.
We include a proof for the purpose of introducing some notation which will be useful in the sequel.
\begin{prop}\label{p:c0}
Let $\rho$ and $G$ be as above.
Then for each orbit $\mathcal{O}$ of $\rho$, there exists a sequence $\{\rho_k\}$
of blow-ups of $\rho$ at $\mathcal{O}$
which converges to $\rho$ in $C^0$--topology. If $\rho$ is minimal then each $\rho_k$ is exceptional.
\end{prop}
\bp Let us fix a finite generating set $S$ of $G$.
For brevity, we may assume \[\mathcal{O}=\rho(G).0.\]
We will choose the following parameters:
\begin{itemize}
\item a positive sequence $\{\ell_y\}_{y\in\mathcal{O}}$ such that $L:=\sum_{y\in\mathcal{O}}\ell_y\le 1$;
\item a family of orientation-preserving homeomorphisms
\[\{\eta_{x,y,g}\co [0,\ell_x]\to[0,\ell_y] \mid {x,y\in\mathcal{O}}\text{ and }g\in G\text{ such that }y=gx\}\]
which is equivariant in the sense that
\[
\eta_{gx,hgx,h}\circ\eta_{x,gx,g}=\eta_{x,hgx,hg}.\]
\end{itemize}
Note that we do \emph{not} require $\eta_{x,y,g}=\eta_{x,y,h}$ even when $gx=y=hx$; however, for the purpose of
the proof of the present proposition, the reader may imagine that $\eta$ is linear:
\[\eta_{x,y,g}(t)=t\cdot \ell_y / \ell_x.\]
We will replace each orbit point $y\in\mathcal{O}\sse S^1$ by a closed interval of length $\ell_y$, whose interior will be denoted by
$I_y$. More formally, we define a measure
\[
d\lambda := (1-L)dm+\sum_{y\in \mathcal{O}} \ell_y d\delta_{y},\]
where $dm$ and $d\delta$ denote the Lebesgue and the Dirac measures, respectively. We let
\[
a_y:=\lambda[0,y),\ b_y:=\lambda[0,y],\]
and define $I_y:=(a_y,b_y)$ for $y\in\mathcal{O}$.
There exists a unique, natural, monotone, surjective, degree--one map
\[H\co S^1\to S^1\] satisfying $H(I_y)=\{y\}$ and $\mu(A)=m(H^{-1}A)$ for each Borel set $A$. That is, $\mu$ is the \emph{image}
(or push-forward) of Lebesgue measure by the map $H$.
Notice that we are treating the unit circle $S^1=\bR/\bZ$ as two different objects. One is the range of $H$, which is the original circle containing $\mathcal{O}$; the other is the domain of $H$, where each $I_y\in[0,1]$ is contained as an open interval.
Since $H$ is 1-1 on $K:=S^1\setminus\coprod_{y\in\mathcal{O}} I_y$, for each $g\in G$ we have a circle homeomorphism
\[
\tilde\rho(g)(t):=
\begin{cases}
a_{gy}+\eta_{y,gy,g}(t-a_y),&\text{ if }t\in\bar I_y,\\
H^{-1}\circ\rho(g)\circ H(t),&\text{ if }t\in K.\end{cases}\]
The equivariance assumption on $\eta$ implies that $\tilde\rho$ is a group action. If $\rho$ is minimal, then $K$ is a Cantor set and $\tilde\rho$ is exceptional.
For each $k\in\bN$ we repeat the above process for the parameters $\{\ell_y^k\}_{y\in\mathcal{O}}$ with $\lim_k \ell_y^k=0$.
For instance, one may choose $\ell_y^k=\ell_y/k$
and arrange $\eta^k_{x,y,g}$ accordingly.
We let $\rho_k$ denote the resulting blow-up, with a semi-conjugacy map $H_k$.
Let $\epsilon>0$.
Since $H_k\to \Id$ uniformly,
for each $s\in S$, and for all $k\gg0$, we obtain
\[
\|\rho_k(s) - \rho(s)\|_\infty
<\|\rho_k(s) - \rho(s)\circ H_k\|_\infty+\epsilon
=\|\rho_k(s) - H_k\circ\rho_k(s)\|_\infty+\epsilon
<2\epsilon.\]
This completes the proof.\ep
\subsection{Exceptional $C^1$--actions}
We will now generalize Herman's construction for a single $C^1$--diffeomorphism to general finitely generated groups (cf. appendix of this
paper).
For this, we follow the proof of Proposition~\ref{p:c0} while making a ``smoother'' choice of the equivariant family $\{\eta_{x,y,g}\}$.
For a countable set $A$ and a real function $f\co A\to\bR$, we will write $\lim_{a\in A}f(a)=L$ if for an arbitrary enumeration $A=\{a_i\}$,
we have $\lim_i f(a_i)=L$. The reader may simply imagine that $\{f(a_i)\}_i$ is monotone decreasing in most cases.
\begin{thm}\label{t:c1}
Let $G$ be a finitely generated group, and let $\rho\co G\to\Diff_+^1(S^1)$ be an action with a choice of an orbit $\mathcal{O}$.
Assume that there exist positive sequences $\{\ell^k_y\}_{y\in\mathcal{O}}$ for $k\in\bN$ such that
\[\lim_{k\to\infty}\sum_{y\in\mathcal{O}} \ell_y^k=0\]
and such that
\[\lim_{k\to\infty} s'(y)\ell^k_y/ \ell^k_{sy}=1=\lim_{y\in \mathcal{O}} s'(y)\ell^k_y/ \ell^k_{sy}\]
for each generator $s$ of $G$.
Then there exists a sequence \[\{\rho_k\co G\to\Diff_+^1(S^1)\}\]
of blow-ups of $\rho$ at $\mathcal{O}$
which converges to $\rho$ in the $C^1$--topology.
\end{thm}
We begin with defining a new equivariant family.
For $a,A>0$, we let
\[\psi(a,A)(t):=-\cot(\pi t/a)/A\co (0,a)\to\bR.\]
We then have
\[\psi(a,A)^{-1}(t)=a/2 + a/\pi \cdot \arctan(At)\]
For $a,A,b,B>0$, we obtain a homeomorphism
\[
\phi(a,b,B/A):=\psi(b,B)^{-1}\circ\psi(a,A)\co [0,a]\to[0,b].\]
A special case of the family $\{\phi(a,b,b/a)\}_{a,b>0}$ is due to Yoccoz and
is used in~\cite{DKN2007,Navas2008GAFA}. See also~\cite{FF2003}, and ~\cite{Jorquera} for an improvement on Farb--Franks' result.
For each $R>0$, we define \[
\xi(R)(t):= \frac{1+R}{1+R^2\cot^2(\pi t)}\cdot\chi_{[0,1]}(t) .\]
For a fixed $R>0$, the function $\xi(R)(x)$ on $x$ is a $C^1$--map supported on $[0,1]$.
\begin{lem}\label{l:Yoccoz}
The following hold for $a,b,c,R,S>0$.
\be
\item $\phi$ is equivariant: $\phi(b,c,S)\circ\phi(a,b,R)=\phi(a,c,RS)$.
\item $\int_\bR \xi(R)(t)dt=1$.
\item For all $t\in[0,a]$ we have that
\[
\phi(a,b,R)'(t) =\frac{b}{aR} \cdot (1-(1-R)\xi(R)(t/a)).\]
\ee
\end{lem}
\bp The proof is straightforward, and we omit the details.\ep
\bp[Proof of Theorem~\ref{t:c1}]
For brevity, fix $k>0$ and let $\ell_y:=\ell^k_y$.
We let $H$ be the semi-conjugacy map determined by the parameter $\{\ell_y\}$ as in Proposition~\ref{p:c0}.
Set
\[
\lambda(g)(t):=g'\circ H(t)
\cdot
\left(1 - \sum_{y\in\mathcal{O}} \left(1 - \frac{\ell_{gy}}{g'(y)\ell_y}\right)\xi\left(\frac{\ell_{gy}}{g'(y)\ell_y}\right)\left(\frac{t-a_y}{\ell_y}\right)\right).\]
For each $t\in I_y=(a_y,b_y)$, we note that
\[\lambda(g)(t)= \phi\left(\ell_y,\ell_{gy},\frac{\ell_{gy}}{g'(y)\ell_y}\right)'
(t-a_y).\]
\begin{claim*}\label{c:c1-blowup}
For each $g\in G$, we have the following.
\be
\item\label{l:lam1}
The map $\lambda(g)\co S^1\to\bR$
is positive and continuous for each $g\in G$.
\item\label{l:lam2}
For each $y\in \mathcal{O}$, we have
$\int_{I_y} \lambda(g) dm = m(I_{gy})$.
\item\label{l:lam3}
For each $x,y\in \mathcal{O}$, we have
$\int_{[a_x,a_y]} \lambda(g) dm = m[a_{gx},a_{gy}]$.
\item\label{l:lam4} $\int_{S^1} \lambda(g)dm = 1$.
\ee
\end{claim*}
\bp[Proof of the Claim]
(\ref{l:lam1})
The positivity of $\lambda$ is obvious.
The uniform convergence theorem,
along with the estimate below
implies that $\lambda(g)$ is continuous:
\[
\lim_{y\in\mathcal{O}}
\left|\left(
1 - \frac{\ell_{gy}}{g'(y)\ell_y}
\right)
\xi
\left(\frac{\ell_{gy}}{g'(y)\ell_y}\right)\left(
\frac{x-a_y}{\ell_y}\right)\right|
\le
\lim_{y\in\mathcal{O}}
\left|
1 - \left(\frac{\ell_{gy}}{g'(y)\ell_y}\right)^2
\right|=0.
\]
(\ref{l:lam2})
Setting $R = (\ell_{gy}/\ell_y)/g'(y)$, we have
$
\int_{I_y}\lambda(g) =
\phi\left(\ell_y,\ell_{gy},R\right)(\ell_y)=\ell_{gy}$.
(\ref{l:lam3})
Recall we have set $K := S^1\setminus\coprod_w I_w$.
We note that
\[\int_{[a_x,a_y]\setminus K} \lambda(g) dm =
\sum_{w\in [x,y)\cap \mathcal{O}} \int_{I_w} \lambda(g)dm
=\sum_{w\in [x,y)\cap \mathcal{O}} \ell_{g(w)}
=\mu\left(g[x,y) \cap\mathcal{O}\right).\]
Recall $d\mu$ is the image of the Lebesgue measure $dm$.
A simple application of Riesz representation theorem as in~\cite[Theorem 1.19]{Mattila1995} shows that
\[
\int_{[a_x,a_y]\cap K} \lambda(g) dm =
\int_{H^{-1}([x,y)\setminus\mathcal{O} )} g'\circ H dm=\int_{[x,y) \setminus\mathcal{O}} g' d\mu
=\mu(g[x,y)\setminus\mathcal{O}).
\]
Summing up the above integrals, we obtain the desired conclusion.
The proof of part (\ref{l:lam4}) is now immediate.
\ep
Consider the equivariant family
\[
\eta_{y,gy,g} := \phi\left(\ell_y,\ell_{gy},\frac{\ell_{gy}}{g'(y)\ell_y}\right)\co [0,\ell_y]\to[0,\ell_{gy}]\]
of $C^2$--diffeormophisms.
From the claim and from
\[
H(x) =\sup\{z\mid \mu[0,z)\le x\},\]
we can see that $\tilde\rho$ resulting from this equivariant family is expressed by
\[
\tilde\rho(g)(x) := a_{g({0})} + \int_{0}^x \lambda(g)\; dx \pmod\bZ.\]
Since $\lambda(g)$ is continuous, we have that $\tilde\rho(G)\le\Diff_+^1(S^1)$.
We now repeat the above process for each $\{\ell_y^k\}_{y\in\mathcal{O}}$ and
\[\eta_{x,y,g}^k:=
\phi\left({\ell_y^k},{\ell^k_{gy}},\frac{\ell^k_{gy}}{g'(y)\ell^k_y}\right)\co \left[0,{\ell^k_y}\right]\to\left[0,{\ell^k_{gy}}\right].\]
We denote the resulting blow-up by $\rho_k$, and the semi-conjugacy map by $H_k$.
We let $I^k_y=H_k^{-1}(y)$ for each $y\in\mathcal{O}$.
It only remains to show that
\[
\|\rho_k(s)'-\rho(s)'\|_\infty\to 0\]
for each generator $s$ of $G$.
Let $\epsilon>0$ be arbitrary,
and let $k$ be sufficiently large.
For $t\in S^1\setminus\coprod_{y\in\mathcal{O}}I^k_y$, we see from the uniform continuity of $s'$ that
\[
|\rho_k(s)'(t)-\rho(s)'(t)|
= |s'\circ H_k(t)-s'(t)|<\epsilon.\]
In the case when $t\in \coprod_{y\in\mathcal{O}} I^k_y$,
we have as $k\to\infty$ that
\[
|\rho_k(s)'(t)-\rho(s)'(t)|
\le \|s'\|_\infty\cdot
\left|1 - \frac{\ell^k_{sy}}{s'(y)\ell^k_y}\right|
\cdot\|\xi\|_\infty
+|s'\circ H_k(t)-s'(t)|\to0,\]
which establishes the theorem.
\ep
\begin{rem}
We do \emph{not} require that $\mathcal{O}$ is a free orbit in Theorem~\ref{t:c1}.
If $gy=y$ and $g'(y)\ne1$ for some $y\in\mathcal{O}$, then
$\tilde\rho(g)\restriction_{I_y}$ could be a nontrivial $C^1$--diffeomorphism acting on the interval $I_y$.
\end{rem}
\begin{rem}
In Theorem~\ref{t:c1},
we may simply assume the existence of a single positive sequence $\{\ell_y\}_{y\in\mathcal{O}}$ instead of $\{\ell_y^k\}_{y\in\mathcal{O}}$ for all $k$. In this case, we will only require the following:
\begin{itemize}
\item $\sum_{y\in\mathcal{O}}\ell_y<\infty$;
\item $\lim_{y\in \mathcal{O}} s'(y)\ell_y/ \ell_{sy}=1$ for each generator $s$;
\item $y\not\in I_y$ for each $y\in\mathcal{O}$,
where here $I_y$ is as defined in the proof of Proposition~\ref{p:c0} by the parameter $\{\ell_y\}$.
\end{itemize}
For $L=\sum_{y\in\mathcal{O}}\ell_y$, the last condition can be rephrased purely in terms of $\{\ell_x\}$,
namely: \[y\not\in\left(\sum_{x\in [0,y)\cap\mathcal{O}}\ell_x/L,\sum_{x\in [0,y]\cap\mathcal{O}}\ell_x/L\right).\]
Then we can simply put $\ell^k_y :=\ell_y/k$ and apply the above proof.
In this case, each $t\in S^1$ satisfies the dichotomy that either
\[
t\in S^1\setminus\coprod_{y\in\mathcal{O}} I_y^k\]
for all sufficiently large $k$, or the following set is infinite:
\[
\{y\in\mathcal{O}\mid t\in I^k_y\text{ for some }k\}.\]
One can deduce the desired $C^1$--convergence from the second bullet point.
\end{rem}
Let $S$ be a fixed finite generating set of a group $G$.
For $g\in G$, we let $\|g\|_S$ denote the length of a shortest word in $S$ representing $g$.
The \emph{spherical growth function} of $G$ is defined as
\[\sigma_{G,S}(n):= \#\{ g\in G\co \|g\|_S=n\}.\]
We often write $\|g\|:=\|g\|_S$ and $\sigma(n):=\sigma_{G,S}(n)$ when the meanings are clear from the context.
It is sometimes useful to consider the \emph{upper spherical growth function}
\[\bar\sigma(n):=\sup_{1\le i\le n}\sigma(i).\]
The functions $\sigma$ and $\bar\sigma$ are sub-multiplicative; moreover, $\bar\sigma$ is monotone increasing.
Recall that a function $f\co\bN\to\bR_+$ is of \emph{sub-exponential growth} if
\[\limsup_{n\to\infty} f(n)^{1/n}=1.\]
The group $G$ is said to be of \emph{sub-exponential growth} if so is the map
\[n\mapsto \#\{ g\in G\co \|g\|_S\le n\},\]
which is easily seen~\cite{dlHarpe2000} to be equivalent to each of the following:
\begin{itemize}
\item $\lim_{n\to\infty}\sigma(n)^{1/n}=1$;
\item $\lim_{n\to\infty} \bar\sigma(n)^{1/n}=1$.\end{itemize}
If $\bar\sigma$ satisfies the following ``ratio--test'':
\[\lim_{n\to\infty} \bar\sigma(n+1)/\bar\sigma(n)=1,\]
then $\bar\sigma$ is of sub-exponential growth.
A partial converse is the following.
\begin{lem}\label{l:sigma-upper}
If $f\co \bN\to[1,\infty)$ is a sub-multiplicative and monotone increasing function such that
\[\lim_{n\to\infty} f(n)^{1/n}=1.\]
Then there exists a monotone increasing function $g\co \bN\to[1,\infty)$
such that
\[
\lim_{n\to\infty} g(n+1)/g(n)=1\]
and such that each $n\in\bN$ satisfies
\[ f(n)\le g(n)\le f(n)^2.\]
\end{lem}
Note that we do not require $g$ is sub-multiplicative.
\begin{exmp}
One may consider $f(n):=\exp\floor{\log_2 (n+1)}$, which can be seen to be sub-multiplicative,
monotone increasing and of sub-exponential growth.
The limit \[\lim_{n\to\infty}\frac{f(n+1)}{f(n)}\] does not exist, however. In this case, we set
\[g(n):=\exp({\log_2 (n+1)}),\]
which satisfies the desired properties in the above lemma.\end{exmp}
\bp[Proof of Lemma~\ref{l:sigma-upper}]
Let us consider the nonnegative, sub-additive, monotone increasing function $F:=\log f$, so that
\[\lim_{n\to\infty} F(n)/n=0.\]
We will then define
\[G(n):=\frac1n\sum_{i=1}^n F(i).\]
By the monotonicity of $F$, we have that $G$ is monotone increasing, and that $G(n)\le F(n)$.
The sub-additivity of $F$ implies that
\[F(n)-G(n) =\frac1n\sum_{i=1}^{n-1} \left(F(n)-F(i)\right)\le\frac{1}{n}\sum_{i=1}^{n-1}F(n-i) \le \frac{n-1}n G(n-1)\le G(n-1).\]
As $n\to\infty$ we finally have
\[G(n+1)-G(n)\le F(n+1)/(n+1)\to0.\]
We then see that $g(n):=\exp(2 G(n))$ is the desired function.\ep
Suppose we have an action $\rho\co G\to\Homeo^+(S^1)$, with a fixed base point $0\in S^1$.
For each point $y\in\rho(G).0$, we define the \emph{orbit length of $y$ at $0$} as \[\|y\|_0:=\min\{j \mid s_j \cdots s_2 s_1(0)=y\text{ for some }s_i\in S\cup S^{-1}\}.\]
In the case the orbit $\rho(G).0$ is free, if $y=g(0)$ then we have $\|y\|_0=\|g\|_S$.
We can also define the \emph{orbit spherical growth function} analogously even when the orbit is not free; however, we will mainly deal with free orbits from now on.
\begin{cor}\label{c:stab0}
Let $G$ be a finitely generated group of sub-exponential growth.
If an action $\rho\co G\to \Diff_+^{1}(S^1)$ with a free orbit $\mathcal{O}:=\rho(G).0$ satisfies that
\[\sup_{g\in G}g'(0)<\infty,\]
then $\rho$ admits a sequence $\{\rho_k\co G\to\Diff_+^1(S^1)\}_{k\ge1}$ of nontrivial blow-ups at $\mathcal{O}$ such that
$\rho_k$ converges to $\rho$ in the $C^1$--topology.
\end{cor}
Note that the group $G$ is not assumed to be abelian.
\bp[Proof of Corollary~\ref{c:stab0}]
By applying Lemma~\ref{l:sigma-upper} to the sub-multiplicative function
\[
\bar\sigma(n)=\sup_{i\le n}\sigma(i)\]
we obtain a monotone increasing function $p(n)\ge1$ such that $p(n+1)/p(n)\to1$ as $n\to\infty$,
and such that
\[\bar\sigma(n)\le p(n)\le \bar\sigma(n)^2.\]
Pick a continuous monotone increasing function $\nu(x)$ such that $\int_1^\infty p/ \nu<\infty$ and \[\lim_{x\to\infty}\nu(x+1)/\nu(x)=1;\] the reader may assume $\nu(x) = x^2p(x)$.
For each $y\in \mathcal{O}$, there uniquely exists $g\in G$ such that $y=g(0)$. Then we set
\begin{equation}\label{eq:lky}\ell^k_{y} := g'(0) /\nu(\|y\|_0+k).\end{equation}
We obtain
\[
\sum_{y\in\mathcal{O}}
\ell^k_y
\le \left(\sup_{g\in G} g'(0)\right)\cdot \sum_{i=0}^\infty
\frac{p(i+k)}{\nu(i+k)}\cdot \frac{\bar\sigma(i)}{p(i+k)}
\to0\text{ as }k\to\infty.\]
For a generator $s$ of $G$, and for $g\in G$, we have
$\abs*{\|sg(0)\|_0-\|g(0)\|_0}\le 1$. This implies for $y=g(0)$ that
\[
\frac{s'(y)\ell_{y}^k}{\ell^k_{sy}}
= \frac{\nu(\|sy\|_0+k)}{\nu(\|y\|_0+k)}
\to 1\]
as $k+\|y\|_0\to\infty$. The conclusion follows from Theorem~\ref{t:c1}.
\ep
\begin{rem}
\be
\item
When $G\cong\bZ$, then the above corollary was proved by Herman~\cite[Theorem X.3.1]{Herman1979}.
In this case, the condition
$\sup_{g\in G}g'(0)<\infty$ holds possibly after replacing $0$ by some point $x_0\in S^1$, as was established by Ma\~n\'e.
\item
If a sub-exponential growth group faithfully acts on $S^1$ with $C^{1+\epsilon}$ diffeomorphisms for some $\epsilon>0$, then the group is actually virtually nilpotent~\cite{Navas2008GAFA}.
\ee
\end{rem}
\subsection{Exceptional $C^{1,\alpha}$--actions}
We are now ready to restate and prove the first of our main results.
\begin{thm}\label{t:c1al}
Let $d\in\bN$, and let $\alpha$ be a concave modulus satisfying
\[
\int_0^1 1/\alpha(x)^ddx<\infty.\]
If $d>1$, we further assume
\begin{equation}\label{eq:alphay}
\sup_{0<y<1} \alpha\!\left( y^{d+1}/\alpha(y)^d\right)/y<\infty.\end{equation}
Suppose that a finitely generated group $G$ admits an action
$\rho\co G\to \Diff_+^{1,\alpha}(S^1)$ with a free orbit $\mathcal{O}:=\rho(G).0$ such that \[\sup_{g\in G} \abs*{\log g'(0)}<\infty.\]
Suppose furthermore that the spherical growth function of $G$ is at most $c n^{d-1}$ for some $c>0$.
Then $\rho$ admits a sequence of nontrivial blow-ups
\[\{\rho_k\co G\to\Diff_+^{1,\alpha}(S^1)\}\]
at $\mathcal{O}$
such that $\rho_k\to\rho$ in the $C^1$--topology.
Moreover, we may require that
\[\sup_k\{[\rho_k(g)]_{\alpha}\}<\infty\] for all $g\in G$.
\end{thm}
\begin{rem}
The proof of Theorem~\ref{t:c1al} is rather technical, and the reader interested in single exceptional
diffeomorphisms may wish to dispense with the
generality afforded by assuming $d\neq 1$, as well as more general group actions.
For the benefit of such readers, we have included a proof of
Theorem~\ref{t:int-alpha} in Section~\ref{s:appendix} below.
This proof also serves as a detailed guide for understanding the proof of Theorem~\ref{t:c1al}.
\end{rem}
Note that a group $G$ satisfying the hypotheses of Theorem~\ref{t:c1al}
is necessarily virtually nilpotent, by Gromov's theorem on groups of polynomial word growth. The reader
may wonder what the precise difference between the spherical growth function and the \emph{word growth function} is,
the latter of which measures the number of words of length at most $n$ with respect to some generating set,
and whose degree is independent of the generating set (cf.~\cite{mann-book}, for instance, for background).
If $G$ is a finitely generated abelian group of rank $d$ then the word growth function is polynomial of degree $d$, and the spherical
growth function is polynomial of degree $d-1$, as is an easy exercise. For a general group $G$ of polynomial word growth, Gromov's
theorem allows us to assume that $G$ is torsion--free and nilpotent, and that if $\gamma_i(G)$ denotes the $i^{th}$ term of the lower central
series of $G$, then $\gamma_i(G)/\gamma_{i+1}(G)$ is torsion--free for all $i$. In this case, for groups of nilpotence class two (i.e.
$\gamma_3(G)=\{1\}$, where here we adopt the convention $\gamma_1(G)=G$), it is also true that if the degree of the word growth
function is $d$ then the spherical growth function has degree $d-1$. This fact can be established by explicit computation~\cite{Stoll1998}; see the proof
of Theorem 4.2 in~\cite{mann-book}. In general, one has a bound which appears to be related in a subtle
way to the error term in computing the exact word growth function. Namely:
\begin{prop}[See~\cite{BLD}, Corollary 11]
Let $G$ be a nilpotent group such that $\gamma_{r+1}(G)=\{1\}$
and with polynomial word growth of degree $d$ with respect to a generating set $S$.
Then there are constants $C_1$ and $C_2$, depending on $S$, such that
\[C_1 n^{d-1}\leq \sigma(n)\leq C_2 n^{d-\epsilon},\] where $\epsilon=1$ for $r=2$ and $\epsilon=2/3r$ otherwise.
\end{prop}
Thus, even for general finitely generated nilpotent group, the precise relationship between the word growth function and the spherical
growth function seems subtle, and thus we elect to state our results in terms of the spherical growth function as opposed to the more
common word growth function.
We return to the discussion of Theorem~\ref{t:c1al}.
In the case of minimal actions, the boundedness of logarithms of derivatives gives a strong restriction on possible actions:
\begin{prop}[{cf. \cite[Chapter IV]{Herman1979} for the case $\Gamma=\bZ$}]\label{p:GoHe}
Let $\Gamma$ be a subgroup of $\Diff_+^1(S^1)$.
If the action of $\Gamma$ is minimal, and if
\[\sup_{g\in \Gamma} \abs*{\log g'(x_0)}<\infty\]
for some $x_0\in S^1$, then $\Gamma$ is $C^1$--conjugate into the rotation group.
\end{prop}
For the proof of this proposition, we employ an equivariant version of the \emph{Gottschalk--Hedlund Lemma}.
For a group $\Gamma$ acting on a space $M$, a \emph{continuous 2--cocycle} is a map
\[c\co \Gamma\times M\to\bR\]
such that $x\mapsto c(g,x)$ is continuous,
and such that \[c(fg,x) = c(g,x)+c(f,gx).\]
\begin{lem}[{\cite[Lemma 3.6.10]{Navas2011}}]
Suppose that a group $\Gamma$ acts minimally on a compact metric space $M$.
Assume that $c\co \Gamma\times M\to\bR$ is a continuous 2-cocycle.
Then the following are equivalent.
\be[(i)]
\item We have $\sup_{g\in\Gamma} |c(g,x_0)|<\infty$ for some $x_0\in M$;
\item There exists a continuous map $\phi\co M\to\bR$ such that
\[c(g,x) = \phi(x) - \phi\circ g(x)\] for all $g\in\Gamma$ and $x\in M$.
\ee
\end{lem}
\begin{rem}
In~\cite{Navas2011}, the above lemma is stated with a finite generation hypothesis for $\Gamma$. However, it is apparent from the proof therein that this hypothesis is unnecessary. See~\cite{CNP13} for a further generalization of the lemma.
\end{rem}
\bp[Proof of Proposition~\ref{p:GoHe}]
Applying the Gottschalk--Hedlund Lemma to
\[c(g,x) := \log g'(x),\]
we obtain a continuous map $\phi\co S^1\to\bR$ such that
\[\log g'(x) = \phi(x) - \phi\circ g(x).\]
As in~\cite[Chapter IV]{Herman1979}, we define $h\in\Diff_+^1(S^1)$ by
\[
h(x) :=\int_0^x e^{\phi(t)+c}dt\pmod\bZ.\]
Here, $c$ is chosen so that $h(1) = 1$. For each $g\in \Gamma$, we can verify
\[h'\circ g(x) \cdot g'(x) = e^{\phi\circ g(x)+c}\cdot e^{\phi(x)-\phi\circ g(x)}=e^{\phi(x)+c}=h'(x).\]
This implies that $(hgh^{-1})'=1$, so that $hgh^{-1}\in\SO(2,\bR)$.
\ep
\bp[Proof of Theorem~\ref{t:c1al}]
As we have noted in Remark~\ref{rem:alpha-diff}, we may assume that $\alpha$ is differentiable.
We retain the notation used in Corollary~\ref{c:stab0}, setting
\[p(x):=x^{d-1},\quad \nu(x) := x^{d+1}\alpha(1/x)^d.\]
Note that $\int_1^\infty p/\nu = \int_0^1 1/ \alpha^d<\infty$.
By convexity, we see that $x/\alpha(x)$ is monotone increasing, and so is $\nu(x)/x$.
We define $\ell_y^k$ as in the equation~\eqref{eq:lky}.
In the proof of Corollary~\ref{c:stab0}, we observed that the hypothesis $\int_1^\infty p/\nu<\infty$ implies that \[\sum_{y\in\mathcal{O}}\ell_y^k\to0.\]
For each generator $s$ of $G$, each point $y\in \mathcal{O}$, and each $k\gg0$, we define
\[A(s,k,y):=\frac1{\alpha(\ell_y^k)}\abs*{1-\frac{\ell^k_{sy}}{s'(y)\ell_y^k}}.\]
Let $\lesssim$ denote an inequality up to bounded errors and scaling.
\begin{claim*}
$\{A(s,k,y)\}_{s,k,y}$ is uniformly bounded.
\end{claim*}
We remind the reader that in Appendix~\ref{s:appendix} , there is a proof of the theorem in the case $d=1$ and $G=\bZ$.
The reader will find the analogous claim in that proof, and may find it helpful to understand the workings of the present argument.
To see this claim in the given generality, we first note \[\alpha(cx)\ge \min(1,c)\alpha(x),\] for $c,x>0$.
Letting $j:=\|sy\|_0+k$, we have
\[
A(s,k,y)
\lesssim\frac{1}{\alpha(1/\nu(j))}\abs*{1- \frac{\nu(j)}{\nu(j\pm1)}}
\le \frac{ \nu'(t)}{\nu(j-1)\alpha(1/\nu(j))}.
\]
Here, $t\in(j-1,j+1)$ is chosen so that \[\nu'(t) = \nu(j\pm1)-\nu(j).\]
We compute
\[
x\nu'(x)=\left(d+1 - d\alpha'(1/x)/(x\alpha(1/x))\right)\nu(x)\in [\nu(x),(d+1)\nu(x)].
\]
Also, note that \[\nu(x/2)= (x/2)^{d+1}\alpha(2/x)^d\ge \nu(x)/2^{d+1}.\]
Assuming $k\gg0$, we see
\[
A(s,k,y)\lesssim \frac{ \nu(j+1)}{j\nu((j+1)/2)\alpha(1/\nu(j))}\lesssim\frac{1}{j\alpha(1/\nu(j))}. \]
If $d\ge 2$, then the equation \eqref{eq:alphay} implies the claim.
If $d=1$, then we recall that $\nu(x)/x$ is increasing and that
\[
\frac{1}{j\alpha(1/\nu(j))}=\frac{\nu(j)^2}{j\nu\circ\nu(j)}\le1.
\]
So the claim is proved.
Note that the claim trivially implies that
$\lim_{x\to\infty}\nu(x+1)/\nu(x)=1$. Hence, we have actions $\{\rho_k\}$ satisfying the conclusion of Corollary~\ref{c:stab0}.
It only remains to show that $\sup_k [\rho_k(s)']_\alpha<\infty$ for each generator $s$ of $G$.
We recall the construction of $\rho_k$ from the proof of Theorem~\ref{t:c1}.
Suppose first that $u,v\in\bar I_y$ for some $y\in \mathcal{O}$.
Then we have
\[
\abs*{\rho_k(s)'(u)-\rho_k(s)'(v)}
\le \abs*{s'(y)} \cdot
\abs*{1-\frac{\ell^k_{sy}}{s'(y)\ell_{s}^k}}
\cdot \|\xi\|_\infty\cdot \frac{|u-v|}{\ell_y^k}.\]
Since $|u-v|\le\ell_y^k$, we have that
\[
|u-v|/\ell_y^k \le \alpha(|u-v|)/\alpha(\ell_y^k).\]
The above claim implies that
\[ \abs*{\rho_k(s)'(u)-\rho_k(s)'(v)}\le C \alpha(|u-v|)\]
for some constant $C$ independent of $s,k$ and $y$. Here, we are using the finite generation hypothesis.
Let us now assume $u,v\not\in\coprod_{y\in\mathcal{O}} I_y$.
We recall the definition of $\mu_k$, given by \[m(H_k^{-1}(A))=\mu_k(A)\] for each Borel set $A$.
For $k\gg0$ and \[L_k=\sum_{y} \ell_y^k<1/2,\] we see
\[ |H_k(u)-H_k(v)|\le |u-v|/ (1-L_k)\le 2 |u-v|.\]
So, by enlarging $C$ if necessary for all $k\gg0$ we have
\begin{align*}
\abs*{\rho_k(s)'(u)-\rho_k(s)'(v)}
&= \abs*{ s'\circ H_k(u)-s'\circ H_k(v)}
\le [s']_\alpha \alpha( H_k(u)-H_k(v))\\
&\le C \alpha(u-v).\end{align*}
Finally, we conclude that $[\rho_k(s)']_\alpha\le 3 C$.
\ep
\subsection{Applications to free abelian groups and Theorem~\ref{t:int-alpha}}
Let us note immediate corollaries of Theorem~\ref{t:c1al} for free abelian groups.
\begin{cor}\label{c:higher-rank}
Let $d\ge1$, and let $\alpha$ be a concave modulus such that \[\int_0^1\frac{1}{\alpha^d}\, dx<\infty.\]
If $d>1$, we further assume
\[\sup_{0<y<1} \alpha\!\left( y^{d+1}/\alpha(y)^d\right)/y<\infty.\]
If $\rho\co \bZ^d\to\Diff_+^{1,\alpha}(S^1)$ is a faithful representation that is $C^1$--conjugate into the rotation group $\SO(2,\bR)$, then there exists a sequence
\[\rho_k\co \bZ^d\to\Diff_+^{1,\alpha}(S^1)\]
of nontrivial blow-ups of $\rho$ such that $\rho_k\to\rho$ in the $C^1$--topology.
\end{cor}
Note that the second inequality above holds for the modulus
\[\alpha(x) =\left(x \cdot \log 1/x\right)^{1/d}\cdot (\log 1/x)^\epsilon.\]
for $\epsilon>0$ as considered by
Deroin--Kleptsyn--Navas~\cite{DKN2007}, and for the stronger modulus
\[\alpha(x) =\left(x\cdot \log 1/x\cdot \log\log1/x\right)^{1/d}\cdot (\log\log 1/x)^\epsilon.\]
Theorem~\ref{t:int-alpha} is now an immediate consequence.
We remark that the method of this section, however, does not extend to the case when $\alpha(x) = x\log (1/x)$;
cf.~\cite[Exercise 4.1.26]{Navas2011} and~\cite[Chapter 12]{KH1995}.
\section{Bounding lengths from below}\label{s:bounding}
In this section, we establish Theorem~\ref{t:alpha-inv} and Theorem~\ref{t:int-alpha-conv}.
For this, we will let $\alpha$ be a given concave modulus, and let \[\rho\co G\to\Diff_+^{1,\alpha}(S^1)\]
be an exceptional action. We write $K$ for the exceptional minimal set of this action, which is the unique, closed, minimal, invariant
subset of $S^1$ under the action of $G$. The complement of $K$ is a union of open intervals in $S^1$, which are permuted
component-wise by the action of $G$, and thus consist of wandering intervals for the action of $G$ on $S^1$. There is a semi-conjugacy
$H\colon S^1\to S^1$ which collapses each wandering interval to a point, and which is at most two--to--one on $K$. The wandering
intervals for the action can obviously be partitioned into at most countably many orbits under the action of $G$. Each endpoint of a wandering
interval lies in $K$, and the action of $K$ is minimal. It follows that if $I$ and $J$ are two wandering intervals, not necessarily in the same
orbit under $G$, then $G.(\partial I)$ accumulates on $\partial J$.
The reader will note that we lose no generality in what follows by assuming that there is only one $G$--orbit of wandering intervals for
the action of $G$, and we will thus adopt this assumption. In this case, the inverse of $H$ is a blow-up of exactly one orbit of a minimal
action, say $\mathcal{O}$.
Each point $y\in\mathcal{O}$ corresponds to a maximal wandering interval \[I_y=(a_y,b_y)\sse S^1.\]
We will continue to denote the minimal set by \[K:=S^1\setminus \coprod_{y\in\mathcal{O}}I_y.\]
We let $\|y\|_0$ and $\sigma(n)$ denote the orbit distance and the spherical growth functions as before, and put $\ell_y:=|I_y|$. Note that we have
\[\ell_{gy}=|I_{gy}|=|gI_y|,\]
where we mean $gy=\bar\rho(g)y$ and $gI_y=\rho(g)I_y$.
\subsection{Endpoint derivatives and the fundamental estimate}
The following lemma is well-known, and is crucial for the content of this section.
\begin{lem}[Fundamental Estimate]\label{lem:fundamental}
Let $\alpha,G,\rho,\mathcal{O},\{\ell_y\}$ be as above, and let $g\in G$. If
\[\lim_{y\in\mathcal{O}} |\ell_{gy}|/|\ell_y|=1,\]
then we have that \[\sup_{y\in\mathcal{O}} \frac{1}{\alpha(\ell_y)}\abs*{1-\frac{\ell_{gy}}{\ell_{y}}}<\infty.\]
\end{lem}
Recall the limit above is taken with respect to an arbitrary enumeration of $\{\ell_y\}$. It is instructive to imagine that $\{\ell_y\}$ is listed in the monotone decreasing order. The reader will note that the limit in the hypotheses of Lemma~\ref{lem:fundamental} continues to make sense
even if there are many orbits of wandering intervals for the action of $G$. Indeed, the limit compares lengths of wandering
intervals in a single orbit only, so no comparisons between distinct orbits need to be made.
\begin{proof}[Proof of Lemma~\ref{lem:fundamental}]
We first claim that $g'(x)=1$ for all $x$ in the minimal set $K$.
By The Mean Value Theorem, for each $y\in \mathcal{O}$ there exists $z_y\in I_y$ satisfying $g'(z_y)=\ell_{gy}/\ell_y$.
In the case when $x\in I_y$ for some $y\in \mathcal{O}$,
then by minimality we obtain a sequence $\{y_i\}\in\mathcal{O}\setminus\{y\}$ such that $\lim_i I_{y_i}\to x$.
It follows that
\[
g'(x) = \lim_i g'(z_{y_i}) = \lim_i \ell_{gy_i}/\ell_{y_i}=1
\]
Since $\{\partial I_y\}_{y\in\mathcal{O}}$ is dense in $K$, our claim is proved.
Let us now fix $y\in\mathcal{O}$, and $w\in\partial I_y$.
The conclusion follows from the inequality
\[\abs*{1-\ell_{gy}/\ell_y}
=
\abs*{s'(w)-s'(z_y)}
\leq \brac*{g'}_\alpha\cdot \alpha(\ell_y).\qedhere\]
\end{proof}
\subsection{Bounding the values of $\alpha(\ell_y)$}
It is evident that passing from the fundamental estimate (Lemma~\ref{lem:fundamental}) to \emph{a priori} bounds on the
values of $\{\ell_y\}_{y\in\mathcal{O}}$ is a natural approach to a converse to Theorem~\ref{t:c1al}, and this is precisely the sort of
control given to us in Theorem~\ref{t:alpha-inv} above. Recall the following result from the introduction.
\begin{thm}\label{t:al-inv}
Let $\alpha$ be a concave modulus, and let
\[\rho\co \bZ^d\to \Diff_+^{1,\alpha}(S^1)\]
be an exceptional action.
Suppose that $\rho$ is semi-conjugate to a faithful action
\[\bar\rho\co\bZ^d\to\Diff_+^1(S^1)\]
by a semi-conjugacy map $H$ such that $\bar\rho$ is $C^1$--conjugate into $\SO(2,\bR)$.
Assume for each generator $s$ of $G$ and $x$ in the exceptional minimal set of $\rho$ we have that
\[ \rho(s)'(x) = \bar\rho(s)'\circ H(x).\]
Then we have that \[\int_0^1 \alpha^{-1}(t)/t^{d+1}\;dt<\infty.\]
\end{thm}
\bp
Set $G:=\bZ^d$.
Let $J$ be a maximal open wandering interval of $\rho$, and let $\mathcal{O}$ be the $\bar\rho(G)$--orbit of $0:=H(J)$.
We may write
\[
\{gJ\}_{g\in G}=\{I_y:=(a_y,b_y)\}_{y\in\mathcal{O}}\]
and $\ell_y:=I_y$.
For each $y=\bar\rho(g).0\in\mathcal{O}$, there exists $z_y\in I_y$ such that \[\rho(s)'(z_y)=\ell_{sy}/\ell_y.\]
Let $s$ be a generator of $G$. Then we have that
\begin{align*}
\brac*{\rho(s)'}_\alpha \cdot\alpha(\ell_y)
&\ge \abs*{\rho(s)'(z_y)-\rho(s)'(a_y)}
\ge \abs*{\frac{\ell_{sy}}{\ell_y}-\bar\rho(s)'(y)}
\\
&\ge \inf_{x\in S^1} \bar\rho(s)'(x) \cdot \abs*{\frac{\ell_{sy}}{\bar\rho(s)'(y)\ell_y}-1}.\end{align*}
Setting $L_y :=\ell_y / \bar\rho(g)'(0)$, we can find a constant $C>0$ such that
\[
\abs*{\frac{L_{sy}}{L_y}-1} \le C\alpha(L_y).\]
Here, we also used the fact
\[
\alpha(\ell_y)\le \max(1,\sup_{h\in G} \bar\rho(h)'(0)) \alpha(L_y).\]
It is easy to show that $h(x)=x(1-C\alpha(x))$ is increasing for small values of $x$, and that in fact $h'(0)\geq 1$.
Recall we let $\|y\|_0$ denote the orbit distance between $0$ and $y$.
\begin{claim*}
There exists a constant $A>0$ such that $\alpha(\ell_y)\ge A/\|y\|_0$.
\end{claim*}
The idea is similar to \cite[Lemma 3.1.3]{Navas2011}, where the case $\alpha(x)=x^{1/d}$ was considered.
We proceed by induction on $i:=\|y\|_0$.
We suppose that $\alpha(\ell_y)\geq A/\|y\|_0$. After possibly requiring $\ell_y\le\epsilon$ for some small $\epsilon>0$, we obtain
\[\ell_{sy}\geq h(\ell_y)\geq h(\alpha^{-1}(A/\|y\|_0))=\alpha^{-1}\left(\frac{A}{\|y\|_0}\right)\left(1-\frac{AC}{\|y\|_0}\right),\] where the second inequality
holds because $\alpha$ is a homeomorphism and because $h$ is increasing near zero. We consider $s,y$ such that $\|sy\|_0=\|y\|_0+1=i+1$. We apply $\alpha$ to this string of inequalities now.
Since $\alpha$ is concave, we have that for
any $0<c<1$, there is an inequality $\alpha(cx)\geq c\alpha(x)$. Thus, after applying $\alpha$ to the right--most hand side, we get a quantity
bounded below by \[\left(1-\frac{AC}{i}\right)\frac{A}{i}.\] If one can show that this quantity is bounded below by $A/(i+1)$, then one obtains
$\alpha(\ell_{sy})\geq A/\|sy\|_0$, the desired conclusion. A straightforward manipulation shows that we obtain
\[\left(1-\frac{AC}{i}\right)\frac{A}{i}\geq \frac{A}{i+1}\] provided that \[i\geq \frac{CA}{1-CA}.\] Thus, letting $A$ be a positive constant so
that $\alpha(\ell_y)\geq A/\|y\|_0$ even for $\ell_y\ge\epsilon$ and so that \[1\geq \frac{CA}{1-CA},\] we obtain the desired claim.
From the claim and from $\sigma(n)\sim n^{d-1}$, we have that
\[
1\ge \sum_{y\in\mathcal{O}} \ell_y
\gtrsim \sum_{n\ge1} n^{d-1} \alpha^{-1}(A/n).\]
We have that
\[
\infty > \sum_{n\ge1} (n+1)^{d-1} \alpha^{-1}(A/n)
\ge \int_1^\infty x^{d-1} \alpha^{-1}(A/x)dx.\]
This implies that $\int_0^1 \alpha^{-1}(t)/t^{d+1} \; dt<\infty$.
\end{proof}
Note that the claim in the proof of Theorem~\ref{t:al-inv} immediately implies Theorem~\ref{t:alpha-inv} from the introduction.
In Theorem~\ref{t:al-inv}, if $\bar\rho$ is actually a representation into the rotation group, then it suffices to assume
\[\lim_{y\in\mathcal{O}}\ell_{sy}/\ell_y=1,\]
where $\ell_y$ is as given in the proof.
From the Fundamental Estimate (Lemma~\ref{lem:fundamental}),
we conclude that $\rho(g)'(x)=1$ for each $x$ in the exceptional minimal set of $\rho$, and hence the same integrability constraints on
$\alpha$ would follow.
We now have Theorem~\ref{t:alpha-inv}
and Corollary~\ref{cor:int-alpha-inv} as immediate consequences.
\begin{que}
In Theorem~\ref{t:al-inv}, does the same conclusion hold without the assumption on the values of $\rho(s)'$ at the minimal set?\end{que}
\begin{exmp}
Let $f$ be an exceptional diffeomorphism such that $\ell_{i+1}/\ell_i\to 1$ for intervals in a maximal wandering set. In this case, let us deduce from Theorem~\ref{t:alpha-inv} that $f'$ is not
$\alpha$--continuous for $\alpha(x)=x\log{1/x}$ (cf.~\cite{Navas2011,KH1995}).
Suppose $f'$ is $\alpha$--continuous. By Theorem~\ref{t:alpha-inv}, there exists a positive constant $A$ such that $\alpha(\ell_i)\ge A/i$ for all $i$.
On the other hand, we have a positive sequence $\{x_i\}_{i\geq 2}$ defined below such that $\sum_i x_i=\infty$ and such that $\alpha(x_i)\leq A/i$. This is a contradiction, since we have
\[\ell_i \ge \alpha^{-1}(A/i) \ge x_i.\]
The sequence $\{x_i\}$ is actually defined as \[x_i=\frac{A}{K\cdot i\log i},\] where $K$ is a positive constant which will be fixed later.
Observe that $\sum_i x_i=\infty$, so it suffices to show that $\alpha(x_i)\leq A/i$ for all $i\geq 2$. Applying $\alpha$, we see
\[\alpha(x_i)=\frac{A}{K\cdot i\log i}\left( \log K+\log i+\log\log i+\log{A^{-1}}\right).\] This simplifies to
\[\frac{A(\log K+\log{A^{-1}})}{K}\cdot\frac{1}{i\log i}+\frac{A}{K\cdot i}+\frac{A\cdot \log\log i}{K\cdot i\log i}.\] Since $1/(i\log i)<1/i$ for $i\geq 3$,
since $(\log\log i)/\log i<1$ for $i\geq 2$,
and since $A$ is a fixed positive constant, it is clear that we can choose a sufficiently large $K$ to make this entire expression less than $A/i$
for $i\geq 2$, which is what we set out to show.
It is not difficult to generalize this last computation to show the same conclusion for the moduli \[\alpha(x)=x(\log{1/x})(\log\log{1/x}),\]
\[\alpha(x)=x(\log{1/x})(\log\log{1/x})(\log\log\log{1/x}),\] and indeed for
\[\alpha(x)=x(\log{1/x})(\log\log{1/x})(\log\log\log{1/x})\cdots(\log^n{1/x}).\]
\end{exmp}
\subsection{A converse to Theorem~\ref{t:alpha-inv}}
We now investigate the degree to which \[\int_0^1 \alpha^{-1}(t)/t^{d+1} dt<\infty\]
is a sufficient condition for the existence of an exceptional $C^{1,\alpha}$
action of $\bZ^d$.
We recall the following result, which was stated in the introduction.
\begin{thm}\label{t:int-comparison}
Let $d$ be a positive integer, and let $\alpha$ be a concave modulus such that
\[\int_0^1\frac{\alpha^{-1}(t)}{t^{d+1}}\, dt<\infty,\] and such that \[M:=\sup_{0<t<1}\frac{\alpha(t)}{t\alpha'(t)}<\infty.\]
If $\rho\co \bZ^d\to\Diff_+^{1,\alpha}(S^1)$ is a faithful representation that is $C^1$--conjugate into the rotation group $\SO(2,\bR)$, then there exists a sequence
\[\rho_k\co \bZ^d\to\Diff_+^{1,\alpha}(S^1)\]
of nontrivial blow-ups of $\rho$
that converges to $\rho$ in the $C^1$--topology.\end{thm}
\bp
We follow our previous arguments for Theorems~\ref{t:c1} and~\ref{t:c1al} closely. We let $\mathcal{O}=\rho(G).0$ be a choice of a free orbit.
Using the spherical growth $\sigma(x)\sim x^{d-1}$ and the orbit distance function $\|y\|_0$,
we set the parameters \[\ell_{g(0)}^k:=g'(0)\alpha^{-1}(1/ (\|y\|_0 + k)).\]
Since $\alpha^{-1}(t)/t^{d+1}$ is integrable near $0$, we have that
\[\sum_{y\in\mathcal{O}}\ell_y^1 \lesssim \sum_n \sigma(n)\alpha^{-1}(1/n)\lesssim \int_1^\infty x^{d-1}\alpha^{-1}(1/x)dx=\int_0^1 \alpha^{-1}(t) / t^{d+1} \; dt <\infty.\]
\begin{claim*}
We have \[\lim_{x\to\infty} \frac{\alpha^{-1}(1/x)}{\alpha^{-1}(1/(x+1))}=1.\]
\end{claim*}
To see the claim, we put
\[ c(x):= \alpha^{-1}(1/x) / \alpha^{-1}(1/(x+1))\ge1\] for $x\gg0$. Then we have some $t=t(x)\in(x,x+1)$ such that
\begin{align*}
& 0< c(x) - 1 = \frac{(\alpha^{-1})'(1/t)}{x(x+1)\alpha^{-1}(1/(x+1))}
=\frac{1}{x(x+1) \alpha^{-1}(1/(x+1)) \cdot \alpha'\circ\alpha^{-1}(1/t)}
\\
&\le
\frac{M\alpha^{-1}(1/x)}{x\alpha^{-1}(1/(x+1))}=M c(x) / x.\end{align*}
The claim follows from the estimate $c(x) <(1- M/x)^{-1}$ for $x\gg0$.
Put $j:=\|y\|_0+k$ for $y\in\mathcal{O}$ and $k>0$.
As in the proof of Theorem~\ref{t:c1al},
\[ A(s,k,y):=\frac1{\alpha(\ell_y^k)}\abs*{1-\frac{\ell^k_{sy}}{s'(y)\ell_y^k}}
\le j \abs*{1- \frac{\alpha^{-1}(1/(j\pm1))}{\alpha^{-1}(1/j)}}
\lesssim\frac{1}{j\alpha'\circ\alpha^{-1}(1/t)\cdot \alpha^{-1}(1/j)}
\]
for some $t\in(j-1,j+1)$.
Using the hypothesis on $\alpha(x)/(x\alpha'(x))$, we estimate
\[
A(s,k,y) \lesssim
\frac{\alpha^{-1}(1/t)}{\alpha^{-1}(1/j)},\]
up to scaling and bounded error.
By the above claim, $A(s,k,y)$ is uniformly bounded as desired.
\ep
Theorem~\ref{t:int-alpha-conv} follows immediately from Theorem~\ref{t:int-comparison}.
\begin{rem}
In the case when $d=1$, one could attempt to prove a converse to Theorem~\ref{t:alpha-inv} by starting with the sequence $\{A/i\}_{i>0}$ and attempting to find
a convergent sequence of interval lengths $\{\ell_i\}_{i\in\bZ}$ such that \[\sup_i\frac{1}{\alpha(\ell_i)}\left(1-\frac{\ell_{i+1}}{\ell_i}\right)=K\]
is finite, whereupon one can apply the method of Theorem~\ref{t:c1al} in order to find the desired exceptional diffeomorphism. A natural
choice of lengths would be \[\ell_i=\alpha^{-1}\left(\frac{A}{|i|+i_0}\right),\] where here $i_0\in\N$ is a suitable shift of indices.
With this choice, the finiteness of the supremum forces the following inequality to hold:
\[\alpha^{-1}\left(\frac{A}{|i|+i_0+1}\right)\geq \alpha^{-1}\left(\frac{A}{|i|+i_0}\right)\left(1-\frac{KA}{|i|+i_0}\right),\] which is equivalent to
\[\frac{A}{|i|+i_0+1}\geq \alpha\left(\alpha^{-1}\left(\frac{A}{|i|+i_0}\right)\left(1-\frac{KA}{|i|+i_0}\right)\right).\] One can then try and pull out
the constant $1-(KA)/(|i|+i_0)$ on the right hand side, but because this constant is less than one, the result will be less than or equal to
the right hand side. In order to get around this difficulty, one can attempt a linear approximation of $\alpha$ at $\alpha^{-1}(A/(|i|+i_0))$
in order to show that \[\frac{A}{|i|+i_0+1}\geq \left(1-\frac{KA}{|i|+i_0}\right)\left(\frac{A}{|i|+i_0}\right),\] possibly up to a
nonzero multiplicative constant.
After some straightforward manipulations, one quickly finds that the hypothesis \[\sup_{x> 0}\frac{\alpha(x)}{x\alpha'(x)}<\infty\] is needed to
make the linear estimate work, and this is precisely one of the hypothesis of Theorem~\ref{t:int-comparison}. This
last supremum is easily seen
not to be finite for arbitrary concave moduli and therefore its finiteness imposes a nontrivial hypothesis on $\alpha$.
As an example, one can consider
$\alpha(x)=1/\log{(1/x)}$.
\end{rem}
\subsection{Remarks on McDuff's Question}
Let $f$ be an exceptional $C^1$--diffeomorphism, and let $\ell_i=|f^i(J)|$ for a wandering interval $J\sse S^1$.
It is evident from the discussion in this section that the assumption $\ell_{i+1}/\ell_i\to 1$ for the lengths of the successive wandering intervals
implies the strong conclusion that $f'=1$ on the endpoints of the wandering intervals. This is a somewhat restrictive phenomenon, and the
methods of this paper are not suited to address the possibility that $f'$ is not identically $1$ on the endpoints. For instance, the fundamental
estimate, the $1$ appearing in the expression \[\left(1-\frac{\ell_{i+1}}{\ell_i}\right)\] must be replaced by the corresponding endpoint derivative.
This is destructive in certain inductive procedures involved in the proof of Theorem~\ref{t:alpha-inv}, since certain products which one
needs to be bounded away from zero become products of derivatives of $f$ at endpoints of a maximal wandering set for $f$.
There is a long-standing open question about exceptional diffeomorphisms of the circle due to D. McDuff~\cite{McDuff1981,athanassopoulos}.
In her work, she considers the
\emph{ratio spectrum} for a maximal wandering set of an exceptional diffeomorphism. Namely, she rearranges the lengths of the intervals
to form a decreasing sequence $\{\lambda_i\}_{i\in\N}$, and considers the possible accumulation points of the set $\lambda_i/\lambda_{i+1}$.
She proves that this set is bounded and that $1$ is an accumulation point, and asks if $1$ is in fact the only accumulation point. One
can naturally strengthen her question as follows:
\begin{que}\label{q:mcduff-strong}
Let $U\subset S^1$ be an open interval meeting the exceptional minimal set of a finitely generated group, and let $\{\lambda_i\}_{i\in \N}$ be the lengths of intervals in a
maximal wandering set lying in $U$, arranged in decreasing order. Does it follow that $\lambda_i/\lambda_{i+1}\to 1$?
\end{que}
Even assuming a positive answer to this strengthened version of McDuff's question, and even after possibly assuming strong Diophantine
properties for the rotation number $\theta$,
it seems impossible to remove the assumptions
$\ell_{i+1}/\ell_i\to 1$ in our results. The reason for this is the mismatch which occurs upon rearranging the lengths of intervals. One may control
the first return time of a wandering interval to $U$ via Diophantine properties, but the index rearrangement map
\[i\mapsto\ell_i\mapsto \lambda_j\mapsto j\] may be so badly behaved that one may not be able to relate $\ell_{i+1}/\ell_i$ to
$\lambda_{j+1}/\lambda_j$ in a way which would be useful for our purposes. Lastly, we note the claim in the proof of Theorem~\ref{t:int-comparison} implies an affirmative answer to McDuff's question for the construction there with parameters given by $\ell_y := \alpha^{-1} (c / (\|y\|_0+k))$.
|
train/arxiv
|
BkiUdQ05qsFAf3AIARFR
| 5 | 1 |
\section{introduction}
Jiao and co-workers recently discovered
superconductivity with a critical temperature $T_c$=4.6 K
in Ta$_4$Pd$_3$Te$_{16}$. \cite{jiao}
These authors concluded that there is evidence for strong electron-electron
interactions in this compound based on an analysis of the specific heat.
Moreover,
Pan and co-workers reported that
$T_c$ increases to over 6 K under pressure
and found a linear contribution to the temperature
dependent thermal conductivity, which
increases with magnetic field similar to cuprate superconductors. \cite{pan}
This implies an electronic thermal conductivity contribution in
the superconducting phase, i.e. line nodes or some part of the Fermi surface
that is not gapped.
These measurements, which extend to $T$=80 mK,
which is well below $T_c$, were taken
as implying an unconventional superconducting
state, perhaps associated with the quasi-one-dimensional features of the
crystal structure. \cite{pan}
The crystal structure (Fig. \ref{struct}) shows a layered structure
with Ta and Pd atoms occurring in one dimensional chains. The structure
differs from those of the superconducting chalcogenides
Ta$_2$PdS$_5$, Nb$_2$Pd$_{0.8}$S$_5$ and Nb$_3$Pd$_{0.7}$Se$_7$
in being a stoichiometric compound with nominally flat two dimensional sheets
in the crystal structure.
\cite{zhang,zhang2,lu,singh-ta2}
We note that there is the possibility
that the layered crystal structure may be misleading from the point
of view of the electronic properties, as
previous electronic structure calculations by Alemany and
co-workers showed important interlayer Te -- Te interactions.
\cite{alemany}
In addition, it should be noted that there are considerable local
distortions around the different Te atoms in the structure.
Here we report electronic structure calculations in relation to
the above observations.
\section{methods}
\begin{figure}
\includegraphics[width=\columnwidth,angle=0]{struct-figure.eps}
\caption{(color online) Crystal structure of Ta$_4$Pd$_3$Te$_{16}$
viewed along the $c$-axis (in spacegroup 12,
$C2/m$, setting 2, unique axis $c$). The unit cell is indicated by the lines.
The atomic positions are from the total energy minimization.
The Te atoms in one layer are labeled according to the inequivalent
Te positions in the structure (see text).}
\label{struct}
\end{figure}
The calculations reported here were performed using standard
density functional theory with the generalized gradient approximation
of Perdew, Burke and Ernzerhof (PBE-GGA). \cite{pbe}
We used the general potential linearized augmented planewave (LAPW)
method \cite{singh-book} as implemented in the WIEN2k code. \cite{wien2k}
We used well converged LAPW basis sets plus local orbitals to treat
the semicore states. The LAPW sphere radii were $R$=2.5 bohr for all atoms,
and the planewave sector cutoff, $k_{max}$ was set according to $R k_{max}$=9.
The monoclinic
lattice parameters were taken from the experimental data of Mar and
Ibers, \cite{mar}
$a$=21.2762 \AA, $b$=19.510 \AA, $c$=3.735 \AA, $\gamma$=128.825$^\circ$
(note we show results using spacegroup 12, $C2/m$ setting 2, unique axis $c$,
while the original paper of Mar and Ibers uses setting 1, unique axis $b$).
The internal atomic coordinates were then determined by
total energy minimization with the PBE-GGA starting from the experimental
structure including relativity at the scalar relativistic level
for the valence bands. The resulting structure, shown in
Fig. \ref{struct}, was used for the electronic
structure calculations. These were performed self-consistently including
spin-orbit for all states.
The shortest Te -- Te distance that couples different layers in the
relaxed structure is 3.61 \AA, which is short enough for significant
interlayer Te -- Te interactions, as were discussed by Alemany and co-workers.
\cite{alemany}
The relaxed structure (Fig. \ref{struct})
is quite complex as seen.
In particular, while nominally the structure could be described as Te
bilayers with metal atoms in the interstitial sites, it is clear that
these bilayers are very strongly distorted. These distortions lead
to interesting structural features, such as short bonded Te zig-zag chains
along the $c$-axis composed of the Te1 and Te8 atoms, as labeled in the
figure, with a Te-Te bond length of only 3.22 \AA.
Also, there are large differences between the different Te sites.
The nearest Te-Te distances for the
different Te atoms run from 3.08 \AA, for the Te3-Te4 bond,
to 3.73 \AA, for Te2. Also the bond valence sums are far from the
nominal value and vary among the different Te, ranging from
2.68 (Te7) to 2.91 (Te8). This implies an important role for Te-Te
bonding.
The experimentally reported crystal structure refinement \cite{mar} shows
similar features including both the range of Te-Te bond lengths
and the variation in bond valence sums.
While this is unusual, it is reminiscent of IrTe$_2$,
which shows a first order structural transition related to
frustrated Te-Te $p$ bonding.
\cite{fang,cao}
\section{electronic structure}
\begin{figure}
\includegraphics[height=\columnwidth,angle=270]{dos.ps}
\caption{(color online) Calculated electronic density of states
and projections of Pd $d$ and Ta $d$ character on a per formula
unit basis. The Fermi energy is at 0 eV.}
\label{dos}
\end{figure}
The calculated electronic density of states and projections of
Pd $d$ and Ta $d$ character are shown in Fig. \ref{dos}.
There is strong hybridization between both types of metal atoms
and the Te evident in this plot. The Ta and Pd $d$
contributions occur in broad peaks above and below the
Fermi level, $E_F$. As seen, the Pd $d$ orbitals are
mostly occupied, while the Ta $d$ orbitals are mostly unoccupied.
In a fully ionic picture, one might assign valences Ta$^{+5}$,
Pd$^{+4}$ in order to obtain the correct stoichiometry with Te$^{-2}$
However Pd$^{4+}$, would seem to be a highly unlikely valence especially
in a telluride.
The Pd density of states with its nearly fully occupied
$d$ orbitals below $E_F$ is clearly not consistent with
a Pd$^{4+}$ state. Therefore important Te-Te p bonding
is expected, consistent with that seen in the density of states.
The calculated value at the Fermi energy is $N(E_F)$=9.6 eV$^{-1}$
per formula unit, which is significantly higher than the value of
5.5 eV$^{-1}$ obtained by Alemany and co-workers. \cite{alemany}
We did calculations for the unrelaxed atomic coordinates
\cite{mar} as well.
However, and find a value similar to but slightly higher
than the value with the relaxed structure.
All results shown here are with the relaxed structure.
The bare linear specific heat coefficient from our calculation is
22.7 mJ/(mol K$^2$). Comparing with the experimental value
of 42.8 mJ/(mol K$^2$), one infers an enhancement of
(1+$\lambda_{tot}$)=1.89 or $\lambda_{tot}$=0.89,
i.e. an intermediate coupling
value. Importantly, as seen from the projections of the density of states,
there is very little transition metal contribution to $N(E_F)$, which
instead derives from Te $p$ states.
The Pd $d$ contribution to $N(E_F)$ based on projection onto the Pd
LAPW spheres is 1.3 eV$^{-1}$ summed over the three Pd atoms, while the
Ta contribution is similar at 1.3 eV$^{-1}$ summed over the four Ta atoms.
This strongly argues against
nearness to magnetism in this compound.
\begin{figure}
\includegraphics[width=\columnwidth,angle=0]{band-figure.eps}
\caption{(color online) Band structure of Ta$_4$Pd$_3$Te$_{16}$
along the lines shown in the right panel. The band crossings
corresponding to the four sheets of Fermi surface are labeled with
Greek letters. The arrow denotes the anticrossing as discussed in the
text. The Fermi energy is at 0 eV.}
\label{bands}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth,angle=0]{fermi-figure3.eps}
\caption{(color online) Fermi surface Ta$_4$Pd$_3$Te$_{16}$.
The four sheets are labeled corresponding to the bands in Fig. \ref{bands}.}
\label{fermi}
\end{figure}
The band structure in the range around the Fermi energy, $E_F$ is given in
Fig. \ref{bands}. As shown, there are four bands crossing $E_F$,
labeled in the plot. These give rise to four sheets of
Fermi surface as depicted in Fig. \ref{fermi}.
The are a 2D hole cylinder (``$\alpha$"), two very 1D and
therefore nested sheets (``$\beta$" and ``$\gamma$")),
which may lead to nearness to
density wave instabilities, and a heavy 3D sheet (``$\delta$").
The 2D cylinder, $\alpha$, and the first 1D sheet, $\beta$ anticross making
the shapes of these surfaces more complex.
Nonetheless the nesting is at $\sim$ 0.1 $2\pi/c$
(from $\beta$) and $\sim$ 0.3 $2\pi/c$ (from $\gamma$),
i.e. along the direction of the metal chains.
The contributions to $N(E_F)$ from these are
1.4 eV$^{-1}$, 1.8 eV$^{-1}$, 1.0 eV$^{-1}$ and 5.4 eV$^{-1}$,
for sheets $\alpha$, $\beta$, $\gamma$ and $\delta$, respectively,
following the
labels of Fig. \ref{bands}.
In addition, there is an anticrossing with another Te
$p$ derived band very close to the Fermi energy
crossing of $\gamma$ as indicated by the
arrow in Fig. \ref{bands}.
This would come to the Fermi energy for very small levels of hole doping.
Such hole doping could come from Ta deficiency.
We emphasize that as is evident from the density of states, all of these
sheets of Fermi surface are derived from Te $p$ states and there is
little transition element $d$ character at the Fermi energy.
This implies that any nearby density wave instability due to nesting would be
a charge density wave (CDW) and not a spin density wave (SDW).
This density wave,
if it actually condensed, would consist of a structural modulation along
the short $c$-axis, metal chain direction, which should be clearly seen
in diffraction.
The mixture of 1D-like, 2D-like and 3D Fermi surfaces leads to
a net anisotropic, but 3D metal. The conductivity anisotropy was
estimated by integrating the transport function $\sigma/\tau$,
where $\tau$ is the scattering rate, taken as constant for all bands
and directions. The eigenvalues of the $\sigma/\tau$ tensor are
3.9 x 10$^{19}$ ($\Omega$ m s)$^{-1}$,
1.5 x 10$^{19}$ ($\Omega$ m s)$^{-1}$ and
10.5 x 10$^{19}$ ($\Omega$ m s)$^{-1}$,
with directions approximately
(note the monoclinic symmetry) in the layers, perpendicular to $c$,
approximately across the layers, and along $c$, respectively.
As noted, the nesting feature is along the $c$-axis direction.
This means that if a density wave condensed, or if there were strong
scattering associated with soft phonons related to the nesting, this
would primarily affect the nested sheets, and therefore the $c$-axis
direction conductivity.
We now turn to the details of the band structure. Because of the
low symmetry, the different bands have mixed orbital and atomic
character. The following is a description of the main characters.
The 2D $\alpha$ sheet is derived from a mixture of different Te $p$
states, and of the four sheets is the sheet with the most Ta $d$ character.
The 1D $\beta$ as mainly Te7 (Fig. \ref{struct}) character. This is
similar to the other 1D ($\gamma$) sheet, which in addition involves
Te1 and Te5 orbitals. The 3D, $\delta$ sheet, which is the sheet that
makes the largest contribution to $N(E_F)$ is mainly from Te3
$p$-orbitals, which are hybridized with Pd $d$ states.
Thus the different sheets of Fermi surface, and in particular
the nested quasi-1D sheets and the other sheets are associated
with different orbitals on different atoms.
\section{Discussion and Conclusions}
As mentioned, Ta$_4$Pd$_3$Te$_{16}$ is a low temperature superconductor,
with experimental signatures of possible unconventional behavior.
One such signature is an inconsistency between the
superconducting $\lambda_{sc}$ inferred from the specific heat jump,
which implies weak coupling behavior, and that inferred from the
enhancement of the low temperature specific heat,
$\gamma=\gamma_{bare}(1+\lambda_{tot})$. \cite{jiao}
In particular, Jiao and co-workers measured the specific heat jump as
$\Delta C/(\gamma T)$=1.4 $\pm$ 0.2 consistent with the weak coupling
BCS value of 1.43, implying low $\lambda_{sc}$.
This included a careful correction for the superconducting volume fraction.
They also inferred $\lambda_{tot}$=2.3 by combining the measured specific
heat $\gamma$ with density of states $N(E_F)$=5.5 eV$^{-1}$ from
the report of Alemany and co-workers. \cite{alemany}
However, we obtain a higher $N(E_F)$=9.6 eV$^{-1}$, which then gives a
correspondingly lower renormalization and $\lambda_{tot}$=0.89. This is
quite reasonable considering the specific heat jump. Therefore it
may not be necessary to consider an enhancement to $\lambda_{tot}$
due to non-phonon channels such as spin fluctuations to reconcile
the $\gamma$ with the specific heat jump.
We note that the transition metal $d$ contribution to $N(E_F)$
is low, which argues against spin-fluctuations.
Also, it should be noted that the Fermi surface consists of four
distinct sheets with contributions from different atoms.
Therefore it is reasonable to expect that the pairing interactions
could be stronger on some sheets than others, which would complicate
the simple interpretation of the specific heat jump using a single
band BCS formula.
From the point of view of superconductivity, nearness to
a CDW can provide an attractive interaction through the electron
phonon interaction. If this is the main
pairing interaction the gap would be largest on the
nested Fermi surface sheets. This type of interaction favors a singlet
state and unless there is an additional repulsive interaction,
such as spin-fluctuations or a strong Coulomb repulsion,
will not lead to a state with any sign changes in the order parameter.
Spin fluctuations associated with nearness to an SDW resulting
from the nested 1D sheets could lead to a specific heat enhancement,
i.e. a larger $\lambda_{tot}$, as discussed by Jiao and co-workers.
\cite{jiao}
As mentioned, this may not be needed considering the value of the
bare $N(E_F)$ from our calculations.
In any case, such an interaction cannot lead to superconductivity on
the nested sheets with the Fermi surface structure of this compound.
The reason is that in a singlet channel, spin fluctuations are
repulsive, which would favor an order parameter that changes sign
between the nested sheets on opposite sides of the zone center, i.e.
a triplet state, while in a triplet channel spin fluctuations are
attractive, which would favor an order parameter that is the same
on opposite sides of the zone center, i.e. not a triplet.
Besides the possibility of pairing due to nearness to a CDW, it is
to be noted that there are other layered tellurides, in particular
Ir$_{1-x}$Pt$_x$Te$_2$ that become superconducting
\cite{pyon}
and which do not have Fermi surface nesting features that
lead to phonon softening.
\cite{cao}
Thus it may well be that Ta$_4$Pd$_3$Te$_{16}$ is also an s-wave
superconductor with pairing from phonons associated with the
Te-Te $p$ bonding and a Fermi surface associated Te $p$ bands.
Recently, Pan and co-workers reported thermal conductivity measurements
on a single crystal sample with a low residual resistivity below
4 $\mu$$\Omega$ cm.
They found a large linear electronic contribution persisting down to
$T$=80 mK, and furthermore that this contribution increases with
field similar to the behavior of a cuprate superconductor.
This implies the presence of ungapped parts of the Fermi surface,
such as line nodes. The data are reminiscent of cuprates and also certain of the
Fe-based superconductors, for which there is other evidence
of line nodes. \cite{hashimoto}
The observed behavior is inconsistent with the behavior of the two gap
superconductor NbSe$_2$. On the other hand, it could be compatible
with a clean multigap superconductor where the gap ratio is larger,
and in fact that data has some similarity to MgB$_2$,
\cite{sologubenko}
which is an s-wave electron phonon superconductor.
Considering the magnitude of the linear thermal conductivity
seen experimentally and using the Wiedemann-Franz relation,
one would have to assume that parts of Fermi surface contributing $\sim$ 30\%
of the conductivity have very small gaps, in order to explain the
data without line nodes. This is possible considering the Fermi surface
structure, in which several different sheets contribute to the
conductivity and to $N(E_F)$.
The fact that these sheets derive from $p$ states associated
with different Te atoms makes more plausible large differences in the coupling
on different sheets, which may then lead to large
differences in the gaps for a clean sample.
It will be of interest to study samples with higher levels
of disorder in order to distinguish these two possibilities.
Specifically, with line nodes, disorder is expected to suppress
the ordering temperature, while in the multiband case disorder
would enhance the gap on the low gap parts of the Fermi surface
and thereby suppress the electronic thermal conductivity at low temperature.
Also, we note that the electronic thermal conductivity has the same
anisotropy as the electronic conductivity from the
ungapped parts of the Fermi surface. We find sheets with
very different anisotropies, 1D sheets, a 2D sheet and a 3D sheet.
Therefore, in a scenario in which there is a small gap on some
part, the particular sheets involved in the electronic
thermal conductivity (i.e. the low gap sheets) could be identified from the
anisotropy of the electronic part of the thermal conductivity in the
superconducting state.
To summarize, electronic structure calculations for
Ta$_4$Pd$_3$Te$_{16}$ show four sheets of Fermi surface
derived primarily from Te $p$ states. Importantly, the transition
metal contribution to the density of states is too low to place the
system near magnetism.
The calculated value of $N(E_F)$=9.6 eV$^{-1}$ is higher than
that from a prior calculation \cite{alemany}, which can resolve
a discrepancy between the specific heat jump and the specific
heat renormalization. The Fermi surface includes
nested 1D-like sections, a 2D-like section and a heavy
3D section, which makes the largest contribution to $N(E_F)$.
The net result is a rather anisotropic but 3D metal.
Thus Ta$_4$Pd$_3$Te$_{16}$ is a multiband superconductor with
an electronic structure derived mostly from Te $p$ states.
\acknowledgments
This work was supported by the U.S. Department of Energy,
Basic Energy Sciences, Materials Sciences and Engineering Division.
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\section*{Reviewer \thereviewer}}
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\begin{document}
\section*{Response to the Reviewers}
\textit{Original Manuscript ID:} \textbf{TMC-2021-03-0227.R1}
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\textit{Original Article Title:} \textbf{Cooperative Task Offloading and Block Mining in Blockchain-based Edge Computing with Multi-agent Deep Reinforcement Learning}
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\textit{To: }The Editor of IEEE Transactions on Mobile Computing
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\textit{Re:} Response to reviewers
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Dear Editor,
Thank you for allowing us to perform a minor revision to our manuscript. We have carefully addressed the remaining comments of Reviewers 1 and 4 with the following key updates:
\begin{itemize}
\item We have enhanced the discussions to better clarify the novelty and contributions of our paper and clearly explain the limitations of existing schemes. The advantages of using the proposed scheme have also been highlighted.
\item We have provided more discussions to explain on why only time consumption is considered in mining utility.
\item We have added more references that are highly related to edge computing and blockchain.
\end{itemize}
We are uploading (1) a revised manuscript with changes highlighted in blue, (2) the revised manuscript without highlights, and (3) the point-by-point responses to the comments (below).
Further, we would like the editor to assign our article to the same set of reviewers for further review. We are looking forward to receiving your comments.
Best regards,
\\
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\textbf{Authors:} \textit{Dinh C. Nguyen (on behalf of all authors)}
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\reviewersection \textbf{Author Should Prepare A Minor Revision}
\begin{point}
The revision looks mostly good to me, however, I still have some concerns that require the authors to address. 1. There are a lot of related references missing. The paper studied the edge computing-enabled blockchain networks. However, the authors even missed the first series of works on ``edge computing meets mobile blockchain". Please carefully check the literature studies and cite them accordingly.
\end{point}
\begin{response}
Thank you very much for your comment. We have added more new references that are highly related to ``edge computing meets mobile blockchain", according to your suggestion. They are references [5], [6], [7] in this revised version.
\end{response}
\begin{point}
The game theory methods that only prove the existence of the NE seems only marginal. What about the uniqueness. Otherwise, it is only sub-optimal solution.
\end{point}
\begin{response}
Thank you very much for your insightful comment. In fact, in game theory, the uniqueness of Nash Equilibrium is a desired property of games, but this cannot be ensured in many cases due to the randomizations of mutual cooperation of players [1]. Even for games in extensive forms there may be multiple Nash Equilibriums. This lead to multiple strategies NE that may be varying in games with different pure strategies. However, according to the related offloading works on game theory [2], [3], the most important thing is to find a strategy for players that must exist at least on NE of its game to satisfy their utility expectation with respect to a given strategy. In our paper, we have already proved the existence an NE given the offloading and mining strategy, and it is feasible to derive an efficient utility optimization solution for our proposed problem.
\\
\textit{Reference:}
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$[1]$ Harsanyi, John C., and Reinhard Selten. 1998. A General Theory of Equilibrium Selection in Games. Cambridge, MA: MIT Press.
\\
$[2]$ Z. Ning, P. Dong, X. Wang, X. Hu, J. Liu, L. Guo, B. Hu, R. Kwok, and V. C. M. Leung, ``Partial Computation Offloading and Adaptive Task Scheduling for 5G-enabled Vehicular Networks," IEEE Transactions on Mobile Computing, pp. 1-1, 2020.
\\
$[3]$ Guo, Hongzhi, and Jiajia Liu, ``Collaborative computation offloading for multiaccess edge computing over fiber-wireless networks," IEEE Transactions on Vehicular Technology, 67, no. 5, pp 4514-4526, 2018.
\end{response}
\begin{point}
The proposed MA-DDPG method needs more elaborations regarding its novelties since it has been applied in many studies. For example, how does the proposed model make it challenging to solve the problem with DQN. This is to highlight the unique technical contributions.
\end{point}
\begin{response}
Thank you very much for your comment. We have enhanced the discussions to clarify better the novelties of the proposed MA-DDPG method. Accordingly, we have pointed out the limitations of existing learning approaches with DQN and highlighted better the advantages of the proposed scheme. We summarize these updates in Section 1.2 of this revised version as follows:
{\color{blue} In distributed blockchain-based MEC systems, traditional single-agent DRL algorithms like DQN [13], [14], [22], [26], [32] face critical challenges caused by diversified and time-varying local environments. In more details, during the training process of DQN, each agent only observes its local information and cannot know the updates from other agents due to non-collaboration. This makes it hard to ensure the stability and convergence of the agents' algorithm [33]. Moreover, this breaks the Markov properties required by the Q-learning algorithm and thus, DQN may not be capable of learning the cooperative offloading policies of EDs. Moreover, the non-cooperative multi-agent DRL solutions [17]-[19] may not be able to learn the cooperative policy; and thus resource usage over the edge network is not efficient which limits the overall offloading performance, e.g., offloading utility.
The proposed MA-DDPG approach enables the efficient learning of the mutual policy among cooperative EDs in the dynamic environment and high-dimensional system state space. Indeed, the proposed MA-DDPG scheme allows EDs to learn mutually the cooperative offloading and mining policy which helps enhance the computation efficiency and thus improves the system utility.}
\end{response}
\reviewersection \textbf{Accept With No Changes}
\begin{point}
The authors have addressed all the reviewers' concerns carefully, therefore, I recommend accept.
\end{point}
\begin{response}
Thank you very much for accepting our paper.
\end{response}
\reviewersection \textbf{Accept With No Changes}
\begin{point}
The authors have addressed the reviewers' comments. The quality of the presentation and the scientific depth of the manuscript have been substantially improved.
\end{point}
\begin{response}
Thank you very much for accepting our paper.
\end{response}
\reviewersection \textbf{Author Should Prepare A Minor Revision}
\begin{point}
In offloading utility, both energy and time consumption are considered, but only time consumption is considered in mining utility, why? Please add explanation.
\end{point}
\begin{response}
Thank you very much for your comment. In terms of block mining in the blockchain, Time to Finality/Verification Latency is a much more important performance factor compared to energy in the evaluation of the efficiency of a mining mechanism in blockchain. Indeed, it is extremely important to achieve an acceptable block verification delay to timely record a block to the blockchain, aiming to prevent the transactions to be arbitrarily changed or reversed that is often caused by the long block verification process. Therefore, in this paper, we consider the time consumption on block verification as the key mining utility evaluation factor. Our design concept is also consistent with the related works on blockchain such as [1] (see Section V-B) and [2] (see Section V-A). For the reviewer's convenience, we have provided a new update to clarify this design concept, as shown in Section 2.1 of this revised version:
\textcolor{blue}{In this paper, we propose a PoR framework and focus on analyzing the block verification latency that is a significant factor in evaluating the efficiency of a blockchain network. }
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\textit{Reference:}
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$[1]$ F. Guo, F. R. Yu, H. Zhang, H. Ji, M. Liu, and V. C. M. Leung, ``Adaptive Resource Allocation in Future Wireless Networks With Blockchain and Mobile Edge Computing," IEEE Transactions on Wireless Communications, vol. 19, no. 3, pp. 1689-1703, Mar. 2020.
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$[2]$ J. Kang, Z. Xiong, D. Niyato, D. Ye, D. I. Kim, and J. Zhao, ``Toward Secure Blockchain-Enabled Internet of Vehicles: Optimizing Consensus Management Using Reputation and Contract Theory," IEEE Transactions on Vehicular Technology, vol. 68, no. 3, pp. 2906-2920, 2019.
\end{response}
\begin{point}
Contributions should be more clear. The paper proposes a novel scheme, but why do people choose the scheme? what's the benefit for society?
\end{point}
\begin{response}
Thank you very much for your comment. We have enhanced the discussions to clarify the paper contributions and explain better the motivations of using the proposed scheme. Accordingly, we have pointed out the limitations of existing learning approaches and highlighted better the advantages of the proposed scheme. The benefits of our scheme for society have also been highlighted. We summarize these updates in Section 1.2 and Section 6 as follows:
{\color{blue}
In distributed blockchain-based MEC systems, traditional single-agent DRL algorithms like DQN [13], [14], [22], [26], [32] face critical challenges caused by diversified and time-varying local environments. In more details, during the training process of DQN, each agent only observes its local information and cannot know the updates from other agents due to non-collaboration. This makes it hard to ensure the stability and convergence of the agents' algorithm [33]. Moreover, this breaks the Markov properties required by the Q-learning algorithm and thus, DQN may not be capable of learning the cooperative offloading policies of EDs. Moreover, the non-cooperative multi-agent DRL solutions [17]-[19] may not be able to learn the cooperative policy; and thus resource usage over the edge network is not efficient which limits the overall offloading performance, e.g., offloading utility.
The proposed MA-DDPG approach enables the efficient learning of the mutual policy among cooperative EDs in the dynamic environment and high-dimensional system state space. Indeed, the proposed MA-DDPG scheme allows EDs to learn mutually the cooperative offloading and mining policy which helps enhance the computation efficiency and thus improves the system utility.
Our proposed approach has potential for future intelligent mobile networks, where EDs are able to build distributed intelligent solutions via our cooperative DRL model for enabling intelligent computation, communications and network control. For example, our scheme can be applied to cooperative edge intelligence scenarios, e.g., mobile edge caching and block mining, cooperative spectrum learning and resource management in next-generation wireless networks [44].}
\end{response}
\end{document}
\section*{Reviewer \thereviewer}}
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\begin{document}
\section*{Response to the Reviewers' Comments}\vspace*{0.8cm}
\begin{tabular}{lp{11cm}}
\textbf{Original Manuscript ID} & {NETWORK-21-00188} \tabularnewline
[0.2cm] \textbf{Original Article Title} & {Intelligent Blockchain-based Edge Computing via Deep Reinforcement Learning: Solutions and Challenges}\tabularnewline
[0.2cm] \textbf{To} & Editor of IEEE Network \tabularnewline
[0.2cm] \textbf{Re} & Response to Reviewers \tabularnewline
[0.2cm] & \tabularnewline
\end{tabular}
\noindent Dear Editor, \\
\noindent Thank you for allowing us to perform a revision to our manuscript. We would also like to thank the anonymous Reviewers for their constructive comments on the previous manuscript version, which have been very helpful in improving the quality of our manuscript. We have carefully addressed the Reviewers' comments with the following key updates:
\begin{itemize}
\item We have enhanced the discussions to better clarify the DRL algorithm design and the training procedure.
\item We have explained better the mining concept and its utility formulation.
\item We have provided more simulations about mining utility and blockchain throughput.
\item We have added more references that are highly related to edge computing and blockchain.
\end{itemize}
\noindent We are uploading (1) a revised manuscript with changes highlighted in blue and (2) the point-by-point responses to the comments (below).\\
\noindent Further, we would like the Editor to assign our article to the same set of reviewers for further review. We are looking forward to receiving your comments. \\
\noindent Best regards,
\\
\\
\textbf{Authors:} \textit{Dinh C. Nguyen (on behalf of all authors)}
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\reviewersection
\begin{point}
The paper proposes a novel cooperative task offloading and blockchain mining (TOBM) scheme for a blockchain-based MEC system, where each edge device not only handles computation tasks but also deals with block mining for improving system utility. The paper concludes with key technical challenges and possible directions for future blockchain based MEC research. 1. Figure 1 of the article is not artistic and some of the text is blocked by the figure, so it is suggested to modify it.
\end{point}
\begin{response}
Thank you very much for your comment. We have updated Fig. 1 in the revised manuscript to make it clearer.
\end{response}
\begin{point}
The indexes in Table 1 are not the normal description of algorithm performance, and some advantages of the scheme proposed by the author may not necessarily represent the overall performance of the algorithms.
\end{point}
\begin{response}
Thank you very much for your comment. We have removed this table as it is not necessary as per your comment. Several performance limitations of existing schemes and our comparison have been described in Section I-A.
\end{response}
\begin{point}
This paper describes the proposed algorithm in steps, which is not clear enough. It is suggested that the author refine this part or use block diagram to describe it.
\end{point}
\begin{response}
Thank you very much for your comment. We have provided a block diagram for the proposed algorithm in Fig. 3(b) of the revised manuscript, as shown in below Figure 1.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.99\linewidth]{Image/Diagram_DDPG.pdf}
\caption{\textcolor{blue}{The MA-DDPG training procedure. }}
\label{MADRL}
\vspace{-0.1in}
\end{figure*}
\end{response}
\begin{point}
The main content of this article is AL aided edge computing and blockchain technology. The author can refer to some articles for details, "Thirty Years of Machine Learning: The Road to Pareto-Optimal Wireless Networks", and "Resource trading in blockchain-based industrial Internet of Things".
\end{point}
\begin{response}
Thank you very much for your comment. We have cited these useful papers as references [3] and [7] in the revised manuscript.
\end{response}
\begin{point}
There is a lack of a thorough analysis and proper design of the holistic system optimization accounting for incentives, offloading, and mining.
\end{point}
\begin{response}
Thank you very much for your insightful comment. Due to the limited space of a magazine paper, we would like to focus on analysing the system utility optimization for offloading and mining in this paper, while the joint consideration with incentives will be reported in future works. In this revised manuscript, we have added a new interesting research direction related to incentives, offloading, and mining, which pave the way for future works. We would summarize our new
update in Section V-C as below:
\textcolor{blue}{In practice, how to encourage EDs, which serve as both offloading and mining nodes, to join computation and data mining for the long term is a critical challenge for B-MEC systems. Although blockchain is able to incentivize EDs via coin payment based on their mining effort, it is still not enough to compensate for the energy resources consumed for computation task offloading. Thus, it is important to jointly consider incentives, offloading and mining in B-MEC system optimization. A possible direction is to design a smart contract-inspired incentive mechanism to accelerate the data offloading and block mining. Another interesting area is to jointly optimize utility with respect to monetary benefits and resource usage of offloading and mining.}
\end{response}
\reviewersection
\begin{point}
This paper proposes a cooperative task offloading and blockchain mining scheme in a blockchain-based mobile edge computing system. A multi-agent deterministic policy gradient approach is applied to determine the task data size, channel condition and transmit power level of each edge device. Simulations are performed to show that the proposed scheme can reduce the blockchain network, energy consumption and task computation time. Here are the detailed comments: How does the IoT user obtains the mining latency of edge devices for voting in Sec. 2?
\end{point}
\begin{response}
Thank you very much for your comment. We have considered the mining latency as the factor to determine the reputation of an ED. An ED that exhibits a lower mining latency will have a better reputation via a predefined mining utility function. Based on the calculated reputation score, each IoT user votes for ED candidates based on their reputation ranking. In the revised manuscript, we have added the following texts in Section II-B:
\textcolor{blue}{In this regard, each IoT user votes for its preferred ED with the most reputation based on its mining latency. Specifically, an ED that exhibits a lower mining latency will have a better reputation via a predefined mining utility function. Based on the calculated reputation score, each IoT user votes for ED candidates based on their reputation ranking.}
\end{response}
\begin{point}
What is the difference in components between the state set and the observation set in Sec. 3?
\end{point}
\begin{response}
Thank you very much for your comment. In fact, at each time slot, an MD does not always have access to all states of the environment, but only observes certain states, which are called observations. While states are complete and detailed information that is relevant to the current task (e.g., which specific wireless channel is occupied by a certain MD for offloading), observations are general information received by an MD (e.g., how many wireless channels are occupied). We have updated Section III-A on the revised manuscript as follows:
\textcolor{blue}{In fact, at each time slot, an MD does not always have access to all states of the environment, but only observes certain states, which is called observations. While states are complete and detailed information that are relevant to the current task (e.g., which specific wireless channel is occupied by a certain MD for offloading), observations are general information received by an MD (e.g., how many wireless channels are occupied).}
\end{response}
\begin{point}
Which learned parameters are downloaded to each edge device in Sec. 3?
\end{point}
\begin{response}
Thank you very much for your comment. There are neural weights of DNNs of the actor and critic. We would summarize our update in Section III-A as below:
\textcolor{blue}{After training at the MEC server, the learned parameters (i.e. neural weights of DNNs of the actor and critic) are downloaded to each ED to execute the model for decision making based on its own locally observed information. }
\end{response}
\begin{point}
How does each agent obtain the observations of all other agents in Sec. 3?
\end{point}
\begin{response}
Thank you very much for your comment. We have adopted the centralized learning and decentralized execution solutions to obtain observations of other agents. In the centralized training step, the information of state-action of all EDs is aggregated by the MEC server to train the DRL model, where each agent can achieve the global view of the learning environment to obtain observations of other agents to collaboratively build the offloading policy. We would summarize our update in Section III-B as below:
\textcolor{blue}{We adopt the centralized learning and decentralized execution solution. In the centralized training step, the information of state-action of all EDs is aggregated by the MEC server to train the DRL model, where each agent can achieve the global view of the learning environment to obtain the observations of other agents for building the collaborative offloading policy.}
\end{response}
\begin{point}
How to execute the model for decision making by using the downloaded learned parameters in Sec.3?
\end{point}
\begin{response}
Thank you very much for your comment. Given downloaded neural weights, each agent can easily compute the policy via its DNN with system utility used as the objective function, and then sample an action. Subsequently, each agent executes the sampled action, i.e., making decision to offload the data or not, in the defined TOBM environment given its states to obtain a reward. We would summarize our update in subsection III-B as below:
\textcolor{blue}{After training at the MEC server, the learned parameters (i.e. neural weights of DNNs of the actor and critic) are downloaded to each ED to execute the model for decision making based on its own locally observed information. Specifically, given the downloaded neural weights, each agent can easily compute the policy via its DNN with system utility used as the objective function, and then sample an action. Subsequently, each agent executes the sampled action, i.e. making decision to offload the data or not, in the defined TOBM environment given its states to obtain a reward.}
\end{response}
\begin{point}
Please compare the proposed scheme with the state-of-the-art cooperative schemes.
\end{point}
\begin{response}
Thank you very much for your comment. We have compared with cooperative schemes with DPoS and cooperative scheme without mining design in Fig. 5(b). Also, we have added new simulation among these cooperative schemes in terms of mining utility in Fig. 6(a) in the revised manuscript as below Figure 2. We summarize this update as below:
\textcolor{blue}{We compare the mining utility of our scheme with other cooperative schemes in Fig. 6(a). Due to a lower mining latency achieved by the proposed consensus design, our scheme yields a better utility compared with its counterparts. The traditional approach without mining design has the highest mining latency, resulting in the lowest utility.}
\begin{figure*}[t!]
\centering
\includegraphics[width=0.99\linewidth]{Image/Image-Mining-Utility-2022-05-15-174603.pdf}
\caption{{Comparison of mining utility with cooperative schemes. }}
\label{Fig:blockchainper}
\vspace{-0.1in}
\end{figure*}
\end{response}
\begin{point}
Why is DDPG selected to optimize the blockchain-based mobile edge computing system utility?
\end{point}
\begin{response}
Thank you very much for your comment. The key reason behind the adaption of DDPG to optimize the system utility is that with this scheme, the actor can directly map states to actions instead of outputting the probability distribution across a discrete action space like DQN, which greatly reduces action sampling complexity. Moreover, given the stochasticity of the policy, there exists high variance of the obtained system reward between different training episodes, where DDPG can come as an efficient solution, by enabling off-policy learning via the joint use of behavior network and target network. We would summarize our update in Section III-B as below:
\textcolor{blue}{The key reason behind the adaption of DDPG to optimize the system utility is that with this scheme, the actor can directly map states to actions instead of outputting the probability distribution across a discrete action space like in other schemes like DQN, which greatly reduces action sampling complexity. Moreover, given the stochasticity of the policy, there exists high variance of the obtained system reward between different training episodes, where DDPG can come as an efficient solution, by enabling off-policy learning via the joint use of behavior network and target network.}
\end{response}
\begin{point}
Some related works are worth citing, e.g., A reinforcement learning and blockchain-based trust mechanism for edge networks, IEEE Trans. Comm., 2020.
\end{point}
\begin{response}
Thank you very much for your comment. We have cited it as reference [8] in this revised version.
\end{response}
\reviewersection
\begin{point}
A new Proof-of-Reputation consensus mechanism based on a lightweight block verification strategy is proposed in this paper. To accommodate the highly dynamic environment and high-dimensional system state space, a distributed deep reinforcement learning-based approach is developed by using a multi-agent deep deterministic policy gradient algorithm. Overall, the paper is well written. Some issues should be further addressed: 1. The comparison between PoR and other consensus schemes including PoW and PoS and their variants in different perspectives such as throughput, security.
\end{point}
\begin{response}
We would like to thank the Reviewer for nicely summarizing our work and the positive feedback. Motivated by your suggestion, we have added a new simulation to compare throughput of these schemes in Figure 3, while the analysis of security will be studied in our future work due to the limited space of this magazine. We summarize this update in Section IV as below.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.99\linewidth]{Image/Image-Throughput-2022-05-15-173950.pdf}
\caption{{Comparison of blockchain throughput with mining schemes. }}
\label{Fig:blockchainper}
\vspace{-0.1in}
\end{figure*}
\textcolor{blue}{Moreover, we investigate the throughput in Fig. 6(b) which is defined as the number of successful transactions per second. We set five transactions per block and make offloading requests ranging from 10 to 200. Compared with DPoS and Proof-of-Work (PoW) schemes, the fast block verification rate of our scheme significantly enhances the throughput, before its performance decreases when the number of requests is higher than 120 since the system cannot handle excessive requests under our configuration.}
\end{response}
\begin{point}
Is the cooperative DRL Algorithm robust to malicious agent who would attack the learning process?
\end{point}
\begin{response}
Thank you very much for your comment. The main purpose of the proposed cooperative DRL algorithm is the adoption of cooperation of multiple edge devices to build a better offfloading policy for better system utility maximization. We also believe that the cooperative learning configuration might be robust to malicious attacks since it allows multiple agents to monitor the shared policy update via observations, where an abnormal update from a certain agent can be detected via the training outcome (e.g., abnormal changes in reward performance at a certain training episode). We would like to summarize this update in subsection III-B in the revised version as below:
\textbf{The cooperation of multi-agents helps improve robustness against malicious attacks since this allows multiple agents to monitor the shared policy update via observations [15].}
\end{response}
\reviewersection
\begin{point}
This paper proposed a novel cooperative task offloading and blockchain mining scheme for blockchain-based MEC systems, where a PoR consensus mechanism was proposed. This paper is novel and the interesting.
\end{point}
\begin{response}
We would like to thank the Reviewer for nicely summarizing our work and the positive feedback.
\end{response}
\begin{point} Please describe more details about PoR, making it more clear.
\end{point}
\begin{response}
Thank you very much for your comment. We have added more descriptions to make it clearer. We would summarize our update in subsection II-B as below:
\textcolor{blue}{In this regard, each IoT user votes its preferred ED with the most reputation based on its mining latency. Specifically, an ED that exhibits a lower mining latency will have a better reputation via a predefined mining utility function. Based on the calculated reputation score, each IoT user votes for ED candidates based on their reputation ranking.}
\textcolor{blue}{Specifically, the block manager first divides the block into a set of equal transaction parts that are assigned to each miner within the miner group along with a unique random number. Next, the miner chooses to associate with one of the miners within its group to implement the verification for its assigned transaction part by allocating its CPU resource. If 51\% of miners respond with positive verification, and the sum of random numbers calculated by all miners is equal to a predefined number, the block manager accepts the verified block and adds it to blockchain. In summary, the mining procedure includes four stages: (1) transmitting unverified block from the block manager to the EMs, (2) verifying the local block at each EM, (3) sharing the verification result among two EMs, and (4) transmitting the verification result back to the manager.}
\end{response}
\begin{point}
Please give out the formulation of offloading utility and mining utility.
\end{point}
\begin{response}
Thank you very much for your comment. We have explained better the formulation of offloading utility and mining utility in Section III in this revised version. We summarize this update as below.
\textcolor{blue}{We here formulate an offloading utility function from the QoE perspective, which is characterized by the task computation time (including local time and offloading time) and energy consumption (including local energy and offloading energy). We define a QoE-aware offloading utility function $J_n^{off}$ to provide a trade-off between the time and energy consumption of the task offloading compared with the local execution at each ED. The offloading utility function reflects the offloading improvement in QoE over the local execution.}
\textcolor{blue}{To this end, we build a mining utility function to characterize the mining efficiency of the proposed scheme. Motivated by [15], we characterize the mining utility $J_n^{mine}$ of each ED $n$ via an exponential function where the mining utility is inversely proportional to the mining latency. Accordingly, an ED that exhibits a lower mining latency has a better mining utility with respect to a CPU resource allocation policy.}
\end{response}
\begin{point}
Please describe more details about MADDPG.
\end{point}
\begin{response}
Thank you very much for your comment. We have added more descriptions to make it more detailed in Section III. We summarize it as below:
\textcolor{blue}{The key reason behind our adaption of DDPG to optimize the system utility is that with this scheme, the actor can directly map states to actions instead of outputting the probability distribution across a discrete action space like DQN, which greatly reduces action sampling complexity. Moreover, given the stochasticity of the policy, there exists high variance of the obtained system reward between different training episodes, where DDPG can come as an efficient solution, by enabling off-policy learning via the joint use of behavior network and target network.}
\textcolor{blue}{We adopt a centralized learning and decentralized execution solution. In the centralized training step, the information of state-action of all EDs is aggregated by the MEC server to train the DRL model, where each agent can achieve the global view of the learning environment to obtain the observations of other agents for building the collaborative offloading policy.} \textcolor{blue}{After training at the MEC server, the learned parameters (i.e. neural weights of DNNs of the actor and critic) are downloaded to each ED to execute the model for decision making based on its own locally observed information. Specifically, given the downloaded neural weights, each agent can easily compute the policy via its DNN with system utility used as the objective function, and then sample an action. Subsequently, each agent executes the sampled action, i.e. making decision to offload the data or not, in the defined TOBM environment given its states to obtain a reward.}
\end{response}
\begin{point}
Please add the simulation about mining utility.
\end{point}
\begin{response}
Thank you very much for your comment. We have added a new simulation result about mining utility in Section IV, as shown in below Figure 4.
\textcolor{blue}{We compare the mining utility of our scheme with other cooperative schemes in Fig.~\ref{Fig:blockchainper}(a). Due to a lower mining latency achieved by the proposed consensus design, our scheme yields a better utility compared with its counterparts. The traditional approach without mining design has the highest mining latency, resulting in the lowest utility. Moreover, we investigate the throughput in Fig.~\ref{Fig:blockchainper}(b) which is defined as the number of successful transactions per second. We set five transactions per block and make offloading requests ranging from 10 to 200. Compared with DPoS and Proof-of-Work (PoW) schemes, the fast block verification rate of our scheme significantly enhances the throughput, before its performance decreases when the number of requests is higher than 120 since the system cannot handle excessive requests under this configuration.
}
\begin{figure*}[t!]
\centering
\includegraphics[width=0.99\linewidth]{Image/Image-Mining-Utility-2022-05-15-174603.pdf}
\caption{{Comparison of mining utility with cooperative schemes. }}
\label{Fig:blockchainper}
\vspace{-0.1in}
\end{figure*}
\end{response}
\end{document}
\section{Introduction}
Recent advances in wireless Internet-of-Things (IoT) have promoted the proliferation of mission-critical applications, e.g., augmented reality and autonomous driving, which rely heavily on edge devices (EDs) to collect data from IoT sensors to serve end users. To meet the ever-growing computation demands of EDs, mobile edge computing (MEC) has been proposed as a promising technique to improve the computation experience of EDs, by offloading computationally-intensive IoT tasks to a nearby MEC server located at a base station (BS) \cite{1}. Multiple EDs can share computation and communication resources of the BS to handle data tasks without device's battery depletion. Task offloading with MEC thus becomes a viable solution to satisfy various EDs' computation demands, {thus enhancing} the \textcolor{black}{quality-of-experience} (QoE) of end users. Furthermore, to provide security in MEC systems, blockchain \cite{3} has emerged as a strong candidate due to its decentralization, immutability, and traceability, which forms blockchain-based MEC (B-MEC) paradigms \cite{4}. Also, blockchain can build {trusted B-MEC schemes} by employing community verification among network entities (e.g., EDs) via mining mechanisms such as Delegated Proof of Stake (DPoS) \cite{add1} without requiring a central authority.
\begin{figure}
\centering
\includegraphics[width=0.95\linewidth]{Image/Overview3.pdf}
\caption{\textcolor{black}{The proposed cooperative task offloading and block mining architecture in the B-MEC system. }}
\label{Overview}
\vspace{-0.1in}
\end{figure}
{In this context, how to ensure high performance, e.g., system utility, for the B-MEC system is a critical challenge. The task offloading process between EDs and the MEC server consumes much energy and latency, while the operation of blockchain results in delays in the offloading due to mining task execution, which would degrade the overall system utility. Hence, it is paramount to simultaneously consider both task offloading and blockchain mining via a joint design and optimization solution, aiming to enhance the system utility of the B-MEC system.}
\subsection{{Existing Solutions for Intelligent Performance Optimization in B-MEC} }
{To achieve intelligent performance optimization in B-MEC systems, different solutions have been proposed in the open literature.} The authors in \cite{4} considered a blockchain-empowered computation offloading scheme where smart devices can offload their computing tasks to the MEC server under the control of blockchain mining for data integrity. Another work in \cite{5} suggested an online computation offloading approach for both data processing and mining tasks in blockchain-empowered MEC with a deep reinforcement learning (DRL) algorithm {\cite{add2}}, \cite{xiao2020reinforcement}. An intelligent offloading framework with actor-critic DRL was also proposed in \cite{6}, while the study in \cite{7} focused on joint optimization of computation offloading and resource allocation using a double-dueling deep Q-network (DQN) for blockchain-enabled MEC systems. \cite{8} developed a computation offloading framework for blockchain-based IoT networks with a multi-agent DRL algorithm \cite{add3}.
Despite these research efforts, there are still several urgent issues to be addressed:
\begin{itemize}
\item \textit{Non-cooperative Offloading:} Most of the existing B-MEC schemes use traditional single-agent DRL algorithms \cite{4,5,6} which \textcolor{black}{exhibit} critical design challenges caused by diversified learning environments. Indeed, each agent only observes its local information in the training without updating the policies of other agents, which makes the learning environment nonstationary \cite{lowe2017multi}. Moreover, non-cooperative multi-agent DRL solutions \cite{7} are not able to learn the mutual offloading policy, which limits the resource utilization for task offloading.
\item \textit{High Blockchain Latency:} The integration of blockchain in MEC potentially results in unnecessary network latency due to block mining \cite{6,7} (i.e., block verification and consensus) which can degrade the overall performance of the B-MEC system.
\item \textit{Lack of Joint Offloading and Mining Design:} In most current B-MEC schemes \cite{6,7,8}, the design and optimization of task offloading and blockchain mining are done separately, leading to a suboptimal performance. To improve {the overall performance of the B-MEC system,} a joint design of task offloading and blockchain mining is highly needed.
\end{itemize}
\subsection{Our Key Contributions}
Motivated by the aforementioned limitations, we propose a novel cooperative DRL solution for joint task offloading and blockchain mining (TOBM), aiming to maximize the overall B-MEC system utility as a sum of the offloading utility and the mining utility. Our main contributions are highlighted as follows:
\begin{itemize}
\item We propose a novel cooperative TOBM scheme for B-MEC to enable a joint design of task offloading and blockchain mining in Section II. To reduce the blockchain network latency, we develop a new Proof-of-Reputation (PoR) mining mechanism via a lightweight block verification solution.
\item We propose a novel cooperative DRL solution in Section III using a multi-agent deep deterministic policy gradient (MA-DDPG) approach \cite{9} to optimize the system utility.
\item {We implement simulations to verify the effectiveness of our proposal in Section IV. We highlight the technical challenges in B-MEC research and discuss several directions for future works in Section V. }
\end{itemize}
\section{{Blockchain-Empowered MEC System}}
\subsection{Overview of Network Architecture}
\textcolor{black}{We consider a {cooperative TOBM architecture} in a B-MEC system, as illustrated in Fig.~\ref{Overview}. {An MEC server co-located at a BS provides} computation services for EDs. We assume that each ED has an IoT data task to be executed locally or offloaded to the MEC server. Furthermore, each ED participates in the block mining by using a PoR consensus mechanism.} The key network components of the B-MEC system are described as follows.
\begin{itemize}
\item \textit{IoT Sensors:} IoT sensors such as cameras, smart meters, and wearables are responsible for sensing physical environments and generating data which need to be processed to serve end users. IoT sensors also act as lightweight blockchain nodes to transmit data to EDs.
\item \textit{Edge Devices:} Each ED such as a laptop or a powerful smartphone manages a group of IoT sensors under its coverage. Based on the QoE requirements, EDs can use their computational capability to process data tasks locally or offload to a nearby MEC server via wireless links. EDs also work as miners to perform block consensus where IoT sensors' users vote to select representative EDs for mining.
\item \textit{MEC Server:} In our considered B-MEC system, there is \textcolor{black}{a single MEC sever} to handle computationally extensive data tasks offloaded from EDs. By analyzing the task profile such as task sizes, channel conditions, and available resources, EDs can make offloading decisions so that the MEC server allocates its resources to execute data tasks under QoE requirements.
\item \textit{Blockchain:} A blockchain network is deployed over the MEC system where each ED acts as a blockchain miner \cite{7}. In this paper, we pay attention to a PoR mining design to solve blockchain latency issues. The proposed PoR scheme allows EDs to join the block mining with mining utility enhancement which helps improve the overall performance of the B-MEC system.
\end{itemize}
\subsection{Task Offloading Model}
We consider a B-MEC system with the set of EDs and the available sub-channels of the BS denoted by $\mathcal{N}$ and $\mathcal{K}$, respectively. It is assumed that each ED $n \in \mathcal{N}$ has an IoT data task including input data and required CPU workload to be executed locally or offloaded to the BS via one of the sub-channels $k \in \mathcal{K}$. Here, the offloading policy is scheduled by a binary variable, which equals 1 (offloading to the MEC server) or 0 (local execution). Each ED $n$ makes offloading decisions based on three main factors: task data size, channel condition, and transmit power level. Moreover, {each ED needs to allocate portion of its computation resource to execute the task locally}. Accordingly, we define four policies to schedule the offloading process of each ED, including offloading decision, channel allocation, transmit power, and computation resource allocation.
\textcolor{black}{We here formulate an offloading utility function from the QoE perspective, which is characterized by the task computation time (including local time and offloading time) and energy consumption (including local energy and offloading energy). We define a QoE-aware offloading utility function $J_n^{off}$ to provide a trade-off between the time and energy consumption of the task offloading compared with the local execution at each ED. The offloading utility function reflects the offloading improvement in QoE over the local execution.} If the offloading computation cost is lower than the local execution cost, the user utility can be positive, implying the user's QoE improvement. However, if offloading too many tasks, EDs may suffer from higher latency due to the traffic congestion, which reduces the user's QoE. As a result, the user offloading utility can be negative.
\subsection{{Blockchain Mining Protocol}}
\label{Subsection:Mining}
\begin{figure}
\centering
\includegraphics [width=0.99\linewidth]{Image/PoR.pdf}
\caption{The proposed PoR consensus in our B-MEC system. }
\label{Fig:PoR}
\vspace{-0.1in}
\end{figure}
\begin{figure*}[t!]
\centering
\begin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[width=0.99\linewidth]{Image/MADRL.pdf}
\caption{{The proposed MA-DDPG architecture. }}
\end{subfigure}%
~
\begin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[width=0.99\linewidth]{Image/Diagram_DDPG.pdf}
\caption{\textcolor{black}{The MA-DDPG training procedure. }}
\end{subfigure}%
\caption{The design of learning framework for the TOBM system. }
\label{MADRL}
\vspace{-0.1in}
\end{figure*}
In the B-MEC system, a crucial component is blockchain consensus that aims to mine the blocks of transactions (i.e., IoT data records) and add them to the blockchain. In traditional consensus mechanisms, e.g., DPoS \cite{add1},\cite{7}, each miner node must implement a repeated verification process across the miner network, which results in unnecessary blockchain latency. Therefore, we here propose a new PoR consensus to solve mining latency issues, including two main parts: miner node formulation and block verification, as illustrated in Fig.~\ref{Fig:PoR}.
\subsubsection{Miner Node Formulation}
In our B-MEC system, IoT sensors' users participate in the delegate selection process to vote the mining candidates among EDs. \textcolor{black}{In this regard, each IoT user votes for its preferred ED with the most reputation based on its mining latency. Specifically, an ED that exhibits a lower mining latency will have a better reputation via a predefined mining utility function. Based on the calculated reputation score, each IoT user votes for ED candidates based on their reputation ranking.} The top EDs with highest reputation scores are selected to become miners to perform consensus. During its time slot of the consensus process, each miner acts as a block manager which is responsible for performing block generation, verification, and aggregating blocks after being verified.
\subsubsection{Lightweight Block Verification}
The block manager first produces an unverified block that contains transactions collected by EDs in a given time. Then, the manager broadcasts this block to all other miners within the miner network for verification. Different from the traditional DPoS scheme which relies on a repeated verification process among miners, here we implement a lightweight verification solution. That is, a miner only verifies once with another node during the consensus process, which significantly reduces the verification latency. \textcolor{black}{Specifically, the block manager first divides the block into a set of equal transaction parts that are assigned to each miner within the miner group along with a unique random number. Next, the miner chooses to associate with one of the miners within its group to implement the verification for its assigned transaction part by allocating its CPU resource. If 51\% of miners respond with positive verification, and the sum of random numbers calculated by all miners is equal to a pre-defined number, the block manager accepts the verified block and adds it to blockchain. In summary, the mining procedure includes four stages: (1) transmitting unverified block from the block manager to the EMs, (2) verifying the local block at each EM, (3) sharing the verification result among two EMs, and (4) transmitting the verification result back to the manager. }
\textcolor{black}{To this end, we build a mining utility function to characterize the mining efficiency of the proposed scheme. Motivated by \cite{10}, we characterize the mining utility $J_n^{mine}$ of each ED $n$ via an exponential function where the mining utility is inversely proportional to the mining latency. Accordingly, an ED that exhibits a lower mining latency has a better mining utility with respect to a CPU resource allocation policy.}
\section{{Proposed Cooperative DRL Solution for System Utility Optimization}}
\subsection{{System Utility Formulation}}
Here, we formulate the system utility \textcolor{black}{for the proposed TOBM scheme} by taking both offloading utility and mining utility into account. {As explained previously,} the offloading utility $J_n^{off}$ reflects the efficiency of task offloading over local execution from the QoE perspective. Meanwhile, the mining utility $J_n^{mine}$ reflects the efficiency of mining blocks in the B-MEC network via the mining latency metric. Therefore, our key objective is to maximize the total system utility, i.e., $J$, as the sum of the offloading utility $J_n^{off}$ and the mining utility $J_n^{mine}$ of all EDs $n \in \mathcal{N}$, with respect to the offloading policies including offloading decision, channel allocation, transmit power, computation resource allocation, and the mining policy with CPU resource allocation.
To apply DRL to \textcolor{black}{the formulated TOBM problem}, we need to convert the objective function from a system utility maximization problem to a reward maximization problem. To do this, we formulate the proposed problem using a multi-agent version of the Markov decision process, also known as a Markov game. This is represented by a tuple of agent set $\mathcal{N}$, state set $\mathcal{S}$, action set $\mathcal{A}$, and observation set $\mathcal{O}$ of all agents \cite{add3}. \textcolor{black}{In fact, at each time slot, an MD does not always have access to all states of the environment, but only observes certain states, which is called observations. While states are complete and detailed information that are relevant to the current task (e.g., which specific wireless channel is occupied by a certain MD for offloading), observations are general information received by an MD (e.g., how many wireless channels are occupied).}
Each ED $n$ is considered as an intelligent agent to learn its optimal policy by observing the local environment formed by the cooperation of EDs and the MEC server, as shown in Fig.~\ref{MADRL}(a). We assume that the considered collaborative task offloading and block mining scheme operates on discrete-time horizon with each time slot $t$ equal and non-overlapping, and the communication parameters remain unchanged during each time slot. Now, we define each item in the tuple at each time slot $t$ as follows:
\begin{itemize}
\item \textbf{\textit{State:}} The environment state $\mathcal{S}(t)$ at time slot $t$ in \textcolor{black}{the proposed TOBM scheme} includes five components: task state, channel state, power state, resource state, and transaction state. Here, the task state is defined as the matrix of input data and required CPU workload of all EDs. The channel state is defined via a matrix of channel condition variables of all BSs at EDs, where each variable equals 1 (occupied channel) or 0 (available channel). The power state consists of transmit power levels of EDs in each sub-channel. Moreover, the resource state contains the states of available computation resource for data task execution, and transaction state includes transaction data size.
\item \textbf{\textit{Action:}} By observing the environment states, each ED takes an action \textcolor{black}{according to the} offloading decision, channel selection, transmit power selection, computation resource allocation, and CPU resource allocation, to complete task execution and block mining at each time slot $t$. Therefore, the action space of each ED is the combination of the above action sets. Accordingly, the action space $\mathcal{A}(t)$ of the cooperative game can be defined as a matrix of action sets of all agents.
\item \textbf{\textit{System Reward Function:}} The system reward at one time slot $t$ is the sum of the rewards of all EDs. The objective of \textcolor{black}{our formulated TOBM problem} is to maximize the overall system utility $J$ as the sum of the offloading utility and the mining utility. Therefore, we define $J$ as our system reward function.
\end{itemize}
\begin{figure*}[t!]
\centering
\begin{subfigure}[t]{0.33\textwidth}
\centering
\includegraphics[width=0.99\linewidth]{Image/0_Convergence1.pdf}
\caption{Average system rewards with different algorithms. }
\end{subfigure}
~
\begin{subfigure}[t]{0.32\textwidth}
\centering
\includegraphics[width=0.99\linewidth]{Image/Offloading_Performance1.pdf}
\caption{Average offloading utility with different numbers of EDs. }
\end{subfigure}%
~
\begin{subfigure}[t]{0.32\textwidth}
\centering
\includegraphics[width=0.99\linewidth]{Image/Blockchain_Performance1.pdf}
\caption{Comparison of block verification latency. }
\end{subfigure}%
\caption{Evaluation of system reward, task offloading, and blockchain performance.}
\label{Offloading_Performance}
\vspace{-0.1in}
\end{figure*}
\subsection{Proposed Cooperative DRL Algorithm}
In the literature, most existing schemes have focused on a conventional single-agent \cite{4,5,6} or an independent multi-agent \cite{add3} setting; however, these solutions are unable to obtain the cooperative policies of EDs because of nonstationary and partially observable environments. Indeed, when the policies of other agents change due to computation mode preference, the ED's observation can be changed (nonstationary) Moreover, in independent multi-agent learning schemes, an agent only has the local information and cannot know the updates from other agents due to non-collaboration. This makes the agents' learning algorithm hard to ensure stability and convergence \cite{lowe2017multi}. Thus, we propose a novel cooperative multi-agent DRL scheme using MA-DDPG \cite{9} for \textcolor{black}{our proposed TOBM scheme}. \textcolor{black}{The key reason behind the adaption of DDPG to optimize the system utility is that with this scheme, the actor can directly map states to actions instead of outputting the probability distribution across a discrete action space like DQN, which greatly reduces action sampling complexity. Moreover, given the stochasticity of the policy, there exists high variance of the obtained system reward between different training episodes, where DDPG can come as an efficient solution, by enabling off-policy learning via the joint use of behavior network and target network.}
The MA-DDPG algorithm employs a deep neural network (DNN) as the non-linear approximator to obtain the optimal policies for agents. Each agent updates its parameters to obtain a optimal policy for maximizing its utility. MA-DDPG is a policy gradient-based off-policy \textit{actor-critic} method where each agent comprises the \textit{actor} to make decisions over time slots with a behavior network and \textcolor{black}{the \textit{critic}} to evaluate the behavior of the \textit{actor}, which helps improve its performance. Specifically, given an episode sample from the memory buffer, the \textit{actor} at each agent updates the behavior network by computing its gradient based on a centralized action-value Q-function. Moreover, the \textit{critic} updates the behavior Q-function for the state-action pair of the \textit{actor} network by minimizing the loss function with the inputs including both local agent's observation and the observations of all other agents. \textcolor{black}{The cooperation of multi-agents helps improve robustness against malicious attacks since this allows multiple agents to monitor the shared policy update via observations \cite{xumisspoke2022}. }
To reduce the computational complexity caused by online training at EDs and solve the nonstationary issues from the concurrently learning process of all EDs, \textcolor{black}{we adopt a centralized learning and decentralized execution solution. In the centralized training step, the information of state-action of all EDs is aggregated by the MEC server to train the DRL model, where each agent can achieve the global view of the learning environment to obtain the observations of other agents for building the collaborative offloading policy.} \textcolor{black}{After training at the MEC server, the learned parameters (i.e. neural weights of DNNs of the actor and critic) are downloaded to each ED to execute the model for decision making based on its own locally observed information. Specifically, given the downloaded neural weights, each agent can easily compute the policy via its DNN with system utility used as the objective function, and then sample an action. Subsequently, each agent executes the sampled action, i.e. making decision to offload the data or not, in the defined TOBM environment given its states to obtain a reward.} The proposed algorithm is illustrated in Fig.~\ref{MADRL}(b).
\section{Performance Evaluation}
In this section, we conduct numerical simulations to evaluate the performance of \textcolor{black}{the proposed TOBM scheme} in a B-MEC system, employing the widely used Shanghai Telecom dataset\footnote{http://www.sguangwang.com/dataset/telecom.zip}. Here, we consider an MEC network with an MEC server and a maximum of 500 mobile phones as EDs distributed over a \textcolor{black}{1~km~$\times$~1~km} area in Shanghai city. \textcolor{black}{The number of channel sub-bands of the BS is set to 30, and each ED has task CPU workloads of [0.8-1.5]~Gcyles and transmit power range of [0-24]~dBm.} A DNN structure with three hidden layers (64, 32 and 32 neurons) \cite{7} is employed with the Adam optimizer for learning simulation. To prove the advantages of the proposed cooperative MA-DDPG scheme, we compare its performance with the state-of-the-art non-cooperative schemes, including DDPG, actor-critic \cite{6} and DQN \cite{add3}.
\textit{Evaluation of Training Performance:} Fig.~\ref{Offloading_Performance}(a) shows the learning curves of the average system reward with the increase of learning episodes for the B-MEC system with 50 EDs. It is clear that our MA-DDPG scheme is more robust and yields the best performance in terms of average system reward, compared to baseline schemes. This is because the proposed scheme allows EDs to learn mutually the cooperative offloading policy which helps reduce the channel congestion and user interference, and enhance computation resource efficiency. Meanwhile, in the DQN and actor-critic schemes, EDs greedily access the wireless channel spectrum to maximize their own utility without collaboration, which increases the possibility of channel collision and thus results in higher offloading latency. Also, DDPG scheme still remains a non-stationary learning issue and its average reward is lower than that of the MA-DDPG scheme.
\begin{figure*}[t!]
\centering
\begin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[width=0.99\linewidth]{Image/SystemUtility_Performance1.pdf}
\caption{{Comparison of system utility with non-cooperative schemes. }}
\end{subfigure}%
~
\begin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[width=0.99\linewidth]{Image/SystemUtility_Performance2.pdf}
\caption{{Comparison of system utility with cooperative schemes. }}
\end{subfigure}%
\caption{Evaluation of overall system utility performance.}
\label{Fig:SystemUtility}
\vspace{-0.1in}
\end{figure*}
\textit{Evaluation of Task Offloading and Blockchain Performance:} Fig.~\ref{Offloading_Performance}(b) indicates the performance of the average offloading utility versus different numbers of EDs. {As can be seen} when the number of EDs is small, the average offloading utility increases with the number of EDs because in this case, the MEC system can support sufficient spectrum and computing resources for handling all tasks of EDs. However, {when exceeding a certain threshold} (i.e., 60 EDs), the offloading utility decreases because the higher the number of offloaded EDs, the higher the competition of resource usage (i.e., channel spectrum). {This in turn increases} the offloading latency, and thus degrades the overall offloading utility. Nevertheless, our MA-DDPG scheme still achieves the best utility performance due to its collaborative offloading policies among EDs compared to other schemes with selfish learning.
Next, we evaluate our proposed PoR consensus scheme and compare it with the traditional DPoS scheme \cite{7} via the verification block latency metric. We set up 10 transactions per block and vary the numbers of mining nodes from 2 to 200. As shown in Fig.~\ref{Offloading_Performance}(c), our proposed PoR scheme requires significantly less time for mining blocks compared to DPoS, due to the optimized block verification procedure. {Although the block verification latency increases with \textcolor{black}{increasing the number of miners}, our scheme still achieves much better performance than DPoS, which verifies the effectiveness of our lightweight blockchain consensus design.}
\textit{Evaluation of the Overall System Utility Performance:}
We evaluate the performance of \textcolor{black}{our proposed TOBM scheme} in terms of the overall system utility as the sum of offloading utility and mining utility. The performances of \textcolor{black}{our cooperative TOBM scheme} with our PoR mining design and other non-cooperative schemes with PoR and DPoS mining are illustrated in Fig.~\ref{Fig:SystemUtility}(a). {Unsurprisingly, our cooperative scheme with a PoR mining design achieves the best overall system utility. The reasons for this observation are two-fold.} First, our offloading scheme with a cooperative MA-DDPG algorithm outperforms other non-cooperative offloading schemes in terms of a better offloading utility, {as evidenced in} Fig.~\ref{Offloading_Performance}(a). Second, our PoR design yields a lower mining latency which consequently improves the mining utility. Moreover, due to better mining utility, our PoR design contributes to better overall system utilities in each non-cooperative offloading scheme, compared to the use of DPoS design.
\textcolor{black}{We compare the system utility performance of our proposed TOBM scheme with the cooperative scheme without mining design \cite{6} and the cooperative scheme with DPoS design \cite{7}. As shown in Fig.~\ref{Fig:SystemUtility}(b), our TOBM scheme with PoR design achieves higher system utility than the cooperative scheme with DPoS design, thanks to the better mining utility of our proposed PoR framework. Moreover, compared with our approach, the cooperative scheme in \cite{6} has the lowest system utility due to the lack of mining design. This simulation result also reveals that a joint design of offloading and mining is of paramount importance to improving the overall system performance in B-MEC systems.}
\begin{figure*}[t!]
\centering
\begin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[width=0.99\linewidth]{Image/Image-Mining-Utility-2022-05-15-174603.pdf}
\caption{{Comparison of mining utility with cooperative schemes. }}
\end{subfigure}%
~
\begin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[width=0.99\linewidth]{Image/Image-Throughput-2022-05-15-173950.pdf}
\caption{{Comparison of blockchain throughput with mining schemes. }}
\end{subfigure}%
\caption{\textcolor{black}{Evaluation of blockchain performance.}}
\label{Fig:blockchainper}
\vspace{-0.1in}
\end{figure*}
\textcolor{black}{We compare the mining utility of our scheme with other cooperative schemes in Fig.~\ref{Fig:blockchainper}(a). Due to a lower mining latency achieved by the proposed consensus design, our scheme yields a better utility compared with its counterparts. The traditional approach without mining design has the highest mining latency, resulting in the lowest utility. Moreover, we investigate the throughput in Fig.~\ref{Fig:blockchainper}(b) which is defined as the number of successful transactions per second. We set five transactions per block and make offloading requests ranging from 10 to 200. Compared with DPoS and Proof-of-Work (PoW) schemes, the fast block verification rate of our scheme significantly enhances the throughput, before its performance decreases when the number of requests is higher than 120 since the system cannot handle excessive requests under this configuration.
}
{\color{black}\section{Research Challenges and Future Perspectives}
\subsection{User Mobility} In realistic B-MEC systems, EDs can move with high speed in wireless networks (e.g., in on-vehicle applications). {This makes their location highly dynamic which has direct impacts on offloading decision making. For example, an ED is likely to execute locally if it moves out of BS coverage, which makes the offloading design ineffective.} Therefore, the user mobility needs to be considered in the offloading formulation in future B-MEC systems, where dynamic characteristics such as {velocity and channel conditions should be taken into account to come up with a robust mobility-aware offloading policy}.
\subsection{Data Privacy}
In this work, {the DRL training performed at the MEC server potentially raises data privacy leakage due to the data exchange during the training.} Federated learning \cite{3} can be an attractive solution to perform collaborative training, where only trained parameters are shared with the MEC server while actual data and user information are stored at local EDs for privacy enhancement. In the B-MEC context, for example, each ED can run a DRL function to learn its offloading and mining policy based on its own local observation. Then, the participating EDs can communicate with the MEC server for DRL model aggregation (e.g., model averaging) to create a new global DRL model without sharing their private data.
\subsection{{Incentive Issues}} \textcolor{black}{In practice, how to encourage EDs, which serve as both offloading and mining nodes, to join computation and data mining for the long term is a critical challenge for B-MEC systems. Although blockchain is able to incentivize EDs via coin payment based on their mining effort, it is still not enough to compensate for the energy resources consumed for computation task offloading. Thus, it is important to jointly consider incentives, offloading and mining in B-MEC system optimization. A possible direction is to design a smart contract-inspired incentive mechanism to accelerate the data offloading and block mining. Another interesting area is to jointly optimize utility with respect to monetary benefits and resource usage of offloading and mining. }
\section{Conclusions}
This article proposed the novel concept of TOBM to assist B-MEC systems. A joint design of offloading and mining was considered, where a PoR consensus mechanism was proposed. Then, a cooperative DRL approach was proposed to solve the TOBM problem, showing a huge system utility improvement over the existing cooperative and non-cooperative schemes. Finally, we highlighted the key research challenges and promising directions for future B-MEC research.
\section*{Reviewer \thereviewer}}
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\begin{document}
\section*{Response to the Reviewers' Comments}\vspace*{0.8cm}
\begin{tabular}{lp{11cm}}
\textbf{Original Manuscript ID} & {1570798355}
\tabularnewline
[0.2cm] \textbf{Original Article Title} & {Latency Optimization for Blockchain-Empowered Federated Learning in Multi-Server Edge Computing}
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[0.2cm] \textbf{To} & Editor of IEEE Network
\tabularnewline
[0.2cm] \textbf{Re} & Response to Reviewers
\tabularnewline
[0.2cm] & \tabularnewline
\end{tabular}
\noindent Dear Guest Editor, \textit{IEEE Journal on Selected Areas in Communications, SI-Intelligent Blockchain}\\
\noindent Thank you for allowing us to perform a revision to our manuscript. We would also like to thank the anonymous Reviewers for their constructive comments on the previous manuscript version, which have been very helpful in improving the quality of our manuscript. We have carefully addressed the Reviewers' comments with the following key updates:
\begin{itemize}
\item ...
\item ...
\end{itemize}
\noindent We are uploading (1) a revised manuscript with changes highlighted in blue and (2) the point-by-point responses to the comments (below).\\
\noindent Further, we would like the Editor to assign our article to the same set of reviewers for further review. We are looking forward to receiving your comments. \\
\noindent Best regards,
\\
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\textbf{Authors:} \textit{Dinh C. Nguyen (on behalf of all authors)}
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\reviewersection
\begin{point}
This paper studies the problem of latency optimization for Blockchain-based federated learning (BFL) in multi-server edge computing. The authors first propose a BFL framework and focus on its latency optimization. They design a reinforcement learning method to decide the offloading and allocation actions according to the current states, with the objective to minimize the system latency. They also conduct some numerical evaluation to demonstrate the effectiveness of their proposed methods.
\begin{enumerate}
\item This paper is rich in content, which utilizes blockchain and federated learning in Multi-Server Edge Computing. They propose a multi-server-assisted BFL architecture, design an offloading strategy and a resource trading strategy for it.
\item Blockchain-based federated learning in multiserver edge computing is complex but practical.
\item The authors provide a convergence analysis of Blockchain-based federated learning.
\end{enumerate}
\end{point}
\begin{response}
Thank you for your appreciating the contributions of our paper. We have also improved the paper as per your comments, as illustrated in the following responses.
\end{response}
\begin{point}
The authors have not made full use of the advantages of blockchain itself. It seems that they simply add blockchain to FL for building trust.
\end{point}
\begin{response}
Thank you very much for your comment. In this revised version, we have explained in details the benefits of blockchain to FL in terms of security provision and robustness. Moreover, we have added new simulations to illustrate the role of blockchain in enabling robust FL training against model poisoning attacks. We summarize these updates as below:
\end{response}
\begin{point}
The main contribution is not clear and hard to follow. It seems that there are too much content, e.g., model training and aggregation in FL, offloading decisions and resource allocation for multiple ESs, RL, and blockchain. The authors really need to give a clear logical structure.
\end{point}
\begin{response}
Thank you very much for your comment. We have improved the contributions of our paper, and given a new subsection in Section I for a clear logical structure of the paper. We summarize these updates as below:
\end{response}
\begin{point}
Lacks of comparison of state-of-the-art works or methods.
\end{point}
\begin{response}
Thank you very much for your comment. We have enhanced the comparison of state-of-the-art methods. Specifically, we have added a new state-of-the-art deep reinforcement learning approach based on Deep Deterministic Policy Gradient (DDPG) and compared with our proposed DRL scheme. Moreover, we have compared our BFL scheme with other related FL approaches in terms of model training with respect to model poisoning attacks. We summarize these updates as below:
\end{response}
\begin{point}
(Detailed comment) First of all, I think the authors have not made full use of the advantages of blockchain itself. It seems that they simply add blockchain to FL for building trust.
\end{point}
\begin{response}
Thank you very much for your comment. In this revised version, we have explained in details the benefits of blockchain to FL in terms of security provision and robustness. Moreover, we have added new simulations to illustrate the role of blockchain in enabling robust FL training against model poisoning attacks. We summarize these updates as below:
\end{response}
\begin{point}
The logical structure of this paper seems confusing and difficult to understand. It seems that there are too much content, e.g., model training and aggregation in FL, offloading decisions and resource allocation for multiple ESs, RL, and blockchain. The authors really need to give a clear logical structure.
\end{point}
\begin{response}
Thank you very much for your comment. We have improved the contributions of our paper, and given a new subsection in Section I for a clear logical structure of the paper. We summarize these updates as below:
\end{response}
\begin{point}
This paper uses A2C reinforcement learning. However, to my best of knowledge, there are many new reinforcement learning methods. Why the authors choose to use A2C? Also, the comparisons of state-of-the-art methods are needed.
\end{point}
\begin{response}
Thank you very much for your comment. We have added a new state-of-the-art deep reinforcement learning approach based on Deep Deterministic Policy Gradient (DDPG) and compared with our proposed DRL scheme. Moreover, given the mixed discrete offloading and continuous allocation action space in our BFL formulation, we adopt a parameterized A2C method, which allows us to train both offloading and resource allocation policies. The use of both actor and critic networks enables efficient policy training, where the actor helps to train the policy, while the critic evaluates the efficiency of the policy training of the actor via training loss minimization. We summarize these updates as below:
\end{response}
\begin{point}
In section II.A, please explain in more detail about ``After the model aggregation process completes, the global model is added to the Blockchain, where mining is executed for secure model sharing".
\end{point}
\begin{response}
Thank you very much for your comment. We have explained it in this revised version. We summarize this update as below:
\end{response}
\begin{point}
There exist many grammatical mistakes. For example:
\begin{enumerate}
\item In section I, ``… on their own datasets, and exchange their model …";
\item In section I, ``each device may be keen in participating …";
\item In section III.B, ``Due to resource constraints, we allows MDs to …";
\item In section IV.A, ``… which can be modelled as a parameterized …";
\item In section IV.E, ``The critic is also involved for training …";
\item In section V.A, ``We trained the agent over 10000 episode with …".
\end{enumerate}
\end{point}
\begin{response}
Thank you very much for your comment. We have carefully checked and revised grammar errors as per your suggestion and other grammar errors throughout the paper.
\end{response}
\reviewersection
\begin{point}
Concern 1: This paper presents the synergy of blockchain and federated learning (blockchain-based FL) in multi-server edge computing to minimize latency. To enhance the intelligence and optimize problems, the author adopted the A2C algorithm. The authors considered data offloading decisions, MDs' transmission power, MDs' computation resource allocation, and hash power allocation to optimize the system latency. Lastly, they conducted the simulation using two datasets and presented the results in different matrics. Generally, the paper is presented interesting and solid ideas in the current intelligent wireless network. This paper presents the current hot issues in the 5G and B5G wireless network environments. The authors mentioned the bottleneck of MDs and edge servers from different perspectives. In particular, they are integrating different technologies to handle or increase the intelligence of the data processing and latency of the system. Generally, the authors understandably present the paper, except when excluding some fundamental points.
\end{point}
\begin{response}
Thank you very much for appreciating the technical contributions of our paper. We have also improved the paper as per your comments, as illustrated in the following responses.
\end{response}
\begin{point}
The paper structure is sound. The authors should present summary-related works in tabular format using different matrics that are more readable. Include the organization of the paper at the end of contributions.Thank you very much for your comment. We have added a table different design features to compare our paper with related works. Further, we have added a subsection to describe the organization of the paper. We summarize this update as below:
\end{point}
\begin{response}
Thank you very much for your comment. We have added a table different design features to compare our paper with related works. Further, we have added a subsection to describe the organization of the paper. We summarize this update as below:
\end{response}
\begin{point}
Do the MDs train the ML model fully or partially?
\end{point}
\begin{response}
Thank you very much for your comment. In this work, the MDs train the ML model fully. We have updated it in this revised version. We summarize this update as below:
\end{response}
\begin{point}
The author considered a multi-objective and multi-agent problem. How can you handle this multi-objective problem?
\end{point}
\begin{response}
Thank you very much for your comment. In this work, we consider a single-agent DRL setting, where a virtual centralized agent interacts with the BFL environment created by interactions of MDs with ESs during model training and block mining. Our DRL scheme aims to optimize the unified system latency consisting of model training latency, consensus latency and block mining latency with respect to learning and latency constraints. To handle the formulated optimization problem, we build a centralized agent which has the comprehensive view of the BFL operations, i.e., model training and block mining. Therefore, the agent can obtain fully system states to make good actions via well-trained data offloading and resource allocation policies in a fashion that the system latency is minimized. We summarize this update as below:
\end{response}
\begin{point}
In the system model, Section B should update to ``Federated learning model" or another appropriate title.
\end{point}
\begin{response}
Thank you very much for your comment. We have updated the title to ``Federated Learning Model" as per your suggestion.
\end{response}
\begin{point}
In the manuscript, there are many symbols and notations. To improve the paper's readability, the author should summarize it in a table.
\end{point}
\begin{response}
Thank you very much for your comment. We have a table of acronyms and a table of notations, as Table II and III in this revised paper.
\end{response}
\begin{point}
The author should include the BFL algorithm rather than the A2C training algorithm.
\end{point}
\begin{response}
Thank you very much for your comment. We have added the BFL algorithm as per your suggestion, as Algorithm 1 in this revised paper.
\end{response}
\begin{point}
There are many consensus mechanisms; which consensus mechanism have you used in your work?
\end{point}
\begin{response}
Thank you very much for your comment. For the model aggregation, we adopt a peer-to-peer-based consensus method. For the block mining, we adopt a Proof-of-work-based consensus menthod. We have highlighted them in the revised paper and update them as below.
\end{response}
\reviewersection
\begin{point}
The authors of this paper studied a new latency optimization problem for Blockchain-based federated learning (BFL) in multi-server edge computing. To assist the machine learning model training for resource-constrained mobile devices, the authors developed an offloading strategy that enables mobile devices to transmit their data to one of the associated edge servers. The authors analyzed the convergence of their BFL under some assumptions The total latency of the BFL system is minimized subject to the constraints on resources. A deep reinforcement learning algorithm was applied to solve their problem. Simulation results are provided to validate the performance of the BFL system. Overall, the analysis is solid, and the results are promising.
\end{point}
\begin{response}
Thank you for your appreciating the technical contributions of our paper. We have also improved the paper as per your comments, as illustrated in the following responses.
\end{response}
\begin{point}
The motivations for using blockchain in the system model are not clear. For example, in the abstract, blockchain seems not essential in system design. Similar issues exist in the main body of this paper.
\end{point}
\begin{response}
Thank you very much for your comment. We have added a subsection in Section II to highlight the motivations of using blockchain in FL. The benefits of blockchain in terms of providing security and FL training robustness are also explained. Moreover, we have implemented new simulations to illustrate the benefits of blockchain in FL. We summarize this update as below:
\end{response}
\begin{point}
There are a lot of assumptions in this work. The reviewer is not sure whether all these assumptions are reasonable or not.
\end{point}
\begin{response}
Thank you very much for your comment. We have updated assumptions and cited highly related works that adopted assumption used in our paper. We summarize this update as below:
\end{response}
\begin{point}
(Detailed comment) This paper is about the joint work of blockchain and federated learning. In the introduction, the authors mentioned that blockchain can mitigate the single-point-of-failure and build trust. However, in the simulations, the advantages are all about federated learning. Thus, it is necessary to show the advantages of blockchain in the simulation sections, such as the high-security level of the system or it can help to resist attacks compared with existing systems without blockchain. If not, this paper is more like a modified work of federated learning, instead of blockchain work.
\end{point}
\begin{response}
Thank you very much for your comment. In this revised version, the benefits of blockchain in terms of providing security and FL training robustness are explained. Moreover, we have implemented new simulations to illustrate the benefits of blockchain in FL against model attacks. We summarize this update as below:
\end{response}
\begin{point}
Reputation is utilized to evaluate the trust level of the blockchain nodes. The authors should clarify why latency can be used for blockchain nodes trust evaluation. If the authors are evaluating the latency instead of the trust level, they should replace the reputation with latency.
\end{point}
\begin{response}
Thank you very much for your comment. Inspired by our previous work, $[34]$ we can adopt the latency matric to evaluate the reputation of a miner. Specifically, the selection of a miner can be based on the ESs' reputation in the previous aggregation round via P2P collaboration, where the key idea of reputation evaluation is quantify each ES's contribution to block generation. Here, we can adopt mining latency as the reputation matric: an ES which generates the fastest block in the previous mining round will have the highest reputation and is selected as the leader for mining coordination in the current mining round. The details of the reputation scheme for blockchain have been presented in our previous work, $[34]$. In this revised paper, we summarize this update as below:
\end{response}
\begin{point}
In Section III-B, please provide the reasons why the latency at the ESs can be ignored. It will be good if there are some comparisons that can be provided in the simulations section. For example, the latency of processing in mobile devices is 100 times or higher than the latency of processing in the edge servers.
\end{point}
\begin{response}
Thank you very much for your comment. We actually considered the data processing latency at the ESs, as shown in Section III. Only the latency of mining at the ESs is ignored since we focus on analyzing the mobile mining from the user perspective in this paper.
\end{response}
\begin{point}
There are so many symbols and acronyms in this paper, it would be friendly to read if the authors can provide a table of symbols and a table of acronyms.
\end{point}
\begin{response}
Thank you very much for your comment. We have a table of acronyms and a table of notations, as Table II and III in this revised paper.
\end{response}
\end{document}
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train/arxiv
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BkiUfj84uzliDEmfhlih
| 5 | 1 |
\section{Introduction}
When some disorders are introduced in Mott insulators or charge-ordered insulators,
disorder-induced electronic states are often observed within the band gaps and are
responsible for the hopping transport in the insulating phase.
\cite{Nakatsuji,Husmann,Coey,Yoshida}
Although it is very important to understand the relationship
between the transport behaviors and the disorder-induced in-gap states,
no unified picture on the relationship is obtained so far.
For example, in Ca$_{2-x}$Sr$_x$RuO$_4$, \cite{Nakatsuji}
NiS$_{2-x}$Se$_x$, \cite{Husmann} and R$_{1-x}$A$_x$MnO$_3$, \cite{Coey}
almost localized electronic states are formed within the band gap
and are responsible for the variable range hopping (VRH) transport.
On the other hand, in lightly-doped La$_{2-x}$Sr$_x$CuO$_4$\ ($x$=0.03),
a sharp peak with clear band dispersion is observed within the Mott gap
while the resistivity shows a VRH behavior. \cite{Yoshida}
The VRH transports and in-gap states suggest that the insulating
and metallic clusters coexist near the metal-insulator transition (MIT) in these materials.
\cite{Pan,Sarma}
Such inhomogeneity plays crucial roles in their remarkable properties
of colossal magnetoresistance and photo-induced MIT in manganites,
\cite{Dagotto, Takubo} and stripe formation in cuprates. \cite{Kivelson}
CuIr$_2$S$_4$ with spinel structure is one of such systems and
shows a MIT at $T_{MI} \sim$ 226K. \cite{first,jphys,optic2,high,optic,nmr,nmr2}
The anomalous hopping transport is also observed in the insulating phase
$\rho \propto $ exp[$-(T/T_0)^{1/2}$], \cite{resist,resist2}
or $\rho \propto$ $A$exp$(-E_a/k_BT)$ +$B$exp[$-(T/T_0)^{1/4}$]. \cite{Cao,hall}
The temperature variation of $\rho \propto $exp[$-(T/T_0)^{1/2}$]
can be described by a Mott VRH conductivity in one dimensional case \cite{Mott}
or by a Efros-Shklovskii VRH conductivity. \cite{Efros}
However, a rather complicated three dimensional charge ordering of Ir$^{3+}$ ($S$=0) and Ir$^{4+}$ ($S$=1/2) sites are indicated in the insulating phase
of CuIr$_2$S$_4$. \cite{nature}
The cubic spinel structure of CuIr$_2$S$_4$ becomes tetragonally elongated
along the $c$-axis and bi-capped hexagonal ring octamers of
Ir$^{3+}$ and Ir$^{4+}$ are formed below $T_{MI}$.
The orbital driven Peierls mechanism
has been suggested, \cite{khomskii}
in which the Ir$^{4+}$ ions are dimerized along $xy$ chains of the $B$-sites.
Moreover, the resistivity of CuIr$_2$S$_4$ is
reduced by x-ray, \cite{xraymit1,xraymit2} visible light, \cite{PRL,vlight}
or electron beam irradiation \cite{tem,tem2} at low temperature ($\sim$ 100 K).
The symmetry of crystal is changed from triclinic to tetragonal
by the irradiation and the photo-induced state has a long lifetime.
It has been proposed that the photo-excitation breaks the Ir$^{3+}$/Ir$^{4+}$
charge ordering and induces the metallic conductivity.
The indication of MIT and the Ir$^{3+}$/Ir$^{4+}$ charge ordering
has been obtained on previous x-ray photoemission and absorption studies
of CuIr$_2$S$_4$. \cite{PRL,Noh,matsuno,xes,xas,cluster}
The Ir 4$f$ core-level spectrum of the insulating phase has two components
with large energy difference, consistent with the charge ordering
of Ir sites. \cite{PRL,Noh}
In contrast, the core-level spectrum has not been changed against
laser irradiation, while the resistivity is reduced.
In this article, we report results of ultra-violet photoemission spectroscopy (UPS)
of CuIr$_2$S$_4$ single crystals combined with laser illumination.
The UPS spectrum shows a clear MIT with band-gap opening $\sim$ 0.09 eV.
However, the UPS spectrum just below the Fermi level ($E_F$) at low temperature ($\sim$ 20 K)
has a peculiar power-law dependence $\sim (E-E_F)^n$, $n \sim$ 1.3 - 1.7,
that is associated with the anomalous conduction.
The systematic spectral change of UPS against laser irradiation has not been observed.
The Ir$^{4+}$-Ir$^{4+}$ dimers in the $xy$ plane of spinel are very robust
against photo-excitation and play important roles in the exotic conduction.
\section{Experiments}
Single crystals of CuIr$_{2}$S$_{4}$ were grown by the bismuth solution method,
described previously in detail. \cite{crystal}
UPS measurements were performed using SCIENTA SES-100 spectrometers
equipped with a He I source ($h\nu$ = 21.2 eV).
The total resolution was 30 meV and the base pressure of
the spectrometer was $1\times10^{-7}$ Pa.
Five CuIr$_2$S$_4$ single crystals were studied for UPS measurements.
The first and second samples named $\#R1$ and $\#R2$ are cleaved at 300 K
\textit{in situ} and then measured at various temperatures.
The third, fourth, and fifth samples named $\#L3$, $\#L4$, and $\#L5$
are cleaved at 20 K \textit{in situ} and then measured at various temperatures.
All photoemission data were collected within 24 hours after the cleaving.
In order to study the effects of visible light excitation,
a Nd:YAG pulsed laser provided optical excitation to the samples
at energies of 2.3 eV (532 nm) with a pulse frequency of 30 Hz
and a pulse width of about 10 ns. The beam was focused to a spot
of 4 mm diameter.
\section{Results and discussions}
\subsection{Temperature dependence}
Figure \ref{v1} shows wide-range UPS of CuIr$_{2}$S$_{4}$ taken at 300 K for sample $\#R1$
and 20 K for sample $\#L3$ immediately after cleaving, respectively.
Compared with the previous studies, \cite{lda,lda2,xes,PRL,matsuno}
structures A, B, and C are assigned to the Ir 5$d$-S 3$p$ antibonding band,
the Cu 3$d$ band, and Ir 5$d$-S 3$p$ bonding band, respectively.
In the near-$E_F$ spectra,
a spectral change across the MIT is observed.
The intensity at $E_{F}$ is substantial in
the spectrum at 300 K (sample $\#R1$, red line, metallic phase),
while the intensity at $E_F$ almost disappears in the spectrum at 20 K
(sample $\#L3$, blue line, insulating phase) [See Fig. \ref{v1} (b)].
The opening of the gap $E_{gap}$ $\sim$ 0.09 eV, obtained by extrapolating
the slope near the valance-band maximum to the base line,
agrees with the previous photoemission results \cite{PRL,Noh,optic}
and is attributed to the Ir$^{3+}$/Ir$^{4+}$ charge ordering and
Ir$^{4+}$-Ir$^{4+}$ dimerization along the $xy$ chains
with the tetragonal distortion.
The spectrum of the insulating phase has a tail above the valence-band maximum
that reaches $E_F$ and forms a kind of $softgap$, interestingly.
This observation directly corresponds to the anomalous hopping transport
observed in the resistivity measurements
$\rho \propto $ exp[$-(T/T_0)^{1/2}$], \cite{resist,resist2}
or $\rho \propto$ $A$exp$(-E_a/k_BT)$ +$B$exp[$-(T/T_0)^{1/4}$]. \cite{Cao,hall}
We have fitted this in-gap spectral feature up to 0.09 eV ($\sim$ $E_{gap}$)
to two types of model functions
$A(E-E_F)^n +\frac{B}{\sqrt{2\pi}C}{\rm exp}[-(\frac{E-E_1}{\sqrt{2}C})^2]$
and $A(E-E_0)^n$ (Fig. \ref{f1}).
The power-law function $A(E-E_F)^n$ is additionally convoluted with a
gaussian function taking into account the instrumental and thermal broadenings ($\sim$ 40 meV).
The gaussian peak $\frac{B}{\sqrt{2\pi}C}{\rm exp}[-(\frac{E-E_1}{\sqrt{2}C})^2]$ is assumed to be a contribution
from surface residual bonds near $E_F$ and it will be discussed later.
The integrated intensity is normalized to unity and
the fitted parameters are shown in Table \ref{t1}.
The exponent of $n$ for sample $\#L3$ is estimated to
be $\sim$ 1.5 and $\sim$ 1.7
by using the functions of $A(E-E_F)^n + \frac{B}{\sqrt{2\pi}C}{\rm exp}[-(\frac{E-E_1}{\sqrt{2}C})^2]$
and $A(E-E_0)^n$, respectively.
\begin{figure}
\begin{center}
\includegraphics[width=7cm,clip]{Fig1.eps}%
\end{center}
\caption{(Color online) UPS of CuIr$_2$S$_4$ taken at 300 K and 20 K immediately after
cleaving. (a) Wide-range spectra. (b) Near-$E_F$ spectra. Inset of (b) shows the expanded spectra near $E_F$.}
\label{v1}
\end{figure}%
\begin{figure}
\begin{center}
\includegraphics[width=7.5cm,clip]{Fig2.eps}%
\end{center}
\caption{(Color online) UPS of CuIr$_2$S$_4$ for sample $\#L3$ taken at 20 K immediately after cleaving.
The solid curves indicate the fitted results with model functions
$A(E-E_F)^n$+$\frac{B}{\sqrt{2\pi}C}{\rm exp}[-(\frac{E-E_1}{\sqrt{2}C})^2]$ and $A(E-E_0)^n$, respectively.}
\label{f1}
\end{figure}%
\begin{table}
\caption{
Best fit parameters of the model functions $A(E-E_F)^n + \frac{B}{\sqrt{2\pi}C}{\rm exp}[-(\frac{E-E_1}{\sqrt{2}C})^2]$
and $A(E-E_0)^n$ for the spectra of sample $\#L3$ of CuIr$_2$S$_4$.
The spectra are normalized using integrated intensity up to 0.09 eV.
}
\begin{center}
\begin{tabular}{ccccc}
\multicolumn{5}{c}{Function : $A(E-E_F)^n + \frac{B}{\sqrt{2\pi}C}{\rm exp}[-(\frac{E-E_1}{\sqrt{2}C})^2]$}\\
\hline\hline
Sample (20 K)& $n$ & $B$ & $C$ (eV) & $E_1$ (eV) \\\hline
$\#L3$ & $1.5\pm0.2$ & $0.23\pm0.03$ & $0.02\pm0.01$ & $0.02\pm0.01$\\
\hline\hline
\\
\multicolumn{5}{c}{Function : $A(E-E_0)^n $}\\
\hline\hline
Sample (20 K) & $n$ & $B$ & $C$ (eV) & $E_1$ (eV) \\\hline
$\#L3$ & $1.7\pm0.2$ & - & - &-\\
\hline\hline
\end{tabular}
\label{t1}
\end{center}
\end{table}
Temperature dependence of the wide-range UPS was
carefully examined for various fractured surfaces.
Figure \ref{v2} (a) shows temperature dependence of UPS for sample $\#R1$,
which was cleaved at 300 K and then measured with decreasing temperature.
Although the spectra show the band-gap opening, a hump structure near $E_F$
remains even at the lowest temperature $\sim$ 20 K in contrast to
the spectra for sample $\#L3$ measured immediately after cleaving at 20 K.
Sample $\#R2$ was also cleaved at 300 K and then measured with decreasing temperature.
However on the spectrum immediately taken after cooling down,
such hump is almost absent [Fig. \ref{v2} (b) blue line].
Moreover after keeping 20 K for 30 minutes, the intense hump (green line) appeared also for this sample $\#R2$.
Even for sample $\#L3$, cleaved at low temperature, the intensity gradually increased with time although the intensity is rather small compared to the samples cleaved at 300K ($\#R1$, $\#R2$). Even at 20 K, the intensity of the hump depends on
the cleavage. While, for sample $\#L4$ cleaved at 20 K, the hump was clearly observed,
its intensity was very small for sample $\#L5$.
The hump structure is probably due to the surface state composed of
unpaired Ir$^{4+}$ at the surface.
Thus the hump structure can be affected by various surface conditions
including treatments such as cleaving, change of temperature or irradiation
from He I source.
We have tried to fit all the spectra taken at 20 K
with the function $A(E-E_F)^n + \frac{B}{\sqrt{2\pi}C}{\rm exp}[-(\frac{E-E_1}{\sqrt{2}C})^2]$
as shown in Fig. \ref{v2} (f) and Table \ref{t2}.
The exponents of $n$ are universally estimated to be $\sim$ $1.3$ - $1.6$ by using this function at low temperature of CuIr$_2$S$_4$, independent of the gaussian contribution, namely the surface contribution. Therefore, we can safely conclude
that, while the hump near $E_F$ is the surface contributions,
the in-gap state with the power law function $A(E-E_F)^n$ with $n$ $\sim$ $1.3$ - $1.6$ is derived from the bulk and is responsible for the VRH transport, $\rho \propto $ exp[$-(T/T_0)^{1/2}$]
(or $\propto$ $A'$exp$(-E_a/k_BT)$ +$B'$exp[$-(T/T_0)^{1/4}$]).
\begin{figure*}
\begin{center}
\includegraphics[width=15cm,clip]{Fig3.eps}%
\end{center}
\caption{(Color online) UPS data of CuIr$_2$S$_4$ taken at various temperatures and for various samples.
(a)-(e) Insets are the expanded spectra near $E_F$. The arrows denote the order of the measurements.
(f) Near $E_F$ region of UPS data for CuIr$_2$S$_4$ taken at 20 K.
The solid curves indicate the fitted results with model functions
$A(E-E_F)^n +\frac{B}{\sqrt{2\pi}C}{\rm exp}[-(\frac{E-E_1}{\sqrt{2}C})^2]$.}
\label{v2}
\end{figure*}%
\begin{table}
\caption{
Best fit parameters of the model function $A(E-E_F)^n + \frac{B}{\sqrt{2\pi}C}{\rm exp}[-(\frac{E-E_1}{\sqrt{2}C})^2]$
for the spectra of sample $\#R1$, $\#R2$, $\#L3$, $\#L4$, and $\#L5$ of CuIr$_2$S$_4$.
The spectra are normalized using integrated intensity up to 0.09 eV.
}
\begin{center}
\begin{tabular}{ccccc}
\multicolumn{5}{c}{Function : $A(E-E_F)^n + \frac{B}{\sqrt{2\pi}C}{\rm exp}[-(\frac{E-E_1}{\sqrt{2}C})^2]$}\\
\hline\hline
Sample (20 K)& $n$ & $B$ & $C$ (eV) & $E_1$ (eV) \\\hline
$\#R1$ & $1.4\pm0.2$ & $0.38\pm0.03$ & $0.03\pm0.01$ & $0.03\pm0.01$\\
$\#R2$ & $1.3\pm0.1$ & $0.22\pm0.03$ & $0.02\pm0.01$ & $0.02\pm0.01$\\
$\#L3$ & $1.5\pm0.2$ & $0.23\pm0.03$ & $0.02\pm0.01$ & $0.02\pm0.01$\\
$\#L4$ & $1.4\pm0.1$ & $0.30\pm0.03$ & $0.03\pm0.01$ & $0.04\pm0.01$\\
$\#L5$ & $1.6\pm0.2$ & $0.24\pm0.03$ & $0.04\pm0.01$ & $0.03\pm0.01$\\
\hline\hline
\end{tabular}
\label{t2}
\end{center}
\end{table}
In the situation forming $softgap$ due to electron-electron Coulomb
repulsion of Efros-Shklovskii type, the spectral
function is usually characterized
by $A(E-E_F)^2$ dependence. \cite{Efros,Massey}
However CuIr$_2$S$_4$ has complicated three-dimensional
charge order and has $hardgap$ $\sim$ 0.09 eV.
Probably, some disorder in the charge-ordered state
is the origin of the in-gap spectral feature and
the VRH transport.
Recently based on transport measurements of Ca$_{2-x}$Sr$_x$RuO$_4$,
Nakatsuji \textit{et al.} proposed that the hopping exponent of $\alpha$ $\sim$ 1/2
is universal feature of the disordered Mott system close to the metal-insulator transition
and reflects the emergence of disorder-induced localized electronic states
in the Mott-Hubbard gap. \cite{Nakatsuji}
The presence of some kind of disorder such as coexisting metallic clusters
in the insulating phase, which can be created near the first order MIT,
gives a strongly localized state and $\alpha$ $\sim$ 1/2.
A distinct in-gap state is also observed in UPS of
Ca$_{2-x}$Sr$_x$RuO$_4$. \cite{Sudayama}
Furthermore, $n$ $\sim$ 1.3 - 1.7 for CuIr$_2$S$_4$ is rather small compared
to the expected value $\sim$ 2 of Efros-Shklovskii type.
The suppressed $n$ $\sim$ 1.5 of $A(E-E_F)^n$
is observed in a recent photoemission study on BaIrO$_3$, which has
a quasi-one-dimensional structure with Ir$_3$O$_{12}$ trimers
and also shows a charge-density wave transition. \cite{Maiti}
The exponent of $n$ = 1.5 for BaIrO$_3$ is attributed to the strong influence
of electron-magnon interaction.
The traveling dimer conduction in low temperature phase of CuIr$_2$S$_4$ is proposed
in Ref. \onlinecite{resist2}.
All Ir$^{4+}$ holes at low temperature fall in the $xy$ orbitals and
form the dimers along the (110) [or (-110)] chains. \cite{khomskii}
When the Ir$^{4+}$-Ir$^{4+}$ dimer bonds are very strong and the system
has good one dimensionality, single hole hopping from an Ir$^{4+}$-Ir$^{4+}$ dimer
to a neighboring Ir$^{3+}$ site along the (110) chain may be suppressed.
Instead, paired hole hopping (bipolaronic hopping) of an Ir$^{4+}$-Ir$^{4+}$ pair
along the (110) chain is favored as shown in Fig. \ref{chain}.
The paired hole conduction may have quasi-one-dimensional and bipolaronic features
in the (110) or (-110) chains even in the three dimensional lattice.
Actually, the UPS line shapes of CuIr$_2$S$_4$
across the MIT resemble those of a bipolaronic material Ti$_4$O$_7$. \cite{Ti4O7}
The spectral weight in the insulating phase of Ti$_4$O$_7$
also obeys the power-law function $A(E-E_0)^n$ with $n$ $\sim$ 2.
Moreover, even in the metallic phase,
the both spectra of CuIr$_2$S$_4$ and Ti$_4$O$_7$ show weak Fermi edges
and broad peaks at $\sim$ 0.75 eV,
around which most of spectral weights are distributed.
Such broad feature is commonly interpreted as the incoherent part
of the spectral function accompanying the quasi-particle excitations
around $E_F$. Probably, the fluctuation of the dimerization may
survive at high temperature metallic phase.
The temperature dependence of the UPS spectra across the MIT
is briefly summarized as follows.
The -Ir$^{3+}$-Ir$^{3+}$-Ir$^{4+}$-Ir$^{4+}$- charge ordering
and Ir$^{4+}$-Ir$^{4+}$ dimerization
along the (110) or (-110) chains
cause the $hard gap$ opening of $\sim$ 0.09 eV, which
also manifests in previous studies. \cite{PRL,xas,matsuno,Noh}
On the other hand, the peculiar in-gap state, which
shows the power-law behavior with exponent $n$ $\sim$ 1.3 - 1.7,
gives the anomalous hopping conductivity at low temperature.
\begin{figure}
\begin{center}
\includegraphics[width=6cm,clip]{Fig4.eps}%
\end{center}
\caption{(Color online) Hopping of holes between an Ir$^{4+}$-Ir$^{4+}$ dimer and a neighboring Ir$^{3+}$ site along the (110) or (-110) chains.
The dimer moves without reducing the number of Ir$^{4+}$-Ir$^{4+}$ bonds.
}
\label{chain}
\end{figure}%
\subsection{Photo-excitation effects}
We have also studied photo-excitation effects on the near-$E_F$ UPS spectra.
No spectral weight increase at $E_F$ was observed for any sample surfaces
by visible light irradiation from the Nd:YAG laser up to 5 mJ/pulse
($8.5\times10^{16}$ cm$^2$photons/pulse) (See, Fig. \ref{l1}),
indicating that the Ir$^{3+}$/Ir$^{4+}$ charge ordered gap
in the insulating phase is very robust against photo-excitation
across the band gap.
On the other hand, the present visible light irradiation also gives the reduction
of the resistance similar to those reported in the x-ray irradiation measurements
[Fig. \ref{l1} (i)]. The weak irradiation
up to 1 mJ/pulse ($1.7\times10^{16}$ cm$^2$photons/pulse)
gives no spectral change near $E_F$ [Fig. \ref{l1} (a)-(e)].
On the other hand, the hump of the in-gap states was decreased
by rather strong irradiations of $\sim 3$ - $5$ mJ/pulse [Fig. \ref{l1} (f)-(h)].
The strong irradiation may cause redistribution of charge at the
surface states.
It has been suggested that the long-range charge ordering is
destroyed by the x-ray. \cite{xraymit1,xraymit2}
Probably, the visible light irradiation destroys
only the phase of -Ir$^{3+}$-Ir$^{3+}$-Ir$^{4+}$-Ir$^{4+}$-
chains in the $xy$ plane of spinel.
When the Ir$^{4+}$-Ir$^{4+}$ dimers in the $xy$ chains
are shifted to the neighboring sites by the irradiation,
the octamer ordering is destroyed (See Fig. \ref{phases}).
This corresponds to the bipolaronic hopping as discussed in Fig. \ref{chain}
and can be regarded as a kind of bipolaronic solid-to-liquid
transition similar to Ti$_4$O$_7$. \cite{Ti4O7}
This picture is consistent with the recent diffraction study of
CuIr$_2$S$_4$ at low temperature, \cite{tem2}
indicating that the long range order (octamer order) is destroyed
but the dimers are preserved locally after x-ray or electron irradiation.
Moreover similar charge ordering as shown in Fig. \ref{phases} (b)
including -Rh$^{3+}$-Rh$^{3+}$-Rh$^{4+}$-Rh$^{4+}$- chains
is observed in the structural study of LiRh$_2$O$_4$,
which also has Rh sites at the $B$ site of spinel and shows
a metal-insulator transition with some lattice distortion. \cite{LiRh2O4}
\begin{figure*}
\begin{center}
\includegraphics[width=15cm,clip]{Fig5.eps}%
\end{center}
\caption{
(Color online)
(a)-(h) UPS of CuIr$_2$S$_4$ before, during, and after visible light
irradiation from the Nd:YAG laser (532 nm) for various samples.
(i) The resistance taken under the same condition as the photoemission
measurements.
}
\label{l1}
\end{figure*}%
\begin{figure}
\begin{center}
\includegraphics[width=8cm,clip]{Fig6.eps}%
\end{center}
\caption{(Color online) (a) Charge ordering of Ir sites of CuIr$_2$S$_4$ sitting in one $xy$ plane.
The lower panel is another $xy$ plane a/4 below upper panel.
The double solid lines denote the Ir$^{4+}$-Ir$^{4+}$ dimerized bonds.
The blue and red lines denote the octamer bonding toward Ir$^{4+}$ in the different planes.
(b) When the Ir$^{4+}$-Ir$^{4+}$ dimers shift by one unit,
the octamer ordering is destroyed and another order may appear.}
\label{phases}
\end{figure}%
\section{Summary}
We have studied the electronic structure of CuIr$_2$S$_4$
single crystals using UPS.
The UPS data shows the band-gap opening of $\sim$ 0.09 eV and
supports the previous report of Ir$^{3+}$/Ir$^{4+}$
charge ordering in the insulating phase.
The observed in-gap state at low temperature is consistent with
the variable range hopping transport.
The UPS measurements under laser irradiation indicate that the Ir$^{3+}$/Ir$^{4+}$ charge
disproportionation by the dimer formation is very robust against the photo-excitation
but the long-range charge ordering would be destroyed.
\section*{Acknowledgments}%
This work was supported by a Grant-In-Aid for Scientific Research
(Grants No.19340092) from the Ministy of Education, Culture, Sports, Science and Technology of Japan.
K. T. acknowledge support from the Japan Society for the Promotion of Science for Young Scientists.
|
train/arxiv
|
BkiUa0zxK6-gDz87NvLv
| 4 | 0.8 |
\section{Moore}
Moore (Muon Object Oriented Reconstruction) is a track reconstruction
package for the Muon Spectrometer.
A detailed description of Moore is available elsewhere~\cite{Moore}.
Moore takes as input collections of digits or clusters inside the
Muon Spectrometer (CSC, MDT, RPC, TGC) and outputs
fitted reconstructed tracks whose parameters are referred
at the first measured point of the Muon Spectrometer.
The reconstruction is performed in several steps and each step is
driven by an ATHENA top-Algorithm module (MooMakeXXX).
Each algorithm is independent, i.e. it retrieves objects created
by the previous modules from StoreGate and it builds a
transient object to be recorded in StoreGate where it is available
for the subsequent algorithms.
The only link between algorithms are the transient objects, in such a way that the
algorithms depend on transient objects but transient objects do not
depend on algorithms.
The decoupling between data and algorithms and the natural step
sequence of algorithm performing the reconstruction gives
the opportunity to plug-in different reconstruction algorithms at
run time.
As it is now the overall reconstruction starts from the searches for $\phi$
regions of activity and builds PhiSegments (MooMakePhiSegments). For each
$\phi$-Segment, the associated MDTs are found and a {\it crude}
RZSegment is built (this is essentially a collection of z hits)
(MooMakeRZSegments).
Inside the MDTs the drift distance is calculated from the drift time,
by applying various corrections: such as the TOF, the second coordinate,
the propagation along the wire, the Lorenz effect.
From the 4 tangential lines the best one is found.
All the MDT segments of the outer station are combined with
those of the Middle layer.
The MDT hits of each combination are added to the phi-hits of the $\phi$
Segment, forming outer track candidates.
All the successfully fitted candidates are kept for further processing
(MooMakeRoads).
The successfull outer track is subsequentely used to associate
inner station MDT hits. A final track is defined as a successfully
fitted collection of trigger hits and MDT hits from at least
two layers (MooMakeTracks).
The parameters of the fitted track are referred to the first measured
point of the Muon Spectrometer.
When dealing with data already selected by the trigger the first two steps
(MooMakePhiSegments) and (MooMakeRZSegments) can be substitute with {\it ad
hoc} makers that seed the track search in the regions selected by the trigger.
\chapter*{Indice\@mkboth {\bf {Indice}} {\bf{Indice}}}
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\section{Introduction}
According to the ATLAS Computing Model, we expect to have four event data
streams produced from the HLT (high level trigger) system: a physics stream
dedicated to the reconstruction of the full ATLAS event sample; an express
line stream to rapidly monitor calibration and reconstruction quality on a
subset of events before the complete reconstruction is run on the physics
stream, and to look for interesting and unusual events; a
pathological stream to study events causing problems for the HLT; and finally
a calibration stream, processed rapidly and used to obtain the calibration
constants for the physics stream reconstruction. The calibration stream
will itself be divided into several sub-streams with different types of events.
In particular, this study is devoted to the calibration stream for the
electrons \cite{nota1}.
Using the High Level Trigger algorithms, described in detail in section \ref{soft}, we have studied the expected rates and purity of the electron stream
as a function of the luminosity of the LHC.
\section{Tools}
In this section all the software tools used are briefly described.
All the selection cuts applied and their meanings are defined.
Moreover, there is a description of the datasets used in the
analysis.
In detail, all the cuts applied for each trigger hypothesis
algorithm are summerized in the table \ref{algo}.
The values of the thresholds are those presented in a recent talk on
the trigger performance \cite{cibran_1}.
\subsection{Software Tools \label{soft}}
The selection at the High Level Trigger (HLT) \cite{hlt} is seeded by the
information obtained at Level~1 (i.e. Regions-of-Interest, RoI). The
level~2 trigger reduces drastically the number of events with a relatively
small latency, using the information contained in the RoI, while the final
trigger level, the event filter, having fewer constraints on the latency,
refines the selections using the full event information.
After the HLT, we obtain typically ``Physics Objects''; these are: muons,
electrons, jets.
In this case we studied the Trigger Menu for the electron objects, in
particular the selection signatures: e25i, selecting single electrons
with a threshold
of 25 GeV, and 2e15i, for di-electron events containing two electron
candidates with at least 15 GeV.
The selection criteria for electrons include a shower-shape analysis in the electromagnetic calorimeter, a search for high $p_T $ tracks and a matching between the clusters and the tracks. The selections applied at each trigger level are
as follows:
\begin{itemize}
\item{ {\bf Level 1}\\
The particles are selected using the Electromagnetic Calorimeter
information, applying cuts on the transvere energy ($E_T$) in the cluster and
isolation criteria
around the clusters (using both hadronic and electromagnetic calorimeters,
with reduced granularity information compared to that available in the HLT).
Each RoI is characterised by: $\eta, \phi, E_T ^{threshold}$ and isolation
criteria.
}
\item{ {\bf Level 2} \\
Starting from the LVL1 region of interest (RoI) of size
$\Delta\eta \times \Delta\phi = 0.2 \times 0.2$, the level 2 algorithms
refine the
selections using the full granularity of the detector.
Electrons are selected using both the calorimeter information and
tracking information. The shower shapes in the calorimeter and the
tracks reconstructed in the inner detector near the calorimeter clusters are
analyzed, applying selection cuts on $E/p$, $\Delta\eta$ and
$\Delta\phi$, as described in detail in table \ref{algo}
(L2 Calo Hypo and L2 Track Hypo algorithms).
}
\item{ {\bf Event Filter} \\
Starting from the Level 2 objects, the Event Filter refines the cuts using more
sophisticated algorithms, and access to full event information in the
calorimeter and the inner detector. The track search is performed in the SCT
and pixel detectors and independently in the TRT. Finally, the objects have
to pass three sets of cuts: EF Calo Hypo, EF Track Hypo and EF Match Hypo
(checking the $E/p$ and spatial matching between the objects found in
the tracking and calorimetry).
}
\end{itemize}
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
{\bf Hypothesis} & {\bf Cuts Applied } & {\bf Cuts
Applied } \\
{\bf Algorithm} & {\bf (e25i) } & {\bf (2e15i) } \\
\hline
L2 Calo Hypo & (E3x7/ E7x7) $>$ 0.88 & (E3x7/ E7x7)$>$ 0.9\\
&(E1-E2)/ (E1+E2)$>$ 0.64 &(E1-E2)/ (E1+E2)$>$ 0.72\\
& ET (em)$>$ 22.0 GeV & ET (em)$>$ 11 GeV \\
& ET (had)$<$ 5.5 GeV & ET (had)$<$ 1 GeV \\ \hline
L2 Track Hypo& PT$>$ 8 GeV &PT$>$ 8 GeV\\
& 0$<ET/PT<$ 3.5& 0.2$<ET/PT<$ 3\\
& $\Delta\eta<0.08$ match & $\Delta\eta<0.07$ match \\
& $\Delta\phi<0.05$ match depending on $\eta$ & $\Delta\phi<0.04$ match depending on $\eta$ \\ \hline
EF Calo Hypo & ET$>$ 23 GeV & ET$>$ 12.5 GeV\\
& $\Delta\eta$, $\Delta\phi <$ 0.099& $\Delta\eta$, $\Delta\phi < 0.2$\\ \hline
EF Track Hypo & $N_{SCThits}>2$, $N_{blayerhits}>0$&${N_{SCThits}}>7$\\
& Impact.Par$<$ 0.5mm& Impact.Par$<$ 1mm \\ \hline
EF Match Hypo & 0.86$<ET/PT<$2.29, $\eta<1.37$ &0.7$<ET/PT<$ 1.7, $\eta <$1.37\\
&0.7$<ET/PT< $2.5, $\eta >$1.37 &0.7$<ET/PT<$ 2.5, $\eta >$1.37\\
&$\Delta\eta$ match$<$0.005, $\Delta\phi$ match$<$0.018& $\Delta\eta$ match$<$0.01, $\Delta\phi$ match$<$0.02 \\ \hline
\end{tabular}
\end{center}
\caption{ \it Hypothesis Algorithms of the electron trigger chain with
the cuts applied at each step of the selection \cite{cibran_1}. \label{algo}}
\end{table}
\subsection{ Preselection Cuts for e25i and 2e15i: Definition of
Reference Electrons \label{def}}
All the analyses have been performed using the trigger selection
hypothesis algorithms corresponding to the HLT trigger chain as described in
\ref{soft}, together with with various preselection criteria. These
preselection criteria remove events which are not of interest, because they
would not pass the level 1 trigger, would not pass offline reconstruction,
or because Monte Carlo truth information
shows they contain no electrons of interest.
At the end of each job, the TrigSteerMonitor prints a table with
the efficiency for each of the algorithms in the sequence. The
efficiencies are calculated with respect to reference electrons
(as defined in the job options). Typically the efficiency could be
calculated with respect to the Monte Carlo electrons, to the LVL1
preselection or to offline reconstructed electrons, either separately
or in combination.
Moreover, as described in the section \ref{mc}, a filter selection
is applied at event generation level to remove events with no chance of
passing the level 1 trigger, and this has to be
taken into account when computing the total cross section.
\subsubsection{Monte Carlo Preselection Cuts \label{pre}}
For the signal samples (single electrons, W and Z decays),
additional preselection cuts have been
applied on the electrons at the MonteCarlo truth level. These select only
electrons that have their momentun in a reasonable interval and that do
not cross the crack region. The requirements are:
\begin{itemize}
\item{ one generated electron in each RoI,}
\item{ two RoIs per event (only for the Z events),}
\item{ Monte Carlo truth $5 < P_T <100$ GeV}
\item{ $|\eta|<1.45$ and $|\eta|>1.55$, $|\eta|<2.45$; these cuts exclude
the crack region.}
\end{itemize}
\subsubsection{Offline Reconstructed Preselection Cuts \label{off}}
To compare the triggered electrons with the sample that would be
reconstructed offline, the offline selection algorithms are also run
on all electron candidates. The first two algorithms
that are run define as an offline electron any cluster-track match,
and then set a series of bits in the IsEM flag variable. For
all electron candidates,
the candidate has to pass a series of cuts based on the shower shape
properties in different compartments of the calorimeter as well as
variables combining ID and calorimeter information. If a cut is not passed,
then a corresponding veto bit is set in the isEM flag. For candidates with an
associated track,
identification cuts based on the tracking information have to be
passed. Thus if isEM=0, then this is a good electron or photon.
\subsubsection{LVL1 Preselection Cuts \label{lvl1}}
The Level1 preselection algorithm simulates the decision of the LVL1
trigger, applying these cuts \cite{lvl1}:
\begin{itemize}
\item{ClusterThreshold = 19.0 GeV (e25i) or 9.0 GeV (2e15i)}
\item{EmRingIsolation = 3.0 GeV (e25i) or 8.0 GeV (2e15i)}
\item{HadRingIsolation = 2.0 GeV (e25i) or 4.0 GeV (2e15i)}
\item{HadCoreIsolation = 2.0 GeV (e25i) or 4.0 GeV (2e15i)}
\end{itemize}
\subsection{Datasets \label{datasets}}
Several different datasets were used to estimate the composition of the
electron stream at low luminosity ($10^{33}\rm cm^{-2}s^{-1}$).
We took samples generated and simulated for the ATLAS Physics Workshop,
in particular:
\begin{itemize}
\item{Single electrons with $E_T=25$ GeV, generated with Pythia, about 1000
events\\
({\tt dataset rome.004022.xxx\_Pt\_25} Rome Production)}
\item{$Z \rightarrow e^{+} e^{-}$ generated with Pythia, about 10000
events\\
({\tt dataset rome.004201.recolum01.ZeeJimmy} Rome Production)}
\item{$W \rightarrow \nu_{e} e^{-}$ generated with Pythia, about 10000
events\\
({\tt dataset rome.004203.recolum01.WenuJimmy} Rome Production)}
\item{QCD di-jets generated with Pythia, about 138k events. This
allows the evaluation of the trigger background \\
({\tt dataset rome.004814.recolum01.JF17\_pythia\_jet\_filter} Rome Production) }.
\end{itemize}
\section{Results}
In this section the results after the trigger selection are shown.
For each sample, the efficiency is calculated at every step
of the trigger chain. Both e25i and 2e15i are used.
The trigger chains have been run on the ESD samples used previously for the
Rome Physics Workshop, the datasets defined in the section \ref{datasets}.
The following efficiencies are defined with respect to both offline
reconstructed electrons and LVL1 confirmation, as described in detail
in section \ref{def}.
The estimation of the rates have been performed for low luminosity,
using a MonteCarlo analysis on the CBNT rootples.
\subsection{Preselection at Generation Level: Cross Section Used \label{mc}}
An estimation of the expected trigger rate and composition of stream is given for low luminosity.
We calculated the event rate starting from the cross section (as derived from
Pythia) and taking into account the geometrical acceptance of the detector,
considering only events that can be reconstructed.
In detail, the cross section of production at LHC of the events analyzed are
reported in table \ref{cs}, with the filter efficiency:
\begin{itemize}
\item{ Electrons should have $p_T$ greater than 15 GeV and $|\eta|<3.0$}
\item{ Two such leptons are required for Z events, and one for W events}
\end{itemize}
To calculate the background trigger rates for electrons,
we are using the Rome dataset 4814 for dijets with ET(hard) = 15 GeV and a
particle level filter cut of $E_T = 17$ GeV. Each dataset contains QCD di-jets as well as physics processes like $W \rightarrow \nu_{e} e^{-}$, $Z \rightarrow e^{+} e^{-}$ and direct photon production,
which have been added to the QCD jets according to their cross
sections. The total cross section is reported in table \ref{cs}.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
{\bf Event} & {\bf cross section } & {\bf Filter $\epsilon$ }& {\bf
cross section after filter selection } \\ \hline
$Z \rightarrow e^{+} e^{-}$& 1603.8 pb & 61\% &978.3 pb \\ \hline
$W \rightarrow \nu_{e} e^{-}$& 17907 pb & 47.9\% & 8577.5 pb \\ \hline
QCD di-jets & 2.3 mb & 14.3 \% & 0.16 mb \\ \hline
\end{tabular}
\end{center}
\caption{ \it The cross-sections for the processes used in this study, together
with the acceptance of the particle level filter (geometrical and Monte Carlo truth cuts) and the final cross-sections used for rate calculations.
\label{cs}}
\end{table}
The rate is calculated using:
\begin{equation}
Rate = \frac{\sigma}{Filter}\frac{N_{sel}}{N_{all}}* L
\label{rate}
\end{equation}
where the filter takes into account the particle level filter applied
at the generation level and reported in table \ref{cs}, while
$N_{sel}$ and $N_{all}$ are the number of events
selected by all the trigger chain and the total number of events respectively.
\subsection{Rates for Low Luminosity}
The rates below have been computed using the TrigESDAnalysis package
to estimate the efficiencies and purities after each trigger
algorithm (Athena Release 10.0.1, TrigESDAnalysis-00-00-05 tag), and
the MonteCarlo analysis using the CBNT Ntuples.
Equation \ref{rate} has been used, with $L=\rm 10^{33}cm^{2}s^{-1}$ and the
cross sections after the filter shown in table \ref{cs}.
\subsubsection{ Single Electron}
For reference, we have estimated the trigger efficiency for a sample of single
electrons with $E_T=25$ GeV (e25i chain), with pile-up added.
The efficiency of preselection (including both LVL1 confirmation and
offline reconstruction isEM) for this sample is about 84\%.
Table \ref{ele} reports the efficiencies of each algorithm
with respect the previous one. These efficiencies are not cumulative.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
{\bf Algorithm} & {\bf Number of Events Single e } & {\bf Efficiency
Single Algo. in {\%} } \\ \hline
Initial & 932 & 100 \\ \hline
ISEM PRES. & 790 & 84 \\ \hline
LVL1 PRES. & 784 & 99 \\ \hline
CALO LVL2 & 772 & 98 \\ \hline
ID LVL2& 736 & 95 \\ \hline
CALO EF & 706 & 95\\ \hline
ID EF & 698 & 98\\ \hline
Matching EF & 676 & 96 \\ \hline
Cumulative Eff & 676/784 & 84 \\ \hline
\end{tabular}
\end{center}
\caption{ \it The values of efficiencies with respect to the
reconstructed events and LVL1 confirmation for each algorithm
separately. The same trigger chain has been used: e25i. The total
number of events are about 1000 single electrons of 25 GeV. The
number of events after each preselection algorithm are reported too.\label{ele}}
\end{table}
Figure \ref{single} shows the $p_T$ and energy
distributions after the preselection cut isEM.
\begin{figure}[!h]
\begin{center}
\begin{tabular}{cc}
\epsfig{figure=fig/elec_pt_new.eps,height=5.5cm,width=5.5cm} &
\epsfig{figure=fig/elect_ene_new.eps ,height=5.5cm,width=5.5cm} \\
\end{tabular}
\end{center}
\caption{\it Electron $p_T$ measured in the tracking (left) and
energy ($E_T$) in the calorimeter (right), after the isEM preselection
cuts applied, for the $E_T=25$ GeV single electron sample. \label{single}}
\end{figure}
\subsubsection{$W \rightarrow e \nu_e$}
The sample W is selected applying the trigger chain: e25i.
The electrons in the crack region have been excluded and only one
electron in each electromagnetic cluster is required too (Number of
primary electrons equal to one). The isEM flag and the LVL1
confirmation are then applied as preselection to define the right sample of
reference electrons.
Table \ref{w} shows the results for the W sample, compared with the
efficiencies derived from the 25 GeV single electron sample.
The efficiencies are cumulative, and calculated with respect to the
electrons obtained after the preselection cuts.
The estimated rate after all selections, using the formula \ref{rate}
and the cross section reported in table \ref{cs}, is about 9.1 Hz at
a luminosity of $L=\rm 10^{33}cm^{-2}s^{-1}$.
In figure \ref{w1}, the $p_T$ distributions of the
electrons from W decay after the preselection and the complete e25i trigger
chain are shown.
The comparison between the electromagnetic energy transverse for the
single electrons and the electrons from W decay is shown in the plot
\ref{eme}, this explains the different efficiency obtained for the
T2CaloHypo algorithm for these events.
The efficiency as a function of $p_T$ of the electrons
from W decay is shown in figure \ref{w2}.
The $\eta$ distributions for single electrons after the Monte Carlo preselections
are shown in figure \ref{w2}.
\begin{figure}[!h]
\begin{center}
\epsfig{figure=w_compa.eps,height=7.5cm,width=11.5cm}
\end{center}
\caption{\it
Electrons $P_T$ spectrum of events $W
\rightarrow \nu_{e} e^{-}$. In black triangles there is the electrons
$P_T$ spectrum of events after all the trigger chain e25i
reconstructed in the tracker, in red squares
the distribution obtained after
the preselection cuts and before the selection of the e25i chain,
these are the ``reference events''.
\label{w1} }
\end{figure}
\begin{figure}[!h]
\begin{center}
\epsfig{figure=fig/effe_mc.eps,height=7.5cm,width=10cm}
\end{center}
\caption{\it
Efficiency with respect to the transverse momentum of the
electrons from events $W
\rightarrow \nu_{e} e^{-}$.
The efficiency is calculated per bin (1GeV), as the ratio of the
MonteCarlo $p_T$ spectrum before and after all the cuts.
\label{w2} }
\end{figure}
\begin{figure}[!h]
\begin{center}
\epsfig{figure=fig/em_new.eps,height=7.5cm,width=11.5cm}
\end{center}
\caption{\it
Transverse electromagnetic energy spectrum from $W
\rightarrow \nu_{e} e^{-}$ events (points) and single electrons (histo).
The threshold defined in the e25i trigger chain is $EmE_T > 22 GeV$,
this explains the different algorithm efficiency between
single electrons and W events.
\label{eme} }
\end{figure}
\begin{figure}[!h]
\begin{center}
\begin{tabular}{cc}
\epsfig{figure=fig/eta_w_new.eps,height=5.5cm,width=5.5cm} &
\epsfig{figure=fig/ele_eta_new.eps,height=5.5cm,width=5.5cm} \\
(a)& (b)\\
\end{tabular}
\end{center}
\caption{\it The (a) plot shows the
the electrons $\eta$ distribution from MonteCarlo
informations (with the constraint of RoI) for the sample: $W
\rightarrow \nu_{e} e^{-}$ while the (b) plot the same distribution
but for the single electrons of 25 GeV. \label{w2}}
\end{figure}
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
{\bf Algorithm} & {\bf Number of W events }& {\bf Efficiency W in {\%} } & {\bf Efficiency
Single e in {\%} } \\ \hline
Initial & 10036 &100 & 100 \\ \hline
ISEM PRES. & 5329 &53 & 84 \\ \hline
LVL1 PRES. & 5101 & 51 & 84 \\ \hline
CALO LVL2 & 4470& 88 & 98 \\ \hline
ID LVL2& 4156& 81 & 94 \\ \hline
CALO EF & 3944& 77 & 90\\ \hline
ID EF & 3904& 76 & 89\\ \hline
Matching EF & 3768&74 & 86 \\ \hline
\end{tabular}
\end{center}
\caption{ \it The values of efficiencies with respect to the
reconstructed events (after LVL preselection) for all the algorithms (ID and Calo) for the
level 2 and the event filter. The cumulative
efficiencies of $W \rightarrow \nu_{e} e^{-}$ and
$E_T=25$ GeV single electron samples are compared, for the e25i trigger
chain. \label{w}}
\end{table}
\newpage
\subsubsection{$Z \rightarrow e^+ e^- $}
The Z sample
is selected applying two independent trigger chains: e25i and 2e15i.
In both cases the electrons in the crack region have been excluded and
only one electron in each electromagnetic cluster is required
(Number of primary electrons equal to one).
The Monte Carlo preselection cuts have been applied before the trigger
chains, see section \ref{pre}.
The trigger steering uses two independent selection chains, and the number
of selected events is the algebraic union of the events selected by
the two chains.
In the table \ref{z}, all the efficiencies are reported for each
algorithm, and finally the total efficiency of selection obtained by
the sum of the e25i and 2e15i trigger chains.
For each algorithm is reported the number of selected events, an event
is defined ``selected'' with respect to the 2e15i chain when there are two
electrons per event while for the e25i chain when there are one or two
electrons per event.
The estimated rate using the formula \ref{rate} and the cross section
after filter reported in table \ref{cs}, is about 0.84 Hz.
The combined efficiency is obtained taking the events that are
selected by e25i or 2e15i trigger chain and excluding double counting from
events selected by both. It is interesting to note that nearly all Z events
are selected by the e25i trigger chain alone. The small fraction of electron
momenta betwen 15 and 25 GeV and the lower efficiency for selecting two
electrons as compared to one mean that the 2e15i trigger adds only a small
number of events not selected by e25i, and 2e15i alone has a significantly
lower efficiency for the overall sample.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
{\bf Algorithm} & {\bf Number of Events } & {\bf Number of Events } \\ \hline
& {\bf e25i } & {\bf 2e15i } \\ \hline
PRES. & 5694 & 5659 \\ \hline
CALO LVL2 & 5543& 4969 \\ \hline
ID LVL2& 5304 & 4969 \\ \hline
CALO EF & 5160 &4940\\ \hline
ID EF & 4993& 4288\\ \hline
Matching EF &4955 & 4003 \\ \hline
Efficiency &87.6\% &70.3\% \\\hline
Combined Eff. & \multicolumn{2}{c|}{88\%} \\\hline
\end{tabular}
\end{center}
\caption{ \it The number of $\rm Z\rightarrow e^{+} e^{-}$
events for each single algorithm of the
trigger chains: 2e15i and e25i, and their matching. The last rows show
the efficiency for each trigger chain and their
combination, the number of events of the match is calculated summing
the events accepted by a trigger chain plus the events of the other
chain not yet included.
Almost all of the events are triggered by e25i trigger chain.
\label{z} }
\end{table}
\begin{figure}[!h]
\begin{center}
\begin{tabular}{cc}
\epsfig{figure=fig/pt_z_gen.eps,height=5.5cm,width=5.5cm} &
\epsfig{figure=zeta2.eps,height=5.5cm,width=5.5cm} \\
(a)& (b)\\
\epsfig{figure=fig/pt_z_15.eps,height=5.5cm,width=5.5cm} &
\epsfig{figure=fig/pt_z_25.eps,height=5.5cm,width=5.5cm} \\
(c)& (d)\\
\end{tabular}
\end{center}
\caption{\it $\rm Z\rightarrow e^{+} e^{-}$ sample. The (a) plot shows the electron $p_T$ from MonteCarlo
information (with the constraint of RoI), while the (b) plot
shows the eta distribution of the electrons from Z decay.
The electron $p_T$ distributions, after the trigger chains: 2e15i
and e25i, are reported in the plots (c) and (d) respectively.\label{zeta}}
\end{figure}
\subsubsection{QCD Di-jets}
The selection of QCD events is done using the same hypothesis algorithms
as for the signal samples, but without the requirement on the number of primary electrons in each cluster.
The electron candidates in this sample represent the background in the calibration stream with respect the other physics events described above.
Each dataset contains QCD di-jets as well as physics processes like $W \rightarrow \nu_{e} e^{-}$, $Z \rightarrow e^{+} e^{-}$, and direct photon
production, so we analyzed in detail the Monte Carlo composition of the stream to define what we obtain after the selection of the relevant physics processes.
Applying the cuts described above the expected rate is about 20 Hz, with an efficiency of 2.9 \% with respect to the preselection and an efficiency of preselection of about 0.004 \%.
The total number of analysed events total events is about 140000, and only
16 events survive all the trigger cuts, with the following composition:
\begin{itemize}
\item{Genuine electrons from W and Z or B hadron decays about 50\%}
\item{Converted photons from $\pi$ or jets about 31\%}
\item{Fake events, for example charge particles with tracks randomly
associated to electromagnetic calorimeter clusters, about 19\%}
\end{itemize}
The selected events in the background sample that contain genuine electrons can
be considered as useful for the electron calibration stream as well.
Figure~\ref{energyrate} shows the rate of accepted events as a function of the
energy of the triggered electron candidate from the Calorimeter EF algorithm. Although
the overall statistics are very low, it can be seen that the electrons
from W and Z boson decays have a somewhat softer $E_T$ spectrum than those
from the QCD background.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
{\bf Algorithm (e25i)} & {\bf Number of Events } & {\bf Rate (Hz) } \\ \hline
Initial & 138532 & --- \\\hline
PRES. & 551 & 640 \\ \hline
CALO LVL2 & 170& 198 \\ \hline
ID LVL2& 33 & 38 \\ \hline
CALO EF & 25 & 29\\ \hline
ID EF & 25& 29\\ \hline
Matching EF &16 & 19 \\ \hline
\end{tabular}
\end{center}
\caption{ \it The number of events for each single algorithm of the
trigger chain 25ei, and their matching for QCD-jets events. \label{jet}}
\end{table}
\begin{figure}[!h]
\begin{center}
\epsfig{figure=rates_new2.eps,height=7.5cm,width=8.5cm}
\caption{\it The plot shows the rates as function of electron energy
after EF.
The rates have been obtained normalized the number of the entries with
respect to the rate with 12 GeV per bin. The QCD events are the
dark histogram, the electrons are represented by the open histogram.\label{energyrate}}
\end{center}
\end{figure}
\section{Different Selections}
In this section we studied applying different cuts at the event filter to
reduce the total electron stream trigger rate to 10 Hz, keeping as
large a fraction as possible of the pure electrons.
The cuts applied at the calorimeter filter level are not yet optimized, and
possible improvements can be achieved modifying the thresholds
applied to the $E/p$ ratio and track parameters. Figure~\ref{ep} shows the
$E/p$ variable for W signal and QCD background events in different $\eta$
ranges. Again, although the statistics for the background sample are very
low, it looks possible to increase the signal purity by tightening the
upper cut on $E/p$.
\begin{figure}[!h]
\begin{center}
\begin{tabular}{cc}
\epsfig{figure=ep_w.eps,height=5.5cm,width=5.5cm} &
\epsfig{figure=ep_w2.eps,height=5.5cm,width=5.5cm} \\
(a)& (b) \\
\epsfig{figure=ep_jet.eps,height=5.5cm,width=5.5cm} &
\epsfig{figure=ep_jet2.eps,height=5.5cm,width=5.5cm} \\
(c)& (d) \\
\end{tabular}
\end{center}
\caption{\it In the (a) plot, the E/p ratio for the W events for high
$\eta$ is
shown, while in the (b) side the same E/p ratio for W but for the
low $\eta$ cuts. In the (c) and (d) plots the same variables but for the
electrons in QCD jets for high and low eta range. \label{ep}}
\end{figure}
In table \ref{cuts}, the effect of applying two different sets of cuts is shown
on the rate of W and Z events in comparison with the rate of QCD jets.
Moreover for the QCD jets, the rate composition has been analysed and
the genuine electron component calculated.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
{\bf EF Match Cuts} & {\bf W Event Rate } & {\bf QCD Jets Rate }& {\bf Z Rate }& {\bf
Total Rate }\\ \hline
0.86$<ET/PT<$2.29, $\eta>1.37$ &9.1Hz & 19Hz(9.5Hz) & 0.84Hz &28.9Hz(19.4Hz)\\
0.7$<ET/PT< $2.5, $\eta <$1.37 & & & &\\
\hline
0.86$<ET/PT<$1.65, $\eta> 1.37$ & 8.5 Hz & 13Hz(7.6Hz) & 0.88Hz & 22.4Hz(17.0Hz)\\
0.7$<ET/PT< $1.4, $\eta <$1.37 & & & &\\
\hline
0.86$<ET/PT< $1.65, $\eta >$1.37 & 7.3Hz & 9Hz (6.8Hz) & 0.87Hz & 17.2Hz(15.0Hz) \\
0.97$<ET/PT<$1.15, $\eta<1.37$ & & & &\\
\hline
\end{tabular}
\end{center}
\caption{ \it New cuts applied at Event Filter level to reduce the
total rate to about 15 Hz. In the QCD column, in parenthesis, the
rate of genuine electrons is reported. The Z rate is almost the
same, the different cuts are applied only on the e25i chain then the
matching of the two independent trigger chains reproduces the same rate.\label{cuts}}
\end{table}
Taking into account the last cuts, we can obtain about more than 17 Hz (Z+W+jet)
trigger rate, where about 15 Hz are genuine electrons useful for the
electron calibration stream. The Z events are less than 1 Hz after the
cuts.
\section{Conclusions and outlook}
After this first analysis on the Rome data sample using the Electron High Level
Trigger chains (e25i and 2e15i) we have estimated the electron trigger
rate expected at low luminosity.
The efficiency and the purity have been calculated, with the standard
thresholds and with modified cuts on the Event Filter, to optimize as
much as possible the ratio purity over efficiency of the obtained
sample. After some limited optimisation, a total rate of 17 Hz has been found,
dominated by genuine electrons from W and Z decay.
For the background, the number of selected events is very small, due to the
limited size of the available background sample. This makes it impossible to
study further cut optimisation to reduce the rate to the target of 10 Hz.
However, it is already apparent that simply increasing the $E_T$ threshold,
while reducing the rate, will not increase the purity of the selected sample.
Much more background event statistics will be required to better
understand the relations between the background composition and the cuts on
the $E/p$ ratio.
More analysis will be performed on the CSC samples. In addition to looking
at the e25i trigger, it will be of interest to look at lower thresholds for
lower luminosity running (e.g. $10^{31}$ and $10^{32} \rm cm^{-2}s^{-1}$). This
is not possible with the present background sample due to the filter cuts
imposed in the event generation.
\clearpage
|
train/arxiv
|
BkiUaqXxK0zjCsHeYYys
| 5 | 1 |
\section{Introduction}
\label{intro}
The phase diagram of strongly interacting matter at finite temperature and
chemical potential has been extensively studied along the past decades. In
the region of very high temperatures and low densities, it is well known
that Quantum Chromodynamics (QCD) predicts the formation of a quark-gluon
plasma (QGP)~\cite{Fukushima:2010bq}. Under these extreme conditions quark
and gluons are expected to be weakly coupled, and the phase diagram can be
explored by means of first-principle perturbative calculations based on
expansions in powers of the QCD coupling constant. Moreover, lattice QCD
(LQCD) calculations indicate that at vanishing chemical potential the
transition from the hadronic phase to the QGP occurs in the form of a smooth
crossover, at a pseudocritical temperature $T_{\rm pc} \sim 150-170$~MeV. On
the other hand, at sufficiently high densities and low temperatures, one
expects to find a ``color-flavor locked'' phase~\cite{Alford:2007xm}, in
which quarks are bounded into color superconducting states analogous to the
Cooper pairs formed by electrons in an ordinary superconductor. At moderate
densities, however, the situation is much more uncertain. The main reason
for this is that first-principle nonperturbative QCD calculations at finite
baryon chemical potential $\mu_B$ are not accessible by Monte Carlo
simulations, due to the presence of a complex fermion determinant in the
corresponding partition function (the so-called ``sign problem''). In this
region, which is not accessible through lattice techniques or first
principles, most of the present theoretical knowledge on the phase
transitions is obtained from the study of effective models for strong
interactions.
Given the important role played by effective models in the understanding of
the QCD phase diagram, it is important to test their reliability. This can
be done by comparing the corresponding predictions with those obtained from
first principle calculations, in situations where the latter are available.
One obvious possibility is to consider the above mentioned case of
strong-interaction matter at finite temperature and vanishing chemical
potential. Another interesting situation is the one in which $\mu_B=0$, but
one has a nonzero isospin chemical potential $\mu_I$. In this case (both at
zero and finite temperature) LQCD simulations are feasible, since the
functional determinant turns out to be real~\cite{Alford:1998sd}. Following
the early work in Refs.~\cite{Kogut:2002tm,Kogut:2002zg}, several groups
have performed LQCD calculations at $\mu_I \neq 0$ using different
techniques, see e.g.\
Refs.~\cite{Kogut:2004zg,deForcrand:2007uz,Cea:2012ev,Detmold:2012wc,Brandt:2017oyy,Brandt:2018bwq}.
One important feature confirmed by these calculations is that at $\mu_I
\gtrsim m_\pi$ one finds the onset of a Bose-Einstein pion condensation
phase, as previously conjectured in Ref.~\cite{Son:2000xc}. For a recent
review on meson condensation triggered by a large isospin imbalance see
Ref.~\cite{Mannarelli:2019hgn}, where references to various theoretical
approaches for the analysis of associated phase transitions can be found.
In this work we consider the properties of quark matter at finite isospin
chemical potential using a particular class of effective theories, viz.\ the
nonlocal Polyakov-Nambu$-$Jona-Lasinio (nlPNJL)
models~\cite{Blaschke:2007np,Contrera:2007wu,Contrera:2009hk,Contrera:2010kz,Hell:2008cc,Hell:2009by}.
In the nlPNJL approach the quarks move in a background color field and
interact through covariant nonlocal chirally symmetric four-point couplings,
which are separable in momentum space. At vanishing $\mu_B$ and finite
temperature these models provide a plausible description of chiral
restoration and deconfinement transitions, in good agreement with LQCD
results~\cite{Dumm:2021vop}. In general, it can be considered that they represent an
improvement over the local Polyakov Nambu$-$Jona-Lasinio (PNJL)
model~\cite{Meisinger:1995ih,Fukushima:2003fw,Megias:2004hj,Ratti:2005jh,
Roessner:2006xn,Mukherjee:2006hq,Sasaki:2006ww}. In fact, nonlocal
interactions arise naturally in the context of several successful approaches
to low-energy quark dynamics, and lead to a momentum dependence in quark
propagators that can be made consistent~\cite{Noguera:2008cm} with lattice
results. Moreover, it can be seen that nonlocal extensions of the NJL model
do not show some of the known inconveniences that are present in the local
theory. Well-behaved nonlocal form factors can regularize the loop integrals
in such a way that anomalies are preserved~\cite{RuizArriola:1998zi} and
charges are properly quantized. In addition, one can avoid the introduction
of various sharp cutoffs to deal with higher order loop
integrals~\cite{Blaschke:1995gr}, improving in this way the predictive power
of the models.
Within the above mentioned framework, the aim of the present work is to
provide a comparison, both at zero and finite temperature, between the
results obtained within the nlPNJL model and those arising from other
theoretical approaches. In particular, we consider the results from the
local NJL model~\cite{He:2005nk,Avancini:2019ego}, its PNJL
extension~\cite{Zhang:2006gu,Sasaki:2010jz}, chiral perturbation theory
(ChPT)~\cite{Adhikari:2019mdk} and recent LQCD
calculations~\cite{Brandt:2017oyy,Brandt:2018bwq}.
This article is organized as follows. In Sec.~\ref{model} we present the
general formalism to describe a two-flavor nonlocal PNJL model at finite
temperature and nonvanishing isospin chemical potential. In
Sec.~\ref{results} we quote and discuss our numerical results, including the
comparison with the outcomes from alternative effective approaches and LQCD
simulations. Finally, in Sec.~\ref{summary} we summarize our results and
present our main conclusions.
\section{Theoretical Formalism}
\label{model}
We start by considering the Euclidean action of a two-flavor quark model
that includes nonlocal scalar and pseudoscalar quark-antiquark currents. One
has
\begin{eqnarray}
S_E &=& \int d^4 x \,\left[
\bar \psi (x) \left(
- i \rlap/\partial + \hat m \right) \psi (x)
\, - \, \frac{G}{2}\, j_a(x) j_a(x) \right] \ ,
\label{action}
\end{eqnarray}
where $\psi = (\psi_u\ \psi_d)^T$ stands for the $u$, $d$ quark field
doublet, and $\hat m = \mbox{diag}(m_u,m_d)$ is the current quark mass
matrix. In what follows we assume that the current masses of $u$ and $d$
quarks are equal, denoting $m_c \equiv m_u=m_d $. The nonlocal currents
$j_a(x)$ in Eq.~(\ref{action}) are given by
\begin{eqnarray}
j_a (x) &=& \int d^4 z \ {\cal G}(z) \ \bar \psi(x+\frac{z}{2}) \
\Gamma_a \ \psi(x-\frac{z}{2}) \ , \label{cuOGE}
\end{eqnarray}
where we have defined $\Gamma_a = ( \openone, i \gamma_5 \vec \tau )$,
$\tau_i$ being Pauli matrices that act on flavor space. The function ${\cal
G}(z)$ is a form factor responsible for the nonlocal character of the
four-point interactions.
To study strong-interaction matter at finite temperature and/or chemical
potential we introduce the partition function of the system, given by
$\mathcal{Z} = \int \mathcal{D} \bar{\psi}\,\mathcal{D}\psi \,\exp[-S_E]$.
As stated, we are interested in dealing with isospin asymmetric matter. This
is effectively implemented by introducing quark chemical potentials $\mu_u$
and $\mu_d$, which in principle can be different from each other. Thus, in
the effective action we perform the replacement
\begin{equation}
\left(\begin{array}{cc} \partial_4 & 0 \\ 0 & \partial_4
\end{array} \right) \rightarrow \left(\begin{array}{cc} \partial_4 -
\mu_u & 0 \\ 0 & \partial_4 - \mu_d \end{array} \right) \ .
\label{kinrep}
\end{equation}
The quark chemical potentials can be written in terms of average and isospin
chemical potentials $\mu$ and $\mu_I$ as
\begin{equation}
\mu_u \ = \ \mu + \dfrac{\mu_I}{2} \ ,\qquad\qquad \mu_d \ = \ \mu -
\dfrac{\mu_I}{2}\ ,
\end{equation}
where $\mu=\mu_B/3$, $\mu_B$ being the baryon chemical potential. For the
nonlocal model under consideration, to obtain the appropriated conserved
currents the replacement in Eq.~(\ref{kinrep}) has to be complemented with a
modification of the nonlocal currents appearing in Eq.~(\ref{cuOGE}),
namely~\cite{GomezDumm:2006vz,Dumm:2010hh}
\begin{eqnarray}
\psi(x-z/2) & \rightarrow & \mathcal{W}(x,x-z/2) \, \psi(x-z/2)\ ,
\nonumber \\
\bar\psi(x+z/2) & \rightarrow & \bar\psi(x+z/2)\, \gamma_0
\,\mathcal{W}(x+z/2,x)\,\gamma_0 \ .
\label{transport}
\end{eqnarray}
In the present case the transport functions $\mathcal{W}$ are simply given
by
\begin{equation}
\mathcal{W}(x,x-z/2)\ = \ \mathcal{W}(x+z/2,x) \ = \ \exp \left(
\frac{z_4}{2}\, \hat \mu \right) \ ,
\end{equation}
where $\hat\mu = {\rm diag}(\mu_u,\mu_d)$.
It is convenient to perform a standard bosonization of the the fermionic
action~\cite{Ripka:1997zb}, introducing auxiliary mesonic fields $\sigma$
and $\pi_i$, $i=1,2,3$, and integrating out the fermion fields. We consider
here the mean field approximation (MFA), in which the bosonic fields are
replaced by their vacuum expectation values (VEV) $\bar\sigma$ and
$\bar\pi_i$. Let us recall that, for $\mu_I=0$, in the chiral limit
($m_c=0$) the action in Eq.~(\ref{action}) is invariant under global ${\rm
U(1)}_B \otimes {\rm SU(2)}_I \otimes {\rm SU(2)}_{IA}$ transformations. The
group U(1)$_B$ is associated to baryon number conservation, while the chiral
group SU(2)$_I \otimes {\rm SU(2)}_{IA}$ corresponds to the symmetries under
isospin and axial-isospin transformations. At zero temperature the
SU(2)$_{IA}$ symmetry is expected to be spontaneously broken by a large
value of $\bar\sigma$ (which leads to large constituent quark masses), while
at high temperatures one expects to have $\bar\sigma=0$, which implies a
restoration of the chiral symmetry. In the presence of finite quark masses
one has an explicit breakdown of SU(2)$_{IA}$ (and also of SU(2)$_I$, if
current $u$ and $d$ quark masses are different to each other), hence the
chiral symmetry is expected to be only partially restored at high $T$. Now,
in the presence of a nonzero isospin chemical potential the full chiral
symmetry group is explicitly broken down to the U(1)$_{I_3} \otimes {\rm
U(1)}_{I_3A}$ subgroup. At $T=0$ it might happen that, similarly to the
$\mu_I=0$ case, U(1)$_{I_3A}$ is spontaneously broken by a large value of
$\bar\sigma$. Moreover, while for finite current quark masses one has
$\bar \pi_3=0$~\cite{Ebert:2006uh}, it can happen that nonvanishing VEVs for
$\pi_1$ and $\pi_2$ be developed, leading to a spontaneous breakdown of the
remaining U(1)$_{I_3}$ symmetry. Since the action is still invariant under
U(1)$_{I_3}$ transformations, without loss of generality one can choose
$\bar\pi_i=\delta_{i1}\bar\Delta$.
We consider the above described general situation in which both $\bar\sigma$
and $\bar\Delta$ can be nonvanishing. At zero temperature, the mean field
thermodynamic potential is found to be given by
\begin{eqnarray}
\Omega^{\rm MFA}(T=0) &=& \frac{\bar \sigma^2 + \bar\Delta^2}{2\ G}
- {\rm Tr} \ln
\begin{pmatrix}
-\rlap/ p_u + M\big(p_u\big) & i \, \gamma_5 \, \rho\big(\bar p\big) \\
i\, \gamma_5 \, \rho\big(\bar p\big) & -\rlap/ p_d + M\big(p_d\big)
\end{pmatrix} \ ,
\label{actionMF}
\end{eqnarray}
where
\begin{eqnarray}
M\big(p\big) \ = \ m_c + g\big(p\big)\, \bar\sigma \ , \qquad \qquad
\rho\big(p\big) \ = \ g\big(p\big) \, \bar\Delta \ .
\label{defsMrhop}
\end{eqnarray}
Here we have defined $p_f^\nu \equiv \left( \vec p ,\, p_4 + i \mu_f \right)$,
with $f=u,d$, and $\bar p = (p_u+p_d)/2$. The function $g(p)$ is the Fourier
transform of the form factor ${\cal G}(z)$ in Eq.~(\ref{cuOGE}).
\hfill
Let us consider the extension of the model to the case of finite
temperature, which can be addressed by using the standard Matsubara
formalism. In order to account for confinement effects, we also include the
coupling of fermions to the Polyakov loop (PL), assuming that quarks move on
a constant color background field $\phi = i g\, G_{4a} \lambda_a/2$, where
$G_{\mu a}$ are SU(3) color gauge fields. We work in the so-called Polyakov
gauge, in which the matrix $\phi$ is given a diagonal representation $\phi =
{\rm diag}(\phi_r,\phi_g,\phi_b) = \phi_3 \lambda_3 + \phi_8 \lambda_8$,
taking the traced Polyakov loop $\Phi=\frac{1}{3} {\rm Tr}\, \exp( i
\phi/T)$ as an order parameter of the confinement/deconfinement transition.
In addition, to account for effective gauge field self-interactions we
introduce a mean field Polyakov-loop potential ${\cal U}$ that depends on
the traced PL, its conjugate $\bar\Phi$ and the temperature. The resulting
scheme is usually referred to as a nonlocal Polyakov-Nambu-Jona-Lasinio
(nlPNJL)
model~\cite{Blaschke:2007np,Contrera:2007wu,Contrera:2009hk,Contrera:2010kz,Hell:2008cc,Hell:2009by}.
Concerning the PL potential, its functional form is usually based on
properties of pure gauge QCD. In this work we consider a potential given by
a polynomial function based on a Ginzburg-Landau
ansatz~\cite{Ratti:2005jh,Scavenius:2002ru}, namely
\begin{eqnarray}
\frac{{\cal{U}}_{\rm poly}(\Phi,\bar \Phi, T)}{T ^4} \ = \ -\,\frac{b_2(T)}{2}\, \bar \Phi \Phi
-\,\frac{b_3}{6}\, \left(\Phi^3 + \bar \Phi^3\right) + \,\frac{b_4}{4}\, \left( \bar \Phi \Phi \right)^2 \ ,
\label{upoly}
\end{eqnarray}
where
\begin{eqnarray}
b_2(T) = a_0 +a_1 \left(\dfrac{T_0}{T}\right) + a_2\left(\dfrac{T_0}{T}\right)^2
+ a_3\left(\dfrac{T_0}{T}\right)^3\ .
\label{pol}
\end{eqnarray}
The parameters $a_i$ and $b_i$ can be fitted to pure gauge lattice QCD
results imposing the presence of a first-order phase transition at the
reference temperature $T_0$, which is a further parameter of the model. In
the absence of dynamical quarks, $T_0$ is the critical temperature for
deconfinement, and from lattice QCD calculations one expects it to be
approximately equal to 270~MeV. However, it has been argued that in the
presence of light dynamical quarks $T_0$ should be rescaled to about 210 and
190~MeV for the case of two and three flavors, respectively, with an
uncertainty of about 30~MeV~\cite{Schaefer:2007pw,Schaefer:2009ui}. The numerical values for
the PL potential parameters are~\cite{Ratti:2005jh}
\begin{equation}
a_0 = 6.75\ ,\quad a_1 = -1.95\ ,\quad a_2 = 2.625\ ,\quad a_3 = -7.44
\ ,\quad b_3 = 0.75\ ,\quad b_4 = 7.5\ .
\end{equation}
In this way, the grand canonical thermodynamic potential of the system is
given by
\begin{eqnarray}
\Omega^{\rm MFA} & = &
\frac{\bar\sigma^2 + \bar\Delta^2}{2\ G} - 2 \, T\ \sum_{n=-\infty}^\infty \sum_{c=r,g,b}
\int \frac{d^3 p}{(2\pi)^3} \ \ln\Big\{ E_{nuc}^2\ E_{ndc}^2 \nonumber \\
& & -\ \rho(\bar p_{nc})^2 \Big[\big(M(p_{nuc})-M(p_{ndc})\big)^2-(\mu_u-\mu_d)^2\Big] \Big\}
+ \,
{\cal{U}}_{\rm poly}(\Phi,\bar \Phi,T)\ ,
\label{granp}
\end{eqnarray}
where we have introduced the definitions $\bar p_{nc} = (p_{nuc}+p_{ndc})/2$
and $E_{nfc}^2=M(p_{nfc})^2 + p_{nfc}^2+\rho(\bar p_{nc})^2$, with $p_{nfc}
\equiv (\vec p,(2n+1)\pi T + i \mu_f +\phi_c)$. As usual in this type of
model, it is seen that $ \Omega^{\rm MFA}$ turns out to be divergent, thus
it has to be regularized. We adopt here a prescription similar as the one
considered e.g.\ in Ref.~\cite{GomezDumm:2004sr}, viz.
\begin{equation}
\Omega^{\rm MFA,\rm reg} \ = \
\Omega^{\rm MFA}\, -\, \Omega^{\rm free}_{\rm q}\,
+\, \Omega^{\rm free,reg}_{\rm q}\, +\, \Omega_0 \ .
\end{equation}
Here the ``free'' potential keeps the interaction with the PL, while $\bar
\sigma$ and $\bar \Delta$ are set to zero. A constant term $\Omega_0$ is
also added so as to fix $\Omega^{\rm MFA,\rm reg} = 0$ at $\mu_B = \mu_I = T
= 0$. For the regularized form of the free piece, the Matsubara sum can be
performed analytically. One has
\begin{eqnarray}
\Omega^{\rm free,reg}_{\rm q} = -2 T \sum_{f=u,d} \, \sum_{c=r,g,b} \,
\sum_{s=\pm 1} \int \frac{d^3 \vec{p}}{(2\pi)^3}\; \mbox{Re}\; \ln \left[ 1
+ \exp\left(-\;\frac{\epsilon_f + s\ (\mu_f + i \phi_c)}{T} \right)\right] \
,
\end{eqnarray}
where $\epsilon_f = \sqrt{\vec p^{\;2} + m_f^2}\,$.
The mean field values $\bar \sigma$ and $\bar \Delta$, as well as the values
of $\phi_3$ and $\phi_8$, can now be obtained from a set of four coupled
``gap equations'' that follow from the minimization of the regularized
thermodynamic potential, namely
\begin{equation}
\frac{\partial \Omega^{\rm MFA,reg}}{\partial \bar \sigma} \ = \ 0\
, \qquad
\frac{\partial \Omega^{\rm MFA,reg}}{\partial \bar \Delta} \ = \ 0\
, \qquad
\frac{\partial \Omega^{\rm MFA,reg}}{\partial \phi_3} \ = \ 0\
, \qquad
\frac{\partial \Omega^{\rm MFA,reg}}{\partial \phi_8} \ = \
0\ .
\label{gapeqs}
\end{equation}
In addition, it is interesting to study the behavior of quark condensates.
As usual, we consider the scalar condensate $\Sigma = \Sigma_u + \Sigma_d$,
where $\Sigma_f = \langle \bar \psi_f \psi_f \rangle$. The corresponding
expressions can be obtained by differentiating $\Omega^{\rm MFA,reg}$ with
respect to the current up and down current quark masses, i.e.
\begin{equation}
\Sigma_f \ = \ \frac{\partial \Omega^{\rm MFA,reg}}{\partial m_f} \ .
\label{Sigma}
\end{equation}
Another relevant quantity is the charged pion condensate $\Pi$, which is
expected to be nonvanishing for $\mu_I \neq 0$. According to our choice
$\bar\pi_i= \delta_{i1} \bar \Delta$, we get
\begin{eqnarray}
\Pi \ = \ \langle \bar \psi i \gamma_5 \tau_1 \psi \rangle \ .
\label{Pi}
\end{eqnarray}
The analytical expression for this condensate can be obtained by taking the
derivative of the regularized thermodynamic potential with respect to an
auxiliary parameter added to $\rho(\bar p)$ in Eq.~(\ref{actionMF}), and
then set to zero after the calculation.
To study the phase transitions, we also introduce the susceptibilities
associated to the $\Sigma$ and $\Pi$ condensates~\cite{Lu:2019diy} and the
Polyakov loop. These are given by
\begin{equation}
\chi_{\rm ch} \ = \ - \frac{\partial\Sigma}{\partial m_c} \ , \qquad
\chi_{\Pi} \ = \ \frac{\partial\Pi}{\partial m_c} \ , \qquad
\chi_\Phi \ = \ \frac{d\Phi}{dT}\ .
\label{chis}
\end{equation}
Finally, from the regularized potential one can calculate various
thermodynamic quantities, such as the energy and entropy densities
$\varepsilon$ and $s$, and the particle number densities $n_I$ and $n_B$.
The corresponding expressions are
\begin{eqnarray}
\varepsilon &=& \Omega^{\rm MFA,reg} + T\, s + n_I\, \mu_I + n_B\, \mu_B\ , \nonumber \\
s &=& -\, \frac{\partial \Omega^{\rm MFA,reg}}{\partial T} \ , \nonumber \\
n_I &=& -\, \frac{\partial \Omega^{\rm MFA,reg}}{\partial \mu_I}
\ , \nonumber \\
n_B &=& -\, \frac{\partial \Omega^{\rm MFA,reg}}{\partial \mu_B} \ .
\label{esn}
\end{eqnarray}
In this work we restrict to the case of $\mu_B = 0$, focusing on the effect
of finite isospin chemical potential $\mu_I$. As stated in the Introduction,
in this situation the results from effective models can be compared with
existing lattice QCD
calculations~\cite{Brandt:2016zdy,Brandt:2017oyy,Brandt:2017zck,Brandt:2018wkp,Brandt:2018bwq},
which do not suffer from the sign problem. Since the thermodynamic potential
turns out to be real, one gets $\Phi = \bar \Phi$, $\phi_8 = 0$, and the
last of Eqs.~(\ref{gapeqs}) is trivially satisfied.
\section{Numerical Results}
\label{results}
To fully define the model it is necessary to specify the form factor
entering the nonlocal fermion current in Eq.~(\ref{cuOGE}). In this work we
consider an exponential momentum dependence for the form factor in momentum
space,
\begin{equation}
g(p) \ = \ \exp (-p^2 / \Lambda^2)\ .
\label{ff}
\end{equation}
This form, which is widely used, guarantees a fast ultraviolet convergence
of quark loop integrals. Notice that the energy scale $\Lambda$, which acts
as an effective momentum cutoff, has to be taken as an additional parameter
of the model. Other functional forms, e.g.\ Lorentzian form factors with
integer~\cite{Dumm:2010hh} or fractional~\cite{Carlomagno:2018tyk} momentum
dependences, have also been considered in the literature.
In any case, it is seen that the form factor choice does not have
in general major impact in the qualitative predictions for the relevant
thermodynamic quantities~\cite{Carlomagno:2013ona}.
Given the form factor shape, the model parameters $m_c$, $G$ and $\Lambda$
can be fixed by requiring that the model reproduce the phenomenological
values of some selected physical quantities. If we take as inputs the pion
mass $m_\pi=138$~MeV, the pion weak decay constant $f_\pi=92.4$~MeV and the
quark condensates $\Sigma_u = \Sigma_d = - (240\ {\rm MeV})^3$, one has $m_c
= 5.67$~MeV, $\Lambda = 752$~MeV and $g = G\Lambda^2 =
20.67$~\cite{GomezDumm:2006vz}.
\subsection{Zero temperature}
\label{zeroT}
At zero temperature the Polyakov loop decouples from the fermions, and the
thermodynamic potential within the nonlocal NJL (nlNJL) model is given by
the expression in Eq.~(\ref{actionMF}), properly regularized. In
Fig.~\ref{fig:1} we show our numerical results for the normalized mean field
condensates $\Sigma/\Sigma_0$ and $\Pi/\Sigma_0$, where
$\Sigma_0\equiv\Sigma(\mu_I=0)$, as functions of the isospin chemical
potential. The solid red lines correspond to the parametrization described
above, which leads to $\Sigma_0 = - $(240~MeV)$^3$. To provide an estimation
of the parametrization dependence, we show with a red shaded band the
results covered by a parameter range such that $\Sigma_0$ lies between
$-$(230~MeV)$^3$ and $-$(250~MeV)$^3$. The right panel of Fig.~\ref{fig:1}
just extends the results given in the left panel, covering a broader range
of values of the scaled isospin chemical potential $\mu_I/m_\pi$. For
comparison, in both panels we include the results obtained from several
alternative approaches. The green band (partially hidden by the red one)
corresponds to the results from the local NJL, for parametrizations leading
to a quark condensate in the range between $-$(240~MeV)$^3$ and
$-$(250~MeV)$^3$. The dashed (green) lines, the dotted (brown) lines and the
dashed-dotted (blue) lines correspond to the results obtained within the
linear sigma model (LSM) in Ref.~\cite{He:2005nk}, the NJL model in
Ref.~\cite{Avancini:2019ego} (where a medium separation regularization
scheme is used) and the Chiral Perturbation Theory (ChPT) approach in
Ref.~\cite{Adhikari:2019mdk}, respectively. In addition, the fat dots denote
the results from lattice QCD obtained in Ref.~\cite{Brandt:2018bwq}.
\begin{figure}[hbt]
\centering{\includegraphics[width=0.75\textwidth]{sig-del-complete}}
\caption{(Color online) Normalized $\Sigma$ and $\Pi$ condensates as
functions of the isospin chemical potential. The solid red line and the red
band correspond to the numerical results obtained within the nlNJL
model. Results from other theoretical approaches (see text) are included for
comparison.}
\label{fig:1}
\end{figure}
As expected, for $\mu_I < m_\pi$ one has $\Sigma = \Sigma_0$ and $\Pi = 0$.
Indeed, for both local and nonlocal NJL models it can be analytically shown
that the onset of the pion condensation at $T=0$ occurs at $\mu_I = m_\pi$.
For larger isospin chemical potentials, as shown in Fig.~\ref{fig:1}, the
chiral condensate decreases monotonically and the charged pion condensate
gets strongly increased. In this way, for $\mu_I\geq m_\pi$ the isospin
symmetry U(1)$_{I_3}$ gets spontaneously broken, while one finds a partial
restoration of the U(1)$_{I_3A}$ symmetry for large values of $\mu_I$. From
the left panel of Fig.~\ref{fig:1} it is also seen that there is an overall
agreement between most theoretical approaches up to $\mu_I \simeq 2m_\pi$.
On the other hand, as shown in the right panel of the figure, for larger
values of $\mu_I$ there is some splitting between the predictions from
different models.
The results for the chiral and pion condensate susceptibilities as functions
of $\mu_I$ are displayed in Fig.~\ref{fig:1b}. It can be seen that the
chiral susceptibility $\chi_{\rm ch}$ (solid line, left panel) is
approximately zero for low values of $\mu_I$, showing a jump to a high value
at $\mu_I = m_\pi$ and remaining relatively large for $\mu_I
> m_\pi$. This signals that at $\mu_I = m_\pi$ one has the onset of
a smooth transition from a phase in which the U(1)$_{I_3A}$ symmetry is
spontaneously broken to a region in which it becomes (partially) restored.
It is found that the height of the jump at $\mu_I = m_\pi$ gets increased if
the current quark mass $m_c$ is reduced. The pion condensate susceptibility
is given by the solid line in the right panel of Fig.~\ref{fig:1b}. It is
seen that $\chi_\Pi$ is zero for low values of $\mu_I$, and has a divergence
at $\mu_I = m_\pi$. This is the signature of a second order phase transition
leading to the appearance of the pion condensate, as shown in
Fig.~\ref{fig:1}. The behavior of the susceptibilities is similar to the one
found in the local NJL model, see Ref.~\cite{Lu:2019diy}
\begin{figure}[hbt]
\centering{}\includegraphics[width=0.8\textwidth]{susc_t0}
\caption{(Color online) Chiral and pion susceptibilities as functions of the
isospin chemical potential. Solid and dashed lines correspond to the results
from nlNJL model calculations and lowest order ChPT expressions,
respectively.}
\label{fig:1b}
\end{figure}
It is interesting to compare the above results with those arising from
Chiral Perturbation Theory. At the lowest order in the chiral expansion, it
is found that for $\mu_I\geq m_\pi$ the condensates satisfy the
relations~\cite{Kogut:2001id}
\begin{equation}
\frac{\Sigma}{\Sigma_0} \ = \ \frac{m_\pi^2}{\mu_I^2} \ ,
\qquad \frac{\Pi}{\Sigma_0} \ = \ \sqrt{1-\frac{m_\pi^2}{\mu_I^2}} \ .
\end{equation}
In this way one has
\begin{equation}
\left(\frac{\Sigma}{\Sigma_0}\right)^2 \, + \, \left(\frac{\Pi}{\Sigma_0}\right)^2 \ = \ 1 \ ,
\label{circle}
\end{equation}
which defines the so-called ``chiral circle''. The relation in
Eq.~(\ref{circle}) is approximately satisfied in local and nonlocal NJL
models, as can be seen from Fig.~\ref{fig:1}. In fact, the agreement is very
good up to $\mu_I \simeq 2m_\pi$, where the prediction from ChPT is
trustable. Moreover, with the aid of the Gell-Mann-Oakes-Renner relation one
can find simple analytical expressions for the susceptibilities, namely
\begin{equation}
\chi_{\rm ch} \ = \ -\,\frac{\Sigma_0}{m_c} \, \frac{m_\pi^2}{\mu_I^2}\ , \qquad
\chi_{\rm \Pi} \ = \ -\,\frac{\Sigma_0}{m_c}\,
\frac{m_\pi^4}{\mu_I^4}\, \frac{1}{\sqrt{1-\frac{m_\pi^4}{\mu_I^4}}} \ ,
\label{suscep}
\end{equation}
where it has been assumed that the ratio $\Sigma_0/f_\pi^2$ is approximately
independent of $m_c$. In Eqs.~(\ref{suscep}), it can be seen that $\chi_\Pi$
diverges at $\mu_I = m_\pi$, while $\chi_{\rm ch}$ is finite and only
becomes divergent in the chiral limit. The behavior of the susceptibilities
as functions of $\mu_I$ obtained from these equations are shown by the
dashed lines in Fig.~\ref{fig:1b}. It can be seen that they match nicely the
results arising from the nlNJL model.
\begin{figure}[hbt]
\centering{}\includegraphics[width=0.8\textwidth]{p-e-nI_vs_muI}
\caption{(Color online) Numerical results for the normalized pressure,
energy density and isospin particle density as functions of the isospin
chemical potential. Besides the local and nonlocal NJL models, the graphs
include the results obtained from the ChPT approach in
Ref.~\cite{Adhikari:2019mdk}, the linear sigma model in
Ref.~\cite{He:2005nk}, and LQCD calculations in
Ref.~\cite{Brandt:2018bwq,Avancini:2019ego}.}
\label{fig:2}
\end{figure}
Next, in Fig.~\ref{fig:2} we show the results obtained within the nlNJL
model for the normalized pressure, energy density and isospin particle
density as functions of $\mu_I/m_\pi$. Results from other theoretical
approaches are also included for comparison (lines and bands for NJL and
nlNJL models are defined in the same way as in Fig.~\ref{fig:1}). In the
left panels we consider a range of $\mu_I$ from $m_\pi$ to $2m_\pi$, for
which LQCD estimations have been obtained in
Refs.~\cite{Brandt:2018bwq,Avancini:2019ego} (short-dashed black lines and
fat dots in the figure). In the right panels we include the results for the
same quantities using a different scale that covers values of the isospin
chemical potential up to $5 m_\pi$. Notice that all three quantities are
zero for $0\leq \mu_I\leq m_\pi$. From the left panels it can be seen that
in general there is a good agreement between the predictions of effective
models ---which do not differ significantly from each other--- and LQCD
results. On the other hand, for larger values of $\mu_I$ the splitting
between the results from various theoretical approaches becomes appreciably
large. Unfortunately, no LQCD results are available up to now within this
enlarged range. The behavior of the studied quantities for the nonlocal
approach (solid red lines, red bands) is found to be qualitatively similar
to the one obtained within the local NJL model (green bands), showing a
monotonic growth when $\mu_I$ gets increased. Notice that the dependence on
the parametrization turns out to be relatively low.
Another interesting magnitude to be analyzed is the interaction energy, or
trace anomaly, $\epsilon-3p$. The behavior of this quantity (normalized by
$\mu_I^4$) as a function of $\mu_I/m_\pi$, is shown in Fig.~\ref{fig:3}. It
is seen that the results obtained within the nlNJL model are similar to
those found in other theoretical approaches. In particular, the so-called
``conformal point'', for which $\epsilon = 3p$, is reached at a value of
$\mu_I/m_\pi$ in the range between 1.75 and 1.77 (depending on the
parametrization), in good agreement with the analytical result $\mu_I/m_\pi
= \sqrt{3}$ arising from leading order ChPT~\cite{Carignano:2016rvs}.
\begin{figure}[hbt]
\includegraphics[width=0.85\textwidth]{e-3p}
\caption{(Color online) Numerical results for the interaction energy as
function of the isospin chemical potential. The graphs include the values
obtained from local and nonlocal NJL models, ChPT~\cite{Adhikari:2019mdk}
and LQCD calculations~\cite{Brandt:2018bwq,Avancini:2019ego}.}
\label{fig:3}
\end{figure}
To conclude this subsection, in Fig.~\ref{fig:4} we plot the numerical
results obtained for the equation of state, i.e.~the behavior of the energy
density as a function of the pressure (here the isospin chemical potential
$\mu_I$ is an underlying parameter). The notation for the curves obtained
within the nonlocal NJL approach and other models, as well as those arising
from lattice QCD calculations, are the same as in Figs.~\ref{fig:2}
and~\ref{fig:3}. Once again, the results from the nonlocal approach are
qualitatively similar to those obtained in the local NJL model, and are
consistent with LQCD results in the low energy region (where LQCD data are
available).
\begin{figure}[hbt]
\vspace*{0.1cm}
\includegraphics[width=0.85\textwidth]{eos}
\caption{(Color online) Numerical results for the equation of state. The
graphs include the results obtained from local and nonlocal NJL models, the
linear sigma model~\cite{He:2005nk}, ChPT~\cite{Adhikari:2019mdk} and LQCD
calculations~\cite{Brandt:2018bwq,Avancini:2019ego}.}
\label{fig:4}
\end{figure}
\subsection{Finite temperature}
\label{finiteT}
We present here our numerical results at finite temperature for the
quantities defined in Sec.~\ref{model}. As discussed above, we include
the interaction between the fermions and a background color field,
considering the Polyakov loop potential in Eq.~(\ref{upoly}). The parameter
$T_0$ entering this potential is taken to be 200~MeV, following the
estimations carried out for the case of two dynamical
quarks~\cite{Schaefer:2007pw,Schaefer:2009ui}.
Let us start by studying the thermal behavior of the normalized mean field
condensates and the traced PL for some representative values of $\mu_I$
within the range $0 \leq \mu_I \leq 2m_\pi$. Our results are shown in
Fig.~\ref{fig:5}. On the left panels we plot the condensates $\Sigma$ and
$\Pi$, normalized by $\Sigma_0$ (solid and dashed lines, respectively),
together with the traced PL $\Phi$ (dashed-dotted lines). The results are
given as functions of the temperature, normalized to the critical
temperature for $\mu_I = 0$, viz.\ $T_c^0 = 174$~MeV. We also include the
curves for the normalized combined quantity $R$, defined by
\begin{equation}
R \ = \ \frac{\sqrt{\Sigma^2+\Pi^2}}{\Sigma_0}\ .
\end{equation}
In the right panels of Fig.~\ref{fig:5} we plot the susceptibilities
associated to the chiral and pion condensates and the traced Polyakov loop
(solid, dashed and dashed-dotted lines, respectively), defined in
Eq.~(\ref{chis}). As usual, the peaks of the curves for $\chi_{\rm ch}$ and
$\chi_\Phi$ are used in order to define the chiral restoration and
deconfinement transition critical temperatures.
{}From the left panels of Fig.~\ref{fig:5} it is seen that for $\mu_I=0$ the
chiral restoration and deconfinement transitions proceed as a smooth
crossovers at temperatures $T \simeq T_c^0$, while the pion condensate
vanishes for all $T$. The situation remains basically the same up to values
of $\mu_I$ approaching $m_\pi$. Then, for a small region of values of
$\mu_I$ just below $m_\pi$ (as shown explicitly for the case of
$\mu_I/m_\pi=0.99$) the pion condensate vanishes for all $T$ except for a
short range of temperatures slightly below the critical value $T_c$ that
characterizes the (almost simultaneous) chiral restoration and deconfinement
crossover transitions. On the other hand, for $\mu_I > m_\pi$, at low
temperatures the pion condensate gets nonzero values, showing the
spontaneous breakdown of isospin symmetry. These values of $\Pi$ are
approximately independent of the temperature up to $T\simeq T_c^0$, where
one finds a second order transition to a U(1)$_{I_3}$ symmetry restored
phase. In addition, it can be seen that these values of $\Pi$ get increased
with $\mu_I$, while the values of the chiral condensate $\Sigma$ decrease,
in such a way that $R$ is approximately constant. We recall that, as
discussed in the previous subsection, from lowest order ChPT one gets at
$T=0$ a constant value $R=1$ for all values of $\mu_I$. Moreover, as noted
in Ref.~\cite{He:2005nk}, the behavior of $R$ as a function of $T$ is very
similar to that found for $\Sigma/\Sigma_0$ when pion condensation is not
considered. Concerning the deconfinement transition, the graphs on the left
panel of Fig.~\ref{fig:5} show that it proceeds as a smooth crossover at an
approximately constant temperature $T \lesssim T_c^0$ for the considered
range of values of the isospin chemical potential.
\begin{figure}[hbt]
\centering{}\includegraphics[width=0.75\textwidth]{nlPNJL240-T}
\caption{(Color online) Left: numerical results for the normalized $\Sigma$
and $\Pi$ condensates, the traced Polyakov loop $\Phi$ and the quantity $R$
as functions of the temperature, for some fixed values of $\mu_I/m_\pi$.
Right: numerical results for the susceptibilities associated to the chiral
and pion condensates and the Polyakov loop, as functions of $T/T_c^0$.}
\label{fig:5}
\end{figure}
Taking now into account the plots in the right panels of Fig.~\ref{fig:5},
it can be seen that the PL susceptibility (green dashed-dotted lines) shows
clear peaks that indicate a crossover-like deconfinement transition at a
temperature slightly lower than $T_c^0$ and approximately independent of
$\mu_I$. In the case of the chiral susceptibility (red solid lines in the
right panels of Fig.~\ref{fig:5}), for $\mu_I=0$ one finds a peak that
defines the critical temperature $T_c^0 = 174$~MeV. Notice that for $\mu_I$
larger than $m_\pi$ the susceptibility $\chi_{\rm ch}$ is relatively large
at low temperatures. This is in agreement with the behavior shown in
Fig.~\ref{fig:1b}, and it can be attributed to the presence of a nonzero
pion condensate. The same effect occurs for values of $\mu_I$ slightly below
$m_\pi$ and temperatures $T\lesssim T_c^0$, owing to the existence of the
already mentioned nonzero value of $\Pi$ in this region (see panels of the
second row in Fig.~\ref{fig:5}). Finally, the pion condensate susceptibility
(dashed lines in the right panels of Fig.~\ref{fig:5}) is also found to be
nonzero in the presence of the pion condensate. Moreover, as expected, it
becomes divergent at the temperatures in which one finds the second order
phase transition into the isospin symmetry restored phase. These
temperatures are slightly lower than $T_c^0$ and basically coincide with the
ones corresponding to the deconfinement transition. For completeness, we
show in Fig.~\ref{fig:5b} the behavior of the $\Sigma$ and $\Pi$
susceptibilities as functions of the isospin chemical potential, for $T=0$
and temperatures slightly below and above $T_c^0$. In fact, it is seen that
the behavior of $\chi_{\rm ch}$ and $\chi_\Pi$ found for $T=0$ (see
Fig.~\ref{fig:1b}) does not change qualitatively up to the critical isospin
symmetry restoration temperature. Notice that for temperatures just below
$T_c^0$ the position of the discontinuity is shifted to $\mu_I/m_\pi$
slightly smaller than 1. It is also worth noticing that the curves for
$T\lesssim T_c^0$ are quite different from the ones obtained in the
framework of the local PNJL model, for which the discontinuity is found to
occur at significantly larger values of $\mu_I/m_\pi$ (see Fig.~3 of
Ref.~\cite{Lu:2019diy}).
\begin{figure}[hbt]
\centering{}\includegraphics[width=0.8\textwidth]{suscep-T}
\caption{(Color online) Chiral (left) and pion (right) condensate
susceptibilities as functions of the isospin chemical potential, for some
representative values of the temperature.}
\label{fig:5b}
\end{figure}
Through the analysis of the quantities in Fig.~\ref{fig:5} one can sketch
the phase diagram in the $\mu_I-T$ plane. This is shown in Fig.~\ref{fig:6},
where the temperature and the isospin chemical potential are normalized to
$T_c^0$ and $m_\pi$, respectively. As expected, for low values of $T$ and
$\mu_I$ the system lies in a ``normal matter'' (NM) phase, i.e.\ a
U(1)$_{I_3A}$ symmetry broken phase in which the scalar quark-antiquark
condensate $\Sigma$ is large and the pion condensate $\Pi$ is zero. By
increasing the temperature one reaches a transition to a ``quark gluon
plasma'' (QGP) phase, in which quarks deconfine and the chiral symmetry
becomes partially restored. Both chiral restoration and deconfinement
transitions occur as smooth crossovers, at approximately a common
temperature that does not depend significantly on $\mu_I$. The corresponding
curves, obtained from the peaks of $\chi_{\rm ch}$ and $\chi_\Phi$
susceptibilities, are shown by the solid and dash-dotted lines in the
figure, respectively. The results are found to be similar to those obtained
from lattice QCD calculations in Ref.~\cite{Brandt:2017oyy}, shown by the
gray and blue bands. On the other hand, for temperatures below the critical
value $T_c^0$, by increasing the isospin chemical potential one reaches a
second order phase transition to a pion-condensate ($\pi$C) region in which
the condensate $\Pi$ is nonvanishing and therefore the U(1)$_{I_3}$ symmetry
is broken. The onset of this phase, shown by the dashed line in
Fig.~\ref{fig:6}, occurs approximately at $\mu_I = m_\pi$ for all
temperature values up to $T_c^0$, in agreement with lattice QCD calculations
(red band in the figure)~\cite{Brandt:2017oyy}. Then, for $\mu_I > m_\pi$,
at a given critical temperature there is a second order phase transition
from the $\pi$C phase to the QGP phase. As discussed above, this critical
temperature is slightly lower than $T_c^0$ and remains approximately
constant for all considered values of $\mu_I$ above the pion mass. The
location of the pseudo-triple point where NM, QGP and $\pi$C phases meet is
found to be in good agreement with the result obtained in lattice QCD
calculations, given by the black square. Concerning the U(1)$_{I_3A}$
symmetry within the $\pi$C phase, it is seen that for a given temperature
$T$ lower than $T_c^0$ the values of the quark condensates decrease steadily
if $\mu_I$ gets increased beyond $m_\pi$. This can be read from the values
of $\Sigma/\Sigma_0$ shown in the left panels of Fig.~\ref{fig:5}. Notice
that for $\mu_I\simeq 1.4\,m_\pi$ the value of $\Sigma$ is found to be
reduced to approximately one half of the $\mu_I=0$ value $\Sigma_0$.
Finally, in Fig.~\ref{fig:6} we also show for comparison the $\pi$C$-$NM
transition curves corresponding to the local PNJL model and leading order
chiral perturbation theory~\cite{Splittorff:2002xn} (thin dashed and
short-dashed lines, respectively). As anticipated in the discussion
concerning Fig.~\ref{fig:5b}, it is seen that there is a substantial
difference between nonlocal and local PNJL-like approaches. This situation
does not change significantly if one considers the NJL model omitting the
interaction with the Polyakov loop.
\begin{figure}[hbt]
\centering{}\includegraphics[width=0.75\textwidth]{pdnlPNJL240.eps}
\caption{(Color online) Phase diagram in the $\mu_I-T$ plane for the
nonlocal PNJL model. NM, QGP and $\pi$C stand for normal matter, quark gluon
plasma and pion condensation phases, respectively. Solid (black), dashed
(red) and dash-dotted (blue) lines correspond to chiral restoration, pion
condensation and deconfinement transitions, while the shaded bands indicate
the transition regions obtained from LQCD results in
Ref.~\cite{Brandt:2017oyy}. The thin dashed and short-dashed lines
indicate the NQM-$\pi$C transition curves arising from the local PNJL model
and leading order ChPT, respectively.} \label{fig:6}
\end{figure}
\section{Conclusions}
\label{summary}
We have analyzed the phase diagram of strongly interacting matter
within a nonlocal two-flavor PNJL model, considering both zero and finite
temperature and nonzero isospin chemical potential. In this context, we have
studied the quark deconfinement and the breakdown/restoration of chiral and
isospin symmetries, together with the corresponding footprints on various
thermodynamic quantities.
At zero temperature, for $\mu_I = m_\pi$ one finds the onset of a phase in
which isospin symmetry is broken by the presence of a nonzero pion
condensate. Up to $\mu_I \simeq 2m_\pi$, one observes a rapid growth of this
condensate, in overall agreement with the predictions from other effective
model analyses and LQCD calculations. The agreement is also good for various
thermodynamic quantities, as the pressure, energy density, isospin particle
density and interaction energy. For larger values of $\mu_I$ (where no LQCD
data are available up to now), although one finds some general agreement in
the qualitative behavior of these quantities, there are significant
quantitative discrepancies between the results from different theoretical
approaches.
In the case of a system at finite temperature, for low values of $\mu_I$ the
pion condensate is absent and one gets, as expected, a transition from the
usual ``normal matter'' (NM) scenario into a quark-gluon plasma (QGP) phase
in which chiral symmetry is restored and quarks are deconfined. This
transition proceeds as a smooth crossover signaled by the behavior of chiral
and Polyakov loop susceptibilities. The critical temperature $T_c^0\simeq
174$~MeV is approximately the same for both chiral restoration and quark
deconfinement. For $T\leq T_c^0$, by increasing the isospin chemical
potential one finds a second order transition into a pion condensation
($\pi$C) phase, in which isospin symmetry is spontaneously broken. The
corresponding critical line $\mu_I(T)$ is a primary result of our analysis.
The critical value $\mu_I = m_\pi$ found at $T=0$ remains approximately
constant up to $T\simeq T^0_c$, reaching a pseudo-triple point in which NM,
QGP and $\pi$C phases coexist. It can be seen that there is a remarkable
agreement between these results and those obtained from lattice QCD
calculations. On the other hand, the $\pi$C-QGP transition occurs at a
temperature of the order of $T_c^0$, which is approximately constant for
$\mu_I > m_\pi$. It is worth noticing that our predictions for the border of
the $\pi$C phase region (and, in particular, for the location of the triple
point) are in good agreement with the available results from lattice QCD,
whereas they differ significantly from the predictions obtained in the
framework of the local PNJL model.
\section{Acknowledgements}
This work has been supported in part by Consejo Nacional de Investigaciones
Cient\'ificas y T\'ecnicas and Agencia Nacional de Promoci\'on Cient\'ifica
y Tecnol\'ogica (Argentina), under Grants No.~PIP17-700 and
No.~PICT17-03-0571, respectively, and by the National University of La Plata
(Argentina), Project No.~X824.
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BkiUejHxK6-gDyrT-mTO
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\section{Introduction}\label{sec:introduction}
A \emph{tree} on a finite set $X$ is a connected acyclic graph with vertex set
$X$. A \emph{rooted tree} is a tree with a distinguished vertex called a
\emph{root}. It is well-known that the number of rooted trees on
$[n]=\{1,2,\dots,n\}$ is $n^{n-1}$, see \cite[5.3.2~Proposition]{EC2}.
Suppose $T$ is a rooted tree with root $r$. For a vertex $v\ne r$ of $T$ there
is a unique path $(u_1,u_2,\dots,u_i)$ from $r=u_1$ to $v=u_i$. Then $u_{i-1}$
is called the \emph{parent} of $v$, and $v$ is called a \emph{child} of
$u_{i-1}$. For two vertices $u$ and $v$, we say that $u$ is a \emph{descendant}
of $v$ if the unique path from $r$ to $u$ contains $v$. Note that every vertex
is a descendant of itself. A \emph{leaf} is a vertex with no children. A rooted
tree is \emph{decreasing} if every nonleaf is greater than its children. The
\emph{maximal decreasing subtree} of $T$, denoted $\MD(T)$, is the maximal
subtree such that it has the same root as $T$ and it is decreasing. If the root
of $T$ has no smaller children, $T$ is called \emph{minimally rooted}.
The notion of maximal decreasing subtree was first appeared in \cite{Chauve2000}
in order to prove the following theorem.
\begin{thm}\cite[Theorem~2.1]{Chauve2000}\label{thm:CDG}
The number of rooted trees on $[n+1]$
such that the root has $\ell$ smaller children equals $\binom{n}{\ell} n^{n-\ell}$.
\end{thm}
Recently, maximal decreasing subtrees reappeared in the study of a certain free
Lie algebra over rooted trees by Bergeron and Livernet \cite{Bergeron2010}. Seo
and Shin \cite{SeoShin} proved some enumeration properties of rooted trees with
fixed size of maximal decreasing subtrees.
We denote by $\TT_{n,k}$ the set of rooted trees on $[n+1]$ whose maximal
decreasing subtrees have $k+1$ vertices. Let $\FF_{n,k}$ denote the set of
functions $f:[n]\to[n]$ such that $[k]\subset f([n])$, where $[0]=\emptyset$.
Equivalently, $\FF_{n,k}$ is the set of words on $[n]$ of length $n$ such that
each of $1,2,\dots,k$ appears at least once. Using the Pr\"ufer code one can
easily see that $\FF_{n,k}$ is in bijection with the set of rooted trees on
$[n+1]$ such that $n+1$ is a leaf and $1,2,\dots,k$ are nonleaves. Thus, we
will consider $\FF_{n,k}$ as the set of such trees.
Seo and Shin \cite{SeoShin} proved the following theorem.
\begin{thm}\cite{SeoShin}\label{thm:SeoShin}
We have
\[
|\TT_{n,k}| = |\FF_{n,k}|.
\]
\end{thm}
In \cite{SeoShin} they showed Theorem~\ref{thm:SeoShin} by finding formulas for
both sides and computing the formulas. In this paper we provide a bijective
proof Theorem~\ref{thm:SeoShin}, which consists of several bijections between
certain objects, see Theorem~\ref{thm:main}. In order to state the objects in
Theorem~\ref{thm:main} we need the following definitions.
An \emph{ordered forest} on a finite set $X$ is an ordered tuple of rooted trees
whose vertex sets form a partition of $X$. We say that an ordered forest
$(T_0,T_1,\dots,T_\ell)$ is \emph{$k$-good} if it satisfies the following
conditions:
\begin{enumerate}
\item If $\ell=0$, then $T_0$ has only one vertex $v$ and we have $v\in [k]$.
\item If $\ell\geq1$, then $T_1,T_2,\dots,T_\ell$ are minimally rooted, and the
number of vertices of $T_0,T_1,\dots,T_i$ contained in $[k]$ is at least $i+1$
when $i=0,1,2,\dots,\ell-1$, and equal to $\ell$ when $i=\ell$.
\end{enumerate}
We now state the main theorem of this paper.
\begin{thm}\label{thm:main}
The following sets have the same cardinality:
\begin{enumerate}
\item the set $\TT_{n,k}$ of rooted trees on $[n+1]$ whose maximal decreasing
subtrees have $k+1$ vertices,
\item the set $\AA_{n,k}$ of cycles of $k+1$ minimally rooted trees such that the
vertex sets of the trees form a partition of $[n+1]$,
\item the set $\BB_{n,k}$ of ordered forests on $[n]$ such that the last $k$ trees
are minimally rooted,
\item the set $\CC_{n,k}$ of sequences of $k$-good ordered forests such that the
vertex sets of the forests form a partition of $[n]$,
\item the set $\FF_{n,k}$ of rooted trees on $[n+1]$ such that $n+1$ is a leaf,
and $1,2,\dots,k$ are nonleaves.
\end{enumerate}
\end{thm}
In Section~\ref{sec:bijections} we find bijections proving
Theorem~\ref{thm:main}. The ideas in the bijections have some applications. In
Section~\ref{sec:some-properties-tt_n} we find a bijective proof of the
following identity, which (finding a bijective proof) is stated as an open
problem in \cite{SeoShin}:
\[
\sum_{k\ge0} \frac{1}{k} |\TT_{n,k}| = n^n.
\]
In Section~\ref{sec:anoth-proof-theor}, we gives another bijective proof of
Theorem~\ref{thm:CDG}.
From now on all trees in this paper are rooted trees.
\section{Bijections}\label{sec:bijections}
In this section we will find four bijections to prove Theorem~\ref{thm:main}. We
assume that $n$ and $k$ are fixed nonnegative integers. We will write cycles
using brackets to distinguish them from sequences. For instance, $[a_1,a_2,a_3]$
is a cycle and $(a_1,a_2,a_3)$ is a sequence, thus $[a_1,a_2,a_3]=[a_2,a_3,a_1]$
and $(a_1,a_2,a_3)\ne(a_2,a_3,a_1)$. For a tree or a forest $T$, we denote by
$V(T)$ the set of vertices in $T$.
\subsection{A bijection $\alpha:\TT_{n,k}\to\AA_{n,k}$}
We will explain the map $\alpha$ by an example. Let $T\in\TT_{19,7}$ be the
following tree.
\[
\pstree{\TR{16}}{
\pstree{\TR{13}}{
\TR{17}
\pstree{\TR{8}}{
\TR{19}
\TR{18}
}
}
\pstree{\TR{12}}{
\pstree{\TR{11}}{
\pstree{\TR{10}}{
\pstree{\TR{15}}{
\pstree{\TR{1}}{
\TR{3}
\pstree{\TR{20}}{
\TR{4}
}
}
}
}
}
\pstree{\TR{7}}{
\pstree{\TR{14}}{
\TR{9} \TR{2}
}
}
\pstree{\TR{5}}{
\TR{6}
}
}
}
\]
Then we can decompose $T$ as follows:
\begin{equation}
\label{eq:1}
T \Leftrightarrow
\left(
\raisebox{1.5cm}{
\pstree{\TR{16}}{
\pstree{\TR{13}}{
\TR{8}
}
\pstree{\TR{12}}{
\pstree{\TR{11}}{
\TR{10}
}
\TR{7}
\TR{5}
}
}
\raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{13}}{
\TR{17}
}
\raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{10}}{
\pstree{\TR{15}}{
\pstree{\TR{1}}{
\TR{3}
\pstree{\TR{20}}{
\TR{4}
}
}
}
}
\raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{8}}{
\TR{19}
\TR{18}
}
\raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{7}}{\pstree{\TR{14}}{\TR{9} \TR{2}}} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{5}}{\TR{6}}
}
\right),
\end{equation}
where the first tree is $\MD(T)$, and the rest are the trees with more than one
vertex in the forest obtained from $T$ by removing the edges in $\MD(T)$. We
now construct a cycle $C$ corresponding to $\MD(T)$ as follows. First, let $C$
be the cycle containing only the maximal vertex $m$, which is the root of
$\MD(T)$. For each remaining vertex $v$, starting from the largest vertex to the
smallest vertex, we insert $v$ in $C$ after the parent of $v$. In the current
example, we get the cycle $[16, 12, 5, 7, 11, 10, 13, 8]$. It is easy to see
that this process is invertible. In fact this is equivalent to the well-known
algorithm called the depth-first search (preorder).
For each element $v$ except the largest element in this cycle, if there is a
tree with root $v$ in \eqref{eq:1} replace $v$ with the tree. We then define
$\alpha(T)$ to be the resulting cycle. It is not hard to see that $\alpha$ is a
bijection. In the current example, we have
\begin{equation}
\label{eq:2}
\alpha(T) = \left[
\raisebox{1.5cm}{
\TR{16} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{12} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{5}}{\TR{6}} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{7}}{\pstree{\TR{14}}{\TR{9} \TR{2}}} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{11}
\raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{10}}{
\pstree{\TR{15}}{
\pstree{\TR{1}}{
\TR{3}
\pstree{\TR{20}}{
\TR{4}
}
}
}
}
\raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{13}}{
\TR{17}
}
\raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{8}}{
\TR{19}
\TR{18}
}
}
\right] \in \BB_{19,7}.
\end{equation}
\subsection{A bijection $\beta:\AA_{n,k}\to\BB_{n,k}$}
In order to define the map $\beta$ we need two bijections $\phi$ and $\rho$ in
the following two lemmas. These bijections will also be used in other
subsections.
\begin{lem}\cite{Chauve2000}\label{lem:CDG}
Let $A\subset [n]$. There is a bijection $\phi$ from the set of minimally
rooted trees on $A$ to the set of rooted trees on $A$ such that $\max(A)$ is a
leaf.
\end{lem}
\begin{proof}
We will briefly describe the bijection $\phi$. See \cite{Chauve2000} for the
details. Consider a minimally rooted tree $T$ on $A$ with root $r$. Let $P$ be
the subtree of $T$ rooted at $\max(A)$ containing all descendants of
$\max(A)$, and let $Q$ be the tree obtained from $T$ by deleting $P$
(including $\max(A)$). We now consider the forest obtained from $P$ by
removing all edges of $\MD(P)$. Suppose this forest has $\ell$ trees
$T_1,T_2,\dots,T_\ell$ rooted at, respectively, $r_1,r_2,\dots,r_\ell$. If
$V(\MD(P)) = \{u_1<u_2<\dots<u_t\}$ and $V(\MD(P))\setminus\{\max(A)\} \cup
\{r\} = \{v_1<v_2<\dots<v_t\}$, let $T'$ be the tree obtained from $\MD(P)$ by
replacing $u_i$ with $v_i$ for all $i$. Then $\phi(T)$ is the tree obtained
from $T'$ by attaching $T_i$ at $r_i$ for $i=1,2,\dots,\ell$ and attaching $Q$
at $r$.
\end{proof}
\begin{lem}\label{lem:leaf}
Let $A\subset[n]$. There is a bijection $\rho$ from the set of rooted trees
on $A$ such that $\max(A)$ is a leaf to the set of ordered forests on
$A\setminus\{\max(A)\}$.
\end{lem}
\begin{proof}
Suppose $T$ is a rooted tree on $A$ such that $\max(A)$ is a leaf. Let
$r=r_1, r_2,\dots, r_{\ell+1}=\max(A)$ be the unique path from the root $r$ of
$T$ to the leaf $\max(A)$. Let $R_1,R_2,\dots,R_{\ell}$ be the rooted trees
with roots $r_1,r_2,\dots,r_{\ell}$ respectively in the forest obtained from
$T$ by removing the edges $r_1r_2,r_2r_3,\dots,r_{\ell}r_{\ell+1}$ and the
vertex $r_{\ell+1}=\max(A)$. We define $\rho(T)=(R_1,R_2,\dots,R_{\ell})$. It
is easy to see that $\rho$ is a desired bijection.
\end{proof}
Let $[T_0,T_1,\dots,T_k]\in \AA_{n,k}$. Since $[T_0,T_1,\dots,T_k]$ is a cycle,
we can assume that $n+1\in T_0$. By Lemmas~\ref{lem:CDG} and \ref{lem:leaf},
the vertex $n+1$ in $\phi(T_0)$ is a leaf, and $\rho(\phi(T_0)) =
(R_1,R_2,\dots,R_{\ell})$ is an ordered forest on $V(T_0)\setminus\{n+1\}$. We
define $\beta([T_0,T_1,\dots,T_k])= (R_1,R_2,\dots,R_{\ell},
T_1,T_2,\dots,T_k)$. Since both $\phi$ and $\rho$ are invertible, $\beta$ is a
bijection.
\begin{example}
Let $\F$ be the cycle in \eqref{eq:2}. Then we can write $\F$ as
\[
\F = \left[
\raisebox{1.5cm}{
\pstree{\TR{10}}{
\pstree{\TR{15}}{
\pstree{\TR{1}}{
\TR{3}
\pstree{\TR{20}}{
\TR{4}
}
}
}
}
\raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{13}}{
\TR{17}
}
\raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{8}}{
\TR{19}
\TR{18}
}
\raisebox{-4pt}{\mbox{ ,\quad}}
\TR{16} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{12} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{5}}{\TR{6}} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{7}}{\pstree{\TR{14}}{\TR{9} \TR{2}}} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{11}
}
\right].
\]
Then
\[
(\rho\circ\phi)
\left(
\raisebox{1.5cm}{
\pstree{\TR{10}}{
\pstree{\TR{15}}{
\pstree{\TR{1}}{
\TR{3}
\pstree{\TR{20}}{
\TR{4}
}
}
}
}
}
\right)
= \rho\left(
\raisebox{1.5cm}{
\pstree{\TR{10}}{
\pstree{\TR{15}}{
\pstree{\TR{1}}{
\TR{3}
\TR{20}
}
}
\TR{4}
}
}
\right)=
\left(
\raisebox{.4cm}{
\pstree{\TR{10}}{\TR{4}} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{15} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{1}}{\TR{3}}
}
\right).
\]
Thus
\[
\beta(\F) = \left(
\raisebox{.8cm}{
\pstree{\TR{10}}{\TR{4}} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{15} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{1}}{\TR{3}} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{13}}{
\TR{17}
}
\raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{8}}{
\TR{19}
\TR{18}
}
\raisebox{-4pt}{\mbox{ ,\quad}}
\TR{16} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{12} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{5}}{\TR{6}} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{7}}{\pstree{\TR{14}}{\TR{9} \TR{2}}} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{11}
}
\right) \in \BB_{19,7}.
\]
\end{example}
\subsection{A bijection $\gamma:\BB_{n,k}\to\CC_{n,k}$}
We call a vertex with label less than or equal to $k$ a \emph{special vertex}.
For two ordered forests $\X$ and $\Y$ whose vertex sets are disjoint and
contained in $[n]$, the pair $(\X,\Y)$ is called a \emph{balanced pair} if the
trees in $\Y$ are minimally rooted and the number of special vertices in $\X$
and $\Y$ is equal to the number of trees in $\Y$.
For two sets $A$ and $B$, the \emph{disjoint union} $A\uplus B$ is just the
union of $A$ and $B$. However, if we write $A\uplus B$, it is always assumed that
$A\cap B=\emptyset$.
\begin{lem}
There is a bijection $f$ from the set of balanced pairs $(\X,\Y)$ to the set
of pairs $(\A,(\X',\Y'))$ of a $k$-good ordered forest $\A$ and a balanced
pair $(\X',\Y')$ such that $V(\X)\uplus V(\Y) = V(\A) \uplus V(\X') \uplus
V(\Y')$.
\end{lem}
\begin{proof}
Suppose $\X=(X_1,X_2,\dots,X_s)$ and $\Y=(Y_1,Y_2,\dots,Y_t)$. We define
$f(\X,\Y)=(\A,(\X',\Y'))$ as follows.
Case 1: If $s\geq1$, and $X_1$ does not contain a special vertex, we define
$\A=(X_1)$, $\X'=(X_2,\dots,X_s)$, and $\Y' = \Y$.
Case 2: If $s\geq1$, and $X_1$ contains at least one special vertex, there is a
unique integer $1\leq j\leq t$ such that $(X_1,Y_1,Y_2,\dots,Y_j)$ is a $k$-good
ordered forest. Then we define $\A=(X_1,Y_1,Y_2,\dots,Y_j)$,
$\X'=(X_2,X_3,\dots,X_s)$, and $\Y'=(Y_{j+1},Y_{j+2},\dots,Y_t)$. Since $\A$ is
a $k$-good ordered forest, there are $j$ special vertices in
$X_1,Y_1,Y_2,\dots,Y_j$. This implies that $(\X',\Y')$ is also a balanced pair.
Case 3: If $s=0$, then $\X=\emptyset$ and there are $t$ special vertices in
$Y_1,Y_2,\dots, Y_t$. Let $U=V(Y_1)\uplus\cdots\uplus V(Y_s)$ and let
$m=\max(U)$. Suppose $Y_i$ contains $m$. We apply the map $\phi$ in
Lemma~\ref{lem:CDG} to $Y_i$. Then $\phi(Y_i)$ is a rooted tree such that $m$ is
a leaf. If we apply the map $\rho$ in Lemma~\ref{lem:leaf} to $\phi(Y_i)$, we
get an ordered forest $\rho(\phi(Y_i)) = (T_1,T_2,\dots, T_\ell)$ on
$V(Y_i)\setminus\{m\}$. Let $\overline{\X}=(T_1,T_2,\dots, T_\ell)$ and
$\overline{\Y} = (Y_1,Y_2,\dots, \widehat{Y_i}, \dots,Y_t)$. Note that the set
of vertices in $\overline{\X}$ and $\overline{\Y}$ is $U\setminus\{m\}$. Let
$s_1<s_2<\cdots<s_t$ be the special vertices in $U$. Suppose $U\setminus\{m\} =
\{ v_1<v_2<\cdots<v_{p}\}$ and $U\setminus\{s_i\} = \{
u_1<u_2<\cdots<u_{p}\}$. Then we define $\X'$ (resp.~$\Y'$) to be the ordered
forest obtained from $\overline{\X}$ (resp.~$\overline{\Y}$) by replacing $v_j$
with $u_j$ for all $j$. We define $\A$ to be the rooted tree with only one
vertex $s_i$. It is clear from the construction that $\A$ is a $k$-good ordered
forest and $(\X',\Y')$ is a balanced pair.
In all cases, we clearly have $V(\X)\uplus V(\Y) = V(\A) \uplus V(\X') \uplus
V(\Y')$. We now show that $f$ is invertible by constructing the inverse map
$g=f^{-1}$. Suppose $\A=(A_1,\dots,A_r)$, $\X'=(X_1,\dots,X_s)$, and
$\Y'=(Y_1,\dots,Y_t)$, where $\A$ is a $k$-good forest and $(\X',\Y')$ is a
balanced pair. Then we define $g(\A,(\X',\Y'))=(\X,\Y)$ as follows.
Case 1: If $r=1$, and $A_1$ does not have a special vertex, we define
$\X=(A_1,X_1,\dots,X_s)$ and $\Y=\Y'$.
Case 2: If $r\ge2$, we define $\X=(A_1,X_1,\dots,X_s)$ and
$\Y=(A_2,\dots,A_r, Y_1,\dots,Y_t)$.
Case 3: If $r=1$, and $A_1$ has a special vertex, then by definition of $k$-good
forests, $A_1$ has only one vertex. Let $U$ be the set of vertices in $\A$,
$\X'$, and $\Y'$, and $m=\max(U)$. Suppose $s_1<\cdots<s_{t+1}$ are the $t+1$
special vertices in $U$, and the unique vertex in $A_1$ is
$s_j$. Let $U\setminus\{m\} = \{ v_1<v_2<\cdots<v_{p}\}$ and $U\setminus\{s_j\}
= \{ u_1<u_2<\cdots<u_{p}\}$. Then we define $\overline{\X}=(T_1,\dots,T_r)$ and
$\overline{\Y}=(R_1,\dots,R_s)$ to be the ordered forests obtained from $\X'$
and $\Y'$ by replacing $u_i$ with $v_i$ for all $i$. Then the set of vertices
in $\X'$ and $\Y'$ is now $U\setminus\{m\}$. Thus we can construct the tree
$T=\rho^{-1}(\X')$ with maximum label $m$, and $R=\phi^{-1}(T)$ is a minimally
rooted tree. We define $\X=\emptyset$ and $\Y=(R_1,\dots,R_{i-1}, R,
R_i,\dots,R_s)$.
It is easy to see that $g$ is the inverse map of $f$.
\end{proof}
Now we are ready to define the map $\gamma$. Suppose $(T_1,T_2,\dots,T_\ell,
T_{\ell+1}, T_{\ell+2}, \dots, T_{\ell+k})\in \BB_{n,k}$. Let
$\X=(T_1,T_2,\dots,T_\ell)$ and $\Y=(T_{\ell+1}, T_{\ell+2}, \dots,
T_{\ell+k})$. Then $(\X,\Y)$ is a balanced pair. We define
$(\X_0,\Y_0),(\X_1,\Y_1),\dots,$ and $\A_1, \A_2,\dots,$ as follows. Let
$(\X_0,\Y_0)=(\X,\Y)$. For $i\geq0$, if $(\X_i,\Y_i)\ne(\emptyset,\emptyset)$,
we define $\A_{i+1}, \X_{i+1}, \Y_{i+1}$ by $f(\X_i, \Y_i) = (\A_{i+1},
(\X_{i+1}, \Y_{i+1}))$. Let $p$ be the smallest integer such that
$\X_p=\Y_p=\emptyset$. Then we define $\gamma(\X,\Y)$ to be
$(\A_1,\A_2,\dots,\A_p)\in \CC_{n,k}$. Since $f$ is invertible, $\gamma$ is a
bijection.
\begin{example}
Let
\[
\F = \left(
\raisebox{.8cm}{
\pstree{\TR{10}}{\TR{4}} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{15} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{1}}{\TR{3}} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{13}}{
\TR{17}
}
\raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{8}}{
\TR{19}
\TR{18}
}
\raisebox{-4pt}{\mbox{ ,\quad}}
\TR{16} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{12} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{5}}{\TR{6}} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{7}}{\pstree{\TR{14}}{\TR{9} \TR{2}}} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{11}
}
\right) \in \BB_{19,7}.
\]
Note that special vertices are less than or equal to $7$. Then
\[
\X=\X_0 = \left(
\raisebox{.8cm}{
\pstree{\TR{10}}{\TR{4}} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{15} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{1}}{\TR{3}}
}
\right), \quad
\Y = \Y_0=\left(
\raisebox{.8cm}{
\pstree{\TR{13}}{
\TR{17}
}
\raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{8}}{
\TR{19}
\TR{18}
}
\raisebox{-4pt}{\mbox{ ,\quad}}
\TR{16} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{12} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{5}}{\TR{6}} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{7}}{\pstree{\TR{14}}{\TR{9} \TR{2}}} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{11}
}
\right),
\]
\[
\A_1 = \left(
\raisebox{.4cm}{
\pstree{\TR{10}}{\TR{4}} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{13}}{
\TR{17}
}
}
\right), \quad
\X_1 = \left(
\raisebox{.4cm}{
\TR{15} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{1}}{\TR{3}}
}
\right), \quad
\Y_1=\left(
\raisebox{.8cm}{
\pstree{\TR{8}}{
\TR{19}
\TR{18}
}
\raisebox{-4pt}{\mbox{ ,\quad}}
\TR{16} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{12} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{5}}{\TR{6}} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{7}}{\pstree{\TR{14}}{\TR{9} \TR{2}}} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{11}
}
\right),
\]
\[
\A_2 = \left(
\raisebox{.1cm}{
\TR{15}}
\right), \quad
\X_2 = \left(
\raisebox{.4cm}{
\pstree{\TR{1}}{\TR{3}}
}
\right), \quad
\Y_2=\left(
\raisebox{.8cm}{
\pstree{\TR{8}}{
\TR{19}
\TR{18}
}
\raisebox{-4pt}{\mbox{ ,\quad}}
\TR{16} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{12} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{5}}{\TR{6}} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{7}}{\pstree{\TR{14}}{\TR{9} \TR{2}}} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{11}
}
\right),
\]
\[
\A_3 = \left(
\raisebox{.4cm}{
\pstree{\TR{1}}{\TR{3}}\raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{8}}{
\TR{19}
\TR{18}
}
\raisebox{-4pt}{\mbox{ ,\quad}}
\TR{16}
}
\right), \quad
\X_3 = \emptyset, \quad
\Y_3=\left(
\raisebox{.8cm}{
\TR{12} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{5}}{\TR{6}} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{7}}{\pstree{\TR{14}}{\TR{9} \TR{2}}} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{11}
}
\right).
\]
In $\Y_3$ the largest vertex $14$ is in the third tree. Using $\phi$ and $\rho$
we get
\[
(\rho \circ \phi) \left(
\raisebox{.8cm}{
\pstree{\TR{7}}{\pstree{\TR{14}}{\TR{9} \TR{2}}}
}
\right) =
\rho \left(
\raisebox{.8cm}{
\pstree{\TR{9}}{
\pstree{\TR{7}}{
\TR{14}
}
\TR{2}
}
}
\right) =
\left(
\raisebox{.4cm}{
\pstree{\TR{9}}{\TR{2}} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{7}
}
\right).
\]
Thus
\[
\overline{\X_3} =
\left(
\raisebox{.4cm}{
\pstree{\TR{9}}{\TR{2}} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{7}
}
\right), \quad
\overline{\Y_3}=\left(
\raisebox{.4cm}{
\TR{12} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{5}}{\TR{6}} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{11}
}
\right).
\]
Since $\X_3$ and $\Y_3$ have 4 special vertices $2,5,6,7$, and $6$ is the third
smallest special vertex, we replace the vertices in
$U\setminus\{14\}$ with those in $U\setminus\{6\}$. Since
\[
\begin{array}{ccccccccccc}
U\setminus\{14\} &=& \{ & 2, & 5, & 6, & 7, & 9, & 11, & 12 &\}, \\
U\setminus\{6\} &=& \{ &2, & 5, & 7, & 9, & 11, & 12, & 14 &\},
\end{array}
\]
we get
\[
\A_4 = \left(
\raisebox{.1cm}{
\TR{6}
}
\right), \quad
\X_4 = \overline{\X_3'}=
\left(
\raisebox{.4cm}{
\pstree{\TR{11}}{\TR{2}} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{9}
}
\right), \quad
\Y_4=\overline{\Y_3'}= \left(
\raisebox{.4cm}{
\TR{14} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{5}}{\TR{7}} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{12}
}
\right),
\]
\[
\A_5 = \left(
\raisebox{.4cm}{
\pstree{\TR{11}}{\TR{2}} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{14}
}
\right), \quad
\X_5=
\left(
\raisebox{.1cm}{
\TR{9}
}
\right), \quad
\Y_5=
\left(
\raisebox{.4cm}{
\pstree{\TR{5}}{\TR{7}} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{12}
}
\right),
\]
\[
\A_6 = \left(
\raisebox{.1cm}{
\TR{9}
}
\right), \quad
\X_6=\emptyset, \quad
\Y_6=
\left(
\raisebox{.4cm}{
\pstree{\TR{5}}{\TR{7}} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{12}
}
\right).
\]
In $\Y_6$, the largest vertex $12$ is in the second tree.
\[
(\rho\circ\phi)(12) = \rho(12) = \emptyset.
\]
Thus
\[
\overline{\X_6}= \emptyset, \quad
\overline{\Y_6}=
\left(
\raisebox{.4cm}{
\pstree{\TR{5}}{\TR{7}}
}
\right).
\]
If we replace the labels in $\{5,7\}$ with $\{5,12\}$ we get
\[
\A_7 = \left(
\raisebox{.1cm}{
\TR{7}
}
\right), \quad
\X_7 =\emptyset, \quad
\Y_7=
\left(
\raisebox{.4cm}{
\pstree{\TR{5}}{\TR{12}}
}
\right).
\]
Since
\[
(\rho\circ\phi)\left(\raisebox{.4cm}{\pstree{\TR{5}}{\TR{12}}}\right) =
\rho\left(\raisebox{.4cm}{\pstree{\TR{5}}{\TR{12}}}\right) = 5,
\]
we have $\overline{\X_7}= (5)$ and $\overline{\Y_6}= \emptyset$.
Replacing $5$ with $12$ we get
\[
\A_8 = \left(
\raisebox{.1cm}{
\TR{5}
}
\right), \quad
\X_8 =
\left(
\raisebox{.1cm}{
\TR{12}
}
\right), \quad
\Y_8=\emptyset.
\]
Finally we get
\[
\A_9 = \left(
\raisebox{.1cm}{
\TR{12}
}
\right), \quad
\X_9 =\emptyset, \quad
\Y_9=\emptyset.
\]
Thus
\[
\gamma(\F)=
\left(
\left(
\raisebox{.4cm}{
\pstree{\TR{10}}{\TR{4}} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{13}}{\TR{17}}}
\right),
\left(
\raisebox{.1cm}{
\TR{15}}
\right),
\left(
\raisebox{.4cm}{
\pstree{\TR{1}}{\TR{3}}\raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{8}}{
\TR{19}
\TR{18}
}
\raisebox{-4pt}{\mbox{ ,\quad}}
\TR{16}
}
\right),
\left(
\raisebox{.1cm}{
\TR{6}
}
\right),
\left(
\raisebox{.4cm}{
\pstree{\TR{11}}{\TR{2}} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{14}
}
\right),
\left(
\raisebox{.1cm}{
\TR{9}
}
\right),
\left(
\raisebox{.1cm}{
\TR{7}
}
\right),
\left(
\raisebox{.1cm}{
\TR{5}
}
\right),
\left(
\raisebox{.1cm}{
\TR{12}
}
\right)
\right)\in \CC_{19,7}.
\]
\end{example}
\subsection{A bijection $\zeta:\CC_{n,k}\to\FF_{n,k}$}
Recall that a special vertex is a vertex with label at most $k$.
\begin{lem}\label{lem:good}
For a fixed set $A\subset [n]$ with $|A|\geq2$, there is a bijection $\psi$
from the set of $k$-good ordered forests on $A$ to the set of rooted trees on
$A$ whose special vertices are nonleaves.
\end{lem}
\begin{proof}
Suppose $\F=(A_1,A_2,\dots,A_p)$ is a $k$-good ordered forest. We first set
all special vertices in $\F$ \emph{active}. Find the smallest vertex $v$ among
the active vertices with minimal distance from the root in $A_1$. Then
exchange the subtrees attached to $v$ and those attached to the root $r$ of
$A_2$, and then attach the resulting tree rooted at $r$ to $v$ as shown below.
\begin{center}
\psset{nodesep=2pt, levelsep=30pt}
\pstree{\TR{$v$}}{
\Tcircle{$T_1$}
\TR{$\cdots$}
\Tcircle{$T_a$}
}
\qquad
\pstree{\TR{$r$}}{
\Tcircle{$U_1$}
\TR{$\cdots$}
\Tcircle{$U_b$}
}
\qquad
\raisebox{-.5cm}{$\Rightarrow$}
\qquad
\pstree{\TR{$v$}}{
\Tcircle{$U_1$}
\TR{$\cdots$}
\Tcircle{$U_b$}
\pstree{\TR{$r$}}{
\Tcircle{$T_1$}
\TR{$\cdots$}
\Tcircle{$T_a$}
}
}
\end{center}
We then make $v$ inactive. Note that $v$ is a nonleaf after this procedure. We
do the same thing with the resulting tree and $A_3$, and proceed until there are
no active special vertices. Since $(A_1,A_2,\dots,A_p)$ is $k$-good, we can
eventually combine all of $A_1,A_2,\dots,A_p$ into a single rooted tree in which
the special vertices are nonleaves. We define $\psi(\F)$ to be the resulting
tree. It is straightforward to check that $\psi$ is invertible.
\end{proof}
Let $(\F_1,\F_2,\dots,\F_h)\in \CC_{n,k}$. For each $k$-good forest $\F_i$ we
define $T_i=\psi(\F_i)$ if $\F_i$ has at least $2$ vertices, and $T_i=X$ if
$\F_i=(X)$ and $X$ has only one vertex. Then we define
$\zeta(\F_1,\F_2,\dots,\F_h)=\rho^{-1}(T_1,\dots,T_h)$. Since $\rho^{-1}$ and
$\psi$ are invertible, $\zeta$ is a bijection.
\begin{example}
Let $(\F_1,\F_2,\dots,\F_h)$ be the following:
\[
\left(
\left(
\raisebox{.4cm}{
\pstree{\TR{10}}{\TR{4}} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{13}}{\TR{17}}}
\right),
\left(
\raisebox{.1cm}{
\TR{15}}
\right),
\left(
\raisebox{.4cm}{
\pstree{\TR{1}}{\TR{3}}\raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{8}}{
\TR{19}
\TR{18}
}
\raisebox{-4pt}{\mbox{ ,\quad}}
\TR{16}
}
\right),
\left(
\raisebox{.1cm}{
\TR{6}
}
\right),
\left(
\raisebox{.4cm}{
\pstree{\TR{11}}{\TR{2}} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{14}
}
\right),
\left(
\raisebox{.1cm}{
\TR{9}
}
\right),
\left(
\raisebox{.1cm}{
\TR{7}
}
\right),
\left(
\raisebox{.1cm}{
\TR{5}
}
\right),
\left(
\raisebox{.1cm}{
\TR{12}
}
\right)
\right)\in \CC_{19,7}.
\]
Then the map $\psi$ sends
\[
\left(
\raisebox{.4cm}{
\pstree{\TR{10}}{\TR{4}} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{13}}{\TR{17}}}
\right)
\mapsto
\raisebox{.4cm}{
\pstree{\TR{10}}{
\pstree{\TR{4}}{
\TR{17}
\TR{13}
}
}
},
\]
\[
\left(
\raisebox{.4cm}{
\pstree{\TR{1}}{\TR{3}}\raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{8}}{
\TR{19}
\TR{18}
}
\raisebox{-4pt}{\mbox{ ,\quad}}
\TR{16}
}
\right)
\mapsto
\left(
\raisebox{.8cm}{
\pstree{\TR{1}}{
\TR{19} \TR{18}
\pstree{\TR{8}}{
\TR{3}
}
}\raisebox{-4pt}{\mbox{ ,\quad}}
\TR{16}
}
\right)
\mapsto
\raisebox{1.2cm}{
\pstree{\TR{1}}{
\TR{19} \TR{18}
\pstree{\TR{8}}{
\pstree{\TR{3}}{
\TR{16}
}
}
}
},
\]
\[
\left(
\raisebox{.4cm}{
\pstree{\TR{11}}{\TR{2}} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{14}
}
\right)
\mapsto
\raisebox{.4cm}{
\pstree{\TR{11}}{\pstree{\TR{2}}{\TR{14}}}
}.
\]
Thus we obtain $(T_1,\dots,T_h)$:
\[
\left(
\raisebox{1.1cm}{
\pstree{\TR{10}}{
\pstree{\TR{4}}{
\TR{17}
\TR{13}
}
} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{15} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{1}}{
\TR{19} \TR{18}
\pstree{\TR{8}}{
\pstree{\TR{3}}{
\TR{16}
}
}
} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{6} \raisebox{-4pt}{\mbox{ ,\quad}}
\pstree{\TR{11}}{\pstree{\TR{2}}{\TR{14}}}\raisebox{-4pt}{\mbox{ ,\quad}}
\TR{9} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{7} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{5} \raisebox{-4pt}{\mbox{ ,\quad}}
\TR{12}
}
\right)
\]
If we add the vertex $n+1$, we obtain $\zeta(\F_1,\F_2,\dots,\F_h)$:
\[
\pstree{\TR{10}}{
\pstree{\TR{15}}{
\pstree{\TR{1}}{
\TR{19} \TR{18}
\pstree{\TR{8}}{
\pstree{\TR{3}}{
\TR{16}
}
}
\pstree{\TR{6}}{
\pstree{\TR{11}}{
\pstree{\TR{9}}{
\pstree{\TR{7}}{
\pstree{\TR{5}}{
\pstree{\TR{12}}{
\TR{20}
}
}
}
}
\pstree{\TR{2}}{
\TR{14}
}
}
}
}
}
\pstree{\TR{4}}{
\TR{17}
\TR{13}
}
}
\]
\end{example}
\section{Some properties of $|\TT_{n,k}|$}\label{sec:some-properties-tt_n}
We denote the cardinality of $|\TT_{n,k}|$ by $a_{n,k}$.
In \cite{SeoShin} Seo and Shin proved the following.
\begin{thm}\cite{SeoShin}
We have
\begin{align}
\label{eq:3}
\sum_{k\ge0} \binom{m+k}{k} a_{n,k} &= (n+m+1)^n, \\
\label{eq:4}
\sum_{k\ge0} \frac{1}{k} a_{n,k} &= n^n.
\end{align}
\end{thm}
We give another proof using generating functions.
\begin{proof}
By Theorem~\ref{thm:SeoShin}, $a_{n,k}$ equals the number of words of length
$n$ on $[n]$ with at least one $i$ for all $i\in[k]$. Thus
\[
a_{n,k} = \left[ \frac{x^n}{n!}\right] (e^x-1)^k e^{(n-k)x}
= \left[ \frac{x^n}{n!} \right] (1-e^{-x})^k e^{nx},
\]
where $\left[ y^n\right]f(y)$ denotes the coefficient of $y^n$ in $f(y)$.
Then we have
\begin{align*}
\sum_{k\ge0} \binom{m+k}{k} a_{n,k} & =
\left[ \frac{x^n}{n!}\right] e^{nx} \sum_{k\ge0} \binom{m+k}{k} (1-e^{-x})^k \\
&= \left[ \frac{x^n}{n!}\right] e^{nx} \frac{1}{(1-(1-e^{-x}))^{m+1}}\\
&= \left[ \frac{x^n}{n!}\right] e^{(n+m+1)x} = (n+m+1)^n,
\end{align*}
where the following binomial theorem \cite[(1.20)]{EC1second} is used:
\[
\frac{1}{(1-x)^n} = \sum_{k\geq0} \binom{n+k-1}{k} x^k.
\]
The second identity is proved similarly:
\begin{align*}
\sum_{k\ge0} \frac{1}{k} a_{n,k}
&=\left[ \frac{x^n}{n!}\right] e^{nx} \sum_{k\ge0} \frac{(1-e^{-x})^k}{k}\\
&=\left[ \frac{x^n}{n!}\right] e^{nx} \ln \frac{1}{1-(1-e^{-x})}\\
&=\left[ \frac{x^n}{n!}\right] x e^{nx}
= n! \left[ x^{n-1}\right] e^{nx} = n!\frac{n^{n-1}}{(n-1)!} = n^n.
\end{align*}
\end{proof}
In \cite{SeoShin} they asked for a bijective proof of \eqref{eq:4}. We can prove
it bijectively using our bijections as follows.
\begin{proof}[Bijective proof of \eqref{eq:4}]
By Theorem~\ref{thm:main}, $a_{n,k}$ is also equal to $|\BB_{n,k}|$, the number
of ordered forests $(T_1,T_2,\dots, T_\ell, T_{\ell+1}, \dots,T_{\ell+k})$ on
$[n]$ such that $T_{\ell+1}, \dots,T_{\ell+k}$ are minimally rooted. Thus
$a_{n,k}/k$ is equal to the number of pairs $(\F, C)$ of an ordered forest
$\F=(T_1,T_2,\dots, T_\ell)$ and a cycle $C=[T_{\ell+1}, \dots,T_{\ell+k}]$ of
$k$ minimally rooted trees such that the vertex sets of $T_1,\dots, T_{\ell+k}$
form a partition of $[n]$. Then, by Theorem~\ref{thm:main}, the number of cycles
of $k$ minimally rooted trees whose vertex sets form a subset $A$ of $[n]$ is
equal to the set of rooted trees $T$ on $A$ with $|\MD(T)|=k$. Thus $a_{n,k}/k$
is equal to the number of ordered forests $(T_1,T_2,\dots, T_\ell, T)$ on $[n]$
with $|\MD(T)|=k$. The sum of $a_{n,k}/k$ for all $k$ is equal to the number of
ordered forests on $[n]$. Suppose $(T_1,T_2,\dots, T_\ell)$ is an ordered forest
on $[n]$ and $r_i$ is the root of $T_i$ for $i\in[\ell]$. By adding the edges
$r_1r_2,r_2r_3,\dots, r_{\ell-1}r_\ell$, we get a rooted tree, say $H$. If we
know the root $r_\ell$ of the last tree, then we can recover the ordered forest
$(T_1,T_2,\dots, T_\ell)$ from $H$. Thus there is a bijection between the set of
ordered forests on $[n]$ and the set of rooted trees on $[n]$ with a choice of
$r_\ell$. The latter set has cardinality $n^n$ by Pr\"ufer code. This proves
\eqref{eq:4}.
\end{proof}
\section{Another proof of Theorem~\ref{thm:CDG}}
\label{sec:anoth-proof-theor}
Using Pr\"ufer code one can easily see that $\binom{n}{\ell} n^{n-\ell}$ is
equal to the number of rooted trees on $\{0,1,2,\dots,n+1\}$ such that $0$ is
the root with $\ell+1$ children and $n+1$ is a leaf. By deleting the root $0$,
such a tree is identified with a forest on $[n+1]$ with $\ell+1$ rooted trees such that
$n+1$ is a leaf. Thus by replacing $n+1$ with $n$, we can rewrite
Theorem~\ref{thm:CDG} as follows.
\begin{thm}\cite{Chauve2000}\label{thm:children}
The number of rooted trees of $[n]$ such that the root has $\ell$ smaller children
equals the number of forests on $[n]$ with $\ell+1$ trees such that $n$ is a leaf.
\end{thm}
\begin{proof}
Let $T$ be a rooted trees of $[n]$ such that the root has $\ell$ smaller
children. We will construct a forest corresponding to $T$. Recall the
bijection $\alpha:\TT_{n,k}\to\AA_{n,k}$. Suppose $T\in \TT_{n-1,k}$,
$\alpha(T)=[T_0,T_1,\dots,T_k]$, $r_i$ is the root of $T_i$ for
$i=0,1,2,\dots,k$. By shifting cyclically we can assume that $r_0$ is the
largest root. Note that, by the construction of $\alpha$, $T$ is rooted at
$r_0$. Also, from the construction of $\alpha$, it is easy to see that the
smaller children of the root $r_0$ in $T$ are exactly the left-to-right maxima
of $r_1,r_2,\dots,r_k$. Suppose $r_{i_1}<r_{i_2}<\cdots<r_{i_\ell}$ are the
smaller children of $r_0$ in $T$. Then $1=i_1<i_2<\cdots<i_\ell\leq k$.
Suppose $n$ is contained in $T_j$. Let $T_1', T_2',\dots, T_k'$ be the
arrangement of the trees $T_0,T_1,\dots,\widehat{T_j}, \dots, T_k$ such that
the word $r_1'r_2'\cdots r_k'$ of the roots of $T_1',\dots,T_k'$ are
order-isomorphic to the word $r_1r_2\cdots r_k$. Notice that $r_{i_1}',
r_{i_2}', \dots,r_{i_\ell}'$ are the left-to-right maxima of $r_1'r_2'\cdots
r_k'$. Thus the following map is invertible:
\begin{equation}
\label{eq:5}
(T_1', T_2',\dots, T_k') \mapsto
\{[T_{i_1}', \dots,T_{i_2-1}'], [T_{i_2}', \dots, T_{i_3-1}'], \dots,
[T_{i_\ell}', \dots, T_k']\}.
\end{equation}
Now we apply the inverse map $\alpha^{-1}$ of the bijection $\alpha$ to each
cycle in \eqref{eq:5}. Then we get a set of rooted trees. Together with $T_j$,
we obtain a forest on $[n]$. Since $T_j$ is the tree containing $n$, we can
recover the original tree $T$ from the forest. This gives a bijection between
the two sets in the theorem.
\end{proof}
The proof of Theorem~\ref{thm:children}, in fact, gives a generalization as
follows.
\begin{cor}
Let $A(n,\ell,k)$ denote the number of rooted trees of $[n]$ such that the
root has $\ell$ smaller children and the minimal decreasing subtree has $k+1$
vertices. Let $B(n,\ell,k)$ denote the number of forests on $[n]$ with
$\ell+1$ trees such that $n$ is a leaf, and the sum of $|\MD(T)|$ for all
trees $T$ in the forest except the one containing $n$ is equal to $k$. Then
\[
A(n,\ell,k) = B(n,\ell,k).
\]
\end{cor}
\begin{proof}
This can be checked by the following observation. Consider a cycle $C$ in
\eqref{eq:5}, and $T=\alpha^{-1}(C)$. Then $|\MD(T)|$ is the number of trees
in $C$. Thus the sum of $|\MD(T)|$ for all cycles $C$ in \eqref{eq:5} is $k$.
\end{proof}
\section*{Acknowledgement}
The author would like to thank Dennis Stanton for helpful discussion and
comments, especially for the idea in the proof using generating functions in
Section~\ref{sec:some-properties-tt_n}.
|
train/arxiv
|
BkiUeYfxK7Dgs_cY6bIu
| 4 | 0.8 |
\section{Introduction}
Kerov, Kirillov, and Reshetikhin introduced a combinatorial object, called a rigged configuration, through Bethe Ansatz analysis of the isotropic Heisenberg spin chain \cite{KKR86}.
They constructed a bijection between rigged configurations and the semistandard Young tableaux \cite{KKR86,KR88}, through which a bijection between rigged configurations and highest weight element of a tensor product of Killirov-Reshetikhin (KR) crystals or highest weight paths was formulated.
The rigged configuration possess a natural statistic and it coincides with the charge introduced by Lascoux and Sch$\ddot{\mathrm{u}}$tzenberger~\cite{LS78}.
On the other hand, the highest weight path carries the statistic called energy by Nakayashiki and Yamada~\cite{NY97}.
The bijection, which is called the rigged configuration bijection, is a bijection such that the charge of a rigged configuration is sent to the energy of the corresponding path.
The bijection of Kerov, Kirillov, and Reshetikhin is a bijection from rigged configurations to the paths of the form $(B^{1,1})^{\otimes L}$ in type $A_{n}^{(1)}$ where $B^{r,s}$ denotes the KR crystal.
Until now, this has been extended in more general setting and in various types~\cite{DS06,KSS02,OS12,OSS03a,OSS03b,OSS03c,OSS13,OSS+17,Sak14,Sch05,Scr16,Scr20,SS06a}.
For nonexceptional types, the generalization of the rigged configuration bijection has been established~\cite{OSS18}.
On the other hand, there remain unsolved problems in exception types.
This paper concerns one of these problems.
For the $G_{2}^{(1)}$ adjoint crystals, the explicit algorithm of the rigged configuration bijection $\Phi$ is not known as pointed out in ~\cite{Scr20} though the crystal structure is very simple (see Fig.~\ref{fig:graph}).
In this paper, we construct a map $\Phi$ from rigged configurations to highest weight elements of $(B^{2,1})^{\otimes L}$ by executing a fundamental procedure $\delta_{\theta}$ repeatedly.
Our result provides an alternate but direct proof of $X=M$ conjecture of~\cite{HKO+99,HKO+02b} in our setting, which has been proved previously by Naoi~\cite{Nao12}.
This paper is organized as follows.
In Section 2, we give the necessary background on KR crystals and paths for $G_{2}^{(1)}$.
Rigged configurations and the bijection $\Phi$ are described in Section 3, where we state our main theorem (Theorem~\ref{th:main}).
In Section 4, we provide an explicit description of the bijection $\delta_{\theta}$ as well as its inverse $\Tilde{\delta}_{\theta}$ for $G_{2}^{(1)}$ adjoint crystals.
Section 5 is devoted to the proof of Theorem~\ref{th:main}.
In Section 6, we explain some forbidden rules in the algorithm.
\section{Affine Algebra $G_{2}^{(1)}$ and the KR Crystal}
\subsection{Affine algebra $G_{2}^{(1)}$}
We consider in this paper the exceptional untwisted affine algebra $G_{2}^{(1)}$.
The Dynkin diagram is depicted in Figure~\ref{fig:Dynkin}.
We follow \cite{Kac90} for the labeling of the Dynkin nodes.
Let $I$ be the index set of the Dynkin nodes and let $\alpha_{i}$, $\alpha_{i}^{\vee}$, $\Lambda_{i}$ $(i\in I)$ be simple roots, simple coroots, fundamental weights, respectively.
Following the notation in \cite{Kac90} we denote the projection of $\Lambda_{i}$ onto the weight space of $G_{2}$ by $\Bar{\Lambda}_{i}$ ($i\in I_{0}:=I\backslash {0}$) and set $\Bar{P}=\bigoplus_{i\in I_{0}}\mathbb{Z}\Bar{\Lambda}_{i}$, $\Bar{P}^{+}=\bigoplus_{i\in I_{0}}\mathbb{Z}_{\geq 0}\Bar{\Lambda}_{i}$.
Let $(A_{i,j})_{i,j\in I}$ stand for the Cartan matrix for $G_{2}^{(1)}$.
The canonical pairing $\left<\;,\;\right> : P^{\vee}\times P \rightarrow \mathbb{Z}$ is given by $\left< \alpha_{i}^{\vee},\alpha_{j}\right>=A_{i,j}$.
\setlength{\unitlength}{12pt}
\begin{figure} \label{fig:Dynkin}
\begin{center}
\begin{picture}(5.5,2)
\put(0,1.5){\circle{0.5}}
\put(2.5,1.5){\circle{0.5}}
\put(5,1.5){\circle{0.5}}
\put(0.3,1.4){\line(1,0){1.9}}
\put(0.3,1.5){\line(1,0){1.9}}
\put(0.3,1.6){\line(1,0){1.9}}
\put(2.8,1.5){\line(1,0){1.9}}
\put(0.75,1){\makebox(1,1){\Large{$<$}}}
\put(-0.5,0){\makebox(1,1){\small{$1$}}}
\put(2,0){\makebox(1,1){\small{$2$}}}
\put(4.5,0){\makebox(1,1){\small{$0$}}}
\end{picture}
\end{center}
\caption{The Dynkin diagram for $G_{2}^{(1)}$ in the Bourbaki labeling.}
\end{figure}
\subsection{KR crystal}
Let $\mathfrak{g}$ be any affine algebra and $U_{q}^{\prime}(\mathfrak{g})$ the corresponding quantized enveloping algebra without the degree operator.
Among finite-dimensional $U_{q}^{\prime}(\mathfrak{g})$-modules there is a distinguished family called Kirillov-Reshetikhin (KR) modules~\cite{Her10,KNT02,Nak03b}.
One of the remarkable properties of KR modules is that they admit crystal bases introduced by Kashiwara~\cite{Kas90,Kas91}.
Such a crystal base is called a KR crystal.
The KR crystal is indexed by $(a,i)\in\mathcal{H}_{0}:=I_{0}\times \mathbb{Z}_{>0}$ and denoted by $B^{a,i}$.
For exceptional types the KR crystal is known to exist when the KR module is irreducible or the index $a$ is adjacent to $0$~\cite{KKM+92b}.
The KR crystal we are interested in in this paper is $G_{2}^{(1)}$ adjoint crystal $B^{2,1}$, which is a level 1 perfect crystal and is constructed in \cite{BFKL06}.
The crystal structure is depicted in Figure~\ref{fig:graph}.
See also \cite{MMO10,MOW12} (note the difference indexing of the Dynkin diagram).
Here vertices in the graph signify elements of $B^{2,1}$ and $b\stackrel{i}{\longrightarrow}b^{\prime}$ stands for $f_{i}b=b^{\prime}$ or equivalently $b=e_{i}b^{\prime}$.
we adopt the anti-Kashiwara convention for the tensor products of crystals as in \cite{BS17}.
Namely, if $B_{1}$ and $B_{2}$ are crystals, then for $b_{1}\otimes b_{2}\in B_{1}\otimes B_{2}$ the action of $e_{i}$ is defined as
\[
e_{i}(b_{1}\otimes b_{2})=
\begin{cases}
e_{i}b_{1}\otimes b_{2} & if \; \varphi_{i}(b_{2})<\varepsilon_{i}(b_{1}), \\
b_{1}\otimes e_{i}b_{2} & if \; \varphi_{i}(b_{2})\geq\varepsilon_{i}(b_{1}),
\end{cases}
\]
where $\varepsilon_{i}(b)=\max\left\{ k\geq0 \relmiddle| e_{i}^{k}b\in B\right\}$ and
$\varphi_{i}(b)=\max\left\{ k\geq0 \relmiddle| f_{i}^{k}b\in B\right\}$.
In what follow in this paper $B=B^{2,1}$.
The set of \emph{classically restricted paths} in $B^{\otimes L}$ of weight $\lambda \in \Bar{P}^{+}$ is by definition
\[
\mathcal{P}(\lambda,L)=\left\{ b\in B^{\otimes L} \relmiddle| \mathrm{wt}(b)=\lambda \text{ and } e_{i}b=0 \text{ for all } i\in I_{0} \right\}.
\]
One may check that the following are equivalent for $b=b_{1}\otimes b_{2}\otimes \cdots \otimes b_{L}\in B^{\otimes L}$ and $\lambda \in \Bar{P}^{+}$.
\begin{itemize}
\item[(1)]
$b$ is a classically restricted path of weight $\lambda \in \Bar{P}^{+}$.
\item[(2)]
$b_{2}\otimes \cdots \otimes b_{L}$ is a classically restricted path of weight $\lambda -\mathrm{wt}(b_{1})$ and $\varepsilon_{i}\leq \left<\alpha_{i}^{\vee},\lambda -\mathrm{wt}(b_{1})\right>$ for all $i\in I_{0}$.
\end{itemize}
The weight function $\mathrm{wt}:B\rightarrow \Bar{P}$ is given by
$\mathrm{wt}(b)=\sum_{i\in I_{0}}(\varphi_{i}(b)-\varepsilon_{i}(b))\Bar{\Lambda}_{i}$.
The weight function $\mathrm{wt}:B^{\otimes L}\rightarrow \Bar{P}$ is defined by
$\mathrm{wt}(b_{1}\otimes\cdots\otimes b_{L})=\sum_{j=1}^{L}\mathrm{wt}(b_{j})$.
In Figure~\ref{fig:graph}, we show the crystal subgraph for $B_{0}$, which is obtained by ignoring the $0$-arrow from the crystal graph for $B^{2,1}$.
All the $0$-arrows are listed below.
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\put(9,0){\framebox(1,1){\small{$3$}}}
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\put(19.5,0.5){\makebox(1,1){\tiny{$0$}}}
\end{picture}
\end{center}
and
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\put(1.5,0.5){\makebox(1,1){\tiny{$0$}}}
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\end{center}
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\begin{figure} \label{fig:graph}
\begin{center}
\begin{picture}(19,7)
\put(0,0){\framebox(1,1){\small{$3$}}}
\put(0,3){\framebox(1,1){\small{$2$}}}
\put(0,6){\framebox(1,1){\small{$1$}}}
\put(3,3){\framebox(1,1){\small{$4$}}}
\put(6,0){\framebox(1,1){\small{$6$}}}
\put(6,6){\framebox(1,1){\small{$5$}}}
\put(9,0){\framebox(1,1){\small{$8$}}}
\put(9,6){\framebox(1,1){\small{$7$}}}
\put(12,0){\framebox(1,1){\small{$10$}}}
\put(12,6){\framebox(1,1){\small{$9$}}}
\put(15,3){\framebox(1,1){\small{$11$}}}
\put(18,0){\framebox(1,1){\small{$14$}}}
\put(18,3){\framebox(1,1){\small{$13$}}}
\put(18,6){\framebox(1,1){\small{$12$}}}
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\put(10,6.5){\vector(1,0){2}}
\put(13,1){\vector(1,1){2}}
\put(13,6){\vector(1,-1){2}}
\put(16,4){\vector(1,1){2}}
\put(18.5,3){\vector(0,-1){2}}
\put(18.5,6){\vector(0,-1){2}}
\put(-0.5,1.5){\makebox(1,1){\tiny{$1$}}}
\put(-0.5,4.5){\makebox(1,1){\tiny{$2$}}}
\put(2,1.5){\makebox(1,1){\tiny{$1$}}}
\put(4,1.5){\makebox(1,1){\tiny{$1$}}}
\put(4,4.5){\makebox(1,1){\tiny{$2$}}}
\put(7.5,-0.5){\makebox(1,1){\tiny{$2$}}}
\put(7.5,6.5){\makebox(1,1){\tiny{$1$}}}
\put(10.5,-0.5){\makebox(1,1){\tiny{$2$}}}
\put(10.5,6.5){\makebox(1,1){\tiny{$1$}}}
\put(14,1.5){\makebox(1,1){\tiny{$1$}}}
\put(14,4.5){\makebox(1,1){\tiny{$2$}}}
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\put(18.5,4.5){\makebox(1,1){\tiny{$1$}}}
\end{picture}
\end{center}
\caption{Crystal graph of $B_{0}$ for $B^{2,1}$.}
\end{figure}
The \emph{energy function} $D: B^{\otimes L}\rightarrow \mathbb{Z}$ gives the grading on $B^{\otimes L}$.
In our case where a path is an element of the tensor product of a single KR crystal it takes a simple form.
Due to the existence of the universal $R$-matrix and the fact that $B\otimes B$ is connected, by \cite{KKM+92a} there is a unique (up to global additive constant) function $H: B\otimes B\rightarrow \mathbb{Z}$ called the \emph{local energy function}, such that
\[
H(e_{i}(b\otimes b^{\prime}))=
\begin{cases}
H(b\otimes b^{\prime})-1 & \text{ if } i=0 \text{ and } e_{0}(b\otimes b^{\prime})=e_{0}b \otimes b^{\prime}, \\
H(b\otimes b^{\prime})+1 & \text{ if } i=0 \text{ and } e_{0}(b\otimes b^{\prime})=b\otimes e_{0}b^{\prime}, \\
H(b\otimes b^{\prime}) & \text{otherwise}.
\end{cases}
\]
We normalize $H$ by the condition $H(\framebox{$1$}\otimes\framebox{$1$})=0$.
The crystal graph of $B_{0}\otimes B_{0}$ decomposes into five connected components as
\begin{equation} \label{eq:decomposition}
B_{0}\otimes B_{0}=B(2\Bar{\Lambda}_{2})\oplus
B(3\Bar{\Lambda}_{1})\oplus B(2\Bar{\Lambda}_{1})\oplus B(\Bar{\Lambda}_{2}) \oplus B(0),
\end{equation}
where $B(\lambda)$ stands for the highest weight $G_{2}$ crystal of weight $\lambda$ and the highest vector are given by $\framebox{$1$}\otimes\framebox{$1$}$ ($\lambda=2\Bar{\Lambda}_{2}$), $\framebox{$2$}\otimes\framebox{$1$}$ ($\lambda=3\Bar{\Lambda}_{1}$), $\framebox{$5$}\otimes\framebox{$1$}$ ($\lambda=2\Bar{\Lambda}_{1}$), $\framebox{$8$}\otimes\framebox{$1$} $ ($\lambda=\Bar{\Lambda}_{2}$), and $\framebox{$14$}\otimes\framebox{$1$}$ ($\lambda=0$).
$H$ is constant on each component, and takes the value $H(\framebox{$1$}\otimes\framebox{$1$})=0$, $H(\framebox{$2$}\otimes\framebox{$1$})=-1$, and $H=-2$ for the rest.
It is easily verified that $H(\framebox{$\emptyset$}\otimes\framebox{$\emptyset$})=-2$ and $H(\framebox{$\emptyset$}\otimes\framebox{$i$})=H(\framebox{$i$}\otimes\framebox{$\emptyset$})=-1$ $(1\leq i\leq 14)$.
With this $H$, the energy function $D$ is defined by
\begin{equation} \label{eq:D}
D(b_{1}\otimes \cdots \otimes b_{L})=\sum_{j=1}^{L}j H(b_{j}\otimes b_{j+1}),
\end{equation}
where $b_{L+1}=\framebox{$1$}$.
\section{Rigged Configuration and the Bijection}
\subsection{Rigged configuration}
Fix $\lambda\in \Bar{P}^{+}$ and a matrix $\boldsymbol{L}=(L_{i}^{(a)})_{(a,i)\in \mathcal{H}_{0}}$ of nonnegative integers.
Let $\nu=(m_{i}^{(a)})_{(a,i)\in\mathcal{H}_{0}}$.
Say that $\nu$ is a $\lambda$-configuration if it satisfies
\begin{equation} \label{eq:admissible}
\sum_{(a,i)\in\mathcal{H}_{0}}i m_{i}^{(a)}\alpha_{a}=\sum_{(a,i)\in\mathcal{H}_{0}}i L_{i}^{(a)}\Bar{\Lambda}_{a}-\lambda
\end{equation}
and $p_{i}^{(a)}\geq 0$ for all $(a,i)\in\mathcal{H}_{0}$, where
\begin{equation} \label{eq:vacancy}
p_{i}^{(a)}=\sum_{j\in \mathbb{Z}_{>0}} L_{j}^{(a)}\min (i,j)-\sum_{b\in I_{0}}\frac{A_{ab}}{\gamma_{b}}\sum_{j\in \mathbb{Z}_{>0}}\min (\gamma_{a}i,\gamma_{b}j)m_{j}^{(b)}
\end{equation}
is the \emph{vacancy numbers} and $\gamma_{a}$ ($a\in I_{0}$) are scaling factors which are given by $\gamma_{1}=1$ and $\gamma_{2}=3$ for $G_{2}$.
Let $\nu$ be an admissible configuration.
We identify $\nu$ with a sequence of partitions $\{ \nu^{(a)}\}_{a\in I_{0}}$ such that $\nu^{(a)}=(1^{m_{1}^{(a)}}2^{m_{2}^{(a)}}\cdots)$.
Let $J=\{ J^{(a,i)} \}_{(a,i)\in \mathcal{H}_{0}}$ be a double sequence of partitions.
Then a rigged configuration (RC) is a pair $(\nu,J)$ subject to the restriction Eq.~\eqref{eq:admissible} and the requirement that $J^{(a,i)}$ be a partition contained in a $m_{i}^{(a)}\times p_{i}^{(a)}$ rectangle.
The set of rigged configurations for fixed $\lambda$ and $\boldsymbol{L}$ is denoted by $\mathrm{RC}(\lambda,\boldsymbol{L})$.
Define the \emph{charge} of a configuration $\nu$ by
\begin{align*}
c(\nu)=&\frac{1}{2}\sum_{a,b\in I_{0}}\frac{A_{ab}}{\gamma_{b}}\sum_{i,j\in \mathbb{Z}_{>0}}\min (\gamma_{a}i,\gamma_{b}j)m_{i}^{(a)}m_{j}^{(b)} \\
&-\sum_{a\in I_{0},j,k\in \mathbb{Z}_{>0}}L_{j}^{(a)}\min (j,k)m_{k}^{(a)}.
\end{align*}
To obtain the charge of a rigged configuration $(\nu, J)$, we add all of the riggings to $c(\nu)$;
\[
c(\nu,J)=c(\nu)+|J|,
\]
where $|J|=\sum_{(a,i)\in \mathcal{H}_{0}}|J^{(a,i)}|$.
We now set
\begin{equation} \label{eq:restriction}
L_{i}^{(a)}=L\delta_{a,2}\delta_{i,1} \quad ((a,i)\in \mathcal{H}_{0}),
\end{equation}
which corresponds to considering paths in $(B^{2,1})^{\otimes L}$.
The set $\mathrm{RC}(\lambda,\boldsymbol{L})$ with the restriction Eq.~\eqref{eq:restriction} is denoted by $\mathrm{RC}(\lambda,L)$.
The vacancy numbers $p_{i}^{(a)}$ and the charge $c(\nu)$ take the following forms.
\begin{align}
p_{i}^{(1)}&=-2\sum_{j\in \mathbb{Z}_{>0}}\min (i,j)m_{j}^{(1)}+\sum_{j\in \mathbb{Z}_{>0}}\min (i,3j)m_{j}^{(2)}, \label{eq:vacancy1} \\
p_{i}^{(2)}&=L+\sum_{j\in \mathbb{Z}_{>0}}\min (3i,j)m_{j}^{(1)}-2\sum_{j\in \mathbb{Z}_{>0}}\min (i,j)m_{j}^{(2)} \label{eq:vacancy2},
\end{align}
and
\begin{align} \label{eq:charge}
c(\nu)=&\sum_{i,j\in \mathbb{Z}_{>0}}\min (i,j)m_{i}^{(1)}m_{j}^{(1)}
-\sum_{i,j\in \mathbb{Z}_{>0}}\min (i,3j)m_{i}^{(1)}m_{j}^{(2)} \\
&+\sum_{i,j\in \mathbb{Z}_{>0}}\min (i,j)m_{i}^{(2)}m_{j}^{(2)} \nonumber
-L\sum_{j\in \mathbb{Z}_{>0}}m_{j}^{(2)}.
\end{align}
\subsection{The bijection from RCs to paths.}
We now describe the bijection $\Phi : \mathrm{RC}(\lambda,L)\rightarrow \mathcal{P}(\lambda,L)$.
Let $(\nu,J)\in \mathrm{RC}(\lambda,L)$.
We shall define a map $\gamma : \mathrm{RC}(\lambda,L)\rightarrow B$ which associates to $(\nu,J)$ an element of $B$.
Define by $\mathrm{RC}_{b}(\lambda,L)$ the elements of $\mathrm{RC}(\lambda,L)$ such that $\gamma (\nu,L)=b$.
We shall define a bijection $\delta_{\theta} : \mathrm{RC}_{b}(\lambda,L)\rightarrow \mathrm{RC}(\lambda-\mathrm{wt}(b),L-1)$.
The disjoint union of these bijections then defines a bijection $\delta_{\theta} : \mathrm{RC}(\lambda,L)\rightarrow \bigsqcup _{b\in B}\mathrm{RC}(\lambda-\mathrm{wt}(b),L-1)$.
The bijection $\Phi$ is defined recursively as follows.
For $b\in B$ let $\mathcal{P}_{b}(\lambda,L)$ be the set of paths in $B^{\otimes L}$ that have $b$ as leftmost tensor factor.
For $L=0$ the bijection $\Phi$ sends the empty rigged configuration (the only element of the set $\mathrm{RC}(\lambda,L)$) to the empty path (the only element of $\mathcal{P}(\lambda,L)$).
Otherwise assume that $\Phi$ has been defined for $B^{\otimes (L-1)}$ and define it for $B^{\otimes L}$ by the commutative diagram
\[
\begin{CD}
\mathrm{RC}_{b}(\lambda,L) @>{\Phi}>> \mathcal{P}_{b}(\lambda,L) \\
@V{\delta_{\theta}}VV @VVV \\
\mathrm{RC}(\lambda-\mathrm{wt}(b),L-1) @>{\Phi}>> \mathcal{P}(\lambda-\mathrm{wt}(b),L-1)
\end{CD}
\]
where the right-hand side vertical map removes the leftmost tensor factor $b$.
Here follows the main theorem of our paper.
\begin{thm} \label{th:main}
$\Phi : \mathrm{RC}(\lambda,L)\rightarrow \mathcal{P}(\lambda,L)$ is a bijection such that
\begin{equation} \label{eq:main}
c(\nu,J)=D(\Phi(\nu,J))\text{ for all }(\nu,J)\in \mathrm{RC}(\lambda,L).
\end{equation}
\end{thm}
\section{The Bijection}
In this section, for $(\nu,J)\in \mathrm{RC}(\lambda,L)$, an algorithm is given which defines $b=\gamma (\nu,J)$, the new smaller rigged configuration $(\Tilde{\nu},\Tilde{J})=\delta (\nu,J)$ such that $(\Tilde{\nu},\Tilde{J})\in \mathrm{RC}(\rho,L-1)$ where $\rho=\lambda -\mathrm{wt}(b)$, and the new vacancy numbers $\Tilde{p}_{i}^{(a)}$ in terms of the old $p_{i}^{(a)}$.
Before describing the algorithm, we give an example of $\Phi$, where we fix notation and give several definitions.
\begin{ex} \label{ex:def}
The algorithm $\Phi$ for an RC of $L=4$ and $\lambda=2\Bar{\Lambda}_{1}$ is depicted at each step $\delta_{\theta}$ below, where partitions $\nu^{(1)}$ and $\nu^{(2)}$ are illustrated as the left and the right Young diagrams in the RC.
A row in a Young diagram is called a \emph{string}.
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so that the first RC corresponds to the path
$\framebox{$7$}\otimes \framebox{$12$}\otimes \framebox{$2$}\otimes \framebox{$1$}$.
The energy of this path is computed to be $-8$.
The computation uses Eq.~\eqref{eq:D} and Eq.~\eqref{eq:Hb}.
The charge of the first RC is computed to be $-8$ using Eq.~\eqref{eq:charge}.
\end{ex}
The down arrows marked in Example~\ref{ex:def} are called \emph{delimiters}, which are marked at each end of the consecutive sequence of three boxes of a string in $\nu^{(1)}$.
The numbers marked on the left of strings are vacancy numbers, which are common for the strings of the same length.
The numbers marked on the right of strings are riggings.
We follow the convention that riggings in strings of the same length are sorted in an increasing order (from the top).
We call a string of length $i$ in $\nu^{(a)}$ \emph{singular}, \emph{q-singular}, and \emph{qq-singular} if its rigging is equal to $p_{i}^{(a)}$, $p_{i}^{(a)}-1$, and $p_{i}^{(a)}-2$, respectively.
Here, ``q-singular'' is the abbreviation of ``quasi singular'' firstly introduced in \cite{Moh12b} and commonly used in the literature~\cite{OSS18,Scr20}.
The ``qq-singular'' strings are firstly introduced in this paper.
A singular or q-singular string is written as a singular/q-singular string.
The other cases are similar.
We say that the string which is not singular is \emph{q-singular at best}.
A string of \emph{qq-singular at best} is defined similarly.
A string which is qq-singular at best but not qq-singular is called \emph{qqq-singular at best}.
The boxes marked by $[n]$ are deleted at each step $\delta_{\theta}$.
The number $n$ indicates the order of marking process.
We call this process the \emph{box marking} and say that the box or the string is marked by $[n]$.
We also say that $[n]$ is marked in a box or a string when the box or the string is marked by $[n]$.
The string marked by $[n]$ is called the \emph{selected $i_{n}$-string} or simply the \emph{$i_{n}$-string}.
The length of the $i_{n}$-string is also denoted by $i_{n}$.
Therefore, the $i_{5}$-string in $\nu^{(1)}$ in the first RC is also a 6-string.
In the first RC in Example~\ref{ex:def}, the $i_{2}$-string coincide with the $i_{3}$-string.
We write $i_{2}=i_{3}$ in this case.
The length of the $i_{5}$-string is larger than that of the $i_{3}$-string.
We write $i_{3}<i_{5}$ in this case.
This notation is generalized in an obvious manner.
Strings in $\nu^{(1)}$ are classified by their types.
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\end{center}
Strings of the first, the second, and the third types are called \emph{type-0}, \emph{type-I}, and \emph{type-II}, respectively.
A string of type-0 or type-I is written as a string of type-0/I.
The other cases are similar.
The \emph{effective length} of a string of length $i$ in $\nu^{(1)}$ is defined by $i^{eff}:=\lceil i/3 \rceil$.
For example, $i_{3}^{eff}=1$ and $i_{5}^{eff}=2$ in the first RC in Example~\ref{ex:def}.
The real length $i$ is recovered as $3i^{eff}$, $3i^{eff}-1$, or $3i^{eff}-2$ if the string is type-0, type-I, or type-II, respectively.
We form a new RC by removing marked boxes, adjusting the values of riggings of the box-deleted strings, and keeping the other the same.
\subsection{Algorithm $\delta_{\theta}$}
The algorithm consists of box marking and adjusting the values of riggings of the box-deleted strings.
We begin with the algorithm of box marking.
\begin{description}
\item[(BM-1)] Marking $[1]$.
Find the singular string of minimum length in $\nu^{(2)}$.
If such a string exists, then mark the rightmost box of the string by $[1]$.
If not, then terminate the algorithm and return $\framebox{$1$}$.
When $[1]$ is marked, we prescribe that boxes of length $(\leq 3(i_{1}-1))$ in $\nu^{(1)}$ cannot be marked.
\item[(BM-2)] Marking $[2]$.
Find the strings of effective length $i_{1}$ in $\nu^{(1)}$.
If such strings exist, then mark the rightmost box of one of such strings by [2] according to the following preferential rule.
\begin{center}
\begin{tabular}{c|ccc}
& type-0 & type-I & type-II \\ \hline
singular & 1 & 2 & 3 \\
q-singular & 4 & 5 & \\
qq-singular & 6 & &
\end{tabular}
\end{center}
For example, firstly find a singular string of length $3i_{1}$.
If such a string exists, then mark the string by $[2]$.
If such strings do not exist, then find the singular string of length $3i_{1}-1$, etc.
If $[2]$ cannot be marked in a string of effective length $i_{1}$, then find the string of minimum length $(\geq 3i_{1}+1)$ which is singular/q-singular/qq-singular.
In this search, the type-II (resp. type-I) qq-singular string must be ignored if there exists a type-I (resp. type-0) singular string of the same effective length (see Example~\ref{ex:BM-2a}).
We prescribe that the type-II qq-singular string of length ($\geq 3i_{1}+1$) is ignored if there exists a type-0 singular string of the same effective length.
We also prescribe that the type-I q-singular (resp. qq-singular) string of length ($\geq 3i_{1}+1$) is ignored if there exists a type-0 singular (resp. q-singular) string of the same effective length (see Example~\ref{ex:BM-2b}).
If the search is successful, then mark the rightmost box of the founded string by $[2]$.
If not, then delete the marked box and return $\framebox{$2$}$.
If [2] is marked, then we prescribe that boxes of north and northwest of the box marked by [2] cannot be marked.
This rule is also applied in any box marking in $\nu^{(1)}$ (not in $\nu^{(2)}$).
\begin{ex} \label{ex:BM-2a}
$L=6$ and $\lambda=2\Bar{\Lambda}_{1}+\Bar{\Lambda}_{2}$.
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which corresponds to the path
$\framebox{$8$}\otimes \framebox{$8$}\otimes \framebox{$1$}\otimes \framebox{$5$}\otimes \framebox{$8$}\otimes \framebox{$1$}$.
The charge of the first RC and the energy of this path coincide, which is computed to be $-19$.
If we selected the qq-singular string for the $i_{2}$-string in the first RC, then we would have
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which corresponds to the path
$\framebox{$4$}\otimes \framebox{$3$}\otimes \framebox{$\emptyset$}\otimes \framebox{$5$}\otimes \framebox{$8$}\otimes \framebox{$1$}$
whose energy is computed to be $-20$.
\end{ex}
\begin{ex} \label{ex:BM-2b}
$L=6$ and $\lambda=2\Bar{\Lambda}_{2}+\Bar{\Lambda}_{2}$ as in Example~\ref{ex:BM-2a}.
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The second RC is not admissible.
\end{ex}
\item[(BM-3)] Marking $[3]$.
\begin{itemize}
\item[(1)] $i_{2}=3i_{1}-2$.
Find the q-singular string of length $3i_{1}$ ignoring q-singular strings of length $3i_{1}-1$ even if they exist.
If such s string of length $3i_{1}$ exists, then mark the rightmost box of the founded string by $[3]$.
If such a string does not exist, then find the string (singular or q-singular) of length $(\geq 3i_{1}+1)$.
If such a string exists, then mark the rightmost box of the founded string by $[3]$.
If not, then delete the marked boxes and return $\framebox{$3$}$.
\item[(2)] $i_{2}=3i_{1}-1$.
\begin{itemize}
\item[(a)] The selected $i_{2}$-string is singular.
Mark the box on the left of $[2]$ by $[3]$.
\item[(b)] The selected $i_{2}$-string is q-singular.
Find the singular/q-singular string of minimum length $(\geq 3i_{1}+1)$.
If such a string exists, then mark the rightmost box of the founded string by $[3]$.
If not, then delete the marked boxes and return $\framebox{$3$}$.
\end{itemize}
\item[(3)] $i_{2}=3i_{1}$.
\begin{itemize}
\item[(a)] The selected $i_{2}$-string is q-singular.
Mark the box on the left of $[2]$ by $[3]$.
\item[(b)] The selected $i_{2}$-string is qq-singular.
Find the singular/q-singular string of minimum length $(\geq i_{2}+1)$.
If such a string exists, then mark the rightmost box of the founded string by [3].
If not, then delete the marked boxes and return $\framebox{$3$}$.
\end{itemize}
\item[(4)] $i_{2}\geq 3i_{1}+1$.
\begin{itemize}
\item[(a)] The selected $i_{2}$-string is singular/q-singular.
Mark the box on the left of [2] by [3].
\item[(b)] The selected $i_{2}$-string is qq-singular.
Follow the same as in case (3-b).
\end{itemize}
\end{itemize}
In cases (3) and (4), if the selected $i_{2}$-string is qq-singular of type-0 and there exists a singular string of length $i_{2}+1$, then discard the previous $i_{2}$-string selection and do the following box marking.
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\begin{ex} \label{ex:BM-3}
$L=5$ and $\lambda=\Bar{\Lambda}_{1}+2\Bar{\Lambda}_{2}$.
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which corresponds to the path
$\framebox{$6$}\otimes \framebox{$3$}\otimes \framebox{$3$}\otimes \framebox{$5$}\otimes \framebox{$1$}$.
The charge of the first RC and the energy of this path coincide, which is computed to be $-14$.
If we overrode the above rule, then we would have
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\put(8,0){\makebox(1,1){{\scriptsize $[1]$}}}
\put(11,4){\makebox(2,1){$\longrightarrow$}}
\put(11,3){\makebox(2,1){$4$}}
\put(15,3){\line(1,0){2}}
\put(15,4){\line(1,0){3}}
\put(15,5){\line(1,0){3}}
\put(15,3){\line(0,1){2}}
\put(16,3){\line(0,1){2}}
\put(17,3){\line(0,1){2}}
\put(18,4){\line(0,1){1}}
\put(14,3){\makebox(1,1){{\scriptsize $0$}}}
\put(14,4){\makebox(1,1){{\scriptsize $2$}}}
\put(17,3){\makebox(1,1){{\scriptsize $0$}}}
\put(18,4){\makebox(1,1){{\scriptsize $1$}}}
\put(21,1){\line(1,0){1}}
\put(21,2){\line(1,0){1}}
\put(21,3){\line(1,0){1}}
\put(21,4){\line(1,0){1}}
\put(21,5){\line(1,0){1}}
\put(21,1){\line(0,1){4}}
\put(22,1){\line(0,1){4}}
\put(20,4){\makebox(1,1){{\scriptsize $1$}}}
\put(22,1){\makebox(1,1){{\scriptsize $0$}}}
\put(22,2){\makebox(1,1){{\scriptsize $0$}}}
\put(22,3){\makebox(1,1){{\scriptsize $0$}}}
\put(22,4){\makebox(1,1){{\scriptsize $0$}}}
\put(24,4){\makebox(2,1){$\longrightarrow$}}
\put(24,3){\makebox(2,1){$1$}}
\end{picture}
\end{center}
which corresponds to the path
$\framebox{$4$}\otimes \framebox{$1$}\otimes \framebox{$7$}\otimes \framebox{$5$}\otimes \framebox{$1$}$.
However, the energy of this path is computed to be $-15$.
\end{ex}
\item[(BM-4)] Marking $[4]$.
\begin{itemize}
\item[(1)] The selected $i_{3}$-string is singular.
The $i_{3}$-string is possibly marked by [2].
\begin{itemize}
\item[(a)] The selected $i_{3}$-string is type-0.
Mark the box on the left of $[3]$ by $[4]$.
\item[(b)] The selected $i_{3}$-string is type-I.
If $i_{3}^{eff}>i_{1}$, then mark the box on the left of $[3]$ by $[4]$.
If $i_{3}^{eff}=i_{1}$, i.e. $i_{3}=3i_{1}-1$, then the $i_{3}$-string is marked by [2].
Find the singular string of minimum length $(\geq i_{3}^{eff})$ in $\nu^{(2)}$.
If such a string exists, then let $l_{4}^{(2)}$ be the length of the founded string.
If not, then set $l_{4}^{(2)}=\infty$.
Find the singular string in $\nu^{(1)}$ of minimum length $(\geq i_{3}+1)$.
Note that the singular strings of length $3i_{1}$ do not exists in this case.
If such a string exists, then let $l_{4}^{(1)}$ be the effective length of the founded string.
If not, then set $l_{4}^{(1)}=\infty$.
If $l_{4}^{(1)}\geq l_{4}^{(2)}$, then mark the rightmost box of the founded singular string of length $l_{4}^{(2)}$ in $\nu^{(2)}$ by [4].
If $l_{4}^{(1)}< l_{4}^{(2)}$, then mark the rightmost box of the founded singular string of effective length $l_{4}^{(1)}$ in $\nu^{(1)}$ by $[4]$.
If $l_{4}^{(1)}=l_{4}^{(2)}=\infty$, then delete the marked boxes and return $\framebox{$4$}$.
\item[(c)] The selected $i_{3}$-string is type-II.
Find the singular string in $\nu^{(2)}$ of length $i_{3}^{eff}-1$ (the effective length reduction).
If such a string exists, then mark the rightmost box of the founded string by $[4]$ (see Examples~\ref{ex:BM-4a} and \ref{ex:BM-4b}).
If not, then find the singular string in $\nu^{(2)}$ of length $i_{3}^{eff}$.
If such a string exists, then mark the rightmost box of the founded string by $[4]$.
If not, then mark the box on the left of $[3]$ by $[4]$.
\end{itemize}
\item[(2)] The selected $i_{3}$-string is q-singular.
Follow the same as in (1-b).
If the selected $i_{4}$-string is type-0 and the selected $i_{3}$-string is type-I with the same effective length, then discard the $i_{3}$-string selection and do the following box marking.
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(5,3)
\put(0,0){\line(1,0){3}}
\put(0,1){\line(1,0){4}}
\put(0,2){\line(1,0){4}}
\put(1,0){\line(0,1){2}}
\put(2,0){\line(0,1){2}}
\put(3,0){\line(0,1){2}}
\put(4,1){\line(0,1){1}}
\put(0,2){\makebox(2,1){$\downarrow$}}
\put(2,1){\makebox(1,1){{\scriptsize $[4]$}}}
\put(3,1){\makebox(1,1){{\scriptsize $[3]$}}}
\end{picture}
\end{center}
This rule guarantees the bijection when the q-singular $i_{5}$-string is in $\nu^{(2)}$ with $i_{4}^{eff}=i_{5}$ and when the singular $i_{5}$-string is in $\nu^{(2)}$ with $i_{4}^{eff}=i_{5}-1$ or $i_{4}^{eff}=i_{5}$ (see Examples~\ref{ex:BM-4c} and \ref{ex:BM-4d}).
\end{itemize}
\begin{ex} \label{ex:BM-4a}
$L=3$ and $\lambda=\Bar{\Lambda}_{1}+\Bar{\Lambda}_{2}$.
The only admissible configuration $\nu$ is
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\begin{center}
\begin{picture}(9,4)
\put(1,2){\line(1,0){4}}
\put(1,3){\line(1,0){4}}
\put(1,2){\line(0,1){1}}
\put(2,2){\line(0,1){1}}
\put(3,2){\line(0,1){1}}
\put(4,2){\line(0,1){1}}
\put(5,2){\line(0,1){1}}
\put(3,3){\makebox(2,1){$\downarrow$}}
\put(0,2){\makebox(1,1){{\scriptsize $1$}}}
\put(8,0){\line(1,0){1}}
\put(8,1){\line(1,0){1}}
\put(8,2){\line(1,0){1}}
\put(8,3){\line(1,0){1}}
\put(8,0){\line(0,1){3}}
\put(9,0){\line(0,1){3}}
\put(7,2){\makebox(1,1){{\scriptsize $0$}}}
\end{picture}
\end{center}
which has the charge $c(\nu)=-5$.
The first two steps of $\Phi$ are depicted as
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\begin{center}
\begin{picture}(25,4)
\put(1,2){\line(1,0){4}}
\put(1,3){\line(1,0){4}}
\put(1,2){\line(0,1){1}}
\put(2,2){\line(0,1){1}}
\put(3,2){\line(0,1){1}}
\put(4,2){\line(0,1){1}}
\put(5,2){\line(0,1){1}}
\put(3,3){\makebox(2,1){$\downarrow$}}
\put(0,2){\makebox(1,1){{\scriptsize $1$}}}
\put(5,2){\makebox(1,1){{\scriptsize $1$}}}
\put(1,2){\makebox(1,1){{\scriptsize $[6]$}}}
\put(2,2){\makebox(1,1){{\scriptsize $[5]$}}}
\put(3,2){\makebox(1,1){{\scriptsize $[3]$}}}
\put(4,2){\makebox(1,1){{\scriptsize $[2]$}}}
\put(8,0){\line(1,0){1}}
\put(8,1){\line(1,0){1}}
\put(8,2){\line(1,0){1}}
\put(8,3){\line(1,0){1}}
\put(8,0){\line(0,1){3}}
\put(9,0){\line(0,1){3}}
\put(7,2){\makebox(1,1){{\scriptsize $0$}}}
\put(9,0){\makebox(1,1){{\scriptsize $0$}}}
\put(9,1){\makebox(1,1){{\scriptsize $0$}}}
\put(9,2){\makebox(1,1){{\scriptsize $0$}}}
\put(8,0){\makebox(1,1){{\scriptsize $[1]$}}}
\put(8,1){\makebox(1,1){{\scriptsize $[4]$}}}
\put(11,2){\makebox(2,1){$\longrightarrow$}}
\put(11,1){\makebox(2,1){$9$}}
\put(15,2){\makebox(1,1){{$\emptyset$}}}
\put(18,2){\line(1,0){1}}
\put(18,3){\line(1,0){1}}
\put(18,2){\line(0,1){1}}
\put(19,2){\line(0,1){1}}
\put(17,2){\makebox(1,1){{\scriptsize $0$}}}
\put(19,2){\makebox(1,1){{\scriptsize $0$}}}
\put(18,2){\makebox(1,1){{\scriptsize $[1]$}}}
\put(21,2){\makebox(2,1){$\longrightarrow$}}
\put(21,1){\makebox(2,1){$2$}}
\end{picture}
\end{center}
which corresponds to the path $\framebox{$9$}\otimes \framebox{$2$}\otimes \framebox{$1$}$ whose energy is $-4$ and
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\begin{center}
\begin{picture}(25,4)
\put(1,2){\line(1,0){4}}
\put(1,3){\line(1,0){4}}
\put(1,2){\line(0,1){1}}
\put(2,2){\line(0,1){1}}
\put(3,2){\line(0,1){1}}
\put(4,2){\line(0,1){1}}
\put(5,2){\line(0,1){1}}
\put(3,3){\makebox(2,1){$\downarrow$}}
\put(0,2){\makebox(1,1){{\scriptsize $1$}}}
\put(5,2){\makebox(1,1){{\scriptsize $0$}}}
\put(3,2){\makebox(1,1){{\scriptsize $[3]$}}}
\put(4,2){\makebox(1,1){{\scriptsize $[2]$}}}
\put(8,0){\line(1,0){1}}
\put(8,1){\line(1,0){1}}
\put(8,2){\line(1,0){1}}
\put(8,3){\line(1,0){1}}
\put(8,0){\line(0,1){3}}
\put(9,0){\line(0,1){3}}
\put(7,2){\makebox(1,1){{\scriptsize $0$}}}
\put(9,0){\makebox(1,1){{\scriptsize $0$}}}
\put(9,1){\makebox(1,1){{\scriptsize $0$}}}
\put(9,2){\makebox(1,1){{\scriptsize $0$}}}
\put(8,0){\makebox(1,1){{\scriptsize $[1]$}}}
\put(11,2){\makebox(2,1){$\longrightarrow$}}
\put(11,1){\makebox(2,1){$4$}}
\put(15,2){\line(1,0){2}}
\put(15,3){\line(1,0){2}}
\put(15,2){\line(0,1){1}}
\put(16,2){\line(0,1){1}}
\put(17,2){\line(0,1){1}}
\put(14,2){\makebox(1,1){{\scriptsize $0$}}}
\put(17,2){\makebox(1,1){{\scriptsize $0$}}}
\put(15,2){\makebox(1,1){{\scriptsize $[3]$}}}
\put(16,2){\makebox(1,1){{\scriptsize $[2]$}}}
\put(20,1){\line(1,0){1}}
\put(20,2){\line(1,0){1}}
\put(20,3){\line(1,0){1}}
\put(20,1){\line(0,1){2}}
\put(21,1){\line(0,1){2}}
\put(19,2){\makebox(1,1){{\scriptsize $0$}}}
\put(21,1){\makebox(1,1){{\scriptsize $0$}}}
\put(21,2){\makebox(1,1){{\scriptsize $0$}}}
\put(20,1){\makebox(1,1){{\scriptsize $[1]$}}}
\put(20,2){\makebox(1,1){{\scriptsize $[4]$}}}
\put(23,2){\makebox(2,1){$\longrightarrow$}}
\put(23,1){\makebox(2,1){$5$}}
\end{picture}
\end{center}
which corresponds to the path $\framebox{$4$}\otimes \framebox{$5$}\otimes \framebox{$1$}$ whose energy is $-5$.
\end{ex}
\begin{ex} \label{ex:BM-4b}
$L=4$ and $\lambda=2\Bar{\Lambda}_{1}+\Bar{\Lambda}_{2}$.
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\begin{center}
\begin{picture}(25,5)
\put(1,2){\line(1,0){1}}
\put(1,3){\line(1,0){4}}
\put(1,4){\line(1,0){4}}
\put(1,2){\line(0,1){2}}
\put(2,2){\line(0,1){2}}
\put(3,3){\line(0,1){1}}
\put(4,3){\line(0,1){1}}
\put(5,3){\line(0,1){1}}
\put(3,4){\makebox(2,1){$\downarrow$}}
\put(0,2){\makebox(1,1){{\scriptsize $0$}}}
\put(0,3){\makebox(1,1){{\scriptsize $2$}}}
\put(2,2){\makebox(1,1){{\scriptsize $0$}}}
\put(5,3){\makebox(1,1){{\scriptsize $2$}}}
\put(1,2){\makebox(1,1){{\scriptsize $[2]$}}}
\put(2,3){\makebox(1,1){{\scriptsize $[6]$}}}
\put(3,3){\makebox(1,1){{\scriptsize $[5]$}}}
\put(4,3){\makebox(1,1){{\scriptsize $[3]$}}}
\put(8,0){\line(1,0){1}}
\put(8,1){\line(1,0){1}}
\put(8,2){\line(1,0){1}}
\put(8,3){\line(1,0){1}}
\put(8,4){\line(1,0){1}}
\put(8,0){\line(0,1){4}}
\put(9,0){\line(0,1){4}}
\put(7,3){\makebox(1,1){{\scriptsize $0$}}}
\put(9,0){\makebox(1,1){{\scriptsize $0$}}}
\put(9,1){\makebox(1,1){{\scriptsize $0$}}}
\put(9,2){\makebox(1,1){{\scriptsize $0$}}}
\put(9,3){\makebox(1,1){{\scriptsize $0$}}}
\put(8,0){\makebox(1,1){{\scriptsize $[1]$}}}
\put(8,1){\makebox(1,1){{\scriptsize $[4]$}}}
\put(11,3){\makebox(2,1){$\longrightarrow$}}
\put(11,2){\makebox(2,1){$9$}}
\put(15,3){\line(1,0){1}}
\put(15,4){\line(1,0){1}}
\put(15,3){\line(0,1){1}}
\put(16,3){\line(0,1){1}}
\put(14,3){\makebox(1,1){{\scriptsize $0$}}}
\put(16,3){\makebox(1,1){{\scriptsize $0$}}}
\put(15,3){\makebox(1,1){{\scriptsize $[2]$}}}
\put(19,2){\line(1,0){1}}
\put(19,3){\line(1,0){1}}
\put(19,4){\line(1,0){1}}
\put(19,2){\line(0,1){2}}
\put(20,2){\line(0,1){2}}
\put(18,3){\makebox(1,1){{\scriptsize $0$}}}
\put(20,2){\makebox(1,1){{\scriptsize $0$}}}
\put(20,3){\makebox(1,1){{\scriptsize $0$}}}
\put(19,2){\makebox(1,1){{\scriptsize $[1]$}}}
\put(22,3){\makebox(2,1){$\longrightarrow$}}
\put(22,2){\makebox(2,1){$3$}}
\end{picture}
\end{center}
which corresponds to the path $\framebox{$9$}\otimes \framebox{$3$}\otimes \framebox{$2$}\otimes \framebox{$1$}$ whose energy is $-7$.
\end{ex}
\begin{ex} \label{ex:BM-4c}
$L=6$ and $\lambda=\Bar{\Lambda}_{1}+\Bar{\Lambda}_{2}$.
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\begin{center}
\begin{picture}(35,8)
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\put(1,5){\line(1,0){5}}
\put(1,6){\line(1,0){6}}
\put(1,7){\line(1,0){6}}
\put(1,4){\line(0,1){3}}
\put(2,4){\line(0,1){3}}
\put(3,4){\line(0,1){3}}
\put(4,5){\line(0,1){2}}
\put(5,5){\line(0,1){2}}
\put(6,5){\line(0,1){2}}
\put(7,6){\line(0,1){1}}
\put(3,7){\makebox(2,1){$\downarrow$}}
\put(0,4){\makebox(1,1){{\scriptsize $2$}}}
\put(0,5){\makebox(1,1){{\scriptsize $1$}}}
\put(0,6){\makebox(1,1){{\scriptsize $1$}}}
\put(3,4){\makebox(1,1){{\scriptsize $1$}}}
\put(6,5){\makebox(1,1){{\scriptsize $0$}}}
\put(7,6){\makebox(1,1){{\scriptsize $1$}}}
\put(2,4){\makebox(1,1){{\scriptsize $[2]$}}}
\put(5,6){\makebox(1,1){{\scriptsize $[4]$}}}
\put(6,6){\makebox(1,1){{\scriptsize $[3]$}}}
\put(10,0){\line(1,0){1}}
\put(10,1){\line(1,0){1}}
\put(10,2){\line(1,0){1}}
\put(10,3){\line(1,0){1}}
\put(10,4){\line(1,0){1}}
\put(10,5){\line(1,0){2}}
\put(10,6){\line(1,0){2}}
\put(10,7){\line(1,0){2}}
\put(10,0){\line(0,1){7}}
\put(11,0){\line(0,1){7}}
\put(12,5){\line(0,1){2}}
\put(9,4){\makebox(1,1){{\scriptsize $0$}}}
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\put(14,4){\makebox(2,1){$8$}}
\put(18,4){\line(1,0){1}}
\put(18,5){\line(1,0){4}}
\put(18,6){\line(1,0){5}}
\put(18,7){\line(1,0){5}}
\put(18,4){\line(0,1){3}}
\put(19,4){\line(0,1){3}}
\put(20,5){\line(0,1){2}}
\put(21,5){\line(0,1){2}}
\put(22,5){\line(0,1){2}}
\put(23,6){\line(0,1){1}}
\put(20,7){\makebox(2,1){$\downarrow$}}
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\put(20,5){\makebox(1,1){{\scriptsize $[5]$}}}
\put(21,5){\makebox(1,1){{\scriptsize $[3]$}}}
\put(26,1){\line(1,0){1}}
\put(26,2){\line(1,0){1}}
\put(26,3){\line(1,0){1}}
\put(26,4){\line(1,0){1}}
\put(26,5){\line(1,0){1}}
\put(26,6){\line(1,0){2}}
\put(26,7){\line(1,0){2}}
\put(26,1){\line(0,1){6}}
\put(27,1){\line(0,1){6}}
\put(28,6){\line(0,1){1}}
\put(25,5){\makebox(1,1){{\scriptsize $0$}}}
\put(25,6){\makebox(1,1){{\scriptsize $1$}}}
\put(27,1){\makebox(1,1){{\scriptsize $0$}}}
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\put(27,5){\makebox(1,1){{\scriptsize $0$}}}
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\put(26,1){\makebox(1,1){{\scriptsize $[1]$}}}
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\put(30,5){\makebox(2,1){$\longrightarrow$}}
\put(30,4){\makebox(2,1){$9$}}
\end{picture}
\end{center}
which corresponds to the path
$\framebox{$8$}\otimes \framebox{$9$}\otimes \framebox{$3$}\otimes \framebox{$5$}\otimes \framebox{$8$}\otimes \framebox{$1$}$.
The charge of the first RC and the energy of this path coincide, which is computed to be $-22$.
If we selected the q-singular string for the $i_{3}$-string in the first RC, then we would have
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\put(1,4){\line(0,1){3}}
\put(2,4){\line(0,1){3}}
\put(3,4){\line(0,1){3}}
\put(4,5){\line(0,1){2}}
\put(5,5){\line(0,1){2}}
\put(6,5){\line(0,1){2}}
\put(7,6){\line(0,1){1}}
\put(3,7){\makebox(2,1){$\downarrow$}}
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\put(3,4){\makebox(1,1){{\scriptsize $1$}}}
\put(6,5){\makebox(1,1){{\scriptsize $0$}}}
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\put(2,4){\makebox(1,1){{\scriptsize $[2]$}}}
\put(5,5){\makebox(1,1){{\scriptsize $[3]$}}}
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\put(10,0){\line(1,0){1}}
\put(10,1){\line(1,0){1}}
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\put(10,4){\line(1,0){1}}
\put(10,5){\line(1,0){2}}
\put(10,6){\line(1,0){2}}
\put(10,7){\line(1,0){2}}
\put(10,0){\line(0,1){7}}
\put(11,0){\line(0,1){7}}
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\put(11,1){\makebox(1,1){{\scriptsize $0$}}}
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\put(11,4){\makebox(1,1){{\scriptsize $0$}}}
\put(12,5){\makebox(1,1){{\scriptsize $0$}}}
\put(12,6){\makebox(1,1){{\scriptsize $0$}}}
\put(10,0){\makebox(1,1){{\scriptsize $[1]$}}}
\put(11,5){\makebox(1,1){{\scriptsize $[5]$}}}
\put(14,5){\makebox(2,1){$\longrightarrow$}}
\put(14,4){\makebox(2,1){$8$}}
\put(18,4){\line(1,0){1}}
\put(18,5){\line(1,0){4}}
\put(18,6){\line(1,0){5}}
\put(18,7){\line(1,0){5}}
\put(18,4){\line(0,1){3}}
\put(19,4){\line(0,1){3}}
\put(20,5){\line(0,1){2}}
\put(21,5){\line(0,1){2}}
\put(22,5){\line(0,1){2}}
\put(23,6){\line(0,1){1}}
\put(20,7){\makebox(2,1){$\downarrow$}}
\put(17,4){\makebox(1,1){{\scriptsize $0$}}}
\put(17,5){\makebox(1,1){{\scriptsize $1$}}}
\put(17,6){\makebox(1,1){{\scriptsize $0$}}}
\put(19,4){\makebox(1,1){{\scriptsize $0$}}}
\put(22,5){\makebox(1,1){{\scriptsize $0$}}}
\put(23,6){\makebox(1,1){{\scriptsize $0$}}}
\put(18,4){\makebox(1,1){{\scriptsize $[2]$}}}
\put(21,5){\makebox(1,1){{\scriptsize $[3]$}}}
\put(22,6){\makebox(1,1){{\scriptsize $[4]$}}}
\put(26,1){\line(1,0){1}}
\put(26,2){\line(1,0){1}}
\put(26,3){\line(1,0){1}}
\put(26,4){\line(1,0){1}}
\put(26,5){\line(1,0){1}}
\put(26,6){\line(1,0){2}}
\put(26,7){\line(1,0){2}}
\put(26,1){\line(0,1){6}}
\put(27,1){\line(0,1){6}}
\put(28,6){\line(0,1){1}}
\put(25,5){\makebox(1,1){{\scriptsize $0$}}}
\put(25,6){\makebox(1,1){{\scriptsize $1$}}}
\put(27,1){\makebox(1,1){{\scriptsize $0$}}}
\put(27,2){\makebox(1,1){{\scriptsize $0$}}}
\put(27,3){\makebox(1,1){{\scriptsize $0$}}}
\put(27,4){\makebox(1,1){{\scriptsize $0$}}}
\put(27,5){\makebox(1,1){{\scriptsize $0$}}}
\put(28,6){\makebox(1,1){{\scriptsize $0$}}}
\put(26,1){\makebox(1,1){{\scriptsize $[1]$}}}
\put(27,6){\makebox(1,1){{\scriptsize $[5]$}}}
\put(30,5){\makebox(2,1){$\longrightarrow$}}
\put(30,4){\makebox(2,1){$8$}}
\end{picture}
\end{center}
which corresponds to the path
$\framebox{$8$}\otimes \framebox{$8$}\otimes \framebox{$7$}\otimes \framebox{$4$}\otimes \framebox{$5$}\otimes \framebox{$1$}$.
However, the energy of this path is computed to be $-23$.
\end{ex}
\begin{ex} \label{ex:BM-4d}
$L=7$ and $\lambda=3\Bar{\Lambda}_{1}+\Bar{\Lambda}_{2}$.
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\begin{center}
\begin{picture}(35,8)
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\put(1,5){\line(1,0){5}}
\put(1,6){\line(1,0){6}}
\put(1,7){\line(1,0){6}}
\put(1,4){\line(0,1){3}}
\put(2,4){\line(0,1){3}}
\put(3,5){\line(0,1){2}}
\put(4,5){\line(0,1){2}}
\put(5,5){\line(0,1){2}}
\put(6,5){\line(0,1){2}}
\put(7,6){\line(0,1){1}}
\put(3,7){\makebox(2,1){$\downarrow$}}
\put(0,4){\makebox(1,1){{\scriptsize $1$}}}
\put(0,5){\makebox(1,1){{\scriptsize $1$}}}
\put(0,6){\makebox(1,1){{\scriptsize $0$}}}
\put(2,4){\makebox(1,1){{\scriptsize $1$}}}
\put(6,5){\makebox(1,1){{\scriptsize $0$}}}
\put(7,6){\makebox(1,1){{\scriptsize $0$}}}
\put(1,4){\makebox(1,1){{\scriptsize $[2]$}}}
\put(5,6){\makebox(1,1){{\scriptsize $[4]$}}}
\put(6,6){\makebox(1,1){{\scriptsize $[3]$}}}
\put(10,0){\line(1,0){1}}
\put(10,1){\line(1,0){1}}
\put(10,2){\line(1,0){1}}
\put(10,3){\line(1,0){1}}
\put(10,4){\line(1,0){1}}
\put(10,5){\line(1,0){1}}
\put(10,6){\line(1,0){3}}
\put(10,7){\line(1,0){3}}
\put(10,0){\line(0,1){7}}
\put(11,0){\line(0,1){7}}
\put(12,6){\line(0,1){1}}
\put(13,6){\line(0,1){1}}
\put(9,5){\makebox(1,1){{\scriptsize $0$}}}
\put(9,6){\makebox(1,1){{\scriptsize $1$}}}
\put(11,0){\makebox(1,1){{\scriptsize $0$}}}
\put(11,1){\makebox(1,1){{\scriptsize $0$}}}
\put(11,2){\makebox(1,1){{\scriptsize $0$}}}
\put(11,3){\makebox(1,1){{\scriptsize $0$}}}
\put(11,4){\makebox(1,1){{\scriptsize $0$}}}
\put(11,5){\makebox(1,1){{\scriptsize $0$}}}
\put(13,6){\makebox(1,1){{\scriptsize $1$}}}
\put(10,0){\makebox(1,1){{\scriptsize $[1]$}}}
\put(11,6){\makebox(1,1){{\scriptsize $[6]$}}}
\put(12,6){\makebox(1,1){{\scriptsize $[5]$}}}
\put(15,5){\makebox(2,1){$\longrightarrow$}}
\put(15,4){\makebox(2,1){$10$}}
\put(19,5){\line(1,0){4}}
\put(19,6){\line(1,0){5}}
\put(19,7){\line(1,0){5}}
\put(19,5){\line(0,1){2}}
\put(20,5){\line(0,1){2}}
\put(21,5){\line(0,1){2}}
\put(22,5){\line(0,1){2}}
\put(23,5){\line(0,1){2}}
\put(24,6){\line(0,1){1}}
\put(21,7){\makebox(2,1){$\downarrow$}}
\put(18,5){\makebox(1,1){{\scriptsize $2$}}}
\put(18,6){\makebox(1,1){{\scriptsize $0$}}}
\put(23,5){\makebox(1,1){{\scriptsize $2$}}}
\put(24,6){\makebox(1,1){{\scriptsize $0$}}}
\put(19,5){\makebox(1,1){{\scriptsize $[6]$}}}
\put(20,5){\makebox(1,1){{\scriptsize $[5]$}}}
\put(21,5){\makebox(1,1){{\scriptsize $[3]$}}}
\put(22,5){\makebox(1,1){{\scriptsize $[2]$}}}
\put(27,1){\line(1,0){1}}
\put(27,2){\line(1,0){1}}
\put(27,3){\line(1,0){1}}
\put(27,4){\line(1,0){1}}
\put(27,5){\line(1,0){1}}
\put(27,6){\line(1,0){1}}
\put(27,7){\line(1,0){1}}
\put(27,1){\line(0,1){6}}
\put(28,1){\line(0,1){6}}
\put(26,6){\makebox(1,1){{\scriptsize $0$}}}
\put(28,1){\makebox(1,1){{\scriptsize $0$}}}
\put(28,2){\makebox(1,1){{\scriptsize $0$}}}
\put(28,3){\makebox(1,1){{\scriptsize $0$}}}
\put(28,4){\makebox(1,1){{\scriptsize $0$}}}
\put(28,5){\makebox(1,1){{\scriptsize $0$}}}
\put(28,6){\makebox(1,1){{\scriptsize $0$}}}
\put(27,1){\makebox(1,1){{\scriptsize $[1]$}}}
\put(27,2){\makebox(1,1){{\scriptsize $[4]$}}}
\put(30,5){\makebox(2,1){$\longrightarrow$}}
\put(30,4){\makebox(2,1){$9$}}
\end{picture}
\end{center}
which corresponds to the path
$\framebox{$10$}\otimes \framebox{$9$}\otimes \framebox{$3$}\otimes \framebox{$1$}\otimes \framebox{$9$}\otimes \framebox{$2$}\otimes \framebox{$1$}$.
The charge of the first RC and the energy of this path coincide, which is computed to be $-24$.
If we selected the q-singular string for the $i_{3}$-string in the first RC, then we would have
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\begin{center}
\begin{picture}(35,8)
\put(1,4){\line(1,0){1}}
\put(1,5){\line(1,0){5}}
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\put(1,7){\line(1,0){6}}
\put(1,4){\line(0,1){3}}
\put(2,4){\line(0,1){3}}
\put(3,5){\line(0,1){2}}
\put(4,5){\line(0,1){2}}
\put(5,5){\line(0,1){2}}
\put(6,5){\line(0,1){2}}
\put(7,6){\line(0,1){1}}
\put(3,7){\makebox(2,1){$\downarrow$}}
\put(0,4){\makebox(1,1){{\scriptsize $1$}}}
\put(0,5){\makebox(1,1){{\scriptsize $1$}}}
\put(0,6){\makebox(1,1){{\scriptsize $0$}}}
\put(2,4){\makebox(1,1){{\scriptsize $1$}}}
\put(6,5){\makebox(1,1){{\scriptsize $0$}}}
\put(7,6){\makebox(1,1){{\scriptsize $0$}}}
\put(1,4){\makebox(1,1){{\scriptsize $[2]$}}}
\put(5,5){\makebox(1,1){{\scriptsize $[3]$}}}
\put(6,6){\makebox(1,1){{\scriptsize $[4]$}}}
\put(10,0){\line(1,0){1}}
\put(10,1){\line(1,0){1}}
\put(10,2){\line(1,0){1}}
\put(10,3){\line(1,0){1}}
\put(10,4){\line(1,0){1}}
\put(10,5){\line(1,0){1}}
\put(10,6){\line(1,0){3}}
\put(10,7){\line(1,0){3}}
\put(10,0){\line(0,1){7}}
\put(11,0){\line(0,1){7}}
\put(12,6){\line(0,1){1}}
\put(13,6){\line(0,1){1}}
\put(9,5){\makebox(1,1){{\scriptsize $0$}}}
\put(9,6){\makebox(1,1){{\scriptsize $1$}}}
\put(11,0){\makebox(1,1){{\scriptsize $0$}}}
\put(11,1){\makebox(1,1){{\scriptsize $0$}}}
\put(11,2){\makebox(1,1){{\scriptsize $0$}}}
\put(11,3){\makebox(1,1){{\scriptsize $0$}}}
\put(11,4){\makebox(1,1){{\scriptsize $0$}}}
\put(11,5){\makebox(1,1){{\scriptsize $0$}}}
\put(13,6){\makebox(1,1){{\scriptsize $1$}}}
\put(10,0){\makebox(1,1){{\scriptsize $[1]$}}}
\put(11,6){\makebox(1,1){{\scriptsize $[6]$}}}
\put(12,6){\makebox(1,1){{\scriptsize $[5]$}}}
\put(15,5){\makebox(2,1){$\longrightarrow$}}
\put(15,4){\makebox(2,1){$10$}}
\put(19,5){\line(1,0){4}}
\put(19,6){\line(1,0){5}}
\put(19,7){\line(1,0){5}}
\put(19,5){\line(0,1){2}}
\put(20,5){\line(0,1){2}}
\put(21,5){\line(0,1){2}}
\put(22,5){\line(0,1){2}}
\put(23,5){\line(0,1){2}}
\put(24,6){\line(0,1){1}}
\put(21,7){\makebox(2,1){$\downarrow$}}
\put(18,5){\makebox(1,1){{\scriptsize $2$}}}
\put(18,6){\makebox(1,1){{\scriptsize $0$}}}
\put(23,5){\makebox(1,1){{\scriptsize $1$}}}
\put(24,6){\makebox(1,1){{\scriptsize $0$}}}
\put(21,5){\makebox(1,1){{\scriptsize $[3]$}}}
\put(22,5){\makebox(1,1){{\scriptsize $[2]$}}}
\put(23,6){\makebox(1,1){{\scriptsize $[4]$}}}
\put(27,1){\line(1,0){1}}
\put(27,2){\line(1,0){1}}
\put(27,3){\line(1,0){1}}
\put(27,4){\line(1,0){1}}
\put(27,5){\line(1,0){1}}
\put(27,6){\line(1,0){1}}
\put(27,7){\line(1,0){1}}
\put(27,1){\line(0,1){6}}
\put(28,1){\line(0,1){6}}
\put(26,6){\makebox(1,1){{\scriptsize $0$}}}
\put(28,1){\makebox(1,1){{\scriptsize $0$}}}
\put(28,2){\makebox(1,1){{\scriptsize $0$}}}
\put(28,3){\makebox(1,1){{\scriptsize $0$}}}
\put(28,4){\makebox(1,1){{\scriptsize $0$}}}
\put(28,5){\makebox(1,1){{\scriptsize $0$}}}
\put(28,6){\makebox(1,1){{\scriptsize $0$}}}
\put(27,1){\makebox(1,1){{\scriptsize $[1]$}}}
\put(30,5){\makebox(2,1){$\longrightarrow$}}
\put(30,4){\makebox(2,1){$6$}}
\end{picture}
\end{center}
which corresponds to the path
$\framebox{$10$}\otimes \framebox{$6$}\otimes \framebox{$5$}\otimes \framebox{$1$}\otimes \framebox{$9$}\otimes \framebox{$2$}\otimes \framebox{$1$}$.
However, the energy of this path is computed to be $-25$.
\end{ex}
\item[(BM-5)] Marking $[5]$.
\begin{itemize}
\item[(1)] The selected $i_{4}$-string is in $\nu^{(1)}$.
Find the singular/q-singular string of minimum length $(\geq i_{4}^{eff})$ in $\nu^{(2)}$.
Suppose that such a string exists.
If the founded string is singular and the length is $1$, then delete the marked boxes and return $\framebox{$\emptyset$}$.
In this case, the box marking of $\nu^{(1)}$ is
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\begin{center}
\begin{picture}(3,3)
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\put(1,2){\line(0,1){1}}
\put(2,2){\line(0,1){1}}
\put(3,2){\line(0,1){1}}
\put(2,0.9){\makebox(1,1){$\uparrow$}}
\put(0,2){\makebox(1,1){{\scriptsize $[4]$}}}
\put(1,2){\makebox(1,1){{\scriptsize $[3]$}}}
\put(2,2){\makebox(1,1){{\scriptsize $[2]$}}}
\put(2,0){\makebox(1,1){{\scriptsize $3$}}}
\end{picture}
\end{center}
If the founded string is q-singular or singular with length $(\geq 2)$, then the rightmost box of the founded string by $[5]$.
If not, then delete the marked boxes and return $\framebox{$6$}$.
\item[(2)] The selected $i_{4}$-string is in $\nu^{(2)}$.
Find the singular or q-singular string of effective length $(\geq i_{4})$ in $\nu^{(1)}$.
If such a string exists and it is not a type-I/II q-singular string of effective length $i_{4}$, then mark the rightmost box of the founded string by $[5]$.
If such a string does not exist, then delete the marked boxes and return $\framebox{$5$}$.
\begin{ex}
$L=4$ and $\lambda=3\Lambda_{1}$.
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\begin{center}
\begin{picture}(18,5)
\put(1,2){\line(1,0){1}}
\put(1,3){\line(1,0){5}}
\put(1,4){\line(1,0){5}}
\put(1,2){\line(0,1){2}}
\put(2,2){\line(0,1){2}}
\put(3,3){\line(0,1){1}}
\put(4,3){\line(0,1){1}}
\put(5,3){\line(0,1){1}}
\put(6,3){\line(0,1){1}}
\put(3,4){\makebox(2,1){$\downarrow$}}
\put(0,2){\makebox(1,1){{\scriptsize $0$}}}
\put(0,3){\makebox(1,1){{\scriptsize $2$}}}
\put(2,2){\makebox(1,1){{\scriptsize $0$}}}
\put(6,3){\makebox(1,1){{\scriptsize $1$}}}
\put(1,2){\makebox(1,1){{\scriptsize $[2]$}}}
\put(5,3){\makebox(1,1){{\scriptsize $[3]$}}}
\put(9,0){\line(1,0){1}}
\put(9,1){\line(1,0){1}}
\put(9,2){\line(1,0){1}}
\put(9,3){\line(1,0){2}}
\put(9,4){\line(1,0){2}}
\put(9,0){\line(0,1){4}}
\put(10,0){\line(0,1){4}}
\put(11,3){\line(0,1){1}}
\put(8,2){\makebox(1,1){{\scriptsize $0$}}}
\put(8,3){\makebox(1,1){{\scriptsize $0$}}}
\put(10,0){\makebox(1,1){{\scriptsize $0$}}}
\put(10,1){\makebox(1,1){{\scriptsize $0$}}}
\put(10,2){\makebox(1,1){{\scriptsize $0$}}}
\put(11,3){\makebox(1,1){{\scriptsize $0$}}}
\put(9,0){\makebox(1,1){{\scriptsize $[1]$}}}
\put(10,3){\makebox(1,1){{\scriptsize $[4]$}}}
\put(13,3){\makebox(2,1){$\longrightarrow$}}
\put(13,2){\makebox(2,1){$5$}}
\end{picture}
\end{center}
which corresponds to the path $\framebox{$5$}\otimes \framebox{$9$}\otimes \framebox{$2$} \otimes \framebox{$1$}$.
The charge of the first RC and the energy of this path coincide, which is computed to be $-8$.
For the rest steps, see Example~\ref{ex:BM-4a}.
If [5] were marked on the left of [3], then we would have
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\begin{center}
\begin{picture}(28,5)
\put(1,2){\line(1,0){1}}
\put(1,3){\line(1,0){5}}
\put(1,4){\line(1,0){5}}
\put(1,2){\line(0,1){2}}
\put(2,2){\line(0,1){2}}
\put(3,3){\line(0,1){1}}
\put(4,3){\line(0,1){1}}
\put(5,3){\line(0,1){1}}
\put(6,3){\line(0,1){1}}
\put(3,4){\makebox(2,1){$\downarrow$}}
\put(0,2){\makebox(1,1){{\scriptsize $0$}}}
\put(0,3){\makebox(1,1){{\scriptsize $2$}}}
\put(2,2){\makebox(1,1){{\scriptsize $0$}}}
\put(6,3){\makebox(1,1){{\scriptsize $1$}}}
\put(1,2){\makebox(1,1){{\scriptsize $[2]$}}}
\put(4,3){\makebox(1,1){{\scriptsize $[5]$}}}
\put(5,3){\makebox(1,1){{\scriptsize $[3]$}}}
\put(9,0){\line(1,0){1}}
\put(9,1){\line(1,0){1}}
\put(9,2){\line(1,0){1}}
\put(9,3){\line(1,0){2}}
\put(9,4){\line(1,0){2}}
\put(9,0){\line(0,1){4}}
\put(10,0){\line(0,1){4}}
\put(11,3){\line(0,1){1}}
\put(8,2){\makebox(1,1){{\scriptsize $0$}}}
\put(8,3){\makebox(1,1){{\scriptsize $0$}}}
\put(10,0){\makebox(1,1){{\scriptsize $0$}}}
\put(10,1){\makebox(1,1){{\scriptsize $0$}}}
\put(10,2){\makebox(1,1){{\scriptsize $0$}}}
\put(11,3){\makebox(1,1){{\scriptsize $0$}}}
\put(9,0){\makebox(1,1){{\scriptsize $[1]$}}}
\put(10,3){\makebox(1,1){{\scriptsize $[4]$}}}
\put(13,3){\makebox(2,1){$\longrightarrow$}}
\put(13,2){\makebox(2,1){$7$}}
\put(17,3){\line(1,0){3}}
\put(17,4){\line(1,0){3}}
\put(17,3){\line(0,1){1}}
\put(18,3){\line(0,1){1}}
\put(19,3){\line(0,1){1}}
\put(20,3){\line(0,1){1}}
\put(16,3){\makebox(1,1){{\scriptsize $3$}}}
\put(20,3){\makebox(1,1){{\scriptsize $3$}}}
\put(17,3){\makebox(1,1){{\scriptsize $[4]$}}}
\put(18,3){\makebox(1,1){{\scriptsize $[3]$}}}
\put(19,3){\makebox(1,1){{\scriptsize $[2]$}}}
\put(23,1){\line(1,0){1}}
\put(23,2){\line(1,0){1}}
\put(23,3){\line(1,0){1}}
\put(23,4){\line(1,0){1}}
\put(23,1){\line(0,1){3}}
\put(24,1){\line(0,1){3}}
\put(22,3){\makebox(1,1){{\scriptsize $0$}}}
\put(24,1){\makebox(1,1){{\scriptsize $0$}}}
\put(24,2){\makebox(1,1){{\scriptsize $0$}}}
\put(24,3){\makebox(1,1){{\scriptsize $0$}}}
\put(23,1){\makebox(1,1){{\scriptsize $[1]$}}}
\put(23,2){\makebox(1,1){{\scriptsize $[5]$}}}
\put(26,3){\makebox(2,1){$\longrightarrow$}}
\put(26,2){\makebox(2,1){$\emptyset$}}
\end{picture}
\end{center}
which corresponds to the path $\framebox{$7$}\otimes \framebox{$\emptyset$}\otimes \framebox{$2$} \otimes \framebox{$1$}$.
However, the energy of this path is computed to be $-6$.
\end{ex}
\end{itemize}
\item[(BM-6)] Marking $[6]$.
\begin{itemize}
\item[(1)] The selected $i_{5}$-string is q-singular.
Find the singular string of minimum length $(\geq i_{5}+1)$.
If such a string exists, then mark the rightmost box of the founded string by $[6]$.
If such a string does not exist, then delete the marked boxes and return $\framebox{$7$}$ when the selected $i_{5}$-string is in $\nu^{(1)}$ and $\framebox{$8$}$ when the selected $i_{5}$-string is in $\nu^{(2)}$.
\item[(2)] The selected $i_{5}$-string is singular.
Mark the box on the left of $[5]$ by $[6]$.
\end{itemize}
According to the marking rule of [4] the box marking such that the selected $i_{4}$-string in $\nu^{(1)}$ is type-II with $i_{4}^{eff}=i_{6}$ cannot occur.
\item[(BM-7)] Marking $[7]$.
\begin{itemize}
\item[(1)] The selected $i_{6}$-string is in $\nu^{(1)}$.
Find the singular string of minimum length $(\geq i_{6}^{eff})$.
If such a string exists, then mark the rightmost unmarked box of the founded string by $[7]$.
If not, then delete the marked boxes and return $\framebox{$9$}$.
Note that it is possible $[7]$ is marked on the left of $[4]$.
\item[(2)] The selected $i_{6}$-string is in $\nu^{(2)}$.
Find the selected $i_{4}$-string with $i_{4}^{eff}=i_{6}$, which is type-0/I.
If the search is successful, then mark [7] on the left of [4].
If not, then follow the box marking rule \textbf{(BM-2)} with replacing $i_{1}$ by $i_{6}$.
If the search is not successful, then delete the marked boxes and return $\framebox{$10$}$.
\end{itemize}
\item[(BM-8)] Marking $[8]$.
\begin{itemize}
\item[(1)] The selected $i_{7}$-string is in $\nu^{(1)}$.
Find the selected $i_{7}$-string which is also marked by [4].
If such a string exists, then mark [8] on the left of [7].
If the selected string is type-I and the rightmost box of the selected string is marked by [3], then prohibit the further box marking of this string (see Example~\ref{ex:BM-8a}).
We say that this string is \emph{inactivated}.
If the selected $i_{7}$-string is not marked by [4], then follow the box marking rule \textbf{(BM-3)} with replacing $i_{1}$ by $i_{6}$ and $i_{2}$ by $i_{7}$ except that [9] cannot be marked in $\nu^{(2)}$.
If the search is not successful, then delete the marked boxes and return $\framebox{$11$}$.
\begin{ex} \label{ex:BM-8a}
$L=5$ and $\lambda=2\Bar{\Lambda}_{1}$.
\setlength{\unitlength}{10pt}
\begin{flushleft}
\begin{picture}(31,6)
\put(1,3){\line(1,0){3}}
\put(1,4){\line(1,0){8}}
\put(1,5){\line(1,0){8}}
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\put(2,3){\line(0,1){2}}
\put(3,3){\line(0,1){2}}
\put(4,3){\line(0,1){2}}
\put(5,4){\line(0,1){1}}
\put(6,4){\line(0,1){1}}
\put(7,4){\line(0,1){1}}
\put(8,4){\line(0,1){1}}
\put(9,4){\line(0,1){1}}
\put(3,5){\makebox(2,1){$\downarrow$}}
\put(6,5){\makebox(2,1){$\downarrow$}}
\put(0,3){\makebox(1,1){{\scriptsize $3$}}}
\put(0,4){\makebox(1,1){{\scriptsize $1$}}}
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\put(9,4){\makebox(1,1){{\scriptsize $1$}}}
\put(3,3){\makebox(1,1){{\scriptsize $[2]$}}}
\put(5,4){\makebox(1,1){{\scriptsize $[8]$}}}
\put(6,4){\makebox(1,1){{\scriptsize $[7]$}}}
\put(7,4){\makebox(1,1){{\scriptsize $[4]$}}}
\put(8,4){\makebox(1,1){{\scriptsize $[3]$}}}
\put(12,0){\line(1,0){1}}
\put(12,1){\line(1,0){1}}
\put(12,2){\line(1,0){1}}
\put(12,3){\line(1,0){2}}
\put(12,4){\line(1,0){3}}
\put(12,5){\line(1,0){3}}
\put(12,0){\line(0,1){5}}
\put(13,0){\line(0,1){5}}
\put(14,3){\line(0,1){2}}
\put(15,4){\line(0,1){1}}
\put(11,2){\makebox(1,1){{\scriptsize $1$}}}
\put(11,3){\makebox(1,1){{\scriptsize $0$}}}
\put(11,4){\makebox(1,1){{\scriptsize $0$}}}
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\put(13,1){\makebox(1,1){{\scriptsize $0$}}}
\put(13,2){\makebox(1,1){{\scriptsize $0$}}}
\put(14,3){\makebox(1,1){{\scriptsize $0$}}}
\put(15,4){\makebox(1,1){{\scriptsize $0$}}}
\put(12,0){\makebox(1,1){{\scriptsize $[1]$}}}
\put(13,4){\makebox(1,1){{\scriptsize $[6]$}}}
\put(14,4){\makebox(1,1){{\scriptsize $[5]$}}}
\put(17,4){\makebox(2,1){$\longrightarrow$}}
\put(17,3){\makebox(2,1){$12$}}
\put(21,3){\line(1,0){2}}
\put(21,4){\line(1,0){4}}
\put(21,5){\line(1,0){4}}
\put(21,3){\line(0,1){2}}
\put(22,3){\line(0,1){2}}
\put(23,3){\line(0,1){2}}
\put(24,4){\line(0,1){1}}
\put(25,4){\line(0,1){1}}
\put(23,5){\makebox(2,1){$\downarrow$}}
\put(20,3){\makebox(1,1){{\scriptsize $0$}}}
\put(20,4){\makebox(1,1){{\scriptsize $1$}}}
\put(23,3){\makebox(1,1){{\scriptsize $0$}}}
\put(25,4){\makebox(1,1){{\scriptsize $1$}}}
\put(21,3){\makebox(1,1){{\scriptsize $[3]$}}}
\put(22,3){\makebox(1,1){{\scriptsize $[2]$}}}
\put(23,4){\makebox(1,1){{\scriptsize $[6]$}}}
\put(24,4){\makebox(1,1){{\scriptsize $[5]$}}}
\put(28,1){\line(1,0){1}}
\put(28,2){\line(1,0){1}}
\put(28,3){\line(1,0){1}}
\put(28,4){\line(1,0){2}}
\put(28,5){\line(1,0){2}}
\put(28,1){\line(0,1){4}}
\put(29,1){\line(0,1){4}}
\put(30,4){\line(0,1){1}}
\put(27,3){\makebox(1,1){{\scriptsize $1$}}}
\put(27,4){\makebox(1,1){{\scriptsize $0$}}}
\put(29,1){\makebox(1,1){{\scriptsize $1$}}}
\put(29,2){\makebox(1,1){{\scriptsize $0$}}}
\put(29,3){\makebox(1,1){{\scriptsize $0$}}}
\put(30,4){\makebox(1,1){{\scriptsize $0$}}}
\put(28,1){\makebox(1,1){{\scriptsize $[1]$}}}
\put(28,4){\makebox(1,1){{\scriptsize $[7]$}}}
\put(29,4){\makebox(1,1){{\scriptsize $[4]$}}}
\end{picture}
\end{flushleft}
which corresponds to the path
$\framebox{$12$}\otimes \framebox{$11$}\otimes \framebox{$1$}\otimes \framebox{$5$}\otimes \framebox{$1$}$.
The charge of the first RC and the energy of this path coincide, which is computed to be $-13$.
If [9] were marked on the left of [8], then the first return would be $13$.
However, $\lambda-\mathrm{wt}(\framebox{$13$})=2\Bar{\Lambda}_{1}-(-3\Bar{\Lambda}_{1}+\Bar{\Lambda}_{2})=5\Bar{\Lambda_{1}}-\Bar{\Lambda_{2}}$, which is not dominant.
\end{ex}
\item[(2)] The selected $i_{7}$-string is in $\nu^{(2)}$.
Find the singular/q-singular string of effective length $i_{7}$ in $\nu^{(1)}$, which is possibly marked by [6], according to the following preferential rule.
\begin{center}
\begin{tabular}{c|cc}
& type-0 & type-I \\ \hline
singular & 1 & 2 \\
q-singular & 3 &
\end{tabular}
\end{center}
If such a string exists, then mark the rightmost unmarked box of the founded string by $[8]$.
If not, then find the singular or q-singular string of minimum length whose effective length $(\geq i_{7}+1)$.
If the search is successful, then mark the rightmost box of the unmarked box of the founded string by $[8]$.
If not, then delete the marked boxes and return $\framebox{$11$}$.
If the selected $i_{8}$-string is type-I and the rightmost box of this string is marked by [5] or [6], then prohibit the further box marking of this string (see Examples~\ref{ex:BM-8b} and \ref{ex:BM-8c}).
We also say that this string is inactivated.
\begin{ex} \label{ex:BM-8b}
$L=5$ and $\lambda=2\Bar{\Lambda}_{1}$ as in Example~\ref{ex:BM-8a}.
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\put(9,4){\line(0,1){1}}
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\put(12,1){\line(1,0){1}}
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\put(26,4){\line(0,1){1}}
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\put(29,1){\line(1,0){1}}
\put(29,2){\line(1,0){1}}
\put(29,3){\line(1,0){1}}
\put(29,4){\line(1,0){2}}
\put(29,5){\line(1,0){2}}
\put(29,1){\line(0,1){4}}
\put(30,1){\line(0,1){4}}
\put(31,4){\line(0,1){1}}
\put(28,3){\makebox(1,1){{\scriptsize $0$}}}
\put(28,4){\makebox(1,1){{\scriptsize $0$}}}
\put(30,1){\makebox(1,1){{\scriptsize $0$}}}
\put(30,2){\makebox(1,1){{\scriptsize $0$}}}
\put(30,3){\makebox(1,1){{\scriptsize $0$}}}
\put(31,4){\makebox(1,1){{\scriptsize $0$}}}
\put(29,1){\makebox(1,1){{\scriptsize $[1]$}}}
\put(29,4){\makebox(1,1){{\scriptsize $[6]$}}}
\put(30,4){\makebox(1,1){{\scriptsize $[5]$}}}
\end{picture}
\end{flushleft}
which corresponds to the path
$\framebox{$12$}\otimes \framebox{$12$}\otimes \framebox{$3$}\otimes \framebox{$2$}\otimes \framebox{$1$}$.
The charge of the first RC and the energy of this path coincide, which is computed to be $-12$.
By the same reasoning as in Example~\ref{ex:BM-8a}, [9] cannot be marked on the left of [8] in the first RC.
\end{ex}
\begin{ex} \label{ex:BM-8c}
$L=5$ and $\lambda=2\Bar{\Lambda}_{1}$ as in Example~\ref{ex:BM-8a}.
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\begin{flushleft}
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\put(9,4){\line(0,1){1}}
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\put(8,4){\makebox(1,1){{\scriptsize $[6]$}}}
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\put(12,1){\line(1,0){1}}
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\put(12,5){\line(1,0){3}}
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\put(12,1){\makebox(1,1){{\scriptsize $[4]$}}}
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\put(17,3){\makebox(2,1){$12$}}
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\put(23,5){\makebox(2,1){$\downarrow$}}
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\put(25,4){\makebox(1,1){{\scriptsize $[3]$}}}
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\put(29,2){\line(1,0){1}}
\put(29,3){\line(1,0){2}}
\put(29,4){\line(1,0){2}}
\put(29,5){\line(1,0){2}}
\put(29,2){\line(0,1){3}}
\put(30,2){\line(0,1){3}}
\put(31,3){\line(0,1){2}}
\put(28,2){\makebox(1,1){{\scriptsize $1$}}}
\put(28,4){\makebox(1,1){{\scriptsize $0$}}}
\put(30,2){\makebox(1,1){{\scriptsize $0$}}}
\put(31,3){\makebox(1,1){{\scriptsize $0$}}}
\put(31,4){\makebox(1,1){{\scriptsize $0$}}}
\put(30,3){\makebox(1,1){{\scriptsize $[1]$}}}
\put(30,4){\makebox(1,1){{\scriptsize $[4]$}}}
\end{picture}
\end{flushleft}
which corresponds to the path
$\framebox{$12$}\otimes \framebox{$7$}\otimes \framebox{$\emptyset$}\otimes \framebox{$2$}\otimes \framebox{$1$}$.
The charge of the first RC and the energy of this path coincide, which is computed to be $-11$.
By the same reasoning as in Example~\ref{ex:BM-8a}, [9] cannot be marked on the left of [8] in the first RC.
\end{ex}
\end{itemize}
\item[(BM-9)] Marking $[9]$.
Find the active singular string that is not inactivated of length $(\geq i_{8})$ in $\nu^{(1)}$.
If the founded string whose length is denoted by $l_{9}^{(1)}$ is type-I and the rightmost box of the founded string is marked by $[2]$, then discard this selection and find the string of length $(> l_{9}^{(1)})$.
If the founded string whose length is denoted by $l_{9}^{(1)}$ is type-II and the rightmost box of the founded string is marked by $[3]$, then discard this selection and find the string of length $(> l_{9}^{(1)})$.
The last two rules prohibit the box marking in a single string across two delimiters.
If the string which can be marked by $[9]$ is founded, then mark the rightmost box of the unmarked box of the founded string by $[9]$.
If not, then delete the marked boxes and return $\framebox{$12$}$.
If the selected $i_{9}$-string is type-0 and the selected $i_{8}$-string is type-I with the same effective length, then discard the $i_{8}$-string selection and do the following box marking.
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\begin{center}
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\put(0,0){\line(1,0){3}}
\put(0,1){\line(1,0){4}}
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\put(2,0){\line(0,1){2}}
\put(3,0){\line(0,1){2}}
\put(4,1){\line(0,1){1}}
\put(0,2){\makebox(2,1){$\downarrow$}}
\put(2,1){\makebox(1,1){{\scriptsize $[9]$}}}
\put(3,1){\makebox(1,1){{\scriptsize $[8]$}}}
\end{picture}
\end{center}
This rule guarantees the bijection when [10] is marked in a singular string with $i_{9}^{eff}=i_{10}$ in $\nu^{(2)}$.
\item[(BM-10)] Marking $[10]$.
Find the singular string of minimum length $(\geq i_{9}^{eff})$ in $\nu^{(2)}$.
If such a string exists, then mark the rightmost unmarked box of the founded string by $[10]$ and delete the marked boxes returning $\framebox{$14$}$.
If not, then delete the marked boxes and return $\framebox{$13$}$.
\end{description}
There are several exceptional rules for box marking.
\begin{ex} \label{ex:boomerang}
$L=4$ and $\lambda=2\Bar{\Lambda}_{2}$.
According to the box marking rules prescribed above, we have
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\put(2,3){\line(0,1){2}}
\put(3,3){\line(0,1){2}}
\put(4,3){\line(0,1){2}}
\put(5,4){\line(0,1){1}}
\put(6,4){\line(0,1){1}}
\put(7,4){\line(0,1){1}}
\put(3,5){\makebox(2,1){$\downarrow$}}
\put(0,3){\makebox(1,1){{\scriptsize $3$}}}
\put(0,4){\makebox(1,1){{\scriptsize $0$}}}
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\put(7,4){\makebox(1,1){{\scriptsize $0$}}}
\put(3,3){\makebox(1,1){{\scriptsize $[2]$}}}
\put(4,4){\makebox(1,1){{\scriptsize $[7]$}}}
\put(5,4){\makebox(1,1){{\scriptsize $[4]$}}}
\put(6,4){\makebox(1,1){{\scriptsize $[3]$}}}
\put(10,0){\line(1,0){1}}
\put(10,1){\line(1,0){1}}
\put(10,2){\line(1,0){1}}
\put(10,3){\line(1,0){1}}
\put(10,4){\line(1,0){2}}
\put(10,5){\line(1,0){2}}
\put(10,0){\line(0,1){5}}
\put(11,0){\line(0,1){5}}
\put(12,4){\line(0,1){1}}
\put(9,3){\makebox(1,1){{\scriptsize $0$}}}
\put(9,4){\makebox(1,1){{\scriptsize $1$}}}
\put(11,0){\makebox(1,1){{\scriptsize $0$}}}
\put(11,1){\makebox(1,1){{\scriptsize $0$}}}
\put(11,2){\makebox(1,1){{\scriptsize $0$}}}
\put(11,3){\makebox(1,1){{\scriptsize $0$}}}
\put(12,4){\makebox(1,1){{\scriptsize $0$}}}
\put(10,0){\makebox(1,1){{\scriptsize $[1]$}}}
\put(10,4){\makebox(1,1){{\scriptsize $[6]$}}}
\put(11,4){\makebox(1,1){{\scriptsize $[5]$}}}
\put(14,4){\makebox(2,1){$\longrightarrow$}}
\put(14,3){\makebox(2,1){$11$}}
\end{picture}
\end{center}
However, $\lambda-\mathrm{wt}(\framebox{$11$})=2\Bar{\Lambda}_{2}-(\Bar{\Lambda}_{1}-\Bar{\Lambda}_{2})=-\Bar{\Lambda}_{1}+3\Bar{\Lambda}_{2}$, which is not dominant.
For the above RC, we must do the following box marking.
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\begin{picture}(16,7)
\put(1,3){\line(1,0){3}}
\put(1,4){\line(1,0){6}}
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\put(3,3){\line(0,1){2}}
\put(4,3){\line(0,1){2}}
\put(5,4){\line(0,1){1}}
\put(6,4){\line(0,1){1}}
\put(7,4){\line(0,1){1}}
\put(3,5){\makebox(2,1){$\downarrow$}}
\put(0,3){\makebox(1,1){{\scriptsize $3$}}}
\put(0,4){\makebox(1,1){{\scriptsize $0$}}}
\put(4,3){\makebox(1,1){{\scriptsize $1$}}}
\put(7,4){\makebox(1,1){{\scriptsize $0$}}}
\put(1,4){\makebox(1,1){{\scriptsize $[9]$}}}
\put(2,4){\makebox(1,1){{\scriptsize $[8]$}}}
\put(3,4){\makebox(1,1){{\scriptsize $[7]$}}}
\put(4,4){\makebox(1,1){{\scriptsize $[4]$}}}
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\put(6,4){\makebox(1,1){{\scriptsize $[2]$}}}
\put(10,0){\line(1,0){1}}
\put(10,1){\line(1,0){1}}
\put(10,2){\line(1,0){1}}
\put(10,3){\line(1,0){1}}
\put(10,4){\line(1,0){2}}
\put(10,5){\line(1,0){2}}
\put(10,0){\line(0,1){5}}
\put(11,0){\line(0,1){5}}
\put(12,4){\line(0,1){1}}
\put(9,3){\makebox(1,1){{\scriptsize $0$}}}
\put(9,4){\makebox(1,1){{\scriptsize $1$}}}
\put(11,0){\makebox(1,1){{\scriptsize $0$}}}
\put(11,1){\makebox(1,1){{\scriptsize $0$}}}
\put(11,2){\makebox(1,1){{\scriptsize $0$}}}
\put(11,3){\makebox(1,1){{\scriptsize $0$}}}
\put(12,4){\makebox(1,1){{\scriptsize $0$}}}
\put(10,0){\makebox(1,1){{\scriptsize $[1]$}}}
\put(10,4){\makebox(1,1){{\scriptsize $[6]$}}}
\put(11,4){\makebox(1,1){{\scriptsize $[5]$}}}
\put(14,4){\makebox(2,1){$\longrightarrow$}}
\put(14,3){\makebox(2,1){$13$}}
\end{picture}
\end{center}
\end{ex}
The first string in $\nu^{(1)}$ is called a \emph{boomerang string}.
Here we summarize the rules of finding boomerang strings, which have a common feature; the box marking restarts in the boomerang string leading to the larger box marking.
\begin{description}
\item[(BS-1)] The selected $i_{2}$-string is type-II q-singular.
The type-0/I singular string of effective length $i_{2}^{eff}$ under the condition that $\nu^{(2)}$ has a singular string of length $i_{2}^{eff}$.
\item[(BS-2)] The selected $i_{2}$-string is type-0/I q-singular or type-I qq-singular.
The following singular strings are boomerang strings for the $i_{2}$-string.
If there exist boomerang strings more than two, then select the string of minimum length.
\begin{itemize}
\item[(1)]
The type-II singular string of effective length $i_{2}^{eff}+1$ under the condition that $\nu^{(2)}$ has a singular string of length $i_{2}^{eff}$ or $i_{2}^{eff}+1$.
\item[(2)]
The type-0/I singular string of effective length $i_{2}^{eff}+1$ under the condition that $\nu^{(2)}$ has a singular string of length $i_{2}^{eff}+1$.
\end{itemize}
\item[(BS-3)] The selected $i_{2}$-string is type-0 qq-singular.
The type-0/I singular string of effective length $i_{2}^{eff}+1$ is a boomerang string if $\nu^{(2)}$ has a singular string of length $i_{2}^{eff}+1$.
See Example~\ref{ex:boomerang}.
Note that we always select the type-II singular string of effective length $i_{2}^{eff}+1$ if it exists.
See Example~\ref{ex:BM-3}.
\item[(BS-4)] The selected $i_{3}$-string with $i_{2}<i_{3}$ is type-II q-singular.
The type-0/I singular string of effective length $i_{3}^{eff}$ is a boomerang string for the $i_{3}$-string if $\nu^{(2)}$ has a singular string of length $i_{3}^{eff}$.
\item[(BS-5)] The selected $i_{3}$-string with $i_{2}<i_{3}$ is type-0/I q-singular.
The singular string of any type and of effective length $i_{3}^{eff}+1$ is a boomerang string for the $i_{3}$-string if $\nu^{(2)}$ has a singular string of length $i_{3}^{eff}+1$.
\item[(BS-6)] The selected $i_{5}$-string in $\nu^{(1)}$ with $i_{3}<i_{5}$ is q-singular and type-I/II.
The singular string of length ($>i_{5}$) and of the effective length $i_{5}^{eff}$ is a boomerang string for the $i_{5}$-string if $\nu^{(2)}$ has a singular string of length $i_{5}^{eff}$.
\end{description}
Here, we summarize the rule of adjusting the values of riggings in $\Tilde{\nu}^{(a)}$ $(a=1,2)$.
The fundamental rule is that we set the box-deleted string in $\Tilde{\nu}^{(a)}$ $(a=1,2)$ to be singular and that we keep the riggings of the box-undeleted strings the same.
\begin{description}
\item[(RA-1)] When $[6]$ is marked in $\nu^{(1)}$ or $\nu^{(2)}$ with $i_{5}< i_{6}$, we set the box-deleted $(i_{6}-n)$-string to be q-singular where $n\geq 1$.
See Example~\ref{ex:BM-8c}.
\item[(RA-2)] When $[3]$ is marked in $\nu^{(1)}$ with $i_{2}< i_{3}$ and if [4] is not marked in $\nu^{(1)}$ and the $i_{3}$-string is not marked by [5], we do as follows according to the type of the selected $i_{3}$-string.
\begin{itemize}
\item[(1)] type-II.
We set the box-deleted $(i_{3}-1)$-string to be q-singular unconditionally.
\item[(2)] type-0/I.
If [4] is marked in $\nu^{(2)}$ with $i_{3}^{eff}=i_{4}$, we set the box-deleted $(i_{3}-1)$-string to be singular.
Otherwise, we set the box-deleted $(i_{3}-1)$-string to be q-singular.
\end{itemize}
\item[(RA-3)] When $[4]$ is marked in $\nu^{(1)}$ with $i_{3}< i_{4}$ and the selected $i_{4}$-string is marked by $[4]$ only, we do as follows according to the type of the selected $i_{4}$-string.
\begin{itemize}
\item[(1)] type-0.
If $[5]$ is marked in a q-singular string in $\nu^{(2)}$ with $i_{4}^{eff}=i_{5}$ or $[5]$ and $[6]$ are marked in the same singular string in $\nu^{(2)}$ with $i_{4}^{eff}=i_{6}-1$, then we set the box-deleted $(i_{4}-1)$-string to be singular.
Otherwise, we set the box-deleted $(i_{4}-1)$-string to be qq-singular.
\item[(2)] type-I.
If $[5]$ is not marked in $\nu^{(2)}$, then we set the box-deleted $(i_{4}-1)$-string to be qq-singular.
If $[5]$ is marked in a q-singular string in $\nu^{(2)}$ with $i_{4}^{eff}=i_{5}$ or [5] and [6] are marked in the same singular string in $\nu^{(2)}$ with $i_{4}^{eff}=i_{6}-1$, then we set the box-deleted $(i_{4}-1)$-string to be q-singular.
Otherwise, we set the box-deleted $(i_{4}-1)$-string to be qq-singular.
\item[(3)] type-II.
We set the box-deleted $(i_{4}-1)$-string to be qq-singular unconditionally.
\end{itemize}
In either case, if the selected $i_{3}$-string is marked by [3] only, we set the box-deleted $(i_{3}-1)$-string to be q-singular.
\item[(RA-4)] When [4] is marked in $\nu^{(1)}$ with $i_{2}< i_{3}=i_{4}$ and the selected $i_{4}$-string is not marked by [7], we do as follows according to the type of the selected $i_{4}$-string.
\begin{itemize}
\item[(1)] type-0.
If [5] is marked in a q-singular string in $\nu^{(2)}$ with $i_{4}^{eff}=i_{5}$ or [5] and [6] are marked in the same singular string in $\nu^{(2)}$ with $i_{4}^{eff}=i_{6}-1$, then we set the box-deleted $(i_{4}-2)$-string to be singular.
Otherwise, we set the box-deleted $(i_{4}-2)$-string to be q-singular.
\item[(2)] type-I/II.
We set the box-deleted $(i_{4}-2)$-string to be q-singular unconditionally.
\end{itemize}
\item[(RA-5)] When [8] is marked in $\nu^{(1)}$ but [9] is not and the selected $i_{8}$-string is marked by [8] only, we set the box-deleted $(i_{8}-1)$-string in $\Tilde{\nu}^{(1)}$ to be q-singular whether [7] is marked in $\nu^{(1)}$ or $\nu^{(2)}$.
\item[(RA-6)] When [9] is marked in $\nu^{(1)}$ and the rightmost box of the selected $i_{9}$-string is marked by [8], i.e., $i_{8}=i_{9}$, we do as follows according to the type of the selected $i_{9}$-string whether [7] is marked in $\nu^{(1)}$ or $\nu^{(2)}$.
\begin{itemize}
\item[(1)] type-0.
If [10] is marked in $\nu^{(2)}$ with $i_{9}^{eff}=i_{10}$, then we set the box-deleted $(i_{9}-2)$-string to be singular.
Otherwise, we set this string to be q-singular.
\item[(2)] type-I/II.
We set the box-deleted $(i_{9}-2)$-string to be q-singular unconditionally.
\end{itemize}
\item[(RA-7)] When [9] are marked in $\nu^{(1)}$ with $i_{8}< i_{9}$, we do as follows according to the type of the $i_{9}$-string whether [7] is marked in $\nu^{(1)}$ or $\nu^{(2)}$.
\begin{itemize}
\item[(1)] type-0.
If [10] is marked in $\nu^{(2)}$ with $i_{9}^{eff}=i_{10}$, then we set the box-deleted $(i_{9}-1)$-string to be singular.
Otherwise, we set this string to be qq-singular.
\item[(2)] type-I.
If [10] is marked in $\nu^{(2)}$ with $i_{9}^{eff}=i_{10}$, then we set the box-deleted $(i_{9}-1)$-string to be q-singular.
Otherwise, we set this string to be qq-singular.
\item[(3)] type-II.
We set the box-deleted $(i_{9}-1)$-string to be qq-singular unconditionally.
\end{itemize}
In either case, if the selected $i_{8}$-string is marked by [8] only, we set the box-deleted $(i_{8}-1)$-string to be q-singular.
\end{description}
Let $A$ be a statement, then $\chi (A)=1$ if $A$ is true and $\chi (A)=0$ if $A$ is false.
The changes of vacancy numbers are defined by $\Delta p_{i}^{(a)}:=\Tilde{p}_{i}^{(a)}-p_{i}^{(a)}$.
Here we compute the changes of vacancy numbers according to the return $b$ of $\delta_{\theta}$.
\begin{description}
\item[(VC-1)] $b= \framebox{$1$}$.
From Eqs.~\eqref{eq:vacancy1} and \eqref{eq:vacancy2}, we have $\Delta p_{i}^{(1)}=0$ $(i\geq 1)$ and $\Delta p_{i}^{(2)}=-1$ $(i\geq 1)$.
\item[(VC-2)] $b= \framebox{$2$}$.
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ are obtained by overwriting
\begin{align*}
&\min (i,3i_{1})(m_{i_{1}}^{(2)}-1)+\min (i,3(i_{1}-1))(m_{i_{1}-1}^{(2)}+1) \\
&-\min (i,3i_{1})m_{i_{1}}^{(2)}-\min (i,3(i_{1}-1))m_{i_{1}-1}^{(2)} \\
&=-\min (i,3i_{1})+\min (i,3(i_{1}-1)) \\
&=-3\chi (i\geq 3i_{1})-2\chi (i=3i_{1}-1)-\chi (i=3i_{1}-2)
\end{align*}
on $\Delta p_{i}^{(1)}$ in \textbf{(VC-1)} and $\Delta p_{i}^{(2)}$ are obtained by overwriting
\begin{align*}
-&2\min (i,i_{1})(m_{i_{1}}^{(2)}-1)-2\min (i,i_{1}-1)(m_{i_{1}-1}^{(2)}+1) \\
&+2\min (i,i_{1})m_{i_{1}}^{(2)}+2\min (i,i_{1}-1)m_{i_{1}-1}^{(2)} \\
&=2\min (i,i_{1})-2\min (i,i_{1}-1)=2\chi (i\geq i_{1})
\end{align*}
on $\Delta p_{i}^{(2)}$ in \textbf{(VC-1)}.
The changed $\Delta p_{i}^{(1)}$ and $\Delta p_{i}^{(2)}$ are listed below.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$-3$ & $(i\geq 3i_{1})$ & 1 & $(i\geq i_{1})$ \\
$-2$ & $(i=3i_{1}-1)$ & & \\
$-1$ & $(i=3i_{1}-2)$ & &
\end{tabular}
\end{center}
\item[(VC-3)] $b= \framebox{$3$}$.
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ are obtained by overwriting $2\chi (i\geq i_{2})$ on
$\Delta p_{i}^{(1)}$ in \textbf{(VC-2)} and $\Delta p_{i}^{(2)}$ are obtained by overwriting $-\chi (3i\geq i_{2})$ on $\Delta p_{i}^{(2)}$ in \textbf{(VC-2)}.
\begin{itemize}
\item[(1)] $i_{2}\leq 3i_{1}-1$.
\begin{center}
\begin{tabular}{cccl|cl}
$\Delta p_{i}^{(1)}$ & $i_{2}$:I & $i_{2}$:II & & $\Delta p_{i}^{(2)}$ & \\ \hline
& $-1$ &$-1$ & $(i\geq 3i_{1})$ & 0 & $(i\geq i_{1})$ \\
& $0$ &$0$ & $(i=3i_{1}-1)$ & & \\
& & $1$ & $(i=3i_{1}-2)$ & &
\end{tabular}
\end{center}
where $i_{2}$:I (resp. $i_{2}$:II) stands for that the $i_{2}$-string is type-I (resp. type-II).
\item[(2)] $i_{2}\geq3i_{1}$.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$-1$ & $(i\geq i_{2})$ & 0 & $(i\geq i_{2}^{eff})$
\end{tabular}
\end{center}
\end{itemize}
\item[(VC-4)] $b= \framebox{$4$}$.
\begin{itemize}
\item[(1)] $i_{2}=i_{3}$.
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ are obtained by overwriting $4\chi (i\geq i_{3})+2\chi (i=i_{3}-1)$ on $\Delta p_{i}^{(1)}$ in \textbf{(VC-2)} and $\Delta p_{i}^{(2)}$ are obtained by overwriting $-2\chi (3i\geq i_{3})-\chi (3i=i_{3}-1)$ on $\Delta p_{i}^{(2)}$ in \textbf{(VC-2)}.
\begin{itemize}
\item[(a)] $i_{3}^{eff}=i_{1}$.
\begin{center}
\begin{tabular}{cccl|cl}
$\Delta p_{i}^{(1)}$ & $i_{3}$:0 & $i_{3}$:I & & $\Delta p_{i}^{(2)}$ & \\ \hline
& $1$ & $1$ & $(i\geq 3i_{1})$ & $-1$ & $(i\geq i_{1})$ \\
& $0$ & $2$ & $(i=3i_{1}-1)$ & & \\
& $-1$ & $1$ & $(i=3i_{1}-2)$ & &
\end{tabular}
\end{center}
\item[(b)] $i_{3}^{eff}>i_{1}$.
\begin{center}
\begin{tabular}{cl|cccl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & $i_{3}$:0/I & $i_{3}$:II \\ \hline
$1$ & $(i\geq i_{3})$ & & $-1$ & $-1$ & $(i\geq i_{3}^{eff})$ \\
$-1$ & $(i=i_{3}-1)$ & & $1$ & $0$ & $(i=i_{3}^{eff}-1)$
\end{tabular}
\end{center}
where $i_{3}$:0/I (resp. $i_{3}$:II) stands for that the $i_{3}$-string is type-0/I (resp. type-II).
\end{itemize}
\item[(2)] $i_{2}<i_{3}$.
In this case, $i_{3}\geq 3i_{1}$.
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ are obtained by overwriting $2\chi (i\geq i_{3})$ on
$\Delta p_{i}^{(1)}$ in \textbf{(VC-3)} and $\Delta p_{i}^{(2)}$ are obtained by overwriting $-\chi (3i\geq i_{3})$ on $\Delta p_{i}^{(2)}$ in \textbf{(VC-3)}.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$1$ & $(i\geq i_{3})$ & $-1$ & $(i\geq i_{3}^{eff})$
\end{tabular}
\end{center}
\end{itemize}
\item[(VC-5)] $b= \framebox{$5$}$.
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ are obtained by overwriting
\[
-3\chi (i\geq 3i_{4})-2\chi (i=3i_{4}-1)-\chi (i=3i_{4}-2)
\]
on $\Delta p_{i}^{(1)}$ in \textbf{(VC-4)} and $\Delta p_{i}^{(2)}$ are obtained by overwriting $2\chi (i\geq i_{4})$ on $\Delta p_{i}^{(2)}$ in \textbf{(VC-4)}.
\begin{itemize}
\item[(1)] $i_{2}=i_{3}$.
\begin{itemize}
\item[(a)] $i_{3}=3i_{4}-1$.
\begin{center}
\begin{tabular}{cccl|cl}
$\Delta p_{i}^{(1)}$ & $i_{1}=i_{4}$ & $i_{1}<i_{4}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
& $-2$ & $-2$ & $(i\geq 3i_{4})$ & $1$ & $(i\geq i_{1})$ \\
& $0$ & $-1$ & $(i=3i_{4}-1)$ & & \\
& $0$ & $-2$ & $(i=3i_{4}-2)$ & &
\end{tabular}
\end{center}
Note that ``$i\geq i_{1}$'' in $\Delta p_{i}^{(2)}$ is not a typo.
\item[(b)] otherwise.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$-2$ & $(i\geq 3i_{4})$ & $1$ & $(i\geq i_{4})$ \\
$-1$ & $(i=3i_{4}-1)$ & & \\
$0$ & $(i=3i_{4}-2)$ & &
\end{tabular}
\end{center}
\end{itemize}
Note that $i_{3}^{eff}<i_{4}$ when the $i_{3}$-string is type-0.
\item[(2)] $i_{2}<i_{3}$.
\begin{itemize}
\item[(a)] $i_{3}^{eff}<i_{4}$.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$-2$ & $(i\geq 3i_{4})$ & $1$ & $(i\geq i_{4})$ \\
$-1$ & $(i=3i_{4}-1)$ & & \\
$0$ & $(i=3i_{4}-2)$ & &
\end{tabular}
\end{center}
\item[(b)] $i_{3}=3i_{4}-1$.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$-2$ & $(i\geq 3i_{4})$ & $1$ & $(i\geq i_{4})$ \\
$-1$ & $(i=3i_{4}-1)$ & & \\
$-2$ & $(i=3i_{4}-2)$ & &
\end{tabular}
\end{center}
\item[(c)] $i_{3}=3i_{4}-2$.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$-2$ & $(i\geq 3i_{4})$ & $1$ & $(i\geq i_{4})$ \\
$-1$ & $(i=3i_{4}-1)$ & & \\
$0$ & $(i=3i_{4}-2)$ & &
\end{tabular}
\end{center}
\end{itemize}
Note that the case when $i_{3}=3i_{4}$ cannot occur.
\end{itemize}
\item[(VC-6)] $b= \framebox{$6$}$.
\begin{itemize}
\item[(1)] $i_{2}=i_{3}=i_{4}$.
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ are obtained by overwriting
\[
6\chi (i\geq i_{4})+4\chi (i=i_{4}-1)+2\chi (i=i_{4}-2)
\]
on $\Delta p_{i}^{(1)}$ in \textbf{(VC-2)} and $\Delta p_{i}^{(2)}$ are obtained by overwriting
\[
-3\chi (3i\geq i_{4})-2\chi (3i=i_{4}-1)-\chi (3i=i_{4}-2)
\]
on $\Delta p_{i}^{(2)}$ in \textbf{(VC-2)}.
\begin{itemize}
\item[(a)] The $i_{4}$-string is type-0 with $i_{4}^{eff}=i_{1}$.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$3$ & $(i\geq i_{4})$ & $-2$ & $(i\geq i_{4}^{eff})$ \\
$2$ & $(i=i_{4}-1)$ & & \\
$1$ & $(i=i_{4}-2)$ & &
\end{tabular}
\end{center}
\item[(b)] The $i_{4}$-string is type-0 with $i_{4}^{eff}>i_{1}$.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$3$ & $(i\geq i_{4})$ & $-2$ & $(i\geq i_{4}^{eff})$ \\
$1$ & $(i=i_{4}-1)$ & $1$ & $(i_{1}\leq i\leq i_{4}^{eff}-1)$ \\
$-1$ & $(i=i_{4}-2)$ & &
\end{tabular}
\end{center}
\item[(c)] The $i_{4}$-string is type-I.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$3$ & $(i\geq i_{4})$ & $-2$ & $(i\geq i_{4}^{eff})$ \\
$1$ & $(i=i_{4}-1)$ & $0$ & $(i=i_{4}^{eff}-1)$\\
$-1$ & $(i=i_{4}-2)$ & &
\end{tabular}
\end{center}
\item[(d)] The $i_{4}$-string is type-II.
\begin{center}
\begin{tabular}{cccl}
$\Delta p_{i}^{(1)}$ & $i_{4}^{eff}=i_{1}+1$ & $i_{4}^{eff}>i_{1}+1$ & \\ \hline
& $3$ & $3$ & $(i\geq i_{4})$ \\
& $1$ & $1$ & $(i=i_{4}-1)$ \\
& $0$ & $-1$ & $(i=i_{4}-2)$
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{cl}
$\Delta p_{i}^{(2)}$ & \\ \hline
$-2$ & $(i\geq i_{4}^{eff})$ \\
$-1$ & $(i=i_{4}^{eff}-1)$
\end{tabular}
\end{center}
\end{itemize}
\item[(2)] $i_{2}<i_{3}=i_{4}$.
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ are obtained by overwriting $4\chi (i\geq i_{4})+2\chi (i=i_{4}-1)$ on $\Delta p_{i}^{(1)}$ in \textbf{(VC-3)} and $\Delta p_{i}^{(2)}$ are obtained by overwriting $-2\chi (3i\geq i_{4})-\chi (3i=i_{4}-1)$ on $\Delta p_{i}^{(2)}$ in \textbf{(VC-3)}.
\begin{center}
\begin{tabular}{cl|cccl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & $i_{4}$:0/I & $i_{4}$:II \\ \hline
$3$ & $(i\geq i_{4})$ & & $-2$ & $-2$ & $(i\geq i_{4}^{eff})$ \\
$1$ & $(i=i_{4}-1)$ & & $0$ & $-1$ & $(i=i_{4}^{eff}-1)$
\end{tabular}
\end{center}
\item[(3)] $i_{3}<i_{4}$.
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ are obtained by overwriting $2\chi (i\geq i_{4})$ on
$\Delta p_{i}^{(1)}$ in \textbf{(VC-4)} and $\Delta p_{i}^{(2)}$ are obtained by overwriting $-\chi (3i\geq i_{4})$ on $\Delta p_{i}^{(2)}$ in \textbf{(VC-4)}.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$3$ & $(i\geq i_{4})$ & $-2$ & $(i\geq i_{4}^{eff})$
\end{tabular}
\end{center}
\end{itemize}
\item[(VC-7)] $b= \framebox{$7$}$.
\begin{itemize}
\item[(1)] $i_{2}=i_{3}=i_{5}$.
In this case, $i_{5}=3i_{4}$.
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ are obtained by overwriting
\begin{align*}
&6\chi (i\geq i_{5})+4\chi (i=i_{5}-1)+2\chi (i=i_{5}-2) \\
&-3\chi (i\geq 3i_{4})-2\chi (i=3i_{4}-1)-\chi (i=3i_{4}-2) \\
=& 3\chi (i\geq i_{5})+2\chi (i=i_{5}-1)+\chi (i=i_{5}-2)
\end{align*}
on $\Delta p_{i}^{(1)}$ in \textbf{(VC-2)} and $\Delta p_{i}^{(2)}$ are obtained by overwriting
\begin{align*}
&-3\chi (3i\geq i_{5})-2\chi (3i=i_{5}-1)-\chi (3i=i_{5}-2)+2\chi (i\geq i_{4}) \\
=&-\chi (i\geq i_{4})
\end{align*}
on $\Delta p_{i}^{(2)}$ in \textbf{(VC-2)}.
\begin{center}
\begin{tabular}{cccl|cl}
$\Delta p_{i}^{(1)}$ & $i_{1}=i_{4}$ & $i_{1}<i_{4}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
& $0$ & $0$ & $(i\geq i_{5})$ & $0$ & $(i\geq i_{4})$ \\
& $0$ & $-1$ & $(i=i_{5}-1)$ & & \\
& $0$ & $-2$ & $(i=i_{5}-2)$ & &
\end{tabular}
\end{center}
\item[(2)] $i_{2}<i_{3}=i_{5}$.
In this case $i_{5}=3i_{4}$.
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ are obtained by overwriting
\begin{align*}
&4\chi (i\geq i_{5})+2\chi (i=i_{5}-1) \\
&-3\chi (i\geq 3i_{4})-2\chi (i=3i_{4}-1)-\chi (i=3i_{4}-2) \\
=& \chi (i\geq i_{5})-\chi (i=i_{5}-2)
\end{align*}
on $\Delta p_{i}^{(1)}$ in \textbf{(VC-3)} and $\Delta p_{i}^{(2)}$ are obtained by overwriting
\[
-2\chi (3i\geq i_{5})-\chi (3i=i_{5}-1)+2\chi (i\geq i_{4})=0
\]
on $\Delta p_{i}^{(2)}$ in \textbf{(VC-3)}.
The changes of vacancy numbers coincide with those in case (1).
\item[(3)] $i_{3}<i_{5}$.
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ are obtained by overwriting $2\chi (i\geq i_{5})$ on
$\Delta p_{i}^{(1)}$ in \textbf{(VC-5)} and the changes of vacancy numbers $\Delta p_{i}^{(2)}$ are obtained by overwriting $-\chi (3i\geq i_{5})$ on $\Delta p_{i}^{(2)}$ in \textbf{(VC-5)}.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$0$ & $(i\geq i_{5})$ & $0$ & $(i\geq i_{5}^{eff})$
\end{tabular}
\end{center}
\end{itemize}
\item[(VC-8)] $b= \framebox{$8$}$.
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ are obtained by overwriting
\[
-3\chi (i\geq 3i_{5})-2\chi (i=3i_{5}-1)-\chi (i=3i_{5}-2)
\]
on $\Delta p_{i}^{(1)}$ in \textbf{(VC-6)} and $\Delta p_{i}^{(2)}$ are obtained by overwriting $2\chi (i\geq i_{5})$ on $\Delta p_{i}^{(2)}$ in \textbf{(VC-6)}.
\begin{itemize}
\item[(1)] $i_{4}^{eff}<i_{5}$.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$0$ & $(i\geq 3i_{5})$ & $0$ & $(i\geq i_{5})$ \\
$1$ & $(i=3i_{5}-1)$ & & \\
$2$ & $(i=3i_{5}-2)$ & &
\end{tabular}
\end{center}
\item[(2)] $i_{2}=i_{3}=i_{4}$ and $i_{4}^{eff}=i_{5}$.
In this case, the $i_{4}$-string is type-0/I.
\begin{center}
\begin{tabular}{cccl|cl}
$\Delta p_{i}^{(1)}$ & $i_{1}=i_{5}$ & $i_{1}<i_{5}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
& $0$ & $0$ & $(i\geq 3i_{5})$ & $0$ & $(i\geq i_{5})$ \\
& $0$ & $-1$ & $(i=3i_{5}-1)$ & & \\
& $0$ & $-2$ & $(i=3i_{5}-2)$ & &
\end{tabular}
\end{center}
\item[(3)] $i_{2}<i_{3}=i_{4}$ and $i_{4}^{eff}=i_{5}$.
\begin{center}
\begin{tabular}{ccccl|cl}
$\Delta p_{i}^{(1)}$ & $i_{4}$:0 & $i_{4}$:I & $i_{4}$:II & & $\Delta p_{i}^{(2)}$ & \\ \hline
& $0$ & $0$ & $0$ & $(i\geq 3i_{5})$ & $0$ & $(i\geq i_{5})$ \\
& $-1$ & $1$ & $1$ & $(i=3i_{5}-1)$ & & \\
& $-2$ & $0$ & $2$ & $(i=3i_{5}-2)$ & &
\end{tabular}
\end{center}
\item[(4)] $i_{3}<i_{4}$ and $i_{4}^{eff}=i_{5}$.
\begin{center}
\begin{tabular}{ccccl|cl}
$\Delta p_{i}^{(1)}$ & $i_{4}$:0 & $i_{4}$:I & $i_{4}$:II & & $\Delta p_{i}^{(2)}$ & \\ \hline
& $0$ & $0$ & $0$ & $(i\geq 3i_{5})$ & $0$ & $(i\geq i_{5})$ \\
& $-1$ & $1$ & $1$ & $(i=3i_{5}-1)$ & & \\
& $0$ & $0$ & $2$ & $(i=3i_{5}-2)$ & &
\end{tabular}
\end{center}
Note that $i_{3}\leq 3i_{5}-2$ when $i_{4}=3i_{5}$ (see \textbf{(BM-4)}).
\end{itemize}
\item[(VC-9)] $b= \framebox{$9$}$.
\begin{itemize}
\item[(1)] $i_{2}=i_{3}=i_{5}=i_{6}$.
In this case, the $i_{6}$-string is type-II and $i_{6}^{eff}=i_{4}+1$.
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ are obtained by overwriting
\begin{align*}
&8\chi (i\geq i_{6})+6\chi (i=i_{6}-1)+4\chi (i=i_{6}-2)+2\chi (i=i_{6}-3) \\
&-3\chi (i\geq 3i_{4})-2\chi (i=3i_{4}-1)-\chi (i=3i_{4}-2) \\
=& 5\chi (i\geq i_{6})+3\chi (i=i_{6}-1)+2\chi (i=i_{6}-2)+\chi (i=i_{6}-3)
\end{align*}
on $\Delta p_{i}^{(1)}$ in \textbf{(VC-2)} and $\Delta p_{i}^{(2)}$ are obtained by overwriting
\begin{align*}
&-4\chi (3i\geq i_{6})-3\chi (3i=i_{6}-1)-2\chi (3i=i_{6}-2) \\
&-\chi (3i=i_{6}-3)+2\chi (i\geq i_{4}) =-2\chi (i\geq i_{4}+1)-\chi (i=i_{4})
\end{align*}
on $\Delta p_{i}^{(2)}$ in \textbf{(VC-2)}.
\begin{center}
\begin{tabular}{cccl|cl}
$\Delta p_{i}^{(1)}$ & $i_{1}=i_{4}$ & $i_{1}<i_{4}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
& $2$ & $2$ & $(i\geq i_{6})$ & $-1$ & $(i\geq i_{4}+1)$ \\
& $0$ & $0$ & $(i=i_{6}-1)$ & $0$ & $(i=i_{4})$ \\
& $0$ & $-1$ & $(i=i_{6}-2)$ & & \\
& $0$ & $-2$ & $(i=i_{6}-3)$ & &
\end{tabular}
\end{center}
\item[(2)] $i_{2}<i_{3}=i_{5}=i_{6}$.
In this case, the $i_{6}$-string is type-II and $i_{6}^{eff}=i_{4}+1$.
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ are obtained by overwriting
\begin{align*}
&6\chi (i\geq i_{6})+4\chi (i=i_{6}-1)+2\chi (i=i_{6}-2) \\
&-3\chi (i\geq 3i_{4})-2\chi (i=3i_{4}-1)-\chi (i=3i_{4}-2) \\
=& 3\chi (i\geq i_{6})+\chi (i=i_{6}-1)-\chi (i=i_{6}-3)
\end{align*}
on $\Delta p_{i}^{(1)}$ in \textbf{(VC-3)} and $\Delta p_{i}^{(2)}$ are obtained by overwriting
\begin{align*}
&-3\chi (3i\geq i_{6})-2\chi (3i=i_{6}-1)-\chi (3i=i_{6}-2) \\
&+2\chi (i\geq i_{4}) =-\chi (i\geq i_{4}+1)
\end{align*}
on $\Delta p_{i}^{(2)}$ in \textbf{(VC-3)}.
The changes of vacancy numbers coincide with those in case (1).
\item[(3)] $i_{3}<i_{5}=i_{6}$.
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ are obtained by overwriting $4\chi (i\geq i_{6})+2\chi (i=i_{6}-1)$ on $\Delta p_{i}^{(1)}$ in \textbf{(VC-5)} and $\Delta p_{i}^{(2)}$ are obtained by overwriting $-2\chi (3i\geq i_{6})-\chi (3i=i_{6}-1)$ on $\Delta p_{i}^{(2)}$ in \textbf{(VC-5)}.
\begin{center}
\begin{tabular}{cl|cccl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & $i_{6}$:0/I & $i_{6}$:II & \\ \hline
$2$ & $(i\geq i_{6})$ & & $-1$ & $-1$ & $(i\geq i_{6}^{eff})$ \\
$0$ & $(i=i_{6}-1)$ & & $1$ & $0$ & $(i=i_{6}^{eff}-1)$
\end{tabular}
\end{center}
\item[(4)] $i_{5}<i_{6}$.
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ are obtained by overwriting $2\chi (i\geq i_{6})$ on
$\Delta p_{i}^{(1)}$ in \textbf{(VC-7)} and $\Delta p_{i}^{(2)}$ are obtained by overwriting $-\chi (3i\geq i_{6})$ on $\Delta p_{i}^{(2)}$ in \textbf{(VC-7)}.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$2$ & $(i\geq i_{6})$ & $-1$ & $(i\geq i_{6}^{eff})$
\end{tabular}
\end{center}
\end{itemize}
\item[(VC-10)] $b= \framebox{$10$}$.
\begin{itemize}
\item[(1)] $i_{5}=i_{6}$.
In this case $i_{6}\geq i_{4}^{eff}+1$.
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ are obtained by overwriting
\[
-6\chi (i\geq 3i_{6})-5\chi (i=3i_{6}-1)-\cdots -\chi (i=3i_{6}-5)
\]
on $\Delta p_{i}^{(1)}$ in \textbf{(VC-6)} and $\Delta p_{i}^{(2)}$ are obtained by overwriting $4\chi (i\geq i_{6})+2\chi (i=i_{6}-1)$ on $\Delta p_{i}^{(2)}$ in \textbf{(VC-6)}.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$-3$ & $(i\geq 3i_{6})$ & $2$ & $(i\geq i_{6})$ \\
$-2$ & $(i=3i_{6}-1)$ & $0$ & $(i=i_{6}-1)$ \\
$-1$ & $(i=3i_{6}-2)$ & & \\
$0$ & $(i=3i_{6}-3)$ & &
\end{tabular}
\end{center}
We omit $\Delta p_{i}^{(1)}$ $(3i_{6}-5\leq i\leq 3i_{6}-4)$.
\item[(2)] $i_{5}<i_{6}$.
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ are obtained by overwriting
\[
-3\chi (i\geq 3i_{6})-2\chi (i=3i_{6}-1)-\chi (i=3i_{6}-2)
\]
on $\Delta p_{i}^{(1)}$ in \textbf{(VC-8)} and $\Delta p_{i}^{(2)}$ are obtained by overwriting $2\chi (i\geq i_{6})$ on $\Delta p_{i}^{(2)}$ in \textbf{(VC-8)}.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$-3$ & $(i\geq 3i_{6})$ & $2$ & $(i\geq i_{6})$ \\
$-2$ & $(i=3i_{6}-1)$ & $0$ & $(i=i_{6}-1)$ \\
$-1$ & $(i=3i_{6}-2)$ & &
\end{tabular}
\end{center}
\end{itemize}
\item[(VC-11)] $b= \framebox{$11$}$.
\begin{itemize}
\item[(1)] The $i_{7}$-string is in $\nu^{(1)}$.
In this case [7] is marked in the rightmost box of the $i_{7}$-string.
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ are obtained by overwriting $2\chi (i\geq i_{7})$ on
$\Delta p_{i}^{(1)}$ in \textbf{(VC-10)} and $\Delta p_{i}^{(2)}$ are obtained by overwriting $-\chi (3i\geq i_{7})$ on $\Delta p_{i}^{(2)}$ in \textbf{(VC-10)}.
\begin{itemize}
\item[(a)] $i_{7}^{eff}=i_{6}$.
\begin{center}
\begin{tabular}{ccccl|cl}
$\Delta p_{i}^{(1)}$ & $i_{7}$:0 & $i_{7}$:I & $i_{7}$:II & & $\Delta p_{i}^{(2)}$ & \\ \hline
& $-1$ & $-1$ & $-1$ & $(i\geq 3i_{6})$ & $1$ & $(i\geq i_{6})$ \\
& & $0$ & $0$ & $(i\geq 3i_{6}-1)$ & & \\
& & & $1$ & $(i\geq 3i_{6}-2)$ & &
\end{tabular}
\end{center}
\item[(b)] $i_{7}^{eff}>i_{6}$.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$-1$ & $(i\geq i_{7})$ & $1$ & $(i\geq i_{7}^{eff})$
\end{tabular}
\end{center}
\end{itemize}
\item[(2)] The $i_{7}$-string is in $\nu^{(2)}$ and $i_{4}=i_{7}$.
We have
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$-1$ & $(i\geq 3i_{7})$ & $1$ & $(i\geq i_{7})$ \\
$0$ & $(i=3i_{7}-1)$ & $0$ & $(i=i_{7}-1)$ \\
$1$ & $(i=3i_{7}-2)$ & & \\
$0$ & $(i=3i_{7}-3)$ & &
\end{tabular}
\end{center}
for all possible box marking patterns of $\nu^{(1)}$,
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\begin{center}
\begin{picture}(18,3)
\put(0,0){\line(1,0){5}}
\put(0,1){\line(1,0){5}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
\put(4,0){\line(0,1){1}}
\put(5,0){\line(0,1){1}}
\put(3,1){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[6]$}}}
\put(2,0){\makebox(1,1){{\scriptsize $[5]$}}}
\put(3,0){\makebox(1,1){{\scriptsize $[3]$}}}
\put(4,0){\makebox(1,1){{\scriptsize $[2]$}}}
\put(0,2){\makebox(1,1){{\small (a)}}}
\put(8,0){\line(1,0){4}}
\put(8,1){\line(1,0){4}}
\put(9,0){\line(0,1){1}}
\put(10,0){\line(0,1){1}}
\put(11,0){\line(0,1){1}}
\put(12,0){\line(0,1){1}}
\put(10,1){\makebox(2,1){$\downarrow$}}
\put(9,0){\makebox(1,1){{\scriptsize $[6]$}}}
\put(10,0){\makebox(1,1){{\scriptsize $[5]$}}}
\put(11,0){\makebox(1,1){{\scriptsize $[3]$}}}
\put(8,2){\makebox(1,1){{\small (b)}}}
\put(15,0){\line(1,0){3}}
\put(15,1){\line(1,0){3}}
\put(16,0){\line(0,1){1}}
\put(17,0){\line(0,1){1}}
\put(18,0){\line(0,1){1}}
\put(16,1){\makebox(2,1){$\downarrow$}}
\put(16,0){\makebox(1,1){{\scriptsize $[6]$}}}
\put(17,0){\makebox(1,1){{\scriptsize $[5]$}}}
\put(15,2){\makebox(1,1){{\small (c)}}}
\end{picture}
\end{center}
where the $i_{6}$-string is always type-II.
We omit the unmarked strings.
In case (a), $\Delta p_{i}^{(1)}$ $(3i_{7}-5 \leq i\leq 3i_{7}-4)$ are obtained by overwriting $2\chi (i=3i_{7}-4)+\chi (i=3i_{7}-5)$ on $\Delta p_{i}^{(1)}$ in \textbf{(VC-2)} and $\Delta p_{i}^{(1)}$ $(i\leq 3i_{7}-6)$ and $\Delta p_{i}^{(2)}$ $(i\leq i_{7}-2)$ are the same as in \textbf{(VC-2)}.
In case (b), $\Delta p_{i}^{(1)}$ $(3i_{7}-5 \leq i\leq 3i_{7}-4)$ are obtained by overwriting $-\chi (i=3i_{7}-5)$ on $\Delta p_{i}^{(1)}$ in \textbf{(VC-3)} and $\Delta p_{i}^{(1)}$ $(i\leq 3i_{7}-6)$ and $\Delta p_{i}^{(2)}$ $(i\leq i_{7}-2)$ are the same as in \textbf{(VC-3)}.
In case (c), $\Delta p_{i}^{(1)}$ $(3i_{7}-5 \leq i\leq 3i_{7}-4)$ are obtained by overwriting $-2\chi (i=3i_{7}-4)-\chi (i=3i_{7}-5)$ on $\Delta p_{i}^{(1)}$ in \textbf{(VC-4)} and $\Delta p_{i}^{(1)}$ $(i\leq 3i_{7}-6)$ and $\Delta p_{i}^{(2)}$ $(i\leq i_{7}-2)$ are the same as in \textbf{(VC-4)}.
\item[(3)] The $i_{7}$-string is in $\nu^{(2)}$ and $i_{4}<i_{7}$.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$-1$ & $(i\geq 3i_{7})$ & $1$ & $(i\geq i_{7})$ \\
$0$ & $(i=3i_{7}-1)$ & & \\
$1$ & $(i=3i_{7}-2)$ & &
\end{tabular}
\end{center}
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ $(i\leq 3i_{7}-3)$ and $\Delta p_{i}^{(2)}$ $(i\leq i_{7}-1)$ are the same as in \textbf{(VC-9)}.
\end{itemize}
\item[(VC-12)] $b= \framebox{$12$}$.
\begin{itemize}
\item[(1)] The selected $i_{7}$-string is in $\nu^{(1)}$ and $i_{4}=i_{7}=i_{8}$.
In this case $i_{5}=i_{6}$.
Possible box marking patterns in $\nu^{(1)}$ are following two (see \textbf{(BM-8)}).
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\begin{center}
\begin{picture}(14,3)
\put(0,0){\line(1,0){6}}
\put(0,1){\line(1,0){6}}
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\put(3,0){\line(0,1){1}}
\put(4,0){\line(0,1){1}}
\put(5,0){\line(0,1){1}}
\put(6,0){\line(0,1){1}}
\put(3,1){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[8]$}}}
\put(2,0){\makebox(1,1){{\scriptsize $[7]$}}}
\put(3,0){\makebox(1,1){{\scriptsize $[4]$}}}
\put(4,0){\makebox(1,1){{\scriptsize $[3]$}}}
\put(5,0){\makebox(1,1){{\scriptsize $[2]$}}}
\put(0,2){\makebox(1,1){{\small (a)}}}
\put(9,0){\line(1,0){5}}
\put(9,1){\line(1,0){5}}
\put(10,0){\line(0,1){1}}
\put(11,0){\line(0,1){1}}
\put(12,0){\line(0,1){1}}
\put(13,0){\line(0,1){1}}
\put(14,0){\line(0,1){1}}
\put(11,1){\makebox(2,1){$\downarrow$}}
\put(10,0){\makebox(1,1){{\scriptsize $[8]$}}}
\put(11,0){\makebox(1,1){{\scriptsize $[7]$}}}
\put(12,0){\makebox(1,1){{\scriptsize $[4]$}}}
\put(13,0){\makebox(1,1){{\scriptsize $[3]$}}}
\put(9,2){\makebox(1,1){{\small (b)}}}
\end{picture}
\end{center}
\begin{center}
\begin{tabular}{cccl|cl}
$\Delta p_{i}^{(1)}$ & $i_{6}=i_{1}+1$ & $i_{6}>i_{1}+1$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
& $1$ & $1$ & $(i\geq 3i_{6})$ & $0$ & $(i\geq i_{6}-1)$ \\
& $2$ & $2$ & $(i=3i_{6}-1)$ & & \\
& $1$ & $1$ & $(i=3i_{6}-2)$ & & \\
& $0$ & $0$ & $(i=3i_{6}-3)$ & & \\
& $0$ & $-1$ & $(i=3i_{6}-4)$ & & \\
& $0$ & $-2$ & $(i=3i_{6}-5)$ & &
\end{tabular}
\end{center}
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ $(i\leq 3i_{6}-6)$ and $\Delta p_{i}^{(2)}$ $(i\leq i_{6}-2)$ are the same as in \textbf{(VC-2)} (case (a)) and \textbf{(VC-3)} (case (b)).
\item[(2)] The selected $i_{7}$-string is in $\nu^{(1)}$ and $i_{4}<i_{7}=i_{8}$.
\begin{itemize}
\item[(a)] $i_{8}^{eff}=i_{6}$.
\begin{center}
\begin{tabular}{ccccl}
$\Delta p_{i}^{(1)}$ & $i_{8}$:0 & $i_{8}$:I & $i_{8}$:II & \\ \hline
& $1$ & $1$ & $1$ & $(i\geq 3i_{6})$ \\
& $0$ & $2$ & $2$ & $(i=3i_{6}-1)$ \\
& & $1$ & $3$ & $(i=3i_{6}-2)$ \\
& & & $2$ & $(i=3i_{6}-3)$
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{cccl}
$\Delta p_{i}^{(2)}$ & $i_{8}$:0/I & $i_{8}$:II & \\ \hline
& $0$ & $0$ & $(i\geq i_{6})$ \\
& $0$ & $-1$ & $(i=i_{6}-1)$
\end{tabular}
\end{center}
\item[(b)] $i_{8}^{eff}>i_{6}$.
\begin{center}
\begin{tabular}{cl|cccl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & $i_{8}$:0/I & $i_{8}$:II & \\ \hline
$1$ & $(i\geq i_{8})$ & & $0$ & $0$ & $(i\geq i_{8}^{eff})$ \\
$2$ & $(i=i_{8}-1)$ & & $0$ & $-1$ & $(i=i_{8}^{eff}-1)$
\end{tabular}
\end{center}
\end{itemize}
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ $(i\leq i_{8}-2)$ and $\Delta p_{i}^{(2)}$ $(i\leq i_{8}^{eff}-2)$ are the same as in \textbf{(VC-10)}.
\item[(3)] The selected $i_{7}$-string is in $\nu^{(1)}$ and $i_{7}<i_{8}$.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$1$ & $(i\geq i_{8})$ & $0$ & $(i\geq i_{8}^{eff})$
\end{tabular}
\end{center}
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ $(i\leq i_{8}-1)$ and $\Delta p_{i}^{(2)}$ $(i\leq i_{8}^{eff}-1)$ are the same as in \textbf{(VC-11)} (1).
\item[(4)] The selected $i_{7}$-string is in $\nu^{(2)}$ and $i_{4}=i_{7}$.
In this case $i_{8}^{eff}=i_{7}$.
The only possible box marking in $\nu^{(1)}$ is
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\begin{center}
\begin{picture}(4,2)
\put(0,0){\line(1,0){4}}
\put(0,1){\line(1,0){4}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
\put(4,0){\line(0,1){1}}
\put(1,1){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[8]$}}}
\put(2,0){\makebox(1,1){{\scriptsize $[6]$}}}
\put(3,0){\makebox(1,1){{\scriptsize $[5]$}}}
\end{picture}
\end{center}
\begin{center}
\begin{tabular}{cccl|cl}
$\Delta p_{i}^{(1)}$ & $i_{7}=i_{1}+1$ & $i_{7}>i_{1}+1$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
& $1$ & $1$ & $(i\geq 3i_{7})$ & $0$ & $(i\geq i_{7}-1)$ \\
& $2$ & $2$ & $(i=3i_{7}-1)$ & & \\
& $1$ & $1$ & $(i=3i_{7}-2)$ & & \\
& $0$ & $0$ & $(i=3i_{7}-3)$ & & \\
& $0$ & $-1$ & $(i=3i_{7}-4)$ & & \\
& $0$ & $0$ & $(i=3i_{7}-5)$ & &
\end{tabular}
\end{center}
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ $(i\leq 3i_{7}-6)$ and $\Delta p_{i}^{(2)}$ $(i\leq i_{7}-2)$ are the same as in \textbf{(VC-4)}.
\item[(5)] The selected $i_{7}$-string is in $\nu^{(2)}$ and $i_{4}<i_{7}$.
\begin{itemize}
\item[(a)] The box marking in $\nu^{(1)}$ is
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(4,2)
\put(0,0){\line(1,0){4}}
\put(0,1){\line(1,0){4}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
\put(4,0){\line(0,1){1}}
\put(1,1){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[8]$}}}
\put(2,0){\makebox(1,1){{\scriptsize $[6]$}}}
\put(3,0){\makebox(1,1){{\scriptsize $[5]$}}}
\end{picture}
\end{center}
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$1$ & $(i\geq 3i_{7})$ & $0$ & $(i\geq i_{7}-1)$ \\
$2$ & $(i=3i_{7}-1)$ & & \\
$1$ & $(i=3i_{7}-2)$ & & \\
$0$ & $(i=3i_{7}-3)$ & &
\end{tabular}
\end{center}
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ $(i\leq 3i_{7}-4)$ and $\Delta p_{i}^{(2)}$ $(i\leq i_{7}-2)$ are the same as in \textbf{(VC-5)}.
\item[(b)] The box marking in $\nu^{(1)}$ is
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(4,2)
\put(0,0){\line(1,0){3}}
\put(0,1){\line(1,0){3}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
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\put(1,0){\makebox(1,1){{\scriptsize $[8]$}}}
\put(2,0){\makebox(1,1){{\scriptsize $[6]$}}}
\end{picture}
\end{center}
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$1$ & $(i\geq 3i_{7})$ & $-$ & \\
$2$ & $(i=3i_{7}-1)$ & & \\
$1$ & $(i=3i_{7}-2)$ & &
\end{tabular}
\end{center}
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ $(i\leq 3i_{7}-3)$ and $\Delta p_{i}^{(2)}$ are the same as in \textbf{(VC-7)}.
\item[(c)] $i_{8}^{eff}>i_{7}$.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$1$ & $(i\geq i_{8})$ & $0$ & $(i\geq i_{8}^{eff})$
\end{tabular}
\end{center}
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ $(i\leq i_{8}-1)$ and $\Delta p_{i}^{(2)}$ $(i\leq i_{8}^{eff}-1)$ are the same as in \textbf{(VC-11)} (2) or (3).
\end{itemize}
\end{itemize}
\item[(VC-13)] $b= \framebox{$13$}$.
\begin{itemize}
\item[(1)] The selected $i_{7}$-string is in $\nu^{(2)}$ with $i_{4}=i_{7}$.
The only possible box marking of the selected $i_{9}$-string is
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(5,2)
\put(0,0){\line(1,0){5}}
\put(0,1){\line(1,0){5}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
\put(4,0){\line(0,1){1}}
\put(5,0){\line(0,1){1}}
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\put(1,0){\makebox(1,1){{\scriptsize $[9]$}}}
\put(2,0){\makebox(1,1){{\scriptsize $[8]$}}}
\put(3,0){\makebox(1,1){{\scriptsize $[6]$}}}
\put(4,0){\makebox(1,1){{\scriptsize $[5]$}}}
\end{picture}
\end{center}
\begin{center}
\begin{tabular}{cccl|cl}
$\Delta p_{i}^{(1)}$ & $i_{7}=i_{1}+1$ & $i_{7}>i_{1}+1$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
& $3$ & $3$ & $(i\geq i_{9})$ & $-1$ & $(i\geq i_{7})$ \\
& $2$ & $2$ & $(i=i_{9}-1)$ & $0$ & $(i=i_{7}-1)$ \\
& $1$ & $1$ & $(i=i_{9}-2)$ & & \\
& $0$ & $0$ & $(i=i_{9}-3)$ & & \\
& $0$ & $-1$ & $(i=i_{9}-4)$ & & \\
& $0$ & $0$ & $(i=i_{9}-5)$ & &
\end{tabular}
\end{center}
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ $(i\leq i_{9}-6)$ and $\Delta p_{i}^{(2)}$ $(i\leq i_{7}-2)$ are the same as in \textbf{(VC-4)}.
\item[(2)] The selected $i_{7}$-string is in $\nu^{(2)}$ with $i_{4}<i_{7}$ and the box marking in $\nu^{(1)}$ is either
\setlength{\unitlength}{10pt}
\begin{center}
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\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$3$ & $(i\geq i_{9})$ & $-1$ & $(i\geq i_{7})$ \\
$2$ & $(i=i_{9}-1)$ & $0$ & $(i=i_{7}-1)$ \\
$1$ & $(i=i_{9}-2)$ & & \\
$0$ & $(i=i_{9}-3)$ & &
\end{tabular}
\end{center}
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ $(i\leq i_{9}-4)$ and $\Delta p_{i}^{(2)}$ $(i\leq i_{7}-2)$ are the same as in \textbf{(VC-5)} (case (a)) and \textbf{(VC-7)} (case (b)).
\item[(3)] The selected $i_{6}$-string is in $\nu^{(2)}$ with $i_{5}=i_{6}$ and the box marking in $\nu^{(1)}$ is either
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\begin{center}
\begin{tabular}{cccl|cl}
$\Delta p_{i}^{(1)}$ & $i_{6}=i_{1}+1$ & $i_{6}>i_{1}+1$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
& $3$ & $3$ & $(i\geq i_{9})$ & $-1$ & $(i\geq i_{6})$ \\
& $2$ & $2$ & $(i=i_{9}-1)$ & $0$ & $(i=i_{6}-1)$ \\
& $1$ & $1$ & $(i=i_{9}-2)$ & & \\
& $0$ & $0$ & $(i=i_{9}-3)$ & & \\
& $0$ & $-1$ & $(i=i_{9}-4)$ & & \\
& $0$ & $-2$ & $(i=i_{9}-5)$ & &
\end{tabular}
\end{center}
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ $(i\leq i_{9}-6)$ and $\Delta p_{i}^{(2)}$ $(i\leq i_{6}-2)$ are the same as in \textbf{(VC-2)} (case (a)) and \textbf{(VC-3)} (case (b)).
\item[(4)] The rightmost box of the $i_{9}$-string is marked by [7].
\begin{center}
\begin{tabular}{cccl|cl}
$\Delta p_{i}^{(1)}$ & $i_{9}^{eff}=i_{6}$ & $i_{9}^{eff}>i_{6}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
& $3$ & $3$ & $(i\geq i_{9})$ & $-1$ & $(i\geq i_{9}^{eff})$ \\
& $2$ & $1$ & $(i=i_{9}-1)$ & & \\
& $1$ & $-1$ & $(i=i_{9}-2)$ & &
\end{tabular}
\end{center}
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ $(i\leq i_{9}-3)$ and $\Delta p_{i}^{(2)}$ $(i\leq i_{9}^{eff}-1)$ are the same as in \textbf{(VC-10)}.
\item[(5)] The rightmost box of the $i_{9}$-string is marked by [8].
\begin{center}
\begin{tabular}{cl|cccl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & $i_{9}$:0/I & $i_{9}$:II & \\ \hline
$3$ & $(i\geq i_{9})$ & & $-1$ & $-1$ & $(i\geq i_{9}^{eff})$ \\
$1$ & $(i=i_{9}-1)$ & & $1$ & $0$ & $(i=i_{9}^{eff}-1)$
\end{tabular}
\end{center}
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ $(i\leq i_{9}-2)$ and $\Delta p_{i}^{(2)}$ $(i\leq i_{9}^{eff}-2)$ are the same as in \textbf{(VC-11)}.
\item[(6)] The $i_{9}$-string is marked by [9] only.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$3$ & $(i\geq i_{9})$ & $-1$ & $(i\geq i_{9}^{eff})$
\end{tabular}
\end{center}
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ $(i\leq i_{9}-1)$ and $\Delta p_{i}^{(2)}$ $(i\leq i_{9}^{eff}-1)$ are the same as in \textbf{(VC-12)}.
\end{itemize}
\item[(VC-14)] $b= \framebox{$14$}$.
\begin{itemize}
\item[(1)] $i_{1}<i_{10}$.
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ are obtained by overwriting
\[
-3\chi (i\geq 3i_{10})-2\chi (i=3i_{10}-1)-\chi (i=3i_{10}-2)
\]
on $\Delta p_{i}^{(1)}$ in \textbf{(VC-13)} and $\Delta p_{i}^{(2)}$ are obtained by overwriting $2\chi (i\geq i_{10})$ on $\Delta p_{i}^{(2)}$ in \textbf{(VC-13)}.
\item[(2)] $i_{1}=i_{10}$.
The only possible box marking of $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right) is
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\end{center}
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
0 & $(i\geq 1)$ & $1$ & $(i\geq i_{1})$ \\
& & 0 & $(i=i_{1}-1)$ \\
& & $-1$ & $(1\leq i\leq i_{1}-2)$ \\
\end{tabular}
\end{center}
\end{itemize}
\item[(VC-$\emptyset$)] $b= \framebox{$\emptyset$}$.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$0$ & $(i\geq 1)$ & $0$ & $(i\geq 1)$
\end{tabular}
\end{center}
\end{description}
\subsection{Inverse algorithm $\Tilde{\delta}_{\theta}$}
The reader may skip this subsection as we do not use it in the following sections.
This is provided just for completeness.
For a given RC $(\Tilde{\nu},\Tilde{J})$ and $b\in B$ the inverse algorithm $\Tilde{\delta}_{\theta}$ of $\delta_{\theta}$ is described in this subsection.
The inverse algorithm consists of adding boxes to strings in $\Tilde{\nu}$ and adjusting the numbers of riggings.
We omit the rule of adjusting the numbers of riggings as it is obvious by the algorithm of $\delta_{\theta}$.
We follow the convention that riggings in strings of the same length are sorted in an decreasing order (from the top).
We assume that $\Tilde{\nu}^{(a)}$ has singular strings of length zero with zero vacancy (and therefore zero rigging).
In the following, strings of length $l$ under some conditions are selected and the $l$-string is referred to the selected string unless stated otherwise.
We denote by $[l]_{eff}$ the effective length of the (selected or unselected) string of length $l$ in $\Tilde{\nu}^{(1)}$.
\begin{flushleft}
Case 1. $b= \framebox{$\emptyset$}$.
\end{flushleft}
Add three boxes to the string of length zero in $\Tilde{\nu}^{(1)}$ and add one box to two strings of length zero in $\Tilde{\nu}^{(2)}$.
\begin{flushleft}
Case 2. $b= \framebox{$2$}$.
\end{flushleft}
Find the singular string of maximum length in $\Tilde{\nu}^{(2)}$ and add one box in this string.
This corresponds to the box marking in this string in $\nu^{(2)}$ by $[1]$.
The augmented string is called the box-added $i_{1}$-string.
This terminology is generalized in an obvious manner.
\begin{flushleft}
Case 3. $b= \framebox{$3$}$.
\end{flushleft}
Find the singular string of maximum length in $\Tilde{\nu}^{(1)}$, add one box in this string, and follow the algorithm of Case 2 such that the length of the box-added $i_{1}$-string does not exceed the effective length of the box-added $i_{2}$-string.
\begin{flushleft}
Case 4. $b= \framebox{$4$}$.
\end{flushleft}
Find the singular/q-singular string of maximum length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$.
We choose the singular string when there exist singular and q-singular strings of the same length.
If the $l^{(1)}$-string is singular, then add two boxes to this string and follow the algorithm of Case 2.
If the $l^{(1)}$-string is q-singular, then add one box to this string and follow the algorithm of Case 3.
\begin{flushleft}
Case 5. $b= \framebox{$5$}$.
\end{flushleft}
Find the singular string of maximum length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$ and add one box in this string.
Then find singular/q-singular string of maximum length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$.
We choose the singular string when there exist singular and q-singular strings of the same length and do the following cases.
\begin{itemize}
\item[(1)] The $l^{(1)}$-string is singular of length $3l^{(2)}+1$.
Add one box to the $l^{(1)}$-string and follow the algorithm, of Case 3.
This corresponds to the following box marking in $\nu^{(1)}$.
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with $i_{3}^{eff}=i_{4}$.
\item[(2)] The $l^{(1)}$-string is singular of length $3l^{(2)}$ or $3l^{(2)}-1$.
Add two boxes to the $l^{(1)}$-string and follow the algorithm of Case 2.
This corresponds to the following box marking in $\nu^{(1)}$
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\end{center}
with $i_{3}^{eff}=i_{4}$.
\item[(3)] The $l^{(1)}$-string is q-singular of length $3l^{(2)}$.
Add one box to the founded string and follow the algorithm of Case 3.
This corresponds to the following box marking in $\nu^{(1)}$
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with $i_{3}^{eff}=i_{4}$.
\end{itemize}
If the box adding not successful, then follow the algorithm of Case 4.
This corresponds to the box marking in $\nu$ with $i_{3}^{eff}<i_{4}$.
\begin{flushleft}
Case 6. $b= \framebox{$6$}$.
\end{flushleft}
Find the singular/q-singular/qq-singular string of maximum length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$.
We choose the singular string when there exists singular and q/qq-singular strings of the same length.
Similarly, we choose the q-singular string when there exists q-singular and qq-singular strings of the same length.
That is, we choose the string of the same length by the following preferential rule
\begin{center}
\begin{tabular}{ccc}
singular & q-singular & qq-singular \\ \hline
1 & 2 & 3
\end{tabular}
\end{center}
In addition, we impose the following preferential rule for the $l^{(1)}$-string.
That is, if the $l^{(1)}$-string is qq-singular and there exists a singular string of length $l^{(1)}-1$ in $\Tilde{\nu}^{(1)}$, then reset the $l^{(1)}$-string to be the $(l^{(1)}-1)$-string.
The box adding goes as follows.
If the $l^{(1)}$-string is singular (resp. q-singular), then add three (resp. two) boxes to the $l^{(1)}$-string and follow the algorithm of Case 2 (resp. Case 3).
If the $l^{(1)}$ is qq-singular, then add one box to the $l^{(1)}$-string and follow the algorithm of Case 4.
\begin{flushleft}
Case 7. $b= \framebox{$7$}$.
\end{flushleft}
Find the singular string of maximum length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$ and do as follows according to the type of the $l^{(1)}$-string.
\begin{itemize}
\item[(1)] type-0.
Find the singular string of length $[l^{(1)}]_{eff}$ in $\Tilde{\nu}^{(2)}$.
If such a string exists, then add three boxes to the $l^{(1)}$-string in $\Tilde{\nu}^{(1)}$ and one box to the founded string in $\Tilde{\nu}^{(2)}$ and follow the algorithm of Case 2.
This corresponds to the following box marking in $\nu^{(1)}$
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\put(3,0){\line(0,1){1}}
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with $i_{5}^{eff}=i_{4}$.
\item[(2)] type-I.
Find the singular string of length $[l^{(1)}]_{eff}-1$ in $\Tilde{\nu}^{(2)}$.
If such a string exists, then add one box to the $l^{(1)}$-string in $\Tilde{\nu}^{(1)}$ and follow the algorithm of Case 5 where the length of the box-added $i_{4}$-string in $\Tilde{\nu}^{(2)}$ is $[l^{(1)}]_{eff}$.
\item[(3)] type-II.
Find the singular string of length $[l^{(1)}]_{eff}-1$ in $\Tilde{\nu}^{(2)}$.
If such a string exists, then add two boxes to the $l^{(1)}$-string in $\Tilde{\nu}^{(1)}$, one box to the founded string in $\Tilde{\nu}^{(2)}$ and follow the algorithm of Case 3.
This corresponds to the following box marking in $\nu^{(1)}$
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\begin{picture}(5,2)
\put(0,0){\line(1,0){4}}
\put(0,1){\line(1,0){4}}
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\put(3,0){\line(0,1){1}}
\put(4,0){\line(0,1){1}}
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with $i_{5}^{eff}=i_{4}$.
\end{itemize}
If the box adding is not successful, then add one box to the $l^{(1)}$-string in $\Tilde{\nu}^{(1)}$ and follow the algorithm of Case 5.
This corresponds to the box marking in $\nu$ with $i_{5}^{eff}>i_{4}$.
\begin{flushleft}
Case 8. $b= \framebox{$8$}$.
\end{flushleft}
Find the singular string of maximum length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$ and add one box to this string.
The resulting string is the box-added $i_{5}$-string.
Then, find the string of maximum length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$ such that it is
\begin{itemize}
\item[(1)] a singular string of type-I/II (resp. type-0) with $[l^{(1)}]_{eff}=l^{(2)}+1$ (resp. $[l^{(1)}]_{eff}=l^{(2)}$) or
\item[(2)] a q-singular string of type-II (resp. type-0) with $[l^{(1)}]_{eff}=l^{(2)}+1$ (resp. $[l^{(1)}]_{eff}=l^{(2)}$) or
\item[(3)] a singular string of type-I/II with $[l^{(1)}]_{eff}=l^{(2)}$ or
\item[(4)] a q-singular string of type-I with with $[l^{(1)}]_{eff}=l^{(2)}$ oir
\item[(5)] a qq-singular string of type-0 with $[l^{(1)}]_{eff}=l^{(2)}$.
\end{itemize}
We choose the string of the same length by the following preferential rule
\begin{center}
\begin{tabular}{ccc}
singular & q-singular & qq-singular \\ \hline
1 & 2 & 3
\end{tabular}
\end{center}
and do as follows according to the type of the $l^{(1)}$-string.
All correspond to the box marking in $\nu$ with $i_{4}^{eff}=i_{5}$.
\begin{itemize}
\item[(1)] type-0.
If the $l^{(1)}$-string is singular (resp. q-singular), then add three (resp. two) boxes to this string and follow the algorithm of Case 2 (resp. Case 3).
If the $l^{(1)}$-string is qq-singular, then add one box to this string and follow the algorithm of Case 4.
\item[(2)] type-I.
If the $l^{(1)}$-string is singular with with $[l^{(1)}]_{eff}=l^{(2)}+1$, then add one box to this string and follow the algorithm of Case 4 ignoring the q-singular string of length $l^{(1)}-1$ in $\Tilde{\nu}^{(1)}$.
This is because the following box marking in $\nu^{(1)}$
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\end{center}
with $i_{4}^{eff}=i_{5}$ is forbidden.
If the selected string is singular (resp. q-singular) with $[l^{(1)}]_{eff}=l^{(2)}$, then add three (resp. two) boxes to this string and follow the algorithm of Case 2 (resp. Case 3).
\item[(3)] type-II.
If the $l^{(1)}$-string is singular (resp. q-singular) with $[l^{(1)}]_{eff}=l^{(2)}+1$, then add two (resp. one) box(es) to this string and follow the algorithm of Case 3 (resp. Case 4).
If the $l^{(1)}$-string is singular with $[l^{(1)}]_{eff}=l^{(2)}$, then add three boxes to this string and follow the algorithm of Case 2.
\end{itemize}
If the box adding is not successful, then follow the algorithm of Case 6 such that the effective length of the box-added $i_{4}$-string is smaller than $l^{(2)}+1$, which corresponds to the box marking in $\nu$ with $i_{4}^{eff}<i_{5}$.
\begin{flushleft}
Case 9. $b= \framebox{$9$}$.
\end{flushleft}
Find the singular string of maximum length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$.
Then, find the singular/q-singular string of maximum length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$.
We choose the singular string when there exist singular and q-singular strings of the same length.
In addition, we impose the following preferential rule for the $l^{(1)}$-string.
\begin{itemize}
\item
If the $l^{(1)}$-string in $\Tilde{\nu}^{(1)}$ is type-I q-singular with $[l^{(1)}]_{eff}=l^{(2)}+1$ and there exists a type-II singular string of the same effective length in $\Tilde{\nu}^{(1)}$, then reset the $l^{(1)}$-string to be the founded type-II singular string.
\item
If the $l^{(1)}$-string is type-I/II q-singular string with $[l^{(1)}]_{eff}=l^{(2)}+1$ and there exists a type-0 singular string of effective length $l^{(2)}$, then reset the $l^{(1)}$-string to be the founded type-0 singular string.
\end{itemize}
The box adding goes as follows.
If the $l^{(1)}$-string is q-singular, then add one box to this string and follow the algorithm of Case 7.
If the $l^{(1)}$-string is singular, then do as follows according to the type of the $l^{(1)}$-string.
\begin{itemize}
\item[(1)] type-0.
If $[l^{(1)}]_{eff}=l^{(2)}$, then add four boxes to this string, one box to the $l^{(2)}$-string, and follow the algorithm of Case 2.
\item[(2)] type-II.
If $[l^{(1)}]_{eff}=l^{(2)}+1$, then add three boxes to this string, one box to the $l^{(2)}$-string, and follow the algorithm of Case 3.
\item[(3)] type-I.
If $[l^{(1)}]_{eff}=l^{(2)}$, then add two boxes to this string, one box to the $l^{(2)}$-string, and follow the algorithm of Case 4 ignoring the q-singular string of length $l^{(1)}-1$ in $\Tilde{\nu}^{(1)}$.
This is due to the rule of \textbf{(BS-5)}.
\end{itemize}
The corresponding box marking in $\nu^{(1)}$ is one of the following three.
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(20,2)
\put(0,0){\line(1,0){5}}
\put(0,1){\line(1,0){5}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
\put(4,0){\line(0,1){1}}
\put(5,0){\line(0,1){1}}
\put(3,1){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[6]$}}}
\put(2,0){\makebox(1,1){{\scriptsize $[5]$}}}
\put(3,0){\makebox(1,1){{\scriptsize $[3]$}}}
\put(4,0){\makebox(1,1){{\scriptsize $[2]$}}}
\put(6,0){\makebox(2,1){$\text{or}$}}
\put(9,0){\line(1,0){4}}
\put(9,1){\line(1,0){4}}
\put(10,0){\line(0,1){1}}
\put(11,0){\line(0,1){1}}
\put(12,0){\line(0,1){1}}
\put(13,0){\line(0,1){1}}
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\put(11,0){\makebox(1,1){{\scriptsize $[5]$}}}
\put(12,0){\makebox(1,1){{\scriptsize $[3]$}}}
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\put(17,0){\line(1,0){3}}
\put(17,1){\line(1,0){3}}
\put(18,0){\line(0,1){1}}
\put(19,0){\line(0,1){1}}
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\end{picture}
\end{center}
where $i_{6}^{eff}=i_{4}$ in the left two and $i_{6}^{eff}=i_{4}+1$ in the rightmost.
If the box adding is not successful, then add two boxes to the $l^{(1)}$-string and follow the algorithm of Case 5.
\begin{flushleft}
Case 10. $b= \framebox{$10$}$.
\end{flushleft}
Find the singular/q-singular string of maximum length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$.
We choose the singular string when there exist singular and q-singular strings of the same length.
If the $l^{(2)}$-string is q-singular, then add one box to this string and follow the algorithm of Case 8.
If the $l^{(2)}$-string is singular, then add two boxes to the $l^{(2)}$-string and find the string of maximum length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$ such that it is
\begin{itemize}
\item[(1)]
a singular string of type-I/II with $[l^{(1)}]_{eff}=l^{(2)}+1$ or a singular string (of any type) with $[l^{(1)}]_{eff}=l^{(2)}$ or
\item[(2)]
a q-singular string of type-II (resp. type-0/I) with $[l^{(1)}]_{eff}=l^{(2)}+1$ (resp. $[l^{(1)}]_{eff}=l^{(2)}$) or
\item[(3)]
a qq-singular string of type-0 with $[l^{(1)}]_{eff}=l^{(2)}$.
\end{itemize}
In addition, we impose the following preferential rule for the $l^{(1)}$-string.
\begin{itemize}
\item
If the $l^{(1)}$-string is type-II q-singular and there exists a type-0 singular string of effective length $[l^{(1)}]_{eff}-1$, then reset the $l^{(1)}$-string to be the founded singular string.
\item
If the $l^{(1)}$-string is qq-singular and there exists a type-I singular string of the same effective length, then reset the $l^{(1)}$-string to be the founded singular string.
\end{itemize}
The box adding goes as follows.
\begin{itemize}
\item[(1)] The $l^{(1)}$-string is singular.
Do the following according to the type of the $l^{(1)}$-string.
\begin{itemize}
\item[(a)] type-0.
Add three boxes to this string and follow the algorithm of Case 2.
\item[(b)] type-I.
If $[l^{(1)}]_{eff}=l^{(2)}$, then add three boxes to this string and follow the algorithm of Case 2.
If $[l^{(1)}]_{eff}=l^{(2)}+1$, then add one box to this string and follow the algorithm of Case 4.
\item[(c)] type-II.
If $[l^{(1)}]_{eff}=l^{(2)}$, then add three boxes to this string and follow the algorithm of Case 2.
If $[l^{(1)}]_{eff}=l^{(2)}+1$, then add two boxes to this string and follow the algorithm of Case 3.
\end{itemize}
\item[(2)] The $l^{(1)}$-string is q-singular.
If the $l^{(1)}$-string is type-0/I (resp. type-II), then add two (resp. one) box(es) to this string and follow the algorithm of Case 3 (resp. Case 4).
\item[(3)] The $l^{(1)}$-string is qq-singular.
Add one box to this string and follow the algorithm of Case 4.
\end{itemize}
If the box adding is not successful, then follow the algorithm of Case 6 such that the effective length of the box-added $i_{4}$-string does not exceed $l^{(2)}$.
\begin{flushleft}
Case 11. $b= \framebox{$11$}$.
\end{flushleft}
Find the singular string of maximum length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$.
Find the singular string of maximum length $l_{s}^{(2)}$ in $\Tilde{\nu}^{(2)}$.
Find the q-singular string of maximum length $l_{q}^{(2)}$ satisfying $[ l^{(1)}+1]_{eff}\geq l_{q}^{(2)}$.
If such a q-singular string does not exist, then set $l_{q}^{(2)}=0$.
If $l_{q}^{(2)}>l_{s}^{(2)}$, then add one box to the $l^{(1)}$-string in $\Tilde{\nu}^{(1)}$, add one box to the $l_{q}^{(2)}$-string in $\Tilde{\nu}^{(2)}$ and follow the algorithm of Case 8.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(7,1)
\put(0,0){\line(1,0){2}}
\put(0,1){\line(1,0){2}}
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\put(5,1){\line(1,0){2}}
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\put(7,0){\line(0,1){1}}
\put(6,0){\makebox(1,1){{\scriptsize $[6]$}}}
\end{picture}
\end{center}
If $l_{s}^{(2)}>l_{q}^{(2)}$, then do as follows.
If $[l^{(1)}+1]_{eff}\geq l_{s}^{(2)}$, the add one box to the $l^{(1)}$-string in $\Tilde{\nu}^{(1)}$, add two boxes to the $l_{s}^{(2)}$ singular string in $\Tilde{\nu}^{(2)}$ and follow the algorithm of Case 6.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(8,1)
\put(0,0){\line(1,0){2}}
\put(0,1){\line(1,0){2}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(1,0){\makebox(1,1){{\scriptsize $[7]$}}}
\put(5,0){\line(1,0){3}}
\put(5,1){\line(1,0){3}}
\put(6,0){\line(0,1){1}}
\put(7,0){\line(0,1){1}}
\put(8,0){\line(0,1){1}}
\put(6,0){\makebox(1,1){{\scriptsize $[6]$}}}
\put(7,0){\makebox(1,1){{\scriptsize $[5]$}}}
\end{picture}
\end{center}
If the box adding is not successful, then do as follows.
Find the singular string of maximum length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$.
Find the singular string of maximum length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$.
If the $l^{(1)}$-string is type-0, $[l^{(1)}+4]_{eff}=l^{(2)}+2$, and there are no q-singular strings of length $l$ ($l^{(1)}+1\leq l\leq l^{(1)}+3$), then add four boxes to the $l^{(1)}$-string, add two boxes to the $l^{(2)}$-string, and follow the algorithm of Case 2.
This corresponds to the following box marking $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(11,2)
\put(0,0){\line(1,0){5}}
\put(0,1){\line(1,0){5}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
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\put(10,0){\makebox(1,1){{\scriptsize $[4]$}}}
\end{picture}
\end{center}
If the box adding is not successful, then do as follows.
Find the singular string of maximum length $l_{1}^{(2)}$ in $\Tilde{\nu}^{(2)}$.
Find the singular string of maximum length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$.
If $l^{(1)}$-string is type-0, $[l^{(1)}+4]_{eff}\leq l^{(2)}+1$, and there are no q-singular strings of length $l$ ($l^{(1)}+1\leq l\leq 3l_{1}^{(2)}$), then find the singular string of length $l_{2}^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $l_{2}^{(2)}+1=[l^{(1)}+4]_{eff}-1$.
If such a singular string exists, then add four boxes to the $l^{(1)}$-string, add one box the $l_{1}^{(2)}$- and $l_{2}^{(2)}$-string and follow the algorithm of Case 2.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right)
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(12,3)
\put(0,1){\line(1,0){5}}
\put(0,2){\line(1,0){5}}
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\put(3,1){\line(0,1){1}}
\put(4,1){\line(0,1){1}}
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\put(8,2){\line(1,0){4}}
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\put(9,0){\makebox(1,1){{\scriptsize $[4]$}}}
\put(11,1){\makebox(1,1){{\scriptsize $[7]$}}}
\end{picture}
\end{center}
where $i_{6}^{eff}=i_{4}+1\leq i_{7}$.
If the box adding is not successful, then do as follows.
Find the singular string of maximum length $l_{1}^{(2)}$ in $\Tilde{\nu}^{(2)}$.
Find the singular string of maximum length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$.
If the $l^{(1)}$-string is type-II, $[l^{(1)}+3]_{eff}\leq l_{1}^{(2)}+1$, and there are no q-singular strings of length $l$ ($l^{(1)}+1\leq l\leq 3l_{1}^{(2)}$), then find the singular string of length $l_{2}^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $l_{2}^{(2)}+1=[l^{(1)}+3]_{eff}-1$.
If such a singular string exists, then add three boxes to the $l^{(1)}$-string, add one box the $l_{1}^{(2)}$- and $l_{2}^{(2)}$-string and follow the algorithm of Case 3.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right)
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(11,3)
\put(0,1){\line(1,0){4}}
\put(0,2){\line(1,0){4}}
\put(1,1){\line(0,1){1}}
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\put(3,1){\line(0,1){1}}
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\put(2,1){\makebox(1,1){{\scriptsize $[5]$}}}
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\put(7,0){\line(1,0){2}}
\put(7,1){\line(1,0){4}}
\put(7,2){\line(1,0){4}}
\put(8,0){\line(0,1){1}}
\put(9,0){\line(0,1){1}}
\put(10,1){\line(0,1){1}}
\put(11,1){\line(0,1){1}}
\put(8,0){\makebox(1,1){{\scriptsize $[4]$}}}
\put(10,1){\makebox(1,1){{\scriptsize $[7]$}}}
\end{picture}
\end{center}
where $i_{6}^{eff}=i_{4}+1\leq i_{7}$.
If the box adding is not successful, then do as follows.
Find the singular string of maximum length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$.
Find the singular string of maximum length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$.
If the $l^{(1)}$-string is type-I and $[l^{(1)}+2]_{eff}=l^{(2)}+2$, then add two boxes to the $l^{(1)}$- and $l^{(2)}$-strings and follow the algorithm of Case 4 ignoring the q-singular string of length $l^{(1)}-1$ in $\Tilde{\nu}^{(1)}$.
The box adding in this case corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(9,2)
\put(0,0){\line(1,0){3}}
\put(0,1){\line(1,0){3}}
\put(1,0){\line(0,1){1}}
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\put(8,0){\line(0,1){1}}
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\put(7,0){\makebox(1,1){{\scriptsize $[7]$}}}
\put(8,0){\makebox(1,1){{\scriptsize $[4]$}}}
\end{picture}
\end{center}
If the box adding is not successful, then do as follows.
Find the singular string of maximum length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$.
Find the singular string of maximum length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$.
If the $l^{(1)}$-string is type-I and $[l^{(1)}+2]_{eff}=l^{(2)}+1$, then add two boxes to the $l^{(1)}$-string and one box to the $l^{(2)}$-strings and follow the algorithm of Case 5.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right)
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(8,2)
\put(0,0){\line(1,0){3}}
\put(0,1){\line(1,0){3}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
\put(1,1){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[6]$}}}
\put(2,0){\makebox(1,1){{\scriptsize $[5]$}}}
\put(6,0){\line(1,0){2}}
\put(6,1){\line(1,0){2}}
\put(7,0){\line(0,1){1}}
\put(8,0){\line(0,1){1}}
\put(7,0){\makebox(1,1){{\scriptsize $[7]$}}}
\end{picture}
\end{center}
where $i_{6}^{eff}=i_{7}$.
If the box adding is not successful, then do as follows.
Find the singular string of maximum length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$.
Find the q-singular string of maximum length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$.
If the $l^{(1)}$-string is type-0 and $[l^{(1)}+1]_{eff}=l^{(2)}+1$, then add one box to the $l^{(1)}$- and the $l^{(2)}$-strings and follow the algorithm of Case 7.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right)
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(7,2)
\put(0,0){\line(1,0){2}}
\put(0,1){\line(1,0){2}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(0,1){\makebox(2,1){$\downarrow$}}
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\put(5,0){\line(1,0){2}}
\put(5,1){\line(1,0){2}}
\put(6,0){\line(0,1){1}}
\put(7,0){\line(0,1){1}}
\put(6,0){\makebox(1,1){{\scriptsize $[7]$}}}
\end{picture}
\end{center}
where $i_{6}^{eff}=i_{7}$.
If the box adding is not successful, then do as follows.
Find the singular string of maximum length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$.
Then, find the string of maximum length $l^{(1)}$ such that the $l^{(1)}$-string is singular with $[l^{(1)}+2]_{eff}<l^{(2)}+1$ or the $l^{(1)}$-string is q-singular with $[l^{(1)}+1]_{eff}<l^{(2)}+1$.
If the $l^{(1)}$-string is singular, then add two boxes to the $l^{(1)}$-string and one box to the $l^{(2)}$-strings and follow the algorithm of Case 5.
If the $l^{(1)}$-string is q-singular, then add one box to the $l^{(1)}$- and the $l^{(2)}$-strings and follow the algorithm of Case 7.
The corresponding box marking in $\nu^{(1)}$ is
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(8,2)
\put(0,0){\line(1,0){3}}
\put(0,1){\line(1,0){3}}
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\put(4,0){\makebox(2,1){$\text{or}$}}
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\put(8,0){\makebox(1,1){{\scriptsize $[6]$}}}
\end{picture}
\end{center}
with $i_{6}^{eff}<i_{7}$.
\begin{flushleft}
Case 12. $b= \framebox{$12$}$.
\end{flushleft}
Find the singular/q-singular string of maximum length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$.
We choose the singular string when there exist singular and q-singular strings of the same length.
In addition, we impose the following preferential rules.
\begin{itemize}
\item
If the $l^{(1)}$-string is q-singular (of any type), then find the type-0 singular string of type-0 of length $l^{(1)\prime}$ whose effective length $[l^{(1)}]_{eff}-1$ in $\Tilde{\nu}^{(1)}$ such that there exists a singular string of length $l^{(2)}$ satisfying $[l^{(1)\prime}+5]_{eff}=l^{(2)}+2$.
If such an $l^{(1)\prime}$-string exists, then reset the $l^{(1)}$-string to be $l^{(1)\prime}$-string.
\item
If the $l^{(1)}$-string is q-singular of type I, then find the type-II of length $l^{(1)\prime}=l^{(1)}-1$ such that there exists a singular string of length $l^{(2)}$ satisfying $[l^{(1)\prime}+4]_{eff}=l^{(2)}+2$.
If such an $l^{(1)\prime}$-string exists, then reset the $l^{(1)}$-string to be $l^{(1)\prime}$-string.
\end{itemize}
The box adding goes as follows.
If the $l^{(1)}$-string is singular of type-0, then find a singular string of length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $[l^{(1)}+5]_{eff}=l^{(2)}+2$.
If such a string exists, then add five boxes to the singular $l^{(1)}$-string, two boxes to the $l^{(2)}$-string, and follow the algorithm of Case 2.
This corresponds to the following box marking $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(12,2)
\put(0,0){\line(1,0){6}}
\put(0,1){\line(1,0){6}}
\put(1,0){\line(0,1){1}}
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\put(3,0){\line(0,1){1}}
\put(4,0){\line(0,1){1}}
\put(5,0){\line(0,1){1}}
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\put(9,1){\line(1,0){3}}
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\put(11,0){\makebox(1,1){{\scriptsize $[5]$}}}
\end{picture}
\end{center}
If the $l^{(1)}$-string is singular of type-II, then find a singular string of length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $[l^{(1)}+4]_{eff}=l^{(2)}+2$.
If such a string exists, then add four boxes to the singular $l^{(1)}$-string, two boxes to the $l^{(2)}$-string, and follow the algorithm of Case 3.
This corresponds to the following box marking $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(11,2)
\put(0,0){\line(1,0){5}}
\put(0,1){\line(1,0){5}}
\put(1,0){\line(0,1){1}}
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\put(3,0){\line(0,1){1}}
\put(4,0){\line(0,1){1}}
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\put(9,0){\makebox(1,1){{\scriptsize $[6]$}}}
\put(10,0){\makebox(1,1){{\scriptsize $[5]$}}}
\end{picture}
\end{center}
If the $l^{(1)}$-string is singular of type-I, then find a singular string of length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $[l^{(1)}+3]_{eff}=l^{(2)}+1$.
If such a string exists, then add three boxes to the singular $l^{(1)}$-string, one box to the $l^{(2)}$-string, and follow the algorithm of Case 5.
This corresponds to the following box marking $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(9,2)
\put(0,0){\line(1,0){4}}
\put(0,1){\line(1,0){4}}
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\put(3,0){\line(0,1){1}}
\put(4,0){\line(0,1){1}}
\put(1,1){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[8]$}}}
\put(2,0){\makebox(1,1){{\scriptsize $[6]$}}}
\put(3,0){\makebox(1,1){{\scriptsize $[5]$}}}
\put(7,0){\line(1,0){2}}
\put(7,1){\line(1,0){2}}
\put(8,0){\line(0,1){1}}
\put(9,0){\line(0,1){1}}
\put(8,0){\makebox(1,1){{\scriptsize $[7]$}}}
\end{picture}
\end{center}
If the $l^{(1)}$-string is singular of type-I, then find a singular string of length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $[l^{(1)}+3]_{eff}=l^{(2)}+2$.
If such a string exists, then add three boxes to the singular $l^{(1)}$-string, two boxes to the $l^{(2)}$-string, and follow the algorithm of Case 4 ignoring the q-singular string of length $l^{(1)}-1$ in $\Tilde{\nu}^{(1)}$.
This corresponds to the following box marking $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(10,2)
\put(0,0){\line(1,0){4}}
\put(0,1){\line(1,0){4}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
\put(4,0){\line(0,1){1}}
\put(1,1){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[8]$}}}
\put(2,0){\makebox(1,1){{\scriptsize $[6]$}}}
\put(3,0){\makebox(1,1){{\scriptsize $[5]$}}}
\put(7,0){\line(1,0){3}}
\put(7,1){\line(1,0){3}}
\put(8,0){\line(0,1){1}}
\put(9,0){\line(0,1){1}}
\put(10,0){\line(0,1){1}}
\put(8,0){\makebox(1,1){{\scriptsize $[7]$}}}
\put(9,0){\makebox(1,1){{\scriptsize $[4]$}}}
\end{picture}
\end{center}
If the $l^{(1)}$-string is singular of type-0/II, then find a singular (resp. q-singular) string of length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $[l^{(1)}+2]_{eff}=l^{(2)}+2$ (resp. $[l^{(1)}+2]_{eff}=l^{(2)}+1$).
If such a string exists, then add two boxes to the $l^{(1)}$-string and follow the algorithm of Case 10 such that $i_{8}^{eff}=i_{6}$.
This corresponds to the following box marking in $\nu^{(1)}$.
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(3,1)
\put(0,0){\line(1,0){3}}
\put(0,1){\line(1,0){3}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
\put(1,0){\makebox(1,1){{\scriptsize $[8]$}}}
\put(2,0){\makebox(1,1){{\scriptsize $[7]$}}}
\end{picture}
\end{center}
If the $l^{(1)}$-string is q-singular of type-0, then find a singular string of length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $[l^{(1)}+2]_{eff}=l^{(2)}+1$.
If such a string exists, then add two boxes to the $l^{(1)}$-string, one box to the $l^{(2)}$-string, and follow the algorithm of Case 7.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(8,2)
\put(0,0){\line(1,0){3}}
\put(0,1){\line(1,0){3}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
\put(0,1){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[8]$}}}
\put(2,0){\makebox(1,1){{\scriptsize $[6]$}}}
\put(6,0){\line(1,0){2}}
\put(6,1){\line(1,0){2}}
\put(7,0){\line(0,1){1}}
\put(8,0){\line(0,1){1}}
\put(7,0){\makebox(1,1){{\scriptsize $[7]$}}}
\end{picture}
\end{center}
If the $l^{(1)}$-string is q-singular of type-I/II, then find a singular string of length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $[l^{(1)}+2]_{eff}=l^{(2)}+2$ or $[l^{(1)}+2]_{eff}=l^{(2)}+1$.
If such a string exists, then add two boxes to the $l^{(1)}$-string and follow the algorithm of Case 11.
This corresponds to the following box marking in $\nu^{(2)}$.
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(8,2)
\put(0,0){\line(1,0){3}}
\put(0,1){\line(1,0){3}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
\put(1,0){\makebox(1,1){{\scriptsize $[7]$}}}
\put(2,0){\makebox(1,1){{\scriptsize $[4]$}}}
\put(4,0){\makebox(2,1){$\text{or}$}}
\put(7,0){\line(1,0){2}}
\put(7,1){\line(1,0){2}}
\put(8,0){\line(0,1){1}}
\put(9,0){\line(0,1){1}}
\put(8,0){\makebox(1,1){{\scriptsize $[7]$}}}
\end{picture}
\end{center}
with $i_{8}^{eff}=i_{7}$ and the $i_{8}$-string in $\nu^{(1)}$ is marked by [8] only.
If the box adding so far is not successful, then do as follows.
If the $l^{(1)}$-string is singular, then add two boxes to the $l^{(1)}$-string and follow the algorithm of Case 10 such that $i_{8}^{eff}>i_{6}$.
If the $l^{(1)}$-string is q-singular, then add one box to the $l^{(1)}$-string and follow the algorithm of Case 11.
\begin{flushleft}
Case 13. $b= \framebox{$13$}$.
\end{flushleft}
Find the singular/q-singular/qq-singular string of maximum length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$
We choose the string of the same length by the following preferential rule
\begin{center}
\begin{tabular}{ccc}
singular & q-singular & qq-singular \\ \hline
1 & 2 & 3
\end{tabular}
\end{center}
In addition, we impose the following preferential rules.
\begin{itemize}
\item
If the $l^{(1)}$-string is q-singular/qq-singular of any type, then find a type-0 singular string of length $l^{(1)\prime}$ in $\Tilde{\nu}^{(1)}$ with $[l^{(1)\prime}]_{eff}=[l^{(1)}]_{eff}-1$ such that there exists a singular string of length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $[l^{(1)\prime}+6]_{eff}=l^{(2)}+2$.
If such an $l^{(1)\prime}$-string exists, then reset the $l^{(1)}$-string to be the $l^{(1)\prime}$-string.
\item
If the $l^{(1)}$-string is qq-singular of type-II, then find a singular string of length $l^{(1)\prime}=l^{(1)}-4$ in $\Tilde{\nu}^{(1)}$ such that there exists a singular string of length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $[l^{(1)\prime}+6]_{eff}=l^{(2)}+2$.
If such an $l^{(1)\prime}$-string exists, then reset the $l^{(1)}$-string to be the $l^{(1)\prime}$-string.
\item
If the $l^{(1)}$-string is q-singular of type-I, then find a singular string of length $l^{(1)\prime}=l^{(1)}-1$ in $\Tilde{\nu}^{(1)}$ such that there exists a singular string of length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $[l^{(1)\prime}+5]_{eff}=l^{(2)}+2$.
If such an $l^{(1)\prime}$-string exists, then reset the $l^{(1)}$-string to be the $l^{(1)\prime}$-string.
\end{itemize}
The box adding goes as follows.
If the $l^{(1)}$-string is qq-singular, then add one box to the $l^{(1)}$-string and follow the algorithm of Case 12.
If the $l^{(1)}$-string is singular of type-0, then find a singular string of length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $[l^{(1)}+6]_{eff}=l^{(2)}+2$.
If such a string exists, then add six boxes to the $l^{(1)}$-string, two boxes to the $l^{(2)}$-string, and follow the algorithm of Case 2.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(13,2)
\put(0,0){\line(1,0){7}}
\put(0,1){\line(1,0){7}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
\put(4,0){\line(0,1){1}}
\put(5,0){\line(0,1){1}}
\put(6,0){\line(0,1){1}}
\put(7,0){\line(0,1){1}}
\put(3,1){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[9]$}}}
\put(2,0){\makebox(1,1){{\scriptsize $[8]$}}}
\put(3,0){\makebox(1,1){{\scriptsize $[7]$}}}
\put(4,0){\makebox(1,1){{\scriptsize $[4]$}}}
\put(5,0){\makebox(1,1){{\scriptsize $[3]$}}}
\put(6,0){\makebox(1,1){{\scriptsize $[2]$}}}
\put(10,0){\line(1,0){3}}
\put(10,1){\line(1,0){3}}
\put(11,0){\line(0,1){1}}
\put(12,0){\line(0,1){1}}
\put(13,0){\line(0,1){1}}
\put(11,0){\makebox(1,1){{\scriptsize $[6]$}}}
\put(12,0){\makebox(1,1){{\scriptsize $[5]$}}}
\end{picture}
\end{center}
If the $l^{(1)}$-string is singular of type-II, then find a singular string of length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $[l^{(1)}+5]_{eff}=l^{(2)}+2$.
If such a string exists, then add five boxes to the $l^{(1)}$-string, two boxes to the $l^{(2)}$-string, and follow the algorithm of Case 3.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(12,2)
\put(0,0){\line(1,0){6}}
\put(0,1){\line(1,0){6}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
\put(4,0){\line(0,1){1}}
\put(5,0){\line(0,1){1}}
\put(6,0){\line(0,1){1}}
\put(2,1){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[9]$}}}
\put(2,0){\makebox(1,1){{\scriptsize $[8]$}}}
\put(3,0){\makebox(1,1){{\scriptsize $[7]$}}}
\put(4,0){\makebox(1,1){{\scriptsize $[4]$}}}
\put(5,0){\makebox(1,1){{\scriptsize $[3]$}}}
\put(9,0){\line(1,0){3}}
\put(9,1){\line(1,0){3}}
\put(10,0){\line(0,1){1}}
\put(11,0){\line(0,1){1}}
\put(12,0){\line(0,1){1}}
\put(10,0){\makebox(1,1){{\scriptsize $[6]$}}}
\put(11,0){\makebox(1,1){{\scriptsize $[5]$}}}
\end{picture}
\end{center}
If the $l^{(1)}$-string is singular of type-I, then find a singular string of length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $[l^{(1)}+4]_{eff}=l^{(2)}+1$.
If such a string exists, then add four boxes to the $l^{(1)}$-string, one box to the $l^{(2)}$-string, and follow the algorithm of Case 5.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(10,2)
\put(0,0){\line(1,0){5}}
\put(0,1){\line(1,0){5}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
\put(4,0){\line(0,1){1}}
\put(5,0){\line(0,1){1}}
\put(1,1){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[9]$}}}
\put(2,0){\makebox(1,1){{\scriptsize $[8]$}}}
\put(3,0){\makebox(1,1){{\scriptsize $[6]$}}}
\put(4,0){\makebox(1,1){{\scriptsize $[5]$}}}
\put(8,0){\line(1,0){2}}
\put(8,1){\line(1,0){2}}
\put(9,0){\line(0,1){1}}
\put(10,0){\line(0,1){1}}
\put(9,0){\makebox(1,1){{\scriptsize $[7]$}}}
\end{picture}
\end{center}
If the $l^{(1)}$-string is singular of type-I, then find a singular string of length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $[l^{(1)}+4]_{eff}=l^{(2)}+2$ .
If such a string exists, then add four boxes to the $l^{(1)}$-string, two boxes to the $l^{(2)}$-string, and follow the algorithm of Case 4 ignoring the q-singular string of length $l^{(1)}-1$ in $\Tilde{\nu}^{(1)}$.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(11,2)
\put(0,0){\line(1,0){5}}
\put(0,1){\line(1,0){5}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
\put(4,0){\line(0,1){1}}
\put(5,0){\line(0,1){1}}
\put(1,1){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[9]$}}}
\put(2,0){\makebox(1,1){{\scriptsize $[8]$}}}
\put(3,0){\makebox(1,1){{\scriptsize $[6]$}}}
\put(4,0){\makebox(1,1){{\scriptsize $[5]$}}}
\put(8,0){\line(1,0){3}}
\put(8,1){\line(1,0){3}}
\put(9,0){\line(0,1){1}}
\put(10,0){\line(0,1){1}}
\put(11,0){\line(0,1){1}}
\put(9,0){\makebox(1,1){{\scriptsize $[7]$}}}
\put(10,0){\makebox(1,1){{\scriptsize $[4]$}}}
\end{picture}
\end{center}
If the box adding is not successful, then do as follows.
If the $l^{(1)}$-string is singular of type-0, then find a singular string of length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $[l^{(1)}+3]_{eff}=l^{(2)}+2$ or a q-singular string of length $l^{(2)}$ satisfying $[l^{(1)}+3]_{eff}=l^{(2)}+1$ .
If such a string exists, then add three boxes to the $l^{(1)}$-string, and follow the algorithm of Case 10.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right)
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(9,2)
\put(0,0){\line(1,0){4}}
\put(0,1){\line(1,0){4}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
\put(4,0){\line(0,1){1}}
\put(1,1){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[9]$}}}
\put(2,0){\makebox(1,1){{\scriptsize $[8]$}}}
\put(3,0){\makebox(1,1){{\scriptsize $[7]$}}}
\put(7,0){\line(1,0){2}}
\put(7,1){\line(1,0){2}}
\put(8,0){\line(0,1){1}}
\put(9,0){\line(0,1){1}}
\put(8,0){\makebox(1,1){{\scriptsize $x$}}}
\end{picture}
\end{center}
where
\setlength{\unitlength}{10pt}
\begin{picture}(1,1)
\put(0,0){\line(1,0){1}}
\put(0,1){\line(1,0){1}}
\put(0,0){\line(0,1){1}}
\put(1,0){\line(0,1){1}}
\put(0,0){\makebox(1,1){{\scriptsize $x$}}}
\end{picture}
is
\setlength{\unitlength}{10pt}
\begin{picture}(2,1)
\put(0,0){\line(1,0){2}}
\put(0,1){\line(1,0){2}}
\put(0,0){\line(0,1){1}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(0,0){\makebox(1,1){{\scriptsize $[6]$}}}
\put(1,0){\makebox(1,1){{\scriptsize $[5]$}}}
\end{picture}
or
\setlength{\unitlength}{10pt}
\begin{picture}(1,1)
\put(0,0){\line(1,0){1}}
\put(0,1){\line(1,0){1}}
\put(0,0){\line(0,1){1}}
\put(1,0){\line(0,1){1}}
\put(0,0){\makebox(1,1){{\scriptsize $[6]$}}}
\end{picture}
with $i_{9}^{eff}=i_{6}$.
If the box adding is not successful, then do as follows.
If the $l^{(1)}$-string is singular of any type, then find a singular string of length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $[l^{(1)}+3]_{eff}-1\geq l^{(2)}+2$ or a q-singular string of length $l^{(2)}$ satisfying $[l^{(1)}+3]_{eff}-1\geq l^{(2)}+1$.
If such a string exists, then add three boxes to the $l^{(1)}$-string, and follow the algorithm of Case 10.
This corresponds to the box marking $\nu^{(1)}$ and $\nu^{(2)}$ depicted above but with $i_{9}^{eff}>i_{6}$.
If the $l^{(1)}$-string is q-singular of type-0, then find a singular string of length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $[l^{(1)}+3]_{eff}=l^{(2)}+1$.
If such a string exists, then add three boxes to the $l^{(1)}$-string, one box to the $l^{(2)}$-string, and follow the algorithm of Case 7.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(9,2)
\put(0,0){\line(1,0){4}}
\put(0,1){\line(1,0){4}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
\put(4,0){\line(0,1){1}}
\put(0,1){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[9]$}}}
\put(2,0){\makebox(1,1){{\scriptsize $[8]$}}}
\put(3,0){\makebox(1,1){{\scriptsize $[6]$}}}
\put(7,0){\line(1,0){2}}
\put(7,1){\line(1,0){2}}
\put(8,0){\line(0,1){1}}
\put(9,0){\line(0,1){1}}
\put(8,0){\makebox(1,1){{\scriptsize $[7]$}}}
\end{picture}
\end{center}
If the $l^{(1)}$-string is q-singular of type-II, then find a singular string of length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $[l^{(1)}+2]_{eff}=l^{(2)}+1$ or $[l^{(1)}+2]_{eff}=l^{(2)}+2$.
If such a string exists, then add two boxes to the $l^{(1)}$-string and follow the algorithm of Case 11 such that the length of the box-added $i_{7}$-string is equal to $[l^{(1)}+2]_{eff}$.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right)
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(9,2)
\put(0,0){\line(1,0){3}}
\put(0,1){\line(1,0){3}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
\put(2,1){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[9]$}}}
\put(2,0){\makebox(1,1){{\scriptsize $[8]$}}}
\put(6,0){\line(1,0){2}}
\put(6,1){\line(1,0){2}}
\put(7,0){\line(0,1){1}}
\put(8,0){\line(0,1){1}}
\put(7,0){\makebox(1,1){{\scriptsize $x$}}}
\end{picture}
\end{center}
where
\setlength{\unitlength}{10pt}
\begin{picture}(1,1)
\put(0,0){\line(1,0){1}}
\put(0,1){\line(1,0){1}}
\put(0,0){\line(0,1){1}}
\put(1,0){\line(0,1){1}}
\put(0,0){\makebox(1,1){{\scriptsize $x$}}}
\end{picture}
is
\setlength{\unitlength}{10pt}
\begin{picture}(2,1)
\put(0,0){\line(1,0){2}}
\put(0,1){\line(1,0){2}}
\put(0,0){\line(0,1){1}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(0,0){\makebox(1,1){{\scriptsize $[7]$}}}
\put(1,0){\makebox(1,1){{\scriptsize $[4]$}}}
\end{picture}
or
\setlength{\unitlength}{10pt}
\begin{picture}(1,1)
\put(0,0){\line(1,0){1}}
\put(0,1){\line(1,0){1}}
\put(0,0){\line(0,1){1}}
\put(1,0){\line(0,1){1}}
\put(0,0){\makebox(1,1){{\scriptsize $[7]$}}}
\end{picture}
with $i_{9}^{eff}=i_{7}$.
If the box adding so far is not successful, then do as follows.
If the $l^{(1)}$-string is q-singular of any type, then find a singular string of length $l^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $[l^{(1)}+2]_{eff}-1\geq l^{(2)}+1$.
If such a string exists, then add two boxes to the $l^{(1)}$-string and follow the algorithm of Case 11 such that the length of the box-added $i_{7}$-string is smaller than $[l^{(1)}+2]_{eff}$.
\begin{flushleft}
Case 14. $b= \framebox{$14$}$.
\end{flushleft}
Find two longest singular strings of length $l_{1}^{(2)}$ and $l_{2}^{(2)}(\leq l_{1}^{(2)})$ in $\Tilde{\nu}^{(2)}$.
If $l_{1}^{(2)}=l_{2}^{(2)}$, then find the singular string of length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$ satisfying $[l^{(1)}+6]_{eff}=l_{1}^{(2)}$.
If such a string exists, then add six boxes to the $l^{(1)}$-string, two boxes to the $l_{1}^{(2)}$- and $l_{2}^{(2)}$-strings, and terminate the box adding.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(13,2)
\put(0,0){\line(1,0){7}}
\put(0,1){\line(1,0){7}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
\put(4,0){\line(0,1){1}}
\put(5,0){\line(0,1){1}}
\put(6,0){\line(0,1){1}}
\put(7,0){\line(0,1){1}}
\put(3,1){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[9]$}}}
\put(2,0){\makebox(1,1){{\scriptsize $[8]$}}}
\put(3,0){\makebox(1,1){{\scriptsize $[7]$}}}
\put(4,0){\makebox(1,1){{\scriptsize $[4]$}}}
\put(5,0){\makebox(1,1){{\scriptsize $[3]$}}}
\put(6,0){\makebox(1,1){{\scriptsize $[2]$}}}
\put(10,0){\line(1,0){3}}
\put(10,1){\line(1,0){3}}
\put(10,2){\line(1,0){3}}
\put(11,0){\line(0,1){2}}
\put(12,0){\line(0,1){2}}
\put(13,0){\line(0,1){2}}
\put(11,0){\makebox(1,1){{\scriptsize $[10]$}}}
\put(11,1){\makebox(1,1){{\scriptsize $[6]$}}}
\put(12,0){\makebox(1,1){{\scriptsize $[1]$}}}
\put(12,1){\makebox(1,1){{\scriptsize $[5]$}}}
\end{picture}
\end{center}
If the box adding is not successful, then do as follows.
Find the singular string of maximum length $l_{1}^{(2)}$ in $\Tilde{\nu}^{(2)}$.
Then find the singular string of length $l_{2}^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $l_{2}^{(2)}=l_{1}^{(2)}-1$ and the type-0 singular string of length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$ satisfying $[l^{(1)}+6]_{eff}=l_{1}^{(2)}+1$.
If such strings exist, then add six boxes to the $l^{(1)}$-string, one box to the $l_{1}^{(2)}$-string, two boxes to the $l_{2}^{(2)}$-string, and follow the algorithm of Case 2.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(13,2)
\put(0,0){\line(1,0){7}}
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\put(7,0){\line(0,1){1}}
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\put(12,1){\makebox(1,1){{\scriptsize $[10]$}}}
\end{picture}
\end{center}
If the box adding is not successful, then do as follows.
Find the singular string of maximum length $l_{1}^{(2)}$ in $\Tilde{\nu}^{(2)}$.
Then find the singular string of length $l_{2}^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $l_{2}^{(2)}=l_{1}^{(2)}-1$ and the type-II singular string of length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$ satisfying $[l^{(1)}+5]_{eff}=l_{1}^{(2)}+1$.
If such strings exist, then add five boxes to the $l^{(1)}$-string, one box to the $l_{1}^{(2)}$-string, two boxes to the $l_{2}^{(2)}$-string, and follow the algorithm of Case 3.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(13,2)
\put(0,0){\line(1,0){6}}
\put(0,1){\line(1,0){6}}
\put(1,0){\line(0,1){1}}
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\put(10,0){\makebox(1,1){{\scriptsize $[6]$}}}
\put(11,0){\makebox(1,1){{\scriptsize $[5]$}}}
\put(11,1){\makebox(1,1){{\scriptsize $[10]$}}}
\end{picture}
\end{center}
If the box adding is not successful, then do as follows.
Find the singular string of maximum length $l_{1}^{(2)}$ in $\Tilde{\nu}^{(2)}$.
Then find the singular string of length $l_{2}^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $l_{2}^{(2)}=l_{1}^{(2)}-1$ and the type-I singular string of length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$ satisfying $[l^{(1)}+4]_{eff}=l_{1}^{(2)}+1$.
If such strings exist, then add four boxes to the $l^{(1)}$-string, one box to the $l_{1}^{(2)}$-string, two boxes to the $l_{2}^{(2)}$-string, and follow the algorithm of Case 4 ignoring the q-singular string of length $l^{(1)}-1$ in $\Tilde{\nu}^{(1)}$.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(10,2)
\put(0,0){\line(1,0){5}}
\put(0,1){\line(1,0){5}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
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\put(4,0){\line(0,1){1}}
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\put(10,0){\line(0,1){2}}
\put(11,0){\line(0,1){2}}
\put(9,0){\makebox(1,1){{\scriptsize $[7]$}}}
\put(10,0){\makebox(1,1){{\scriptsize $[4]$}}}
\put(10,1){\makebox(1,1){{\scriptsize $[10]$}}}
\end{picture}
\end{center}
In this case, if there exists a type-II singular string of length $l^{(1)\prime}=l^{(1)}-1$ in $\Tilde{\nu}^{(1)}$, then discard the box adding above, reset the $l^{(1)}$-string to be the $l^{(1)\prime}$-string, and follow the previous box marking.
If the box adding is not successful, then do as follows.
Find the singular string of maximum length $l_{1}^{(2)}$ in $\Tilde{\nu}^{(2)}$.
Then find the singular string of length $l_{2}^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $l_{2}^{(2)}=l_{1}^{(2)}$ and the type-I singular string of length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$ satisfying $[l^{(1)}+4]_{eff}=l_{1}^{(2)}+1$.
If such strings exist, then add four boxes to the $l^{(1)}$-string, one box to the $l_{1}^{(2)}$- and $l_{2}^{(2)}$-strings, and follow the algorithm of Case 5.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(10,2)
\put(0,0){\line(1,0){5}}
\put(0,1){\line(1,0){5}}
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\put(3,0){\line(0,1){1}}
\put(4,0){\line(0,1){1}}
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\put(9,0){\makebox(1,1){{\scriptsize $[7]$}}}
\put(9,1){\makebox(1,1){{\scriptsize $[10]$}}}
\end{picture}
\end{center}
If the box adding is not successful, then do as follows.
Find the singular string of maximum length $l_{1}^{(2)}$ in $\Tilde{\nu}^{(2)}$.
Then find the singular string of length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$ satisfying $[l^{(1)}+3]_{eff}=l_{1}^{(2)}+1$.
If such strings exist, then add three boxes to the $l^{(1)}$-string, one box to the $l_{1}^{(2)}$-string, and follow the algorithm of Case 10.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right)
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(9,1)
\put(0,0){\line(1,0){4}}
\put(0,1){\line(1,0){4}}
\put(1,0){\line(0,1){1}}
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\put(8,0){\makebox(1,1){{\scriptsize $[10]$}}}
\end{picture}
\end{center}
with $i_{9}^{eff}=i_{10}$. When the box-added $i_{9}$-string is type-0, the length of the box-added $i_{6}$-string is smaller than or equal to $i_{9}^{eff}$ and when the box-added $i_{9}$-string is type-I/II, the length of the box-added $i_{6}$-string is strictly smaller than $i_{9}^{eff}$.
If the box adding is not successful, then do as follows.
Find the singular string of maximum length $l_{1}^{(2)}$ in $\Tilde{\nu}^{(2)}$.
Then find the type-II singular string of length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$ satisfying $[l^{(1)}+2]_{eff}=l_{1}^{(2)}+1$.
If such strings exist, then add two boxes to the $l^{(1)}$-string, one box to the $l_{1}^{(2)}$-string, and follow the algorithm of Case 11.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right)
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(8,2)
\put(0,0){\line(1,0){3}}
\put(0,1){\line(1,0){3}}
\put(1,0){\line(0,1){1}}
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\put(6,0){\line(1,0){2}}
\put(6,1){\line(1,0){2}}
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\put(8,0){\line(0,1){1}}
\put(7,0){\makebox(1,1){{\scriptsize $[10]$}}}
\end{picture}
\end{center}
with $i_{9}^{eff}=i_{10}$.
If the box adding is not successful, then do as follows.
Find the singular string of maximum length $l_{1}^{(2)}$ in $\Tilde{\nu}^{(2)}$.
Then find the type-I singular string of length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$ satisfying $[l^{(1)}+1]_{eff}=l_{1}^{(2)}+1$.
If such strings exist, then add one boxes to the $l^{(1)}$-string, one box to the $l_{1}^{(2)}$-string, and follow the algorithm of Case 12.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right)
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(7,2)
\put(0,0){\line(1,0){2}}
\put(0,1){\line(1,0){2}}
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\put(7,0){\line(0,1){1}}
\put(6,0){\makebox(1,1){{\scriptsize $[10]$}}}
\end{picture}
\end{center}
with $i_{9}^{eff}=i_{10}$.
In this case, if there exists a type-II singular string of length $l^{(1)\prime}=l^{(1)}-1$ in $\Tilde{\nu}^{(1)}$, then discard the box adding above, reset the $l^{(1)}$-string to be the $l^{(1)\prime}$-string, and follow the previous box marking.
If the box adding so far is not successful, do as follows.
Let $l_{1}^{(2)}$ and $l_{2}^{(2)}(\leq l_{1}^{(2)})$ be the two largest length of singular strings in $\Tilde{\nu}^{(2)}$.
Find the q-singular/qq-singular string of maximum length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$ such that it is
\begin{itemize}
\item[(1)]
a q-singular string of type-0 with $[l^{(1)}+3]_{eff}=l_{1}^{(2)}+1=l_{2}^{(2)}+1$ or
\item[(2)]
a q-singular string of type-0/I with $[l^{(1)}+2]_{eff}=l_{1}^{(2)}+1$ or
\item[(3)]
a q-singular string of type-II with $[l^{(1)}+1]_{eff}=l_{1}^{(2)}+1$ or
\item[(4)]
a qq-singular string of type-0 with $[l^{(1)}+1]_{eff}=l_{1}^{(2)}+1$.
\end{itemize}
We choose the q-singular string when there exist q-singular and qq-singular strings of the same length.
In addition, we impose the following preferential rules.
\begin{itemize}
\item
If the $l^{(1)}$-string is q-singular/qq-singular of type-0 or q-singular of type-I, then find a type-0 singular string of length $l^{(1)\prime}$ in $\Tilde{\nu}^{(1)}$ with $[l^{(1)\prime}]_{eff}=[l^{(1)}]_{eff}-1$ such that there exist singular strings of length $l_{1}^{(2)}$ and $l_{2}^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $[l^{(1)\prime}+6]_{eff}=l_{1}^{(2)}+1=l_{2}^{(2)}+2$.
If such an $l^{(1)\prime}$-string exists, then reset the $l^{(1)}$-string to be the $l^{(1)\prime}$-string.
\item
If the $l^{(1)}$-string is q-singular of type-II, then find a string of length $l^{(1)\prime}=l^{(1)}-4$ in $\Tilde{\nu}^{(1)}$ such that there exist singular strings of length $l_{1}^{(2)}$ and $l_{2}^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $[l^{(1)\prime}+6]_{eff}=l_{1}^{(2)}+1=l_{2}^{(2)}+2$.
If such an $l^{(1)\prime}$-string exists, then reset the $l^{(1)}$-string to be the $l^{(1)\prime}$-string.
\item
If the $l^{(1)}$-string is q-singular of type-0, then find a string of length $l^{(1)\prime}=l^{(1)}-2$ in $\Tilde{\nu}^{(1)}$ such that there exist singular strings of length $l_{1}^{(2)}$ and $l_{2}^{(2)}$ in $\Tilde{\nu}^{(2)}$ satisfying $[l^{(1)\prime}+5]_{eff}=l_{1}^{(2)}+1=l_{2}^{(2)}+2$.
If such an $l^{(1)\prime}$-string exists, then reset the $l^{(1)}$-string to be the $l^{(1)\prime}$-string.
\end{itemize}
If the box adding is not successful, then do as follows.
If the $l^{(1)}$-string is singular, then do as before.
Find two longest singular strings of length $l_{1}^{(2)}$ and $l_{2}^{(2)}$.
If $l_{1}^{(2)}=l_{2}^{(2)}$, then find the type-0 q-singular string of length $l^{(1)}(\geq 3)$ in $\Tilde{\nu}^{(1)}$ satisfying $[l^{(1)}+3]_{eff}=l_{1}^{(2)}+1$.
If such a string exists, then add three boxes to the $l^{(1)}$-string, one box to the $l_{1}^{(2)}$- and $l_{2}^{(2)}$-strings, and follow the algorithm of Case 7.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(9,2)
\put(0,0){\line(1,0){4}}
\put(0,1){\line(1,0){4}}
\put(1,0){\line(0,1){1}}
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\put(8,1){\makebox(1,1){{\scriptsize $[10]$}}}
\end{picture}
\end{center}
If the box adding is not successful, then do as follows.
Find the singular strings of length $l_{1}^{(2)}$ in $\Tilde{\nu}^{(2)}$.
Then find the type-0/I q-singular string of length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$ satisfying $[l^{(1)}+2]_{eff}=l_{1}^{(2)}+1$.
If such a string exists, then add two boxes to the $l^{(1)}$-string, one box to the $l_{1}^{(2)}$-string, and follow the algorithm of Case 11.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(8,1)
\put(0,0){\line(1,0){3}}
\put(0,1){\line(1,0){3}}
\put(1,0){\line(0,1){1}}
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\put(6,0){\line(1,0){2}}
\put(6,1){\line(1,0){2}}
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\put(8,0){\line(0,1){1}}
\put(7,0){\makebox(1,1){{\scriptsize $[10]$}}}
\end{picture}
\end{center}
with $i_{9}^{eff}=i_{10}$.
If the box adding is not successful, then do as follows.
Find the singular strings of length $l_{1}^{(2)}$ in $\Tilde{\nu}^{(2)}$.
Then find the type-II q-singular string of length $l^{(1)}$ in $\Tilde{\nu}^{(1)}$ satisfying $[l^{(1)}+1]_{eff}=l_{1}^{(2)}+1$.
If such a string exists, then add one box to the $l^{(1)}$-string, one box to the $l_{1}^{(2)}$-string, and follow the algorithm of Case 12.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(8,2)
\put(0,0){\line(1,0){3}}
\put(0,1){\line(1,0){3}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
\put(0,1){\makebox(2,1){$\downarrow$}}
\put(2,0){\makebox(1,1){{\scriptsize $[9]$}}}
\put(6,0){\line(1,0){2}}
\put(6,1){\line(1,0){2}}
\put(7,0){\line(0,1){1}}
\put(8,0){\line(0,1){1}}
\put(7,0){\makebox(1,1){{\scriptsize $[10]$}}}
\end{picture}
\end{center}
with $i_{9}^{eff}=i_{10}$.
If the box adding is not successful, then do as follows.
Find the singular strings of length $l_{1}^{(2)}$ in $\Tilde{\nu}^{(2)}$.
Then find the type-0 qq-singular string of length $l^{(1)}(\geq 4)$ in $\Tilde{\nu}^{(1)}$ satisfying $[l^{(1)}+1]_{eff}=l_{1}^{(2)}+1$.
If such a string exists, then add one box to the $l^{(1)}$-string, one box to the $l_{1}^{(2)}$-string, and follow the algorithm of Case 12.
This corresponds to the following box marking in $\nu^{(1)}$ (left) and $\nu^{(2)}$ (right)
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(7,2)
\put(0,0){\line(1,0){2}}
\put(0,1){\line(1,0){2}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(0,1){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[9]$}}}
\put(5,0){\line(1,0){2}}
\put(5,1){\line(1,0){2}}
\put(6,0){\line(0,1){1}}
\put(7,0){\line(0,1){1}}
\put(6,0){\makebox(1,1){{\scriptsize $[10]$}}}
\end{picture}
\end{center}
with $i_{9}^{eff}=i_{10}$.
If the box adding is not successful, then do as follows.
Find the singular string of maximum length in $\Tilde{\nu}^{(2)}$.
Then add one box to this string and follow the algorithm of Case 13.
This corresponds to the box marking in $\nu$ depicted above but with $i_{9}^{eff}<i_{10}$.
\section{Proof of Theorem~\ref{th:main}}
Theorem~\ref{th:main} is proved in this section.
The following notation is used.
Let $(\nu,J)\in \mathrm{RC}(\lambda,L)$, $b=\gamma (\nu,J)$, $\rho=\lambda-\mathrm{wt}(b)$, and $(\Tilde{\nu},\Tilde{J})=\delta_{\theta} (\nu,J)$.
For $(\nu,J)\in \mathrm{RC}(\lambda,L)$, we define $\Delta c(\nu,J)=c(\nu,J)-c(\delta_{\theta} (\nu,J))$.
The following lemma is essentially the same as~\cite[Lemma 5.1]{OSS03a}.
\begin{lem} \label{lem:stat}
To prove that Eq.~\eqref{eq:main} holds, it suffices to show that it holds for $L=1$ and that for $L\geq 2$ with $\Phi (\nu,J)=b_{1}\otimes \cdots \otimes b_{L}$, we have
\begin{equation} \label{eq:dc}
\Delta c(\nu,J)=-\alpha_{1}^{(2)}+\chi (b_{1}=\emptyset)
\end{equation}
and
\begin{equation} \label{eq:Ha}
H(b_{1}\otimes b_{2})=\Tilde{\alpha} _{1}^{(2)}-\alpha_{1}^{(2)}+\chi (b_{1}=\emptyset)-\chi (b_{2}=\emptyset),
\end{equation}
where $\alpha _{1}^{(2)}$ and $\Tilde{\alpha}_{1}^{(2)}$ are the lengths of the first columns in $\nu ^{(2)}$ and $\Tilde{\nu}^{(2)}$ respectively, and $\delta_{\theta} (\nu,J)=(\Tilde{\nu},\Tilde{J})$.
\end{lem}
There are five things that must be verified:
\begin{itemize}
\item[(I)] $\rho$ is dominant.
\item[(II)] $(\Tilde{\nu},\Tilde{J})\in \mathrm{RC}(\rho,L-1)$.
\item[(III)] $b$ can be appended to $(\Tilde{\nu},\Tilde{J})$.
\item[(IV)] \eqref{eq:dc} in Lemma~\ref{lem:stat} holds.
\item[(V)] \eqref{eq:Ha} in Lemma~\ref{lem:stat} holds.
\end{itemize}
Parts (I) and (II) show that $\delta_{\theta}$ is well-defined.
Part (III) shows that $\delta_{\theta}$ has an inverse.
Parts (IV) and (V) suffice to prove that $\Phi$ preserves statistics.
We omit the proof of (III) as it is very similar to the proof of well-definedness.
We need several preliminary lemmas on the convexity and nonnegativity of the vacancy numbers $p_{i}^{(a)}$.
\begin{lem} \label{lem:large_i}
For large $i$, we have
\[
p_{i}^{(a)}=\lambda_{a}
\]
where $\lambda_{a}$ is defined by $\lambda=\sum_{a\in I_{0}}\lambda_{a}\Bar{\Lambda}_{a}$.
\end{lem}
\begin{proof}
This follows from the formula for the vacancy number Eq.~\eqref{eq:vacancy} and the constraint Eq.~\eqref{eq:admissible}.
\end{proof}
Direct calculations show that
\begin{align}
-p_{3i-2}^{(1)}+2p_{3i-1}^{(1)}-p_{3i}^{(1)}&=-2m_{3i-1}^{(1)}, \label{eq:ddp1a} \\
-p_{3i-1}^{(1)}+2p_{3i}^{(1)}-p_{3i+1}^{(1)}&=-2m_{3i}^{(1)}+m_{i}^{(2)}, \label{eq:ddp1b}\\
-p_{3i}^{(1)}+2p_{3i+1}^{(1)}-p_{3i+2}^{(1)}&=-2m_{3i+1}^{(1)}, \label{eq:ddp1c}
\end{align}
and
\begin{align}
-p_{i-1}^{(2)}+2p_{i}^{(2)}-p_{i+1}^{(2)}=&3m_{3i}^{(1)}+2(m_{3i-1}^{(1)}+m_{3i+1}^{(1)})
+m_{3i-2}^{(1)}+m_{3i+2}^{(1)} \label{eq:ddp2} \\
-&2m_{i}^{(2)}. \nonumber
\end{align}
In particular, these equations imply the convexity condition
\begin{equation} \label{eq:convexity}
p_{i}^{(a)}\geq \frac{1}{2}(p_{i-1}^{(a)}+p_{i+1}^{(a)})\quad \text{if } m_{i}^{(a)}=0.
\end{equation}
The following two lemmas follow immediately from Lemma~\ref{lem:large_i} and the convexity condition Eq.~\eqref{eq:convexity}.
\begin{lem} \label{lem:vacancy01}
Let $\nu$ be a configuration and let $n$ be $0$ or $1$.
The following are equivalent:
\begin{itemize}
\item[(1)]
$p_{i}^{(a)}\geq n$ for all $i\in \mathbb{Z}_{>0}$, $a\in I_{0}$;
\item[(2)]
$p_{i}^{(a)}\geq n$ for all $i\in \mathbb{Z}_{>0}$, $a\in I_{0}$ such that $m_{i}^{(a)}>0$.
\end{itemize}
\end{lem}
\begin{lem} \label{lem:vacancy23}
Let $\nu$ be a configuration and let $n$ be $2$ or $3$.
Let $p_{i_{0}}^{(a)}\geq n$.
The following are equivalent:
\begin{itemize}
\item[(1)]
$p_{i}^{(a)}\geq n$ for all $i\geq i_{0}$, $a\in I_{0}$;
\item[(2)]
$p_{i}^{(a)}\geq n$ for all $i\geq i_{0}$, $a\in I_{0}$ such that $m_{i}^{(a)}>0$.
\end{itemize}
\end{lem}
\begin{lem} \label{lem:vacancy1}
Suppose that
\[
p_{i}^{(a)}\geq 1, \ldots, p_{j}^{(a)}\geq 1 \quad (i<j)
\]
with $m_{i}^{(a)}=\cdots =m_{j}^{(a)}=0$.
If $p_{k}^{(a)}=1$ for some $k$ $(i\leq k\leq j)$, then
\[
p_{i}^{(a)}= 1, \ldots, p_{j}^{(a)}= 1 \quad (i<j).
\]
\end{lem}
\begin{proof}
This follows immediately from the convexity condition Eq.~\eqref{eq:convexity}.
\end{proof}
\begin{proof}[Proof of (I)]
Here we show $\rho=\lambda-\mathrm{wt}(b)$ is dominant with $l\geq 1$.
Let us assume that $\nu^{(1)}\neq \emptyset$ and $\nu^{(2)}\neq \emptyset$ and let $l^{(a)}$ be the largest part in $\nu^{(a)}$ $(a=1,2)$.
In particular, $m_{l^{(1)}}^{(1)}\neq 0$ and $m_{l^{(2)}}^{(2)}\neq 0$.
The proof for the case when $\nu^{(1)}= \emptyset$ or $\nu^{(2)}= \emptyset$ is much simpler and we omit the details.
\begin{flushleft}
Case 0. $b= \framebox{$\emptyset$}$.
\end{flushleft}
Since $\mathrm{wt}(\framebox{$\emptyset$})=0$, there is nothing to prove.
\begin{flushleft}
Case 1. $b= \framebox{$1$}$.
\end{flushleft}
Since $\mathrm{wt}(\framebox{$1$})=\Bar{\Lambda}_{2}$, we must show that $\lambda_{2}\geq 1$.
Suppose that $\lambda_{2}=0$ so that $p_{\infty}^{(2)}=0$.
The convexity condition implies $p_{l^{(2)}}^{(2)}=0$ so that $\nu^{(2)}$ has a singular string of length $l^{(2)}$, which contradicts $b= \framebox{$1$}$.
Hence $\lambda_{2}\geq 1$.
\begin{flushleft}
Case 2. $b= \framebox{$2$}$.
\end{flushleft}
Since $\mathrm{wt}(\framebox{$2$})=3\Bar{\Lambda}_{1}-\Bar{\Lambda}_{2}$, we must show that $\lambda_{1}\geq 3$.
We first suppose that
\[
\lambda_{1}=-2\sum_{j=1}^{l^{(1)}}j m_{j}^{(1)}+3\sum_{j=1}^{l^{(2)}}j m_{j}^{(2)}=0.
\]
Then
\[
p_{l^{(1)}}^{(1)}=-2\sum_{j=1}^{l^{(1)}}j m_{j}^{(1)}+\sum_{j=1}^{l^{(2)}}\min(l^{(1)},3j) m_{j}^{(2)}
=\sum_{j=1}^{l^{(2)}}\left( \min(l^{(1)},3j)-3j \right) m_{j}^{(2)}.
\]
If $l^{(1)}<3l^{(2)}$, then $p_{l^{(1)}}^{(1)}<0$.
Hence $l^{(1)}\geq 3l^{(2)}$, which yields $p_{l^{(1)}}^{(1)}=0$.
This implies that $\nu^{(1)}$ has a singular string of effective length $(\geq l^{(2)})$, which contradicts $b= \framebox{$2$}$.
Secondly, we suppose that $\lambda_{1}=1$.
Then
\[
p_{l^{(1)}}^{(1)}=1+\sum_{j=1}^{l^{(2)}}\left( \min(l^{(1)},3j)-3j \right) m_{j}^{(2)}.
\]
If $m_{l^{(2)}}^{(2)}=1$, then $p_{l^{(1)}}^{(1)}<0$ when $l^{(1)}\leq 3l^{(2)}-2$.
If $m_{l^{(2)}}^{(2)}\geq 2$, then $p_{l^{(1)}}^{(1)}<0$ when $l^{(1)}\leq 3l^{(2)}-1$.
Hence $l^{(1)}\geq 3l^{(2)}-1$ if $m_{l^{(2)}}^{(2)}=1$ and $l^{(1)}\geq 3l^{(2)}$ if $m_{l^{(2)}}^{(2)}\geq 2$.
Simple calculations yield two possible cases:
\begin{itemize}
\item[(1)] $m_{l^{(2)}}^{(2)}=1$, $l^{(1)}=3l^{(2)}-1$, and $p_{l^{(1)}}^{(1)}=0$.
\item[(2)] $m_{l^{(2)}}^{(2)}\geq 1$, $l^{(1)}\geq 3l^{(2)}$, and $p_{l^{(1)}}^{(1)}=1$.
\end{itemize}
Case (1) contradicts $b= \framebox{$2$}$ because $\nu^{(1)}$ has a type-I singular string of effective length $l^{(2)}$.
Case (2) also contradicts $b= \framebox{$2$}$ because $\nu^{(1)}$ has a string (singular or q-singular) of length $(\geq 3l^{(2)})$.
Hence $\lambda_{1}\geq 2$.
Thirdly, we suppose that $\lambda_{1}=2$.
By the similar calculations as above yield four possible cases:
\begin{itemize}
\item[(1)] $m_{l^{(2)}}^{(2)}=1$, $l^{(1)}=3l^{(2)}-2$, and $p_{l^{(1)}}^{(1)}=0$.
\item[(2)] $m_{l^{(2)}}^{(2)}=1$, $l^{(1)}=3l^{(2)}-1$, and $p_{l^{(1)}}^{(1)}=1$.
\item[(3)] $m_{l^{(2)}}^{(2)}=2$, $l^{(1)}=3l^{(2)}-1$, and $p_{l^{(1)}}^{(1)}=0$.
\item[(4)] $m_{l^{(2)}}^{(2)}\geq 1$, $l^{(1)}\geq 3l^{(2)}$, and $p_{l^{(1)}}^{(1)}=2$.
\end{itemize}
Case (1) (resp. (3)) implies that $\nu^{(1)}$ has a type-II (resp. I) singular string of effective length $l^{(2)}$.
Case (2) implies that $\nu^{(1)}$ has a type-I string (singular or q-singular) of effective length $l^{(2)}$.
Case (4) implies that $\nu^{(1)}$ has a string (singular or q-singular or qq-singular) of length $(\geq 3l^{(2)})$.
All contradict $b= \framebox{$2$}$ by the algorithm of $\delta_{\theta}$.
Hence, $\lambda_{1}\geq 3$.
\begin{flushleft}
Case 3. $b= \framebox{$3$}$.
\end{flushleft}
In this case, after $[2]$ was marked in $\nu^{(1)}$ the box marking has terminated.
Since $\mathrm{wt}(\framebox{$3$})=\Bar{\Lambda}_{1}$, we must show that $\lambda_{1}\geq 1$.
Suppose that $\lambda_{1}=0$.
As in Case 2, it follows that $\nu^{(1)}$ has a singular string of length ($\leq 3i_{1}$) contradicting $b= \framebox{$3$}$.
Hence $\lambda_{1}\geq 1$.
Note that even if [2] was marked in this singular string, [3] can also be marked in this string.
\begin{flushleft}
Case 4. $b= \framebox{$4$}$.
\end{flushleft}
In this case, after $[3]$ was marked in $\nu^{(1)}$ the box marking has terminated.
Since $\mathrm{wt}(\framebox{$4$})=-\Bar{\Lambda}_{1}+\Bar{\Lambda}_{2}$, we must show that $\lambda_{2}\geq 1$.
We first suppose that
\[
\lambda_{2}=L+\sum_{j=1}^{l^{(1)}}j m_{j}^{(1)}-2\sum_{j=1}^{l^{(2)}}j m_{j}^{(2)}=0.
\]
Then we have $p_{l^{(2)}+1}^{(2)}$ by convexity and
\[
p_{l^{(2)}}^{(2)}=\sum_{j=1}^{l^{(1)}}\left( \min(3l^{(2)},j)-j \right) m_{j}^{(1)}.
\]
If $l^{(1)}>3l^{(2)}$, then $p_{l^{(2)}}^{(2)}<0$.
Hence $l^{(1)}\leq 3l^{(2)}$, which yields $p_{l^{(2)}}^{(2)}=0$ so that all $l^{(2)}$-strings in $\nu^{(2)}$ are singular.
If [3] is marked in a string of of effective length $(<l^{(2)})$, then [4] can be marked in a singular string of length $l^{(2)}$ in $\nu^{(2)}$.
This is a contradiction.
Note that [1] must have been marked in a singular string of length $(< l^{(2)})$ in this case.
Therefore, it suffices to consider the case when [3] had been marked in a string of effective length $l^{(2)}$ in $\nu^{(1)}$.
\begin{itemize}
\item[(1)] $l^{(1)}=3l^{(2)}$.
We show that $m_{l^{(2)}}^{(2)}\geq 2$.
Suppose that $m_{l^{(2)}}^{(2)}=1$.
From Eq.~\eqref{eq:ddp2}, we have
\begin{equation} \label{eq:Icase4}
-p_{l^{(2)}-1}^{(2)}=3m_{3l^{(2)}}^{(1)}+2m_{3l^{(2)}-1}^{(1)}+m_{3l^{(2)}-2}^{(1)}-2.
\end{equation}
However, this does not hold because $m_{3l^{(2)}}^{(1)}=m_{l^{(1)}}^{(1)}> 0$.
Hence $m_{l^{(2)}}^{(2)}\geq 2$ so that [4] can be marked in an $l^{(2)}$-string in $\nu^{(2)}$ even if [1] is already marked in another $l^{(2)}$-string in $\nu^{(2)}$, which contradicts $b= \framebox{$4$}$.
\item[(2)] $l^{(1)}=3l^{(2)}-1$.
Suppose that $m_{l^{(2)}}^{(2)}=1$.
Equation~\eqref{eq:Icase4} still holds, from which we have that $p_{l^{(2)}-1}^{(2)}=0$ because $m_{l~{(1)}}^{(1)}=m_{3l^{(2)}-1}^{(1)}>0$.
Suppose that $m_{j}^{(2)}\neq 0$ for some $j (\leq l^{(2)}-1)$.
Then, $p_{j}^{(2)}\geq 1$ as otherwise [1] would have been marked in a singular string of length ($\leq l^{(2)}-2$) contradicting $b= \framebox{$4$}$.
However, $p_{j}^{(2)}\geq 1$ implies that $p_{l^{(2)}-1}^{(2)}\geq 1$ by convexity, which in turn contradicts Eq.~\eqref{eq:Icase4}.
Therefore it suffices to consider the case when $m_{j}^{(2)}=0$ $(j\leq l^{(2)}-1)$.
We show that $m_{l^{(2)}}^{(2)}\geq 2$.
Suppose that $m_{l^{(2)}}^{(2)}=1$.
Then from Eq.~\eqref{eq:vacancy1}, we have $p_{l^{(1)}}^{(1)}<0$ so that $m_{l^{(2)}}^{(2)}\geq 2$, which implies that [4] can be marked in an $l^{(2)}$-string in $\nu^{(2)}$ even if [1] is already marked in another $l^{(2)}$-string in $\nu^{(2)}$, which contradicts $b= \framebox{$4$}$.
\item[(3)] $l^{(1)}=3l^{(2)}-2$.
In this case, [1] been marked in a singular string of length ($\leq l^{(2)}-1$) in $\nu^{(2)}$.
Therefore, [4] can be marked in an $l^{(2)}$-string in $\nu^{(2)}$, which contradicts $b= \framebox{$4$}$.
\end{itemize}
All cases (a), (b), and (c) contradict $b= \framebox{$4$}$.
Hence $\lambda_{2}\geq 1$.
\begin{flushleft}
Case 5. $b= \framebox{$5$}$.
\end{flushleft}
In this case, after $[4]$ was marked in $\nu^{(2)}$ the box marking has terminated.
Since $\mathrm{wt}(\framebox{$5$})=2\Bar{\Lambda}_{1}-\Bar{\Lambda}_{2}$, we must show that $\lambda_{1}\geq 2$.
We first suppose that $\lambda_{1}=0$.
Then we have $l^{(1)}\geq 3l^{(2)}$ and $p_{l^{(1)}}^{(1)}=0$ as in Case 2.
Note that all $l^{(1)}$-strings in $\nu^{(1)}$ are singular.
\begin{itemize}
\item[(1)]
$l^{(1)}=3l^{(2)}$.
Note that $[3]$ is not marked in an $l^{(1)}$-string (singular) in $\nu^{(1)}$.
Otherwise, $[4]$ would be marked in the same string because it is singular.
Therefore, $[5]$ can be marked in an $l^{(1)}$-string in $\nu^{(1)}$.
\item[(2)]
$l^{(1)}=3l^{(2)}+1$.
Note that $[3]$ can be marked in an $l^{(1)}$-string in $\nu^{(1)}$.
In this case, the effective length is reduced by one ($l^{(2)}+1\rightarrow l^{(2)}$) and [4] is marked in an $l^{(2)}$-string in $\nu^{(2)}$ followed by the box marking of an $l^{(1)}$-string in $\nu^{(1)}$ by $[5]$.
\item[(3)]
$l^{(1)}\geq 3l^{(2)}+2$.
$[3]$ is not marked in the $l^{(1)}$-string.
Otherwise, $[4]$ could not be marked in $\nu^{(2)}$ because the effective length of $l^{(1)}$-strings is larger than $l^{(2)}$.
It is obvious that $[5]$ can be marked in the $l^{(1)}$-string.
\end{itemize}
In three cases above, $[5]$ can be marked in the $l^{(1)}$-string contradicting $b= \framebox{$5$}$.
Next, we suppose that $\lambda_{1}=1$.
As in Case 2, we have two possible cases:
\begin{itemize}
\item[(1)] $m_{l^{(2)}}^{(2)}=1$, $l^{(1)}=3l^{(2)}-1$, and $p_{l^{(1)}}^{(1)}=0$.
\item[(2)] $m_{l^{(2)}}^{(2)}\geq 1$, $l^{(1)}\geq 3l^{(2)}$, and $p_{l^{(1)}}^{(1)}=1$.
\end{itemize}
Note that $l^{(1)}$-strings in $\nu^{(1)}$ are singular or q-singular.
Here we assume that $[3]$ is marked in an $l^{(1)}$-string as otherwise $[5]$ could be marked in this string contradicting $b= \framebox{$5$}$.
Note that case (1) must be excluded.
This is shown as follows.
If $[1]$ were marked in a singular row of length $l^{(2)}$, then $[4]$ could not be marked in $\nu^{(2)}$ because $m_{l^{(2)}}^{(2)}=1$.
If $[1]$ were marked in a singular row of length $(< l^{(2)})$, then $[4]$ would be marked on the left of the box marked by $[3]$ in the $l^{(1)}$-string.
Thus, case (b) survives.
If one of the $l^{(1)}$-strings is singular, the same arguments as those in the case of $\lambda_{1}=0$ hold and lead to a contradiction.
Therefore all the $l^{(1)}$-strings must be q-singular so that $[4]$ is not marked in an $l^{(1)}$-string.
This implies $[5]$ can be marked in an $l^{(1)}$-string contradicting $b= \framebox{$5$}$.
Hence we have $\lambda_{1}\geq 2$.
\begin{flushleft}
Case 6. $b= \framebox{$6$}$.
\end{flushleft}
In this case, after $[4]$ was marked in $\nu^{(1)}$ the box marking has terminated.
Since $\mathrm{wt}(\framebox{$6$})=-3\Bar{\Lambda}_{1}+2\Bar{\Lambda}_{2}$, we must show that $ \lambda_{2}\geq 2$.
We first suppose that $\lambda_{2}=0$.
As in Case 4 this results in a contradiction.
Secondly we suppose that $\lambda_{2}=1$.
Then $p_{l^{(2)}+1}^{(2)}=1$ by convexity and
\[
p_{l^{(2)}}^{(2)}=1+\sum_{j=1}^{l^{(1)}}\left( \min(3l^{(2)},j)-j \right) m_{j}^{(1)}.
\]
We have following two possible cases in a similar fashion as in Case 2:.
\begin{itemize}
\item[(1)]
$m_{l^{(1)}}^{(1)}=1$, $l^{(1)}=3l^{(2)}+1$, and $p_{l^{(2)}}^{(2)}=0$.
\item[(2)]
$m_{l^{(1)}}^{(1)}\geq 1$, $l^{(1)}\leq 3l^{(2)}$, and $p_{l^{(2)}}^{(2)}=1$.
\end{itemize}
Let us show that we have a contradiction in both cases.
In case (1), we show that $m_{l^{(2)}}^{(2)}\geq 2$, which in turn leads to a contradiction.
Suppose that $m_{l^{(2)}}^{(2)}=1$.
From Eq.~\eqref{eq:ddp2}, we have
\[
-p_{l^{(2)}-1}^{(2)}-1=3m_{3l^{(2)}}^{(1)}+2(m_{3l^{(2)}+1}^{(1)}+m_{3l^{(2)}-1}^{(1)})+m_{3l^{(2)}-2}^{(1)}-2.
\]
However, this does not hold because $m_{l^{)1)}}^{(1)}=m_{3l^{(2)}+1}^{(1)}>0$.
Therefore, $m_{l^{(2)}}^{(2)}\geq 2$.
If the $l^{(1)}$-string is q-singular at best, then [4] must have been marked in a string of length ($\leq 3l^{(2)}$) so that [5] can be marked in one of the $l^{(2)}$-strings in $\nu^{(2)}$ even if [5] cannot be marked in a string of length ($\leq l^{(2)}-1$).
If the $l^{(1)}$-string is singular and [3] was marked in a string of length ($\leq 3l^{(2)}$), then [4] would not be the $l^{(1)}$-string in $\nu^{(1)}$ because the effective length of the $l^{(1)}$-string is larger than $l^{(2)}$ so that [4] must have been marked in a string of length ($\leq 3l^{(2)}$) in $\nu^{(1)}$ which leads to a contradiction as before.
Thus, it remains to consider the case when the $l^{(1)}$-string is singular and [3] has been marked in this string.
We have two possible ways of box marking in $\nu^{(1)}$.
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In both cases, the effective length is reduced by one ($l^{(2)}+1\rightarrow l^{(2)}$) and $[4]$ is marked in a singular string of length $l^{(2)}$ in $\nu^{(2)}$, which is also a contradiction.
Hence, $\lambda_{2}\geq 2$.
In case (2), we also show that $m_{l^{(2)}}^{(2)}\geq 2$ leading to a contradiction.
Suppose that $m_{l^{(2)}}^{(2)}=1$.
From Eq.~\eqref{eq:ddp2}, we have
\[
-p_{l^{(2)}-1}^{(2)}+1=3m_{3l^{(2)}}^{(1)}+2m_{3l^{(2)}-1}^{(1)}+m_{3l^{(2)}-2}^{(1)}-2\geq 1,
\]
which forces that $p_{l^{(2)}-1}^{(2)}=0$, which implies that $m_{l^{(2)}-1}^{(2)}\neq 0$ by convexity.
Therefore [1] must be marked in a singular string of length ($\leq l^{(2)}-1$) in $\nu^{(2)}$ and hence [5] can be marked in a string in a string (singular or q-singular) of length $l^{(2)}$ in $\nu^{(2)}$ even if [5] cannot be marked in a string of length ($\leq l^{(2)}-1$), which contradicts $b= \framebox{$6$}$.
Hence $m_{l^{(2)}}^{(2)}\geq 2$.
The remaining argument is similar to case (1) and results in a contradiction.
Hence, $\lambda_{2}\geq 2$.
\begin{flushleft}
Case 7. $b= \framebox{$7$}$.
\end{flushleft}
Since $\mathrm{wt}(\framebox{$7$})=0$, there is nothing to prove.
\begin{flushleft}
Case 8. $b= \framebox{$8$}$.
\end{flushleft}
Since $\mathrm{wt}(\framebox{$8$})=0$, there is nothing to prove.
\begin{flushleft}
Case 9. $b= \framebox{$9$}$.
\end{flushleft}
In this case, after $[6]$ was marked in $\nu^{(1)}$ the box marking has terminated.
Since $\mathrm{wt}(\framebox{$9$})=-2\Bar{\Lambda}_{1}+\Bar{\Lambda}_{2}$, we must show that $\lambda_{2}\geq 1$.
The proof is analogous to that in Case 4.
\begin{flushleft}
Case 10. $b= \framebox{$10$}$.
\end{flushleft}
In this case, after $[6]$ was marked in $\nu^{(2)}$ the box marking has terminated.
Since $\mathrm{wt}(\framebox{$10$})=3\Bar{\Lambda}_{1}-2\Bar{\Lambda}_{2}$, we must show that $\lambda_{1}\geq 3$.
We first suppose that $\lambda_{1}=0$.
Then we have that $l^{(1)}\geq 3l^{(2)}$ and $p_{l^{(1)}}^{(1)}=0$ as in Case 2.
Note that all $l^{(1)}$-strings in $\nu^{(1)}$ are singular.
If $[6]$ is marked in a singular string of length $(< l^{(2)})$, then it is obvious that $[7]$ can be marked in an $l^{(1)}$-string even if [7] cannot be marked in a string of length ($\leq l^{(1)}-1$).
So we assume that $[6]$ is marked in an $l^{(2)}$-string and the selected $l^{(2)}$-string is singular.
If $[4]$ is marked in a singular string of length $(< l^{(1)})$, it is obvious that $[7]$ can be marked in an $l^{(1)}$-string.
If $[4]$ were marked in a singular string of length $(>3l^{(2)})$, then $[5]$ (and therefore $[6]$) would not be marked in $\nu^{(2)}$.
So it suffices to consider the case when $l^{(1)}=3l^{(2)}$ and $[4]$ is marked in an $l^{(1)}$-string.
We have two possible ways of box marking in $\nu^{(1)}$.
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In both cases, $[7]$ can be marked in the string marked by $[4]$, which contradicts $b= \framebox{$10$}$.
Secondly, we suppose that $\lambda_{1}=1$.
As in Case 2, we have two possible cases.
\begin{itemize}
\item[(1)] $m_{l^{(2)}}^{(2)}=1$, $l^{(1)}=3l^{(2)}-1$, and $p_{l^{(1)}}^{(1)}=0$,
\item[(2)] $m_{l^{(2)}}^{(2)}\geq 1$, $l^{(1)}\geq 3l^{(2)}$, and $p_{l^{(1)}}^{(1)}=1$.
\end{itemize}
As before it suffices to consider the case when $[4]$ is marked in an $l^{(1)}$-string and the selected $l^{(1)}$- and $l^{(2)}$-strings are singular.
By the same arguments as in Case 2, both cases (1) and (2) result in a contradiction.
Thirdly, we suppose that $\lambda_{1}=2$.
As in Case 2, we have four possible cases.
\begin{itemize}
\item[(1)] $m_{l^{(2)}}^{(2)}=1$, $l^{(1)}=3l^{(2)}-2$, and $p_{l^{(1)}}^{(1)}=0$,
\item[(2)] $m_{l^{(2)}}^{(2)}=1$, $l^{(1)}=3l^{(2)}-1$, and $p_{l^{(1)}}^{(1)}=1$,
\item[(3)] $m_{l^{(2)}}^{(2)}=2$, $l^{(1)}=3l^{(2)}-1$, and $p_{l^{(1)}}^{(1)}=0$,
\item[(4)] $m_{l^{(2)}}^{(2)}\geq 1$, $l^{(1)}\geq 3l^{(2)}$, and $p_{l^{(1)}}^{(1)}=2$.
\end{itemize}
As before it suffices to consider the case when $[4]$ is marked in an $l^{(1)}$-string and the selected $l^{(1)}$- and $l^{(2)}$-strings are singular.
By the same arguments as in Case 2, these cases result in a contradiction.
Hence, $\lambda_{1}\geq 3$.
\begin{flushleft}
Case 11. $b= \framebox{$11$}$.
\end{flushleft}
In this case, after $[7]$ was marked in $\nu^{(1)}$ or $\nu^{(2)}$ the box marking has terminated.
Since $\mathrm{wt}(\framebox{$11$})=\Bar{\Lambda}_{1}-\Bar{\Lambda}_{2}$, we must show that $\lambda_{1}\geq 1$.
Suppose that $\lambda_{1}=0$.
Then we have that $l^{(1)}\geq 3l^{(2)}$ and $p_{l^{(1)}}^{(1)}=0$ as in Case 2.
Note that all $l^{(1)}$-strings in $\nu^{(1)}$ are singular.
\begin{itemize}
\item[(1)]
$[7]$ is marked in $\nu^{(2)}$.
If [7] is marked in a singular string of length $(< l^{(2)})$, then it is obvious that [8] can be marked in an $l^{(1)}$-string in $\nu^{(1)}$ ((6) must have been marked in a string of length $(<3l^{(2)})$).
So we assume that [7] is marked in an $l^{(2)}$-string and that the string is singular.
If [6] is marked in a singular string of length $(<3l^{(2)})$ (i.e., it is not an $l^{(1)}$-string), it is obvious that [8] can be marked in an $l^{(1)}$-string.
If [6] were marked in a singular string of length $(>3l^{(2)})$, then [7] would not be marked in $\nu^{(2)}$.
So it suffices to consider the case when $l^{(1)}=3l^{(2)}$ and [6] is marked in an $l^{(1)}$-string and [7] is marked in an $l^{(2)}$-string and both selected strings are singular.
We have three possible ways of box marking in $\nu^{(1)}$.
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\end{picture}
\end{center}
It is obvious that [8] can be marked on the left of the box marked by [6].
Note that the box marking depicted below
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\begin{picture}(4,5)
\put(0,2){\line(1,0){2}}
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\end{picture}
\end{center}
cannot occur because the $l^{(1)}$-string is a boomerang string for the string marked by [3] (q-singular) in the above figure (see \textbf{(BS-4)}).
\item[(2)]
[7] is marked in $\nu^{(1)}$.
If [7] is marked in a string of length $(< l^{(1)})$, then it is obvious that [8] can be marked in a singular $l^{(1)}$-string.
So we assume that [7] is marked in an $l^{(1)}$-string.
\begin{itemize}
\item[(a)]
$l^{(1)}=3l^{(2)}$.
We have three possible ways of box marking in $\nu^{(1)}$.
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The box marking of
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\put(4,3){\makebox(1,1){{\scriptsize $[3]$}}}
\put(3,0){\makebox(3,1){{\scriptsize $3l^{(2)}$}}}
\end{picture}
\end{center}
cannot occur because $l^{(1)}$-string is a boomerang string for the string marked by [2] (qq-singular) in the above figure (see \textbf{(BS-3)}).
In all cases, [8] can be marked on the left of the box marked by [7].
\item[(b)]
$l^{(1)}=3l^{(2)}+1$.
The only possible way of marking [7] is
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\put(0,2){\line(1,0){3}}
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\put(2,2){\makebox(1,1){{\scriptsize $[7]$}}}
\put(1,0){\makebox(3,1){{\scriptsize $3l^{(2)}+1$}}}
\end{picture}
\end{center}
where [4] must have been marked in a string of length $(\leq 3l^{(2)}-2)$ as otherwise [7] could be marked in the left of the box marked by [4].
It is obvious [8] can be marked on the left of the box marked by [7].We omit the details.
\item[(c)]
$l^{(1)}\geq 3l^{(2)}+2$.
It is obvious that [8] can be marked on the left of the box marked by [7].
\end{itemize}
\end{itemize}
Both cases (1) and (2) contradict $b= \framebox{$11$}$ and therefore $\lambda_{1}\geq 1$.
\begin{flushleft}
Case 12. $b= \framebox{$12$}$.
\end{flushleft}
Since $\mathrm{wt}(\framebox{$12$})=-\Bar{\Lambda}_{1}$, there is nothing to prove.
\begin{flushleft}
Case 13. $b= \framebox{$13$}$.
\end{flushleft}
In this case, after [9] was marked in $\nu^{(1)}$ the box marking has terminated.
Since $\mathrm{wt}(\framebox{$13$})=-3\Bar{\Lambda}_{1}+\Bar{\Lambda}_{2}$, we must show that $\lambda_{2}\geq 1$.
The proof is analogous to that in Case 4.
\begin{flushleft}
Case 14. $b= \framebox{$14$}$.
\end{flushleft}
Since $\mathrm{wt}(\framebox{$14$})=-\Bar{\Lambda}_{2}$, there is nothing to prove.
\end{proof}
\begin{proof}[Proof of (II)]
\mbox{}
\begin{flushleft}
Case 1. $b= \framebox{$1$}$.
\end{flushleft}
Since $\Delta{p}_{i}^{(2)}=-1$ $(i\geq 1)$ (see \textbf{(VC-1)}), we must show that $p_{i}^{(2)}\geq 1$ $(i\geq 1)$.
If $m_{i}^{(2)}=0$ $(i\geq 1)$, then $\nu^{(2)}=\emptyset$ so that we assume $m_{j}^{(2)}\neq 0$ for some $j$.
Since the $j$-string is q-singular at best, $p_{j}^{(2)}\geq 1$ so that $p_{i}^{(2)}\geq 1$ $(i\geq 1)$ by Lemma~\ref{lem:vacancy01}.
Hence the admissibility in the new RC is guaranteed.
\begin{flushleft}
Case 2. $b= \framebox{$2$}$.
\end{flushleft}
Firstly we show that $p_{i}^{(2)}\geq 1$ $(i\leq i_{1}-1)$.
In the proof of Case 1, we assumed that $m_{j}^{(2)}\neq 0$ for some $j$.
In the following we assume that $i_{1}\geq 2$ and $m_{i}^{(2)}= 0$ $(i\leq i_{1}-1)$.
Suppose that $p_{i_{1}-1}^{(2)}=0$.
By convexity we have $p_{i}^{(2)}=0$ $(i\leq i_{1})$.
By Eq.~\eqref{eq:ddp2}, we have
\[
0=3m_{3(i_{1}-1)}^{(1)}+2(m_{3(i_{1}-1)-1}^{(1)}+m_{3(i_{1}-1)+1}^{(1)})
+m_{3(i_{1}-1)-2}^{(1)}+m_{3(i_{1}-1)+2}^{(1)}
\]
so that $m_{3i_{1}-1}^{(1)}=\cdots =m_{3i_{1}-5}^{(1)}=0$.
Similarly we have $m_{i}^{(1)}=0$ $(i\leq 3i_{1}-1)$ so that
\begin{align*}
p_{3(i_{1}-1)}^{(1)}=&-2\sum_{j\geq 3i_{1}}\min (3(i_{1}-1),j)m_{j}^{(1)}+\sum_{j\geq i_{1}}\min (3(i_{1}-1),3j)m_{j}^{(2)} \\
=&-6(i_{1}-1)\sum_{j\geq 3i_{1}}m_{j}^{(1)}+3(i_{1}-1)\sum_{j\geq i_{1}}m_{j}^{(2)}.
\end{align*}
Since $p_{3(i_{1}-1)}^{(1)}\geq 0$, we have
$\sum_{j\geq 3i_{1}}m_{j}^{(1)}\leq \frac{1}{2}\sum_{j\geq i_{1}}m_{j}^{(2)}$
which yields
\[
p_{i_{1}}^{(2)}-p_{i_{1}-1}^{(2)}=3\sum_{j\geq 3i_{1}}m_{j}^{(1)}-2\sum_{j\geq i_{1}}m_{j}^{(2)}
\leq -\frac{1}{2}\sum_{j\geq i_{1}}m_{j}^{(2)}<0.
\]
The last inequality is due to the fact that $m_{i_{1}}^{(2)}>0$.
Therefore $p_{i_{1}}^{(2)}<p_{i_{1}-1}^{(2)}=0$, which is a contradiction.
Hence $p_{i_{1}-1}^{(2)}\geq 1$ and therefore $p_{i}^{(2)}\geq 1$ $(i\leq i_{1}-1)$ by convexity.
Note that $m_{i}^{(2)}= 0$ $(i\leq i_{1}-1)$.
Secondly we show that $p_{3i_{1}}^{(1)}\geq 3$, $p_{3i_{1}-1}^{(1)}\geq 2$, and $p_{3i_{1}-2}^{(1)}\geq 1$.
We assume that $m_{3i_{1}}^{(1)}=m_{3i_{1}-1}^{(1)}=m_{3i_{1}-2}^{(1)}=0$.
Otherwise the proof is much easier.
If $m_{3i_{1}+1}^{(1)}\neq 0$, then $(3i_{1}+1)$-strings in $\nu^{(1)}$ must be qqq-singular at best so that $p_{3i_{1}+1}^{(1)}\geq 3$.
By convexity we have $p_{3i_{1}}^{(1)}\geq 3$, $p_{3i_{1}-1}^{(1)}\geq 2$, and $p_{3i_{1}-2}^{(1)}\geq 1$.
Therefore we further assume that $m_{3i_{1}+1}^{(1)}= 0$.
Let us show that $p_{3i_{1}-2}^{(1)}\geq 1$.
From Eq.~\eqref{eq:ddp1b}, we have
\begin{equation} \label{eq:IIcase2}
-p_{3i_{1}-1}^{(1)}+2p_{3i_{1}}^{(1)}-p_{3i_{1}+1}^{(1)}=m_{i_{1}}^{(2)}>0.
\end{equation}
Suppose that $p_{3i_{1}-2}^{(1)}=0$, then $p_{3i_{1}-1}^{(1)}=p_{3i_{1}}^{(1)}=0$ by convexity.
This contradicts Eq.~\eqref{eq:IIcase2}.
Hence $p_{3i_{1}-2}^{(1)}\geq 1$.
This implies $p_{3i_{1}-1}^{(1)}\geq 1$, $p_{3i_{1}}^{(1)}\geq 1$, and $p_{3i_{1}+1}^{(1)}\geq 1$ by convexity.
However, $p_{3i_{1}}^{(1)}= 1$ contradicts Eq.~\eqref{eq:IIcase2} so that $p_{3i_{1}}^{(1)}\geq 2$ and hence $p_{3i_{1}-1}^{(1)}\geq 2$ by convexity.
Now we show that $p_{3i_{1}}^{(1)}=2$ is not the case.
Suppose that $p_{3i_{1}}^{(1)}=2$.
Then $(p_{3i_{1}-1}^{(1)},p_{3i_{1}+1}^{(1)})=(2,2)$ or $(3,1)$ by convexity, which contradicts Eq.~\eqref{eq:IIcase2}.
Hence $p_{3i_{1}}^{(1)}\geq 3$.
If $m_{j}^{(1)}>0$ for some $j(\geq 3i_{1}+1)$, then $p_{j}^{(1)}\geq 3$ so that $p_{i}^{(1)}\geq 3$ $(i\geq 3i_{1})$ by Lemma~\ref{lem:vacancy23}.
If not, then $p_{i}^{(1)}\geq 3$ $(i\geq 3i_{1})$ by convexity.
We have shown that $p_{i}^{(1)}\geq 3$ $(i\geq 3i_{1})$, $p_{3i_{1}-1}^{(1)}\geq 2$, $p_{3i_{1}-2}^{(1)}\geq 1$, and $p_{i}^{(2)}\geq 1$ $(i\leq i_{1}-1)$.
Therefore, the admissibility of the new RC is guaranteed (see \textbf{(VC-2)}).
\begin{flushleft}
Case 3. $b= \framebox{$3$}$.
\end{flushleft}
Although it suffices to show that $p_{i}^{(1)}\geq 1$ $(i\geq 3i_{1})$ when $i_{2}=3i_{1}-2$ or $i_{2}=3i_{1}-1$ and $p_{i}^{(1)}\geq 1$ $(i\geq i_{2})$ when $i_{2}\geq 3i_{1}$ (see \textbf{(VC-3)}), we show that $p_{i}^{(1)}\geq 2$ $(i\geq \max (i_{2},3i_{1}))$.
We assume that $m_{i}^{(1)}=0$ $(i\geq i_{2}+1)$.
Otherwise, the proof is much easier.
From Eq.~\eqref{eq:ddp1b}, we have
\begin{equation} \label{eq:IIcase3}
-p_{3i_{1}-1}^{(1)}+2p_{3i_{1}}^{(1)}-p_{3i_{1}+1}^{(1)}=m_{i_{1}}^{(2)}>0.
\end{equation}
\begin{itemize}
\item[(1)] $i_{2}=3i_{1}-2$.
From Eq.~\eqref{eq:IIcase3}, we have $p_{3i_{1}}^{(1)}\geq 1$.
The convexity relation yields $p_{3i_{1}-1}^{(1)}\geq 1$ and $p_{3i_{1}+1}^{(1)}\geq 1$.
If $p_{3i_{1}}^{(1)}= 1$, then Eq.~\eqref{eq:IIcase3} leads to a contradiction so that $p_{3i_{1}}^{(1)}\geq 2$.
\item[(2)] $i_{2}=3i_{1}-1$.
Since the selected $i_{2}$-string is q-singular, $p_{3i_{1}-1}^{(1)}\geq 1$ so that Eq.~\eqref{eq:IIcase3} yields $p_{3i_{1}}^{(1)}\geq 2$ as in case (1).
\item[(3)] $i_{2}\geq 3i_{1}$.
Since the selected $i_{2}$-string is qq-singular, $p_{i_{2}}^{(1)}\geq 2$.
\end{itemize}
Therefore, $p_{i}^{(1)}\geq 2$ $(i\geq \max (i_{2},3i_{1}))$.
In particular, $\Tilde{p}_{i}^{(1)}\geq 1$ $(i\geq \max (i_{2},3i_{1}))$.
\begin{flushleft}
Case 4. $b= \framebox{$4$}$.
\end{flushleft}
It is easily verified that $p_{i}^{(1)}+\Delta p_{i}^{(1)}\geq 0$ $(i\leq i_{3}-1)$.
Since $\Delta p_{i}^{(2)}=-1$ $(i\geq i_{3}^{eff})$ we must show that $p_{i}^{(2)}\geq 1$ $(i\geq i_{3}^{eff})$.
Since [4] or [5] are not marked in $\nu^{(2)}$, $p_{j}^{(2)}\geq 1$ for $j(\geq i_{1}+1)$ such that $m_{j}^{(2)}\neq 0$.
Therefore it suffices to show that $p_{i_{3}^{eff}}^{(2)}\geq 1$.
\begin{itemize}
\item[(1)] $i_{3}^{eff}=i_{1}$.
If $m_{i_{1}}^{(2)}\geq 2$, then $p_{i_{1}}^{(2)}\geq 1$.
Assume that $m_{i_{1}}^{(2)}=1$.
In Case 2, we have shown that $p_{i_{1}-1}^{(2)}\geq 1$.
From Eq.~\eqref{eq:ddp2}, we have
\begin{align*}
&-p_{i_{1}-1}^{(2)}+2p_{i_{1}}^{(2)}-p_{i_{1}+1}^{(2)} \\
=&3m_{3i_{1}}^{(1)}+2(m_{3i_{1}-1}^{(1)}+m_{3i_{1}+1}^{(1)})
+m_{3i_{1}-2}^{(1)}+m_{3i_{1}+2}^{(1)}-2.
\end{align*}
Suppose that $p_{i_{1}}^{(2)}=0$.
Then the left-hand side is smaller than $0$.
On the other hand, the right-hand side is greater than or equal to $0$ because $m_{3i_{1}}^{(1)}\neq 0$ or $m_{3i_{1}-1}^{(1)}\neq 0$.
This is a contradiction so that $p_{i_{1}}^{(2)}\geq 1$.
\item[(2)] $i_{3}^{eff}>i_{1}$.
The argument is similar to case (1) and we have $p_{i}^{(2)}\geq 1$ $(i\geq i_{3}^{eff})$.
\end{itemize}
When $i_{2}<i_{3}$, the box-deleted $(i_{3}-1)$-string in $\Tilde{\nu}^{(1)}$ is set to be q-singular by \textbf{(RA-2)}.
Therefore, we must check that
\begin{equation} \label{eq:IIcase4}
p_{i_{3}-1}^{(1)}+\Delta p_{i_{3}-1}^{(1)}\geq 1.
\end{equation}
\begin{itemize}
\item[(1)] $i_{3}=3i_{1}$.
In this case, $i_{2}=3i_{1}-2$.
If $m_{3i_{1}-1}^{(1)}=0$, then $p_{3i_{1}-1}^{(2)}\geq 1$ by convexity because $p_{3i_{1}}^{(1)}\geq 1$.
If $m_{3i_{1}-1}^{(1)}\neq 0$, then $p_{3i_{1}-1}^{(2)}\geq 1$ by the box marking of this case.
By \textbf{(VC-3)}, $\Delta p_{3i_{1}-1}^{(1)}=0$.
Hence Eq.~\eqref{eq:IIcase4} is satisfied.
\item[(2)] $i_{3}\geq 3i_{1}+1$.
As shown in Case 3, $p_{i_{3}-1}^{(1)}\geq 2$ and $\Delta p_{i_{3}-1}^{(1)}=-1$ by \textbf{(VC-3)}.
Hence Eq.~\eqref{eq:IIcase4} is satisfied.
\end{itemize}
\begin{flushleft}
Case 5. $b= \framebox{$5$}$.
\end{flushleft}
We must show that $p_{i}^{(1)}\geq 2$ $(i\geq 3i_{4})$ and $p_{3i_{4}-1}^{(1)}\geq 1$.
In order to show that $p_{i}^{(1)}\geq 2$ $(i\geq 3i_{4})$, it suffices to show $p_{3i_{4}}^{(1)}\geq 2$ by noting Lemma~\ref{lem:vacancy23}.
Furthermore, we must show that $p_{3i_{4}-2}^{(1)}\geq 2$ when the $i_{3}$-string is type-I with $i_{3}\geq 3i_{1}+2$ and $i_{3}^{eff}=i_{4}$; $i_{3}=3i_{4}-1$ (see \textbf{(VC-5)}).
Firstly, let us show that $p_{3i_{4}}^{(1)}\geq 2$ and $p_{3i_{4}-1}^{(1)}\geq 1$.
If $m_{3i_{4}}^{(1)}\neq 0$, then $p_{3i_{4}}^{(1)}\geq 2$ as otherwise [5] would be marked in the $3i_{4}$-string in $\nu^{(1)}$.
Suppose that $m_{3i_{4}}^{(1)}= 0$.
Then, from Eq.~\eqref{eq:ddp1b} we have
\begin{equation} \label{eq:IIcase5}
-p_{3i_{4}-1}^{(1)}+2p_{3i_{4}}^{(1)}-p_{3i_{4}+1}^{(1)}=m_{i_{4}}^{(2)}\geq 1.
\end{equation}
Hence $p_{3i_{4}}^{(1)}\geq 1$.
We show that $p_{3i_{4}}^{(1)}= 1$ is not the case.
Suppose that $p_{3i_{4}}^{(1)}= 1$.
Then, $(p_{3i_{4}-1}^{(1)},p_{3i_{4}+1}^{(1)})=(0,0), (0,1)$ or $(1,0)$ by Eq.~\eqref{eq:IIcase5} and by convexity.
If $p_{3i_{4}+1}^{(1)}=0$, then $m_{3i_{4}+1}^{(1)}=0$ since otherwise [5] would be marked in the $(3i_{4}+1)$-string if $m_{3i_{4}+1}^{(1)}\neq 0$.
However, the convexity relation
\[
p_{3i_{4}+1}^{(1)}\geq \frac{1}{2}(p_{3i_{4}}^{(1)}+p_{3i_{4}+2}^{(1)})=\frac{1}{2}(1+p_{3i_{4}+2}^{(1)})
\]
implies $p_{3i_{4}+1}^{(1)}\geq 1$.
This contradicts our assumption so that $(p_{3i_{4}-1}^{(1)},p_{3i_{4}+1}^{(1)})=(0,1)$.
Since [5] is not marked in a $(3i_{4}-1)$-string, $m_{3i_{4}-1}^{(1)}=0$.
From Eq.~\eqref{eq:ddp1a}, we have
\[
-p_{3i_{4}-2}^{(1)}+2p_{3i_{4}-1}^{(1)}-p_{3i_{4}}^{(1)}=-p_{3i_{4}-2}^{(1)}-1=0.
\]
However this does not hold.
Hence $p_{3i_{4}}^{(1)}\geq 2$.
If $m_{3i_{4}-1}^{(1)}\neq 0$, then $p_{3i_{4}-1}^{(1)}\geq 1$ because [5] is not marked in the $(3i_{4}-1)$-string in $\nu^{(1)}$.
If not, then $p_{3i_{4}-1}^{(1)}\geq 1$ by convexity.
Secondly, we show that $p_{3i_{4}-2}^{(1)}\geq 2$ when the $i_{3}$-string is type-I with $i_{3}\geq 3i_{1}+2$ and $i_{3}^{eff}=i_{4}$; $i_{3}=3i_{4}-1$.
\begin{itemize}
\item[(1)] $i_{2}=i_{3}$.
It suffices to consider the case when $m_{3i_{1}-1}^{(1)}=\cdots =m_{3i_{4}-2}^{(1)}=0$ by noting Lemma~\ref{lem:vacancy23}.
Since $p_{3i_{4}-1}^{(1)}\geq 1$, we have $p_{3i_{1}-1}^{(1)}\geq 1, \ldots, p_{3i_{4}-2}^{(1)}\geq 1$ by convexity.
We show that $p_{3i_{4}-2}^{(1)}=1$ is not the case.
Suppose that $p_{3i_{4}-2}^{(1)}=1$.
Then $p_{3i_{1}-1}^{(1)}=\cdots =p_{3i_{4}-2}^{(1)}=1$ by Lemma~\ref{lem:vacancy1}.
Hence
\[
0=p_{3i_{4}-2}^{(1)}-p_{3i_{4}-3}^{(1)}=-2\sum_{j\geq i_{3}}m_{j}^{(1)}+\sum_{j\geq i_{4}}m_{j}^{(2)}
\]
and
\[
0=p_{3i_{1}}^{(1)}-p_{3i_{1}-1}^{(1)}=-2\sum_{j\geq i_{3}}m_{j}^{(1)}+\sum_{j\geq i_{1}}m_{j}^{(2)}
\]
so that $\sum_{j\geq i_{4}}m_{j}^{(2)}=\sum_{j\geq i_{1}}m_{j}^{(2)}$, which is a contradiction because $m_{i_{1}}^{(2)}\neq 0$.
Hence $p_{3i_{4}-2}^{(1)}\geq 2$.
\item[(2)] $i_{2}<i_{3}$.
We first suppose that he box marking in $\nu^{(1)}$ is
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\put(0,0){\line(1,0){2}}
\put(0,1){\line(1,0){3}}
\put(0,2){\line(1,0){3}}
\put(1,0){\line(0,1){2}}
\put(2,0){\line(0,1){2}}
\put(3,1){\line(0,1){1}}
\put(0,2){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[2]$}}}
\put(2,1){\makebox(1,1){{\scriptsize $[3]$}}}
\end{picture}
\end{center}
Then $p_{3i_{4}-2}^{(1)}=p_{i_{2}}^{(1)}\geq 2$, where the $i_{2}$ (resp. $i_{3}$)-string is qq-singular (resp. q-singular).
Next, suppose that $i_{2}\leq 3i_{4}$.
It suffices to consider the case when $m_{i}^{(1)}=0$ $(i_{2}+1\leq i \leq 3i_{4}-2)$ as in Case (1).
If $i_{2}\geq 3i_{1}$, then the selected $i_{2}$-string is qq-singular and $p_{i_{2}}^{(1)}\geq 2$ so that $p_{3i_{4}-2}^{(1)}\geq 2$.
Therefore we further assume that $i_{2}\leq 3i_{1}-1$.
Suppose that $p_{3i_{4}-2}^{(1)}=1$.
Then $p_{i_{2}+1}^{(1)}=\cdots =p_{3i_{4}-2}^{(1)}=1$ by Lemma~\ref{lem:vacancy1}.
If $i_{2}=3i_{1}-1$, then $p_{i_{2}}^{(1)}\geq 1$.
This is the equality because the convexity relation
\[
1=p_{i_{2}+1}^{(1)}\geq \frac{1}{2}(p_{i_{2}}^{(1)}+p_{i_{2}+2}^{(1)})=\frac{1}{2}(p_{i_{2}}^{(1)}+1)
\]
implies $p_{i_{2}}^{(1)}=1$.
Hence, $p_{3i_{1}-1}^{(1)}=\cdots =p_{3i_{4}-2}^{(1)}=1$.
The remaining argument is the same as in case (a) and we have $p_{3i_{4}-2}^{(1)}\geq 2$.
\end{itemize}
\begin{flushleft}
Case 6. $b= \framebox{$6$}$.
\end{flushleft}
It is easily verified that $p_{i}^{(1)}+\Delta p_{i}^{(1)}\geq 0$ $(i\leq i_{4}-1)$.
We must show that $p_{i_{4}^{eff}}^{(2)}\geq 2$; $p_{i}^{(2)}\geq 2$ $(i\geq i_{4}^{eff})$ follows from this.
Furthermore, we must show that $p_{i_{4}^{eff}-1}^{(2)}\geq 1$ when the $i_{4}$-string is type-II with $i_{2}=i_{3}=i_{4}$ or $i_{2}<i_{3}=i_{4}$ (see \textbf{(VC-6)}).
Firstly, we show that $p_{i_{4}^{eff}}^{(2)}\geq 2$.
\begin{itemize}
\item[(1)] The $i_{4}$-string is type-0/I.
If $m_{i_{4}^{eff}}^{(2)}\neq 0$, then the $i_{4}^{eff}$-string in $\nu^{(2)}$ must be qq-singular at best so that $p_{i_{4}^{eff}}^{(2)}\geq 2$.
Now we assume that $m_{i_{4}^{eff}}^{(2)}= 0$.
From Eq.~\eqref{eq:ddp2}, we have
\begin{align} \label{eq:IIcase6a}
&-p_{i_{4}^{eff}-1}^{(2)}+2p_{i_{4}^{eff}}^{(2)}-p_{i_{4}^{eff}+1}^{(2)} \\
=&3m_{3i_{4}^{eff}}^{(1)}+2(m_{3i_{4}^{eff}-1}^{(1)}+m_{3i_{4}^{eff}+1}^{(1)})
+m_{3i_{4}^{eff}-2}^{(1)}+m_{3i_{4}^{eff}+2}^{(1)}. \nonumber
\end{align}
Obviously, $p_{i_{4}^{eff}}^{(2)}\geq 1$ because $m_{3i_{4}^{eff}}^{(1)}$ or $m_{3i_{4}^{eff}-1}^{(1)}$ is not $0$.
We show that $p_{i_{4}^{eff}}^{(2)}= 1$ is not the case.
Suppose that $p_{i_{4}^{eff}}^{(2)}= 1$.
If $m_{i_{4}^{eff}+1}^{(2)}\neq 0$, then $p_{i_{4}^{eff}+1}^{(2)}\geq 2$.
If $m_{i_{4}^{eff}+1}^{(2)}= 0$, then $p_{i_{4}^{eff}+1}^{(2)}\geq 1$ by convexity.
In either case, the left-hand side of Eq.~\eqref{eq:IIcase6a} is smaller than or equal to $1$ while the right-hand side is greater than or equal to $2$, which is a contradiction.
Hence $p_{i_{4}^{eff}}^{(2)}\geq 2$.
\item[(2)] The $i_{4}$-string is type-II.
If $m_{i_{4}^{eff}-1}^{(2)}\neq 0$, then [4] would be marked in $\nu^{(2)}$ contradicting $b= \framebox{$6$}$ so that $m_{i_{4}^{eff}-1}^{(2)}=0$.
The remaining argument is similar to case (1).
\end{itemize}
Secondly, we show that $p_{i_{4}^{eff}-1}^{(2)}\geq 1$ when the $i_{4}$-string is type-II with $i_{2}=i_{3}=i_{4}$ or $i_{2}<i_{3}=i_{4}$.
We consider two cases (1) $m_{i_{4}^{eff}-1}^{(2)}=0$ and (2) $m_{i_{4}^{eff}-1}^{(2)}\neq 0$ separately.
\begin{itemize}
\item[(1)] $m_{i_{4}^{eff}-1}^{(2)}=0$.
If $m_{i_{4}^{eff}}^{(2)}\neq 0$, then $p_{i_{4}^{eff}}^{(2)}\geq 2$ and therefore $p_{i_{4}^{eff}-1}^{(2)}\geq 1$ by convexity.
This is because [5] is not marked in the $i_{4}^{eff}$-string in $\nu^{(2)}$.
Now, we assume that $m_{i_{4}^{eff}}^{(2)}=0$.
From Eq.~\eqref{eq:ddp2}, we have
\begin{align} \label{eq:IIcase6b}
&-p_{i_{4}^{eff}-1}^{(2)}+2p_{i_{4}^{eff}}^{(2)}-p_{i_{4}^{eff}+1}^{(2)} \\
=&3m_{3i_{4}^{eff}}^{(1)}+2(m_{3i_{4}^{eff}-1}^{(1)}+m_{3i_{4}^{eff}+1}^{(1)})+m_{3i_{4}^{eff}-2}^{(1)}+m_{3i_{4}^{eff}+2}^{(1)}. \nonumber
\end{align}
Suppose that $p_{i_{4}^{eff}-1}^{(2)}=0$.
Then $p_{i_{4}^{eff}}^{(2)}=0$ by convexity, which contradicts Eq.~\eqref{eq:IIcase6b} because $m_{i_{4}}^{(1)}=m_{3i_{4}^{eff}-2}^{(1)}\neq 0$.
Hence $p_{i_{4}^{eff}-1}^{(2)}\geq 1$.
\item[(2)] $m_{i_{4}^{eff}-1}^{(2)}\neq 0$.
The $i_{4}$-string is
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Since the marking [2] and [3] causes the effective length reduction $i_{4}^{eff}\rightarrow i_{4}^{eff}-1$, the $(i_{4}^{eff}-1)$-string in $\nu^{(2)}$ must be q-singular at best.
Otherwise [4] would be marked in this string.
Hence $p_{i_{4}^{eff}-1}^{(2)}\geq 1$.
\end{itemize}
When $i_{3}<i_{4}$, the box-deleted $(i_{4}-1)$-string in $\Tilde{\nu}^{(1)}$ is set to be qq-singular by \textbf{(RA-3)}, we must check that
\[
p_{i_{4}-1}^{(1)}+\Delta p_{i_{4}-1}^{(1)}\geq 2.
\]
Here, $\Delta p_{i_{4}-1}^{(1)}=1$ (see \textbf{(VC-4)}) and it is obvious that $p_{i_{4}-1}^{(1)}\geq 1$ so that the above inequality is satisfied.
When $i_{2}<i_{3}=i_{4}$, the box-deleted $(i_{4}-2)$-string in $\Tilde{\nu}^{(1)}$ is set to be q-singular by \textbf{(RA-4)}, we must check that
\begin{equation} \label{eq:IIcase6c}
p_{i_{4}-2}^{(1)}+\Delta p_{i_{4}-2}^{(1)}\geq 1.
\end{equation}
If $i_{4}\geq 3i_{1}+2$, then $p_{i_{4}-2}^{(1)}\geq 2$ (see Case 3) and $\Delta p_{i_{4}-2}^{(1)}=-1$ (see \textbf{(VC-3)}) so that Eq.~\eqref{eq:IIcase6c} is satisfied.
If $i_{4}=3i_{1}+1$, then $\Delta p_{i_{4}-2}^{(1)}=\Delta p_{3i_{1}-1}^{(1)}=0$ (see \textbf{(VC-3)}).
Let us show that $p_{3i_{1}-1}^{(1)}\geq 1$.
Since this is obvious if $m_{3i_{1}}^{(1)}\neq 0$ or $m_{3i_{1}-1}^{(1)}\neq 0$, we assume that $m_{3i_{1}}^{(1)}=0$ and $m_{3i_{1}-1}^{(1)}=0$.
From Eq.~\eqref{eq:ddp1b}, we have
\[
-p_{3i_{1}-1}^{(1)}+2p_{3i_{1}}^{(1)}-p_{3i_{1}+1}^{(1)}=m_{i_{1}}^{(2)}>0
\]
so that $p_{3i_{1}}^{(1)}\geq 1$ and therefore $p_{3_{1}-1}^{(1)}\geq 1$ by convexity.
Hence, Eq.~\eqref{eq:IIcase6c} is satisfied.
\begin{flushleft}
Case 7. $b= \framebox{$7$}$.
\end{flushleft}
It is not hard to check that $p_{i}^{(1)}+\Delta p_{i}^{(1)}\geq 0$ $(i\leq i_{5}-1)$.
\begin{flushleft}
Case 8. $b= \framebox{$8$}$.
\end{flushleft}
It is is not hard to check that $p_{i}^{(1)}+\Delta p_{i}^{(1)}\geq 0$ $(i\leq 3i_{5}-1)$.
\begin{flushleft}
Case 9. $b= \framebox{$9$}$.
\end{flushleft}
It is not hard to check that $p_{1}^{(1)}+\Delta p_{i}^{(1)}\geq 0$ $(i\leq i_{6}-1)$.
Since $\Delta p_{i}^{(2)}=-1$ $(i\geq i_{6}^{eff})$ (see \textbf{(VC-9)}), we must show that $p_{i_{6}^{eff}}^{(2)}\geq 1$; $p_{i_{6}^{i}}^{(2)}\geq 1$ $(i\geq i_{6}^{eff})$ follows from this.
If $m_{i_{6}^{eff}}^{(2)}\neq 0$, then $p_{i_{6}^{eff}}^{(2)}\geq 1$.
So we assume that $m_{i_{6}^{eff}}^{(2)}= 0$.
From Eq.~\eqref{eq:ddp2}, we have
\begin{align*}
&-p_{i_{6}^{eff}-1}^{(2)}+2p_{i_{6}^{eff}}^{(2)}-p_{i_{6}^{eff}+1}^{(2)} \\
=&3m_{3i_{6}^{eff}}^{(1)}+2(m_{3i_{6}^{eff}-1}^{(1)}+m_{3i_{6}^{eff}+1}^{(1)})
+m_{3i_{6}^{eff}-2}^{(1)}+m_{3i_{6}^{eff}+2}^{(1)}.
\end{align*}
Suppose that $p_{i_{6}^{eff}}^{(2)}=0$.
Then $p_{i_{6}^{eff}-1}^{(2)}=p_{i_{6}^{eff}+1}^{(2)}=0$ by convexity.
The left-hand side is $0$ while the right-hand side is positive because $m_{i_{6}}^{(1)}\neq 0$, which is a contradiction.
Hence, we have $p_{i_{6}^{eff}}^{(2)}\geq 1$.
When $i_{5}<i_{6}$, we must check that $p_{i_{6}-1}^{(1)}\geq 1$ because the box-deleted $(i_{6}-1)$-string in $\Tilde{\nu}^{(1)}$ is set to be q-singular by \textbf{(RA-1)} and $\Delta p_{i_{6}-1}^{(1)}=0$ (see \textbf{(VC-7)}).
However this is obvious because $i_{5}$-string is q-singular and [6] is not marked in a string of length $i$ $(i_{5}\leq i\leq i_{6}-1)$.
We omit the proof in the case when $b= \framebox{$i$}$ with $10\leq i\leq 14$.
The proof is similar to those in the previous cases.
When $b= \framebox{$\emptyset$}$, there is nothing to prove.
\end{proof}
\begin{proof}[Proof of (IV)]
Let $\Tilde{\nu}=(\Tilde{m}_{i}^{(a)})_{(a,i)\in\mathcal{H}_{0}}$.
\begin{flushleft}
Case 0. $b= \framebox{$\emptyset$}$.
\end{flushleft}
In this case, $\Tilde{m}_{1}^{(2)}=m_{1}^{(2)}-2$ and $\Tilde{m}_{3}^{(1)}=m_{3}^{(1)}-1$.
\begin{align*}
\Delta c(\nu)=&3\left( (m_{3}^{(1)})^{2}-(\Tilde{m}_{3}^{(1)})^{2}\right)+2\sum_{j\neq 3}\min (3,j)(m_{3}^{(1)}-\Tilde{m}_{3}^{(1)})m_{j}^{(1)} \\
-&3(m_{3}^{(1)}m_{1}^{(2)}-\Tilde{m}_{3}^{(1)}\Tilde{m}_{1}^{(2)})-\sum_{j\neq 1}\min (3,3j)(m_{3}^{(1)}-\Tilde{m}_{3}^{(1)})m_{j}^{(2)} \\
&-\sum_{i\neq 3}\min (i,3)m_{i}^{(1)}(m_{1}^{(2)}-\Tilde{m}_{1}^{(2)}) \\
+&(m_{1}^{(2)})^{2}-(\Tilde{m}_{1}^{(2)})^{2}+4\sum_{j\neq 1} \min (1,j)(m_{1}^{(2)}-\Tilde{m}_{1}^{(2)})m_{j}^{(2)} \\
-&L\sum_{j\geq 1}m_{j}^{(2)}+(L-1)\left( \sum_{j\geq 1}m_{j}^{(2)}-2 \right) \\
=&\sum_{j\geq 1}m_{j}^{(2)}-\alpha_{1}^{(2)}-2L+1.
\end{align*}
The change of the sum of riggings is
\[
\Delta |J|=J^{(1,3)}+2J^{(2,1)}=p_{3}^{(1)}+2p_{1}^{(2)},
\]
where
\[
p_{3}^{(1)}=-6\sum_{j\geq 3}m_{j}^{(1)}-4m_{2}^{(1)}-2m_{1}^{(1)}+3\sum_{j\geq 1}m_{j}^{(2)}
\]
and
\[
p_{1}^{(2)}=L+3\sum_{j\geq 3}m_{j}^{(1)}+2m_{2}^{(1)}+m_{1}^{(1)}-2\sum_{j\geq 1}m_{j}^{(2)}.
\]
Hence altogether $\Delta c(\nu,J)=-\alpha_{1}^{(2)}+1$.
\begin{flushleft}
Case 1. $b= \framebox{$1$}$.
\end{flushleft}
Let $(\Tilde{\nu},\Tilde{J})=\delta _{b}(\nu,J)$.
Then
\[
\Delta c(\nu)=c(\nu)-c(\Tilde{\nu})=-\sum_{i}m_{i}^{(2)}=-\alpha_{1}^{(2)}.
\]
The riggings are unchanged so that $\Delta c(\nu,J)=-\alpha_{1}^{(2)}$.
In what follows we assume that $i_{1}>1$.
The proof in the case when $i_{1}=1$ is similar.
We write $\Delta _{i\rightarrow j}c(\nu)=c(\Tilde{\nu}_{i})-c(\Tilde{\nu}_{j})$, $\Delta _{i\rightarrow j} |J|=|\Tilde{J}_{i}|-|\Tilde{J}_{j}|$, and $\Delta _{i\rightarrow j}c(\nu,J)=c(\Tilde{\nu}_{i},\Tilde{J}_{i})-c(\Tilde{\nu}_{j},\Tilde{J}_{j})$, where $\delta_{b_{i}}(\nu,J)=(\Tilde{\nu}_{i},\Tilde{J}_{i})$ and $\delta_{b_{j}}(\nu,J)=(\Tilde{\nu}_{j},\Tilde{J}_{j})$ with
$b_{i}= \framebox{$i$}$ and $b_{j}= \framebox{$j$}$ $(i<j)$.
We also write $\Delta_{1}c(\nu,J)=c(\nu,J)-c(\Tilde{\nu},\Tilde{J})$ where $(\Tilde{\nu},\Tilde{J})=\delta _{b}(\nu,J)$ with $b= \framebox{$1$}$.
\begin{flushleft}
Case 2. $b= \framebox{$2$}$.
\end{flushleft}
In this case, $\Tilde{m}_{i_{1}}^{(2)}=m_{i_{1}}^{(2)}-1$ and $\Tilde{m}_{i_{1}-1}^{(2)}=m_{i_{1}-1}^{(2)}+1$.
\[
\Delta_{1\rightarrow 2}c(\nu)=-3\sum_{j\geq 3i_{1}}m_{j}^{(1)}-2m_{3i_{1}-1}^{(1)}-m_{3i_{1}-2}^{(1)}+2\sum_{j\geq i_{1}}m_{j}^{(2)}-1
\]
and
\[
\Delta_{1\rightarrow 2}|J|=p_{i_{1}}^{(2)}-(p_{i_{1}-1}^{(2)}+\Delta p_{i_{1}-1}^{(2)}),
\]
where $\Delta p_{i_{1}-1}^{(2)}=-1$ by \textbf{(VC-1)}.
The direct calculation yields
\[
\Delta_{1\rightarrow 2}|J|=3\sum_{j\geq 3i_{1}}m_{j}^{(1)}+2m_{3i_{1}-1}^{(1)}+m_{3i_{1}-2}^{(1)}-2\sum_{j\geq i_{1}}m_{j}^{(2)}+1.
\]
Hence $\Delta_{1\rightarrow 2} c(\nu,J)=0$ so that $\Delta c(\nu,J)=\Delta_{1}c(\nu,J)+\Delta_{1\rightarrow 2}c(\nu,J)=-\alpha_{1}^{(2)}$.
As easily verified that $\Delta_{i\rightarrow j}c(\nu,J)$ does not contain terms involving $m_{i}^{(a)}$.
Let us write
\[
\Delta_{i\rightarrow j}|J|=\sum_{a}\sum_{k}(p_{i_{k}}^{(a)}+\Delta r_{i_{k}}^{(a)}-(p_{i_{k}-n_{k}}^{(a)}+\Delta p_{i_{k}-n_{k}}^{(a)}+\Delta r_{i_{k}-n_{k}}^{(a)})),
\]
where $\Delta r_{k}^{(a)}$ is the rigging adjustment for the selected $i_{k}$-string in $\nu^{(a)}$, i.e., $\Delta r_{i_{k}}^{(a)}=0,-1$, and $-2$ when the selected $i_{k}$-string in $\nu^{(a)}$ is singular, q-singular, and qq-singular, respectively so that $p_{i_{k}}^{(a)}+\Delta r_{i_{k}}^{(a)}$ is the value of the rigging of the selected $i_{k}$-string in $\nu^{(a)}$ and $n_{k}$ is the numbers of box marking in the $i_{k}$-string in $\nu^{(a)}$, $p_{i_{k}-n_{k}}^{(a)}+\Delta p_{i_{k}-n_{k}}^{(a)}=\Tilde{p}_{i_{k}-n_{k}}^{(a)}$, and $\Delta r_{i_{k}-n_{k}}^{(a)}$ is the rigging adjustment for the $\Tilde{p}_{i_{k}-n_{k}}^{(a)}$-string in $\Tilde{\nu}^{(a)}$ so that $p_{i_{k}-n_{k}}^{(a)}+\Delta p_{i_{k}-n_{k}}^{(a)}+\Delta r_{i_{k}-n_{k}}^{(a)}$ is the value of rigging of the box-deleted $\Tilde{p}_{i_{k}-n_{k}}^{(a)}$-string in $\Tilde{\nu}^{(a)}$.
Then,
\[
\Delta_{i\rightarrow j}c(\nu,J)=t-\sum_{a}\sum_{k}(\Delta r_{i_{k}}^{(a)}-(\Delta p_{i_{k}-n_{k}}^{(a)}+\Delta r_{i_{k}-n_{k}}^{(a)})),
\]
where $t$ is the ``constant'' term in $\Delta_{i\rightarrow j}c(\nu)$, which does not contain $m_{l}^{(a)}$.
\begin{flushleft}
Case 3. $b= \framebox{$3$}$.
\end{flushleft}
In this case one caution is in order.
When $i_{2}^{eff}=i_{1}$, we must consider the changes of $m_{i_{2}}^{(1)}$, $m_{i_{2}-1}^{(1)}$, $m_{i_{1}}^{(2)}$, and $m_{i_{1}-1}^{(2)}$ simultaneously in $c(\nu)$.
\begin{itemize}
\item[(1)] $i_{2}^{eff}=i_{1}$.
We compute $\Delta_{1\rightarrow 3}c(\nu,J)$.
\begin{align*}
\Delta _{1\rightarrow 3}c(\nu,J)=-1&+\Delta r_{i_{1}}^{(2)}-(\Delta p_{i_{1}-1}^{(2)}+\Delta r_{i_{1}-1}^{(2)}) \\
&+\Delta r_{i_{2}}^{(1)}-(\Delta p_{i_{2}-1}^{(1)}+\Delta r_{i_{2}-1}^{(1)}),
\end{align*}
where $\Delta p_{i_{1}-1}^{(2)}=-1$ by \textbf{(VC-1)} and $\Delta r_{i_{1}}^{(2)}=\Delta r_{i_{2}-1}^{(1)}=0$.
The values of the rigging adjustment $\Delta r_{i_{2}}^{(1)}$ and $\Delta p_{i_{2}-1}^{(1)}$ are listed below.
In this table the type is refereed to the type of the $i_{2}$-string.
\begin{center}
\begin{tabular}{c|c|c}
type & $\Delta r_{i_{2}}^{(1)}$ & $\Delta p_{i_{2}-1}^{(1)}$ \\ \hline
0 & $-2$ & $-2$ \\
I & $-1$ & $-1$ \\
II & $0$ & $0$
\end{tabular}
\end{center}
The values of $\Delta r_{i_{2}}^{(1)}$ is due to the fact that the selected $i_{2}$-string is singular (resp. q-singular) when it is type-II (resp. type-I) and is qq-singular when it is type-0.
The values of $\Delta p_{i_{2}-1}^{(1)}$ come from \textbf{(VC-2)}.
Hence altogether $\Delta_{1\rightarrow 3}c(\nu,J)=0$ so that $\Delta c(\nu,J)=\Delta_{1}c(\nu,J)+\Delta_{1\rightarrow 3}c(\nu,J)=-\alpha_{1}^{(2)}$.
\item[(2)] $i_{1}<i_{2}^{eff}$.
Since $\Delta_{1\rightarrow 3}c(\nu,J)=\Delta_{1\rightarrow 2}c(\nu,J)+\Delta_{2\rightarrow 3}c(\nu,J)$, we compute $\Delta_{2\rightarrow 3}c(\nu,J)$.
\begin{equation*}
\Delta _{1\rightarrow 3}c(\nu,J)=-1+\Delta r_{i_{2}}^{(1)}-(\Delta p_{i_{2}-1}^{(1)}+\Delta r_{i_{2}-1}^{(1)}),
\end{equation*}
where $\Delta r_{i_{2}}^{(1)}=-2$ because the selected $i_{2}$-string in $\nu^{(1)}$ is qq-singular and $\Delta r_{i_{2}-1}^{(1)}=0$.
By \textbf{(VC-2)}, $\Delta p_{i_{2}-1}^{(1)}=-3$.
Hence altogether $\Delta_{2\rightarrow 3}c(\nu,J)=0$ so that $\Delta c(\nu,J)=\Delta_{1}c(\nu,J)+\Delta_{1\rightarrow 2}c(\nu,J)+\Delta_{2\rightarrow 3}c(\nu,J)=-\alpha_{1}^{(2)}$.
\end{itemize}
\begin{flushleft}
Case 4. $b= \framebox{$4$}$.
\end{flushleft}
\begin{itemize}
\item[(1)] $i_{2}=i_{3}$.
In this case, $\Tilde{m}_{i_{3}}^{(1)}=m_{i_{3}}^{(1)}-1$ and $\Tilde{m}_{i_{3}-2}^{(1)}=m_{i_{3}-2}^{(1)}+1$.
As before, we must consider the changes of $m_{i_{3}}^{(1)}$, $m_{i_{3}-2}^{(1)}$, $m_{i_{1}}^{(2)}$ and $m_{i_{1}-1}^{(2)}$ simultaneously in $c(\nu)$ when $i_{2}\leq 3i_{1}+1$.
\begin{itemize}
\item[(a)] $i_{2}\leq 3i_{1}+1$.
We compute $\Delta_{1\rightarrow 4}c(\nu,J)$.
\begin{align*}
\Delta _{1\rightarrow 4}c(\nu,J)=t&+\Delta r_{i_{1}}^{(2)}-(\Delta p_{i_{1}-1}^{(2)}+\Delta r_{i_{1}-1}^{(2)}) \\
&+\Delta r_{i_{3}}^{(1)}-(\Delta p_{i_{3}-2}^{(1)}+\Delta r_{i_{3}-2}^{(1)}),
\end{align*}
where $t=-2$ when the $i_{3}$-string is type-II $(i_{3}=3i_{1}+1)$ and $t=-1$ otherwise.
As before $\Delta r_{i_{1}}^{(2)}=\Delta r_{i_{1}-1}^{(2)}=0$, $\Delta p_{i_{1}-1}^{(2)}=-1$, and $\Delta r_{i_{3}-2}^{(1)}=0$ because the box-deleted $(i_{3}-2)$-string is set to be singular.
The values of $\Delta r_{i_{3}}^{(1)}$ and $\Delta p_{i_{3}-2}^{(1)}$ (see \textbf{(VC-2)}) are listed below.
In this table the type is refereed to the type of the $i_{3}$-string.
Note that the $i_{3}$-string is singular when it is type-I ($i_{3}=3i_{1}-1$) while it is q-singular when the type is not I.
\begin{center}
\begin{tabular}{c|c|c}
type & $\Delta_{i_{3}}^{rig}$ & $\Delta p_{i_{3}-2}^{(1)}$ \\ \hline
0 & $-1$ & $-1$ \\
I & $0$ & $0$ \\
II & $-1$ & $-2$
\end{tabular}
\end{center}
Hence altogether $\Delta_{1\rightarrow 4}c(\nu,J)=0$ so that $\Delta c(\nu,J)=\Delta_{1}c(\nu,J)+\Delta_{1\rightarrow 4}c(\nu,J)=-\alpha_{1}^{(2)}$.
\item[(b)] $i_{2}\geq 3i_{1}+2$.
Since $\Delta_{1\rightarrow 4}c(\nu,J)=\Delta_{1\rightarrow 2}c(\nu,J)+\Delta_{2\rightarrow 4}c(\nu,J)$, we compute $\Delta_{2\rightarrow 4}c(\nu,J)$.
\begin{equation*}
\Delta _{1\rightarrow 3}c(\nu,J)=-2+\Delta r_{i_{3}}^{(1)}-(\Delta p_{i_{3}-2}^{(1)}+\Delta r_{i_{3}-2}^{(1)}),
\end{equation*}
where $\Delta p_{i_{3}-2}^{(1)}=-3$ by \textbf{(VC-2)}, $\Delta r_{i_{3}-2}^{(1)}=0$, and $\Delta r_{i_{3}}^{(1)}=-1$ because the selected $i_{3}$-string is q-singular.
Hence $\Delta_{2\rightarrow 4}c(\nu,J)=0$ so that $\Delta c(\nu,J)=\Delta_{1}c(\nu,J)+\Delta_{1\rightarrow 2}c(\nu,J)+\Delta_{2\rightarrow 4}c(\nu,J)=-\alpha_{1}^{(2)}$.
\end{itemize}
\item[(2)] $i_{2}<i_{3}$.
\begin{itemize}
\item[(a)] $i_{3}^{eff}=i_{1}$.
We compute $\Delta_{1\rightarrow 4}c(\nu,J)$.
In this case $i_{2}=3i_{1}-2$ and $i_{3}=3i_{1}$.
The $i_{2}$-string is singular and the $i_{3}$-string is q-singular (see \textbf{(BM-3)}); $\Delta r_{i_{2}}^{(1)}=0$ and $\Delta r_{i_{3}}^{(1)}=-1$.
\begin{align*}
\Delta _{1\rightarrow 4}c(\nu,J)=-1&+\Delta r_{i_{1}}^{(2)}-(\Delta p_{i_{1}-1}^{(2)}+\Delta r_{i_{1}-1}^{(2)}) \\
&+\Delta r_{i_{2}}^{(1)}-(\Delta p_{i_{2}-1}^{(1)}+\Delta r_{i_{2}-1}^{(1)}) \\
&+\Delta r_{i_{3}}^{(1)}-(\Delta p_{i_{3}-1}^{(1)}+\Delta r_{i_{3}-1}^{(1)}),
\end{align*}
where $\Delta r_{i_{1}}^{(2)}=\Delta r_{i_{1}-1}^{(2)}=0$, $\Delta p_{i_{1}-1}^{(2)}=-1$ as before.
By \textbf{(RA-2)} $\Delta r_{i_{2}-1}^{(1)}=-1$ and $\Delta r_{i_{3}-1}^{(1)}=-1$.
By \textbf{(VC-3)} $\Delta p_{i_{2}-1}^{(1)}=0$ and $\Delta p_{i_{3}-1}^{(1)}=0$.
Hence altogether $\Delta_{1\rightarrow 4}c(\nu,J)=0$ so that $\Delta c(\nu,J)=\Delta_{1}c(\nu,J)+\Delta_{1\rightarrow 4}c(\nu,J)=-\alpha_{1}^{(2)}$.
\item[(b)] $i_{3}^{eff}>i_{1}$.
We compute $\Delta_{3\rightarrow 4}c(\nu,J)$.
\begin{equation*}
\Delta _{3\rightarrow 4}c(\nu,J)=-1+\Delta r_{i_{3}}^{(1)}-(\Delta p_{i_{3}-1}^{(1)}+\Delta r_{i_{3}-1}^{(1)}),
\end{equation*}
where $\Delta p_{i_{3}-1}^{(1)}=-1$ by \textbf{(VC-3)} and $\Delta r_{i_{3}}^{(1)}=\Delta r_{i_{3}-1}^{(1)}=-1$.
Hence $\Delta_{3\rightarrow 4}c(\nu,J)=0$ so that $\Delta c(\nu,J)=\Delta_{1}c(\nu,J)+\Delta_{1\rightarrow 3}c(\nu,J)+\Delta_{3\rightarrow 4}c(\nu,J)=-\alpha_{1}^{(2)}$.
\end{itemize}
\end{itemize}
The verifications of $\Delta c(\nu,J)=-\alpha_{1}^{(2)}$ for $b= \framebox{$i$}$ $(5\leq i\leq 14)$ are similar and we omit the details.
\end{proof}
Define the following subsets of $B_{0}\otimes B_{0}$.
\begin{align*}
S_{0}=&\left\{ \framebox{$1$}\otimes\framebox{$1$}, \framebox{$14$}\otimes\framebox{$14$} \right\} \bigsqcup \left\{\framebox{$i$}\otimes\framebox{$j$} \relmiddle| i=1,2;2\leq j \leq 14 \right\} \\
&\bigsqcup \left\{\framebox{$i$}\otimes\framebox{$j$} \relmiddle| i=3,4,6; 6\leq j \leq 14; j\neq 7 \right\} \\
&\bigsqcup \left\{\framebox{$i$}\otimes\framebox{$j$} \relmiddle| i=5,8,10; 10\leq j \leq 14 \right\} \\
&\bigsqcup \left\{\framebox{$i$}\otimes\framebox{$j$} \relmiddle| i=7,9,11,12,13; j=13,14 \right\},
\end{align*}
\begin{align*}
S_{1}=&\left\{ \framebox{$2$}\otimes\framebox{$1$} \right\}
\bigsqcup \left\{\framebox{$i$}\otimes\framebox{$j$} \relmiddle| i=3,4,6;1\leq j \leq 7; j\neq 6 \right\} \\
&\bigsqcup \left\{\framebox{$i$}\otimes\framebox{$j$} \relmiddle| i=5,8,10; 2\leq j \leq 9 \right\} \\
&\bigsqcup \left\{\framebox{$i$}\otimes\framebox{$j$} \relmiddle| i=7,9,11,12,13; 6\leq j \leq 12; j\neq 7 \right\} \\
&\bigsqcup \left\{\framebox{$14$}\otimes\framebox{$j$} \relmiddle| 10 \leq j \leq 13 \right\},
\end{align*}
and
\begin{align*}
S_{2}=&\left\{\framebox{$i$}\otimes\framebox{$1$} \relmiddle| i=5,8,10 \right\} \\
&\bigsqcup \left\{\framebox{$i$}\otimes\framebox{$j$} \relmiddle| i=7,9,11,12,13; 1\leq j \leq 7; j\neq 6 \right\} \\
&\bigsqcup \left\{\framebox{$14$}\otimes\framebox{$j$} \relmiddle| 1 \leq j \leq 9 \right\}.
\end{align*}
The subset $S_{0}$ (resp. $S_{1}$) is $B(2\Bar{\Lambda}_{2})$ (resp. $B(3\Bar{\Lambda}_{1})$) in Eq.~ \eqref{eq:decomposition} and the subset $S_{2}$ is the disjoint union of $B(2\Bar{\Lambda}_{1})$, $B(\Bar{\Lambda}_{2})$, and $B(0)$ in Eq.~\eqref{eq:decomposition} so that
\begin{equation} \label{eq:Hb}
H(b_{1}\otimes b_{2})=
\begin{cases}
0 & \text{if } b_{1}\otimes b_{2} \in S_{0}, \\
-1 & \text{if } b_{1}\otimes b_{2} \in S_{1}, \\
-2 & \text{otherwise}.
\end{cases}
\end{equation}
\begin{proof}[Proof of (V)]
The proof is reduced to showing the following lemma.
\end{proof}
\begin{lem}
$b_{1}\otimes b_{2}$ belongs to $S_{2}$ or takes the form of $\framebox{$\emptyset$}\otimes b$ if and only if $\alpha_{1}^{(2)}-\Tilde{\alpha}_{1}^{(2)}=2$, where $b=\framebox{$j$}$ with $1\leq j \leq 14$ or $j=\emptyset$.
$b_{1}\otimes b_{2}$ belongs to $S_{1}$ if and only if $\alpha_{1}^{(2)}-\Tilde{\alpha}_{1}^{(2)}=1$.
$b_{1}\otimes b_{2}$ belongs to $S_{0}$ or takes the form of $b\otimes \framebox{$\emptyset$}$ with $b=\framebox{$i$}$ with $1\leq i \leq 14$ if and only if $\alpha_{1}^{(2)}-\Tilde{\alpha}_{1}^{(2)}=0$.
\end{lem}
\begin{proof}
Assertions are the direct consequences of the following three lemmas.
\end{proof}
\begin{lem} \label{lem:da=2}
If $\alpha_{1}^{(2)}-\Tilde{\alpha}_{1}^{(2)}=2$. then $b_{1}\otimes b_{2}$ belongs to $S_{2}$ or it takes the form of $\framebox{$\emptyset$}\otimes b$ with $b=\framebox{$i$}$ with $1\leq i \leq 14$ or $i=\emptyset$.
\end{lem}
\begin{lem} \label{lem:da=1}
If $\alpha_{1}^{(2)}-\Tilde{\alpha}_{1}^{(2)}=1$. then $b_{1}\otimes b_{2}$ belongs to $S_{1}$.
In particular, it does not take the form of $b\otimes \framebox{$\emptyset$}$ or $\framebox{$\emptyset$}\otimes b$.
\end{lem}
\begin{lem} \label{lem:da=0}
If $\alpha_{1}^{(2)}-\Tilde{\alpha}_{1}^{(2)}=0$. then $b_{1}\otimes b_{2}$ belongs to $S_{0}$ or takes the form of $b\otimes \framebox{$\emptyset$}$ with $b=\framebox{$i$}$ with $1\leq i \leq 14$.
\end{lem}
We give the proof of Lemma \ref{lem:da=2} only.
Proofs of Lemma \ref{lem:da=1} and Lemma \ref{lem:da=0} are similar.
\begin{proof}[Proof of Lemma \ref{lem:da=2}]
Since $\alpha_{1}^{(2)}-\Tilde{\alpha}_{1}^{(2)}=2$, $i\geq 5$ but $i\neq 6$.
It is obvious that $b_{1}= \framebox{$\emptyset$}$ is also possible.
\begin{flushleft}
Case 1. $b_{1}= \framebox{$\emptyset$}$.
\end{flushleft}
From \textbf{(VC-$\emptyset$)}, it is obvious that $1\leq j\leq 14$ or $j=\emptyset$.
\begin{flushleft}
Case 2. $b_{1}= \framebox{$5$}$
\end{flushleft}
Since $b_{1}= \framebox{$5$}$, after [4] was marked in $\nu^{(2)}$ the box marking has terminated.
Since $\alpha_{1}^{(2)}-\Tilde{\alpha}_{1}^{(2)}=2$, two strings of length one are marked by [1] and [4] in $\nu^{(2)}$ and they are deleted in $\Tilde{\nu}^{(1)}$.
Since $\Delta p_{i}^{(2)}=1$ $(i\geq 1)$ (see \textbf{(VC-5)}), all strings of $\Tilde{\nu}^{(2)}$ are not singular so that $j=1$.
In particular $j\neq \emptyset$.
\begin{flushleft}
Case 3. $b_{1}= \framebox{$7$}$.
\end{flushleft}
Since $b_{1}= \framebox{$7$}$, after [5] was marked in $\nu^{(1)}$ the box marking has terminated.
We claim that $1\leq j \leq 7$ but $j\neq 6$ and $j\neq \emptyset$.
To do so, we show that neither [4] nor [6] can be marked in $\Tilde{\nu}^{(1)}$.
From \textbf{(VC-7)}, the changes of vacancy numbers are summarized as
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$0$ & $(i\geq i_{5})$ & $0$ & $(i\geq i_{5}^{eff})$ \\
& & $1$ & $(1\leq i\leq i_{5}^{eff}-1)$
\end{tabular}
\end{center}
Note that strings of length ($\geq i_{5}$) in $\nu^{(1)}$ and therefore in $\Tilde{\nu}^{(1)}$ are q-singular at best.
Otherwise [6] would be marked in $\nu^{(1)}$ contradicting $b_{1}= \framebox{$7$}$.
The selected $i_{1}$- and $i_{4}$-strings of length one are deleted in $\Tilde{\nu}^{(2)}$ so that it is obvious from the table above that [1] can be marked in a string of length ($\geq i_{5}^{eff}$) if possible.
If [1] cannot be marked in $\Tilde{\nu}^{(2)}$, then $b_{2}=\framebox{$1$}$.
If [1] is marked in $\Tilde{\nu}^{(2)}$, then the box marking in $\Tilde{\nu}^{(1)}$ is possible only for strings of length ($\geq 3i_{5}^{eff}-2$).
Therefore, in order to mark [4] or [6] there must be a singular string of length $(\geq 3i_{5}^{eff})$ in $\Tilde{\nu}^{(1)}$.
However, such a string does not exist because string of length $(\geq i_{5})$ in $\Tilde{\nu}^{(1)}$ are q-singular at best.
Hence the claim follows.
\begin{flushleft}
Case 4. $b_{1}= \framebox{$8$}$.
\end{flushleft}
Since $b_{1}= \framebox{$8$}$, after [5] is marked in $\nu^{(2)}$ the box marking has terminated.
Two strings of length one are marked by [1] and [5] in $\nu^{(2)}$.
Only one 1-string (the selected $i_{1}$-string) in $\nu^{(2)}$ is singular.
Strings of length $(\geq 2)$ in $\nu^{(2)}$ are not singular.
Otherwise [6] would be marked in $\nu^{(2)}$ contradicting $b_{1}= \framebox{$8$}$.
Since $\Delta p_{i}^{(2)}= 0$ $(i\geq 1)$ (see \textbf{(VC-8)}), all strings of $\Tilde{\nu}^{(2)}$ are still not singular so that $j=1$ and $j\neq \emptyset$.
\begin{flushleft}
Case 5. $b_{1}= \framebox{$9$}$.
\end{flushleft}
Since $b_{1}= \framebox{$9$}$, after [6] was marked in $\nu^{(1)}$ the box marking has terminated.
We claim that $1\leq j \leq 7$ but $j\neq 6$ and $j\neq \emptyset$.
To do so, we show that neither [4] nor [6] can be marked in $\Tilde{\nu}^{(1)}$.
From \textbf{(VC-9)}, we have two cases.
\begin{itemize}
\item[(1)] $i_{5}=i_{6}$.
\begin{itemize}
\item[(a)] The selected $i_{6}$-string in $\nu^{(1)}$ is type-II.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
2 & $(i\geq i_{6})$ & $-1$ & $(i\geq i_{6}^{eff})$ \\
0 & $(i=i_{6}-1)$ & 0 & $(i=i_{6}^{eff}-1)$ \\
& & 1 & $(1\leq i\leq i_{6}^{eff}-2)$
\end{tabular}
\end{center}
Since the selected $i_{1}$- and $i_{4}$-strings are deleted in $\Tilde{\nu}^{(2)}$, it is obvious from the table above that [1] can be marked in a string of length ($\geq i_{6}^{eff}-1$) if possible.
If [1] cannot be marked in $\Tilde{\nu}^{(2)}$, then $b_{2}=\framebox{$1$}$.
If [1] is marked in $\Tilde{\nu}^{(2)}$, then the box marking in $\Tilde{\nu}^{(1)}$ is possible only for strings of length ($\geq 3(i_{6}^{eff}-1)-2=i_{6}-3$).
Therefore, in order to mark [4] or [6] there must a singular string of length ($\geq i_{6}-1$) in $\Tilde{\nu}^{(1)}$.
However such a string does not exist in $\Tilde{\nu}^{(1)}$.
This is shown as follows.
The $(i_{6}-1)$-strings (type-0) in $\nu^{(1)}$ are qq-singular at best.
Otherwise [5] would be marked in an $(i_{6}-1)$-string.
Since $\Delta p_{i_{6}-1}^{(1)}=0$ and $\Delta p_{i}^{(1)}=2$ $(i\geq i_{6})$, strings of length $(\geq i_{6}-1)$ in $\Tilde{\nu}^{(1)}$ are qq-singular at best.
\item[(b)] The selected $i_{6}$-string in $\nu^{(1)}$ is type-0/I.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
2 & $(i\geq i_{6})$ & $-1$ & $(i\geq i_{6}^{eff})$ \\
0 & $(i=i_{6}-1)$ & 1 & $(1\leq i\leq i_{6}^{eff}-1)$
\end{tabular}
\end{center}
From the table above, box marking in $\Tilde{\nu}^{(1)}$ is possible only for strings of length ($\geq 3i_{6}^{eff}-2$).
The remaining argument is the same as in (a).
\end{itemize}
\item[(2)] $i_{5}<i_{6}$.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
2 & $(i\geq i_{6})$ & $-1$ & $(i\geq i_{6}^{eff})$
\end{tabular}
\end{center}
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ $(i\leq i_{6}-1)$ and $\Delta p_{i}^{(2)}$ $(i\leq i_{6}^{eff}-1)$ are the same as those in Case 3.
Note that the box-deleted $(i_{6}-1)$-string in $\Tilde{\nu}^{(1)}$ is set to be q-singular by \textbf{(RA-1)}.
The argument is the same as that in Case 3.
\end{itemize}
\begin{flushleft}
Case 6. $b_{1}= \framebox{$10$}$.
\end{flushleft}
Since $b_{1}= \framebox{$10$}$, after [6] was marked in $\nu^{(2)}$ the box marking has terminated.
\begin{itemize}
\item[(1)] $i_{5}=i_{6}=2$.
The effective length of the $i_{4}$-string in $\nu^{(1)}$ is one.
Otherwise, [7] would be marked on the left of [4] in $\nu^{(1)}$.
Therefore, only one 1-string (the $i_{1}$-string) in $\nu^{(2)}$ is singular.
Otherwise [5] would be marked in the singular 1-string in $\nu^{(2)}$ and the box marking would terminate.
Since $\Delta p_{1}^{(2)}=0$ (see \textbf{(VC-10)}), strings of length one in $\Tilde{\nu}^{(2)}$ are not singular.
Furthermore, $\Delta p_{i}^{(2)}=2$ $(i\geq 2)$ (see \textbf{(VC-10)}), strings of length ($\geq 2$) in $\Tilde{\nu}^{(2)}$ are also not singular.
Hence $j=1$.
In particular, $j\neq \emptyset$.
\item[(2)] $i_{5}=1$ and $i_{6}\geq 2$.
Since strings of length $i$ $(1\leq i \leq i_{6}-1)$ in $\nu^{(1)}$ except the $i_{1}$-string are not singular and $\Delta p_{i}^{(2)}=0$ $(1\leq i \leq i_{6}-1)$ (see \textbf{(VC-10)}), strings of length $i$ $(2\leq i \leq i_{6}-1)$ in $\Tilde{\nu}^{(1)}$ are also not singular.
Furthermore, $\Delta p_{i}^{(2)}=2$ $(i\geq i_{6})$ (see \textbf{(VC-10)}) so that all strings in $\Tilde{\nu}^{(2)}$ are not singular.
Hence $j=1$.
In particular, $j\neq \emptyset$.
\end{itemize}
\begin{flushleft}
Case 7. $b_{1}= \framebox{$11$}$.
\end{flushleft}
Since $b_{1}= \framebox{$11$}$, after [7] was marked in $\nu^{(1)}$ or $\nu^{(2)}$ the box marking has terminated.
We claim that $1\leq j \leq 7$ but $j\neq 6$ and $j\neq\emptyset$.
To do so, we show that neither [4] nor [6] can be marked in $\Tilde{\nu}^{(1)}$.
Suppose that [7] is marked in $\nu^{(1)}$.
Note that [7] is always marked in the string which is unmarked so far.
The changes of vacancy numbers $\Delta p_{i}^{(2)}$ in Case 6 are changed to $\Delta p_{i}^{(2)}=1$ only for $i\geq i_{7}^{eff}$ in this case.
Hence $j=1$ and $j\neq \emptyset$ as in Case 6.
So we assume that [6] is always marked in $\nu^{(1)}$ and [7] is always marked in $\nu^{(2)}$.
Furthermore if $i_{6}^{eff}<i_{7}$, then $\Delta p_{i}^{(1)}=-1$ $(i\geq 3i_{7})$, $\Delta p_{3i_{7}-1}^{(1)}=0$, and $\Delta p_{3i_{7}-2}^{(1)}=1$ from \textbf{(VC-11)}.
Here we note the fact that [8] cannot be marked in $\nu^{(1)}$.
This fact implies that strings of length $3i_{7}-2$ in $\nu^{(1)}$ are singular at best, strings of length $3i_{7}-1$ in $\nu^{(1)}$ are q-singular at best, and strings of length ($\geq 3i_{7}$) are qq-singular at best by the algorithm $\delta_{\theta}$.
Hence strings of length ($\geq 3i_{7}-2$) in $\Tilde{\nu}^{(1)}$ are q-singular at best.
The singularity of strings of length ($\leq 3i_{7}-3$) is the same as in Case 5.
Hence $1\leq j\leq 7$ but $j\neq 6$ and $j\neq \emptyset$.
So we assume that $i_{6}^{eff}=i_{7}$.
From \textbf{(VC-11)}, we have following two cases.
\begin{itemize}
\item[(1)] $i_{5}=i_{6}$.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$-1$ & $(i\geq 3i_{7})$ & 1 & $(i\geq i_{7})$ \\
0 & $(i=i_{6}+1)$ & $0$ & $(i=i_{7}-1)$ \\
1 & $(i=i_{6})$ & 1 & $(1\leq i \leq i_{7}-2)$ \\
0 & $(i=i_{6}-1)$ & &
\end{tabular}
\end{center}
Note that the $i_{6}$-string must be type-II.
Otherwise [8] would be marked on the left of [6].
Since the selected $i_{1}$- and $i_{4}$-strings are deleted in $\Tilde{\nu}^{(2)}$, it is obvious that [1] can be marked in a string of length ($\geq i_{7}-1$).
Therefore if [1] is marked in $\Tilde{\nu}^{(2)}$, then the box marking is possible only for strings of length ($\geq 3(i_{7}-1)-2=i_{6}-3$).
Therefor, in order to mark [4] or [6] there must be a singular string of length ($\geq i_{6}-1$) in $\Tilde{\nu}^{(1)}$.
However, such a string does not exist in $\Tilde{\nu}^{(1)}$.
This is shown as follows.
The $(i_{6}-1)$-strings in $\nu^{(1)}$ are qq-singular at best.
Otherwise [5] would be marked in an $(i_{6}-1)$-string.
Furthermore, since [8] cannot be marked in $\nu^{(1)}$, strings of length $3i_{7}-1(=i_{6}+1)$ are q-singular at best and strings of length ($\geq 3i_{7}$) are qq-singular at best.
Hence, from the table above, strings of length ($\geq i_{6}-1$) are not singular so that the box marking of [4] or [6] in $\Tilde{\nu}^{(1)}$ is impossible.
\item[(2)] $i_{5}<i_{6}$.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$-1$ & $(i\geq 3i_{7})$ & 1 & $(i\geq i_{7})$ \\
0 & $(i=i_{6}+1)$ & $0$ & $(i_{5}^{eff}\leq i\leq i_{7}-1)$ \\
1 & $(i=i_{6})$ & 1 & $(1\leq i \leq i_{5}^{eff}-1)$ \\
0 & $(i=i_{6}-1)$ & &
\end{tabular}
\end{center}
Note that the $i_{6}$-string must be type-II.
Otherwise [8] would be marked on the left of [6].
Since the selected $i_{1}$- and $i_{4}$-strings are deleted in $\Tilde{\nu}^{(2)}$, it is obvious that [1] can be marked in a string of length ($\geq i_{5}^{eff}$).
Therefore if [1] is marked in $\Tilde{\nu}^{(2)}$, then the box marking is possible only for strings of length ($\geq 3i_{5}^{eff}-2$) and in order to [4] or [6] there must be a singular string of length ($\geq 3i_{5}^{eff}$).
However, such a string does not exist in $\Tilde{\nu}^{(1)}$.
This is shown as follows.
Strings of length $i$ ($i_{5}\leq i\leq i_{6}-1$) in $\nu^{(1)}$ are q-singular at best and their singularity does not change in $\Tilde{\nu}^{(1)}$.
Furthermore, since [8] cannot be marked in $\nu^{(1)}$, strings of length $3i_{7}-1(=i_{6}+1)$ are q-singular at best and strings of length ($\geq 3i_{7}$) are qq-singular at best.
Hence from the table above, strings of length ($\geq i_{5}$) in $\Tilde{\nu}^{(1)}$ are not singular so that the box marking of [4] or [6] in $\Tilde{\nu}^{(1)}$ is impossible.
\end{itemize}
\begin{flushleft}
Case 8. $b_{1}= \framebox{$12$}$.
\end{flushleft}
Since $b_{1}= \framebox{$12$}$, after [8] was marked in $\nu^{(1)}$ the box marking has terminated.
We claim that $1\leq j \leq 7$ but $j\neq 6$ and $j\neq \emptyset$.
To do so, we show that neither [4] nor [6] can be marked in $\Tilde{\nu}^{(1)}$.
Firstly, we suppose that [8] is marked in the string which is unmarked so far in $\nu^{(1)}$, where [7] is marked in $\nu^{(1)}$ or $\nu^{(2)}$.
From \textbf{(VC-12)} (5), $\Delta p_{i}^{(1)}=1$ $(i\geq i_{8})$ and $\Delta p_{i}^{(2)}=0$ $(i\geq i_{8}^{eff})$.
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ ($i\leq i_{8}-1$) and $\Delta p_{i}^{(2)}$ ($i\leq i_{8}^{eff}-1$) are the same as those in Case 7.
The box-deleted $(i_{8}-1)$-string in $\Tilde{\nu}^{(1)}$ is set to be q-singular by the rule of rigging adjustment \textbf{(RA-5)}.
Therefore, the box marking of [4] or [6] in $\Tilde{\nu}^{(1)}$ is impossible as in Case 7.
Secondly, we suppose that [8] is marked in a marked string.
From \textbf{(VC-12)}, we have following four cases.
\begin{itemize}
\item[(1)] The $i_{7}$-string is in $\nu^{(1)}$ with $i_{8}=i_{7}=i_{4}$ ($i_{5}=i_{6}=2$) or the $i_{7}$-string is in $\nu^{(2)}$ with $i_{4}=i_{7}(=2)$.
The box marking in $\nu^{(1)}$ is one of the following three.
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\put(4,1){\line(0,1){1}}
\put(5,1){\line(0,1){1}}
\put(2,2){\makebox(2,1){$\downarrow$}}
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\end{center}
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$1$ & $(i\geq 6)$ & $0$ & $(i\geq 1)$ \\
$2$ & $(i=5)$ & & \\
$1$ & $(i=4)$ & & \\
$0$ & $(i=3)$ & &
\end{tabular}
\end{center}
Since 3-strings in $\nu^{(1)}$ are not singular, it is obvious that the box marking of [4] or [6] in $\Tilde{\nu}^{(1)}$ is impossible.
\item[(2)] $i_{8}=i_{7}\neq i_{4}$.
\begin{itemize}
\item[(a)] $i_{8}^{eff}=i_{6}$.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$1$ & $(i\geq 3i_{6})$ & $0$ & $(i\geq i_{6})$
\end{tabular}
\end{center}
\item[(b)] $i_{8}^{eff}>i_{6}$.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$1$ & $(i\geq i_{8})$ & $0$ & $(i\geq i_{8}^{eff})$
\end{tabular}
\end{center}
\end{itemize}
The changes of vacancy numbers $\Delta p_{i}^{(2)}$ $(i\leq i_{6}-1)$ (case (a)) and $\Delta p_{i}^{(2)}$ $(i\leq i_{8}^{eff}-1)$ (case (b)) are the same as those in Case 6 so that the box marking in $\Tilde{\nu}^{(1)}$ is possible only for strings of length $(\geq 3i_{6}-2)$ (case (a)) and $(\geq 3i_{8}^{eff}-2)$ (case (b)).
It is obvious that the box marking of [4] or [6] in $\Tilde{\nu}^{(1)}$ is impossible.
\item[(3)] $i_{8}=i_{6}=i_{5}\neq i_{3}$.
In this case, the $i_{8}$-string is type-I and $i_{8}^{eff}=i_{7}$; $i_{8}=3i_{7}-1$.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$1$ & $(i\geq i_{8}+1)$ & $0$ & $(i\geq i_{7}-1)$ \\
$2$ & $(i=i_{8})$ & $1$ & $(1\leq i\leq i_{7}-2)$ \\
$1$ & $(i=i_{8}-1)$ & & \\
$0$ & $(i=i_{8}-2)$ & &
\end{tabular}
\end{center}
The box marking in $\Tilde{\nu}^{(1)}$ is possible only for strings of length $(\geq 3(i_{7}-1)-2=i_{8}-4)$.
Therefore, in order to mark [4] or [6] there must be a singular string of length ($\geq i_{8}-2$) in $\Tilde{\nu}^{(1)}$.
However, such a string does not exist.
This is shown as follows.
The $(i_{8}-2)$-strings in $\nu^{(1)}$ are not singular as otherwise [4] would be marked in this string.
Therefore, strings of length ($\geq i_{8}-2$) in $\Tilde{\nu}^{(1)}$ are not singular.
\item[(4)] $i_{8}=i_{6}\neq i_{5}$.
In this case, the $i_{8}$-string is type-I and $i_{8}^{eff}=i_{7}$; $i_{8}=3i_{7}-1$.
Note that the box-deleted $(i_{8}-2)$-string is set to be q-singular by \textbf{(RA-1)}.
\begin{itemize}
\item[(a)] $i_{3}=i_{5}(=3)$.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$1$ & $(i\geq i_{8}+1)$ & $0$ & $(i\geq i_{7}-1)$ \\
$2$ & $(i=i_{8})$ & $1$ & $(1\leq i\leq i_{7}-2)$ \\
$1$ & $(i=i_{8}-1)$ & &
\end{tabular}
\end{center}
\item[(b)] $i_{3}<i_{5}$.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$1$ & $(i\geq i_{8}+1)$ & $0$ & $(i\geq i_{5}^{eff})$ \\
$2$ & $(i=i_{8})$ & $1$ & $(1\leq i\leq i_{5}^{eff}-1)$ \\
$1$ & $(i=i_{8}-1)$ & &
\end{tabular}
\end{center}
\end{itemize}
In both cases, it is obvious that the box marking of [4] or [6] in $\Tilde{\nu}^{(1)}$ is impossible.
\end{itemize}
\begin{flushleft}
Case 9. $b_{1}= \framebox{$13$}$.
\end{flushleft}
Since $b_{1}= \framebox{$13$}$, after [9] was marked in $\nu^{(1)}$ the box marking has terminated.
We claim that $1\leq j \leq 7$ but $j\neq 6$ and $j\neq \emptyset$.
The proof is very similar to Case 8 and we omit the details.
\begin{flushleft}
Case 10. $b_{1}= \framebox{$14$}$.
\end{flushleft}
We claim that $1\leq j \leq 9$ and $j\neq \emptyset$.
\begin{itemize}
\item[(1)] $i_{1}<i_{10}$.
\begin{itemize}
\item[(a)] $i_{9}^{eff}< i_{10}$.
\begin{center}
\begin{tabular}{cl|cl}
$\Delta p_{i}^{(1)}$ & & $\Delta p_{i}^{(2)}$ & \\ \hline
$0$ & $(i\geq 3i_{10})$ & 1 & $(i\geq i_{10})$ \\
1 & $(i=3i_{10}-1)$ & & \\
2 & $(i=3i_{10}-2)$ & & \\
3 & $(i_{9}\leq i \leq 3i_{10}-3)$ & &
\end{tabular}
\end{center}
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ $(i\leq 3(i_{10}-1))$ are the same as in Case 13.
Therefore, [4] or [6] can be marked in a singular string of length ($\geq 3i_{10}$) in $\Tilde{\nu}^{(1)}$.
If [4] is marked in such a string, then $j\geq 6$.
Although [5] can be marked in a q-singular string of length ($\geq i_{10}$) in $\Tilde{\nu}^{(2)}$, [6] cannot so that $j=8$ is possible but $j\ngeq 10$.
In particular, $j\neq \emptyset$.
If [6] is marked in such a string, then $j\geq 9$.
Since [7] cannot be marked in $\Tilde{\nu}^{(2)}$ because strings of length ($\geq i_{10}$) in $\Tilde{\nu}^{(2)}$ are q-singular at best so that $j\ngeq 11$.
Hence altogether we have $1\leq j\leq 9$ and $j\neq \emptyset$.
\item[(b)] $i_{9}^{eff}= i_{10}$.
Possible ways of box marking in $\nu^{(1)}$ are following three;
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(17,3)
\put(0,0){\line(1,0){4}}
\put(0,1){\line(1,0){4}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
\put(4,0){\line(0,1){1}}
\put(0,1){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[9]$}}}
\put(2,0){\makebox(1,1){{\scriptsize $[8]$}}}
\put(3,0){\makebox(1,1){{\scriptsize $[7]$}}}
\put(0,2){\makebox(1,1){{\small (i)}}}
\put(8,0){\line(1,0){3}}
\put(8,1){\line(1,0){3}}
\put(9,0){\line(0,1){1}}
\put(10,0){\line(0,1){1}}
\put(11,0){\line(0,1){1}}
\put(9,0){\makebox(1,1){{\scriptsize $[9]$}}}
\put(10,0){\makebox(1,1){{\scriptsize $[8]$}}}
\put(8,2){\makebox(1,1){{\small (ii)}}}
\put(15,0){\line(1,0){2}}
\put(15,1){\line(1,0){2}}
\put(16,0){\line(0,1){1}}
\put(17,0){\line(0,1){1}}
\put(16,0){\makebox(1,1){{\scriptsize $[9]$}}}
\put(15,2){\makebox(1,1){{\small (iii)}}}
\end{picture}
\end{center}
We omit the unmarked or marked strings except the selected $i_{9}$-string.
In either case, $\Delta p_{i}^{(2)}=1$ $(i\geq i_{10})$.
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ are obtained by overwriting
\[
-3\chi (i\geq 3i_{10})-2\chi (i=3i_{10}-1)-\chi (i=3i_{10}-2)
\]
on $\Delta p_{i}^{(1)}$ in Case 13 where $\Delta p_{i}^{(1)}\leq 3$ and in particular $\Delta p_{i}^{(1)}=3$ $(i\geq i_{9})$.
Therefore, $\Delta p_{i}^{(1)}\leq 0$ for some strings of effective length $i_{10}$ and $\Delta p_{i}^{(1)}=0$ $(i\geq 3i_{10})$.
In case (i), the selected $i_{9}$-string is type-0 and the box-deleted $(i_{9}-3)$-string in $\Tilde{\nu}^{(1)}$ is set to be singular but [4] or [6] cannot be marked in this string as in Case 9 so that [4] or [6] can be marked in only a singular string of effective length ($\geq i_{10}$) in $\Tilde{\nu}^{(1)}$.
The remaining argument is the same as in case (a) so that $1\leq j\leq 9$ and $j\neq \emptyset$.
In cases (ii) and (iii), the box-deleted $(i_{9}-1)$- or $(i_{9}-2)$-string is set to be q/qq-singular by \textbf{(RA-6)} or \textbf{(RA-7)} so that [4] or [6] cannot marked in this string.
Similarly, we have $1\leq j\leq 9$ and $j\neq \emptyset$.
\end{itemize}
\item[(2)] $i_{1}=i_{10}$.
From \textbf{(VC-14)}, it is obvious that $j=1$.
\end{itemize}
\end{proof}
\section{Verifications of the Rules of Forbidden Box Marking}
In this section we verify the rules of forbidden box marking.
They are necessary to ensure Eq.~\eqref{eq:dc}.
In \textbf{(BM-2)}, we excluded the box marking in $\nu^{(1)}$ depicted below.
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(11,3)
\put(0,0){\line(1,0){3}}
\put(0,1){\line(1,0){4}}
\put(0,2){\line(1,0){4}}
\put(1,0){\line(0,1){2}}
\put(2,0){\line(0,1){2}}
\put(3,0){\line(0,1){2}}
\put(4,1){\line(0,1){1}}
\put(0,2){\makebox(2,1){$\downarrow$}}
\put(2,0){\makebox(1,1){{\scriptsize $[2]$}}}
\put(3,1){\makebox(1,1){{\scriptsize $[3]$}}}
\put(5,1){\makebox(2,1){$\text{and}$}}
\put(8,0){\line(1,0){2}}
\put(8,1){\line(1,0){3}}
\put(8,2){\line(1,0){3}}
\put(9,0){\line(0,1){2}}
\put(10,0){\line(0,1){2}}
\put(11,1){\line(0,1){1}}
\put(8,2){\makebox(2,1){$\downarrow$}}
\put(9,0){\makebox(1,1){{\scriptsize $[2]$}}}
\put(10,1){\makebox(1,1){{\scriptsize $[3]$}}}
\end{picture}
\end{center}
where $i_{2}^{eff}>i_{1}$ and the $i_{2}$ (resp. $i_{3}$)-string is qq-singular (resp. singular).
Suppose that [4] is not marked in $\nu^{(2)}$.
Then,
\begin{align*}
\Delta _{2\rightarrow 4}c(\nu,J)=-2&+\Delta r_{i_{2}}^{(1)}-(\Delta p_{i_{2}-1}^{(1)}+\Delta r_{i_{2}-1}^{(1)}) \\
&+\Delta r_{i_{3}}^{(1)}-(\Delta p_{i_{3}-1}^{(1)}+\Delta r_{i_{3}-1}^{(1)}),
\end{align*}
where $\Delta r_{i_{2}}^{(1)}=-2$, $\Delta r_{i_{3}}^{(1)}=0$, $\Delta r_{i_{2}-1}^{(1)}=0$ and $\Delta r_{i_{3}-1}^{(1)}=-1$ by \textbf{(RA-2)}.
The changes of vacancy numbers are $\Delta p_{i_{2}-1}^{(1)}=-3$ and $\Delta p_{i_{3}-1}^{(1)}=-1$ (see \textbf{(VC-2)} and \textbf{(VC-3)}).
Therefore $\Delta _{2\rightarrow 4}c(\nu,J)=-1$ so that $\Delta c(\nu,J)\neq \alpha_{1}^{(2)}$.
We excluded the box marking in $\nu^{(1)}$ depicted below.
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(4,3)
\put(0,0){\line(1,0){3}}
\put(0,1){\line(1,0){4}}
\put(0,2){\line(1,0){4}}
\put(1,0){\line(0,1){2}}
\put(2,0){\line(0,1){2}}
\put(3,0){\line(0,1){2}}
\put(4,1){\line(0,1){1}}
\put(0,2){\makebox(2,1){$\downarrow$}}
\put(2,0){\makebox(1,1){{\scriptsize $[2]$}}}
\put(3,1){\makebox(1,1){{\scriptsize $[3]$}}}
\end{picture}
\end{center}
where $i_{2}^{eff}>i_{1}$ and the $i_{2}$ (resp. $i_{3}$)-string is qq-singular (resp. q-singular).
Suppose that [4] is marked in $\nu^{(2)}$ with $i_{3}^{eff}=i_{4}$.
Then,
\begin{align*}
\Delta _{2\rightarrow 5}c(\nu,J)=-1&+\Delta r_{i_{2}}^{(1)}-(\Delta p_{i_{2}-1}^{(1)}+\Delta r_{i_{2}-1}^{(1)}) \\
&+\Delta r_{i_{3}}^{(1)}-(\Delta p_{i_{3}-1}^{(1)}+\Delta r_{i_{3}-1}^{(1)}) \\
&+\Delta r_{i_{4}}^{(2)}-(\Delta p_{i_{4}-1}^{(2)}+\Delta r_{i_{4}-1}^{(2)}),
\end{align*}
where $\Delta r_{i_{2}}^{(1)}=-2$, $\Delta r_{i_{3}}^{(1)}=-1$, $\Delta r_{i_{4}}^{(2)}=0$, $\Delta r_{i_{2}-1}^{(1)}=0$, $\Delta r_{i_{3}-1}^{(1)}=0$ by \textbf{(RA-2)}, and $\Delta r_{i_{4}-1}^{(2)}=0$,
The changes of vacancy numbers $\Delta p_{i_{2}-1}^{(1)}$ and $\Delta p_{i_{3}-1}^{(1)}$ are obtained by overwriting
\[
-3\chi (i\geq 3i_{5})-2\chi (i=3i_{5}-1)-\chi (i=3i_{5}-2)
\]
on the values of previous case.
That is, $\Delta p_{i_{2}-1}^{(1)}=-4$ and $\Delta p_{i_{3}-1}^{(1)}=-3$.
By \textbf{(VC-3)} and \textbf{(VC-4)} $\Delta p_{i_{4}-1}^{(2)}=0$.
Therefore, $\Delta _{2\rightarrow 5}c(\nu,J)=3$ so that $\Delta c(\nu,J)\neq \alpha_{1}^{(2)}$.
We excluded the box marking (a) in $\nu^{(1)}$ depicted below and prescribe that the box marking must be (b).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(12,4)
\put(0,0){\line(1,0){2}}
\put(0,1){\line(1,0){4}}
\put(0,2){\line(1,0){4}}
\put(1,0){\line(0,1){2}}
\put(2,0){\line(0,1){2}}
\put(3,1){\line(0,1){1}}
\put(4,1){\line(0,1){1}}
\put(0,2){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[2]$}}}
\put(2,1){\makebox(1,1){{\scriptsize $[4]$}}}
\put(3,1){\makebox(1,1){{\scriptsize $[3]$}}}
\put(0,3){\makebox(1,1){{\small (a)}}}
\put(8,0){\line(1,0){2}}
\put(8,1){\line(1,0){4}}
\put(8,2){\line(1,0){4}}
\put(9,0){\line(0,1){2}}
\put(10,0){\line(0,1){2}}
\put(11,1){\line(0,1){1}}
\put(12,1){\line(0,1){1}}
\put(8,2){\makebox(2,1){$\downarrow$}}
\put(9,1){\makebox(1,1){{\scriptsize $[4]$}}}
\put(10,1){\makebox(1,1){{\scriptsize $[3]$}}}
\put(11,1){\makebox(1,1){{\scriptsize $[2]$}}}
\put(8,3){\makebox(1,1){{\small (b)}}}
\end{picture}
\end{center}
where $i_{2}^{eff}>i_{1}$ and the first (resp. second) string is singular (resp. qq-singular).
It is easily verified that (a) and (b) yields the same RC unless [5] and [6] are marked in $\nu^{(2)}$ with $i_{5}=i_{6}$ and $i_{4}^{eff}=i_{6}$.
The changes of vacancy numbers in (a) are $\Delta p_{i_{2}-1}^{(1)}=-3$ and $\Delta p_{i_{4}-2}^{(1)}=-1$ (see \textbf{(VC-2)} and \textbf{(VC-3)}).
Now suppose that [5] and [6] are marked in $\nu^{(2)}$ with $i_{5}=i_{6}$ and $i_{4}^{eff}=i_{6}$.
In case (a), the new vacancy changes $\Delta p_{i_{2}-1}^{(1)}$ and $\Delta p_{i_{4}-2}^{(1)}$ are obtained by overwriting
\begin{align*}
&-6\chi (i\geq 3i_{6})-5\chi (i=3i_{6}-1)-4\chi (i=3i_{6}-2) \\
&-3\chi (i=3i_{6}-3)-2\chi (i=3i_{6}-4)-\chi (i=3i_{6}-5)
\end{align*}
on the old ones.
That is, $\Delta p_{i_{2}-1}^{(1)}=-6$ and $\Delta p_{i_{4}-2}^{(1)}=-5$.
$\Delta p_{i_{6}-2}^{(2)}=0$, which is unchanged (see \textbf{(VC-3)}).
$\Delta_{2\rightarrow 10}c(\nu,J)$ is computed to be
\begin{align*}
-2&+\Delta r_{i_{2}}^{(1)}-(\Delta p_{i_{2}-1}^{(1)}+\Delta r_{i_{2}-1}^{(1)}) \\
&+\Delta r_{i_{4}}^{(1)}-(\Delta p_{i_{4}-2}^{(1)}+\Delta r_{i_{4}-2}^{(1)}) \\
&+\Delta r_{i_{6}}^{(2)}-(\Delta p_{i_{6}-2}^{(2)}+\Delta r_{i_{6}-2}^{(2)}),
\end{align*}
where $\Delta r_{i_{2}}^{(1)}=-2$ and other rigging adjustments are zero so that $\Delta_{2\rightarrow 10}c(\nu,J)=7$.
In this case, we must do the following box marking in $\nu^{(1)}$.
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\begin{center}
\begin{picture}(7,2)
\put(0,0){\line(1,0){7}}
\put(0,1){\line(1,0){7}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(3,0){\line(0,1){1}}
\put(4,0){\line(0,1){1}}
\put(5,0){\line(0,1){1}}
\put(6,0){\line(0,1){1}}
\put(7,0){\line(0,1){1}}
\put(3,1){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[9]$}}}
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\put(3,0){\makebox(1,1){{\scriptsize $[7]$}}}
\put(4,0){\makebox(1,1){{\scriptsize $[4]$}}}
\put(5,0){\makebox(1,1){{\scriptsize $[3]$}}}
\put(6,0){\makebox(1,1){{\scriptsize $[2]$}}}
\end{picture}
\end{center}
Suppose that $i_{6}\geq i_{1}+2$.
Then $\Delta p_{i_{9}-6}^{(1)}=-3$ and $\Delta p_{i_{6}-2}^{(2)}=1$,
\[
\Delta_{2\rightarrow 13}c(\nu,J)=-2-\Delta p_{i_{9}-6}^{(1)}-\Delta p_{i_{6}-2}^{(2)}=0
\]
so that $\Delta c(\nu,J)=\Delta_{1}c(\nu,J)+\Delta_{1\rightarrow 2}c(\nu,J)+\Delta_{2\rightarrow 13}c(\nu,J)=-\alpha_{1}^{(2)}$.
The computation in the case when $i_{6}=i_{1}+1$ is similar.
In this case, we must consider the changes of $m_{i_{1}}^{(2)}$ and $m_{i_{1}-1}^{(2)}$ as well.
We also excluded the box marking (a) in $\nu^{(1)}$ depicted below and prescribed that the box marking must be (b).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(12,4)
\put(0,0){\line(1,0){3}}
\put(0,1){\line(1,0){4}}
\put(0,2){\line(1,0){4}}
\put(1,0){\line(0,1){2}}
\put(2,0){\line(0,1){2}}
\put(3,0){\line(0,1){2}}
\put(4,1){\line(0,1){1}}
\put(0,2){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[3]$}}}
\put(2,0){\makebox(1,1){{\scriptsize $[2]$}}}
\put(3,1){\makebox(1,1){{\scriptsize $[4]$}}}
\put(0,3){\makebox(1,1){{\small (a)}}}
\put(8,0){\line(1,0){3}}
\put(8,1){\line(1,0){4}}
\put(8,2){\line(1,0){4}}
\put(9,0){\line(0,1){2}}
\put(10,0){\line(0,1){2}}
\put(11,0){\line(0,1){2}}
\put(12,1){\line(0,1){1}}
\put(8,2){\makebox(2,1){$\downarrow$}}
\put(9,1){\makebox(1,1){{\scriptsize $[4]$}}}
\put(10,1){\makebox(1,1){{\scriptsize $[3]$}}}
\put(11,1){\makebox(1,1){{\scriptsize $[2]$}}}
\put(8,3){\makebox(1,1){{\small (b)}}}
\end{picture}
\end{center}
where $i_{2}^{eff}>i_{1}$ and the first (resp. second) string is singular (resp. q-singular).
The verification of this case is similar.
In \textbf{(BM-3)}, we excluded the following box marking (see Example~\ref{ex:BM-3}).
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(3,3)
\put(0,0){\line(1,0){2}}
\put(0,1){\line(1,0){3}}
\put(0,2){\line(1,0){3}}
\put(1,0){\line(0,1){2}}
\put(2,0){\line(0,1){2}}
\put(3,1){\line(0,1){1}}
\put(1,2){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[2]$}}}
\put(2,1){\makebox(1,1){{\scriptsize $[3]$}}}
\end{picture}
\end{center}
where the $i_{2}$-string is qq-singular and the $i_{3}$-string is singular.
Suppose that [4] is not marked in $\nu^{(2)}$.
Then
\[
\Delta_{3\rightarrow 4}c(\nu,J)=-1+\Delta r_{i_{3}}^{(1)}-(\Delta p_{i_{3}-1}^{(1)}+\Delta r_{i_{3}-1}^{(1)}),
\]
where $\Delta r_{i_{3}}^{(1)}=0$, $\Delta r_{i_{3}-1}^{(1)}=-1$ by \textbf{(RA-2)}, and $\Delta p_{i_{3}-1}^{(1)}=-1$ (see \textbf{(VC-3)}) so that $\Delta_{3\rightarrow 4}c(\nu,J)=1$, which yields $\Delta c(\nu,J)\neq \alpha_{1}^{(2)}$.
In \textbf{(BM-4)}, we excluded the following box marking.
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(4,3)
\put(0,0){\line(1,0){3}}
\put(0,1){\line(1,0){4}}
\put(0,2){\line(1,0){4}}
\put(1,0){\line(0,1){2}}
\put(2,0){\line(0,1){2}}
\put(3,0){\line(0,1){2}}
\put(4,1){\line(0,1){1}}
\put(0,2){\makebox(2,1){$\downarrow$}}
\put(2,0){\makebox(1,1){{\scriptsize $[3]$}}}
\put(3,1){\makebox(1,1){{\scriptsize $[4]$}}}
\end{picture}
\end{center}
where $i_{3}^{eff}>i_{1}$ and the first (resp. second) string is singular (resp. q-singular).
Suppose that [5] is marked in a q-singular string in $\nu^{(2)}$ with $i_{4}^{eff}=i_{5}$.
Then
\begin{align*}
\Delta _{3\rightarrow 8}c(\nu,J)=-1&+\Delta r_{i_{3}}^{(1)}-(\Delta p_{i_{3}-1}^{(1)}+\Delta r_{i_{3}-1}^{(1)}) \\
&+\Delta r_{i_{4}}^{(1)}-(\Delta p_{i_{4}-1}^{(1)}+\Delta r_{i_{4}-1}^{(1)}) \\
&+\Delta r_{i_{5}}^{(2)}-(\Delta p_{i_{5}-1}^{(2)}+\Delta r_{i_{5}-1}^{(2)}),
\end{align*}
where $\Delta r_{i_{3}}^{(1)}=-1$, $\Delta r_{i_{4}}^{(1)}=0$, $\Delta r_{i_{5}}=-1$, $\Delta r_{i_{3}-1}=-1$ by \textbf{(RA-3)}, $\Delta r_{i_{4}-1}^{(1)}=0$ by \textbf{(RA-3)}, and $\Delta r_{i_{5}-1}^{(2)}=0$.
If [5] is not marked in $\nu^{(2)}$, then $\Delta p_{i_{3}-1}^{(1)}=-1$ and $\Delta p_{i_{4}-1}^{(1)}=1$.
When [5] is marked in $\nu^{(2)}$ with $i_{4}^{eff}=i_{5}$, the new vacancy changes in $\nu^{(1)}$ are obtained by overwriting
\[
-3\chi (i\geq 3i_{5})-2\chi (i=3i_{5}-1)-\chi (i=3i_{5}-2)
\]
on old ones.
Therefore, $\Delta p_{i_{3}-1}^{(1)}=-2$ and $\Delta p_{i_{4}-1}^{(1)}=-1$.
$\Delta p_{i_{5}-1}^{(2)}=-1$, which is unchanged (see \textbf{(VC-3)} and \textbf{(VC-4)}).
Hence, $\Delta _{3\rightarrow 8}c(\nu,J)=2$, which yields $\Delta c(\nu,J)\neq \alpha_{1}^{(2)}$.
In \textbf{(BM-5)}, we excluded the following box marking.
\setlength{\unitlength}{10pt}
\begin{center}
\begin{picture}(14,3)
\put(0,0){\line(1,0){2}}
\put(0,1){\line(1,0){2}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(0,1){1}}
\put(0,1){\makebox(2,1){$\downarrow$}}
\put(1,0){\makebox(1,1){{\scriptsize $[5]$}}}
\put(0,2){\makebox(1,1){{\small (a)}}}
\put(5,0){\line(1,0){3}}
\put(5,1){\line(1,0){3}}
\put(6,0){\line(0,1){1}}
\put(7,0){\line(0,1){1}}
\put(8,0){\line(0,1){1}}
\put(5,1){\makebox(2,1){$\downarrow$}}
\put(7,0){\makebox(1,1){{\scriptsize $[5]$}}}
\put(5,2){\makebox(1,1){{\small (b)}}}
\put(11,0){\line(1,0){3}}
\put(11,1){\line(1,0){3}}
\put(12,0){\line(0,1){1}}
\put(13,0){\line(0,1){1}}
\put(14,0){\line(0,1){1}}
\put(11,1){\makebox(2,1){$\downarrow$}}
\put(12,0){\makebox(1,1){{\scriptsize $[5]$}}}
\put(13,0){\makebox(1,1){{\scriptsize $[3]$}}}
\put(11,2){\makebox(1,1){{\small (c)}}}
\end{picture}
\end{center}
where $i_{5}^{eff}=i_{4}$.
In cases (a) and (b),
\begin{align*}
\Delta _{5\rightarrow 7}c(\nu,J)=-1&+\Delta r_{i_{4}}^{(2)}-(\Delta p_{i_{4}-1}^{(2)}+\Delta r_{i_{4}-1}^{(2)}) \\
&+\Delta r_{i_{5}}^{(1)}-(\Delta p_{i_{5}-1}^{(1)}+\Delta r_{i_{5}-1}^{(1)}),
\end{align*}
where $\Delta r_{i_{4}}^{(2)}=0$, $\Delta r_{i_{5}}^{(1)}=-1$, $\Delta r_{i_{4}-1}^{(2)}=0$, and $\Delta r_{i_{5}-1}^{(1)}=0$.
In case (a) , the box marking by [4] and [5] does not affect the change of vacancy number of the $(i_{5}-1)$-string in $\nu^{(1)}$ and that of the $(i_{4}-1)$-string in $\nu^{(2)}$ so that $\Delta p_{i_{4}-1}^{(2)}=-1$ and $\Delta p_{i_{5}-1}^{(1)}=-1$ (see \textbf{(VC-4)}) and therefore $\Delta _{5\rightarrow 7}c(\nu,J)=-2$, which yields $\Delta c(\nu,J)\neq \alpha_{1}^{(2)}$.
In case (b), $\Delta p_{i_{4}-1}^{(2)}=-1$ (see \textbf{(VC-4)}) and $\Delta p_{i_{5}-1}^{(1)}=0$ (see \textbf{(VC-5)}) so that $\Delta _{5\rightarrow 7}c(\nu,J)=-1$, which yields $\Delta c(\nu,J)\neq \alpha_{1}^{(2)}$.
In case (c),
\begin{align*}
\Delta _{3\rightarrow 7}c(\nu,J)=-1&+\Delta r_{i_{4}}^{(2)}-(\Delta p_{i_{4}-1}^{(2)}+\Delta r_{i_{4}-1}^{(2)}) \\
&+\Delta r_{i_{5}}^{(1)}-(\Delta p_{i_{5}-2}^{(1)}+\Delta r_{i_{5}-2}^{(1)}),
\end{align*}
where $\Delta r_{i_{4}}^{(2)}=0$, $\Delta r_{i_{5}}^{(1)}=-1$, $\Delta r_{i_{4}-1}^{(2)}=0$, and $\Delta r_{i_{5}-2}^{(1)}=0$.
Since the box marking by [3], [4], and [5] and the deletion these boxes do not affect the change of the vacancy number of the $(i_{5}-2)$-string in $\Tilde{\nu}^{(1)}$ so that $\Delta p_{i_{5}-2}^{(1)}=-1$ (see \textbf{(VC-2)}).
$\Delta p_{i_{4}-1}^{(2)}$ is obtained by overwriting
\[
-2\chi (3i\geq i_{5})-\chi (3i=i_{5}-1)+2\chi (i\geq i_{4})
\]
on $\Delta p_{i_{4}-1}^{(2)}$ in \textbf{(VC-2)}.
Again it is unchanged and $\Delta p_{i_{4}-1}^{(2)}=0$.
Hence $\Delta _{3\rightarrow 7}c(\nu,J)=-1$, which yields $\Delta c(\nu,J)\neq \alpha_{1}^{(2)}$.
In \textbf{(BM-8)}, we excluded the following box marking in $\nu^{(1)}$.
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\end{center}
where $i_{9}^{eff}=i_{6}$ ($i_{9}=3i_{6}-1$) (see Example~\ref{ex:BM-8a}).
\begin{align*}
\Delta _{3\rightarrow 13}c(\nu,J)=-1&+\Delta r_{i_{6}}^{(2)}-(\Delta p_{i_{6}-2}^{(2)}+\Delta r_{i_{6}-2}^{(2)}) \\
&+\Delta r_{i_{9}}^{(1)}-(\Delta p_{i_{9}-5}^{(1)}+\Delta r_{i_{9}-5}^{(1)}),
\end{align*}
where $\Delta r_{i_{6}}^{(2)}=\Delta r_{i_{6}-2}^{(2)}=\Delta r_{i_{9}}^{(1)}=\Delta r_{i_{9}-5}^{(1)}=0$.
The change of vacancy number $\Delta p_{i_{9}-5}^{(1)}$ is obtained by overwriting
\begin{align*}
&10\chi (i\geq i_{9})+8\chi (i=i_{9}-1)+\cdots +2\chi (i=i_{9}-4) \\
&-6\chi (i\geq 3i_{6})-5\chi (i=3i_{6}-1)-\cdots -\chi (i=3i_{6}-5)
\end{align*}
on $\Delta p_{i_{9}-5}^{(1)}$ in \textbf{(VC-3)} so that $\Delta p_{i_{9}-5}^{(1)}=-1$.
The change of vacancy number $\Delta p_{i_{6}-2}^{(2)}$ is obtained by overwriting
\begin{align*}
&-5\chi (3i\geq i_{4})-4\chi (3i=i_{9}-1)-\cdots -2\chi (3i=i_{9}-4) \\
&+4\chi (i\geq i_{6})+2\chi (i=i_{6}-1)
\end{align*}
on $\Delta p_{i_{6}-2}^{(2)}$ in \textbf{(VC-3)} so that $\Delta p_{i_{6}-2}^{(2)}=0$.
Hence $\Delta _{3\rightarrow 13}c(\nu,J)=-1$, which yields $\Delta c(\nu,J)\neq \alpha_{1}^{(2)}$.
The verification of the following forbidden box marking in $\nu^{(1)}$
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\put(3,0){\makebox(1,1){{\scriptsize $[6]$}}}
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with $i_{9}^{eff}=i_{7}$ (see Example~\ref{ex:BM-8b} or \ref{ex:BM-8c}) is similar.
In \textbf{(BM-9)}, we excluded the following box marking in $\nu^{(1)}$.
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\put(3,1){\makebox(1,1){{\scriptsize $[9]$}}}
\end{picture}
\end{center}
where the first (resp. second) string is singular (resp. q-singular).
If [7] is marked in $\nu^{(1)}$, then $i_{7}\leq i_{8}-2$ and if [7] is marked in $\nu^{(2)}$, then $i_{8}^{eff}>i_{7}$ (see the preferential rule of \textbf{(BM-8)} (2)).
Suppose that [10] is marked in a singular string in $\nu^{(2)}$ with $i_{9}^{eff}=i_{10}$.
Then,
\begin{align*}
\Delta _{11\rightarrow 14}c(\nu,J)=-1&+\Delta r_{i_{8}}^{(1)}-(\Delta p_{i_{8}-1}^{(1)}+\Delta r_{i_{8}-1}^{(1)}) \\
&+\Delta r_{i_{9}}^{(1)}-(\Delta p_{i_{9}-1}^{(1)}+\Delta r_{i_{9}-1}^{(1)}) \\
&+\Delta r_{i_{10}}^{(2)}-(\Delta p_{i_{10}-1}^{(2)}+\Delta r_{i_{10}-1}^{(2)}),
\end{align*}
where $\Delta r_{i_{8}}^{(1)}=-1$, $\Delta r_{i_{9}}^{(1)}=0$, $\Delta r_{i_{10}}^{(2)}=0$, $\Delta r_{i_{8}-1}^{(1)}=-1$ by \textbf{(RA-7)}, $\Delta r_{i_{9}-1}^{(1)}=0$ by \textbf{(RA-7)}, and $\Delta r_{i_{10}-1}^{(2)}=0$.
The changes of vacancy numbers $\Delta p_{i}^{(1)}$ are obtained by overwriting
\[
-3\chi (i\geq 3i_{10})-2\chi (i=3i_{10}-1)-\chi (i=3i_{10}-2)
\]
on $\Delta p_{i}^{(1)}$ in \textbf{(VC-13)} so that $\Delta p_{i_{8}-1}^{(1)}=-2$ and $\Delta p_{i_{9}-1}^{(1)}=-1$.
By \textbf{(VC-12)} and \textbf{(VC-13)}, $\Delta p_{i_{10}-1}^{(2)}=0$.
Hence, $\Delta _{11\rightarrow 14}c(\nu,J)=2$, which yields $\Delta c(\nu,J)\neq \alpha_{1}^{(2)}$.
The verifications of rules of boomerang strings are similar and we omit the details.
\subsection*{Acknowledgements}
The author would like to express his gratitude to Professor Masato Okado for helpful discussions.
He also would like to thank Professor Travis Scrimshaw for helpful comments.
This work was partly supported by Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849).
|
train/arxiv
|
BkiUbaA5ixsDL_dM-xgW
| 5 | 1 |
\section{Introduction}
Pure spinor superfield theory \cite{Cederwall:2013vba} provides a
solution to the long-standing problem of covariant quantisation of (Brink--Schwarz) superparticles
\cite{Brink:1981nb,Casalbuoni:1976tz}
or (Green--Schwarz) superstrings
\cite{Green:1983wt} with manifest supersymmetry, or roughly equivalently, to the problem of finding off-shell superspace formulations of maximally supersymmetric field theories, including supergravity.
Concretely, the difficulties with space-time supersymmetric particles and strings manifest themselves
as a mixture of first and second class constraints in the same spinor.
This is the famous $\kappa$-symmetry
\cite{deAzcarraga:1982njd,Siegel:1983hh,Bengtsson:1984rw}, which is necessary for the superparticle/superstring action to describe the dynamics of a ${1\over2}$-BPS object.
In the present overview, we will not start with these superparticle or -string actions. Rather, the introduction of pure spinor variables will be motivated by the structure of the (on-shell) multiplets of maximal super-Yang--Mills theory (SYM) and supergravity (SG) in their traditional treatment on superspace.
The relation of the pure spinor formulation to the Green--Schwarz superstring is explained in ref. \cite{Berkovits:2004tw}.
The basics of the formalism is laid out in Section \ref{PureSpinorSuperfieldSection}. In Section \ref{SGSection} it is applied to supergravity, with maximal supergravity as main focus. A brief account of the pure spinor superstring theory of Berkovits is given in Section \ref{StringSection}. Quantum theory is sketched in Section \ref{QuantumSection}, and some convergence results for
loop diagrams are restated. Finally, some remarks are made in Section \ref{RemarksSection} concerning possible refinement and development of the formalism.
The technical level of the presentation is kept at a minimum. Instead, we aim at collecting results from the sources in the reference list and present them as concisely and coherently as possible, while emphasising concepts rather than techniques.
\section{Pure spinor superfield theory\label{PureSpinorSuperfieldSection}}
Before going into a more precise derivation of pure spinor superfield formulations of specific supersymmetric models, we would like to sketch what lies at the heart of the formalism.
The supersymmetry algebra (which of course is a subalgebra of the super-Poincar\'e algebra) takes the generic form
$\{Q_\alpha,Q_\beta\}=2\gamma^a_{\alpha\beta}\*_a$. Here, $\alpha$ is some (possibly multiple) spinor index, and
$Q_\alpha={\*\over\*\theta^\alpha}+(\gamma^a\theta)_\alpha\*_a$. Covariant fermionic derivatives
$D_\alpha={\*\over\*\theta^\alpha}-(\gamma^a\theta)_\alpha\*_a$ satisfy $\{Q_\alpha,D_\beta\}=0$. They anticommute among themselves as $\{D_\alpha,D_\beta\}=-2\gamma^a_{\alpha\beta}\*_a$ --- flat superspace in endowed with torsion
$T_{\alpha\beta}{}^a=2\gamma^a_{\alpha\beta}$.
Suppose we introduce a bosonic spinor $\lambda^\alpha$ subject to the constraint
\begin{align}
(\lambda\gamma^a\lambda)=0\label{PureEq}\;.
\end{align}
Then we may form a fermionic operator
\begin{align}
Q=\lambda^\alpha D_\alpha\;,\label{FieldQ}
\end{align}
which, thanks to the constraint on $\lambda$ is nilpotent: $Q^2=0$.
It seems meaningful to consider the cohomology of $Q$, acting on functions of $x$, $\theta$ and $\lambda$.
This cohomology is guaranteed to be supersymmetric, since $Q$ anticommutes with the supersymmetry generators. It thus describes some supermultiplet.
As it turns out, any linear supermultiplet in any dimension may be obtained this way.
In the case of on-shell multiplets the virtue of the formalism is even greater, since it seems to offer a natural way to an off-shell formulation by relaxation of the linear ``equation of motion'' $Q\Psi=0$.
The correspondence will be made more precise below, first for $D=10$ super-Yang--Mills theory and later for $D=11$ supergravity.
A word of caution: We will refer to a spinor $\lambda$ subject to the constraint \eqref{PureEq} as a ``pure spinor''.
This is a slight misuse of the mathematical terminology. A pure spinor, in the sense of Cartan \cite{Cartan:1938jua}, is a chiral spinor in even dimension $D=2n$, constrained to lie in the minimal $Spin(2n)$ orbit of the spinor module. This implies that, if the Dynkin label of the spinor module in question is $(0\ldots01)$, monomials of degree of homogeneity $p$ in $\lambda$ belong to the single module
$(0\ldots0p)$. The concept of a pure spinor is not defined in other cases, neither for odd dimensions or for extended supersymmetry. In certain cases, our constrained spinors coincide with Cartan pure spinors. This happens notably in $D=10$. There, the spinor bilinears are a vector $(10000)$ and a self-dual 5-form $(00002)$, and the constraint in the vector immediately puts $\lambda$ in the minimal orbit. In other situations, for example $D=11$ which we will encounter later, where the symmetric spinor bilinears are a vector, a $2$-form and a $5$-form, the vector constraints puts $\lambda$ in (a completion of) an intermediate orbit, which is not the minimal one.
\subsection{From superspace to pure spinor superspace}
Although we ultimately aim at addressing supergravity, the introduction of pure spinor superspace is much simpler in the setting
of super-Yang--Mills theory \cite{Brink:1976bc}, first treated in superspace in ref. \cite{Siegel:1978yi}.
Flat $(10|16)$-dimensional superspace, appropriate for $D=10$ super-Yang--Mills, has coordinates $Z^M=(x^m,\theta^\mu)$. There is no metric on superspace, but a super-vielbein $E_M{}^A$. The Lorentz frame indices $A=(a,\alpha)$ consist of a Lorentz vector and a chiral spinor. The non-vanishing superspace torsion is
\begin{align}
T_{\alpha\beta}{}^a=2\gamma^a_{\alpha\beta}\;,
\end{align}
where the components are converted to Lorentz frame using the inverse vielbein:
$T_{BC}{}^A=(E^{-1})_C{}^N(E^{-1})_B{}^MT_{MN}{}^A$.
Let us now recollect some known facts about $D=10$ super-Yang--Mills.
A connection on superspace, taking values in the adjoint of some gauge group, is written
$A=dZ^MA_M=dZ^ME_M{}^AA_A$.
There is {\it a priori}\ two superfields, $A_a(x,\theta)$ (bosonic, dimension $1$) and $A_\alpha(x,\theta)$
(fermionic, dimension $\fr2$)\footnote{As is standard, dimension is in terms of powers of inverse length.}.
The field strength is $F=dA+A\wedge A$, and due to the presence of torsion we have
\begin{align}
F_{\alpha\beta}=2D_{(\alpha}A_{\beta)}+2A_{(\alpha}A_{\beta)}+2\gamma^a_{\alpha\beta}A_a\;.\label{F0Eq}
\end{align}
The symmetric product of two spinors can be decomposed into a vector
$F_a=\fraction{16}\gamma_a^{\alpha\beta}F_{\alpha\beta}$
and a self-dual $5$-form
$F_{abcde}=\fraction{2\cdot5!\cdot16}\gamma_{abcde}^{\alpha\beta}F_{\alpha\beta}$.
Obviously, from eq. \eqref{F0Eq}, setting $F_a=0$ expresses $A_a$ in terms of $A_\alpha$, leaving only the latter as an independent superfield. This goes under the name of ``conventional constraint''. Note that it is natural, since $F_{\alpha\beta}$ has dimension $1$, and there are no physical and gauge-covariant fields of this dimension in the supermultiplet we want to derive
(the spinor $\chi^\alpha$ has dimension $\Fr32$ and the field strength $F_{ab}$ dimension $2$).
For this reason, it is tempting to also set the $5$-form part $F_{abcde}$ to zero.
However, doing this turns out to put the theory on shell. It indeed describes the on-shell $D=10$ super-Yang--Mills multiplet.
The constraint in question is physical, rather than conventional.
Details can be found {\it e.g.}\ in refs. \cite{Nilsson:1985cm,Cederwall:2001bt}.
It is of course also well known that the supersymmetry transformations of the component fields work (``close'', modulo gauge transformations) only when the equations of motion are satisfied. There is no set of auxiliary fields that remedies this.
This observation prompted Nilsson \cite{Nilsson:1985cm} to first draw the (correct) conclusion that in order to go off-shell one needs to relax the $5$-form part of $F_{\alpha\beta}=0$. And this is what pure spinor superfield theory naturally does, as we will see. Indeed, the equations of motion in the pure spinor superfield description of $D=10$ super-Yang--Mills theory will be
$\gamma_{(5)}^{\alpha\beta}F_{\alpha\beta}=0$.
This structure was found when searching for deformations of the equations of motion for maximally supersymmetric super-Yang--Mills \cite{Cederwall:2001bt,Cederwall:2001td,Cederwall:2001dx}.
Introduce a bosonic spinor $\lambda^\alpha$, subject to the constraint $(\lambda\gamma^a\lambda)=0$.
A function $\Psi$ of $x$, $\theta$ and $\lambda$, expanded in powers in $\lambda$, is\footnote{There is no other way of dealing with the $\lambda$ dependence, since no scalar is encountered at any power of $\lambda$.}
\begin{align}
\Psi(x,\theta,\lambda)=C(x,\theta)+\lambda^\alpha A_\alpha(x,\theta)
+\fr2\lambda^\alpha\lambda^\beta B_{\alpha\beta}(x,\theta)+\ldots
\end{align}
Now, acting with the ``BRST operator'' $Q$ gives
\begin{align}
Q\Psi=\lambda^\alpha D_\alpha C+\lambda^\alpha\lambda^\beta D_\alpha A_\beta+\ldots
\end{align}
The linearised equations of motion are encoded as $\Psi$ being $Q$-closed, and gauge transformations correspond to $Q$-exact functions.
It also immediately follows that, for the specific choice $\Psi=\lambda^\alpha A_\alpha$,
\begin{align}
Q\Psi+\Psi^2=\lambda^\alpha\lambda^\beta F_{\alpha\beta}\;.
\end{align}
The full non-linear equations of motion are encoded as
\begin{align}
Q\Psi+\Psi^2=0\label{SYMEOM}\;.
\end{align}
One should think of $\lambda$ as a ghost variable, which explains it being bosonic, although it is a spinor.
Then $\Psi$ should also be assigned ghost number $1$, so that $A_\alpha$ has ghost number $0$.
The above shows that the cohomology of $Q$ in the ghost number $0$ sector is precisely the linear super-Yang--Mills multiplet. One should also make sure that there is no essential cohomology in other ghost numbers (powers of $\lambda$). This can be done as follows.
In order to investigate the cohomology, we will do it in two steps: first we find the zero-mode ({\it i.e.}, $x$-independent) cohomology. It will correspond to fields in in a component formulation. Then, in the second step, these fields will, in the full cohomology, be related by differential operators constructed from ${\*\over\*x}$. The procedure is not presented as a mathematical proof here; a fuller account can be found in refs. \cite{Cederwall:2013vba,Eager:2021wpi,CederwallJonssonPalmkvistSaberi}.
The zero-mode cohomology of $Q$ is the cohomology of $\lambda^\alpha{\*\over\*\theta^\alpha}$. Had $\lambda$ been unconstrained, the only cohomology would have been the constant one. Now, when $\lambda$ is constrained, the problem is algebraic, and the result is reflected in the partition function of $\lambda$. Encode the power of $\lambda$ in a variable $t$. Then the partition function, taking values in the representation ring, is
\begin{align}
Z(t)=(00000)\oplus(00001)t\oplus(00002)t^2\oplus(00003)t^3+\ldots\label{LambdaPartition1}
\end{align}
It is straightforward to factor out the dependence of an unconstrained spinor, which we write as
\begin{align}
(1-t)^{-(00001)}=(00000)\oplus(00001)t\oplus\vee^2(00001)t^2\oplus\vee^3(00001)t^3+\ldots
\end{align}
($\vee^p$ is the $p$-fold totally symmetric product). We then obtain
\begin{align}
&Z(t)=(1-t)^{-(00001)}\\
&\otimes\left[
(00000)\ominus(10000)t^2\oplus(00010)t^3\ominus(00001)t^5\oplus(10000)t^6\ominus(00000)t^8
\right]\;.\nn\label{LambdaPartition2}
\end{align}
Each term inside the square brackets represents a component field. They are, in order of appearance,
the ghost $c$, the fields in the physical multiplet $A_a$ and $\chi^\alpha$, the antifields $\chi^{\star}_\alpha$ and $A^{\star a}$, and the ghost antifield $c^\star$. Each of them appears in the zero-mode cohomology as some function of $\theta$ and $\lambda$, for example the physical gauge field appears as $(\lambda\gamma^a\theta)A_a$.
Going back to the full cohomology of $Q$, it will relate the component fields, now $x$-dependent, with differential operators.
The proper mathematical tool for this procedure is that of homotopy transfer, see ref. \cite{Eager:2021wpi}.
It is straightforward to show that the action of $Q$ is indeed that of the BRST operator of the component fields and antifields of the super-Yang--Mills theory. The cohomology consists precisely of the linearised physical (on-shell) fields, and a ghost zero-mode.
It looks tempting to try to derive eq. \eqref{SYMEOM} from a Chern--Simons like action. This can indeed be done, leading to the appropriate off-shell formulation, but requires the machinery of the following subsection.
It is also clear from the nature of the cohomology that such an action should be regarded as a Batalin--Vilkovisky (BV) action
\cite{Batalin:1981jr}, containing ghosts, fields and their antifields.
\subsection{Non-minimal variables, integration, BV actions}
In order to write an action that reproduces the equations of motion of the previous subsection,
one needs an integration over the pure spinor $\lambda$. In addition, it should (when one also includes integration over $\theta$) pick up the top zero-mode cohomology, {\it i.e.}, the top component of the component super-Yang--Mills BRST complex, corresponding to the ghost antifield. This cohomology sits at $\lambda^3\theta^5$. One is in the seemingly problematic situation of needing a residue-like measure, in the sense of picking a certain component, while on the other hand having a series expansion that contains only positive powers of $\lambda$. Such a measure is clearly degenerate, and not useful.
This problem was solved in ref.
\cite{Berkovits:2005bt}, using what is known as a non-minimal set of variables.
In addition to the pure spinor $\lambda^\alpha$, one introduces a conjugate pure spinor $\bar\lambda_\alpha$. In order not to disturb the cohomology, an equal number of additional fermions $r_\alpha$, which are pure with respect to $\bar\lambda$:
$(\bar\lambda\gamma^a r)=0$. We identify $r_\alpha$ as $d\bar\lambda_\alpha$, and products of $r$'s with wedge product of $d\bar\lambda$'s. Then, the modified non-minimal BRST operator
\begin{align}
Q=(\lambda D)+\bar\*\;,
\end{align}
where $\bar\*=d\bar\lambda_\alpha{\*\over\*\bar\lambda_\alpha}$ is the Dolbeault operator,
has the same cohomology as the minimal one previously considered.
The pure spinor space is a (non-compact) Calabi--Yau space \cite{Cederwall:2011yp}. It possesses a holomorphic top form, in this case an $11$-form $\Omega$. The schematic form of this Calabi--Yau form is
\begin{align}
\Omega\sim\lambda^{-3}(d\lambda)^{11}\;.
\end{align}
For detailed expressions, see refs. \cite{Berkovits:2005bt,Cederwall:2011yp,Cederwall:2013vba}.
Remember that the pure spinor field $\Psi$ now depends on $x$, $\theta$, $\lambda$, $\bar\lambda$ and $d\bar\lambda$. The last dependence is seen as $\Psi$ being an antiholomorphic cochain. One may try an integration measure
\begin{align}
\int [dZ] f=\int d^{10}x\int d^{16}\theta\int \Omega\wedge f\;,
\end{align}
where the last integral is over the pure spinor Calabi--Yau space.
This measure is non-degenerate, and carries ghost number $-3$ as desired, due to the $\lambda^{-3}$ in $\Omega$. However, the cohomologies we encountered have representatives which are $0$-forms, so any pair of such functions seems to have vanishing scalar product. On the other hand, the pure spinor space is a non-compact c\^one, so integrals na\"ively diverge at large radius.
This ``$0\times\infty$'' structure can be regularised \cite{Berkovits:2005bt} to yield finite results.
The trick is to observe that the behaviour on pure spinor space is topological, and to insert a $Q$-invariant regularisation
\cite{Marnelius:1990eq,Berkovits:2005bt}
$e^{-t\{Q,\chi\}}$ for some fermion $\chi$. Such a regulator will give $t$-independent results.
If one chooses $\chi=\theta^\alpha\bar\lambda_\alpha$, one gets a factor
\begin{align}
e^{-t\{Q,\chi\}}=e^{-t((\lambda\bar\lambda)+(\theta d\bar\lambda))}\;.\label{SimpleRegEq}
\end{align}
The first factor makes integrals convergent at large radius. The second one contains terms up to $\theta^{11}(d\bar\lambda)^{11}$. When integrated with a $0$-form, it will pick up a component at $\theta^5$.
This regulated measure is exactly what is needed.
We can think of it as an operator that localises the integral to the vicinity of the tip $\lambda=0$ of the pure spinor c\^one.
Alternatively, the basis for cohomology can be chosen to include such factors, and then no regularisation of the measure is necessary.
Now, a Chern--Simons-like BV action for $D=10$ can be written as \cite{Berkovits:2001rb,Movshev:2003ib}
\begin{align}
S=\int[dZ]\mathrm{tr\,}(\fr2\Psi Q\Psi+\fr3\Psi^3)\;.
\end{align}
Note that the action only contains a cubic interaction term, while the component $F^2$ contains quartic interactions.
A component action can be derived by homotopy transfer \cite{Eager:2021wpi}, or put more mundanely, the higher order interactions arise from repeated use of the equations of motion. See also ref. \cite{Berkovits:2018gbq}.
This property, that the supersymmetric action is of lower order than the component action, becomes even more pronounced when we turn to supergravity in the following Section.
The superfield $\Psi$ is self-conjugate with respect to the BV anti-bracket:
\begin{align}
(A,B)=\int A{\overleftarrow\delta\over\delta\Psi(Z)}[dZ]{\overrightarrow\delta\over\delta\Psi(Z)}B\;.
\end{align}
Then it is straightforward to show that the classical master equation $(S,S)=0$ is satisfied.
\subsection{Other models}
Any supermultiplet can be derived as the cohomology in a pure spinor superfield.
In many cases, the zero-mode cohomology is such that the corresponding fields define a component BV complex, and an integration can be defined. Situations where this does not happen is {\it e.g.}\ when self-dual tensors are contained in the supermultiplet, such as the $N=(2,0)$ multiplet in $D=6$ or type IIB supergravity.
If the multiplet has an off-shell formulation with some auxiliary fields,
this off-shell multiplet is found as the cohomolgy of $Q$ \cite{Cederwall:2008zv,Cederwall:2017ruu}.
In such cases, the pure superspinor complex only contains ghosts and fields, and anti-fields to these are found in a conjugate pure spinor superfield \cite{Cederwall:2017ruu}.
In many cases one needs to use pure spinor superfields in non-trivial modules
\cite{Cederwall:2011vy,Cederwall:2008vd,Cederwall:2008xu,Cederwall:2017cez,Cederwall:2020dui,Cederwall:2021ejp} (see for example the field $\Phi^a$ in Section
\ref{Geometry4FormSection}). In the language of ref. \cite{Eager:2021wpi}, they belong to sections of some sheaf over the pure spinor space.
Higher derivative deformations of supersymmetric models, for example supersymmetric Born--Infeld theory may be given simple (polynomial) actions in pure spinor superfield theory \cite{Cederwall:2011vy,Chang:2014nwa}.
Supergravity, in particular in $11$ dimensions, will be addressed in the following Section.
\subsection{Pure spinor partition functions and superalgebras}
Given the usefulness of pure spinors for the description of supermultiplets in general, it seems meaningful to pursue a deeper mathematical investigation of the algebraic properties of pure spinor space itself. Functions on pure spinor space can be thought of
algebraically as power series in $\lambda$, modulo the ideal generated by $(\lambda\gamma^a\lambda)$.
The partition function of eqs. \eqref{LambdaPartition1}, \eqref{LambdaPartition2} can indeed be understood as the partition function of the on-shell super-Yang--Mills multiplet by factoring out also a level $2$ vector:
\begin{align}
Z(t)=&(1-t)^{-(00001)}\otimes(1-t^2)^{(10000)}\\
&\otimes
\left[(00000)\oplus\bigoplus\limits_{n=0}^\infty\bigl((n0010)t^{3+2n}\ominus(n1000)t^{4+2n}\bigr)\right]\;.\nn
\end{align}
The factor in the square represents the ghost zero-mode and the on-shell $n$'th derivative of the fermion and the field strength.
This is for the $D=10$ super-Yang--Mills example. Similar statements hold for any multiplet.
The first two factors are cancelled by the partition functions for functions of $\theta$ and $x$. In this way it becomes clear that
the pure spinor entirely encodes a full supermultiplet.
The investigation of the partition function of a pure spinor through the ghost structure associated to the bilinear constraint was initiated by Chesterman \cite{Chesterman:2002ey}, and refined by Berkovits and Nekrasov
\cite{Berkovits:2005hy}.
Consider the BRST operator $q$ for the pure spinor constraint.
It will (generically) involve an infinite number of ghosts due to the infinite reducibility of the constraint.
One may think of $q$ as the coalgebra differential of a superalgebra, and the content of the algebra as a vector space may be deduced from a continued factorisation of the partition function
\begin{align}
Z(t)=\prod\limits_{n=1}^\infty(1-t^n)^{R_n}\;.
\end{align}
One has to remember that statistics are switched and modules are conjugated when going from the ghosts (coalgebra elements) to the superalgebra.
The superalgebra in question,
which is our definition of the Koszul dual to the functions of a pure spinor, will always be some deformation of the direct sum of the supersymmetry algebra (levels $1$ and $2$) and the freely generated algebra on the supermultiplet (levels $n\geq3$)
\cite{CederwallJonssonPalmkvistSaberi}.
The Koszul duality can be interpreted as a denominator formula for the superalgebra.
In cases where the superfield is not a scalar, this is expected to generalise to character formulas for representations of the superalgebra.
When the constraint puts $\lambda$ in a minimal orbit, the superalgebra is a Lie superalgebra, more precisely a Borcherds superalgebra \cite{Cederwall:2015oua}.
For the particular case of $D=10$ super-Yang--Mills theory, the corresponding Borcherds superalgebra in fact exactly encodes the structure of {\it interacting} super-Yang--Mills theory
\cite{Movshev:2003ib}. This is a quite amazing and unexpected result, since all that is described by the cohomology is
the linear multiplet.
It is not yet clear what the corresponding statement is for other theories, {\it e.g.}\ $D=11$ supergravity, but partial results exist
\cite{Jonsson:2021lwa,CederwallJonssonPalmkvistSaberi}. There, the superalgebra is not a Lie superalgebra, but and $L_\infty$ algebra involving (at least) a $3$-bracket and a $4$-bracket.
\section{$D=11$ supergravity\label{SGSection}}
We will not turn to $D=11$ supergravity \cite{Cremmer:1978km}.
The pure spinor superfield formulation of this model can be derived from its traditional superspace
\cite{Wess:1977fn,Brink:1978iv} formulation \cite{Brink:1980az,Cremmer:1980ru} in much the same way as the super-Yang-Mills theory was in Section \ref{PureSpinorSuperfieldSection}, however with some additional ingredients.
Recall the component field content of the $D=11$ supergravity multiplet: the metric $g_{mn}$, a $3$-form $C$ with a $4$-form field strength $H=dC$, and the gravitino field $\chi_m{}^\alpha$ with field strength $\psi_{mn}{}^\alpha$.
An essential feature, that was used as a guideline for the construction of the supersymmetric action, is that supersymmetry demands the presence of a Chern--Simons term $\int C\wedge H\wedge H$.
In what follows, we will use $11$-dimensional ``pure spinors'' $\lambda^\alpha$. A Dirac spinor in $D=11$ has $32$ components.
The symmetric spinor bilinears consist of a vector, a 2-form and a 5-form, constructed with
$\gamma^a_{\alpha\beta}$, $\gamma^{ab}_{\alpha\beta}$ and $\gamma^{abcde}_{\alpha\beta}$.
A spinor subject to $(\lambda\gamma^a\lambda)=0$ is not necessarily in a minimal orbit, which would require also
$(\lambda\gamma^{ab}\lambda)=0$. Rather, the pure spinor space consists of a ``generic'', $23$-dimensional part, complemented by the $16$-dimensional minimal orbit, which is a singular subspace, and the zero orbit, the tip of the c\^one. The space is sketched in Figure \ref{11coneFig}.
\begin{figure}
\begin{center}
\includegraphics[width=.5\linewidth]{cone.pdf}
\begin{narrow}[both=1cm]
\caption{A sketch of the space of $D=11$ pure spinors.
\label{11coneFig}}
\end{narrow}
\end{center}
\end{figure}
\subsection{Geometry vs. $4$-form\label{Geometry4FormSection}}
There are two different versions of superfields that can describe the on-shell (linearised) supergravity multiplet.
One relies on the standard description of superspace geometry, where one introduces a dynamical super-vielbein
$E_M{}^A$. Then, conventional constraints are used to eliminate all components except the lowest-dimensional ones,
$E_\mu{}^a$ in a controlled and covariant way \cite{Cederwall:2000ye,Cederwall:2000fj,Cederwall:2004cg}.
Note that physical fields then are described by $1$-forms (in fermionic indices), but with an extra index $a$, while superdiffeomorphisms can be thought of as sitting in a superfield $\xi^a$ with the bosonic diffeomorphism parameters as leading components. The situation reminds of the treatment of super-Yang--Mills theory in the previous Section, although all fields have an extra index $a$. It can indeed be verified that the linearised (around Minkowski space) multiplet is described by the cohomology of $Q=(\lambda D)$ on a field $\Phi^a(x,\theta,\lambda)$.
The field is in addition required to have a ``shift symmetry''
\cite{Cederwall:2009ez,Cederwall:2010tn,Cederwall:2011vy}
$\Phi^a\simeq\Phi^a+(\lambda\gamma^a\varrho)$ for an arbitrary parameter
$\varrho^\alpha(x,\theta,\lambda)$. (The shift symmetry ties together the index structure with the cohomology, and is also directly responsible for the presence of the fermionic diffeomorphism ghosts in the zero-mode cohomology.)
The above is one way to relate the on-shell linearised supergravity multiplet to pure spinor superfield cohomology. Since it is geometrical, it carries no information about the gauge symmetry of the $3$-form $C$, which indeed only appears through its field strength $H$ in the dimension $1$ torsion.
The other way of reproducing the linearised multiplet is through a scalar field. The full ghost system for the $C$ field contains a ghost, a ghost-for-ghost and a ghost-for-ghost-for-ghost. The latter is a fermionic $0$-form. We can think of it as the $\theta$- and $\lambda$-independent zero-mode cohomology of a pure spinor superfield $\Psi$ of ghost number $3$ and dimension $-3$.
A careful calculation of the zero-mode cohomology gives at hand that it indeed contains the mentioned ghosts, together with the super-diffeomorphism ghosts (at $\lambda^2$), the physical fields (at $\lambda^3$) and all corresponding antifields.
We refer to refs. \cite{Cederwall:2009ez,Cederwall:2010tn,Cederwall:2013vba} for the detailed calculations.
Now we are in a situation where the traditional supergeometric approach gives the full non-linear equations of motion, but does not account for the full ghost structure. The scalar field $\Psi$, on the other hand, is more fundamental in that it contain all ghosts, and also the potential $C$, but it is {\it a priori}\ unclear how to go beyond the linearised level.
Importantly, in order to write down an action containing the Chern--Simons term, $\Psi$ is needed.
\subsection{BV action}
Before giving the form of the full non-linear action, we need to understand integration, regularisation etc. in a way analogous to the $10$-dimensional case. We will refrain from detailed expressions.
The pure spinor space is $23$-dimensional. We again introduce non-minimal variables $\bar\lambda$ and $d\bar\lambda$, and
include the Dolbeault operator $\bar\*$ in $Q$. The top cohomology of the third order ghost antifield now sits at
$\lambda^7\theta^9$. A measure based on this cohomology has the correct ghost number $-7$ for an action with $\int\Psi Q\Psi$, where $\Psi$ carries ghost number $3$. The pure spinor space is again Calabi--Yau, with $\Omega\sim\lambda^{-7}(d\lambda)^{23}$.
A completely analogous regularisation will contain $\theta^{23}(d\bar\lambda)^{23}$, so the $\theta$ integration will effectively pick out a term with $\theta^9$ ($9=32-23$), as desired.
A linearised action
\begin{align}
S_2=\fr2\int[dZ]\Psi Q\Psi
\end{align}
reproduces the on-shell multiplet correctly.
How are interactions constructed as additional terms in a BV action?
One starting point may be to look at the Chern--Simons term $\int C\wedge H\wedge H$.
It must contain at least one field $\Psi$, but the remaining factors can in principle be formed from $\Phi^a$, containing the field strength $H$. The concrete task now becomes to find an expression for $\Phi^a$ in terms of $\Psi$, such that cohomology maps to cohomology. This means that one needs to find a bosonic operator $R^a$ of ghost number $-2$ and dimension $2$ which commutes with $Q$ modulo terms of the type $(\lambda\gamma^a\varrho)$.
The procedure is similar to that of finding a $b$ operator, used in gauge fixing (see Section \ref{QuantumSection}).
Such an operator was constructed in ref. \cite{Cederwall:2009ez} using non-minimal variables. It takes a somewhat complicated form, beginning as
\begin{align}
R^a=((\lambda\gamma_{cd}\lambda)(\bar\lambda\gamma^{cd}\bar\lambda))^{-1}(\bar\lambda\gamma^{ab}\bar\lambda)\*_b
+\ldots
\end{align}
One can then use $\Psi$ as the fundamental field, and write
$\Phi^a=R^a\Psi$.
A term
\begin{align}
S_3=\fr6\int[dZ](\lambda\gamma_{ab}\lambda)\Psi R^a\Psi R^b\Psi
\end{align}
is then guaranteed to fulfil $(S_2,S_3)=0$, {\it i.e.}, work as a linear deformation of $S_2$ \cite{Cederwall:2009ez}.
Note that the factor $(\lambda\gamma_{ab}\lambda)$ serves several purposes:
It contracts the indices on a pair of fermionic fields.
It ensures the correct ghost number and dimensions of the interaction term.
And finally, it ensures the invariance under the shift symmetry, thanks to the Fierz identity
$(\gamma^b\lambda)_\alpha(\lambda\gamma_{ab}\lambda)=0$, which holds for pure spinors.
It can be verified, using explicit expressions for the cohomologies, that the Chern--Simons term is correctly reproduced by $S_3$.
In order to construct a complete action, the master equation $(S,S)=0$ must be checked, not only to linear order in $S_3$ as above. It turns out \cite{Cederwall:2010tn} that only a minor modification is needed: a $4$-point coupling which is almost of the same form as $S_3$. It relies on yet another operator, $T$, of ghost number $-3$ and dimension $3$. The field $T\Psi$ then carries ghost number and dimension $0$, and its ghost number $0$ part can be thought of as containing the trace of the linearised gravity field.
Then, the action
\begin{align}
S=\int[dZ]\left(\fr2\Psi Q\Psi+\fr6(\lambda\gamma_{ab}\lambda)(1-\Fr32T\Psi)\Psi R^a\Psi R^b\Psi\right)
\label{FullSGActionEQ}
\end{align}
turns our to satisfy $(S,S)=0$ to all orders,
It is striking, but ideal from the point of view of perturbative calculations, that a model containing gravity becomes polynomial around Minkowski space. A detailed understanding, {\it e.g.}\ through homotopy transfer, of how the non-polynomial nature of geometry around Minkowski space arises, is still lacking. Neither does the construction offer any direct clues concerning how to proceed to other backgrounds. Some remarks concering these issues are given in the concluding Section.
\subsection{Twisting}
Pure spinor superfields provide a good framework for twisting supersymmetric theories, and to find all possible twistings
\cite{Eager:2018dsx,Saberi:2021weg}. This is because any point in the space of spinors obeying
$(\lambda\gamma^a\lambda)=0$ provides a nilpotent operator $\lambda^\alpha D_\alpha$. (Note that here $\lambda$ is not a variable, but takes some specific value.)
The list of possible twistings can be read off from the stratification of pure spinor space in different orbits under the Lorentz group, forming subspaces of pure spinor space.
In supergravity, supersymmetry is local, and twisting is performed by giving an expectation value to a superdiffeomorphism ghost
\cite{Costello:2016mgj}. A treatment in the pure spinor formalism is favourable, since these ghosts are naturally present.
Among other theories, the twistings of $D=11$ supergravity has been thus examined
\cite{Raghavendran:2021qbh,Saberi:2021weg,Eager:2021ufo}.
The minimal twist leads to the $SL(5)$ supersymmetric model of ref.
\cite{Cederwall:2021ejp}.
\section{Superstrings\label{StringSection}}
The covariant quantisation of space-time supersymmetric string theory remained elusive for a long time, until
Berkovits constructed the pure spinor superstring
\cite{Berkovits:2000fe,Berkovits:2000ph,Berkovits:2000nn,Berkovits:2017ldz}.
The variables used are the same as displayed above for $D=10$ super-Yang--Mills theory.
In both the left- and right-moving sectors of the world sheet, one introduces in addition to the superspace coordinates $X$ (self-conjugate) and $\theta$, with its conjugate $p$, a pure spinor $\lambda$ and its conjugate $\omega$.
The variable $\lambda$ has the same chirality as $\theta$. In type IIA superstring theory this chirality is opposite for left- and right-movers, and in type IIB the same.
The left-moving BRST operator reads
\begin{align}
Q=\oint\lambda^\alpha d_\alpha\;,\label{StringQ}
\end{align}
where
\begin{align}
d_\alpha=p_\alpha+\*X^a(\gamma_a\theta)_\alpha+\fr8(\gamma^a\theta)_\alpha(\theta\gamma_a\*\theta)\;,
\end{align}
with the operator product expansion
\begin{align}
d_\alpha(z)d_\beta(\zeta)={1\over z-\zeta}\gamma_{a\alpha\beta}\Pi^a+\hbox{(regular)}\;,
\end{align}
where $\Pi^a=\*X^a-(\theta\gamma^a\*\theta)$ is the momentum conjugate to $X$ in the Green--Schwarz superstring.
This implies $Q^2=0$. Again, it can of course be extended with non-minimal variables.
Notably the list of fields above is complete, including ghosts. There is no Virasoro ghost pair $(b,c)$ (and, unlike the Neveu--Schwarz--Ramond superstring, no super-Virasoro ghosts $(\beta,\gamma)$).
This of course also happens for the superparticle.
All ``coordinates'' are world-sheet scalars.
The cancellation of the conformal anomaly requires no Virasoro ghost, but simply reads
$c=10-2\cdot 16+2\cdot 11=0$. This may seem as a simplification, but also has its price in making {\it e.g.}\ gauge fixing more complicated, see Section \ref{QuantumSection}.
Integration over pure spinor variables follows the same principles as for the super-Yang--Mills theory.
\section{Quantum theory\label{QuantumSection}}
The procedures sketched in this Section
focus on principles and qualitative results.
The issue of gauge fixing and the $b$ operator is discussed in somewhat more detail, since this is one of the
points where the formalism becomes complicated and simplifications are desired.
The physical fields are ``hidden'' within a structure which exhibits many qualitatively simple features. Their extraction from that structure is more complicated \cite{Eager:2021wpi,Berkovits:2018gbq}. If one wants to use the formalism to derive precise quantitative results, much work is involved (see refs. below).
\subsection{Gauge fixing\label{GaugeFixingSection}}
The non-minimal variables open for a possibility to construct operators with negative ghost number.
The so-called ``$b$ ghost'', or $b$ operator, is the standard example (see also the negative ghost number operators of Section
\ref{SGSection}).
It is called $b$ because it assumes the r\^ole of the conjugate to the ghost $c$ for world-line reparametrisations or world-sheet conformal transformations (see the cancellation of the conformal anomaly in Section \ref{StringSection}). In pure spinor superfield theory it is a composite operator. This is because ``$p^2=0$'' (in a superparticle action) is only a derived linearised equation of motion, a consequence (after gauge fixing) of $Q\Phi=0$, not a constraint associated with a world-line symmetry.
In order to perform perturbative quantum calculations gauge fixing is necessary. The ``kinetic operator'' $Q$ is of course not invertible. If one can find an operator $b$ such that $\{Q,b\}=\Box$ and chooses the Siegel gauge \cite{Siegel:1984ogw}
\begin{align}
b\Psi=0\;,
\end{align}
the propagator $G$ can be written as
\begin{align}
G={b\over\Box}\;.
\end{align}
In the following we will write the field theory $b$ operator. The one for string theory is very similar (in the same way as the $Q$'s of eqs. \eqref{FieldQ} and \eqref{StringQ} are), just containing a small number of more terms with derivatives.
As mentioned, this is one of the instances where things become complicated in the pure spinor formalism.
The $b$ operator in $D=10$ was constructed in ref. \cite{Berkovits:2001rb} using non-minimal variables, and reads
\begin{align}
b&=b_0+b_1+b_2+b_3\nn\\
&=-\fr2(\lambda\bar\lambda)^{-1}(\bar\lambda\gamma^aD)\*_a\nn\\
&+\fraction{16}(\lambda\bar\lambda)^{-2}(\bar\lambda\gamma^{abc}d\bar\lambda)\left[N_{ab}\*_c
-\fraction{24}(D\gamma_{abc}D)\right]\\
&+\fraction{64}(\lambda\bar\lambda)^{-3}(d\bar\lambda\gamma^{abc}d\bar\lambda)(\bar\lambda\gamma_aD)N_{bc}\nn\\
&-\fraction{1024}(\lambda\bar\lambda)^{-4}(\bar\lambda\gamma^{ab}{}_id\bar\lambda)(d\bar\lambda\gamma^{cdi}d\bar\lambda)N_{ab}N_{cd}\;,\nn
\end{align}
where $N=(\lambda\omega)$ and $N_{ab}=(\lambda\gamma_{ab}\omega)$ are invariant operators, in the sense that they respect the pure spinor constraint $(\lambda\gamma^a\lambda)=0$.
The $b$ operator in $D=11$ \cite{Cederwall:2012es,Berkovits:2017xst} is somewhat more complicated. We will not display the full expression, but note that it is singular on the $16$-dimensional subspace (like the negative ghost number operators encountered in Section \ref{SGSection}), and begins as
\begin{align}
b=((\lambda\gamma_{de}\lambda)(\bar\lambda\gamma^{de}\bar\lambda))^{-1}
(\bar\lambda\gamma_{ab}\bar\lambda)(\lambda\gamma^{ab}\gamma^cD)\*_c+\ldots
\end{align}
There is also a possibility to find a $b$ operator that acts within functions of the minimal variables
\cite{CederwallBOperatorUnpublished}, using the principles of ref. \cite{Cederwall:2010zz}.
On (holomorphic) functions of a $D=10$ pure spinor $\lambda$, the `ìnvariant derivative operator''
\begin{align}
\tilde\omega_\alpha=\omega_\alpha-\fraction{4(N+3)}(\gamma^a\lambda)_\alpha(\omega\gamma_a\omega)
\end{align}
acts exactly like $\omega_\alpha$ between monomials $\lambda^{\alpha_1}\ldots\lambda^{\alpha_p}$, and annihilates the ideal generated by $(\lambda\gamma^a\lambda)$.
The minimal $b'$ operator reads
\begin{align}
b'=\fraction{(N+4)(N+5)(N+6)}\left[\fr2(N^2+9N+15)(\tilde\omega\gamma^aD)\*_a
-\fraction{128}N^{ab}(\tilde\omega D^3_{ab})\right]\;,\label{BPrimeEq}
\end{align}
where $D^3_{ab}$ is the antisymmetric product of three $D$'s in
$(01001)$,
\begin{align}
(D^3)_{ab}^\alpha=(\gamma^i)^{\alpha[\beta}(\gamma_{abi})^{\gamma\delta]}D_\beta D_\gamma D_\delta\;.
\end{align}
It can be shown explicitly that $b$ and $b'$ differ by a $Q$-exact expression.
Using $b'$, it is seen directly that Siegel gauge implies Lorenz gauge for the Yang--Mills connection. Namely, acting on
a ghost number $0$ field $\Psi=\lambda^\alpha A_\alpha$,
$b'\Psi=\fraction{16}\*^a(D\gamma_a A)$.
A similar minimal $b$ operator should exist for any supersymmetric theory with local symmetries, {\it e.g.}\ $D=11$ supergravity, but has not been constructed.
It may rightly be claimed that gauge fixing, in the form presented here, is rudimentary,
and more or less implemented at a first-quantised level.
A proper field-theoretic BV gauge fixing
\cite{Henneaux:1985kr}, involving a gauge fixing fermion, has not been developed in pure spinor superfield theory.
\subsection{Perturbative results}
The construction sketched above gives a recipe for calculating scattering amplitudes in the pure spinor formalism.
Any diagram --- which will contain a large number of component field diagrams --- should be saturated with appropriate vertex operators \cite{Aisaka:2009yp} representing external states.
There is a remaining issue of regularisation at $\lambda=0$ which was addressed and solved in ref.
\cite{Berkovits:2006vi}. This is due to the $b$ operators in propagators containing negative powers of $(\lambda\bar\lambda)$ that ultimately risk to make integrals divergent.
Explicit evaluation of the regulated integrals is in general extremely complicated.
Results exist for superstring theory on Minkowski space \cite{Berkovits:2004dt,Berkovits:2006bk,Berkovits:2006vi,Aisaka:2009yp,Mafra:2011nv}
and on anti-de Sitter space \cite{Berkovits:2000yr,Berkovits:2004xu}.
The degree of convergence of loop diagrams in pure spinor superfield theory is generically much better than for loop diagrams in a component formulation or with superfields manifesting some fraction of supersymmetry.
Typical behaviour is the vanishing of bubbles and triangles in off-shell diagrams, {\it i.e.}, as subdiagrams of any diagram.
For maximal super-Yang--Mills theory in $D=4$ \cite{Bjornsson:2010wu,Ben-Shahar:2021doh}, power counting is enough to demonstrate perturbative finiteness.
In maximal supergravity in $D=4$ \cite{Cederwall:2012es,Karlsson:2014xva,Grassi:2011uf,Anguelova:2004pg}, power counting shows finiteness up to $6$ loops, and possibly a divergence at $7$ loops, see also refs.
\cite{Vanhove:2010nf,Bern:2009kd,Bern:2018jmv}. The precise statement is that an $L$-loop diagram is convergent in $D$ dimensions if $D<2+{14\over L}$, while for super-Yang--Mills theory it reads
$D<4+{6\over L}$.
\section{Remarks\label{RemarksSection}}
Some final remarks, concerning shortcomings of the present status of pure spinor superfield theory, and some desirable developments.
The classical theory of pure spinor superfields exhibits a striking simplicity.
Quantum calculations tend to become cumbersome, although in principle well defined, mainly due to the complicated expression for the $b$ operator used in gauge fixing, and the regularisation it brings along.
It remains an open question if these calculations can be simplified, either by finding a replacement for the $b$ operator, or by some completely different means.
One approach, which has not been properly explored, would be to use the minimal (holomorphic) version of the $b$ operator,
$b'$ of eq. \eqref{BPrimeEq}. The construction will certainly extend to other negative ghost number operators
\cite{Cederwall:2010zz,CederwallBOperatorUnpublished}.
Possibly, in such a framework, the r\^ole of the non-minimal variables can be limited to integration, with the
``simple'' regularisation of eq. \eqref{SimpleRegEq},
and the complicated regularisations at $\lambda=0$ may be avoided.
An urgent question for supergravity is the lack of manifest background invariance of the action
\eqref{FullSGActionEQ}.
This is of course usual in string theory and string field theory
\cite{Zwiebach:1992ie} (see however ref. \cite{Sen:1994kx}), but one should be able to do better in a supergravity theory.
Indeed, even if the basis of the construction is in supergeometry, the geometric picture is lost in the final form.
There is some hope for ``re-geometrisation'', and for an understanding how to deform the model to non-flat backgrounds.
It relies on deforming the algebra which is Koszul dual to functions of
$11$-dimensional pure spinors \cite{CederwallJonssonPalmkvistSaberi},
in a manner similar to ref.
\cite{Figueroa-OFarrill:2015tan}.
As mentioned in Section \ref{GaugeFixingSection}, a proper field-theoretic BV gauge fixing procedure has not been developed for pure spinor superfield theory. There is no doubt that this can be done. It is probably one of the most important points on which the framework should be developed.
The whole idea about the formalism presented is to manifest as much symmetry as possible.
It is well known that dimensional reductions of $D=11$ enjoys U-duality, and that this symmetry
can be ``geometrised'' within the context of
exceptional geometry
\cite{Cederwall:2017fjm,Cederwall:2019bai,Hohm:2019wql,Butter:2018bkl}.
Can the pure spinor framework be extended to accomodate for these symmetries?
Such a task may be very difficult, due to the infinite reducibility of local symmetries in extended geometry, since the pure spinor superfields always are based on the lowest-dimensional ghost field.
Indeed, already double supergeometry \cite{Cederwall:2016ukd,Butter:2022gbc} contains infinite reducibility in the Ramond-Ramond sector.
\newpage
\bibliographystyle{utphysmod2}
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\section{Introduction}
In \cite{KS} B. Khanedani and T. Suwa
introduced an index for singular holomorphic foliations on complex compact surfaces called the \textit{Variational index}.
In \cite{LS} D. Lehmann and T. Suwa generalized the variational index for higher dimensional holomorphic foliations. In particular, they showed that if $V$ is a complex subvariety invariant by a holomorphic foliation $\mathscr{F}$ of dimension $k\geq 1$ on a $n$-dimensional complex compact manifold $X$, then
$$
c_1^{n-k}(\det(N^*\mathscr{F})) \cdot [V] = (-1)^{n-k}\sum_{\lambda} \operatorname{Res}_{ c_1^{n-k}}(\mathscr{F}; S_\lambda),
$$
where $S_\lambda$ is a connected component of $S(\mathscr{F},V):=({\rm Sing}(\mathscr{F})\cap V) \cup {\rm Sing}(V) $ (here ${\rm Sing}(\mathscr{F})$ and ${\rm Sing}(V)$ denote the singular sets of $\mathscr{F}$ and $V$, respectively), $ [V]$ is the integration current of $V$ and $N^*\mathscr{F}$ is the conormal sheaf of $\mathscr{F}$. In the case $X$ is a complex surface and $S(\mathscr{F},V)$ is an isolated set, then for each $p\in S(\mathscr{F},V)$,
$$-\operatorname{Res}_{ c_1}(\mathscr{F}; p)=\operatorname{Var}(\mathscr{F},V,p),$$
where $\operatorname{Var}(\mathscr{F},V,p)$ denotes the Variational index of $\mathscr{F}$ along $V$ at $p$, as defined by Khanedani-Suwa in \cite{KS}.
\par M. Brunella in \cite{Br4} studied the Khanedani-Suwa variational index and its relations with the GSV and the Camacho-Sad indices. See also \cite[II, Proposition 1.2.1]{Mc}.
M. McQuillan, in his proof of the Green-Griffiths conjecture (for a projective surface $X$ with $c_1^2(X)> c_2(X)$ ), \cite{Mc}, showed that
if $ X$ is a complex surface of general type and $\mathscr{F}$ is a holomorphic foliation on $X$, then $\mathscr{F}$ has no entire leaf which is Zariski dense. See \cite{Dem, PS, GPR,Den} for more details about the Green-Griffiths conjecture and generalizations. M. Brunella in \cite{Br2} provided an alternative proof of McQuillan's result by showing that if $[T_f]$ is the Ahlfors current associated to a Zariski dense entire curve $f :\mathbb{C} \to X$ which is tangent to $\mathscr{F}$, then
$$
c_1(N^*\mathscr{F}) \cdot [T_f] = \sum_{p\in {\rm Sing}(\mathscr{F}) \cap \operatorname{Supp}(T_f)} \frac{1}{2\pi i}[T_f] (\chi_{U_p}d(\phi_p \beta_p)) \leq 0,
$$
where $\chi_{U_p}$ denotes the characteristc function of a neighborhood $U_p$ of $p\in {\rm Sing}(\mathscr{F}) \cap \operatorname{Supp}(T_f)$, see section \ref{secao-var} for more details.
\par To continue we consider $\mathscr{F}$ a singular holomorphic foliation of dimension $k\geq 1$ on a complex compact manifold $X$ of dimension at least two. We recall that a positive closed current $T$ in $X$ is \emph{invariant} by $\mathscr{F}$ if $T_{|\mathscr{F}} \equiv 0$, that is, $T(\eta)=0$ for every test form $\eta$ vanishing along the leaves of $\mathscr{F}$, so that $T(\eta)$ depends only on the restriction of $\eta$ to the leaves.
\par In \cite{Br3} M. Brunella proved a more general variational index type Theorem for positive closed currents of bidimension $(1,1)$ invariant by one-dimensional holomorphic foliations, with isolated singularities, on complex compact manifolds. More precisely, he showed that if $T$ is an invariant positive closed current of bidimension $(1,1)$, then
$$
c_1(\det(N^*\mathscr{F})) \cdot [T] = \sum_{p\in {\rm Sing}(\mathscr{F}) \cap \operatorname{Supp}(T)} \frac{1}{2\pi i}[T] (\chi_{U_p}d(\phi_p \beta_p)).
$$
Compare this formula with the so called \textit{asymptotic Chern Class} of a foliation on a complex surface introduced in \cite{CLS}.
Moreover, M. Brunella showed, in the same work, that a generic one-dimensional holomorphic foliation on complex projective spaces has no invariant measure.
In \cite[Corollary 1.2]{K} L. Kaufmann showed that there is no diffuse foliated cycle directed by embedded Lipschitz laminations of dimension $k\geq n/2$ on ${\mathbb{P}}^n$.
\par We denote the class of a closed current $T$ of bidimension $(p,p)$ in the cohomology group $H^{n-p,n-p}(X)$ by $[T]$. In order to provide a generalization of the above results, we define the residue of $\mathscr{F}$ relative to $T$ along a connected component of the singular set of $\mathscr{F}$, (see for instance Def. \ref{res} in Sect. \ref{secao-var}.). In this work we prove the following result.
\begin{Theorem} \label{teorema}
Let $\mathscr{F}$ be a holomorphic foliation, of dimension $k\geq 1$, on a complex compact manifold $X$ with $\dim({\rm Sing}(\mathscr{F}))\leq k-1$. Write ${\rm Sing}(\mathscr{F}) = \displaystyle\bigcup_{\lambda} Z_\lambda$, a decomposition into connected components and let $U_{\lambda}$ be a regular neighborhood of $Z_\lambda$. For $p \geq k$, if $T$ is a positive closed current of bidimension $(p,p)$ invariant by $\mathscr{F}$, then
$$
c_1^{p-k+1}(\det (N^*{\mathscr{F}}))\cdot [T]=\sum_{Z_\lambda\subset \operatorname{Supp}(T)\cap {\rm Sing}(\mathscr{F})} \operatorname{Res}(\mathscr{F}, T, Z_\lambda).
$$
\end{Theorem}
\par A compact non-empty subset
$\mathcal{M} \subset X$ is said to be a \textit{ minimal set} for ${\mathscr{F}}$ if
the following properties are satisfied
\begin{enumerate}
\item[(i)] $\mathcal{M}$ is invariant by ${\mathscr{F}}$;
\item[(ii)] $\mathcal{M} \cap {\rm Sing}({\mathscr{F}})= \emptyset$;
\item[(iii)] $\mathcal{M}$ is minimal with respect to these properties.
\end{enumerate}
\par The problem of existence of minimal sets for codimension one holomorphic foliations on $\mathbb{P}^n$ was considered by Camacho--Lins Neto--Sad in \cite{CLS}.
To our knowledge, this problem remains open for $n=2$. If ${\mathscr{F}}$ is
a codimension one holomorphic foliation on $\mathbb{P}^n$, with $n\geq 3$,
Lins Neto \cite{Ln} proved that ${\mathscr{F}}$ has no minimal sets.
M. Brunella posed in \cite{Br1} the following question:
\begin{conjecturen}\label{conje}
Let $X$ be a compact connected complex manifold of dimension $n\geq 3$, and let ${\mathscr{F}}$
be a codimension one holomorphic foliation on $X$ such that
$N{\mathscr{F}}$ is ample. Then every leaf of ${\mathscr{F}}$
accumulates to ${\rm Sing}(\mathscr{F})$.
\end{conjecturen}
\par In \cite{perrone}, Brunella--Perrone
proved the above Conjecture for codimension-one holomorphic
foliations on a projective manifold with cyclic Picard group. In \cite{C-P} the natural conjecture has been stated:
\begin{conjecturen}[Generalized Brunella's conjecture]\label{conje1}
Let $X$ be a compact connected complex manifold of dimension $n\geq
3$, and let ${\mathscr{F}}$ be a holomorphic foliation of codimension $r<n$ on $X$ such that
$\det(N{\mathscr{F}})$ is ample. Then every
leaf of ${\mathscr{F}}$ accumulates to ${\rm Sing}(\mathscr{F})$, provided $n\geq 2r+1$.
\end{conjecturen}
The main result in \cite{C-P} suggests that the property of accumulation of the leaves of a foliation $\mathscr{F}$ to the its singular set
(\textit{or nonexistence of minimal sets of ${\mathscr{F}}$} ) depends on the existence of strongly $q$-convex spaces which contains the singularities of $\mathscr{F}$.
In \cite{CLS} it was proved that there is no invariant measure with support on a nontrivial minimal set of a foliation on $\mathbb{P}^2$. We observe that in $\mathbb{P}^n$ we have that $\det(N_{\mathscr{F}})$ is ample for every foliation $\mathscr{F}$. The following Corollary \ref{inter-sing} generalize the result in \cite[Theorem 2]{CLS}.
\begin{Corollary}\label{inter-sing}
Let $\mathscr{F}$ be a holomorphic foliation, of dimension $k\geq 1$, on a projective manifold $X$ such that $\dim({\rm Sing}(\mathscr{F}))\leq k-1$ and $\det(N{\mathscr{F}})$ ample.
Suppose that $h^{n-p,n-p} (X ) = 1$, for some $p \geq k$. If $T$ is a
positive closed current of bidimension $(p,p)$ invariant by $\mathscr{F}$, then $\operatorname{Supp}(T)\cap {\rm Sing}(\mathscr{F})\neq \emptyset $. In particular, there is no invariant positive closed current of bidimension $(p,p)$ with support on a nontrivial minimal set of $\mathscr{F}$.
\end{Corollary}
Compare Corollary \ref{inter-sing} with \cite[Corollary 5.5]{K}.
Since $h^{n-p,n-p} ({\mathbb{P}}^n ) = 1$, this result holds for foliations on ${\mathbb{P}}^n $, in particular if $V\subset \mathbb{P}^n$ is an $\mathscr{F}$-invariant complex subvariety, then $V\cap {\rm Sing}(\mathscr{F})\neq \emptyset.$ This is the Esteves--Kleiman's result \cite[12, Proposition 3.4, pp. 12]{EK}.
We can also apply Theorem \ref{teorema} to the Ahlfors currents associated to $f: \mathbb{C}^k\to X$ a holomorphic map of generic maximal rank which
is a leaf of the foliation $\mathscr{F}$. Fix a K\"ahler form $\omega$ on $X$. On $\mathbb{C}^k$ we take the homogeneous metric form
$$\omega_0:=dd^c \ln |z|^2,$$ and denote by
$$\sigma=d^c \ln |z|^2 \wedge \omega_0^{k-1}$$
the Poincar\'e form. Consider $\eta \in A^{1,1}(X)$ and for any $r>0$ define
$$T_{f,r}(\eta)=\int_0^r \frac{dt}{t} \int_{B_t} f^*\eta \wedge \omega_0^{k-1},$$
where $B_t \subset \mathbb{C}^k$ is the ball of radius $t$. Then we consider the positive currents $\Phi_r \in A^{1,1}(X)'$ defined by
$$\Phi_r(\eta):=\frac{T_{f,r}(\eta)}{T_{f,r}(\omega)}.$$
This gives a family of positive currents of bounded mass from which we can extract a subsequence $\Phi_{r_n}$ which converges to a current $ [T_f] \in A^{1,1}(X)'$ called the
Ahlfors current of $f$, see \cite[Claim 2.1]{GPR}.
This construction has been generalized in \cite{BS} by Burns--Sibony and \cite{DeT} by De Th\'elin. In order to associate to $f: \mathbb{C}^k\to X$
a positive closed current of any bidimension $(s, s)$, $1 \leq s \leq k $ (also called Ahlfors currents) it is necessary to impose some technical conditions.
We obtain another consequence of Theorem \ref{teorema} as follows:
\begin{Corollary}\label{inter-Ahlfors}
Let $\mathscr{F}$ be a holomorphic foliation, of dimension $k\geq 1$, on a projective manifold $X$ such that $\dim({\rm Sing}(\mathscr{F}))\leq k-1$ and $\det(N{\mathscr{F}})$ ample.
Let $f: \mathbb{C}^k\to X$ be a holomorphic map of generic maximal rank which
is a leaf of the foliation.
Suppose that $h^{n-p,n-p} (X ) = 1$, for some $p \geq k$, and that there exists an Ahlfors current of bidimension $(p,p)$ associated to $f$.
Then $\overline{f( \mathbb{C}^k)}\cap {\rm Sing}(\mathscr{F})\neq \emptyset $.
\end{Corollary}
\subsection*{Acknowledgments}
MC was partially supported by CNPq, CAPES and FAPEMIG. A. F-P was partially supported by CNPq grant number 427388/2016-3.
\section{Singular holomorphic foliations }
Let $X$ be a connected compact complex manifold of dimension $n$. To define a (singular) holomorphic foliation $\mathscr{F}$ on $X$ we adopt the following point of view: such a $\mathscr{F}$ is determined by a coherent subsheaf $N^\ast \mathscr{F}$ of the cotangent sheaf $T^\ast X = \Omega^1_X$ of $X$ which satisfies
\begin{itemize}
\item[1)] integrability: $d N^\ast \mathscr{F} \subset N^\ast \mathscr{F} \wedge \Omega^1_X$ and
\item[2)] $\Omega^1_X / N^\ast \mathscr{F}$ is torsion free.
\end{itemize}
The generic rank of $N^\ast \mathscr{F}$ is the codimension of $\mathscr{F}$, the dual $(N^\ast \mathscr{F})^\ast = N\mathscr{F}$ is the normal sheaf of the foliation and the singular locus of $\mathscr{F}$ is
\begin{equation}\label{sing}
{\rm Sing}(\mathscr{F})= \{p \in X : (\Omega^1_X / N^\ast \mathscr{F})_p \,\mathrm{is\, not\, a\, free} \,\mathscr{O}_p -\mathrm{module}\}.
\end{equation}
Condition 2 above implies codim$({\rm Sing}(\mathscr{F})) \geq 2$.
Remark that, on $X \setminus {\rm Sing}(\mathscr{F})$, we have an exact sequence of holomorphic vector bundles
$$
0 \longrightarrow N^\ast \mathscr{F} \longrightarrow \Omega^1_X \longrightarrow T^\ast \mathscr{F} \longrightarrow 0
$$
and, dualizing
$$
0 \longrightarrow T\mathscr{F} \longrightarrow TX \longrightarrow N\mathscr{F} \longrightarrow 0,
$$
where $T\mathscr{F}$ is called the tangent bundle of $\mathscr{F}$, of dimension $k=(n - \operatorname{codim}( \mathscr{F}))$. Also, since the singular set has codimension greater than $1$ we have the adjunction formula
$$
KX = K\mathscr{F} \otimes \det (N^\ast{\mathscr{F}}),
$$
where $K\mathscr{F}=\det(T\mathscr{F})^*$ denotes the canonical bundle of $\mathscr{F}$.
If $\mathscr{F}$ has codimension $(n-k)$ then, by taking the $(n-k)$-th wedge product of the inclusion
$$N^\ast \mathscr{F} \longrightarrow \Omega^1_X,$$ we get a $(n-k)$-form $\omega$ with coefficients in the line bundle $(\bigwedge^{n-k} N\mathscr{F})^{\ast \ast} = \det (N\mathscr{F})$.
\subsection{Holomorphic foliations on complex projective spaces}
Let $\omega \in \mathrm H^0({\mathbb{P}}^n,\Omega_{{\mathbb{P}}^n}^{n-k}(m))$ be the twisted $(n-k)$-form induced by a holomorphic foliation $\mathscr{F}$ of dimension $k$ on ${\mathbb{P}}^n$.
Take a generic non-invariant linearly embedded subspace $i:L\simeq {\mathbb{P}}^{n-k} \hookrightarrow {\mathbb{P}}^n$. We have an induced non-trivial section
$i^*\omega \in \mathrm H^0(L ,\Omega_{L}^{n-k}(m)) \simeq \mathrm H^0( {\mathbb{P}}^{n-k} , \mathcal{O}_{{\mathbb{P}}^{n-k}}(k-n-1+m)),$
since $\Omega_{{\mathbb{P}}^{n-k}}^{n-k}=\mathcal{O}_{{\mathbb{P}}^{n-k}}(k-n-1)$ . The {\it degree} of $\mathscr{F}$ is defined by $$\deg(\mathscr{F}):=\deg(Z(i^*\omega))=k-n-1+m.$$
In particular, $\omega \in \mathrm H^0({\mathbb{P}}^n,\Omega_{{\mathbb{P}}^n}^k(\deg(\mathscr{F})+n-k+1))$. That is, $\det(N\mathscr{F})=\mathcal{O}_{{\mathbb{P}}^n}(\deg(\mathscr{F})+n-k+1)$ is ample.
A holomorphic foliation, of degree $d$, can be induced by a polynomial $(n-k)$-form on ${\mathbb{C}}^{n+1}$ with homogeneous coefficients of degree $d+1$, see for instance \cite{CMM}.
\section{Variational residue and proof of Theorem \ref{teorema}}\label{secao-var}
Hence, a holomorphic foliation of dimension $k$ is given by a family $(\{V_\mu\}, \{\omega_\mu\})_{\mu \in \Lambda}$, where $\mathcal{V} =\{V_\mu\}_{\mu \in \Lambda}$ is an open cover of $X$ by Stein open sets, $\omega_\mu$ is an integrable holomorphic $(n-k)$-form defined in $V_\mu$ and locally decomposable in $V_\mu \setminus {\rm Sing}({\mathscr{F}})$. This means that, for each $p \in V_\mu$, there is an open neighborhood $V_p \subset V_\mu$ of $p$ such that
$$\omega_{\mu|V_{p}} = \omega_1^\mu \wedge \dots \wedge \omega_{n-k}^\mu,$$ where $\omega_j^\mu$ is a holomorphic 1-form and $d \omega_j^\mu \wedge \omega_\mu =0$ for $1 \leq j \leq n-k$.
The integrability condition tells us that, in $V_\mu \setminus {\rm Sing}({\mathscr{F}})$, there is a $C^\infty$ $1$-form $\alpha_\mu$ satisfying:
\\
\\
\noindent(i) $d\omega_\mu = \alpha_\mu \wedge \omega_\mu$, for all $\mu \in \Lambda$. $\alpha_\mu$ is not unique, but its restriction to the leaves of $\mathscr{F}$ is, provided $\omega_\mu$ is fixed.
\\
\\
\noindent(ii) $\alpha_\mu$ is of type $(1,0)$ since $\omega_\mu$ is holomorphic and ${\alpha_\mu}_{|\mathscr{F}}$ is holomorphic. This last fact follows from: if we assume that around a regular point the foliation $\mathscr{F}$ is generated by $\partial/\partial z_i$, $i=1, \dots, k$, then $\iota_{\partial/\partial z_i}(d\omega_\mu) = \left(\iota_{\partial/\partial z_i}\alpha_\mu\right) \omega_\mu$. In particular, if $k=1$ then $\alpha_{\mu|\mathscr{F}}$ is closed and $d\alpha_{\mu|\mathscr{F}}=0$.
\\
\\
\noindent(iii) In the overlapping $V_{\mu\nu}$ we have $\omega_\mu=f_{\mu\nu}\omega_\nu$, with $f_{\mu\nu} \in {\mathscr{O}}^{\ast}(V_{\mu\nu})$ and the cocycle $\{f_{\mu\nu}\}_{\mu, \nu \in \Lambda}$ determines the line bundle $\det (N{\mathscr{F}})$. Hence
\begin{equation}\label{nor1}
\left(\alpha_\mu - \alpha_\nu - \dfrac{df_{\mu\nu}}{f_{\mu\nu}}\right) \wedge \omega_\mu= 0.
\end{equation}
This shows that $\alpha_\mu - \alpha_\nu - \dfrac{df_{\mu\nu}}{f_{\mu\nu}}$ is a $C^\infty$ local section of the conormal bundle $ N^\ast \mathscr{F}$ of the regular foliation ${\mathscr{F}}_{| X \setminus {\rm Sing}({\mathscr{F}})}$. Since the sheaf of smooth sections of $N^\ast \mathscr{F}$ is acyclic, we have that there exist $C^\infty$ 1-forms $\gamma_\mu$ in $V_\mu$ satisfying: $\gamma_\mu$ is a local section of $N^\ast \mathscr{F}$ and
$$\alpha_\mu - \alpha_\nu - \dfrac{df_{\mu\nu}}{f_{\mu\nu}} = \gamma_\mu - \gamma_\nu,$$
so that
$$\alpha_\mu - \gamma_\mu = \alpha_\nu - \gamma_\nu + \dfrac{df_{\mu\nu}}{f_{\mu\nu}}.$$
Call $\beta_\mu =\alpha_\mu - \gamma_\mu,$
hence
\begin{equation}\label{nor2}
\beta_\mu =\beta_\nu + \dfrac{df_{\mu\nu}}{f_{\mu\nu}}, \; d \beta_\mu = d \beta_\nu \; \mathrm{in} \; V_{\mu \nu},\; d\omega_\mu= \beta_\mu \wedge \omega_\mu \; \mathrm{and} \; d \beta_\mu \wedge \omega_\mu =0.
\end{equation}
By the second equality in \ref{nor2}, the 2-forms $\{d\beta_\mu\}$ piece together and we have a global $C^\infty$ 2-form on $X \setminus {\rm Sing}({\mathscr{F}})$ which we denote by $d \beta$.
We shall briefly digress on the geometric meaning of this smooth 2-form $d\beta$ (see \cite{C-C} 6.2.4): the first equality in \ref{nor2} tells us that the 1-forms $\{ \beta_\mu\}$ behave as connection matrices of $\det (N {\mathscr{F}})$, in $V_\mu$, for some connection. In this case it's natural to consider the basic connections (in the sense of Bott, see \cite{C-L}).
Fix a $C^{\infty}$ decomposition $$TX_{|X \setminus {\rm Sing}({\mathscr{F}})} = N{\mathscr{F}} \oplus T{\mathscr{F}},$$
where $N_{\mathscr{F}}$ and $T_{\mathscr{F}}$ are the normal and tangent bundles, respectively, of the regular foliation ${\mathscr{F}}_{| X \setminus {\rm Sing}({\mathscr{F}})}$.
Let $V_\mu$ be the domain of a local trivialization of $N{\mathscr{F}}$ and $\{v_1^\mu, \dots, v_{n-k}^\mu\}$ be a local frame for $N{{\mathscr{F}}_{|V_{\mu}}}$ such that $\omega_\mu (v_1^\mu, \dots, v_{n-k}^\mu) \equiv 1$. For a suitable basic connection $\nabla$ and $\zeta$ any section of $T{{\mathscr{F}}_{|V_\mu}}$, we have that
$$\beta_\mu(\zeta)=-{\mathrm{tr}} (\theta^\mu)(\zeta)$$
if, and only if, $d\omega_\mu = \beta_\mu \wedge \omega_\mu$,
where $\theta^\mu$ is the connection matrix in $V_\mu$ of $\nabla$ relative to the frame $\{v_1^\mu, \dots, v_{n-k}^\mu\}$. In particular, the 1-forms $\{\beta_\mu\}$ piece together to give a well defined global form $\beta$ on $X \setminus {\rm Sing}({\mathscr{F}})$.
It follows that $d\beta =-{\mathrm{tr}}(K_\nabla) = - c_1(K_\nabla)$ where $K_\nabla=\{K_\nabla^\mu\}_{\mu \in \Lambda}$ is the curvature form of $\nabla$ and the class $d\beta = -c_1({N}{\mathscr{F}})= c_1( \det N^\ast{\mathscr{F}})$.
\begin{Definition}\label{res}
\em{Let $\mathscr{F}$ be a singular foliation of dimension $k\geq 1$, as above, and consider
$${\rm Sing}({\mathscr{F}})= \displaystyle\bigcup_{\lambda} Z_\lambda$$
a decomposition of its singular locus into connected components. For $p \geq k$, suppose $T$ is a positive closed current of bidimension $(p,p)$ which is invariant by $\mathscr{F}$. The residue of $\mathscr{F}$ relative to $T$ along $Z_\lambda$ is
$$
\operatorname{Res}(\mathscr{F}, T, Z_\lambda) = \dfrac{ T\left(d(\varphi_{_{\lambda}} \beta)^{p-k+1}\wedge \chi_{_{Z_\lambda}} \upsilon_{_{Z_\lambda}} \right)}{\mathrm{vol}(Z_\lambda )} \cdot [Z_\lambda],
$$
where $\chi_{_{Z_\lambda}} $ denotes the characteristic function, $\upsilon_{_{Z_\lambda}}$ is a volume element of $Z_\lambda$ and $\varphi_\lambda : X \longrightarrow \mathbb{R}$ is a $C^\infty$ function satisfying $\varphi_\lambda^{-1}(0) =Z_\lambda$, $0 < \varphi_\lambda \leq 1$ in $X \setminus Z_\lambda$ and $\varphi_\lambda =1$ in $X \setminus {{U_{\lambda}}}$.}
\end{Definition}
Now, we are able to prove Theorem \ref{teorema}:
\noindent\textbf{Theorem}. \emph{Let $\mathscr{F}$ be a holomorphic foliation of dimension $k$ on a complex compact manifold $X$ with $\dim({\rm Sing}(\mathscr{F}))\leq k-1$. Write ${\rm Sing}(\mathscr{F}) = \displaystyle\bigcup_{\lambda\in L} Z_\lambda$, a decomposition into connected components and let $U_{\lambda}$ be a regular neighborhood of $Z_\lambda$. For $p \geq k$, if $T$ is a positive closed current of bidimension $(p,p)$ invariant by $\mathscr{F}$ then,
$$
c_1^{p-k+1}(\det (N^*{\mathscr{F}}))\cdot [T]=\sum_{Z_\lambda\subset \operatorname{Supp}(T)\cap {\rm Sing}(\mathscr{F})} \operatorname{Res}(\mathscr{F}, T, Z_\lambda).
$$
}
\begin{proof} In order to show geometrically that $$c_1^{p-k+1}(\det (N^*{\mathscr{F}})) \cdot [T]$$ localizes at $\operatorname{Supp}(T)\cap {\rm Sing}(\mathscr{F})$ we will use the concept of regular neighborhood.
\par An open set ${U_{\lambda}}$, $Z_\lambda \subset {U_{\lambda}} \subset X$, is a \emph{regular neighborhood} of $Z_\lambda$ provided $\overline{{U_{\lambda}}}$ is a (real) $C^0$ manifold of dimension $n$ with boundary $\partial {U_{\lambda}}$.
Regular neighborhoods can be obtained in the following way: take a Whitney stratification $\mathcal S$ of $Z_\lambda$ and let $W_\lambda$ be any open set containing $Z_\lambda$. By the proof of Proposition 7.1 of \cite{Ma}, we can construct a family of tubular neighborhoods $\{T_{S, \rho_{S}}\}$, with $|T_{S, \rho_{S}}| \subset W_\lambda$ ($\rho_S$ is the tubular function), of the strata $S$ of $\mathcal{S}$, satisfying the commutation relations which give control data for $\mathcal{S}$: if $S$ and $S'$ are strata with $S < S'$
then
$$\begin{array}{c}
\pi_S \pi_{S'} (p)= \pi_S(p) \\
\rho_S\pi_{S'} (p)= \rho_S(p).
\end{array}
$$
This allows for the construction of ${U_{\lambda}}$ as a subset of $W_\lambda$ and, by shrinking $W_\lambda$, we may assume ${U_{\lambda}} \cap {U_{{\tilde{\lambda}}}}= \emptyset$ for ${{\lambda}} \neq {{{\tilde{\lambda}}}}$. We call $\{U_\lambda\}_{\lambda \in L}$ a \emph{system of regular neighborhoods} of $Z$.
Let $\{U_\lambda\}_{\lambda \in L}$be a system of regular neighborhoods of ${\rm Sing}(\mathscr{F})$. Since
$$d(\varphi_{_{\lambda}} \beta) = d \varphi_\lambda \wedge \beta + \varphi_\lambda d\beta = \varphi_\lambda d\beta
$$
outside $U_\lambda$, $d \beta_{|\mathscr{F}}=0$ in $X \setminus {\rm Sing}(\mathscr{F})$ and $T$ is $\mathscr{F}$-invariant, we have
$$\left(T \wedge \chi_{_{Z_\lambda}} \upsilon_{_{Z_\lambda}}\right)\left(d(\varphi_{_{\lambda}} \beta)^{p-k+1} \right) =0$$ in $X \setminus U_\lambda$. By reducing the tubular function of $U_\lambda$ we conclude that
$$
\operatorname{Supp} \left(T \wedge \chi_{_{Z_\lambda}} \upsilon_{_{Z_\lambda}}\right)\left(d(\varphi_{_{\lambda}} \beta)^{p-k+1} \right) \subseteq Z_\lambda
$$
which gives
$$
T \left(d(\varphi_{_{\lambda}} \beta)^{p-k+1} \wedge \chi_{_{Z_\lambda}} \upsilon_{_{Z_\lambda}}\right) = \mu_{_{Z_\lambda}} \left[{Z_\lambda}\right](\upsilon_{_{Z_\lambda}})
$$
and
$$
\mu_{_{Z_\lambda}} = \operatorname{Res}(\mathscr{F}, T, Z_\lambda).
$$
Since $d(\varphi_{_{\lambda}}\beta)$ represents $c_1(\det N^\ast{\mathscr{F}})$, we get
$$
c_1^{p-k+1}(\det (N^*{\mathscr{F}}))\cdot [T]=\sum_{Z_\lambda\subset \operatorname{Supp}(T)\cap {\rm Sing}(\mathscr{F})} \operatorname{Res}(\mathscr{F}, T, Z_\lambda).
$$
\end{proof}
\subsection{ Proof of Corollaries \ref{inter-sing} and
\ref{inter-Ahlfors} }
It is enough to prove the Corollary \ref{inter-sing}. The result
is a straightforward consequence of Theorem \ref{teorema}. In fact, suppose by contradiction that
$T$ is a closed positive current of bidimension $(p,p)$ invariant by $\mathscr{F}$ and that $\operatorname{Supp}(T)\cap {\rm Sing}(\mathscr{F}) =\emptyset $. Then, it follows from Theorem \ref{teorema} that
$$
c_1^{p-k+1}(\det (N^*{\mathscr{F}}))\cdot [T]=0.
$$
Since $h^{n-p,n-p} (X ) = 1$ and $\det (N^*{\mathscr{F}})$ is ample, then $[T]=b \cdot c_1^{n-p}(\det (N{\mathscr{F}})) \in H^{n-p,n-p} (X )$, for some $b>0$.
Therefore, we have
$$c_1^{p-k+1}(\det (N^*{\mathscr{F}}))\cdot [T]= (-1)^{p-k+1} b\cdot c_1^{n-k+1}(\det (N{\mathscr{F}}))\neq 0.$$
This is a contradiction.
|
train/arxiv
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BkiUdtY4eIOjR8afhCdh
| 5 | 1 |
\section{Introduction}
Quantized neural networks (QNNs) are neural networks that represent their weights and compute their activations using low-bit integer variables. QNNs significantly improve the latency and computational efficiency of inferencing the network for two reasons.
First, the reduced size of the weights and activations allows for a much more efficient use of memory bandwidth and caches. Second, integer arithmetic requires less silicon area and less energy to execute than floating-point operations. Consequently, dedicated hardware for running QNNs can be found in GPUs, mobile phones, and autonomous driving computers.
Adversarial attacks are a well-known vulnerability of neural networks that raise concerns about their use in safety-critical applications \cite{szegedy2013intriguing,goodfellow2014explaining}. These attacks are norm-bounded input perturbations that make the network misclassify samples, despite the original samples being classified correctly and the perturbations being barely noticeable by humans. For example, most modern image classification networks can be fooled when changing each pixel value of the input image by a few percent.
Consequently, researchers have tried to train networks that are provably robust against such attacks.
The two most common paradigms of training robust networks are adversarial training \cite{madry2017towards}, and abstract interpretation-based training \cite{mirman2018differentiable,wong2018provable}.
Adversarial training and its variations perturb the training samples with gradient descent-based adversarial attacks before feeding them into the network \cite{madry2017towards,zhang2019theoretically,wu2020adversarial,lechner2021adversarial}. While this improves the robustness of the trained network empirically, it provides no formal guarantees of the network's robustness due to the incompleteness of gradient descent-based attacking methods, i.e., gradient descent might not find all attacks.
Abstract interpretation-based methods avoid this problem by overapproximating the behavior of the network in a forward pass during training. In particular, instead of directly training the network by computing gradients with respect to concrete samples, these algorithms compute gradients of bounds obtained by propagating abstract domains.
While the learning process of abstract interpretation-based training is much less stable than a standard training procedure, it provides formal guarantees about the network's robustness. The interval bound propagation (IBP) method \cite{gowal2019scalable} effectively showed that the learning process with abstract interpretation can be stabilized when gradually increasing the size of the abstract domains throughout the training process.
Previous work has considered adversarial training and IBP for floating-point arithmetic neural networks, however robustness of QNNs has received comparatively much less attention. Since it was demonstrated by~\cite{giacobbe2020many} that neural networks may become vulnerable to adversarial attacks after quantization even if they have been verified to be robust prior to quantization, one must develop specialized training and verification procedures in order to guarantee robustness of QNNs. Previous works have proposed several robustness {\em verification} procedures for QNNs~\cite{giacobbe2020many,baranowski2020smt,henzinger2021scalable}, but none of them consider algorithms for {\em learning} certifiably robust QNNs. Furthermore, the existing verification procedures are based on constraint solving and cannot be run on GPU or other accelerating devices, making it much more challenging to use them for verifying large QNNs.
In this work, we present the first abstract interpretation training method for the discrete semantics of QNNs. We achieve this by first defining abstract interval arithmetic semantics that soundly over-approximate the discrete QNN semantics, giving rise to an end-to-end differentiable representation of a QNN abstraction. We then instantiate quantization-aware training techniques within these abstract interval arithmetic semantics in order to obtain a procedure for training certifiably robust QNNs.
Next, we develop a robustness verification procedure which allows us to formally verify QNNs learned via our IBP-based training procedure. We prove that our verification procedure is {\em complete}, meaning that for any input QNN it will either prove its robustness or find a counterexample. This contrasts the case of abstract interpretation verification procedures for neural networks operating over real arithmetic which are known to be incomplete~\cite{mirman2021fundamental}. The key advantage of our training and verification procedures for QNNs is that it can make use of GPUs or other accelerator devices. In contrast, the existing verification methods for QNNs are based on constraint solving so cannot be run on GPU.
Finally, we perform an experimental evaluation showing that our method outperforms existing state-of-the-art certified $L_\infty$-robust QNNs.
We also elaborate on the limitations of our training method by highlighting how the low precision of QNNs makes IBP-based training difficult.
We summarize our contribution in three points:
\begin{itemize}
\item We introduce the first learning procedure for learning robust QNNs. Our learning procedure is based on quantization-aware training and abstract interpretation method
\item We develop the first \emph{complete} robustness verification algorithm for QNNs (i.e., one that always terminates with the correct answer) that runs entirely on GPU or other neural network accelerator devices and make it publicly available.
\item We experimentally demonstrate that our method advances the state-of-the-art on certifying $L_\infty$-robustness of QNNs.
\end{itemize}
\section{Related Work}
\paragraph{Abstract interpretations for neural networks}
Abstract interpretation is a method for overapproximating the semantics of a computer program in order to make its formal analysis feasible~\cite{cousot1977abstract}. Abstract interpretation executes program semantics over abstract domains instead of concrete program states. The method has been adapted to the robustness certification of neural networks by computing bounds on the outputs of neural networks \cite{wong2018provable, GehrMDTCV18,tjeng2019evaluating}.
For instance, polyhedra \cite{,katz2017reluplex,ehlers2017formal,GehrMDTCV18,singh2019abstract, tjeng2019evaluating}, intervals \cite{gowal2019scalable}
, hybrid automata \cite{xiang2018output}, zonotopes \cite{singh2018fast}, convex relaxations \cite{dvijotham2018dual,ZhangCXGSLBH20,wang2021beta}, and polynomials \cite{zhang2018efficient} have been used as abstract domains in the context of neural network verification.
Abstract interpretation has been shown to be most effective for verifying neural networks when directly incorporating them into gradient descent-based training algorithms by optimizing the obtained output bounds as the loss function \cite{mirman2018differentiable,wong2018provable,gowal2019scalable,ZhangCXGSLBH20}.
Most of the abstract domains discussed above exploit the piecewise linear structure of neural networks, e.g., linear relaxations such as polytopes and zonotopes. However, linear relaxations are less suited for QNNs due to their piecewise-constant discrete semantics.
\paragraph{Verification of quantized neural networks}
The earliest work on the verification of QNNs has focused on binarized neural networks (BNNs), i.e., 1-bit QNNs \cite{hubara2016binarized}. In particular, \cite{narodytska2018verifying} and \cite{cheng2018verification} have reduced the problem of BNN verification to boolean satisfiability (SAT) instances. Using modern SAT solvers, the authors were able to verify formal properties of BNNs.
\citep{jia2020efficient} further improve the scalability of BNNs by specifically training networks that can be handled by SAT-solvers more efficiently.
\citep{amir2021smt} developed a satisfiability modulo theories (SMT) approach for BNN verification.
Verification for many bit QNNs was first reported in \citep{giacobbe2020many} by reducing the QNN verification problem to quantifier-free bit-vector satisfiability modulo theory (QF\_BV SMT). The SMT encoding was further improved in \cite{henzinger2021scalable} by removing redundancies from the SMT formulation.
\citep{baranowski2020smt} introduced fixed-point arithmetic SMT to verify QNNs. The works of \cite{sena2021verifying,sena2022automated} have studied SMT-based verification for QNNs as well. Recently, \cite{mistry2022milp} proposed encoding of the QNN verification problem into a mixed-integer linear programming (MILP) instance.
IntRS \cite{lin2021integer} considers the problem of certifying adversarial robustness of quantized neural networks using randomized smoothing. IntRS is limited to $L_2$-norm bounded attacks and only provides statistical instead of formal guarantees compared to our approach.
\paragraph{Decision procedures for neural network verification}
Early work on the verification of floating-point neural networks has employed off-the-shelf tools and solvers. For instance, \cite{pulina2012challenging,katz2017reluplex,ehlers2017formal} employed SMT-solvers to verify formal properties of neural networks. Similarly, \cite{tjeng2019evaluating} reduces the verification problem to mixed-integer linear programming instances.
Procedures better tailored to neural networks are based on branch and bound algorithms \cite{bunel2018unified}. In particular, these algorithms combine incomplete verification routines (bound) with divide-and-conquer (branch) methods to tackle the verification problem.
The speedup advantage of these methods comes from the fact that the bounding methods can be easily implemented on GPU and other accelerator devices \cite{gpupoly,wang2021beta}.
\paragraph{Quantization-aware training} There are two main strategies for training QNNs: post-training quantization and quantization-aware training \cite{krishnamoorthi2018quantizing}. In post-training quantization, a standard neural network is first trained using floating-point arithmetic, which is then translated to a quantized representation by finding suitable fixed-point format that makes the quantized interpretation as close to the original network as possible. Post-training quantization usually results in a drop in the accuracy of the network with a magnitude that depends on the specific dataset and network architecture.
To avoid a significant reduction in accuracy caused by the quantization in some cases, {\em quantization-aware training (QAT)} models the imprecision of the low-bit fixed-point arithmetic already during the training process, i.e., the network can adapt to a quantized computation during training.
The rounding operations found in the semantics of QNNs are non-differentiable computations. Consequently, QNNs cannot be directly trained with stochastic gradient descent.
Researchers have come up with several ways of circumventing the problem of non-differentiable rounding.
The most common approach is the straight-through gradient estimator (STE) \cite{bengio2013estimating,hubara2017quantized}. In the forward pass of a training step, the STE applies rounding operations to computations involved in the QNN, i.e., the weights, biases, and arithmetic operations. However, in the backward pass, the rounding operations are removed such that the error can backpropagate through the network.
The approach of \citep{gupta2015deep} uses stochastic rounding that randomly selects one of the two nearest quantized values for a given floating-point value.
Relaxed quantization \cite{louizos2019relaxed} generalizes stochastic rounding by replacing the probability distribution over the nearest two values with a distribution over all possible quantized values.
DoReFA-Net \cite{zhou2016dorefa} combines the straight-through gradient estimator and stochastic rounding to train QNN with high accuracy. The authors observed that quantizing the first and last layer results in a significant drop in accuracy, and therefore abstain from quantizing these two layers.
Instead of having a fixed pre-defined quantization range, i.e., fixed-point format, more recent QAT schemes allow learning the quantization range.
PACT \cite{choi2018pact} treats the maximum representable fixed-point value as a free variable that is learned via stochastic gradient descent using a straight-through gradient estimation.
The approach of \citep{jacob2018quantization} keeps a moving average of the values ranges during training and adapts the quantization range according to the moving average.
LQ-Nets \cite{zhang2018lq} learn an arbitrary set of quantization levels in the form of a set of coding vectors. While this approach provides a better approximation of the real-valued neural network than fixed-point-based quantization formats, it also prevents the use of efficient integer arithmetic to run the network.
MobileNet \cite{howard2019searching} is a specialized network architecture family for efficient inference on the ImageNet dataset and employs quantization as one technique to achieve this target.
HAWQ-V3 \cite{yao2021hawq} dynamically assigns the number of bits of each layer to either 4-bit, 8-bit, or 32-bit depending on how numerically sensitive the layer is.
EfficientNet-lite \cite{tan2019efficientnet} employs a neural architecture search to automatically find a network architecture that achieves high accuracy on the ImageNet dataset while being fast for inference on a CPU.
\section{Preliminaries}\label{sec:prelims}
\paragraph{Quantized neural networks (QNNs)} Feedforward neural networks are functions $f_{\theta}: \mathbb{R}^n\rightarrow\mathbb{R}^m$ that consist of several sequentially composed layers $f_{\theta} = l_1\circ\dots\circ l_s$, where layers are parametrized by the vector $\theta$ of neural network parameters. Quantization is an interpretation of a neural network $f_{\theta}$ that evaluates the network over a fixed point arithmetic and operates over a restricted set of bitvector inputs~\citep{smith1997scientist}, e.g.~$4$ or $8$ bits. Formally, given an admissible input set $\mathcal{I}\subseteq \mathcal{R}^n$, we define an interpretation map
\[ \sem{\cdot}_{\mathcal{I}}: (\mathbb{R}^n\rightarrow\mathbb{R}^m) \rightarrow (\mathcal{I}\rightarrow\mathbb{R}^m), \]
which maps a neural network to its interpretation operating over the input set $\mathcal{I}$. For instance, if $\mathcal{I}=\mathbb{R}^n$ then $\sem{f_{\theta}}_{\mathbb{R}}$ is the idealized real arithmetic interpretation of $f_{\theta}$, whereas $\semfloat{f}$ denotes its floating-point 32-bit implementation~\citep{kahan1996ieee}. Given $k\in\mathbb{N}$, the k-bit {\em quantization} is then an interpretation map $\semink{\cdot}{k}$ which uses $k$-bit fixed-point arithmetic. We say that $\semink{f_{\theta}}{k}$ is a $k$-bit {\em quantized neural network (QNN)}.
The semantics of the QNN $\semink{f_{\theta}}{k}$ are defined as follows. Let $[\mathbb{Z}]_k=\{0,1\}^k$ be the set of all bit-vectors of bit-width $k$. The QNN $\semink{f_{\theta}}{k}$ then also consists of sequentially composed layers $\semink{f_{\theta}}{k}= l_1\circ\dots\circ l_s$, where now each layer is a function $l_i:[\mathbb{Z}]_k^{n_i}\rightarrow [\mathbb{Z}]_k^{n_{i+1}}$ that operates over $k$-bit bitvectors and is defined as follows:
\begin{align}
x'_i &= \sum_{j=1}^{n_i} w_{ij} x_j + b_i,\label{eq:sum}\\
x''_i &= \text{round}(x'_i,M_i) = \lfloor x'_i\cdot M_i \rfloor, \qquad \text{and} \label{eq:round} \\
y_i &= \sigma_i(\min\{2^{N_i}-1, x''_i \}),\label{eq:relun}
\end{align}
Here, $w_{i,j}\in [\mathbb{Z}]_k^{n_i}$ and $b_i\in [\mathbb{Z}]_k^{n_i}$ for each $1\leq j\leq n_i$ and $1\leq i\leq n_0$ denote the weights and biases of $f$ which are also bitvectors of appropriate dimension. Note that it is a task of the training procedure to ensure that trained weights and biases are bitvectors, see below.
In eq.~\eqref{eq:sum}, the linear map defined by weights $w_{i,j}$ and biases $b_i$ is applied to the input values $x_j$. Then, eq.~\eqref{eq:round} multiplies the result of eq.~\eqref{eq:sum} by $M_i$ and takes the floor of the obtained result. This is done in order to scale the result and round it to the nearest valid fixed-point value, for which one typically uses $M_i$ of the form $2^{-k}$ for some integer $k$. Finally, eq.~\eqref{eq:relun} applies an activation function $\sigma_i$ to the result of eq.~\eqref{eq:round} where the result is first ``cut-off'' if it exceeds $2^{N_i}-1$, i.e., to avoid integer overflows, and then passed to the activation function. We restrict ourselves to {\em monotone} activation functions, which will be necessary for our IBP procedure to be correct. This is still a very general assumption which includes a rich class of activation function, e.g.~ReLU, sigmoid or tanh activation functions. Furthermore, similarly to most quantization-aware training procedures our method assumes that it is provided with {\em quantized} versions of these activation functions that operate over bit-vectors.
\paragraph{Adversarial robustness for QNNs} We now formalize the notion of adversarial robustness for QNN classifiers. Let $\semink{f_{\theta}}{k}: [\mathbb{Z}]_k^{n}\rightarrow [\mathbb{Z}]_k^{m}$ be a $k$-bit QNN.
It naturally defines a classifier with $m$ classes by assuming that it assigns to an input $x\in [\mathbb{Z}]_k^{n}$ a label of the maximal output neuron on input value $x$, i.e.~$y = \mathsf{class}(x) = \mathsf{argmax}_{1\leq i\leq m} \semink{f_{\theta}}{k}(x)[i]$ with $\semink{f_{\theta}}{k}(x)[i]$ being the value of the $i$-th output neuron on input value $x$. If the maximum is attained at multiple output neurons, we assume that $\mathsf{argmax}$ picks the smallest index $1\leq i\leq m$ for which the maximum is attained.
Intuitively, a QNN is adversarially robust at a point $x$ if it assigns the same class to every point in some neighbourhood of $x$. Formally, given $\epsilon>0$, we say that $\semink{f_{\theta}}{k}$ is {\em $\epsilon$-adversarially robust} at point $x$ if
\[ \forall x'\in [\mathbb{Z}]_k^{n}.\, ||x-x'||_\infty < \epsilon \Rightarrow \mathsf{class}(x') = \mathsf{class}(x), \]
where $||\cdot||_\infty$ denotes the $L_\infty$-norm. Then, given a finite dataset $\mathcal{D} = \{(x_1,y_1),\dots,(x_{|\mathcal{D}|},y_{|\mathcal{D}|})\}$ with $x_i\in[\mathbb{Z}]_k^{n}$ and $y_i\in[\mathbb{Z}]_k^{m}$ for each $1\leq i\leq |\mathcal{D}|$, we say that $\semink{f_{\theta}}{k}$ is {\em $\epsilon$-adversarially robust} with respect to the dataset $\mathcal{D}$ if it is $\epsilon$-adversarially robust at each datapoint in $\mathcal{D}$.
\section{Quantization-aware Interval Bound Propagation}\label{sec:training}
In this section, we introduce an end-to-end differentiable abstract interpretation method for training certifiably robust QNNs. We achieve this by extending the interval bound propagation (IBP) method of~\cite{gowal2019scalable} to the discrete semantic of QNNs.
Our resulting quantization-aware interval bound propagation method (QA-IBP) trains an interval arithmetic abstraction of a QNN via stochastic gradient descent by propagating upper and lower bounds for each layer instead of concrete values.
First, we replace each layer $l_i$ with two functions $\underline{l_i},\, \overline{l_i}$: $[\mathbb{Z}]_k^{n_i} \rightarrow [\mathbb{Z}]_k^{n_0}$ defined as follows:
\begin{align}
\mu_j &= \frac{\overline{x_j} + \underline{x_j}}{2} & r_j &= \frac{\overline{x_j} - \underline{x_j}}{2}\\
\mu_i &= \sum_{j=1}^{n_i} w_{ij} \mu_j + b_i & r_i &= \sum_{j=1}^{n_i} | w_{ij} |r_j \\
\underline{x'_i} &= \mu_i - r_i & \overline{x'_i} &= \mu_i + r_i \label{eq:ibpsum}\\
\underline{x''_i} &= \text{round}(\underline{x'_i},k_i) = \lfloor \underline{x'_i}\cdot M_i \rfloor \\
\overline{x''_i} &= \text{round}(\overline{x'_i},k_i) = \lfloor \overline{x'_i}\cdot M_i \rfloor \label{eq:ibpround}\\
\underline{y_i} &= \max\{0,\min\{2^{N_i}-1, \underline{x''_i} \}\} \\
\overline{y_i} &= \max\{0,\min\{2^{N_i}-1, \overline{x''_i} \}\}.
\end{align}
As with standard QNNs, $w_{i,j}\in [\mathbb{Z}]_k^{n_i}$ and $b_i\in [\mathbb{Z}]_k^{n_i}$ for each $1\leq j\leq n_i$ and $1\leq i\leq n_0$ denote the weights and biases of $f$ which are also bit-vectors of appropriate dimension.
By the sequential composition of all layers of the QNN we get the IBP representation of the QNN in the form of two functions $\underline{\semink{f_{\theta}}{k}}$ and $\overline{\semink{f_{\theta}}{k}}$. For a given input sample $x$ and $\epsilon>0$ we can use the IBP representation of $\semink{f_{\theta}}{k}$ to potentially prove the adversarial robustness of the QNN. In particular, the input sample defines an abstract interval domain $(\underline{x},\overline{x})$ with $\underline{x} = x-\epsilon$ and $\overline{x} = x + \epsilon$.
Next, we propagate the abstract domains through the IBP representation of the network to obtain output bounds $\underline{y} =\underline{\semink{f_{\theta}}{k}}(\underline{x},\overline{x})$ and $\overline{y} =\overline{\semink{f_{\theta}}{k}}(\underline{x},\overline{x})$.
Finally, we know that $\semink{f_{\theta}}{k}$ is {\em $\epsilon$-adversarially robust} at point $x$ if
\begin{equation}\label{eq:ibprobust}
\underline{y_i} > \overline{y_j}, \qquad\text{for } i=\mathsf{class}(x) \text{ and } \forall j\neq \mathsf{class}(x).
\end{equation}
\paragraph{End-to-end differentiation}
We modify the IBP representation of QNNs described above to allow an end-to-end differentiation necessary for a stochastic gradient descent-based learning algorithm. In particular, first, we apply the straight-through gradient estimator to propagate the error backward through the rounding operations in Eq. \ref{eq:ibpround}. We do this by replacing the non-differentiable rounding operation with the fake quantization function
\begin{align}
\text{fake\_quant}(x_i) &:= \text{round}(x_i,k_i)\\
\frac{\partial\ \text{fake\_quant}(x_i)}{\partial x_i} &:= 1.
\end{align}
We also add fake quantization operations around the weights and biases in Eq. \ref{eq:ibpsum}. The modified training graph is visualized in Figure \ref{fig:fakequant}.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{qat_ibp.pdf}
\caption{Illustration of how the interval bound propagation training graph is affected by quantization-aware training. \textbf{A)} Standard IBP inference graph. \textbf{B)} IBP inference path with fake quantization operations inserted to model quantized weights, biases, and computations.}
\label{fig:fakequant}
\end{figure}
For a single training sample $(x,j)$, we define the per-sample training loss
\begin{align}
L(\underline{y},\overline{y},j) = \sum_{i\neq j}(\overline{y_i} - \underline{y_j})\mathds{1}[\underline{y_j}-\overline{y_i}\leq 0],
\end{align}
where $\underline{y}, \overline{y}$ are the QA-IBP output bounds QNN with respect to the input domain $(x-\epsilon,x+\epsilon)$ and $j$ corresponds to the label, i.e., class $j$.
The loss term encourages the QA-IBP to produce output bounds that prove adversarial robustness of the QNN $\semink{f_{\theta}}{k}$ with respect to the input $x$ and adversarial radius $\epsilon$.
\paragraph{Existence of robust QNNs} We conclude this section by presenting an interesting result on the existence of robust QNNs. In particular, given $\epsilon>0$ and a finite dataset of $1$-dimensional bit-vectors whose any two distinct datapoints are at least $2\epsilon$ away, we prove that there exists a QNN with ReLU activations that is $\epsilon$-robust with respect to the dataset. Furthermore, we provide an upper bound on the number of neurons that the QNN must contain. We assume $2\epsilon$ distance simply for the $\epsilon$-neighbourhoods of datapoints to be disjoint so that robustness cannot impose contradicting classification conditions.
Note that this result {\em does not} hold for real arithmetic feed-forward neural networks with ReLU or any other continuous activation functions. Indeed, it was observed in~\cite[Corollary~5.12]{MirmanBV21} that a dataset $\{(-2,-1), (0,1), (2,-1)\}$ cannot be $1$-robustly classified by a feed-forward neural network which uses continuous activation functions. The intuition behind this impossibility result for real arithmetic networks is that a classifier would have to be a continuous function that correctly classifies all points in some open neighbourhoods of $x=-1$ and $x=1$.
\begin{theorem}\label{thm:existence}
Let $\epsilon>0$ and let $\mathcal{D} = \{(x_1,y_1),\dots,(x_{|\mathcal{D}|},y_{|\mathcal{D}|})\}$ with $x_i\in[\mathbb{Z}]_k^1$ and $y_i\in[\mathbb{Z}]_k^1$ for each $1\leq i\leq |\mathcal{D}|$. Suppose that $||x_i - x_j||_{\infty}\geq 2\epsilon$ for each $i\neq j$. Then, there exists a QNN $\semink{f_{\theta}}{k}$ which is $\epsilon$-robust with respect to the dataset $\mathcal{D}$ and which consists of $\mathcal{O}(|\mathcal{D}| \cdot \lceil\epsilon\cdot 2^k + 1 \rceil)$ neurons.
\end{theorem}
The proof of Theorem~\ref{thm:existence} is provided in the Appendix. It starts with an observation that the set $\mathcal{B}_{\epsilon}(x_i)=\{x'\in[\mathbb{Z}]_k^1 \mid ||x'-x_i||_{\infty}<\epsilon\}$ is finite and consists of at most $\lceil 2\epsilon\cdot 2^k+1 \rceil$ bit-vectors for each $1\leq i\leq |\mathcal{D}|$. Hence, constructing a QNN $\semink{f_{\theta}}{k}$ that is $\epsilon$-robust with respect to the dataset $\mathcal{D}$ is equivalent to constructing a QNN that correctly classifies $\mathcal{D}' = \cup_{i=1}^{|\mathcal{D}|} \{(x',y_i) \mid x'\in \mathcal{B}_{\epsilon}(x_i)\}$
which consists of $\mathcal{O}(|\mathcal{D}| \cdot \lceil\epsilon\cdot 2^k + 1 \rceil)$ datapoints. In the Appendix, we then design the QNN $\semink{f_{\theta}}{k}$ that correctly classifies a dataset and consists of at most linearly many neurons in the dataset size. Our construction also implies the following corollary.
\begin{corollary}
Let $\mathcal{D} = \{(x_1,y_1),\dots,(x_{|\mathcal{D}|},y_{|\mathcal{D}|})\}$ with $x_i\in[\mathbb{Z}]_k^1$ and $y_i\in[\mathbb{Z}]_k^1$ for each $1\leq i\leq |\mathcal{D}|$. Then, there exists a QNN $\semink{f_{\theta}}{k}$ which correctly classifies the dataset, i.e.~$\semink{f_{\theta}}{k}(x_i) = y_i$ for each $1\leq i\leq |\mathcal{D}|$, and which consists of $\mathcal{O}(|\mathcal{D}|)$ neurons.
\end{corollary}
\section{A Complete Decision Procedure for QNN Verification}\label{sec:verification}
In the previous section, we presented a quantization-aware training procedure for QNNs with robustness guarantees which was achieved by extending IBP to quantized neural network interpretations. We now show that IBP can also be used towards designing a {\em complete} verification procedure for already trained feed-forward QNNs. By completeness, we mean that the procedure is guaranteed to return either that the QNN is robust or to produce an adversarial attack.
There are two important novel aspects of the verification procedure that we present in this section. First, to the best of our knowledge this is the first complete robustness verification procedure for QNNs that is applicable to networks with non-piecewise linear activation functions. Existing constraint solving based methods that reduce verification to SMT-solving are complete but they only support piecewise linear activation fucntions such as ReLU~\citep{krizhevsky2010convolutional}. These could in theory be extended to more general activation functions by considering more expressive satisfiability modulo theories~\citep{smt2018handbook}, however this would lead to inefficient verification procedures and our experimental results in the following section already demonstrate the significant gain in scalability of our IBP-based methods as opposed to SMT-solving based methods for ReLU networks. Second, we note that while our IBP-based verification procedure is complete for QNNs, in general it is known that existing IBP-based verification procedures for real arithmetic neural networks are {\em not complete}~\citep{mirman2021fundamental}. Thus, our result leads to an interesting contrast in IBP-based robustness verification procedures for QNNs and for real arithmetic neural networks.
\paragraph{Verification procedure} We now describe our robustness verification procedure for QNNs. Its pseudocode is shown in Algorithm~\ref{alg:algorithm}. Since verifying $\epsilon$-robustness of a QNN with respect to some finite dataset $\mathcal{D}$ and $\epsilon>0$ is equivalent to verifying $\epsilon$-robustness of the QNN at each datapoint in $\mathcal{D}$, Algorithm~\ref{alg:algorithm} only takes as inputs a QNN $\semink{f_{\theta}}{k}$ that operates over bit-vectors of bit-width $k$, a single datapoint $x\in [\mathbb{Z}]_k^{n}$ and a robustness radius $\epsilon>0$. It then returns either \texttt{ROBUST} if $\semink{f_{\theta}}{k}$ is verified to be $\epsilon$-robust at $x\in [\mathbb{Z}]_k^{n}$, or \texttt{VULNERABLE} if an adversarial attack $||x'-x||_{\infty} < \epsilon$ with $\mathsf{class}(x') = \mathsf{class}(x)$ is found.
The algorithm proceeds by initializing a stack $D$ of abstract intervals to contain a single element $\{(x-\epsilon\cdot \mathbf{1}, x+\epsilon \cdot \mathbf{1})\}$, where $\mathbf{1}\in [\mathbb{Z}]_k^{n}$ is a unit bit-vector of bit-width $k$. Intuitively, $D$ contains all abstract intervals that may contain concrete adversarial examples but have not yet been processed by the algorithm. The algorithm then iterates through a loop which in each loop iteration processes the top element of the stack. Once the stack is empty and the last loop iteration terminates, Algorithm~\ref{alg:algorithm} returns \texttt{ROBUST}.
In each loop iteration, Algorithm~\ref{alg:algorithm} pops an abstract interval $(\underline{x},\overline{x})$ from $D$ and processes it as follows. First, it uses IBP for QNNs that we introduced before to propagate $(\underline{x},\overline{x})$ in order to compute an abstract interval $(\underline{y},\overline{y})$ that overapproximates the set of all possible outputs for a concrete input point in $(\underline{x},\overline{x})$. The algorithm then considers three cases. First, if $(\underline{y},\overline{y})$ does not violate Equation~\eqref{eq:ibprobust} which characterizes violation of robustness by a propagated abstract interval, Algorithm~\ref{alg:algorithm} concludes that the abstract interval $(\underline{x},\overline{x})$ does not contain an adversarial example and it proceeds to processing the next element of $D$. Second, if $(\underline{y},\overline{y})$ violates Equation~\eqref{eq:ibprobust}, the algorithm uses projected gradient descent restricted to $(\underline{x},\overline{x})$ to search for an adversarial example and returns \texttt{VULNERABLE} if found. Note that the adversarial attack is generated with respect to the quantization-aware representation of the network, thus ensuring that the input space corresponds to valid quantized inputs. Third, if $(\underline{y},\overline{y})$ violates Equation~\eqref{eq:ibprobust} but the adversarial attack could not be found by projected gradient descent, the algorithm refines the abstract interval $(\underline{x},\overline{x})$ by splitting it into two smaller subintervals. This is done by identifying
\[ i^{\ast} = \mathsf{argmax}_{1\leq i\leq n} (\overline{x}[i] - \underline{x}[i]) \]
and splitting the abstract interval $(\underline{x},\overline{x})$ along the $i^{\ast}$-th dimension into two abstract subintervals $(\underline{x'},\overline{x'}),(\underline{x''},\overline{x''})$, which are both added to the stack $D$.
\paragraph{Correctness, termination and completeness} The following theorem establishes that Algorithm~\ref{alg:algorithm} is complete, that it terminates on every input and that it is a complete robustness verification procedure. The proof is provided in the Appendix.
\begin{theorem}\label{thm:verification}
If Algorithm~\ref{alg:algorithm} returns \texttt{ROBUST} then $\semink{f_{\theta}}{k}$ is $\epsilon$-robust at $x\in [\mathbb{Z}]_k^{n}$. On the other hand, if Algorithm~\ref{alg:algorithm} returns \texttt{VULNERABLE} then there exists an adversarial attack $||x'-x||_{\infty} < \epsilon$ with $\mathsf{class}(x') \neq \mathsf{class}(x)$. Therefore, Algorithm~\ref{alg:algorithm} is correct. Furthermore, Algorithm~\ref{alg:algorithm} terminates and is guaranteed to return an output on any input. Since Algorithm~\ref{alg:algorithm} is correct and it terminates, we conclude that it is also complete.
\end{theorem}
\begin{algorithm}[t]
\caption{QNN robustness verification procedure}
\label{alg:algorithm}
\begin{algorithmic}[1]
\State \textbf{Input} QNN $\semink{f_{\theta}}{k}$, datapoint $x\in [\mathbb{Z}]_k^{n}$, robustness radius $\epsilon>0$
\State \textbf{Output} \texttt{ROBUST} or \texttt{VULNERABLE}
\State $D \leftarrow \{(x-\epsilon\cdot \mathbf{1},x+\epsilon\cdot \mathbf{1})\}$
\While{$D \neq \{\}$}
\State $(\underline{x},\overline{x}) \leftarrow$ pop item from $D$
\State $(\underline{y},\overline{y}) \leftarrow \underline{\semink{f_{\theta}}{k}}(\underline{x},\overline{x}), \overline{\semink{f_{\theta}}{k}}(\underline{x},\overline{x})$ propagated via IBP
\If{$(\underline{y},\overline{y})$ violates Equation (\ref{eq:ibprobust})}
\State Try to generate adversarial example using projected gradient descent restricted to $(\underline{x},\overline{x})$
\If{adversarial example found}
\State \textbf{return} \texttt{VULNERABLE}
\Else
\State $(\underline{x'},\overline{x'}),(\underline{x''},\overline{x''}) \leftarrow$ partition of $(\underline{x},\overline{x})$ into two abstract subintervals
\State $D \leftarrow D \cup \{(\underline{x'},\overline{x'}),(\underline{x''},\overline{x''})\}$
\EndIf
\EndIf
\EndWhile
\State \textbf{return} \texttt{ROBUST}
\end{algorithmic}
\end{algorithm}
\section{Experiments}\label{sec:exp}
We perform an experimental evaluation to assess the effectiveness of our quantization-aware interval bound propagation (QA-IBP).
In particular, we first use our training procedure to train two QNNs. We then use our complete verification procedure to verify robustness of trained QNNs and we compare our method to the existing verification methods for QNNs of \cite{giacobbe2020many} and \cite{henzinger2021scalable}.
Our full experimental setup and code can be found on GitHub \footnote{\url{https://github.com/mlech26l/quantization_aware_ibp}}.
\begin{table*}
\centering
\begin{tabular}{l|ccc|ccc}\toprule
Method & \multicolumn{3}{c}{MNIST} & \multicolumn{3}{c}{Fashion-MNIST} \\
& $\epsilon=0$ & $\epsilon=1$ & $\epsilon=4$ & $\epsilon=0$ & $\epsilon=1$ & $\epsilon=4$ \\\midrule
QF\_BV SMT \cite{giacobbe2020many} & 97.1\% & 92\% & 0\% & 85.6\% & 44\% & 0\% \\
QF\_BV SMT \cite{henzinger2021scalable} & 97.1\% & \textbf{99\%}$^*$ & 53\% & 85.6\% & 76\% & 36\% \\
QA-IBP (ours) & 99.2\% & \textbf{98.8\%} & \textbf{95.6\%} & 86.3\% & \textbf{80.0\%} & \textbf{59.8\%}\\\bottomrule
\end{tabular}
\caption{Certified robust accuracy of a convolutional neural network trained with QA-IBP compared to existing methods for certifying $L_\infty$-robustness of quantized neural networks reported in the literature. $^*$ Note that \cite{henzinger2021scalable} certified only a subset of the test set due to a high per-sample runtime of their approach. Due to this choice they reported a higher $\epsilon=1$ robust accuracy than clean accuracy.}
\label{tab:experiment}
\end{table*}
We train two CNNs with QA-IBP using an 8-bit quantization scheme for both weights and activations on the MNIST \cite{lecun1998gradient} and Fashion-MNIST \cite{xiao2017online} datasets. We quantize all layers; however, we note that our approach is also compatible with mixing quantized and non-quantized layers. Our MNIST network consists of five convolutional layers, followed by two fully-connected layers. Our Fashion-MNIST network contains three convolutional layers and two fully-connected layers. We apply a fixed pre-defined fixed-point format on both weights and activations. Further details on the network architectures can be found in the Appendix.
We use the Adam optimizer \cite{kingma2014adam} with a learning rate of $10^{-4}$ with decoupled weight decay of $10^{-4}$ \cite{loshchilov2018decoupled} and a batch size of 512. We train our networks for a total of 5,000,000 gradient steps on a single GPU.
We linearly scale the value of $\epsilon$ during the QA-IBP training from 0 to 4 and apply the elision of the last layer optimization as reported in \cite{gowal2019scalable}.
We pre-train our networks for 5000 steps in their non-IBP representation with a learning rate of $5\cdot 10^{-4}$.
After training, we certify all test samples using Algorithm \ref{alg:algorithm} with a timeout of 20s per sample. We certify for $L_\infty$-robustness with radii 1 and 4 to match the experimental setup of \cite{henzinger2021scalable}.
We report and compare the certified robust accuracy of the trained networks to the values reported in \cite{henzinger2021scalable}.
The results in Table \ref{tab:experiment} demonstrate that our QA-IBP significantly improves the certified robust accuracy for QNNs on both datasets.
Regarding the runtime of the verification step, both \cite{giacobbe2020many} and \cite{henzinger2021scalable} certified a smaller model compared to our evaluation. In particular, \cite{giacobbe2020many} report a mean runtime of their 8-bit model to be over 3 hours, and \cite{henzinger2021scalable} report the mean runtime of their 6-bit model to be 90 and 49 seconds for MNIST/Fashion-MNIST, respectively.
Conversely, our method was evaluated with a timeout of 20 seconds and tested on larger networks than the two existing approaches, showing the efficiency of our approach.
\subsection{Ablation analysis}
In this section, we assess the effectiveness of Algorithm \ref{alg:algorithm} for certifying robustness compared to an incomplete verification based on the QA-IBP output bounds alone.
In particular, we perform an ablation and compare Algorithm \ref{alg:algorithm} to an incomplete verification baseline. Our baseline consists of checking the bounds obtained by QA-IBP as an incomplete verifier combined with projected gradient descent (PGD) as an incomplete falsifier, i.e., we try to certify robustness via QA-IBP and simultaneously try to generate an adversarial attack using PGD.
In particular, this baseline resembles a version of Algorithm \ref{alg:algorithm} without any branching into subdomains.
We carry our ablation analysis on the two networks of our first experiment. We report the number of samples where the verification approach could not determine whether the network is robust on the sample or not. We use a timeout of 20s per instance when running our algorithm.
\begin{table}[]
\centering
\begin{tabular}{c|ccc}\toprule
& \multicolumn{3}{c}{CIFAR-10} \\
Weight decay & $\epsilon=0$ & $\epsilon=1$ & $\epsilon=4$ \\\midrule
$1\cdot 10^{-4}$ & 30.0\% & 28.4\% & \textbf{22.4\%} \\
$5\cdot 10^{-5}$ & 39.6\% & \textbf{34.1\%} & 20.6\% \\
$1\cdot 10^{-5}$ & \textbf{83.1\%} & 0\% & 0\% \\\bottomrule
\end{tabular}
\caption{Robust accuracy of our convolutional neural network trained with QA-IBP on the CIFAR-10 dataset with various values for the weight decay. The results express the \textit{robustness-accuracy tradeoff} \cite{tsipras2018robustness}, i.e., the empirical observation made for non-quantized neural networks that we can have a high robustness or a high accuracy but not both at the same time.}
\label{tab:cifar10}
\end{table}
The results shown in Table \ref{tab:timeout} indicate that our algorithm is indeed improving on the number of samples for which robustness could be decided.
However, the improvement is relatively small, suggesting that the QA-IBP training stems from most of the observed gains.
\begin{table*}[]
\centering
\begin{tabular}{l|cc|cc}\toprule
Method & \multicolumn{2}{c}{MNIST} & \multicolumn{2}{c}{Fashion-MNIST} \\
& $\epsilon=1$ & $\epsilon=4$ & $\epsilon=1$ & $\epsilon=4$ \\\midrule
IBP + Projected gradient descent & \textbf{0.35\%} & 3.51\% & 3.96\% & 19.23\% \\
Algorithm \ref{alg:algorithm} & \textbf{0.35\%} & \textbf{3.47\%} & \textbf{3.93\%} & \textbf{18.51\%} \\\bottomrule
\end{tabular}
\caption{Percentage of samples where the robustness of the network can be determined, i.e., certified or falsified, by the method (lower is better). Algorithm timeout was set to 20s per instance. }
\label{tab:timeout}
\end{table*}
\subsection{Limitations}
In this section, we aim to scale our QA-IBP beyond the two gray-scale image classification tasks studied before to the CIFAR-10 dataset \cite{krizhevsky2010convolutional}.
Our setup consists of the same convolutional neural network as used for the Fasion-MNIST.
We run our setup with several values for the weight decay rate. Similar to above, we linearly scale $\epsilon$ from 0 to 4 during training and report the certified robust accuracy obtained by QA-IBP with an $\epsilon$ of 1 and 4 and the clean accuracy.
The results in Table \ref{tab:cifar10} express the robustness-accuracy tradeoff, i.e., the observed antagonistic relation between clean accuracy and robustness, which has been extensively studied in non-quantized neural networks \cite{zhang2019theoretically,bubeck2021universal}.
Depending on the weight decay value, different points on the trade-off were observed. In particular, our training procedure either obtains a network that has an acceptable accuracy but no robustness or a certifiable robust network with a significantly reduced clean accuracy.
We also trained larger models on all datasets (MNIST, Fashion-MNIST, and CIFAR-10), but observed the same behavior of having a high accuracy but no robustness when training, i.e., as in row with $1\cdot 10^{-5}$ in Table \ref{tab:cifar10}.
We found the underlying reason for this behavior to be activations of internal neurons that clamp to the minimum and maximum value of the quantization range in their QA-IBP representation but not in their standard representation. Consequently, the activation gradients become zero during QA-IBP due to falling in the constant region of the activation function.
This effect is specific to quantized neural networks due to the upper bound on the representative range of values.
Our observation hints that future research needs to look into developing dynamic quantization ranges or weight decay schedules that can adapt to both the standard and the QA-IBP representation of a QNN.
\section{Conclusion}
In this paper, we introduced quantization-aware interval bound propagation (QA-IBP), the first method for training certifiably robust QNNs. We also present a theoretical result on the existence and upper bounds on the needed size of a robust QNN for a given dataset of $1$-dimensional datapoints. Moreover, based on our interval bound propagation method, we developed the first complete verification algorithm for QNNs that may be run on GPUs.
We experimentally showed that our training scheme and verification procedure advance the state-of-the-art on certifying $L_\infty$-robustness of QNNs.
We also demonstrated the limitations of our method regarding training stability and convergence. Particularly, we found that the boundedness of the representable value range of QNNs compared to standard networks leads to truncation of the abstract domains, which in turn leads to gradients becoming zero. Our results suggest that dynamic quantization schemes that adapt their quantization range to the abstract domains instead of the concrete activation values of existing quantization schemes may further improve the certified robust accuracy of quantized neural networks.
Nonetheless, our work serves as a new baseline for future research. Promising directions on how to improve upon QA-IBP and potentially overcome its numerical challenges is to adopt advanced quantization-aware training techniques. For instance, dynamical quantization ranges \cite{choi2018pact, jacob2018quantization}, mixed-precision layers \cite{zhou2016dorefa,yao2021hawq}, and automated architecture search \cite{tan2019efficientnet} have shown promising results for standard training QNNs and might enhance QA-IBP-based training procedures as well.
Moreover, further improvements may be feasible by adapting recent advances in IBP-based training methods for non-quantized neural networks \cite{muller2022certified} to our quantized IBP variant.
\FloatBarrier
\section{Introduction}
Quantized neural networks (QNNs) are neural networks that represent their weights and compute their activations using low-bit integer variables. QNNs significantly improve the latency and computational efficiency of inferencing the network for two reasons.
First, the reduced size of the weights and activations allows for a much more efficient use of memory bandwidth and caches. Second, integer arithmetic requires less silicon area and less energy to execute than floating-point operations. Consequently, dedicated hardware for running QNNs can be found in GPUs, mobile phones, and autonomous driving computers.
Adversarial attacks are a well-known vulnerability of neural networks that raise concerns about their use in safety-critical applications \cite{szegedy2013intriguing,goodfellow2014explaining}. These attacks are norm-bounded input perturbations that make the network misclassify samples, despite the original samples being classified correctly and the perturbations being barely noticeable by humans. For example, most modern image classification networks can be fooled when changing each pixel value of the input image by a few percent.
Consequently, researchers have tried to train networks that are provably robust against such attacks.
The two most common paradigms of training robust networks are adversarial training \cite{madry2017towards}, and abstract interpretation-based training \cite{mirman2018differentiable,wong2018provable}.
Adversarial training and its variations perturb the training samples with gradient descent-based adversarial attacks before feeding them into the network \cite{madry2017towards,zhang2019theoretically,wu2020adversarial,lechner2021adversarial}. While this improves the robustness of the trained network empirically, it provides no formal guarantees of the network's robustness due to the incompleteness of gradient descent-based attacking methods, i.e., gradient descent might not find all attacks.
Abstract interpretation-based methods avoid this problem by overapproximating the behavior of the network in a forward pass during training. In particular, instead of directly training the network by computing gradients with respect to concrete samples, these algorithms compute gradients of bounds obtained by propagating abstract domains.
While the learning process of abstract interpretation-based training is much less stable than a standard training procedure, it provides formal guarantees about the network's robustness. The interval bound propagation (IBP) method \cite{gowal2019scalable} effectively showed that the learning process with abstract interpretation can be stabilized when gradually increasing the size of the abstract domains throughout the training process.
Previous work has considered adversarial training and IBP for floating-point arithmetic neural networks, however robustness of QNNs has received comparatively much less attention. Since it was demonstrated by~\cite{giacobbe2020many} that neural networks may become vulnerable to adversarial attacks after quantization even if they have been verified to be robust prior to quantization, one must develop specialized training and verification procedures in order to guarantee robustness of QNNs. Previous works have proposed several robustness {\em verification} procedures for QNNs~\cite{giacobbe2020many,baranowski2020smt,henzinger2021scalable}, but none of them consider algorithms for {\em learning} certifiably robust QNNs. Furthermore, the existing verification procedures are based on constraint solving and cannot be run on GPU or other accelerating devices, making it much more challenging to use them for verifying large QNNs.
In this work, we present the first abstract interpretation training method for the discrete semantics of QNNs. We achieve this by first defining abstract interval arithmetic semantics that soundly over-approximate the discrete QNN semantics, giving rise to an end-to-end differentiable representation of a QNN abstraction. We then instantiate quantization-aware training techniques within these abstract interval arithmetic semantics in order to obtain a procedure for training certifiably robust QNNs.
Next, we develop a robustness verification procedure which allows us to formally verify QNNs learned via our IBP-based training procedure. We prove that our verification procedure is {\em complete}, meaning that for any input QNN it will either prove its robustness or find a counterexample. This contrasts the case of abstract interpretation verification procedures for neural networks operating over real arithmetic which are known to be incomplete~\cite{mirman2021fundamental}. The key advantage of our training and verification procedures for QNNs is that it can make use of GPUs or other accelerator devices. In contrast, the existing verification methods for QNNs are based on constraint solving so cannot be run on GPU.
Finally, we perform an experimental evaluation showing that our method outperforms existing state-of-the-art certified $L_\infty$-robust QNNs.
We also elaborate on the limitations of our training method by highlighting how the low precision of QNNs makes IBP-based training difficult.
We summarize our contribution in three points:
\begin{itemize}
\item We introduce the first learning procedure for learning robust QNNs. Our learning procedure is based on quantization-aware training and abstract interpretation method
\item We develop the first \emph{complete} robustness verification algorithm for QNNs (i.e., one that always terminates with the correct answer) that runs entirely on GPU or other neural network accelerator devices and make it publicly available.
\item We experimentally demonstrate that our method advances the state-of-the-art on certifying $L_\infty$-robustness of QNNs.
\end{itemize}
\section{Related Work}
\paragraph{Abstract interpretations for neural networks}
Abstract interpretation is a method for overapproximating the semantics of a computer program in order to make its formal analysis feasible~\cite{cousot1977abstract}. Abstract interpretation executes program semantics over abstract domains instead of concrete program states. The method has been adapted to the robustness certification of neural networks by computing bounds on the outputs of neural networks \cite{wong2018provable, GehrMDTCV18,tjeng2019evaluating}.
For instance, polyhedra \cite{,katz2017reluplex,ehlers2017formal,GehrMDTCV18,singh2019abstract, tjeng2019evaluating}, intervals \cite{gowal2019scalable}
, hybrid automata \cite{xiang2018output}, zonotopes \cite{singh2018fast}, convex relaxations \cite{dvijotham2018dual,ZhangCXGSLBH20,wang2021beta}, and polynomials \cite{zhang2018efficient} have been used as abstract domains in the context of neural network verification.
Abstract interpretation has been shown to be most effective for verifying neural networks when directly incorporating them into gradient descent-based training algorithms by optimizing the obtained output bounds as the loss function \cite{mirman2018differentiable,wong2018provable,gowal2019scalable,ZhangCXGSLBH20}.
Most of the abstract domains discussed above exploit the piecewise linear structure of neural networks, e.g., linear relaxations such as polytopes and zonotopes. However, linear relaxations are less suited for QNNs due to their piecewise-constant discrete semantics.
\paragraph{Verification of quantized neural networks}
The earliest work on the verification of QNNs has focused on binarized neural networks (BNNs), i.e., 1-bit QNNs \cite{hubara2016binarized}. In particular, \cite{narodytska2018verifying} and \cite{cheng2018verification} have reduced the problem of BNN verification to boolean satisfiability (SAT) instances. Using modern SAT solvers, the authors were able to verify formal properties of BNNs.
\citep{jia2020efficient} further improve the scalability of BNNs by specifically training networks that can be handled by SAT-solvers more efficiently.
\citep{amir2021smt} developed a satisfiability modulo theories (SMT) approach for BNN verification.
Verification for many bit QNNs was first reported in \citep{giacobbe2020many} by reducing the QNN verification problem to quantifier-free bit-vector satisfiability modulo theory (QF\_BV SMT). The SMT encoding was further improved in \cite{henzinger2021scalable} by removing redundancies from the SMT formulation.
\citep{baranowski2020smt} introduced fixed-point arithmetic SMT to verify QNNs. The works of \cite{sena2021verifying,sena2022automated} have studied SMT-based verification for QNNs as well. Recently, \cite{mistry2022milp} proposed encoding of the QNN verification problem into a mixed-integer linear programming (MILP) instance.
IntRS \cite{lin2021integer} considers the problem of certifying adversarial robustness of quantized neural networks using randomized smoothing. IntRS is limited to $L_2$-norm bounded attacks and only provides statistical instead of formal guarantees compared to our approach.
\paragraph{Decision procedures for neural network verification}
Early work on the verification of floating-point neural networks has employed off-the-shelf tools and solvers. For instance, \cite{pulina2012challenging,katz2017reluplex,ehlers2017formal} employed SMT-solvers to verify formal properties of neural networks. Similarly, \cite{tjeng2019evaluating} reduces the verification problem to mixed-integer linear programming instances.
Procedures better tailored to neural networks are based on branch and bound algorithms \cite{bunel2018unified}. In particular, these algorithms combine incomplete verification routines (bound) with divide-and-conquer (branch) methods to tackle the verification problem.
The speedup advantage of these methods comes from the fact that the bounding methods can be easily implemented on GPU and other accelerator devices \cite{gpupoly,wang2021beta}.
\paragraph{Quantization-aware training} There are two main strategies for training QNNs: post-training quantization and quantization-aware training \cite{krishnamoorthi2018quantizing}. In post-training quantization, a standard neural network is first trained using floating-point arithmetic, which is then translated to a quantized representation by finding suitable fixed-point format that makes the quantized interpretation as close to the original network as possible. Post-training quantization usually results in a drop in the accuracy of the network with a magnitude that depends on the specific dataset and network architecture.
To avoid a significant reduction in accuracy caused by the quantization in some cases, {\em quantization-aware training (QAT)} models the imprecision of the low-bit fixed-point arithmetic already during the training process, i.e., the network can adapt to a quantized computation during training.
The rounding operations found in the semantics of QNNs are non-differentiable computations. Consequently, QNNs cannot be directly trained with stochastic gradient descent.
Researchers have come up with several ways of circumventing the problem of non-differentiable rounding.
The most common approach is the straight-through gradient estimator (STE) \cite{bengio2013estimating,hubara2017quantized}. In the forward pass of a training step, the STE applies rounding operations to computations involved in the QNN, i.e., the weights, biases, and arithmetic operations. However, in the backward pass, the rounding operations are removed such that the error can backpropagate through the network.
The approach of \citep{gupta2015deep} uses stochastic rounding that randomly selects one of the two nearest quantized values for a given floating-point value.
Relaxed quantization \cite{louizos2019relaxed} generalizes stochastic rounding by replacing the probability distribution over the nearest two values with a distribution over all possible quantized values.
DoReFA-Net \cite{zhou2016dorefa} combines the straight-through gradient estimator and stochastic rounding to train QNN with high accuracy. The authors observed that quantizing the first and last layer results in a significant drop in accuracy, and therefore abstain from quantizing these two layers.
Instead of having a fixed pre-defined quantization range, i.e., fixed-point format, more recent QAT schemes allow learning the quantization range.
PACT \cite{choi2018pact} treats the maximum representable fixed-point value as a free variable that is learned via stochastic gradient descent using a straight-through gradient estimation.
The approach of \citep{jacob2018quantization} keeps a moving average of the values ranges during training and adapts the quantization range according to the moving average.
LQ-Nets \cite{zhang2018lq} learn an arbitrary set of quantization levels in the form of a set of coding vectors. While this approach provides a better approximation of the real-valued neural network than fixed-point-based quantization formats, it also prevents the use of efficient integer arithmetic to run the network.
MobileNet \cite{howard2019searching} is a specialized network architecture family for efficient inference on the ImageNet dataset and employs quantization as one technique to achieve this target.
HAWQ-V3 \cite{yao2021hawq} dynamically assigns the number of bits of each layer to either 4-bit, 8-bit, or 32-bit depending on how numerically sensitive the layer is.
EfficientNet-lite \cite{tan2019efficientnet} employs a neural architecture search to automatically find a network architecture that achieves high accuracy on the ImageNet dataset while being fast for inference on a CPU.
\section{Preliminaries}\label{sec:prelims}
\paragraph{Quantized neural networks (QNNs)} Feedforward neural networks are functions $f_{\theta}: \mathbb{R}^n\rightarrow\mathbb{R}^m$ that consist of several sequentially composed layers $f_{\theta} = l_1\circ\dots\circ l_s$, where layers are parametrized by the vector $\theta$ of neural network parameters. Quantization is an interpretation of a neural network $f_{\theta}$ that evaluates the network over a fixed point arithmetic and operates over a restricted set of bitvector inputs~\citep{smith1997scientist}, e.g.~$4$ or $8$ bits. Formally, given an admissible input set $\mathcal{I}\subseteq \mathcal{R}^n$, we define an interpretation map
\[ \sem{\cdot}_{\mathcal{I}}: (\mathbb{R}^n\rightarrow\mathbb{R}^m) \rightarrow (\mathcal{I}\rightarrow\mathbb{R}^m), \]
which maps a neural network to its interpretation operating over the input set $\mathcal{I}$. For instance, if $\mathcal{I}=\mathbb{R}^n$ then $\sem{f_{\theta}}_{\mathbb{R}}$ is the idealized real arithmetic interpretation of $f_{\theta}$, whereas $\semfloat{f}$ denotes its floating-point 32-bit implementation~\citep{kahan1996ieee}. Given $k\in\mathbb{N}$, the k-bit {\em quantization} is then an interpretation map $\semink{\cdot}{k}$ which uses $k$-bit fixed-point arithmetic. We say that $\semink{f_{\theta}}{k}$ is a $k$-bit {\em quantized neural network (QNN)}.
The semantics of the QNN $\semink{f_{\theta}}{k}$ are defined as follows. Let $[\mathbb{Z}]_k=\{0,1\}^k$ be the set of all bit-vectors of bit-width $k$. The QNN $\semink{f_{\theta}}{k}$ then also consists of sequentially composed layers $\semink{f_{\theta}}{k}= l_1\circ\dots\circ l_s$, where now each layer is a function $l_i:[\mathbb{Z}]_k^{n_i}\rightarrow [\mathbb{Z}]_k^{n_{i+1}}$ that operates over $k$-bit bitvectors and is defined as follows:
\begin{align}
x'_i &= \sum_{j=1}^{n_i} w_{ij} x_j + b_i,\label{eq:sum}\\
x''_i &= \text{round}(x'_i,M_i) = \lfloor x'_i\cdot M_i \rfloor, \qquad \text{and} \label{eq:round} \\
y_i &= \sigma_i(\min\{2^{N_i}-1, x''_i \}),\label{eq:relun}
\end{align}
Here, $w_{i,j}\in [\mathbb{Z}]_k^{n_i}$ and $b_i\in [\mathbb{Z}]_k^{n_i}$ for each $1\leq j\leq n_i$ and $1\leq i\leq n_0$ denote the weights and biases of $f$ which are also bitvectors of appropriate dimension. Note that it is a task of the training procedure to ensure that trained weights and biases are bitvectors, see below.
In eq.~\eqref{eq:sum}, the linear map defined by weights $w_{i,j}$ and biases $b_i$ is applied to the input values $x_j$. Then, eq.~\eqref{eq:round} multiplies the result of eq.~\eqref{eq:sum} by $M_i$ and takes the floor of the obtained result. This is done in order to scale the result and round it to the nearest valid fixed-point value, for which one typically uses $M_i$ of the form $2^{-k}$ for some integer $k$. Finally, eq.~\eqref{eq:relun} applies an activation function $\sigma_i$ to the result of eq.~\eqref{eq:round} where the result is first ``cut-off'' if it exceeds $2^{N_i}-1$, i.e., to avoid integer overflows, and then passed to the activation function. We restrict ourselves to {\em monotone} activation functions, which will be necessary for our IBP procedure to be correct. This is still a very general assumption which includes a rich class of activation function, e.g.~ReLU, sigmoid or tanh activation functions. Furthermore, similarly to most quantization-aware training procedures our method assumes that it is provided with {\em quantized} versions of these activation functions that operate over bit-vectors.
\paragraph{Adversarial robustness for QNNs} We now formalize the notion of adversarial robustness for QNN classifiers. Let $\semink{f_{\theta}}{k}: [\mathbb{Z}]_k^{n}\rightarrow [\mathbb{Z}]_k^{m}$ be a $k$-bit QNN.
It naturally defines a classifier with $m$ classes by assuming that it assigns to an input $x\in [\mathbb{Z}]_k^{n}$ a label of the maximal output neuron on input value $x$, i.e.~$y = \mathsf{class}(x) = \mathsf{argmax}_{1\leq i\leq m} \semink{f_{\theta}}{k}(x)[i]$ with $\semink{f_{\theta}}{k}(x)[i]$ being the value of the $i$-th output neuron on input value $x$. If the maximum is attained at multiple output neurons, we assume that $\mathsf{argmax}$ picks the smallest index $1\leq i\leq m$ for which the maximum is attained.
Intuitively, a QNN is adversarially robust at a point $x$ if it assigns the same class to every point in some neighbourhood of $x$. Formally, given $\epsilon>0$, we say that $\semink{f_{\theta}}{k}$ is {\em $\epsilon$-adversarially robust} at point $x$ if
\[ \forall x'\in [\mathbb{Z}]_k^{n}.\, ||x-x'||_\infty < \epsilon \Rightarrow \mathsf{class}(x') = \mathsf{class}(x), \]
where $||\cdot||_\infty$ denotes the $L_\infty$-norm. Then, given a finite dataset $\mathcal{D} = \{(x_1,y_1),\dots,(x_{|\mathcal{D}|},y_{|\mathcal{D}|})\}$ with $x_i\in[\mathbb{Z}]_k^{n}$ and $y_i\in[\mathbb{Z}]_k^{m}$ for each $1\leq i\leq |\mathcal{D}|$, we say that $\semink{f_{\theta}}{k}$ is {\em $\epsilon$-adversarially robust} with respect to the dataset $\mathcal{D}$ if it is $\epsilon$-adversarially robust at each datapoint in $\mathcal{D}$.
\section{Quantization-aware Interval Bound Propagation}\label{sec:training}
In this section, we introduce an end-to-end differentiable abstract interpretation method for training certifiably robust QNNs. We achieve this by extending the interval bound propagation (IBP) method of~\cite{gowal2019scalable} to the discrete semantic of QNNs.
Our resulting quantization-aware interval bound propagation method (QA-IBP) trains an interval arithmetic abstraction of a QNN via stochastic gradient descent by propagating upper and lower bounds for each layer instead of concrete values.
First, we replace each layer $l_i$ with two functions $\underline{l_i},\, \overline{l_i}$: $[\mathbb{Z}]_k^{n_i} \rightarrow [\mathbb{Z}]_k^{n_0}$ defined as follows:
\begin{align}
\mu_j &= \frac{\overline{x_j} + \underline{x_j}}{2} & r_j &= \frac{\overline{x_j} - \underline{x_j}}{2}\\
\mu_i &= \sum_{j=1}^{n_i} w_{ij} \mu_j + b_i & r_i &= \sum_{j=1}^{n_i} | w_{ij} |r_j \\
\underline{x'_i} &= \mu_i - r_i & \overline{x'_i} &= \mu_i + r_i \label{eq:ibpsum}\\
\underline{x''_i} &= \text{round}(\underline{x'_i},k_i) = \lfloor \underline{x'_i}\cdot M_i \rfloor \\
\overline{x''_i} &= \text{round}(\overline{x'_i},k_i) = \lfloor \overline{x'_i}\cdot M_i \rfloor \label{eq:ibpround}\\
\underline{y_i} &= \max\{0,\min\{2^{N_i}-1, \underline{x''_i} \}\} \\
\overline{y_i} &= \max\{0,\min\{2^{N_i}-1, \overline{x''_i} \}\}.
\end{align}
As with standard QNNs, $w_{i,j}\in [\mathbb{Z}]_k^{n_i}$ and $b_i\in [\mathbb{Z}]_k^{n_i}$ for each $1\leq j\leq n_i$ and $1\leq i\leq n_0$ denote the weights and biases of $f$ which are also bit-vectors of appropriate dimension.
By the sequential composition of all layers of the QNN we get the IBP representation of the QNN in the form of two functions $\underline{\semink{f_{\theta}}{k}}$ and $\overline{\semink{f_{\theta}}{k}}$. For a given input sample $x$ and $\epsilon>0$ we can use the IBP representation of $\semink{f_{\theta}}{k}$ to potentially prove the adversarial robustness of the QNN. In particular, the input sample defines an abstract interval domain $(\underline{x},\overline{x})$ with $\underline{x} = x-\epsilon$ and $\overline{x} = x + \epsilon$.
Next, we propagate the abstract domains through the IBP representation of the network to obtain output bounds $\underline{y} =\underline{\semink{f_{\theta}}{k}}(\underline{x},\overline{x})$ and $\overline{y} =\overline{\semink{f_{\theta}}{k}}(\underline{x},\overline{x})$.
Finally, we know that $\semink{f_{\theta}}{k}$ is {\em $\epsilon$-adversarially robust} at point $x$ if
\begin{equation}\label{eq:ibprobust}
\underline{y_i} > \overline{y_j}, \qquad\text{for } i=\mathsf{class}(x) \text{ and } \forall j\neq \mathsf{class}(x).
\end{equation}
\paragraph{End-to-end differentiation}
We modify the IBP representation of QNNs described above to allow an end-to-end differentiation necessary for a stochastic gradient descent-based learning algorithm. In particular, first, we apply the straight-through gradient estimator to propagate the error backward through the rounding operations in Eq. \ref{eq:ibpround}. We do this by replacing the non-differentiable rounding operation with the fake quantization function
\begin{align}
\text{fake\_quant}(x_i) &:= \text{round}(x_i,k_i)\\
\frac{\partial\ \text{fake\_quant}(x_i)}{\partial x_i} &:= 1.
\end{align}
We also add fake quantization operations around the weights and biases in Eq. \ref{eq:ibpsum}. The modified training graph is visualized in Figure \ref{fig:fakequant}.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{qat_ibp.pdf}
\caption{Illustration of how the interval bound propagation training graph is affected by quantization-aware training. \textbf{A)} Standard IBP inference graph. \textbf{B)} IBP inference path with fake quantization operations inserted to model quantized weights, biases, and computations.}
\label{fig:fakequant}
\end{figure}
For a single training sample $(x,j)$, we define the per-sample training loss
\begin{align}
L(\underline{y},\overline{y},j) = \sum_{i\neq j}(\overline{y_i} - \underline{y_j})\mathds{1}[\underline{y_j}-\overline{y_i}\leq 0],
\end{align}
where $\underline{y}, \overline{y}$ are the QA-IBP output bounds QNN with respect to the input domain $(x-\epsilon,x+\epsilon)$ and $j$ corresponds to the label, i.e., class $j$.
The loss term encourages the QA-IBP to produce output bounds that prove adversarial robustness of the QNN $\semink{f_{\theta}}{k}$ with respect to the input $x$ and adversarial radius $\epsilon$.
\paragraph{Existence of robust QNNs} We conclude this section by presenting an interesting result on the existence of robust QNNs. In particular, given $\epsilon>0$ and a finite dataset of $1$-dimensional bit-vectors whose any two distinct datapoints are at least $2\epsilon$ away, we prove that there exists a QNN with ReLU activations that is $\epsilon$-robust with respect to the dataset. Furthermore, we provide an upper bound on the number of neurons that the QNN must contain. We assume $2\epsilon$ distance simply for the $\epsilon$-neighbourhoods of datapoints to be disjoint so that robustness cannot impose contradicting classification conditions.
Note that this result {\em does not} hold for real arithmetic feed-forward neural networks with ReLU or any other continuous activation functions. Indeed, it was observed in~\cite[Corollary~5.12]{MirmanBV21} that a dataset $\{(-2,-1), (0,1), (2,-1)\}$ cannot be $1$-robustly classified by a feed-forward neural network which uses continuous activation functions. The intuition behind this impossibility result for real arithmetic networks is that a classifier would have to be a continuous function that correctly classifies all points in some open neighbourhoods of $x=-1$ and $x=1$.
\begin{theorem}\label{thm:existence}
Let $\epsilon>0$ and let $\mathcal{D} = \{(x_1,y_1),\dots,(x_{|\mathcal{D}|},y_{|\mathcal{D}|})\}$ with $x_i\in[\mathbb{Z}]_k^1$ and $y_i\in[\mathbb{Z}]_k^1$ for each $1\leq i\leq |\mathcal{D}|$. Suppose that $||x_i - x_j||_{\infty}\geq 2\epsilon$ for each $i\neq j$. Then, there exists a QNN $\semink{f_{\theta}}{k}$ which is $\epsilon$-robust with respect to the dataset $\mathcal{D}$ and which consists of $\mathcal{O}(|\mathcal{D}| \cdot \lceil\epsilon\cdot 2^k + 1 \rceil)$ neurons.
\end{theorem}
The proof of Theorem~\ref{thm:existence} is provided in the Appendix. It starts with an observation that the set $\mathcal{B}_{\epsilon}(x_i)=\{x'\in[\mathbb{Z}]_k^1 \mid ||x'-x_i||_{\infty}<\epsilon\}$ is finite and consists of at most $\lceil 2\epsilon\cdot 2^k+1 \rceil$ bit-vectors for each $1\leq i\leq |\mathcal{D}|$. Hence, constructing a QNN $\semink{f_{\theta}}{k}$ that is $\epsilon$-robust with respect to the dataset $\mathcal{D}$ is equivalent to constructing a QNN that correctly classifies $\mathcal{D}' = \cup_{i=1}^{|\mathcal{D}|} \{(x',y_i) \mid x'\in \mathcal{B}_{\epsilon}(x_i)\}$
which consists of $\mathcal{O}(|\mathcal{D}| \cdot \lceil\epsilon\cdot 2^k + 1 \rceil)$ datapoints. In the Appendix, we then design the QNN $\semink{f_{\theta}}{k}$ that correctly classifies a dataset and consists of at most linearly many neurons in the dataset size. Our construction also implies the following corollary.
\begin{corollary}
Let $\mathcal{D} = \{(x_1,y_1),\dots,(x_{|\mathcal{D}|},y_{|\mathcal{D}|})\}$ with $x_i\in[\mathbb{Z}]_k^1$ and $y_i\in[\mathbb{Z}]_k^1$ for each $1\leq i\leq |\mathcal{D}|$. Then, there exists a QNN $\semink{f_{\theta}}{k}$ which correctly classifies the dataset, i.e.~$\semink{f_{\theta}}{k}(x_i) = y_i$ for each $1\leq i\leq |\mathcal{D}|$, and which consists of $\mathcal{O}(|\mathcal{D}|)$ neurons.
\end{corollary}
\section{A Complete Decision Procedure for QNN Verification}\label{sec:verification}
In the previous section, we presented a quantization-aware training procedure for QNNs with robustness guarantees which was achieved by extending IBP to quantized neural network interpretations. We now show that IBP can also be used towards designing a {\em complete} verification procedure for already trained feed-forward QNNs. By completeness, we mean that the procedure is guaranteed to return either that the QNN is robust or to produce an adversarial attack.
There are two important novel aspects of the verification procedure that we present in this section. First, to the best of our knowledge this is the first complete robustness verification procedure for QNNs that is applicable to networks with non-piecewise linear activation functions. Existing constraint solving based methods that reduce verification to SMT-solving are complete but they only support piecewise linear activation fucntions such as ReLU~\citep{krizhevsky2010convolutional}. These could in theory be extended to more general activation functions by considering more expressive satisfiability modulo theories~\citep{smt2018handbook}, however this would lead to inefficient verification procedures and our experimental results in the following section already demonstrate the significant gain in scalability of our IBP-based methods as opposed to SMT-solving based methods for ReLU networks. Second, we note that while our IBP-based verification procedure is complete for QNNs, in general it is known that existing IBP-based verification procedures for real arithmetic neural networks are {\em not complete}~\citep{mirman2021fundamental}. Thus, our result leads to an interesting contrast in IBP-based robustness verification procedures for QNNs and for real arithmetic neural networks.
\paragraph{Verification procedure} We now describe our robustness verification procedure for QNNs. Its pseudocode is shown in Algorithm~\ref{alg:algorithm}. Since verifying $\epsilon$-robustness of a QNN with respect to some finite dataset $\mathcal{D}$ and $\epsilon>0$ is equivalent to verifying $\epsilon$-robustness of the QNN at each datapoint in $\mathcal{D}$, Algorithm~\ref{alg:algorithm} only takes as inputs a QNN $\semink{f_{\theta}}{k}$ that operates over bit-vectors of bit-width $k$, a single datapoint $x\in [\mathbb{Z}]_k^{n}$ and a robustness radius $\epsilon>0$. It then returns either \texttt{ROBUST} if $\semink{f_{\theta}}{k}$ is verified to be $\epsilon$-robust at $x\in [\mathbb{Z}]_k^{n}$, or \texttt{VULNERABLE} if an adversarial attack $||x'-x||_{\infty} < \epsilon$ with $\mathsf{class}(x') = \mathsf{class}(x)$ is found.
The algorithm proceeds by initializing a stack $D$ of abstract intervals to contain a single element $\{(x-\epsilon\cdot \mathbf{1}, x+\epsilon \cdot \mathbf{1})\}$, where $\mathbf{1}\in [\mathbb{Z}]_k^{n}$ is a unit bit-vector of bit-width $k$. Intuitively, $D$ contains all abstract intervals that may contain concrete adversarial examples but have not yet been processed by the algorithm. The algorithm then iterates through a loop which in each loop iteration processes the top element of the stack. Once the stack is empty and the last loop iteration terminates, Algorithm~\ref{alg:algorithm} returns \texttt{ROBUST}.
In each loop iteration, Algorithm~\ref{alg:algorithm} pops an abstract interval $(\underline{x},\overline{x})$ from $D$ and processes it as follows. First, it uses IBP for QNNs that we introduced before to propagate $(\underline{x},\overline{x})$ in order to compute an abstract interval $(\underline{y},\overline{y})$ that overapproximates the set of all possible outputs for a concrete input point in $(\underline{x},\overline{x})$. The algorithm then considers three cases. First, if $(\underline{y},\overline{y})$ does not violate Equation~\eqref{eq:ibprobust} which characterizes violation of robustness by a propagated abstract interval, Algorithm~\ref{alg:algorithm} concludes that the abstract interval $(\underline{x},\overline{x})$ does not contain an adversarial example and it proceeds to processing the next element of $D$. Second, if $(\underline{y},\overline{y})$ violates Equation~\eqref{eq:ibprobust}, the algorithm uses projected gradient descent restricted to $(\underline{x},\overline{x})$ to search for an adversarial example and returns \texttt{VULNERABLE} if found. Note that the adversarial attack is generated with respect to the quantization-aware representation of the network, thus ensuring that the input space corresponds to valid quantized inputs. Third, if $(\underline{y},\overline{y})$ violates Equation~\eqref{eq:ibprobust} but the adversarial attack could not be found by projected gradient descent, the algorithm refines the abstract interval $(\underline{x},\overline{x})$ by splitting it into two smaller subintervals. This is done by identifying
\[ i^{\ast} = \mathsf{argmax}_{1\leq i\leq n} (\overline{x}[i] - \underline{x}[i]) \]
and splitting the abstract interval $(\underline{x},\overline{x})$ along the $i^{\ast}$-th dimension into two abstract subintervals $(\underline{x'},\overline{x'}),(\underline{x''},\overline{x''})$, which are both added to the stack $D$.
\paragraph{Correctness, termination and completeness} The following theorem establishes that Algorithm~\ref{alg:algorithm} is complete, that it terminates on every input and that it is a complete robustness verification procedure. The proof is provided in the Appendix.
\begin{theorem}\label{thm:verification}
If Algorithm~\ref{alg:algorithm} returns \texttt{ROBUST} then $\semink{f_{\theta}}{k}$ is $\epsilon$-robust at $x\in [\mathbb{Z}]_k^{n}$. On the other hand, if Algorithm~\ref{alg:algorithm} returns \texttt{VULNERABLE} then there exists an adversarial attack $||x'-x||_{\infty} < \epsilon$ with $\mathsf{class}(x') \neq \mathsf{class}(x)$. Therefore, Algorithm~\ref{alg:algorithm} is correct. Furthermore, Algorithm~\ref{alg:algorithm} terminates and is guaranteed to return an output on any input. Since Algorithm~\ref{alg:algorithm} is correct and it terminates, we conclude that it is also complete.
\end{theorem}
\begin{algorithm}[t]
\caption{QNN robustness verification procedure}
\label{alg:algorithm}
\begin{algorithmic}[1]
\State \textbf{Input} QNN $\semink{f_{\theta}}{k}$, datapoint $x\in [\mathbb{Z}]_k^{n}$, robustness radius $\epsilon>0$
\State \textbf{Output} \texttt{ROBUST} or \texttt{VULNERABLE}
\State $D \leftarrow \{(x-\epsilon\cdot \mathbf{1},x+\epsilon\cdot \mathbf{1})\}$
\While{$D \neq \{\}$}
\State $(\underline{x},\overline{x}) \leftarrow$ pop item from $D$
\State $(\underline{y},\overline{y}) \leftarrow \underline{\semink{f_{\theta}}{k}}(\underline{x},\overline{x}), \overline{\semink{f_{\theta}}{k}}(\underline{x},\overline{x})$ propagated via IBP
\If{$(\underline{y},\overline{y})$ violates Equation (\ref{eq:ibprobust})}
\State Try to generate adversarial example using projected gradient descent restricted to $(\underline{x},\overline{x})$
\If{adversarial example found}
\State \textbf{return} \texttt{VULNERABLE}
\Else
\State $(\underline{x'},\overline{x'}),(\underline{x''},\overline{x''}) \leftarrow$ partition of $(\underline{x},\overline{x})$ into two abstract subintervals
\State $D \leftarrow D \cup \{(\underline{x'},\overline{x'}),(\underline{x''},\overline{x''})\}$
\EndIf
\EndIf
\EndWhile
\State \textbf{return} \texttt{ROBUST}
\end{algorithmic}
\end{algorithm}
\section{Experiments}\label{sec:exp}
We perform an experimental evaluation to assess the effectiveness of our quantization-aware interval bound propagation (QA-IBP).
In particular, we first use our training procedure to train two QNNs. We then use our complete verification procedure to verify robustness of trained QNNs and we compare our method to the existing verification methods for QNNs of \cite{giacobbe2020many} and \cite{henzinger2021scalable}.
Our full experimental setup and code can be found on GitHub \footnote{\url{https://github.com/mlech26l/quantization_aware_ibp}}.
\begin{table*}
\centering
\begin{tabular}{l|ccc|ccc}\toprule
Method & \multicolumn{3}{c}{MNIST} & \multicolumn{3}{c}{Fashion-MNIST} \\
& $\epsilon=0$ & $\epsilon=1$ & $\epsilon=4$ & $\epsilon=0$ & $\epsilon=1$ & $\epsilon=4$ \\\midrule
QF\_BV SMT \cite{giacobbe2020many} & 97.1\% & 92\% & 0\% & 85.6\% & 44\% & 0\% \\
QF\_BV SMT \cite{henzinger2021scalable} & 97.1\% & \textbf{99\%}$^*$ & 53\% & 85.6\% & 76\% & 36\% \\
QA-IBP (ours) & 99.2\% & \textbf{98.8\%} & \textbf{95.6\%} & 86.3\% & \textbf{80.0\%} & \textbf{59.8\%}\\\bottomrule
\end{tabular}
\caption{Certified robust accuracy of a convolutional neural network trained with QA-IBP compared to existing methods for certifying $L_\infty$-robustness of quantized neural networks reported in the literature. $^*$ Note that \cite{henzinger2021scalable} certified only a subset of the test set due to a high per-sample runtime of their approach. Due to this choice they reported a higher $\epsilon=1$ robust accuracy than clean accuracy.}
\label{tab:experiment}
\end{table*}
We train two CNNs with QA-IBP using an 8-bit quantization scheme for both weights and activations on the MNIST \cite{lecun1998gradient} and Fashion-MNIST \cite{xiao2017online} datasets. We quantize all layers; however, we note that our approach is also compatible with mixing quantized and non-quantized layers. Our MNIST network consists of five convolutional layers, followed by two fully-connected layers. Our Fashion-MNIST network contains three convolutional layers and two fully-connected layers. We apply a fixed pre-defined fixed-point format on both weights and activations. Further details on the network architectures can be found in the Appendix.
We use the Adam optimizer \cite{kingma2014adam} with a learning rate of $10^{-4}$ with decoupled weight decay of $10^{-4}$ \cite{loshchilov2018decoupled} and a batch size of 512. We train our networks for a total of 5,000,000 gradient steps on a single GPU.
We linearly scale the value of $\epsilon$ during the QA-IBP training from 0 to 4 and apply the elision of the last layer optimization as reported in \cite{gowal2019scalable}.
We pre-train our networks for 5000 steps in their non-IBP representation with a learning rate of $5\cdot 10^{-4}$.
After training, we certify all test samples using Algorithm \ref{alg:algorithm} with a timeout of 20s per sample. We certify for $L_\infty$-robustness with radii 1 and 4 to match the experimental setup of \cite{henzinger2021scalable}.
We report and compare the certified robust accuracy of the trained networks to the values reported in \cite{henzinger2021scalable}.
The results in Table \ref{tab:experiment} demonstrate that our QA-IBP significantly improves the certified robust accuracy for QNNs on both datasets.
Regarding the runtime of the verification step, both \cite{giacobbe2020many} and \cite{henzinger2021scalable} certified a smaller model compared to our evaluation. In particular, \cite{giacobbe2020many} report a mean runtime of their 8-bit model to be over 3 hours, and \cite{henzinger2021scalable} report the mean runtime of their 6-bit model to be 90 and 49 seconds for MNIST/Fashion-MNIST, respectively.
Conversely, our method was evaluated with a timeout of 20 seconds and tested on larger networks than the two existing approaches, showing the efficiency of our approach.
\subsection{Ablation analysis}
In this section, we assess the effectiveness of Algorithm \ref{alg:algorithm} for certifying robustness compared to an incomplete verification based on the QA-IBP output bounds alone.
In particular, we perform an ablation and compare Algorithm \ref{alg:algorithm} to an incomplete verification baseline. Our baseline consists of checking the bounds obtained by QA-IBP as an incomplete verifier combined with projected gradient descent (PGD) as an incomplete falsifier, i.e., we try to certify robustness via QA-IBP and simultaneously try to generate an adversarial attack using PGD.
In particular, this baseline resembles a version of Algorithm \ref{alg:algorithm} without any branching into subdomains.
We carry our ablation analysis on the two networks of our first experiment. We report the number of samples where the verification approach could not determine whether the network is robust on the sample or not. We use a timeout of 20s per instance when running our algorithm.
\begin{table}[]
\centering
\begin{tabular}{c|ccc}\toprule
& \multicolumn{3}{c}{CIFAR-10} \\
Weight decay & $\epsilon=0$ & $\epsilon=1$ & $\epsilon=4$ \\\midrule
$1\cdot 10^{-4}$ & 30.0\% & 28.4\% & \textbf{22.4\%} \\
$5\cdot 10^{-5}$ & 39.6\% & \textbf{34.1\%} & 20.6\% \\
$1\cdot 10^{-5}$ & \textbf{83.1\%} & 0\% & 0\% \\\bottomrule
\end{tabular}
\caption{Robust accuracy of our convolutional neural network trained with QA-IBP on the CIFAR-10 dataset with various values for the weight decay. The results express the \textit{robustness-accuracy tradeoff} \cite{tsipras2018robustness}, i.e., the empirical observation made for non-quantized neural networks that we can have a high robustness or a high accuracy but not both at the same time.}
\label{tab:cifar10}
\end{table}
The results shown in Table \ref{tab:timeout} indicate that our algorithm is indeed improving on the number of samples for which robustness could be decided.
However, the improvement is relatively small, suggesting that the QA-IBP training stems from most of the observed gains.
\begin{table*}[]
\centering
\begin{tabular}{l|cc|cc}\toprule
Method & \multicolumn{2}{c}{MNIST} & \multicolumn{2}{c}{Fashion-MNIST} \\
& $\epsilon=1$ & $\epsilon=4$ & $\epsilon=1$ & $\epsilon=4$ \\\midrule
IBP + Projected gradient descent & \textbf{0.35\%} & 3.51\% & 3.96\% & 19.23\% \\
Algorithm \ref{alg:algorithm} & \textbf{0.35\%} & \textbf{3.47\%} & \textbf{3.93\%} & \textbf{18.51\%} \\\bottomrule
\end{tabular}
\caption{Percentage of samples where the robustness of the network can be determined, i.e., certified or falsified, by the method (lower is better). Algorithm timeout was set to 20s per instance. }
\label{tab:timeout}
\end{table*}
\subsection{Limitations}
In this section, we aim to scale our QA-IBP beyond the two gray-scale image classification tasks studied before to the CIFAR-10 dataset \cite{krizhevsky2010convolutional}.
Our setup consists of the same convolutional neural network as used for the Fasion-MNIST.
We run our setup with several values for the weight decay rate. Similar to above, we linearly scale $\epsilon$ from 0 to 4 during training and report the certified robust accuracy obtained by QA-IBP with an $\epsilon$ of 1 and 4 and the clean accuracy.
The results in Table \ref{tab:cifar10} express the robustness-accuracy tradeoff, i.e., the observed antagonistic relation between clean accuracy and robustness, which has been extensively studied in non-quantized neural networks \cite{zhang2019theoretically,bubeck2021universal}.
Depending on the weight decay value, different points on the trade-off were observed. In particular, our training procedure either obtains a network that has an acceptable accuracy but no robustness or a certifiable robust network with a significantly reduced clean accuracy.
We also trained larger models on all datasets (MNIST, Fashion-MNIST, and CIFAR-10), but observed the same behavior of having a high accuracy but no robustness when training, i.e., as in row with $1\cdot 10^{-5}$ in Table \ref{tab:cifar10}.
We found the underlying reason for this behavior to be activations of internal neurons that clamp to the minimum and maximum value of the quantization range in their QA-IBP representation but not in their standard representation. Consequently, the activation gradients become zero during QA-IBP due to falling in the constant region of the activation function.
This effect is specific to quantized neural networks due to the upper bound on the representative range of values.
Our observation hints that future research needs to look into developing dynamic quantization ranges or weight decay schedules that can adapt to both the standard and the QA-IBP representation of a QNN.
\section{Conclusion}
In this paper, we introduced quantization-aware interval bound propagation (QA-IBP), the first method for training certifiably robust QNNs. We also present a theoretical result on the existence and upper bounds on the needed size of a robust QNN for a given dataset of $1$-dimensional datapoints. Moreover, based on our interval bound propagation method, we developed the first complete verification algorithm for QNNs that may be run on GPUs.
We experimentally showed that our training scheme and verification procedure advance the state-of-the-art on certifying $L_\infty$-robustness of QNNs.
We also demonstrated the limitations of our method regarding training stability and convergence. Particularly, we found that the boundedness of the representable value range of QNNs compared to standard networks leads to truncation of the abstract domains, which in turn leads to gradients becoming zero. Our results suggest that dynamic quantization schemes that adapt their quantization range to the abstract domains instead of the concrete activation values of existing quantization schemes may further improve the certified robust accuracy of quantized neural networks.
Nonetheless, our work serves as a new baseline for future research. Promising directions on how to improve upon QA-IBP and potentially overcome its numerical challenges is to adopt advanced quantization-aware training techniques. For instance, dynamical quantization ranges \cite{choi2018pact, jacob2018quantization}, mixed-precision layers \cite{zhou2016dorefa,yao2021hawq}, and automated architecture search \cite{tan2019efficientnet} have shown promising results for standard training QNNs and might enhance QA-IBP-based training procedures as well.
Moreover, further improvements may be feasible by adapting recent advances in IBP-based training methods for non-quantized neural networks \cite{muller2022certified} to our quantized IBP variant.
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\section{Emergence of Mass}
\label{intro}
The natural energy-scale for strong interactions is characterised by the proton mass:
\begin{equation}
m_p \approx 1\,{\rm GeV} \approx 2000\,m_e\,,
\end{equation}
where $m_e$ is the mass of the electron. In the Standard Model, $m_e$ is rightly attributed to the Higgs boson; but what is the source of the enormous enhancement to produce $m_p$? This is the crux: the source of the vast majority of visible mass in the Universe is unknown. Followed logically to its origin, this question leads to an appreciation that the existence of our Universe depends critically on, \emph{inter alia}, the following empirical facts:
(\emph{i}) the proton is massive, \emph{i.e}.\ the mass-scale for strong interactions is vastly different to that of electromagnetism;
(\emph{ii}) the proton is absolutely stable, despite being a composite object constituted from three valence-quarks;
and (\emph{iii}) the pion, responsible for long-range interactions between nucleons, is unnaturally light (not massless), possessing a lepton-like mass despite being a strongly interacting composite object built from a valence-quark and valence-antiquark.
These are basic emergent features of Nature; and QCD must somehow explain them and many other high-level phenomena with enormous apparent complexity.
The Lagrangian of chromodynamics is simple:
\begin{subequations}
\label{QCDdefine}
\begin{align}
{\mathpzc L}_{\rm QCD} & = \sum_{j=u,d,s,\ldots}
\bar{q}_j [i\gamma^\mu D_{\mu} - m_j] q_j - \tfrac{1}{4} G^a_{\mu\nu} G^{a\mu\nu},\\
\label{gluonSI}
D_{\mu} & = \partial_\mu + i g \tfrac{1}{2} \lambda^a A^a_\mu\,,\\
G^a_{\mu\nu} & = \partial_\mu A^a_\nu + \partial_\nu A^a_\mu +
\underline{\textcolor[rgb]{0.00,0.0,00}{i g f^{abc}A^b_\mu A^c_\nu}}.
\label{gluonSIU}
\end{align}
\end{subequations}
Here: $\{q_j\}$ are the quark fields, with $j$ their flavor label and $m_j$ their Higgs-generated current-quark masses; and $\{ A_\mu^a, a=1,\ldots, 8 \}$ are the gluon fields, with $\{\tfrac{1}{2} \lambda^a\}$ the generators of the SU$(3)$ (color/chromo) gauge-group in the fundamental representation. In comparison with quantum electrodynamics (QED), the \emph{solitary} difference is the term describing gluon self-interactions, marked as the underlined piece in Eq.\,\eqref{gluonSIU}. Somehow, ${\mathpzc L}_{\rm QCD}$ -- this one line, along with two definitions -- is responsible for the origin, mass, and size of almost all visible matter in the Universe. That being true, then QCD is quite possibly the most remarkable fundamental theory ever invented.
The only apparent mass-scales in Eq.\,\eqref{QCDdefine} are the current-quark masses, generated by the Higgs boson; but focusing on the valence quarks which define nucleons, \emph{i.e}.\ $u$, $d$, this scale is more-than 100-times smaller than $m_p$. No amount of ``staring'' at ${\mathpzc L}_{\rm QCD}$ can reveal the source of that enormous amount of ``missing mass''; yet, it is there. This contrasts starkly with QED wherein, \emph{e.g}.\ the scale in the spectrum of the hydrogen atom is set by $m_e$, a prominent feature of ${\mathpzc L}_{\rm QED}$ that is generated by the Higgs boson.
The emergence of mass is seen even more keenly by considering that even treated as a classical theory, chromodynamics is a non-Abelian local gauge field theory. As with all such theories formulated in four spacetime dimensions, no energy-scale exists in the absence of Lagrangian masses for the matter fields. (The absence of such masses defines the chiral limit.) There is no dynamics in a scale-invariant theory, only kinematics: the theory looks the same at all length-scales; hence, there can be no clumps of anything. Bound states are therefore impossible and, accordingly, our Universe cannot exist.
A \emph{spontaneous} breaking of symmetry, as realised via the Higgs mechanism, does not solve this problem: the masses of the neutron and proton, the kernels of all visible matter, are roughly 100-times larger than the Higgs-generated current-masses of the light $u$- and $d$-quarks, the main building blocks of protons and neutrons.
Consequently, the questions of how does a mass-scale appear and why does it have the value we observe are inseparable from the question of how did the Universe come into being.
Modern quantum field theories are built upon Poincar\'e invariance, so consider the energy-momentum tensor in classical chromodynamics, $T_{\mu\nu}$. Conservation of energy and momentum is a consequence of spacetime translational invariance, elements in the set of Poincar\'e transformations; hence, $\partial_\mu T_{\mu\nu} = 0$. Suppose now that one performs a global scale transformation on the coordinates and fields in classical chromodynamics. In the absence of fermion masses, \emph{viz}.\ suppressing the Higgs mechanism, the classical action is invariant under the scale transformation and the dilation current is conserved:
\begin{equation}
{\cal D}_\mu = T_{\mu\nu} x_\nu \; | \;
\partial_\mu {\cal D}_\mu = 0 = [\partial_\mu T_{\mu\nu} ] x_\nu + T_{\mu\nu} \delta_{\mu\nu} = T_{\mu\mu}\,.
\label{SIcQCD}
\end{equation}
Evidently, the energy-momentum tensor must be traceless in a scale invariant theory.
Classical chromodynamics is not a meaningful framework for many reasons; amongst them the fact that strong interactions are empirically known to be characterised by a large mass-scale, $m_p \approx 1\,$GeV. In quantising the theory, regularisation and renormalisation of ultraviolet divergences introduces a mass-scale. This is ``dimensional transmutation'': all quantities, including the field operators themselves, become dependent on a mass-scale. It entails violation of Eq.\,\eqref{SIcQCD}, \emph{i.e}.\ the appearance of the chiral-limit ``trace anomaly'':
\begin{equation}
\label{SIQCD}
T_{\mu\mu} = \beta(\alpha(\zeta)) \tfrac{1}{4} G^{a}_{\mu\nu}G^{a}_{\mu\nu} =: \Theta_0 \,,
\end{equation}
where $\beta(\alpha(\zeta))$ is QCD's $\beta$-function, $\alpha(\zeta)$ is the associated running-coupling, and $\zeta$ is the renormalisation scale. Eq.\,\eqref{SIQCD} indicates that a mass-scale related to the resolving power of a given measurement is introduced via quantisation, \emph{viz}.\ the scale \emph{emerges} as an integral part of the theory's quantum definition.
There is another aspect of chromodynamics that should be highlighted; namely, the classical Lagrangian still defines a non-Abelian local gauge theory. Accordingly, the concept of local gauge invariance persists; but without a mass-scale, there is no notion of confinement. For example, three quarks can be prepared in a colour-singlet combination and colour rotations will keep the three-body system neutral; but the quarks involved need not have any proximity to one another. Indeed, proximity is meaningless because all lengths are equivalent in a scale invariant theory. Hence, the question of ``Whence a mass-scale?'' is equivalent to ``Whence a confinement scale?''.
Knowing a trace anomaly exists does not actually deliver a great deal: it only indicates that there is a mass-scale. The crucial issue is whether one can compute and/or understand the magnitude of that scale.
One can certainly measure the size of the scale anomaly, for consider the expectation value of the energy-momentum tensor in the proton:
\begin{equation}
\label{EPTproton}
\langle p(P) | T_{\mu\nu} | p(P) \rangle = - P_\mu P_\nu\,,
\end{equation}
where the right-hand-side follows from the equations-of-motion for a one-particle proton state. In the chiral limit,
\begin{align}
\label{anomalyproton}
\langle p(P) | T_{\mu\mu} | p(P) \rangle = - P^2 & = m_p^2
= \langle p(P) | \Theta_0 | p(P) \rangle\,;
\end{align}
namely, there is a clear sense in which it is possible to conclude that the entirety of the proton mass is produced by gluons. The trace anomaly is measurably large; and that property must logically owe to gluon self-interactions, which are also responsible for asymptotic freedom.
This is a valid conclusion. After all, what else could be responsible for a mass-scale in QCD? QCD is all about gluon self-interactions; and it's gluon self-interactions that (potentially) enable one to rigorously (nonperturbatively) define the expectation value in Eq.\,\eqref{anomalyproton}. On the other hand, it's only a sensible conclusion when the operator and the wave function are defined at a resolving-scale $\zeta \gg m_p$, \emph{viz}.\ when one employs a parton-model basis.
There is also another issue, which can be exposed by returning to Eq.\,\eqref{EPTproton} and replacing the proton by the pion:
\begin{equation}
\label{EPTpion}
\langle \pi(q) | T_{\mu\nu} | \pi(q) \rangle = - q_\mu q_\nu\,
\stackrel{\rm chiral\,limit}{\Rightarrow} \langle \pi(q) | \Theta_0 | \pi (q) \rangle = 0
\end{equation}
because the pion is a massless Nambu-Goldstone (NG) mode. Equation\,\eqref{EPTpion} could mean that the scale anomaly vanishes trivially in the pion state, \emph{viz}.\ that gluons and their self-interactions have no impact within a pion because each term in the practical computation of the operator expectation value vanishes when evaluated in the pion. However, that is a difficult way to achieve Eq.\,\eqref{EPTpion}. It is easier to imagine that Eq.\,\eqref{EPTpion} owes to cancellations between different operator-component contributions. Of course, such precise cancellation should not be an accident. It could only arise naturally because of some symmetry and/or symmetry-breaking pattern.
Equations\,\eqref{anomalyproton} and \eqref{EPTpion} present a quandary, which highlights that no understanding of the source of the proton's mass can be complete unless it simultaneously explains the meaning of Eq.\,\eqref{EPTpion}. Moreover, any discussion of confinement, fundamental to the proton's absolute stability, is impossible before this conundrum is resolved. The explanation of these features of Nature must lie in the dynamics responsible for the emergence of $m_p$ as the natural mass-scale for nuclear physics; and one of the most important goals in modern science is to explain and elucidate the entire array of empirical consequences of this dynamics.
\section{Gluons are Massive}
\label{SecGluonMass}
Gluons must be the key; after all, their self interactions separate QCD from QED. Gluons are supposed to be massless. This is true in perturbation theory; but it is a feature that is not preserved nonperturbatively. Beginning with a pioneering effort almost forty years ago \cite{Cornwall:1981zr}, continuum and lattice studies of QCD's gauge sector have been increasing in sophistication and reliability; and today it is known that the gluon propagator saturates at infrared momenta \cite{Boucaud:2011ug, Strauss:2012dg, Binosi:2014aea, Aguilar:2015bud, Binosi:2016nme, Gao:2017uox, Cyrol:2017ewj, Rodriguez-Quintero:2018wma, Binosi:2019ecz}:
\begin{equation}
\label{eqGluonMass}
\Delta(k^2\simeq 0) = 1/m_g^2.
\end{equation}
Thus, the long-range propagation characteristics of gluons are dramatically affected by their self-interactions. Importantly, one may associate a renormalisation-group-invariant gluon mass-scale with this effect: $m_0 \approx 0.5\,$GeV\,$\approx m_p/2$, and summarise a large body of work by stating that gluons, although acting as massless degrees-of-freedom on the perturbative domain, actually possess a running mass, whose value at infrared momenta is characterised by $m_0$.
Asymptotic freedom ensures that QCD's ultraviolet behaviour is controllable; but the emergence of a gluon mass reveals a new frontier within the Standard Model because the existence of a running gluon mass, large at infrared momenta, has an impact on all analyses of the bound-state problem. For instance, it could be a harbinger of gluon saturation \cite{Accardi:2012qut, NAP25171X}.
Furthermore, $m_0>0 $ entails that QCD dynamically generates its own infrared cutoff, so that gluons with wavelengths $\lambda \gtrsim \sigma :=1/m_0 \approx 0.5\,$fm decouple from the strong interaction, hinting at a dynamical realisation of confinement \cite{Brodsky:2015aia}.
\begin{figure}[t]
\centerline{\includegraphics[width=0.45\textwidth]{F1NStar.pdf}}
\caption{\label{FigwidehatalphaII}
Dot-dashed [blue] curve: process-in\-de\-pen\-dent running-coupling $\alpha_{\rm PI}(k^2)$ \cite{Binosi:2016nme}: the shaded [blue] band bracketing this curve combines a 95\% confidence-level window based on existing lQCD results for the gluon two-point function with an error of 10\% in the continuum analysis of relevant ghost-gluon dynamics.
Solid [black] curve, updated result \cite{Rodriguez-Quintero:2018wma}.
Also depicted, world's data on the process-dependent coupling $\alpha_{g_1}$, defined via the Bjorken sum rule, the sources of which are listed elsewhere \cite{Binosi:2016nme}.
The shaded [yellow] band on $k>1\,$GeV represents $\alpha_{g_1}$ obtained from the Bjorken sum by using QCD evolution to extrapolate high-$k^2$ data into the depicted region \cite{Deur:2008rf}; and, for additional context, the dashed [red] curve is the effective charge obtained in a light-front holographic model \cite{Deur:2016tte}.
The k-axis scale is linear to the left of the vertical dashed line and logarithmic otherwise.
}
\end{figure}
\section{Process-Independent Effective Charge}
There are many other consequences of the intricate nonperturbative nature of QCD's gauge-sector dynamics. Important amongst them is the generation of a process-independent running coupling, $\alpha_{\rm PI}(k^2)$ -- see Refs.\,\cite{Binosi:2016nme, Rodriguez-Quintero:2018wma}. Depicted as the solid [black] curve in Fig.\,\ref{FigwidehatalphaII}, this is a new type of effective charge, which is an analogue of the Gell-Mann--Low effective coupling in QED because it is completely determined by the gauge-boson propagator. The result in Fig.\,\ref{FigwidehatalphaII} is a parameter-free prediction, capitalising on analyses of QCD's gauge sector undertaken using continuum methods and informed by numerical simulations of lattice-QCD (lQCD).
As a unique process-independent effective charge, $\alpha_{\rm PI}$ appears in every one of QCD's dynamical equations of motion, including the gap equation, setting the strength of all interactions. $\alpha_{\rm PI}$ therefore plays a crucial role in understanding the dynamical origin of light-quark masses in the Standard Model even in the absence of a Higgs coupling, as described in the next section.
\begin{figure}[t]
\centerline{\includegraphics[clip,width=0.43\textwidth]{F2NStar.pdf}}
\caption{\label{gluoncloud}
Renormalisation-group-invariant dressed-quark mass function, $M(p)$ in Eq.\,\eqref{Spgen}: \emph{solid curves} -- continuum nonperturbative results \cite{Bhagwat:2003vw, Bhagwat:2006tu}; ``data'' -- numerical simulations of lQCD \protect\cite{Bowman:2005vx}.
The current-quark of perturbative QCD evolves into a constituent-quark as its momentum becomes smaller. The constituent-quark mass arises from a cloud of low-momentum gluons attaching themselves to the current-quark. This is DCSB, the essentially nonperturbative effect that generates a quark \emph{mass} \emph{from nothing}; namely, it occurs even in the absence of a Higgs mechanism.
The size of $M(0)$ is a measure of the magnitude of the QCD scale anomaly in $n=1$-point Schwinger functions \cite{Roberts:2016vyn}.
Moreover, experiments on $Q^2\in [0,12]\,$GeV$^2$ at the modern Thomas Jefferson National Accelerator Facility (JLab) will be sensitive to the momentum dependence of $M(p)$ within a domain that is here indicated approximately by the shaded region.
}
\end{figure}
\section{Dynamical Chiral Symmetry Breaking}
The emergence of a gluon mass-scale in the Standard Model drives an enormous array of phenomena. Crucial amongst them is dynamical chiral symmetry breaking (DCSB), which is most readily apparent in the dressed-quark propagator:
\begin{equation}
\label{Spgen}
S(p) = 1/[i \gamma\cdot p A(p^2) + B(p^2)] = Z(p^2)/[i\gamma\cdot p + M(p^2)]\,.
\end{equation}
$S(p)$ can be predicted in QCD using nonperturbative continuum and lattice techniques. The quantity $M(p^2)$ in Eq.\,\eqref{Spgen} is the dressed-quark mass-function, whose computed behaviour is depicted and explained in Fig.\,\ref{gluoncloud}. Evidently, even in the absence of a Higgs mechanism, when current-masses vanish, quarks acquire a running mass whose value in the infrared is approximately $m_p/3$. This is the scale required to support the constituent quark model and all its successes. It follows that DCSB can be identified as the source for more than 98\% of the visible mass in the Universe, \emph{viz}.\ DCSB is fundamentally connected with the \emph{origin of mass from nothing}.
One must insist that chiral symmetry breaking in the absence of a Higgs mechanism is ``dynamical,'' as distinct from spontaneous, because nothing is added to QCD in order to effect this remarkable outcome and there is no change of variables in ${\mathpzc L}_{\rm QCD}$ that will make it apparent. Instead, through the act of quantising the classical chromodynamics of massless gluons and quarks, a large mass-scale is generated in both the gauge and matter sectors.
DCSB is empirically revealed very clearly in properties of the pion. In fact \cite{Roberts:2016vyn}, the key to understanding Eq.\,\eqref{EPTpion} is a set of Goldberger-Treiman-like (GT) relations \cite{Qin:2014vya, Binosi:2016rxz}, the best known of which states:
\begin{equation}
m \simeq 0 \; \left| \; f_\pi E_\pi(k;0) = B(k^2)
\right. ,
\end{equation}
where $E_\pi$ is the leading piece of the pion's Bethe-Salpeter amplitude (wave function), and $B$ is the scalar piece of the dressed-quark self-energy, Eq.\,\eqref{Spgen}. This equation is exact in chiral QCD and expresses the fact that the Nambu-Goldstone theorem is fundamentally an expression of equivalence between the quark one-body problem and the two-body bound-state problem in QCD's color-singlet flavor-nonsinglet pseudoscalar channel. Consequently and enigmatically:\\[0.5ex]
\hspace*{1em}\parbox[t]{0.44\textwidth}{\emph{The properties of the nearly-massless pion are the cleanest expression of the mechanism that is responsible for (almost) all the visible mass in the Universe}.}
\smallskip
\hspace*{-\parindent}The manner by which this can be exploited at modern and planned facilities, in order to chart, \emph{inter alia}, the origin and distribution of mass, is canvassed elsewhere \cite{Horn:2016rip, Denisov:2018unj, Aguilar:2019teb}.
With the GT relations in hand, one can construct an algebraic proof \cite{Qin:2014vya, Binosi:2016rxz}, that at any and each order in a symmetry-preserving truncation of those equations in quantum field theory necessary to describe a pseudoscalar bound state, there is a precise cancellation between the mass-generating effect of dressing the valence-quark and -antiquark which constitute the system and the attraction generated by the interactions between them, \emph{i.e}.
\begin{align}
M^{\rm dressed}_{\rm quark} + M^{\rm dressed}_{\rm antiquark}+ U^{\rm dressed}_{\rm quark-antiquark\;interaction} \stackrel{\rm chiral\;limit}{\equiv} 0\,.
\label{EasyOne}
\end{align}
This guarantees the ``disappearance'' of the scale anomaly in the chiral-limit pion.
An analogy with quantum mechanics thus arises: the mass of a QCD bound-state is the sum of the mass-scales characteristic of the constituents plus a (negative and sometimes large) binding energy.
Since QCD's interactions are universal, similar cancellations must take place within the proton. However, in the proton channel there is no symmetry that requires the cancellations to be complete. Hence, the proton's mass has a value that is typical of the magnitude of scale breaking in QCD's one body sectors, \emph{viz}.\ the dressed-gluon and -quark mass-scales.
The perspective just described may be called the ``DCSB paradigm''. It provides a basis for understanding why the mass-scale for strong interactions is vastly different to that of electromagnetism, why the proton mass expresses that scale, and why the pion is nevertheless unnaturally light. In this picture, no significant mass-scale is possible in QCD unless one of commensurate size is expressed in the dressed-propagators of gluons and quarks.
\section{Faddeev Equation for Baryons}
\label{FEBaryons}
The Faddeev equation was introduced almost sixty years ago \cite{Faddeev:1960su}. It treats the quantum mechanical problem of three-bodies interacting via pairwise potentials by reducing it to a sum of three terms, each of which describes a solvable scattering problem in distinct two-body subsystems. An analogous approach to the three-valence-quark (baryon) bound-state problem in quantum chromodynamics (QCD) is
explained in Refs.\,\cite{Cahill:1988dx, Reinhardt:1989rw, Efimov:1990uz}. In this case, owing to DCSB and the importance of symmetries \cite{Binosi:2016rxz}, a Poincar\'e-covariant quantum field theory generalisation of the Faddeev equation is required. Like the Bethe-Salpeter equation for mesons, it is natural to consider such a Faddeev equation as one of the tower of QCD's Dyson-Schwinger equations (DSEs), which are being used to develop a systematic continuum approach to the strong-interaction bound-state problem \cite{Roberts:2015lja, Eichmann:2016yit}.
The first direct treatment of the Poincar\'e-covariant Faddeev equation for the nucleon is described in Ref.\,\cite{Eichmann:2009qa}. Following that approach, Refs.\,\cite{Qin:2018dqp, Qin:2019hgk} calculated the spectrum of ground-state $J=1/2^+$, $3/2^+$ $(qq^\prime q^{\prime\prime})$-baryons, where $q, q^\prime, q^{\prime\prime} \in \{u,d,s,c,b\}$, their first positive-parity excitations and parity partners. Introducing two parameters, to compensate for deficiencies of the leading-order truncation when used for excited hadrons, a description of the known spectrum of 39 such states was obtained, with a mean-absolute-relative-difference between calculation and experiment of $3.6\,(2.7)$\%. This is exemplified in Fig.\,\ref{810MassComparison}. The framework was subsequently used to predict the masses of 90 states not yet seen empirically.
\begin{figure}[!t]
\begin{center}
\includegraphics[clip,width=0.95\linewidth]{F3NStar.pdf}
\end{center}
\caption{\label{810MassComparison}
Masses of pseudoscalar and vector mesons, and ground-state positive-parity octet and decuplet baryons calculated using continuum (Cont$^{\rm m}$ -- squares, red) \cite{Qin:2019hgk} and lattice \cite{Durr:2008zz} methods in QCD compared with experiment (PDG -- black bars, with decay-widths of unstable states shaded in grey) \cite{Tanabashi:2018oca}. The continuum study did not include isospin symmetry breaking effects, which are evidently small, as highlighted by the empirically determined $\Sigma$-$\Lambda$ mass difference.
}
\end{figure}
Since solving the Faddeev equation yields Poincar\'e-covariant wave functions for all these systems, information about their internal structure is also revealed. For instance, considering $J=3/2^+$ states, each has a complicated rest-frame angular-momentum profile. The ground-states are primarily $S$-wave in character, but each possesses $P$-, $D$- and $F$-wave components: the $P$-wave is large in $\{u,d,s\}$-quark baryons. Turning to the first positive-parity excitations, each has a large $D$-wave component, which grows with increasing current-quark mass, but they otherwise exhibit features consistent with radial excitations.
Whilst a complete treatment of the Poincar\'e-covariant Faddeev equation is now possible using modern hardware, it remains a complex task. Hence, for the development of insights in a wide array of baryon problems, it remains common to treat the equation in a quark-diquark approximation, where the diquark correlations are nonpointlike and dynamical \cite{Segovia:2015ufa}. This amounts to a simplified treatment of the scattering problem in the two-body subchannels (as explained, \emph{e.g}.\ in Ref.\,\cite[Sec.\,II.A.2]{Hecht:2002ej}), which is founded on an observation that the same interaction which describes colour-singlet mesons also generates diquark correlations in the colour-antitriplet $(\bar 3)$ channel \cite{Cahill:1987qr, Maris:2002yu, Bi:2015ifa}. Whilst the diquarks do not survive as asymptotic states, \emph{viz}.\ they are absent from the strong interaction spectrum \cite{Bender:1996bb, Bhagwat:2004hn}, the attraction between the quarks in the $\bar 3$ channel sustains a system in which two quarks are always correlated as a colour-$\bar 3$ pseudoparticle, and binding within the baryon is effected by the iterated exchange of roles between the bystander and diquark-participant quarks. Contemporary studies indicate that diquark correlations are an important component of all baryons; and owing to the dynamical character of the diquarks, it is typically the lightest allowed diquark correlation which defines the most important component of a baryon's Faddeev amplitude \cite{Yin:2019bxe}.
\section{$\bf \Delta$ Baryons}
The quark+diquark picture of baryons has yielded a unified picture of the four lightest $(I,J^P) = (1/2,1/2^+)$ baryon isospin doublets \cite{Chen:2017pse}. Turning to $(3/2,3/2^+)$ states with no net strangeness, the $\Delta^{+,0}$ baryons are conventionally viewed as the lightest isospin- and spin-flip excitations of the proton and neutron (nucleons, $N$), respectively. Hence, since nucleons are the basic elements of all nuclei, developing a detailed understanding of the $\Delta$-baryons is of fundamental importance. Without this, hadron physics remains at a level akin to atomic physics based only on knowledge of the hydrogen atom's ground state.
In this connection, given the simplicity of electromagnetic probes, it is worth using virtual photons to chart $\Delta$-resonance structure; and using intense, energetic electron-beams at the JLab, $\gamma^\ast p \to \Delta^+$ data are now available for $0 \leq Q^2 \lesssim 8\,$GeV$^2$ \cite{Aznauryan:2011ub, Aznauryan:2011qj, Aznauryan:2012ba}.
These data have stimulated much theoretical analysis and speculation about, \emph{inter alia}:
the role that resonance electroproduction experiments can play in exposing nonperturbative aspects of QCD, such as the nature of confinement and DCSB;
the relevance of perturbative QCD to processes involving moderate momentum transfers;
and hadron shape deformation.
Deformation can be characterised by a given hadron's spectroscopic quadrupole moment:
\begin{equation}
\label{deform}
Q = \frac{3 J_z^2 - J(J+1)}{(J+1)(2 J+3)} Q_0 ,
\end{equation}
where $Q_0$, the state's intrinsic deformation, is a measure of the $D$-wave component of its rest-frame-projected Faddeev wave function. The difference between $Q$ and $Q_0$ expressed in Eq.\,\eqref{deform} represents the averaging of the nonspherical charge distribution due to its rotational motion as seen in the laboratory/rest-frame. Plainly, $Q=0$ when $J=1/2$, \emph{i.e}.\ there is no measurable quadrupole moment for an isolated $J=1/2$ bound-state.
It is worth remarking here that just above the $\Delta$-baryon level lies the nucleon's first positive-parity excitation, \emph{viz}.\ the Roper resonance, $N(1440)\,1/2^+$. Discovered in 1963 \cite{Roper:1964zza}, its characteristics were long the source of puzzlement because, \emph{e.g}.\ constituent-quark potential models typically (and erroneously) produce a spectrum in which this excitation lies above the first negative-parity state $N(1535)\,1/2^-$ \cite{Capstick:2000qj, Crede:2013sze, Giannini:2015zia}. This has now changed following: acquisition and analysis of high-precision proton-target exclusive electroproduction data with single- and double-pion final states, on a large energy domain and with momentum-transfers out to $Q^2\approx 5\,$GeV$^2$; development of a dynamical reaction theory capable of simultaneously describing all partial waves extracted from available, reliable data; and formulation and application of a Poincar\'e covariant approach to the continuum bound-state problem in relativistic quantum field theory. Today, it is widely accepted that the Roper is, at heart, the first radial excitation of the nucleon, consisting of a well-defined dressed-quark core that is augmented by a meson cloud, which both reduces the Roper's core mass by approximately 20\% and contributes materially to the electroproduction form factors at low-$Q^2$ \cite{Burkert:2017djo, Chen:2018nsg}.
Notably, a similar pattern of energy levels is found in the spectrum of $\Delta$-baryons, \emph{viz}.\ contradicting quark-model predictions,
the first positive-parity excitation, $\Delta(1600)\,3/2^+$, lies below the negative parity $\Delta(1700)\,3/2^-$, with the splitting being roughly the same as that in the nucleon sector.
This being so and given the Roper-resonance example, it is likely that elucidating the nature of the $\Delta(1600)$-baryon will require both (\emph{i}) data on its electroproduction form factors which extends well beyond the meson-cloud domain and (\emph{ii}) predictions for these form factors to compare with that data.
The data exist \cite{Trivedi:2018rgo, Burkert:2019opk}; and can be analysed with this aim understood. This is especially important now that theoretical predictions are available \cite{Lu:2019bjs}.
The Poincar\'e-covariant wave functions for the $\Delta(1232)$ and $\Delta(1600)$ are depicted in Ref.\,\cite[Fig.\,2]{Lu:2019bjs}, from which it is plain that, in their rest-frames, both the ground-state and first positive-parity excitation are primarily $S$-wave in character. Notably, the $\Delta(1232)$ mass is almost insensitive to non-$S$-wave components; and its quadrupole moment is large in magnitude and negative, indicating oblate intrinsic deformation.
Turning to the $\Delta(1600)$ and considering its mass, $P$-wave components generate noticeable repulsion, $D$-waves produce some attraction, and $F$-waves can be neglected. Interestingly, in this quark+diquark approach, too, when comparing the ground state with the first positive-parity excitation, some of the $S$-wave strength is shifted into $P$- and $D$-wave contributions just as in the direct Faddeev equation solution described in Sec.\,\ref{FEBaryons}. The intrinsic quadrupole moment of the $\Delta(1600)$ is just 45\% of that of the $\Delta(1232)$. By this measure, the $\Delta(1600)$ is still oblate, but possesses less quadrupole deformation than the $\Delta(1232)$.
\section{$\mathbf{\gamma^\ast + p \to \Delta(1232)}$, $\mathbf{\Delta(1600)}$ }
Regarding the $\gamma^\ast p \to \Delta^+(1232)$ transition, on $Q^2 \gtrsim 0.5 m_p^2$, \emph{i.e}.\ outside the meson cloud domain for this process, the magnetic dipole and Coulomb quadrupole form factors reported in Ref.\,\cite{Lu:2019bjs} agree well with available data. Consistent with the data, too, the electric quadrupole form factor is very small in magnitude; hence, it is particularly sensitive to the diquark content and quark-diquark angular-momentum structure of the baryons involved, and also to meson-baryon final-state-interactions (MB\,FSIs) on a larger domain than the other form factors. These remarks are supported by the following observations: the role played by higher partial waves in the wave functions increases with momentum transfer (something also observed in meson form factors), here generating destructive interference; agreement with data on $G_M^\ast$ is impossible without the higher partial waves; and the effect of such components is very large in $G_E^\ast$, with the complete result for $G_E^\ast$ exhibiting a zero at $Q^2 \approx 4 m_p^2$, which is absent in the S-wave-only result(s).
\begin{figure}[t]
\centerline{%
\includegraphics[clip, width=0.42\textwidth]{F4A.pdf}}
\vspace*{-8ex}
\centerline{%
\includegraphics[clip, width=0.42\textwidth]{F4B.pdf}}
\vspace*{-8ex}
\centerline{%
\includegraphics[clip, width=0.42\textwidth]{F4C.pdf}}
\caption{\label{D1600TFFs}
\emph{Top panel} -- Magnetic dipole $\gamma^\ast p\to \Delta^+(1600)$ transition form factor;
\emph{middle} -- electric quadrupole; and
\emph{bottom}: Coulomb quadrupole.
Data from Ref.\,\cite{Tanabashi:2018oca}; and the conventions of Ref.\,\cite{Jones:1972ky} are employed.
All panels:
solid [black] curve, complete result;
long-dashed [blue] curve, result obtained when $\Delta(1600)$ is reduced to $S$-wave state;
dashed [blue] curve, both the proton and $\Delta(1600)$ are reduced to $S$-wave states;
dotted [green] curve, obtained by enhancing proton's axial-vector diquark content;
shaded [grey] band, light-front relativistic Hamiltonian dynamics (LFRHD) \cite{Capstick:1994ne};
dot-dashed [brown] curve, light-front relativistic quark model (LFRQM) with unmixed wave functions \cite{Aznauryan:2015zta};
and dot-dot-dashed [orange] curve, LFRQM with configuration mixing \cite{Aznauryan:2016wwmO}.
}
\end{figure}
Drawn from Ref.\,\cite{Lu:2019bjs}, predictions for the $\gamma^\ast p\to \Delta^+(1600)$ transition form factors are displayed in Fig.\,\ref{D1600TFFs}. Empirical results are here only available at the real-photon point: $G_M^\ast(Q^2=0)$, $G_E^\ast(Q^2=0)$. Evidently, the quark model results -- [shaded grey band] \cite{Capstick:1994ne}, dot-dashed (brown) curve \cite{Aznauryan:2015zta} and dot-dot-dashed [orange] curve \cite{Aznauryan:2016wwmO}) -- are very sensitive to the wave functions employed for the initial and final states. Furthermore, inclusion of relativistic effects has a sizeable impact on transitions to positive-parity excited states \cite{Capstick:1994ne}.
The quark+diquark Faddeev equation prediction is the solid [black] curve in each panel of Fig.\,\ref{D1600TFFs}. In this instance, every transition form factor is of unique sign on the domain displayed. Notably, the mismatches with the empirical results for $G_M^\ast(0)$, $G_E^\ast(0)$ are commensurate in relative sizes with those in the $\Delta(1232)$ case, suggesting that MB\,FSIs are of similar importance in both channels.
Axial-vector diquark contributions interfere constructively with MB\,FSIs \cite{Hecht:2002ej}; hence, regarding form factors, one can mimic some meson-cloud effects by modifying the axial-vector diquark content of the participating hadrons. Accordingly, to illustrate the potential impact of MB\,FSIs, the transition form factors were also computed using an enhanced axial-vector diquark content in the proton. This was achieved by setting $m_{1^+} = m_{0^+} = 0.85\,$GeV, values with which the proton's mass is practically unchanged. The procedure produced the dotted [green] curves in Fig.\,\ref{D1600TFFs}; better aligning the $x\simeq 0$ results with experiment and suggesting thereby that MB\,FSIs will improve the Faddeev equation predictions.
The short-dashed [blue] curve in Fig.\,\ref{D1600TFFs} is the result obtained when only rest-frame $S$-wave components are retained in the wave functions of the proton and $\Delta(1600)$; and the long-dashed [blue] curve is that computed with a complete proton wave function and a $S$-wave-projected $\Delta(1600)$. Once again, the higher partial-waves have a visible impact on all form factors, with $G_E^\ast$ being most affected: the higher waves produce a change in sign.
This reemphasises one of the conclusions from the quark model studies, \emph{viz}.\ data on the $\gamma^\ast p\to \Delta^+(1600)$ transition form factors will be sensitive to the structure of the $\Delta^+(1600)$.
It is interesting to observe that whilst all $\gamma^\ast p \to \Delta^+(1232)$ transition form factors are larger in magnitude than those for $\gamma^\ast p \to \Delta^+(1600)$ in some neighbourhood of $Q^2=0$, this ordering is reversed on $Q^2 \gtrsim 2 m_p^2$. One can thus argue that the $\gamma^\ast p \to \Delta^+(1600)$ transition is more localised in configuration space.
It is also notable that $R_{SM} \sim G_C^\ast/G_M^\ast$ is qualitatively similar for both $\gamma^\ast p \to \Delta^+$ transitions; but $R_{EM} \sim G_E^\ast/G_M^\ast $ is markedly different, being of opposite sign on $Q^2 \lesssim 4 m_p^2$ and uniformly larger in magnitude for the $\Delta(1600)$. These observations again highlight the sensitivity of the electric quadrupole form factor to the degree of intrinsic deformation of the $\Delta$-baryons.
\section{Summary and Outlook}
\subsection{Particulars}
Poincar\'e-covariance entails that no realistic hadron wave function can be purely $S$-wave in character. Hence, all $J\neq 0$ hadrons possess intrinsic deformation; and for baryons, such deformation can be exposed via measurements of nucleon-to-resonance transition form factors. Of course, in relativistic quantum field theory, the magnitude and mixing between orbital angular momentum and spin is frame and renormalisation-scale dependent. This is also true of the interpretations of these quantities because the natural degrees of freedom evolve with scale. Consequently, even though Poincar\'e-invariant form factors are the same for all observers, their interpretation via subsystem degrees-of-freedom can change with frame and scale. All these remarks extend to deformation. A close connection with quantum mechanics notions of probability distributions can be made via light-front projections of Poincar\'e-covariant wave functions; but unless the studied quantity is protected by symmetry, even conclusions drawn from such projections are scale dependent.
\subsection{Broader View}
$N^\ast$ physics has made great strides in this millennium.
For instance, continuum and lattice QCD approaches now appear to qualitatively confirm the constituent quark model spectrum; at least, they produce a spectrum of baryons that possesses a richness which cannot be explained by a two-body model; and the associated ``missing resonance'' problem is being addressed, with empirical gaps being filled.
Novel images of $N^\ast$ structure have been drawn through synergistic efforts between experiment and theory, with the availability of electroproduction data on a large $Q^2$ domain playing an especially important role. A signature example here is the emerging understanding of the Roper resonance as the first radial excitation of the nucleon's dressed-quark core augmented by a meson cloud whose presence has a material impact on its low-$Q^2$ properties. With this picture, the Roper becomes ``normal'', being much like the $\Delta$-baryon in many ways.
Such experiment-theory interplay is proving critical in validating modern QCD theory predictions for the momentum-dependence of the running-coupling, -masses, and other basic Schwinger functions that are computable using continuum and lattice methods.
The next decade will see completion of the analysis of existing electroproduction data available on $Q^2\in [0,5]\,$GeV$^2$ and the gathering of new results out to $Q^2 \approx 12\,$GeV$^2$. The new large-$Q^2$ data will probe deeply into the quark-core domain of $N^\ast$ structure, enabling empirical flavour separation of transition form factor contributions and, thereby, stringent tests of theory predictions regarding the role of diquark correlations within baryons. Furthermore, completed and new analyses of data on electroexcitation amplitudes of baryon parity partners will enable validation of predictions concerning the role of DCSB (emergent mass) in the spectrum and structure of nucleon resonances and help resolve questions relating to chiral symmetry restoration in highly excited systems.
Studies of these types, both ongoing and anticipated, will reveal facts about the running-coupling and -masses in QCD at length-scales associated with more-than 98\% of the observable mass of a typical hadron. They are therefore crucial in developing answers to some of the Standard Model's most fundamental questions; namely, what is mass, what is confinement, and how are they related?
\section*{Acknowledgements}
I am grateful to R.\,W.~Gothe for convening the session in which this perspective was presented.
The material contained herein is the product of collaborations with many people, to all of whom I am greatly indebted.
Related discussions can be found in contributions to this volume from, \emph{e.g}.\
V.~Burkert,
C.~Chen,
G.~Eichmann,
C.\,S.~Fischer,
R.\,W.~Gothe,
V.~Mokeev
and
J.~Rodr{\'{\i}}guez-Quintero.
Work partly supported by
Jiangsu Province \emph{Hundred Talents Plan for Professionals}.
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train/arxiv
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BkiUbAE241xg-EsBtVTO
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\section{Introduction}
Heavy electron materials exhibit a number of interesting correlated electron phenomena, including unusual broken symmetry ground states, quantum criticality and non-Fermi liquid behavior, which arise from the interactions between a lattice of nearly-localized 4f electrons and itinerant conduction electron states.\cite{doniach,hewson1997kondo} When the 4f states are weakly hybridized with the itinerant states, the materials tend to exhibit long-range antiferromagnetism mediated by RKKY interactions; in the opposite limit the long-range order disappears, the resulting itinerant quasiparticles have enhanced effective masses, and the system typically is unstable towards unconventional superconductivity.\cite{zachreview,ColemanHFdeath,ColemanQCreview} {The emergence of a heavy-fermion fluid in close proximity to an antiferromagnetic instability of localized moments remains an active area of experimental and theoretical research.} Several Ce-based compounds happen to exhibit a level of hybridization that places them close to the quantum critical (QC) boundary between long-range antiferromagnetism and superconductivity. As a result small perturbations induced by doping or pressure can result in dramatic changes to the ground state properties.\cite{tuson,ParkDropletsNature2013} These compounds offer an ideal testing ground to investigate the interplay between the hybridization and the emergent states of the strongly correlated system.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{xtal2.eps}
\includegraphics[width=\linewidth]{Ir115spectra44degV3.eps}
\caption{(a) Unit cell indicating the three indium sites and the field orientation. (b) Projection of the unit cell in the $ab$ plane. (c)
A representative frequency swept spectrum of CeIrIn$_5$ at 11.7T for $\theta=44^{\circ}$ and temperatures from 6K to 80K; the magnitude is normalized by temperature. The temperature-dependent Knight shifts for the In(1), In(2A) and In(2B) sites are scaled over the raw spectral data. \label{spectra}}
\end{figure}
CeIrIn$_5$ is an excellent example of a system close to a QC boundary; while it is superconducting below 0.4K, the normal state exhibits antiferromagnetic fluctuations and non-Fermi liquid behavior.\cite{ceirindiscovery, ZhengIr115PRL} {Thus this compound can provide vital information about the emergence of the coherent heavy-fermion fluid near a QC boundary.} Dynamical mean field theory (DMFT) calculations indicate that CeIrIn$_5$ undergoes a crossover from localized to itinerant electron behavior with decreasing temperature, accompanied by changes to the Fermi surface.\cite{HauleCeIrIn5,Choi2011} Experimental evidence is provided by resistivity, specific heat, and nuclear magnetic resonance (NMR) Knight shift measurements which are well-described by a two-fluid picture of heavy fermion behavior.\cite{YangPinesNature}
Recent calculations have shown that this hybridization-driven crossover is strongly anisotropic in this material.\cite{haule2010dynamical} {Here we provide direct experimental evidence for such hybridization anisotropy, which may play a key role in stabilizing the unconventional superconductivity in this family of heavy fermions.}
{The DMFT calculations indicate that} since the local 4f states are anisotropic, the hybridization is dominated by the orbital overlap between the Ce 4f and the out-of-plane In(2) electron orbitals (see Fig. \ref{spectra}). This hybridization should be manifest in the spin susceptibility $\chi_{c\!f}$, describing the correlations between the itinerant and local moment electron spins. This quantity can be directly probed via Knight shift experiments.\cite{Curro2004,ShirerPNAS2012} We have conducted detailed angular-dependent studies of the In(1) and In(2) Knight shifts, and find that $\chi_{c\!f}$ indeed depends on the orientation of the applied magnetic field with respect to the crystal axes. The temperature dependence of this correlation function is determined by the Kondo lattice coherence temperature, $T^*$, which we find to be largest along the Ce-In(2) bond axis.
\section{Knight Shift Measurements}
High quality single crystals of CeIrIn$_5$ were synthesized using the standard flux method described in Ref. \onlinecite{Moshopoulou200125}. Characterization with powder X-ray diffraction showed the samples were pure with a small amount of In flux.\cite{Shockley2011} A large single crystal with dimensions 3mm $\times$ 3mm $\times$ 1mm was chosen for the NMR studies. NMR measurements were performed in an Oxford high-homogeneity NMR magnet at a fixed field of 11.7T. All spectra were obtained using a standard Hahn echo pulse sequence.\cite{slichter1990principles} The orientation of the sample was controlled by a single-axis goniometer, and the sample was mounted such that the applied field was directed at an angle $\theta$ from $(001)$, in the plane spanned by (100) and (001), as shown in Fig. \ref{spectra}(a). For each angle, a full spectrum including several different satellite transitions of the $^{115}$In ($I=9/2$) was obtained using an automated tuning system integrated with the NMR spectrometer. The quadrupolar nature of this isotope enabled us to extract the orientation of the field, and hence the Knight shift, as described in detail in the Appendix. There are four In(2) sites per unit cell, and when $\theta>0$ these four sites split into two inequivalent sites depending on whether the field is parallel or perpendicular to the face of the unit cell (see Fig. \ref{spectra}(b)). We refer to these two In(2) sites as In(2A) and In(2B). Characteristic spectra of the In(1), In(2A) and In(2B) sites are shown in Fig. \ref{spectra}(c) for $\theta = 44^{\circ}$ at several different temperatures. Each of the three sites clearly exhibits different temperature dependent behavior. Detailed spectra were also measured at various rotation angles in order to observe any anisotropy in the temperature dependence.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{KvsT.eps}
\caption{(a) Knight shift at $\theta = 44^{\circ}$ (solid points) and bulk susceptibility $\chi(T) = \chi_c\cos^2(\theta) + \chi_a\sin^2(\theta)$ versus temperature (solid line). (b) $K(2A)$ and $K(2B)$ versus $K(1)$, and $K(2A)$ versus $K(2B)$ with temperature implicit. Solid lines are fits to the high temperature ($T>T^*$). $T^*$ is the temperature below which the linear relationship between these quantities breaks down. \label{KvsK}}
\end{figure}
For a spin 1/2 nucleus, the resonance frequency is given by $\omega = \gamma H_0 (1+ K(\theta,\phi))$, where $\gamma$ is the gyromagnetic ratio, $H_0$ is the magnetic field, and $K(\theta,\phi) = \mathbf{H}_0\cdot \mathbb{K}\cdot\mathbf{H_0}/H_0^2$. Here $\mathbb{K}$ is the Knight shift tensor, with principal axes lying along the unit cell directions. In general, the Knight shift arises because of the hyperfine coupling between the nuclear and electron spins of the material, which gives rise to an effective hyperfine field at the nuclear site in addition to the external field, thus shifting the resonance frequency. Hyperfine couplings can arise from on-site Fermi contact interactions, as well as via transferred couplings to electron spins located on neighboring atoms. The exact values of these couplings depend on details of the electronic structure of the material, are different for each site, and are generally difficult to compute. However, it is useful to consider an effective hyperfine interaction that is appropriate for heavy fermion materials: $\mathcal{H}_{hyp} = \mathbf{\hat{I}}\cdot (\mathbb{A}\cdot \mathbf{S}_c + \mathbb{B}\cdot \mathbf{S}_f)$, where $\mathbb{A}$ and $\mathbb{B}$ are temperature-independent hyperfine coupling tensors to the conduction electron and local moment spins, $\mathbf{S}_c$ and $\mathbf{S}_f$.\cite{curro2004scaling} $\mathbf{\hat{I}}$ is the nuclear spin on the ligand site, in this case either the In(1), the In(2A), or the In(2B). In the paramagnetic state, the spins are polarized by the external field, and the Knight shift is given by $\mathbb{K} = \mathbb{K}_0 + \mathbb{A}\cdot{{\chi}}_{cc} + (\mathbb{A}+\mathbb{B})\cdot\chi_{c\!f} + \mathbb{B}\cdot\chi_{f\!f}$, where $\chi_{ij} = \langle S_{i}S_{j}\rangle$ are the components of the total susceptibility $\chi = \chi_{cc} + 2\chi_{cf} + \chi_{f\!f}$, and $\mathbb{K}_0$ is the temperature-independent orbital shift tensor. For sufficiently large temperatures $\chi_{f\!f}$ is the dominant contribution thus $\mathbb{K} \approx \mathbb{K}_0 + \mathbb{B}\cdot\chi$. As a result, $K_{\alpha}$ is linearly proportional to $\chi_{\alpha}$, where $\alpha = (a,b,c)$ are the principal directions of the tensor. Furthermore, since the shift of each site is proportional to $\chi$, each shift is also proportional to the shifts of the other sites, as shown in detail in the Appendix. This linear dependence is evident in Figs. \ref{KvsK} for $T > T^*$.
\begin{figure}[h]
\includegraphics[width=\linewidth]{CeIrIn5_KvsKplots_allangles.eps}
\caption{Clockwise from the top right pane: $K(2)$ ($K(2A) = K(2B) = K(2)$ for this orientation) vs. $K(1)$ for $\theta=0^{\circ}$ (purple inverse triangles), $K(2B)$ vs. $K(1)$ for $\theta=28^{\circ}$ (green diamonds), $K(2A)$ and $K(2B)$ vs. $K(1)$ and $K(2A)$ vs. $K(2B)$ for $\theta=44^{\circ}$ (pink and dark red and red circles, respectively), and $K(2A)$ and $K(2B)$ vs. $K(1)$ and $K(2A)$ vs. $K(2B)$ for $\theta=82^{\circ}$ (light blue, dark blue and blue squares, respectively). Solid lines are fits to the high temperature portion as described in the text. $T^*$ is indicated by the grey arrows. \label{KvsKv2}}
\end{figure}
Below the coherence temperature, $T^*$, the conduction and local moment spin degrees of freedom become entangled, and $\chi_{c\!f}$ grows in magnitude relative to $\chi_{\!f\!f}$. As a result, $K_{\alpha}$ is no longer proportional to $\chi_{\alpha}$, as seen in Fig. \ref{KvsK}. $T^*$ is a material-dependent crossover temperature that depends on the hybridization and intersite couplings between the $S_f$ spins in the Kondo lattice.\cite{YangPinesNature,YangPinesPNAS2012,HauleCeIrIn5,jiang2014universal} $T^*$ can be measured experimentally via independent measurements of $K_{\alpha}$ and $\chi_{\alpha}$: when $K_{\alpha}$ is plotted versus $\chi_{\alpha}$ with temperature as an implicit variable, the linear relationship breaks down at $T^*$, as observed in Fig. \ref{KvsK} at $\theta = 44^{\circ}$. Several other pairs of shifts and angles are shown in Fig. \ref{KvsKv2}, and in each case there is a clear break in this linear relationship at low temperatures.
\begin{figure}
\centering
\includegraphics[width=0.98\linewidth]{CeIrIn5_Kcf_allanglesV2.eps}
\caption{$K_{cf}/K_{cf}^0$ versus $T$ for various angles (colors and symbols are defined as in Fig. \ref{KvsK}). The solid lines are fits as described in the text. INSET: $K_{cf}$ versus $T$. \label{Kcf}}
\end{figure}
In order to discern the influence of hybridization anisotropy, it is important to measure $T^*$ as a function of angle. Our previous studies of the In(1) site in CeIrIn$_5$ indicated that $T^* \sim 40$ K, and did not appear to vary significantly with field orientation or magnitude.\cite{ShirerPNAS2012,CeIrIn5HighFieldNMR} However, the precision of the $T^*$ measurement is limited for the In(1) site because the coupling constants $A_a = B_a$ in the plane. Therefore the magnitude of the Knight shift anomaly gradually decreases with angle and vanishes for $\mathbf{H}_0 \parallel (100)$. This problem can be circumvented by measuring the Knight shifts of both of the In(2) sites and the In(1) site. This approach is superior because all of the Knight shift measurements can be acquired simultaneously at the same crystalline orientation without the need for separate measurements of $\chi$.\cite{CeIrIn5HighFieldNMR} The behavior below $T^*$ is governed by the temperature dependence of the correlation function $\chi_{c\!f}(T)$. As the the conduction electron and local moments become entangled, this quantity grows in magnitude and can be extracted from the Knight shift below $T^*$. To do so, we fit the high temperature data ($T > T^*$) for each pair $(K_1, K_2)$ of Knight shifts to $K_1 = a + b K_2$, and then plot $K_{c\!f}(\theta, T) = K_1(T,\theta) - a-b K_2(T,\theta)$ versus temperature in the inset of Fig. \ref{Kcf}. As shown in the Appendix, this quantity is proportional to $\chi_{c\!f}(\theta,T)$ and becomes non-zero below $T^*$. The constants $a$ and $b$ depend on the ratios of hyperfine couplings of the various pairs of sites and are unimportant for our analysis.\cite{CeIrIn5HighFieldNMR} We have confirmed that these constants are consistent for three different data sets.
\section{Anisotropy}
As seen in Fig. \ref{Kcf}, $K_{c\!f}$ vanishes above $T^*$, but grows in magnitude with decreasing temperature below this temperature. This data clearly indicate that the onset temperature, $T^*$, depends on the angle $\theta$. {This angular variation is model independent, and can be discerned both in the plots of $K_i$ versus $K_j$ in Figs. \ref{KvsK} and \ref{KvsKv2}. For concreteness }we fit the temperature dependence of $K_{c\!f}$ to the two-fluid expression\cite{YangDavidPRL}:
\begin{equation}
K_{c\!f}(T) = K_{c\!f}^0(1-T/T^*)^{3/2}[1+\ln(T^*/T)],
\label{eqn:Kcf}
\end{equation}
and plot $K_{c\!f}^0$ and $T^*$ versus angle $\theta$ in Figs. \ref{TstarVSangle}(a) and \ref{TstarVSangle}(b). $K_{c\!f}^0$ is proportional to a complex ratio of the hyperfine couplings {and anisotropic g-factors of the material}, and the angular dependence of this quantity seen in Fig. \ref{TstarVSangle}(a) reflects the anisotropies of these couplings. {The main panel of Fig. \ref{Kcf} shows $K_{cf}(T)$ normalized by $K_{c\!f}^0$, which removes any anisotropies introduced by the hyperfine couplings and g-factors. The onset temperature of the anomaly, $T^*$, varies with angle. Here $T^*$ is unrelated to the hyperfine couplings and reflects intrinsic properties of the electronic degrees of freedom of the Kondo lattice. As seen in Figs. \ref{Kcf} and \ref{TstarVSangle}(b), as the field angle rotates from the $(001)$ direction, $T^*$ increases from 40K to nearly 50K at $44^{\circ}$ and then reaches a minimum of 26 K for the $(100)$ direction. In order to parameterise this anisotropy, we fit this angular dependence to the form $T^*(\theta) = T^*_0 + T^*_2\cos(2\theta) + T^*_4\cos(4\theta)$, which qualitatively reproduces the hybridization function calculated in Ref. \onlinecite{haule2010dynamical}. We find $T^*_0 = 42(2)$ K, $T^*_2= -7(2)$ K and $T^*_4 = 7(2)$ K, shown in Fig. \ref{TstarVSangle}(c). These results reveal that the heavy electron fluid, which emerges from the collective hybridization of the lattice of 4f sites with the conduction electrons, is anisotropic in this material. This result suggests that the hybridization is not isotropic and has four-fold symmetry.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{TstarVSangleV6.eps}
\caption{(Color online) (a) $K_{cf}^0$ and (b) $T^*$ versus angle as determined from the fits shown in Fig. \ref{Kcf}, as determined by plotting $K(2B)$ vs. $K(1)$ (pink), $K(2A)$ vs. $K(1)$ (blue) and $K(2A)$ vs. $K(2B)$ (green). Dashed lines are guides to the eye, and the solid line in (b) is a fit as described in the text. (c) $T^*(\theta)$ shown as a polar plot, relative to the $(001)$ (vertical) and $(001)$ (horizontal) directions. The dotted red lines indicate the Ce-In(2) directions. (d) $\chi_{c\!f}(T,\theta)$ and $T^*/J_z$ (solid line) versus $\theta$ for the two-spin model discussed in the text. \label{TstarVSangle}}
\end{figure}
\subsection{Hybridization}
A recent analysis of data in a broad range of heavy fermion materials indicated that $T^*$ is proportional to the intersite RKKY exchange interaction, which itself is proportional to $J^2$, where the Kondo coupling $J$ is a function of the hybridization.\cite{YangPinesNature} Therefore, anisotropy in the hybridization should be reflected in the experimentally measured quantity, $T^*$. In order to discern how an anisotropic hybridization can give rise to anisotropy in the susceptibility $\chi_{cf}$, it is instructive to consider a generalization of the two-spin model introduced in Ref. \onlinecite{ShirerPNAS2012}. We consider an anisotropic coupling between two free spins, $\mathbf{S}_c$ and $\mathbf{S}_f$: $\mathcal{H} = J_{\perp}({S}_c^x{S}_f^x+{S}_c^y{S}_f^y)+J_z{S}_c^z{S}_f^z$, where $J_{z,\perp}$ is the coupling between the spins derived from the anisotropic hybridization parallel (perpendicular) to the z-axis. This model is the single-site limit of the periodic Anderson model in the limit of large on-site repulsion, $U$, relative to the hybridization, $V$, such that $J_\alpha = 4V_{\alpha}^2/U$.\cite{jiang2014universal} In this case, the susceptibilities $\chi_{cc}$, $\chi_{cf}$ and $\chi_{ff}$ are exactly solvable. For the isotropic case $J_{\perp}=J_z$, the $\chi_{ij}$ are all isotropic and scale as $T/T^*$, where $T^* = J_z/k_B$. When $J_{\perp}\neq J_z$, these susceptibilities become anisotropic tensors, such that the susceptibility becomes angular dependent: $\chi_{cf}(T,\theta) = \chi_{cf}^z(T)\cos^2(\theta) + \chi_{cf}^{\perp}(T)\sin^2(\theta)$, shown in Fig. \ref{TstarVSangle}(d) for the case $J_{\perp}=0.2 J_z$. We fit this quantity to Eq. \ref{eqn:Kcf} for several values of $\theta$ and the solid line in Fig. \ref{TstarVSangle}(d) shows the fitted values of $T^*(\theta)$. Clearly $T^*$ is anisotropic, although this model not sophisticated enough to capture the four-fold variation observed in Fig. \ref{TstarVSangle}(b). A model including multiple sites would represent the full lattice better and be more likely to resemble the experimental measures.
\subsection{Crystalline Electric Field}
\begin{figure}[h]
\includegraphics[width=\linewidth]{CEF_KandChiV2.eps}
\caption{Susceptibility ($\chi^{CEF}$, solid lines) and Knight shifts ($K^{CEF}$, dotted lines) versus temperature calculated in the CEF model for various field orientations, as described in the text. INSET: $K^{CEF}$ versus $\chi^{CEF}$ (solid lines) for the same orientations. Dashed lines are fits to the high temperature data points. \label{fig:CEFKandChi}}
\end{figure}
An alternative interpretation of Knight shift anomalies is that the hyperfine coupling constants depend on the particular crystalline electric field (CEF) doublets.\cite{KnightShiftAnomalyCEF,Curro2001} The strong spin-orbit coupling combined with CEF interactions at the Ce ions give rise to a temperature-dependent anisotropic $g$-factor. The Ce$^{3+}$ ions in this material experience a CEF that splits the $J=5/2$ ground state multiplet into three doublets, with excited states energies $\Delta_1 = 6.7$ meV and $\Delta_2=29$ meV above the ground state \cite{CEF115study,WillersCEF115s}. In order to explore the possible role of the CEF in the anisotropy we observe in $K_{cf}$, we have computed $K_{cf}^{CEF}$ and $\chi^{CEF}$ as a function of field orientation using the hyperfine coupling model discussed in Refs. \onlinecite{CeIrIn5HighFieldNMR,KnightShiftAnomalyCEF,Curro2001}. In this scenario, the hyperfine coupling between the In(1) site and the Ce spin depends on the particular CEF doublet; thus when the temperature $T \lesssim \Delta_1/3 k_B$, the thermal population of the excited states is significantly reduced and the effective hyperfine field changes. As a result, the Knight shift differs from the susceptibility below the anomaly temperature $T^*_{CEF} \sim \Delta_1/k_B$. Here we computed $\chi_{c,ab}^{CEF}$ and $K_{c,ab}^{CEF}$ using the same CEF parameters and hyperfine couplings as in Ref. \onlinecite{CeIrIn5HighFieldNMR} in a field of 11.7 T. These quantities are shown in Fig. \ref{fig:CEFKandChi}. Note that these calculations do not accurately capture the behavior of the real material because this model neglects the role of hybridization of the Ce 4f states. Nevertheless, there is a clear anisotropy in the magnitudes of both $K^{CEF}$ and $\chi^{CEF}$, which reflects the anisotropy of the $g$-factor of the Ce. We have also assumed isotropic hyperfine couplings in this calculation, but relaxing this assumption would simply modify the relative scale factors of the Knight shifts shown in Fig. \ref{fig:CEFKandChi}.
\begin{figure}[h]
\includegraphics[width=\linewidth]{CEF_KcfV4.eps}
\caption{(a) $K_{cf}^{CEF}$ versus $T$ for several different field orientations (see legend in Fig. \ref{fig:CEFKandChi}) using the fits to the high-temperature $K^{CEF}$ versus $\chi^{CEF}$ relationship. (b) $T^*_{CEF}$ extracted by fitting the theoretical $K_{cf}^{CEF}$ results to Eq. 1 in the main text as a function of angle. (c) $K_{cf}^{CEF}/K_{cf}^0$ versus $T$ for various angles. The dotted line is a fit to Eq. 1 in the main text. \label{fig:CEFKcf}}
\end{figure}
We then fit the high temperature portion ($T>100$ K) of $K^{CEF}(\theta)$ versus $\chi^{CEF}(\theta)$ for various values of $\theta$, and plotted $K_{cf}^{CEF}(\theta) = K^{CEF}(\theta) - a - b\chi^{CEF}(\theta)$, shown in Fig. \ref{fig:CEFKcf}(a). $\chi^{CEF}(\theta)$ grows in magnitude below $T_{CEF}^* \approx 55 K \sim 0.7 \Delta_1$, and varies strongly with $\theta$. For concreteness, we fit
$K_{cf}^{CEF}(\theta)$ to Eq. \ref{eqn:Kcf} to extract $T^*_{CEF}$, shown in Fig. \ref{fig:CEFKcf}(b). However, the fits, shown for $\theta = 0$ and $90^{\circ}$ as dotted lines in Fig. \ref{fig:CEFKcf}(a), are poor and do not capture well the detailed temperature dependence of the theoretical CEF curves. The fitted $T^*$ increases from 64 K to 120 K and then down to 98 K as the field varies from the $c$-axis to the $ab$-plane. Fig. \ref{fig:CEFKcf}(c) shows $K_{cf}^{CEF}(\theta)$ normalized by $K_{cf}^0$ in order to remove the anisotropy of the $g$-factor.
Although the angular dependence of $T^*$ extracted from these fits qualitatively reproduces the experimental observations, the temperature dependence of $K_{cf}^{CEF}$ does not match well with the experimental data shown in Fig. \ref{Kcf}. From this data alone, it is not possible to rule out this model, although previous work indicates that the field dependence of the Knight shift anomaly is not captured by the CEF model.\cite{CeIrIn5HighFieldNMR} It is possible that both CEF and hybridization effects could be playing a role in determining the anisotropic behavior we observe.
\section{Discussion}
An anisotropic coherence temperature has important implications in the context of the two-fluid model, and can explain various observations. For example, in compounds such as CeCoIn$_5$ and CeCu$_2$Si$_2$, evidence suggests that the temperature onset of the Knight shift anomaly differed when the field was parallel or perpendicular to the tetragonal $c-$axis.\cite{curro2004scaling} Our experiments on CeIrIn$_5$ suggest that hybridization anisotropy may also be present in these materials, and call for further measurements. Anisotropy may also explain the variations in $T^*$ measured by different experimental techniques such as NMR, resistivity, specific heat or Hall measurements.\cite{YangPinesNature} If there is a magnetic field present to break the symmetry, then experiments that couple to the susceptibility will reveal this anisotropy. On the other hand, for measurements in the absence of a field, such as resistivity or specific heat, the heavy electron fluid will have an isotropic response with a slightly different temperature scaling. For example, in the isotropic case of the two-spin model discussed above, both the susceptibility and the specific heat scale as $T/T^*$, where $T^* = J_z/k_B =J_{\perp}/k_B$. However, when $J_{\perp}\neq J_z$ the susceptibility scales as $T/T^*(\theta)$ and the specific heat scales as $T/T_C^*$, where $T_C^*$ is isotropic. In this case, both $T^*(\theta)$ and $T_C^*$ are more complicated functions of $J_z$ and $J_{\perp}$.
The anisotropy we observe in the static susceptibility will also be reflected in the dynamic susceptibility of the Kondo liquid, which may play a role in the emergence of superconductivity in this material. Anisotropic spin fluctuations have been shown to give rise to d-wave superconductivity and enhance $T_c$ in for 2D fluctuations.\cite{MonthouxPinesReview} Superconductivity appears to be fairly common in certain heavy fermion families, such as the CeMIn$_5$ series, but not in other Ce-based heavy fermion families. Our observations suggest the reason for the stability of superconductivity in the CeMIn$_5$ series may arise from the particular orbital overlap between the In(2) and the Ce sites in this structure, giving rise to the anisotropy in $T^*$ we observe experimentally.
In summary, we have found evidence that the coherence temperature $T^*$ as measured by the Knight shift is anisotropic in CeIrIn$_5$, reflecting an anisotropic collective hybridization in the Kondo lattice among multiple sites. Our results demonstrate that the NMR Knight shift is a vital new tool to explore and quantify this anisotropy, and suggests that the In(2) sites in this compound play a key role in the development of the heavy electron fluid. Detailed calculations, for example Quantum Monte Carlo simulations, should be carried out in order to test the effects of anisotropic hybridization and discern whether the four-fold symmetry we observe arises from collective hybridization among multiple sites.
\section{Acknowledgements}
We thank Y-F. Yang, D. Pines, M. Jiang and R. Scalettar for enlightening discussions. A.C. Shockley would also like to thank F. Bert, I. Mukhamedshin and P. Mendels. Work at UC Davis was supported by the NSF under Grant No.\ DMR-1005393.
\section{Appendix}
\subsection{Spectrum Analysis}
Each spectrum, covering up to a range of 40 MHz, contains nine In(1) transitions plus up to eighteen In(2) transitions, depending on the orientation of the field with respect to the crystal. The resonance frequencies are determined by the NMR Hamiltonian: $\mathcal{H}_n = \gamma \hbar \mathbf{\hat{I}}\cdot\left(\mathbb{I}+\mathbb{K}\right)\cdot\mathbf{H}_0 +\mathcal{H}_Q$, where $\mathbb{K}$ is the Knight shift tensor, $\gamma$ is the gyromagnetic ratio, $\mathbf{H}_0$ is the external applied field, and $\mathcal{H}_Q$ is the quadrupolar Hamiltonian. The latter is given by:
\begin{equation}
\mathcal{H}_Q = \frac{\hbar}{6}\left[\omega_{zz}(3\hat{I}_z^2 - \hat{I}^2) + (\omega_{xx} - \omega_{yy}) (\hat{I}_x^2 - \hat{I}_y^2)\right],
\end{equation}
where $\left(\omega_{xx},\omega_{yy},\omega_{zz}\right)$ are the eigenvalues of the electric field gradient (EFG) tensor, with eigenvectors directed along the $x,y,z$ directions. For the In(1) site, $\omega_{zz} = 6.07$ MHz along (001), and $\omega_{xx} = \omega_{yy} = -3.04$ MHz along (100) and (010). For the In(2) $\omega_{zz} = 18.17$ MHz along (100), $\omega_{xx} = -13.26$ MHz along (010), and $\omega_{yy} = -4.91$ MHz along (001). $\mathcal{H}_n$ was diagonalized numerically and the resonance frequencies were fit to the spectral data with the shift $K_{\alpha\!\alpha}$, the polar angle $\theta$, and the azimuthal angle, $\phi$, left as variable parameters. The In(1) site has axial symmetry, therefore there are nine equally-spaced satellite transitions whose frequencies only depend on $\theta$. For each orientation of the crystal, we fit the positions of the In(1) peaks in order to extract the angle $\theta$. The azimuthal angle $\phi$ describes the orientation of $\mathbf{H}_0$ relative to (100). By analyzing the satellites of the In(2) we found $\phi = 0 \pm 2^{\circ}$ for each rotation of the goniometer.
\subsection{Relationship between shifts of different sites}
The hyperfine interaction is given by $\mathcal{H}_{\rm hf} = \mathbf{\hat{I}}\cdot[A \mathbf{S}_c + B \mathbf{S}_f$], where $A$ and $B$ are the hyperfine couplings to the itinerant electron spins, $\mathbf{S}_c$, and to the local moment spins, $\mathbf{S}_f$. In this case the Knight shift of each site is given by:
\begin{equation}
K_{i} = K_i^0 + A_i\chi_{cc} + (A_i+B_i)\chi_{c\!f} + B_i\chi_{f\!f}
\label{eqn:Knightshift}
\end{equation}
where $i$ corresponds to In(1), In(2A) or In(2B), $K_i^0$ is a temperature independent orbital term, and the components of the susceptibility are given by $\chi_{\alpha\beta}$. The bulk susceptibility is $\chi = \chi_{cc} + 2\chi_{c\!f} + \chi_{f\!f}$. For $T>T^*$, $\chi_{c\!f}$ and $\chi_{cc}$ can be neglected, therefore $K_i = K_i^0 + B_i\chi$. In this case $K_i$ is also linearly proportional to $K_j$: $K_i = a + b K_j$, where
\begin{eqnarray}
a &=& K_i^0 - (B_i/B_j) K_j^0\\
b &=& B_i/B_j.
\label{eqn:constants}
\end{eqnarray}
These relationships enables us to extract $\chi_{c\!f}$ using just two pairs of Knight shifts without the need for independent measurements of $\chi$.
Using Eqs. \ref{eqn:Knightshift} and \ref{eqn:constants} we find:
\begin{eqnarray}
\nonumber K_{c\!f}(T) &=& K_i(T) - a-b K_j(T) \\
&=& \left(A_i - \frac{B_i}{B_j}A_j\right)(\chi_{c\!f}(T) + \chi_{cc}(T)).
\end{eqnarray}
Since the hyperfine couplings are temperature independent, and $\chi_{cc}$ can be neglected, this quantity is proportional to $\chi_{c\!f}$ and becomes non-zero below $T^*$.
|
train/arxiv
|
BkiUawbxK6nrxrSHdJ9m
| 5 | 1 |
\section{Introduction}
Jigsaw puzzles are a popular form of entertainment, first produced around 1760 by John Spilsbury, a Londonian engraver and mapmaker. Given $n$ different non-overlapping tiles of an image, the objective is to reconstruct the original image, taking advantage of both the shape and chromatic information of each piece. Despite the popularity and vast distribution of jigsaw puzzles, their assembly is not trivial computationally, as this problem was proven to be NP-hard~\cite{journals/aai/Altman89,springerlink:10.1007/s00373-007-0713-4}. Nevertheless, a computational jigsaw solver may have applications in many real-world applications, such as biology~\cite{journals/science/MarandeB07}, chemistry~\cite{oai:xtcat.oclc.org:OCLCNo/ocm45147791}, literature~\cite{conf/ifip/MortonL68}, speech descrambling~\cite{Zhao:2007:PSA:1348258.1348289}, archeology~\cite{journals/tog/BrownTNBDVDRW08,journals/KollerL06}, image editing~\cite{bb43059}, and the recovery of shredded documents or photographs~\cite{cao2010automated,marques2009reconstructing,justino2006reconstructing,conf/icip/DeeverG12}. Regardless, as noted in~\cite{GolMalBer04}, research of the topic may be justified solely due to its intriguing nature.
Most recently proposed solvers employ greedy strategies. Greedy algorithms are known to be problematic when encountering
local optima. Moreover, such solvers rarely offer a backtrack option, i.e., a possibility to cancel a piece assignment which seemed correct at first but then turned to be globally incorrect. Hence, state-of-the-art solvers are very successful on some images, but perform poorly on others. The enormous search space of the problem, containing many local optima, seems most suitable for a genetic algorithm (GA)-based solver. The use of genetic algorithms in the field was first attempted in 2002 by Toyama \textit{et al.}~\cite{bb58987} but its successful performance was limited to 64-piece puzzles, probably due to the inherent difficulty in designing a crossover operator for the problem~\cite{sholomon2013genetic}. More recently, Sholomon \textit{et al.}~\cite{sholomon2013genetic} presented another GA-based solver which can handle up to 22,755-piece puzzles. Nevertheless, their solver can handle only puzzles with (1) known piece orientations, (2) known image dimensions, and (3) pieces of a single image.
In this work we propose a novel GA-based solver, relaxing most previous assumptions. First, we assume no \textit{a priori} knowledge regarding piece location or orientation. Second, we assume that the image dimensions (i.e., the number of row and column tiles) are unknown. Finally, we allow the input piece set to contain pieces from either a single image or from multiple images. In the case of a ``mixed-bag'' puzzle, the solver concurrently solves all puzzles, unmixing their pieces along the process. We set a new state-of-the-art by solving the largest and most complex puzzle ever (i.e., 22,755 pieces, which is twice the size of the current state-of-the-art without making any assumptions) and achieving the highest accuracy ever reported, even with respect to solvers that assume known image dimensions.
\begin{figure*}
\centering
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{pieces_middle_start.jpg}
\caption{}
\label{fig:res_20k__gen_00000000}
\end{subfigure}%
~
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{pieces_middle.jpg}
\caption{}
\label{fig:res_20k__gen_00000001}
\end{subfigure}
~
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{pieces_middle_90.jpg}
\caption{}
\label{fig:res_20k__gen_00000002}
\end{subfigure}
~
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{pieces_middle_180.jpg}
\caption{}
\label{fig:res_20k__gen_00000003}
\end{subfigure}
~
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{pieces_middle_270.jpg}
\caption{}
\label{fig:res_20k__gen_00000004}
\end{subfigure}
\caption{ Example of a relative relation assignment: (a) Pieces 4 and 7, (b)--(e) all possible image configurations of assigning the relative relation (4.b, 7.c). Note that the two pieces are always in the same relative configuration despite the different orientations.}
\label{fig:relassign}
\end{figure*}
\section{Related Work}
Freeman and Garder~\cite{bb47278} were the first to tackle computationally the jigsaw problem, in 1964. Their solver handled up to 9-piece puzzles, using only piece shape. Kosiba \textit{et al.}~\cite{kosiba1994automatic} were the first to facilitate the use of image content. Subsequent research has been confined to color-based square-piece puzzles, instead of the earlier shape-based variants. Cho \textit{et al.}~\cite{conf/cvpr/ChoAF10} presented a probabilistic puzzle solver that can handle up to 432 pieces, given some a priori knowledge of the puzzle (e.g., anchor pieces). Their results were improved a year later by Yang \textit{et al.}~\cite{yang2011particle}, who presented a so-called particle filter-based solver. A major contribution to the field was made recently by Pomeranz \textit{et al.}~\cite{conf/cvpr/PomeranzSB11} who introduced, for the first time, a fully automated jigsaw puzzle solver that can handle square-piece puzzles of up to 3,000 pieces, without a priori knowledge of the image. Their solver treats puzzles with unknown piece location but with known orientation. Gallagher~\cite{conf/cvpr/Gallagher12} was the first to handle puzzles with both unknown piece location and orientation, i.e., each piece might be misplaced and/or rotated by 90, 180 or 270 degrees. The latter solver was tested on 432- and 1,064-piece puzzles and a single 9,600-piece image. Sholomon \textit{et al.} ~\cite{sholomon2013genetic} successfully solved a 22,755-piece puzzle with only piece location unknown. Thus, as far as we know, current state-of-the-art algorithms can solve correctly a 22,755-piece puzzle with unknown piece location and a 9,600-piece puzzle with unknown piece location and orientation. All of the above solvers make use of the image dimensions during the solution process. We believe our work provides for the first time a solver capable of perfectly reconstructing a 22,755-piece puzzle with unknown piece location and orientation and no knowledge of the original image dimensions.
\section{Puzzle Solving}
In its most basic form, every puzzle solver requires an estimation function to evaluate the compatibility of adjacent pieces and a strategy for placing the pieces (as accurately as possible). We propose using genetic algorithms as a piece placement strategy, aimed at achieving an optimal global score with respect to compatibilities of adjacent pieces in the resulting image. The proposed GA elements (e.g., chromosome representation, crossover operator, and fitness function) are required to tackle several non-trivial issues. First, the eventual solution needs to be valid, i.e., every puzzle piece should appear once and only once, without missing and/or duplicate pieces. Second, the resulting image dimensions should strive to meet those of the (unknown) dimensions of the original image, avoid undesirable cases (e.g., pseudo-linear configurations) which contrive to be optimal solutions. Third, in the spirit of a gradual, evolutionary improvement over time, puzzle segments assembled correctly, up to an exact location and orientation, should be tracked, inherited to child chromosomes, and undergo proper translation and rotation (see Figure~\ref{fig:posIndp}).
\begin{figure}
\centering
\begin{subfigure}[t]{0.20\textwidth}
\centering
\includegraphics[width=\textwidth, height=3cm]{pos_ind_100.jpeg}
\caption{Original Image}
\label{fig:pos_ind_100}
\end{subfigure}
~
\begin{subfigure}[t]{0.20\textwidth}
\centering
\includegraphics[width=\textwidth, height=3cm]{pos_ind_001.jpeg}
\caption{Chromosome}
\label{fig:pos_ind_001}
\end{subfigure}%
\caption{ Characteristics sought by crossover; chromosome shown correctly assembled a number of puzzle segments, most of which are incorrectly located (with respect to correct absolute location) or oriented (notice tower orientation versus flags); crossover operator should exploit correctly assembled segments and allow them to be translated and rotated in a child chromosome. }
\label{fig:posIndp}
\end{figure}
\begin{figure*}
\centering
\begin{subfigure}[t]{0.20\textwidth}
\centering
\includegraphics[width=\textwidth]{toyassign_0.jpg}
\caption{}
\label{fig:res_20k__gen_00000000}
\end{subfigure}%
~
\begin{subfigure}[t]{0.20\textwidth}
\centering
\includegraphics[width=\textwidth]{toyassign_1.jpg}
\caption{1.b -- 2.d}
\label{fig:res_20k__gen_00000001}
\end{subfigure}
~
\begin{subfigure}[t]{0.20\textwidth}
\centering
\includegraphics[width=\textwidth]{toyassign_2.jpg}
\caption{4.d -- 8.c}
\label{fig:res_20k__gen_00000002}
\end{subfigure}
~
\begin{subfigure}[t]{0.20\textwidth}
\centering
\includegraphics[width=\textwidth]{toyassign_3.jpg}
\caption{8.d -- 7.c}
\label{fig:res_20k__gen_00000003}
\end{subfigure}
~
\begin{subfigure}[t]{0.20\textwidth}
\centering
\includegraphics[width=\textwidth]{toyassign_4.jpg}
\caption{3.a -- 4.c}
\label{fig:res_20k__gen_00000004}
\end{subfigure}
~
\begin{subfigure}[t]{0.20\textwidth}
\centering
\includegraphics[width=\textwidth]{toyassign_5.jpg}
\caption{2.c -- 7.a}
\label{fig:res_20k__gen_00000004}
\end{subfigure}
~
\begin{subfigure}[t]{0.20\textwidth}
\centering
\includegraphics[width=\textwidth]{toyassign_6.jpg}
\caption{2.a -- 6.b}
\label{fig:res_20k__gen_00000004}
\end{subfigure}
~
\begin{subfigure}[t]{0.20\textwidth}
\centering
\includegraphics[width=\textwidth]{toyassign_7.jpg}
\caption{5.b -- 1.a}
\label{fig:res_20k__gen_00000004}
\end{subfigure}
\caption{ Different types of relative relation assignment; each sub-caption describes the corresponding assignment made, also marked by a red line, (c) rotation of pieces, (e) explicit assignment (4.c, 3.a) leading to an implicit assignment (3.d, 7.d), and (f) merge of two piece groups, during which the entire left group is rotated. Note how in (d) the assignment (2.a, 4.c) is geometrically infeasible due to a collision between pieces 1 and 7. }
\label{fig:toyassign}
\end{figure*}
\vfill\eject
\subsection{Chromosome Representation}
We propose for each chromosome to be equivalent to a complete solution of the jigsaw puzzle problem, i.e., a suggested placement of all the pieces. Since the image dimensions are unknown, it is not clear how to store a piece configuration in a two-dimensional array (according to piece locations and orientations), in accordance with the actual image. Instead, we store, for each chromosome, only the relative placement of neighboring pieces. Since a piece orientation is unknown, for each given piece we denote its edges as $a,b,c,d$, starting clockwise from a random edge. To denote the relative placement of two pieces we may say, for example, that edge $p_{i}.b$ (i.e., edge $b$ of piece $p_{i}$) is placed next to edge $p_{j}.c$, thus encoding both the relative spatial location and orientation of the pieces. Figure~\ref{fig:relassign} depicts the labeling of piece edges and their relative placement. Each chromosome is represented by an $(n \times 4)$ matrix (i.e., a matrix whose dimensions are the number of pieces times the number of piece edges), where a matrix entry $x_{i,j} (i = 1..n, j = 1..4)$ is the corresponding piece edge adjacent to $x_{i}.j$ (e.g., $x_{i}.j = p_{2}.c$) or \textit{``none''}, if no edge is placed next to it (e.g., at the puzzle's boundary). This representation lends itself more easily to subsequent relative-placement evaluations and crossover operations.
\subsection{Chromosome Evaluation}
We use the {\em dissimilarity} measure below, which was presented in various previous works~\cite{conf/cvpr/PomeranzSB11,conf/cvpr/ChoAF10,sholomon2013genetic}. This measure relies on the premise that adjacent jigsaw pieces in the original image tend to share similar colors along their abutting edges, and thus, the sum (over all neighboring pixels) of squared color differences (over all color bands) should be minimal. Assuming that pieces $p_{i}$, $p_{j}$ are represented in normalized L*a*b* space by a $K \times K \times 3$ matrix, where $K$ is the height/width of a piece (in pixels), their dissimilarity, where $p_{j}$ is ``to the right'' of $p_{i}$, for example, is
\begin{equation} \label{eq:dissimilarity}
D(x_{i}.b,x_{j}.d)=\sqrt{\sum_{k=1}^{K}\sum_{ch=1}^{3}(x_{i}(k,K,ch)-x_{j}(k,1,ch))^{2}}.
\end{equation}
when $ch$ denotes a spectral channel. Obviously, to maximize the compatibility of two pieces, their dissimilarity should be minimized.
We compute the compatibility of all possible edges for all possible pieces, summing up to 16 edges per piece pair. Notice that computing the dissimilarity between piece edges (e.g., between $p_{1}.b$ and $p_{2}.c$) is invariant to their final piece rotation. For example, assume $b$ is the right edge of piece $p_1$ and $c$ is the left edge of piece $p_2$, with respect to some initial piece orientation. The compatibility of $p_{1}.b$ to $p_{2}.c$ is similar whether or not the pieces are rotated, i.e., $p_{2}$ is to the right of $p_{1}$ (if the pieces are not rotated), or $p_{2}$ is below $p_{1}$ if they are rotated (clockwise) by 90 degrees. In any event, the final chromosome fitness is the sum of pairwise dissimilarities over all adjacent edges.
Representing a chromosome, as suggested, by an $(n \times 4)$ matrix, where a matrix entry $x_{i,j} (i = 1..n, j = 1..4)$ corresponds to a single piece edge, we define its fitness as:
\begin{equation} \label{eq:fitness}
\sum_{i=1}^{n}\sum_{j=1}^{4}D(p_{i}.j,x_{i,j})
\end{equation}
where $D$ is the dissimilarity of the two given edges (edge $j$ of piece $p_{i}$ and the edge located at $x_{i,j}$). This value should, of course, be minimized.
\makeatletter
\setlength{\@fptop}{0pt}
\makeatother
\begin{figure*}[t!]
\centering
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth, height=3cm]{res_20k__gen_00000000.jpeg}
\caption{Generation 1}
\label{fig:res_20k__gen_00000000}
\end{subfigure}%
~
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth, height=3cm]{res_20k__gen_00000001.jpeg}
\caption{Generation 2}
\label{fig:res_20k__gen_00000001}
\end{subfigure}
~
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth, height=3cm]{res_20k__gen_00000002.jpeg}
\caption{Generation 3}
\label{fig:res_20k__gen_00000002}
\end{subfigure}
~
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth, height=3cm]{res_20k__gen_00000003.jpeg}
\caption{Generation 4}
\label{fig:res_20k__gen_00000003}
\end{subfigure}
~
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth, height=3cm]{res_20k__gen_00000004.jpeg}
\caption{Generation 5}
\label{fig:res_20k__gen_00000004}
\end{subfigure}
~
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth, height=3cm]{res_20k__gen_00000100.jpeg}
\caption{Final Generation (100)}
\label{fig:res_20k__gen_00000100}
\end{subfigure}
~
\caption{ Solution process of a 22,755-piece puzzle (largest puzzle solved with unknown piece orientation and image dimensions): (a)--(e) Best chromosomes achieved by the GA in the first to fifth generation, and (f) final generation. Note that every chromosome (image) contains all puzzle tiles, so its image results in varying dimensions (down scaled, for display purposes). The original image was perfectly reconstructed. }
\label{fig:res20k}
\end{figure*}
Special care should be taken in the case of $x_{i,j} = none$, i.e., a piece edge with no adjacent pieces. Intuitively, \textit{``none''} piece edges should be highly discouraged. In principle, only boundary pieces of the original image should contain \textit{none} edges. Most boundary pieces should have only a single \textit{none} edge, whereas the four corner pieces are expected to have two \textit{none} edges each. (Ideally, no piece should have more than two \textit{none} edges.) Assigning a dissimilarity of ``0'' to a \textit{none} edge might cause the GA to converge to a non-rectangular image. On the other hand, assigning an extremely high value might lead to cases where image shape might take precedence over image content. Having tested the algorithm with many different values, we picked the dissimilarity of a \textit{none} edge (i.e., $D(p_{i}.j,none)$) to be twice the average of all dissimilarities (i.e., the average of all 16 pairwise dissimilarities over all piece pairs). It appears that this measure highly encourages the GA to reach a correct reconstruction of the original image, with respect to both image dimensions and content.
\subsection{Crossover Operator}
The most involved element of the proposed GA is the crossover operator. Considering the proposed representation and in light of the chosen fitness function, one can grasp the major role undertaken by the crossover operator. The operator must verify the validity of each newly created chromosome. Traditionally, this means that each puzzle piece appears once and only once in the child chromosome, and no piece is missing or duplicated. The inherent difficulty surrounding merely this condition is likely to have delayed the derivation of an effective GA-based solver for the problem. Meeting this condition still assures no validity. Since chromosomes contain only relative relations (e.g., edge $p_{1}.b$ is adjacent to edge $p_{2}.c$), the operator must also verify that all relative relation assignments result in a geometrically feasible image, i.e., that all pieces must be placed properly with no overlaps. Moreover, the operator should verify that all prospective characteristics (e.g., correctly assembled puzzle segments) discovered by the parents may be inherited by their offsprings, and allow these segments to be translated in space and rotated inside the offspring (see, e.g., Figure~\ref{fig:posIndp}).
We propose a novel specialized crossover operator to address the aforementioned challenges. In contrast to previous works and in accordance with the chosen representation, this operator is based on relative relations of pieces edges. The assignment of a relative relation, e.g., ``edge $p_{1}.b$ is adjacent to edge $p_{2}.c$'', is the basic building block of the operator. An example of such an assignment can be viewed in Figure~\ref{fig:relassign}. To maintain consistency, each assignment is double since it immediately implies the commutative relation (i.e., that edge $p_{2}.c$ is adjacent to edge $p_{1}.b$). Each crossover consists of exactly $n-1$ double relative relation assignments, resulting in a single connected component (albeit not necessarily rectangular). Examples of such images can be viewed in Figure~\ref{fig:res20k}. (Note that since every chromosome contains all the puzzle pieces, its image results in varying dimensions. This is not depicted, as the figure is down scaled for display purposes.) The following paragraphs describe the complete crossover procedure. First, we describe the intrinsics of the relative relation assignment and then provide the full details of creating a child chromosome.
\subsubsection{The Relative Relation Assignment}
The crossover procedure starts with no relative relations; each puzzle piece is detached and isolated from the others. Each assignment of a relative relation between two piece edges falls inside one of three possible scenarios. The first option is for the relation to be between two detached pieces. Naturally, this is always the case of the first assignment. The double relation between the two edges is recorded and the pieces are inserted into a newly created {\em ``piece group''}. Each such group is recorded inside a matrix, tracking the spatial relations of its pieces and verifying the geometrical validity of the represented sub-image. The second possible scenario is that one of the edges belongs to such a piece group, while the other is of a detached piece. Again, the relation is recorded and the detached piece is inserted, at the correct location, into the already existing group. Notice that by inserting a piece inside a group, implicit relations might be set, e.g., inserting a piece to the left of a piece results in it being inserted below another piece, as depicted in Figure~\ref{fig:toyassign}. All such implicit relations are recorded at the end of the crossover operation.
\begin{figure}
\centering
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{assignerr_1.jpg}
\caption{Valid assignment}
\label{fig:pos_ind_100}
\end{subfigure}
~
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{assignerr_0.jpg}
\caption{Geometrically infeasible assignment}
\label{fig:pos_ind_001}
\end{subfigure}%
\caption{ Geometrically valid and invalid relative relation assignments. }
\label{fig:assignerr}
\end{figure}
Unlike the first two cases, the third case requires extra care. This is the case of the two edges originating from two different piece groups. Notice that setting a relation between two edges belonging to the same group is prohibited, since it is either redundant (the pieces already conform implicitly to the relation) or results in an invalid image. The two piece groups should then be merged to a single connected component. Unfortunately, as can be seen in Figure~\ref{fig:assignerr}, some piece formations cannot be merged. The operator tries to merge the smaller group to the larger one, starting from the requested edge and continuing with all other recorded relations in the group. If a collision occurs between two pieces, the entire assignment of this particular relative relation is deemed illegal and is discarded, leaving the two groups disjoint. In case of a success, the new double relation is recorded and the two groups are merged into a single group. Notice that such a merge might change the orientation of all pieces inside the group being merged.
\subsubsection{A Multi-Phased Algorithm}
Having described the relative relation assignment, we now proceed with the assignment selection algorithm. Given two parent chromosomes, the operator initiates a multi-phased algorithm, described in Figure~\ref{fig:crossoverOverview}. Note that the algorithm continuously attempts to assign edges, until $n-1$ successful assignments are made. Each assignment could fail if one of the edges has already been assigned during the above group merging; otherwise, it could fail as a result of the mutation process, with some low probability. The algorithm first tries to assign all common relations, i.e., all relative relations appearing in both parents. Second, the algorithm tries to assign relations which appear in at least one of the parents and are also {\em best buddies}. To understand this phase, we briefly review the concept of a best-buddy piece, first introduced by Pomeranz \textit{et al.}~\cite{conf/cvpr/PomeranzSB11}. Two pieces are said to be best buddies if each piece considers the other as its most compatible piece, according to the dissimilarity score. We generalize this notion for piece edges, i.e., two edges are considered best-buddy edges if each edge considers the other as its most compatible edge out of all existing edges. The edges $x_{i}.a$ and $x_{j}.d$ are said to best buddies if
\begin{align}
\forall e_{k} \in Edges, \; D(x_{i}.a,x_{j}.d) \leq D(x_{i}.a,e_{k})\notag \\
\text{and \quad\quad\quad\quad\quad\quad\quad\quad}\\
\forall e_{p} \in Edges, \; D(x_{j}.d,x_{i}.a) \leq D(x_{j}.d,e_{p}) \notag
\end{align}
where $Edges$ is the set of all piece edges (i.e., sides $a$, $b$, $c$, and $d$ of all $n$ given pieces). Next, we introduce a greedy element. We compute, in advance, the most compatible edge for each of the $4 \times n$ edges and then try to assign most compatible edges in a random order. Finally, we try to assign two random edges until completing $n-1$ successful assignments.
Upon termination, after merging all piece groups, the crossover results in a single matrix, containing all $n-1$ pieces. The matrix records the absolute location and orientation of each piece in the resulting image. As mentioned earlier, the operator now scans the matrix, and records all relative relations created, both explicitly and implicitly, to the new chromosome. Note that this chromosome represents a geometrically valid images, containing each puzzle piece exactly once.
\begin{figure}[htbp]
\begin{center}
\hrule
\bigskip
\begin{description}
\item \textbf{Until} $(n - 1)$ relative relations are assigned \textbf{do}
\begin{enumerate}
\item Try assigning all {\em common} relative relations.
\item Try assigning all {\em best-buddy} relative relations.
\item Try assigning all {\em most-compatible} relative relations.
\item Try assigning {\em random} relative relations.
\end{enumerate}
\end{description}
\bigskip
\hrule
\bigskip
\caption{Crossover overview}
\label{fig:crossoverOverview}
\end{center}
\end{figure}
\subsection{Rationale}
The proposed genetic algorithm is based entirely on the concept of assigning relative relations between piece edges. This concept stems from the intuitive understanding that some puzzle segments are ``easier'' than others. A human solver will usually assemble first disjoint but distinct elements (e.g., animals, humans, vehicles, etc.), gradually joining them together. The assembly of more difficult parts (skies, sea, woods, and other background scenes) would be deferred to a later stage, when there are less pieces to choose from, which makes each decision easier. We mimic this behavior by allowing the solver to concurrently assemble different puzzle segments (piece groups), forcing it to merge them only at later stages. This advantage should be constructive especially in the case of puzzle pieces belonging to multiple images (i.e., a ``mixed bag'' of pieces), as the solver may handle each sub-image separately, as opposed to tackling the entire image of multiple excessive pieces.
Correctly assembled segments in the parents are identified either by their mutual appearance in both parents, or by a greedy consideration (due to a very high compatibility of certain edges). Only the relative relation between the edges is inherited, as the segments might be easily rotated and translated in space during the crossover procedure.
\subsection{GA Parameters}
We use the standard roulette-wheel selection. In each generation we start by copying the best four chromosomes. The following are the parameters used: \\
\\
population size = 300 \\
number of generations = 100 \\
probability of not using a shared relation = 0.001 \\
\begin{figure*}
\centering
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth, height=3cm]{res_mix_small__gen_00000000.jpeg}
\caption{Generation 1}
\label{fig:res_mix_small__gen_00000000}
\end{subfigure}%
~
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth, height=3cm]{res_mix_small__gen_00000001.jpeg}
\caption{Generation 2}
\label{fig:res_mix_small__gen_00000001}
\end{subfigure}
~
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth, height=3cm]{res_mix_small__gen_00000002.jpeg}
\caption{Generation 3}
\label{fig:res_mix_small__gen_00000002}
\end{subfigure}
~
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth, height=3cm]{res_mix_small__gen_00000003.jpeg}
\caption{Generation 4}
\label{fig:res_mix_small__gen_00000003}
\end{subfigure}
~
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth, height=3cm]{res_mix_small__gen_00000004.jpeg}
\caption{Generation 5}
\label{fig:res_mix_small__gen_00000004}
\end{subfigure}
~
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth, height=3cm]{res_mix_small__gen_00000100.jpeg}
\caption{Final Generation (100)}
\label{fig:res_mix_small__gen_00000100}
\end{subfigure}
~
\caption{ Solution process of a ``mixed-bag'' puzzle, composed of four different 432-piece puzzles: (a)--(e) Best chromosomes achieved by the GA from first to fifth generation, and (f) final generation. All original four images were perfectly reconstructed. GA has no knowledge that given pieces belong to different images. }
\label{fig:mixSmall}
\end{figure*}
\section{Results}
To evaluate the accuracy results of each assembled image, we adopt the {\em neighbor comparison} measure used in all previous works, i.e., the fraction of correct pairwise piece adjacencies (with respect to the original image). In all experiments we used a standard tile size of 28 $\times$ 28 pixels.
We tested our solver against previously published benchmark sets~\cite{conf/cvpr/PomeranzSB11,sholomon2013genetic,conf/cvpr/ChoAF10} containing 20-image sets of 432-, 540-, 805-, 5,015-, and 10,375-piece puzzles. We report in Table~\ref{tab:scoresAll} the average accuracy results per set, as well as the number of puzzles reconstructed perfectly. The results obtained for the set of 432-piece puzzles can be compared to the ones achieved by~\cite{conf/cvpr/Gallagher12} on the same image set (see Table~\ref{tab:scoresGal}). We stress that the solver in~\cite{conf/cvpr/Gallagher12} assumes known image dimensions to improve the accuracy results (described there as the ``trimming'' and ``filling'' stages), while ours does not. According to Table 6 in \cite{conf/cvpr/Gallagher12}, their algorithm results in an average of 90.4\%, i.e., we achieve a significant improvement of over 4\% with less assumptions.
\begin{table}
\centering
\begin{tabular}{ |c||c|c| } \hline
\# Pieces & Neighbor & Perfect\\ \hline \hline
432 & 94.88\% & 11 \\ \hline
540 & 94.08\% & 8 \\ \hline
805 & 94.12\% & 6 \\ \hline
5,015 & 94.90\% & 7 \\ \hline
10,375 & 98.03\% & 4 \\ \hline
\end{tabular}
\caption{Accuracy results under neighbor comparison on different benchmark sets. For each set we report the average accuracy obtained by the GA and number of puzzles perfectly reconstructed (out of 20). }
\label{tab:scoresAll}
\end{table}
\begin{table}
\centering
\begin{tabular}{ |c||c|c| } \hline
& Neighbor & Perfect\\ \hline \hline
Gallagher~\cite{conf/cvpr/Gallagher12} & 90.4\% & 9 \\ \hline
GA & 94.88\% & 11 \\ \hline
\end{tabular}
\caption{Performance comparison for puzzles with 432 pieces. Note that ~\cite{conf/cvpr/Gallagher12} assumes known image dimensions to improve accuracy results, while our solver does not. }
\label{tab:scoresGal}
\end{table}
The largest puzzle that has been attempted so far with pieces of unknown orientation is a 9,600-piece puzzle. We attempted a 22,755-piece puzzle (i.e., more than twice larger), with unknown piece orientation and unknown image dimensions. As can be seen in Figure~\ref{fig:res20k}, perfect reconstruction was achieved. The solution process required a meager 3.5 hours on a modern PC (compared to 23.5 hours reported in~\cite{conf/cvpr/Gallagher12} for the 9,600 piece-puzzle).
Finally, we applied the solver to ``mixed-bag'' puzzles, i.e., puzzles containing pieces from multiple images. Of course, the solver is unaware of the image dimensions, and of the fact that multiple images are involved. We created a mixed puzzle by combining four different 432-piece puzzles. Figure~\ref{fig:mixSmall} depicts the gradual assembly of the puzzle until a perfect reconstruction of all four images is achieved. Next, we created another mixed puzzle by combining 16,405 pieces from seven different images; three 5,015-piece puzzles and four smaller ones. To the best of our knowledge, this is the most complex mixed-bag that has been solved, in terms of both the puzzle size and the number of mixed images. Despite containing a much larger number of pieces (relatively to single-image puzzles solved previously), the GA succeeds in fully reconstructing all the seven puzzles. Figure~\ref{fig:mixHuge} shows the reassembled puzzles.
\begin{figure*}
\centering
\begin{subfigure}[t]{0.33\textwidth}
\centering
\includegraphics[width=\textwidth]{res_mix_5k_init.jpeg}
\caption{}
\label{fig:intro_10375_gen_00000000}
\end{subfigure}
\begin{subfigure}[t]{0.33\textwidth}
\centering
\includegraphics[width=\textwidth]{res_mix_5k.jpeg}
\caption{}
\label{fig:intro_10375_gen_00000000}
\end{subfigure}
\caption{ Perfect reconstruction of a ``mixed-bag'' puzzle. This puzzle is constructed of 16,405 pieces from seven different images; three 5,015-piece puzzles and four smaller puzzles. As far as we know, this is the largest and most complex mixed puzzle that has been solved, in terms of size and number of puzzles. }
\label{fig:mixHuge}
\end{figure*}
\section{Conclusions}
In this paper we presented, for the first time, a GA-based solver capable of handling puzzles with (1) pieces of unknown orientation, (2) unknown image dimensions, and (3) pieces originating from multiple images. Our solver sets a new state-of-the-art in terms of the accuracy achieved and the complexity of the puzzles handled. We improved significantly results obtained by previous works, making almost no assumptions. Specifically, we successfully tackled puzzles of more than twice the size that has been attempted before with the same relaxed assumptions~\cite{conf/cvpr/Gallagher12}. Finally, we showed how to assemble mixed puzzles, i.e., puzzles with pieces from multiple different images. As far as we know, the mixed puzzle solved in this paper is the largest and most complex in terms of the total number of pieces and the number of mixed images.
|
train/arxiv
|
BkiUe-s5qhLA14d4gJ_h
| 5 | 1 |
\section*{\hfil #1\hfil}}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{cite,setspace}
\newcommand{\Ex}{ \mathbb{E} }
\begin{document}
\title{\vspace{-0.5cm}\LARGE Matched Multiuser Gaussian Source-Channel \\Communications via Uncoded Schemes}
\author{Chao Tian, Jun Chen, Suhas Diggavi and Shlomo Shamai}
\maketitle
\abovecaptionskip=0.2cm
\abovedisplayskip=3pt
\belowdisplayskip=4pt
\begin{abstract}
We investigate whether uncoded schemes are optimal for Gaussian sources on multiuser Gaussian channels. Particularly, we consider two problems: the first is to send correlated Gaussian sources on a Gaussian broadcast channel where each receiver is interested in reconstructing only one source component (or one specific linear function of the sources) under the mean squared error distortion measure; the second is to send vector Gaussian sources on a Gaussian multiple-access channel, where each transmitter observes a noisy combination of the source, and the receiver wishes to reconstruct the individual source components (or individual linear functions) under the mean squared error distortion measure. It is shown that when the channel parameters match certain general conditions, the induced distortion tuples are on the boundary of the achievable distortion region, and thus optimal. Instead of following the conventional approach of attempting to characterize the achievable distortion region, we ask the question whether and how a match can be effectively determined. This decision problem formulation helps to circumvent the difficult optimization problem often embedded in region characterization problems, and also leads us to focus on the critical conditions in the outer bounds that make the inequalities become equalities, which effectively decouples the overall problem into several simpler sub-problems.
\end{abstract}
\section{Introduction}
\label{sec:intro}
Although the source-channel separation architecture is asymptotically optimal in the point-to-point communication setting \cite{Shannon:48} as well as several classes of multiuser communication settings (see {\em e.g.,} \cite{Tian:14} and references therein), uncoded schemes have several particularly attractive properties. Firstly, they have very simple encoders and decoders; secondly, they belong to the so-called zero-delay codes, which can avoid the long delay required to approach the asymptotic performance in the separation-based schemes; lastly, they are in fact optimal in some settings, while the separation-based schemes are not (see {\em e.g.}, \cite{Goblick65}).
It was shown in \cite{Gastpar:03} that uncoded schemes are optimal when certain matching conditions involving the source probability distribution, the channel transition probability distribution, the channel cost function and the distortion measure function are satisfied. Though the focus in \cite{Gastpar:03} was mainly on the point-to-point setting, recent results \cite{Gastpar:08,Lapidoth:10BC,Lapidoth:10MAC,Tian:11} suggest that the concept of matching indeed carries over to the multiuser case. In fact, in multiuser settings, matching may occur naturally when the distortion measure, the channel cost function and source distribution are all fixed, and the channel parameters, which represent physically meaningful quantities, satisfy certain conditions. In this work, we consider such matching, particularly, when the sources and the channels are Gaussian, the channel constraints are on the expected average signal power, the distortion measure is the mean squared error (MSE), and only the channels parameters, such as the channel amplification factors and the additive noise powers, are allowed to vary.
In this context, of interest is whether for a fixed source and fixed coding parameters, the distortion vector such induced is on the boundary of the achievable distortion region (and thus optimal). More specifically, we seek to answer the following questions:
\begin{itemize}
\item Is there a set of (explicitly) computable conditions that can be used to certify a fixed uncoded scheme to be optimal for a given source and channel pair?
\item If so, is there a non-trivial set of channels that satisfy such conditions for a given source and uncoded scheme pair?
\end{itemize}
We shall refer to this kind of channels as ``matched channels''; a dual question is to ask for ``matched sources'', however in the context of problems considered here, the dual question is notationally more involved, and thus we choose to investigate the problems from the perspective of ``matched channels''. One can also ask for ``matched distortion measures'', similarly as the approach taken in \cite{Gastpar:03}, however in the Gaussian setting we consider here, fixing the MSE distortion is practically more important and well-motivated. The set of matched channels should be distinguished from the complete set of channels for which the given uncoded scheme is optimal. The former may be a strict subset of the latter, since the former may be only sufficient for an uncoded scheme to be optimal, which usually depends on the specific outer bounding technique employed. Characterizing the latter region is naturally more difficult than answering the questions we posed above.
The two questions posed above are in essence the two facets of the same question. Since we only provide conditions for matching, or in other words, sufficient conditions for the scheme to be optimal, the set of matched channels may in fact be empty. A trivial condition to answer the first question is simply an impossible one such that we would never be able to certify a channel to be matched. Thus the second question is important, and we show indeed for the two problems considered here, there are non-trivial channels that match the source and the uncoded scheme.
Traditionally, research in information theory asks for characterization of certain achievable region, for which we first derive an expression for an outer bound, and derive an expression for an inner bound, and then make comparison of them. This approach can be challenging because it usually involves optimization over a set of parameters, and solving such an optimization problem explicitly can be difficult. It is not clear whether the obstacle mainly stems from the intractable nature of the underlying communication problem, or it is mainly caused by the embedded optimization problem.
The aforementioned difficulty motivates the formulation of the first question, which is a decision problem instead of an optimization problem. An analogy of this situation can be found in computer science algorithm research, where instead of asking whether an optimization problem can be solved in polynomial time, the alternative question is asked whether a decision ({\em e.g.}, regarding a solution is above a threshold) can be made in polynomial time. Our problem formulation naturally leads to a different approach in the investigation. Instead of focusing on comparison of the inner bounds and outer bounds using their expressions, we focus on the necessary conditions that the outer bound becomes tight, {\em i.e.,} the conditions when the information inequalities hold with strict equality. With fixed source and fixed coding parameters, the coding vector can be substituted into the conditions, and the necessary and sufficient conditions for such equality can be derived. The outer bounds naturally provide certain ``decoupled'' conditions, which significantly simplifies the overall task. Though this approach may have inherently been used by many researchers in the past, its effectiveness becomes particularly evident in our investigation of uncoded schemes in the joint source channel communication setting.
In the rest of the paper, we focus specifically on two joint source channel coding problems using the approach outlined above. The first problem is to send correlated Gaussian sources on a Gaussian broadcast channel where each receiver is interested in reconstructing only one source component
(or equivalently one specific linear function of the source) under the MSE distortion measure. The second problem is to send vector Gaussian sources on a Gaussian multiple-access channel, where each transmitter observes a noisy combination of the source, {\em i.e.,} a case of the vector CEO problem, and the receiver wishes to reconstruct the source components (or equivalently linear functions of the source components) under the MSE distortion measure. General conditions for matching are derived, which either include or generalize well-known existing results on the optimality of uncoded schemes in the multiuser setting. Particularly notable are the following cases:
\begin{itemize}
\item The first problem generalizes the two-user case considered in \cite{Lapidoth:10BC} and \cite{Tian:11} to the $M$-user case, for which we show that the uncoded scheme is optimal for a large set of sources and channels; our results reveal that uncoded scheme can still be optimal when some source components are negatively correlated.
\item The results on the second problem includes as specials cases the symmetric scalar Gaussian CEO problem \cite{Gastpar:08}, the problem of sending bivariate Gaussian sources on a Gaussian multiple-access channel \cite{Lapidoth:10MAC}, and sending remote (noisy) bivariate Gaussians on a Gaussian multiple-access channel \cite{LapidothWang:11}. Our results reveal that in addition to the symmetric case considered in \cite{Gastpar:08}, uncoded scheme is also optimal when the sensor observation quality is proportional to the channel quality. These results also allow the sensor observations to have more general correlation structure and the observations to be noisy, thus extending the results in\cite{Lapidoth:10MAC} and \cite{LapidothWang:11}. When viewed from the perspective of computation, our result also provides new characterizations for the problem of computing linear functions of Gaussian random variables on the Gaussian multiple-access channels considered in \cite{NazerGastpar:07} and \cite{Vishwanath:12}.
\end{itemize}
Notationally, we write for a source $S$ at time $n$ as $S[n]$, and a length-$N$ vector as $S^N$. For a set of coefficients $(\alpha_1,\alpha_2,\ldots,\alpha_M)$, we sometimes write it in a (column) vector form as $\bar{\alpha}$. For a real matrix $\Sigma$, we write its transpose as $\Sigma^t$. The positive semidefinite order is denoted as $\succeq$.
\section{Correlated Gaussian Sources on a Gaussian Broadcast Channel}
In this section we consider the problem of broadcasting correlated Gaussian sources on a Gaussian broadcast channel, which can be described as follows; see also Fig. \ref{fig:BC} for an illustration. Let the zero-mean Gaussian source be $(S_1[n],S_2[n],\ldots,S_M[n])$ with covariance matrix $\Sigma_{S_1,S_2,\ldots,S_M}$, which is assumed to be full rank.
The channel is given by
\begin{align}
Y_m[n]=X[n]+Z_m[n],
\end{align}
where $(Z_1,Z_2,\ldots,Z_M)$ are zero-mean additive noises which are mutually independent, with variances $\sigma^2_{Z_1}\geq \sigma^2_{Z_2}\geq \ldots \geq \sigma^2_{Z_M}$, respectively; the channel input must satisfy an average power constraint $\frac{1}{N}\sum_{n=1}^N\Ex X^2\leq P$. The transmitter encodes the length-$N$ source vector $(S^N_1,S^N_2,\ldots,S^N_M)$ into a length-$N$ channel vector $X^N$, and the $m$-th receiver reconstructs from the channel output vector $Y^N_m$ the source vector ${S}^N_m$ as $\hat{S}^N_m$, resulting in a distortion $D_m=\frac{1}{N}\sum_{n=1}^N\Ex(S_m[n]-\hat{S}_m[n])^2$. We omit the formal problem definition using encoding and decoding function, which is standard and can be obtained by extending that in, for example, \cite{Tian:11}.
The uncoded scheme of interest has the form
\begin{align}
X[n]=\sum_{m=1}^M \alpha_m S_m[n],
\end{align}
such that
\begin{align}
\Ex[X^2[n]]=P.
\end{align}
In other words, at each time instance, the channel input is simply a linear combination of the source components with coefficients $(\alpha_1,\alpha_2,\ldots,\alpha_M)$, such that the resulting signal has an expected variance that is equal to the power constraint $P$.
We shall assume $\alpha_m\neq 0$. The decoders simply estimate $S_m[n]$ as $\hat{S}_m[n]=\Ex(S_m[n]|Y_m[n])$, at each time instance $n=1,2,\ldots,N$ at decoder $m=1,2,\ldots,M$.
Notice that the problem can be equivalently formulated as computation of linear functions of Gaussian sources on the broadcast channel, however this alternative formulation is notationally more involved.
\begin{figure}[tb]
\centering
\includegraphics[width=13cm]{figs/BC}
\vspace{-0.5cm}
\caption{Sending correlated Gaussian random variables on a Gaussian broadcast channel. \label{fig:BC}}
\end{figure}
Define
\begin{align}
\bar{\beta}=(\beta_1,\beta_2,\ldots,\beta_M)^t\triangleq\frac{1}{P}\Sigma_{S_1,S_2,\ldots,S_M}\bar{\alpha}.
\label{eqn:alphatobeta}
\end{align}
The main result is summarized in the following theorem, which gives a matching condition in a positive semidefinite form.
\begin{theorem}
\label{theorem:BC}
A Gaussian broadcast channel is said to be matched to a given source and an uncoded scheme with non-zero parameters $\bar{\alpha}$, and the distortion vector induced by the given scheme is on the boundary of the achievable distortion region thus optimal, if
\begin{align}
\Sigma^{(0)}\triangleq\Sigma_{V_1,V_2,\ldots, V_M} - \Sigma_{S_1,S_2,\ldots,S_M}+P\bar{\beta}\bar{\beta}^t
\succeq 0,
\label{eqn:semidefinite}
\end{align}
where the entries of the symmetric matrix $\Sigma_{V_1,V_2,\ldots, V_M}$ are specified as
\begin{align}
\gamma_{j,m}&=\beta_m\beta_j\frac{P\sigma^2_{Z_m}}{P+\sigma^2_{Z_m}},\quad 1\leq j<m,\quad m=2,3,\ldots,M,\\
\gamma_{m,m}&=\alpha^{-1}_m\left[\beta_m(\sum_{j=1}^{m-1}\alpha_j\beta_j)\frac{P\sigma^2_{Z_m}}{P+\sigma^2_{Z_m}}+\beta_m\sum_{j=m+1}^M\alpha_j\beta_j\frac{P\sigma^2_{Z_j}}{P+\sigma^2_{Z_j}}\right],\quad m=1,2,\ldots,M.
\end{align}
\end{theorem}
This theorem establishes a condition that is sufficient to guarantee a distortion vector induced by an uncoded scheme to be on the boundary of the achievable distortion region, and thus an optimal solution. The matrix $\Sigma_{V_1,V_2,\ldots, V_M}$ may seems mysterious at the first sight, however, the reason for introducing this matrix will become clear shortly.
This theorem clearly answers our first question regarding conditions that can be used to certify whether a given uncoded scheme is optimal. In fact, it also provides clues on the second question regarding whether there exist non-trivial channels that such matching is possible. Indeed, in Section \ref{sec:Cholesky} and Section \ref{sec:BCchannels} we establish several properties of matched channels, through which an answer to the second question is given. Before presenting those results, the proof of this theorem is presented next in two parts: the critical conditions in a novel outer bound are outlined in Section \ref{sec:BCOuter}, and then these conditions for the bound to hold with equality in the uncoded scheme are analyzed in Section \ref{sec:BCinner}. The proof details for the outer bound are relegated to the Appendix.
\subsection{Extracting the Critical Conditions from the Outer Bound}
\label{sec:BCOuter}
In order to obtain the matching condition, we first derive a novel outer bound for this problem. An important technique in the derivation of this outer bound is the introduction of certain appropriate random variables outside of the original problem. This approach is partly motivated by our previous work \cite{TianDiggaviShamai:09,Song:12,Khezeli:14(1),Khezeli:14(2)}, which can further be traced back to Ozarow \cite{Ozarow:80}. Consider $M$ zero-mean Gaussian random variables $(W_1,W_2,\ldots,W_M)$, independent of everything else, with covariance matrix $\Sigma_{W_1,W_2,\ldots,W_M}$, and write
\begin{align}
U_m[n]=S_m[n]+W_m[n].
\end{align}
The outer bound will be written as a necessary condition that any distortion vector has to satisfy. For this purpose, we bound the following quantity for any encoding and decoding functions:
\begin{align}
E(\Sigma_{W_1,W_2,\ldots,W_M})\triangleq \sum_{m=1}^M (\sigma^2_{Z_{m}}-\sigma^2_{Z_{m+1}})\exp\left[\frac{2}{N}\sum_{j=1}^mI(U^N_j;Y^N_j|U^N_1,U^N_2,\ldots,U^N_{j-1})\right],
\end{align}
where $\sigma^2_{Z_{M+1}}\triangleq 0$ for notational simplicity. An almost identical quantity was used in \cite{TianDiggaviShamai:09} to obtain an approximate characterization for the distortion region of the Gaussian broadcast problem with bandwidth mismatch. We shall upper-bound this quantity using the channel properties and lower-bound it using the source reconstruction requirements, then combine them to obtain an eventual outer bound.
This quantity can be upper-bounded as given in the Appendix as
\begin{align}
E(\Sigma_{W_1,W_2,\ldots,W_M})\leq P+\sigma^2_{Z_1},\label{eqn:BCchannel}
\end{align}
with equality holds if and only if
\begin{align}
h(Y^N_M|S^N_1,S^N_2,\ldots,S^N_M)&=h(Y^N_M|U^N_1,U^N_2,\ldots,U^N_M),\label{eqn:degerate}\\
h(Y^N_1)&=\frac{N}{2}\log 2\pi e( P+\sigma^2_{Z_1}),\label{eqn:BCpower}
\end{align}
and the following condition stemming from the entropy power inequalities hold with equality
\begin{align}
&\exp\left[\frac{2}{N}h(Y^N_m|U^N_m,U^N_{m-1},\ldots,U^N_1)\right]=\exp\left[\frac{2}{N}h(Y^N_{m+1}|U^N_m,U^N_{m-1},\ldots,U^N_1)\right]+2\pi e[\sigma^2_{Z_m}-\sigma^2_{Z_{m+1}}],\nonumber\\
&\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad m=1,2,\ldots,M.\label{eqn:epi}
\end{align}
The conditions in (\ref{eqn:epi}) are standard, as Bergmans \cite{Bergmans:74} also used the entropy power inequality to establish the Gaussian broadcast channel capacity, and in general a Gaussian codebook suffices to make them equalities. The condition (\ref{eqn:BCpower}) intuitively requires that the power is fully utilized.
The condition (\ref{eqn:degerate}) is however rather peculiar, which essentially requires the noisy source $(U^N_1,U^N_2,\ldots,U^N_M)$ to be as useful as the real source $(S^N_1,S^N_2,\ldots,S^N_M)$ in determining the channel output $Y^N_M$.
The quantity $E(\Sigma_{W_1,W_2,\ldots,W_M})$ can also be lower-bounded as given in the Appendix, where its individual summands are bounded as
\begin{align}
\exp\left[\frac{2}{N}\sum_{j=1}^mI(U^N_j;Y^N_j|U^N_1,U^N_2,\ldots,U^N_{j-1})\right]\geq \frac{|\Sigma_{S_1,S_2,\ldots,S_m}+\Sigma_{W_1,W_2,\ldots,W_m}|}{\Pi^m_{j=1}(D_j+\sigma^2_{W_j})},\label{eqn:BCsource}
\end{align}
with equality holds if and only if
\begin{align}
h(U^N_m|Y^N_m)&=\frac{N}{2}\log [2\pi e(D_m+\sigma^2_{W_m} )],\quad m=1,2\ldots,M,\label{eqn:BCcondition1}\\
h(U^N_m|Y^N_m,U^N_1,U^N_2,\ldots,U^N_{m-1})&=h(U^N_m|Y^N_m),\quad m=2,3,\ldots,M.\label{eqn:BCcondition2}
\end{align}
The conditions in (\ref{eqn:BCcondition1}) are standard which can be viewed as requiring the codes to achieve the given distortions with equality, however the conditions in (\ref{eqn:BCcondition2}) are peculiar which essentially require all the information $(Y^N_m,U^N_1,U^N_2,\ldots,U^N_{m-1})$ on $U^N_m$ to be from $Y^N_m$.
Combining (\ref{eqn:BCchannel}) and (\ref{eqn:BCsource}), we obtain an outer bound, or outer bounds since each set of the auxiliary random variables $(W_1,W_2,\ldots,W_M)$ provides one specific outer bound. In the approach we shall take, the precise form of this outer bound is less important than the extracted matching conditions (\ref{eqn:degerate}), (\ref{eqn:BCpower}), (\ref{eqn:epi}), (\ref{eqn:BCcondition1}) and (\ref{eqn:BCcondition2}). In fact, the conditions (\ref{eqn:BCpower}), (\ref{eqn:epi}) and (\ref{eqn:BCcondition1}) can be satisfied simply by choosing a proper jointly Gaussian coding scheme, yet the conditions (\ref{eqn:degerate}) and (\ref{eqn:BCcondition2}) are the effectual non-trivial conditions. Note that from the problem setting and taking into consideration the fact that physical degradedness is equivalent to stochastic degradedness in the broadcast setting, we have the Markov string
\begin{align}
&Y^N_1\leftrightarrow Y^N_2 \leftrightarrow\ldots\leftrightarrow Y^N_M\leftrightarrow X^N \leftrightarrow (S^N_1,S^N_2,\ldots,S^N_M)\nonumber\\
&\qquad\leftrightarrow (U^N_1,U^N_2,\ldots,U^N_M)\leftrightarrow (U^N_1,U^N_2\leftrightarrow\ldots\leftrightarrow U^N_{M-1})\leftrightarrow\ldots\leftrightarrow U^N_1,\label{eqn:markovstring}
\end{align}
where we have utilized the fact that physical degradation is equivalent to stochastic degradation in broadcast channels. This Markov string is not sufficient to guarantee (\ref{eqn:degerate}) and (\ref{eqn:BCcondition2}), and thus they require special attention.
\subsection{The Forward Matching Condition}
\label{sec:BCinner}
We first introduce some additional notation and make a few observations. Notice that due to the power constraint, the coefficient vector $\bar{\alpha}\triangleq (\alpha_1,\alpha_2,\ldots,\alpha_M)^t$ should satisfy
\begin{align}
\bar{\alpha}^t \Sigma_{S_1,S_2,\ldots,S_M}\bar{\alpha}=P,
\end{align}
and it follows that
\begin{align}
\sum_{m=1}^M\alpha_m\beta_m=1.
\end{align}
Due to the jointly Gaussian distribution of the uncoded scheme, we can write
\begin{align}
U_m=\beta_m X+ (S_m-\beta_m X)+W_m,\quad m=1,2,\ldots,M,
\end{align}
where the three components are mutually independent, since $\beta_m X=\Ex[S_m|X]$; we have also omitted the time index $[n]$ to simplify the notation. It follows that the covariance matrix of $(U_1,U_2,\ldots,U_M)$ given $Y_m$ can be decomposed as follows
\begin{align}
\Sigma_{U_1, U_2,\ldots, U_M|Y_m}=& \sigma^2_{X|Y_m}\bar{\beta}\bar{\beta}^t
+\Sigma_{S_1,S_2,\ldots,S_M|X}+\Sigma_{W_1,W_2,\ldots,W_M},
\end{align}
where
\begin{align}
\sigma^2_{X|Y_m}=\frac{P\sigma^2_{Z_m}}{P+\sigma^2_{Z_m}},\qquad m=1,2,\ldots,M.
\end{align}
Let $V_m\triangleq(S_m-\beta_m X)+W_m$ for $m=1,2,\ldots,M$.
With the above observations, we now return to the derivation of the forward matching conditions. As mentioned earlier, we need to substitute the random vectors specified by the uncoded scheme, {\em i.e.,} assigning $X[n]=\sum_{m=1}^M \alpha_m S_m[n]$, into the critical conditions (\ref{eqn:degerate}), (\ref{eqn:BCpower}), (\ref{eqn:epi}), (\ref{eqn:BCcondition1}) and (\ref{eqn:BCcondition2}) in order to identify the matching conditions. It is straightforward to see that (\ref{eqn:BCpower}), (\ref{eqn:epi}) and (\ref{eqn:BCcondition1}) indeed hold with equality due to the jointly Gaussian distribution of the uncoded scheme, and the chosen coefficients. Thus we only need to focus on (\ref{eqn:degerate}) and (\ref{eqn:BCcondition2}),
which in the context of the uncoded scheme are equivalent to the following single-letter forms
\begin{align}
h(Z_M)&=h(Y_M|U_1,U_2,\ldots,U_M),\label{eqn:BCasifnonoise}\\
h(U_m|Y_m,U_1,U_2,\ldots,U_{m-1})&=h(U_m|Y_m),\quad m=2,3,\ldots,M.\label{eqn:BCindep}
\end{align}
To satisfy the condition (\ref{eqn:BCindep}) with the jointly Gaussian uncoded scheme, for any $m=2,3,\ldots,M$, we must have $\Ex[V_mV_j]+\beta_m\beta_j\sigma^2_{X|Y_m}=0$ for $j=1,2,\ldots,m-1$. This specifies all the off-diagonal terms of $\Sigma_{V_1,V_2,\ldots,V_M}$, as
\begin{align}
\Ex[V_mV_j]=\gamma_{m,j}=-\beta_m\beta_j\sigma^2_{X|Y_m},\quad 1\leq j<m,\quad m=2,3,\ldots,M. \label{eqn:crossterms}
\end{align}
It remains to determine the diagonal entries of $\Sigma_{V_1,V_2,\ldots,V_M}$.
Since
\begin{align}
\sum_{m=1}^M\alpha_m S_m=X=\sum_{m=1}^M\alpha_m\beta_m X,
\end{align}
in order to satisfy the condition (\ref{eqn:BCasifnonoise}) with equality,
we must have
\begin{align}
\sum_{m=1}^M \alpha_mV_m=\sum_{m=1}^M \alpha_mW_m=0, \label{eqn:sumzero}
\end{align}
where we have taken into account of the fact that $\Sigma_{S_1,S_2,\ldots,S_M}$ is full rank. Thus for any $m=1,2,\ldots,M$,
\begin{align}
\sum_{j=1}^M\alpha_j\Ex[V_mV_j]=\Ex[V_m\sum_{j=1}^M\alpha_jV_j]=0.
\end{align}
It follows that $\gamma_{m,m}=\sigma^2_{V_m}$ can be determined from
\begin{align}
\alpha_m\sigma^2_{V_m}=-\sum_{j=1}^{m-1}\alpha_j\Ex[V_mV_j]-\sum_{j=m+1}^M\alpha_j\Ex[V_mV_j]=\beta_m\sigma^2_{X|Y_m}\sum_{j=1}^{m-1}\alpha_j\beta_j+\beta_m\sum_{j=m+1}^M\alpha_j\beta_j\sigma^2_{X|Y_j},
\end{align}
since $\alpha_m\neq 0$.
Thus the conditions (\ref{eqn:BCasifnonoise}) and (\ref{eqn:BCindep}) being equalities uniquely specify the matrix $\Sigma_{V_1,V_2,\ldots,V_M}$. Conversely, as long as the matrix $\Sigma^{(0)}$ is positive semidefinite, the conditions (\ref{eqn:BCasifnonoise}) and (\ref{eqn:BCindep}) hold with equality and the corresponding auxiliary random variables $(W_1,W_2,\ldots,W_M)$ can be found, and the outer bound derived previously is thus tight. This is exactly the matching condition given in Theorem \ref{theorem:BC}.
\textbf{Remark: }The outer bounds conditions (\ref{eqn:degerate}) and (\ref{eqn:BCcondition2}) in the context of the uncoded scheme provide two constraints on the matrix $\Sigma_{V_1,V_2,\ldots,V_M}$. Their effects on the matrix $\Sigma_{V_1,V_2,\ldots,V_M}$ are largely decoupled: the condition required by (\ref{eqn:BCcondition2}) being equal determines the off-diagonal entries of $\Sigma_{V_1,V_2,\ldots,V_M}$, while the condition (\ref{eqn:degerate}) determines its diagonal entries. This decoupling effect is particularly helpful in deriving the matching conditions. In the second problem we consider in the next section, {\em i.e.,} the multiple access channel problem, this decoupling effect is even more pronounced.
\subsection{Cholesky Factorization and a Necessary Condition For Matching}
\label{sec:Cholesky}
The condition given Theorem \ref{theorem:BC} is in a positive semidefinite form, however, due to the specific problem structure, it can also be represented as a set of recursive conditions, which is discussed in this section. This alternative representation also leads to a necessary condition for matching to hold, which plays an instrumental role for several results given in Section \ref{sec:BCchannels}, where we answer the second question regarding the existence of non-trivial set of matched channels.
Determining whether a matrix is positive semidefinite is equivalent to computing the LDL decomposition, and checking whether the resultant diagonal matrix in the decomposition has non-negative entries. The matrix $\Sigma^{(0)}$ is positive semidefinite if and only if the diagonal matrix in the LDL decomposition has only non-negative entries. Computationally this can be accomplished with the Cholesky factorization \cite{matrixcomputation} on the matrix $\Sigma^{(0)}$. Here we provide an intuitive description of the Cholesky factorization in the context of the problem being considered, and its conceptual interpretation as the recursive thresholding determination for the channel to yield a matching.
In the first step of the Cholesky factorization, we use symmetric column and row Gaussian elimination to eliminate all the entries of the $M$-th column and the $M$-th row, except the diagonal entry. Denote the resulting upper-left $(M-1)\times(M-1)$ matrix after this first step as $\Sigma^{(1)}$. A necessary condition for the matrix $\Sigma^{(0)}$ to be positive definite is that the lower right entry of the matrix $\Sigma^{(0)}$ is strictly positive, or all the entries on the last column are zero. Notice that the condition only involves $\sigma^2_{X|Y_M}$ or the channel noise power $\sigma^2_{Z_M}$, which yields a necessary condition on $\sigma^2_{Z_M}$ in the form of $\sigma^2_{Z_M}\geq f^{(0)}(P,\bar{\alpha})$.
Continue the Cholesky factorization on $\Sigma^{(1)}$, and a similar necessary condition is its lower right entry is strictly positive,
or the entries on the $(M-1)$-th row of $\Sigma^{(1)}$ are zero. Similarly as the previous step, the condition on $\sigma^2_{Z_{M-1}}$ is found to be in the form that $\sigma^2_{Z_{M-1}}\geq f^{(1)}(P,\bar{\alpha},\sigma^2_{Z_M})$.
Continuing this process will yield a set of conditions in the form of
\begin{align}
\sigma^2_{Z_{m}}\geq f^{(M-m)}(P,\bar{\alpha},\sigma^2_{Z_M},\sigma^2_{Z_M},\sigma^2_{Z_{M-1}},\ldots,\sigma^2_{Z_{m+1}}),\quad m=M,M-1,\ldots,1.
\end{align}
The matrix $\Sigma^{(0)}$ is positive definite if and only if all such threshold conditions are satisfied.
Notice that the threshold function $f^{(M-m)}(P,\bar{\alpha},\sigma^2_{Z_M},\sigma^2_{Z_M},\sigma^2_{Z_{m-1}})$ for $\sigma^2_{Z_{m}}$ depends on the channel noise power values $(\sigma^2_{Z_M},\sigma^2_{Z_{M-1}},\ldots,\sigma^2_{Z_{m+1}})$, but not on $(\sigma^2_{Z_{1}},\sigma^2_{Z_{2}},\ldots,\sigma^2_{Z_{m-1}})$. Thus these functions $f^{(m)}(\cdot)$, $m=M,M-1,\ldots,1$ can be viewed as a recursive threshold checking (or determination) procedure, and the channel noise power $\sigma^2_{Z_m}$ needs to be chosen to be larger than the threshold determined by $(\sigma^2_{Z_M},\sigma^2_{Z_{M-1}},\ldots,\sigma^2_{Z_{m+1}})$ in every step to yield a matching. Given the above observation, it is natural to speculate that if a channel is matched, then any more noisy channel also induces a match. This intuition is in fact correct, and the statement is made more rigorous in the next section as Corollary \ref{coro1}.
One necessary condition for a matching to exist is that the matrix $\Sigma_{V_1,V_2,\ldots, V_M}$ is positive semidefinite. We can thus apply the Cholesky factorization technique on this particular matrix to obtain a necessary condition for matching to exist.
\begin{lemma}
For the matrix $\Sigma_{V_1,V_2,\ldots, V_M}$ constructed previously to be positive semidefinite (with $\sigma^2_{Z_M}>0$), it must be true that $\alpha_i\beta_i\geq 0$, $i=1,2,\ldots,M$.
\end{lemma}
Note that this condition is essentially independent of the channel, as long as the channel is not perfect. This lemma is proved in the Appendix.
\subsection{Properties and Existence of Matched Channels}
\label{sec:BCchannels}
With Lemma 1, we can establish several properties of the set of matched channels, given next as corollaries to Theorem \ref{theorem:BC}. Their proofs are provided in the Appendix. These properties essentially provide an answer to the second question posed earlier, and we shall also further illustrate such sources and channels using an example.
\begin{corollary}
\label{coro1}
If an uncoded scheme is matched on $(\sigma^2_{Z_1},\sigma^2_{Z_2},\ldots,\sigma^2_{Z_M})$, then it is matched and thus optimal on any channel with noise powers $\sigma^2_{Z^+_1}\geq \sigma^2_{Z^+_2}\geq \ldots\geq \sigma^2_{Z^+_M}$ where $\sigma^2_{Z^+_m}\geq \sigma^2_{Z_m}$, $m=1,2,\ldots,M$.
\end{corollary}
The corollary reveals a property of matched channels: once a channel is matched, any channel with more noise is also a matched channel and thus the uncoded scheme is optimal. The next corollary states, from the perspective of only the source and the uncoded scheme parameters, the necessary and sufficient condition for matching to exist.
\begin{corollary}
\label{coro2}
Matching (on a broadcast channel with finite noise powers) exists, if and only if $\alpha_i\beta_i>0$ and the matrix $\Pi\Sigma_{S_1,S_2,\ldots,S_M}\Pi$ has its largest eigenvalue being 1 with multiplicity 1, where $\Pi$ is a diagonal matrix with diagonal entries being
\begin{align}
\left(\sqrt{\frac{\alpha_1}{\sum_{i=1}^M\rho_{1,i}\alpha_i}},\sqrt{\frac{\alpha_2}{\sum_{i=1}^M\rho_{2,i}\alpha_i}},\ldots,\sqrt{\frac{\alpha_M}{\sum_{i=1}^M\rho_{M,i}\alpha_i}}\right).
\end{align}
Moreover, if the above condition holds, then any channel with $\sigma^2_{Z_1}\geq \sigma^2_{Z_2}\ldots \sigma^2_{Z_M}\geq \sigma^2_Z$ is a matched channel, where $\sigma^2_Z=\frac{\lambda_2P}{1-\lambda_2}$, and $\lambda_2$ is the second largest eigenvalue of the matrix $\Pi\Sigma_{S_1,S_2,\ldots,S_M}\Pi$.
\end{corollary}
\textbf{Remark:} If the entries of $\mbox{diag}(\bar{\alpha})\Sigma_{S_1,S_2,\ldots,S_M}\mbox{diag}(\bar{\alpha})$ are strictly positive, then matching is always possible.
This follows from the fact that the matrix $\Pi\Sigma_{S_1,S_2,\ldots,S_M}\Pi$ has positive entries, and $\bar{v}^t_1=(\sqrt{\alpha_1\beta_1}, \sqrt{\alpha_2\beta_2}, ..., \sqrt{\alpha_M\beta_M})$ is its positive eigenvector, such that $1$ is its largest eigenvalue with multiplicity 1 (by Perron-Frobenius Theorem \cite{matrixanalysis}).
Different from the case discussed in the previous remark, the next corollary gives another sufficient condition for matching to occur when the sources and the coding parameters satisfy the same positive correlation condition.
\begin{corollary}
\label{coro3}
Let the entries of the matrix $\mbox{diag}(\bar{\alpha})\Sigma_{W_1,W_2,\ldots,W_M}\mbox{diag}(\bar{\alpha})$ be strictly positive. Denote the entries of $\Sigma_{S_1,S_2,\ldots,S_M}$ as $\rho_{i,j}$, and define
\begin{align}
\sigma^2_{Z^*_m}\triangleq \max_{j<m} \frac{\beta_j\beta_m}{\rho_{j,m}}P^2-P, \quad m=2,3,\ldots,M.
\end{align}
Any channel with $\sigma^2_{Z_1}\geq \sigma^2_{Z_2}\geq \ldots\geq \sigma^2_{Z_M}$ such that $\sigma^2_{Z_m}\geq \sigma^2_{Z^*_m}$ for $m=2,3,\ldots,M$ is a matched channel.
\end{corollary}
\textbf{Remark:} $\sigma^2_{Z^*_m}$ as defined above may in fact be negative. However this does not cause any discrepancy due to the requirement $\sigma^2_{Z_1}\geq \sigma^2_{Z_2}\geq \ldots\geq \sigma^2_{Z_M}$. As a reality check, notice that
\begin{align}
\sum_{i=1}^M(\alpha_i)(\beta_i\beta_m P)=P\beta_m=\sum_{i=1}^M(\alpha_i)(\rho_{i,m}),
\end{align}
but $\beta_M^2P<\sigma^2_{S_M}$ unless $\alpha_1=\alpha_2=\ldots=\alpha_{M-1}=0$, which however would contradict our assumption.
It thus follows
\begin{align}
\max_{j<M} \frac{\beta_j\beta_mP}{\rho_{j,M}}> 1,
\end{align}
and thus $\sigma^2_{Z^*_M}>0$ always holds under the condition in the corollary.
\textbf{Remark:} For the symmetric case where $\sigma^2_{S_i}=\sigma^2_S$, $\sigma^2_{S_iS_j}=\rho\sigma^2_S$, $\alpha_i=\alpha$ and $\Ex[S_i|X]=\beta X$, for $i=1,2,\ldots,M$. A necessary and sufficient condition for matching is simply
\begin{align}
\frac{\sigma^2_{Z_M}}{P+\sigma^2_{Z_M}}\geq \frac{1-\rho}{1+(M-1)\rho}.
\label{eqn:symmetric}
\end{align}
To see this, notice that
\begin{align}
P=\alpha^2M\sigma^2_S[1+(M-1)\rho].
\end{align}
and $\beta$ can be computed as
\begin{align}
\beta =\frac{1}{\alpha M}.
\end{align}
Checking the first condition in the Cholesky factorization, it is easily verified that (\ref{eqn:symmetric}) is a necessary condition for matching. However, from Corollary \ref{coro3}, it is seen that it is sufficient to choose any $\sigma^2_{Z_m}\geq \sigma^2_{Z^*_m}$, where
$\sigma^2_{Z^*_m}=\frac{\beta^2}{\rho \sigma^2_{S}}P^2-P$, $m=2,3,\ldots,M$. This is exactly condition (\ref{eqn:symmetric}).
\subsection{An Example: A Source with Three Components}
\begin{figure}[tb]
\centering
\includegraphics[width=8cm]{figs/rho1rho2}
\caption{Determining the $(\rho_1,\rho_2)$ pairs for which matching is possible, given in shade. \label{fig:rho1rho2}}
\end{figure}
Let us consider a source with three components whose covariance matrix is either
\begin{align}
\Sigma_{S_1,S_2,S_3}=\begin{pmatrix}
1& \rho_{1}&\rho_{2}\\
\rho_{1}&1&\rho_{1}\\
\rho_{2}&\rho_{1}&1
\end{pmatrix},\label{eqn:cov1}
\end{align}
or
\begin{align}
\Sigma_{S_1,S_2,S_3}=\begin{pmatrix}
1& \rho_{2}&\rho_{1}\\
\rho_{2}&1&\rho_{1}\\
\rho_{1}&\rho_{1}&1
\end{pmatrix},\label{eqn:cov2}
\end{align}
and further assume that the coefficients are chosen $\alpha_1=\alpha_2=\alpha_3=1$ in the uncoded scheme.
In addition to the constraint that the matrix $\Sigma_{S_1,S_2,S_3}$ must be positive definite, for a matching to exist, the condition in Corollary \ref{coro2} must be satisfied.
It can be shown that the eigenvalues of $\Pi\Sigma_{S_1,S_2,S_3}\Pi$ are
\begin{align}
\lambda_1=1,\quad \lambda_2=\frac{-2\rho_1^2 + \rho_2 + 1}{2\rho_1^2+2\rho_1\rho_2+3\rho_1 + \rho_2 + 1},\quad \lambda_3=\frac{1-\rho_2}{\rho_1 + \rho_2 + 1},
\end{align}
and we must have $\lambda_2<1$ and $\lambda_3<1$. In the appendix, we show that the valid choices are the $(\rho_1,\rho_2)$ pairs such that
\begin{align}
\rho_2< 1,\quad 0< \rho_1< 1,\quad \rho_1+2\rho_2>0,\quad \rho_2> 2\rho_1^2-1. \label{eqn:source2}
\end{align}
The corresponding region is plotted in Fig \ref{fig:rho1rho2}. Notice that the two matrices are equivalent for the purpose of determining whether matching is possible, thus the region in Fig \ref{fig:rho1rho2} is valid for both cases.
Next let us fix a $(\rho_1,\rho_2)$ pair, and consider the region of $(\frac{P\sigma^2_{Z_2}}{P+\sigma^2_{Z_2}},\frac{P\sigma^2_{Z_3}}{P+\sigma^2_{Z_3}})$ pairs such that matching occurs. The tradeoffs can be computed explicitly, and are illustrated in Fig. \ref{fig:N1N2} for $(\rho_1,\rho_2)=(\frac{1}{2},\frac{1}{6})$. The circles in the plots give the channels specified by Corollary \ref{coro2}. The channels given by Corollary \ref{coro3} can be computed directly (given as the dots), which is loose in the first case, but on the lower boundary (and it is an extreme point) for the second case. Since $\sigma^2_{Z_3}\geq \sigma^2_{Z_2}$, we also include this boundary in the plot. For the first case, the boundary $\frac{P\sigma^2_{Z_3}}{P+\sigma^2_{Z_3}}<P$ is also shown, while for the second, the lower bound $y\geq\frac{16}{15}$ required by the function $f^{(0)}(P,\bar{\alpha})$ in the first step of the Cholesky factorization is shown.
The corresponding channels that matching occurs are those inside the \lq\lq{}fan\rq\rq{} regions. Note that there is a tension between the noise powers $\sigma^2_{Z_2}$ and $\sigma^2_{Z_3}$ for matching to occur with for the fixed source and uncoded scheme.
\begin{figure}[tb]
\centering
\includegraphics[width=16cm]{figs/N1N2_2cases}
\caption{Illustration of the regions of matched channel parameters when $(\rho_1,\rho_2)=(\frac{1}{2},\frac{1}{6})$ for the two covariance matrices (\ref{eqn:cov1}) and (\ref{eqn:cov2}), respectively. \label{fig:N1N2}}
\end{figure}
\section{Vector Gaussian CEO on a Gaussian Multiple-Access Channel}
In this section we consider the problem of sending correlated Gaussian sources on a Gaussian multiple-access channel, where the transmitters observe noise linear combinations of the source components; see also Fig. \ref{fig:MAC} for an illustration.
A zero-mean vector Gaussian source $(S_1[n],S_2[n],\ldots,S_M[n])$ has a covariance matrix $\Sigma_{S_1,S_2,\ldots,S_M}$. There are a total of $L$ sensors, whose observations are $(T_1[n],T_2[n],\ldots,T_L[n])$, respectively, with covariance matrix $\Sigma_{T_1,T_2,\ldots,T_L}$. The sources and observations are jointly Gaussian. Each sensor observes $T^N_\ell$, encodes it under an average transmission power constraint $P_\ell$, $\ell=1,2,\ldots,L$.
The channel output is given as
\begin{align}
Y[n]=Z[n]+\sum_{\ell=1}^L\delta_\ell X_\ell[n],
\end{align}
where $\delta_\ell>0$. The receiver wishes to reconstruct $(S^N_1,S^N_2,\ldots,S^N_M)$ using channel output $Y^N$ to minimize the individual MSE measure, which achieves MSE distortion $D_m$ for $S_m$, {\em i.e.}, $D_m=\frac{1}{N}\sum_{n=1}^N\Ex(S_m[n]-\hat{S}_m[n])^2$.
Notice that due to the jointly Gaussian distribution, we can write
\begin{align}
\Ex[S_m|T_1,T_2,\ldots,T_L]\triangleq \tilde{S}_m=\sum_{\ell=1}^L\gamma_{m,\ell}T_\ell,\quad m=1,2,\ldots,M.
\end{align}
The parameters $\gamma_{m,\ell}$ can be conveniently written in a matrix form $\Gamma$, and computed as
\begin{align}
{\Gamma}=\Sigma_{(S_1,S_2,\ldots,S_M),(T_1,T_2,\ldots,T_L)}\Sigma^{-1}_{T_1,T_2,\ldots,T_L},
\end{align}
where $\Sigma_{(S_1,S_2,\ldots,S_M),(T_1,T_2,\ldots,T_L)}$ is the cross-covariance matrix between vectors $(S_1,S_2,\ldots,S_M)$ and $(T_1,T_2,\ldots,T_L)$.
Notice that the problem can be equivalently formulated as computation of linear functions of Gaussian sources on the multiple-access channel. In this alternative setting, the functions to be computed are $(S_1,S_2,\ldots,S_M)$, which can be represented as noisy linear functions of the sensor observations $(T_1,T_2,\ldots,T_L)$. This alternative formulation is notationally more involved in the current problem setting, but we shall explore this connection in a separate work.
We assume $M\leq L$, and consider the case that the matrices $\Sigma_{S_1,S_2,\ldots,S_M}$, $\Sigma_{T_1,T_2,\ldots,T_L}$, $\Sigma_{\tilde{S}_1,\tilde{S}_2,\ldots,\tilde{S}_M}$ and $\Sigma_{(S_1,S_2,\ldots,S_M),(T_1,T_2,\ldots,T_L)}$ all have full (row) rank, which hold in general except certain degenerate cases. Denote the entries of $\Sigma_{T_1,T_2,\ldots,T_L}$ as $\psi_{i,j}$. The uncoded scheme we consider is
\begin{align}
X_\ell[n]=\eta_\ell\sqrt{\frac{P_\ell}{\psi_{\ell,\ell}}}T_\ell[n],\quad \ell=1,2,\ldots,L,\label{eqn:Xform}
\end{align}
where $\eta_\ell$ is either $+1$ or $-1$ to be specified next. In other words, each sensor sends its noisy observation directly using the full power, but it can choose whether to negate its observations. The $m$-th receiver estimates $S_m[n]$ as $\hat{S}_m[n]=\Ex[S_m[n]|Y[n]]$.
Define
\begin{align}
&\bar{\alpha}\triangleq\left[\Sigma_{(S_1,S_2,\ldots,S_M),(T_1,T_2,\ldots,T_L)}\Sigma^t_{(S_1,S_2,\ldots,S_M),(T_1,T_2,\ldots,T_L)}\right]^{-1}\nonumber\\
&\qquad\cdot\Sigma_{(S_1,S_2,\ldots,S_M),(T_1,T_2,\ldots,T_L)}\Sigma_{T_1,T_2,\ldots,T_L}\left(\delta_1 \eta_1\sqrt{\frac{P_1}{\psi_{1,1}}},\delta_2 \eta_2\sqrt{\frac{P_2}{\psi_{2,2}}},\ldots,\delta_L \eta_L\sqrt{\frac{P_L}{\psi_{L,L}}}\right)^t,\label{eqn:alphas}
\end{align}
and we assume $\alpha_m\neq 0$, $m=1,2,\ldots,M$, which is true in general except certain degenerate cases. Our main result on this problem is summarized in the following theorem.
\begin{figure}[tb]
\centering
\includegraphics[width=16cm]{figs/MAC}
\vspace{-0.5cm}
\caption{Sending correlated Gaussian random variables on a Gaussian multiple-access channel with noise observations. \label{fig:MAC}}
\end{figure}
\begin{theorem}
\label{theorem:MAC}
A Gaussian multiple-access channel is said to be matched to a given Gaussian source and an uncoded scheme with parameters $\bar{\eta}$, and the distortion vector induced by the given scheme is on the boundary of the achievable distortion region and thus optimal, if
\begin{enumerate}
\item $\eta_\ell\eta_{\ell'}\psi_{\ell,\ell'}\geq 0,\quad 1\leq \ell<\ell'\leq L$;
\item The vector $\left(\delta_1 \eta_1\sqrt{\frac{P_1}{\psi_{1,1}}},\delta_2 \eta_2\sqrt{\frac{P_2}{\psi_{2,2}}},\ldots,\delta_L \eta_L\sqrt{\frac{P_L}{\psi_{L,L}}}\right)\Sigma_{T_1,T_2,\ldots,T_L}$ is in the row space of the matrix $\Sigma_{(S_1,S_2,\ldots,S_M),(T_1,T_2,\ldots,T_L)}$;
\item $\sigma^2_Z\geq \frac{\lambda_2P}{1-\lambda_2}$, where $\lambda_2$ is the second largest eigenvalue of the matrix $\Pi\Sigma_{\tilde{S}_1,\tilde{S}_2,\ldots,\tilde{S}_M}\Pi$, where $\Pi$ is a diagonal matrix with diagonal entries
\begin{align}
\left(\sqrt{\frac{\alpha_1}{\sum_{i=1}^M\rho_{1,i}\alpha_i}},\sqrt{\frac{\alpha_2}{\sum_{i=1}^M\rho_{2,i}\alpha_i}},\ldots,\sqrt{\frac{\alpha_M}{\sum_{i=1}^M\rho_{M,i}\alpha_i}}\right),
\end{align}
and $\rho_{m,j}$'s are the entries of the matrix
\begin{align}
\Sigma_{\tilde{S}_1,\tilde{S}_2,\ldots,\tilde{S}_M}=\Sigma_{(S_1,S_2,\ldots,S_M),(T_1,T_2,\ldots,T_L)}\Sigma^{-1}_{T_1,T_2,\ldots,T_L}\Sigma^t_{(S_1,S_2,\ldots,S_M),(T_1,T_2,\ldots,T_L)}.
\end{align}
\end{enumerate}
\end{theorem}
These conditions can be intuitively explained as follows: condition one guarantees that the channel inputs from all transmitters coherently add up; condition two stems from the requirement that the noise observations should serve the same role as the underlying source for the chosen power constraints and amplification factors, {\em i.e.,} as if the observation noise does not exist; condition three is similar to the effect in the previous problem where once a channel is matched, a more noisy channel will also induce a match.
When all $\psi_{\ell,\ell\rq{}}\geq 0$, we can simply choose $\eta_\ell=+1$ (or $-1$) for all $\ell$ to satisfy the first condition. However, when some of the terms $\psi_{\ell,\ell\rq{}}$ are negative, a simple algorithmic approach can be used to determine whether there exists a valid assignment of $\eta_\ell$. In fact this condition is completely source dependent, and the choice of $\{\eta_\ell,\ell=1,2,\ldots,L\}$ is unique up to a negation (assuming any component $T_\ell$ is not completely independent of the others), and thus can be considered fixed for a given source observation covariance matrix.
The proof of this theorem also has two parts given in Section \ref{sec:outerMAC} and Section \ref{sec:innerMAC}. This theorem answers the first question regarding the conditions to certify whether an uncoded scheme is optimal in this communication problem. The answer to the second question for this problem turns out to be simpler than that in the broadcast case, and we discuss in Section \ref{sec:MACexample} as special case examples several problems previously considered in the literature.
\subsection{Extracting the Critical Conditions from the Outer Bound}
\label{sec:outerMAC}
Define
\begin{align}
\Ex(S_m-\tilde{S}_m)^2=\Delta_m,\quad m=1,2,\ldots,M,
\end{align}
and thus
\begin{align}
\Ex\tilde{S}^2_m=\sigma^2_{S_m}-\Delta_m,\quad m=1,2,\ldots,M.
\end{align}
The reason to introduce $\tilde{S}_m$'s is that in the remote coding setting, the distortion can be decomposed into two independent parts: the first part is due to encoding the observable part of the underlying sources, which are $\tilde{S}$ random variables, with a distortion, and the second is due to the inherent noisy nature of the observations which induces a fixed distortion $\Delta_m$. Thus encoding the source $S_m$ to distortion $D_m$ is equivalent to encoding the equivalent source $\tilde{S}_m$ to distortion $D_m-\Delta_m$.
We can now derive an outer bound by combining the approach used in the broadcast problem with a technique based on Witsenhausen's bound \cite{Witsenhausen}. Again consider $M$ auxiliary zero-mean Gaussian random variables $(W_1,W_2,\ldots,W_M)$ with covariance matrix $\Sigma_{W_1,W_2,\ldots,W_M}$, which are independent of everything else, and write
\begin{align}
U_m[n]=\tilde{S}_m[n]+W_m[n],\quad m=1,2,\ldots,M.
\end{align}
Notice the Markov string
\begin{align}
(U^N_1,U^N_2,\ldots,U^N_M)\leftrightarrow (\tilde{S}^N_1,\tilde{S}^N_2,\ldots,\tilde{S}^N_M)\leftrightarrow (T^N_1,T^N_2,\ldots,T^N_L)\leftrightarrow (X^N_1,X^N_2,\ldots,X^N_L)\leftrightarrow Y,
\end{align}
and we can write using data processing inequality that
\begin{align}
I(X^N_1,X^N_2,\ldots,X^N_L;Y^N)\geq I(U^N_1,U^N_2,\ldots,U^N_M;Y^N),\label{eqn:dataprocessing}
\end{align}
where equality holds if and only if
\begin{align}
h(Y^N|X^N_1,X^N_2,\ldots,X^N_L)=h(Y^N|U^N_1,U^N_2,\ldots,U^N_M).\label{eqn:MACindep}
\end{align}
Following the exact steps as in \cite{Gastpar:08} (see also \cite{Lapidoth:10MAC}) and applying Witsenhausen's bound \cite{Witsenhausen}, we can obtain
\begin{align}
I(X^N_1,X^N_2,\ldots,X^N_L;Y^N)\leq \frac{N}{2}\log \left(1+\frac{P}{\sigma^2_Z}\right),
\label{eqn:channelcondition}
\end{align}
where
\begin{align}
P\triangleq \sum_{\ell=1}^L\delta^2_\ell P_\ell+2\sum_{\ell=1}^L\sum_{\ell'=\ell+1}^L\rho^*_{\ell,\ell'}\delta_\ell\delta_{\ell'}\sqrt{P_\ell P_{\ell'}}.
\end{align}
and $\rho^*_{\ell,\ell'}=|\psi_{\ell,\ell'}(\psi_{\ell,\ell}\psi_{\ell',\ell'})^{-\frac{1}{2}}|$. This inequality intuitively says that the mutual information between the channel inputs and the output is upper bounded by the capacity of a point-to-point channel, whose power constraint is equal to the resultant signal power when all the inputs on the multiple-access channel are coherently added. We will not attempt to further simplify this condition at this point, since in the context of the uncoded scheme, it has a particularly simple form.
The right hand side of (\ref{eqn:dataprocessing}) can be bounded similarly as in the broadcast problem. Here the equivalent source is $({\tilde{S}_1,\tilde{S}_2,\ldots,\tilde{S}_M})$, and the distortion vectors are $(D_1-\Delta_1,D_2-\Delta_2,\ldots,D_M-\Delta_M)$, and moreover, $\sigma^2_{Z_m}=\sigma^2_{Z}$ for $m=1,2,\ldots,M$. We thus arrive at
\begin{align}
I(U^N_1,U^N_2,\ldots,U^N_M;Y^N)\geq\frac{N}{2}\log\frac{|\Sigma_{\tilde{S}_1,\tilde{S}_2,\ldots,\tilde{S}_M}+\Sigma_{W_1,W_2,\ldots,W_M}|}{\Pi^M_{m=1}(D_m-\Delta_m+\sigma^2_{W_m})},
\label{eqn:sameasBC}
\end{align}
where equality holds if and only if
\begin{align}
h(U^N_m|Y^N)&=\frac{N}{2}\log [2\pi e(D_m-\Delta_m+\sigma^2_{W_m} )],\quad m=1,2,\ldots,M,\label{eqn:MACcondition1}\\
h(U^N_m|Y^N,U^N_1,U^N_2,\ldots,U^N_{m-1})&=h(U^N_m|Y^N),\quad m=2,3,\ldots,M.\label{eqn:MACcondition2}
\end{align}
An outer bound is then obtained by combining (\ref{eqn:dataprocessing}), (\ref{eqn:channelcondition}) and (\ref{eqn:sameasBC}). Again the precise form of this outer bound is less important than the extracted matching conditions (\ref{eqn:MACindep}), (\ref{eqn:channelcondition}) being equality, (\ref{eqn:MACcondition1}) and (\ref{eqn:MACcondition2}). The condition (\ref{eqn:channelcondition}) being equality and the condition (\ref{eqn:MACcondition1}) can be satisfied simply by choosing a proper jointly Gaussian coding scheme, and the conditions (\ref{eqn:MACindep}) and (\ref{eqn:MACcondition2}) are almost identical to (\ref{eqn:degerate}) and (\ref{eqn:BCcondition2}) in the broadcast case.
\subsection{The Forward Matching Conditions}
\label{sec:innerMAC}
Since the uncoded scheme takes single letter encoding function, (\ref{eqn:channelcondition}) being equality is equivalent to
\begin{align}
I(X_1,X_2,\ldots,X_L;Y)=\frac{1}{2}\log \left(1+\frac{P}{\sigma^2_Z}\right).
\end{align}
Because in the uncoded scheme the channel input $X$ is given in (\ref{eqn:Xform}), the equality holds as long as
\begin{align}
\eta_\ell\eta_{\ell'}\psi_{\ell,\ell'}\geq 0,\quad 1\leq \ell<\ell'\leq L.
\end{align}
This yields the first condition stated in Theorem \ref{theorem:MAC}.
The conditions (\ref{eqn:MACindep}) and (\ref{eqn:MACcondition2}) in the context of uncoded scheme are equivalent to
\begin{align}
h(Z)&=h(Y|U_1,U_2,\ldots,U_M),\label{eqn:MACasinnonoise}\\
h(U_m|Y,U_1,U_2,\ldots,U_{m-1})&=h(U_m|Y),\quad m=2,3,\ldots,M.\label{eqn:MACcondition2singleletter}
\end{align}
Denote
\begin{align}
\tilde{X}=\sum_{\ell=1}^L\delta_\ell X_\ell.
\end{align}
For (\ref{eqn:MACasinnonoise}) to hold with equality, two conditions must hold
\begin{align}
\Ex[\tilde{X}|\tilde{S}_1,\tilde{S}_2,\ldots,\tilde{S}_M]=\tilde{X},\label{eqn:dataprocessing1}
\end{align}
and
\begin{align}
\Ex[\tilde{X}|U_1,U_2,\ldots,U_M]=\tilde{X}.\label{eqn:dataprocessing2}
\end{align}
Let us consider the first condition (\ref{eqn:dataprocessing1}). Due to the jointly Gaussian distribution, there exists a set of coefficients $(\alpha_1,\alpha_2,\ldots,\alpha_M)$ such that
\begin{align}
\Ex[\tilde{X}|\tilde{S}_1,\tilde{S}_2,\ldots,\tilde{S}_M]=\sum_{m=1}^M\alpha_m\tilde{S}_m=\sum_{m=1}^M\alpha_m\sum_{\ell=1}^L\gamma_{m,\ell}T_\ell.
\end{align}
However notice that
\begin{align}
\tilde{X}=\sum_{\ell=1}^L\delta_\ell \eta_\ell\sqrt{\frac{P_\ell}{\psi_{\ell,\ell}}}T_\ell,
\end{align}
thus the condition (\ref{eqn:dataprocessing1}) is equivalent to the fact that the vector
\begin{align}
\left(\delta_1 \eta_1\sqrt{\frac{P_1}{\psi_{1,1}}},\delta_2 \eta_2\sqrt{\frac{P_2}{\psi_{2,2}}},\ldots,\delta_L \eta_L\sqrt{\frac{P_L}{\psi_{L,L}}}\right)
\end{align}
is in the row space of the matrix ${\Gamma}$. Equivalently, the vector
$$\left(\delta_1 \eta_1\sqrt{\frac{P_1}{\psi_{1,1}}},\delta_2 \eta_2\sqrt{\frac{P_2}{\psi_{2,2}}},\ldots,\delta_L \eta_L\sqrt{\frac{P_L}{\psi_{L,L}}}\right)\Sigma_{T_1,T_2,\ldots,T_L}$$
needs to be in the row space of the matrix $\Sigma_{(S_1,S_2,\ldots,S_M),(T_1,T_2,\ldots,T_L)}$. This leads to the second condition stated in Theorem \ref{theorem:MAC}.
When this condition is satisfied, the coefficients $\bar{\alpha}$ can be determined exactly as in (\ref{eqn:alphas}).
The conditions (\ref{eqn:MACcondition2singleletter}) and (\ref{eqn:dataprocessing2}) are now identical to the broadcast case with $\tilde{S}_1,\tilde{S}_2,\ldots,\tilde{S}_M$ being the sources and $\tilde{X}$ being the channel input. By Corollary 2, such a channel is matched when the second largest eigenvalue of the matrix $\Pi\Sigma_{\tilde{S}_1,\tilde{S}_2,\ldots,\tilde{S}_M}\Pi$ is less than $\frac{\sigma^2_Z}{P+\sigma^2_Z}$, or in other words, the noise power must be above or equal to the given threshold stated in Theorem \ref{theorem:MAC}.
\textbf{Remark:} The first condition in Theorem \ref{theorem:MAC} generally has a unique solution if it can be satisfied, up to a negation of all the signs of the channel input signals. The second condition can almost always be satisfied by choosing appropriate a $(\delta_1,\delta_2,\ldots,\delta_L)$ vector, except a few special cases where an all positive solution does not exist (recall we have assumed $\delta_\ell>0$, and thus only all positive solutions are valid). If the third condition is satisfied for certain source-channel-code triple, then it is satisfied for any more noisy channels. It is seen that the critical conditions in the outer bound derivation essentially decouples the matching problem into several simpler ones, leading to the three largely independent conditions given in Theorem \ref{theorem:MAC}.
\subsection{Matched Channels in Special Case Scenarios}
\label{sec:MACexample}
In the multiple-access setting, the conditions for matching in Theorem \ref{theorem:MAC} are already rather simple, and there is no need to further investigate the properties of matched channels as in the broadcast case. Next we consider two special cases in the general problem setting which extend those considered in \cite{Gastpar:08} and \cite{Lapidoth:10MAC}, respectively.
\subsubsection{The Scalar CEO Problem}
\begin{figure}[tb]
\centering
\includegraphics[width=14cm]{figs/CEO}
\vspace{-0.5cm}
\caption{The scalar Gaussian CEO problem on a Gaussian multiple-access channel. \label{fig:CEO}}
\end{figure}
Consider a zero-mean scalar Gaussian source $S[n]$ with variance $\sigma^2_S$. There are a total of $L$ sensors, whose observations are
\begin{align}
T_\ell[n]=d_\ell S[n]+Z\rq{}_\ell[n], \ell=1,2,\ldots,L,
\end{align}
where $d_\ell\geq 0$ (without loss of generality) and $Z\rq{}_\ell[n]$\rq{}s are the zero-mean independent additive noise with variance $\sigma^2_{Z\rq{}}$. This special case is depicted in Fig. \ref{fig:CEO}.
It is clear that the first condition in Theorem \ref{theorem:MAC} is satisfied by $\eta_\ell=1$ for all $\ell=1,2,\ldots,L$.
The second condition for this case is equivalent to
\begin{align}
\left(\delta_1 \sqrt{\frac{P_1}{\psi_{1,1}}},\delta_2 \sqrt{\frac{P_2}{\psi_{2,2}}},\ldots,\delta_L \sqrt{\frac{P_L}{\psi_{L,L}}}\right)\Sigma_{T_1,T_2,\ldots,T_L}\propto (d_1,d_2,\ldots,d_L),
\end{align}
where $\propto$ here means a component-wise proportional relation.
In other words, the uncoded scheme is optimal if
\begin{align}
\delta_\ell\sqrt{P_\ell(d^2_\ell\sigma^2_S+\sigma^2_{Z\rq{}})}+\sum_{\ell'\neq \ell}\delta_{\ell'}\sqrt{\frac{P_{\ell\rq{}}}{d^2_{\ell'}\sigma^2_S+\sigma^2_{Z\rq{}}}}d_\ell d_{\ell'}\sigma^2_S\propto d_\ell.
\end{align}
However, the LHS of the above condition can be simplified to
\begin{align}
\delta_m\sigma^2_{Z\rq{}}\sqrt{\frac{P_\ell}{d^2_\ell\sigma^2_S+\sigma^2_{Z'}}}+d_\ell\sum_{\ell'=1}^M\delta_{\ell'}d_{\ell'}\sqrt{\frac{P_{\ell'}}{d^2_{\ell'}\sigma^2_S+\sigma^2_{Z'}}}\sigma^2_S,
\end{align}
where the second term is proportional to $d_\ell$, and the first term is proportional to $d_\ell$ if and only if
\begin{align}
\frac{P_\ell\delta^2_\ell}{(d^2_\ell\sigma^2_S+\sigma^2_{Z'})d^2_\ell}=\mbox{const},\quad \ell=1,2,\ldots,L.
\label{eqn:proportional}
\end{align}
It remains to check the third condition, however in this case $M=1$, and the second eigenvalue of the matrix $\Pi\Sigma_{\tilde{S}_1,\tilde{S}_2,\ldots,\tilde{S}_M}\Pi$ can be viewed as zero, thus any noise power $\sigma^2_Z$ will allow a matching. Summarizing the above analysis, it is seen that for the scalar CEO problem on a Gaussian multiple-access channel, as long as the condition (\ref{eqn:proportional}) holds, the uncoded scheme is optimal. Conversely, for any noisy observation qualities, there always exists a matched channel by choosing the values of $\delta_\ell$ properly.
The condition (\ref{eqn:proportional}) corresponds to a proportional quality requirement: the quality of the observations need to match the transmission powers and the transmission amplification factors. Gastpar \cite{Gastpar:08} showed that when all the sensors have the same observation quality, the same power and the same amplification factor, the uncoded scheme is optimal. Our result thus generalizes it to the proportional case.
\subsubsection{Correlated Gaussian on a Gaussian Multiple-Access Channel}
Consider the case when $M=L$, and we shall assume that the first condition in Theorem \ref{theorem:MAC} can be satisfied. The second condition is also satisfied trivially since the matrix $\Sigma_{(S_1,S_2,\ldots,S_M),(T_1,T_2,\ldots,T_L)}$ is full rank in our problem setting. Thus only the last condition needs to be checked in this case. Equivalently, when $\lambda_2$ is strictly less than $1$, there always exists a noise power $\sigma^2_Z$ such that the channel is matched and thus the uncoded scheme is optimal.
\begin{figure}[tb]
\centering
\includegraphics[width=16cm]{figs/LPCAL}
\vspace{-0.5cm}
\caption{Sending correlated Gaussians on a Gaussian multiple-access channel. \label{fig:LOCAL}}
\end{figure}
Lapidoth and Tinguely \cite{Lapidoth:10MAC} previously considered the case that $T_m=S_m$, $m=1,2,\ldots,M$; see Fig. \ref{fig:LOCAL}. It was shown that for covariance matrix $\Sigma_{S_1,S_2,\ldots,S_M}$ with strictly positive entries, there always exists a noise power $\sigma^2_Z$ such that the uncoded scheme is optimal. Our result generalizes theirs to the case that the observations can be noisy linear combinations, and the covariance matrix $\Sigma_{S_1,S_2,\ldots,S_M}$ does not necessarily all have strictly positive entries.
\section{Conclusion}
We considered the problem of determining whether a given uncoded scheme is optimal for multiuser joint source-channel coding. It was shown that for both broadcast and multiple-access in the Gaussian setting, matching occurs naturally under certain general conditions. Our approach differs from the more conventional approach in that instead of attempting to find explicit outer bound and inner bound then compare them, our focus is on the critical conditions that make the outer bound hold with equality. This approach has a decoupling effect which tremendously simplifies the overall task. As future work, we plan to extend and generalize this approach to explore matching in other channel networks.
\section*{Appendix}
\begin{proof}[Proof of the Outer Bound in Theorem \ref{theorem:BC}]
To upper-bound $E(\Sigma_{W_1,W_2,\ldots,W_M})$, first recall the Markov string
\begin{align}
&Y^N_1\leftrightarrow Y^N_2 \leftrightarrow\ldots\leftrightarrow Y^N_M\leftrightarrow X^N \leftrightarrow (S^N_1,S^N_2,\ldots,S^N_M)\nonumber\\
&\qquad\leftrightarrow (U^N_1,U^N_2,\ldots,U^N_M)\leftrightarrow (U^N_1,U^N_2\leftrightarrow\ldots\leftrightarrow U^N_{M-1})\leftrightarrow\ldots\leftrightarrow U^N_1.\label{eqn:markovstring}
\end{align}
We start by writing the following:
\begin{align}
&\sum_{j=1}^m I(U^N_j;Y^N_j|U^N_1,U^N_2,\ldots,U^N_{j-1})\nonumber\\
&=\sum_{j=1}^m \left[I(U^N_1,U^N_2,\ldots,U^m_j;Y^n_j)-I(U^N_1,U^N_2,\ldots,U^m_{j-1};Y^n_j)\right]\nonumber\\
&=\sum_{j=1}^m \left[h(Y^N_j|U^N_1,U^N_2,\ldots,U^N_{j-1})-h(Y^N_j|U^N_1,U^N_2,\ldots,U^N_j)\right]\nonumber\\
&=\sum_{j=1}^m h(Y^N_j|U^N_1,U^N_2,\ldots,U^N_{j-1})-\sum_{j=1}^m h(Y^N_j|U^N_1,U^N_2,\ldots,U^N_j).
\end{align}
Since physical degradedness is equivalent to stochastic degradedness in the broadcast setting, {\em i.e.,} $Z_j$ can be assumed to be decomposable into two independent components as $Z_{j+1}+\Delta Z_j$, we can apply the entropy power inequality \cite{CoverThomas} for $j=1,2,\ldots,M-1$,
\begin{align}
&\exp\left[\frac{2}{N}h(Y^N_j|U^N_1,U^N_2,\ldots,U^N_j)\right]\nonumber\\
&\geq \exp\left[\frac{2}{N}h(Y^N_{j+1}|U^N_1,U^N_2,\ldots,U^N_j)\right]+\exp\left[\log(2\pi e (\sigma^2_{Z_j}-\sigma^2_{Z_{j+1}}))\right]\nonumber\\
&=\exp\left[\frac{2}{n}h(Y^N_{j+1}|U^N_1,U^N_2,\ldots,U^m_j)\right]+2 \pi e(\sigma^2_{Z_j}-\sigma^2_{Z_{j+1}}). \label{eqn:applyentropypower}
\end{align}
For $j=M$, it is clear that
\begin{align}
&\exp\left[\frac{2}{N}h(Y^N_M|U^N_1,U^N_2,\ldots,U^N_M)\right]\geq \exp\left[\frac{2}{N}h(Y^N_M|S^N_1,S^N_2,\ldots,S^N_M)\right]= 2 \pi e\sigma^2_{Z_M},
\end{align}
with equality if and only if
\begin{align}
h(Y^N_M|U^N_1,U^N_2,\ldots,U^N_M)=h(Y^N_M|S^N_1,S^N_2,\ldots,S^N_M).\label{eqn:BCasifnonoisea}
\end{align}
It now follows that
\begin{align}
&E(\Sigma_{W_1,W_2,\ldots,W_M})\nonumber\\
&=\sum_{m=1}^M (\sigma^2_{Z_{m}}-\sigma^2_{Z_{m+1}}) \exp\left[\frac{2}{N}\sum_{j=1}^m I(U^N_j;Y^N_j|U^N_1,U^N_2,\ldots,U^N_{j-1})\right]\nonumber\\
&\leq \sum_{m=1}^M (\sigma^2_{Z_{m}}-\sigma^2_{Z_{m+1}}) \frac{\exp\left[\frac{2}{N}\sum_{j=1}^m h(Y^N_j|U^N_1,U^N_2,\ldots,U^N_{j-1})\right]}{\prod_{j=1}^m\left[\exp\left(\frac{2}{N}h(Y^N_{j+1}|U^N_1,U^N_2,\ldots,U^N_j)\right)+2 \pi e (\sigma^2_{Z_j}-\sigma^2_{Z_{j+1}})\right]},
\end{align}
where for convenience $\exp\left[\frac{2}{N}h(Y^N_{M+1}|S^N_1,S^N_2,\ldots,S^N_M)\right]\triangleq 0$.
We upper-bound this summation, by considering the summands in the reversed order, \textit{i.e.}, $m=M,M-1,\ldots,1$.
Starting with the summands when $m=M-1$ and $m=M$, we have
\begin{align}
&(\sigma^2_{Z_{M-1}}-\sigma^2_{Z_{M}}) \frac{\exp\left[\frac{2}{N}\sum_{j=1}^{M-1} h(Y^N_j|U^N_1,U^N_2,\ldots,U^N_{j-1})\right]}{\prod_{j=1}^{M-1}\left[\exp\left(\frac{2}{N}h(Y^N_{j+1}|U^N_1,U^N_2,\ldots,U^N_j)\right)+2 \pi e(\sigma^2_{Z_j}-\sigma^2_{Z_{j+1}})\right]}\nonumber\\
&\qquad\qquad+\sigma^2_{Z_{M}} \frac{\exp\left[\frac{2}{N}\sum_{j=1}^{M} h(Y^N_j|U^N_1,U^N_2,\ldots,U^N_{j-1})\right]}{\prod_{j=1}^{M}\left[\exp\left(\frac{2}{N}h(Y^N_{j+1}|U^N_1,U^N_2,\ldots,U^N_j)\right)+2 \pi e(\sigma^2_{Z_j}-\sigma^2_{Z_{j+1}})\right]}\nonumber\\
&=\frac{\exp\left[\frac{2}{N}\sum_{j=1}^{M-1} h(Y^N_j|U^N_1,U^N_2,\ldots,U^N_{j-1})\right]}{\prod_{j=1}^{M-1}\left[\exp\left(\frac{2}{N}h(Y^N_{j+1}|U^N_1,U^N_2,\ldots,U^m_j)\right)+2 \pi e (\sigma^2_{Z_j}-\sigma^2_{Z_{j+1}})\right]}\nonumber\\
&\qquad\qquad\cdot\left[(\sigma^2_{Z_{M-1}}-\sigma^2_{Z_{M}}) +\sigma^2_{Z_{M}}\frac{\exp\left[\frac{2}{N}h(Y^N_M|U^N_{M-1})\right]}{2 \pi e\sigma^2_{Z_{M}}}\right]\nonumber\\
&=\frac{1}{2 \pi e}\frac{\exp\left[\frac{2}{N}\sum_{j=1}^{M-1} h(Y^N_j|U^N_1,U^N_2,\ldots,U^m_{j-1})\right]}{\Pi_{j=1}^{M-2}\left[\exp\left(\frac{2}{N}h(Y^N_{j+1}|U^N_1,U^N_2,\ldots,U^N_j)\right)+2 \pi e(\sigma^2_{Z_{M-1}}-\sigma^2_{Z_{M}})\right]}.\label{eqn:firststep}
\end{align}
Continuing this line of reduction, we finally arrive at when $m=1$
\begin{align}
&E(\Sigma_{W_1,W_2,\ldots,W_M})\nonumber\\
&\leq (\sigma^2_{Z_{2}}-\sigma^2_{Z_{1}})\frac{\exp\left[\frac{2}{N} h(Y^N_1)\right]}{\exp\left(\frac{2}{N}h(Y^N_{2}|U^N_1)\right)+2 \pi e(\sigma^2_{Z_{2}}-\sigma^2_{Z_{1}})}+\frac{1}{2 \pi e}\frac{\exp\left[\frac{2}{N}\sum_{j=1}^{2} h(Y^N_j|U^N_1,\ldots,U^N_{j-1})\right]}{\exp\left(\frac{2}{N}h(Y^N_{2}|U^N_1)\right)+2 \pi e(\sigma^2_{Z_{2}}-\sigma^2_{Z_{1}})}\nonumber\\
&=\frac{\exp\left[\frac{2}{N} h(Y^N_1)\right]}{\exp\left(\frac{2}{N}h(Y^N_{2}|U^N_1))\right)+2 \pi e(\sigma^2_{Z_{2}}-\sigma^2_{Z_{1}})}\left[(\sigma^2_{Z_{2}}-\sigma^2_{Z_{1}})+\frac{\exp \left[\frac{2}{N}h(Y^N_2|U^N_1)\right]}{2\pi e}\right]\nonumber\\
&=\frac{\exp\left[\frac{2}{N} h(Y^N_1)\right]}{2\pi e}\leq P+\sigma^2_{Z_1},\label{eqn:BCchannelA}
\end{align}
where the last inequality is by the concavity of the $\log(\cdot)$ function and the given power constraint. The chain of inequalities in (\ref{eqn:BCchannelA}) holds with equality holds if and only if
\begin{align}
h(Y^N_1)=\frac{N}{2}\log 2\pi e( P+\sigma^2_{Z_1}),
\end{align}
as well as (\ref{eqn:BCasifnonoisea}) and the entropy power inequalities hold with equality.
We next lower bound $E(\Sigma_{W_1,W_2,\ldots,W_M})$. By the rate-distortion theorem \cite{CoverThomas}
\begin{align}
I(U^N_1;Y^N_1)\leq \frac{N}{2}\log\frac{\sigma^2_{S_1}+\sigma^2_{U_1}}{D+\sigma^2_{U_1}},
\end{align}
with equality holds if and only if
\begin{align}
h(U^N_1|Y^N_1)=\frac{N}{2}\log [2\pi e(D_1+\sigma^2_{W_1} )].
\end{align}
Furthermore,
\begin{align}
&I(U^N_m;Y^N_m|U^N_1,U^N_2,\ldots,U^N_{m-1})\nonumber\\
&=h(U^N_m|U^N_1,U^N_2,\ldots,U^N_{m-1})-h(U^N_m|Y^N_m,U^N_1,U^N_2,\ldots,U^N_{m-1})\nonumber\\
&=\frac{N}{2}\log\frac{|2\pi e(\Sigma_{S_1,S_2,\ldots,S_m}+\Sigma_{W_1,W_2,\ldots,W_m})|}{|2\pi e(\Sigma_{S_1,S_2,\ldots,S_{m-1}}+\Sigma_{W_1,W_2,\ldots,W_{m-1}})|}-h(U^N_m|Y^N_m,U^N_1,U^N_2,\ldots,U^N_{m-1})\nonumber\\
&\geq \frac{N}{2}\log\frac{|2\pi e(\Sigma_{S_1,S_2,\ldots,S_m}+\Sigma_{W_1,W_2,\ldots,W_m})|}{|2\pi e(\Sigma_{S_1,S_2,\ldots,S_{m-1}}+\Sigma_{W_1,W_2,\ldots,W_{m-1}})|}-h(U^N_m|Y^N_m)\label{eqn:BCcondition2a}\\
&\geq \frac{N}{2}\log\frac{|2\pi e(\Sigma_{S_1,S_2,\ldots,S_m}+\Sigma_{W_1,W_2,\ldots,W_m})|}{|2\pi e(\Sigma_{S_1,S_2,\ldots,S_{m-1}}+\Sigma_{W_1,W_2,\ldots,W_{m-1}})|}-\frac{N}{2}\log [2\pi e(D_m+\sigma^2_{W_m} )]\label{eqn:BCcondition3a}\\
&=\frac{N}{2}\log\frac{|\Sigma_{S_1,S_2,\ldots,S_m}+\Sigma_{W_1,W_2,\ldots,W_m}|}{|(\Sigma_{S_1,S_2,\ldots,S_{m-1}}+\Sigma_{W_1,W_2,\ldots,W_{m-1}})|[D_m+\sigma^2_{W_m}]},\nonumber
\end{align}
where (\ref{eqn:BCcondition2a}) is because conditioning reduces entropy, and (\ref{eqn:BCcondition3a}) is because Gaussian distribution maximizes the entropy for random variables with the same variance, together with the concavity of the $\log$ function. For (\ref{eqn:BCcondition2a}) to hold with equality, we must have
\begin{align}
h(U^N_m|Y^N_m,U^N_1,U^N_2,\ldots,U^N_{m-1})=h(U^N_m|Y^N_m),\quad m=2,3,\ldots,M,
\end{align}
and for (\ref{eqn:BCcondition3a}) to hold with equality requires
\begin{align}
h(U^N_m|Y^N_m)=\frac{N}{2}\log [2\pi e(D_m+\sigma^2_{W_m} )],\quad m=2,3,\ldots,M.
\end{align}
It follows that
\begin{align}
\exp\left[\frac{2}{N}\sum_{j=1}^mI(U^N_m;Y^N_m|U^N_1,U^N_2,\ldots,U^N_{m-1})\right]\geq \frac{|\Sigma_{S_1,S_2,\ldots,S_m}+\Sigma_{W_1,W_2,\ldots,W_m}|}{\Pi^m_{j=1}(D_j+\sigma^2_{W_j})}.\label{eqn:BCsourceA}
\end{align}
Combining (\ref{eqn:BCchannelA}) and (\ref{eqn:BCsourceA}), we reach an outer bound
\begin{align}
\sum_{m=1}^M (\sigma^2_{Z_{m}}-\sigma^2_{Z_{m+1}})\frac{|\Sigma_{S_1,S_2,\ldots,S_m}+\Sigma_{W_1,W_2,\ldots,W_m}|}{\Pi^m_{j=1}(D_j+\sigma^2_{W_j})}\leq P+\sigma^2_{Z_1}.
\end{align}
\end{proof}
\begin{proof}[Proof of Lemma 1]
For simplicity, let us define $B^{(0)}_m\triangleq \sigma^2_{X|Y_m}$ for $m=1,2,\ldots,M$.
It is clear that
\begin{align}
B^{(0)}_1\geq B^{(0)}_2\geq\ldots\geq B^{(0)}_M>0.
\end{align}
Recall $\alpha_m\neq 0$. In the $k$-th step of the Cholesky decomposition $k=0,1,\ldots,M-1$, we claim that $\alpha_{M-k}\beta_{M-k}\geq 0$ and $\sum_{i=1}^{M-k-1} \alpha_i\beta_i\geq0$.
Moreover, we claim the matrix partially diagonalized, denoted as $\Sigma^{(k)}_{V_1,V_2,\ldots, V_M}$, has entries in the following form:
\begin{itemize}
\item $\gamma^{(k)}_{i,j}=0$, $j>M-k$ and $i\neq j$; by symmetry, $\gamma_{i,j}=0$, $i> M-k$ and $i\neq j$.
\item $\gamma^{(k)}_{m,m}=\frac{\beta_m}{\alpha_m}B^{(k)}_m\sum_{j=1}^{m-1}\alpha_j\beta_j$, $m> M-k$;
\item $\gamma^{(k)}_{i,j}=-\beta_i\beta_j B^{(k)}_j$, $j\leq M-k$ and $i<j$; by symmetry $\gamma_{i,j}=-\beta_i\beta_j B^{(k)}_i$, $i\geq M-k$ and $i>j$;
\item $\gamma^{(k)}_{m,m}=\frac{\beta_m}{\alpha_m}\left[B^{(k)}_m\sum_{j=1}^{m-1}\alpha_j\beta_j+\sum_{j=m+1}^{M-k}\alpha_j\beta_jB^{(k)}_j\right]$, $m\leq M-k$.
\end{itemize}
where the terms $(B^{(k)}_1,B^{(k)}_2,\ldots,B^{(k)}_M)$ are determined recursively as
\begin{align}
B^{(k)}_m=B^{(k-1)}_m,\quad m> M-k,
\end{align}
and
\begin{align}
B^{(k+1)}_m=\left\{
\begin{array}{cc}
B^{(k)}_m+\frac{\alpha_{M-k}\beta_{M-k}}{\sum_{i=1}^{M-k-1}\alpha_i\beta_i}B^{(k)}_{M-k},&\beta_m\sum_{i=1}^{M-k-1}\alpha_i\beta_i\neq 0\\
B^{(k)}_m,& \mbox{otherwise}
\end{array}
\right.\quad m\leq M-k.
\end{align}
for which
\begin{align}
B^{(k)}_1\geq B^{(k)}_2\geq\ldots\geq B^{(k)}_M.
\end{align}
Clearly the claim is true when $k=0$. Suppose it is also true for $k=k^*$, and we wish to prove the claim for $k=k^*+1$.
It is clear that due to the positive semidefinite requirement for the degenerate case when
\begin{align}
\gamma^{(k^*)}_{M-k^*,M-k^*}=\frac{\beta_{M-k^*}}{\alpha_{M-k^*}}\sum_{j=1}^{{M-k^*}-1}\alpha_j\beta_j=0,
\end{align}
we must have for $i< M-k^*$
\begin{align}
\gamma^{(k^*)}_{i, M-k^*}=\beta_i\beta_{M-k^*}=0,
\end{align}
and this Cholesky step can essentially be skipped, and $(B^{(k)}_1,B^{(k)}_2,\ldots,B^{(k)}_M)$ does not need to be updated. It is easy to check the recursive formula $\gamma^{(k^*+1)}_{m,m}$ for $m\leq M-k^*-1$ is indeed valid for this case. If $\beta_{M-k^*}=0$, then clearly $\sum_{i=1}^{M-k^*-2} \alpha_i\beta_i=\sum_{i=1}^{M-k^*-1} \alpha_i\beta_i\geq0$ due to the assumption in the induction. If $\beta_{M-k^*}\neq 0$ then $\beta_i=0$ for $i< M-k^*$, and thus $\sum_{i=1}^{M-k^*-2} \alpha_i\beta_i=0$. Thus the induction is valid in this case.
If $\gamma^{(k^*)}_{M-k^*,M-k^*}\neq 0$, then due to the assumption in the induction we have
\begin{align}
\alpha_{M-k^*}\beta_{M-k^*}>0,\qquad \sum_{j=1}^{{M-k^*}-1}\alpha_j\beta_j>0.
\end{align}
First observe that due to the assumption in the induction, we have
\begin{align}
B^{(k^*+1)}_1\geq B^{(k^*+1)}_2\geq\ldots\geq B^{(k^*+1)}_M>0.
\end{align}
Using the Cholesky factorization, we have for any $j\leq M-k^*-1$ and $i<j$
\begin{align}
\gamma^{(k^*+1)}_{i,j}&=\gamma^{(k^*)}_{i,j}-\beta_i\beta_{M-k^*}B^{(k^*)}_{M-k^*}\frac{\alpha_{M-k^*}\beta_j}{\sum_{t=1}^{{M-k^*-1}}\alpha_t\beta_t}\nonumber\\
&=-\beta_i\beta_j B^{(k^*)}_j-\beta_i\beta_{M-k^*}B^{(k^*)}_{M-k^*}\frac{\alpha_{M-k^*}\beta_j}{\sum_{t=1}^{{M-k^*-1}}\alpha_t\beta_t}\nonumber\\
&=-\beta_i\beta_j\left[B^{(k^*)}_j+\frac{\alpha_{M-k^*}\beta_{M-k^*}}{\sum_{t=1}^{{M-k^*-1}}\alpha_t\beta_t}B^{(k^*)}_{M-k^*}\right]\nonumber\\
&=-\beta_i\beta_jB^{(k^*+1)}_j.
\end{align}
Similarly for $m\leq M-k^*-1$
\begin{align}
\gamma^{(k^*+1)}_{m,m}&=\gamma^{(k^*)}_{m,m}-\beta^2_m\beta_{M-k^*}B^{(k^*)}_{M-k^*}\frac{\alpha_{M-k^*}}{\sum_{t=1}^{{M-k^*-1}}\alpha_t\beta_t}\nonumber\\
&=\frac{\beta_m}{\alpha_m}\left[B^{(k^*)}_m\sum_{j=1}^{m-1}\alpha_j\beta_j+\sum_{j=m+1}^{M-k^*}\alpha_j\beta_jB^{(k^*)}_j\right]-\beta^2_m\beta_{M-k^*}B^{(k^*)}_{M-k^*}\frac{\alpha_{M-k^*}}{\sum_{j=1}^{{M-k^*-1}}\alpha_j\beta_j}\nonumber\\
&=\frac{\beta_m}{\alpha_m}B^{(k^*)}_m\sum_{j=1}^{m-1}\alpha_j\beta_j+\frac{\beta_m}{\alpha_m}\sum_{j=m+1}^{M-k^*-1}\alpha_j\beta_jB^{(k^*)}_j\nonumber\\
&\qquad+\alpha_{M-k^*}\beta_{M-k^*}\frac{\beta_m}{\alpha_m}\frac{\sum_{j=1}^{{M-k^*-1}}\alpha_j\beta_j-\alpha_m\beta_m}{\sum_{j=1}^{{M-k^*-1}}\alpha_j\beta_j}B^{(k^*)}_{M-k^*}\nonumber\\
&=\frac{\beta_m}{\alpha_m}\left[B^{(k^*)}_m+\frac{\sum_{j=1}^{m-1}\alpha_{M-k^*}\beta_{M-k^*}}{\sum_{t=1}^{{M-k^*-1}}\alpha_t\beta_t}B^{(k^*)}_{M-k^*}\right]\sum_{j=1}^{m-1}\alpha_j\beta_j\nonumber\\
&\qquad+\frac{\beta_m}{\alpha_m}\sum_{j=m+1}^{M-k^*-1}\alpha_j\beta_j\left[B^{(k^*)}_j+\frac{\sum_{j=1}^{m-1}\alpha_{M-k^*}\beta_{M-k^*}}{\sum_{t=1}^{{M-k^*-1}}\alpha_t\beta_t}B^{(k^*)}_{M-k^*}\right]\nonumber\\
&=\frac{\beta_m}{\alpha_m}\left[B^{(k^*+1)}_m\sum_{j=1}^{m-1}\alpha_j\beta_j+\sum_{j=m+1}^{M-k^*-1}\alpha_j\beta_jB^{(k^*+1)}_j\right].
\end{align}
Suppose ${\alpha_{M-k^*-1}}{\beta_{M-k^*-1}}<0$, this implies that
\begin{align}
\sum_{j=1}^{{M-k^*}-2}\alpha_j\beta_j=\sum_{j=1}^{{M-k^*}-1}\alpha_j\beta_j-{\alpha_{M-k^*-1}}{\beta_{M-k^*-1}}>0
\end{align}
which however contradicts with the positive semidefinite requirement that
\begin{align}
\frac{\beta_{M-k^*-1}}{\alpha_{M-k^*-1}}\sum_{j=1}^{{M-k^*}-2}\alpha_j\beta_j\geq 0.
\end{align}
If ${\alpha_{M-k^*-1}}{\beta_{M-k^*-1}}=0$, then from the assumption in the induction, we have $\sum_{j=1}^{{M-k^*}-2}\alpha_j\beta_j=\sum_{j=1}^{{M-k^*}-1}\alpha_j\beta_j\geq 0$ thus this case does not cause any problem.
If ${\alpha_{M-k^*-1}}{\beta_{M-k^*-1}}\geq0$, then it also follows that $\sum_{j=1}^{{M-k^*}-2}\alpha_j\beta_j\geq 0$. The lemma is proved.
\end{proof}
\begin{proof}[Proof of Corollary \ref{coro1}]
It suffices to consider the case that $\sigma^2_{Z^+_m}= \sigma^2_{Z_m}$, $m=1,2,\ldots,m^*-1,m^*+1,\ldots,M$, and $\sigma^2_{Z^+_m}= \sigma^2_{Z_m}+\sigma^2_{\Delta Z}$. Denote $\Delta P=\frac{P\sigma^2_{Z^+_m}}{P+\sigma^2_{Z^+_m}}-\frac{P\sigma^2_{Z_m}}{P+\sigma^2_{Z_m}}$, and matrices constructed for the two channels as $\Sigma_{V_1,V_2,\ldots, V_M}$ and $\Sigma^*_{V_1,V_2,\ldots, V_M}$, respectively.
It is clear that
\begin{align}
\Sigma^*_{V_1,V_2,\ldots, V_M}-\Sigma_{V_1,V_2,\ldots, V_M}=\begin{pmatrix}
\frac{\beta_1}{\alpha_1}\alpha_{m^*}\beta_{m^*}\Delta P&0&\ldots&-\beta_1\beta_{m^*}\Delta P\\
0&\frac{\beta_2}{\alpha_2}\alpha_{m^*}\beta_{m^*}\Delta P&\ldots&-\beta_2\beta_{m^*}\Delta P\\
...\\
-\beta_1\beta_{m^*}\Delta P& -\beta_2\beta_{m^*}\Delta P&\ldots&\frac{\beta_{m^*}}{\alpha_{m^*}}\Delta P\sum_{i=1}^{m^*-1}\alpha_i\beta_i,
\end{pmatrix}.\label{eqn:incremental}
\end{align}
However, it is easily seen that this matrix is positive semidefinite since the first $m^*-1$ diagonal terms are non-negative, and we can remove all the other terms through symmetric elimination, {\em i.e.,} the Cholesky factorization.
It follows that
\begin{align}
&\Sigma^*_{V_1,V_2,\ldots, V_M}-\Sigma_{S_1,S_2,\ldots,S_M}+P\bar{\beta}\bar{\beta}^t
\nonumber\\
&=[\Sigma^*_{V_1,V_2,\ldots, V_M}-\Sigma_{V_1,V_2,\ldots, V_M}]
+\left[\Sigma_{V_1,V_2,\ldots, V_M}-\Sigma_{S_1,S_2,\ldots,S_M}+P\bar{\beta}\bar{\beta}^t
\right]
\end{align}
is positive semidefinite since it is a summation of two positive semidefinite matrices.
\end{proof}
\begin{proof}[Proof of Corollary \ref{coro2}]
First note that the entries in matrix $\Pi$,
\begin{align}
\sum_{i=1}^M\rho_{j,i}\alpha_i=P\beta_j,\quad j=1,2,\ldots,M.
\end{align}
For the ``only if'' direction, it follows from Corollary 1 that matching must hold for the degraded channel with noise power $\sigma^2_{Z_1'}=\sigma^2_{Z_2'}=\ldots=\sigma^2_{Z_M'}\triangleq \sigma^2_{Z}=\sigma^2_{Z_1}$. The requirement (\ref{eqn:semidefinite}) implies
\begin{align}
\frac{P\sigma^2_{Z}}{P+\sigma^2_{Z}}\mbox{diag}\left(\frac{\beta_1}{\alpha_1},\frac{\beta_2}{\alpha_2},\ldots,\frac{\beta_M}{\alpha_M}\right)+\left(P-\frac{P\sigma^2_{Z}}{P+\sigma^2_{Z}}\right)\bar{\beta}\bar{\beta}^t
\succeq \Sigma_{S_1,S_2,\ldots,S_M}. \label{eqn:matchingspecial}
\end{align}
This implies that $\beta_i>0$, since otherwise the left hand side is rank deficient. Together with Lemma 1, it follows that $\alpha_i\beta_i>0$. Next multiply both sides from left and from right by $\Pi$, it follows that
\begin{align}
\frac{\sigma^2_{Z}}{P+\sigma^2_{Z}} I_M+\frac{P}{P+\sigma^2_{Z}}\begin{pmatrix}
\sqrt{\alpha_1\beta_1}\\
\sqrt{\alpha_2\beta_2}\\
...\\
\sqrt{\alpha_M\beta_M}
\end{pmatrix}\begin{pmatrix}
\sqrt{\alpha_1\beta_1}& \sqrt{\alpha_2\beta_2}& ...& \sqrt{\alpha_M\beta_M}
\end{pmatrix}\succeq D\Sigma_{S_1,S_2,\ldots,S_M}D.
\end{align}
Notice that $\bar{v}^t_1=(\sqrt{\alpha_1\beta_1}, \sqrt{\alpha_2\beta_2}, ..., \sqrt{\alpha_M\beta_M})$ is in fact an eigenvector corresponding to eigenvalue $1$ for the matrix $\Pi\Sigma_{S_1,S_2,\ldots,S_M}\Pi$, easily verified using (\ref{eqn:alphatobeta}). We can write the eigen decomposition of the matrix $\Pi\Sigma_{S_1,S_2,\ldots,S_M}\Pi$ as
\begin{align}
\Pi\Sigma_{S_1,S_2,\ldots,S_M}\Pi=\bar{v}_1\bar{v}^t_1+\sum_{i=2}^M\lambda_i\bar{v}_i\bar{v}^t_i,
\end{align}
where $\lambda_2,\lambda_3,\ldots,\lambda_M$ are the other eigenvalues of $\Pi\Sigma_{S_1,S_2,\ldots,S_M}\Pi$, and $\bar{v}_2,\bar{v}_3,\ldots,\bar{v}_M$ are the corresponding eigenvectors. It follows that
\begin{align}
\frac{\sigma^2_{Z}}{P+\sigma^2_{Z}} I_M\succeq \frac{\sigma^2_{Z}}{P+\sigma^2_{Z}}\bar{v}_1\bar{v}^t_1+\sum_{i=2}^M\lambda_i\bar{v}_i\bar{v}^t_i, \label{eqn:Nlarge}
\end{align}
which implies $\lambda_i\leq \frac{\sigma^2_{Z}}{P+\sigma^2_{Z}}$, $i=2,3,\ldots,M$.
For the ``if'' direction, we choose a $\sigma^2_{Z}$ such that (\ref{eqn:Nlarge}) holds, which is always possible when $\sigma^2_{Z}$ is sufficiently large. This implies that for the channel $\sigma^2_{Z_1}=\sigma^2_{Z_2}=\ldots=\sigma^2_{Z_M}=\sigma^2_{Z}$, condition (\ref{eqn:matchingspecial}) holds, and thus it is a matched channel.
\end{proof}
\begin{proof}[Proof of Corollary \ref{coro3}]
The entries of matrix $\mbox{diag}(\bar{\alpha})\Sigma_{W_1,W_2,\ldots,W_M}\mbox{diag}(\bar{\alpha})$, denoted as $\phi_{i,j}$, is given (by symmetry only the upper-triangle entries need to be specified)
\begin{align}
\phi_{j,m}=\alpha_j\beta_j\alpha_m\beta_m\frac{P^2}{P+\sigma^2_{Z_m}}-\alpha_j\alpha_m\rho_{j,m},\quad j<m,
\end{align}
and
\begin{align}
\phi_{m,m}=\alpha_m\beta_m\sum_{j=1}^{m-1}\alpha_j\beta_j\frac{P\sigma^2_{Z_m}}{P+\sigma^2_{Z_m}}+\alpha_m\beta_m\sum_{j=m+1}^M\alpha_j\beta_j\frac{P\sigma^2_{Z_j}}{P+\sigma^2_{Z_j}}-\alpha^2_m\rho_{m,m}+\alpha^2_m\beta^2_mP.
\end{align}
A necessary and sufficient condition for matching is that the matrix $\mbox{diag}(\bar{\alpha})\Sigma_{W_1,W_2,\ldots,W_M}\mbox{diag}(\bar{\alpha})$ is positive semidefinite.
Observe that
\begin{align}
\sum_{j=1}^M\phi_{j,m}=\alpha_m\beta_mP\sum_{i=1}^{M}\alpha_j\beta_j - \alpha_m\beta_mP=0.
\end{align}
If all the off-diagonal entries of $\mbox{diag}(\bar{\alpha})\Sigma_{W_1,W_2,\ldots,W_M}\mbox{diag}(\bar{\alpha})$ are non-positive, then the matrix is diagonally dominant, and the diagonal entries are all positive, which implies that it is a positive semidefinite matrix. Thus as long as
\begin{align}
\alpha_j\beta_j\alpha_m\beta_m\frac{P^2}{P+\sigma^2_{Z_m}}\leq\alpha_j\alpha_m\rho_{j,m},\quad j<m \label{eqn:allcondition}
\end{align}
and
\begin{align}
\alpha_j\beta_j\alpha_m\beta_m\frac{P^2}{P+\sigma^2_{Z_j}}\leq\alpha_j\alpha_m\rho_{j,m},\quad j>m\label{eqn:allcondition2}
\end{align}
the positive semidefinite condition is satisfied.
It is thus sufficient to have
\begin{align}
\sigma^2_{Z_m}\geq \max_{j< m} \frac{\beta_j}{\rho_{j,m}}\beta_mP^2-P,\quad m=2,3\ldots, M.
\end{align}
Together with Corollary 1, this implies the statement given in the corollary is indeed true.
\end{proof}
\begin{proof}[Proof of (\ref{eqn:source2})]
Since the matrix $\Sigma_{S_1,S_2,S_3}$ is positive definite, we have $-1<\rho_1<1$ and $-1<\rho_2<1$.
Since $\alpha_m=1$, we must also have $\beta_m>0$, $0<\lambda_2<1$ and $0<\lambda_3<1$ for matching to occur.
The first condition gives that
\begin{align}
\rho_1+\rho_2+1>0\quad\mbox{and}\quad 2\rho_1+1>0,
\end{align}
but the latter two require a few more steps. Notice that the condition $0<\lambda_3<1$ implies that
\begin{align}
\rho_1+2\rho_2>0.
\end{align}
If $\rho_2>2\rho^2_1-1$, then $0<\lambda_2<1$ implies
\begin{align}
2\rho_1\rho_2> -4\rho^2_1-3\rho_1.
\end{align}
If $\rho_1>0$, this yields a condition already implied by $\rho_1+2\rho_2>0$; on the other hand, $\rho_1\leq 0$ is an impossible case.
It can be verified that $\rho_2<2\rho^2_1-1$ is also an impossible case. Thus we must have $\rho_2>2\rho^2_1-1$ and $\rho_1>0$ simultaneously, from which we obtained the set of conditions given in (\ref{eqn:source2}).
\end{proof}
\bibliographystyle{IEEEbib}
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| 4 | 0.8 |
\section{Introduction}
First, let us recall some facts about a $G$-structures and their
prolongations. There are two definitions of $G$--structures. The first reads that a $G$--structure is
a principal bundle $ P \rightarrow M$ with structure group $G$
together with a soldering form $\theta.$ The second reads that it is a
reduction of the frame bundle $P^1 M$ to the Lie group $G.$ In
the latter case, the soldering form $\theta$ is induced from a canonical soldering form on
the frame bundle.
Now let $\mathfrak g$ be the Lie algebra of the Lie group $G$ and let $\mathbb V$ be a vector space. From the structure theory we know that there is
a $G$-invariant complement ${\mathcal D}$ of $\partial (\mathfrak g \otimes
\mathbb V^*)$ in $ \mathbb V \otimes \wedge^2 \mathbb V^*,$ where
$\partial $ is the operator of alternation, see \cite{ko}. Let us recall that the torsion of a linear connection lies in the space
$\mathbb V \otimes \wedge^2 \mathbb V^*.$
The almost Clifford and almost Cliffordian structures are $G$-structures based on
Clifford algebras. Two most important examples are an almost
hypercomplex geometry and an almost quaternionic geometry, which are based
on Clifford algebra $\mathcal Cl(0,2)$. An important geometric property of almost hypercomplex structure reads that there is no nontrivial $G$-invariant
subspace ${\mathcal D}$ in $ \mathbb V \otimes \wedge^2 \mathbb V^*,$ because the first prolongation $\mathfrak g^{(1)}$ of the Lie algebra $\mathfrak g$
vanishes. For almost quaternionic structure, the situation is more
complicated, because $\mathfrak g^{(1)} = \mathbb V^*,$ see \cite{am}.
For these reasons, in the latter case, there exists a distinguished class of linear connections compatible with the structure. Our goal is to describe some of
these connections for almost Cliffordian $G$--structures based on Clifford algebras $\mathcal Cl(s,t)$ generally.
\section{Clifford algebras}
The pair $(\mathbb V,Q)$, where $\mathbb V$ is a vector space of dimension $n$ and $Q$ is a quadratic form is
called a quadratic vector space.
To define Clifford algebras in coordinates, we start by choosing a basis $e_i,\ i=1,...,n$ of $\mathbb V$
and by $I_i, \ i=1,...,n$ we denote the image of $e_i$ under the inclusion $\mathbb V \hookrightarrow \mathcal Cl(\mathbb V,Q)$.
Then the elements $I_i$ satisfy the relation
$$ I_j I_k + I_k I_j = -2 B_{jk} 1,$$
where $1$ is the unity in the Clifford algebra and $B$ is a bilinear form
obtained from $Q$ by polarization. In a quadratic finite dimensional real vector space it is always
possible to choose a basis $e_i$ for which the matrix of the bilinear form $B$
has the form
$$\begin{pmatrix}
O_r & & \\
& E_s & \\ & & -E_t
\end{pmatrix},\ r+s+t=n,
$$
where $E_k$ denotes the $k \times k$ identity matrix and $O_k$
the $k \times k$ zero matrix. Let us restrict to the case
$r=0$, whence $B$ is nondegenerate. Then $B$ defines inner product of signature
$(s,t)$ and we call the corresponding Clifford algebra $\mathcal Cl(s,t)$.
For example, $\mathcal Cl(0,2)$ is generated by $I_1,I_2$, satisfying
$I_1^2=I_2^2=-E$ with $I_1 I_2 = -I_2 I_1$, i.e.
$\mathcal Cl(0,2) $ is isomorphic to $\mathbb H$.
Following the classification of the Clifford algebra, Boot periodicity reads that
${\mathcal C} l(0,n) \cong {\mathcal C} l(0,q) \otimes \mbox{$\mathbb R$} (16p)$, where $n=8p+q$, $q=0,\dots,7$ and
$\mbox{$\mathbb R$} (N)$ denotes the $N \times N$ matrices with coefficients in $\mbox{$\mathbb R$}$. To determine
explicit matrix representations we use the periodicity conditions
\begin{align*}
{\mathcal C} l(0,n) &\cong {\mathcal C} l(n-2,0) \otimes {\mathcal C} l(0,2),\\
{\mathcal C} l(n,0) &\cong {\mathcal C} l(0,n-2) \otimes {\mathcal C} l(2,0),\\
{\mathcal C} l(s,t) &\cong {\mathcal C} l(s-1,t-1)\otimes {\mathcal C} l(1,1)
\end{align*}
together with the explicit matrix representations of ${\mathcal C} l(0,2)$, ${\mathcal C} l(2,0)$, ${\mathcal C} l(1,0)$ and
${\mathcal C} l (0,1)$. More precisely, the identification of Clifford algebra ${\mathcal C} l(s,t),$ where either $s$ or $t$ is greater than 2, can be obtained by one of the following possibilities:
\begin{itemize}
\item[a)]
If $s>t$ then
$${\mathcal C} l(s,t)\cong{\mathcal C} l(s-t,0)\otimes\overset{t}{\otimes}{\mathcal C} l(1,1).$$
Further, to classify the algebra ${\mathcal C} l(s-t,0)$,
the following four cases are possible:
\begin{align*}
{\mathcal C} l(4p,0) &\cong \overset{p}{\otimes}({\mathcal C} l(0,2) \otimes {\mathcal C} l(2,0)),\\
{\mathcal C} l(4p+1,0)&\cong {\mathcal C} l(1,0)\otimes\overset{p}{\otimes}({\mathcal C} l(0,2) \otimes {\mathcal C} l(2,0)),\\
{\mathcal C} l(4p+2,0)&\cong{\mathcal C} l(2,0)\otimes\overset{p}{\otimes}({\mathcal C} l(0,2) \otimes {\mathcal C} l(2,0)),\\
{\mathcal C} l(4p+3,0)&\cong{\mathcal C} l(0,1)\otimes {\mathcal C} l(2,0)\otimes\overset{p}{\otimes}({\mathcal C} l(0,2) \otimes {\mathcal C} l(2,0)).
\end{align*}
\item[b)]
If $s<t$ then
$${\mathcal C} l(s,t)\cong{\mathcal C} l(0,t-s)\otimes\overset{s}{\otimes}{\mathcal C} l(1,1).$$
To classify the algebra ${\mathcal C} l(0,t-s)$,
the following four cases are possible:
\begin{align*}
{\mathcal C} l(0,4p) &\cong \overset{p}{\otimes}({\mathcal C} l(2,0) \otimes {\mathcal C} l(0,2)),\\
{\mathcal C} l(0,4p+1)&\cong{\mathcal C} l(0,1)\otimes\overset{p}{\otimes}({\mathcal C} l(2,0) \otimes {\mathcal C} l(0,2)),\\
{\mathcal C} l(0,4p+2)&\cong{\mathcal C} l(0,2)\otimes\overset{p}{\otimes}({\mathcal C} l(2,0) \otimes {\mathcal C} l(0,2)),\\
{\mathcal C} l(0,4p+3)&\cong{\mathcal C} l(1,0)\otimes {\mathcal C} l(0,2)\otimes\overset{p}{\otimes}({\mathcal C} l(2,0) \otimes {\mathcal C} l(0,2)).
\end{align*}
\item[c)]
If $s=t$ then
$${\mathcal C} l(s,t)\cong\overset{s}{\otimes}{\mathcal C} l(1,1).$$
\end{itemize}
For example
$$ {\mathcal C} l(3,0) \cong {\mathcal C} l(0,1) \otimes {\mathcal C} l(2,0),$$
where the matrix representation of ${\mathcal C} l (0,1)$ is given by the matrices
$$\begin{pmatrix}
1&0 \\ 0&1
\end{pmatrix} \text{ and }
\begin{pmatrix}
\phantom{-}0&1 \\ -1&0
\end{pmatrix}
$$
and the matrix representation of ${\mathcal C} l(2,0)$ is given by the matrices \Small
$$E,
I_1 =
\begin{pmatrix}
\phantom{-}0& -1 & \phantom{-}0 & \phantom{-}0\\
-1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1\\
\phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0\\
\end{pmatrix},
I_2 =
\begin{pmatrix}
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\\
1 & 0 & 0 & 0\\
0 & 1& 0 & 0\\
\end{pmatrix},$$
$$I_3=I_1I_2 =
\begin{pmatrix}
\phantom{-}0& \phantom{-}0 & \phantom{-}0 &-1\\
\phantom{-}0 & \phantom{-}0 & -1 & \phantom{-}0\\
\phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0\\
\phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0\\
\end{pmatrix},
$$ \normalsize
where $E$ is an identity matrix.
Now, the matrix representation of ${\mathcal C} l(3,0)$ is given by
$$
\begin{pmatrix}
E & 0 \\ 0 &E
\end{pmatrix},
\begin{pmatrix}
I_1 & 0 \\ 0 &I_1
\end{pmatrix},
\begin{pmatrix}
I_2 & 0 \\ 0 &I_2
\end{pmatrix},
\begin{pmatrix}
I_3 & 0 \\ 0 &I_3
\end{pmatrix},
$$
$$
\begin{pmatrix}
0&E \\ -E & 0
\end{pmatrix},
\begin{pmatrix}
0&I_1 \\ -I_1 & 0
\end{pmatrix},
\begin{pmatrix}
0& I_2 \\ -I_2 & 0
\end{pmatrix},
\begin{pmatrix}
0& I_3 \\ -I_3 & 0
\end{pmatrix},
$$
for explicit description see \cite{hv}.
We now focus on the algebra $\mathcal O:=\mathcal Cl(s,t)$,
i.e. the algebra generated by
complex unities $I_i, i=1, \dots ,t$
and product unites $J_j,\ j= 1, \dots ,s $, which are anti commuting,
i.e. $I_i^2=-E$, $J_j^2=E$ and $K_i K_j = -K_j K_i$, $i \neq j,$ where $K\in \{I_i,J_j \}$.
On the other hand, this algebra is generated by elements
$F_i$, $i=1,\dots ,k$ as a vector space.
We chose a basis $F_i$, $i=1,\dots ,k$, such that
$F_1=E$, $F_i=I_{i-1}$ for $i=2,\dots,t+1$,
$F_j=J_{j-t-1}$ for $j=t+2,\dots,s+t+1$ and by all different multiples of $I_i$ and $J_j$ of length $2,...,s+t$. Let us note that both complex and product unities can be found among these multiple generators.
\begin{lemma} \label{l1} Let $F_1,\dots ,F_k$ denote the $k=2^{s+t}$
elements of the matrix representation of Clifford algebra ${\mathcal C} l(s,t)$
on $\mathbb R^{k}$. Then there exists a real
vector $X \in \mathbb R^k$ such that the dimension of a linear span
$\langle F_i X | i=1,\dots,k\rangle$ equals to $k$.
\end{lemma}
\begin{proof}
Let us suppose, without loss of generality, that $F_1,\dots ,F_k$ are the elements constructed
by means of Boot periodicity as above.
Then, by induction we prove that the matrix
$F= \sum_{i=1}^k a_i F_i, \ a_i\in\mbox{$\mathbb R$},$ is a square matrix that has exactly one entry $a_i$ in each
column and each row. For ${\mathcal C} l(1,0)$ and ${\mathcal C} l(0,1)$ we have
$$F=\begin{pmatrix}
a_0 & a_1 \\
a_1 & a_0
\end{pmatrix},
F=\begin{pmatrix}
a_0 & a_1 \\
-a_1 & a_0
\end{pmatrix},
$$
respectively. For the rest of the generating cases ${\mathcal C} l (2,0)$, ${\mathcal C} l (0,2)$ and ${\mathcal C} l (1,1)$ the matrix $F$ can be obtained in very similar way, e.g. for ${\mathcal C} l (2,0)$ we have
$$F=\begin{pmatrix}
a_0 & -a_1 & a_2 & -a_3\\
-a_1 & a_0 & -a_3 & a_2\\
a_2 & a_3 & a_0 & a_1\\
a_3 & a_2 & a_1 & a_0
\end{pmatrix}.$$
We now restrict to Clifford algebras of type $ {\mathcal C} l(s,0),\ s>2,$ and show the induction step by means of the periodicity condition
$$ {\mathcal C} l(s,0) \cong {\mathcal C} l(0,s-2)\otimes {\mathcal C} l(2,0) .$$
The rest of the cases according to the Clifford algebra identification above can be proved similarly and we leave it to the reader.
Let $G_i$, $i=1,\dots,l$ denote the $l$ elements of the matrix representation of Clifford algebra ${\mathcal C} l(0,s-2)$ with the
required property, i.e. the matrix
$G= \sum_{i=1}^l g_i G_i$ is a square matrix with exactly one entry $g_i$ in each
column and each row, i.e.
$$ G:=
\begin{pmatrix}
g_{\sigma_1(1)} & \dots & g_{\sigma_1(l)}\\
\vdots & & \vdots \\
g_{\sigma_l(1)} & \dots & g_{\sigma_l(l)}
\end{pmatrix},
$$
where $\sigma_i$ are all perumtations of $\{1,\dots,l\}$.
The matrix of ${\mathcal C} l(2,0)$ is
$$H:=\begin{pmatrix}
a_1 & -a_2 & a_3 & -a_4\\
-a_2 & a_1 & -a_4 & a_3\\
a_3 & a_4 & a_1 & a_2\\
a_4 & a_3 & a_2 & a_1
\end{pmatrix}.$$
Then the matrix for the representation
of Clifford algebra
${\mathcal C} l(s,0)$ is composed as follows:
$$ F:=
\begin{pmatrix}
g_{\sigma_1(1)}H & \dots & g_{\sigma_1(l)}H\\
\vdots & & \vdots \\
g_{\sigma_l(1)}H & \dots & g_{\sigma_l(l)}H
\end{pmatrix}.
$$
Finally, if the matrix $G$ has exactly one $g_i$ in each column and each row, the
matrix $F$ is a square matrix with exactly one $a_jg_i$, where $j=1,\dots,4,i=1,\dots ,l$ in each column and each row.
Now, let $F= \sum_{i=1}^k b_i F_i$ be a $k \times k$ matrix constructed as above and let $e_i$ denote the standard basis of $\mathbb R^{k}$. Then the vector
$$v_i:=Fe_i^T$$
is the $i$-th column of the matrix $F$ and thus it is composed of $k$ different entries $b_i$.
If the dimension of $\langle F_i X | i=1\dots,k\rangle$ was less then $k$, then the vector $v$
has to be zero and thus all $b_i$ have to be zero.
\end{proof}
\begin{defin} \label{g2}
Let $P^1 M$ be a bundle of linear frames
over $M$ (the fiber bundle $P^1 M$ is a principal bundle over $M$
with the structure group $GL(n, \mathbb{R})$). Reduction of the bundle $P^1 M$
to the subgroup $G \subset GL(n, \mathbb{R})$ is called a
{\em $G$-structure}\index{$G$-structure}.
\end{defin}
\begin{defin}
If $M$ is an $km$--dimensional manifold, where $k=2^{s+t}$ and
$m \in \mathbb N$ then an almost Clifford manifold
is given by a reduction of the structure group
$GL(km, \mathbb R)$ of the principal frame bundle of $M$
to
$$GL(m, {\mathcal O}) = \{ A \in GL(km, \mathbb R) | A I_i = I_i A , A J_j = J_j A \} ,$$
where $\mathcal O$ is an arbitrary Cliffford algebra.
\end{defin}
In other words, an almost Clifford manifold
is a smooth manifold equipped with the set of anti commuting affinors
$I_i, i=1, \dots ,t,\ I_i^2=-E$ and $J_j,\ j= 1, \dots ,s,\ J_j^2=E $ such that
the free associative unitary algebra generated by
$\langle I_i,J_j,E \rangle$ is isomorphically equivalent to $\mathcal O$.
In particular, on the elements of this reduced bundle one can define affinors in the form of
$F_1, \dots, F_{k}$ globally.
\section{$A$-planar curves and morphisms}
The concept of planar curves is a generalization of a geodesic on
a smooth manifold equipped with certain structure.
In \cite{ms} authors proved a set of facts about structures
based on two different affinors. Following \cite{hs06, h11}, a manifold equipped with an
affine connection and a set of affinors $A=\{ F_1 , \dots, F_l\}$
is called an $A$-structure and a curve satisfying $\nabla_{\dot{c}}
\dot{c} \in \langle F_1(\dot{c}) , \dots, F_l(\dot{c}) \rangle$ is called an
$A$-planar curve.
\begin{defin}
Let $M$ be a smooth manifold such that $\dim (M) = m.$ Let $A$ be a
smooth $\ell$-dimensional $(\ell<m)$ vector subbundle in $ T^* M \otimes
TM$ such that the identity affinor $E=id_{TM}$ restricted to
$T_x M$ belongs to\\ $A_x M \subset T_x^* M \otimes T_xM $ at each
point $x \in M.$ We say that $M$ is equipped with an
$\ell$-dimensional {\em $A$--structure.}
\end{defin}
It is easy to see that an almost Clifford structure is not an $A$--structure,
because the affinors in the form of $F_0, \dots, F_{\ell} \in A $
have to be defined only locally.
\begin{defin}
The $A$--structure where $A$ is a Clifford
algebra $\mathcal O$ is called an almost Cliffordian manifold.
\end{defin}
Classical concept of $F$--planar curves defines
the {\em $F$-planar} \index{$F$-planar} curve
as the curve
$c : \mathbb R \rightarrow M$ satisfying the condition
$$ \nabla_{\dot{c}} \dot{c} \in \langle \dot{c} , F(\dot{c}) \rangle,$$
where $F$ is an arbitrary affinor.
Clearly, geodesics are $F$-planar curves for all affinors,
because $\nabla_{\dot{c}} {\dot{c}} \in \langle \dot{c} \rangle
\subset \langle \dot{c},F(\dot{c}) \rangle.$
Now, for any tangent vector $X \in T_x M$ we shall write $A_x(X)$ for
the vector subspace
$$ A_x (X) = \lbrace F_i(X) | F_i \in A_x M \rbrace \subset T_x M$$
and call it the {\em $A$--hull of the vector $X$}. Similarly,
{\em $A$--hull of a vector field} is a subbundle in $TM$ obtained pointwise.
For example, $A$--hull of an almost quaternionic structure is
$$ A_x (X) = \lbrace aX+bI(X)+cJ(X)+dK(X) | a,b,c,d \in \mathbb R \rbrace.$$
\begin{defin}
Let $M$ be a smooth manifold equipped with an $A$--structure and a linear connection $\nabla$.
A smooth curve $c : \mathbb R \to M$ is said to be {\em $A$--planar} if $$\nabla_{\dot{c}} {\dot{c}} \in A(\dot{c}).$$
\end{defin}
One can easily check that the class of connections
\begin{equation}
[\nabla]_{A} = \nabla + \sum_{i=1}^{\dim A} \Upsilon_i \otimes F_i,
\end{equation}
where $\Upsilon_i$ are one forms on $M$,
share the same class of $A$--planar curves, but we have to describe them more carefully for Cliffordian
manifolds.
\begin{thm}
Let $M$ be a smooth manifold equipped with an almost Cliffordian structure, i.e. an $A$--structure,
where $A={\mathcal C} l(s,t),$ $\dim (M) \geq 2(s+t) $,
and let $\nabla$ be a linear connection such that $\nabla A =0$.
The class of connections $[\nabla]$ preserving $A$, sharing the same torsion and $A$--planar curves
is isomorphic to $T^*M$ and the isomorphism has the following form:
\begin{equation} \label{aconn}
\Upsilon \mapsto \nabla + \sum_{i=1}^{k}\epsilon_i (\Upsilon \circ F_i) \odot F_i,
\end{equation}
where $\langle F_1, \dots, F_k \rangle = A$, $k=2^{s+t},$ as a vector space, $\epsilon_i\in \{\pm 1\}$ and $\Upsilon$ is a one form on $M$.
\end{thm}
\begin{proof}
First, let us consider the difference tensor
$$P(X,Y)=\bar{\nabla}_{X} (Y)
- {\nabla}_{X} (Y) $$
and one can see that its value
is symmetric in each tangent space because both connections share the same torsion.
Since both $\nabla$ and $\bar\nabla$ preserve $F_i$, $i=1,\dots,k,$
the difference tensor $P$ is Clifford
linear in the second variable. By symmetry it is thus Clifford bilinear and we can procesed by induction.
Let $X=\dot{c}$ and the deformation $P(X,X)$ equals $\sum_{i=1}^{k} \Upsilon_i(X)F_i(X)$
because $c$ is $A$-planar with respect to $\nabla$ and $\bar \nabla$
In this case we shall verify
First, for $s=1,t=0$
\begin{align*}
P(X,X) &= a(X)X+b(JX)JX, \\
P(X,X)&=J^2P(X,X)=P(JX,JX) = a(JX)JX+b(X)X.
\end{align*}
The difference of the first row and the
second rows implies $a(X)=b(X)$ and $a(JX)=b(JX)$
because we can suppose that $X,JX$ are linearly independent.
For $s=0,t=1$
\begin{align*}
P(X,X) &= a(X)X+b(IX)IX, \\
-P(X,X) &= I^2P(X,X)=P(IX,IX) = a(IX)IX-b(X)X.
\end{align*}
The sum of the first row and the
second row implies $a(X)=b(X)$ and $a(IX)=-b(IX)$
because we can suppose that $X,IX$ are linearly independent.
Let us suppose that the property holds for a Clifford algebra
${\mathcal C} l(s,t)$, $k=2^{s+t}$ i.e.
\begin{align*}
P(X,X) &= \sum_{i=1}^{k} \epsilon_i(\Upsilon( F_i(X))) F_i(X),
\end{align*}
where $\epsilon_i \in \{ \pm 1 \}$.
For ${\mathcal C} l(s,t+1)$ we have
\begin{align*}
P(X,X) &= \sum_{i=1}^{k}\epsilon_i(\Upsilon( F_i(X))) F_i(X)
+ \sum_{i=1}^{k} (\xi_i ( F_iS(X))) F_iS(X),
\end{align*} and
\begin{align*}
S^2P(X,X) &= \sum_{i=1}^{k} \epsilon_i(\Upsilon( F_i(SX))) F_i(SX) +
\sum_{i=1}^{k} (\xi_i ( F_i(X))) F_i(X) ,
\end{align*}
The sum of the first row and the
second rows implies
$$ \epsilon_i\Upsilon( F_i(X))) = -\xi_i (F_i X) \text{ and } \epsilon_i\Upsilon( F_i(S X))) = -\xi_i (F_i S X), $$
because we can suppose that $F_i X$ are linearly independent.
The case of ${\mathcal C} l(s+1,t)$ is calculated in the same way.
Now, $ P(X,X) = \sum_{i=1}^{k} \epsilon_i(\Upsilon( F_i(X))) F_i(X) $ an one
shall compute
\begin{align*}
P(X,Y) &= \frac{1}{2}(
\sum_{i=1}^{k} \epsilon_i\Upsilon( F_i(X+Y)) F_i(X+Y) -
\sum_{i=1}^{k}\epsilon_i \Upsilon ( F_i(X)) F_i(X) \\
&-
\sum_{i=1}^{k} \epsilon_i\Upsilon( F_i(Y)) F_i(Y)).
\end{align*}
by polarization.
Assuming that vectors $F_i(X),F_i(Y)$, $i=1,\dots, k$ are linearly
independent we compare the coefficients of
$X$ in the expansions of $P(sX,tY)=stP(X,Y)$ as above to get
$$
s\Upsilon(sX+tY)-s\Upsilon(sX)=st(\Upsilon(X+Y)-\Upsilon(X)).
$$
Dividing by $s$ and, putting $t=1$ and taking the limit $s\to 0$, we conclude
that
$\Upsilon(X+Y)=\Upsilon(X)+\Upsilon(Y)$.
We have proved that the form $\Upsilon$ is linear in $X$ and
$$ (X,Y) \to \sum_{i=1}^{k} \epsilon_i(\Upsilon( F_i(X))) F_i(Y) + \sum_{i=1}^{k}\epsilon_i
(\Upsilon ( F_i(Y))) F_i(X)$$
is a symmetric complex bilinear map which agrees with $P(X,Y)$ if both arguments coincide, it always agrees with $P$ by polarization and
$\bar{\nabla}$ lies in the projective equivalence class $[\nabla]$.
\end{proof}
\section{$\mathcal D$--connections}
Let $\mathbb V = \mbox{$\mathbb R$}^n, $ $G \subset GL (\mathbb V)
= GL(n, \mbox{$\mathbb R$})$ be a Lie group with Lie algebra $\mathfrak g$ and $M$ be a
smooth manifold of dimension $n.$
\begin{defin}
The {\em first prolongation
$\mathfrak{g}^{(1)}$} of $\mathfrak{g}$ is a space of symmetric
bilinear mappings $t: \mathbb{V} \times \mathbb{V} \rightarrow
\mathbb{V} $ such that, for each fixed $v_1 \in \mathbb{V},$ the
mapping $v \in \mathbb{V} \mapsto t(v,v_1) \in \mathbb{V}$ is in
$\mathfrak{g}.$
\end{defin}
\begin{example}\label{1}A {\em complex structure} $(M,I), I^2=-E$, is a $G$-structure
where $G=GL(n,\mathbb{C})$ with Lie algebra $\mathfrak{g} = \lbrace A \in
\mathfrak{gl}(2n, \mathbb{R}) | AI=IA \rbrace$. The first
prolongation $\mathfrak{g}^{(1)}$ is a space of symmetric
bilinear mappings $$ \mathfrak{g}^{(1)} =\{ t | t: \mathbb{V}
\times \mathbb{V} \rightarrow \mathbb{V}, t(IX,Y)=It(X,Y),
t(Y,X)=t(X,Y) \}.$$
On the other hand, a {\em product structure} $(M,P), P^2=E$ is a $G$-structure
where $G=GL(n,\mbox{$\mathbb R$}) \oplus GL(n,\mbox{$\mathbb R$}) $ with Lie algebra $\mathfrak{g} =
\mathfrak g(n,\mbox{$\mathbb R$}) \oplus \mathfrak g(n,\mbox{$\mathbb R$}) $. The first
prolongation $\mathfrak{g}^{(1)}$ is a space of symmetric
bilinear mappings $$ \mathfrak{g}^{(1)} =\{ t | t: \mathbb{V}_1\oplus \mathbb{V}_2
\times \mathbb{V}_1\oplus \mathbb{V}_2 \rightarrow \mathbb{V}_1\oplus \mathbb{V}_2,
t(\mathbb{V}_i,\mathbb{V}_i)\in \mathbb{V}_i,t(\mathbb{V}_2,\mathbb{V}_1) =0 \}.$$
\end{example}
\begin{lemma}
Let $M$ be a $(km)$--dimensional Clifford manifold based on
Clifford algebra $\mathcal O = \mathcal Cl(s,t),\ k=2^{s+t}$,
$s+t>1$, $m\in \mathbb N,$ i.e. manifold equipped with $G$--structure,
where
$$G= GL (m, \mathcal O) = \{ B \in GL(km, \mbox{$\mathbb R$}) | B I_i = I_i B, \: B J_j = J_j B\},$$
and $I_i$ and $J_j$ are algebra generators of $\mathcal O$.
Then the first prolongation $\mathfrak g^{(1)}$ of Lie algebra $\mathfrak g$ of Lie group $G$ vanishes.
\end{lemma}
\begin{proof}
Lie algebra $\mathfrak g$ of a Lie group $G$ is of the form
$$\mathfrak g=\mathfrak{gl}( m, \mathcal O ) = \{ B \in \mathfrak{gl}(km, \mbox{$\mathbb R$}) | B I_i = I_i B, \: B J_j = J_j B \},$$
where $I_i$ and $J_j$ are generators of $\mathcal O$, i.e.
$K_{\bar i} K_{\bar j} = - K_{\bar j} K_{\bar i}$ for $K_{\bar i}, K_{\bar j} \in \{ I_i,J_j \} $,
${\bar i} \neq {\bar j}$.
For $t \in \mathfrak g^{(1)}$ and $K_{\bar i} \neq K_{\bar j}$ we have equations
$$ t(K_{\bar i}X,K_{\bar j}X) = K_{\bar i} K_{\bar j} t(X,X), $$
$$ t(K_{\bar i}X,K_{\bar j}X) = t(K_{\bar j}X,K_{\bar i}X) = K_{\bar j} K_{\bar i} t(X,X) =
-K_{\bar i} K_{\bar j} t(X,X),$$ which lead to
$t(X,X)=0$. Finally from polarization
$$t(X,Y) =\frac{1}{2} (t(X+Y,X+Y) - t(X,X) -t(Y,Y))=0.\eqno\qedhere$$
\end{proof}
Let us shorty note, that the Example \ref{1} covers Clifford manifold for $\mathcal O = \mathcal Cl(0,1)$ and $\mathcal O = \mathcal Cl(1,0)$.
Next, suppose that there is a $G$-invariant complement $\mathcal{D}$ to
$\partial(\mathfrak{g} \otimes \mathbb{V}^{\ast})$ in $\mathbb{V}
\otimes \wedge^2 \mathbb{V}^{\ast}:$
$$ \mathbb{V} \otimes \wedge^2 \mathbb{V}^{\ast}
= \partial (\mathfrak{g} \otimes \mathbb{V}^{\ast}) \oplus
\mathcal{D}, $$
where
$$ \partial : \mathrm{Hom} (\mathbb{V}, \mathfrak{g}) = \mathfrak{g} \otimes \mathbb{V}^{\ast}
\rightarrow \mathbb{V} \otimes \wedge^2 \mathbb{V}^{\ast}$$
is the Spencer operator of alternation.
\begin{defin}
Let $\pi: P \rightarrow M $ be a $G$-structure. A connection
$\omega$ on $P$ is called a {\em $\mathcal{D} - connection$} if its torsion
function
$$ t^{\omega} : P \rightarrow \mathbb{V} \otimes \wedge^2 \mathbb{V}^{\ast} = \partial (\mathfrak{g} \otimes \mathbb{V}^{\ast})
\oplus \mathcal{D} $$ has values in $\mathcal{D}.$
\end{defin}
\begin{thm}{\cite{am}}
\begin{enumerate}
\item Any $G$-structure $\pi : P \rightarrow M$ admits a
$\mathcal{D}$-connection $\nabla.$ \item Let $\omega,
\bar{\omega},$ be two $\mathcal{D}$-connections. Then the
corresponding operators of covariant derivative $\nabla,
\bar{\nabla}$ are related by
$$ \bar{\nabla}= \nabla + S,$$
where $S$ is a tensor field such that for any $x \in M,\ S_x$
belongs to the first prolongation $\mathfrak{g}^{(1)}$ of the
Lie algebra $\mathfrak{g}.$
\end{enumerate}
\end{thm}
\begin{defin}
We say that a connected linear Lie group $G$ with Lie algebra
$\mathfrak{g}$ is of type $k$ if its $k$-th prolongation vanishes,
i.e. $\mathfrak{g}^{(k)}=0$ and $\mathfrak{g}^{(k-1)} \neq 0.$ In
this sense, any $G$-structure with Lie group $G$ of type $k$ is called a {\em G-structure of
type k.} \end{defin}
\begin{thm}{\cite{am}}
Let $\pi: P \rightarrow M$ be a $G$-structure of type 1 and
suppose that there is given a $G$-equivariant decomposition
$$ \mathbb{V} \otimes \wedge^2 \mathbb{V} = \partial
(\mathfrak{g} \otimes \mathbb{V}^*) \oplus \mathcal{D}.$$
Then there exists a unique connection, whose torsion tensor(calculated
with respect to a coframe $p \in P$) has values in $\mathcal{D}
\subset \mathbb{V} \otimes \wedge^2 \mathbb{V}^*.$
\end{thm}
\begin{cor}
Let $M$ be a smooth manifold equipped with a $G$--structure, where
$G= GL (n, \mathcal O)$, $ \mathcal O =\mathcal Cl(s,t)$, $s+t>1,$ i.e. an almost Clifford manifold. Then the
$G$--structure is of type 1 and there exists a unique $\mathcal D$--connection.
\end{cor}
\section{An almost Clifordian manifold}
One can see that an almost Cliffordian manifold $M$
is given as a $G$--structure provided that there is a reduction of the structure group of the principal frame bundle of $M$
to
$$G:=GL(m, \mathcal O) GL(1, \mathcal O) =
GL(m, \mathcal O) \times_{Z(GL(1, \mathcal O))} GL(1, \mathcal O),$$
where $Z(G)$ is a center of $G$.
The action of $G$ on $T_x M$ looks like
$$QXq, \text{ where } Q \in GL(m, \mathcal O), q \in GL(1, \mathcal O),$$
where the right action of $GL(1, \mathcal O)$ is blockwise.
In this case the tensor fields in the form $F_1, \dots, F_{k}$
can be defined only locally. It is easy to see that the Lie algebra $\mathfrak{gl}(m, {\mathcal O})$ of a Lie group $GL(m, {\mathcal O})$ is of the form
$$\mathfrak{gl}(m, {\mathcal O}) = \{ A \in \mathfrak{gl}(km, \mathbb R) | A I_i = I_i A ,
A J_j = J_j A\} $$
and
the Lie algebra $\mathfrak{g}$ of a Lie group $GL(m, {\mathcal O})GL(1, \mathcal O)$ is of the form
$$\mathfrak{g} = \mathfrak{gl}(m, {\mathcal O}) \oplus
\mathfrak{gl}(1, \mathcal O).$$
\noindent Let us note that the case of ${\mathcal C} l(0,3)$ was studied in a detailed way in \cite{b2}.
\begin{remark} \label{lema1} Let $\mathcal O$ be the Clifford algebra ${\mathcal C} l(0,2)$. For any one--form $\xi$ on $\mathbb V$ and any $X,Y \in \mathbb V$, the elements of the form
\begin{align*}
S^{\xi}(X,Y) &= - \xi(X)Y -\xi(Y)X
+ \xi(I_1X)I_1Y + \xi(I_1Y)I_1X
+ \xi(I_2X)I_2Y \\
&+ \xi(I_2Y)I_2X+ \xi(I_1I_2X)I_1I_2Y + \xi(I_1I_2Y)I_1I_2X
\end{align*}
belong to the
first prolongation $\mathfrak g^{(1)}$ of the Lie algebra $\mathfrak g$ of the
Lie group $GL (m, \mathcal O) GL(1, \mathcal O)$.
\end{remark}
\begin{proof}
We fix $X \in \mathbb V$ and define
$ S^{\xi}_X:= S^{\xi}(X,Y): \mathbb V \to \mathbb V$.
We have to prove that $ S^{\xi}_X(I_i Y)= I_i S^{\xi}_X( Y) + \sum_{l=1}^{4} a_l F_l (Y)$,
for $i=1,2$ and $S^{\xi}_X( Y)=S^{\xi}_Y(X)$. We compute directly for any $X$ and for $I_1$
\begin{align*}
S_X^{\xi}(I_1Y)
&= - \xi(X)I_1Y - \xi(I_1Y)X - \xi(I_1X)Y - \xi(Y)I_1X
- \xi(I_2X)I_1I_2Y \\
& - \xi(I_1I_2Y)I_2X + \xi(I_1I_2X)I_2Y + \xi(I_2Y)I_1I_2X \\
&=- \xi(I_1Y)X
- \xi(Y)I_1X
- \xi(I_1I_2Y)I_2X\\
&+ \xi(I_2Y)I_1I_2X
+ \sum_{l=1}^{4} a_l F_l (Y).
\end{align*}\
On the other hand,
\begin{align*}
I_1 S_X^{\xi}(Y) &= - \xi(X)I_1Y - \xi(Y)I_1X
- \xi(I_1X)Y - \xi(I_1Y)X + \xi(I_2X)I_1I_2Y \\
&+ \xi(I_2Y)I_1I_2X - \xi(I_1I_2X)I_2Y - \xi(I_1I_2Y)I_2X\\
&= - \xi(Y)I_1X - \xi(I_1Y)X
+ \xi(I_2Y)I_1I_2X\\
&- \xi(I_1I_2Y)I_2X
+ \sum_{l=1}^{4} a_l F_l (Y)
\end{align*}
and
\begin{align*}
S_X^{\xi}(I_1Y) - I_1 S_X^{\xi}(Y)&=\\
= &-\xi(I_1Y)X
- \xi(Y)I_1X
- \xi(I_1I_2Y)I_2X
+ \xi(I_2Y)I_1I_2X\\
-&( - \xi(Y)I_1X
- \xi(I_1Y)X
+ \xi(I_2Y)I_1I_2X
- \xi(I_1I_2Y)I_2X)\\
+ &\sum_{l=1}^{4} \bar{a}_l F_l (Y) =
\sum_{l=1}^{4} \bar{a}_l F_l (Y).
\end{align*}
By the same process for $I_2$ we obtain
\begin{align*}
S_X^{\xi}(I_2Y) - I_2S_X^{\xi}(Y)&=\\
= &- \xi(I_2Y)X
+ \xi(I_1I_2Y)I_1X
- \xi(Y)I_2X
- \xi(I_1Y)I_1I_2X\\
-&( - \xi(Y)I_2X
- \xi(I_1Y)I_1I_2X
- \xi(I_2Y)X + \xi(I_1I_2Y)I_1X)\\
+ &\sum_{l=1}^{4} \bar{a}_l F_l (Y) =
\sum_{l=1}^{4} \bar{a}_l F_l (Y).
\end{align*}
Finally, we have to prove the symmetry, but this is obvious.
\end{proof}
\begin{lemma} \label{koef}
Let ${\mathcal C} l(s,t)$ be the Clifford algebra, $n=s+t,$ and let us denote by $F_i$ the affinors obtained from the generators of ${\mathcal C} l(s,t).$ Then there exist $\varepsilon_i\in\{\pm 1\},\ i=1,...,n$ such that for $A\in V^\ast,$ the tensor $S^A\in V\times V\rightarrow V$ defined by
\begin{equation}\label{SAdef}
S^A(X,Y)=\sum_{i=1}^n \varepsilon_i A(F_iX)F_iY,\ X,Y\in V,
\end{equation}
satisfies the identity
\begin{equation}\label{SAident}
S^A(I_jX,Y)-I_jS^A(X,Y)=0
\end{equation}
for all algebra generators $I_j$ of ${\mathcal C} l(s,t).$
\end{lemma}
\begin{proof}
Let us consider the gradation of the Clifford algebra ${\mathcal C} l(s,t)={\mathcal C} l^0\oplus{\mathcal C} l^1\oplus ...\oplus{\mathcal C} l^n$ with respect to the generators of ${\mathcal C} l(s,t).$ Then we can define gradually: for $E\in{\mathcal C} l^0$ we choose $\varepsilon=1$. If the identity \eqref{SAident} should be satisfied for the terms in \eqref{SAdef}, then it must hold
$$\varepsilon_0A(I_jX)Y=\varepsilon_iA(I_jX)I_jI_jY\ \text{for all}\ I_j,$$
i.e.
\begin{equation*}
\varepsilon_i=
\begin{cases} \phantom{-}1 & \text{for $I_j^2=1$,}
\\
-1 &\text{for $I_j^2=-1$.}
\end{cases}
\end{equation*}
For $F_i\in{\mathcal C} l^v$ the following equality holds:
$$\varepsilon_iA(F_iI_jX)F_iY=\varepsilon_kA(F_kX)I_jF_kY$$
and thus $F_i=I_jF_k.$ Note that $F_k$ can be an element of both ${\mathcal C} l^{v+1}$ and ${\mathcal C} l^{v-1}.$ W.l.o.g. we choose $I_j$ such that $F_k\in{\mathcal C} l^{v+1}.$ Now two possibilities can appear:
Either \begin{equation}\label{eq1} F_iI_j=F_k,\end{equation} which leads to $I_jF_kI_j=F_k$ and thus $\varepsilon _k=\varepsilon_i,$ or
\begin{equation}\label{eq2} F_iI_j=-F_k,\end{equation} which leads to $I_jF_kI_j=-F_k$ and thus $\varepsilon _k=-\varepsilon_i.$
This concludes the definition of $\varepsilon_i$ such that the identity \eqref{SAident} holds. To prove the consistency, we have to show that the value of $\varepsilon_k$ does not depend on $I_j,$ i.e. for the generators $I$ such that $I^2=1$ and $J$ such that $J^2=-1,$ the resulting coefficient $\varepsilon_k$ obtained after two consequent steps of the algorithm with the alternate use of both $I$ and $J$, does not depend on the order.
Thus let us consider the following cases:
\begin{itemize}
\item[(a)] $F_i=IF_k,$ which results into the possibilities $IF_kI=F_k,$ see \eqref{eq1}, which leads to $\varepsilon_k=\varepsilon_i,$ or $IF_kI=-F_k,$ see \eqref{eq2}, which leads to $\varepsilon_k=-\varepsilon_i.$
\item[(b)] $F_j=JF_k,$which similarly leads to either $JF_kJ=F_k$ implying $\varepsilon_k=\varepsilon_j,$ or
$JF_kJ=-F_k$ implying $\varepsilon_k=-\varepsilon_j.$
\end{itemize}
Applying the processes (a) and (b) alternately we obtain:
$$JIF_kJ=\begin{cases}-IF_k & \Rightarrow \varepsilon_l=-\varepsilon_i\\
\phantom{-}IF_k & \Rightarrow\varepsilon_l=\varepsilon_i\end{cases}$$
for $F_l=JIF_k$ and
$$IJF_kI=\begin{cases}-JF_k & \Rightarrow \varepsilon_l=-\varepsilon_j\\
\phantom{-}JF_k & \Rightarrow\varepsilon_l=\varepsilon_j\end{cases}$$
for $F_l=IJF_k.$ Obviously, the corresponding cases give the same result of $\varepsilon_k.$
\end{proof}
\begin{thm} Let $\mathcal O$ be the Clifford algebra ${\mathcal C} l(s,t)$.
For any one--form $\xi$ on $\mathbb V$ and any $X,Y \in \mathbb V$, the elements of the form
$$ S_X^{\xi} (Y) = \sum_{i=1}^{k} \epsilon_i (\xi(F_iX)F_iY + \xi(F_i Y)F_i X),\ k=2^{s+t},$$
where the coefficients $\epsilon_i$ depend on the type of $\mathcal O,$
belong to the first prolongation $\mathfrak g^{(1)}$ of the Lie algebra $\mathfrak g$ of the
Lie group $GL (m, \mathcal O)GL(1, \mathcal O)$.
\end{thm}
\begin{proof}
One can easily see that
$S^\xi$ is symmetric and we have to prove the second condition, i.e.
$S_X^{\xi}I_iY -I_iS_X^{\xi}Y \in \mathcal O(Y)$, i.e.
\begin{align*}
S_X^{\xi}I_iY -I_iS_X^{\xi}Y &=
\sum_{j=1}^{k} \bar{\epsilon}_j \xi(F_jX)F_j Y
+\sum_{j=1}^{k} \bar{\epsilon}_j \xi(F_jY)F_jX \\
&- \sum_{j=1}^{k} \bar{\epsilon}_j I_i\xi(F_jX)F_jY
-\sum_{j=1}^{k} \bar{\epsilon}_j I_i\xi(F_jY)F_jTX.
\end{align*}
From Lemma \ref{koef} we have
$$
S_X^{\xi}I_iY -I_iS_X^{\xi}Y =
\sum_{j=1}^{k} \bar{\epsilon}_j \xi(F_jX)F_j Y
-\sum_{j=1}^{k} \bar{\epsilon}_j I_i\xi(F_jY)F_jTX
= \sum_{j=0}^{k} \psi_i F_j Y.$$
\end{proof}
\begin{cor}
Let $M$ be an almost Cliffordian manifold based on Clifford algebra
$\mathcal O= {\mathcal C} l(s,t)$, where $\dim(M) \geq 2(s+t)$, i.e. smooth manifold equipped
with $G$--structure, where $G= GL (n, \mathcal O)GL(1, \mathcal O)$
or equivalently $A$--structure where $A=\mathcal O$. Then the class of $\mathcal D$--connections
preserving $A$ and sharing the same $A$--planar curves is
isomorphic to $(\mbox{$\mathbb R$}^{km})^*.$
\end{cor}
\section*{Acknowledgment}
This work was supported by the European Regional Development Fund in the IT4Innovations Center of Excelence project CZ.1.05/1.1.00/02.0070.
|
train/arxiv
|
BkiUdI85qYVBWXyiroQP
| 5 | 1 |
\section{Introduction}
\vspace{5mm}
Let $(\Omega,$ ${\rm F},$ ${\sf P})$ be a complete probability space, let
$\{{\rm F}_t, t\in[0,T]\}$ be a nondecreasing right-continous
family of $\sigma$-subfields of ${\rm F},$
and let ${\bf w}_t$ be a standard $m$-dimensional Wiener
stochastic process, which is
${\rm F}_t$-measurable for any $t\in[0, T].$ We assume that the components
${\bf w}_{t}^{(i)}$ $(i=1,\ldots,m)$ of this process are independent. Consider
an Ito stochastic differential equation in the integral form
\begin{equation}
\label{1.5.2}
{\bf x}_t={\bf x}_0+\int\limits_0^t {\bf a}({\bf x}_{\tau},\tau)d\tau+
\int\limits_0^t B({\bf x}_{\tau},\tau)d{\bf w}_{\tau},\ \ \
{\bf x}_0={\bf x}(0,\omega),\ \ \ \omega\in\Omega.
\end{equation}
\vspace{3mm}
\noindent
Here ${\bf x}_t$ is some $n$-dimensional stochastic process
satisfying to equation (\ref{1.5.2}).
The nonrandom functions ${\bf a}: \Re^n\times[0, T]\to\Re^n$,
$B: \Re^n\times[0, T]\to\Re^{n\times m}$
guarantee the existence and uniqueness up to stochastic
equivalence of a solution
of equation (\ref{1.5.2}) \cite{1}. The second integral
on the right-hand side of (\ref{1.5.2}) is
interpreted as an Ito stochastic integral.
Let ${\bf x}_0$ be an $n$-dimensional random variable, which is
${\rm F}_0$-measurable and
${\sf M}\{\left|{\bf x}_0\right|^2\}<\infty$
(${\sf M}$ denotes a mathematical expectation).
We assume that
${\bf x}_0$ and ${\bf w}_t-{\bf w}_0$ are independent when $t>0.$
One of the effective approaches
to numerical integration of
Ito stochastic differential equations is an approach based on the Taylor--Ito
expansion \cite{2}-\cite{3}.
The most important feature of such
expansion is a presence in them of the so-called iterated
stochastic Ito integrals, which play the key
role for solving the
problem of numerical integration of Ito stochastic differential equations
and has the
following form
\begin{equation}
\label{ito}
J[\psi^{(k)}]_{T,t}=\int\limits_t^T\psi_k(t_k) \ldots \int\limits_t^{t_{2}}
\psi_1(t_1) d{\bf w}_{t_1}^{(i_1)}\ldots
d{\bf w}_{t_k}^{(i_k)}\ \ \ (i_1,\ldots,i_k = 0, 1,\ldots,m),
\end{equation}
\vspace{3mm}
\noindent
where every $\psi_l(\tau)$ $(l=1,\ldots,k)$ is a continuous non-random
function
on $[t,T],$ ${\bf w}_{\tau}^{(i)}$
($i=1,\ldots,m$) are independent standard Wiener processes, and
${\bf w}_{\tau}^{(0)}=\tau$.
In this article we pay special
attention to the case $k=2,$\ \ $i_1, i_2=1,\ldots,m$,\ \
$\psi_1(\tau),$ $\psi_2(\tau)\equiv 1$.
This case corresponds to
the so-called Milstein method \cite{Mi2}, \cite{3} of
numerical integration of Ito stochastic differential equations
of form (\ref{1.5.2}), which
has the order of strong convergence 1.0 under
the specific conditions \cite{Mi2}, \cite{3}.
The Milstein method has the following form \cite{Mi2}, \cite{3}
$$
{\bf y}_{p+1}={\bf y}_p+\sum_{i_{1}=1}^{m}B_{i_{1}}
I_{\tau_{p+1},\tau_p}^{(i_{1})}+\Delta{\bf a}
+\sum_{i_{1},i_{2}=1}^{m}G_{i_{1}}
B_{i_{2}}\hat I_{\tau_{p+1},\tau_p}^{(i_{1}i_{2})},
$$
\vspace{3mm}
\noindent
where $\Delta=T/N$ $(N>1)$ is a constant (for simplicity)
step of integration,\
$\tau_p=p\Delta$ $(p=0, 1,\ldots,N)$,\
$$
G_i = \sum^ {n} _ {j=1} B_{ji} ({\bf x}, t)
{\partial \over \partial {\bf x}_j}\ \ \
(i=1,\ldots,m),
$$
\vspace{3mm}
\noindent
$B_i$ is an $i$th column of the matrix function $B$ and $B_{ij}$
is an $ij$th element
of the matrix function
$B$, ${\bf a}_i$ is an $i$th element of the
vector function ${\bf a}$, and ${\bf x}_i$
is an
$i$th element of the column ${\bf x}$,
the columns
$B_{i_{1}},$\ ${\bf a},$\ $G_{i_{1}}B_{i_{2}}$
are calculated in the point $({\bf y}_p,p),$
$$
I_{\tau_{p+1},\tau_p}^{(i_{1})}=
\int\limits_{\tau_p}^{\tau_{p+1}}d{\bf w}_{\tau}^{(i_1)},
$$
\vspace{3mm}
\noindent
$\hat I_{\tau_{p+1},\tau_p}^{(i_{1}i_{2})}$ is an approximation of the
following double (iterated) stochastic Ito integral
$$
I_{\tau_{p+1},\tau_p}^{(i_{1}i_{2})}=
\int\limits_{\tau_p}^{\tau_{p+1}}\int\limits_{\tau_p}^{s}
d{\bf w}_{\tau}^{(i_1)}d{\bf w}_{s}^{(i_2)}.
$$
\vspace{3mm}
The Levy stochastic area $A^{(i_1i_2)}_{T,t}$ is defined as follows
\cite{1c}
\vspace{2mm}
$$
A_{T,t}^{(i_1 i_2)}=
\frac{1}{2}\left(I_{T,t}^{(i_1 i_2)}-I_{T,t}^{(i_2 i_1)}\right).
$$
\vspace{4mm}
It is clear that
\begin{equation}
\label{2121}
I_{T,t}^{(i_1 i_2)}=\frac{1}{2}I_{T,t}^{(i_1)}I_{T,t}^{(i_2)}+
A_{T,t}^{(i_1 i_2)}\ \ \ \hbox{w. p. 1},
\end{equation}
\vspace{3mm}
\noindent
where $i_1\ne i_2.$
From (\ref{2121}) it follows that the problem of numerical
simulation of double (iterated) stochastic Ito integral
$I_{T,t}^{(i_1 i_2)}$ is equivalent to the problem of numerical
simulation of the Levy stochastic area.
There are some methods for representation of the Levy stochastic area
(see, for example, \cite{2}--\cite{3}). In this article we consider another
representation of the Levy stochastic area, which is based
on the Legendre polynomials and simpler than its
existing analogues.
\vspace{5mm}
\section{Approach to Expansion of Iterated stochastic Ito integrals,
Based on Generalized Multiple
Fourier Series}
\vspace{5mm}
Consider an
iterated stochastic Ito integrals (\ref{ito}).
Define the following function on a hypercube $[t, T]^k$
$$
K(t_1,\ldots,t_k)=
\begin{cases}
\psi_1(t_1)\ldots \psi_k(t_k),\ &\ t_1<\ldots<t_k\\
~\\
0,\ &\ \hbox{\rm otherwise}
\end{cases},\ \ \ t_1,\ldots,t_k\in[t, T],\ \ \ k\ge 2,
$$
\vspace{6mm}
\noindent
and
$K(t_1)\equiv\psi_1(t_1),$ $t_1\in[t, T].$
Suppose that $\{\phi_j(x)\}_{j=0}^{\infty}$
is a complete orthonormal system of functions in
the space $L_2([t, T])$.
The function $K(t_1,\ldots,t_k)$ is sectionally continuous in the
hypercube $[t, T]^k.$
At this situation it is well known that the generalized
multiple Fourier series
of $K(t_1,\ldots,t_k)\in L_2([t, T]^k)$ is converging
to $K(t_1,\ldots,t_k)$ in the hypercube $[t, T]^k$ in
the mean-square sense, i.e.
\begin{equation}
\label{sos1z}
\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm lim}\cr
$\stackrel{}{{}_{p_1,\ldots,p_k\to \infty}}$\cr
}} }\biggl\Vert
K(t_1,\ldots,t_k)-
\sum_{j_1=0}^{p_1}\ldots \sum_{j_k=0}^{p_k}
C_{j_k\ldots j_1}\prod_{l=1}^{k} \phi_{j_l}(t_l)\biggr
\Vert_{L_2([t, T]^k)}=0,
\end{equation}
\vspace{2mm}
\noindent
where
\vspace{-2mm}
\begin{equation}
\label{ppppa}
C_{j_k\ldots j_1}=\int\limits_{[t,T]^k}
K(t_1,\ldots,t_k)\prod_{l=1}^{k}\phi_{j_l}(t_l)dt_1\ldots dt_k
\end{equation}
\vspace{2mm}
\noindent
is the Fourier coefficient and
$$
\left\Vert f\right\Vert_{L_2([t, T]^k)}=\left(\int\limits_{[t,T]^k}
f^2(t_1,\ldots,t_k)dt_1\ldots dt_k\right)^{1/2}.
$$
\vspace{4mm}
Consider the partition $\{\tau_j\}_{j=0}^N$ of $[t,T]$ such that
\begin{equation}
\label{1111}
t=\tau_0<\ldots <\tau_N=T,\ \ \ \
\Delta_N=
\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm max}\cr
$\stackrel{}{{}_{0\le j\le N-1}}$\cr
}} }\Delta\tau_j\to 0\ \ \hbox{if}\ \ N\to \infty,\ \ \ \
\Delta\tau_j=\tau_{j+1}-\tau_j.
\end{equation}
\vspace{2mm}
{\bf Theorem 1}\ \cite{2006}-\cite{200a}, \cite{301a}-\cite{arxiv-12},
\cite{arxiv-24}-\cite{arxiv-9}. {\it Suppose that
every $\psi_l(\tau)$ $(l=1,\ldots, k)$ is a continuous non-random function on
$[t, T]$ and
$\{\phi_j(x)\}_{j=0}^{\infty}$ is a complete orthonormal system
of continous functions in $L_2([t,T]).$ Then
\vspace{2mm}
$$
J[\psi^{(k)}]_{T,t}=
\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm l.i.m.}\cr
$\stackrel{}{{}_{p_1,\ldots,p_k\to \infty}}$\cr
}} }\sum_{j_1=0}^{p_1}\ldots\sum_{j_k=0}^{p_k}
C_{j_k\ldots j_1}\Biggl(
\prod_{l=1}^k\zeta_{j_l}^{(i_l)}-
\Biggr.
$$
\vspace{3mm}
\begin{equation}
\label{tyyy}
-\Biggl.
\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm l.i.m.}\cr
$\stackrel{}{{}_{N\to \infty}}$\cr
}} }\sum_{(l_1,\ldots,l_k)\in {\rm G}_k}
\phi_{j_{1}}(\tau_{l_1})
\Delta{\bf w}_{\tau_{l_1}}^{(i_1)}\ldots
\phi_{j_{k}}(\tau_{l_k})
\Delta{\bf w}_{\tau_{l_k}}^{(i_k)}\Biggr),
\end{equation}
\vspace{5mm}
\noindent
where
\vspace{-2mm}
$$
{\rm G}_k={\rm H}_k\backslash{\rm L}_k,\ \ \
{\rm H}_k=\{(l_1,\ldots,l_k):\ l_1,\ldots,l_k=0,\ 1,\ldots,N-1\},
$$
$$
{\rm L}_k=\{(l_1,\ldots,l_k):\ l_1,\ldots,l_k=0,\ 1,\ldots,N-1;\
l_g\ne l_r\ (g\ne r);\ g, r=1,\ldots,k\},
$$
\vspace{4mm}
\noindent
${\rm l.i.m.}$ is a limit in the mean-square sense,\ \
$i_1,\ldots,i_k=0,1,\ldots,m,$\ \
$C_{j_k\ldots j_1}$ is the Fourier coefficient
{\rm (\ref{ppppa}),}
every
\vspace{-1mm}
\begin{equation}
\label{rr23}
\zeta_{j}^{(i)}=
\int\limits_t^T \phi_{j}(s) d{\bf w}_s^{(i)}
\end{equation}
\vspace{2mm}
\noindent
is a standard Gaussian random variable
for various
$i$ or $j$ {\rm(}if $i\ne 0${\rm),}
$\Delta{\bf w}_{\tau_{j}}^{(i)}=
{\bf w}_{\tau_{j+1}}^{(i)}-{\bf w}_{\tau_{j}}^{(i)}$
$(i=0, 1,\ldots,m),$\
$\left\{\tau_{j}\right\}_{j=0}^{N}$ is a partition of
$[t,T],$ which satisfies the condition {\rm (\ref{1111})}.
}
\vspace{2mm}
In order to evaluate the significance of Theorem 1 for practice we will
demonstrate its transformed particular cases for
$k=1,\ldots,5$ \cite{2006}-\cite{2013}, \cite{arxiv-1}
\begin{equation}
\label{a1}
J[\psi^{(1)}]_{T,t}
=\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm l.i.m.}\cr
$\stackrel{}{{}_{p_1\to \infty}}$\cr
}} }\sum_{j_1=0}^{p_1}
C_{j_1}\zeta_{j_1}^{(i_1)},
\end{equation}
\vspace{3mm}
\begin{equation}
\label{a2}
J[\psi^{(2)}]_{T,t}
=\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm l.i.m.}\cr
$\stackrel{}{{}_{p_1,p_2\to \infty}}$\cr
}} }\sum_{j_1=0}^{p_1}\sum_{j_2=0}^{p_2}
C_{j_2j_1}\biggl(\zeta_{j_1}^{(i_1)}\zeta_{j_2}^{(i_2)}
-{\bf 1}_{\{i_1=i_2\ne 0\}}
{\bf 1}_{\{j_1=j_2\}}\biggr),
\end{equation}
\vspace{5mm}
$$
J[\psi^{(3)}]_{T,t}=
\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm l.i.m.}\cr
$\stackrel{}{{}_{p_1,\ldots,p_3\to \infty}}$\cr
}} }\sum_{j_1=0}^{p_1}\sum_{j_2=0}^{p_2}\sum_{j_3=0}^{p_3}
C_{j_3j_2j_1}\biggl(
\zeta_{j_1}^{(i_1)}\zeta_{j_2}^{(i_2)}\zeta_{j_3}^{(i_3)}
-\biggr.
$$
\begin{equation}
\label{a3}
-\biggl.
{\bf 1}_{\{i_1=i_2\ne 0\}}
{\bf 1}_{\{j_1=j_2\}}
\zeta_{j_3}^{(i_3)}
-{\bf 1}_{\{i_2=i_3\ne 0\}}
{\bf 1}_{\{j_2=j_3\}}
\zeta_{j_1}^{(i_1)}-
{\bf 1}_{\{i_1=i_3\ne 0\}}
{\bf 1}_{\{j_1=j_3\}}
\zeta_{j_2}^{(i_2)}\biggr),
\end{equation}
\vspace{6mm}
$$
J[\psi^{(4)}]_{T,t}
=
\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm l.i.m.}\cr
$\stackrel{}{{}_{p_1,\ldots,p_4\to \infty}}$\cr
}} }\sum_{j_1=0}^{p_1}\ldots\sum_{j_4=0}^{p_4}
C_{j_4\ldots j_1}\biggl(
\prod_{l=1}^4\zeta_{j_l}^{(i_l)}
\biggr.
-
$$
$$
-
{\bf 1}_{\{i_1=i_2\ne 0\}}
{\bf 1}_{\{j_1=j_2\}}
\zeta_{j_3}^{(i_3)}
\zeta_{j_4}^{(i_4)}
-
{\bf 1}_{\{i_1=i_3\ne 0\}}
{\bf 1}_{\{j_1=j_3\}}
\zeta_{j_2}^{(i_2)}
\zeta_{j_4}^{(i_4)}-
$$
$$
-
{\bf 1}_{\{i_1=i_4\ne 0\}}
{\bf 1}_{\{j_1=j_4\}}
\zeta_{j_2}^{(i_2)}
\zeta_{j_3}^{(i_3)}
-
{\bf 1}_{\{i_2=i_3\ne 0\}}
{\bf 1}_{\{j_2=j_3\}}
\zeta_{j_1}^{(i_1)}
\zeta_{j_4}^{(i_4)}-
$$
$$
-
{\bf 1}_{\{i_2=i_4\ne 0\}}
{\bf 1}_{\{j_2=j_4\}}
\zeta_{j_1}^{(i_1)}
\zeta_{j_3}^{(i_3)}
-
{\bf 1}_{\{i_3=i_4\ne 0\}}
{\bf 1}_{\{j_3=j_4\}}
\zeta_{j_1}^{(i_1)}
\zeta_{j_2}^{(i_2)}+
$$
$$
+
{\bf 1}_{\{i_1=i_2\ne 0\}}
{\bf 1}_{\{j_1=j_2\}}
{\bf 1}_{\{i_3=i_4\ne 0\}}
{\bf 1}_{\{j_3=j_4\}}
+
{\bf 1}_{\{i_1=i_3\ne 0\}}
{\bf 1}_{\{j_1=j_3\}}
{\bf 1}_{\{i_2=i_4\ne 0\}}
{\bf 1}_{\{j_2=j_4\}}+
$$
\begin{equation}
\label{a4}
+\biggl.
{\bf 1}_{\{i_1=i_4\ne 0\}}
{\bf 1}_{\{j_1=j_4\}}
{\bf 1}_{\{i_2=i_3\ne 0\}}
{\bf 1}_{\{j_2=j_3\}}\biggr),
\end{equation}
\vspace{6mm}
$$
J[\psi^{(5)}]_{T,t}
=\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm l.i.m.}\cr
$\stackrel{}{{}_{p_1,\ldots,p_5\to \infty}}$\cr
}} }\sum_{j_1=0}^{p_1}\ldots\sum_{j_5=0}^{p_5}
C_{j_5\ldots j_1}\Biggl(
\prod_{l=1}^5\zeta_{j_l}^{(i_l)}
-\Biggr.
$$
$$
-
{\bf 1}_{\{i_1=i_2\ne 0\}}
{\bf 1}_{\{j_1=j_2\}}
\zeta_{j_3}^{(i_3)}
\zeta_{j_4}^{(i_4)}
\zeta_{j_5}^{(i_5)}-
{\bf 1}_{\{i_1=i_3\ne 0\}}
{\bf 1}_{\{j_1=j_3\}}
\zeta_{j_2}^{(i_2)}
\zeta_{j_4}^{(i_4)}
\zeta_{j_5}^{(i_5)}-
$$
$$
-
{\bf 1}_{\{i_1=i_4\ne 0\}}
{\bf 1}_{\{j_1=j_4\}}
\zeta_{j_2}^{(i_2)}
\zeta_{j_3}^{(i_3)}
\zeta_{j_5}^{(i_5)}-
{\bf 1}_{\{i_1=i_5\ne 0\}}
{\bf 1}_{\{j_1=j_5\}}
\zeta_{j_2}^{(i_2)}
\zeta_{j_3}^{(i_3)}
\zeta_{j_4}^{(i_4)}-
$$
$$
-
{\bf 1}_{\{i_2=i_3\ne 0\}}
{\bf 1}_{\{j_2=j_3\}}
\zeta_{j_1}^{(i_1)}
\zeta_{j_4}^{(i_4)}
\zeta_{j_5}^{(i_5)}-
{\bf 1}_{\{i_2=i_4\ne 0\}}
{\bf 1}_{\{j_2=j_4\}}
\zeta_{j_1}^{(i_1)}
\zeta_{j_3}^{(i_3)}
\zeta_{j_5}^{(i_5)}-
$$
$$
-
{\bf 1}_{\{i_2=i_5\ne 0\}}
{\bf 1}_{\{j_2=j_5\}}
\zeta_{j_1}^{(i_1)}
\zeta_{j_3}^{(i_3)}
\zeta_{j_4}^{(i_4)}
-{\bf 1}_{\{i_3=i_4\ne 0\}}
{\bf 1}_{\{j_3=j_4\}}
\zeta_{j_1}^{(i_1)}
\zeta_{j_2}^{(i_2)}
\zeta_{j_5}^{(i_5)}-
$$
$$
-
{\bf 1}_{\{i_3=i_5\ne 0\}}
{\bf 1}_{\{j_3=j_5\}}
\zeta_{j_1}^{(i_1)}
\zeta_{j_2}^{(i_2)}
\zeta_{j_4}^{(i_4)}
-{\bf 1}_{\{i_4=i_5\ne 0\}}
{\bf 1}_{\{j_4=j_5\}}
\zeta_{j_1}^{(i_1)}
\zeta_{j_2}^{(i_2)}
\zeta_{j_3}^{(i_3)}+
$$
$$
+
{\bf 1}_{\{i_1=i_2\ne 0\}}
{\bf 1}_{\{j_1=j_2\}}
{\bf 1}_{\{i_3=i_4\ne 0\}}
{\bf 1}_{\{j_3=j_4\}}\zeta_{j_5}^{(i_5)}+
{\bf 1}_{\{i_1=i_2\ne 0\}}
{\bf 1}_{\{j_1=j_2\}}
{\bf 1}_{\{i_3=i_5\ne 0\}}
{\bf 1}_{\{j_3=j_5\}}\zeta_{j_4}^{(i_4)}+
$$
$$
+
{\bf 1}_{\{i_1=i_2\ne 0\}}
{\bf 1}_{\{j_1=j_2\}}
{\bf 1}_{\{i_4=i_5\ne 0\}}
{\bf 1}_{\{j_4=j_5\}}\zeta_{j_3}^{(i_3)}+
{\bf 1}_{\{i_1=i_3\ne 0\}}
{\bf 1}_{\{j_1=j_3\}}
{\bf 1}_{\{i_2=i_4\ne 0\}}
{\bf 1}_{\{j_2=j_4\}}\zeta_{j_5}^{(i_5)}+
$$
$$
+
{\bf 1}_{\{i_1=i_3\ne 0\}}
{\bf 1}_{\{j_1=j_3\}}
{\bf 1}_{\{i_2=i_5\ne 0\}}
{\bf 1}_{\{j_2=j_5\}}\zeta_{j_4}^{(i_4)}+
{\bf 1}_{\{i_1=i_3\ne 0\}}
{\bf 1}_{\{j_1=j_3\}}
{\bf 1}_{\{i_4=i_5\ne 0\}}
{\bf 1}_{\{j_4=j_5\}}\zeta_{j_2}^{(i_2)}+
$$
$$
+
{\bf 1}_{\{i_1=i_4\ne 0\}}
{\bf 1}_{\{j_1=j_4\}}
{\bf 1}_{\{i_2=i_3\ne 0\}}
{\bf 1}_{\{j_2=j_3\}}\zeta_{j_5}^{(i_5)}+
{\bf 1}_{\{i_1=i_4\ne 0\}}
{\bf 1}_{\{j_1=j_4\}}
{\bf 1}_{\{i_2=i_5\ne 0\}}
{\bf 1}_{\{j_2=j_5\}}\zeta_{j_3}^{(i_3)}+
$$
$$
+
{\bf 1}_{\{i_1=i_4\ne 0\}}
{\bf 1}_{\{j_1=j_4\}}
{\bf 1}_{\{i_3=i_5\ne 0\}}
{\bf 1}_{\{j_3=j_5\}}\zeta_{j_2}^{(i_2)}+
{\bf 1}_{\{i_1=i_5\ne 0\}}
{\bf 1}_{\{j_1=j_5\}}
{\bf 1}_{\{i_2=i_3\ne 0\}}
{\bf 1}_{\{j_2=j_3\}}\zeta_{j_4}^{(i_4)}+
$$
$$
+
{\bf 1}_{\{i_1=i_5\ne 0\}}
{\bf 1}_{\{j_1=j_5\}}
{\bf 1}_{\{i_2=i_4\ne 0\}}
{\bf 1}_{\{j_2=j_4\}}\zeta_{j_3}^{(i_3)}+
{\bf 1}_{\{i_1=i_5\ne 0\}}
{\bf 1}_{\{j_1=j_5\}}
{\bf 1}_{\{i_3=i_4\ne 0\}}
{\bf 1}_{\{j_3=j_4\}}\zeta_{j_2}^{(i_2)}+
$$
$$
+
{\bf 1}_{\{i_2=i_3\ne 0\}}
{\bf 1}_{\{j_2=j_3\}}
{\bf 1}_{\{i_4=i_5\ne 0\}}
{\bf 1}_{\{j_4=j_5\}}\zeta_{j_1}^{(i_1)}+
{\bf 1}_{\{i_2=i_4\ne 0\}}
{\bf 1}_{\{j_2=j_4\}}
{\bf 1}_{\{i_3=i_5\ne 0\}}
{\bf 1}_{\{j_3=j_5\}}\zeta_{j_1}^{(i_1)}+
$$
\begin{equation}
\label{a5}
+\Biggl.
{\bf 1}_{\{i_2=i_5\ne 0\}}
{\bf 1}_{\{j_2=j_5\}}
{\bf 1}_{\{i_3=i_4\ne 0\}}
{\bf 1}_{\{j_3=j_4\}}\zeta_{j_1}^{(i_1)}\Biggr),
\end{equation}
\vspace{6mm}
\noindent
where ${\bf 1}_A$ is the indicator of the set $A$.
It was shown in \cite{2007-2}-\cite{2013}, \cite{arxiv-1}
that Theorem 1 is valid for convergence
in the mean of degree $2n$ ($n\in \mathbb{N}$).
Moreover, the complete orthonormal in $L_2([t,T])$ systems of Haar and
Rademacher--Walsh functions also
can be applied in Theorem 1
\cite{2006}-\cite{2013}.
The generalization of Theorem 1 for
complete orthonormal with weigth $r(x)\ge 0$ system
of functions in the space $L_2([t,T])$ can be found in
\cite{2018}, \cite{arxiv-26b}.
Recently, Theorem 1 has been applied
to expansion and mean-square
approximation of iterated stochastic Ito integrals
with respect to the infinite-dmensional $Q$-Wiener process
\cite{31a}, \cite{200a}. These results can be directly
applied to construction of
high-order strong numerical methods for non-commutative
semilinear stochastic partial differential equations
\cite{31a}, \cite{200a}.
Note that we obtain the following useful possibilities
of the approach, based on Theorem 1.
1. There is an obvious formula (see (\ref{ppppa})) for calculation
of expansion coefficients
of iterated stochastic Ito integral (\ref{ito}) with any
fixed multiplicity $k$.
2. We have possibilities for exact calculation of the mean-square
error of approximation
of iterated stochastic Ito integral (\ref{ito})
\cite{2017}-\cite{2018}, \cite{17a}, \cite{arxiv-2}.
3. Since the used
multiple Fourier series is a generalized in the sense
that it is built using various complete orthonormal
systems of functions in the space $L_2([t, T])$, then we
have new possibilities
for approximation --- we may
use not only trigonometric functions as in \cite{2}-\cite{3}
but Legendre polynomials.
4. As it turned out \cite{2006}-\cite{200a},
\cite{301a}-\cite{arxiv-12},
\cite{arxiv-24}-\cite{arxiv-9} it is more convenient to work
with Legendre po\-ly\-no\-mi\-als for building of approximations
of iterated stochastic Ito integrals (\ref{ito}).
Approximations based on the Legendre polynomials essentially simpler
than their analogues based on the trigonometric functions.
Another advantages of the application of Legendre polynomials
in the framework of the mentioned question are considered
in \cite{29a}, \cite{301a}.
5. The approach to expansion of iterated
stochastic Ito integrals, based on the Karhunen--Loeve expansion
of the Wiener process as well as the approach from \cite{rr} (we will
discuss this work in Sect. 5)
lead to
iterated application of the operation of limit
transition (the operation of limit transition
is implemented only once in Theorem 1 (see above))
starting from
second multiplicity (in the general case)
and third multiplicity (for the case
$\psi_1(s), \psi_2(s), \psi_3(s)\equiv 1;$
$i_1, i_2, i_3=1,\ldots,m$)
of iterated stochastic Ito integrals.
Multiple series from Theorem 1 (the operation of limit transition
is implemented only once) are more convenient
for approximation than the iterated ones
(iterated application of operation of the limit
transition),
since partial sums of multiple series converge for any possible case of
convergence to infinity of their upper limits of summation
(let us denote them as $p_1,\ldots, p_k$).
For example, for more simple and convenient for practice case
when $p_1=\ldots=p_k=p\to\infty$. For iterated series it is obviously
not the case. However, in \cite{2}
the authors unreasonably use
the condition $p_1=p_2=p_3=p\to\infty$
within the application of the mentioned approach,
based on the Karhunen--Loeve expansion of the Wiener process \cite{Mi2}.
Note that rightness of formulas (\ref{a1})--(\ref{a5})
can be
verified
by the fact that if
$i_1=\ldots=i_5=i=1,\ldots,m$
and $\psi_1(s),\ldots,\psi_5(s)\equiv \psi(s)$ in (\ref{a1})--(\ref{a5}),
then we can obtain from (\ref{a1})--(\ref{a5}) the
following equalities which are right w. p. 1
\vspace{-1mm}
$$
J[\psi^{(1)}]_{T,t}
=\frac{1}{1!}\delta_{T,t},
$$
$$
J[\psi^{(2)}]_{T,t}
=\frac{1}{2!}\left(\delta^2_{T,t}-\Delta_{T,t}\right),\
$$
$$
J[\psi^{(3)}]_{T,t}
=\frac{1}{3!}\left(\delta_{T,t}^3-3\delta_{T,t}\Delta_{T,t}\right),
$$
$$
J[\psi^{(4)}]_{T,t}
=\frac{1}{4!}\left(\delta^4_{T,t}-6\delta_{T,t}^2\Delta_{T,t}
+3\Delta^2_{T,t}\right),\
$$
$$
J[\psi^{(5)}]_{T,t}
=\frac{1}{5!}\left(\delta^5_{T,t}-10\delta_{T,t}^3\Delta_{T,t}
+15\delta_{T,t}\Delta^2_{T,t}\right),
$$
\vspace{2mm}
\noindent
where
\vspace{-3mm}
$$
\delta_{T,t}=\int\limits_t^T\psi(s)d{\bf w}_s^{(i)},\ \ \
\Delta_{T,t}=\int\limits_t^T\psi^2(s)ds,
$$
\vspace{2mm}
\noindent
which can be independently
obtained using the Ito formula and Hermite polynomials.
The cases
$k=2, 3$ and $p_1=p_2=p_3=p$ are considered in
\cite{2011-2}-\cite{2013}.
\vspace{5mm}
\section{New Representation of the Levy Stochastic Area,
Based on the Legendre Polynomials}
\vspace{5mm}
Let us consider (\ref{a2}) for the case
$i_1\ne i_2$,\ \ $\psi_1(s),$ $\psi_2(s)\equiv 1$. At that we suppose that
$\{\phi_j(x)\}_{j=0}^{\infty}$ is a complete orthonormal sys\-tem
of Legendre polynomials in the space $L_2([t,T]).$ Then
\begin{equation}
\label{zorro}
I_{T,t}^{(i_1 i_2)}
=
\frac{T-t}{2}\left(\zeta_0^{(i_1)}\zeta_0^{(i_2)}+\sum_{i=1}^{\infty}
\frac{1}{\sqrt{4i^2-1}}\left(
\zeta_{i-1}^{(i_1)}\zeta_{i}^{(i_2)}-
\zeta_i^{(i_1)}\zeta_{i-1}^{(i_2)}\right)\right),
\end{equation}
\vspace{2mm}
\noindent
where
\begin{equation}
\label{jq2}
I_{T,t}^{(i_1i_2)}=\int\limits_t^T\int\limits_t^s
d{\bf w}_{\tau}^{(i_1)}d{\bf w}_s^{(i_2)}\ \ \ (i_1,i_2=1,\ldots,m),
\end{equation}
\vspace{2mm}
\noindent
$\zeta_j^{(i)}$
are independent standard Gaussian random variables
(for various
$i$ or $j$), which have the following form
\vspace{-3mm}
$$
\zeta_j^{(i)}=\int\limits_t^T
\phi_j(s)d{\bf w}_s^{(i)},
$$
\vspace{2mm}
\noindent
where
\vspace{-3mm}
\begin{equation}
\label{fi}
\phi_i(s)=\sqrt{\frac{2i+1}{T-t}}P_i\left(\left(s-t-\frac{T-t}{2}\right)
\frac{2}{T-t}\right),\ \ \ i=0, 1, 2,\ldots,
\end{equation}
\vspace{5mm}
\noindent
and $P_i(x)$ $(i=0, 1, 2,\ldots)$ is the Legendre polynomial.
\vspace{2mm}
From (\ref{zorro}) we obtain
\vspace{1mm}
$$
\frac{T-t}{2}\sum_{i=1}^{\infty}
\frac{1}{\sqrt{4i^2-1}}\left(
\zeta_{i-1}^{(i_1)}\zeta_{i}^{(i_2)}-
\zeta_i^{(i_1)}\zeta_{i-1}^{(i_2)}\right)
=\frac{1}{2}\left(I_{T,t}^{(i_1 i_2)}-I_{T,t}^{(i_2 i_1)}\right).
$$
\vspace{4mm}
Then,
the new representation of the Levy stochastic area, which is based
on the Legendre polynomials,
has the following form
\begin{equation}
\label{l100}
A_{T,t}^{(i_1 i_2)}=
\frac{T-t}{2}\sum_{i=1}^{\infty}
\frac{1}{\sqrt{4i^2-1}}\left(
\zeta_{i-1}^{(i_1)}\zeta_{i}^{(i_2)}-
\zeta_i^{(i_1)}\zeta_{i-1}^{(i_2)}\right).
\end{equation}
\vspace{5mm}
\section{The Classical Representation of the Levy Stochastic Area}
\vspace{5mm}
Let us consider (\ref{a2}) for the case
$i_1\ne i_2$,\ \ $\psi_1(s),$ $\psi_2(s)\equiv 1$. At that we suppose that
$\{\phi_j(x)\}_{j=0}^{\infty}$ is a complete orthonormal system
of trigonometric functions in $L_2([t,T]).$ Then
\vspace{1mm}
$$
I_{T,t}^{(i_1 i_2)}=
\frac{1}{2}(T-t)\Biggl(
\zeta_{0}^{(i_1)}\zeta_{0}^{(i_2)}
+\frac{1}{\pi}
\sum_{r=1}^{\infty}\frac{1}{r}\biggl(
\zeta_{2r}^{(i_1)}\zeta_{2r-1}^{(i_2)}-
\zeta_{2r-1}^{(i_1)}\zeta_{2r}^{(i_2)}+\biggr.\Biggr.
$$
\begin{equation}
\label{ajjja}
+\biggl.\Biggl.
\sqrt{2}\left(\zeta_{2r-1}^{(i_1)}\zeta_{0}^{(i_2)}-
\zeta_{0}^{(i_1)}\zeta_{2r-1}^{(i_2)}\right)\biggr)\Biggr),
\end{equation}
\vspace{3mm}
\noindent
where we use the same notations as in (\ref{zorro}), but $\phi_j(s)$
has the following form
\vspace{1mm}
\begin{equation}
\label{rre}
\phi_j(s)=\frac{1}{\sqrt{T-t}}
\begin{cases}
1,\ &\ {\rm if}\ j=0\\
~\\
\sqrt{2}{\rm sin}(2\pi r(s-t)/(T-t)),\ &\ {\rm if}\ j=2r-1\\
~\\
\sqrt{2}{\rm cos}(2\pi r(s-t)/(T-t)),\ &\ {\rm if}\ j=2r
\end{cases},\ \ \ r=1,\ 2,\ldots
\end{equation}
\vspace{3mm}
From (\ref{ajjja}) we obtain
\vspace{1mm}
$$
\frac{T-t}{2\pi}
\sum_{r=1}^{\infty}\frac{1}{r}\left(
\zeta_{2r}^{(i_1)}\zeta_{2r-1}^{(i_2)}-
\zeta_{2r-1}^{(i_1)}\zeta_{2r}^{(i_2)}+
\sqrt{2}\left(\zeta_{2r-1}^{(i_1)}\zeta_{0}^{(i_2)}-
\zeta_{0}^{(i_1)}\zeta_{2r-1}^{(i_2)}\right)\right)
=
$$
\vspace{2mm}
$$
=\frac{1}{2}\left(I_{T,t}^{(i_1 i_2)}-I_{T,t}^{(i_2 i_1)}\right).
$$
\vspace{5mm}
Then,
the representation of the Levy stochastic area, which is based
on the trigonometric functions,
has the following form
\vspace{1mm}
\begin{equation}
\label{l200}
{\hat A}_{T,t}^{(i_1 i_2)}=
\frac{T-t}{2\pi}
\sum_{r=1}^{\infty}\frac{1}{r}\left(
\zeta_{2r}^{(i_1)}\zeta_{2r-1}^{(i_2)}-
\zeta_{2r-1}^{(i_1)}\zeta_{2r}^{(i_2)}+
\sqrt{2}\left(\zeta_{2r-1}^{(i_1)}\zeta_{0}^{(i_2)}-
\zeta_{0}^{(i_1)}\zeta_{2r-1}^{(i_2)}\right)\right).
\end{equation}
\vspace{4mm}
Note that Milstein G.N. proposed in \cite{Mi2} the method of expansion
of iterated stochastic Ito integrals of second multiplicity
based on trigonometric Fourier expansion of the Brownian bridge
process (version of the so-called Karunen--Loeve expansion)
$$
{\bf w}_t-\frac{t}{\Delta}{\bf w}_{\Delta},\ \ \
t\in[0,\Delta],\ \ \ \Delta>0,
$$
\vspace{2mm}
\noindent
where ${\bf w}_t$ is a standard
vector Wiener process with independent components
${\bf w}^{(i)}_t$,\ $i=1,\ldots,m.$
The trigonometric Fourier expansion of the Brownian bridge
process has the form
\begin{equation}
\label{6.5.2}
{\bf w}_t^{(i)}-\frac{t}{\Delta}{\bf w}_{\Delta}^{(i)}=
\frac{1}{2}a_{i,0}+\sum_{r=1}^{\infty}\left(
a_{i,r}{\rm cos}\frac{2\pi rt}{\Delta} +b_{i,r}{\rm sin}
\frac{2\pi rt}{\Delta}\right),
\end{equation}
\vspace{2mm}
\noindent
where
\vspace{-1mm}
$$
a_{i,r}=\frac{2}{\Delta} \int\limits_0^{\Delta}
\left({\bf w}_s^{(i)}-\frac{s}{\Delta}{\bf w}_{\Delta}^{(i)}\right)
{\rm cos}\frac{2\pi rs}{\Delta}ds,\
$$
$$
b_{i,r}=\frac{2}{\Delta} \int\limits_0^{\Delta}
\left({\bf w}_s^{(i)}-\frac{s}{\Delta}{\bf w}_{\Delta}^{(i)}\right)
{\rm sin}\frac{2\pi rs}{\Delta}ds,
$$
\vspace{2mm}
\noindent
$r=0, 1,\ldots,$\ \ $i=1,\ldots,m.$
It is easy to demonstrate \cite{Mi2} that the random variables
$a_{i,r}, b_{i,r}$
are Gaussian ones and they satisfy the following relations
$$
{\sf M}\left\{a_{i,r}b_{i,r}\right\}=
{\sf M}\left\{a_{i,r}b_{i,k}\right\}=0,
$$
$$
{\sf M}\left\{a_{i,r}a_{i,k}\right\}=
{\sf M}\left\{b_{i,r}b_{i,k}\right\}=0,
$$
$$
{\sf M}\left\{a_{i_1,r}a_{i_2,r}\right\}=
{\sf M}\left\{b_{i_1,r}b_{i_2,r}\right\}=0,
$$
$$
{\sf M}\left\{a_{i,r}^2\right\}=
{\sf M}\left\{b_{i,r}^2\right\}=\frac{\Delta}{2\pi^2 r^2},
$$
\vspace{3mm}
\noindent
where $i,\ i_1, i_2=1,\ldots,m,$\ \ $r\ne k,$\ \ $i_1\ne i_2.$
According to (\ref{6.5.2}) we have
\begin{equation}
\label{6.5.7}
{\bf w}_t^{(i)}={\bf w}_{\Delta}^{(i)}\frac{t}{\Delta}+
\frac{1}{2}a_{i,0}+
\sum_{r=1}^{\infty}\left(
a_{i,r}{\rm cos}\frac{2\pi rt}{\Delta}+b_{i,r}{\rm sin}
\frac{2\pi rt}{\Delta}\right),
\end{equation}
\vspace{2mm}
\noindent
where the series
converges in the mean-square sense.
Expansion (\ref{ajjja}) has been obtained in \cite{Mi2} using
(\ref{6.5.7}).
\vspace{5mm}
\section{New Simple Method for Obtainment of Representations of the Levy
Stochastic Area}
\vspace{5mm}
It is well known that the idea of representing of the Wiener
process as a functional series with random coefficients that are
independent standard Gaussian random variables, with using of a
complete orthonormal system of trigonometric functions in $L_2([0, T])$
goes back to the works of Wiener \cite{7b} (1924) and Levy
\cite{7c} (1951).
The specified series was used in \cite{7b} and \cite{7c}
for construction of the Brownian motion process (Wiener process).
A little later, Ito and McKean in \cite{7d} (1965) used for
this purpose the complete orthonormal system of
Haar functions in $L_2([0, T])$.
Let ${\bf w}_{\tau},$ $\tau\in[0, \hat T]$ be an $m$-dimestional
Wiener process with independent components
${\bf w}_{\tau}^{(i)}$ $(i=1,\ldots,m).$
We have
$$
{\bf w}_s^{(i)}-{\bf w}_t^{(i)}=
\int\limits_t^s d{\bf w}_{\tau}^{(i)}=
\int\limits_t^T {\bf 1}_{\{\tau<s\}} d{\bf w}_{\tau}^{(i)},
$$
\vspace{2mm}
\noindent
where
$$
{\bf 1}_{\{\tau<s\}}=
\begin{cases}
1,\ &\ \tau<s\\
~\\
0,\ &\ \hbox{\rm otherwise}
\end{cases},\ \ \ \tau, s\in [t, T],\ \ \ 0\le t<T\le \hat T.
$$
\vspace{5mm}
Consider the Fourier expansion of ${\bf 1}_{\{\tau<s\}}$ at the
interval $[t, T]$ (see, for example, \cite{7e})
\begin{equation}
\label{eee}
{\bf 1}_{\{\tau<s\}}=\sum_{j=0}^{\infty}\int\limits_t^T
{\bf 1}_{\{\tau<s\}}\phi_j(\tau)d\tau \cdot \phi_j(\tau)=
\sum_{j=0}^{\infty}\int\limits_t^s
\phi_j(\tau)d\tau \cdot \phi_j(\tau),
\end{equation}
\vspace{2mm}
\noindent
where $\{\phi_j(\tau)\}_{j=0}^{\infty}$ is a complete
orthonormal system of functions in the space $L_2([t, T])$
and the series on the right-hand side of (\ref{eee}) converges
in the mean-square sence, i.e.
$$
\int\limits_t^T\left({\bf 1}_{\{\tau<s\}}-
\sum_{j=0}^{m}\int\limits_t^s
\phi_j(\tau)d\tau \cdot \phi_j(\tau)\right)^2 d\tau\to 0\ \ \
\hbox{if}\ \ \ m\to\infty.
$$
\vspace{5mm}
Let $\left({\bf w}_s^{(i)}-{\bf w}_t^{(i)}\right)^m$ be
the mean-square approximation of the process
${\bf w}_s^{(i)}-{\bf w}_t^{(i)}$,
which has the following form
\begin{equation}
\label{jq1}
\left({\bf w}_s^{(i)}-{\bf w}_t^{(i)}\right)^m=
\int\limits_t^T
\left(\sum_{j=0}^{m}\int\limits_t^s
\phi_j(\tau)d\tau \cdot \phi_j(\tau)\right)
d{\bf w}_{\tau}^{(i)}=\sum_{j=0}^{m}\int\limits_t^s
\phi_j(\tau)d\tau \cdot \int\limits_t^T\phi_j(\tau)
d{\bf w}_{\tau}^{(i)}.
\end{equation}
\vspace{4mm}
Moreover
\vspace{-1mm}
$$
{\sf M}\left\{\Biggl(
{\bf w}_s^{(i)}-{\bf w}_t^{(i)}-
\left({\bf w}_s^{(i)}-{\bf w}_t^{(i)}\right)^m\Biggr)^2\right\}=
$$
$$
={\sf M}\left\{\left(
\int\limits_t^T\left(
{\bf 1}_{\{\tau<s\}}-
\sum_{j=0}^{m}\int\limits_t^s
\phi_j(\tau)d\tau \cdot \phi_j(\tau)\right)
d{\bf w}_{\tau}^{(i)}\right)^2\right\}=
$$
\begin{equation}
\label{t1}
=\int\limits_t^T
\left({\bf 1}_{\{\tau<s\}}-
\sum_{j=0}^{m}\int\limits_t^s
\phi_j(\tau)d\tau \cdot \phi_j(\tau)\right)^2 d\tau\to 0\ \ \
\hbox{if}\ m\to\infty.
\end{equation}
\vspace{4mm}
In \cite{rr} it was proposed to use an expansion similar
to (\ref{jq1}) for construction of expansions of double stochastic Ito
integrals (\ref{jq2}). At that,
to obtain the mentioned expansion of (\ref{jq2}), the truncated
expansions (\ref{jq1}) of components of the Wiener
process ${\bf w}_s$ have been
iteratively substituted in the single integrals \cite{rr}.
This procedure leads to the calculation
of coefficients of the double Fourier series,
which is a time-consuming task for not too complex problem
of expansion of stochastic Ito integral (\ref{jq2}).
In contrast to \cite{rr} we subsitute the truncated expansion
(\ref{jq1}) only one time and only into the innermost integral
in (\ref{jq2}).
This procedure leads to the simple calculation
of the coefficients
\vspace{-1mm}
$$
\int\limits_t^s
\phi_j(\tau)d\tau,\ \ \ (j=0, 1, 2,\ldots)
$$
\vspace{2mm}
\noindent
of the usual (not double) Fourier series.
Moreover, we use the Legendre polynomials for construction
of the expansion of (\ref{jq2}). For the first time the
Legendre polynomials have been applied in the
framework of the mentioned problem in the author's paper
\cite{300a} (1997) (see also \cite{2006}-\cite{200a}, \cite{400a}-\cite{29})
while in the papers of other author's these polynomials
have not been considered as the basis functions for construction
of expansions of iterated stochastic Ito integrals.
\vspace{2mm}
{\bf Theorem 2.}\ {\it Let
$\phi_j(\tau)$ $(j=0, 1, \ldots )$ be a complete
orthonormal system of functions in the space $L_2([t, T]).$
Let
\vspace{-1mm}
\begin{equation}
\label{l1}
\int\limits_t^T
\left({\bf w}_s^{(i_1)}-{\bf w}_t^{(i_1)}\right)^m
d{\bf w}_s^{(i_2)}=
\sum_{j=0}^{m}\int\limits_t^T\phi_j(\tau)
d{\bf w}_{\tau}^{(i_1)}
\int\limits_t^T\int\limits_t^s
\phi_j(\tau)d\tau d{\bf w}_{\tau}^{(i_2)}
\end{equation}
\vspace{2mm}
\noindent
be an approximation of iterated stochastic Ito integral
$$
\int\limits_t^T
\int\limits_t^s
d{\bf w}_{\tau}^{(i_1)}d{\bf w}_{s}^{(i_2)}\ \ \ (i_1\ne i_2),
$$
\vspace{2mm}
\noindent
where $i_1,i_2=1,\ldots,m$.
Then
$$
\int\limits_t^T
\int\limits_t^s
d{\bf w}_{\tau}^{(i_1)}d{\bf w}_{s}^{(i_2)}
=\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm l.i.m.}\cr
$\stackrel{}{{}_{m\to \infty}}$\cr
}} }
\int\limits_t^T
\left({\bf w}_s^{(i_1)}-{\bf w}_t^{(i_1)}\right)^m
d{\bf w}_s^{(i_2)},
$$
\vspace{3mm}
\noindent
where $i_1, i_2=1,\ldots,m$.}
\vspace{2mm}
{\bf Proof.} Using the standard properties of Ito stochastic
integral as well as (\ref{t1}) and the property of orthonormality
of functions
$\phi_j(\tau)$ $(j=0, 1, \ldots )$ at the interval $[t, T],$
we obtain
\vspace{1mm}
$$
{\sf M}\left\{\left(
\int\limits_t^T
\int\limits_t^s
d{\bf w}_{\tau}^{(i_1)}d{\bf w}_{s}^{(i_2)}-
\int\limits_t^T
\left({\bf w}_s^{(i_1)}-{\bf w}_t^{(i_1)}\right)^m
d{\bf w}_s^{(i_2)}\right)^2\right\}=
$$
\vspace{2mm}
$$
=
\int\limits_t^T
{\sf M}\left\{\Biggl(
{\bf w}_s^{(i_1)}-{\bf w}_t^{(i_1)}-
\left({\bf w}_s^{(i_1)}-{\bf w}_t^{(i_1)}\right)^m\Biggr)^2\right\}ds=
$$
\vspace{2mm}
$$
=
\int\limits_t^T\int\limits_t^T
\left({\bf 1}_{\{\tau<s\}}-
\sum_{j=0}^{m}\int\limits_t^s
\phi_j(\tau)d\tau \cdot \phi_j(\tau)\right)^2d\tau ds=
$$
\vspace{2mm}
\begin{equation}
\label{l5}
=\int\limits_t^T\left((s-t)-
\sum_{j=0}^{m}\left(\int\limits_t^s
\phi_j(\tau)d\tau\right)^2\right) ds.
\end{equation}
\vspace{6mm}
Because of continuity
and nondecreasing
of members of the functional sequence
\vspace{-1mm}
$$
u_n(s)=\sum_{j=0}^{m}\left(\int\limits_t^s
\phi_j(\tau)d\tau\right)^2
$$
\vspace{2mm}
\noindent
and because of the
property
of continuity of the limit function
$u(s)= s-t$
according to Dini
theorem
we have a uniform convergence
$u_n(s)$ to $u(s)$ at the interval $[t, T]$.
Then from this fact as well as from (\ref{l5}) we obtain
\vspace{2mm}
\begin{equation}
\label{rez1}
\int\limits_t^T
\int\limits_t^s
d{\bf w}_{\tau}^{(i_1)}d{\bf w}_{s}^{(i_2)}
=\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm l.i.m.}\cr
$\stackrel{}{{}_{m\to \infty}}$\cr
}} }
\int\limits_t^T
\left({\bf w}_s^{(i_1)}-{\bf w}_t^{(i_1)}\right)^m
d{\bf w}_s^{(i_2)}.
\end{equation}
\vspace{5mm}
Let $\{\phi_j(\tau)\}_{j=0}^{\infty}$ be a complete
orthonormal system of Legendre polynomials in the space $L_2([t, T]),$
which has the form (\ref{fi}).
Then
\begin{equation}
\label{l2}
\int\limits_t^s
\phi_j(\tau)d\tau=
\frac{T-t}{2}\left(\frac{\phi_{j+1}(s)}{\sqrt{(2j+1)(2j+3)}}-
\frac{\phi_{j-1}(s)}{\sqrt{4j^2-1}}\right)\ \ \ \hbox{for}\ \ \ j\ge 1.
\end{equation}
\vspace{4mm}
Denote (see Theorem 1)
\vspace{-1mm}
$$
\zeta_j^{(i)}=\int\limits_t^T\phi_j(\tau)
d{\bf w}_{\tau}^{(i)}\ \ \ (i=1,\ldots,m).
$$
\vspace{2mm}
From (\ref{l1}) and (\ref{l2}) we obtain
$$
\int\limits_t^T
\left({\bf w}_s^{(i_1)}-{\bf w}_t^{(i_1)}\right)^m
d{\bf w}_s^{(i_2)}=
\frac{1}{\sqrt{T-t}}\zeta_0^{(i_1)}
\int_t^T(s-t){\bf w}_s^{(i_2)}+
$$
\vspace{1mm}
$$
+
\frac{T-t}{2}\sum_{j=1}^m
\zeta_j^{(i_1)}
\left(\frac{1}{\sqrt{(2j+1)(2j+3)}}\zeta_{j+1}^{(i_2)}-
\frac{1}{\sqrt{4j^2-1}}\zeta_{j-1}^{(i_2)}\right)=
$$
\vspace{4mm}
$$
=
\frac{T-t}{2}\zeta_0^{(i_1)}\left(
\zeta_0^{(i_2)}+\frac{1}{\sqrt{3}}\zeta_1^{(i_2)}\right)
+
$$
\vspace{1mm}
$$
+
\frac{T-t}{2}\sum_{j=1}^m
\zeta_j^{(i_1)}
\left(\frac{1}{\sqrt{(2j+1)(2j+3)}}\zeta_{j+1}^{(i_2)}-
\frac{1}{\sqrt{4j^2-1}}\zeta_{j-1}^{(i_2)}\right)=
$$
\vspace{4mm}
$$
=\frac{T-t}{2}\left(\zeta_0^{(i_1)}\zeta_0^{(i_2)}+\sum_{j=1}^{m}
\frac{1}{\sqrt{4j^2-1}}\left(
\zeta_{j-1}^{(i_1)}\zeta_{j}^{(i_2)}-
\zeta_j^{(i_1)}\zeta_{j-1}^{(i_2)}\right)\right)+
$$
\vspace{1mm}
\begin{equation}
\label{l6}
+
\frac{T-t}{2}\zeta_m^{(i_1)}\zeta_{m+1}^{(i_2)}
\frac{1}{\sqrt{(2m+1)(2m+3)}}.
\end{equation}
\vspace{6mm}
Then from (\ref{rez1}) and (\ref{l6}) we obtain
\vspace{1mm}
$$
\int\limits_t^T
\int\limits_t^s
d{\bf w}_{\tau}^{(i_1)}d{\bf w}_{s}^{(i_2)}
=\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm l.i.m.}\cr
$\stackrel{}{{}_{m\to \infty}}$\cr
}} }
\int\limits_t^T
\left({\bf w}_s^{(i_1)}-{\bf w}_t^{(i_1)}\right)^m
d{\bf w}_s^{(i_2)}=
$$
\vspace{2mm}
\begin{equation}
\label{l10}
=
\frac{T-t}{2}\left(\zeta_0^{(i_1)}\zeta_0^{(i_2)}+\sum_{j=1}^{\infty}
\frac{1}{\sqrt{4j^2-1}}\left(
\zeta_{j-1}^{(i_1)}\zeta_{j}^{(i_2)}-
\zeta_j^{(i_1)}\zeta_{j-1}^{(i_2)}\right)\right).
\end{equation}
\vspace{4mm}
From (\ref{l10}) it follows that the equality (\ref{l100}) is true.
Not difficult to see that the relation (\ref{l200})
also can be obtained by the method from this section.
Let $\{\phi_j(\tau)\}_{j=0}^{\infty}$ be a complete
orthonormal trigonomertic system of functions in the space $L_2([t, T]),$
which has the form (\ref{rre}).
We have
\vspace{-2mm}
\begin{equation}
\label{rre11}
\int\limits_t^s
\phi_j(\tau)d\tau=
\frac{T-t}{2\pi r}\left\{
\begin{matrix}
\phi_{2r-1}(s),\ &\ j=2r\cr\cr
\sqrt{2}\phi_0(s)-\phi_{2r}(s),\ &\ j=2r-1
\end{matrix}
\right.\ ,
\end{equation}
\vspace{4mm}
\noindent
where $j\ge 1$ and $r=1, 2,\ldots.$
From (\ref{l1}) and (\ref{rre11}) we obtain
$$
\int\limits_t^T
\left({\bf w}_s^{(i_1)}-{\bf w}_t^{(i_1)}\right)^m
d{\bf w}_s^{(i_2)}=
\frac{1}{\sqrt{T-t}}\zeta_0^{(i_1)}
\int_t^T(s-t){\bf w}_s^{(i_2)}+
$$
\vspace{1mm}
$$
+
\frac{T-t}{2}\sum_{r=1}^m
\frac{1}{\pi r}
\left(\left(\zeta_{2r}^{(i_1)}\zeta_{2r-1}^{(i_2)}-
\zeta_{2r-1}^{(i_1)}\zeta_{2r}^{(i_2)}\right)+
\sqrt{2}\zeta_{0}^{(i_2)}\zeta_{2r-1}^{(i_1)}\right)=
$$
\vspace{4mm}
$$
=
\frac{1}{\sqrt{T-t}}\zeta_0^{(i_1)}
\frac{(T-t)^{3/2}}{2}\left(\zeta_0^{(i_2)}-
\frac{\sqrt{2}}{\pi}\sum_{r=1}^{\infty}\frac{1}{r}
\zeta_{2r-1}^{(i_2)}\right)+
$$
\vspace{1mm}
$$
+
\frac{T-t}{2}\sum_{r=1}^m
\frac{1}{\pi r}
\left(\left(\zeta_{2r}^{(i_1)}\zeta_{2r-1}^{(i_2)}-
\zeta_{2r-1}^{(i_1)}\zeta_{2r}^{(i_2)}\right)+
\sqrt{2}\zeta_{0}^{(i_2)}\zeta_{2r-1}^{(i_1)}\right)=
$$
\vspace{4mm}
$$
=\frac{1}{2}(T-t)\Biggl(
\zeta_{0}^{(i_1)}\zeta_{0}^{(i_2)}
+\frac{1}{\pi}
\sum_{r=1}^{m}\frac{1}{r}\biggl(
\zeta_{2r}^{(i_1)}\zeta_{2r-1}^{(i_2)}-
\zeta_{2r-1}^{(i_1)}\zeta_{2r}^{(i_2)}+\biggr.\Biggr.
$$
\vspace{1mm}
$$
+\biggl.\Biggl.
\sqrt{2}\left(\zeta_{2r-1}^{(i_1)}\zeta_{0}^{(i_2)}-
\zeta_{0}^{(i_1)}\zeta_{2r-1}^{(i_2)}\right)\biggr)\Biggr)-
$$
\vspace{1mm}
\begin{equation}
\label{zq1}
-\frac{T-t}{\pi \sqrt{2}}
\zeta_{0}^{(i_1)}\sum_{r=m+1}^{\infty}\frac{1}{r}
\zeta_{2r-1}^{(i_2)}.
\end{equation}
\vspace{6mm}
From (\ref{zq1}) and (\ref{rez1}) we obviously obtain
(\ref{ajjja}).
\vspace{5mm}
\section{Convergence in the
Mean of Degree $2n$ of and With Probability 1}
\vspace{5mm}
Let us denote
\vspace{-1mm}
\begin{equation}
\label{555}
A_{T,t}^{(i_1 i_2)q}=
\frac{T-t}{2}\sum_{i=1}^{q}
\frac{1}{\sqrt{4i^2-1}}\left(
\zeta_{i-1}^{(i_1)}\zeta_{i}^{(i_2)}-
\zeta_i^{(i_1)}\zeta_{i-1}^{(i_2)}\right),
\end{equation}
\vspace{1mm}
\begin{equation}
\label{abc}
{\hat A}_{T,t}^{(i_1 i_2)q}=
\frac{T-t}{2\pi}
\sum_{r=1}^{q}\frac{1}{r}\left(
\zeta_{2r}^{(i_1)}\zeta_{2r-1}^{(i_2)}-
\zeta_{2r-1}^{(i_1)}\zeta_{2r}^{(i_2)}+
\sqrt{2}\left(\zeta_{2r-1}^{(i_1)}\zeta_{0}^{(i_2)}-
\zeta_{0}^{(i_1)}\zeta_{2r-1}^{(i_2)}\right)\right).
\end{equation}
\vspace{6mm}
Then, from (\ref{2121}) we obtain
\vspace{-1mm}
\begin{equation}
\label{bb1}
I_{T,t}^{(i_1 i_2)q}=\frac{T-t}{2}\zeta_{0}^{(i_1)}\zeta_{0}^{(i_2)}+
A_{T,t}^{(i_1 i_2)q},
\end{equation}
\begin{equation}
\label{bb2}
I_{T,t}^{(i_1 i_2)q}=
\frac{T-t}{2}\zeta_{0}^{(i_1)}\zeta_{0}^{(i_2)}+
{\hat A}_{T,t}^{(i_1 i_2)q}.
\end{equation}
\vspace{4mm}
Not difficult to demonstrate \cite{Mi2} that from (\ref{ajjja})
we can get the another representation for ${\hat A}_{T,t}^{(i_1 i_2)q}$
\vspace{2mm}
$$
{\hat A}_{T,t}^{(i_1 i_2)q}
=\frac{T-t}{2\pi}\Biggl(
\sum_{r=1}^{q}\frac{1}{r}\left(
\zeta_{2r}^{(i_1)}\zeta_{2r-1}^{(i_2)}-
\zeta_{2r-1}^{(i_1)}\zeta_{2r}^{(i_2)}+
\right.\Biggr.
$$
\vspace{2mm}
$$
+\Biggl.\left.\sqrt{2}\left(\zeta_{2r-1}^{(i_1)}\zeta_{0}^{(i_2)}-
\zeta_{0}^{(i_1)}\zeta_{2r-1}^{(i_2)}\right)\right)
+\sqrt{2}\left(\frac{\pi^2}{6}-\sum_{r=1}^q\frac{1}{r^2}\right)^{1/2}\left(
\xi_q^{(i_1)}\zeta_0^{(i_2)}-\zeta_0^{(i_1)}\xi_q^{(i_2)}\right)\Biggr),
$$
\vspace{5mm}
\noindent
where
\vspace{-3mm}
$$
\xi_q^{(i)}=\left(\frac{\pi^2}{6}-
\sum\limits_{r=1}^q\frac{1}{r^2}\right)^{-1/2}
\sum_{r=q+1}^{\infty}
\frac{1}{r}\zeta_{2r-1}^{(i)},
$$
\vspace{4mm}
\noindent
and $\zeta_0^{(i)},$ $\zeta_{2r}^{(i)},$
$\zeta_{2r-1}^{(i)},$ $\xi_q^{(i)}$ ($r=1,\ldots,q,$\
$i=1,\ldots,m$) are independent standard Gaussian random variables.
From (\ref{555}) and (\ref{abc}) we obtain
\vspace{2mm}
$$
{\sf M}\left\{\left(A_{T,t}^{(i_1 i_2)}- A_{T,t}^{(i_1 i_2)q}
\right)^2\right\}=
\frac{(T-t)^2}{2}\Biggl(\frac{1}{2}-\sum_{i=1}^q
\frac{1}{4i^2-1}\Biggr)=
$$
\vspace{2mm}
$$
=
\frac{(T-t)^2}{2}
\sum\limits_{i=q+1}^{\infty}\frac{1}{4i^2-1}
\le \frac{(T-t)^2}{2}\int\limits_{q}^{\infty}
\frac{1}{4x^2-1}dx
=
$$
\vspace{2mm}
\begin{equation}
\label{teac}
=-\frac{(T-t)^2}{8}{\rm ln}\left|
1-\frac{2}{2q+1}\right|
\le C_1\frac{(T-t)^2}{q},
\end{equation}
\vspace{6mm}
$$
{\sf M}\left\{\left({\hat A}_{T,t}^{(i_1 i_2)}-{\hat A}_{T,t}^{(i_1 i_2)q}
\right)^2\right\}
=
\frac{3(T-t)^{2}}{2\pi^2}
\left(\frac{\pi^2}{6}-\sum\limits_{r=1}^q\frac{1}{r^2}\right)
=
$$
\vspace{2mm}
$$
=
\frac{3(T-t)^{2}}{2\pi^2}\sum\limits_{r=q+1}^{\infty}\frac{1}{r^2}
\le\frac{3(T-t)^{2}}{2\pi^2}\int\limits_q^{\infty}
\frac{dx}{x^2}
=
$$
\vspace{2mm}
\begin{equation}
\label{teac0}
=\frac{3(T-t)^{2}}{2\pi^2 q}
\le C_2 \frac{(T-t)^2}{q},
\end{equation}
\vspace{6mm}
\noindent
where constants $C_1,$ $C_2$ does not depent on $q.$
\vspace{2mm}
In \cite{2011-2}-\cite{2013}, \cite{arxiv-1} it was shown that
\vspace{2mm}
$$
{\sf M}\left\{\left(J[\psi^{(k)}]_{T,t}
-J[\psi^{(k)}]^{p_1,\ldots,p_k}_{T,t}\right)^{2n}\right\}\le
$$
\vspace{3mm}
$$
\le
(k!)^{2n}
\biggl(n(2n-1)\biggr)^{n \cdot (k-1)}(2n-1)!!\ \times
$$
\begin{equation}
\label{sss}
\times\
\left(
\int\limits_{[t,T]^k}
K^2(t_1,\ldots,t_k)
dt_1\ldots dt_k -\sum_{j_1=0}^{p_1}\ldots
\sum_{j_k=0}^{p_k}C^2_{j_k\ldots j_1}
\right)^n,
\end{equation}
\vspace{3mm}
\noindent
where
\vspace{1mm}
$$
J[\psi^{(k)}]^{p_1,\ldots,p_k}_{T,t}=
\sum_{j_1=0}^{p_1}\ldots\sum_{j_k=0}^{p_k}
C_{j_k\ldots j_1}\Biggl(
\prod_{l=1}^k\zeta_{j_l}^{(i_l)}-
\Biggr.
$$
\vspace{3mm}
$$
-\Biggl.
\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm l.i.m.}\cr
$\stackrel{}{{}_{N\to \infty}}$\cr
}} }\sum_{(l_1,\ldots,l_k)\in {\rm G}_k}
\phi_{j_{1}}(\tau_{l_1})
\Delta{\bf w}_{\tau_{l_1}}^{(i_1)}\ldots
\phi_{j_{k}}(\tau_{l_k})
\Delta{\bf w}_{\tau_{l_k}}^{(i_k)}\Biggr)
$$
\vspace{8mm}
\noindent
is the prelimit expression in (\ref{tyyy}).
For the case $k=2,$ $i_1\ne i_2$, and $\psi_1(s),$ $\psi_2(s)\equiv 1$
from (\ref{sss}) we obtain
\vspace{2mm}
\begin{equation}
\label{yyy}
{\sf M}\left\{\left(I_{T,t}^{(i_1 i_2)}-I_{T,t}^{(i_1 i_2)q}
\right)^{2n}\right\}\le C_{n,2}\left(\frac{(T-t)^2}{2}
\left(\frac{1}{2}-\sum_{i=1}^q
\frac{1}{4i^2-1}\right)\right)^n\ \to 0\ \ \ {\rm if}\ \ \ q\to\infty,
\end{equation}
\vspace{3mm}
\begin{equation}
\label{yyy2}
{\sf M}\left\{\left(I_{T,t}^{(i_1 i_2)}-I_{T,t}^{(i_1 i_2)q}
\right)^{2n}\right\}\le C_{n,2}
\left(\frac{3(T-t)^{2}}{2\pi^2}\left(\frac{\pi^2}{6}-
\sum\limits_{r=1}^q\frac{1}{r^2}\right)
\right)
^n\ \to 0\ \ \ {\rm if}\ \ \ q\to\infty,
\end{equation}
\vspace{6mm}
\noindent
where
$$
C_{n,k}=(k!)^{2n}
\biggl(n(2n-1)\biggr)^{n \cdot (k-1)}(2n-1)!!
$$
\vspace{2mm}
At that $I_{T,t}^{(i_1 i_2)q}$ in (\ref{yyy})
has the form (\ref{bb1}) while $I_{T,t}^{(i_1 i_2)q}$
in (\ref{yyy2}) has the form (\ref{bb2}).
From (\ref{teac}), (\ref{teac0}), (\ref{yyy}), (\ref{yyy2}) we obtain
\vspace{1mm}
$$
{\sf M}\left\{\left(A_{T,t}^{(i_1 i_2)}-A_{T,t}^{(i_1 i_2)q}
\right)^{2n}\right\}\ \to 0\ \ \ {\rm if}\ \ \ q\to\infty.
$$
\vspace{1mm}
$$
{\sf M}\left\{\left(\hat A_{T,t}^{(i_1 i_2)}-\hat A_{T,t}^{(i_1 i_2)q}
\right)^{2n}\right\}\ \to 0\ \ \ {\rm if}\ \ \ q\to\infty,
$$
\vspace{4mm}
Let us address now to the convergence with
probability 1 for $A_{T,t}^{(i_1 i_2)q}$.
First, note the well-known fact.
\vspace{2mm}
{\bf Lemma 1.}\ {\it If for the sequence of random variables
$\xi_q$ and for some
$\alpha>0$ the number series
$$
\sum\limits_{q=1}^{\infty}{\sf M}\left\{|\xi_q|^{\alpha}\right\}
$$
\vspace{3mm}
\noindent
converges, then the sequence $\xi_q$ converges to zero w.
p. {\rm 1}.}
\vspace{2mm}
From (\ref{teac}) and (\ref{yyy}) $(n=2)$ we obtain
\vspace{1mm}
$$
{\sf M}\left\{\left(I_{T,t}^{(i_1 i_2)}-I_{T,t}^{(i_1 i_2)q}
\right)^{4}\right\}={\sf M}
\left\{\left(A_{T,t}^{(i_1 i_2)}-A_{T,t}^{(i_1 i_2)q}
\right)^{4}\right\}\le \frac{K}{q^2},
$$
\vspace{4mm}
\noindent
where constant $K$ does not depend on $q.$
Since the series
\vspace{-1mm}
$$
\sum\limits_{q=1}^{\infty}\frac{K}{q^2}
$$
\vspace{2mm}
\noindent
converges, then according to Lemma 1
we obtain that
$A_{T,t}^{(i_1 i_2)}-A_{T,t}^{(i_1 i_2)q}\to 0$\ if\ $q\to \infty$
w. p. 1. Then
$A_{T,t}^{(i_1 i_2)q}\to A_{T,t}^{(i_1 i_2)}$\ if\ $q\to \infty$
w. p. 1.
\vspace{7mm}
|
train/arxiv
|
BkiUd545qdmB642cwHh4
| 4 | 0.8 |
\section{Introduction}
\PARstart{C}{ooperative} communication \cite{1} has recently
attracted much attention due to it can achieve a larger rate
region, compared with traditional networks. Many cooperative
protocols have been proposed in the literature. These protocols
are usually classified into three categories: decode-and-forward
(DF) protocol, where the relay decodes and re-encodes the signals
transmitted by the source; amplify-and-forward (AF) protocol, in
which a relay simply amplifies its received signals;
compress-and-forward (CF) protocol, where the relay compresses the
signals from the source, and forwards these compressed soft
information to the destination. For DF protocol, it has been
widely researched based on LDPC codes \cite{2}, and suffers a loss
of performance when the relay can't be guaranteed to recover the
source information. The relay always can assist the source to
convey information with amplified soft information when AF
protocol is employed, but the protocol is suboptimal. CF protocol,
jointing source-channel coding, is a form of Wyner-Ziv (WZ) coding
\cite{3} in case of lossy compression and Slepian-Wolf coding
\cite{4} in the lossless case. It takes advantages of the
statistical dependence of the relay's and destination's channel
output, and achieves higher rate than DF and AF.
\\
\indent So far, most of the researches concerning CF remain at the
theoretical level and the realization of CF is tackled in only a
few papers, \cite{5}\cite{6}\cite{7}. These papers achieve
Slepian-Wolf compression at the relay by taking the syndrome of an
LDPC code \cite{5}, or by using an Irregular Repeat Accumulator
(IRA) code \cite{6} that combines Slepian-Wolf coding with channel
coding on the relay-to-destination channel. However these
approaches are suitable only when the source-relay channel output
is binary and the relay does lossless (rather than lossy)
compression. Otherwise, e.g. the Gaussian relay channel is
considered in \cite{7}, significant capacity losses will result.
\\
\indent It is known that LDPC codes are good channel codes and
recent work has also shown that LDGM codes are good source codes.
Some near-ideal encoding/decoding algorithms with LDGM codes have
been proposed \cite{8}\cite{9}, furthermore nested LDGM-LDPC codes
often are used to guarantee both channel coding and source coding
performance. In \cite{10}, such nested codes are used to approach
capacity in dirty paper coding, where good channel coding ensures
low error probability and good source coding guarantees good
shaping of the transmitted signal.
\\
\indent To focus on the effective compression of the received data
at the relay, we consider a three terminal cooperative system
where the source-destination and source-relay links are both
binary erasure channel (BEC), and the relay-destination link is
orthogonal to them. As the received signal by the relay through
the BEC is 3-ary, the aforementioned syndrome methods are
insufficient, making the proposed nested codes necessary.
\\
\indent This paper is organized as follows. Section II gives a
brief introduction to the system model involving a relay channel.
Section III describes the compression and decoding algorithms
based on nested LDGM-LDPC codes. Section IV presents the degree
distribution optimization method necessary for good performance.
Experiment results are given in Section V to verify the
effectiveness of the proposed algorithms and optimization methods.
Finally, Section VI concludes the paper.
\section{system model}
The full-duplex single relay system is shown in Fig.1. It
comprises a source S, a destination D and a relay node R. The S-R
and S-D links, both with erasure probability $\varepsilon$, form a
binary erasure broadcast channel. The R-D link, with a capacity
denoted by $C_{rd}$, is orthogonal to the S-D and S-R links.
Denoting an erased bit by $E$, the relay system is described by
four random variables $x_s$, $x_r$, $y_r$, $y_d$ and one
conditional probability distribution $p(y_r,y_d|x_s)$,which is
shown in Table I.
\begin{figure}
\begin{center}
\includegraphics[width=2in]{sf_fig1}
\caption{The single relay system} \label{fig.1}
\end{center}
\end{figure}
\begin{table}[!hbp]
\caption{Conditional probability distribution of
$p(y_r,y_d|x_s=0)$, When $p(y_r,y_d|x_s=1)$, the variables value:
0 and 1 exchange} \label{table_1} \centering
\begin{tabular}{cccc}
\hline $y_d,y_r$ & 0& 1 & $E$ \\
\hline
0& $(1-\varepsilon)^2$ &0 &$(1-\varepsilon)\varepsilon$ \\
1&0&0&0\\
$E$ &$(1-\varepsilon)\varepsilon$ &0&$\varepsilon^2$\\
\hline
\end{tabular}
\end{table}
\indent In every block a message $w$ which is random variable
uniformly distributed on $[1,2^{nR})$ is encoded into sequence
$\boldsymbol{x_s}=(x_{s1},x_{s2},\cdots,x_{sn})$ at the source S,
and transmitted through BEC.
$\boldsymbol{y_r}=(y_{r1},y_{r2},\cdots,y_{rn})$,
$\boldsymbol{y_d}=(y_{d1},y_{d2},\cdots,y_{dn})$ is received by R
and D respectively. At R $\boldsymbol{y_r}$ is decoded or
loselessly compressed, and then re-encoded into
$\boldsymbol{x_r}=(x_{r1},x_{r2},\cdots,x_{rn'})$ to implement DF
or CF respectively. When $C_{rd}$ is sufficiently large,
$\boldsymbol{y_r}$ could be decoded with the compressed signals
$\boldsymbol{x_r}$ and the side information $\boldsymbol{y_d}$.
$w$ could be recovered at D by jointly decoding $\boldsymbol{y_r}$
and $\boldsymbol{y_d}$. Then the achievable rate \cite{11} of the
system is given by\\
1) If $\boldsymbol{y_r}$ is decoded, then
\begin{equation}
R \leq I(x_s;y_r|x_r)=1-\varepsilon
\end{equation}
2) If $\boldsymbol{y_r}$ is compressed and encoded into
$\boldsymbol{x_r}$, with large $C_{rd}$ the compression can be
lossless, in which case
\begin{equation}
R \leq I(x_s;y_ry_d)=1-\varepsilon^2
\end{equation}
It can be seen from (1) that, when $C_{rd}$ is large, the S-R link
become the bottleneck of DF mode. However CF can achieve a higher
rate in (2), which is actually the cut-set bound \cite{11} on the
capacity of the relay channel. On the other hand, in CF mode the
relay node is unable to decode $\boldsymbol{y_r}$ , so the
$C_{rd}$ must be large enough to transmit all the information
about the erased positions.\\
\indent In this paper, we focus on lossless CF, and since it
already achieves capacity, the remaining task is to minimize
$C_{rd}$ the necessary. As $\boldsymbol{y_d}$ is available at D
and is correlated with $\boldsymbol{y_r}$, it can be used as side
information to reduce $\boldsymbol{y_r}$'s encoding rate from
$H(y_r)$ into $H(y_r|y_d)$ through Slepian-Wolf coding. In the
next section, nested LDGM-LDPC codes will be designed to realize
the compression and binning necessary for Slepian-Wolf coding.
\section{The nested LDGM-LDPC code for compression}
Considering that LDPC as channel code achieves
capacity-approaching performance with low-complexity iterative
decoding manner, we encode the source messages with an LDPC code,
denoted by factor graph \cite{12} $C_1$, which is shown in Fig.2.
The circles denote the variable nodes q, representing $n$ bits
codeword of LDPC codes, and the black squares denote the check
(function) nodes s, representing $k$ parity check equations. The
number of edges connected to one node is denoted as the degree of
the node. The rate $R_0=\frac{n-k}{n}$ could approach cut-set
bound in (2) by optimizing the degree distribution of $C_1$, which
will is discussed in the next section.
\begin{figure}
\begin{center}
\includegraphics[width=0.8in]{sf_fig2}
\caption{The LDPC code $C_1$} \label{fig.2}
\end{center}
\end{figure}
\indent With $C_1$ the message $w$ at S is encoded into a binary
sequence $\boldsymbol{x_s}\in\{0,1\}^n$ and transmitted over BEC.
Then the received sequence $\boldsymbol{y_r}$ by R is relayed to D
via CF. We design a nested LDGM-LDPC construction $C_2$, to deal
with the compression problem at R, whose factor graph is shown in
Fig.3. The circles are also the variable nodes and black squares
are the factor nodes. As each is regarded as a ternary symbol, we
firstly map $\boldsymbol{y_r}$ into a binary sequence
$\boldsymbol{c}$ by
\begin{eqnarray}
\hat{y}_{ri}=\varphi(c_{2i-1}\oplus v_{2i-1} \oplus
\zeta_{2i-1},c_{2i}\oplus v_{2i} \oplus \zeta_{2i}), \nonumber \\
\varphi(00)=0,\varphi(01)=1,\varphi(1*)=E,i=1,2,\cdots,n,
\end{eqnarray}
Where $*$ denotes "don't care" positions that can be encoded into
either 0 or 1. $\boldsymbol{v}$ is a pseudo-random dither sequence
which is added to assure uniform distribution.
$\boldsymbol{\zeta}$ is normally an all-zero sequence, but in
practice erroneous decimation of the b-nodes will inevitably occur
and cause contradictions, which must be corrected by flipping the
bits in corresponding to the c-nodes with contradictions. Here the
factor nodes connected to $\boldsymbol{y_r}$ and $\boldsymbol{c}$
represent the mapping in (3). Then with LDGM part of $C_2$ , the
binary sequence containing $*$ is quantized into a shorter one
$\boldsymbol{b}$ by
\begin{eqnarray}
c=bG,b\in\{0,1\}^m,m=nR_b
\end{eqnarray}
where $G$ is the generation matrix and $R_b$ is optimized to be
slightly larger than $2-\varepsilon$ , so that LDGM coding with
$R_b$ is lossless. After that, we use the LDPC part of $G_2$ to
compress $\boldsymbol{b}$ into $\boldsymbol{p}$, with
\begin{eqnarray}
p=bH^T,p\in\{0,1\}^t,t=nR_p
\end{eqnarray}
where $H$ is the sparse parity check matrix and $R_p$ also
optimized is slightly larger than $H(y_r|y_d)$. $H(y_r|y_d)$ is
the Slepian-Wolf theoretical limit.\\
\begin{figure}
\begin{center}
\includegraphics[width=2.5in]{sf_fig3}
\caption{The nested LDGM-LDPC codes $C_2$} \label{fig.3}
\end{center}
\end{figure}
\indent Until now the nested LDGM-LDPC code has been used to
compress $\boldsymbol{y_r}$ into $\boldsymbol{p}$ (as well as the
flipped positions $\boldsymbol{\zeta}$) at the rate approximately
equal to $H(y_r|y_d)$. Assuming that $C_{rd}$ is sufficient for
transmitting these information, D can decode $\boldsymbol{y_r}$
from the side information $\boldsymbol{y_r}$ and the compressed
information $\boldsymbol{p}$ losslessly. The iterative belief
propagation \cite{13} decoding process is also executed from
Fig.3. Finally, combining $\boldsymbol{y_r}$ with
$\boldsymbol{y_d}$, D recovers the message $w$, again by the BP
algorithm.
\section{Code Optimization $C_1$ and $C_2$}
When LDPC, LDGM and nested codes are designed, it is always
critical to optimize the degree distributions so that BP converges
well, which can be visualized on the EXIT chart \cite{14} as a
gap. So we optimize the degree distribution using the EXIT and EBP
curve (Extended BP curve which is another form of EXIT
chart)\cite{15} considering both the encoding process and decoding
process also from the source node, relay node and destination
node.\\
\indent The design of $C_1$ involves just LDPC optimization over
a BEC with erasure probability $\varepsilon^2$ , which means that
the prior information at q-nodes acquired from channel output
becomes $I_{q,pri}=1-\varepsilon^2$. S-regular, q-irregular LDPC
code \cite{13} is designed to achieve good performance. Let $d_s$
be the left-degree of all s-nodes, and denote $v_{qd}$ as the
fraction of edges connected to q-nodes with the right-degree $d$.
$I_{qs}$, $I_{sq}$ represent the average mutual information (MI)
in every q-to-s, s-to-q message at a certain iteration
respectively. Then the optimization the degree distribution
$v_{qd}$ of q-nodes is a linear programming problem, which is
\begin{align}
&max R_0=1-\frac{1/d_s}{\sum_{d}v_{qd}/d}\nonumber\\
&s.t.\sum_{d}v_{qd}=1,I_{qs}>I_{qs}^{-}+\vartriangle_{qs},I_{sq}\in[0,1]
\end{align}
in which
\begin{eqnarray}
I_{qs}=1-(1-I_{q,pri})\sum_{d}v_{qd}(1-I_{sq})^{d-1}
\end{eqnarray}
\begin{eqnarray}
I_{qs}^{-}=(I_{sq})^{\frac{1}{d_s-1}}
\end{eqnarray}
where superscript "-" refers to last iteration. EXIT curves at
q-nodes and s-nodes called q-curve and s-curve are given by (7)
and (8). $\vartriangle_{qs}$ is added to ensure that there are
some gap between the matched EXIT curves, so that BP algorithm
won't get stuck.\\
\indent For the nested LDGM-LDPC code $C_2$, the LDGM part is
essentially dictated by good encoder-side performance at the relay
node, so we optimize it first. As Ternary symbol $y_{ri}$ is
encoded into two bits $c_{2i-1}$,$c_{2i}$ in c-nodes, to simplify
analysis, we assume all b-nodes have the left-degree $d_b$ and the
two bits $c_{2i-1}$,$c_{2i}$ connected to the same $y_{ri}$-node
have the same right-degree, called the c-degree of the $y_r$-node.
Now we only have to optimize the c-degree distribution of the
$y_r$-nodes, represented by $v_{cd}$, the fraction of edges
connected to $y_r$-nodes with c-degree $d$ from the edge
perspective. Besides, the optimization is needed with the
constraint of the monotonic condition \cite{9}, which makes sure
that encoding can proceed with a vanishing fraction of
contradictions and thus flipped bits. Thus the optimization
problem is summarized as
\begin{align}
&\min_{v_{cd}}R_{b}=\frac{2}{d_b\sum_{d}v_{cd}/d}\nonumber\\
&s.t.\sum_{d}v_{cd}=1,I_{bc,pri}|_{I_{bc}=0}\geq 0,
\frac{dI_{bc,pri}}{dI_{bc}}\geq 0, I_{bc}\in[0,1]
\end{align}
in which
\begin{eqnarray}
&I_{bc,pri}=1-(1-I_{bc})/(1-I_{cb})^{d_b-1}
\end{eqnarray}
\begin{eqnarray}
&I_{cb}=I_{yc}\sum_dv_{cd}(I_{bc})^{d-1}
\end{eqnarray}
Where $I_{bc}$, $I_{cb}$ denotes the average MI in every b-to-c,
c-to-b message at a certain iteration respectively, and
$I_{yc}=1-0.5\varepsilon$ represents the priors average MI of the
$y_r$-nodes with c-degree, which is acquired from the mapping in
(3). $I_{bc,pri}$ denotes the priors average MI of the b-nodes at
fixed points(i.e. $I_{bc,pri}$ making the average MI
$I_{bc}=I_{bc}^{-}$), and it should be 0 when $I_{bc}=0$ and
increase monotonically as $I_{bc}$ increase from 0 to 1.\\
\indent With the degree distribution of LDGM part fixed, we
optimize the degree distribution of the LDPC part of $C_2$ to
achieve decoder-side performance. During the iterative decoding of
$\boldsymbol{y_r}$ at the destination, the average MI of the
message from $y_r$- to c-nodes, denoted as $I_{yc,d}$ for those
with c-degree $d$, varies between $I_{c0}=I(c;y_d)$ and
$I_{c1}=I(c_{2i-1};y_{di}|c_{2i})+I(c_{2i};y_{di}|c_{2i-1})$,
according to the incoming messages from c- and -nodes, that is
\begin{eqnarray}
I_{yc,d}=I_{c0}(1-I_{bc}^d)+I_{c1}I_{bc}^d
\end{eqnarray}
Making $I_{cb}$ in (11) become
\begin{eqnarray}
I_{cb}=\sum_dv_{cd}I_{yc,d}(I_{bc})^{d-1}
\end{eqnarray}
Let $I_{bc,ext}$ denote the extrinsic MI of b-node at fixed
points, derived only from $I_{cb}$, which is
\begin{eqnarray}
I_{bc,ext}=1-(1-I_{cb})^{d_b}
\end{eqnarray}
Thus the decoder-side EBP curve of the LDGM part formed by
$I_{bc,pri}$ vs. $I_{bc,ext}$ is derived from (10) and (14). The
LDPC part is designed to make the EBP curve of LDPC part
$I_{bp,pri}$ vs. $I_{bp,ext}$ match that of LDGM part, so that
$\boldsymbol{y_r}$ can be decoded. In other words, suppose the EBP
curve is plotted with $I_{bp,ext}$ in the horizontal axis and
$I_{bc,ext}$ in the vertical axis, then EBP curve of the LDPC part
should lie below that of the LDGM part, with a small gap between
them. The gap assures that iterative decoding does not get stuck.
Thus let $v_{pd}$ and $v_{bd}$ denotes the fraction of edges
connected to p-node and b-node with the left-degree and
right-degree $d$ respectively. The degree distributions are
optimized to achieve the minimal rate, which is
\begin{align}
&\min_{v_{pd},v_{bd}}R_{p}=R_b-R_{bp}=R_b-(1-\frac{\sum_{d}v_{pd}/d}{\sum_{d}v_{bd}/d})\nonumber\\
&s.t.\sum_{d}v_{pd}=1,\sum_{d}v_{bd}=1,I_{pb}^{+}>I_{pb}+\vartriangle_{pb},I_{pb}\in[0,1]\nonumber\\
&I_{bp,ext}=1-\sum_{d}v_{bd}(1-I_{pb})^d \nonumber\\
&I_{bc,pri}=I_{bp,ext},I_{bp,pri}=I_{bc,ext}\nonumber\\
&I_{bp}=1-(1-I_{bp,pri})\sum_{d}v_{bd}(1-I_{pb})^{d-1}\nonumber\\
&I_{pb}^{+}=\sum_{d}v_{pd}(I_{bp})^{d-1}
\end{align}
Where $I_{bc,ext}$ is derived from the EBP curve of LDGM part at
the decoder-side with the corresponding $I_{bc,pri}$ known.
$I_{bp}$, $I_{pb}$ denotes the average MI in every b-to-p, p-to-b
message at a certain iteration respectively, and superscript "+"
refers to the next iteration. $\vartriangle_{pb}$ is designed to
keep the gap in the EBP curves of the LDGM and LDPC part to make
the BP converge.
\section{Numerical Results}
In this section we evaluate the performance of our optimization
for degree distribution and LDGM-LDPC encoding and decoding
process. Let the erasure probabilities of the S-R and S-D links be
$\varepsilon=0.5$, the R-D link be ideal with $C_{rd}$ at least
1.25 bit/sym, and the block length $n=10^5$. Since the cut-set
bound in (2) is $I(x_s;y_dy_d)=0.75$ bit/sym, the code rate of
$C_1$ is $R_0\leq0.75$ bit/sym. The degree distribution of $C_1$
is optimized to achieve the maximal rate $R_0$. With
$2-\varepsilon=1.5$ bit/sym and $H(y_r|y_d)=1.25$ bit/sym, the
nested code $C_2$ has $R_b\geq1.5$ bit/sym, $R_p\geq1.25$
bit/sym.\\
\indent Optimizing the degree distribution of LDPC code by (6)
with $I_{q,pri}=0.75$, $d_s=16$, $R_0=0.742$ is acquired, and the
optimized degree distribution $v_{qd}$ is shown in Table II. With
$d_b=6$, $I_{yc}=0.75$, the degree distribution of LDGM part of
$C_2$ is optimized by (9), and $R_b=1.5019$ is acquired. The
optimized degree distribution $v_{cd}$ of LDGM part at encoding
side is represented in Table III.
\begin{table}[!hbp]
\caption{The degree distribution of LDPC code $C_1$ at the
source} \label{table_2} \centering
\begin{tabular}{cccccccccc} \hline $d$ &
$v_{qd}$& $d$ & $v_{qd}$& $d$ & $v_{qd}$& $d$ &
$v_{qd}$& $d$ & $v_{qd}$ \\
\hline
2&0.2467&5&0.0154&8&0.0679&13&0.0027&21&0.0689\\
3&0.1768&5&0.0473&9&0.046 &17&0.0067&24&0.0835\\
4&0.0479&6&0.0712&10&0.018&19&0.0412&27&0.0598\\
\hline
\end{tabular}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[width=3in]{sf_fig4}
\caption{The EBP curve of nested LDGM-LDPC codes in $C_2$ with gap
at destination node} \label{fig.4}
\end{center}
\end{figure}
\begin{table}[!hbp]
\caption{The degree distribution $v_{cd}$ of LDGM part in $C_2$}
\label{table_3} \centering
\begin{tabular}{cccccccccc} \hline $d$ &
$v_{cd}$& $d$ & $v_{cd}$& $d$ & $v_{cd}$& $d$ &
$v_{cd}$& $d$ & $v_{cd}$ \\
\hline
1&0.002 &6&0.026 &11&0.0081&21&0.0033&37&0.0016\\
2&0.5987&7&0.0161 &13&0.0074&24&0.0025&17&0.0013\\
3&0.1598&8&0.0126 &15&0.0056&27&0.0021&19&0.0012\\
4&0.0175&9&0.0099 &17&0.0047&30&0.0019&19&0.001\\
3&0.0408&10&0.0089&19&0.0043&33&0.0018& & \\
\hline
\end{tabular}
\end{table}
\indent With $I_{c0}=0.0472$ and $I_{c1}=0.2028$, The EBP curve of
LDGM part at the decoder-side is shown in the dashed curve of
Fig.4. Then by (15), $R_p=1.2696$ is acquired and the optimized
degree distributions $v_{pd}$, $v_{bd}$ of LDPC part at the
decoder-side is shown in Table IV. The EBP curve of LDPC part is
shown in the solid curve of Fig.4 with some gap. It can be seen
that The EBP curve of LDPC part indeed lies below that of LDGM
part and both of them match well, which assure that the iterative
BP algorithm converge.\\
\begin{table}[!hbp]
\caption{The degree distribution $v_{bd}$,$v_{pd}$ of LDPC part in
$C_2$} \label{table_4} \centering
\begin{tabular}{cccccccccc}
\hline $d$ &$v_{bd}$& $d$ & $v_{bd}$& $d$ & $v_{bd}$& $d$ &
$v_{bd}$& $d$ & $v_{pd}$ \\
\hline
1&0.0039&4&0.0173 &7&0.1917&21&0.0947&2&0.6087\\
2&0.6505&7&0.0009 &13&0.0303&24&0.0108&17&0.3913\\
\hline
\end{tabular}
\end{table}
\begin{table}[!hbp]
\caption{BER performance at destination: BEC channel; single-relay
system} \label{table_1} \centering
\begin{tabular}{ccc}
\hline Relay Protocol &Designed Rate & BER \\
\hline
CF&$R=0.472$&$1.357\times10^{-5}$\\
DF&$R=0.49$ &$3.265\times10^{-5}$\\
\hline
\end{tabular}
\end{table}
\indent With optimized degree distribution of $C_1$ and $C_2$, the
simulation of encoding and decoding processes is executed. The
experiment result shows that the iteration count of BP decoding
the LDGM part in $C_2$ is about 200, and LDPC decoding in
destination is only about 150. However, when erasure probability
$\varepsilon$ decreases, which denotes the channel capacity
increases, the iteration count will decrease. With Monte Carlo
simulation, The BER performance at destination node for relay
system in BEC under CF and DF is shown in Table V. Simulation
shows that under CF the BER is about $10^5$ in most blocks, and
some blocks even recover source information correctly, which sees
that CF is much better than DF ($R\leq0.5$, by only optimizing the
degree distribution of $C_1$ to realize DF). Besides, $R_0=0.472$
is close to CF theoretical limit 0.75.\\
\indent Some remarks on the design of $C_1$ and $C_2$ are in
order. Firstly, when the degree distribution of LDPC in $C_1$ and
LDGM in $C_2$ is optimized, the degree of s-nodes and b-nodes
should be chosen carefully. Here reasonable choice of $d_s$ ranges
from 16 to 20 and $d_b=6$. Secondly, we should leave a uniform gap
between the EXIT curves of LDPC codes in $C_1$ and between EBP
curves of the LDGM and LDPC parts in $C_2$ to make iterative
decoding converge with a reasonable number of iterations rather
than getting stuck. Besides, in order not to cut down the designed
rate, we need assure the gap $\vartriangle_{qs}$ and
$\vartriangle_{pb}$ should not be larger than 0.01 in every
iteration. E.g. this gap of the EBP curves of the nested LDGM-LDPC
part in $C_2$ is designed with $\vartriangle_{pb}=0.004$, which is
shown in Fig.4. The decoding process of $\boldsymbol{y_r}$ will
not get stuck and with the flipped position known in the
destination, $\boldsymbol{y_r}$ could be decoded correctly.\\
\indent Thirdly, there will inevitably be some incorrect
decimation in the LDGM quantization process, which cause
contradictions that must be corrected by flipping some bits in
$\boldsymbol\zeta$. This $\boldsymbol\zeta$ must also be
transmitted to the destination using a fraction of $C_{rd}$, so
that it can perform decoding correctly. We found that the number
of flipped positions is about 600, and $R_p$ is 1.2696bit/sym, so
we require $C_{rd}=R_p+2*H_{2}(600/2*10^5)=1.8385$ bit/sym (here
$H_{2}(p)=p\log_{2}p+(1-p)\log_{2}(1-p)$). We could see that even
with the flipped positions transmitted the required channel
capacity $C_{rd}$ of R-D link can still be lower than $H(y_r)=1.5$
bit/sym. In our future work, we will study the lossy compression
of $\boldsymbol{y_r}$, so that we can get lower $C_{rd}$.\\
\indent It is also observed that the performance of the proposed
practical CF scheme improves as the block length increases at the
cost of larger memory consumption and coding delay.
\section{conclusion}
In this paper, a first practical CF scheme for a type of relay
system based on nested LDGM-LDPC has been proposed, and methods
for optimizing the degree distributions have been described.
Simulation results show that the nested LDGM-LDPC codes can
perform Slepian-Wolf compression of the relay's ternary received
signals when the relay system is BEC. The performance of our
scheme approaches the CF theoretical cut-set bound, while previous
schemes are either limited to binary signals or suboptimal. Our
work shows nested LDGM-LDPC codes for practical CF scheme is
sufficient.\\
\indent It is apparently straightforward to extend the proposed
scheme to realize lossy compression, which would offer better
performance achieved at a lower relay-destination channel capacity
. The design will be considered in our future work. We will also
try to optimize the gap between the BEP curves of the LDGM and
LDPC parts of the nested code, so that its decoding can converge
more quickly and reliably.
|
train/arxiv
|
BkiUd2c5qU2ApzQrUB-w
| 5 | 1 |
\section{Introduction}
\label{sec1}
Content-based image retrieval (CBIR) is a core research area in medical image analysis, with numerous studies across many different image modalities \citep{dermaXAI, 9296383, HAQ2021101847}. CBIR supports clinicians in retrieving relevant images from a large database compared to a query image, which reduces labor-intensive manual search and aids in diagnosis. For instance, a physician might want to investigate how patients in a large database with a similar disease as a new patient, such as liver metastasis, were diagnosed. The information from the previous diagnoses can then be used to determine the proper treatment for the new patient. In analysis of CT image of the liver, CBIR have been an active and important area of medical image analysis for many years \citep{1403458,liverhandcraft2013,9043172}. CBIR has the potential to make labour intensive tasks in the clinical workflow more time efficient, as illustrated in Section \ref{sec:use-case}.
\begin{figure*}[ht]
\centering
\includegraphics[width=0.95\textwidth]{content-based-image-retrieval-chart.pdf}
\caption{Illustration of content-based image retrieval.}
\label{fig:cbirc}
\end{figure*}
Currently, deep learning-based CBIR, or deep CBIR, constitute the state-of-the-art of CBIR \citep{guidedCBIR,9043172,HAQ2021101847}, due to its high precision and efficiency. However, deep CBIR suffers from some critical limitations. First (1), current deep CBIR for CT liver images rely on labeled data for training \citep{9043172}. Obtaining labeled data can be costly and time-consuming, which therefore limits the usability of deep CBIR systems. However, recent works have shown how self-supervised learning can leverage unlabeled data to improved CBIR systems \citep{8982468,s22062188}, but such approaches have not been explored in the context of CBIR of CT liver images. Second (2), deep CBIR suffer from a fundamental lack of explainability. This can have detrimental effects in a clinical setting, since deep learning-based systems are known to exploit confounding factors and artifacts to make their predictions. For instance, \cite{gautamisbi} showed that a deep-learning-based system learned to use tokens and artifacts in X-ray images to makes its predictions instead of clinically relevant features. These tokens and artifacts would not be present for new patients, and such a system would not work as intended if put into clinical practice. Therefore, it is not advisable to blindly trust the retrieved images from the deep CBIR system without investigating what input features influence the retrieval process through an explainability analysis.
A promising direction to address the first limitation is learning from unlabeled data through self-supervision. Recent self-supervised learning frameworks have shown remarkable results, in some cases even rivalling supervised learning \citep{simclr, swav, Simsiam}. In a nutshell, contemporary self-supervised approaches train a feature extractor that extract informative representations by exploiting known invariances in the data. These representations can then be used for other tasks, such as CBIR by taking similarities between the new representation to retrieve similar images. These self-supervised approaches have been show to improve performance in the context of chest X-ray \citep{HowTransferableAreSS,bigMedicalSS} and dermatology classification \citep{bigMedicalSS}, organ andcardiac segmentation \citep{hansen2022anomaly}, and whole heart segmentation \citep{dong2021self}, but have yet to be developed for CBIR of CT liver images.
In this paper, we propose a clinically motivated self-supervised framework for CBIR of CT liver images. Our proposed framework incorporates domain knowledge that exploits known properties of the liver, which leads to improved performance compared to well-known self-supervised baselines. Concretely, a novel Houndsfield unit clipping strategy that removes non-liver pixels from the input and allows the system to focus on the liver is incorporated into the self-supervised training. While the focus in this paper is on the liver in CT images, our proposed framework could also be used to focus on other organs by altering how the Houndsfield units are clipped.
For the second limitation, great improvements have been made in the field of explainable artificial intelligence (XAI) over the last couple of years, and numerous studies have shown how XAI can improve the reliability and trustworthiness of deep learning-based systems in healthcare \citep{guidedCBIR, gautamisbi}. However, the majority of these improvements have been in algorithms that can explain models which produce decisions, such as classification or similarity scores. When learning from unlabeled data through e.g. self-supervised learning, such a score or similarity measure might not be available and standard XAI techniques cannot be applied. But the recent field of representation learning explainability \citep{wickstrom2021relax} aims at explaining vector representations, and can therefore tackle the lack of explainability in deep CBIR. But such a representation learning explainability analysis has not been performed in the context of CBIR of CT liver images.
Our contributions are:
\begin{itemize}
\item A clinically motivated self-supervised framework specifically designed to extract liver specific features.
\item A novel explainability analysis that explains the representations produced in the feature extraction process.
\item Thorough evaluation on real-world datasets.
\item A case-study where images from the same patient are retrieved across different examinations.
\end{itemize}
\section{Related work}
\subsection{Content-based image retrieval}
The goal of content-based image retrieval (CBIR) is to find similar images from a large-scale database, given a query image. CBIR is an active area of research that span numerous medical imaging domains, such as X-ray \citep{HAQ2021101847, guidedCBIR}, dermatology \citep{dermaXAI, Ballerini2010}, mammography \citep{Jiang2014}, and histopathology \citep{Peng2019, Zheng2019}. An illustration of a CBIR system in the context of CT liver images is shown in Figure \ref{fig:cbirc}.
\subsection{Content-based image retrieval of CT liver images}
CBIR of CT liver images have been extensively studied. Early studies relied on handcrafting features based on certain properties in the images. Gabor filters have been used to extract texture information \citep{1403458}. Texture information have also been combined with density information in the context of focal liver lesion retrieval \citep{liverhandcraft2013}. Histogram-based features extraction have been explored to retrieve CT scans with similar liver lesions. Manifold learning have been utilized to facilitate CBIR of CT liver images \citep{Mirasadi2019}. Lastly, a Bayesian approach has been studied in connection with multi-labeled CBIR of CT liver images \citep{9296383}.
Recently, deep learning-based feature extraction have improved performance significantly in CBIR of CT liver images. The most straight forward approach for deep CBIR is to train a neural network for the task of CT liver image classification and use the intermediate features prior to the classification layer for calculating similarities. This has been demonstrated to produce good results when the network was trained for the task of focal liver lesions detection \citep{9043172}. However, all these approaches need labeled data to train the deep learning-based feature extractor.
\subsection{Self-supervised learning}
Learning from unlabeled data is a fundamental problem in machine learning. Recently, self-supervised learning have shown promising results in computer vision \citep{simclr, Simsiam}, natural language processing \citep{BERT, GPT}, and time series analysis \citep{franceschi, WICKSTROM202254}. Furthermore, recent studies have also demonstrated that self-supervised learning can improve performance across several imaging domains in medical image analysis \citep{bigMedicalSS, HowTransferableAreSS,hansen2022anomaly,dong2021self}.
For computer vision, there are three main approaches to self-supervised learning. First, contrastive self-supervised learning is performed by sampling positive pairs and negative samples and learning a representation where the positive pairs are mapped in close proximity and far from the negative samples. The SimCLR framework \citep{simclr} is one of the most widely used approaches in this category. Second, clustering-based self-supervised learning utilizes clustering algorithms to produce pseudo-labels which in turn are used to learn a useful representation of the data. DeepCluster \citep{DeepCluster} and the SwAV framework \citep{swav} are two of the most widely used clustering-based self-supervised approaches in the literature. Lastly, siamese self-supervised approaches learns how to produce a useful representation by maximizing agreement between positive pairs of samples. The two main contemporary approaches in siamese self-supervised approaches is the SimSiam framework \citep{Simsiam} and the BYOL framework \citep{byol}.
\subsection{Explainability}
Explainability is of vital importance for machine learning systems in healthcare. Without it, clinicians cannot fully trust the algorithms decision and the system becomes less reliable. Many recent studies have shown how explainability can be incorporated into deep learning systems for medical image analysis, ranging from diabetic retinopathy \citep{DRxai}, dermatology \citep{dermaXAI, 9246575}, X-ray \citep{xrayXAI}, and endoscopic images \citep{WICKSTROM2020101619, endocopyXAI}.
Most of the widely used explainability techniques typically leverage the classification or similarity score to ascertain input feature importance \citep{guidebackprop,Schulz2020Restricting,Plummer2020}, and such approaches have been explored in the context of deep CBIR. For models trained for classification tasks, explanations through gradient information have been shown to both provide new insights and improve performance for X-ray images \citep{guidedCBIR}. For models trained to output a similarity score, it has been shown how the similarity score can be used to provide explanations \citep{CvprDong2019, Plummer2020}. Similarity score explanations have been explored for X-ray images \citep{9706900}. Lastly, it has been shown that explanation by examples can be effective in histopathological images \citep{Peng2019}.
In the unlabeled setting where only the feature extraction model is available, these techniques are not applicable. In such cases, it is desirable to explain the vector representation of an image, since the decision is not available. Representation learning explainability is a very recent field of XAI, that has yet to be developed for CBIR. In this work, we leverage the RELAX framework \citep{wickstrom2021relax} to explain the feature extractors trained using self-supervised learning. RELAX is the first method that allows for representation learning explainability and has been shown to provide superior performance to competing alternatives \citep{wickstrom2021relax}.
\section{A clinically motivated self-supervised approach for CT liver images}
In this section, we present our proposed clinically-motivated self-supervised approach and the SimSiam framework for self-supervised learning.
\subsection{A clinically motivated self-supervised approach for CT liver images}
We propose to incorporate clinical knowledge into our self-supervised framework to learn more clinically relevant features. In self-supervised learning, known invariances in the data are used to train a feature extractor that extracts relevant features from the input images. For instance, the liver can occur on both the left and right hand side of an image, depending on which direction the patient is inspected. Therefore, the feature extractor should be invariant to horizontal flips in the images, and this invariance can be learned by incorporating horizontal flipping into the self-supervised learning procedure. Identifying these invariances is crucial to make the self-supervised system work properly and focus on clinically relevant features in the input images. Our motivation is based on the knowledge that the pixel intensities of the liver lay within a certain range for CT images. A standard pre-processing step is to clip the pixel intensities of the CT images \citep{Li2018}, such that unimportant pixels are removed prior to learning. The pixel intensities of CT images represent a physical quantity, namely the Houndfield unit. The same clipping is usually applied to all images. However, if this clipping was incorporated into the self-supervised learning procedure, the network could be guided to learn which feature are liver features and which ones are not. In a sense, we are exploiting the knowledge that the liver should be invariant to pixel intensity clipping for a certain range of clipping.
Based on this motivation, we propose a Houndsfield clipping strategy where the pixel values for the same image are clipped and scaled based on different ranges of Houndsfield units. Figure \ref{fig:clip} shows how our proposed clipping scheme affects an image. The leftmost image has no clipping applied, and illustrates why it is important to remove some pixel intensities in order to highlight relevant structures in the images. The middle images show the narrow clipping strategy between 50 and 150 Houndsfield units. Notice how only the liver and some other organs are now visible in the image. The rightmost image shows the wide clipping strategy between -200 and 300 Houndsfield units. In this case, some redundant structures are removed, but more organs are left visible compared to the middle image. The images considered in this paper are intra venous contrast enhanced images taken in the portal venous phase. These two ranges were chosen based on the following. First, it is known that the liver typically has Houndsfield units in the range 50-60 \citep{Tisch2019}. Furthermore, we have collected all pixel intensities for the liver in the Decathlon dataset. These values are shown in Figure \ref{fig:pixel-i}, and illustrates how the narrow clip will remove some of the liver pixels but keep the main proportion, while the wide clip will keep almost all liver pixels apart from some outliers. Our proposed framework for learning representations that focus on liver features is shown in Figure \ref{fig:fwork}. Each image is clipped with the wide and narrow range, before the data augmentation is applied. Afterwards, we follow the SimSiam approach described below. During testing, we use the wide clipping to ensure that most liver pixels are kept in the images.
\begin{figure}[htb]
\centering
\includegraphics[width=0.975\columnwidth]{clip-example.pdf}
\caption{Effect of Houndsfield unit clipping on CT liver images. From left to right, no clipping, narrow clip (50, 150), and wide clip (-200, 300).}
\label{fig:clip}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=0.95\columnwidth]{pixel-intensities.png}
\caption{Distribution of pixel intensity values for liver pixel from the Decathlon dataset and the two clipping strategies used in our proposed framework.}
\label{fig:pixel-i}
\end{figure}
\begin{figure*}[ht]
\centering
\includegraphics[width=0.95\textwidth]{ct-liver-framework-figure.pdf}
\caption{Illustration of proposed self-supervised framework.}
\label{fig:fwork}
\end{figure*}
\subsection{SimSiam framework}
In this work, we build on the SimSiam framework. The main motivation for this choice is that both contrastive and clustering-based self-supervised approaches requires a large batch size during training to provide high quality representations \citep{simclr, swav}. This can be computationally challenging, especially if the medical images in question are large. However, the siamese-approaches \citep{Simsiam} are less sensitive to the batch size used during training. Furthermore, we opt for the SimSiam approach over BYOL to avoid training both a student and a teacher network used in BYOL, again to avoid additional computational overhead.
Let $\mathbf{X}\in \mathbb{R}^{H\times W}$ represent an input image with height $H$ and width $W$ and $f$ a feature extractor that transforms $\mathbf{X}$ into a new $d$-dimensional representation $\mathbf{h}\in \mathbb{R}^{d}$, that is $f(\mathbf{X})=\mathbf{h}$. Next, two views $\mathbf{X}_1$ and $\mathbf{X}_2$ are constructed by augmenting the original image. The task performed in SimSiam to learn a useful representation, is to maximize the similarity between the two views. The representation $h$ is the new representation that can be used for downstream tasks, such as the CBIR. However, the loss is not computed directly on the output of the feature extractor $f$. Instead, a multilayer perceptron-based projection head $g$ transforms $\mathbf{h}$ into a new representation $\mathbf{z}$, that is $g(\mathbf{h})=\mathbf{z}$, where the loss is computed. This projector is a crucial component in most self-supervised frameworks \citep{simclr, moco}, as it avoid dimensional collapse in the representation $h$ \citep{jing2022understanding}, which is the one that will be used for downstream tasks such as CBIR. The learning is performed by minimizing the negative cosine similarity between the two views:
\begin{equation}\label{eq:dist}
D(\mathbf{z}_1, \mathbf{z}_2) = -\frac{\mathbf{z}_1}{\lVert \mathbf{z}_1 \rVert_2}\cdot\frac{\mathbf{h}_2}{\lVert \mathbf{h}_2 \rVert_2},
\end{equation}
where $\lVert \cdot \rVert_2$ denotes the $\ell_2$-norm.
\begin{equation}\label{eq:loss1}
L = D(\mathbf{z}_1, \mathbf{h}_2) + D(\mathbf{z}_2, \mathbf{h}_1)
\end{equation}
An important component of the the SimSiam framework is a stop-gradient $\text{{\fontfamily{tnr}\selectfont
(stopgrad)}}$ operation, which is incorporate in Equation \ref{eq:loss1} as follows:
\begin{equation}\label{eq:loss2}
L = \frac{1}{2}D(\mathbf{z}_1, \text{{\fontfamily{tnr}\selectfont
stopgrad}}(\mathbf{h}_2)) + \frac{1}{2}D(\mathbf{z}_2, \text{{\fontfamily{tnr}\selectfont
stopgrad}}(\mathbf{h}_1))
\end{equation}
The stop-gradient operation is applied to the projector network, such that the encoder on $\mathbf{X}_2$ no gradient from $\mathbf{h}_2$ in the first term, but it recieves gradients from $\mathbf{z}_2$ (and similarly for $\mathbf{X}_1$). The stop-grad operation allows SimSiam to mimic a teacher-student setup, but avoid the need to store two networks. Furthermore, it has been shown that the stop-grad operation is critical to avoid the problem of complete collapse in the representations \citep{collapse}.
\paragraph{Data augmentation} The prior knowledge inject through the data augmentation is of paramount importance to ensure that the models learns relevant features. The data augmentation used in SimSiam is similar to the standard approach in recent self-supervised learning \citep{moco,simclr}:
\begin{enumerate}
\item Crop with a random proportion from [0.2, 1.0], and resize to a fixed size.
\item Flip horizontally with a probability of 0.5.
\item Color augmentation is performed by randomly adjusting the brightness, contrast, saturation, and hue of each image with a strength of [0.4, 0.4, 0.4, 0.1]
\item Randomly convert image to gray scale version with a probability 0.2.
\end{enumerate}
Note that the input images are converted to pseudo RBG images by stacking the input image 3 times along the channel axis. Prior works have shown that the augmentation scheme listed above can lead to increased performance across several medical image related tasks \citep{bigMedicalSS, HowTransferableAreSS, hansen2022anomaly, dong2021self}, albeit not in the context of CBIR of CT liver images. However, these augmentations are selected with natural images in mind, and do not take into account the properties of CT liver images. Our proposed Houndsfield unit clipping scheme takes into account the particular characteristics of CT images of the liver, which we hypothesize can improve the self-supervised framework.
\section{Explaining representations}
Explainability is a critical component for creating trustworthy and reliable deep learning-based systems. For deep CBIR, we want to know what information the feature extractor is using to create the representation that the retrieval is based on. This requires explaining the vector representations produced by the feature extractor, which can not be accomplished with standard explainability techniques since they require a classification or similarity score to create the explanation. However, the recent field of representation learning explainability address the problem of explaining representations \citep{wickstrom2021relax}. In this work, we leverage the RELAX \citep{wickstrom2021relax} framework to explain the representations used in the CBIR system.
\subsection{RELAX}\label{sec:relax}
RELAX is an occlusion-based explainability framework that provides input feature importance in relation to a vector representation, as opposed to a classification or similarity score. The core idea of RELAX is to evaluate how the representation of an image changes as parts of the image are removed using a mask. Let $\mathbf{M}\in [0, 1]^{H\times W}$ represent a stochastic mask used for removing parts of the image. Next, $\bar{\mathbf{h}} = f(\mathbf{X} \odot \mathbf{M})$, where $\odot$ denotes element-wise multiplication, is the representation of a masked version of $\mathbf{X}$ and $s(\mathbf{h}, \bar{\mathbf{h}})$ is a similarity measure between the unmasked and the masked representation. The intuition behind RELAX is that when informative parts are masked out, the similarity between the two representations should be low, and vice versa for non-informative parts. Finally, the importance $R_{ij}$ of pixel $(i,j)$ is defined as:
\begin{equation}\label{eq:relax}
\bar{R}_{ij} = \frac{1}{N}\sum\limits_{n=1}^N s(\mathbf{h}, \bar{\mathbf{h}}_n)M_{ij}(n).
\end{equation}
Here, $\bar{\mathbf{h}}_n$ is the representation of the image masked with mask $n$, and
$M_{ij}(n)$ the value of element $(i, j)$ for mask $n$. The similarity measure used in the cosine similarity, as proposed in prior works \cite{wickstrom2021relax}. The RELAX framework is illustrated in Figure \ref{fig:relax}.
The mask generation is a crucial component in RELAX. In this work, we follow the strategy used in previous studies \citep{Petsiuk2018rise, wickstrom2021relax}. Binary masks of size $h\times w$, where $h<H$ and $w<W$, are generated, where each element of the mask is sampled from a Bernoulli distribution with probability $p$. To produce smooth and spatially coherent masks, the small masks are upsampled using bilinear interpolation to the same size as the input image. Furthermore, the number of masks required to obtain reliable estimates of importance is an important hyperparameter. In this work, we generate 3000 masks to obtain an explanation for a single image, as suggested in a prior work \citep{wickstrom2021relax}.
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{relax-illustration.pdf}
\caption{Illustration of RELAX. A feature extractor produces a new representation of an input image, and RELAX determines what input features are important for the representation.}
\label{fig:relax}
\end{figure}
\begin{figure*}[ht]
\centering
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{map.png}
\label{fig:unn-map}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{acc.png}
\label{fig:unn-acc}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{rr.png}
\label{fig:unn-rr}
\end{subfigure}
\caption{From left to right, mean average precision, knn accuracy, and relevance rank scores versus epochs across 5 training runs on the test images from the Decathlon dataset The plot show how performance increase with training time, and that the proposed framework learns faster with better results.}
\label{fig:scores}
\end{figure*}
\section{Evaluation}
We introduce the set of scores utilized to provided quantitative evaluation of our proposed framework.
\subsection{Evaluating quality of CBIR}
A standard approach to evaluate the quality of a CBIR system is to measure the class-consistency in the top retrieved images \citep{guidedCBIR, LI201866}. One of the most common approaches to evaluate the class-consistency is through mean average precision (MAP):
\begin{equation}
\text{MAP} = \frac{1}{N}\sum\limits_{n=1}^N\frac{1}{K}\sum\limits_{k=1}^K\text{precision(k)}_n,
\end{equation}
where $N$ is the number of test samples (query images), $K$ is the top-$K$ retrieved images for each query image, and precision is defined as:
\begin{equation}
\text{precision(k)} = \frac{\lvert \text{relevant images} \cap \text{k-retrieved images}\rvert}{\lvert \text{k-retrieved images} \rvert}.
\end{equation}
MAP evaluates the precision of the retrieved images across several values of K, which makes it robust towards fluctuations among the top retrieved images.
\subsection{Evaluating quality of representations}
The most widespread approach for evaluating the representation produced by a self-supervised learning framework is to train a simple classifier on the learned representations \citep{simclr,swav,moco}. The motivation for this, is that a simple classifier is highly dependent on the representation it is given in order to perform the desired task. In this work, we follow recent studies that use a k-nearest neighbors (KNN) classifier \citep{caron2021emerging, swav} to evaluate the quality of the representation. We opt for a KNN classifier over a linear classifier as it does not require any training, which can lead to ambiguities in the results \citep{Kolesnikov2019CVPR}, and has minimal hyperparameters to tune.
\subsection{Evaluating the quality of explanations}
Great improvements have been made in the field of XAI over the last couple of years. In contrast, the field of evaluation for explanations is still under active development \citep{doshivelez2017rigorous}. However, recent advances have introduced new methods for providing quantitative evaluation of explanations. In this work, we use the relevance rank accuracy score (RR) \citep{ARRAS202214}. RR measures how many of the top-$M$ relevant pixels lies within the ground truth segmentation mask. It can be considered a proxy for how well the explanation agrees with a human explanation for a given images. Let $R_M$ denote the $M$ most relevant pixels in an explanation, and $S$ the segmentation mask for the liver. RR can then be defined as:
\begin{equation}
\text{RR} = \frac{1}{N}\sum\limits_n^N \frac{\lvert R_M(n) \cap S(n) \rvert}{\lvert S(n) \rvert}.
\end{equation}
The RR is computed using the Quantus toolbox \citep{hedstrom2022quantus}.
\begin{figure*}[ht]
\centering
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{unn-map.png}
\label{fig:map}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{unn-acc.png}
\label{fig:acc}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{unn-rr.png}
\label{fig:rr}
\end{subfigure}
\caption{From left to right, mean average precision, knn accuracy, and relevance rank scores versus epochs across 5 training runs on the test images from the UNN dataset. The plot show how performance increase with training time, and that the proposed framework learns faster with better results.}
\label{fig:unn-scores}
\end{figure*}
\section{Data}
In this section, we present the data used to evaluated our proposed framework.
\subsection{Decathlon data}
The medical segmentation decathlon is a biomedical image analysis challenge where several tasks and modalities are considered \citep{antonelli2021medical}. One of the datasets in this challenge is a CT liver dataset acquired from the IRCAD Hopitaux Universitaires and consists of 201 contrast-ehanced CT liver images from patietns with mostly cancers and metastaic liver disease. However, we exclude 70 of these images as they do not include label information. Using every slide from each volume is computationally intractable. Therefore, we construct a slice-wise dataset as follows. From each volume, we sample 5 slices with no liver and 5 slices with liver. We construct the training set from the first 100 volumes and the test set from the remaining 31 volumes. This results in a balanced dataset with 1310 training images and 310 test images.
\subsection{UNN data}
The UNN dataset is from an extensive database of CT scans from The University Hospital of North Norway (UNN). It is under development through a close collaboration between UiT, The Arctic University of Norway, and UNN. The database contains CT volumes of 376 patients surgically treated for rectal cancer from 2006 to 2011 in North Norway. The examinations were conducted for diagnostic and routine follow-up purposes. The full dataset consists of CT with coronal, sagittal, and axial slices of mainly the thorax, abdomen, and pelvis. Examinations were conducted with different scanners and protocols at eight different hospitals in North Norway in the period 2005 to 2020.
From the full dataset a subset of 3347 axial volumes from 368 patients was selected based on descriptive keywords and DICOM metadata to limit it to contain mostly volumes of the liver and
abdomen. This subset is similar to the CT liver partition of the medical segmentation decathlon dataset in terms of image resolutions and contents, but more diverse in terms of image quality,
contrast enhancement levels, and artifacts because it is only curated using keywords and metadata, and not by manual assessment.
From the UNN subset 10 randomly selected volumes from 10 different and randomly selected patients with liver tumors were manually labeled with segmentation masks of the liver and metastatic regions by a clinician (co-author K.R.) to be used in our study. In addition, two volumes from a patient that had been treated with liver surgery to remove a metastatic liver segment were included. One volume was before the surgery, and one after the surgery. The study of these pre- and post-operative images is conducted as a use-case of cross-examination CBIR.
\section{Experiments}
We present the results of the experimental evaluation of our proposed framework. All models were trained with a batch size of 32 and for 250 epochs. Optimization was carried out using stochastic gradient descent with momentum=0.9, weight decay=0.0001, and learning rate=0.05 * batch size / 256, as used in the SimSiam framework \citep{Simsiam}. As in previous works \citep{simclr, Simsiam}, a ResNet50 \citep{7780459} was used as the feature extractor, with the output of the average pooling layer as the final representation. For both the KNN classifier and the MAP we set K=5. Code is available at \url{https://github.com/Wickstrom/clinical-self-supervised-CBIR-ct-liver.git}.
\subsection{Quantitative results}
Table \ref{tab:dec} and \ref{tab:unn} presents the MAP, accuracy of a 5NN classifier, and the RR on the test data from the Decathlon and UNN datasets. The results show that the proposed framework outperforms the standard self-supervised approach across most scores. Furthermore, self-supervised learning greatly improves upon simply using the feature extractor trained on the Imagenet dataset. Also, the improvements are transferable across datasets, as the feature extractors trained on the Decathlon data also leads to improved performance in the UNN data.
Figure \ref{fig:scores} and \ref{fig:unn-scores} presents the evolution of MAP, accuracy of a 5NN classifier, and the RR on the test data from the Decathlon and UNN datasets across training. The plots highlight how the scores improve as training progresses and stabilizes. However, an interesting observation is that the MAP and KNN accuracy achieves its highest value earlier in training on the UNN dataset.
\begin{table}[ht]
\centering
\caption{Mean and std of mean average precision, knn accuracy and relevance rank score across 5 training runs on the test images from the Decathlon dataset. Results show that the proposed framework outperforms the baselines. Bold numbers indicates the highest performing model.}
\begin{tabular}{l|ccc}
\toprule
{pretraining} & MAP & ACC & RR \\
\midrule
IN & 79.4 & 80.3 & 5.00 \\
IN + SS (baseline) & 87.7 $\pm$ 1.0 & 88.8 $\pm$ 2.0 & 13.6 $\pm$ 2.5 \\
IN + SS (ours) & \textbf{89.1} $\pm$ \textbf{1.3} & \textbf{90.6} $\pm$ \textbf{1.4} & \textbf{16.2} $\pm$ \textbf{3.1} \\
\bottomrule
\end{tabular}
\label{tab:dec}
\end{table}
\begin{table}[ht]
\centering
\caption{Mean and std of mean average precision, knn accuracy and relevance rank score across 5 training runs on test images from the UNN dataset. Results show that the proposed framework outperforms the baselines. Bold numbers indicates the highest performing model.}
\begin{tabular}{l|ccc}
\toprule
{pretraining} & MAP & ACC & RR \\
\midrule
IN & 80.7 & 83.0 & 4.34 \\
IN + SS (baseline) & 84.1 $\pm$ 2.3 & 86.8 $\pm$ 1.6 & \textbf{17.5} $\pm$ \textbf{4.0}\\
IN + SS (ours) & \textbf{85.1} $\pm$ \textbf{1.7} & \textbf{87.6} $\pm$ \textbf{1.9} & \textbf{17.5} $\pm$ \textbf{3.0} \\
\bottomrule
\end{tabular}
\label{tab:unn}
\end{table}
\begin{figure*}[ht]
\centering
\includegraphics[width=0.9\textwidth]{ct-liver-imagenet-example.pdf}
\caption{\textbf{Example (1):} CBIR example from Decathlon dataset with feature extractor pretrained on Imagenet dataset. Top row shows, from left to right, the query and the top 5 retrieved images. Bottom row shows the important features for the representation of each image, with important features in red and less important in blue. Some of the retrieved images do not contain the liver, and the explainability analysis shows that the feature extractor is focusing on the spine and rib cage instead of the organs. This information is important to understand why non-relevant images are retrieved, and would not be available without the explainability analysis.}
\label{fig:in}
\end{figure*}
\subsection{Explaining representations - qualitative results}
The relevance rank scores in Table \ref{tab:dec} and \ref{tab:unn} show that the proposed framework utilizes liver features in the image to a larger degree than the baseline approaches. However, the scores are far from perfect, which means that other parts of the image are also being used. Also, the feature extractor that is only trained on the Imagenet dataset has a very low relevance rank score, meaning that it is putting little attention on the liver. All of these observations can be investigated through XAI. In this section, we illuminate these observations through a new explainability analysis for CBIR by leveraging the RELAX framework that was described in Section \ref{sec:relax}. We show 4 qualitative examples, where the first example shows explanations for the feature extractor trained using Imagenet, and the remaining three examples shows explanations for the feature extractor trained using the proposed framework. In all examples, we show a query from the test set and the 5 retrieved images by CBIR system. Additionally, we show the explanation for the query and retrieved images. The explanation show which features in the input are the most important for the representation of the image, where important pixels are highlighted in red and non-important pixels in blue.
\textbf{Example 1: the feature extractor pretrained on Imagenet focuses on hard edges such as the spine.} Figure \ref{fig:in} displays an example where 2 of the 5 the retrieved images do not contain parts of the liver. When inspecting the explanations, it is clear that the feature extractor is not focusing on the liver, but rather on the tailbone. We hypothesize that since the feature extractor has never been presented with CT images, it utilizes prominent features with hard edges such as spine, as opposed to organs with softer boundaries. The behaviour discovered in this example is important, as it might also result in unexpected or poor retrievals for other queries.
\textbf{Example 2: the feature extractor trained using the proposed framework focuses on liver features.} Figure \ref{fig:ss} shows an example where all the retrieved images contain liver. Additionally, it is evident that the feature extractor is putting more emphasis on the liver for all the images, which illustrates how the proposed self-supervised framework has enabled the feature extractor to focus on clinically relevant features.
\begin{figure*}[ht]
\centering
\includegraphics[width=0.9\textwidth]{ct-liver-ss-example.pdf}
\caption{\textbf{Example (2):} CBIR example from Decathlon dataset with feature extractor pretrained using the proposed self-supervised framework. Top row shows, from left to right, the query and the top 5 retrieved images. Bottom row shows the important features for the representation of each image, with important features in red and less important in blue. All retrieved images contain the liver, and the explainability analysis shows that the feature extractor is focusing on the liver.}
\label{fig:ss}
\end{figure*}
\textbf{Example 3: the feature extractor trained using the proposed framework uses features from organs that often co-occur with the liver.} Figure \ref{fig:ss2} displays an example where CBIR system retrieves 5 images that contain the liver, but where the explainability analysis shows that it not focusing on part of the images where the liver is present. Instead, it puts attention on the kidneys, which are quite prominent in all images. The kidneys often occur together with the liver in many CT images, and it also has similar pixel intensities as the liver (in terms of Houndsfield units). Therefore, it is not surprising that the feature extractor has learned to utilize both liver and kidney features, which also explains the behaviour in this example. Such insights would not be obtainable without conducting the explainability analysis.
\begin{figure*}[ht]
\centering
\includegraphics[width=0.9\textwidth]{ct-liver-ss-example-2.pdf}
\caption{\textbf{Example (3):} CBIR example from Decathlon dataset with feature extractor pretrained using the proposed self-supervised framework. Top row shows, from left to right, the query and the top 5 retrieved images. Bottom row shows the important features for the representation of each image, with important features in red and less important in blue. All retrieved images contain the liver, but the explainability analysis reveals that the focus is on the kidneys, not the liver.}
\label{fig:ss2}
\end{figure*}
\textbf{Example 4: the feature extractor trained using the proposed framework focuses on liver features, also for images from a different dataset.} Lastly, Figure \ref{fig:unn} shows and example from the UNN dataset. This example illustrates that also on this new and unseen dataset, the feature extractor is basing the representation of these images features associated with the liver.
\begin{figure*}[ht]
\centering
\includegraphics[width=0.9\textwidth]{ct-liver-unn-example.pdf}
\caption{\textbf{Example (4):} CBIR example from UNN dataset with feature extractor pretrained using the proposed self-supervised framework. Top row shows, from left to right, the query and the top 5 retrieved images. Bottom row shows the important features for the representation of each image, with important features in red and less important in blue. All retrieved images contain the liver and the feature extractor is focusing on the liver, which illustrates that the feature extractor trained on the Decathlon dataset transfers well to the UNN dataset.}
\label{fig:unn}
\end{figure*}
\subsection{Case study: cross-examination CBIR}\label{sec:use-case}
A typical scenario in clinical practice is comparing a newly conducted examination with one ore more previous examinations. For instance, one might want to compare a particular slice from the new examination with a selection of slices from one or several previous examinations. Such a comparison can help physicians understand how a diseases has progress since the previous examination, such as the development of liver metastasis. But when conducting such a comparison, the physician must manually inspect the new examination, and potentially several previous examinations. The CT scans are often taken with different settings across examinations, and it is therefore not possible to simply select the same slice from different examinations, as this can image completely different parts of the patient. A precise and reliable CBIR system could make such a cross-examination more efficient, by automating the retrieval process for the physician.
A typical scenario in clinical practice is comparing a newly conducted examination with one or more previous examinations. Such as development, progress, or effect of treatment of liver metastasis, the status of liver cirrhosis, auto-immune diseases in the liver, or any morphological or anatomical changes in the liver over the course. One might want to compare a region of interest in the slice from the current examination with previously conducted examinations. The comparison result will be applicable for evaluating the disease over the long course. In routine clinical practice, the selection of slices from the previous examinations is made manually to achieve the comparison, which is very time-consuming. CT scans taken over time are often conducted in an inconsistent sequence and contain unlike body regions in the same array of slices; therefore, selecting the identical arrays of slices from the different examinations is inaccurate. A precise and transparent CBIR system could make a cross-examination more efficient through the automatic retrieval process.
Figure \ref{fig:case-study} displays an example of such a cross-examination. The query is selected from a recent examination, and the retrieved images are from the previous examination of the same patient. This patient is from the UNN dataset and was selected since liver metastasis has been developed between the two examinations. The query was selected by an experienced physician (co-author K.R.), which also selected five images to examine from the previous examination. Ideally, the CBIR system should align well with the image selected by the physician. In this example, the CBIR system produces a successful retrieval, as it identifies the same images as the physician.
However, an interesting observation is that the CBIR retrieved are not sorted in the same manner as the physician's retrievals. Probably this deviation is due to the CBIR system being trained on single slices without a sense of spatial coherence. Future works could address this by incorporating neighboring samples as positive pairs in the self-supervised training.
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{use-case-example.pdf}
\caption{An example of cross-examination CBIR. The query is from a recent examination, and the retrieved images are from a prior examination from the same patient. The goal of such a study is to investigate the development of liver metastasis. The query and retrieved images in the top row are selected by an experienced physician, and the bottom row are the retrieved images from the CBIR system. The CBIR successfully retrieves the same images as the physician, but lacks the spatial coherence to order the retrieved images.}
\label{fig:case-study}
\end{figure*}
\section{Conclusion}
We propose a clinically motivated self-supervised framework for CBIR of CT liver images. Our proposed framework exploits the properties of the liver to learn more clinically relevant features, which results in show leads to improved performance. Moreover, we leverage the RELAX framework to provide the first representation learning explainability analysis in the context of CBIR of CT liver images. Our analysis provides new insights into the feature extraction process and shows how self-supervised learning can provide feature extractors that extract more clinically relevant features compared to feature extractors trained on non-CT liver images. Our experimental evaluation also shows how the proposed framework generalizes to new datasets, and we present a clinically relevant user study. In future works, we intend to investigate how the proposed approach can be extended to extract features specific to other organs based on clipping strategies catered specifically to the desired organ. We believe that the proposed framework can play an essential role in constructing reliable CBIR that can effectively utilize unlabeled data.
\section*{Declaration of competing interest}
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
influence the work reported in this paper.
\section*{Acknowledgments}
This work was supported by The Research Council of Norway
(RCN), through its Centre for Research-based Innovation funding
scheme [grant number 309439] and Consortium Partners; RCN
FRIPRO [grant number 315029]; RCN IKTPLUSS [grant number
303514]; and the UiT Thematic Initiative.
\bibliographystyle{model2-names.bst}\biboptions{authoryear}
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| 5 | 1 |
\section{Introduction}
Chiral perturbation theory(ChPT) is a useful method to solve low-energy mesons interactions.
A significant problem is to establish the chiral Lagrangian (CL).
Until now, the monomials of CL have been obtained to the $O(p^6)$ order \cite{weinberg,GS1,GS2,p61,p62,p6p,p6a1,p6a2,tensor1,U3,ourf},
including both the normal and anomalous parts, two and three flavours, and the special unitary and unitary groups.
The $O(p^6)$ order monomials seem accurate enough to describe the present experiments.
In constrast, the coefficients for each of the monomials, called the low-energy constants (LECs),
also play an important role of ChPT.
Concrete numerical results need them.
However ChPT does not provide these LECs, and hence they need to be obtained from other theories.
Because of non-perturbative effects, exact values for the LECs are hard to obtain, especially at the $O(p^6)$ order.
At the $O(p^4)$ order, although exact results have not been obtained, values from different methods are close.
Their signs and magnitudes are the same.
At the $O(p^6)$ order, there are various methods to obtain LECs, the more common ones being
resonance chiral theory \cite{tpf,res2}, sum rules \cite{sumrules}, lattice QCD \cite{lattr,latt}
holographic theory \cite{Colangelo:2012ipa}, QCD \cite{our5}, and global fit \cite{gfit1}.
Of course, LECs can also be extracted from experimental data.
Some methods can obtain a value for a single LEC; while others produce values for combinations of LECs.
Each method has its advantages and disadvantages, but some values do come with large errors because of non-perturbative QCD processes.
Several years ago, values of a few LECs obtained by different methods were scattered in various references.
A large number of LECs remained unknown in value. To taking advantage of ChPT, LECs are needed for numerical calculations.
Our motivation was to explore a method to obtain all the LECs in a single calculation.
We hoped the method would be model-independent, as had been developed from the underlying QCD theory.
Nevertheless at that time, the theoretical analysis was poor, and we only used simple methods that involved some rough approximations \cite{our5,oura2}.
Although expedient, preliminary results at least were obtained.
Given that the approximations are too rough, the results did yield a gauge-invariant, nonlocal, dynamical-quark (GND model) model \cite{GND}.
Ref. \cite{gfit1} details checks of our $O(p^6)$ LECs ($C_i$) via a global fit of the $O(p^4)$ LECs ($L_i$) to the $O(p^6)$ order.
The $\chi^2$ divided by the degrees of freedom ($\chi^2/$dof) is $4.13/4$.
If all LECs are multiplied by 0.27, $\chi^2/$dof=1.20/3 and hence the $C_i$ values appeared to be too large.
In statistical theory, when dof is large enough (a typical choice is dof$>$30), $\chi^2/$dof$\sim1$.
However at only 4, the dof is too small and it is not sufficiently large to assess the reliability of the calculation by $\chi^2/$dof.
Because of the $\chi^2$ distribution, $\chi^2/$dof is also not as small as it should be.
Furthermore the $O(p^4)$ and $O(p^6)$ LECs are independent.
Evaluating the confidence level of the $C_i$ by $L_i$ is not suitable, especially when dof is small.
A more believable assessment can be garnered by comparing experiment data.
However the absolute values of our $L_i$ are too large, and may include some systematic errors in the calculation,
such as those associated with the large $N_C$ limit.
These systematic errors must exist at all orders.
With the passing of time, it is now the moment improve the $O(p^6)$ LECs.
While tedious, these needs to be improved step by step.
This paper research analyses the origin of the systematic errors and tries to remedy them as precisely as possible.
This paper is organized as follows: In Sec. \ref{rev}, we review our method to obtain the CL from QCD
and introduce a more precise approximation. In Sec. \ref{cat}, a concrete method to calculate LECs is introduced.
In Sec. \ref{results}, we list our results for the LECs, both normal and anomalous, up to and including order $O(p^6)$.
In Sec. \ref{comp}, we compare our results with others in the literature, and check for new predictions.
Section \ref{summ} concludes with a summary.
\section{Review and Improvements over Previous Calculations of LECs}\label{rev}
In a previous work, we obtained the action of the chiral theory in the large $N_c$ limit.
Drived from first principles QCD, it takes the form \cite{WQ0,WQ1}
\begin{eqnarray}
S_{\rm eff}
&=&-iN_c{\rm Tr}\ln[i\partial\!\!\!/+J_{\Omega}-\Pi_{\Omega c}]
+N_c\int d^{4}xd^{4}x'
\Phi^{\sigma\rho}_{\Omega c}(x,x')\Pi^{\sigma\rho}_{\Omega c}(x,x')
+N_c\sum^{\infty}_{n=2}{\int}d^{4}x_1\cdots d^4x_{n}'\nonumber\\
&&\times
\frac{(-i)^{n}(N_c g_s^2)^{n-1}}{n!}\bar{G}^{\sigma_1\cdots\sigma_n}_{\rho_1
\cdots\rho_n}(x_1,x'_1,\cdots,x_n,x'_n)
\Phi^{\sigma_1\rho_1}_{\Omega c}(x_1 ,x'_1)\cdots
\Phi^{\sigma_n\rho_n}_{\Omega c}(x_n ,x'_n)\nonumber\\
&&+iN_c \int d^4x~{\rm tr}_{lf}\{\Xi_c(x)[-i\sin\frac{\vartheta_c(x)}{N_f}
+\gamma_5\cos\frac{\vartheta_c(x)}{N_f}]\Phi_{\Omega,c}^T(x,x)\},
\label{Snorm}
\end{eqnarray}
in which $J_{\Omega}$ is the external source $J$ including currents and densities
after making a Goldstone-field-dependent chiral rotation $\Omega$:
\begin{eqnarray}
J_{\Omega}=[\Omega P_R+\Omega^{\dag}P_L][J+i\slashed{\partial}][\Omega P_R+\Omega^\dag P_L]
=\slashed{v}_\Omega+\slashed{a}_\Omega\gamma_5-s_\Omega+ip_\Omega\gamma_5,\hspace{0.3cm}
J=\slashed{v}+\slashed{a}\gamma_5-s+ip\gamma_5,\hspace{0.3cm}U=\Omega^2\;.\label{JOmega}
\end{eqnarray}
where $\Phi_{\Omega c}$ and $\Pi_{\Omega c}$ are, respectively, the two-point rotated-quark
Green's functions and the interaction part of two-point rotated-quark
vertices in the presence of external sources; $\Phi_{\Omega c}$ is defined by
\begin{eqnarray}
&&\Phi_{\Omega c}^{\sigma\rho}(x,y)\equiv
\frac{1}{N_c}\langle\overline{\psi}_{\Omega}^{\sigma}(x)\psi_{\Omega}^{\rho}(y)\rangle
=-i[(i\slashed{\partial}+J_{\Omega}-\Pi_{\Omega c})^{-1}]^{\rho\sigma}(y,x),\hspace{0.3cm}
\psi_{\Omega}(x)\equiv[\Omega(x)P_L+\Omega^\dag(x)P_R]\psi(x),\label{PhiPi}\\
&&\Pi_{\Omega c}^{\sigma\rho}(x,y)=-\tilde{\Xi}^{\sigma\rho}(x)\delta^4(x-y)
-\sum^{\infty}_{n=1}{\int}d^{4}x_1\cdots{d^4}x_{n}
d^{4}x_{1}'\cdots{d^4}x_{n}'\frac{(-i)^{n+1}(N_c g_s^2)^n}{n!}\nonumber\\
&&\hspace{0.2cm}\times\overline{G}^{\sigma\sigma_1\cdots\sigma_n}_{\rho\rho_1
\cdots\rho_n}(x,y,x_1,x'_1,\cdots,x_n,x'_n)
\Phi_{\Omega c}^{\sigma_1\rho_1}(x_1 ,x'_1)\cdots
\Phi_{\Omega c}^{\sigma_n\rho_n}(x_n ,x'_n),\label{fineqNc}
\end{eqnarray}
with subscript $_c$ denoting the classical field and $\psi(x)$ the
light quark fields. $\bar{G}^{\sigma_1\cdots\sigma_n}_{\rho_1
\cdots\rho_n}(x_1,x'_1,\cdots,x_n,x'_n)$ is the effective gluon $n$-point
Green's function and $g_s$ is the coupling constant of QCD.
The $\Phi_{\Omega c}$ and $\Pi_{\Omega c}$ are related by the first equation of \eqref{PhiPi} and determined by
\begin{eqnarray}
&&[\Phi_{\Omega c}+\tilde{\Xi}]^{\sigma\rho}+\sum^{\infty}_{n=1}{\int}d^4x_1d^4x'_1\cdots{d^4}x_n
d^4x'_n\frac{(-i)^{n+1}(N_c g_s^2)^n}{n!}\overline{G}^{\sigma\sigma_1\cdots\sigma_n}_{\rho\rho_1
\cdots\rho_n}(x,y,x_1,x'_1,\cdots,x_n,x'_n)\nonumber\\
&&\times \Phi_{\Omega c}^{\sigma_1\rho_1}(x_1 ,x'_1)\cdots
\Phi_{\Omega c}^{\sigma_n\rho_n}(x_n,x'_n)=O(\frac{1}{N_c}),\label{SDE}
\end{eqnarray}
where $\vartheta_c$ is the phase angle of the $U(1)$ factor,
and $\Xi_c$ and $\tilde{\Xi}$ are two parameters in the calculation, defined by Eqs. (21) and (65) in Ref.\cite{WQ0}.
In this work, they have little importance and are neglected.
Eq. (\ref{SDE}) is the Schwinger-Dyson equation (SDE) in the presence of the rotated external source.
In Ref.\cite{WQ1}, we have assumed an approximate solution of (\ref{SDE}) given by
\begin{eqnarray}
\Pi^{\sigma\rho}_{\Omega c}(x,y)=[\Sigma(\overline{\nabla}^2_x)]^{\sigma\rho}\delta^4(x-y),\hspace{3cm}
\overline{\nabla}^{\mu}_x=\partial^{\mu}_x-iv_{\Omega}^{\mu}(x)\;,\label{defsig}
\end{eqnarray}
where $\Sigma$ is the quark self-energy which satisfies SDE (\ref{SDE}) with vanishing rotated external source.
Under the ladder approximation, this SDE in Euclidean space-time is reduced to the standard form
\begin{eqnarray}
\Sigma(p^2)-3C_2(R)\int\frac{d^4q}{4\pi^3}\frac{\alpha_s[(p-q)^2]}{(p-q)^2}
\frac{\Sigma(q^2)}{q^2+\Sigma^2(q^2)}=0\;, \label{eq0}
\end{eqnarray}
where $\alpha_s(p^2)$ is the running coupling constant of QCD which depends on $N_C$ and the number of quark flavours,
and $C_2(R)$ is the second-order Casimir operator of the quark representation $R$.
In this work, the quarks belonging to the $SU(N_C)$ fundamental representation,
and therefore $C_2(R)=(N_c^2-1)/2N_c$; in the large $N_C$ limit, the second term is neglected.
In our previous article, because of the computational complexity, we did not calculate all terms in Eq. \eqref{Snorm},
but introduced some approximations to reduced the last three terms and leave only the first.
The approximations are rough, and the results reproduce those of the GND model \cite{GND}.
The last three terms could have introduced some systematic errors.
The absolute values of the previous results seem too large when compared with others in the literature.
The purpose of this paper is to remedy these errors partially
and produce a more credible result.
For now, we have no idea how to solve the last term in \eqref{Snorm} and hence omit it,;
we shall only focus on the second and third terms.
The infinite sum in both the third terms in \eqref{Snorm} and the second term in \eqref{fineqNc}
appear similar but they are not equal.
To find the source of the systematic errors and to maintain a manageable calculation
we only retain the two-point contributions,
the terms with $n=2$ in \eqref{Snorm} and $n=1$ in \eqref{fineqNc}.
When we simplify SDE from \eqref{SDE} to \eqref{eq0}, we also include $n=1$ in \eqref{SDE}.
The present approximation and the approximation for SDE are all to first order,
omitting all higher order contributions; they are then at the same level of accuracy.
The other terms describe mesonics interactions, which we consider as less important when compared previous terms.
The final results will substantiate this decision. Hence we shall add an additional effective action,
\begin{eqnarray}
\Delta S_{\rm eff}&\sim&\frac{1}{2}N_c\int d^{4}xd^{4}x'
\Phi^{\sigma\rho}_{\Omega c}(x,x')\Pi^{\sigma\rho}_{\Omega c}(x,x')\label{alag0}\\
&\sim&-\frac{i}{2}N_c\mathrm{Tr}
[(i\slashed{\partial}+J_{\Omega}
-\Sigma(\overline{\nabla}^2))^{-1}\Sigma(\overline{\nabla}^2)].\label{alag}
\end{eqnarray}
the approximation of which has been used in \eqref{defsig}.
With the same considerations for the anomalous parts, we only calculated the first term in \eqref{Snorm},
although using a different method to introduce the fifth dimensional integral \cite{oura2}.
We need not repeat this as the additional effective Lagrangian is the same as \eqref{alag}.
Therefore, for the present study, we calculate \eqref{alag} including both the normal and anomalous parts.
\section{Calculation of the Additional Terms}\label{cat}
To calculate the additional terms, we first use the Wick rotation to change \eqref{alag} to Euclidean spacetime as \cite{our5,oura2},
and then expand it as a Taylor series,
\begin{eqnarray}
\Delta S_{\rm eff}
&=&-\frac{1}{2}N_c\mathrm{Tr}
[(D+\Sigma(-\overline{\nabla}^2))^{-1}\Sigma(-\overline{\nabla}^2)]\\
&=&-\frac{1}{2}N_c\mathrm{tr}
[(-i\slashed{k}+D+\Sigma((k+i\overline{\nabla})^2))^{-1}\Sigma((k+i\overline{\nabla})^2)]\\
&=&-\frac{1}{2}N_c\int d^4x\int\frac{d^4k}{(2\pi)^4}\mathrm{tr}[(\Sigma_k X+i\slashed{k}X)\sum_{n=0}^{\infty}(-1)^n[(D+\Sigma_{1})(\Sigma_k X+i\slashed{k}X)]^n
\Sigma((k+i\overline{\nabla})^2)],\label{dels}
\end{eqnarray}
where ``Tr" includes the traces over coordinate space, spinor space and flavour space,
``tr" includes only the traces over spinor and flavour spacec,
$D\equiv\overline{\slashed{\nabla}}-i\slashed{a}_\Omega\gamma_5-s_\Omega+ip_\Omega\gamma_5$,
and $\Sigma_1\equiv\Sigma((k+i\overline{\nabla})^2)-\Sigma(k^2)$.
To a given order ($O(p^2)$, $O(p^4)$ or $O(p^6)$), after the tracing over the spinor space, the series are
\begin{eqnarray}
\Delta S_{\rm eff}^{2n}=\int d^4x\sum_{k=1}^m a_k \langle O_{\Omega,k}\rangle \label{soomega}
\end{eqnarray}
where $a_k$ are coefficients, $O_{\Omega,k}$ are monomials with flavour indices including
$\overline{\nabla}^\mu,a_\Omega^\mu,s_\Omega$ $p_\Omega$, and ``$\langle\ldots\rangle$" means trace over flavour space.
In \eqref{soomega}, we have used the basic relations to simplify the results,
including trace relations, the Einstein summation convention, and for the anomalous terms also including the Schouten identity.
Nevertheless, not all of the $O_{\Omega,k}$ are independent.
Generally, the number of $O_{\Omega,k}$ is larger than the number of linear independent terms $O_l$.
Specifically, the relationship between the two is given by
\begin{eqnarray}
\langle O_l\rangle=\sum_{k=1}^m A_{lk}\langle O_{\Omega,k}\rangle,\hspace{0.2cm}l=1,2,3,\cdots,M\label{lrs0}
\end{eqnarray}
where $M$ is the number of LECs up to a given order.\footnote{The building blocks of $O_l$ and $O_{\Omega}$ are different.
Their relations can be found in Table XV in Ref.\cite{our5}}
This implies that generally $A_{lk}$ is not a square matrix.
The reduced row echelon form of $A_{lk}$ is
\begin{eqnarray}
A_{lk}\to B_{lk}=
\left(\begin{array}{cccccccccc}
1 & C_{12} & O & C_{14} & O & \cdots & \cdots & \cdots \\
& & 1 & C_{24} & O & \cdots & \cdots & \cdots \\
& & & \cdots & \cdots & \cdots & \cdots & \cdots \\
& & & & 1 & \cdots & \cdots & \cdots
\end{array}\right),\label{lrs1}
\end{eqnarray}
where the bottom-left corner contains just zero elements wiht $O$ representing a zero matrix of the appropriate dimension
$C$ representing possibly a non-zero matrix and ``$\cdots$" also nonzero matrices.
The rank of $A_{lk}$ or $B_{lk}$ is equal to the number of independent linear bases
with each nonzero row-vector in $B_{lk}$ corresponding to a linear basis in $\langle O_{\Omega}\rangle$.
We select those $O_{\Omega,k'}$ that are independent,
and set $B_{k'k'}=1$ and $B_{k'k'}$ to be the first non-zero elements in the $k'$th row in $B_{lk}$.
All dependent $O_{\Omega,k}$ can be replaced by $O_{\Omega,k'}$.
Without using the Cayley-Hamilton relations, the LECs $K_l$ for arbitrary $N_f$ flavours are defined as
\begin{eqnarray}
S_{\rm eff}^{2n}=\int d^4x\sum_{l=1}^M K_l \langle O_l\rangle=\int d^4x\sum_{l=1}^M\sum_{k=1}^m K_l A_{lk}\langle O_{\Omega,k}\rangle.\label{dlecs}
\end{eqnarray}
In the second equation, we have has used \eqref{lrs0}.
Comparing \eqref{soomega} and \eqref{dlecs}, because all the relations in $N_f$ flavours have been used
the coefficients in front of $\langle O_{\Omega,k}\rangle$ need to be equal.
\begin{eqnarray}
a_k=\sum_{l=1}^M K_l A_{lk},\hspace{0.3cm}k=1,2,3,\ldots m.\label{rkk}
\end{eqnarray}
In \eqref{rkk}, there are $m$ equations in $M$ variables with $m>M$.
We select the $M$ independent $O_{\Omega,k'}$ to solve the equations,
leaving the remaining $m-M$ equations as constraints. Hence the additional LECs are
\begin{eqnarray}
\Delta K_{l}=\sum_{k'} a_{k'}A^{-1}_{k'l}.
\end{eqnarray}
Replacing $\Delta K_{l}$ in \eqref{rkk}, the second equal sign must hold.
These the are the constraint equations.
Finally, using the Cayley-Hamilton relations, all the LECs can be obtained for three and two flavours.
\section{Results}\label{results}
\subsection{Order $O(p^2)$ and $O(p^4)$}
To the $O(p^2)$ order, the additional analytical LECs are
\begin{eqnarray}
\Delta F_0^2&=&\int\frac{d^4k}{(2\pi)^4}[
4 \Sigma_k^2 X^2 -6 \Sigma_k^4 X^3],\notag\\
\Delta F_0^2B_0&=&\int\frac{d^4k}{(2\pi)^4}[2 \Sigma_k X
-4 \Sigma_k^3 X^2 ],\notag\\
X&\equiv&\frac{1}{k^2+\Sigma_k^2},\hspace{0.3cm}\Sigma_k\equiv\Sigma(k^2).\label{ap2}
\end{eqnarray}
The analytical results in \eqref{ap2} and the higher order in the following are all set in Euclidean spacetime.
A more precise result, with all terms on the right hand side of \eqref{Snorm}, has been given in \cite{WQ0}.
This is just the Pagels-Stokar formula for $F_0^2$
and also the $\Lambda\to\infty$ results in our previous work \cite{our5}.
It seems that only the first term in \eqref{Snorm} is sufficient to give the $O(p^2)$ results,
the other terms do not contribute at the $O(p^2)$ order.
In considering the accuracy, we use the original results in \cite{our5} at the $O(p^2)$ order.
The method described in \cite{WQ0}, however, is not easily extendible to higher order,
and there is no proof that indicates that the other terms in \eqref{Snorm} make no contribution to the LECs.
Hence at higher order, we consider \eqref{dels}.
At the $O(p^4)$ order, the additional analytical results are
\begin{eqnarray}
\Delta C_{2}&=&\int\frac{d^4k}{(2\pi)^4}\bigg[- \Sigma_k'^2 X
+11 \Sigma_k^2\Sigma_k'^2 X^2
-34 \Sigma_k^4\Sigma_k'^2 X^3
-2 \Sigma_k^4 X^4
+40 \Sigma_k^6\Sigma_k'^2 X^4
+4 \Sigma_k^6 X^5
-16 \Sigma_k^8\Sigma_k'^2 X^5
\bigg],\notag\\
\Delta C_{3}&=&\int\frac{d^4k}{(2\pi)^4}\bigg[-\frac{1}{2} \Sigma_k'^2 X
+\frac{49}{6} \Sigma_k^2\Sigma_k'^2 X^2
-\frac{2}{3} \Sigma_k^2 X^3
-\frac{83}{3} \Sigma_k^4\Sigma_k'^2 X^3
-\frac{5}{3} \Sigma_k^4 X^4
+\frac{100}{3} \Sigma_k^6\Sigma_k'^2 X^4\notag\\
&&+\frac{10}{3} \Sigma_k^6 X^5
-\frac{40}{3} \Sigma_k^8\Sigma_k'^2 X^5\bigg],\notag\\
\Delta C_{4}&=&\int\frac{d^4k}{(2\pi)^4}\bigg[-\frac{26}{3} \Sigma_k^2 X^3
+\frac{118}{3} \Sigma_k^4 X^4
-\frac{104}{3} \Sigma_k^6 X^5
\bigg],\notag\\
\Delta C_{5}&=&\int\frac{d^4k}{(2\pi)^4}\bigg[\frac{14}{3} \Sigma_k^2 X^3
-\frac{52}{3} \Sigma_k^4 X^4
+\frac{44}{3} \Sigma_k^6 X^5
\bigg],\notag\\
\Delta C_{6}&=&\int\frac{d^4k}{(2\pi)^4}\bigg[6 \Sigma_k^2 X^2
-8 \Sigma_k^4 X^3
\bigg],\notag\\
\Delta C_{7}&=&\int\frac{d^4k}{(2\pi)^4}\bigg[2 \Sigma_k^2 X^2
\bigg],\notag\\
\Delta C_{8}&=&\int\frac{d^4k}{(2\pi)^4}\bigg[-4 \Sigma_k X^2
+34 \Sigma_k^3 X^3
-36 \Sigma_k^5 X^4
\bigg],\notag\\
\Delta C_{9}&=&\int\frac{d^4k}{(2\pi)^4}\bigg[ \Sigma_k'^2 X
-\frac{2}{3} \Sigma_k\Sigma_k' X^2
-3 \Sigma_k^2\Sigma_k'^2 X^2
-\frac{2}{3} \Sigma_k^2 X^3
+\frac{2}{3} \Sigma_k^3\Sigma_k' X^3
+2 \Sigma_k^4\Sigma_k'^2 X^3
\bigg],\notag\\
\Delta C_{10}&=&\int\frac{d^4k}{(2\pi)^4}\bigg[-2i \Sigma_k'^2 X
+\frac{82i}{3} \Sigma_k^2\Sigma_k'^2 X^2
+\frac{20i}{3} \Sigma_k^2 X^3
-\frac{268i}{3} \Sigma_k^4\Sigma_k'^2 X^3
-\frac{52i}{3} \Sigma_k^4 X^4
+\frac{320i}{3} \Sigma_k^6\Sigma_k'^2 X^4\notag\\
&&+\frac{32i}{3} \Sigma_k^6 X^5
-\frac{128i}{3} \Sigma_k^8\Sigma_k'^2 X^5\bigg],\notag\\
\Delta C_{11}&=&\int\frac{d^4k}{(2\pi)^4}\bigg[-2 \Sigma_k X^2
+6 \Sigma_k^3 X^3
\bigg].
\end{eqnarray}
The definitions of $C_i$ can be found in Eq. (19) of \cite{GND},
and the relations between $C_i$ and common $L_i$ can be found in Eq. (24) in \cite{GND}.
To obtain numerical results, we use the same quark self-energy $\Sigma_k$ as in \cite{WQ1,our5}
with the running coupling constant $\alpha_s(p^2)$ of \cite{alphas}.
To complete the integral, two other input parameters are needed.
One is $F_0$, for which we choose $F_0=87$MeV; the other is a cutoff $\Lambda$
which comes from the calculation of the first term in \eqref{Snorm}.
The details can be found in \cite{our5} and therefore we have chosen $\Lambda=$1.0$^{+0.1}_{-0.1}$GeV.
The three-flavour numerical results are listed in the second row in Table \ref{tp4r}.
Those LECs that depend on $\Lambda$ are expressible as
\begin{eqnarray}
L_{\Lambda=1\mathrm{GeV}}\bigg|^{L_{\Lambda=1.1\mathrm{GeV}}-L_{\Lambda=1\mathrm{GeV}}}_{L_{\Lambda=0.9\mathrm{GeV}}-L_{\Lambda=1\mathrm{GeV}}}.
\end{eqnarray}
The superscript and subscript indicate how the LECs are sensitive to $\Lambda$.
For two flavours, we give the usual $\bar{l}_i$, $i=1,2,3,4,5,6$,
\begin{eqnarray}
l_i=\frac{1}{32\pi^2}\gamma_i(\bar{l}_i\!+\!\ln\frac{M^2_\pi}{\mu^2}),
\end{eqnarray}
where the $\gamma_i$ are given in Ref.\cite{GS1}. These results are also listed in Table \ref{tp4r}.
\begin{table*}[!h]
\extrarowheight 5pt
\caption{The $p^4$ order LECs. $\Lambda_{\text{QCD}}$ is in unit of GeV, and $L_1,\ldots,L_{10},l_7$ are in units $10^{-3}$.}\label{tp4r}
\resizebox{\textwidth}{!}{\begin{tabular}{cccccccccccc}
\hline\hline $N_f=3$ & $\Lambda_\mathrm{QCD}$ & $L_1$
& $L_2 $ & $L_3$ & $L_4$ & $L_5$ & $L_6$ & $L_7$ & $L_8$ & $L_9$ & $L_{10}$\\
\hline New & 453$^{-6}_{+12}$ & $0.92^{+0.03}_{-0.04}$ & $1.84^{+0.05}_{-0.08}$ & $-4.94^{-0.14}_{+0.21}$ & $0^{+0}_{-0}$ & $1.26^{+0.01}_{-0.06}$ & $0^{+0}_{-0}$ & $-0.42^{+0.04}_{-0.05}$ & $0.84^{-0.05}_{+0.04}$ & $6.53^{+0.24}_{-0.37}$ & $-5.43^{-0.29}_{+0.44}$ \notag\\
Old \cite{our5}&453$^{-6}_{+12}$& 1.23$^{+0.03}_{-0.04}$ &
2.46$^{+0.05}_{-0.08}$ & -6.85$^{-0.15}_{+0.21}$ & 0$^{+0}_{-0}$ &
1.48$^{-0.01}_{-0.03}$ & 0$^{+0}_{-0}$ & -0.51$^{+0.05}_{-0.06}$ &
1.02$^{-0.06}_{+0.06}$&8.86$^{+0.24}_{-0.37}$ &-7.40$^{-0.29}_{+0.44}$\\
Ref.\cite{GS2}:& &$0.9\pm 0.3$&$1.7\pm 0.7$&-$4.4\pm 2.5$&$0\pm
0.5$&$2.2\pm 0.5$&$0\pm 0.3$&-$0.4\pm 0.15$&$1.1\pm 0.3$&$7.4\pm
0.7$&-$6.0\pm 0.7$\\
Ref.\cite{Pich:2008xj}:& &$0.4\pm 0.3$&$1.4\pm 0.3$&-$3.5\pm 1.1$&$-0.3\pm0.5$
&$1.4\pm 0.5$&$-0.2\pm 0.3$&-$0.4\pm 0.2$&$0.9\pm 0.3$&$6.9\pm
0.7$&-$5.5\pm 0.7$\\
Ref.\cite{Colangelo:2012ipa}:& &$0.5$&$1.0$&-$3.2$&
& & & & &$6.3$&-$6.3$\\
Ref.\cite{gfit1}& &$0.88\pm 0.09$&$0.61\pm 0.20$&-$3.04\pm 0.43$&$0.75\pm
0.75$&$0.58\pm 0.13$&$0.29\pm 0.85$&-$0.11\pm 0.15$&$0.18\pm 0.18$&$5.93\pm
4.3$&-$4.06\pm 0.39$\\
\hline $N_f=2$&$\Lambda_\mathrm{QCD}$&$\bar{l}_1$
&$\bar{l}_2$ &$\bar{l}_3$ &$\bar{l}_4$ & $\bar{l}_5$ & $\bar{l}_6$ & $l_7$ & & &\\
\hline New & 465$^{-6}_{+12}$ & $-2.33^{-0.17}_{+0.24}$ & $6.85^{+0.09}_{-0.14}$ & $2.33^{+0.28}_{-0.36}$ & $4.24^{+0.00}_{-0.04}$ & $13.55^{+0.53}_{-0.80}$ & $15.60^{+0.44}_{-0.67}$ & $3.61^{-0.55}_{+0.80}$ & & &\notag\\
Old \cite{our5} &465$^{-6}_{+12}$&
-4.77$^{-0.17}_{+0.24}$ & 8.01$^{+0.09}_{-0.14}$ & 1.97$^{+0.29}_{-0.35}$
& 4.34$^{-0.01}_{-0.02}$ & 17.35$^{+0.53}_{-0.80}$ &
19.98$^{+0.44}_{-0.67}$ &$4.18^{-0.65}_{+0.97}$
\footnote{There exists a mistake in \cite{our5} for $l_7$, this is a correction.} &&&\\
Ref.\cite{GS1}:& &$-2.3\pm 3.7$&$6.0\pm 1.3$&$2.9\pm 2.4$&$4.3\pm 0.9$&$13.9\pm 1.3$&$16.5\pm 1.1$& $O(5)$\\
\hline\hline
\end{tabular}}
\end{table*}
Comparing the ``new" and ``old" results in Table \ref{tp4r},
the new absolute values of $L_i(\bar{l}_i)$ are, as expected, smaller than the older ones, except for $\bar{l}_3$.
For comparison, we also list the other results obtained by different methods:
\cite{GS1,GS2} are the first results from experimental data;
\cite{Pich:2008xj} gives the LECs from resonance chiral theory;
\cite{Colangelo:2012ipa} gives the LECs from a class of holographic theories;
\cite{gfit1} gives the $L_1,\ldots,L_8$ from the global fit of the $O(p^6)$ LECs,
and $L_{9}$ and $L_{10}$ are given in \cite{pkef,C87-3}, respectively.
These are the usual methods to obtain LECs at present.
On the whole, most of the old results are larger than the others, but the new results are closer.
The only one new result larger than the old one is $\bar{l}_3$, but is also much closer than the older one.
Our new results are much closer to the experimental results,
and most results are within the error uncertainties of the resonance results.
Nevertheless they appear a bit far from the results from the global fit.
One possible reason is that the global-fitted results do not maintain the large $N_C$ limit,
but $L_4$ and $L_6$ have fits that are not very small, the effect of which propagate throughout the calculation and decrease the values of other LECs.
So far, our calculation remains valid only in the large $N_C$ limit, and therefore they are not very closed to the global fit results.
These observations indicate that our approximation in \eqref{alag0} is reasonable.
Although we only selected $n=2$ in \eqref{Snorm} and $n=1$ in \eqref{fineqNc}, the tendency is clear.
The second and the third terms in \eqref{Snorm} carry the systematic error in our original calculations.
To date Table \ref{tp4r} gives the leading order corrections.
Hence, we believe that when we extend the calculations to the $O(p^6)$ order, the results will be more credible.
\begin{table*}[!h]
\extrarowheight 4pt
\caption{$\tilde{Y}_n$ and their relations to $Y_i$.}\label{np6}
\begin{eqnarray}
\begin{array}{llllllll}
\hline\hline n & \tilde{Y}_n & \text{relations} & n & \tilde{Y}_i & \text{relations}\\
\hline
1 & \langle u^{\mu}u_{\mu}h^{\nu\lambda}h_{\nu\lambda} \rangle & Y_{1} & 35 & \langle i f_{+}^{\mu\nu}u^{\lambda}u_{\mu}u_{\lambda}u_{\nu} +i f_{+}^{\mu\nu}u_{\mu}u^{\lambda}u_{\nu}u_{\lambda} \rangle & Y_{68}\\
2 & \langle h^{\mu\nu}u^{\lambda}h_{\mu\nu}u_{\lambda} \rangle & Y_{3} & 36 & \langle u^{\mu}u_{\mu}f_{+}^{\nu\lambda}f_{+\nu\lambda} \rangle & Y_{71}\\
3 & \langle h^{\mu\nu}u^{\lambda}h_{\mu\lambda}u_{\nu} + h^{\mu\nu}u_{\nu}{h_{\mu}}^{\lambda}u_{\lambda} \rangle & Y_{5} & 37 & \langle f_{+}^{\mu\nu}u^{\lambda}f_{+\mu\nu}u_{\lambda} \rangle\\
4 & \langle u^{\mu}u_{\mu}u^{\nu}u_{\nu}\chi_{+} \rangle & Y_{7} & 38 & \langle f_{+}^{\mu\nu}{f_{+\mu}}^{\lambda}u_{\nu}u_{\lambda} \rangle & Y_{75}\\
5 & \langle u^{\mu}u_{\mu}u^{\nu}\chi_{+}u_{\nu} \rangle & Y_{11} & 39 & \langle f_{+}^{\mu\nu}{f_{+\mu}}^{\lambda}u_{\lambda}u_{\nu} \rangle & Y_{76}\\
6 & \langle \chi_{+}u^{\mu}u^{\nu}u_{\mu}u_{\nu} \rangle & Y_{13} & 40 & \langle f_{+}^{\mu\nu}u^{\lambda}f_{+\mu\lambda}u_{\nu} + f_{+}^{\mu\nu}u_{\nu}{f_{+\mu}}^{\lambda}u_{\lambda} \rangle & Y_{78}\\
7 & \langle \chi_{+}h^{\mu\nu}h_{\mu\nu} \rangle & Y_{17} & 41 & \langle \chi_{+}f_{+}^{\mu\nu}f_{+\mu\nu} \rangle & Y_{81}\\
8 & \langle u^{\mu}u_{\mu}\chi_{+}\chi_{+} \rangle & Y_{19} & 42 & \langle i f_{+}^{\mu\nu}\chi_{+}u_{\mu}u_{\nu} +i f_{+}^{\mu\nu}u_{\mu}u_{\nu}\chi_{+} \rangle & Y_{83}\\
9 & \langle \chi_{+}u^{\mu}\chi_{+}u_{\mu} \rangle & Y_{23} & 43 & \langle i f_{+}^{\mu\nu}u_{\mu}\chi_{+}u_{\nu} \rangle & Y_{85}\\
10 & \langle \chi_{+}\chi_{+}\chi_{+} \rangle & Y_{25} & 44 & \langle f_{-}^{\mu\nu}{h_{\nu}}^{\lambda}u_{\lambda}u_{\mu} + f_{-}^{\mu\nu}u_{\mu}u^{\lambda}h_{\nu\lambda} \rangle & Y_{86}\\
11 & \langle i \chi_{-}h^{\mu\nu}u_{\mu}u_{\nu} +i \chi_{-}u^{\mu}u^{\nu}h_{\mu\nu} \rangle & Y_{28} & 45 & \langle f_{-}^{\mu\nu}u_{\mu}{h_{\nu}}^{\lambda}u_{\lambda} + f_{-}^{\mu\nu}u^{\lambda}h_{\nu\lambda}u_{\mu} \rangle & Y_{89}\\
12 & \langle {h^{\lambda}}_{\lambda}h^{\mu\nu}u_{\mu}u_{\nu} + {h^{\lambda}}_{\lambda}u^{\mu}u^{\nu}h_{\mu\nu} \rangle & Y_{28}-\frac{2}{N_f}Y_{30} & 46 & \langle u^{\mu}u_{\mu}f_{-}^{\nu\lambda}f_{-\nu\lambda} \rangle & Y_{90}\\
13 & \langle i h^{\mu\nu}u_{\mu}\chi_{-}u_{\nu} \rangle & Y_{31} & 47 & \langle f_{-}^{\mu\nu}u^{\lambda}f_{-\mu\nu}u_{\lambda} \rangle & Y_{92}\\
14 & \langle h^{\mu\nu}u_{\mu}{h^{\lambda}}_{\lambda}u_{\nu} \rangle & Y_{31}-\frac{1}{N_f}Y_{30} & 48 & \langle f_{-}^{\mu\nu}{f_{-\mu}}^{\lambda}u_{\nu}u_{\lambda} \rangle & Y_{94}\\
15 & \langle u^{\mu}u_{\mu}\chi_{-}\chi_{-} \rangle & Y_{33} & 49 & \langle f_{-}^{\mu\nu}{f_{-\mu}}^{\lambda}u_{\lambda}u_{\nu} \rangle & Y_{95}\\
16 & \langle i u^{\mu}u_{\mu}{h^{\nu}}_{\nu}\chi_{-} +i u^{\mu}u_{\mu}\chi_{-}{h^{\nu}}_{\nu} \rangle & -2Y_{33}+\frac{2}{N_f}Y_{34} & 50 & \langle f_{-}^{\mu\nu}u^{\lambda}f_{-\mu\lambda}u_{\nu} + f_{-}^{\mu\nu}u_{\nu}{f_{-\mu}}^{\lambda}u_{\lambda} \rangle & Y_{97}\\
17 & \langle u^{\mu}u_{\mu}{h^{\nu}}_{\nu}{h^{\lambda}}_{\lambda} \rangle & -Y_{33}+\frac{2}{N_f}Y_{34}-\frac{1}{N_f^2}Y_{36} & 51 & \langle i f_{+}^{\mu\nu}{f_{-\nu}}^{\lambda}h_{\mu\lambda} -i f_{+}^{\mu\nu}{h_{\mu}}^{\lambda}f_{-\nu\lambda} \rangle & Y_{100}\\
18 & \langle u^{\mu}\chi_{-}u_{\mu}\chi_{-} \rangle & Y_{37} & 52 & \langle i f_{+}^{\mu\nu}{f_{-\nu}}^{\lambda}f_{-\mu\lambda} -i f_{+}^{\mu\nu}{f_{-\mu}}^{\lambda}f_{-\nu\lambda} \rangle & Y_{101}\\
19 & \langle i u^{\mu}{h^{\nu}}_{\nu}u_{\mu}\chi_{-} \rangle & -Y_{37}+\frac{1}{N_f}Y_{34} & 53 & \langle \chi_{+}f_{-}^{\mu\nu}f_{-\mu\nu} \rangle & Y_{102}\\
20 & \langle u^{\mu}{h^{\nu}}_{\nu}u_{\mu}{h^{\lambda}}_{\lambda} \rangle & -Y_{37}+\frac{2}{N_f}Y_{34}-\frac{1}{N_f^2}Y_{36} & 54 & \langle f_{+}^{\mu\nu}f_{-\mu\nu}\chi_{-} - f_{+}^{\mu\nu}\chi_{-}f_{-\mu\nu} \rangle & Y_{104}\\
21 & \langle \chi_{-}\chi_{-}\chi_{+} \rangle & Y_{39} & 55 & \langle i f_{+}^{\mu\nu}f_{-\mu\nu}{h^{\lambda}}_{\lambda} -i f_{+}^{\mu\nu}{h^{\lambda}}_{\lambda}f_{-\mu\nu} \rangle & -Y_{104}\\
22 & \langle i {h^{\mu}}_{\mu}\chi_{-}\chi_{+} +i {h^{\mu}}_{\mu}\chi_{+}\chi_{-} \rangle & -2Y_{39}+\frac{2}{N_f}Y_{41} & 56 & \langle i f_{-}^{\mu\nu}\chi_{-}u_{\mu}u_{\nu} -i f_{-}^{\mu\nu}u_{\mu}u_{\nu}\chi_{-} \rangle & Y_{105}\\
23 & \langle {h^{\mu}}_{\mu}{h^{\nu}}_{\nu}\chi_{+} \rangle & -Y_{39}+\frac{2}{N_f}Y_{41}-\frac{1}{N_f^2}Y_{42} & 57 & \langle f_{-}^{\mu\nu}{h^{\lambda}}_{\lambda}u_{\mu}u_{\nu} - f_{-}^{\mu\nu}u_{\mu}u_{\nu}{h^{\lambda}}_{\lambda} \rangle & Y_{105}\\
24 & \langle i \chi_{-}\chi_{+}^{\mu}u_{\mu} +i \chi_{-}u^{\mu}\chi_{+\mu} \rangle & Y_{43} & 58 & \langle f_{-}^{\mu\nu}\chi_{+\mu}u_{\nu} + f_{-}^{\mu\nu}u_{\nu}\chi_{+\mu} \rangle & Y_{107}\\
25 & \langle {h^{\nu}}_{\nu}\chi_{+}^{\mu}u_{\mu} + {h^{\nu}}_{\nu}u^{\mu}\chi_{+\mu} \rangle & Y_{43}-\frac{2}{N_f}Y_{44} & 59 & \langle \nabla^{\mu}f_{-}^{\nu\lambda}\nabla_{\mu}f_{-\nu\lambda} \rangle & Y_{109}\\
26 & \langle \chi_{+}^{\mu}\chi_{+\mu} \rangle & Y_{47} & 60 & \langle i \nabla^{\lambda}f_{+}^{\mu\nu}h_{\mu\lambda}u_{\nu} -i \nabla^{\lambda}f_{+}^{\mu\nu}u_{\nu}h_{\mu\lambda} \rangle & Y_{110}\\
27 & \langle u^{\mu}u_{\mu}u^{\nu}u_{\nu}u^{\lambda}u_{\lambda} \rangle & Y_{49} & 61 & \langle i \nabla^{\mu}{f_{+\mu}}^{\nu}{f_{-\nu}}^{\lambda}u_{\lambda} -i \nabla^{\mu}{f_{+\mu}}^{\nu}u^{\lambda}f_{-\nu\lambda} \rangle & Y_{111}\\
28 & \langle u^{\mu}u_{\mu}u^{\nu}u^{\lambda}u_{\lambda}u_{\nu} \rangle & Y_{52} & 62 & \langle i \nabla^{\mu}{f_{+\mu}}^{\nu}{h_{\nu}}^{\lambda}u_{\lambda} -i \nabla^{\mu}{f_{+\mu}}^{\nu}u^{\lambda}h_{\nu\lambda} \rangle & Y_{112}\\
29 & \langle u^{\mu}u_{\mu}u^{\nu}u^{\lambda}u_{\nu}u_{\lambda} \rangle & Y_{54} & 63 & \langle i \chi_{-}^{\mu}\nabla_{\mu}{h^{\nu}}_{\nu} \rangle & Z_{1}
\footnote{$Z_1=-\frac{1}{2}Y_{19}-\frac{1}{2}Y_{23}+\frac{1}{N_f}Y_{24}-\frac{1}{2}Y_{33}+\frac{1}{N_f}Y_{34}
-\frac{1}{2}Y_{37}-\frac{1}{2}Y_{39}
+\frac{1}{N_f}Y_{41}-\frac{1}{2N_f^2}Y_{42}+\frac{1}{2}Y_{43}-\frac{1}{N_f}Y_{44}+\frac{1}{N_f}Y_{46}-Y_{47}+4Y_{113}$}\\
30 & \langle u^{\mu}u^{\nu}u^{\lambda}u_{\mu}u_{\nu}u_{\lambda} \rangle & Y_{58} & 64 & \langle \nabla^{\mu}{h^{\nu}}_{\nu}\nabla_{\mu}{h^{\lambda}}_{\lambda} \rangle & Z_{2}
\footnote{$Z_2=-\frac{1}{2}Y_{19}-\frac{1}{2}Y_{23}+\frac{1}{N_f}Y_{24}-Y_{33}+\frac{2}{N_f}Y_{34}-Y_{37}-Y_{39}
+\frac{2}{N_f}Y_{41}-\frac{1}{N_f^2}Y_{42}+Y_{43}-\frac{2}{N_f}Y_{44}+\frac{1}{N_f}Y_{46}-Y_{47}+4Y_{113}$}\\
31 & \langle u^{\mu}u^{\nu}u^{\lambda}u_{\mu}u_{\lambda}u_{\nu} \rangle & Y_{60} & 65 & \langle f_{+}^{\mu\nu}u_{\mu}\chi_{-\nu} + f_{+}^{\mu\nu}\chi_{-\mu}u_{\nu} \rangle & Z_{3}
\footnote{$Z_3=-Y_{66}-Y_{67}+Y_{68}+\frac{1}{2}Y_{71}-\frac{1}{2}Y_{73}+Y_{75}-2Y_{76}+\frac{1}{2}Y_{78}
-\frac{1}{2}Y_{83}-Y_{85}
+\frac{1}{2}Y_{90}-\frac{1}{2}Y_{92}
-Y_{94}+\frac{1}{2}Y_{97}-\frac{1}{2}Y_{100}+\frac{1}{2}Y_{101}-\frac{1}{4}Y_{104}+Y_{110}+Y_{112}$}\\
32 & \langle i f_{+}^{\mu\nu}u^{\lambda}u_{\lambda}u_{\mu}u_{\nu} +i f_{+}^{\mu\nu}u_{\mu}u_{\nu}u^{\lambda}u_{\lambda} \rangle & Y_{64} & 66 & \langle \chi_{-}^{\mu}\chi_{-\mu} \rangle & (2.15)\text{ in \cite{p62}}\\
33 & \langle i f_{+}^{\mu\nu}u^{\lambda}u_{\mu}u_{\nu}u_{\lambda} \rangle & Y_{66} & 67 & \langle i f_{+}^{\mu\nu}{f_{+\nu}}^{\lambda}f_{+\mu\lambda} \rangle & (2.15)\text{ in \cite{p62}}\\
34 & \langle i f_{+}^{\mu\nu}u_{\mu}u^{\lambda}u_{\lambda}u_{\nu} \rangle & Y_{67} & 68 & \langle \nabla^{\mu}f_{+}^{\nu\lambda}\nabla_{\mu}f_{+\nu\lambda} \rangle & (2.15)\text{ in \cite{p62}}\\
\hline\hline
\end{array}\notag
\end{eqnarray}
\end{table*}
\subsection{Order $O(p^6)$}
Because our method only applies in the large $N_C$ limit, for simplicity, in the $O(p^6)$ order,
we only need to calculate the large $N_C$ limit terms.
Without the equations of motion, in large $N_C$ limit, the CL is
\begin{eqnarray}
\mathscr{L}_6=\sum_{n=1}^{68}\tilde{K}_n\tilde{Y}_n.
\end{eqnarray}
These $\tilde{Y}_n$ and their relationship to $Y_i$ defined in \cite{p62} are listed in Table \ref{np6},
$\tilde{K}_n$ are some coefficients.
As for the $p^2$ and $p^4$ orders, expanding \eqref{dels} to the $p^6$ order,
with the method in Sec. \ref{cat}, treating $\tilde{Y}_n$ as $\langle O_{\Omega,k}\rangle$, $Y_n$ as $\langle O_{l}\rangle$ in \eqref{dlecs},
we can obtain the $\Delta\tilde{K}_i$ coefficients. We have listed the values in \eqref{dkt} in Appendix \ref{tki}.
The final $O(p^6)$ LECs are listed in Table \ref{cp6}.
\begin{eqnarray}
C_{\Lambda=1\mathrm{GeV}}\bigg|^{C_{\Lambda=1.1\mathrm{GeV}}-C_{\Lambda=1\mathrm{GeV}}}_{C_{\Lambda=0.9\mathrm{GeV}}-C_{\Lambda=1\mathrm{GeV}}},\hspace{1cm}
c_{\Lambda=1\mathrm{GeV}}\bigg|^{c_{\Lambda=1.1\mathrm{GeV}}-c_{\Lambda=1\mathrm{GeV}}}_{c_{\Lambda=0.9\mathrm{GeV}}-c_{\Lambda=1\mathrm{GeV}}}\;.\notag
\end{eqnarray}
Because of the new relation given in Ref. \cite{p6p}, we remove $c_{37}$ as previously.
Unlike the $O(p^4)$ order, some absolute values of the new LECs are smaller than the old ones, such as $C_1$ and $C_{4}$;
some are almost unchanged, such as $C_3$ and $C_{12}$; some are larger than the old ones, such as $C_{52}$ and $C_{65}$;
and some even change signs, such as $C_{22}$ and $C_{69}$.
These arise because of the choice of the independent terms in \cite{p62} and the complex relations in \eqref{rkk}.
The calculations are too highly complicated. To avoid possible mistakes,
the expansion in \eqref{dels} and most of the other calculations are done by computer.
To check the correctness of our results, we examined them in various ways.
First, some terms in Table \ref{np6} have two parts,
which we calculated separately. $C$, $P$, and Hermitian invariance constrains the two parts of the coefficient
as being equal or with a sign difference.
Our analytical results for the separate parts must give the same coefficients. Second,
if we switch off the quark self-energy, all the LECs, except the contact terms', must vanish\cite{our5}.
This places a strong restriction on our results.
Third, because of the strict constraint conditions in \eqref{rkk},
we have $109-68=41$ constraint conditions, with 109 being the number of $O_{\Omega}$ in \eqref{dlecs}.
They also impose strong restrictions on our results.
With all the above assessments, we are confident of the reliability of our numerical results for $O(p^6)$ LECs.
Our choice $F_0=87$ MeV is the leading order value of $F_\pi$.
The relation between $F_0$ and $F_\pi$ is given in Ref. \cite{piond} to the $O(p^6)$ order.
With our results listed in Tables \ref{tp4r} and \ref{cp6}, the numerical results to the $O(p^6)$ order yield $F_\pi=92.76^{+0.01}_{-0.06}$ MeV.
Comparing with the previous result $F_\pi=92.97^{+0.00}_{-0.04}$,
the new result is a bit closer to that in PDG2014 \cite{pdg2014} $F_\pi=92.21$ MeV.
From this point of view, the additional terms in \eqref{alag} improve the results slightly.
{\extrarowheight 5pt
\begin{longtable}{crrcrrcrrcrrcrrcrr}
\caption{\label{cp6}The $p^6$ order LECs. They are in units of $10^{-3}$ GeV$^{-2}$.
The value $\equiv0$ means that the LECs vanish at the lagre $N_C$ limit.}\\
\hline\hline $i$ & new $C_i$~ & old $C_i$~ & $j$ & new $c_j$~ & old $c_j$~ &
$i$ & new $C_i$~ & old $C_i$~ & $j$ & new $c_j$~ & old $c_j$~~~\\
\hline\endfirsthead
\hline\hline $i$ & new $C_i$~ & old $C_i$~ & $j$ & new $c_j$~ & old $c_j$~ &
$i$ & new $C_i$~ & old $C_i$~ & $j$ & new $c_j$~ & old $c_j$~~~\\
\hline\endhead
\hline\hline
\endfoot
\hline\endlastfoot
\hline 1 & $2.98^{+0.07}_{-0.13}$ & $3.79^{+0.10}_{-0.17}$ & 1 & $3.04^{+0.07}_{-0.12}$ & $3.58^{+0.09}_{-0.15}$ & 46 & $-1.67^{-0.05}_{+0.09}$ & $-0.60^{-0.02}_{+0.04}$ & 26 & $-4.82^{-0.15}_{+0.24}$ & $-1.14^{-0.05}_{+0.07}$ \\
2 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & & 47 & $3.10^{+0.09}_{-0.15}$ & $0.08^{+0.01}_{-0.00}$ & & & \\
3 & $-0.05^{+0.01}_{-0.01}$ & $-0.05^{+0.01}_{-0.01}$ & 2 & $-0.09^{+0.01}_{-0.01}$ & $-0.03^{+0.01}_{-0.01}$ & 48 & $4.74^{+0.10}_{-0.17}$ & $3.41^{+0.06}_{-0.10}$ & & & \\
4 & $2.13^{+0.06}_{-0.10}$ & $3.10^{+0.09}_{-0.15}$ & 3 & $2.15^{+0.06}_{-0.10}$ & $2.89^{+0.08}_{-0.13}$ & 49 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & \\
5 & $-1.28^{+0.10}_{-0.13}$ & $-1.01^{+0.08}_{-0.11}$ & 4 & $0.82^{-0.04}_{+0.03}$ & $1.21^{-0.07}_{+0.06}$ & 50 & $10.54^{+0.89}_{-1.29}$ & $8.71^{+0.78}_{-1.12}$ & 27 & $16.70^{+1.61}_{-2.30}$ & $13.57^{+1.41}_{-2.00}$ \\
6 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & & 51 & $-9.83^{+0.28}_{-0.24}$ & $-11.49^{+0.18}_{-0.09}$ & 28 & $5.51^{+1.22}_{-1.62}$ & $0.93^{+0.98}_{-1.25}$ \\
7 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & & 52 & $-6.39^{-0.77}_{+1.07}$ & $-5.04^{-0.67}_{+0.93}$ & & & \\
8 & $2.10^{-0.15}_{+0.16}$ & $2.31^{-0.16}_{+0.18}$ & & & & 53 & $-5.36^{-0.71}_{+1.05}$ & $-11.98^{-0.87}_{+1.33}$ & 29 & $-4.77^{-0.66}_{+0.98}$ & $-11.01^{-0.81}_{+1.23}$ \\
9 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & & 54 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & \\
10 & $-0.65^{+0.05}_{-0.06}$ & $-1.05^{+0.08}_{-0.09}$ & 5 & $-0.68^{+0.05}_{-0.06}$ & $-0.98^{+0.07}_{-0.09}$ & 55 & $10.45^{+0.80}_{-1.22}$ & $16.79^{+0.96}_{-1.49}$ & 30 & $9.86^{+0.75}_{-1.14}$ & $15.72^{+0.89}_{-1.38}$ \\
11 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & & 56 & $4.45^{+0.03}_{-0.18}$ & $19.34^{+0.52}_{-0.98}$ & 31 & $3.27^{-0.04}_{-0.07}$ & $17.57^{+0.42}_{-0.82}$ \\
12 & $-0.34^{+0.01}_{-0.01}$ & $-0.34^{+0.02}_{-0.01}$ & 6 & $-0.35^{+0.02}_{-0.01}$ & $-0.33^{+0.01}_{-0.01}$ & 57 & $4.72^{+1.36}_{-1.85}$ & $7.92^{+1.34}_{-1.85}$ & 32 & $4.24^{+1.32}_{-1.78}$ & $7.18^{+1.28}_{-1.76}$ \\
13 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & & 58 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & \\
14 & $-0.87^{+0.14}_{-0.21}$ & $-0.83^{+0.12}_{-0.19}$ & 7 & $-1.83^{+0.25}_{-0.35}$ & $-1.72^{+0.25}_{-0.35}$ & 59 & $-14.59^{-1.01}_{+1.55}$ & $-22.49^{-1.21}_{+1.89}$ & 33 & $-13.69^{-0.94}_{+1.44}$ & $-21.19^{-1.12}_{+1.76}$ \\
15 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & 8 & $0.91^{-0.11}_{+0.13}$ & $0.86^{-0.12}_{+0.15}$ & 60 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & \\
16 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & & 61 & $2.42^{-0.19}_{+0.22}$ & $2.88^{-0.22}_{+0.26}$ & 34 & $2.40^{-0.19}_{+0.22}$ & $2.84^{-0.22}_{+0.26}$ \\
17 & $0.17^{+0.01}_{-0.04}$ & $0.01^{-0.01}_{-0.01}$ & 9 & $-0.74^{+0.13}_{-0.18}$ & $-0.84^{+0.12}_{-0.17}$ & 62 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & \\
18 & $-0.60^{+0.07}_{-0.09}$ & $-0.56^{+0.09}_{-0.11}$ & & & & 63 & $2.48^{-0.21}_{+0.25}$ & $2.99^{-0.24}_{+0.30}$ & & & \\
19 & $-0.27^{+0.09}_{-0.13}$ & $-0.48^{+0.09}_{-0.13}$ & 10 & $-0.22^{+0.07}_{-0.11}$ & $-0.37^{+0.07}_{-0.10}$ & 64 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & \\
20 & $0.17^{-0.02}_{+0.03}$ & $0.18^{-0.03}_{+0.04}$ & 11 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & 65 & $-2.82^{+0.18}_{-0.20}$ & $-2.43^{+0.15}_{-0.16}$ & 35 & $2.16^{-0.23}_{+0.31}$ & $3.39^{-0.32}_{+0.41}$ \\
21 & $-0.06^{+0.01}_{-0.01}$ & $-0.06^{+0.01}_{-0.01}$ & & & & 66 & $0.80^{+0.04}_{-0.07}$ & $1.71^{+0.07}_{-0.12}$ & 36 & $0.80^{+0.04}_{-0.07}$ & $1.57^{+0.06}_{-0.10}$ \\
22 & $-0.35^{+0.20}_{-0.26}$ & $0.27^{+0.19}_{-0.25}$ & 12 & $-0.41^{+0.20}_{-0.26}$ & $0.15^{+0.18}_{-0.24}$ & 67 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & \\
23 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & & 68 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & \\
24 & $0.87^{+0.02}_{-0.04}$ & $1.62^{+0.04}_{-0.07}$ & & & & 69 & $0.52^{+0.00}_{-0.01}$ & $-0.86^{-0.04}_{+0.06}$ & 38 & $0.60^{+0.00}_{-0.01}$ & $-0.68^{-0.03}_{+0.05}$ \\
25 & $-3.03^{-0.41}_{+0.59}$ & $-5.98^{-0.49}_{+0.72}$ & 13 & $-3.02^{-0.39}_{+0.56}$ & $-5.39^{-0.45}_{+0.66}$ & 70 & $1.66^{-0.11}_{+0.11}$ & $1.73^{-0.08}_{+0.07}$ & 39 & $1.53^{-0.12}_{+0.12}$ & $1.81^{-0.08}_{+0.07}$ \\
26 & $2.71^{+0.35}_{-0.54}$ & $3.35^{+0.29}_{-0.47}$ & 14 & $3.39^{+0.36}_{-0.56}$ & $4.17^{+0.30}_{-0.49}$ & 71 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & \\
27 & $-1.35^{+0.13}_{-0.15}$ & $-1.54^{+0.15}_{-0.18}$ & 15 & $-2.39^{+0.19}_{-0.23}$ & $-2.71^{+0.21}_{-0.25}$ & 72 & $-1.80^{+0.12}_{-0.11}$ & $-3.30^{+0.05}_{-0.00}$ & 40 & $-1.64^{+0.13}_{-0.13}$ & $-3.17^{+0.05}_{-0.02}$ \\
28 & $0.18^{+0.00}_{-0.01}$ & $0.30^{+0.01}_{-0.01}$ & & & & 73 & $0.15^{+0.48}_{-0.62}$ & $0.50^{+0.43}_{-0.56}$ & 41 & $0.07^{+0.47}_{-0.61}$ & $0.30^{+0.42}_{-0.54}$ \\
29 & $-0.99^{-0.21}_{+0.24}$ & $-3.08^{-0.26}_{+0.32}$ & 16 & $-0.60^{-0.19}_{+0.21}$ & $-2.22^{-0.22}_{+0.27}$ & 74 & $-3.34^{-0.11}_{+0.19}$ & $-5.07^{-0.16}_{+0.27}$ & 42 & $-3.26^{-0.10}_{+0.17}$ & $-4.74^{-0.14}_{+0.24}$ \\
30 & $0.37^{+0.01}_{-0.02}$ & $0.60^{+0.02}_{-0.03}$ & & & & 75 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & \\
31 & $-0.46^{+0.07}_{-0.13}$ & $-0.63^{+0.05}_{-0.09}$ & 17 & $-0.92^{+0.15}_{-0.23}$ & $-1.10^{+0.12}_{-0.19}$ & 76 & $-1.15^{-0.26}_{+0.34}$ & $-1.44^{-0.23}_{+0.31}$ & 43 & $-1.11^{-0.25}_{+0.33}$ & $-1.29^{-0.23}_{+0.30}$ \\
32 & $0.17^{-0.02}_{+0.03}$ & $0.18^{-0.03}_{+0.04}$ & 18 & $0.42^{-0.06}_{+0.08}$ & $0.43^{-0.07}_{+0.08}$ & 77 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & \\
33 & $-0.05^{-0.02}_{+0.05}$ & $0.09^{-0.00}_{+0.03}$ & 19 & $0.29^{-0.07}_{+0.13}$ & $0.41^{-0.06}_{+0.10}$ & 78 & $8.82^{+0.80}_{-1.21}$ & $17.51^{+1.02}_{-1.59}$ & 44 & $8.19^{+0.74}_{-1.12}$ & $16.16^{+0.94}_{-1.45}$ \\
34 & $0.66^{-0.18}_{+0.29}$ & $1.59^{-0.10}_{+0.17}$ & 20 & $0.74^{-0.18}_{+0.28}$ & $1.56^{-0.10}_{+0.17}$ & 79 & $5.86^{-0.14}_{+0.12}$ & $-0.56^{-0.30}_{+0.40}$ & 45 & $6.09^{-0.13}_{+0.11}$ & $0.26^{-0.26}_{+0.34}$ \\
35 & $0.10^{-0.09}_{+0.12}$ & $0.17^{-0.12}_{+0.17}$ & 21 & $0.22^{-0.14}_{+0.19}$ & $0.29^{-0.18}_{+0.24}$ & 80 & $1.01^{-0.05}_{+0.04}$ & $0.87^{-0.04}_{+0.03}$ & 46 & $1.09^{-0.05}_{+0.05}$ & $0.85^{-0.04}_{+0.02}$ \\
36 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & & 81 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & \\
37 & $-0.60^{+0.07}_{-0.09}$ & $-0.56^{+0.09}_{-0.11}$ & & & & 82 & $-4.58^{-0.24}_{+0.38}$ & $-7.13^{-0.32}_{+0.51}$ & 47 & $-4.26^{-0.22}_{+0.35}$ & $-6.73^{-0.29}_{+0.47}$ \\
38 & $0.47^{-0.04}_{+0.02}$ & $0.41^{-0.08}_{+0.07}$ & 22 & $-1.27^{+0.19}_{-0.26}$ & $-1.32^{+0.18}_{-0.25}$ & 83 & $-1.74^{+0.17}_{-0.22}$ & $0.07^{+0.20}_{-0.27}$ & 48 & $-1.71^{+0.17}_{-0.21}$ & $-0.22^{+0.18}_{-0.25}$ \\
39 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & 23 & $0.91^{-0.11}_{+0.13}$ & $0.86^{-0.12}_{+0.15}$ & 84 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & \\
40 & $-4.98^{-0.14}_{+0.25}$ & $-6.35^{-0.18}_{+0.32}$ & 24 & $0.00^{-0.01}_{+0.03}$ & $-4.84^{-0.14}_{+0.25}$ & 85 & $-0.96^{+0.04}_{-0.03}$ & $-0.82^{+0.03}_{-0.02}$ & 49 & $-1.07^{+0.05}_{-0.04}$ & $-0.78^{+0.03}_{-0.01}$ \\
41 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & & 86 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & \\
42 & $1.88^{+0.03}_{-0.06}$ & $0.60^{-0.00}_{+0.00}$ & & & & 87 & $4.79^{+0.29}_{-0.46}$ & $7.57^{+0.37}_{-0.60}$ & 50 & $4.35^{+0.26}_{-0.42}$ & $7.18^{+0.34}_{-0.55}$ \\
43 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & & 88 & $-1.69^{-0.68}_{+0.93}$ & $-5.47^{-0.73}_{+1.03}$ & 51 & $-1.36^{-0.65}_{+0.89}$ & $-4.85^{-0.69}_{+0.97}$ \\
44 & $1.73^{+0.07}_{-0.14}$ & $6.32^{+0.20}_{-0.36}$ & 25 & $4.88^{+0.16}_{-0.27}$ & $6.03^{+0.19}_{-0.33}$ & 89 & $17.27^{+1.11}_{-1.77}$ & $34.74^{+1.61}_{-2.62}$ & 52 & $15.84^{+1.00}_{-1.60}$ & $32.19^{+1.46}_{-2.37}$ \\
45 & $\equiv 0~~~~~ $ & $\equiv 0~~~~~ $ & & & & 90 & $2.32^{-0.44}_{+0.55}$ & $2.44^{-0.38}_{+0.46}$ & 53 & $2.28^{-0.44}_{+0.55}$ & $2.51^{-0.37}_{+0.46}$ \\
\hline\hline
\end{longtable}
}
\subsection{Anomaly}
Following the same procedures as for the normal parts, the anomalous LECs can also be revised.
The $O(p^4)$ CL are Wess-Zumino terms, that had been obtained from the first term in \eqref{Snorm} \cite{oura1,oura2}.
The additional terms in \eqref{alag} must vanish to the $O(p^4)$ order.,
thereby imposing another requirement.
We checked our calculations and verified this requirement.
From another point of view, we checked that the terms with $n=2$ in \eqref{Snorm} and $n=1$ in \eqref{fineqNc} were suitable.
To the $O(p^6)$ order, without the equations of motion, in the large $N_C$ limit,
the $n$-flavor CL is
\begin{eqnarray}
\mathscr{L}^W_6=\sum_{n=1}^{23}\tilde{K}^W_n\tilde{O}^W_n.
\end{eqnarray}
These $\tilde{O}^W_n$ and their relations with $O^W_I$ in Ref. \cite{p6a2} are listed in Table VI in Ref. \cite{oura2}.
The analytical results for $\Delta\tilde{K}^W_n$ are listed in \eqref{dkwt} in Appendix \ref{tobi}.
$C^W_3$ $C^W_{18}$ and $c^W_{12}$ vanish in the large $N_C$ limit.
The nonzero numerical results are listed in the second column in Table \ref{ap63} for three flavors and
in the second row in Table \ref{ap62} for two flavours.
\begin{eqnarray}
C^W_{\Lambda=1\mathrm{GeV}}\bigg|^{C^W_{\Lambda=1.1\mathrm{GeV}}-C^W_{\Lambda=1\mathrm{GeV}}}_{C^W_{\Lambda=0.9\mathrm{GeV}}-C^W_{\Lambda=1\mathrm{GeV}}},\hspace{1cm}
c^W_{\Lambda=1\mathrm{GeV}}\bigg|^{c^W_{\Lambda=1.1\mathrm{GeV}}-c^W_{\Lambda=1\mathrm{GeV}}}_{c^W_{\Lambda=0.9\mathrm{GeV}}-c^W_{\Lambda=1\mathrm{GeV}}}\;.\notag
\end{eqnarray}
\begin{table*}[!h]
\caption{The nonzero values of the $p^6$ order anomalous LECs $C^W_i$ in three flavors.
They are in units of $10^{-3}$ GeV$^{-2}$.
The forth column to the eighth column contain the results given in \cite{Strandberg:2003zf}: (I)-ChPT, (II)-VMD, (III)-ChPT
(extrapolation), (IV)-CQM, (V)-CQM (extrapolation).}\label{ap63}
\resizebox{\textwidth}{!}{
\begin{tabular}{crrcccccrcccc}
\hline\hline $n$& new~~~~ & old\cite{oura2}~~ & (I)\cite{Strandberg:2003zf} & (II)\cite{Strandberg:2003zf}
& (III)\cite{Strandberg:2003zf} & (IV)\cite{Strandberg:2003zf} & (V)\cite{Strandberg:2003zf}
& \cite{Colangelo:2012ipa} & \cite{PiDecay,olecs1,olecs2,olecs3,olecs4,hola1}\\
\hline 1 & $2.90^{+0.49}_{-0.69}$ & $4.97^{+0.55}_{-0.79}$& & & & & & &\\
2 & $-1.79^{+0.09}_{-0.11}$ & $-1.43^{+0.10}_{-0.12}$& $-0.32\pm10.4$ && $0.78\pm 12.7$ & $4.96\pm9.70$ & $-0.074\pm13.3$& &\\
4 & $-1.89^{+0.19}_{-0.24}$ & $-0.96^{+0.22}_{-0.29}$& $0.28\pm9.19$ & &$0.67\pm 10.9$ & $6.32\pm6.09$ & $-0.55\pm9.05$& &\\
5 & $1.56^{+0.29}_{-0.41}$ & $3.26^{+0.34}_{-0.49}$& $28.5\pm28.83$ & &$9.38\pm 152.2$ & $33.05\pm28.66$& $34.51\pm41.13$& &\\
6 & $0.72^{+0.02}_{-0.04}$ & $0.91^{+0.03}_{-0.04}$& & & & & & &\\
7 & $2.02^{-0.23}_{+0.29}$ & $1.68^{-0.24}_{+0.31}$& $0.013\pm1.17$ & & & $0.51\pm0.06$ & & &$0.1\pm1.2$\\
& & & $20.3\pm18.7$ & & & && &$1.0^a$\\
& & & & & & && &$0.35\pm0.07$\\
8 & $0.52^{+0.01}_{-0.03}$ & $0.41^{+0.01}_{-0.02}$& $0.76\pm0.18$ & & & & & &$0.58\pm0.20$\\
& & & & & & & & &$5.0^a$\\
9 & $1.21^{-0.03}_{+0.02}$ & $1.15^{-0.03}_{+0.03}$& & & & & & &\\
10 & $-0.14^{-0.00}_{+0.01}$ & $-0.18^{-0.01}_{+0.01}$& & & & & & &\\
11 & $-1.39^{+0.07}_{-0.09}$ & $-1.15^{+0.08}_{-0.10}$&$ -6.37\pm4.54$ & & & $-0.00143\pm0.03$& &&$0.68\pm0.21$\\
12 & $-4.05^{-0.12}_{+0.20}$ & $-5.13^{-0.15}_{+0.25}$& & & & & & -2.1&\\
13 & $-6.81^{-0.20}_{+0.33}$ & $-6.37^{-0.18}_{+0.31}$& $-74.09\pm55.89$& -20.00 & $-8.44\pm69.9$ & $14.15\pm 15.22$ & $-7.46\pm19.62$& -8.8&\\
14 & $-2.48^{-0.07}_{+0.12}$ & $-2.00^{-0.06}_{+0.10}$& $29.99\pm11.14$ & -6.01 & $0.72\pm15.3$ & $10.23\pm7.56$ & $-0.58\pm9.77$& -1.3&\\
15 & $2.35^{+0.07}_{-0.11}$ & $4.17^{+0.12}_{-0.20}$& $-25.3\pm 23.93$& 2.00 & $-3.10\pm28.6$ & $19.70\pm7.49$ & $8.89\pm9.72$& 4.4&\\
16 & $1.79^{+0.05}_{-0.09}$ & $3.58^{+0.10}_{-0.17}$& & & & && -0.2&\\
17 & $0.71^{+0.02}_{-0.03}$ & $1.98^{+0.06}_{-0.10}$& & & & && -0.1&\\
19 & $0.56^{+0.02}_{-0.03}$ & $0.29^{+0.01}_{-0.01}$& & & & && -7.0&\\
20 & $-1.02^{-0.03}_{+0.05}$ & $1.82^{+0.05}_{-0.09}$& & & & && -0.4&\\
21 & $3.11^{+0.09}_{-0.15}$ & $2.48^{+0.07}_{-0.12}$& & & & && 2.6&\\
22 & $4.72^{+0.14}_{-0.23}$ & $5.01^{+0.14}_{-0.24}$& $6.52\pm0.78$ & 8.01 & & $3.94\pm0.43$ && 7.9&$5.4\pm0.8$\\
& & & $5.07\pm0.71$ & & & $3.94 \pm0.43$ && &6.71,6.21,4.45$^b$\\
23 & $2.80^{+0.08}_{-0.14}$ & $2.74^{+0.08}_{-0.13}$& & & & && 0.9&\\
\hline\hline
\end{tabular}
}
\begin{flushleft}
$^a$ This result is just the absolute value given in \cite{olecs3}.
$^b$ These result are in \cite{hola1} by different inputs.
\end{flushleft}
\end{table*}
\begin{table*}[!h]
\extrarowheight 4pt
\caption{The nonzero values of the $p^6$ order anomalous LECs $c^W_i$ in two flavors.
They are in units of $10^{-3}$ GeV$^{-2}$.}\label{ap62}
\resizebox{\textwidth}{!}{\begin{tabular}{ccccccccccccc}
\hline\hline &$c_1^W$ & $c_2^W$ & $c_3^W$& $c_4^W$& $c_5^W$& $c_6^W$& $c_7^W$& c$_8^W$& $c_9^W$& $c_{10}^W$& $c_{11}^W$& $c_{13}^W$\\
\hline\text{new}&$-1.81^{+0.09}_{-0.11}$ & $-1.61^{+0.08}_{-0.09}$ & $3.61^{-0.19}_{+0.22}$ & $0.80^{-0.04}_{+0.04}$ & $-1.40^{+0.07}_{-0.09}$ & $0.52^{+0.26}_{-0.35}$ & $-0.41^{-0.01}_{+0.02}$ & $0.21^{+0.01}_{-0.01}$ & $6.49^{+0.18}_{-0.30}$ & $-6.15^{-0.17}_{+0.29}$ & $4.63^{+0.13}_{-0.22}$ & $-9.25^{-0.26}_{+0.43}$ \notag\\
\text{old}\cite{oura2}&$-1.46^{+0.10}_{-0.12}$ & $-1.25^{+0.09}_{-0.11}$ & $2.96^{-0.20}_{+0.25}$ & $0.63^{-0.04}_{+0.05}$ & $-1.17^{+0.08}_{-0.10}$ & $0.77^{+0.26}_{-0.36}$ & $-0.04^{-0.00}_{+0.00}$ & $0.02^{+0.00}_{-0.00}$ & $8.19^{+0.23}_{-0.38}$ & $-8.73^{-0.24}_{+0.41}$ & $4.85^{+0.13}_{-0.23}$ & $-9.70^{-0.27}_{+0.45}$\notag\\
\hline\hline
\end{tabular}}
\end{table*}
\section{Comparisons}\label{comp}
In this section, we shall gather the LECs to the $O(p^6)$ order given in the literature to provide a means to assess our new results.
It is just only an update of \cite{our5} but also includes some new results.
Usually, these LECs are given as dimensionless parameters with the convention of $C_i^r\equiv C_iF_0^2$ or $c_i^r\equiv c_iF_0^2$ .
The following values of the physical constants come from the central values of PDG2014 \cite{pdg2014},
\begin{eqnarray}
m_{\pi^\pm}=139.57018\text{MeV},m_{\pi^0}=134.9766\text{MeV},F_\pi=92.21\text{MeV},
m_{K^\pm}=493.677\text{MeV}.\label{expd}
\end{eqnarray}
Some of these values are needed in certain parts of the calculations.
\subsection{$\pi\pi$ and $\pi K$ scattering}
In $\pi\pi$ scattering there exist six additional constants, $r^r_i,i=1,2,3,4,5,6$.
Their relationship to LECs can be found in Eq. (5.3) in \cite{p6rn}.
Ref. \cite{p6rn} also gives their values obtained using two theoretical methods,
resonance-saturation (RS)\cite{Bijnens:pp} and pure dimensional analysis
(ND) which only accounts for the order of magnitude.
Ref. \cite{pp3,pp4} also give the $r^r_i$ parameters,
all of which are listed in Table. \ref{pp2}.
Our new $r^r_{4,5,6}$ only change slightly, the error bars from the references being also small.
They are nonetheless in good agreement. However, in the references, $r^r_{1,2,3}$ have large error bars,
and hence our new results can potentially change signs.
\begin{table*}[!h]
\extrarowheight 4pt
\caption{The obtained values for the combinations of the $p^6$ order LECs
from $\pi\pi$ scattering and our work. The coefficients in the table are in units of $10^{-4}$.}\label{pp2}
\begin{eqnarray*}
\begin{array}{lcccccc}
\hline\hline & r^r_1 & r^r_2 & r^r_3&r^r_4&r^r_5 & r^r_6\\
\hline \mbox{RS in Ref.\cite{p6rn}} & -0.6&1.3&-1.7&-1.0&1.1&0.3\\
\mbox{ND in Ref.\cite{p6rn}} & 80&40&20&3&6&2\\
\mbox{set C (n=5)\cite{pp3}}&-14\pm17\pm3&22\pm16\pm4&-3\pm1\pm3&-0.22\pm0.13\pm0.05&0.9\pm0.1\pm0.5&0.25\pm0.01\pm0.05\\
\mbox{set C (n=3)\cite{pp3}}&-20\pm17\pm3&7\pm10\pm4&-4\pm1\pm3&0.13\pm0.13\pm0.05&0.9\pm0.1\pm0.5&0.25\pm0.01\pm0.05\\
\mbox{Ref. \cite{pp4}}&&18&0.9&-1.9&&\\
\mbox{Old\cite{our5}} & -9.32^{-2.62}_{+3.51} & 8.93^{+3.12}_{-4.27} & -3.06^{-0.81}_{+1.11} &
-0.12^{+0.22}_{-0.29}&0.87^{+0.04}_{-0.06}&0.42^{+0.02}_{-0.03}\\
\text{new} & 0.11^{-2.50}_{+3.27} & -2.84^{+2.95}_{-3.94} & 1.03^{-0.69}_{+0.92} & -0.63^{+0.21}_{-0.28} & 0.37^{+0.02}_{-0.04} & 0.28^{+0.01}_{-0.02}\\
\hline\hline
\end{array}
\end{eqnarray*}
\end{table*}
Furthermore, Ref. \cite{pk} introduces some coefficients, such as $c^+_{20}$, $c^+_{01}$, $c^+_{10}$, in $\pi K$ Scattering.
Table \ref{pkb} lists all these coefficients and some LECs which were obtained in Ref. \cite{pk1} from different models.
Whereas the new results on the left-hand side of Table \ref{pkb} are smaller and approach their results,
results on the right-hand side of of Table \ref{pkb} seem worse.
This may be because of propagation errors from the complex relations between $c^+$ and $C_i$.
\begin{table*}[!h]
\extrarowheight 4pt
\caption{The obtained values for the combinations of the $p^6$ order LECs
from $\pi K$ scattering and our work.
The LECs in the l.h.s. of the
table are in units of $10^{-4}\mathrm{GeV}^{-2}$.}\label{pkb}
\begin{eqnarray*}
\begin{array}{lcccc|lccc}
\hline\hline & C_1+4C_3 & C_2 & C_4+3C_3 & C_1+4C_3+2C_2&&
c_{20}^+\frac{m_\pi^4}{F_\pi^4}&c_{01}^+\frac{m_\pi^2}{F_\pi^4}&c_{10}^-\frac{m_\pi^3}{F_\pi^4}\\
\hline \mbox{input~}c^+_{30},c^+_{11},c^-_{20} & 20.7\pm4.9 & -9.2\pm4.9 & 9.9\pm2.5 & 2.3\pm10.8&&&\\
\mbox{input~}c^+_{30},c^+_{11},c^-_{01} & 28.1\pm4.9 & -7.4\pm4.9 & 21.0\pm2.5 & 13.4\pm10.8
&\mbox{Dispersive}&\hspace*{-1cm}0.024\pm0.006&2.07\pm0.10&0.31\pm0.01\\
\pi \pi \mbox{~amplitude} & & & 23.5\pm2.3 & 18.8\pm7.2&&&\\
\mbox{Resonance model} & 7.2 & -0.5 & 10.0 & 6.2&\mbox{Resonance~model}
&0.003&3.8&0.09\\
\mbox{Old \cite{our5}} & 35.9^{+1.3}_{-2.1} & 0.0^{+0.0}_{-0.0} & 29.5^{+1.1}_{-1.9}
& 35.9^{+1.3}_{-2.1}& \mbox{Old \cite{our5}}&\hspace*{-0.5cm}0.006^{-0.002}_{+0.003}&-0.159^{+0.133}_{-0.178}&0.020^{+0.037}_{-0.050}\\
\mbox{New} & 27.8^{+1.1}_{-1.8} & 0.0^{+0.0}_{-0.0} & 19.8^{+0.9}_{-1.4}
& 27.8^{+1.1}_{-1.8}& \mbox{New}&\hspace*{-0.5cm}0.016^{-0.002}_{+0.002}&-0.474^{+0.142}_{-0.186}&-0.082^{+0.036}_{-0.047}\\
\hline\hline
\end{array}
\end{eqnarray*}
\end{table*}
\subsection{Form factors}
Ref. \cite{p6rn} also estimates the expressions of the vector form factor, the scalar form factor,
two form factors of $\pi(p)\to e\nu\gamma(q)$ with LECs,
and \cite{Kl3,Kpi} give some LECs using measurements of the pion scalar form factor, $K_{l3}$
and the $\pi K$ form factors.
Ref. \cite{pp3} gives one form factor from $\pi\pi$-scattering.
All of them are listed in Table \ref{ff}.
Ref. \cite{kpif} extrapolates the lattice data on the scalar $K\pi$ form factor to obtain some LECs;
the results are listed in Table \ref{kpi}.
Two of these results, $r^r_{S2}$ and $2C_{12}^r+C_{34}^r$ , seem better, as their signs have been changed.
Furthermore in Fig.\ref{Vecf}, we compare the experimental data \cite{FNAL,NA7,OLYA,ALEPH,DM1,TOF,NA72} for the vector form factors collected in Figure 4
and 5 in \cite{piond} with our old and new results. In obtaining our numerical predictions,
we have exploited the formula given by Eq.(3.16) in \cite{piond} which especially depends on $O(p^6)$
LECs through $r_{V1}^r$ and $r_{V2}^r$ defined in \cite{p6rn}, and we
input the formula with the old and new $O(p^4)$ and $O(p^6)$ LECs.
From Fig.\ref{Vecf}, we see that at both $O(p^4)$ and $O(p^6)$,
for the space-like form factors, the new line is higher than the old ones, whereas
for the time-like form factors, the new line are lower than old ones.
The new results are slightly worse than the old ones.
Nevertheless, considering the experimental errors, they are all consistent with the experimental data.
\begin{table*}[!h]
\extrarowheight 4pt
\caption{The obtained values for the combinations of the $p^6$ order LECs
appear in vector and scalar form factor of pion.
the coefficients in the table are in units of $10^{-4}$.}\label{ff}
\begin{eqnarray*}
&\begin{array}{cccc|ccccc|cccc}
\hline\hline & \text{New} & \mbox{Old\cite{our5}} & \mbox{Ref.\cite{p6rn}} & & \text{New}&\mbox{Old\cite{our5}} & \mbox{Ref.\cite{p6rn}}& \mbox{Ref.\cite{pp3}}& & \text{New}&\mbox{Old\cite{our5}}
& \mbox{Ref.\cite{p6rn}}\\
\hline r^r_{V1} & -1.60^{+0.32}_{-0.41} & -2.13^{+0.30}_{-0.39} & -2.5 & r^r_{S2}& -0.86^{+0.00}_{+0.00}& 0.07^{+0.05}_{-0.08}&-0.3&1\pm4\pm1 &r_{A1}^r& 1.29^{+0.05}_{-0.06}&1.14^{+0.07}_{-0.09}&-0.5\\
r^r_{V2} & 1.10^{+0.07}_{-0.10}& 2.23^{+0.10}_{-0.16} & 2.6 & r^r_{S3}& 0.21^{-0.01}_{+0.01}&0.20^{-0.01}_{+0.01}&0.6&&r_{A2}^r& -0.72^{-0.07}_{+0.10}&-0.38^{-0.06}_{+0.08}&1.1\\
\hline\hline
\end{array}&\notag\\
&\begin{array}{cccc|cccc}
\hline\hline & \text{New}&\mbox{Old\cite{our5}}&\mbox{Ref.\cite{Kl3}}&& \text{New}& \mbox{Old\cite{our5}}&\mbox{Ref.\cite{Kpi}}\\
\hline C_{12}^r& -0.026^{+0.001}_{-0.001}&-0.026^{+0.001}_{-0.001}&-0.1&C^r_{12}& -0.026^{+0.001}_{-0.001}&
-0.026^{+0.001}_{-0.001}&(0.3\pm 5.4)\times 10^{-3}\\
2C_{12}^r\!\!+C_{34}^r& -0.001^{-0.012}_{+0.020}&
0.068^{-0.006}_{+0.010}&-0.10\pm 0.17&C_{12}^r\!\!+C_{34}^r& 0.025^{-0.013}_{+0.021}&0.094^{-0.007}_{+0.011}&(3.2\pm 1.5)\times 10^{-2}\\
\hline\hline
\end{array}&
\end{eqnarray*}
\end{table*}
\begin{table*}[!h]
\extrarowheight 4pt
\caption{The LECs come from extrapolating the lattice data on the scalar $K\pi$ form factor.
They are in units of $10^{-4}$GeV$^{-2}$.}\label{kpi}
\begin{eqnarray*}
\begin{array}{lcccccc}
\hline\hline &C_{12}&C_{34}&C_{14}&2C_{17}\\
\hline \mbox{Ref.\cite{kpif}} & 5.74\pm0.95& 1.07\pm0.96&0.71\pm1.42&1.92\pm3.36\\
\mbox{Old\cite{our5}}& -0.34^{+0.02}_{-0.01} & 1.59^{-0.10}_{+0.17} & -0.83^{+0.12}_{-0.19} & 0.03^{-0.02}_{-0.02}\\
\text{New}& -0.34^{+0.01}_{-0.01} & 0.66^{-0.18}_{+0.29} & -0.87^{+0.14}_{-0.21} & 0.35^{+0.03}_{-0.08}\\
\hline\hline
\end{array}
\end{eqnarray*}
\end{table*}
\begin{figure}[h]
\begin{minipage}[b]{\textwidth}
\includegraphics[scale=0.6]{vec1}\includegraphics[scale=0.6]{vec2}
\end{minipage}
\caption{The space like and time like data for the vector form factor with the old and new results.} \label{Vecf}
\end{figure}
\subsection{Photon-photon collisions}
Ref. \cite{pigamma,pigamma1} introduce some parameters by $\gamma\gamma\to\pi^0\pi^0$ and $\gamma\gamma\to\pi^+\pi^-$.
These parameters are also all related to LECs. They are listed in Table \ref{gg}.
These results are nearly unchanged, with $a^r_2$ and $b^r$ still having opposite signs.
\begin{table*}[!h]
\extrarowheight 4pt
\caption{The obtained values for the combinations of the $p^6$ order LECs
appear in photon-photon collisions.}\label{gg}
\begin{eqnarray*}
\begin{array}{cccc|cccc}
\hline\hline & \text{New}& \mbox{Old\cite{our5}} & \mbox{Ref.\cite{pigamma}} & & \text{New}&\mbox{Old\cite{our5}} & \mbox{Ref.\cite{pigamma1}}\\
\hline a^r_1 & -8.15^{-0.85}_{+1.17}& -5.65^{-0.91}_{+1.23} & -14\pm 5 & a^r_1& -5.46^{-0.12}_{+0.07}& -5.86^{-0.49}_{+0.58}&-3.2\\
a^r_2 & 3.94^{+0.03}_{-0.06}& 3.79^{+0.02}_{-0.05} & 7\pm 3 & a^r_2& -1.73^{-0.09}_{+0.16}&-0.98^{-0.07}_{+0.12}&0.7\\
b^r & 1.87^{+0.06}_{-0.10}& 1.66^{+0.05}_{-0.09} & 3\pm 1 & b^r& -0.20^{-0.02}_{+0.02}&-0.23^{-0.01}_{+0.02}&0.4\\
\hline\hline
\end{array}
\end{eqnarray*}
\end{table*}
\subsection{Radiative pion decay}
Ref. \cite{PiDecay} gives a group of LECs to the $O(p^6)$ order. They are listed in Table \ref{rpd}.
Most of the new results are fitted better than the old ones, with the exception of $C^r_{88}$.
\begin{table*}[!h]
\extrarowheight 4pt
\caption{The obtained values for the combinations of the $p^6$ order LECs
from pion radiative decay and our work.
The coefficients in the table are in units of $10^{-5}$.}\label{rpd}
\begin{eqnarray*}
\begin{array}{lcccccc}
\hline\hline &C^r_{12}&C^r_{13}&C^r_{61}&C^r_{62}&2C^r_{63}-C^r_{65}&C^r_{64}\\
\hline \mbox{Ref.\cite{PiDecay}} & -0.6\pm 0.3&0\pm 0.2&1.0\pm 0.3& 0\pm 0.2&1.8\pm 0.7 &0\pm 0.2\\
\hline
\mbox{Old\cite{our5}}&-0.26^{+0.01}_{-0.01}&0.0^{+0.0}_{-0.0}&2.18^{-0.17}_{+0.20}&0.0^{+0.0}_{-0.0}&6.36^{-0.42}_{+0.56}
&0.0^{+0.0}_{-0.0}\\
\text{New}& -0.26^{+0.01}_{-0.01} & 0.00^{+0.00}_{+0.00} & 1.83^{-0.14}_{+0.17} & 0.00^{+0.00}_{+0.00} & 5.89^{-0.45}_{+0.53} & 0.00^{+0.00}_{+0.00}\\
\hline\hline
&C^r_{78}&C^r_{80}&C^r_{81}&C^r_{82}& C^r_{87}&C^r_{88}\\
\hline \mbox{Ref.\cite{PiDecay}}&10.0\pm 3.0&1.8\pm 0.4&
0\pm 0.2& -3.5\pm 1.0&3.6\pm 1.0 &-3.5\pm 1.0\\
\hline
\mbox{Old\cite{our5}}&13.26^{+0.77}_{-1.20}&0.66^{-0.03}_{+0.02}&0.0^{+0.0}_{-0.0}&-5.39^{-0.24}_{+0.39}&5.73^{+0.28}_{-0.45}
&-4.14^{-0.55}_{+0.78}\\
\text{New}& 6.68^{+0.60}_{-0.91} & 0.77^{-0.04}_{+0.03} & 0.00^{+0.00}_{+0.00} & -3.47^{-0.18}_{+0.29} & 3.63^{+0.22}_{-0.35} & -1.28^{-0.52}_{+0.70}\\
\hline\hline
\end{array}
\end{eqnarray*}
\end{table*}
\subsection{holography}
Ref. \cite{Colangelo:2012ipa} gives almost all LECs without scalar and pseudoscalar fields in the large $N_C$ limit
from a class of holographic theories, the results are listed in Table \ref{hol}.
The new results still produce some large differences in these LECs, but some LECs retain the same sign,
such as $C_{69}$ and $C_{79}-\frac{1}{2}C_{90}$. Some differences may come from the error associated with $C_{90}$,
the source of which needs to be further checked.
\begin{table*}[!h]
\extrarowheight 4pt
\caption{The $p^6$ order LECs from Cosh holographic models \cite{Colangelo:2012ipa} and our work.
They are in units of $10^{-3}$GeV$^{-2}$.}\label{hol}
\begin{eqnarray*}
\begin{array}{lccccccccc}
\hline\hline &C_{1}&C_{3}&C_{4}&C_{40}&C_{42}&C_{44}&C_{46}&C_{47}\\
\hline \mbox{Ref.\cite{Colangelo:2012ipa}} & -0.3&0.3&0&0.2&2.2&-5.5&-3.2&6.2\\
\mbox{Old\cite{our5}}& 3.79^{+0.10}_{-0.17} & -0.05^{+0.01}_{-0.01} & 3.10^{+0.09}_{-0.15} & -6.35^{-0.18}_{+0.32} & 0.60^{-0.00}_{+0.00} & 6.32^{+0.20}_{-0.36} & -0.60^{-0.02}_{+0.04} & 0.08^{+0.01}_{-0.00}\\
\text{New}& 2.98^{+0.07}_{-0.13} & -0.05^{+0.01}_{-0.01} & 2.13^{+0.06}_{-0.10} & -4.98^{-0.14}_{+0.25} & 1.88^{+0.03}_{-0.06} & 1.73^{+0.07}_{-0.14} & -1.67^{-0.05}_{+0.09} & 3.10^{+0.09}_{-0.15}\\
\hline\hline
&C_{48}&C_{50}+C_{90}&C_{51}+C_{90}&C_{52}-C_{90}&C_{53}-\frac{1}{2}C_{90}&C_{55}+\frac{1}{2}C_{90}&C_{56}-C_{90}&C_{57}+2C_{90}\\
\hline \mbox{Ref.\cite{Colangelo:2012ipa}}&5.8&19.1&5.2&-11.6&-8.8&16.7&7.1&17.2\\
\mbox{Old\cite{our5}}&3.41^{+0.06}_{-0.10} & 11.16^{+0.40}_{-0.66} & -9.04^{-0.20}_{+0.38} & -7.48^{-0.29}_{+0.47} & -13.21^{-0.68}_{+1.10} & 18.01^{+0.77}_{-1.26} & 16.89^{+0.89}_{-1.44} & 12.80^{+0.58}_{-0.92}\\
\text{New}& 4.74^{+0.10}_{-0.17} & 12.86^{+0.45}_{-0.75} & -7.50^{-0.15}_{+0.30} & -8.71^{-0.33}_{+0.53} & -6.52^{-0.49}_{+0.78} & 11.61^{+0.58}_{-0.95} & 2.13^{+0.47}_{-0.73} & 9.36^{+0.48}_{-0.76}\\
\hline\hline
&C_{59}-\frac{1}{2}C_{90}&C_{66}&C_{69}&C_{70}-\frac{1}{2}C_{90}&C_{72}+\frac{1}{2}C_{90}&C_{73}+C_{90}&C_{74}&C_{76}-\frac{1}{2}C_{90}\\
\hline \mbox{Ref.\cite{Colangelo:2012ipa}}&-20.1&-0.3&0.3&5.3&-4.7&-4.4&-19.0&11.1\\
\mbox{Old\cite{our5}}&-23.71^{-1.02}_{+1.66} & 1.71^{+0.07}_{-0.12} & -0.86^{-0.04}_{+0.06} & 0.51^{+0.11}_{-0.16} & -2.08^{-0.14}_{+0.23} & 2.94^{+0.05}_{-0.10} & -5.07^{-0.16}_{+0.27} & -2.66^{-0.05}_{+0.08}\\
\text{New}& -15.75^{-0.79}_{+1.28} & 0.80^{+0.04}_{-0.07} & 0.52^{+0.00}_{-0.01} & 0.49^{+0.11}_{-0.16} & -0.64^{-0.10}_{+0.16} & 2.48^{+0.04}_{-0.07} & -3.34^{-0.11}_{+0.19} & -2.31^{-0.04}_{+0.07}\\
\hline\hline
&C_{78}+\frac{1}{2}C_{90}&C_{79}-\frac{1}{2}C_{90}&C_{87}&C_{88}-C_{90}&C_{89}\\
\hline \mbox{Ref.\cite{Colangelo:2012ipa}}&16.1&4.1&6.8&-5.2&29.2\\
\mbox{Old\cite{our5}}&18.74^{+0.83}_{-1.36} & -1.78^{-0.11}_{+0.17} & 7.57^{+0.37}_{-0.60} & -7.91^{-0.35}_{+0.57} & 34.74^{+1.61}_{-2.62}\\
\text{New}& 9.99^{+0.58}_{-0.93} & 4.70^{+0.08}_{-0.15} & 4.79^{+0.29}_{-0.46} & -4.01^{-0.24}_{+0.38} & 17.27^{+1.11}_{-1.77}\\
\hline\hline
\end{array}
\end{eqnarray*}
\end{table*}
\subsection{Other results}
Aside from the above results, there are more LECs given using these different method.
Most of them are from resonance approximations, but our results do not rely on the assumption of the presence of resonances.
In this subsection, we list the values we gathered from the literature and compare with ours.
Table \ref{tp} lists some LECs from resonance estimates in \cite{tpf};
Table \ref{res1} lists other resonance-estimated LECs in \cite{res2};
Table \ref{sumr} lists the LECs from sum rules in \cite{sumrules};
$C^r_{87}$ values were obtained from many references, and are listed Table \ref{C87};
the other LECs obtained from different models in different references are listed in Table \ref{za}.
Except for $C^r_{14}+C^r_{15}$ in Table \ref{za}, all the LECs retain their signs and the same orders of magnitudes,
and almost all new results are in closer correspondence with others.
In Table \ref{lncrr}, we checked the relations for the large $N_C$ limit given in Ref. \cite{lecr}.
\begin{eqnarray}
C_{20}=-3C_{21}=C_{32}=\frac{1}{6}C_{35},\hspace{1cm}C_{24}=6C_{28}=3C_{30}\;.
\end{eqnarray}
Because we only calculate a part of the large-$N_C$ expression in \eqref{Snorm},
not all of the LECs satisfy the relations.
For the anomalous parts, we collect the results in Table \ref{ap63}.
The anomalous results are less than the normal ones,
and the differences between each are slightly larger than the normal ones.
\begin{table*}[!h]
\extrarowheight 4pt
\caption{The obtained values for the $p^6$ order LECs in Ref.\cite{tpf} and our works
They are in units of $10^{-3}\text{ GeV}^{-2}$.}\label{tp}
\begin{eqnarray*}
\begin{array}{ccccccc}
\hline\hline
&C_{14}&C_{19}&C_{38}&C_{61}&C_{80}&C_{87}\\
\hline
\mbox{Ref.\cite{tpf}} & -4.3 &-2.8&1.2&1.9&1.9&7.6\\
\mbox{Old\cite{our5}}&-0.83^{+0.12}_{-0.19}&-0.48^{+0.09}_{-0.13}&0.41^{-0.08}_{+0.07}&2.88^{-0.22}_{+0.26}&0.87^{-0.04}_{+0.03}
&7.57^{+0.37}_{-0.60}\\
\text{New}& -0.87^{+0.14}_{-0.21} & -0.27^{+0.09}_{-0.13} & 0.47^{-0.04}_{+0.02} & 2.42^{-0.19}_{+0.22} & 1.01^{-0.05}_{+0.04} & 4.79^{+0.29}_{-0.46}\\
\hline\hline
\end{array}
\end{eqnarray*}
\end{table*}
\begin{table*}[!h]
\extrarowheight 4pt
\caption{The obtained values for the $p^6$ order LECs from resonance Lagrangian given by Ref.\cite{res2} and our work
The coefficients in the table are in units of $10^{-4}/F_0^2$.}\label{res1}
\begin{eqnarray*}
\begin{array}{cccccccc}
\hline\hline & C_{78} & C_{82} & C_{87} & C_{88} & C_{89} & C_{90}\\
\hline
\mbox{Lowest Meson Dominance} & 1.09 & -0.36 & 0.40 & -0.52 &1.97&0.0\\
\mbox{Resonance Lagrangian I } & 1.09 & -0.29 & 0.47 & -0.16&2.29&0.33\\
\mbox{Resonance Lagrangian II} & 1.49 & -0.39 & 0.65 & -0.14&3.22&0.51\\
\hline
\mbox{Old\cite{our5}} & 1.326^{+0.077}_{-0.120} & -0.539^{-0.024}_{+0.039} & 0.573^{+0.028}_{-0.045} & -0.414^{-0.055}_{+0.078}&
2.630^{+0.122}_{-0.198}&0.185^{-0.029}_{+0.035}\\
\mbox{New}& 0.668^{+0.060}_{-0.091} & -0.347^{-0.018}_{+0.029} & 0.363^{+0.022}_{-0.035} & -0.128^{-0.052}_{+0.070} & 1.307^{+0.084}_{-0.134} & 0.176^{-0.033}_{+0.041}\\
\hline\hline
\end{array}
\end{eqnarray*}
\end{table*}
\begin{table*}[!h]
\extrarowheight 4pt
\caption{The LECs come from sum rules in \cite{sumrules}.
They are in units of $10^{-3}\mathrm{GeV}^{-2}$.}\label{sumr}
\begin{eqnarray*}
\begin{array}{cccccccc}
\hline\hline & C_{12}+C_{61}+C_{80} & C_{12}-C_{61}+C_{80} & C_{61} & C_{12}+C_{80}\\
\hline w_{DK}\mbox{\cite{sumrules}}& 2.48\pm0.19 & -0.55\pm0.21 & 1.51\pm0.19 & 0.97\pm0.11\\
\hat{w}\mbox{\cite{sumrules}}& 2.48\pm0.18 & -0.46\pm0.19 & 1.47\pm0.17 & 1.01\pm0.10\\
\mbox{Old\cite{our5}} & 3.41^{-0.25}_{+0.28} & -2.36^{+0.20}_{-0.24} & 2.88^{-0.22}_{+0.26} & 0.53^{-0.02}_{+0.02}\\
\mbox{New}& 3.09^{-0.22}_{+0.25} & -1.75^{+0.16}_{-0.19} & 2.42^{-0.19}_{+0.22} & 0.67^{-0.03}_{+0.03}\\
\hline\hline
\end{array}
\end{eqnarray*}
\end{table*}
\begin{table*}[!h]
\extrarowheight 4pt
\caption{The obtained values for the $p^6$ order LECs $C^r_{87}$.
They are in units of $10^{-5}$.}\label{C87}
\begin{eqnarray*}
\begin{array}{cccccc}
\hline\hline & \mbox{New} & \mbox{Old\cite{our5}}& \mbox{Ref.\cite{C87-1}} & \mbox{Ref.\cite{C87-2}}&\mbox{Ref.\cite{C87-3}}\\
\hline C^r_{87} & 3.63^{+0.22}_{-0.35} & 5.73^{+0.28}_{-0.45} & 3.1\pm 1.1 & 4.3\pm 0.4 & 3.70\pm 0.14\\
\hline\hline
\end{array}
\end{eqnarray*}
\end{table*}
\begin{table*}[!h]
\extrarowheight 4pt
\caption{Some LECs from different references.}\label{za}
\begin{eqnarray*}
\begin{array}{cccccccc}
\hline\hline & 10^5(2C^r_{63}-C^r_{65}) & 10^6C^r_{38} & 10^5(C^r_{88}-C^r_{90}) & (C_{14}+C_{15})10^3\text{GeV}^2& (C_{15}+2C_{17})10^3\text{GeV}^2\\
\hline & 1.8\pm0.7\text{\cite{rkd}} & 2\pm6\text{\cite{C38-1}} & -4.6\pm0.4\text{\cite{C88C90-1}} & 0.37\pm0.08\text{\cite{latt}} &1.29\pm0.16\text{\cite{latt}} \\
&&8\pm5\text{\cite{C38-2}} & -4.5\pm0.5\text{\cite{C88C90-2}}\\
\mbox{Old\cite{our5}} & 6.36^{-0.48}_{+0.57} & 3.08^{-0.62}_{+0.56} & -5.99^{-0.27}_{+0.43}& -0.83^{+0.12}_{-0.19} & 0.03^{-0.02}_{-0.02}\\
\mbox{New}& 5.89^{-0.45}_{+0.53} & 3.57^{-0.34}_{+0.17} & -3.04^{-0.18}_{+0.29}& -0.87^{+0.14}_{-0.21} & 0.35^{+0.03}_{-0.08}\\
\hline\hline
\end{array}
\end{eqnarray*}
\end{table*}
\begin{table*}[!h]
\extrarowheight 4pt
\caption{The obtained values for the $p^6$ order LECs from our work
The coefficients in the table are in units of $10^{-3}\mathrm{GeV}^{-2}$.}\label{lncrr}
\begin{eqnarray*}
\begin{array}{ccccc|ccc}
\hline\hline & C_{20}& -3C_{21} & C_{32}& \frac{1}{6}C_{35}&C_{24}&6C_{28}&3C_{30}\\
\hline \mbox{Old\cite{our5}} &0.18^{-0.03}_{+0.04}&
0.18^{-0.03}_{+0.03}&0.18^{-0.03}_{+0.04}&0.028^{-0.020}_{+0.028}&1.62^{+0.04}_{-0.07}&1.80^{+0.06}_{-0.06}&1.80^{+0.06}_{-0.09}\\
\mbox{New}&0.17^{-0.02}_{+0.03} & 0.17^{-0.02}_{+0.03} & 0.17^{-0.02}_{+0.03} & 0.02^{-0.01}_{+0.02} & 0.87^{+0.02}_{-0.04} & 1.10^{+0.03}_{-0.05} & 1.10^{+0.03}_{-0.05}\\
\hline\hline
\end{array}
\end{eqnarray*}
\end{table*}
\section{Summary}\label{summ}
In this research, we updated our original LECs to the $O(p^6)$ order, including two and three flavours, and normal and anomalous ones.
The new contributions come from $n=2$ in \eqref{Snorm} and $n=1$ in \eqref{fineqNc}.
This is one small step beyond the GND model.
As a check, the $O(p^4)$-order absolute values have decreased, and are closer to others,
so our updates are plausible. Up to the $O(p^6)$ order, the absolute values of the LECs exhibited varying changes or remained unchanged.
We also compared these LECs with others.
Most of them are much closer than the old values, but some combinations of LECs fare badly.
The combinations found in references are directly from phenomenological data, and they can be more precise.
However, we have obtained LECs separately.
On the whole, the new LECs values are better than old ones;
one possible reason for the differences is propagation of errors.
In this method, more precise results needs a more detailed analysis of \eqref{Snorm},
which remains as work for the future.
\section*{Acknowledgments}
Jiang thanks Professor Rui-Jing Lu for the helpful discussions..
This work was supported by the National Science Foundation of China (NSFC) under Grants No. 11205034 and No. 11475092;
the Natural Science Foundation of Guangxi Grants No. 2013GXNSFBA019012 and No. 2013GXNSFFA019001;
the Specialized Research Fund for the Doctoral Program of High Education of China No. 20110002110010, and the Tsinghua University Initiative Scientific Research Program No. 20121088494.
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\section{Introduction}
\subsection{Review of sums of two squares in arithmetic progressions}
Let $S$ be the set of positive integers expressible as a sum of two perfect squares. We denote by $\mathbf{1}_{S}$ the indicator function of $S$. It is multiplicative and for a prime $p$ we have \begin{equation}\label{eq:characterization}
\mathbf{1}_{S}(p^k)=0 \text{ if and only if }p\equiv 3 \bmod 4 \text{ and }2 \nmid k.
\end{equation}
Landau \cite{Landau1908} proved that \[ \# (S \cap [1,x]) \sim \frac{Kx}{\sqrt{\log x}}\]
where $K=\prod_{p\equiv 3 \bmod 4}(1-p^{-2})^{-1/2}/\sqrt{2}\approx 0.764$ is the Landau-Ramanujan constant. See \cite[Lec.~IV]{Hardy1940} for Hardy's account of Ramanujan's unpublished work on this problem.
Landau's method yields an asymptotic expansion in descending powers of $\log x$, which gives an error term $O_k(x/(\log x)^{k+1/2})$ for each $k\ge 1$.\footnote{A more complicated main term, leading to a significantly better error term (conjecturally $O_{\varepsilon}(x^{1/2+\varepsilon})$, but no better than that), is described e.g. in \cite[App.~B]{Gorodetsky2021} (cf. \cite{Ramachandra1976}, \cite[p.~187]{Montgomery2007}, \cite[Thm.~2.1]{David2021}).}
Prachar \cite{Prachar1953} proved that sums of two squares are equidistributed in arithmetic progressions, in the following sense. If $(a,q)=1$ then
\[ S(x;q,a) :=\#\{ n \in S: n \le x, \, n \equiv a \bmod q \} \sim \frac{(4,q)}{(2,q)q}\prod_{\substack{p\mid q\\p \equiv 3 \bmod 4}} \left(1+\frac{1}{p}\right)\frac{Kx}{\sqrt{\log x}}\]
as $x \to \infty$ as long as $a \equiv 1 \bmod (4,q)$; see Iwaniec's work on the half-dimensional sieve and the reference therein for results allowing $q$ to vary with $x$ \cite{Iwaniec1976}. The condition $a \equiv 1 \bmod (4,q)$ is necessary: otherwise $(a,q)=1$ and $a \not\equiv 1 \bmod (4,q)$ imply $a \equiv 3 \bmod 4$. However, $S$ is disjoint from $3 \bmod 4$.
\subsection{Main theorem and corollary}
Here we consider a Chebyshev's bias phenomenon for $S$. We ask, what can be said about the size of the set
\begin{equation}\label{eq:bias}
\{ n \le x: S(n;q,a)>S(n;q,b)\}
\end{equation}
for distinct $a,b \bmod q$ with $a\equiv b \equiv 1 \bmod (4,q)$ and $(a,q)=(b,q)=1$? These conditions guarantee that $S(n;q,a)\sim S(n;q,b) \to \infty$ as $n \to \infty$, so it is sensible to study \eqref{eq:bias}. We let $\chi_{-4}$ be the unique non-principal Dirichlet character modulo $4$. Motivated by numerical evidence (based on $n \le 10^8$) showing $S(n;3,1)-S(n;3,2)$ and $S(n;5,1)-S(n;5,3)$ are positive much more frequently than not, we were led to discover and prove the following.
\begin{thm}\label{thm:bias}
Fix a positive integer $q $. Assume that the Generalized Riemann Hypothesis (GRH) holds for the Dirichlet L-functions $L(s,\chi)$ and $L(s,\chi \chi_{-4})$ for all Dirichlet character $\chi$ modulo $q$.
Then, whenever $a,b$ satisfy $a\equiv b \equiv 1 \bmod (4,q)$, $(a,q)=(b,q)=1$ and \begin{equation}\label{eq:lincond} C_{q,a,b}:=\sum_{\substack{\chi \bmod q\\ \chi^2 = \chi_0}} (\chi(a)-\chi(b)) \left(1-\frac{\chi(2)}{\sqrt{2}}\right)^{-1/2}\sqrt{L\left(\frac{1}{2},\chi\right)L\left(\frac{1}{2},\chi \chi_{-4}\right)}>0
\end{equation}
we have, as $x \to \infty$,
\[ \# \{n \le x: S(n;q,a) > S(n;q,b)\} = x(1+o(1)). \]
\end{thm}
Here (and later) $\chi_0$ is the principal character modulo $q$. Observe that $\chi(a)=\chi(b)=1$ for $\chi=\chi_0$ as well as for $\chi=\chi_0\chi_{-4}$ (if $4 \mid q$), so these two characters may be omitted from the sum \eqref{eq:lincond}. As we explain in Remark~\ref{rem:nonneg} below, $L(1/2,\chi)L(1/2,\chi\chi_{-4})$ is non-negative under the conditions of Theorem~\ref{thm:bias}; the squareroot we take in \eqref{eq:lincond} is the non-negative one.
\begin{remark}\label{rem:nonneg}
Under GRH for (non-principal) real $\chi$, we have $L(1/2,\chi)\ge 0$ since otherwise there is a zero of $L(s,\chi)$ in $(1/2,1)$ by the intermediate value theorem. Similarly, $L(1/2,\chi \chi_{-4}) \ge 0$ if $\chi \neq \chi_0 \chi_{-4}$.
Conrey and Soundararajan proved, unconditionally, that for a positive proportion of quadratic characters $\chi$ we have $L(s,\chi)> 0$ on $(1/2,1)$ \cite{Conrey2002}, implying $L(1/2,\chi) \ge 0$.
Chowla's Conjecture \cite{Chowla1965} states that $L(1/2,\chi) \neq 0$ for all real Dirichlet characters. It was studied extensively, see e.g. Soundararajan \cite{Sound2000} and the references therein.
\end{remark}
If $\chi(a)=1$ for all real characters modulo $q$ then $a$ is a quadratic residue modulo $q$, and vice versa.
Let us specialize $a$ to be a quadratic residue modulo $q$ and $b$ to be a nonquadratic residue. We observe that (on GRH)
$(\chi(a)-\chi(b))\sqrt{L(1/2,\chi)L(1/2,\chi \chi_{-4})}$ is non-negative for each real $\chi \bmod q$ that is not $\chi_0$ or $\chi_0 \chi_{-4}$. Hence, under GRH and our assumptions on $a$ and $b$, a necessary and sufficient condition for \eqref{eq:lincond} to hold is that $L(1/2,\chi)L(1/2,\chi\chi_{-4}) \neq 0$ for some real $\chi \bmod q$ with $\chi(b)=-1$. One way to guarantee this is to assume Chowla's Conjecture. We state this as the following corollary.
\begin{cor}\label{cor:quad}
Suppose that $GRH$ holds for $\chi$ and $\chi \chi_{-4}$ as $\chi$ varies over all Dirichlet characters modulo $q$.
Let $a$ and $b$ be quadratic and nonquadratic residues modulo $q$, respectively, with $(a,q)=(b,q)=1$ and $a \equiv b \equiv 1 \bmod (4,q)$. If $L(1/2,\chi)L(1/2,\chi\chi_{-4}) \neq 0$ for some $\chi \bmod q$ with $\chi(b)=-1$ then \begin{equation}\label{eq:sn ineq}S(n;q,a)>S(n;q,b)
\end{equation}
holds for a density-1 set of integers. In particular, if Chowla's Conjecture holds then \eqref{eq:sn ineq} holds for a density-1 set of integers.
\end{cor}
It would be interesting to try and establish the positivity of \eqref{eq:lincond}, possibly in a statistical sense, without hypotheses like Chowla's Conjecture.
For a given Dirichlet character $\chi$, one can computationally verify that $L(1/2,\chi)$ is non-zero, and in fact compute all zeros of $L(s,\chi)$ up to a certain height, see Rumely \cite{Rumley1993} which in particular shows $L(1/2,\chi) \neq 0$ for characters of conductor $\le 72$. Nowadays computing $L(1/2,\chi)$ is a one-line command in \emph{Mathematica}, and so the verification of \eqref{eq:lincond} is practical for fixed $q$, $a$ and $b$.
We expect the expression in \eqref{eq:lincond} to be non-zero as long as $\chi(a)\neq\chi(b)$ for some real character, or equivalently, if $a/b$ is nonquadratic residue modulo $q$. It is instructive to consider the following two possibilities for $a$ and $b$ separately:
\begin{itemize}
\item Suppose $a$, $b$ and $a/b$ are all nonquadratic residues, a situation that could occur only if the modulus $q$ is composite. Although the expression $C_{q,a,b}$ should be non-zero and give rise to a bias, it seems the sign is very difficult to predict. Interestingly, for primes, as we shall review below, there is no bias in this case.
\item If exactly one of $a$ and $b$ is a quadratic residue then Corollary~\ref{cor:quad} tells us the direction of the bias (if it exists) is towards the quadratic residue. A sufficient condition for the bias to exist is Chowla's Conjecture.
\end{itemize}
\subsection{Comparison with primes}
Chebyshev's bias was originally studied in the case of primes, that is, replacing $S$ by the set of primes. Letting $\pi(x;q,a)$ be the numbers of primes up to $x$ lying in the arithmetic progression $a \bmod q$, Chebyshev famously observed in 1853 that $\pi(x;4,3)>\pi(x;4,1)$ happens more often than not \cite[pp.~697-698]{Tchebychef}.
Littlewood \cite{Littlewood1914} showed that $\pi(x;4,3)-\pi(x;4,1)$ changes sign infinitely often. Knapowski and Tur\'{a}n \cite{Knapowski1962} conjectured that $\pi(x;4,3)>\pi(x;4,1)$ holds 100\% of the time in natural density sense. This was refuted, under GRH, by Kaczorowski \cite{Kaczorowski1992,Kaczorowski1995}, who showed (conditionally) that $\{ x:\pi(x;4,3)>\pi(x;4,1) \}$ does not have a natural density, and that its upper natural density is strictly less than 1.
Rubinstein and Sarnak \cite{Rubinstein1994} studied the set $\{ x : \pi(x;q,a)>\pi(x;q,b)\}$ where $a \not\equiv b \bmod q$ and $(a,q)=(b,q)=1$.
They showed, under GRH and the Grand Simplicity Hypothesis (GSH) that this set has logarithmic density strictly between $0$ and $1$. Additionally, the logarithmic density is greater than $1/2$ if and only if $a$ is a nonquadratic residue and $b$ is a quadratic residue. In particular, no bias is present at all if both $a$ and $b$ are nonquadratic residues, as opposed to the sums of two squares analogue.
GSH asserts that the multiset of $\gamma \ge 0$ such that $L(1/2+i\gamma,\chi)=0$, for $\chi$ running over primitive Dirichlet characters, is linearly independent over $\mathbb{Q}$; here $\gamma$ are counted with multiplicity. It implies Chowla's Conjecture (since $0$ is linearly dependent) and that zeros of $L(s,\chi)$ are simple. As opposed to Chowla, it is very hard to gather evidence for GSH, even for individual L-functions. However, see Best and Trudgian for such evidence in the case of $\zeta$ \cite{Best2015}. In the literature, this hypothesis also goes under the name Linear Independence (LI).
\subsection{Strong biases}
Chebyshev's bias was studied in various settings and for various sets, e.g. \cite{Ng2000,Moree2004,Fiorilli2014,Fiorilli20142,Devin2020,Bailleul2021,Devin20212}, in particular for products of a fixed number of primes \cite{Dummit2016,Ford2010,Meng2018,Devin2021}.
As far as we are aware, Theorem~\ref{thm:bias} is the first instance where a set of integers of arithmetic interest -- in this case sums of two squares -- is shown to exhibit a complete Chebyshev's bias, that is, a bias that holds for a natural density-$1$ set of integers:
\[ \# \{ n \le x: M(n;q,a)>M(n;q,b)\} = x(1+o(1))\]
where $M(n;q,a)$ counts elements up to $n$ in a set $M\subseteq \mathbb{N}$ that are congruent to $a$ modulo $q$.
A key issue here is the natural density: Meng \cite{Meng2020} has a related work about a bias that holds for a logarithmic density-$1$ set of integers (see \S\ref{sec:martin conj}). Recently, Devin proposed a conjecture \cite[Conj.~1.2]{Devin20212} on a bias in logarithmic density $1$.
See Fiorilli \cite{Fiorilli20142,Fiorilli2014} for biases, in logarithmic density, that come arbitrarily close to $1$, and Fiorilli and Jouve \cite{fiorilli2020unconditional} for complete biases in `Frobenius sets' of primes (that generalize arithmetic progressions).
We also mention a very strong bias was proved by Dummit, Granville and Kisilevsky \cite{Dummit2016} who take Chebyshev's observation to a different direction. They show that substantially more than a quarter of the odd integers of the form $pq$ up to $x$, with $p$, $q$ both prime, satisfy $p \equiv q \equiv 3 \bmod 4$.
In the function field setting, complete biases were established in various special situations (e.g. for low degree moduli): \cite[Cor.~4.4]{Cha2008}, \cite[Thm.~1.5]{Cha2016}, \cite[Ex.~6]{Devin2021}. See also the work of Porritt \cite{Porritt2020} on which we elaborate in \S\ref{sec:generating}.
\section*{Acknowledgements}
We thank James Maynard and Zeev Rudnick for helpful discussions, Dor Elboim for teaching me about the Hankel contour, Dan Carmon for a typographic correction, Tom Bloom for pointing out the works of Kaczorowski and Andrew Granville for useful comments that substantially improved the presentation. We are thankful for the referee's thorough reading of the manuscript and valuable comments.
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 851318).
\section{Origin of the bias, computational evidence and a variation}
\subsection{Review of the original bias}
Fix a modulus $q$. All the constants below might depend on $q$. We give an informal explanation for the origin of the bias. It is instructive to start with the case of primes. By orthogonality of characters,
\[ \pi(x;q,a)-\pi(x;q,b) = \frac{1}{\phi(q)} \sum_{\chi_0 \neq \chi \bmod q} \overline{(\chi(a)-\chi(b))}\sum_{n \le x} \mathbf{1}_{n \text{ is a prime}} \chi(n).\]
The generating function of primes was studied by Riemann \cite{Riemann1859}, who showed that
\[ \sum_{n \ge 1} \frac{ \mathbf{1}_{n \text{ is a prime}}}{n^s} = \sum_{k \ge 1} \frac{\mu(k)}{k} \log \zeta(sk)\]
for $\Re s > 1$. Here $\mu$ is the M\"obius function, $\zeta$ is the Riemann zeta function and the logarithm is chosen so that $\log \zeta(s)$ is real if $s$ is real and greater than $1$.
More generally, given a Dirichlet character $\chi$ we have
\[ \sum_{n \ge 1} \frac{\mathbf{1}_{n \text{ is a prime}}\chi(n) }{n^s} = \sum_{k \ge 1} \frac{\mu(k)}{k} \log L(sk,\chi^k).\]
We may also write this $L$-function identity in terms of arithmetic functions:
\begin{align}\label{eq:primes} \mathbf{1}_{n \text{ is a prime}}\chi(n) &= \frac{\Lambda(n)\chi(n)}{\log n} +\sum_{k \ge 2} \frac{\mu(k)\Lambda(n^{1/k})\chi(n)}{\log n} \mathbf{1}_{n \text{ is a kth power}} \\
&=\frac{\Lambda(n)\chi(n)}{\log n} - \frac{1}{2} \mathbf{1}_{n =p^2, \, p\text{ prime}} \chi^2(p) + \alpha(n)\chi(n),
\end{align}
where $\Lambda$ is the von Mangoldt function and $\alpha$ is supported only on cubes and higher powers and its sum is negligible for all practical purposes.
Under GRH we can show that (see \cite[Lem.~2.1]{Rubinstein1994}) \[\sum_{n \le x} \frac{\chi(n)\Lambda(n)}{\log n} = \frac{\sum_{n \le x}\chi(n)\Lambda(n)}{\log x} + O\left( \frac{\sqrt{x}}{(\log x)^2} \right)\]
and (still under GRH) we can use the explicit formula to show that $\sum_{n \le x} \chi(n)\Lambda(n)$ is typically of order $\asymp \sqrt{x}$, in the sense that
\begin{equation}\label{eq:l2 primes} \frac{1}{X} \int_{X}^{2X} \left| \frac{\sum_{n \le x} \chi(n)\Lambda(n)}{\sqrt{x}}\right|^2 dx \ll 1,
\end{equation}
see \cite[Thm.~13.5]{Montgomery2007}. Under Linear Independence, one can show that the random variable
\[e^{-y/2}\sum_{n \le e^y} \chi(n)\Lambda(n)\]
has a limiting distribution with expected value $0$ (here $y$ is chosen uniformly at random between $0$ and $Y$, and $Y \to \infty$). The exponential change of variables leads to the appearance of logarithmic density. To summarize, $(\log x /\sqrt{x})\sum_{n \le x} \chi(n) \Lambda(n) /\log n$ (with $x=e^y$) has expectation $0$ and order of magnitude $\asymp 1$. The bias comes from the term $- \mathbf{1}_{n=p^2, \, p\text{ prime}}\chi^2(p)/2$. Indeed,
\[ -\frac{1}{2}\sum_{n \le x} \mathbf{1}_{n=p^2, \, p\text{ prime}} \chi^2(p)= -\frac{1}{2} \sum_{p \le \sqrt{x}} \chi^2(p).\]
If $\chi$ is a non-real character, $\chi^2$ is non-principal and GRH guarantees this sum is $o(\sqrt{x}/\log x$). However, if $\chi$ is real, this sum is of the same order of magnitude as $\sum_{n \le x} \chi(n) \Lambda(n)/\log n$, namely it is
\[ -\frac{1}{2}\sum_{n \le x} \mathbf{1}_{n=p^2, \, p\text{ prime}} \chi^2(p) = -\frac{1}{2} (\pi(\sqrt{x})+O(1)) \sim -\frac{\sqrt{x}}{\log x}\]
using the Prime Number Theorem.
Rubinstein and Sarnak replaced $\sum_{n \le x} \chi(n) \Lambda(n)/\log n$ with $\sum_{n \le x} \chi(n) \Lambda(n)/\log x$ using partial summation. This is advantageous as we have a nice explicit formula for the sum of $\chi(n)\Lambda(n)$. However, one can work directly with $\chi(n) \Lambda(n)/\log n$, whose generating function is $\log L(s,\chi)$, and this was done by Meng \cite{Meng2018} in his work on Chebyshev's bias for products of $k$ primes. Meng's approach is more flexible because it works even when the generating function has singularities which are not poles. So while $-L'(s,\chi)/L(s,\chi)=\sum_{n \ge 1} \Lambda(n)\chi(n)/n^s$ is meromorphic with simple poles at zeros of $L(s,\chi)$, which leads to the explicit formula by using the Residue theorem, Meng's approach can deal with $-\log L(s,\chi)$ directly although it does not have poles, rather it has essential singularities.
Meng's work applies in particular to $k=1$ and $k=2$, thus generalizing Rubinstein and Sarnak as well as Ford and Sneed \cite{Ford2010}.
\subsection{The generating function of sums of two squares}\label{sec:generating}
Let us now return to sums of two squares. Let $a,b$ be residues modulo $q$ with $(a,q)=(b,q)=1$ and $a\equiv b \equiv 1 \bmod (4,q)$. Orthogonality of characters shows
\begin{equation}\label{eq:sdiff}
S(x;q,a)-S(x;q,b) = \frac{1}{\phi(q)} \sum_{\chi_0 \neq \chi \bmod q} \overline{(\chi(a)-\chi(b))}\sum_{n \le x} \mathbf{1}_{S}(n) \chi(n).
\end{equation}
We want to relate $\sum_{n \le x}\mathbf{1}_{S}(n) \chi(n)$ to L-functions and their zeros, and obtain an analogue of the explicit formula for primes. The generating function of sums of two squares was studied by Landau \cite{Landau1908}, who showed that for $\Re s > 1$,
\begin{equation}\label{eq:landform} \sum_{n \ge 1} \frac{ \mathbf{1}_{S}(n)}{n^s} = \sqrt{\zeta(s)L(s,\chi_{-4})} H(s)
\end{equation}
where $H$ has analytic continuation to $\Re s >1/2$. Here the square root is chosen so that $\sqrt{\zeta(s)}$ and $\sqrt{L(s,\chi_{-4})}$ are real and positive for $s$ real and greater than $1$. This representation of the generating function plays a crucial role in the study of the distribution of sums of two squares, see e.g. \cite{Gorodetsky2021}.
Later, Shanks \cite[p.~78]{Shanks1964} and Flajolet and Vardi \cite[pp.~7-9]{flajolet1996zeta} (cf. \cite[Lem.~2.2]{Gorodetsky2021}) proved independently the identity
\begin{equation}\label{eq:geniden}
\sum_{n \ge 1} \frac{\mathbf{1}_{S}(n)}{n^s} = \sqrt{\zeta(s)L(s,\chi_{-4})(1-2^{-s})^{-1}} \prod_{k \ge 1} \left(\frac{(1-2^{-2^{k}s})\zeta(2^k s)}{L(2^ks,\chi_{-4})}\right)^{2^{-k-1}},
\end{equation}
and their proof can yield an analogue of \eqref{eq:geniden} with a twist by $\chi(n)$.
Shanks and Flajolet and Vardi were interested in efficient computation of the constant $K$, and this identity leads to
\[ K=\frac{1}{\sqrt{2}}\prod_{k \ge 1}\left(\frac{\zeta(2^k)(1-2^{-2^k})}{L(2^k,\chi_{-4})}\right)^{2^{-k-1}}.\]
Since both sides of \eqref{eq:geniden} enjoy Euler products, this identity can be verified by checking it locally at each prime; one needs to check $p=2$, $p\equiv 1 \bmod 4$ and $p \equiv 3 \bmod 4$ separately using \eqref{eq:characterization}. For the purpose of this paper we do not need the terms corresponding to $k>1$ in \eqref{eq:geniden}. What we need is stated and proved in Lemma~\ref{lem:analytic}, namely that
\begin{equation}\label{eq:simpiden} F(s,\chi):=\sum_{n \ge 1} \frac{\mathbf{1}_{S}(n) \chi(n)}{n^s} = \sqrt{L(s,\chi)L(s,\chi \chi_{-4})} \sqrt[4]{\frac{L(2s,\chi^2)}{L(2s,\chi^2 \chi_{-4})}} G(s,\chi)
\end{equation}
for $G$ which is analytic and non-vanishing in $\Re s > 1/4$ and bounded in $\Re s \ge 1/4 +\varepsilon$ for each $\varepsilon>0$. The important feature of this formula is that it allows us to analytically continue $F(s,\chi)$ to the left of $\Re s =1/2$ (once we remove certain line segments), as opposed to \eqref{eq:landform} whose limit is $\Re s > 1/2$. See the discussion at the end \S\ref{sec:analyticcont}.
Recently, a formula very similar to \eqref{eq:simpiden} was used by Porritt \cite{Porritt2020} in his study of character sums over polynomials with $k$ prime factors, and $k$ tending to $\infty$. We state his formula in the integer setting. Let $\Omega$ be the additive function counting prime divisors with multiplicity. He showed that, for complex $z$ with $|z|<2$, we have \cite[Eq.~(4)]{Porritt2020}
\[ \sum_{n \ge 1} \frac{z^{\Omega(n)}\chi(n)}{n^s} = L(s,\chi)^z L(2s,\chi^2)^{\frac{z^2-z}{2}} E_{z}(s,\chi)\]
for $E_z(s,\chi)$ which is analytic in $\Re s >\max\{1/3, \log_2 |z|\}$. He then proceeds to apply a Selberg-Delange type analysis, leading to an explicit formula for a polynomial analogue of $\sum_{n \le x, \, \Omega(n)=k} \chi(n)$ where $k$ grows like $a\log \log x$ for $a \in (0,2^{1/2})$ (in the polynomial world, $q$ and $q^{1/2}$ replace $2$ and $2^{1/2}$, where $q$ is the size of the underlying finite field). His results show a strong Chebyshev's bias once $a>1.2021\ldots$ \cite[Thm.~4]{Porritt2020}.
\subsection{Analyzing singularities}
We shall analyze each of the sums in \eqref{eq:sdiff}. We first observe that we do not need to analyze the sums corresponding to $\chi$ or $\chi\chi_{-4}$ being principal, because these characters do not contribute to \eqref{eq:sdiff} (as $\chi_0(a)=\chi_0(b)$).
Assume GRH and let $\chi$ be a non-principal character such that $\chi \chi_{-4}$ is also non-principal. We apply a truncated Perron's formula to $\sum_{n \le x}\mathbf{1}_{S}(n)\chi(n)$ (Corollary~\ref{cor:perrons}). We then want to shift the contour to $\Re s = 1/2 -c$ ($c=1/10$, say) and apply the Residue theorem.
We cannot do it, because $L(s,\chi)$ and $L(s,\chi\chi_{-4})$ have zeros on $\Re s= 1/2$ so $F(s,\chi)$, which involves the square root of $L(s,\chi)L(s,\chi\chi_{-4})$, cannot be analytically continued to $\Re s \ge 1/2 - c$. Let us analyze the zeros and poles, in the half-plane $\Re s > 1/4$, of $L(s,\chi)$, $L(s,\chi \chi_{-4})$, $L(2s,\chi^2)$ and $L(2s,\chi^2 \chi_{-4})$. These are the functions which appear in \eqref{eq:simpiden} and dictate the region to which we may analytically continue $F(s,\chi)$.
\begin{itemize}
\item We have zeros of $L(s,\chi)$, $L(s,\chi\chi_{-4})$ on $\Re s = 1/2$ (only) by GRH. We do not have poles at $s=1$ because we assume $\chi$, $\chi\chi_{-4}$ are non-principal.
\item Under GRH, $L(2s,\chi^2)$ and $L(2s,\chi^2 \chi_{-4})$ have no zeros in $\Re s >1/4$.
\item If $\chi^2$ is principal then $L(2s,\chi^2)$ has a simple pole at $s=1/2$. Similarly, if $\chi^2 \chi_{-4}$ is principal then $L(2s,\chi^2 \chi_{-4})$ has a simple pole at $s=1/2$. If $\chi^2$ and $\chi^2 \chi_{-4}$ are non-principal then these L-functions have no poles.
\end{itemize}
We call these zeros and poles `singularities of $F$'. They all lie on $\Re s = 1/2$. We construct an open set $A_{\chi}$ by taking the half-plane $\Re s > 1/4$ are removing the segments $\{\sigma+it:1/4<\sigma \le 1/2\}$ for every singularity $1/2+it$. This domain is simply connected and $L(s,\chi)$, $L(s,\chi\chi_{-4})$, $L(2s,\chi^2)$ and $L(2s,\chi^2\chi_{-4})$ have no poles or zeros there. Hence, they have well-defined logarithms there and we may analytically continue $F(s,\chi)$ to $A_{\chi}$.
Although we cannot literally shift the contour to the left of $\Re s = 1/2$, we can move to a contour which stays in $A_{\chi}$ and is to the left of $\Re s = 1/2$ `most of the time'. Specifically, we shall use truncated Hankel loop contours going around the singularities, joined to each other vertically on $\Re s = 1/2-c$, as in Meng \cite{Meng2018,Meng2020}. The precise contour is described in \S\ref{sec:contour}. See Figure~\ref{fig:Contour} for depiction.
A truncated Hankel loop contour around a singularity $\rho$ of $F(s,\chi)$ is a contour $\mathcal{H}_{\rho}$ traversing the path depicted in Figure~\ref{fig:Hankel}. It is parametrized in \eqref{eq:hankel}.
\begin{figure}
\centering
\begin{minipage}{.5\textwidth}
\centering
\begin{tikzpicture}
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\def2*\delta+2*\epsilon{2*0.2+2*0.02}
\defasin(\epsilon/\radius){asin(0.02/0.15)}
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\draw[line width=0.6pt,->](-0.1,0) -- (2*10,0);
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(10*0.2,-10*1.25+10*5*2*\delta+2*\epsilon+2*10*0.2+2*10*0.02)--(10*1.5,-10*1.25+10*5*2*\delta+2*\epsilon+2*10*0.2+2*10*0.02)
(10*1.5,10*0*2*\delta+2*\epsilon-10*1.25)--(10*0.2,10*0*2*\delta+2*\epsilon-10*1.25);
\foreach \x in {0,...,5}
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mark=at position 0.07 with{\arrow[line width =1pt]{>}},
mark=at position 0.22 with{\arrow[line width =1pt]{>}},
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(10*0.2,10*\x*2*\delta+2*\epsilon-10*1.25)--(10*0.2,-10*1.25+10*0.2+10*\x*2*\delta+2*\epsilon)--(10*0.42,-10*1.25+10*0.2+10*\x*2*\delta+2*\epsilon)
(10*0.42,-10*1.25+10*0.2+10*\x*2*\delta+2*\epsilon) arc[start angle=180+asin(\epsilon/\radius), end angle=540-asin(\epsilon/\radius),radius=10*0.15]
(10*0.42,-10*1.25+10*0.2+2*10*0.02+10*\x*2*\delta+2*\epsilon)--(10*0.2,-10*1.25+10*0.2+2*10*0.02+10*\x*2*\delta+2*\epsilon)--(10*0.2,-10*1.25+10*\x*2*\delta+2*\epsilon+2*10*0.2+2*10*0.02);}
\end{tikzpicture}
\captionof{figure}{Contour of integration}
\label{fig:Contour}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
\begin{tikzpicture}
\def10{10}
\def0.15{0.15}
\def0.02{0.02}
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\def0.2{0.2}
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\def1pt{1pt}
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\defcos(\n1){cos(asin(\epsilon/\radius))}
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mark=at position 0.07 with{\arrow[line width =1pt]{>}},
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(10*0.2,0)--(10*0.42,0)
(10*0.42,0) arc[start angle=180+asin(\epsilon/\radius), end angle=540-asin(\epsilon/\radius),radius=10*0.15]
(10*0.42,2*10*0.02)--(10*0.2,2*10*0.02);
\draw[line width=1pt](10*0.42+10*0.15,10*0.02)--++(45:10*0.15);
\coordinate (H) at (10*0.42+10*0.15,10*0.02); \node[below] at (H){\small\bf {$\rho$}};
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\end{tikzpicture}
\captionof{figure}{Truncated Hankel loop contour around $\rho$ with radius $r$}
\label{fig:Hankel}
\end{minipage}
\end{figure}
Given a character $\chi$ and a singularity $\rho$ of $F(s,\chi)$ we let
\[ f(\rho,\chi,x) := \frac{1}{2\pi i} \int_{\mathcal{H}_{\rho}} F(s,\chi)x^s \frac{ds}{s}\]
be the Hankel contour integral around $\rho$. By analyticity, the value of $f(\rho,\chi,x)$ is independent of $r$ (once $r$ is small enough) and for our purposes we choose $r=o(1/\log x)$.
We end up obtaining
\begin{equation}\label{eq:bothcases} \sum_{n \le x} \mathbf{1}_{S}(n) \chi(n) \approx \sum_{\substack{\rho:\, L(\rho,\chi)L(\rho,\chi \chi_{-4})=0\\ \text{or }\rho=\frac{1}{2}}} f(\rho,\chi,x).
\end{equation}
See Lemma~\ref{lem:perronshifted} for a statement formalizing \eqref{eq:bothcases} (in practice we sum only over zeros up to a certain height). If $F(s,\chi) \sim C(s-\rho)^{m}$ asymptotically as $s \to \rho^+$ (i.e. as $s-\rho$ tends to $0$ along the positive part of the real line) for one of the singularities $\rho$, it can be shown that
\begin{equation}\label{eq:loopsize}
f(\rho,\chi,x) \sim \frac{C}{\Gamma(-m)}\frac{x^{\rho}}{\rho}(\log x)^{-1-m}
\end{equation}
as $x \to \infty$. An informal way to see this is
\[ \int_{\mathcal{H}_{\rho}} F(s,\chi) x^s \frac{ds}{s} \approx \frac{C}{\rho}\int_{\mathcal{H}_{\rho}} (s-\rho)^m x^s ds = \frac{Cx^{\rho}}{\rho} (\log x)^{-1-m} \int_{(\log x)(\mathcal{H}_{\rho}-\rho)} z^m e^z dz\]
and the last integral gives \eqref{eq:loopsize} by Hankel's original computation \cite{Tenenbaum2015}.\footnote{To justify the first passage it suffices to show $\int_{\mathcal{H}_{\rho}} |G(s)x^s| |ds| =o(x^{\Re \rho}(\log x)^{-1-m})$ for $G(s) = F(s,\chi)/s - C(s-\rho)^m/\rho$, which is possible in our case.}.
In particular, from analyzing $\sqrt{L(s,\chi)L(s,\chi\chi_{-4})}$ we see that we have
\[ f(\rho,\chi,x) \sim c_{\rho}\frac{x^{\rho}}{\rho}(\log x)^{-1-\frac{m_\rho}{2}}, \qquad c_{\rho} = \frac{\lim_{s \to \rho^+} F(s,\chi)(s-\rho)^{-\frac{m_{\rho}}{2}}}{\Gamma(-\frac{m_{\rho}}{2})}, \]
for any given $\rho \neq 1/2$ where $m_{\rho}$ is the multiplicity of $\rho$ in $L(s,\chi)L(s,\chi\chi_{-4})$. As $m_{\rho} \ge 1$, we are led to think of $\sum_{ \rho \neq 1/2} f(\rho,\chi,x)$ as a quantity of order $\asymp \sqrt{x}(\log x)^{-3/2}$. However, we are not able to use \eqref{eq:loopsize} in order to bound $\sum_{\rho \neq 1/2} f(\rho,\chi,x)$ efficiently. We proceed via a different route and show
\begin{equation}\label{eq:average} \frac{1}{X}\int_{X}^{2X} \left|\frac{\sum_{\rho \neq 1/2} f(\rho,\chi,x)}{\sqrt{x}(\log x)^{-3/2}} \right|^2 dx \ll 1
\end{equation}
without making use of \eqref{eq:loopsize}. It follows that $\sum_{\rho\neq 1/2} f(\rho,\chi,x)\ll \sqrt{x}(\log x)^{-3/2}$ most of the time. This estimate is analogous to \eqref{eq:l2 primes} and its proof is similar too. We believe this sum is $ \ll_{\varepsilon}\sqrt{x}(\log x)^{\varepsilon-3/2}$ always but are not able to show this. This is similar to how GRH can show $\sum_{n \le x} \Lambda(n)\chi(n) \ll \sqrt{x} (\log x)^2$, but the true size in this question is expected to be $\ll_{\varepsilon} \sqrt{x} (\log x)^{\varepsilon}$. Here one should think of $f(\rho,\chi,x)$ as an analogue of the expression $-x^{\rho}/\rho$ from the explicit formula.
Finally, let us analyze the contribution of the (possible) singularity at $1/2$. If $L(1/2,\chi)L(1/2,\chi \chi_{-4})=0$ or if $\chi^2 \chi_{-4}$ is principal we can show (e.g. using \eqref{eq:loopsize}) that $f(1/2,\chi,x) \ll \sqrt{x}(\log x)^{-5/4}$, see Lemma~\ref{lem:hankelhalf}. The constant $5/4$ arises from analyzing $\sqrt{L(s,\chi)L(s,\chi\chi_{-4})}$ and $\sqrt[4]{L(2s,\chi^2)/L(2s,\chi^2 \chi_{-4})}$ and applying \eqref{eq:loopsize} with $m \ge 1/4$. It follows from \eqref{eq:bothcases} and \eqref{eq:average} that
\[ \frac{1}{X}\int_{X}^{2X} \left| \frac{\sum_{n \le x}\mathbf{1}_{S}(n)\chi(n)}{\sqrt{x}(\log x)^{-5/4}} \right|^2 dx \ll 1\]
unless $\chi^2$ is principal and $L(1/2,\chi)L(1/2,\chi \chi_{-4})\neq 0$. This remaining case leads to the bias. Indeed, we have
\[ F(s,\chi) \sim C\left(s-\frac{1}{2}\right)^{-1/4}\]
as $s\to \frac{1}{2}^{+}$ because of the fourth root in $\sqrt[4]{L(2s,\chi^2)}$, and this allows us to show (Lemma~\ref{lem:asymphalf}) that
\[ f(1/2,\chi,x) \asymp \sqrt{x}(\log x)^{-3/4}. \]
This is bigger than the typical contribution of all the other singularities by $(\log x)^{1/2}$.
In one line, the bias comes from the fact that the value of $m$ in the asymptotic relation $F(s,\chi) \sim C(s-\rho)^m$ ($s \to \rho^+$, $\rho$ a singularity) is minimized when $\chi^2$ is principal, $\rho=1/2$ and $L(1/2,\chi)L(1/2,\chi \chi_{-4})\neq0$, in which case $m=-1/4$.
\subsection{Computational evidence}
We examine the bias for $q \in \{3,5,15\}$. For $q=3$ we only have two relevant residues modulo $q$, namely $1 \bmod 3$ -- quadratic, and $2 \bmod 3$ -- nonquadratic. We have $L(1/2,\chi) \approx 0.480$ for the unique non-principal Dirichlet character modulo $3$ and $L(1/2,\chi \chi_{-4}) \approx 0.498$. Under GRH, Corollary~\ref{cor:quad} predicts $S(x;3,1)>S(x;3,2)$ for almost all $x$. Up to $10^8$, $96.8\%$ of the time $S(x;3,1)>S(x;3,2)$. See Figure~\ref{fig:mod3} for the (quite oscillatory) graph of $S(x;3,1)-S(x;3,2)$ up to $10^7$.
\begin{figure}[h!]
\caption{Graph of $S(x;3,1)-S(x;3,2)$ up to $10^7$.}
\centering
\includegraphics[width=15cm]{mod3.png}
\label{fig:mod3}
\end{figure}
For $q=5$ we have $4$ possible residues: $1$ and $4 \bmod 5$, both quadratic, and $2$ and $3 \bmod 5$, nonquadratic. We have $L(1/2,\chi)\approx 0.231$ and $L(1/2,\chi \chi_{-4})\approx 1.679$ for $\chi$, the unique non-principal quadratic character modulo $5$.
Corollary~\ref{cor:quad} predicts, under GRH, that $S(x;5,1)$ and $S(x;5,4)$ are almost always greater than $S(x;5,2)$ and $S(x;5,3)$.
The value of $C_{5,a,b}$ is simply $\chi(a)-\chi(b)$ times $(1+1/\sqrt{2})^{-1/2}\sqrt{L(1/2,\chi)L(1/2,\chi \chi_{-4})}$. For $(a,b) \in \{ (1,2),(1,3),(4,2),(4,3)\}$ its value is $\approx 0.7309$, since $\chi(a)=1$ and $\chi(b)=-1$. For $(a,b) \in \{ (1,2),(1,3),(4,2),(4,3)\}$, we find that the percentage of integers $n \le 10^7$ with $S(n;5,a)>S(n;5,b)$ is $96.1\%$, $95.2\%$, $95.3\%$ and $94.6\%$, respectively.
For $q=15$ we have the quadratic residues $1,4$ and the nonquadratic residues $2,7,8,11,13,14$. We expect $S(x;5,a)-S(x;5,b)$ to be positively biased for $a \in \{1,4\}$ and $b \in \{2,7,8,11,13,14\}$. Moreover, we expect a bias also for $S(x;5,a)-S(x;5,b)$ whenever $a\neq b \in \{2,7,8,11,13,14\}$ and $a/b \not\equiv 4 \bmod 15$, which means $(a,b) \neq (2,8)$, $(7,13)$, $(11,14)$. The direction of the bias in this case is harder to predict.
See Table~\ref{tab:1} for a table of the values of $C_{15,a,b}$ and Table~\ref{tab:2} for $\#\{n \le 10^7: S(n;15,a)>S(n;15,b)\}/10^7$ (in percentages).
We omit pairs with $a \ge b$ due to symmetry and pairs $a,b$ with $a/b$ being a quadratic residue.
\begin{table}[H]
\centering
\begin{tabular}{cccccccccc}
& & \multicolumn{8}{ c }{$b$} \\
& & 1 & 4 & 2 & 7 & 8 & 11 & 13 & 14 \\
\cline{3-10}
\multicolumn{1}{ c }{\multirow{7}{*}{$a$} } &
\multicolumn{1}{ c| }{1} & & & 1.427 & 9.698 & 1.427 &9.931 & 9.698 & 9.931
\\
\multicolumn{1}{ c }{} &
\multicolumn{1}{ c| }{4} & & & 1.427 & 9.698 & 1.427 & 9.931 & 9.698 &9.931 \\
\multicolumn{1}{ c }{} &
\multicolumn{1}{ c| }{2} & & & & 8.271 & & 8.504 & 8.271 & 8.504 \\
\multicolumn{1}{ c }{} &
\multicolumn{1}{ c| }{7} & & & & & -8.271 & 0.233 & & 0.233 \\
\multicolumn{1}{ c }{} &
\multicolumn{1}{ c| }{8} & & & & & & 8.504 & 8.271 & 8.504\\
\multicolumn{1}{ c }{} &
\multicolumn{1}{ c| }{11} & & & & & & & -0.233 &\\
\multicolumn{1}{ c }{} &
\multicolumn{1}{ c| }{13} & & & & & & & & 0.233
\end{tabular}
\caption{Values of $C_{15,a,b}$.}
\label{tab:1}
\end{table}
\begin{table}[H]
\centering
\begin{tabular}{cccccccccc}
& & \multicolumn{8}{ c }{$b$} \\
& & 1 & 4 & 2 & 7 & 8 & 11 & 13 & 14 \\
\cline{3-10}
\multicolumn{1}{ c }{\multirow{7}{*}{$a$} } &
\multicolumn{1}{ c| }{1} & & & 93.99 & 99.99 & 86.12 & 99.98 & 99.99 & 99.99
\\
\multicolumn{1}{ c }{} &
\multicolumn{1}{ c| }{4} & & & 96.28 & 99.97 & 90.72 & 99.99 & 99.99 & 99.96 \\
\multicolumn{1}{ c }{} &
\multicolumn{1}{ c| }{2} & & & & 99.90 & & 99.85 & 99.93 & 99.90 \\
\multicolumn{1}{ c }{} &
\multicolumn{1}{ c| }{7} & & & & & 0.03 & 57.99 & & 57.99 \\
\multicolumn{1}{ c }{} &
\multicolumn{1}{ c| }{8} & & & & & & 99.52 & 99.96 & 99.99\\
\multicolumn{1}{ c }{} &
\multicolumn{1}{ c| }{11} & & & & & & & 40.19 &\\
\multicolumn{1}{ c }{} &
\multicolumn{1}{ c| }{13} & & & & & & & & 59.23
\end{tabular}
\caption{Percentage of $n \le 10^7$ with $S(n;15,a)>S(n;15,b)$.}
\label{tab:2}
\end{table}
We see that the two tables are correlated. This is not a coincidence: the proof of Theorem~\ref{thm:bias} actually shows that $S(x;q,a)-S(x;q,b) = \phi(q)^{-1}C_{q}C_{q,a,b}\sqrt{x}/(\log x)^{3/4} + E(x)$ where $C_q$ is positive constant depending only on $q$ and $E(x)$ is a function which, on average, is smaller than $\sqrt{x}/(\log x)^{3/4}$. Concretely, $(1/X)\int_{X}^{2X} |E(x)|^2 dx \ll X/(\log X)^{5/2}$. So, for most values of $x$, $S(x;q,a)-S(x;q,b)$ is proportional to $C_{q,a,b}$.
\subsection{Martin's conjecture}\label{sec:martin conj}
Let $\omega$ be the additive function counting prime divisors (without multiplicity). In \cite{Meng2020}, Meng states a conjecture of Greg Martin, motivated by numerical data, saying that
\[ \left\{ x: \sum_{\substack{n \le x\\ n \equiv 1 \bmod 4}} \omega(n) < \sum_{\substack{n \le x\\ n \equiv 3 \bmod 4}} \omega(n)\right\}\]
contains all sufficiently large $x$. Meng assumed GRH and GSH to prove that this set has logarithmic density $1$. He also obtains results for other moduli under Chowla's Conjecture, and studies an analogous problem with the completely additive function $\Omega$. Meng writes: `In order to prove the full conjecture, one may need to formulate new ideas and introduce more powerful tools to bound the error terms of the summatory functions' \cite[Rem.~4]{Meng2020}. We are able to prove a natural density version of Meng's result, making progress towards Martin's conjecture. We do not assume GSH.
\begin{thm}\label{thm:omega}
Fix a positive integer $q $. Assume that GRH holds for the Dirichlet L-functions $L(s,\chi)$ for all Dirichlet character $\chi$ modulo $q$.
Then, whenever $a,b$ satisfy $(a,q)=(b,q)=1$ and \begin{equation}\label{eq:lincondomega} D_{q,a,b}:=\sum_{\substack{\chi \bmod q\\ \chi^2 = \chi_0}} (\chi(a)-\chi(b)) L\left( \frac{1}{2},\chi\right)>0
\end{equation}
we have, as $x \to \infty$,
\[ \# \left\{n \le x: \sum_{\substack{ m \le n \\ m \equiv a \bmod q}} \omega(m)< \sum_{\substack{ m \le n \\ m \equiv b \bmod q}} \omega(m)\right\} = x(1+o(1)) \]
and
\[ \# \left\{n \le x: \sum_{\substack{ m \le n \\ m \equiv a \bmod q}} \Omega(m)> \sum_{\substack{ m \le n \\ m \equiv b \bmod q}} \Omega(m)\right\} = x(1+o(1))\]
\end{thm}
The proof is given in \S\ref{sec:omega}. If $a$ is a quadratic residue modulo $q$ and $b$ is not, a sufficient condition for $D_{q,a,b}$ to be positive is Chowla's Conjecture.
\section{Preparatory lemmas}\label{sec:prep}
Given a Dirichlet character $\chi$ modulo $q$ we write
\[ F(s,\chi) = \sum_{n \in S} \frac{\chi(n)}{n^s} = \prod_{p \not \equiv 3 \bmod 4}(1-\chi(p)p^{-s})^{-1} \prod_{p \equiv 3 \bmod 4}(1-\chi^2(p)p^{-2s})^{-1}.\]
This converges absolutely for $\Re s > 1$. For $\Re s = 1$ (or smaller) it does not, because such convergence implies $\sum_{p \not \equiv 3 \bmod 4,\, p \nmid q}1/p$ converges. Observe that $F(s,\chi)$ does not vanish for $\Re s > 1$.
We shall use the convention where $\sigma$ and $t$ denote the real and complex parts of $s \in \mathbb{C}$.
\subsection{Perron}
\begin{lem}[Effective Perron's formula]\cite[p.~219, Thm.~2.3]{Tenenbaum2015}\label{lem:effective0}
Let $F(s)=\sum_{n \ge 1} a_n/n^s$ be a Dirichlet series with abscissa of absolute convergence $\sigma_a<\infty$. For $\kappa>\max\{0,\sigma_a\}$, $T \ge 1$ and $x \ge 1$ we have
\begin{equation}
\sum_{n \le x} a_n = \frac{1}{2\pi i} \int_{\kappa-iT}^{\kappa+iT} F(s)x^s \frac{ds}{s} + O\left(x^{\kappa}\sum_{n \ge 1} \frac{|a_n|}{n^{\kappa}(1+T|\log(x/n)|)}\right),
\end{equation}
with an absolute implied constant.
\end{lem}
This lemma leads to the following, which is a variation on \cite[p.~220,~Cor.~2.4]{Tenenbaum2015}.
\begin{cor}\label{cor:effbdd}
Suppose $|a_n| \le 1$. Let $F(s)=\sum_{n \ge 1} a_n/n^s$. Then, for $x,T \gg 1$,
\begin{equation}
\sum_{n \le x} a_n = \frac{1}{2\pi i} \int_{1+\frac{1}{\log x}-iT}^{1+\frac{1}{\log x}+iT} F(s)x^s \frac{ds}{s} + O\left(1 + \frac{x \log(xT)}{T}\right),
\end{equation}
with an absolute implied constant.
\end{cor}
\begin{proof}
Since $|a_n|\le 1$ we have $\sigma_a \le 1$ so that we may apply Lemma~\ref{lem:effective0} with $\kappa = 1 + 1/\log x$. The contribution of $n \le x/2$ to the error term is
\[\ll x \sum_{n \le x/2} \frac{1}{Tn} \ll \frac{x \log x}{T}.\]
The contribution of $n \ge 2x$ to the error is
\[ \ll x \sum_{k \ge 1} \sum_{n \in [2^k x, 2^{k+1} x)} \frac{1}{n^{\kappa}Tk} \ll x \sum_{k \ge 1} \frac{2^{-\frac{k}{\log x}}}{T} \ll \frac{x \log x}{T}.\]
Finally, if $n \in (x/2,2x)$, the contribution is
\[ \ll \sum_{n \in (x/2,2x)} \frac{1}{1+T|\log(x/n)|} \ll 1+\frac{x \log T}{T}\]
where the second inequality follows e.g. by the argument in \cite[p.~220, Cor.~2.4]{Tenenbaum2015}.
\end{proof}
As a special case of this corollary we have
\begin{cor}\label{cor:perrons}
Let $\chi$ be a Dirichlet character. We have
\begin{equation}\label{eq:applicableperron} \sum_{\substack{n \le x\\ n \in S}}\chi(n) = \frac{1}{2\pi i}\int_{1+\frac{1}{\log x}-iT}^{1+\frac{1}{\log x}+iT} F(s,\chi)x^s \frac{ds}{s} + O\left( 1+\frac{x\log (xT)}{T} \right).
\end{equation}
\end{cor}
\begin{remark}\label{rem:modif}
Since $|F(s,\chi)| \le \sum_{n \ge 1} n^{-1-1/\log x} \ll \log x$ when $\Re s = 1+1/\log x$, we see that perturbing the parameter $T$ (appearing in the range of integration) by $O(1)$ incurs an error of $O(x \log x/T)$ which is absorbed in the existing error term.
\end{remark}
\subsection{Analytic continuation}\label{sec:analyticcont}
Given a Dirichlet series $G(s)$ associated with a multiplicative function $g(n)$, which converges absolutely for $\Re s > 1$ and does not vanish there, we define the $k$th root of $G$ as
\[ G^{1/k}(s)=\exp\left( \frac{\log G(s)}{k}\right)\]
for each positive integer $k$, where the logarithm is chosen so that $\arg G(s) \to 0$ as $s \to \infty$.
The function $G^{1/k}$ is also a Dirichlet series, since
\[ G^{1/k}(s) = \prod_{p} \exp\left( \frac{\log \sum_{i \ge 0} \frac{g(p^i)}{p^{is}}}{k}\right). \]
\begin{lem}\label{lem:analytic}
Let $\chi$ be a Dirichlet character modulo $q$. We have, for $\Re s > 1$,
\[ F(s,\chi) = \sqrt{L(s,\chi)L(s,\chi \chi_{-4})} \sqrt[4]{\frac{L(2s,\chi^2)}{L(2s,\chi^2 \chi_{-4})}} G(s,\chi)\]
for $G$ which is analytic and non-vanishing in $\Re s > 1/4$ and bounded in $\Re s \ge 1/4 +\varepsilon$.
If $\chi$ is real then
\begin{equation}\label{eq:Ghalf} G\left( \frac{1}{2},\chi\right)= \left(1-\frac{\chi(2)}{\sqrt{2}}\right)^{-1/2}\left(1-\frac{\mathbf{1}_{2 \nmid q}}{2}\right)^{1/4}\prod_{\substack{p\equiv 3 \bmod 4\\ p \nmid q}} \left(1-\frac{1}{p^2}\right)^{-1/4}.
\end{equation}
\end{lem}
\begin{proof}
We have, for $\Re s > 1$,
\begin{align} \log G(s,\chi) &= \log F(s,\chi) - \frac{\log L(s,\chi)}{2} - \frac{\log L(s,\chi \chi_{-4})}{2} + \frac{\log L(2s,\chi^2 \chi_{-4})}{4} - \frac{\log L(2s,\chi^2)}{4} \\
\label{eq:logGdef}&= \frac{-\log(1-\chi(2)2^{-s})}{2} + \frac{\log(1-\chi^2(2)2^{-2s})}{4} + \sum_{p \equiv 3 \bmod 4} g_p(s,\chi)
\end{align}
for
\[ g_p(s,\chi) = \frac{-\log(1-\chi^2(p)p^{-2s})}{2} + \frac{\log(1-\chi^2(p)p^{-2s})-\log(1+\chi^2(p)p^{-2s})}{4}.\]
Each $g_p$ is analytic in $\Re s > 0$. Fix $c>0$. For $\Re s \ge c$ we have
\[ g_p(s,\chi) = \frac{\chi^4(p)}{4}p^{-4s} + O_c(p^{-6c}) = O_c(p^{-4c})\]
by Taylor expanding $\log(1+x)$. In particular, if $c=1/4+\varepsilon$ we have
\[ \left|\sum_{p \equiv 3 \bmod 4} g_p(s,\chi)\right| \ll \sum_{p \equiv 3 \bmod 4} p^{-4c} \ll \sum_{n \ge 1} n^{-1-4\varepsilon} < \infty \]
and so $G$ may be extended to $\Re s > 1/4$ via \eqref{eq:logGdef}.
As each $g_p$ is analytic in $\Re s > 1/4$, and $\sum_{p \equiv 3 \bmod 4} g_p(s,\chi)$ converges uniformly in $\Re s \ge 1/4+\varepsilon$ for every choice of $\varepsilon>0$, it follows that $\log G$ is analytic in $\Re s > 1/4$ and so is $G$. The formula for $G(1/2,\chi)$ for real $\chi$ follows from evaluating $g_p$ at $s=1/2$ and observing $\chi^2 =\chi_0$.
\end{proof}
Assuming GRH for $L(s,\chi)$ where $\chi$ is non-principal, $L(s,\chi)^{1/k}$ may be analytically continued to the region
\begin{equation}\label{eq:analyticreg} \mathbb{C} \setminus (\{ \sigma+it : L(1/2+it,\chi)=0 \text{ and }\sigma \le 1/2 \} \cup \{ \sigma: \sigma \le -1\})
\end{equation}
and is non-zero there (because of trivial zeros of $L(s,\chi)$ we have to remove $\{\sigma:\sigma\le -1\}$). For $\chi$ principal we have a singularity at $s=1$ and so
$L(s,\chi)^{1/k}$ may be analytically continued to
\begin{equation}\label{eq:analyticreg2} \mathbb{C} \setminus \left(\{ \sigma+it : L(1/2+it,\chi)=0 \text{ and }\sigma \le 1/2 \} \cup \{ \sigma: \sigma \le 1\}\right).
\end{equation}
Hence, given $\chi$ which is non-principal and such that $\chi \chi_{-4}$ is non-principal, we have the following.
Under GRH for $\chi$, $\chi \chi_{-4}$, $\chi^2$ and $\chi^2 \chi_{-4}$, we may continue $F(s,\chi)$ analytically to
\begin{equation}\label{eq:analyticcomplex}\left\{ s \in \mathbb{C}: \Re s > \frac{1}{4}\right\} \setminus \{ \sigma+it : L(1/2+it,\chi)L(1/2+it,\chi \chi_{-4})=0 \text{ and }\sigma \le 1/2 \}
\end{equation}
if both $\chi^2$ and $\chi^2 \chi_{-4}$ are non-principal; otherwise we may continue it to
\begin{equation}\label{eq:analyticreal}\left\{ s \in \mathbb{C}: \Re s > \frac{1}{4}\right\} \setminus \left(\{ \sigma+it : L(1/2+it,\chi)L(1/2+it,\chi \chi_{-4})=0 \text{ and }\sigma \le 1/2 \} \cup \{ \sigma: \sigma \le 1/2\}\right).
\end{equation}
\subsection{L-function estimates}
We quote three classical bounds on $L$-functions from the book of Montgomery and Vaughan \cite{Montgomery2007}.
\begin{lem}\label{lem:upper}\cite[Thm.~13.18 and Ex.~8 at \S13.2.1]{Montgomery2007}
Let $\chi$ be a Dirichlet character. Under GRH for $L(s,\chi)$, there exists a constant $A$ depending only on $\chi$ such that the following holds. Uniformly for $\sigma \ge 1/2$ and $|t| \ge 1$,
\[ |L(s,\chi)| \le \exp\left( A \frac{\log (|t|+4)}{\log \log (|t|+4)}\right).\]
\end{lem}
\begin{lem}\label{lem:lower}\cite[Thm.~13.23]{Montgomery2007}
Let $\chi$ be a Dirichlet character. Suppose $|t| \gg 1$. Under GRH for $L(s,\chi)$, there exists a constant $A$ depending only on $\chi$ such that the following holds. Uniformly for $\sigma \ge 1/2 + 1/\log \log (|t|+4)$ and $|t| \ge 1$,
\[ \left| \frac{1}{L(s,\chi)}\right| \le \exp\left( A \frac{\log(|t|+4)}{\log \log (|t|+4)}\right).\]
\end{lem}
\begin{lem}\label{lem:log}\cite[Cor.~13.16 and Ex.~6(c) at \S13.2.1]{Montgomery2007}
Let $\sigma \in (1/2,1)$ be fixed. Let $\chi$ be a primitive Dirichlet character. We have, as $|t| \to \infty$, \[|\log L(s,\chi)| \ll_{\sigma} \frac{(\log(|t|+4) )^{2-2\sigma}}{\log \log (|t|+4)}.\]
\end{lem}
The following is a consequence of the functional equation.
\begin{lem}\label{lem:fe}\cite[Cor.~10.10]{Montgomery2007}
Let $\chi$ be a Dirichlet character and $\varepsilon \in (0,1)$. We have $|L(s,\chi)| \asymp |L(1-s,\overline{\chi})| (|t|+4)^{1/2-\sigma}$ uniformly for $\varepsilon \le \sigma \le 1/2$ and $|t|\ge 1$, where the implied constants depend only on $\chi$ and $\varepsilon$.
\end{lem}
These four lemmas are originally stated for primitive characters. However, if $\chi$ is induced from a primitive character $\psi$, then in $\Re s >0$ the ratio $L(s,\chi)/L(s,\psi)$ is equal to the finite Euler product $\prod_{p:\, \chi(p)=0}(1-\psi(p)/p^s)$. This product is bounded away from $0$ and from $\infty$ when $\Re s \ge \varepsilon$, so we can convert results for $L(s,\psi)$ to results for $L(s,\chi)$ as long as we restrict our attention to $\sigma \ge \varepsilon$.
\subsection{Contour choice}\label{sec:contour}
Let $\chi$ be a non-principal Dirichlet character modulo $q$. Fix $c \in (0,1/8)$ (say, $c=1/10$). Let $T \gg 1$.
We want to use Cauchy's Integral Theorem to shift the vertical contour appearing in \eqref{eq:applicableperron} to the left of $\Re s = 1/2$ (namely to $\Re s = 1/2 - c$), at the `cost' of certain horizontal contributions. As we want to avoid zeros of $L(s,\chi)L(s,\chi \chi_{-4})$ and poles and zeros of $L(2s,\chi^2)/L(2s,\chi^2\chi_{-4})$ (which by GRH can only occur at $s=1/2$), we will use (truncated) Hankel loop contours to go around the relevant zeros and poles; the integrals over these loops will be the main contribution to our sum. It will also be convenient for $1/2- iT$ and $1/2+iT$ to avoid zeros of $L(s,\chi)L(s,\chi\chi_{-4})$; this is easy due to Remark~\ref{rem:modif}, showing that changing $T$ by $O(1)$ does not increase the error term arising from applying Perron's (truncated) formula. We replace the range $[1+1/\log x - iT,1+1/\log x+iT]$ with $[1+1/\log x-iT'', 1+1/\log x + i T']$ where $T-1 \le T',T'' \le T$ and \[L\left(\frac{1}{2}+it,\chi\right)L\left(\frac{1}{2}+it,\chi \chi_{-4}\right) \neq 0\]
for every $t \in [T',T) \cup (-T,-T'']$.
Let \begin{equation}\label{eq:zeros}
\gamma_1<\gamma_2<\ldots < \gamma_m
\end{equation}
be the imaginary parts of the zeros of $L(s,\chi)L(s,\chi \chi_{-4})$ on $\sigma = 1/2$ with $t\in (-T,T)$ (without multiplicities), and, if either $\chi^2$ or $\chi^2 \chi_{-4}$ is principal, we include the number $0$ (if it is not there already). Let $r \in (0,1)$ be a parameter that will tend to $0$ later. Consider the contour
\begin{equation}\label{eq:contour} I_1 \cup \bigcup_{j=1}^{m} (J_j \cup \mathcal{H}_{1/2+i\gamma_j}) \cup J_{m+1} \cup I_2
\end{equation}
where $I_1$ transverses the horizontal segment
\[ I_1 = \left\{ \sigma - i T'' : \frac{1}{2}-c \le \sigma \le 1+\frac{1}{\log x}\right\}\]
from right to left, $I_2$ transverses the horizontal segment
\[ I_2 = \left\{ \sigma + i T' : \frac{1}{2}-c \le \sigma \le 1+\frac{1}{\log x}\right\}\]
from left to right, $J_j$ transverses the following vertical segment from its bottom point to the top:
\[ J_j = \left\{ \frac{1}{2}-c +it: \gamma_{j-1} \le t \le \gamma_{j} \right\}\]
where
\[ \gamma_0 := -T'', \qquad \gamma_{m+1}:= T',\]
and finally each $\mathcal{H}_{\rho}$ traverses the following truncated Hankel loop contour in an anticlockwise fashion:
\begin{multline}\label{eq:hankel}
\left\{ s \in \mathbb{C}: \frac{1}{2}-c \le \Re s \le \frac{1}{2}-r, \, \Im s =\Im \rho, \, \arg(s-\rho)=-\pi\right\} \\ \cup \left\{ s\in \mathbb{C}: |s-\rho|=r, \, -\pi < \arg(s-\rho)<\pi\right\} \cup \left\{ \frac{1}{2}-c \le \Re s \le \frac{1}{2}-r, \Im s= \Im \rho, \, \arg(s-\rho)=\pi \right\}
\end{multline}
where in our case $c=1/10$ and $r=o(1/\log x)$. We refer the reader to Tenenbaum \cite[pp.~179-180]{Tenenbaum2015} for background on the Hankel contour and its truncated version.
If $r$ is small enough, the contour in \eqref{eq:contour} does not intersect itself.
If both $\chi^2$ and $\chi^2 \chi_{-4}$ are non-principal characters and the corresponding L-functions satisfy GRH
observe $\sqrt[4]{L(2s,\chi^2)/L(2s,\chi^2\chi_{-4})}$ is analytic in $\Re s > 1/2 - 2c > 1/4$ and so is $F(s,\chi)$ by Lemma~\ref{lem:analytic}.
If $\chi^2$ is principal then $\chi^2 \chi_{-4}$ cannot be principal. Similarly, if $\chi^2 \chi_{-4}$ is principal then $\chi$ is a non-real Dirichlet character of order $4$ and $\chi^2$ cannot be principal. In both cases, $\sqrt[4]{L(2s,\chi^2)/L(2s,\chi^2\chi_{-4})}$ has an algebraic singularity at $s=1/2$, which we avoid already as we inserted $0$ to the list \eqref{eq:zeros} if it is not there already.
In any case, by Cauchy's Integral Theorem,
\begin{equation}\label{eq:cauchy}
\frac{1}{2\pi i}\int_{1+\frac{1}{\log x}-iT''}^{1+\frac{1}{\log x}+iT'} F(s,\chi)x^s \frac{ds}{s} = \frac{1}{2\pi i}\left( \int_{I_1} + \sum_{j=1}^{m}\left(\int_{J_j} + \int_{\mathcal{H}_{1/2+i\gamma_j}}\right) + \int_{J_{m+1}} + \int_{I_2} \right) F(s,\chi)x^s \frac{ds}{s}.
\end{equation}
\begin{lem}\label{lem:perronshifted}
Let $\chi$ be a non-principal Dirichlet character. Assume GRH holds for the following four characters:
\begin{equation}\label{eq:fourchars}
\chi, \qquad \chi \chi_{-4},\qquad \chi^2, \qquad \chi^2 \chi_{-4}.
\end{equation}
Let $c \in (0,1/8)$ be a fixed constant. Let $T \gg 1$. We have
\begin{multline} \sum_{\substack{n \le x \\ n \in S}} \chi(n) = \sum_{j=1}^{m} \frac{1}{2\pi i}\int_{\mathcal{H}_{1/2+i\gamma_j}} F(s,\chi)x^s \frac{ds}{s} \\+ O\left( 1 + \frac{x\log (xT)}{T}+\frac{(x+\max\{x,T\}^{1/2}+x^{\frac{1}{2}-c}T^{c+1})\exp\left( \frac{A \log T}{\log \log T}\right)}{T}\right).
\end{multline}
The implied constant and $A$ depend only on $\chi$ and $c$.
\end{lem}
\begin{proof}
By \eqref{eq:applicableperron}, Remark~\ref{rem:modif} and
\eqref{eq:cauchy}, it suffices to upper bound $\int_{I_j}|F(s,\chi)| |x^s| |ds|/|s|$ and $\int_{J_j}|F(s,\chi)| |x^s| |ds|/|s|$. We first treat $I_j$, and concentrate on $I_2$ as the argument for $I_1$ is analogous. We have, using Lemma~\ref{lem:analytic},
\begin{align} \int_{I_2}|F(s,\chi)| |x^s| \frac{|ds|}{|s|} &\ll \frac{1}{T}\int_{\frac{1}{2}-c}^{1+\frac{1}{\log x}} \sqrt{|L(\sigma+iT',\chi)L(\sigma+iT', \chi \chi_{-4})|} \sqrt[4]{\frac{|L(2\sigma+2iT',\chi^2)|}{|L(2\sigma+2iT',\chi^2\chi_{-4})|}} x^{\sigma}d\sigma\\
&\ll \frac{1}{T}\int_{\frac{1}{2}-c}^{1+\frac{1}{\log x}} \sqrt{|L(\sigma+iT',\chi)L(\sigma+iT', \chi \chi_{-4})|} \sqrt[4]{\frac{|L(2\sigma+2iT',\chi^2)|}{|L(2\sigma+2iT',\chi^2\chi_{-4})|}} x^{\sigma}d\sigma.
\end{align}
It is now convenient to consider $\sigma \ge 1/2$ and $\sigma \le 1/2$ separately.
If $\sigma \ge 1/2$ we bound all the relevant $L$-functions using Lemmas~\ref{lem:upper} and \ref{lem:lower}, obtaining that this part of the integral contributes
\[ \ll \frac{\exp\left( \frac{A \log T}{\log \log T}\right)}{T} \int_{\frac{1}{2}}^{1+\frac{1}{\log x}} x^{\sigma} d\sigma \ll \frac{x\exp\left( \frac{A \log T}{\log \log T}\right)}{T}\]
where $A$ is a constant large enough depending on $\chi$.
For $\sigma$ below $1/2$ we first apply Lemma~\ref{lem:fe} to the $L$-functions of $\chi$ and $\chi \chi_{-4}$ to reduce to the situation where the real parts of the variables inside the $L$-functions are $\ge 1/2$. Then we apply Lemmas~\ref{lem:upper} and \ref{lem:lower} as before, obtaining that this part of the integral contributes
\[ \ll \frac{\exp\left( \frac{A \log T}{\log \log T}\right)}{T} \int_{\frac{1}{2}-c}^{\frac{1}{2}} T^{1/2-\sigma}x^{\sigma} d\sigma \ll \frac{\max\{x,T\}^{1/2}\exp\left( \frac{A \log T}{\log \log T}\right)}{T}.\]
It follows that
\[ \left(\int_{I_1} + \int_{I_2}\right)|F(s,\chi)| |x^s| \frac{|ds|}{|s|} \ll \frac{(x+\max\{x,T\}^{1/2})\exp\left( \frac{A \log T}{\log \log T}\right)}{T}. \]
We turn to the contribution of $J_j$. We have
\begin{align} \sum_{j=1}^{m+1} \int_{J_j} |F(s,\chi)| |x^s| \frac{|ds|}{|s|} &\ll x^{\frac{1}{2}-c}\int_{-T-1}^{T+1} \left| F\left( \frac{1}{2}-c+it,\chi\right)\right| \frac{dt}{|t|+1}\\
& \ll x^{\frac{1}{2}-c}\left( 1 + \sum_{2^k \le 2T}2^{-k} \left(\int_{2^{k-1}}^{2^{k}}+\int_{-2^{k}}^{-2^{k-1}}\right) \left| F\left( \frac{1}{2}-c+it,\chi\right)\right|dt\right)
\end{align}
where
\begin{align} \left| F\left( \frac{1}{2}-c+it,\chi\right)\right| &= \sqrt{\left|L\left(\frac{1}{2}-c+it,\chi\right)L\left(\frac{1}{2}-c+it, \chi \chi_{-4}\right)\right|} \sqrt[4]{\frac{|L(1-2c+2it,\chi^2)|}{|L(1-2c+2it,\chi^2\chi_{-4})|}} \\
&\ll (|t|+4)^{c} \exp\left( A \frac{\log(|t|+4)}{\log \log (|t|+4)}\right) \ll T^{c} \exp\left( \frac{A\log T}{\log \log T}\right),
\end{align}
where we used Lemmas~\ref{lem:upper} and \ref{lem:fe} to bound the $L$-functions of $\chi$ and $\chi \chi_{-4}$ and Lemma~\ref{lem:log} for the other two L-functions. This leads to
\[ \sum_{j=1}^{m+1} \int_{J_j} |F(s,\chi)| |x^s| \frac{|ds|}{|s|} \ll x^{\frac{1}{2}-c} T^{c} \exp\left( \frac{A\log T}{\log \log T}\right)\]
and concludes the proof.
\end{proof}
\section{Hankel calculus}
In this section, $\mathcal{H}_{\rho}$ is the Hankel contour described in \eqref{eq:hankel}, going around $\rho$ in an anticlockwise fashion.
\begin{lem}\label{lem:ratio}
Let $\chi$ be a Dirichlet character. Assume GRH for $L(s,\chi)$. Given a non-trivial zero $\rho=1/2+i\gamma$ of $L(s,\chi)$ we have
\[ \max_{s \in \mathcal{H}_{\rho}} \left| \frac{L(s,\chi)}{s-\rho}\right| \ll (|\gamma|+1)^{c+o(1)}.\]
Here the $o(1)$ exponent goes to $0$ as $\gamma$ goes to $\infty$ (and it might depend on $\chi$), and the implied constant is absolute.
\end{lem}
\begin{proof}
Since we can write $L(s,\chi)$ as $L(\rho,\chi)$ plus an integral of $L'(z,\chi)$ over a line segment connecting $s$ and $\rho$, it follows that the maximum we try to bound is
\[\ll \max_{\substack{|s-\rho| \le r, \text{ or }\\ s=\sigma+i\gamma \text{ with}\\\frac{1}{2}-c \le \sigma \le \frac{1}{2}}} |L'(s,\chi)|. \]
By Cauchy's integral formula, Lemma~\ref{lem:upper} and Lemma~\ref{lem:fe},
\[ |L'(s,\chi)| \ll \int_{|z-s|=\frac{1}{\log (|\gamma|+1)}} \frac{|L(z,\chi)|}{|z-s|}|dz| \ll (|\gamma|+1)^{\max\{0,\frac{1}{2}-\Re s\} + o(1)},\]
implying the desired bound.
\end{proof}
\begin{lem}\label{lem:hankelabsolute}
Let $\chi$ be a non-principal Dirichlet character. Assume GRH for the characters in \eqref{eq:fourchars}. Given a non-trivial zero $\rho=1/2+i\gamma \neq 1/2$ of $L(s,\chi)L(s,\chi \chi_{-4})$ we have
\[\int_{\mathcal{H}_{\rho}} |F(s,\chi)||x^s| |ds| \ll \sqrt{x} (|\gamma|+1)^{c+o(1)}\left( (\log x)^{-3/2} + r^{3/2} x^r \right).\]
Here the $o(1)$ exponent goes to $0$ as $\gamma$ goes to $\infty$ (and might depend on $\chi$), and the implied constant depends only on $\chi$ (it is independent of $r$).
\end{lem}
\begin{proof}
We have, for $s$ on the contour,
\[ |F(s,\chi)| \ll \sqrt{|L(s,\chi)L(s,\chi \chi_{-4})|} (|\gamma|+1)^{o(1)} \]
by Lemmas~\ref{lem:analytic}, \ref{lem:upper} and \ref{lem:lower}.
Integrating $\sqrt{|L(s,\chi)L(s,\chi \chi_{-4})|}|x^{s}|$ over the circle part of the contour contributes
\[ \ll rx^{\frac{1}{2}+r}\int_{-\pi}^{\pi} \sqrt{\left|L\left(\rho+re^{i\theta},\chi\right)L\left(\rho+re^{i\theta},\chi\chi_{-4}\right)\right|}d\theta.\]
Writing $L(s,\chi)L(s,\chi \chi_{-4})$ as $L(s,\chi)L(s,\chi \chi_{-4})/(s-\rho)$ times $s-\rho$ and appealing to Lemmas~\ref{lem:ratio}, \ref{lem:upper} and \ref{lem:fe}, we find that this is
\[ \ll rx^{\frac{1}{2}+r} (|\gamma|+1)^{c+o(1)}\int_{-\pi}^{\pi} \sqrt{|re^{i\theta}|}d\theta \ll r^{3/2}x^{\frac{1}{2}+r} (|\gamma|+1)^{c+o(1)}.\]
Integrating $\sqrt{|L(s,\chi)L(s,\chi \chi_{-4})|}|x^{s}|$ over one of the segment parts of the contour contributes
\[ \ll \sqrt{x}\int_{-c}^{0} \sqrt{\left|L\left(\rho+t,\chi\right)L\left(\rho+t,\chi\chi_{-4}\right)\right|}x^{t}dt.\]
Again writing $L(s,\chi)L(s,\chi \chi_{-4})$ as $L(s,\chi)L(s,\chi \chi_{-4})/(s-\rho)$ times $s-\rho$ and appealing to Lemma~\ref{lem:ratio}, we can bound this contribution by
\begin{equation}
\ll \sqrt{x} (|\gamma|+1)^{c+o(1)} \int_{-\infty}^{0} \sqrt{|t|}x^t dt \ll \sqrt{x} (|\gamma|+1)^{c+o(1)} (\log x)^{-3/2},
\end{equation}
concluding the proof.
\end{proof}
\begin{lem}\label{lem:l2bound}
Let $\chi$ be a non-principal Dirichlet character. Assume GRH for the characters in \eqref{eq:fourchars}. For any pair $\rho_1=1/2+i\gamma_1$, $\rho_2=1/2+i\gamma_2$ of non-trivial zeros different from $1/2$ we have
\begin{equation} \frac{1}{X}\int_{X}^{2X} \int_{\mathcal{H}_{\rho_1}}\int_{\mathcal{H}_{\rho_2}} F(s_1,\chi) \overline{F(s_2,\chi)} \frac{x^{s_1}}{s_1}\frac{x^{\overline{s_2}}}{\overline{s_2}}ds_1 \overline{ds_2} dx\ll \frac{X \left( (\log X)^{-3/2} + r^{3/2} X^r \right)^2}{((|\gamma_1|+1) (|\gamma_2|+1))^{1-c+o(1)}(1+|\gamma_1-\gamma_2|)}
\end{equation}
where implied constants depend only on $\chi$.
\end{lem}
\begin{proof}
We first integrate by the $x$-variable and then take absolute values, obtaining that the integral is
\begin{align} &\ll \int_{\mathcal{H}_{\rho_1}}\int_{\mathcal{H}_{\rho_2}} |F(s_1,\chi)| |F(s_2,\chi)| \frac{X^{\Re(s_1+s_2)}}{|s_1| |s_2| |s_1 + \overline{s_2}+1|}|ds_1| | ds_2| \\
&\ll \frac{1}{|\rho_1 \rho_2|(1+|\gamma_1-\gamma_2|)}\int_{\mathcal{H}_{\rho_1}}\int_{\mathcal{H}_{\rho_2}} |F(s_1,\chi)| |F(s_2,\chi)| X^{\Re(s_1+s_2)}|ds_1| | ds_2| \\
& = \frac{1}{|\rho_1 \rho_2|(1+|\gamma_1-\gamma_2|)}\left( \int_{\mathcal{H}_{\rho_1}} |F(s_1,\chi)| X^{\Re(s_1)} |ds_1| \right) \left( \int_{\mathcal{H}_{\rho_2}} |F(s_2,\chi)| X^{\Re(s_2)} |ds_2| \right) .
\end{align}
The result now follows from Lemma~\ref{lem:hankelabsolute}.
\end{proof}
\begin{lem}\label{lem:hankelhalf}
Let $\chi$ be a non-principal Dirichlet character. Assume GRH for the characters in \eqref{eq:fourchars}. If $\chi^2 \chi_{-4}$ is principal, or if $\chi^2$ is principal as well as $L(1/2,\chi)L(1/2,\chi \chi_{-4})=0$ then
\[ \int_{\mathcal{H}_{1/2}} F(s,\chi)\frac{x^s}{s} ds \ll \sqrt{x} \left( x^r r^{5/4} + \frac{1}{(\log x)^{5/4}}\right).\]
The implied constant depends only on $\chi$ (it is independent of $r$).
\end{lem}
\begin{proof}
This is a variation on Lemma~\ref{lem:hankelabsolute}. If $\chi^2 \chi_{-4}$ is principal we have
\[ |F(s,\chi)| \ll \sqrt[4]{|2s-1|}\]
on $\mathcal{H}_{1/2}$, where the implied constant depends on $\chi$ only. Similarly, if $\chi^2$ is principal and $L(1/2,\chi)L(1/2,\chi \chi_{-4})=0$ then
\[ |F(s,\chi)| \ll \sqrt{\left| s- \frac{1}{2}\right|} \sqrt[4]{\frac{1}{|2s-1|}}\le \sqrt[4]{|2s-1|}\]
by Lemma~\ref{lem:ratio}. In both cases the integral is
\[ \ll \sqrt{x}\int_{\mathcal{H}_{1/2}} \sqrt[4]{\left| s- \frac{1}{2}\right|}x^{\max\{\Re s - \frac{1}{2},0\}} |ds|. \]
The contribution of $\Re s \ge 1/2$ is $\ll \sqrt{x}x^{r} r^{5/4}$, while the contribution of $\Re s \le 1/2$ is
\[ \ll \sqrt{x} \int_{-\infty}^{0} \sqrt[4]{|t|}x^{t} dt \ll \frac{\sqrt{x}}{(\log x)^{5/4}}, \]
concluding the proof.
\end{proof}
Let $C_q$ be the following positive constant, depending only on $q$:
\begin{equation}\label{eq:cqdef} C_{q} = \frac{2\pi^{-1/4}}{\Gamma(1/4)} \prod_{p \equiv 3 \bmod 4} \left(1-\frac{1}{p^2}\right)^{-1/4} \prod_{\substack{p \mid q \\ p \equiv 3 \bmod 4}} \left(1 - \frac{1}{p}\right)^{1/2}.
\end{equation}
\begin{lem}\label{lem:asymphalf}
Let $\chi$ be a non-principal Dirichlet character modulo $q$. Assume GRH for the characters in \eqref{eq:fourchars}. If $\chi^2$ is principal and $L(1/2,\chi)L(1/2,\chi\chi_{-4})\neq 0$ then
\begin{multline}\frac{1}{2\pi i}\int_{\mathcal{H}_{1/2}} F(s,\chi)\frac{x^s}{s} ds = C_{q}\frac{\sqrt{x}}{(\log x)^{3/4}} \left(1-\frac{\chi(2)}{\sqrt{2}}\right)^{-1/2}\sqrt{L\left(\frac{1}{2},\chi\right)L\left(\frac{1}{2},\chi \chi_{-4}\right)} \left( 1+O\left( x^{-c/2}\right)\right)
\\ + O\left( \sqrt{x}\left( x^r r^{7/4} + \frac{1}{(\log x)^{7/4}} \right) \right)
\end{multline}
where the implied constants depend only on $\chi$.
\end{lem}
\begin{proof}
Let $M(s,\chi) = \sqrt{L(s,\chi)L(s,\chi\chi_{-4})} \sqrt[4]{L(2s,\chi^2)(2s-1)/L(2s,\chi^2 \chi_{-4})}G(s,\chi)/s$ where $G$ is defined in Lemma~\ref{lem:analytic}. On $\mathcal{H}_{1/2}$ we have $F(s,\chi)/s=M(s,\chi)(2s-1)^{-1/4}$. We define $M$ at $s=1/2$ by its limit there, which exists as $L(s,\chi_0) = \zeta(s)\prod_{p\mid q}(1-p^{-s})$ has a simple pole at $s=1$. In fact, $M$ is analytic in a neighborhood of $s=1/2$ by our assumption on $\chi$ and $\chi_{-4}$. We have
\[ M\left(\frac{1}{2},\chi\right) = 2\sqrt{L\left(\frac{1}{2},\chi\right)L\left(\frac{1}{2},\chi \chi_{-4}\right)} \prod_{p \mid q}\left(1-\frac{1}{p}\right)^{1/4}L(1,\chi_0 \chi_{-4})^{-1/4}G\left( \frac{1}{2},\chi\right). \]
The expression $L(1,\chi_0\chi_{-4})$ may be simplified as $L(1,\chi_{-4}) \prod_{p \mid q}(1-\chi_{-4}(p)/p) = \prod_{p \mid q}(1-\chi_{-4}(p)/p)\pi/ 4$.
Our integral is
\[\frac{1}{2\pi i}\left(M\left( \frac{1}{2},\chi \right) \int_{\mathcal{H}_{1/2}} (2s-1)^{-1/4}x^s ds + \int_{\mathcal{H}_{1/2}} (2s-1)^{-1/4}x^s \left( M(s,\chi)- M\left(\frac{1}{2},\chi\right)\right) ds\right).\]
The second integral here is small, namely $\ll\sqrt{x} \left( x^r r^{7/4} + (\log x)^{-7/4}\right)$, by an argument parallel to Lemma~\ref{lem:hankelhalf}. It suffices to show that
\[ \frac{1}{2\pi i}\int_{\mathcal{H}_{1/2}} \left(s-\frac{1}{2}\right)^{-1/4} x^s ds = \frac{1}{\Gamma(1/4)} \frac{\sqrt{x}}{(\log x)^{3/4}} \left( 1+O\left( x^{-c/2}\right)\right).\]
Making the change of variables $\left(s-\frac{1}{2}\right)\log x =y$, this boils down to Hankel's $\Gamma$-function representation, see e.g. \cite[p.~180,~Cor.~0.28]{Tenenbaum2015}.
\end{proof}
\section{Proof of Theorem~\ref{thm:bias}}
\subsection{Character sum estimates}
\begin{proposition}\label{prop:charsums}
Let $\chi$ be a non-principal Dirichlet character modulo $q$. Assume GRH for the characters in \eqref{eq:fourchars}. If $\chi^2 \neq \chi_0$ we have
\[ \frac{1}{X} \int_{X}^{2X}\left|\sum_{\substack{n \le x\\ n \in S}} \chi(n)\right|^2 dx \ll \frac{X}{(\log X)^3},\]
while if $\chi^2 =\chi_0$ we have
\[ \frac{1}{X} \int_{X}^{2X}\left|\sum_{\substack{n \le x\\ n \in S}} \chi(n) - C_q \frac{\sqrt{x}}{(\log x)^{3/4}} \left(1-\frac{\chi(2)}{\sqrt{2}}\right)^{-1/2}\sqrt{L\left(\frac{1}{2},\chi\right)L\left(\frac{1}{2},\chi\chi_{-4}\right)}\right|^2 dx \ll \frac{X}{(\log X)^{5/2}},\]
where $C_q$ is defined in \eqref{eq:cqdef}. The implied constants depend only on $q$.
\end{proposition}
\begin{proof}
By Lemma~\ref{lem:perronshifted} with $T = X^{3/4} \asymp x^{3/4}$ and $c=1/10$ we have, uniformly for $x \in [X,2X]$,
\[\sum_{\substack{n \le x \\ n \in S}} \chi(n) = \sum_{j=1}^{m} \frac{1}{2\pi i}\int_{\mathcal{H}_{1/2+i\gamma_j}} F(s,\chi)x^s \frac{ds}{s}+ O\left( x^{\frac{1}{2}-\frac{1}{100}}\right)
\]
for any non-principal $\chi$. The function $F$ is defined in the first line of \S\ref{sec:prep} and is analytic in the set \eqref{eq:analyticreg} or in the set \eqref{eq:analyticreg2}, depending on $\chi$. The $m=m_{\chi}$ contours $\mathcal{H}_{1/2+i\gamma_j}=\mathcal{H}_{1/2+i\gamma_j,\chi}$ are defined in \S\ref{sec:contour}. They are Hankel loop contours going anticlockwise around zeros $1/2+i\gamma_j$ of $L(s,\chi)L(s,\chi\chi_{-4})$ up to height $T$ (exclusive), as well as around $s=1/2$ in case $\chi^2$ or $\chi^2 \chi_{-4}$ is principal.
Let us write
\[\sum_{\substack{n \le x \\ n \in S}} \chi(n) = S_1(x) + S_2(x) + O\left(x^{\frac{1}{2}-\frac{1}{100}}\right)\]
where
$S_1(x)$ is the contribution of Hankel loops not going around $1/2$:
\[ S_1(x) = \sum_{\substack{1 \le j \le m\\ \gamma_j \neq 0}} \frac{1}{2\pi i} \int_{\mathcal{H}_{1/2+i\gamma_j}} F(s,\chi) x^s \frac{ds}{s},\]
and $S_2$ is the contribution of the loop around $s=1/2$, in case such a loop exists:
\[ S_2(x) = \begin{cases}\frac{1}{2\pi i} \int_{\mathcal{H}_{1/2}} F(s,\chi) x^s \frac{ds}{s}& \text{if }L(1/2,\chi)L(1/2,\chi\chi_{-4})=0 \text{ or } \chi_0 \in \{\chi^2\chi_{-4}, \chi^2\},\\0 & \text{otherwise.}\end{cases}\]
We shall take $r=o(1/\log X)$ in all the definitions of the loops. If $L(1/2,\chi)L(1/2,\chi \chi_{-4})=0$ or $\chi^2 \chi_{-4}= \chi_0$ we have, by Lemma~\ref{lem:hankelhalf}, the pointwise bound
\[ S_{2}(x) \ll \frac{\sqrt{x}}{(\log x)^{5/4}}.\]
If $\chi^2=\chi_0$ and $L(1/2,\chi)L(1/2,\chi \chi_{-4})\neq 0$, we have by Lemma~\ref{lem:asymphalf} the following asymptotic relation:
\[ S_{2}(x) = C_{q} \frac{\sqrt{x}}{(\log x)^{3/4}} \left(1-\frac{\chi(2)}{\sqrt{2}}\right)^{-1/2}\sqrt{L\left(\frac{1}{2},\chi\right)L\left(\frac{1}{2},\chi \chi_{-4}\right)} + O\left( \frac{\sqrt{x}}{(\log x)^{7/4}}\right).\]
In all cases,
\[ S_2(x) = \mathbf{1}_{\chi^2 = \chi_0}C_q \frac{\sqrt{x}}{(\log x)^{3/4}} \left(1-\frac{\chi(2)}{\sqrt{2}}\right)^{-1/2}\sqrt{L\left(\frac{1}{2},\chi\right)L\left(\frac{1}{2},\chi\chi_{-4}\right)} + O\left(\frac{\sqrt{x}}{(\log x)^{5/4}}\right).\]
It now suffices to show that
$(1/X)\int_{X}^{2X} \left|S_1(x)\right|^2 dx \ll X/(\log X)^3$.
We have, by Lemma~\ref{lem:l2bound},
\begin{align}\label{eq:l2s1} \frac{1}{X} \int_{X}^{2X} \left|S_1(x)\right|^2 dx &\ll \frac{1}{X} \int_{X}^{2X} \left| \sum_{\substack{1 \le j \le m \\ \gamma_j \neq 0}} \int_{\mathcal{H}_{1/2+i\gamma_j}} F(s,\chi) x^s \frac{ds}{s} \right|^2 \\
&\ll \frac{X}{(\log X)^3} \sum_{\substack{\gamma_1,\gamma_2 \neq 0:\\ L(1/2+i\gamma_j,\chi)=0\text{ or}\\L(1/2+i\gamma_j,\chi \chi_{-4})=0}} \frac{1}{|\gamma_1 \gamma_2|^{1-1/5}(1+|\gamma_1-\gamma_2|)}.
\end{align}
The sum over zeros converges by a standard argument, see \cite[Thm.~13.5]{Montgomery2007} where this is proved in the case of zeros of the Riemann zeta function. The only input needed is that between height $T$ and $T+1$ there are $\ll \log T$ zeros, which is true for any Dirichlet L-function, see \cite[Thm.~10.17]{Montgomery2007}.
\end{proof}
\subsection{Conclusion of proof}
Suppose $a,b$ satisfy $a\equiv b \equiv 1 \bmod (4,q)$ and $(a,q)=(b,q)=1$. Suppose the constant $C_{q,a,b}$ appearing in \eqref{eq:lincond} is positive.
Consider $X \gg 1$ which will tend to $\infty$. By orthogonality of characters we write
\begin{equation}\label{eq:ortho}
S(x;q,a) - S(x;q,b) = \frac{1}{\phi(q)} \sum_{\chi_0 \neq \chi \bmod q} \overline{(\chi(a)-\chi(b))} \sum_{\substack{n \le x\\ n \in S}}\chi(n)
\end{equation}
for each $x \in [X,2X]$.
By Proposition~\ref{prop:charsums} and Cauchy-Schwarz, we can write
\begin{equation}\label{eq:finalexp} S(x;q,a)-S(x;q,b) = \frac{C_q}{\phi(q)} \frac{\sqrt{x}}{(\log x)^{3/4}} C_{q,a,b} + T(x)
\end{equation}
where
\[ \frac{1}{X} \int_{X}^{2X} \left| T(x)\right|^2 dx \ll \frac{X}{(\log X)^{5/2}}.\]
We see that in an $L^2$-sense, $T(x)$ is smaller (by a power of $\log x$) than the term of order $\sqrt{x}/(\log x)^{3/4}$ in \eqref{eq:finalexp}. To make this precise, we use Chebyshev's inequality:
\[ \mathbb{P}_{x \in [X,2X]}\left( |T(x)| \ge \frac{\sqrt{X}}{\Psi(X)(\log X)^{3/4}} \right) \ll \frac{\Psi^2(X)}{\log X} = o(1)\]
for any function $\Psi$ tending to $\infty$ slower than $(\log X)^{1/2}$.
Here $x$ is a number chosen uniformly at random between $X$ and $2X$. It follows that $\mathbb{P}_{x \in [X,2X]}\left(S(x;q,a)>S(x;q,b)\right) \sim 1$
which finishes the proof. \qed
\section{Martin's conjecture}\label{sec:omega}
\subsection{Preparation}
Let $F_{\omega}(s,\chi) = \sum_{n \ge 1} \chi(n)\omega(n)/n^s$ for $\Re s >1$.
By Lemma~\ref{lem:effective0} with $\kappa = 1+1/\log x$ and $\omega(n)\ll \log n$,
\begin{equation}\label{eq:perronomega} \sum_{n \le x} \chi(n) \omega(n) = \frac{1}{2\pi i} \int_{1+\frac{1}{\log x}-iT}^{1+\frac{1}{\log x}+iT} F_{\omega}(s,\chi)x^s \frac{ds}{s} + O\left( \log x+ \frac{x \log x \log (xT)}{T}\right)
\end{equation}
for all $T \gg 1$. We have $\omega = 1 * \mathbf{1}_{\text{Primes}}$. For $\Re s >1$ this identity leads to
\begin{equation}\label{eq:omega gen} F_{\omega}(s,\chi) = L(s,\chi) \left( \log L(s,\chi) - \frac{1}{2}\log L(2s,\chi^2) + G_{\omega}(s,\chi)\right)
\end{equation}
where $G_{\omega}$ may be analytically continued to $\Re s > 1/3$, and is bounded in $\Re s \ge 1/3 + \varepsilon$ (cf. \cite[Eq.~(2.3)]{Meng2020}). If $\chi$ is non-principal, GRH for $\chi$ and $\chi^2$ implies that $F_{\omega}$ can be analytically continued to
\begin{equation}\label{eq:fomega analytic}
\left\{ s \in \mathbb{C}: \Re s > \frac{1}{3}\right\} \setminus \{ \sigma+it : L(1/2+it,\chi)=0 \text{ and }\sigma \le 1/2 \}
\end{equation}
if $\chi^2$ is non-principal, and
\begin{equation}\label{eq:fomega analytic2}
\left\{ s \in \mathbb{C}: \Re s > \frac{1}{3}\right\} \setminus \left(\{ \sigma+it : L(1/2+it,\chi)=0 \text{ and }\sigma \le 1/2 \} \cup \{ \sigma : \sigma \le 1/2\}\right)
\end{equation}
if $\chi^2$ is principal.
Almost the same analysis applies for $F_{\Omega}(s,\chi) = \sum_{n \ge 1} \chi(n)\Omega(n)/n^s$, with the only change being the following variation on \eqref{eq:omega gen}:
\[ F_{\Omega}(s,\chi) = L(s,\chi) \left( \log L(s,\chi) + \frac{1}{2}\log L(2s,\chi^2) + G_{\Omega}(s,\chi)\right)\]
where $G_{\Omega}$ may be analytically continued to $\Re s > 1/3$ and is bounded in $\Re s \ge 1/3+\varepsilon$. This is a consequence of $\Omega = 1 * \mathbf{1}_{\text{Prime Powers}}$.
The following lemma is essentially \cite[Lem.~3]{Meng2020}, and its proof is similar to the proof of Lemma~\ref{lem:perronshifted}.
\begin{lem}\label{lem:contourshiftomega}
Let $\chi$ be a non-principal character. Assume GRH holds for $\chi$ and $\chi^2$. Let $c \in (0,1/6)$ be a fixed constant. Let $T\gg 1$. We have
\begin{multline}\label{eq:perronshiftedomega} \sum_{n \le x} \chi(n)\omega(n) = \sum_{j=1}^{m} \frac{1}{2\pi i}\int_{\mathcal{H}_{1/2+i\gamma_j}} F_{\omega}(s,\chi)x^s \frac{ds}{s} \\+ O\left( \log x + \frac{x\log x \log(xT)}{T}+\frac{(x+\max\{x,T\}^{1/2}+x^{\frac{1}{2}-c}T^{c+1})\exp\left( \frac{A \log T}{\log \log T}\right)}{T}\right)
\end{multline}
where the list $\{1/2+i\gamma_j\}_{j=1}^{m}$ consists of the distinct non-trivial zeros of $L(s,\chi)$ with $-T'' \le t \le T'$ where $T',T''$ depend only on $T$ and $\chi$ are satisfy $T',T''=T+O(1)$. If $\chi^2$ is principal we include $1/2$ in the list.
Here $\mathcal{H}_{\rho}$ is the truncated Hankel loop contour defined in \eqref{eq:hankel}, and it has radius $r$ which is chosen to be sufficiently small (in terms of $T$, $x$ and the list of $\gamma_j$s). The implied constant and $A$ depend only on $\chi$ and $c$.
\end{lem}
\begin{proof}
The proof is similar to that of Lemma~\ref{lem:perronshifted}, the main difference being the appearance of the factor $\log L(s,\chi)$ because of $\eqref{eq:omega gen}$. We need to be careful because $\log L(s,\chi)$ may be large even if $L(s,\chi)$ is small. We need to explain why the contribution of $\log L(s,\chi)$ may be absorbed into $\exp(A \log T/\log \log T)$. We shall show that $\log L(s,\chi) = O(\log T)$ holds on the relevant contour. Recall that $\arg L(s,\chi)$ is defined via $\log L(s,\chi) = \log |L(s,\chi)| + i\arg L(s,\chi)$. We have $\arg L(s,\chi)=O(\log(|t|+4))$ uniformly in $t$ and $\sigma \in [1/6,2]$, see \cite[Lem.~12.8]{Montgomery2007}. Hence our focus will be on bounding $\log |L(s,\chi)|$. By Lemmas~\ref{lem:upper} and \ref{lem:fe} we have $\log |L(\sigma+it,\chi)| \le C\log(|t|+4)$ for $|t| \gg 1$ and $\sigma \in [1/6,2]$, so that we have an easy upper bound on $\log |L(s,\chi)|$, and the focus is truly on lower bounding $\log |L(s,\chi)|$.
We want to shift the contour in \eqref{eq:perronomega} to $\Re s = 1/2 - c$ and avoid logarithmic singularities using Hankel loops. Before we do so, we replace the endpoints of the integral, namely $1+1/\log x \pm iT$, with $1+1/\log x + iT'$ and $1+1/\log x - iT''$, where $T',T'' = T+O(1)$ and the bound $\log |L(s,\chi)| \ge -C\log (|t|+4))$ holds uniformly on $\Im s = T'$ and $\Im s = -T''$ with $\sigma \in [1/6,2]$. Changing the endpoints does not affect the error term in \eqref{eq:perronomega} due to a simple variation on Remark~\ref{rem:modif}. The existence of such $T'$ and $T''$ is exactly the content of \cite[Thm.~13.22]{Montgomery2007}.
Lemmas~\ref{lem:upper}-\ref{lem:fe} allow us to bound both the vertical and horizontal contributions of $L(s,\chi)$ and $\log L(2s,\chi^2)$. The horizontal contribution of $\log L(s,\chi)$ is small due to the choice of $T'$ and $T''$.
To bound the vertical contribution of $\log L(s,\chi)$ we use \cite[Ex.~1 at \S12.1.1]{Montgomery2007} which says that for $\Re s \ge 1/6$, $\log L(s,\chi)=\sum_{\rho:\, |\gamma-t|\le 1}\log(s-\rho)+O(\log (|t|+4))$ unconditionally. Applying this with $\rho=1/2-c+it$ this is $ O_c(\log |t|)$ since all the zeros satisfying $|\gamma-t|\le 1$ are non-trivial and lie on $\Re s= 1/2$, and there are $\ll \log|t|$ zeros between height $t-1$ and $t+1$.
\end{proof}
The following lemma is implicit in \cite[pp.~110-111]{Meng2020}.
\begin{lem}\label{lem:mrho}
Let $\chi$ be a non-principal character and suppose GRH holds for $\chi$ and $\chi^2$.
Let $\rho=1/2+i\gamma \neq 1/2$ be a non-trivial zero of $L(s,\chi)$. Let $m_{\rho,\chi}$ be the multiplicity of $\rho$ in $L(s,\chi)$. We have
\[ \int_{\mathcal{H}_{\rho}} F_{\omega}(s,\chi)x^s \frac{ds}{s} = m_{\rho,\chi}\int_{\mathcal{H}_{\rho}} L(s,\chi) \log (s-\rho)x^s \frac{ds}{s}.\]
\end{lem}
\begin{proof}
Since
\[ F_{\omega}(s,\chi) = L(s,\chi) \log L(s,\chi) - \frac{1}{2}L(s,\chi)\log L(2s,\chi^2) + L(s,\chi)G_{\omega}(s,\chi)\]
and $L(s,\chi)\log L(2s,\chi^2)$, $L(s,\chi)G_{\omega}(s,\chi)$ are analytic in an open set containing $\mathcal{H}_{\rho}$, it follows that
\[ \int_{\mathcal{H}_{\rho}} F_{\omega}(s,\chi)x^s \frac{ds}{s} = \int_{\mathcal{H}_{\rho}}L(s,\chi)\log L(s,\chi)x^s \frac{ds}{s}\]
by Cauchy's Integral Theorem. We may write $\log L(s,\chi)$ as
\[ \log L(s,\chi) = m_{\rho,\chi} \log (s-\rho) + H_{\rho}(s,\chi)\]
for a function $H_{\rho}$\footnote{An estimate for $H_{\rho}(s,\chi)$ on $\mathcal{H}_{\rho}$ may be obtained, see \cite[Eq.~(2.15)]{Meng2020}.} which is analytic in an open set containing the loop, since $L(s,\chi)/(s-\rho)^{m_{\rho,\chi}}$ has a removable singularity at $s=\rho$. By Cauchy's Integral Theorem, $H_{\rho}(s,\chi)$ does not contribute to the Hankel contour integral, giving the conclusion.
\end{proof}
Lemmas~\ref{lem:contourshiftomega} and \ref{lem:mrho} hold as stated for $\Omega$ in place of $\omega$ as well.
We have the following lemma, a `logarithmic' analogue of Lemma~\ref{lem:hankelabsolute}.
\begin{lem}\label{lem:omegaabsolute}
Let $\chi$ be a non-principal character and suppose GRH holds for $\chi$.
Let $\rho=1/2+i\gamma$ be a non-trivial zero of $L(s,\chi)$. Let
\[ I_{\rho}:= \int_{H_{\rho}} |L(s,\chi)| |\log(s-\rho)| x^{\Re s} |ds|.\]
Then
\[ I_{\rho} \ll \sqrt{x}(|\gamma|+1)^{c+o(1)} \left( \frac{\log \log x}{\log^2 x}+\log (r^{-1}) r^2 x^r \right).\]
\end{lem}
\begin{proof}
We write $L(s,\chi)$ as $L(s,\chi)/(s-\rho)$ times $(s-\rho)$, and use Lemma~\ref{lem:ratio} to bound $L(s,\chi)/(s-\rho)$ by $(|\gamma|+1)^{c+o(1)}$. We now consider separately $|s-\rho|=r$ and $s=\rho+t$, $-c\le t\le -r$.
\end{proof}
The following is an $\omega$-analogue of Lemmas~\ref{lem:hankelhalf} and \ref{lem:asymphalf}.
\begin{lem}\label{lem:hankelhalf2}
Let $\chi$ be a non-principal Dirichlet character. Assume GRH holds for $\chi$ and $\chi^2$.
\begin{enumerate}
\item If $L(1/2,\chi) = 0$ then
\[ \int_{\mathcal{H}_{1/2}} F_{\omega}(s,\chi)\frac{x^s}{s} ds \ll \sqrt{x} \left( \frac{\log \log x}{\log^2 x}+\log (r^{-1}) r^2 x^r \right).\]
\item If $\chi^2$ is principal and $L(1/2,\chi)\neq 0$ then
\begin{equation}\label{eq:omegasymp} \frac{1}{2\pi i}\int_{\mathcal{H}_{1/2}} F_{\omega}(s,\chi)\frac{x^s}{s} ds = -L\left(\frac{1}{2},\chi\right) \frac{\sqrt{x}}{\log x} + O\left(\frac{\sqrt{x}}{(\log x)^2}\right).
\end{equation}
\end{enumerate}
The implied constants depend only on $\chi$.
\end{lem}
\begin{proof}
The first part is a minor modification of the proof of Lemma~\ref{lem:omegaabsolute}. The second part is \cite[Eq.~(2.28)]{Meng2020}.
\end{proof}
Lemma~\ref{lem:hankelhalf2} holds for $\Omega$ in place of $\omega$, with the only difference being a sign change in \eqref{eq:omegasymp}.
\subsection{Proof of Theorem~\ref{thm:omega}}
We shall prove the theorem in the case of $\omega$; the proof for $\Omega$ is analogous.
Suppose $a,b$ satisfy $(a,q)=(b,q)=1$. Suppose the constant $D_{q,a,b}$ appearing in \eqref{eq:lincondomega} is positive.
Consider $X \gg 1$ which will tend to $\infty$.
By orthogonality of characters we write
\begin{equation}\label{eq:orthoomega}
\sum_{\substack{n \le x\\ n \equiv a \bmod q}}\omega(n) -\sum_{\substack{n \le x\\ n \equiv b \bmod q}}\omega(n) = \frac{1}{\phi(q)} \sum_{\chi_0 \neq \chi \bmod q} \overline{(\chi(a)-\chi(b))} \sum_{n \le x}\chi(n)\omega(n)
\end{equation}
for each $x \in [X,2X]$.
By \eqref{eq:perronshiftedomega} with $T = X^{3/4} \asymp x^{3/4}$ and $c=1/10$ we have, uniformly for $x \in [X,2X]$,
\[\sum_{n \le x} \chi(n)\omega(n) = \sum_{j=1}^{m} \frac{1}{2\pi i}\int_{\mathcal{H}_{1/2+i\gamma_j}} F_{\omega}(s,\chi)x^s \frac{ds}{s}+ O\left( x^{\frac{1}{2}-\frac{1}{100}}\right)
\]
for any non-principal $\chi$.
For any pair $\rho_1=1/2+i\gamma_1$, $\rho_2=1/2+i\gamma_2$ of non-trivial zeros of $L(s,\chi)$ different from $1/2$ we have, from Lemmas~\ref{lem:mrho} and \ref{lem:omegaabsolute},
\begin{multline}
\frac{1}{X}\int_{X}^{2X}\int_{\mathcal{H}_{\rho_1}} F_{\omega}(s_1,\chi)x^{s_1} \frac{ds_1}{s_1} \overline{ \int_{\mathcal{H}_{\rho_2}} F_{\omega}(s_2,\chi) x^{s_2} \frac{ds_2}{s_2}} \\ \ll \frac{m_{\rho_1,\chi}m_{\rho_2,\chi}}{|\gamma_1\gamma_2|^{1-c+o(1)}\left(1+|\gamma_1-\gamma_2|\right)} X\left( \frac{\log \log X}{\log^2 X}+\log r^{-1} r^2 x^r \right)^2
\end{multline}
in analogy with Lemma~\ref{lem:l2bound}. We take $r=o(1/\log X)$. Since $m_{\rho} = O(\log (|\rho|+1))$ \cite[Thm.~10.17]{Montgomery2007} and
\[\sum_{\substack{\gamma_1,\gamma_2 \neq 0:\\ L(1/2+i\gamma_j,\chi)=0}}\frac{1}{|\gamma_1 \gamma_2|^{1-1/5}(1+|\gamma_1-\gamma_2|)}\]
converges \cite[Thm.~13.5]{Montgomery2007}, it follows that in an $L^2$-sense, the contribution of $\rho\neq 1/2$ to \eqref{eq:orthoomega} is $O(\sqrt{x}\log \log x/(\log x)^2)$; this step corresponds to \eqref{eq:l2s1}. By Lemma~\ref{lem:hankelhalf2}, the contribution of loops around $s=1/2$ is
\[ -\frac{\sqrt{x}}{\log x}\frac{1}{\phi(q)} \sum_{\substack{\chi \bmod q\\ \chi^2 = \chi_0}} \overline{\chi(a)-\chi(b)} L\left( \frac{1}{2},\chi\right) + O\left( \frac{\sqrt{x} \log \log x}{\log ^2 x}\right) = -\frac{\sqrt{x}}{\log x} \left( \frac{D_{q,a,b}}{\phi(q)}+o(1)\right). \]
As in the proof of Theorem~\ref{thm:bias}, Chebyshev's inequality allows us to conclude the following. The probability that for a number $x$ chosen uniformly at random from $[X,2X]$, $\sum_{n \le x, \, n \equiv a \bmod q} \omega(n) < \sum_{n \le x, \, n \equiv b \bmod q} \omega(n)$ tends to $1$ with $X$. This finishes the proof. \qed
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\section{Introduction}\label{intro}
Given a graph $G$ with a weight $J_{ij}$ for each edge $(i,j)$, and variables, or
\emph{spins}, $\mathbf{S}=\{S_i=\pm1|i\in V\}$ for each vertex, we can consider the energy
function defined by
\[H=H(\mathbf{S})=-\sum_{\text{edges}~(i,j)}J_{ij}S_i S_j.\]
Here the edges are considered to be undirected, so that if $(i,j)$ is in the sum, then
$(j,i)$ is not. We imagine that there is frustration in the model, i.e., many small
loops where the product of the weights $J_{ij}$ is negative. This typically arises when
$J_{ij}$ is itself chosen randomly, such as in the Edwards--Anderson spin glass.
This is the most straightforward case and so convenient for descriptions and
illustrations. More generally, there could be more than two possible values, or
\emph{spin states}, for $S_i$ and there could be \emph{external fields} --- terms in
$H(\mathbf{S})$ depending on individual $S_i$. In general, $H(\mathbf{S})$ can be the sum
of arbitrary functions of the form $J_{ij}(S_i,S_j)$ and $h_i(S_i)$ for the purposes of
the method described here. The empirical runs described later will involve 16 states.
Two classical problems are considered here, the first of finding \textbf{ground states} or
minimum energy states, and the second of \textbf{simulating} or \textbf{sampling from} the
Gibbs/Boltzmann distribution.
Given an inverse temperature, $\beta$, the Gibbs/Boltzmann distribution over sets of spin
configurations is given as usual by
\begin{align*}
P(\mathbf{S})&=Z(\beta)^{-1}e^{-\beta H(\mathbf{S})},\qquad\text{with}\\
Z(\beta)&=\sum_\mathbf{S} e^{-\beta H(\mathbf{S})}.
\end{align*}
Sampling from this distribution includes as a special case the problem of sampling the
ground states ($\beta=\infty$), and if there is frustration then finding a ground state is
known to be NP-hard for a general graph. Nevertheless, as seen in \cite{selbydwave1} and
\cite{selbydwave2}, ground states can be found quite efficiently for moderately large
graphs by searching over a covering set of low treewidth subgraphs.
The graph property \emph{treewidth} can be understood as the exponential complexity of
using dynamic programming to compute a locally-defined quantity. More precisely, this
statement applies to a graph $G$ and a property $\phi$ that is a collection of functions
$\phi_H$ for subgraphs $H\subset G$. Each $\phi_H$ is defined on the set of possible spin
states on $\partial H$, the set of vertices adjacent to $H$ but not in $H$. Suppose for
$H\subset H'\subset G$, $\phi_{H'}(\mathbf{S}_{\partial H'})$ is equal to a simple combining
function on the collection $\phi_H(\mathbf{S}_{\partial H})$, where $\mathbf{S}_{\partial H}$ ranges
over the spin configurations on $\partial H$ that are compatible with the spin
configuration $\mathbf{S}_{\partial H'}$. Suppose also that there are $s$ spin states for each
vertex, there are $m$ edges in $G$ and the treewidth of $G$ is $w$, then standard dynamic
programming using the tree decomposition will compute the property $\phi_G$ in $O(m\cdot
s^{w+1})$ steps. In what follows, we shall make use of two different choices of $\phi_H$:
the partition function $Z(\beta)$ for $H$, and a choice of random spins in $H$ according
to the Gibbs distribution.
There is a great deal of literature devoted to algorithms on families of graphs with
bounded treewidth, but here it is assumed that the desired graph of study does not have
bounded treewidth and the approach is to approximate it using bounded treewidth subgraphs.
The most common method of sampling Ising-type models generally involves Markov chain Monte
Carlo (MCMC), updating one spin at a time according to a random process dependent on its
immediate neighbours. More efficient methods using cluster updates, such as those of
Swendsen--Wang or Wolff \cite{Wolff}, are appropriate in models without frustration, but
do not work well in frustrated models. The method described here uses a different cluster
update method that works whether or not there is frustration.
A version of the subgraph sampling method was developed independently in 2013 by Decelle
and Krzakala \cite{decelle}, termed belief-propagation-guided Monte Carlo sampling. The
description given there apparently only considers the possibility of using trees for the
covering subgraphs (and provides a method for producing such trees at random). I believe
it is important to use a more general set of subgraphs, as described here, to be able to
sample (or find ground states) with large and difficult models. Evidence for this is given
in Section~\ref{comparison3}.
The sampling method described here is also similar to an earlier technique described by
Hamze and de Freitas \cite{hamze}, which focuses on a Markov Random Field with
observations attached. The method of \cite{hamze} only considers trees for subgraphs, and
only an exact partition of the graph by trees. As shown in Section~\ref{correct}, this
restriction is not necessary for the purposes of obtaining detailed balance, though under
these conditions the authors manage to prove rigorous bounds showing the efficacy of their
method.
The sampling method in \cite{fix} is similar to the one considered here, except that all
subgraphs of a given treewidth are allowed, not just induced subgraphs. I believe this is
significantly different from the method of induced subgraphs considered here and in
\cite{decelle}, for reasons given in Section~\ref{description}.
The primary method of comparison in this preprint will be wall-clock time, using a single
thread on a specific reference computer (an Intel Core i7-3930K CPU running at 3.20GHz),
except for the first comparison in Section~\ref{comparison1} where the method being
compared with was described by another party, and timings were not immediately
available. It may seem unusual to use something platform-dependent and
implementation-dependent like wall time rather than counting spin flips for example, but
not all spin flips are alike and I believe it is necessary to take seriously the facts (i)
that a spin flip isn't necessarily well-defined and (ii) that simpler methods are often
much more easily optimised. If a simple count of spin flips were used as a basis for
comparison, then this would tend to favour subgraph-based methods (at least as far as
sampling is concerned, if not ground state finding). This is because conventional methods
of flipping single spins can be executed efficiently without using any arithmetic
operations (which is how the comparison code used here works), whereas subgraph-based
sampling methods necessarily involve some kind of calculations to keep track of the
$Z$-values.
Each comparison here will involve combining subgraph-based sampling with parallel
tempering (also known as ``exchange Monte Carlo'', see, e.g.,~\cite{hukushima}) and
comparing this with single-site update methods also combined with parallel tempering. It
is well known that parallel tempering can significantly improve performance and we apply
it in both cases to ensure a useful comparison between the best available algorithms.
\section{Description of sampling method}\label{description}
Take a collection of induced subgraphs $T_1,\ldots, T_m$ of the graph $G$. Recall that an
induced subgraph is the restriction of $G$ to a particular subset of vertices, and
contains all the edges of $G$ between those vertices. In other words, if both of the
endpoints of an edge of $G$ are in some $T_i$ then the edge itself must be in $T_i$. We
also require that $\cup T_i=G$, i.e., every vertex and edge is represented in some $T_i$.
We consider induced subgraphs only, as this ensures that when we sweep over a subgraph the
spins of all its neighbouring vertices can be held fixed. It is possible to ignore this
restriction and still use the method for a non-induced subgraph, fixing the spin value
from the pre-updated graph to use as the neighbouring value during the subgraph
update. However, then monotonicity would be lost: at $\beta=\infty$ an induced subgraph
update is guaranteed not to increase the global energy, but this is no longer true for
non-induced subgraphs. I believe this will lead to poorer performance for non-induced
subgraphs as the $\beta=\infty$ point is the most difficult and the difficulty of
obtaining a ground state is a guide to the difficulty of obtaining low temperature / high
$\beta$ samples. The approach of \cite{fix} is to choose the best maximal subgraph of a
given treewidth (not necessarily induced, so at treewidth 1 this is a spanning tree),
trying to minimise the weight of the edges not in the subgraph, thereby making the
approximation as good as possible.
The idea is that $T_i$ should be chosen to be easy to solve exactly (in the sense of
finding its ground states, or calculating its conditional Gibbs distribution). A good
choice would be to only use subgraphs of a given treewidth. (It is not necessary to take
all subgraphs of a given treewidth.) For simplicity, the examples here are all illustrated
by the case of treewidth 1, i.e., $T_i$ will all be trees, but it should be remembered
that $T_i$ will have some non-trivial treewidth $w$ in general.
Given an induced subgraph, $T$, and a spin configuration, $\mathbf{S}_{G\setminus T}=\{S_i|i\in
G\setminus T\}$ defined on the remainder of $G$, we can condition $P()$ on
$\mathbf{S}_{G\setminus T}$ to get a distribution, $P_T(\cdot|\mathbf{S}_{G\setminus T})$ over the spin
configurations, $\mathbf{S}_T=\{S_i|i\in T\}$, over $T$.
The Monte Carlo step is to take $T$ to be a random $T_i$ from our fixed collection and
then replace $\mathbf{S}_T$ with a random configuration chosen according to the conditional
distribution $P_T(\cdot|\mathbf{S}_{G\setminus T})$. This defines a sampling procedure, subject to the
usual Monte Carlo caveats regarding burn-in and waiting for independent samples.
\vskip5pt
We need to show
\begin{itemize}
\item that this operation has the correct invariant distribution,
\item that this operation is efficient to perform, and
\item that the performance in terms of equilibration time can be better than traditional methods.
\end{itemize}
We'll take these in turn. The first two items are simple to demonstrate, but the last item
is more open-ended and will be evaluated in different ways in separate sections of this
preprint.
\subsection{Correctness of invariant distribution}\label{correct}
To show detailed balance, consider the flux from configuration of spins $\mathbf{S}$ to
$\mathbf{S}'$. Let us write $\mathbf{S}\Delta\mathbf{S}'$ to mean $\{i\in V|S_i\neq S'_i\}$. Then the flux
$\mathbf{S}\to\mathbf{S}'$ is equal to
\[P(\mathbf{S})\frac{1}{m}\sum_{T|\mathbf{S}\Delta\mathbf{S}'\subset T}P_T(\mathbf{S}'_T|\mathbf{S}_{G\setminus T}),\]
where the sum is over subgraphs in our fixed collection that include $\mathbf{S}\Delta\mathbf{S}'$. This is equal to
\[P(\mathbf{S})\frac{1}{m}\sum_{T|\mathbf{S}\Delta\mathbf{S}'\subset T}\frac{P(\mathbf{S}')}{\sum_{\mathbf{S}''|\mathbf{S}''_{G\setminus T}=\mathbf{S}_{G\setminus T}}P(\mathbf{S}'')}.\]
But $\mathbf{S}\Delta\mathbf{S}'\subset T$ is the same as saying $\mathbf{S}_{G\setminus T}=\mathbf{S}'_{G\setminus T}$, so the condition on $\mathbf{S}^{\prime\prime}$ in the sum in the denominator, $\mathbf{S}''_{G\setminus T}=\mathbf{S}_{G\setminus T}$, is equivalent to $\mathbf{S}^{\prime\prime}_{G\setminus T}=\mathbf{S}'_{G\setminus T}$, and the whole expression is thus symmetric under interchanging $\mathbf{S}$ and $\mathbf{S}'$, proving detailed balance.
\subsection{Efficiency of calculation}\label{efficient}
For clarity we'll scrutinise the case where the $T_i$ all have treewidth 1, that is they
are trees. In general, if $T_i$ had treewidth $w$ then the time taken to draw a random
instance from $P_T(\cdot|\mathbf{S}_{G\setminus T})$, or to calculate
$\sum_{\mathbf{S}^{\prime\prime}|\mathbf{S}^{\prime\prime}_{G\setminus T}=\mathbf{S}_{G\setminus T}}P(\mathbf{S}^{\prime\prime})$, would be roughly
$s^{w+1}|\mathcal{E}(T,G)|$ elementary operations, where $\mathcal{E}(T,G)$ is the set of
edges with at least one end in $T$ and $s$ is the number of (spin) states of each
vertex. For illustration we show how it works with $s=2$, $w=1$, though results reported
in later sections use up to $s=16$, $w=2$.
We proceed inductively over the vertices of $T$ from its leaves inwards in the usual
manner of dynamic programming based on a tree decomposition. After $r$ steps, we have a
collection, $U_1,\ldots,U_m$ of connected subtrees of $T$ of total size $\sum_i|U_i|=r$,
with $T\setminus\cup_i U_i$ being connected. For example:
\begin{figure}[h]
\includegraphics{treesampling6.eps}
\caption{Three steps in the inductive evaluation and sample-choosing of a subgraph}
\label{treedecomp}
\end{figure}
The above vertex numbering shows one of many possible orders in which the vertices may be
processed, and three successive steps are illustrated. The rule is a vertex may be
processed only when at most one of its neighbours is unprocessed. Note that the rest of
the graph $G\setminus T$ is suppressed in the above picture, but must be taken into
account as a ``background field" in the inductive calculation as it contributes to the
conditional probability of the tree by interactions along edges between $T$ and
$G\setminus T$.
There are two quantities that need to be maintained inductively, for each value $S_b$ of
the boundary spins of $U_i$ (the boundary of $U_i$ being vertices adjacent to $U_i$ that
are not themselves in $U_i$; in this tree case there is only one such vertex):
\begin{itemize}
\item The total $Z$ value for $U_i$ given $S_b$ and $\mathbf{S}_{G\setminus T}$.
\item A choice of spins of the vertices in $U_i$, representing a sample of the distribution $P()$ conditioned on $S_b$ and $\mathbf{S}_{G\setminus T}$.
\end{itemize}
These two quantities are easy to maintain inductively when a vertex is added to a $U_i$. To illustrate the process, the example of the above pictures is traced in some detail. For $U_3$ in the first picture there will be, for each of the two possible values, $\pm1$, of $S_7$
\begin{itemize}
\item the value $Z_{U_3}$ defined as:
\[Z_{U_3}(S_7)=\sum_{S_4,S_5,S_6}e^{-\beta H_{U_3}(S_4,S_5,S_6, S_7, \mathbf{S}_{G\setminus T})}\]
where $H_{U_3}$ contains the contribution to the energy from edges meeting the vertices
$4$, $5$ and $6$ of $U_3$. These edges will join vertices $4,5,6$ to each other, to vertex
$7$, and to $G\setminus T$, but they can't meet the other vertices of $T$ by
construction. It should be remembered that the above expression for $Z_{U_3}$ is just a
definition, not the method by which it was calculated.
\item A choice of spins $\mathcal{C}_{U_3}(S_7)=(S_4,S_5,S_6)$
\end{itemize}
Moving from the first of the above pictures to the second, vertex $7$ is incorporated into
$U_3$ and $8$ is the new boundary vertex. It is seen that $\Delta_H=H_{U'_3}-H_{U_3}$
only depends on $S_7$ and $S_8$ (and spins over $G\setminus T$, which we suppress), and
no other vertices of $T$. So
\begin{itemize}
\item the new value $Z_{U'_3}$ is easily calculated as
\[Z_{U'_3}(S_8)=\sum_{S_7=\pm1}e^{-\beta\Delta_H(S_7,S_8)}Z_{U_3}(S_7),\]
\item and the new choice of spins is given by
\[
\mathcal{C}_{U_3'}(S_8)=\begin{cases} S_7=-1, (S_4,S_5,S_6)=\mathcal{C}_{U_3}(-1)&\text{with prob.}\;Z_{U'_3}(S_8)^{-1}Z_-\\
S_7=+1, (S_4,S_5,S_6)=\mathcal{C}_{U_3}(+1)&\text{with prob.}\;Z_{U'_3}(S_8)^{-1}Z_+\\
\end{cases}
\]
where $Z_{\pm}$ are the summands in the above expression for $Z_{U'_3}(S_8)$.
\end{itemize}
When the new vertex involves fusing more than one $U_i$, the expression for the new $Z$
will involve products of the old $Z$s. In the example above, in the transition from the
second to third picture, $Z_{U'_2}$ is built up of the sum of terms of the form
$e^{-\beta\Delta_H}Z_{U_2}Z_{U'_3}$.
In practice this algorithm can be easily and compactly implemented, with the spin choice
being represented by a pointer. This is a small difference from the approach of
\cite{decelle} where a second reverse pass is used to construct the sample values for the
spins of the subgraph (which will be a tree). Some care is needed for a fast
implementation since straightforward methods of dealing with the inevitable large
numerical range of values involve computing logarithms and exponentials and generating
random numbers in the inner loop, or the loop one level outside it, which would be rather
slow. These can be avoided, for example by using look-up tables for the exponentials,
rescaling the $Z$-values when necessary, and storing and judiciously reusing random
numbers. See Appendix~\ref{AppendixB} for further discussion of numerical aspects. The
code used to produce the results below is available from \cite{mygithub}.
\section{Chimera graphs}\label{chimera}
Experiments in this preprint were carried out on Chimera graphs (\cite{dwavechimera},
\cite{boixo1}) of various sizes. These were chosen because originally the aim was to
compare (\cite{selbydwave1}, \cite{selbydwave3}) classical optimisation with that of
D-Wave hardware \cite{boixo1} --- a quantum device whose current implementation is based
on a Chimera graph. This preprint will not contain any comparisons aginst D-Wave hardware
as its aim is look at classical optimisation and simulation techniques in their own right.
An $n\times n$ Chimera graph, $C_n$, consists of $8n^2$ vertices arranged as $n^2$
complete bipartite graphs $K_{4,4}$. We shall use the notation $N=8n^2$
throughout. Writing the bipartite decomposition of $K_{4,4}$ as $A\cup B$, the $n^2$
graphs of the form $A$ are pointwise connected horizontally in rows, and the graphs of the
form $B$ connected vertically. An $8\times8$ Chimera graph is illustrated in
Fig.~\ref{chimeraexample}. The separate 4-vertex $A$ and $B$ graphs can be thought of as
\textbf{``big vertices''} (a term used throughout this preprint) in a simpler collapsed
graph with $2n^2$ vertices.
The Chimera graph is highly non-planar in that its genus is at least $(4/3)n^2+O(n)$,
which follows from the fact that the complete graph $K_{4n+1}$ can be minor-embedded into
$C_n$. This means that matching techniques \cite{planar} that enable ground states of
planar, and by extension low genus, graphs to be found in polynomial time are not
applicable here. On the other hand, while the Chimera graph is not simply two
dimensional, it isn't fully three dimensional either. In \cite{katzgraber} it is shown
that in a certain scaling limit, $C_n$ behaves like a planar graph in that it has no
positive critical temperature. This result is apparently at odds with the fact that
$K_{4n+1}$ is minor-embeddable in $C_n$ and should have a positive critical temperature
because a complete graph is effectively infinite dimensional. However, these facts can be
reconciled because $|K_{4n+1}|/|C_n|\sim\sqrt{2/N}$ and \cite{katzgraber} is not concerned
with features on the scale of $1/\sqrt{N}$. Furthermore \cite{katzgraber} considers a
uniform weight distribution on the edges of $C_n$ which would not translate to a uniform
weight distribution on the edges of $K_{4n+1}$.
The treewidth of $C_n$ is $4n$, or $n$ in terms of big vertices. In practice, $C_n$ can be
exhaustively searched using a simple treewidth-based method to find either ground states
or perfect Gibbs samples at inverse temperature $\beta$, for $n$ up to 8 on a normal
desktop computer. Larger sizes rapidly present problems in terms of memory as well as time
for this method. It is possible that branch-and-cut methods could be combined with
heuristic methods to increase beyond $n=8$ the largest $C_n$ that can be exhaustively
searched, but that is not explored here.
\begin{figure}[h]
\includegraphics{C8+resized.eps}
\caption{$8\times8$ Chimera graph shown in collapsed view with 128 ``big vertices'', the inset showing a portion of the full 512-vertex graph}
\label{chimeraexample}
\end{figure}
Following \cite{ronnow}, the class of random instances \textbf{Range $\mathbf{r}$} is defined by
choosing each edge coupling randomly from the set $\{-r,-r+1,\ldots,-1,1,\ldots,r\}$. The
instances considered here are Range 1 in Section~\ref{comparison1} and Range 7 in other
sections.
\section{Comparison - equilibration time and Binder ratio}\label{comparison1}
In \cite{katzgraber} the authors study, inter alia, the critical behaviour of bimodal
($J_{ij}=\pm1$, same as Range 1) spin-glass on the Chimera graph. They make a prediction
for its universality class, and test this by showing that the expectation of the Binder
ratio as a function of suitably-scaled temperature is independent of the size $N$ of the
underlying graph (figure 2, upper, of \cite{katzgraber}).
This gives us an opportunity to compare the subgraph-based sampling methods described here
with the more standard, though well-tuned, Monte Carlo methods used in the paper.
There are now two levels of probability space: the space of random $J_{ij}$ (known as the
``disorder"), and for a given choice of $J_{ij}$, the space of spins, $\mathbf{S}$ (averaging
over which can be referred to as taking a ``thermal average").
In \cite{katzgraber}, parallel tempering is used, which greatly improves convergence in
this sort of problem. It is a kind of Monte Carlo meta-method and can be used ``on top" of
another Monte Carlo method. \cite{katzgraber} uses a standard Monte Carlo sweep as the
subroutine. For comparison we also use parallel tempering, but instead base it on top of
subgraph-based sampling.
The comparison we examine is with $p=0.5$, $N=512$ in the notation of
\cite{katzgraber}. That is, the graph is the Chimera graph of order 8 and $J_{ij}$ are IID
$\pm1$. (For avoidance of doubt, it is assumed here that the Hamiltonian given there in
formula (1) as $-\sum_{i,j=1}^N J_{ij}S_iS_j$ was intended as $-\sum_{i<j}J_{ij}S_iS_j$,
otherwise the undirected edge weights $J_{ij}+J_{ji}$ would effectively be taken from the
set $\{-1, 0, 1\}$, not $\{-1, 1\}$ as they are meant to be.)
The choice of temperatures used here covers a similar range (0.2 - 2) to that specified in
table 1 of \cite{katzgraber} (0.212 - 1.632). The temperature choice here was decided upon
by trying to make the exchange probabilities between adjacent temperatures in the parallel
tempering method all equal to 0.6, which ended up requiring 25 temperatures for the range
0.2 - 2. (As it turned out, these probabilities got a little higher than that for the
bottom two temperatures. In retrospect, this exchange probability of 0.6 may have been a
bit higher than optimal.)
For a given disorder, \cite{katzgraber} takes two independent random spin configurations,
$\mathbf{S}$ and $\mathbf{S}'$ and defines the spin overlap $q=(1/N)\sum_iS_iS'_i$. Then the quantity of
interest is its excess kurtosis, the Binder ratio
\[g_q=\frac{1}{2}\left(3-\frac{\langle q^4\rangle}{\langle q^2\rangle^2}\right)\]
Here $\langle.\rangle$ denotes thermal average and disorder average. The interest is in
the thermal average, as it is trivial to sample $J_{ij}$ to obtain the disorder average.
At very low temperatures, assuming the ground state is close to unique, we might expect
$q$ to take the values close to $+1$ or $-1$ according to whether $\mathbf{S}'$ happened to hit
the same ground state as $\mathbf{S}$ or its negation. This would make $g_q$ close to 1. At high
temperatures, $S_i$ will be independently $\pm1$, which makes
$q\approx2(1/N)B(N,1/2)-1\approx N(0,1/N)$ and so $g_q\approx0$. This is what we see, with
$g_q$ apparently decreasing smoothly at intermediate temperatures indicating the lack of a
phase transition at $T>0$, at least in the scaling limit as $N\to\infty$.
The subgraph-based method used here in this experiement was the Gibbs analogue of the
method known as strategy 3 in Appendix B of \cite{selbydwave1} and as PT-TW1 in Sections
\ref{comparison2} and \ref{comparison3}. It uses the collapsed Chimera graph of 128
aggregate vertices with 16 spin states each, and there is a prescribed set of 32 trees.
The relevant question in this simulation is how many steps the Monte Carlo process takes
to equilibrate, i.e., for the associated Markov chain to reach something close to an
equilibrium distribution. For each disorder, a pair of spin states is initialised
uniformly at random at each temperature. Then $R$ exchange parallel tempering steps are
performed, which involves $R$ tree-steps for each temperature. At that point the states
are deemed to be in thermal equilibrium and they are run for $R$ further Monte Carlo steps
during which the thermal averages $\langle q^2\rangle$ and $\langle q^4\rangle$ are
evaluated.
This whole process (including disorder average) was repeated for increasing values of $R$,
($250, 500, 1000, 2000, \ldots$) until the final values of $g_q$ appeared to stabilise
within acceptable error margins (up to 0.01 absolute). It turned out this required
$R=1000$, i.e., 1000 tree-steps. As far as I can gather, in terms of elementary
operations, such a tree-step should be approximately comparable to about 100 standard
Monte Carlo sweeps, when both are optimised, making about 100,000 sweep-equivalents for
equilibration. This compares with the $2^{21}$, or about 2,000,000 sweeps required in
\cite{katzgraber}, so it appears there might be a significant advantage with this
approach; but see also the discussion below.
\begin{figure}[h]
\input{Comparison-Katzgraber.tex}
\caption{Log Binder ratio vs rescaled temperature at N=512 after 5964 anneals --- comparison with \cite{katzgraber}.}
\label{brcomp1}
\end{figure}
\begin{figure}[h]
\input{Comparison-Katzgraber-errorsx10.tex}
\caption{Log Binder ratio vs rescaled temperature at N=512 after 5964 anneals --- errors multiplied by 10.}
\label{brcomp2}
\end{figure}
The graph in Fig.~\ref{brcomp1} shows the results of the method described here applied to
the bimodal (Range 1) example problem of~\cite{katzgraber} with $N=512$. The graph in
Fig.~\ref{brcomp2} shows the errors artificially amplified by a factor of 10, since they
are too small to be seen clearly at their true scale. As can be seen, there is good
agreement with the results given in Fig.~2 of ~\cite{katzgraber}, with similar
uncertainties. In both cases the results are averages over 5964 disorders. These graphs
serve as a check that the method described here is functioning correctly. They do not
compare performance, because that is hidden in the number of equilibration steps required
for each disorder.
Returning to the performance comparison, as alluded to above, there are some problems with
this comparison. In particular, we are making the following assumptions:
\begin{itemize}
\item that there is an equivalence between sweeps and tree-steps
\item that the accuracy (used as a termination criterion) is comparable
\item that the parallel tempering algorithms are tuned in the same way, or if not,
that the impact of any differences is negligible
\end{itemize}
The next sections describe how we have attempted to construct new comparisons that compare
the two approaches on as fair a basis as possible.
\section{Comparison of equilibration time using ground state values as the termination criterion}\label{comparison2}
To make a much more careful and controlled comparison of subgraph-update-based sampling
with site-update-based sampling, we choose a particular problem, and try to simulate it
using versions of these two methods that are as nearly identical as possible. As a matter of
notation:
\begin{itemize}
\item \textbf{SGS} shall mean subgraph-update-based sampling in general, as described in
Section~\ref{description}. In the example below, we shall consider \textbf{PT-TW1},
parallel tempering using treewidth 1 in the ``big vertex'' graph.
\item \textbf{SSS} shall mean single-site-update-based sampling. In other words, the
traditional method whereby each spin variable is updated depending only on its immediate
neighbours. (To be clear, this category includes multispin methods, such as that
described in \cite{isakov}. Even though these methods operate on more than one spin at a
time, the outcome is the same as operating on spins individually.) The comparison below
will use \textbf{PT-TW0}, parallel tempering using a conventional update based on
immediate neighbours, in terms of big vertices.
\end{itemize}
The particular problem set chosen is that of the Chimera graphs of sizes $6\times6$,
$8\times8$, $10\times10$ and $12\times12$ (number of spins 288, 512, 800 and 1152
respectively). For these graphs, the couplings on the edges are chosen uniformly from the
14 possibilities $\pm1, \pm2, \ldots \pm7$, there are no external fields and each spin can
be $+1$ or $-1$. This mimics the ``Range 7" (harder) example set from \cite{ronnow}.
In fact the $J_{ij}$ used were half the values stated above, i.e., chosen from
$\pm\frac12, \pm\frac22, \ldots, \pm\frac72$, so that the energy quantum, the smallest
possible change in energy due to a change in $S_i$ for a given disorder $J_{ij}$, is $1$
rather than $2$. Of course this scaling factor doesn't fundamentally change anything
because $\beta$ can always be rescaled, but we state it explicitly to allow the reader to
interpret the numbers in what follows, where we mention specific values of $\beta$ and
maximum allowable absolute errors in energy.
To make an interesting and fair comparison we would ideally compare the best SGS-based
method against the best SSS-based method, in so far as that makes sense. Of course, we
don't necessarily know the best methods, but parallel tempering currently stands out as
one of the best methods known for equilibration of such frustrated models.
\subsection{Parallel Tempering parameters}\label{ptparams}
For each problem size, ($6\times6$, $8\times8$, $10\times10$ and $12\times12$), we choose
100 random disorders, and for each of these we determine the time required for the SGS-
and SSS-based methods to equilibrate to a reasonable accuracy. The principal statistic
comparing SGS with SSS is simply the respective totals of these times, though other
comparisons may be considered.
It may be argued that fixing the required accuracy for each disorder does not match the
likely underlying objective, which is to get a good expectation over disorders. It may be,
for example, that it is more efficient to spend less time trying to equilibrate the most
difficult disorders, allowing them to be less accurate, on the grounds that they will be
compensated for by the majority of more accurate disorders. I do not believe this kind of
optimisation technique would help a great deal for two reasons. First, by the nature of
the exponential decay of the subleading eigenvalue of the Monte Carlo iteration, there
should be a fairly sharp change from an inaccurate result to an accurate one as the number
of equilibration steps crosses the characteristic threshold for that particular
disorder. That means that one can't afford to use much less than the proper number of
equilibration steps, otherwise the results would be very inaccurate and swamp the accurate
results from other disorders. Second, scrutinising the results here, though there are
certain disorders that are considerably harder than the others, these still don't
represent an overwhelming proportion of the total equilibration steps expended over all
100 disorders.
The set of temperatures is determined by fixing an effective absolute zero at
$\beta=\beta^*=20$ and aiming for a constant transition acceptance rate, $0.25$, between
neighbouring $\beta$s. The value $\beta=20$ is sufficiently large (for the class of
disorders considered here) that there is only a negligible chance of a state at this
inverse temperature not being the ground state. \cite{katzgraberemc} describes such a
constant acceptance rate as in general ``not too bad, but not optimal". The temperature
set here is determined in advance of the actual simulation using a method based on the
average energy and heat capacity at a range of different temperatures. The acceptance
rates during actual simulations match the target value reasonably well, almost always
lying between $0.2$ and $0.3$.
Having fixed the maximum $\beta$, and having fixed the spacing between the $\beta$s by
deciding on the transition acceptance rate, the remaining parameter to be decided is the
minimum $\beta$, or equivalently the number of temperatures. This is determined by trying
a range of different possible values on a trial set of disorders and seeing which number
of temperatures requires the fewest steps on average to equilibrate. It is found that SGS
requires slightly fewer temperatures than SSS for a given problem size, thus SSS will end
up with some ``bonus information" about high temperature states. However, it is not given
credit for this, and all that is compared is the time required to estimate the ground
state energy (and so presumably low temperature states) to a given accuracy. The
justification for this is that it is assumed that the main interest lies in colder
temperatures (strong coupling, high $\beta$), since higher temperatures are relatively
easy to simulate using any sensible method. Full details of the sets of temperatures used
are in Appendix~\ref{AppendixA}.
\subsection{Determination of Equilibration}
Equilibration is determined using the following method. Starting from a set of uniformly
random states at each temperature, $n$ Monte Carlo steps are performed. Each such step
consists of doing a single-temperature step at each temperature and then attempting to
perform exchanges. After that, $n$ more steps are performed during which any observables
can be measured and the energy at $\beta^*$ is averaged to make a sample value $E_1$. This
whole process is repeated $25$ times starting each time from random initial states, and
the average value $E=(E_1+\ldots+E_{25})/25$ is formed. If this is within a chosen
threshold (taken to be $0.2$ here) of the smallest energy observed at any point so far,
$E_\text{min}$, then it is deemed that $n$ steps are sufficient for equilibration. (It is
possible to simultaneously test, for each $m$, whether the smaller number of steps $m$
would have sufficed in not much more time than it takes just to test $n$ itself.)
This procedure relies on a number of assumptions. First, that $E_\text{min}$ is the true
ground state energy. Empirical evidence strongly suggests that it is, but for the purposes
of comparison it may not matter too much if it isn't, provided that the same value of
$E_\text{min}$ is used for both SSS and SGS, and this can be checked. Second that
$\beta^*$ is the hardest value of $\beta$ to equilibrate and the states at other
temperatures will have equilibrated if the state at $\beta^*$ has. Even if this turns out
not to be an accurate assumption, then at least SSS and SGS are still being compared on an
equal basis. In any case, it is assumed that the lower temperatures are the objects of
interest and the higher temperatures are a means to this end. The third assumption is that
the number of restarts ($25$) is sufficiently large to get a good estimate of the required
number of equilibration steps. The number $25$ was chosen to limit the total computer time
for all experiments to a week or so, but in fact it is on the small side and there is a
noticeable variance in the estimate for the required number of equilibration
steps. However, when the estimate is averaged over 100 disorders, this variance becomes
tolerably small.
\subsection{Timing}
The aim is to compare wall time, though for practical reasons we break this down into the
product of the number of parallel tempering (PT) steps and the time per PT-step. A PT-step
includes a sweep at each temperature and the attempting to do an exchange for each pair of
neighbouring temperatures. The time for such a step is liable to change if low-level
spin-flip algorithms are optimised, if the computing device changes, or if there are other
processes running on the computer. Separating out the wall time into a
platform-independent step count and a simple low-level timing frees us up to do the
step-counting runs in any environment with code at different stages of optimisation, means
that we only have to measure the timings once under controlled conditions, and enables us
to consider the effects of further optimisations. It is hoped that the ratio
$t_\text{PT-TW1}/t_\text{PT-TW0}$ of times per step for the respective methods is fairly
robust and won't vary too much across different platforms, though it is a significant
assumption that this ratio is stable under further optimisation of the respective
low-level spin-flip algorithms. The times $t_\text{TW1}$ and $t_\text{TW0}$ were measured
on a reference computer (an Intel Core i7-3930K CPU @ 3.20GHz) where both spin-flip
algorithms (TW1, TW0) underwent a similar degree of effort in optimisation. They work in
an analogous way as far as it makes sense for them to do so. TW0 is optimised further in
the following way, eliminating all arithmetic operations for a given $\beta$, it only
requires a simple lookup of the neighbours of a spin to get the probability that the new
spin should be up or down, whereas for TW1 it appears to be actually necessary to
accumulate Z-values. The arithmetic involved in TW1 can, however, be reduced to a few
simple multiplications, additions and divisions, with no exponentials or logarithms
necessary in the inner loops, due to a fortunate way in which the required numerical range
is locally bounded: see Appendix~\ref{AppendixB} for further details. The level of
optimisation used here is not as great as with the fastest multispin implementations
described in \cite{isakov}, though the code is general enough to work just as well with
arbitrary weights $J_{ij}$. In the language of \cite{isakov}, the TW0 code used here
achieves about $0.16$ spin-flips per nanosecond using a single thread.
The timings for the implementation and computer used are given in Appendix~\ref{AppendixA}.
\subsection{Results of equilibration comparison}
\begin{table}[h]
\begin{center}
\begin{tabular}{r r r r r r r r r}
Chimera size & $N$ & $n_\text{TW0}$ & $t^\text{eq}_\text{TW0}/\text{s}$ &
$n_\text{TW1}$ & $t^\text{eq}_\text{TW1}/\text{s}$ &
$t^\text{eq}_\text{TW0}/t^\text{eq}_\text{TW1}$ \\
\hline
$6\times6$ & $288$ & $3.76\times10^3$ & $0.0549$ & $68.5$ & $0.0126$ & $4.36$ \\
$8\times8$ & $512$ & $2.24\times10^4$ & $0.804$ & $303$ & $0.136$ & $5.91$ \\
$10\times10$ & $800$ & $1.49\times10^5$ & $10.7$ & $1.24\times10^3$ & $1.26$ & $8.44$ \\
$12\times12$ & $1152$ & $6.00\times10^5$ & $85.8$ & $3.25\times10^3$ & $6.05$ & $14.2$
\end{tabular}
\end{center}
\caption{Results. $n_\text{TW0}$ and $n_\text{TW1}$ denote the number of equilibration steps required, and the last column gives the time advantage of TW1 over TW0.}
\label{t2}
\end{table}
Table~\ref{t2} shows a modest but potentially useful speed improvement (last column) which
appears to increase with problem size. The factor of 14 is small enough that it could
potentially be erased by a better implementation of TW0, but on the face of it it is worth
having, and may increase for larger problem sizes.
\section{Comparison - finding ground states using parallel tempering}\label{comparison3}
Four methods are compared here, named GS-TW1, GS-TW2, PT-TW0 and PT-TW1. GS-TW1 and GS-TW2
are specialised ground state finding algorithms described in Appendix B of
\cite{selbydwave1} (where GS-TW1 corresponds to strategies 3 and 13, and GS-TW2
corresponds to strategies 4 and 14) and reviewed in subsection \ref{gsmethod} below, using
treewidth 1 and 2 subgraph neighbourhoods respectively as measured in big vertices,
roughly equivalent to treewidth 4 and 8 in terms of single spins (though slightly more
powerful than that, due to handling half of the $K_{4,4}$ as a single unit).
PT-TW0 and PT-TW1 are parallel tempering each with a fixed set of temperatures (for each
graph size), where the Monte Carlo move updates, respectively, the spin variables of a
subgraph of treewidth 0 (conventional update based on immediate neighbours) or one of
treewidth 1 (tree-based Monte Carlo move as described in Section~\ref{description}). The
treewidths 0 and 1 are in terms of big vertices and are roughly equivalent to treewidths 0
and 4 in terms of single spins (though, as before, slightly more powerful than that).
\subsection{Ground state finding method GS-TWw ($w=0,1,2,\ldots$)}\label{gsmethod}
This is the basic method we use to search for low energy states or ground states, subject
to adjustments as noted below. $E(S)$ denotes the energy of state $S$. As noted above,
treewidth $w$ refers to big vertices, so is approximately equivalent to treewidth $4w$ on
individual spins.
\begin{enumerate}
\item Set $A=\{\}$, a multiset, and randomise the current state (configuration of spin variables), $\mathbf{S}$.
\item Randomly perturb $\mathbf{S}$ and let $B=0$.
\item Let $E_0=E(\mathbf{S})$. Repeatedly loop through each subgraph in the treewidth $w$
subgraph collection and change $\mathbf{S}$ by performing subgraph updates at
$\beta=\infty$ until you've done a sweep that doesn't lower the energy. Let
$E=E(\mathbf{S})$.
\item If $E<E_0$ then let $A=A\cup\{E\}$, $B=0$.
\item Increase $B$ by $|\{a\in A|a\le E\}|$. If $B/|A|$ is
below a certain threshold then go back to step 3, otherwise go to step 2.
\end{enumerate}
It is worth remembering that even at $\beta=\infty$ (finding the minimum energy of the
subgraph) the subgraph update is random because it has to make choices between equal
energy substates as it runs.
The subgraph collections used are fixed and chosen by hand, as illustrated in
Appendix~\ref{AppendixC}.
The idea of the $B$ counter is that $|\{a\in A|a\le E\}|/|A|$ approximates the
probability, $p$, that a random state being considered has lower energy than the current
one, and the above method tries to spend time proportional to $1/p$ considering states
which are $p$ of the way up the energy distribution. In that way, it tries to spend a
longer time working on the more promising states.
This is an idealised non-terminating version of the algorithm, finding lower and lower
energies indefinitely. The time-to-solve (TTS) of GS-TWw is defined by the average time
it takes this version to find the ground state energy, as if there is an oracle that knows
the answer and can halt the solver when it has found it. The solver is not required to
state any particular level of confidence in its answer, only to arrive at it.
In practice, the above algorithm is modified to terminate at or below some target energy
$E_t$. It is then called with successively lower values of $E_t$, each time returning the
lowest energy found. The timing evaluator then looks something like this:
\begin{enumerate}
\item Let $E_t=\infty$
\item Let $n=0$, $\text{timer}=0$
\item Repeat until $n=500$:
\item \hskip3em Call above algorithm, modified to terminate at or below energy $E_t$. Let $E$ be the minimum energy obtained during this run.
\item \hskip3em If $E<E_t$, let $E_t=E$ and goto step 2
\item \hskip3em Let $n=n+1$
\item Result is $n$ instances of energy $E_t$ found in the time interval $T$ since $\text{timer}$ was last set to 0, and the estimated TTS is $T/n$.
\end{enumerate}
The random number generator is never reset, so that runs should be independent. If the
final energy obtained is $E_{\min}$, then a state with energy $E_{\min}$ will have been
encountered $n+1$ times during the process. The first occurrence has to be discarded for
timing purposes since it was obtained before it was known that $E_{\min}$ was the minimum
energy, and so runs were being interrupted and restarted.
Waiting for $500$ minimum energy states serves two purposes. First, it makes the estimated
TTS more accurate, both statistically and also by overcoming the accuracy limitations of
the CPU timer on the computer for fast cases. Second, it allows one to be reasonably
confident that it has found the actual ground state rather than just a low energy
state. Experimentally it appears that the energy landscape is sufficiently well-behaved
that during the course of using the above algorithm to look for an energy $E>E_0$, where
$E_0$ is the ground state energy, there is a reasonable probability (at least $0.05$ or so
for Range 7, and usually much lower) that it will encounter an energy less than $E$, and
these probabilities are independent for different runs. This means that finding $500$
independent minimum energies should ensure that there is only a small chance
$0.95^{500}\approx7\times10^{-12}$ of having missed the true ground state. Of course this
is by no means a rigorous argument and there remains the possibility of hidden bad cases
arising with larger $n$. However, this regularity hypothesis has been verified for
$n\le8$ where exact ground states can be found, and for $n>8$ it is at least still valid
to compare the various solvers by requiring each to find the same lowest energy state
found by any of the solvers.
\subsection{Results of ground state finding}
The comparison was made on Range 7 instances, which are harder and perhaps better for
comparison purposes than Range 1 instances as they don't have a large artificial
degeneracy. GS-TW1 and GS-TW2 were run for $n=4,5,\ldots,16$. For $n<10$, 1000 instances
were used; for $n=10$ and $n=11$, 500 instances were used; for $n=12$ and $n=13$, 250
instances were used; and for $n>13$, 100 instances were used. PT-TW0 and PT-TW1 were run
for even values of $n$ from $4$ to $16$ inclusive. For $n<10$, 1000 instances were used;
for $n=12$, 250 instances were used; and for $n>12$, 100 instances were used.
The results are shown in Fig.~\ref{range7graph}. The graphs of log-time vs linear size are
approximately linear (though perhaps slightly concave) with different gradients in each
case. PT-TW1 outperforms PT-TW0 by some margin, and the gap appears to widen with problem
size. Interestingly, PT-TW1 crosses over GS-TW1, showing the effectiveness of parallel
tempering as a meta-technique. GS-TW2 is initially worse than GS-TW1 for the easier
problems, but crosses over at approximately $n=13$ ($N=1352$), showing the value of using
moderately high treewidth (GS-TW2 has a treewidth of 8 in terms of individual spins).
\begin{figure}[h]
\input{range7.tex}
\caption{Comparison of four methods of finding ground states}
\label{range7graph}
\end{figure}
\section{Conclusions and discussion}
The above results show some evidence for an advantage of using subgraph-based methods over
traditional spin-flip methods, and also some evidence that this advantage increases with
problem size. The question arises as to how representative these results are of more
general problems on more general graphs.
It is possible that a constant factor of this advantage could be erased by improvements to
the low-level SSS spin-flip, such as using a GPU, that might not be applicable to the SGS
case. On the other hand, if the problem were generalised slightly, for example by using
more than two spin states, then the extra complexity may hit SSS harder than SGS.
It is possible that a dynamically adaptive method of choosing temperatures, as mentioned
in \cite{katzgraberemc}, would help SSS more than SGS, because it might be especially
helpful with the difficult disorders for which SGS has a greater advantage over SSS. It is
hard to make a confident guess at the differential advantage, and so this needs to be
tested.
On the other hand, the Chimera graph is in fact a relatively easy graph to simulate
because it is somewhat sparse and locally-connected. Since the advantage of SGS over SSS
appears to be larger with the more difficult disorders and larger problem sizes, it is
possible that a more difficult graph altogether would show the advantage of SGS over
SSS considerably more strongly, though this can't be taken for granted as the subgraphs
used would also become more restricted. An interesting next experiment would be to see how
well SGS methods, using different treewidths, perform on a 3D spin glass.
It would be interesting to apply the methods described here to Markov Random Fields,
possibly giving more efficient inference in certain difficult cases. This was the
motivating factor in \cite{hamze} and \cite{fix}.
\begin{appendices}
\section{Further details of equilibration comparison}\label{AppendixA}
\begin{table}[h]
\begin{center}
\begin{tabular}{|p{7em}|p{10em}|p{14em}|}
\hline
Chimera size & $\beta$s used by SSS only & $\beta$s used by SSS and SGS\\
\hline
$6\times6$ & 0.256 0.296 0.346 & 0.414 0.507 0.645 0.882 1.375 2.598 20.000 \\
$8\times8$ & 0.285 0.322 & 0.363 0.414 0.478 0.559 0.669 0.840 1.121 1.646 2.894 20.000 \\
$10\times10$ & 0.310 & 0.341 0.376 0.419 0.472 0.539 0.622 0.736 0.892 1.134 1.531 2.222 3.507 6.166 20.000 \\
$12\times12$ & 0.250 0.269 0.289 & 0.310 0.337 0.367 0.399 0.439 0.489 0.552 0.630 0.727 0.850 1.018 1.264 1.626 2.169 3.110 4.850 20.000 \\
$14\times14$ & 0.244 0.259 0.275 0.292 0.310 & 0.333 0.358 0.385 0.419 0.455 0.495 0.545 0.608 0.677 0.763 0.871 1.006 1.190 1.442 1.812 2.417 3.724 20.000 \\
\hline
\end{tabular}
\end{center}
\caption{Temperature sets used}
\label{t3}
\end{table}
\begin{table}[h]
\begin{center}
\begin{tabular}{r r r r r r r}
Chimera size & $N$ & $t_\text{SSS}/\mu\text{s}$ & $nt_\text{SSS}$ & $t_\text{SGS}/\mu\text{s}$ & $nt_\text{SGS}$ & $t_\text{SGS}/t_\text{SSS}$\\
\hline
$6\times6$ & $288$ & $14.6$ & $10$ & $184$ & $7$ & $12.6$ \\
$8\times8$ & $512$ & $35.9$ & $12$ & $449$ & $10$ & $12.5$ \\
$10\times10$ & $800$ & $71.7$ & $15$ & $1020$ & $14$ & $14.2$ \\
$12\times12$ & $1152$ & $143$ & $20$ & $1860$ & $17$ & $13.0$ \\
$14\times14$ & $1568$ & $229$ & $23$ & $2800$ & $18$ & $12.2$ \\
\end{tabular}
\end{center}
\caption{Setup and low-level reference timings. $nt_X$ is the number of temperatures used for method $X$.}
\label{t1}
\end{table}
\section{Numerical considerations}\label{AppendixB}
Given the basic method of Section~\ref{description} to compute the $Z$-values of the
partial subgraphs, there are still a number of ways of implementing it. If the values of
$\log(Z)$ are stored instead of $Z$ then it will be necessary to perform logarithms and
exponentials in the inner loop, which will be slow. (It would not be possible, in all but
the simplest cases, to use look-up tables to eliminate this, since the number of entries
in such a table would be exponential in the number of neighbouring vertices of the
subgraph.)
Instead of storing $\log(Z)$, the implementation here stores the $Z$-values themselves.
This means that it is possible to avoid all logarithms and exponentials in the inner loop
by using a look-up table of $e^{-\beta n}$ for the relatively manageable number of
different $\beta$s and $n$ arising from energy differences from a single vertex. (A slight
enhancement is to use look-up tables to precalculate values of $n$ arising from a small
neighbourhood of a vertex, rather than from the single vertex itself.) Using $Z$-values in
this way is quite convenient and means that the inner loop calculation involves only a few
lookups, multiplications and additions. However, there is a potential problem that the
values of $Z$ can become too large or small for a particular floating point
representation. For speed, one would ideally like to be able to use a native
representation, typically IEE 754 binary64 (double) or 80-bit extended precision format
native to the x86 processor family.
Recall from Section~\ref{description}, sampling a spin configuration on a subgraph $T$
proceeds by constructing intermediate partition functions, $Z_H(\mathbf{S}_{\partial H})$, of a
series of subgraphs $H\subset T$. The partition function $Z_H(\mathbf{S}_{\partial H})$ depends
on the spin configuration on $\partial H$, where $\partial H$ is the ``boundary within
$T$'', the set of vertices of $T$ adjacent to $H$ that are not in $H$ (and also on
$\mathbf{S}_{G\setminus T}$, though this latter dependency is suppressed as $\mathbf{S}_{G\setminus T}$
is constant during the sampling process associated with $T$). The question arises: how
much precision (size of mantissa) and range (size of exponent) is required to store
$Z_H(\mathbf{S}_{\partial H})$ adequately?
The precision presents no problem because all $Z$-values are \emph{positive}-linear
combinations of other $Z$-values. At each stage of the algorithm a random choice of spins
will be made based on the relative values of $Z_H(\mathbf{S}_{\partial H})$, that is the spin
configuration $\mathbf{S}_{\partial H}$ is chosen with probability $Z_H(\mathbf{S}_{\partial
H})/\sum_{\mathbf{S}'_{\partial H}}Z_H(\mathbf{S}'_{\partial H})$. Thus to make an accurate random
choice, it is sufficient to know $Z_H(\mathbf{S}_{\partial H})$ to a modest relative accuracy of,
say, $10^{-9}$. Because these $Z$-values are positive-linear combinations of other
$Z$-values from other subgraphs, it is sufficient to know these other $Z$-values to a
relative accuracy of $10^{-9}$.
Turning to the required size of the exponent of the floating point representation, we
first observe that the overall normalisation of the $Z$-values doesn't matter. That is,
$Z_H(\mathbf{S}_{\partial H})$ can be multiplied by an overall factor $\lambda_H$, depending on
$H$ but not on $\mathbf{S}_{\partial H}$. This means that we are not worried if all of the
$Z$-values simultaneously become very large or small as $H$ grows. However, on the face of
it, it is possible that for a given $H$ the relative values for different spin
configurations, $Z_H(\mathbf{S}_{\partial H})/Z_H(\mathbf{S}'_{\partial H})$, might be too big or small,
which would present a problem no matter how the $Z$-values were rescaled.
Fortunately at this point we are rescued by a convenient fact. The range of $Z$-values for
a given $H$ is limited by construction to a locally-computable constant and does not grow
with the size of the partial subgraph it is defined on.
To see this, let us first define $E(\mathbf{S}_H)$ to be the total energy of the spin
configuration $\mathbf{S}_H$ taken along edges within $H$ and along edges between $H$ and
$G\setminus T$, and $E(\mathbf{S}_{\partial H};\mathbf{S}_H)$ to be the total energy of the spin
configurations $\mathbf{S}_{\partial H}$ and $\mathbf{S}_H$ taken along edges joining $\partial H$ to
$H\cup \partial H\cup (G\setminus T)$. Then
\[Z_H(\mathbf{S}_{\partial H})=\sum_{\mathbf{S}_H}e^{-\beta (E(\mathbf{S}_H)+E(\mathbf{S}_{\partial H};\mathbf{S}_H))}.
\]
If $M(H)$ is defined by
\[
M(H)=\max_{\mathbf{S}_H, \mathbf{S}_{\partial H}, \mathbf{S}'_{\partial H}}|E(\mathbf{S}_{\partial H};\mathbf{S}_H)-E(\mathbf{S}'_{\partial H};\mathbf{S}_H))|,
\]
then for two different boundary spin configurations $\mathbf{S}_{\partial H}$ and
$\mathbf{S}'_{\partial H}$, we have
\[
e^{-\beta M(H)}\le Z(\mathbf{S}'_{\partial H})/Z(\mathbf{S}_{\partial H})\le e^{\beta M(H)}.
\]
Note also that $E(\mathbf{S}_{\partial H};\mathbf{S}_H)=E(\mathbf{S}_{\partial H};\mathbf{S}_{H\cap\partial \partial
H})$, so $E(\mathbf{S}_{\partial H};\mathbf{S}_H)$ does not notice what goes on deep within $H$. The
choice of subgraphs will typically be designed to minimise the boundary $\partial H$, and
for the examples of $T$ in this preprint, $M(H)$ is defined by a local calculation
involving only a few vertices.
The maximum $\beta$ that can safely be used with this simple floating point representation
of $Z$ is determined by the maximum value of $M(H)$ over subgraphs $H$ of $T$ that are
used in the tree decomposition, the maximum degree of a node in the tree decomposition and
the size of the exponent in the floating point representation. The actual calculations of
$M(H)$ are quite tedious and omitted here, but in the examples used in this preprint it
was found that by using extended precision floating point representation, $\beta^*$ (the
largest inverse temperature that it was necessary to use; see Section~\ref{ptparams}) was
comfortably smaller than the maximum value of $\beta$ not causing calculations to
overflow.
\section{Subgraphs used for GS-TW1 and GS-TW2}\label{AppendixC}
\begin{figure}[h]
\centering
\includegraphics{C8S3.eps}
\caption{A maximal induced tree (shown in dark) on the ``big vertex'' graph used for
method GS-TW1. Including the versions of this subgraph with horizontals exchanged for
verticals, there are 32 possible sets of this form. Coverage = 100/128 = 78.1\% of
``big vertices''.}
\label{C8S3}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics{C8S4.eps}
\caption{A maximal induced treewidth-2 subgraph used for method GS-TW2. Coverage = 114/128 = 89.1\% of ``big vertices''.}
\label{C8S4}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics{C8S4p1.eps}
\caption{Another maximal induced treewidth-2 subgraph used for method GS-TW2. Coverage = 107/128 = 83.6\% of ``big vertices''.}
\label{C8S4p1}
\end{figure}
\end{appendices}
|
train/arxiv
|
BkiUeRrxK6-gDz87Q9kI
| 5 | 1 |
\section{\label{sec:motivationtorus}Motivation: Open-Orbit Vortices}
Schwarz's early simulations \cite{schwarz85} apparently produced sustained
homogeneous turbulence. However, Buttke \cite{buttkecomment87} could only
reproduce the results by using deliberately inadequate spatial resolution.
Buttke subsequently argued \cite{buttke88} that since LIA does not allow vortex
stretching, it cannot describe superfluid turbulence. However, Schwarz
\cite{schwarzreply87} attributed the difference in computational results to an
artifact of the simulation geometry. With periodic boundary conditions, a vortex
line can cross the entire volume and close on itself. Energy loss will
increase the radius of curvature of such an ``open-orbit" vortex until the
vortex is entirely straight and interacts with the applied velocity field only
through an overall translation. No further energy is transfered between the
applied field and the vortex. The LIA contribution to the velocity of a straight
open-orbit vortex also vanishes, so only a reconnection can disrupt its
stability. In extreme cases the entire system can degenerate to an open-orbit
state, where all the vortices align and straighten into a clearly
non-homogeneous state that can persist indefinitely.
Figures~\ref{fig:vortextanglefullbiotvslia}c and
\ref{fig:vortextanglefullbiotvslia}f show such an open-orbit state. To prevent
the open-orbit state from developing, Schwarz \cite{schwarz88} inserted an
occasional mixing step, in which half the vortices, selected at random, were
rotated by 90$^\circ$ about the direction of the applied flow. Buttke's
simulations, lacking this artificial mixing procedure, gave quite different
results.
\begin{figure}
\centering
\includegraphics{figure1.pdf}
\caption{Projections of vortex tangles from the present work, with applied velocity perpendicular to (a-c) or within (d-f) the viewing plane. All computations use $v_{ns}=11$ cm/s, in a periodic cube with side length $50 \mu$m. The first row shows full Biot-Savart law calculation at time 0.142 s. The second and third rows show the LIA calculation at times 0.9 s and 0.142 s, respectively. In frames (b) and (e) the tangle has begun to exhibit anisotropy; in frames (c) and (f) the system has reached an open-orbit state, with the vortices almost entirely straight and parallel.
}
\label{fig:vortextanglefullbiotvslia}
\end{figure}
Various other methods of avoiding the open-orbit state have been used.
Schwarz attributed the state to the periodic boundary conditions since it did
not occur when he used the much more computationally expensive real-wall
boundaries; however, the resulting tangles are not statistically identical to
those generated with periodic boundary conditions and a mixing
step\citep{schwarz88}. Aarts\cite{aartsdiss} does not include mixing, but
instead focuses on the time domain after the vortex line length equilibrates
but before the system degenerates into the open-orbit state. Adachi {\em et
al.} \cite{adachi10} find that retaining the nonlocal terms instead of using
the LIA eliminates the need for any special accommodations to fix or avoid the
open-orbit state, suggesting that the LIA rather than the boundary conditions
may be at the root of the problem. Yet, as
Nemirovskii\cite{nemirovskiireview} notes, it is unclear why the nonlocal
term should be so effective, since nothing in the Biot-Savart integral
prevents the types of reconnections that produce open-orbit vortices.
Kondaurova {\em et al.}\cite{kondaurova08} suggest the vortex
reconnection condition as another possible source of trouble. Instead
of basing reconnection solely on the locations of neighboring vortices,
they consider the velocities of nearby vortex segments and carry out a
reconnection only if the segments would cross through each other during
the time step. In a study comparing several reconnection algorithms,
Baggaley\cite{baggaley12recon} finds that the reconnection details have
little statistical effect on the vortex tangle in a nonlocal (Biot-Savart)
calculation. However, results from LIA calculations do depend heavily
on when and how reconnections are carried out. Once again this hints at
a sickness in the LIA.
Here we propose yet another scheme for calculating the vortex motion between
reconnections. We include only those portions of the Biot-Savart integral where
$|\vec{s}_o-\vec{s}|$ is smaller than a nonlocal interaction distance $d_{NL}$,
and we vary $d_{NL}$ from 0 to interactions that span the
computational volume. Our results confirm that the LIA does not suffice for
modeling superfluid turbulence, but we find that only a small region of nonlocal
interaction is needed to reach accurate results. Our calculations also shed
light on some of the behaviors observed in previous computational work.
\section{\label{sec:reproducetorus}Results from
the Full Biot-Savart Integral and the LIA}
We first present data using the full Biot-Savart integral and the LIA to show
that our code is in good agreement with previous simulations and experiment. We
run simulations at temperature $T\approx1.6$~K, which corresponds to
$\alpha=0.1$. We ignore $\alpha'$ since it is an order of magnitude smaller,
and we use $v_n$ as our driving velocity ($v_s=0$) throughout this work to
eliminate uniform vortex translation. Except as otherwise noted, we use a
periodic cube with side length $D=50 \mu$m. Our equation of motion is
integrated using a Runge-Kutta-Fehlberg method (RKF54) with an adaptive time
step. Our point spacing is also adaptive, depending on the local radius of
curvature within the range $R/12\leq l\leq R/5$. As the vortex grows and a
particular point spacing exceeds the upper limit, we add a new point along a
circular arc determined by the points that neighbor this overlarge spacing, as
described by Schwarz\cite{schwarz85}. In practice the time step is usually
about $2\times 10^{-6}$ s, and the spacing between points is
about $8\times 10^{-5}$ cm. We carry out a reconnection when vortices approach
within $2R_{min}/\ln(R_{min}/a_0)$, where $R_{min}$ is the smaller radius of
curvature at the points in question. The separation must also be below an
absolute cutoff, generally 10 $\mu$m. For our homogeneous vortex tangles, the
absolute level typically exceeds the curvature-based cutoff by more than an
order of magnitude and plays little role in the dynamics. As we discuss
below, the absolute cutoff becomes relevant in the untangled open-orbit state
produced with the LIA. We perform the reconnections themselves so as
not to increase the total vortex line length, by adjusting the position
along the vortex of the points involved in the reconnection\cite{owenthesis}.
We shall compare the LIA and full Biot-Savart calculations in multiple
ways, including through visual
inspection (as in Figure~\ref{fig:vortextanglefullbiotvslia}), using the
line length density given by
\begin{equation}
L = \frac{1}{V}\int\mathrm{d}\xi, \label{eq:ldens}
\end{equation}
where $V$ is the system volume, and with measures of the anisotropy
given by~\citep{schwarz88}:
\begin{align}
&I_{\parallel}=\frac{1}{V L}\int\left[1-\left(\hat{s}^{\,\prime}\cdot
\hat{r}_{\parallel}\right)^2\right]d\xi, \label{eq:Ipardef}\\
&I_{\perp}=\frac{1}{V L}\int\left[1-\left(\hat{s}^{\,\prime}\cdot
\hat{r}_{\perp}\right)^2\right]d\xi, \label{eq:Iperpdef}\\
&I_{\ell}\hat{r}_{\parallel}=\frac{1}{V L^{3/2}}\int\hat{s}^{\,\prime}
\times\vec{s}^{\,\prime\prime}d\xi. \label{eq:Ielldef}
\end{align}
Here $\hat{r}_{\parallel}$ and $\hat{r}_{\perp}$ are unit vectors parallel and
perpendicular to the $\vec{v}_{ns}$ direction.
As long as both directions perpendicular to the flow
velocity are equivalent, the following relation is true
regardless of tangle geometry: $I_{\parallel}/2+I_{\perp}=1$.
Note also that
if a vortex tangle is completely isotropic, $I_{\parallel}=I_{\perp}=2/3$ and
$I_{\ell}=0$. If vortices lie entirely within planes normal to $\vec{v}_{ns}$,
$I_{\parallel}=1$, $I_{\perp}=1/2$, and $I_{\ell}$ will depend on the
structure of the vortices.
Figure~\ref{fig:ldensfullbiotvsliatime} compares
$L(t)$ and $I_{\parallel}(t)$ for calculations with the
LIA and full Biot-Savart law. The data agree well with those of
Adachi {\em et al.}\cite{adachi10}, with small differences arising from different system sizes and driving velocities.
\begin{figure
\centering
\includegraphics{figure2}
\caption{Results from LIA and full Biot-Savart
calculations for (a) $L(t)$ and (b) $I_\parallel(t)$,
with $v_{ns}=11$~cm/s.
}
\label{fig:ldensfullbiotvsliatime}
\end{figure}
Clearly the LIA results deviate significantly from those of the fully
nonlocal calculation. The presence of nonlocal interactions reduces
the vortex line length density. As Adachi {\em et al.}\cite{adachi10}
explain, the nonlocal interaction is strongest immediately before and
after a reconnection, when vortices are closest. The nonlocal term
tends to repel two parallel vortex segments and attract antiparallel
segments. Consequently, fewer parallel reconnections occur when the
nonlocal interaction is included. Furthermore, reconnections between
antiparallel segments produce highly curved regions that retreat away from
the reconnection site quickly. By contrast, reconnections of parallel
vortices result in more gently curved segments that do not retreat
quickly. The net result of these effects is that the nonlocal interaction
increases the average intervortex separation and correspondingly decreases
the line length density.
Figure \ref{fig:vortextanglefullbiotvslia} shows several snapshots
from these simulations. The first row is a homogeneous tangle from
the full Biot-Savart calculation. The LIA tangle appears similar for
$0.02<t<0.08$~s, a nearly steady-state regime. Aarts \cite{aartsdiss}
uses this regime to evaluate properties of the vortex
tangle. At later times, the system collapses into the open-orbit state,
with vortices aligning perpendicular to the driving velocity. The second
and third rows of Figure \ref{fig:vortextanglefullbiotvslia} illustrate
configurations during the collapse and after its completion. How quickly
the open-orbit state forms depends strongly on the exact parameters used
for a simulation. For example, it appears more quickly at high driving
velocities, where the growth of line length in the plane perpendicular to
the applied velocity helps to nucleate the open-orbit state. In some
cases the intermediate, partially collapsed state can continue for
significant times, possibly indefinitely. The density of the open-orbit
state is an artificial value, determined by the absolute cutoff distance
used for reconnections. While this value plays almost no role for a highly
tangled state, it becomes the {\em de facto} reconnection distance as
vortices form open orbits and straighten. In the final state, neighboring
vortices are spaced far enough apart to avoid further reconnections.
We demonstrate homogeneity of the full Biot-Savart calculation
directly by finding the average line length
density in different constituent volumes within the system
(Equation~\ref{eq:ldens}).
Figure~\ref{fig:homogldenstime} shows a sample evolution of $L(t)$.
We use the eight $\mathbb{R}^3$ octants as the volumes to calculate each curve.
\begin{figure}
\centering
\includegraphics{figure3}
\caption{Evolution of the vortex line length density, $L(t)$,
within each octant of the periodic cube. The curve marked by black asterisks gives the line length per total system
volume, $D^3$. Each other curve gives $L(t)$ for
one of the eight $\mathbb{R}^3$ octants. The applied velocity is
$v_n=12$~cm/s.
}
\label{fig:homogldenstime}
\end{figure}
Even after steady-state turbulence has set in, we expect some variation
in $L(t)$ among the octants, but the average equilibrated values should
be the same. Figure~\ref{fig:homogldens} shows these average
equilibrated $L$ values for each octant, for a range of driving
velocities. Time averages were done over the equilibrated time domain.
We use the bracket notation $\langle X\rangle$ to denote the time average,
as opposed to averages over the length of the vortex at a fixed time
for which we use the notation $\bar X$. From Figure~\ref{fig:homogldens}, it
is clear that on this scale our system is homogeneous at every velocity.
\begin{figure}
\centering
\includegraphics{figure4}
\caption{Average
equilibrated line length density, $\langle L\rangle$,
within each octant of the periodic cube, at multiple velocities.
}
\label{fig:homogldens}
\end{figure}
Scaling arguments provide an additional test of homogeneous turbulence.
Simple dimensional analysis shows that the line length density
in homogeneous turbulence should depend on the driving velocity
as\citep{schwarz88}
\begin{equation}
\langle L\rangle=c_L^2(v_{ns}/\beta)^2, \label{eq:ldensv}
\end{equation}
where $c_L$ is temperature-dependent.
As noted previously, $\beta$ depends logarithmically on the average local
radius of curvature, which decreases with increasing velocity. To
keep $c_L$ independent of velocity, we do not combine $\beta$ with $c_L$.
As shown in Figure~\ref{fig:ldensbeta}, our simulations closely follow
this scaling law. The best-fit value of $c_L$ is 0.106.
\begin{figure}
\centering
\includegraphics{figure5}
\caption{Average equilibrated line length density, $\langle L\rangle$,
for the total system volume, at multiple velocities.
The linear fit to our data yields a slope of $c_L=0.106$.
}
\label{fig:ldensbeta}
\end{figure}
Another scaling check of our simulations comes through the mutual friction
force density due to interactions between the normal fluid and superfluid
components. This force results in the interaction term of the vortex equation
of motion ($\vec v_{fric}$),
\begin{equation}
\vec{F}_{sn}=-\frac{\rho_s\kappa}{V}\hat{s}^{\,\prime}\times\vec{v}_{fric},
\label{eq:mutualfriction}
\end{equation}
where $V$ is the system volume. When averaged along the entire
vortex, only $\bar{F}_{sn}$ parallel to the driving velocity is non-negligible.
Schwarz \cite{schwarz88} also developed a scaling relation for the
equilibrated force, $\langle\bar{F}_{sn}\rangle\propto v_{ns}^3/\beta^2$.
We construct a new quantity:
\begin{equation}
\bar{\Gamma} := \frac{\bar{F}_{sn}}{\rho_s\kappa\alpha v_{ns}},
\label{eq:gammafric}
\end{equation}
which we can fit in the form
$\beta\langle\bar{\Gamma}\rangle^{1/2}=c_F v_{ns}+y_0$,
as shown in Figure~\ref{fig:slopegammafric}. The
linear fit shows a slope of $c_F=0.088$ with no $y$-intercept, as
theory predicts.
\begin{figure}
\centering
\includegraphics{figure6}
\caption{Left axis (red circles):
The quantity $\langle\bar{\Gamma}\rangle$ taken from the mutual
friction force density, $\bar{F}_{sn}$ along $\hat{v}_{ns}$.
The linear fit of this data yields a slope of $c_F=0.088$.
Right axis (blue diamonds): The average vortex velocity, $\bar{v}_L$.
The linear fit of this data yields a slope of $c_v=0.070$.
}
\label{fig:slopegammafric}
\end{figure}
Yet another quantity used to describe homogeneous superfluid turbulence is
the average vortex velocity relative to the applied superfluid velocity
$\vec{v}_s$:
\begin{equation}
\bar{v}_L=\left[\frac{1}{V L}\int \dot{s}\mathrm{d}\xi\right]-\vec{v}_s.
\label{eq:avgvortexvelocity}
\end{equation}
As for $\bar{F}_{sn}$, only the component of $\bar v_L$ in the
direction of the driving velocity is non-negligible. This time the quantity in
question requires no scaling with $\beta$: $\bar{v}_L/v_{ns}=b(T)$,
where $b(T)$ is a temperature-dependent parameter
\citep{brewer82,schwarz88}. Our data for $\bar v_L$
are shown in Figure~\ref{fig:slopegammafric}.
The anisotropy parameters from Equations~\ref{eq:Ipardef} through
\ref{eq:Ielldef} are also consistent with previous simulations:
$\langle I_\parallel\rangle=0.761$, $\langle I_\perp\rangle=0.620$,
$\langle I_\ell\rangle=0.646$.
These values show no significant trend over our range of $v_{ns}$.
Schwarz \cite{schwarz88} points out that under the LIA,
the average mutual friction force density $\bar{F}_{sn}$ along
$\hat{v}_{ns}$ can be written as:
\begin{equation}
\bar{F}_{sn}=\rho_s\kappa\alpha(c_L^2I_\parallel
- c_L^3I_\ell)v_{ns}^3/\beta^2. \label{eq:mutualfrictionanisotropy}
\end{equation}
As we have defined it, then:
\begin{equation*}
\bar{\Gamma}=(c_L^2I_\parallel - c_L^3I_\ell)
\left(\frac{v_{ns}}{\beta}\right)^2.
\end{equation*}
We can see that the quantity $c_F$, obtained earlier
from the mutual friction force density data, is equal to
$(c_L^2 I_\parallel-c_L^3I_\ell)^{1/2}$.
Equation~\ref{eq:mutualfrictionanisotropy} is only approximate when nonlocal
contributions are included through the Biot-Savart law; hence we expect our
calculations to produce slightly different values from those of Schwarz
\cite{schwarz88} and Aarts \cite{aartsdiss}, who used the LIA. Additionally,
Schwarz \cite{schwarz88} states that, when neglecting $\alpha'$, the slope
parameter we call $c_v$ is equal to $c_L I_\ell$, again assuming the LIA. With
these relations, we compare characteristics of homogeneous turbulence in our
own simulations and in previous works, in Table~\ref{tab:homogturbquantities}.
We use $c_v$ or $c_L I_\ell$, depending on which can be derived from the data
shown in earlier works. Similarly, we use $c_F$ or $(c_L^2
I_\parallel-c_L^3 I_l)^{1/2}$. Schwarz \cite{schwarz88} does
show that his theoretical calculations of $I_\perp/I_\parallel$,
$(c_L^2I_\parallel-c_L^3I_\ell)^{1/2}$, and $c_LI_\ell$ match experiment.
\begin{table*}
\centering
\begin{tabular}{|c | c | c | c c | c c |}
\hline
Source (T=1.6~K) & $c_L$ & $I_\perp/I_\parallel$ &
$c_F$ & $(c_L^2I_\parallel - c_L^3I_\ell)^{1/2}$ & $c_v$ & $c_L I_\ell$\\
\hline
Present Work & 0.106 & 0.815 & 0.088 & 0.088 & 0.070 & 0.069 \\
Aarts\cite{aartsdiss} & 0.11 & - & 0.095$^\dagger$ & - & 0.045$^\dagger$
& - \\
Schwarz\cite{schwarz88} & 0.137 & 0.775$^\ddagger$ & - & 0.116$^\dagger$ & - & 0.063$^\dagger$ \\
\hline
\end{tabular}
\caption{Comparison of
quantities characterizing homogeneous turbulence.
$\dagger$: value calculated from other reported quantities;
$\ddagger$: value estimated from published figure.
}
\label{tab:homogturbquantities}
\end{table*}
Both Adachi {\em et al.}\cite{adachi10} and Kondaurova {\em et
al.}\cite{kondaurova08} report $\gamma=c_L/\langle\beta\rangle$
rather than $c_L$ itself. Doing so requires one to insert an
ad hoc $y$-intercept to maintain a decent linear fit. When
we do this, we get a value of $\gamma=137.3$~s/cm$^2$,
where Adachi {\em et al.}\cite{adachi10} report a value of
$\gamma=109.6$~s/cm$^2$. The experimental value is $\gamma=93$~s/cm$^2$
~\citep{childers76,brewer82}. We can extract the value for
$I_\perp/I_\parallel\approx0.8$ from Adachi {\em et al.}\cite{adachi10},
which also matches our quantity from Table~\ref{tab:homogturbquantities}
of $0.815$. Kondaurova {\em et al.}\cite{kondaurova08} find
$\gamma=280$~s/cm$^2$, which is extremely high. In more recent work by
the same lead author\cite{kondaurova14}, $\gamma$ ranges from 105.8 to
120.2 s/cm$^2$ at 1.6 K, depending on the reconnection criterion used. The
difference may be from the larger sample volume and much lower applied
fluid velocities used for the later calculations.
A final test is the rate of vortex reconnections, an important process
in carrying energy down to smaller length scales. Previous simulations
and analysis show that the rate of vortex reconnections is related to
the line length density by $\dot{n}=C\kappa\langle L\rangle^{5/2}$,
where $\dot{n}$ is the reconnection rate per unit volume and $C$
is a dimensionless constant with a value of approximately 0.1-0.5
~\citep{tsubota00,barenghi04,nemirovskii06,kondaurova14}. Once again the
earlier Kondaurova {\em et al.}\cite{kondaurova08} results are outliers,
with an overlarge prefactor of $C=2.47$. Figure~\ref{fig:reconnections}
gives our data on the vortex reconnection rate for a series of trials. We
find an exponent of 2.47 and $C=0.42$, within the normal range.
\begin{figure}
\centering
\includegraphics{figure7}
\caption{A plot of $\ln(\dot{N})$ versus $\ln(\langle L\rangle)$, finding
the exponent of $\dot{N}=\mathrm{d}N/\mathrm{d}t\propto\langle L\rangle^m$,
$m=2.47$.
}
\label{fig:reconnections}
\end{figure}
Our full Biot-Savart calculation matches previous calculations of homogeneous
superfluid turbulence through multiple comparisons.
\section{\label{sec:limbiottorus}Nonlocal Distance, $d_{NL}$}
In this section, we investigate further the finding by Adachi {\em et
al.}\cite{adachi10} that the LIA approximation is inadequate for producing
homogeneous turbulence. We saw in Figure~\ref{fig:ldensfullbiotvsliatime}
that despite the fact that $L$ retains an approximate $\langle
L\rangle\propto v^2$ relationship \citep{schwarz88,aartsdiss},
and even in a time domain before the open-orbit vortex state
dominates, $L$ still deviates significantly between the LIA and
full Biot-Savart calculations. The two methods do produce equally
isotropic systems, however. For certain limited objectives, such
as Schwarz's\citep{schwarz82,schwarz88} efforts to identify the
attributes necessary for a given behavior, the LIA may be an acceptable
approximation.
Here we investigate whether some degree of the computation-saving
power found in the LIA could be retained by truncating the nonlocal
interaction at a distance which we denote by $d_{NL}$. For each point on
the vortex, we only include contributions to the Biot-Savart integral,
Equation~\ref{eq:toruseqofmotion}, from vortex segments that are within
a distance $d_{NL}$ of the evaluation point. Our results for $\langle
L\rangle^{1/2}(v_{ns})$ are displayed in
Figure~\ref{fig:ldensvarylimbiot}, for a range of $d_{NL}$ values.
\begin{figure}
\centering
\includegraphics{figure8}
\caption{Effects of nonlocal distance, $d_{NL}$, on
the equilibrated line length density, $\langle L\rangle$. Each curve
is a series of trials at the $d_{NL}$, listed in the legend. The side
length for the sample region is $D=50 \mu$m, and
the largest $d_{NL}$ allows interactions throughout the volume.
Averages at each $v_{ns}$ used the initial equilibrated
range, similar to Aarts\cite{aartsdiss}.
}
\label{fig:ldensvarylimbiot}
\end{figure}
For a sample volume with side $D=50 \mu$m, Biot-Savart interactions among
all pairs of vortices occur at $d_{NL}=\sqrt{3}D/2 = 43.3 \mu$m. As
$d_{NL}$ decreases from this maximum value, the data at first remain
remarkably similar to those using the maximum $d_{NL}$. Eventually the
data begin to deviate, with the line-length density increasing above its
value from the full calculation. The onset of this divergence is even more
apparent in Figure~\ref{fig:rmsdevldens}, which plots the root-mean-square
(RMS) deviation of $\langle L\rangle$ from the full Biot-Savart value:
\begin{equation}
\delta_{rms}(\langle L\rangle)
=\left\langle \left( \frac{\langle L\rangle-\langle L_{BS}\rangle}
{\langle L_{BS}\rangle}\right)^2 \right\rangle_{v_{ns}}^{1/2}.
\label{eq:rmsdev}
\end{equation}
Here $\langle\ldots\rangle_{v_{ns}}$ represents the average over trials
with different $v_{ns}$ values but the same $d_{NL}$. The quantity in
Equation~\ref{eq:rmsdev} acts as a measure of the error in our calculation
due to the truncated nonlocal interaction. Figure~\ref{fig:rmsdevldens}
shows how this quantity varies with $d_{NL}$.
\begin{figure
\centering
\includegraphics{figure9}
\caption{RMS fractional deviation of
$\langle L \rangle$ from the full Biot-Savart calculation
for several nonlocal distances. Box side is $50 \mu$m.
The trials contributing to each point are from
Figure~\ref{fig:ldensvarylimbiot}, and use the initial
equilibrated time domain for calculating the average of $L(t)$.
}
\label{fig:rmsdevldens}
\end{figure}
Truncated nonlocal distances of at least $8.33 \mu$m show good agreement
($\lesssim5\%$) with the full Biot-Savart calculation, but a clear
departure arises by $6.25 \mu$m.
These time averages all use the initial equilibrated time domain before the
characteristic drop in $L(t)$, seen in
Figure~\ref{fig:ldensfullbiotvsliatime}, i.e. before the degeneration into the
open-orbit state occurs at the lower $d_{NL}$ values. However, the subjectivity
in selecting the time domain over which to average $L(t)$ could potentially
affect the $d_{NL}$ needed for agreement with the full Biot-Savart calculation.
An alternative approach, for conditions that result in the open-orbit state, is
to calculate the standard deviation of $L(t)$ over the entire time domain after
the initial equilibration. The drop in $L(t)$ upon entering the open-orbit
state sharply increases the standard deviation. Similarly, since from
Figure~\ref{fig:ldensfullbiotvsliatime}, $I_\parallel$ only deviates from the
full Biot-Savart value outside of the initial equilibrated range, we can also
use this quantity and its standard deviation as indicators of an adequate
$d_{NL}$. There is still a limited sample of data, since we cannot run trials
forever, and we still have to pick out a time when the data start to
equilibrate. Figure~\ref{fig:rmsdevallparams} shows the RMS fractional
deviation of $\sigma(L(t))$, $\sigma(I_\parallel(t))$, $\langle L\rangle$, and
$\langle I_\parallel\rangle$ (making the appropriate substitutions into
Equation~\ref{eq:rmsdev}) for the trials from Figure~\ref{fig:rmsdevldens}
using the entire equilibrated time domain.
\begin{figure}
\centering
\includegraphics{figure10}
\caption{RMS fractional deviation of
the standard deviations $\sigma(L(t))$ (red circles)
and $\sigma(I_\parallel(t))$ (blue diamonds)
from the full Biot-Savart calculation plotted versus
nonlocal distance. Inset: RMS fractional deviation of the
averages $\langle L\rangle$ (red circles)
and $\langle I_\parallel\rangle$ (blue diamonds) from
the full Biot-Savart calculation plotted versus nonlocal distance.
Averages use the entire equilibrated time domain.
Comparing $\langle L\rangle$ in the inset here
to the data in Figure~\ref{fig:rmsdevldens} shows the effect of
changing the time domain used for the averaging.
}
\label{fig:rmsdevallparams}
\end{figure}
From the inset of Figure~\ref{fig:rmsdevallparams} we see
that the critical $d_{NL}$ value is $\lesssim 8.33 \mu$m, the
same whether we use the entire equilibrated time domain in the
averages of $L(t)$ and $I_\parallel(t)$ or simply the initial
time domain, like in Figure~\ref{fig:rmsdevldens}. The main part of
Figure~\ref{fig:rmsdevallparams} shows that the standard deviations of
$L(t)$ and $I_\parallel(t)$ are very good parameters for measuring this
critical $d_{NL}$, showing a pronounced increase in error for nonlocal
distances below this same critical value. We can use a small fraction of
the nonlocal interaction volume and still get a very good approximation
of the Biot-Savart integral for homogeneous superfluid turbulence.
While Figures \ref{fig:rmsdevldens} and \ref{fig:rmsdevallparams} show
averaged results over trials at several values of driving velocity,
Figure~\ref{fig:devvsvelocity} breaks out the behavior of line length
as a function of driving velocity for several nonlocal distances $d_{NL}$.
\begin{figure
\includegraphics{figure11}
\caption{Fractional deviation of
(a) $\langle L\rangle$, and (b) $\sigma(L(t))$,
from the full Biot-Savart calculation plotted versus
driving velocity. Each curve is a different
nonlocal distance, $d_{NL}$, indicated by the legend.
Averages use the entire equilibrated time
domain.
}
\label{fig:devvsvelocity}
\end{figure}
For a fixed $d_{NL}$, deviations of $\langle L\rangle$ from the full nonlocal
calculation are more pronounced at the lowest velocities. At high velocity
$\langle L\rangle$ is often an average of an initial equilibrated level above
that of the full Biot-Savart calculation, and a later level below the
Biot-Savart value. The overall $\langle L\rangle$ depends on the amount of time
spent in each regime and has little real significance; it may even be
coincidentally close to the correct value. For $\sigma(L(t))$ the reverse is
true, with deviations larger at higher velocities. The slight enhancement of
$\langle L\rangle$ in the initially equilibrated state has little effect on
$\sigma(L(t))$, but the collapse to the open-orbit state drastically increases
its value. $\langle I_\parallel\rangle$ and $\sigma(I_\parallel(t))$, not shown
in the figure, also deviate more at higher velocities. As shown in
Figure~\ref{fig:ldensfullbiotvsliatime}~(b), $\langle I_\parallel\rangle$
matches that of the full Biot-Savart calculations until onset of the open-orbit
state, which occurs only for our higher velocities. The abrupt shift in $\langle
I_\parallel\rangle$ also increases $\sigma(I_\parallel(t)$. Because of the
different aspects measured by these quantities, calculating several of them is
useful for finding any discrepancies from the full Biot-Savart calculations.
From a practical perspective, regardless of the underlying cause of any
difference, we want to choose an interaction distance such that every driving
velocity we use produces turbulent behavior matching the full Biot-Savart law.
Our previous averages over all velocities at one $d_{NL}$ pick out deviations at
any $v_{ns}$, and larger number of trials contributiong to each curve reduces
the scatter so that the critical interaction distance becomes clear.
We next examine how this critical interaction distance varies with system size.
We simulated turbulence in three other system sizes, with other parameters held
constant. $\langle L\rangle$ data for all three
system sizes are shown in Figure~\ref{fig:slopeldensallDvals}.
\begin{figure}
\includegraphics{figure12}
\caption{$\beta\langle L\rangle^{1/2}$ data
for full Biot-Savart calculations at several system sizes $D$.
Averages use the entire equilibrated time domain.
}
\label{fig:slopeldensallDvals}
\end{figure}
The good agreement in $\langle L\rangle$ for
different system sizes is another mark of homogeneity. Since
$\langle L\rangle$ is an intensive quantity, with a homogeneous system it must
be independent of system size. Figure \ref{f:boxsize} illustrates the line
length in the initial equilibrated region, for the three sizes tested at applied
velocity of 9 cm/s. The line length density changes slightly with cell size, but
for a given cell size it remains constant for interaction distances at least
$12.5 \mu$m. For a fixed velocity, the necessary interaction distance does not
depend on system size.
\begin{figure}
\includegraphics{figure13}
\caption{Square root of line length density as function of
interaction distance $d_{NL}$, for three box sizes $D$. In each
case $\alpha=0.1$ and $v_{ns}=9$ cm/s.
}
\label{f:boxsize}
\end{figure}
\section{\label{sec:discussion}Discussion}
Similar calculations with other friction coefficients $\alpha$ and applied
velocities $v_{ns}$ shed light on why the non-local interaction need only be
retained for a portion of the cell volume. Our trials with smaller $\alpha$ are
related to lower temperatures, although an attempt to model real behavior of
superfluid helium for these values of $\alpha$ would have to include the
additional $\alpha^\prime$ term that we neglect. The critical interaction
distance is as low as $4 \mu$m, for the lowest friction and highest velocity. We
unfortunately cannot keep $v_{ns}$ entirely fixed while varying $\alpha$. At the
velocities of 9-11 cm/s used in much of this work, turbulence is unsustainable
for $\alpha=0.01$, when less energy is drawn from the driving velocity field. On
the other hand, velocities of 20 cm/s or 50 cm/s, which work for $\alpha=0.01$,
would take impractically long with $\alpha=0.1$.
\begin{figure}
\includegraphics{figure14}
\caption{Excess line length as a function of interaction distance, for
several combinations of friction parameter $\alpha$ and applied
velocity $v_{ns}$. For each curve $\langle L\rangle^{1/2}$ for the
largest $d_{NL}$ is subtracted. Main graph: expanded view near critical
interaction distance. Inset: entire range. The legend identifies each
curve by $\alpha$ and $v_{ns}$, the latter measured in cm/s.
}
\label{f:lowTall}
\end{figure}
Figure \ref{f:lowTall} shows results from several trials with lower
friction coefficient. The critical interaction distance changes noticeably
among the trials, generally decreasing as $v_{ns}$ increases. We
suggest that a key issue is the relationship of $d_{NL}$ to the typical
intervortex spacing. Clearly the non-local term has minimal effect
if the calculation volume includes only a single, relatively straight
vortex segment along which the velocity field is being calculated. To
significantly alter the LIA simulation, $d_{NL}$ must be larger than
either the typical vortex separation or the typical vortex radius of
curvature. In the former case, the Biot-Savart calculation would add an
interaction between neighboring vortices too distance for an immediate
reconnection. In the latter case, the time development of a single
contorted filament would change. This is probably most relevant for
highly curved vortices, where self-reconnections are especially likely.
The typical vortex separation $\ell$ can be estimated from the line length
through $\ell=\langle L\rangle^{-1/2}$. For straight, parallel vortices
in a square lattice, this formula gives exactly the nearest-neighbor
separation. The calculation is not exact for other lattices, let alone
for a tangle of curved vortices, but $\ell$ does give a sense of the
intervortex distance.
Table \ref{t:critdist} shows the critical interaction distance along with
the typical vortex separation and the average radius of curvature. The
critical $d_{NL}$ does seem related to $\langle R\rangle$. In fact, the
turn-ups for the different curves of Figure \ref{f:lowTall} are ordered
exactly by the $\langle R\rangle$ values. The relationship to $\ell$ is
less clear. This is hardly surprising, given that the relation of $\ell$
to the nearest-neighbor vortex spacing is inexact and depends on features
like the typical radius of curvature in the tangle. In any case, the
necessary $d_{NL}$ apparently has the same order of magnitude as $\ell$.
\begin{table}
\label{t:critdist}
\begin{tabular}{|lcccc|}
\hline
$\alpha$\hspace*{.45in} & $v_{ns}$ (cm/s)\hspace*{.15in} & critical $d_{NL}$ &
$\hspace*{.2in}\ell$\hspace*{.2in} & \hspace*{.1in}$\langle R\rangle$\hspace*{.1in}\\
\hline
0.01 & 50 & 4 & 5.8 & 3.7\\
0.01 & 20 & 6 & 13.1 & 6.6\\
0.036 & 17 & 6 & 7.5 & 4.8\\
0.036 & 12 & 6 & 10.6 & 6.3\\
0.036 & 10 & 7 & 12.5 & 7.5\\
0.1 & 9 & 9 & 8.9 & 6.3\\
\hline
\end{tabular}
\caption{Onset of deviation from non-local calculation, compared to the
typical intervortex distance and the average radius of curvature. All
lengths are given in $\mu$m. The critical $d_{NL}$ is identified as the
smallest $d_{NL}$ with results indistinguishable by our measures from
the full Biot-Savart calculation. In each case we used 1 $\mu$m steps
in $d_{NL}$ near the critical value.}
\end{table}
Finding the critical $d_{NL}$ comparable to the typical vortex separation
suggests that the key effect of the nonlocal interaction is to draw
together nearby vortex segments until they are within the reconnection
distance. In the LIA, neighboring vortices are invisible to each
other unless they happen to drift close enough to reconnect. If an
extra, small attraction from their Biot-Savart interaction results
in a reconnection that would not otherwise have occured, it has a
significant impact on the subsequent motion. This finding makes sense
in light of the open-orbit problem. An occasional open-orbit vortex
need not doom a simulation if further reconnections free the vortex
from its open-orbit state. Only a collective alignment of open-orbit
vortices persists indefinitely. The occasional rotation step that
Schwarz\cite{schwarz88} used to sustain a vortex tangle operates exactly
by ensuring reconnections. Our findings explain why the non-local
calculation prevents the simultaneous degeneration of vortices into the
open-orbit state. As Nemirovskii notes\cite[p. 152]{nemirovskiireview},
the addition of non-local contributions cannot prevent the reconnections
that lead to open-orbit vortices. Rather, the non-local term enables
{\em additional} reconnections that disrupt the system from settling
into a collective open-orbit configuration.
Beyond the intervortex spacing, the nonlocal contribution is apparently
negligible. This is consistent with the rapid fall-off of the Biot-Savart
law. In addition, for a random tangle the number of vortex segments a
given distance from a calculation point increases as the square of the
distance, and the multiple interactions partially cancel, further reducing
the total. The unimportance of distant non-local terms has been remarked
on previously. A complete Biot-Savart calculation with periodic boundary
conditions would require tiling space with the computational volume
and adding contributions from each copy of the tangle. Although Ewald
summation provides a rapidly-converging technique \cite{toukmaji}, it
has not been applied to vortex-filament simulations. However, Kondaurova
{\em et al.}\cite{kondaurova14} find that surrounding the main region
with 26 replicas has negligible effect on the results. Our own ``full"
Biot-Savart calculation uses the main volume, with just a few of the
closest image vortices included as well. While it has been standard
practice to exclude very distant contributions, we now argue that
much of the contribution from within the main volume can also be safely
omitted. Yet another effort, in a similar direction, is Baggaley's recent
tree method\cite{baggaley12tree}. Baggaley essentially averages groups
of more distance vortex segments before evaluating the Biot-Savart law,
and finds good agreement with the full Biot-Savart calculations of Adachi
{\em et al.}\cite{adachi10}. This is to be expected since Baggaley's
method has minimal averaging for very close vortex segments. Those are
often treated the same as in the Adachi calculation. Averaging over more
distant segments has a negligible effect on the statistical properties
of the tangle, as does our truncating the Biot-Savart law to omit these
segments entirely.
Reducing the interaction distance to such a degree can provide
considerable savings in computation time. With interactions omitted
beyond $8.33 \mu$m in a box of side $50 \mu$m, the interaction volume
for each vortex segment is about $2\%$ of the entire box: when the
system is homogeneous, each segment will have only $1/50$ of the other
vortex segments contributing to its nonlocal velocity. Since other
calculations must be performed for each point in the tangle, the end
result is that simulations with the reduced interaction distance take
$1/3$ to $1/10$ of the computation time. The savings could be greater for
larger computational volumes or higher driving velocities. A question of
practical interest is how small an interaction distance can safely be
used. Our results suggest that ensuring interactions from neighboring
vortices should be adequate. Furthermore, a possible $d_{NL}$ need not
be tested against the full Biot-Savart calculation. It is enough to
obtain equivalent results from a few trials with different $d_{NL}$,
all of which can be small compared to the entire cell size.
\begin{figure}[b]
\includegraphics{figure15}
\caption{For comparison with Figure~\ref{fig:rmsdevallparams}:
Figure (a) has system size $D=25 \mu$m, Figure (b)
has $D=100 \mu$m. RMS fractional deviation
from the full Biot-Savart calculation of the standard deviations
$\sigma(L(t))$ (red circles)
and $\sigma(I_\parallel(t))$ (blue diamonds), plotted versus
nonlocal distance. Insets: RMS fractional deviation from
the full Biot-Savart calculation of $\langle L \rangle$ (red circles)
and $\langle I_\parallel\rangle$ (blue diamonds), plotted versus
nonlocal distance.
Time averages use the entire time domain after initial equilibration.
}
\label{fig:rmsdevdiffr0}
\end{figure}
As noted previously, in many cases a single $d_{NL}$ may be desired
across a range of velocities. Usable velocities vary with cell size,
as shown in Figure \ref{fig:slopeldensallDvals}. When the velocity is
too low, tangles are not self-sustaining. When it is too high, the total
vortex length and hence the computational time become prohibitive. As
a practical matter, the necessary interaction distance then becomes
dependent on cell size as well. For the present calculations with
$\alpha=0.1$, we find that $d_{NL}=D/6$ is appropriate for several cell
sizes. Figure \ref{fig:rmsdevdiffr0} illustrates how deviations from the
full non-local calculation begin for two cell sizes, with the velocity
ranges of Figure \ref{fig:slopeldensallDvals}.
\section{\label{sec:conclusion}Conclusion}
Adachi {\em et al.}\cite{adachi10} show the importance of the nonlocal
interaction for homogeneous superfluid turbulence behavior, especially
when simulating this system using periodic boundaries. By using a
truncated Biot-Savart integral, we have found it possible to regain much
of the time saved with the localized induction approximation while still
accurately modeling the statistical behavior of the system. We find that
the nonlocal interaction is only important up to distances of about the
intervortex spacing.
Finally, we address the longstanding question of whether the trouble
with obtaining reproducible results on vortex tangles lies with the LIA,
the reconnection procedure, or the periodic boundary conditions. Our
answer is that the LIA is the main culprit. Its sicknesses cause
unusual sensitivity to other details of the simulation, as shown
previously\cite{baggaley12recon} in the case of reconnection requirements.
Worse, its results can deviate quantitatively from those of other
techniques, even when no qualitatively obvious problem arises. However,
the reason behind the problems with the LIA is primarily its inaccurate
treatment of neighboring vortices, rather than the lack of stretching
that causes difficulty for classical fluid simulations. An appropriate
admixture of a limited non-local term appears to resolve the trouble
with the pure LIA.
\underline{Acknowledgement:} One of us (O. M. Dix) acknowledges support
from the U.S. Department of Education through a GAANN fellowship.
|
train/arxiv
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BkiUeJHxK1UJ-2RwXGTM
| 5 | 1 |
\section{Introduction}
It has been argued in a very recent paper \cite{GGPR06} that
purely bosonic solutions preserving $31$ out of $32$
supersymmetries, hence describing {\it BPS preon} states
\cite{BPS01}, do not exist for IIB supergravity. Using the moving
$G$-frame method of \cite{BPS03} (Sec. 1.2), we rederive this
result here (Sec. 2). Then, we apply the same technique to the IIA
case and also show that preonic solutions do not exist in type IIA
supergravity (Sec. 3). Nevertheless, the concluded absence of
preonic solutions could be modified if the `stringy'
$(\alpha^\prime)^3$-corrections to the dilatino transformation
rule were made explicit and taken into account (Sec. 5).
For $D=11$ supergravity, the existence of BPS preonic solutions is
not ruled out even at the classical level ({\it i.e.}, ignoring
$(\alpha^\prime)^3$-corrections), although the above negative
results for type IIA supergravity already set strong restrictions
(Sec. 4) to be satisfied by these solutions.
\subsection{Basic notions and notation}
In eleven-dimensional supergravity \cite{CJS78} the only fermionic
field is the gravitino, $\check{\psi}^{\;\check{\alpha}}=
dx^{\check \mu} \check{\psi}_{\check{\mu}}^{\;\check{\alpha}} =
dx^{\mu}\check{\psi}_{{\mu}}^{\;\check{\alpha}} +
dx^{\#}\check{\psi}_{{\#}}^{\;\check{\alpha}} $ (${\check \mu} =
(\mu;\#), \; \mu=0,1,\dots,9$). In contrast, the ten-dimensional
type II supergravities \cite{IIBsugra,IIAsugra} contain, in
addition to two sixteen-component `spin 3/2' gravitini, two `spin
1/2' dilatini fields $\check{\chi}_{\check{\alpha}}$. We use the
czek superscript $\check{\alpha}$ to denote the type II indices of
the $32$-component reducible spinors. In the IIB case
$\check{\alpha}$ is the double index $\check{\alpha}=(\alpha, I)$,
where $I=1,2\,$ labels the two Majorana-Weyl (MW) spinors of the
same chirality and $\alpha=1, \ldots 16$. In the IIA case, where
both chiralities are present, $\check{\alpha}$ denotes the
Majorana spinor index and thus $\check{\alpha}=1,\ldots , 32$.
In particular, for the dilatino of type IIA supergravity we write
\begin{eqnarray}\label{IIADIL}
\hbox{IIA} \; :\qquad \qquad \check{\chi}^{\check{\alpha}}:=
({\chi}^{{\alpha}1}\; , \; {\chi}^2_{{\alpha}})\; , \qquad
\check{\alpha}=1, \ldots ,32\; , \quad \alpha=1, \ldots , 16 \; .
\end{eqnarray}
while in type IIB supergravity the $32$-component dilatino field
decomposes into two MW spinors of the same chirality,
\begin{eqnarray}\label{IIBDIL}
\hbox{IIB} \; : \qquad \check{\chi}_{\check{\alpha}}:=
({\chi}^1_{{\alpha}}\; , \; {\chi}^2_{{\alpha}})\; , \quad
\check{\alpha}=(\alpha\,, I)\; , \quad I=1,2 \; , \quad \alpha=1,
\ldots , 16 \; . \quad
\end{eqnarray}
Notice that in the IIB case the position of the index
$\check{\alpha}$ cannot be changed since the two MW spinors are of
the same chirality and there is no $16\times16$ charge conjugation
matrix in the MW spinor representation. In contrast, in type IIA a
$32\times 32$ charge conjugation matrix exists; it is
anti-diagonal in the Weyl-like realization used here and exchanges
the 1 and 2 MW components in (\ref{IIADIL}).
In this condensed 32-component notation, the supersymmetry
transformation rules for the gravitini and dilatini fermionic
fields can be written in compact form for both IIA and IIB cases
as
\begin{eqnarray}
\label{susyDIL}
\delta_{susy} \check{\psi}_a^{\check{\alpha}}= {\cal
D}_a\check{\varepsilon}{}^{\,\check{\alpha}}:= D_a
\check{\varepsilon}{}^{\,\check{\alpha}}-
\check{\varepsilon}{}^{\check{\beta}} {\check t}_{a\,
\check{\beta}}{}^{\check{\alpha}}\; , \qquad \delta_{susy}
\check{\chi} = \check{\varepsilon} M\; , \qquad
\end{eqnarray}
where $D=d-\omega$ is the Lorentz covariant derivative and ${\cal
D}=D-{\check t}$ is the generalized covariant derivative which
includes, besides the (suitable) spin connection $\omega:= {1\over
4}\omega^{ab}{\check\Gamma}_{ab}$, the additional tensorial IIA or
IIB ${\check t}$ contributions. The transformation rules for the
dilatino are algebraic and are characterized by a $32\times 32$
matrix $M_{\check{\beta}}{}^{\check{\alpha}}$. The form that this
matrix takes will be crucial for the discussion below.
In the IIA case, and ignoring inessential bilinear fermionic
contributions, the terms in $\delta_{susy} \check{\chi}$ (Eq.
(\ref{susyDIL}), see {\it e.g.} \cite{IIAsugra} and \cite{MMS03}
and refs therein) are determined by the matrix
\begin{eqnarray}\label{M=IIA}
\hbox{IIA} \; : \qquad M_{\check{\beta}}{}^{\check{\alpha}}=
\left(\matrix{ {3\over 8} e^{\Phi} R\!\!\!\!/^{(2)} + {1\over 8}
R\!\!\!\!/^{(4)} & {1\over 2} \partial\!\!\!\!/\Phi- {1\over 4}
H\!\!\!\!/^{(3)} \cr {1\over 2} \tilde{\partial\!\!\!\!/}\Phi +
{1\over 4} \tilde{H}\!\!\!\!/^{(3)} & - {3\over 8} e^{\Phi}
\tilde{R}\!\!\!\!/^{(2)} + {1\over 8} \tilde{R}\!\!\!\!/^{(4)}
}\right) \; . \qquad
\end{eqnarray}
in terms of all the possible IIA fluxes (on-shell field
strengths), namely\footnote{$\sigma^a= \sigma^a_{\alpha\beta}$,
${\tilde \sigma}^a= {\sigma^a}^{\alpha\beta}$, $a=0,1,\dots,9\,$;
$\;\sigma^a{\tilde\sigma}^b+\sigma^b{\tilde\sigma}^a= 2 \eta^{ab}
={\tilde\sigma}^a\sigma^b+{\tilde\sigma}^b \sigma^a$. The sigma
matrices with one and five (three) vector indices are symmetric
(antisymmetric) with respect to the spinor ones. The transposition
of untilded sigma matrices with four and two vector indices,
respectively, converts them into the corresponding tilded and
minus tilded ones.},
\begin{eqnarray}
\label{fluxesIIA}
R_2:=dC_1\; , \qquad R_4:=dC_3-C_1\wedge H_3\; , \qquad H_3:=dB_2 \; \qquad and
\qquad d\Phi \; : \qquad
\end{eqnarray}
\begin{eqnarray}
\label{fl-NSNS} \cases{ H\!\!\!\!/^{(3)}= {1\over 3!}
H_{abc}\sigma^{abc} \; , \qquad \sigma^{abc}:=
({\sigma}^{[a}\tilde{\sigma}^{b}\sigma^{c]})_{\alpha\beta}\; , \cr
\tilde{H}\!\!\!\!/^{(3)}= {1\over 3!} H_{abc}\tilde{\sigma}^{abc}
\; , \qquad \tilde{\sigma}^{abc}:= (\tilde{\sigma}^{[a}\sigma^
{b}\tilde{\sigma}^{c]})^{\alpha\beta}}
\; , \qquad
\cases{ \partial\!\!\!\!/\Phi := \partial_a\Phi
\sigma^a_{\alpha\beta}\; , \cr \tilde{\partial}\!\!\!\!/\Phi :=
\partial_a\Phi \tilde{\sigma}^{a\, \alpha\beta}\; ,}\qquad
\\ \label{fl-RRIIA} \cases{ R\!\!\!\!/^{(2)}:= {1\over 2!}
R_{ab}(\sigma^{ab})= - \tilde{R}\!\!\!\!/^{(2)}{}^T \; , \qquad
\sigma^{ab}:= (\sigma^{[a}\tilde{\sigma}^{b]}){}_\alpha{}^\beta \;
, \qquad \tilde{\sigma}^{ab}:=
(\tilde{\sigma}^{[a}\sigma^{b]}){}^\beta{}_\alpha \; ,\cr
R\!\!\!\!/^{(4)}= {1\over 4!}
R_{abcd}\sigma^{abcd}{}_\alpha{}^\beta =
(\tilde{R}\!\!\!\!/^{(4)}){}^\beta{}_\alpha\; , \qquad
\sigma^{abcd}:=
(\sigma^{[a}\tilde{\sigma}^{b}{\sigma}^{c}\tilde{\sigma}^{d]}){}_\alpha{}^\beta
\; . \quad }
\end{eqnarray}
The type IIB matrix $M$, in contrast, is given by (see
\cite{IIBsugra})
\begin{eqnarray}\label{M=IIB}
\hbox{IIB} \; : \qquad M_{\check{\beta}\check{\alpha}}=
\left(\matrix{ {1\over 2} {\partial\!\!\!\!/}\Phi + {1\over 4}
{H}\!\!\!\!/^{(3)} & - {1\over 2} e^{\!^{\Phi}} R\!\!\!\!/^{(1)} +
{1\over 4} e^{^{{1\over 2}\Phi}} R\!\!\!\!/^{(3)} \cr
{1\over 2}
e^{\!^{\Phi}} R\!\!\!\!/^{(1)} + {1\over 4} e^{^{{1\over 2}\Phi}}
R\!\!\!\!/^{(3)} & {1\over 2} {\partial\!\!\!\!/}\Phi - {1\over 4}
{H}\!\!\!\!/^{(3)} }\right) \; , \qquad
\end{eqnarray}
and involves the one-form and the three-form fluxes of type IIB
supergravity,
\begin{eqnarray}\label{fl-1,3IIB}
\;\; R_1:=dC_0\; , \quad R_3:=dC_2-C_0H_3\; , \qquad
H_3:=dB_2 \; \qquad and \qquad d\Phi \; , \qquad
\end{eqnarray}
but not the self dual five-form flux $R_5$,
\begin{eqnarray}\label{fl-5IIB}
R_5:=dC_4-C_2\wedge H_3\; , \qquad R_5=*R_5 \; \Leftrightarrow
\cases{ R\!\!\!\!/^{(5)} =0 \; , \cr
\tilde{R}\!\!\!\!/^{(5)}\not=0 \; . }
\end{eqnarray}
When only purely bosonic solutions are considered,
$\check{\psi}=0$, $\check{\chi}=0$, the parameter associated with
the preserved supersymmetry obeys a differential equation and an
algebraic one, namely ${\cal D}\check{\varepsilon}=0$ and
$\check{\varepsilon}M=0$. Usually, to describe a solution
preserving $k$ supersymmetries (a $\nu=k/32$ state), one uses $k$
independent bosonic Killing spinors $\epsilon_I^{\check\alpha}$
($I=1,\ldots , k$,
$\check{\varepsilon}=\kappa^I\epsilon_I^{\check\alpha}$ with
arbitrary constant fermionic $\kappa^I$) that satisfy the
following differential (from $\delta_{susy}
\check{\psi}_a^{\check{\alpha}}=0$) and algebraic (from
$\delta_{susy} \check{\chi}=0$ ) Killing equations
\begin{eqnarray}\label{KEqGR}
&& {\cal D}\check{\epsilon}_I := D
\check{\epsilon}_I - \check{\epsilon}_I {\check t}\, = 0\; ,
\qquad
\\ \label{KEqDIL} && \check{\epsilon}_I \, M\;
=0 \qquad (I=1, \ldots , k)\; , \qquad
\end{eqnarray}
which guarantee that the solution remains bosonic and hence
invariant after a gravitino and dilatino supersymmetry
transformation.
The conclusion of \cite{GGPR06} on the absence of a preonic
solution of type IIB supergravity is based on the algebraic
equation (\ref{KEqDIL}) and uses (\ref{KEqGR}) to close the
argument. We now recover this result below by using the moving
$G$-frame method of \cite{BPS03}.
\subsection{The moving $G$-frame method and preonic spinors}
A preonic state \cite{BPS01} preserves all supersymmetries but
one; it is a $\nu=31/32$ supersymmetric BPS state. As a result,
it can be characterized by one bosonic spinor
$\check{\lambda}_{\check{\alpha}}$ orthogonal to all the $31$
bosonic Killing spinors $\check{\epsilon}_I^{\;\check{\alpha}}$ in
(\ref{KEqGR}),
\begin{eqnarray}
\label{PreonL} \check{\epsilon}_I \, \check{\lambda} = \;
\check{\epsilon}_I^{\; \check{\alpha}}
\check{\lambda}_{\check{\alpha}} =0 \; , \qquad I=1, \ldots , 31\;
. \qquad
\end{eqnarray}
As it was noticed in \cite{BPS03}, when the generalized holonomy
group of supergravity \cite{Duff+Stelle91,Duff03} is a subgroup
of $SL(32,\mathbb{R})$ (which is the case for both $D=11$
\cite{Hull03} and type II $D=10$ supergravities \cite{P+T03}), the
spinor characterizing a BPS preonic state obeys the differential
equation
\begin{eqnarray}
\label{Dprl} {\cal D} \check{\lambda} := D \check{\lambda} +
\check{t} \check{\lambda} =0 \; , \qquad
\end{eqnarray}
where ${\check t}$ is the same tensorial part of the generalized
connection in Eqs. (\ref{KEqGR}) and (\ref{susyDIL}). Notice that
if ${\check t}\not= 0$ (the case of non-vanishing fluxes), Eq.
(\ref{Dprl}) is not equivalent to the Killing equation
(\ref{KEqGR}) even for the type IIA case where the $32\times 32$
charge conjugation matrix does exist.
Applied to the present problem, the moving $G$-frame method
\cite{BPS03} implies that Eq. (\ref{KEqDIL}), looked at as an
equation for the matrix $M$, is solved when $k=31$ by
\begin{eqnarray}
\label{KEqDIL31} M = \check{\lambda} \otimes \check{s} \qquad i.e.
\qquad \cases{IIA\; : \; M_{\check{\beta}}{}^{\check{\alpha}}=
\check{\lambda}_{\check{\beta}} \; \check{s}^{\check{\alpha}} \; ,
\cr IIB\; : \; M_{\check{\beta}\check{\alpha}}=
\check{\lambda}_{\check{\beta}} \; \check{s}_{\check{\alpha}} \;
,} \qquad
\end{eqnarray}
where $\check{s}_{\check{\alpha}}$ is a certain spinor. The
algebraic structure of the matrix $M$ implies a series of
restrictions on the preonic spinor
$\check{\lambda}_{\check{\beta}}$. At the same time, Eq.
(\ref{KEqDIL31}) imposes a series of restrictions on the fluxes
involved in the matrix $M$.
Eq. (\ref{KEqDIL31}) will be the basic equation in our analysis of
the absence of preons among the bosonic solutions of type II
supergravities.
\section{Absence of preons in type IIB supergravity}
In the type IIB case the matrix $M$ has the form of Eq.
(\ref{M=IIB}), and Eq. (\ref{KEqDIL31}) implies the following
relations for the one- and three-form fluxes
\begin{eqnarray}
\label{Eq-IIB-pr1}
\begin{matrix}
{ & {1\over 2} \partial\!\!\!\! / \Phi + {1\over
4}H\!\!\!\!/^{(3)} = \; \lambda^1_\alpha s_\beta^1 \; , &
\hbox{(\ref{Eq-IIB-pr1}a)} & \cr & - {1\over 2} e^{\!^{\Phi}}
R\!\!\!\!/^{(1)} + {1\over 4} e^{^{{1\over 2}\Phi}}
R\!\!\!\!/^{(3)}= \lambda^1_\alpha s_\beta^2 \; , &
\hbox{(\ref{Eq-IIB-pr1}b)} & \cr
& + {1\over 2}
e^{\!^{\Phi}} R\!\!\!\!/^{(1)} + {1\over 4} e^{^{{1\over 2}\Phi}}
R\!\!\!\!/^{(3)} = \; \lambda^2_\alpha s_\beta^1 \; , &
\hbox{(\ref{Eq-IIB-pr1}c)} & \cr \qquad & {1\over 2}
{\partial\!\!\!\!/}\Phi - {1\over 4} {H}\!\!\!\!/^{(3)} = \;
\lambda^2_\alpha s_\beta^2 \; . & \hbox{(\ref{Eq-IIB-pr1}d)} &
\qquad } \end{matrix}
\end{eqnarray}
These fluxes then can be expressed through the IIB preonic spinor
$\check{\lambda}_{\check{\alpha}}:= (\lambda^1_{{\alpha}}\; , \;
{\lambda}^2_{{\alpha}})$ and an arbitrary spinor
$\check{s}_{\check{\beta}}:= ({s}^1_{{\beta}}\; , \;
{s}^2_{{\beta}})$. Furthermore, the consistency of Eqs.
(\ref{Eq-IIB-pr1}) imposes a set of algebraic equations on these
two spinors. They follow from the fact that the fluxes enter into
Eqs. (\ref{Eq-IIB-pr1}) through matrices which possess definite
symmetry properties,
\begin{eqnarray}\label{IIBsym-asym}
(\partial\!\!\!\! / \Phi)^T = +\partial\!\!\!\! / \Phi \; , \qquad
(H\!\!\!\!/^{(3)})^T=- H\!\!\!\!/^{(3)} , \qquad
(R\!\!\!\!/^{(3)})^T=- R\!\!\!\!/^{(3)} , \qquad
(R\!\!\!\!/^{(1)})^T=+R\!\!\!\!/^{(1)} . \qquad
\end{eqnarray}
These lead to the algebraic constraints
\begin{eqnarray}
\label{Eq-IIBls}
\begin{matrix}
{ \hbox{(a)} \qquad & \lambda^1_{[\alpha} s_{\beta]}^1 +
\lambda^2_{[\alpha} s_{\beta]}^2 =0 \; , \qquad & \qquad
\hbox{(b)} \qquad & - \lambda^1_{[\alpha} s_{\beta]}^2 +
\lambda^2_{[\alpha} s_{\beta]}^1 =0 \; , & \cr
\hbox{(c)} \qquad & \lambda^1_{(\alpha} s_{\beta )}^1 - \lambda^2_{(\alpha} s_{\beta)}^2 =0 \; ,
\qquad & \qquad
\hbox{(d)} \qquad & \lambda^1_{(\alpha} s_{\beta )}^2 +
\lambda^2_{(\alpha} s_{\beta )}^1 =0 \; . & }
\end{matrix}
\end{eqnarray}
A straightforward algebra shows that Eqs. (\ref{Eq-IIBls}) have
only trivial solutions. This means that either the preonic or the
auxiliary spinor is zero,
\begin{eqnarray} \label{lorS=0IIB}
\hbox{IIB} \; : \qquad \lambda^1_{\alpha}=\lambda^2_{\alpha}=0
\qquad or \qquad
s_{\beta}^1 = s_{\beta}^2=0 \; . \qquad
\end{eqnarray}
In both cases the matrix $M=0$ and, hence, all the fluxes except
the five-form flux (Eq.(\ref{fl-5IIB})) are equal to zero,
$R_1=d\Phi=R_1=R_3=0$. Nevertheless, the fact that the solution
$s_{\beta}^1=s_{\beta}^2=0$ of (\ref{Eq-IIBls}) allows for a
non-vanishing preonic spinor
$(\lambda^1_{\alpha},\lambda^2_{\alpha})$ might give hope, at this
stage, of finding a nontrivial and unique solution ${\check
\lambda}$ to Eq. (\ref{Dprl}) and $k=31$ solutions
$\check{\epsilon}_I$ for Eq. (\ref{KEqGR}). This possibility is
ruled out by looking at Eq. (\ref{KEqGR}). For simplicity let us
begin by discussing Eq. (\ref{Dprl}). When only the five-form flux
is non-vanishing, Eq. (\ref{Dprl}) would acquire the relatively
simple form of
\begin{eqnarray} \label{KEq-l12-IIB}
\qquad R_1=R_3=H_3=d\Phi=0 \;: \quad \qquad \left\{
\begin{matrix} {D_b\lambda^1_{\alpha}= - {1\over 16} (\sigma_b
R\!\!\!\!/^{(5)})_{\alpha}{}^{\beta}\lambda^2_{\beta} \; , \cr
D_b\lambda^2_{\alpha}= \; \; {1\over 16} (\sigma_b
R\!\!\!\!/^{(5)})_{\alpha}{}^{\beta}\lambda^1_{\beta}\; . }
\end{matrix}\right.
\end{eqnarray}
Now one observes that, if $(\lambda^1_{\alpha},
\lambda^2_{\alpha})$ is a solution of Eq.(\ref{KEq-l12-IIB}),
$(-\lambda^2_{\alpha}, \lambda^1_{\alpha})$ provides another one.
As a result, the number of solutions of Eqs. (\ref{KEq-l12-IIB})
is always even. The same is true of the Killing equation
(\ref{KEqGR}) since it has the same structure. Hence with
vanishing one- and three-form fluxes one can only have an even
number of preserved supersymmetries. These might include
two-preonic solutions (preserving $30$ supersymmetries) besides
those preserving all $32$ supersymmetries, but not a preonic
solution. The authors of \cite{GGPR06} then concluded that preonic
solutions do not exist for type IIB supergravity.
We now apply our $G$-frame approach, used above to rederive the
IIB result of \cite{GGPR06}, to show that preonic solutions are
also absent in type IIA supergravity.
\section{Absence of preons in type IIA supergravity}
The crucial point is that in the IIA case the matrix $M$, Eq.
(\ref{M=IIA}), receives contributions from all IIA fluxes, Eq.
(\ref{fluxesIIA}). Hence if $M$ is zero, all IIA fluxes are zero,
the generalized covariant derivative ${\cal D}$ becomes the
Lorentz covariant derivative $D$ and the generalized holonomy
group reduces to $SO(1,9)$, for which the number of possible
preserved supersymmetries is known (see \cite{Duff03, Bry00}).
As we shall see presently, $M$ is indeed zero if we assume the
existence of $31$ Killing spinors. In type IIA supergravity the
preonic $\check{\lambda}_{\check{\alpha}}$ and auxiliary
$\check{s}^{\check{\alpha}}$ spinors are $32$-component $D=10$
Majorana spinors,
\begin{eqnarray}\label{IIAprl}
\hbox{IIA} \; : \qquad \check{\lambda}_{\check{\alpha}}:=
({\lambda}^1_{{\alpha}}\; , \; {\lambda}^{{\alpha}2})\; , \qquad
\check{s}^{\check{\alpha}}:= ({s}^{{\alpha}1}\; , \;
{s}^2_{{\alpha}})\; , \qquad \alpha=1,\ldots, 16\; . \qquad
\end{eqnarray}
Eq.(\ref{KEqDIL31}) can be split into four equations for the
($16\times16$)-component blocks
\begin{eqnarray} \label{M=lsIIA}
\begin{matrix} { & {3\over
8} e^{\!^{\Phi}} R\!\!\!\!/^{(2)} + {1\over 8} R\!\!\!\! / ^{(4)}=
{\lambda}^1_{{\alpha}} {s}^{{\beta}1}\; , & \quad
\hbox{(\ref{M=lsIIA}a)} \quad &
{1\over 2} \partial\!\!\!\! / \Phi - {1\over 4} H\!\!\!\!/^{(3)} =
{\lambda}^1_{{\alpha}} {s}^2_{\beta}\; , & \quad
\hbox{(\ref{M=lsIIA}b)} & \cr & {1\over 2} \tilde{
\partial\!\!\!\! / } \Phi + {1\over 4}\tilde{ H\!\!\!\!/^{(3)} }=
{\lambda}^{{\alpha}2} {s}^{{\beta}1}\; , & \quad
\hbox{(\ref{M=lsIIA}c)} \quad & -{3\over 8} e^{\!^{\Phi}}
\tilde{ R\!\!\!\!/ }^{(2)} + {1\over 8}
\tilde{R\!\!\!\! / }^{(4)}=
{\lambda}^{{\alpha}2} {s}^2_{{\beta}}\; . & \quad
\hbox{(\ref{M=lsIIA}d)} & }
\end{matrix} \;
\end{eqnarray}
We now notice that $\tilde{ R\!\!\!\!/ }^{(2)}= - ({R\!\!\!\!/
}^{(2)})^T$, $\tilde{ R\!\!\!\!/ }^{(4)}= +( {R\!\!\!\!/
}^{(4)})^T$ and that, accordingly, the {\it l.h.s.}'s of Eqs.
(\ref{M=lsIIA}a) and (\ref{M=lsIIA}d) are equal among themselves.
Hence, the {\it r.h.s.}'s of these equations are also equal,
${\lambda}^1_{{\alpha}} {s}^{{\beta}1}= {\lambda}^{{\beta}2}
{s}^2_{{\alpha}}$. This equation identifies the components of
$\check{\lambda} $ and $\check{s}$ up to a factor $a$,
\begin{eqnarray}\label{lbd=as}
{s}^{{\alpha}1} = a {\lambda}^{{\alpha}2}\; , \qquad
{s}^2_{{\alpha}}= a {\lambda}^1_{{\alpha}}\; . \qquad
\end{eqnarray}
Then, decomposing Eq. (\ref{M=lsIIA}a) or (\ref{M=lsIIA}d) into
their irreducible parts ({\it i.e.}, identifying the coefficients
of the matrices $\sigma^{ab}{}_{\alpha}{}^{\beta}$,
$\sigma^{abcd}{}_{\alpha}{}^{\beta}$ {\it and}
$\delta_{\alpha}{}^{\beta}$, one finds the expressions for the RR
fluxes in terms of preonic spinors as well as an orthogonality
condition between $\lambda^1$ and $\lambda^2$,
\begin{eqnarray}\label{R2=...}
e^\Phi R_{ab}= - {a\over 6} \lambda^2 \sigma_{ab}\lambda^1 \; ,
\qquad
R_{abcd}= {a\over 2} \lambda^2 \sigma_{abcd}\lambda^1 \; , \qquad
{\lambda}^{{\alpha}2}
{\lambda}^1_{{\alpha}}\; =0 . \qquad
\end{eqnarray}
Substituting (\ref{lbd=as}) for the $s$ spinors in
(\ref{M=lsIIA}b) and (\ref{M=lsIIA}c), these equations can be
rewritten in the form
\begin{eqnarray} \label{dP=all}
{1\over 2} \partial\!\!\!\! / \Phi - {1\over 4} H\!\!\!\!/^{(3)}
= a{\lambda}^1_{{\alpha}} {\lambda}^1_{\beta}\; ,
\qquad
\hbox{(\ref{dP=all}a)} \qquad {1\over 2} \tilde{
\partial\!\!\!\! / } \Phi + {1\over 4}\tilde{ H\!\!\!\!/^{(3)} }=
a{\lambda}^{{\alpha}2} {\lambda}^{{\beta}2}\; . \qquad
\hbox{(\ref{dP=all}b)} \qquad
\end{eqnarray}
The {\it r.h.s.}'s of Eqs. (\ref{dP=all}) are symmetric, while the
{\it l.h.s.}'s contain the antisymmetric matrices $ H\!\!\!\!/ =-
(H\!\!\!\!/ )^T$ and $\tilde{ H\!\!\!\!/ }=-(\tilde{ H\!\!\!\!/
})^T$ which, hence, should be equal to zero. This implies the
vanishing of the NS-NS flux $H_3$ for a hypothetical preonic
solution of type IIA supergravity, $ H_{abc}=0$. Then one arrives
at
\begin{eqnarray} \label{dP=allV}
{1\over 2} \sigma^a_{{\alpha}\beta}D_a \Phi =
a{\lambda}^1_{{\alpha}} {\lambda}^1_{\beta}\; ,
\qquad
{1\over 2} \tilde{\sigma
}{}^{a{\alpha}\beta} D_a \Phi = a{\lambda}^{{\alpha}2}
{\lambda}^{{\beta}2}\; . \qquad
\end{eqnarray}
Since we are in ten dimensions these equations imply, besides
$D_a\Phi \sim \lambda^1 {\tilde \sigma}_a\lambda^1$,
\begin{eqnarray} \label{ls5l=0}
a {\lambda}^1 \tilde{\sigma}^{a_1\ldots a_5} {\lambda}^1=0 \; ,
\qquad
a{\lambda}^{2}{\sigma}^{a_1\ldots a_5} {\lambda}^{2}=0 \; . \qquad
\end{eqnarray}
Eqs. (\ref{dP=allV}) or (\ref{ls5l=0}) imply the {\it absence of
BPS preons among the bosonic solutions of type IIA supergravity}.
Indeed, for non-vanishing $a$ ($a\not=0$) Eqs.(\ref{ls5l=0}) have
only trivial\footnote{A simple way to prove it from Eq.
(\ref{dP=allV}) is to notice that this equation implies $D_a\Phi
\propto \lambda^1\tilde{\sigma}_a\lambda^1$ and that, hence,
$D_a\Phi$ is a light-like ten-vector, $D_a\Phi D^a\Phi=0$. Then
one may choose the Lorentz frame where $D_a\Phi \propto
(1,0,\ldots , 0, \pm 1)$; in it, $D_a\Phi \sigma^a_{\alpha\beta}
\propto (\sigma^0_{\alpha\beta} \pm \sigma^{9}_{\alpha\beta}) =
2\sum_{p}\delta^p_{\alpha}\delta^p_{\beta}$, where $p=1, \ldots ,
8$. In this frame, the first equation in (\ref{dP=allV}) reads
$D_0 \Phi\sum_{p} \delta^p_{\alpha}\delta^p_{\beta} =
a{\lambda}^1_{{\alpha}} {\lambda}^1_{\beta}\,$, which immediately
implies that $a\not=0$ is only possible if half of the sixteen
components of ${\lambda}^1_{\beta}$ are zero,
${\lambda}^1_{\beta}={\lambda}_q \delta^q_{\beta}$. Taking this in
account, the above equation reduces to $D_0 \Phi \delta_{qp}=
a\lambda_q \, \lambda_p $ with $p,q=1, \ldots ,8 \,$, which for
$a\not=0$ only admits the trivial solution
$\lambda^1=0=\lambda^2$. } solutions, $\lambda^1=0=\lambda^2$.
This may correspond to the case of a fully supersymmetric solution
of supergravity (preserving the $32$ supersymmetries), but not to
a preonic one. The other possibility, $a=0$, also implies the
vanishing of the $M$ matrix (\ref{M=IIA}) and hence of all type
IIA supergravity fluxes, $R_2=0=R_4$, $H_3=0=d\Phi$, and thus the
generalized connection in the Killing equation (\ref{KEqGR})
reduces to the spin-connection, ${\cal D}=D$. In such a case it is
known (see \cite{Duff03, Bry00}) that the Killing spinor equation
$D\check{\epsilon}=0$ may have either $32$ or up to $16$
solutions. Thus a solution preserving $31$ supersymmetries, a BPS
preonic solution, is not allowed.
This completes the proof of the absence of BPS preonic,
$\nu=31/32$ supersymmetric bosonic solutions in type II
supergravities {\it i.e.}, in the {\it classical} approximation to
the type II string theories.
\section{The case of D=11 supergravity}
It is known that the $D=10$ type IIA supergravity can be obtained
by dimensional reduction from $D=11$ supergravity {\it i.e.}, its
solutions can be identified with solutions of $D=11$ supergravity
that are independent of one of the coordinates.
In particular, the type IIA dilatino
$\check{\chi}^{\check{\alpha}}$, Eq. (\ref{IIADIL}), originates
from the 11-th component $\check{\psi}_{\#}^{\check{\alpha}}$ of
the $D=11$ gravitino
$\check{\psi}_{\check{\mu}}^{\check{\alpha}}=(\check{\psi}_{{\mu}}^{\check{\alpha}},
\check{\psi}_{\#}^{\check{\alpha}})$; schematically,
\begin{eqnarray}\label{chi=psi11}
\check{\chi}^{\check{\alpha}}=\check{\psi}_{\#}^{\check{\alpha}}\; . \qquad
\end{eqnarray}
The type IIA supersymmetry transformations can also be obtained
from those of $D=11$ by dimensional reduction . This implies, in
particular, that the IIA $M$-matrix (\ref{M=IIA}) comes from the
eleventh component of the $D=11$ generalized connection;
schematically,
\begin{eqnarray}
\label{M=t11}
M_{\check{\beta}}{}^{\check{\alpha}} = {(\omega+\check{t})}_{\#\;
\check{\beta}}{}^{\check{\alpha}} \; . \qquad
\end{eqnarray}
This observation provides a starting point to probe the existence
of BPS preonic solutions in $D=11$ supergravity or, more
precisely, among the purely bosonic solutions of the classical
$D=11$ supergravity \cite{CJS78}. It was shown in \cite{JG+SP02}
that the existence of $k$ Killing {\it spinors} ($k=31$ for
preonic solutions) implies the existence of $k(k+1)/2$ Killing
{\it vectors},
\begin{eqnarray}
\label{KillIJ}
K^{\check{a}}_{IJ}:= \check{\epsilon}^{\;\check{\alpha}}_I
\Gamma^{\check{a}}_ {\check{\alpha}\check{\beta}}
\check{\epsilon}^{\;\check{\beta}}_J \; , \qquad
\end{eqnarray}
such that both the metric and the field strength $F_4=dA_3$ of the
three-form gauge field $A_3$ are invariant under `translations'
along the directions of $K^{\check a}_{IJ}$,
\begin{eqnarray}
\label{Kill2IJ} \delta_{K_{IJ}}g_{{\check\mu}{\check\nu}}=
2D_{(\check{\mu}} K_{\check{\nu})IJ} =0 \; , \qquad
\delta_{K_{IJ}} F_4:= {\cal L}_{K_{IJ}} F_4=0 \; .
\end{eqnarray}
This actually implies that any supersymmetric solution of $D=11$
can be considered (at least locally) as a solution of $D=10$ type
IIA supergravity lifted (`oxidized') to $D=11$. Thus, because of
the above negative result for the existence of preonic solutions
in type IIA supergravity, the only remaining possibility to have
BPS preonic solutions in the D=11 case requires that they result
from the `oxidization' of a less supersymmetric solution of the
$D=10$ type IIA supergravity.
If the lifting to $D=11$ has to produce more supersymmetries, we
need that one or more Killing spinors
$\epsilon^{\check{\alpha}}_{\tilde{I}}$ have non-vanishing
derivative in the direction of a Killing vector, schematically,
$\partial_{\#}\epsilon^{\check{\alpha}}_I \not=0$. In this way,
the set of $D=11$ Killing equations
${\cal D}_{\#}\epsilon^{\check{\alpha}}_I:=
D_{\#}\epsilon^{\check{\alpha}}_I -
\epsilon^{\check{\beta}}_I\check{t}_{\#\check{\beta}}{}^{\check{\alpha}}=0$
will no longer reduce (see Eq. (\ref{M=t11})) to the algebraic
equation (\ref{KEqDIL}). As a result, the arguments from the
discussion of the type IIA case would not apply in $D=11$ to
exclude the existence of a preonic solution.
A Killing spinor $\epsilon^{\check{\alpha}}_J$ can be
characterized \cite{JG+SP02} by means of three differential forms:
a Killing vector one-form $K_{1\; JJ}:=
e_{\check{a}}K^{\check{a}}_{JJ}$, a two-form $\Omega_{2\; JJ}$ and
a five-form $\Sigma_{5\; JJ}$. These forms are the diagonal
elements of the symmetric bilinear matrix forms with tensorial
components defined in Eq. (\ref{KillIJ}) and by
\begin{eqnarray}\label{2,5-IJ}
\Omega^{\check{a}_1\check{a}_2}_{IJ}:=
\check{\epsilon}^{\;\check{\alpha}}_I
\Gamma^{\check{a}_1\check{a}_2}_ {\check{\alpha}\check{\beta}}
\check{\epsilon}^{\;\check{\beta}}_J \; , \qquad
\Sigma^{\check{a}_1\ldots \check{a}_5}_{IJ}:=
\check{\epsilon}^{\;\check{\alpha}}_I \Gamma^{\check{a}_1\ldots
\check{a}_5}_ {\check{\alpha}\check{\beta}}
\check{\epsilon}^{\;\check{\beta}}_J \; . \qquad
\end{eqnarray}
The independence of a Killing spinor on a coordinate $x^{\#}$ in
some direction would also imply the independence of its associated
Killing vector $K_{JJ}$ (Eq. (\ref{KillIJ})), of the two-form
$\Omega_{JJ}$ and of the five-form $\Sigma_{JJ}$ (Eq.
(\ref{Kill2IJ})) on that direction. As the direction $x^{\#}$
should be characterized by one of the Killing vectors, the result
of \cite{JG+SP02}, stating that ${\cal L}_K\Omega_2=0$ and ${\cal
L}_K\Sigma_5=0$, implies the independence of the two- and the
five-form on $x^{\#}$. However, the Lie derivative of a Killing
vector with respect to another Killing vector, ${\cal L}_K
K^\prime_1$, may still be nonzero when there are two or more
Killing vectors. Thus, at present we cannot conclude that all
Killing spinors $\epsilon^{\check \alpha}_I$ are independent of
$x^\#$ so that, albeit rather exotic, the possibility of a $\nu=
31/32$ supersymmetric solution in $D=11$ supergravity remains
open.
\section{Could preonic BPS solutions still exist?}
The established absence of preonic solutions in type II
supergravities, {\it i.e.} for the {\it classical} approximations
to type II string theories, does not preclude the preonic
conjecture of \cite{BPS01}. At the time it was made, solutions
preserving more than $16$ out of the $32$ supersymmetries were not
known except for the fully supersymmetric ones (see
\cite{FiPa03}). It was already mentioned in \cite{BPS01} that a
kind of `BPS preon conspiracy' could produce that only composites
of some number of preons (but not the preons themselves) could be
found (`observed') as supergravity solutions.
On account of the fundamental role played by preons in the
classification of BPS states \cite{BPS01}, it is tempting to
speculate that the fact that type II supergravities do not have
preonic solutions rather points out at a need for their
modification. The most natural refinement to try is to take into
account stringy, $(\alpha^\prime)^3$-corrections to the
supergravity equations and to the supersymmetry transformation
rules of the supergravity fields. Preonic solutions in a `stringy
corrected' type IIA supergravity would be allowed if the
corrections modified Eqs.(\ref{dP=allV}) by adding some terms
$\propto \sigma_{abcdf}$ and $\propto \tilde{\sigma}_{abcdf}$.
Schematically, the `required' modification would have to be of the
form
\begin{eqnarray}
\label{dP=allVq?} a{\lambda}^1_{{\alpha}} {\lambda}^1_{\beta} -
{1\over 2} \sigma^a_{{\alpha}\beta}D_a \Phi =0 \quad &\mapsto&
\qquad a{\lambda}^1_{{\alpha}} {\lambda}^1_{\beta} - {1\over 2}
\sigma^a_{{\alpha}\beta}D_a \Phi = Q^-_{abcde}
\sigma^{abcde}_{{\alpha}\beta} \; , \qquad \nonumber
\\ a{\lambda}^{2{\alpha}} {\lambda}^{2\beta} - {1\over 2}
\tilde{\sigma}^{a\, {\alpha}\beta}D_a \Phi = 0 \quad &\mapsto&
\qquad a{\lambda}^{2{\alpha}} {\lambda}^{2\beta} - {1\over 2}
\tilde{\sigma}^{a\, {\alpha}\beta}D_a \Phi = Q^+_{abcde}
\tilde{\sigma}^{abcde\;{\alpha}\beta} \; \qquad
\end{eqnarray}
for some $Q_{abcde}\propto (\alpha^\prime)^3$ (clearly, the
$\propto \sigma_a$ contribution could also be changed, but this is
not essential for the present schematic discussion). Such a
modification (\ref{dP=allVq?}) of Eq. (\ref{dP=allV}) might result
from the associated additions to the dilatino transformation rules
(\ref{susyDIL}) (of the type $\propto \tilde{\sigma}_{abcdf}$ plus
other terms not essential for our discussion). In terms of the $M$
matrix, this modification would imply
\begin{eqnarray} \label{M=IIA+Q} M\; \mapsto \; M + \left(
\matrix{ 0 & Q^-_{abcde} \sigma^{abcde}_{{\alpha}\beta} \cr
{Q}^+_{abcde} \tilde{\sigma}^{abcde\; {\alpha}\beta} & 0}\right)=
M + Q^{\pm}_{abcde} \Gamma^{abcde} {1\over 2}(1\pm\Gamma^{11}) \;
.
\end{eqnarray}
Direct calculations of the stringy corrections
\cite{alphaprime,PVW2000,GrossWitten86} to the supersymmetry
transformation rules have been hampered by the lack of a covariant
technique to calculate higher order loop amplitudes in superstring
theory\footnote{Such a technique has been recently proposed in the
framework of Berkovits's pure spinor approach \cite{NBloops} to
the covariant description of the quantum superstring.}.
Nevertheless, bosonic string calculations allowed to find stringy
corrections to the Einstein equation \cite{GrossWitten86}. The
influence of these corrections on the supersymmetric vacua and
their relevance for their supersymmetric properties \cite{CFPSS86}
was used to find corrections to the gravitino supersymmetry
transformation properties. As the discussions of the
$\alpha^\prime$ modifications have also been extended to the
eleven-dimensional theory \cite{PVW2000,M-thQC}\footnote{The
contributions to the generalized connection ({\it i.e.} to the
supersymmetry transformation rules for the gravitino) were
calculated for a particular background and, then, conjectured to
hold in general \cite{M-thQC} on grounds of their universal
form.}, one can obtain the `corrected' transformation rules for
the type IIA dilatino\footnote{For the heterotic string case, the
simplest possible corrections to the ($N=1$) dilatino $\chi$
transformation rules (see Eq. (23) in \cite{CFPSS86}) consist in
modifying (`renormalizing') the dilaton $\Phi$ appearing in the
standard supersymmetry transformations.} by dimensional reduction
from those of the $D=11$ gravitino and thus derive the expression
of the matrix $M$ in Eq.(\ref{M=IIA}) that incorporates the
`stringy corrected' counterpart of Eq. (\ref{M=t11}).
In this perspective it looks promising that the $D=11$ generalized
connection $\check{t}_{\tilde{\mu}} = (\check{t}_{{\mu}},
\check{t}_{10})$ ({\it cf.} (\ref{susyDIL}); here
$\tilde{\mu}=(\mu;\#)$ = $0,\ldots 9; 10$) considered in
\cite{M-thQC}, contains the terms
$\check{Q}^{{\tilde{\mu}}_1\ldots
\tilde{\mu}_6}\Gamma_{\tilde{\mu}_1\ldots \tilde{\mu}_6}$; their
dimensional reduction would produce, among others, the
contribution $\check{Q}^{\mu_1\ldots \mu_5\,
10}\Gamma_{{\mu}_1\ldots {\mu}_5} \Gamma^{11}$ which is of the
needed type, see Eq. (\ref{M=IIA+Q}) (the
$\Gamma^{10}\equiv\Gamma^{\#}$ in $D=11$ is the $\Gamma^{11}$ in
$D=10$ ).
To summarize, although it has been shown that a $\nu=31/32$
preonic solution is not allowed in the {\it classical} type II
supergravities (in \cite{GGPR06} for type IIB and here for type
IIA), a conclusive analysis with quantum stringy corrections,
providing a more precise description of string/M-theory, remains
to be done. If preons were found to exist when quantum corrections
are taken into account, it would be only natural on account of
their special role as the `quarks of M-theory' \cite{BPS01}
\footnote{Let us recall \cite{BPS01} that the potential relevance
of BPS preons derives from their quark-like role in the
classification of BPS states according to the number of preserved
supersymmetries: all BPS states of M-theory preserving $k$
supersymmetries can be considered as composites of $32-k$ BPS
preons (the statement is true for any $D$ with 32 replaced by the
corresponding spinor dimension).}. Preons would only be `seen' by
looking at the `quantum solutions' of string theory, an
approximation of which is provided by supergravity with stringy
corrections.
As far as the study of `classical' supergravity is concerned, the
natural next step is to clarify the level of the mentioned `preon
conspiracy' \cite{BPS01} in the classical $D=10$ supergravity {\it
i.e.}, whether it is possible to find two-preonic $\nu=30/32$
supersymmetric solutions, preserving all but two supersymmetries,
or whether the `counterpart' of the colourless quark states in the
case of preons should include no less than four preonic
constituents corresponding to the highest non-fully
supersymmetric states up to now found, the $\nu=28/32$ states of
the IIB case \cite{BeRo03}.
As for $D=11$ supergravity, although we have not been able to
reach a definite conclusion on the existence of $\nu=31/32\,$
supersymmetric solutions, we have presented here their
characteristic properties: such a $D=11$ BPS preonic solution
should have Killing directions, both for the metric $g$ and the
gauge field strength $F_4$, such that at least one of its $31$
Killing spinors depends on the coordinates corresponding to these
directions.
Finally, we conclude by mentioning that all searches for preonic
solutions, including this one, have been concerned with purely
bosonic solutions, a restriction that does not follow from
\cite{BPS01}.
\bigskip
\bigskip
{\bf Acknowledgments}. The authors wish to thank J. Gauntlett and
D. Sorokin for useful discussions. This work has been partially
supported by research grants from the Ministerio de Educaci\'on y
Ciencia (FIS2005-02761) and EU FEDER funds, the Generalitat
Valenciana (ACOMP06/187, GV05/102), the Ukrainian State Fund for
Fundamental Research (N383), the INTAS (2005/2006-7928) and by the
EU `Forces Universe' network (MRTN-CT-2004-005104).
{\small
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train/arxiv
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BkiUdCc5qYVBSIiKs2TE
| 5 | 1 |
\section{Introduction and background}\label{intro}
\noindent Let $(\ensuremath{\mathbb{X}}, \Vert\cdot\Vert)$ be an infinite-dimensional Banach space, and let $\mathcal{B}=(\ensuremath{\mathbf{e}}_n)_{n=1}^\infty$ be a semi-normalized basis for $\ensuremath{\mathbb{X}}$ with biorthogonal functionals
$(\ensuremath{\mathbf{e}}_n^*)_{n=1}^\infty$.
The basis $\mathcal{B}$ is \textit {quasi-greedy} if for any $x\in \ensuremath{\mathbb{X}}$ the corresponding series expansion,
\[
x=\sum_{n=1}^\infty \ensuremath{\mathbf{e}}_n^*(x)\ensuremath{\mathbf{e}}_n
\]
converges in norm after reordering it so that the sequence $(|\ensuremath{\mathbf{e}}_n^*(x)|)_{n=1}^\infty$ is decreasing. Wojtaszczyk showed \cite{Wo2000} that a basis $(\ensuremath{\mathbf{e}}_n)_{n=1}^\infty$ of $\ensuremath{\mathbb{X}}$ is quasi-greedy if and only if the greedy operators $\mathcal{G}_N\colon \ensuremath{\mathbb{X}}\to \ensuremath{\mathbb{X}}$ defined by
\[
x=\sum_{j=1}^{\infty}\ensuremath{\mathbf{e}}_j^{\ast}(x)\ensuremath{\mathbf{e}}_{j} \mapsto \mathcal{G}_{N}(x)=\sum_{j\in \Lambda_{N}(x)}\ensuremath{\mathbf{e}}_j^{\ast}(x)\ensuremath{\mathbf{e}}_{j},
\]
where $\Lambda_{N}(x)$ is any $N$-element set of indices such that
\[
\min\{|\ensuremath{\mathbf{e}}_{j}^{\ast}(x)| \colon j\in \Lambda_{N}(x)\}\ge \max\{|\ensuremath{\mathbf{e}}_{j}^{\ast}(x)| \colon j\not\in \Lambda_{N}(x)\},
\]
are uniformly bounded, i.e.,
\begin{equation}\label{WoCondition}
\Vert \mathcal{G}_N(x)\Vert\le C \Vert x\Vert, \quad x \in \ensuremath{\mathbb{X}}, \, N\in\ensuremath{\mathbb{N}}.
\end{equation}
for some constant $C$ independent of $x$ and $N$. Note that the operators $(\mathcal{G}_{N})_{N=1}^{\infty}$ are neither linear nor continuous, so this is not just the Uniform Boudedness Principle!
Obviously, If \eqref{WoCondition} holds then there is a (possibly different) constant $\tilde{C}$ such that
\begin{equation}\label{SupWoCondition}
\Vert x-\mathcal{G}_N(x)\Vert\le \tilde{C} \Vert x\Vert, \quad x \in \ensuremath{\mathbb{X}}, \, N\in\ensuremath{\mathbb{N}}.
\end{equation}
We will denote by $C_w$ the smallest constant such that \eqref{WoCondition} holds, and by $C_t$ the least constant in \eqref{SupWoCondition}. It is rather common (cf.\ \cites{GHO2013,DKKT2003}) and convenient to define the {\it quasi-greedy constant} $C_{qg}$ of the basis as
\begin{equation*}\label{WojtCharacterization}
C_{qg}=\max\{ C_w, C_t \}.
\end{equation*}
If $\mathcal{B}$ is a quasi-greedy basis and $C$ is a constant such that $C_{qg}\le C$ we will say that $\mathcal{B}$ is {\it $C$-quasi-greedy}.
Recall also that a basis $(\ensuremath{\mathbf{e}}_n)_{n=1}^\infty$ in a Banach space $\ensuremath{\mathbb{X}}$ is {\it unconditional} if for any $x\in \ensuremath{\mathbb{X}}$ the series $\sum_{n=1}^\infty \ensuremath{\mathbf{e}}_n^*(x)\ensuremath{\mathbf{e}}_n$ converges in norm to $x$ regardless of the order in which we arrange
the terms. The property of being unconditional is easily seen to be equivalent to that of being {\it suppression unconditional}, which means that the natural projections onto any subsequence of the basis
\[
P_A(x)=\sum_{n\in A} \ensuremath{\mathbf{e}}_n^*(x) \ensuremath{\mathbf{e}}_n,\quad A\subset \mathbb N,
\]
are uniformly bounded, i.e., there is a constant $K$ such that for all $x=\sum_{n=1}^{\infty} \ensuremath{\mathbf{e}}_n^*(x) \ensuremath{\mathbf{e}}_n$ and all $A\subset \ensuremath{\mathbb{N}}$,
\begin{equation}\label{unifbound}
\left\Vert\sum_{n\in A} \ensuremath{\mathbf{e}}_n^*(x) \ensuremath{\mathbf{e}}_n \right\Vert \le K \left\Vert\sum_{n=1}^{\infty} \ensuremath{\mathbf{e}}_n^*(x) \ensuremath{\mathbf{e}}_n \right\Vert.
\end{equation}
The smallest $K$ in \eqref{unifbound} is the {\it suppression unconditional constant} of the basis, and will be denoted by $K_{su}$. Notice that
\[
K_{su}=\sup \{ \Vert P_A \Vert \colon A\subseteq \ensuremath{\mathbb{N}} \text{ is finite} \}=\sup \{ \Vert P_A \Vert \colon A\subseteq \ensuremath{\mathbb{N}} \text{ is cofinite}\}.
\]
If a basis $\mathcal{B}$ is unconditional and $K$ is a constant such that $ K_{su}\le K$ we will say that $\mathcal{B}$ is {\it $K$-suppression unconditional}.
Quasi-greedy bases are not in general unconditional; in fact, most classical spaces contain conditional quasi-greedy bases. Wojtaszczyk gave in \cite{Wo2000} a general construction (improved in \cite{GW2014}) to produce quasi-greedy bases in some Banach spaces. His method yields the existence of conditional quasi-greedy bases in separable Hilbert spaces, in the spaces $\ell_{p}$ and $L_{p}[0,1]$ for $1<p<\infty$, and in the Hardy space $H_1$.
Dilworth and Mitra showed in \cite{DilworthMitra2001} that $\ell_1$ also has a conditional quasi-greedy basis. In spite of that, quasi-greedy bases preserve some vestiges of unconditionality and, for instance, they are {\it unconditional for constant coefficients} (see \cite{Wo2000}).
Conversely, unconditional bases are always quasi-greedy. To be precise, if $\mathcal{B}$ is $K$-suppression unconditional then $\mathcal{B}$ is
$K$-quasi-greedy. In particular, unconditional bases with $K_{su}=1$ are quasi-greedy with $C_{w}=1$. Our aim is to show the converse of this statement, thus characterizing $1$-quasi-greedy bases. The related problem of characterizing bases that are $1$-greedy was solved in \cite{AlbiacWojt2006}. This question is relevant since the optimality in the constants of greedy-like bases seems to improve the properties of the corresponding basis. Indeed, in the
''isometric case" greedy bases gain in symmetry (they are invariant under greedy permutations instead of merely democratic). Our result reinforces this pattern by showing that ''isometric'' quasi-greedy basis are not merely unconditional for constant coefficients but unconditional.
\section{The Main Theorem and its Proof}
\noindent As a by-product of their research on unconditionality-type properties of quasi-greedy bases, Garrig\'os and Wojtaszczyk \cite{GW2014} have shown that
bases in Hilbert spaces with $C_w=1$ are orthogonal. A direct proof of their result can be obtained as follows.
Let
$\mathcal{B}=(\ensuremath{\mathbf{e}}_n)_{n=1}^\infty$ be a basis in a (real or complex) Hilbert space with $C_w=1$. Then, if $|\varepsilon|=1$, $0< t < 1$, and $i\not=j$,
\[
\Vert \ensuremath{\mathbf{e}}_i \Vert^2 \le \Vert \ensuremath{\mathbf{e}}_i +\varepsilon t \ensuremath{\mathbf{e}}_j\Vert^2=\Vert \ensuremath{\mathbf{e}}_i\Vert^2+2 t \Re(\varepsilon \langle \ensuremath{\mathbf{e}}_i,\ensuremath{\mathbf{e}}_j \rangle) + t^2 \Vert \ensuremath{\mathbf{e}}_j\Vert^2.
\]
Simplifying,
\[
-2 \Re(\varepsilon \langle \ensuremath{\mathbf{e}}_i,\ensuremath{\mathbf{e}}_j \rangle) \le t \Vert \ensuremath{\mathbf{e}}_j\Vert^2.
\]
Choosing $\varepsilon$ such that $ \varepsilon\langle \ensuremath{\mathbf{e}}_i,\ensuremath{\mathbf{e}}_j \rangle =-| \langle \ensuremath{\mathbf{e}}_i,\ensuremath{\mathbf{e}}_j \rangle|$ and letting $t$ tend to zero we obtain $ | \langle \ensuremath{\mathbf{e}}_i,\ensuremath{\mathbf{e}}_j \rangle|=0$.
A strengthening of this argument leads to the following generalization of Garrig\'os-Wojtaszczyk's result.
\begin{theorem}\label{1QGTheorem} A quasi-greedy basis $(\ensuremath{\mathbf{e}}_{n})_{n=1}^{\infty}$ in a Banach space $\ensuremath{\mathbb{X}}$ is quasi-greedy with $C_w=1$ if and only if it is unconditional with suppression unconditional constant $K_{su}=1$.
\end{theorem}
\begin{proof}
We need only show that if $x$ and $y$ are vectors finitely supported in $(\ensuremath{\mathbf{e}}_{n})_{n=1}^{\infty}$ with disjoint supports then $\Vert x \Vert \le \Vert x+y\Vert$. This readily implies that $(\ensuremath{\mathbf{e}}_{n})_{n=1}^{\infty}$ is unconditional with suppression unconditional constant $K_{su}=1$.
Suppose that this in not the case and that we can pick $x$, $y\in \ensuremath{\mathbb{X}}$ finitely and disjointly supported in $(\ensuremath{\mathbf{e}}_{n})_{n=1}^{\infty}$ with $\Vert x+y\Vert <\Vert x \Vert$.
Consider the function $\varphi\colon\ensuremath{\mathbb{R}}\to[0,\infty)$ defined by
\[
\varphi(t)=\Vert x + t y\Vert.
\]
Using the definition, it is straightforward to check that $\varphi$ is a convex function on the entire real line. Moreover, $\varphi(0)=\Vert x\Vert$ and, by assumption, $\varphi(1)<\Vert x\Vert$. Therefore, $\varphi(t)<\Vert x \Vert $
for all $0<t<1$. Choosing $t\in (0,1)$ small enough we have $x=\mathcal{G}_N(x+ty)$, where $N$ is the cardinal of the support of $x$. Consequently, for such a $t$,
\[
\Vert x+t y \Vert =\varphi(t)
<\Vert x \Vert =\Vert \mathcal{G}_N(x+ty) \Vert \le \Vert x+t y \Vert,
\]
where we used the hypothesis on the quasi-greedy constant of the basis to obtain the last inequality. This absurdity proves the result.
\end{proof}
We close with some consequences of Theorem~\ref{1QGTheorem}, which need no further explanation.
\begin{corollary} Suppose $\mathcal{B}=(\ensuremath{\mathbf{e}}_{n})_{n=1}^{\infty}$ is a basis in a Banach space $\ensuremath{\mathbb{X}}$ with
$C_w=1$. Then $C_t=1$; in particular $\mathcal{B}$ is $1$-quasi-greedy.
\end{corollary}
\begin{corollary} If a basis $(\ensuremath{\mathbf{e}}_{n})_{n=1}^{\infty}$
in a Banach space $\ensuremath{\mathbb{X}}$ is $1$-quasi-greedy then it is $1$-suppression unconditional.\end{corollary}
\begin{corollary} Suppose $\mathcal{B}=(\ensuremath{\mathbf{e}}_{n})_{n=1}^{\infty}$ is a basis in a Banach space $(\ensuremath{\mathbb{X}}, \Vert\cdot\Vert)$. Then $\ensuremath{\mathbb{X}}$ admits an equivalent norm $\VERT\cdot\VERT$ so that $\mathcal{B}$ is $1$-quasi-greedy in the space $(\ensuremath{\mathbb{X}}, \VERT\cdot\VERT)$ if and only if $\mathcal{B}$ is unconditional.
\end{corollary}
\subsection*{Acknowledgement} F. Albiac acknowledges the support of the Spanish Ministry for Economy and Competitivity Grant {\it Operators, lattices, and structure of Banach spaces},
reference number MTM2012-31286.
\begin{bibsection}
\begin{biblist}
\bib{AlbiacWojt2006}{article}{
author={Albiac, F.},
author={Wojtaszczyk, P.},
title={Characterization of 1-greedy bases},
journal={J. Approx. Theory},
volume={138},
date={2006},
number={1},
pages={65--86},
issn={0021-9045},
}
\bib{DKKT2003}{article}{
author={Dilworth, S. J.},
author={Kalton, N. J.},
author={Kutzarova, D.},
author={Temlyakov, V. N.},
title={The thresholding greedy algorithm, greedy bases, and duality},
journal={Constr. Approx.},
volume={19},
date={2003},
number={4},
pages={575--597},
}
\bib{DilworthMitra2001}{article}{
author={Dilworth, S. J.},
author={Mitra, D.},
title={A conditional quasi-greedy basis of $l_1$},
journal={Studia Math.},
volume={144},
date={2001},
number={1},
pages={95--100},
}
\bib{GHO2013}{article}{
author={Garrig{\'o}s, G.},
author={Hern{\'a}ndez, E.},
author={Oikhberg, T.},
title={Lebesgue-type inequalities for quasi-greedy bases},
journal={Constr. Approx.},
volume={38},
date={2013},
number={3},
pages={447--470},
}
\bib{GW2014}{article}{
author={Garrig{\'o}s, G.},
author={Wojtaszczyk, P.},
title={Conditional quasi-greedy bases in Hilbert and Banach spaces},
journal={Indiana Univ. Math. J.},
volume={63},
date={2014},
number={4},
pages={1017--1036},
}
\bib{Wo2000}{article}{
author={Wojtaszczyk, P.},
title={Greedy algorithm for general biorthogonal systems},
journal={J. Approx. Theory},
volume={107},
date={2000},
number={2},
pages={293--314},
}
\end{biblist}
\end{bibsection}
\end{document}
|
train/arxiv
|
BkiUbQ825V5ha7jY7vFR
| 5 | 1 |
\section{Introduction}
\label{sec:intro}
Generalized linear multilevel models (GLMMs), also known as hierarchical or mixed models, are a powerful and more flexible extension to the widely popular class of Generalized linear models (GLMs)~\cite{mcculloch2005generalized,gelman2006data}.
The key distinction between the two is that the former models, from a Bayesian perspective, are characterized by a hierarchy of priors and hyperpriors imposed over its parameters that are effectively learned from the data, making them \emph{adaptively} regularised.
This enables modelling of group-structured data by allowing the model parameters to vary across levels of one or more grouping variables.
To be clear on the terminology used, we define the terms we use for referring to group--structured data.
With \emph{grouping} we refer to one way of categorizing the data into different \emph{levels}, and with
level we mean the different instances present in a grouping.
By learning level-specific parameters that deviate from the global population parameters, a GLMM
can learn the differences between levels as well as partially pool their individual parameters at the same time. In particular, partial pooling helps estimating
the parameters of levels with little data.
However, despite their important benefits for many scientific fields and the commonality of group--structured data, GLMMs are still underused in practice where GLMs still dominate.
This can be possibly explained by the GLMMs' higher modelling complexity and required modelling choices: in the absence of domain knowledge
it is difficult to know which predictors have heterogeneous coefficients and in
which groupings.
Ignoring multilevel structure typically results in biased and badly calibrated estimates that fail to fully capture the underlying relationship of the predictors on the outcome.
The three main approaches of naively dealing with group--structured data are:
\begin{itemize}[topsep=0pt,itemsep=2pt,partopsep=4pt, parsep=2pt]
%
%
%
\item pooling the data of each level to a single point, or
\item modelling all observations as a single level, or
\item building separate models for each of the levels.
\end{itemize}
The first approach ignores within--group correlations and dependencies that in most cases are crucial, and often confound the underlying relationships.
The second approach ignores the grouping altogether and fails to learn differences between the levels.
The third approach ignores the similarity of the levels, often ending up overfitting the models in each level, particularly when a level contains few observations.
It is also possible to represent the grouping as one--hot encoded variables, which is closely related to building separate models per group but often computationally inefficient.
In the commonly used \texttt{R}'s formula syntax, an example of a model without group heterogeneity is expressed as \texttt{y \texttildelow \, x1 + x2}, meaning that a response $y$ is determined by two predictors $x_1$ and $x_2$. If we have two group indicators, \texttt{g1} and \texttt{g2}, a model where the regression coefficients are assumed to vary over both group structures is \texttt{y \texttildelow \, x1 + x2 + (x1 + x2 | g1) + (x1 + x2 | g2)}. As the number of predictors and grouping variables increase, there is a combinatorial explosion in the number of possible models.
We propose an approach for quickly assessing which group heterogeneities are big enough so that they should be included in the model. For example, the approach could recommend to construct a GLMM with the syntax
\texttt{y \texttildelow \, x1 + x2 + (x1 | g2)} which is expected to have similar predictive performance as the most complex model.
In this paper, we aim to answer one of the most relevant questions for building a GLMM: which coefficients vary within groups, i.e. across different levels?
To the best of our knowledge, there are no existing methods that help in answering this question in a robust and automatic manner. %
In this work, we address this issue by proposing a method to perform assessment of group heterogeneity
directly based on the data.
This approach avoids explicitly comparing different models, which
can be time--consuming and prone to selection-induced bias~\cite{cawley2010over,wang2015difficulty}.
By looking specifically at group heterogeneity we aim at telling which coefficients are \emph{different enough} across the levels of a grouping so that they should be taken into account as group coefficients on the target variable.
As the main contribution of this work, we propose a method to assess group heterogeneity
by interpreting the strength of the interaction between numerical predictors and categorical dummy variables as an indicator of group heterogeneity.
We measure this interaction similarly as the interaction of two numerical predictors,
meaning that the joint relationship of two variables with the outcome is different than the
sum of their separate relationships. To perform this with dummy variables, we utilize
the smooth predictions given by
a Gaussian process model, and the KL-diff$^2$ interaction ranking method~\cite{paananen2019ranking}.
We perform extensive experiments showing how the method works in both simulations and real--world data.
The remainder of this article is structured as follows.
In Section~\ref{sec:previouswork} we discuss previous work addressing similar questions.
In Section~\ref{sec:kldiff} we describe
our proposed approach for assessing group heterogeneity.
In Section~\ref{sec:experiments} we present both simulations and real world data experiments showing how the method performs in different cases.
Finally, in Section~\ref{sec:conclusion} we summarize the contributions and impact of our work.
\section{Previous Work}
\label{sec:previouswork}
We are not aware of any literature references discussing group heterogeneity assessment explicitly.
Therefore, in this section we discuss methods that are related and, even though they were not designed for this, can be used for this purpose.
As discussed earlier, the interaction of a numerical predictor and a dummy variable can be
interpreted as group heterogeneity if the used model can express these interactions.
In this work, we build on top of the interaction assessment method KL-diff$^2$~\cite{paananen2019ranking}, which we discuss in Section~\ref{sec:kldiff}.
There are similar methods that find pairwise interactions based on the predictions of a model, like H-statistic, and partial dependence values~\cite{friedman2008predictive,greenwell2018simple}.
These methods have been compared to KL-diff$^2$ and shown to perform worse~\cite{paananen2019ranking}.
Even though this is not a method to evaluate interactions per se, analysis of variance (ANOVA) tables can be used for the purpose of interaction assessment.
However, the modeller first needs to build a full linear model considering already all pairwise interactions, which has a quadratic cost $O(p^2)$.
Furthermore, in the case of evaluating group heterogeneity, the user needs to know beforehand which grouping factors and predictors to study, which may not be obvious in all cases.
Because of this, the computational cost is often prohibitive and therefore cannot be regarded as a general method for evaluating interactions.
A recently proposed method finds an equivalent Gaussian process (GP) representation for a given (generalized) linear model to take advantage of the nice properties of the GP's kernel regarding scalability with respect to the number of predictors~\cite{agrawal2019kernel}.
Then, after evaluating the interactions in GP space, they are brought back into the original linear model's space by undoing the transformation.
However, it does not seem to be a general method for finding the equivalent kernel of any linear model.
Therefore, the usefulness of this method largely depends on the problem and model at hand and thus renders it difficult to use as a general purpose solution.
\section{group heterogeneity assessment}
\label{sec:kldiff}
In this section, we briefly describe the KL-diff method, which uses a Gaussian process
surrogate model to assess the strength of predictor variables and their interactions by differentiating
through the Kullback-Leibler (KL) divergence of the model's
predictive distribution~\cite{paananen2019ranking}.
The first subsection presents the methodology for assessing the relevance of individual predictors and pairwise interactions.
The second subsection presents our proposal for applying these methods to assess group heterogeneity.
\subsection{KL-diff for ranking predictors and interactions}
After observing data $\mathcal{D} = \{ {X}, {y} \}$, where ${X} \in \mathbb{R}^{N\times D}$ represents the matrix of predictors and ${y} \in \mathbb{R}^N$ the vector of responses, let us denote the posterior predictive distribution of a model $\mathcal{M}$ at some predictor value ${x}^*$ as
\begin{equation*}
p (y^{*}) \vcentcolon = p (y^{*} | {\theta}^{*}) \vcentcolon = p (y^{*} | {x}^{*},\mathcal{D}, \mathcal{M}) ,
\end{equation*}
where ${\theta}^* = \{ \theta_1^*, \ldots, \theta_{n_p}^* \}$ is the vector of $n_p$ parameters of the distribution that depends on ${x}^*$, $\mathcal{D}$, and $\mathcal{M}$.
We will refer to the posterior predictive distribution corresponding to ${x}^*$ simply as $p(y^*)$.
KL-diff is a local predictive relevance measure for the predictors that takes into account both the mean prediction and its uncertainty through the posterior predictive distribution.
This concept is formalized as the derivative of a dissimilarity measure, such as Kullback-Leibler
divergence, between two predictive distributions with respect to a predictor.
The method evaluates the derivatives of the KL divergence from the predictive distribution at point ${x}^*$ to the predictive distribution at point ${x}^{**}$ when both points coincide: ${x}^* = {x}^{**}$.
At this point, the first derivative with respect to any predictor $x^*_d$ is zero because the KL divergence has its global minimum of $0$ when both distributions are identical.
Therefore, the authors suggest to use the square root of the second derivative~\cite{paananen2019ranking}.
Based on the predictive distribution that we defined earlier, $p(y^*)$, of a model $\mathcal{M}$, the KL-diff relevance measure with respect to a single predictor variable $x_d$ is defined as
\begin{align*}
& \text{KL-diff}({x}^*, x_d, \mathcal{M}) = \sqrt{\dfrac{\partial^2 \mathcal{D}_{\text{KL}} [p(y^*)\| p(y^{**})]}{(\partial x^{**}_d)^2}\bigg\rvert_{{x}^{**}={x}^*}} \nonumber\\
& = \sqrt{\sum_{k=1}^{n_p}\sum_{l=1}^{n_p}\dfrac{\partial^2 \mathcal{D}_{\text{KL}} [p(y^{*})\| p(y^{*})]}{\partial\theta_k^*\partial\theta_l^*}\dfrac{\partial\theta_k^*}{\partial x_d^*}\dfrac{\partial\theta_l^*}{\partial x_d^*}},
\end{align*}
where $\mathcal{D}_{\text{KL}} [p(y^*) \| p(y^{**})]$ refers to the KL divergence from $p(y^*)$ to $p(y^{**})$ and $\{\theta_1^*, \ldots, \theta_{n_p}^*\}$ are the parameters of $p(y^*)$.
The KL-diff measures with respect to all predictors $x_d$ averaged over the $N$ training
observations can be used as a ranking of their predictive relevance.
If we can use the partial derivatives of the mean prediction with respect to a predictor as a local measure of its relevance, it is natural then that the cross--derivatives of two predictors measures the strength of the joint interaction between both predictors~\cite{paananen2019ranking}.
By computing the cross--derivatives of the Kullback-Leibler divergence between the predictive distributions, an analogous measure of interaction strength is obtained.
Following the same notation as in the previous section, the KL-diff$^2$ with respect to the predictors $x_d$ and $x_e$ is
\begin{align*}
& \text{KL-diff}^2 ({x^*}, (x_d, x_e), \mathcal{M}) = \sqrt{\dfrac{\partial^4 \mathcal{D}_{\text{KL}} [p(y^*)\| p(y^{*})]}{(\partial x^{*}_d)^2(\partial x_e^{*})^2}} \nonumber\\
& \approx \left[ 2 \sum_{k=1}^{n_p}\sum_{l=1}^{n_p}\dfrac{\partial^2 \mathcal{D}_{\text{KL}} [p(y^*)\| p(y^{*})]}{\partial\theta_k^*\partial\theta_l^*} \dfrac{\partial^2\theta_k^*}{\partial x_d^* \partial x_e^*}\dfrac{\partial^2\theta_l^*}{\partial x_d^* \partial x_e^*} \right]^{\frac{1}{2}} .
\end{align*}
The KL-diff$^2$ values represent local measures of pairwise interaction strength of
the predictors $x_d$ and $x_e$.
\subsection{Interpreting interactions as group heterogeneity}
\label{sec:multilevel}
Remember that our goal in this work is to identify and assess group heterogeneity in terms of how a given predictor coefficient varies across the levels of a grouping.
The main task is then to identify the predictors whose coefficients have the largest variation between the different levels of one or more grouping factors.
Let us now have some data $\mathcal{D} = \{{X}, {y}\}$ with the coefficients of some of the predictors $x_d$ in ${X}$ varying according to some group structure.
In the simplest setting, with a single grouping variable, we append a dummy vector $g = \{g_1, \ldots, g_N\}$ to the data matrix ${X}$, where $N$ is the number of observations, and where each of the entries $g_i$ is an integer indicating the level to which the $i$th observation belongs.
This formulation naturally generalizes to any number of grouping variables $K$ and denoting individual dummy vectors as $g_k = \{g_{k1}, \ldots, g_{kN}\}$.
With the added dummy vectors, a Gaussian process with a squared exponential covariance function can model complex interactions between the
numerical and dummy variables if they are present in the data.
Then, by applying KL-diff$^2$, we can assess the interaction strength between all $x_d$ predictor variables and all $g_k$ dummy variables to identify potential group heterogeneity.
We note that while it is not our main goal, we can also assess group heterogeneity in terms
of varying intercepts between levels. This can be done by assessing the individual relevances
of the dummy predictors using KL-diff.
The computational cost of running KL-diff$^2$ requires just computing the analytical derivatives of the KL divergence between the predictive distributions.
For Gaussian process models with commonly used likelihoods for numerical, binary or count data, these derivatives
can be computed in analytical form~\cite{paananen2019ranking}.
The method is thus readily applicable to evaluate group heterogeneity for most standard GLMMs.
The proposed methodology is not restricted to Gaussian process models,
but they were chosen because of their several suitable properties.
Firstly, they are able to represent a wide range of linear and nonlinear functions and model high--order interactions.
Secondly, the smoothness properties of the prediction function can be guaranteed and controlled
by the properties of the Gaussian process covariance function.
Thirdly, Gaussian processes can be used in both regression and classification tasks, and
provide well--calibrated predictive uncertainty estimates.
In the Gaussian process covariance function, the predictors are usually assumed to be real-valued.
In Bayesian optimisation, it can be beneficial to modify the Gaussian process covariance function
such that the learned prediction function is discontinuous for integer-valued
variables~\cite{garrido2020dealing}.
However, in this work
we specifically \emph{want} to keep the prediction function continuous even for the integer-valued dummy variables.
That way the model
can learn a smooth prediction function that includes interactions between the numerical predictors and dummy variables arising
from group heterogeneity.
\section{EXPERIMENTS}
\label{sec:experiments}
In this section, we validate the group heterogeneity assessment method discussed in Section~\ref{sec:multilevel} by performing both simulated and real data experiments.
The focus of the experiments is on assessing the performance of the method in terms of correctly identified group coefficients.
However, it is also important to study the false discovery rate, as it can impact the user's budget for model iterations.
Given the lack of alternative methods for directly assessing group heterogeneity, and therefore of baselines as such, we will focus the discussion on the robustness of the method itself (e.g., by studying false versus true discovery rates). In Section~\ref{sec:bikedata} we show that the method finds similar models
as explicitly comparing model candidates using cross-validation.
In all examples, we use squared exponential covariance function in the Gaussian process surrogate model.
For computational reasons, we optimize the hyperparameters of the surrogate model by maximizing the log marginal likelihood, which enables representing the predictive distribution of the Gaussian process in analytical form.
Given the high sample complexity of exact inference in Gaussian processes, this is a commonly used approach that still gives sufficiently well calibrated predictive uncertainty estimates in many cases.
\subsection{Simulated experiments}
\label{sec:simulations}
We perform extensive simulations for regression and binary classification data sets.
For both cases we employ the same data generation process apart from transforming the target $y$.
The predictors $x_d \in {X}$ are normally and independently distributed with mean $0$ and standard deviation $1$.
We sample the population coefficients $b_d$ and intercept $a$ from an uncorrelated multivariate normal distribution.
Additionally, we apply a $0.4$ sparsity factor, so we effectively set $40$\% of the coefficients to be exactly zero.
Then, each data point is randomly assigned to a level for each group with equal probabilities for the levels.
Like the population coefficients, we also sample the group coefficients from an uncorrelated multivariate Gaussian with group-specific parameters.
The linear predictor is then generated as a sum of both population and group terms. %
The data generation process is summarized below:
\begin{align*}
x_{id} & \sim \text{Normal}(0, 1) , \, d = 1,\ldots,D, \, i = 1,\ldots,N \\
%
b_d & \sim \text{Normal}(\mu_f, \sigma^2_f),\qquad a \sim \text{Normal}(\mu_{b,f}, \sigma^2_{b,f}) \\
z_d & \sim \text{Bernoulli}(p=0.6) \\
g_{ik} & \sim \text{DiscreteUniform}(1, L) , \, k = 1,\ldots,K \\ %
b_{lkd} & \sim \text{Normal}(\mu_{g_{k}}, \sigma^2_{g_{k}}),\qquad a_{lk} \sim \text{Normal}(\mu_{b,g_{k}}, \sigma^2_{b,g_{k}}) \\
\mu_i & = a + \sum_{d=1}^D z_db_dx_{id} + \sum_{k=1}^K\sum_{l=1}^L\left[a_{lk} + \sum_{d=1}^D z_db_{lkd}x_{g_k = l,d} \right] \\
y_i & \sim \pi(f(\mu_i), \phi_i),
\end{align*}
where $\pi$ is the family's distribution, $f$ is an inverse-link function and $\phi$ the family's specific dispersion parameter, $N$ is the number of observations, $D$ the number of numerical predictors in
the data, $K$ is the number of grouping variables, and $L$ is the number of levels in each group.
For each family, we sample different parameters $\mu_f, \sigma^2_f$ for the population coefficients and $\mu_{g_{k}}, \sigma^2_{g_{k}}$ for the group coefficients, and $\mu_{b,f}, \sigma^2_{b,f}$ for the population intercept and $\mu_{b,g_{k}}, \sigma^2_{b,g_{k}}$ for the group intercepts.
In Table~\ref{tab:family_settings}, we show the chosen family-specific values for these parameters.
They were chosen so that the non-zero coefficients are strong and therefore we mitigate the issues of identifying present but very small coefficients (with true values close to $0$).
For the Bernoulli family, we use the probit link function, and generate the data so that the classes are not clearly separable.
\begin{table}[tbp]
\centering
\caption{Values for the family--specific parameters}
\begin{tabular}[tbp]{rrr}
\toprule
Parameter & Gaussian & Bernoulli \\
\midrule
$\mu_f$, $\sigma^2_f$ & $5$, $10$ & $0$, $2$ \\
$\mu_{b,f}$, $\sigma^2_{b,f}$ & $0$, $20$ & $0$, $4$ \\
$\mu_{g_{k}}$, $\sigma^2_{g_{k}}$ & $0$, $5$ & $0$, $3$ \\
$\mu_{b,g_{k}}$, $\sigma^2_{b,g_{k}}$ & $0$, $5$ & $0$, $3$ \\
\bottomrule
\end{tabular}
\label{tab:family_settings}
\end{table}
Within this simulation setting, we systematically varied and fully crossed several factors as shown in Table~\ref{tab:settings}.
Within each simulation condition, we repeat the experiments for $50$ data realizations to account for randomness in the data generation process.
\begin{table}[tbp]
\centering
\caption{Varied simulation settings}
\begin{tabular}[tbp]{lr}
\toprule
Factor & Values \\
\midrule
$N$: Number of observations & $300$ \\
$D$: Number of predictor variables & $\left\{ 5, 10, 15, 20 \right\}$ \\
$K$: Number of grouping factors & $\left\{ 1, 2, 3 \right\}$ \\
$L$: Number of levels in a group & $5$ \\
$s$: Sparsity & $0.4$ \\
$\mathcal{L}$: Likelihood family & Gaussian, Bernoulli \\
\bottomrule
\end{tabular}
\label{tab:settings}
\end{table}
In order to select the relevant heterogeneous group coefficients using the proposed method, we have to apply a decision rule defining which of the returned interactions to choose.
Given that KL-diff$^2$ returns a continuous measure indicating the strength of the interaction, we need to decide which of the returned indicators are strong enough to be chosen.
In this work, we will select the top-$T$ interactions per grouping variable after sorting them, where $T = \lfloor tD\rfloor$, with $t \in [0, 1] \in \mathbb{R}$ and $D$ is the number of possible group coefficients.
In practice, the threshold could be chosen with some selection criterion such as a cross-validation score.
The selection criterion is left for future research.
\subsubsection{Gaussian simulations}
\begin{figure*}[htb]
\centering
\centerline{\includegraphics[scale=0.8]{figs/pct_roc_uncertainty_gaussian_new_0_4_sparsity_noprior.pdf}}
\caption{ROC curves for recovered group coefficients as we increase the selection threshold for a model with 40\% sparsity for different numbers of predictors $D$ and grouping variables $K$ in the data. The dashed black line indicates chance selection.}
\label{fig:res_gaussian_t3}
\end{figure*}
Because the Gaussian observation model allows for exact inference under a Gaussian process prior, the predictive distribution has a closed form expression that we use to differentiate following the method described in Section~\ref{sec:kldiff}.
In Figure~\ref{fig:res_gaussian_t3}, we show the results for the Gaussian observation model for different number of grouping variables (as specified in Table~\ref{tab:settings}) and as we increase the interaction selection threshold.
The dots represent the mean estimate and the bars show the $95$\% intervals, both on the y- and on the x-axis.
The figure shows the receiver operating characteristic (ROC) curve where we show the true positive rate as a function of the false positive rate.
For every point in the figure the performance of the method is above chance selection (dashed black line). For just a single grouping variable,
selection is excellent even with many predictors.
As expected, as we increase the number of grouping variables, the problem gets harder because each
level in the groups explains a smaller part of the total variance.
Given that we have fixed the sparsity in our model to $40\%$, the true number of group coefficients is around $60\%$.
The sparsity affects the performance of the method to some extent, and more thorough analysis of the effect of sparsity is left for further research.
There are several additional factors that may impact the performance of our method.
First, because we are randomly sampling the group coefficients, some of them may end up being too small to be detectable.
Second, our method is not performing feature selection in terms of a \emph{true} model.
We are evaluating the strength of the interactions through the predictive distribution, and therefore we can only detect group coefficients that make a difference in the predictions.
Because the difficulty depends on the signal-to-noise ratio and the number of observations in the data,
by adding more observations or reducing noise, all the ROC curves get closer to optimal performance.
To keep the experiment concise, the number of levels in each group was held constant. We also did
additional experiments assessing the effect of the number of levels on the method's ability
to detect group heterogeneity. Based on these experiments, having only 2 or 3 levels will make
it difficult to detect heterogeneity, but more than that is enough to detect it.
As for large numbers of levels, even tens of levels did not make the problem more difficult
as long as the number of observations per level is not very small.
\subsubsection{Bernoulli simulations}
In the case of a Bernoulli observational model we no longer have an analytical expression for the predictive distribution as the required integral cannot be solved in closed form.
Likewise, we cannot perform exact inference to find the posterior, and we must turn to approximate inference methods.
In order to use KL-diff$^2$, we use the Laplace approximation to the posterior of Gaussian process latent values.
As before, we optimize the hyperparameters by maximizing the log marginal likelihood, which allows for a closed form solution of the predictive distribution.
Even though we are computing the interaction strengths through an approximate posterior, the results look quite similar to the Gaussian ones.
Due to lack of space and similarity with the previous results, we have omitted the plots of the Bernoulli
results, but the reader can inspect these in the supplementary material located in our github repository
at~\href{https://github.com/topipa/group-heterogeneity-paper}{github.com/topipa/group-heterogeneity-paper}.
\subsubsection{Case study: Usage of rental bicycles} \label{sec:bikedata}
In this section we demonstrate the group heterogeneity assessment framework in a real data set\footnote{http://archive.ics.uci.edu/ml/datasets/bike+sharing+dataset}. The data set
includes the number of daily rental bike uses over two years together with weather information.
The target variable is \texttt{dailyuses}, the number of bike uses per day.
The numerical predictors are temperature, humidity, and wind speed.
We have five grouping variables:
day of the week, month, season, weather category, and a public holiday indicator.
The total data set thus contains $3$ numerical predictor
variables and $5$ grouping variables, and $731$ observations.
As such, there are already more than $100$ possible predictor-grouping combinations.
We fit a Gaussian process model on the data set and compute the KL-diff$^2$ interaction values between
the numerical and dummy predictors as described earlier.
The eventual goal is to assess the heterogeneity
of the group coefficients for building a GLMM.
To this end, we build
a base model that includes population-level slopes for all the numerical
variables, and varying intercepts for all the grouping variables:
\texttt{dailyuses \texttildelow \, temperature + humidity + windspeed + (1 | month) + (1 | day\_of\_week) + (1 | season) +\\ (1 | weather) + (1 | holiday)}.
Based on the KL-diff$^2$ evaluation, we choose the grouping variable that has the largest total interaction with
the three numerical predictors, which turns out to be the month indicator.
We construct three separate models, each having a single numerical predictor varying over months.
All of the models are fitted using the \texttt{rstanarm} \texttt{R} package~\cite{rstanarm}.
For the three extended models, we
evaluate the group heterogeneity by computing the standard deviation of the posterior means of the slopes between different months, and the predictive performance of the model compared to the base model with leave-one-out cross-validation and the expected log predictive density (elpd) utility~\cite{Vehtari+Gelman+Gabry:2017_practical}.
The results are presented in Table~\ref{tab:bikedata} together with the
KL-diff$^2$ heterogeneity values. The results show that the method
identifies the relevant predictors that have heterogeneous coefficients
between different months, which is present in the resulting
models as both heterogeneity of the slopes between levels, and predictive performance improvement
compared to the base model.
Adding varying coefficients from the other grouping variables does not improve
the predictive performance of the model according to the cross-validation assessment.
Thus, based on the KL-diff$^2$ evaluation, a recommended model formula is
\texttt{dailyuses \texttildelow \, temperature + humidity + windspeed + (temperature + humidity + windspeed | month) + \\ (1 | day\_of\_week) + (1 | season) + \\ (1 | weather) + (1 | holiday)}.
\begin{table}[tbp]
\centering
\caption{Group heterogeneity results for the rental bike data}
\begin{tabular}[tbp]{lrrr}
\toprule
Predictor & KL-diff$^2$ & Slope std. & elpd difference\\
& & between levels & to base model \\
\midrule
temperature & 0.33 & 0.63 & $54 \pm 10$ \\
humidity & 0.27 & 0.23 & $36 \pm 9$ \\
wind speed & 0.12 & 0.07 & $3 \pm 3$ \\
\bottomrule
\end{tabular}
\label{tab:bikedata}
\end{table}
\section{CONCLUSION}
\label{sec:conclusion}
In this work, we propose a new approach to identify heterogeneous group coefficients from data by using a Gaussian process surrogate model and interaction strength estimates given by the KL-diff$^2$ method~\cite{paananen2019ranking}.
We achieve this by interpreting interactions between numerical predictors and dummy grouping variables as indications for group heterogeneity.
We demonstrate that the proposed method provides a highly useful tool for practitioners who often shy away from hierarchical and multilevel models due to their higher complexity as compared to generalized linear models or other related inference methods.
Quite commonly, the main issue before fitting multilevel models is to assess what the relevant multilevel structure exists in the data (conditioned on the chosen model class).
By tuning the selection threshold, users of the new method gain extra flexibility to either ensure that all the relevant structure is captured or that the resulting model does not exceed their computational resources.
After identifying the most relevant heterogeneous group coefficients, the user can go and fit the model again explicitly including those coefficients, making multilevel modelling a more accessible option that brings many benefits over standard non--multilevel models.
\section{ACKNOWLEDGEMENTS}
We thank Academy of Finland (grants 298742 and 313122),
Finnish Center for Artificial Intelligence,
and Technology Industries of Finland Centennial Foundation (grant 70007503; Artificial Intelligence for Research and Development)
for support of this research.
We also acknowledge the computational resources provided by the Aalto Science-IT project.
\bibliographystyle{IEEEbib}
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\section*{Introduction}
We start by recalling the \emph{local Langlands correspondence for $\GL_n$} over non-archimedean local fields. Let $F$ be a nonarchimedean local field, write $\ka$ for its residue field, and fix a prime number $\ell\neq\pchar\ka$. The local Langlands correspondence for $\GL_n$ over $F$ posits a collection of \emph{canonical} bijections
\begin{align*}
\left\{
\begin{tabular}{c}
isomorphism classes of irreducible \\
smooth representations of $\GL_n(F)$ over $\ov\bQ_\ell$
\end{tabular}
\right\}\longleftrightarrow\left\{
\begin{tabular}{c}
isomorphism classes of $n$-dimensional continuous \\
Frobenius-semisimple representations of $W_F$ over $\ov\bQ_\ell$
\end{tabular}
\right\},
\end{align*}
where $W_F$ denotes the Weil group of $F$ with respect to a fixed separable closure of $F$, and $n$ ranges over all positive integers. By canonical, we mean they are characterized by recovering local class field theory when $n=1$, being compatible with central characters and duals, and preserving $L$-functions and $\eps$-factors of pairs.
To prove such a correspondence, work of Bernstein--Zelevinsky allows us to reduce to the case of cuspidal representations on the automorphic side and irreducible representations on the Galois side. More precisely, they show {\cite[10.3]{Zel80}} that one can uniquely reconstruct bijections as above from their restrictions to cuspidal representations
\begin{align}\label{eq:llc}
\left\{
\begin{tabular}{c}
isomorphism classes of irreducible \\
cuspidal representations of $\GL_n(F)$ over $\ov\bQ_\ell$
\end{tabular}
\right\}\longleftrightarrow\left\{
\begin{tabular}{c}
isomorphism classes of irreducible continuous \\
$n$-dimensional representations of $W_F$ over $\ov\bQ_\ell$
\end{tabular}
\right\}\tag{$\diamond$}.
\end{align}
As alluded to above, work of Henniart \cite[Theorem 1.2]{Hen85} indicates that there is at most one collection of bijections as in Equation (\ref{eq:llc}) satisfying our canonicity requirements. This ensures the \emph{uniqueness} of the local Langlands correspondence for $\GL_n$ over $F$.
As for its \emph{existence}, such bijections were first proved for function fields by Laumon--Rapoport--Stuhler \cite[(15.7)]{LRS93} and for $p$-adic fields by Harris--Taylor \cite[Theorem A]{HT01} and Henniart \cite[1.2]{Hen00}. When $F$ is a $p$-adic field, Scholze gave a new proof and characterization \cite[Theorem 1.2]{Sch13} of the local Langlands correspondence for $\GL_n$ over $F$, simplifying arguments of Harris--Taylor.
By adapting the methods of Scholze's proof, the goal of this paper is to give a new proof and characterization of the local Langlands correspondence for $\GL_n$ when $F$ is a function field. Thus, let us henceforth assume that $p\deq\pchar{F}>0$. Grothendieck's $\ell$-adic monodromy theorem \cite[Appendix]{ST68} implies that irreducible continuous $n$-dimensional representations of $W_F$ over $\ov\bQ_\ell$ are smooth, so by fixing a field isomorphism $\ov\bQ_\ell=\bC$, Equation (\ref{eq:llc}) remains unchanged if we replace $\ov\bQ_\ell$ with $\bC$. Unless otherwise specified, all subsequent representations shall be taken over $\bC$.
We begin by motivating this new characterization of the local Langlands correspondence for $\GL_n$ over $F$, as in the work of Scholze. Since $n$-dimensional semisimple continuous representations of $W_F$ are determined by their traces, it would be natural to characterize the correspondence via a trace condition. More precisely, one would like to construct a map $\pi\mapsto\rho(\pi)$ from isomorphism classes of irreducible smooth representations of $\GL_n(F)$ to isomorphism classes of $n$-dimensional semisimple continuous representations of $W_F$ satisfying the following condition: for all $\tau$ in $W_F$, there exists a test function $f_\tau$ in $C^\infty_c(\GL_n(F))$ such that
\begin{align*}
\tr(f_\tau|\pi) = \tr\left(\tau|\rho(\pi)\right)
\end{align*}
for all irreducible smooth representations $\pi$ of $\GL_n(F)$.
However, this is too much to ask for, as noted by Scholze in \cite{Sch13}. To see this, note that we want $\rho(\pi)$ to be (a Tate twist of) the Weil representation corresponding to $\pi$ under the local Langlands correspondence. But if this were the case, then the above equation implies that $f_\tau$ has nonzero trace on \emph{every} component of the Bernstein center of $\GL_n(F)$. This is impossible because $f_\tau$ is locally constant.\footnote{However, such an $f_\tau$ \emph{does} exist as an element of the Bernstein center and hence as a distribution. See Proposition \ref{ss:functiondesire}.}
We smooth out this issue by convolving $f_\tau$ with a cut-off function. To elaborate, let us introduce some notation: write $q$ for $\#\ka$, and write $v:W_F\rar\bZ$ for the unramified homomorphism sending geometric $q$-Frobenii to $1$. For any $\tau$ in $W_F$ with $v(\tau)>0$ and $h$ in $C^\infty_c(\GL_n(\cO))$, we construct a test function $f_{\tau,h}$ in $C^\infty_c(\GL_n(F))$ satisfying the following theorem.
\begin{thmA}Let $n$ be a positive integer, and let $\pi$ be any irreducible smooth representation of $\GL_n(F)$.
\begin{enumerate}[(i)]
\item There exists a unique $n$-dimensional semisimple continuous representation $\rho(\pi)$ of $W_F$ satisfying the following property:
\begin{align*}
\mbox{for all }\tau\mbox{ in }W_F\mbox{ with }v(\tau)>0\mbox{ and }h\mbox{ in }C^\infty_c(\GL_n(\cO))\mbox{, we have }\tr(f_{\tau,h}|\pi) = \tr\left(\tau|\rho(\pi)\right)\tr(h|\pi).
\end{align*}
Write $\rec(\pi)$ for $\rho(\pi)(\frac{1-n}2)$, where $(s)$ denotes the $s$-th Tate twist.
\item Suppose $\pi$ is isomorphic to a subquotient of the normalized parabolic induction of
\begin{align*}
\pi_1\otimes\dotsb\otimes\pi_t,
\end{align*}
where the $\pi_i$ are irreducible smooth representations of $\GL_{n_i}(F)$ such that $n_1+\dotsb+n_t=n$. Then
\begin{align*}
\rec(\pi) = \rec(\pi_1)\oplus\dotsb\oplus\rec(\pi_t).
\end{align*}
\end{enumerate}
\end{thmA}
Theorem A.(i) indicates that $f_{\tau,h}$ satisfies the trace compatibility property we would expect from the convolution $f_\tau* h$. As our notation suggests, we shall see in Theorem C that $\pi\mapsto\rec(\pi)$ equals the usual local Langlands correspondence for $\GL_n$ over $F$ (after forgetting the monodromy operator). Thus this gives a new characterization of the local Langlands correspondence in this case.
How can we find such an $f_{\tau,h}$? Let $\ov\ka$ be a fixed separable closure of $\ka$. The Deligne--Carayol conjecture (which was proven for function fields by Boyer \cite[Theorem 3.2.4]{Boy99} and for $p$-adic fields by Harris--Taylor \cite[Theorem B]{HT01}) indicates that, roughly speaking, the local Langlands correspondence for $\GL_n$ as in Equation (\ref{eq:llc}) can be found in the cohomology of deformation spaces of certain $1$-dimensional formal $\cO$-modules over $\ov\ka$. By the Dieudonn\'e equivalence, $1$-dimensional formal $\cO$-modules correspond to special examples of \emph{$1$-dimensional effective minuscule local shtukas}. The deformation spaces of general $1$-dimensional effective minuscule local shtukas can be pieced together from those of $1$-dimensional formal $\cO$-modules (see \S\ref{s:deformationspaces}), and this mirrors how general irreducible smooth representations of $\GL_n(F)$ can be pieced together from cuspidal ones via parabolic induction (see \S\ref{s:parabolicinduction}).
Therefore, to find a test function $f_{\tau,h}$ satisfying Theorem A, we study deformation spaces of $1$-dimensional effective minuscule local shtukas. We start by parameterizing these objects as follows. Write $r\deq v(\tau)$, write $F_r$ for the $r$-th degree unramified extension of $F$, write $\cO_r$ for its ring of integers, and write $\ka_r$ for its residue field. Then the Cartan decomposition for $\GL_n(F_r)$ shows that isomorphism classes of $1$-dimensional effective minuscule local shtukas over $\ka_r$ correspond to elements
\begin{align*}
\de\in\GL_n(\cO_r)\diag(\vpi,1,\dotsc,1)\GL_n(\cO_r)
\end{align*}
up to $\GL_n(\cO_r)$-$\sg$-conjugacy, where $\vpi$ is a uniformizer of $F$. Form the deformation space of the corresponding $1$-dimensional effective minuscule local shtuka with Drinfeld level-$m$ structure, and write $R^i\psi_{\de,m}$ for the $i$-th $\ell$-adic cohomology group of the adic generic fiber of this deformation space. As $m$ varies, we use these cohomology groups to construct representations
\begin{align*}
R^i\psi_\de\deq\dirlim_m R^i\psi_{\de,m}\mbox{ and }[R\psi_\de]\deq\sum_{i=0}^\infty(-1)^iR^i\psi_\de,
\end{align*}
which have commuting actions of $W_{F_r}$ and $\GL_n(\cO)$. From here, we can first define a function $\phi_{\tau,h}$ in $C^\infty_c(\GL_n(F_r))$ by sending
\begin{align*}
\de\mapsto
\begin{cases}
\tr(\tau\times h|[R\psi_\de]) & \mbox{if }\de\mbox{ is in }\GL_n(\cO_r)\diag(\vpi,1,\dotsc,1)\GL_n(\cO_r),\\
0 & \mbox{otherwise,}
\end{cases}
\end{align*}
and then we let $f_{\tau,h}$ be a transfer of $\phi_{\tau,h}$ to $\GL_n(F)$.
Now that we have our test function $f_{\tau,h}$, let's discuss the proof of Theorem A. After constructing $\rho(\pi)$ in special cases, we use the geometry of our deformation spaces, along with work of Bernstein--Zelevinsky \cite{Zel80} on the local automorphic side, to inductively piece together $\rho(\pi)$ in the general case. The process of piecing together $\rho(\pi)$ amounts entirely to local deformation theory and nonarchimedean harmonic analysis. This proceeds as in \cite{Sch13}, except the necessary harmonic analysis is more difficult in characteristic $p$. Therefore we carefully give the argument and supply references for the relevant harmonic analysis in characteristic $p$.
We use global techniques to construct $\rho(\pi)$ in special cases. But instead of Shimura varieties as in \cite{Sch13}, we use moduli spaces of \emph{$\sD$-elliptic sheaves}, the latter of which is an equi-characteristic analogue of abelian varieties equipped with certain endomorphisms. For good reduction, these moduli spaces were first considered by Laumon--Rapoport--Stuhler \cite{LRS93} in their proof of the local Langlands correspondence for $\GL_n$ over $F$, and we shall also crucially use a version of these moduli spaces with bad reduction (which corresponds to using Drinfeld level structures) as considered by Boyer \cite{Boy99} in his proof of the Deligne--Carayol conjecture for $F$.
We explicitly define $\rho(\pi)$ as a multiplicity space in the cohomology of the moduli space of $\sD$-elliptic sheaves. To show that $\rho(\pi)$ satisfies our desired trace compatibility condition, we adapt Scholze's version of the Langlands--Kottwitz method to compute traces of Frobenius and Hecke operators on the aforementioned cohomology groups. These traces are related to our test function $f_{\tau,h}$ via a Serre--Tate theorem for $\sD$-elliptic sheaves, which equates deformations of $\sD$-elliptic sheaves to deformations of certain associated local shtukas. For this, it is crucial to carry out the Langlands--Kottwitz method at bad reduction, which is new in positive characteristic.
This global work yields the following construction of $\ell$-adic Galois representations associated with certain discrete automorphic representations. Let $\bf{F}$ be a global function field, write $\bA$ for its ring of adeles, and let $\{x_1,x_2,\infty\}$ be three distinct places of $\bf{F}$. Write $G_{\bf{F}}$ for the absolute Galois group of $\bf{F}$ with respect to a fixed separable closure of $\bf{F}$.
\begin{thmB}
Let $\Pi$ be an irreducible discrete automorphic representation of $\GL_n(\bA)$ whose components at $x_1$, $x_2$, and $\infty$ are either irreducible $L^2$ representations or Speh modules. Then there exists a unique $n$-dimensional semisimple continuous representation $R(\Pi)$ of $G_\bf{F}$ over $\ov\bQ_\ell$ such that, for all places $o$ of $\bf{F}$ not lying in $\{x_1,x_2,\infty\}$, the restriction of $R(\Pi)$ to $W_{\bf{F}_o}$ satisfies
\begin{align*}
\res{R(\Pi)}_{W_{\bf{F}_o}} = \rho(\Pi_o),
\end{align*}
where we identify $\ov\bQ_\ell$ with $\bC$.
\end{thmB}
We finish the construction of $\rho(\pi)$ by finding a $\Pi$ as above such that $\pi$ is isomorphic to $\Pi_o$, and then applying Theorem B. In general, this is more difficult in characteristic $p$ because the trace formula is not as developed, but we circumvent this using M\oe glin--Waldspurger's description of the discrete automorphic spectrum. This concludes our construction of $\rho(\pi)$ in special cases and thus our proof of Theorem A.
From here, we prove the following bijectivity result.
\begin{thmC}
The map $\pi\mapsto\rec(\pi)$ yields a bijection from isomorphism classes of irreducible cuspidal representations of $\GL_n(F)$ to isomorphism classes of $n$-dimensional irreducible continuous representations of $W_F$.
\end{thmC}
The key ingredient is an explicit calculation of the inertia invariants of nearby cycles due to Scholze \cite[Theorem 5.3]{Sch13b}, which relies on a case of Grothendieck's purity conjecture as proved by Thomason \cite[Corollary 3.9]{Tho84}. We use this calculation to deduce that if $\rec(\pi)$ is unramified, then $\pi$ must be as well. By passing to the Galois side and using our work on local-global compatibility, we show that after applying cyclic base change \cite[(II.1.4)]{HL11} finitely many times, the representation $\pi$ becomes unramified. Passing to this unramified representation ultimately allows us to prove Theorem C. As in \cite{Sch13}, this argument bypasses the need to appeal to Henniart's numerical local Langlands \cite[Theorem 1.2]{Hen88}.
Finally, we show that $\pi\mapsto\rec(\pi)$ satisfies the usual canonicity requirements of the local Langlands correspondence for $\GL_n$.
\begin{thmD}
The bijections
\begin{align*}
\rec:\left\{
\begin{tabular}{c}
isomorphism classes of irreducible \\
cuspidal representations of $\GL_n(F)$
\end{tabular}
\right\}\rar^\sim\left\{
\begin{tabular}{c}
isomorphism classes of $n$-dimensional \\
irreducible continuous representations of $W_F$
\end{tabular}
\right\}
\end{align*}
satisfy the following properties:
\begin{enumerate}[(i)]
\item for all irreducible cuspidal representations of $\GL_1(F)$, that is, smooth characters $\chi:F^\times\rar\bC^\times$, we have
\begin{align*}
\rec(\chi) = \chi\circ\Art^{-1},
\end{align*}
where $\Art$ denotes the Artin isomorphism $\Art:F^\times\rar^\sim W^\ab_F$ that sends uniformizers to geometric $q$-Frobenii.
\item for all irreducible cuspidal representations $\pi$ of $\GL_n(F)$ and smooth characters $\chi:F^\times\rar\bC^\times$, we have
\begin{align*}
\rec(\pi\otimes(\chi\circ\det)) = \rec(\pi)\otimes\rec(\chi).
\end{align*}
\item for all irreducible cuspidal representations $\pi$ of $\GL_n(F)$ with central character $\om_\pi:F^\times\rar\bC^\times$, we have
\begin{align*}
\rec(\om_\pi) = \det\circ\rec(\pi)\mbox{ and }\rec(\pi^\vee) = \rec(\pi)^\vee.
\end{align*}
\item for all irreducible cuspidal representations $\pi$ of $\GL_n(F)$ and $\pi'$ of $\GL_{n'}(F)$, we have
\begin{align*}
L(\pi\times\pi',s) = L(\rec(\pi)\otimes\rec(\pi'),s)\mbox{ and } \eps(\pi\times\pi',\psi,s) = \eps(\rec(\pi)\otimes\rec(\pi'),\psi,s)
\end{align*}
for all nontrivial continuous homomorphisms $\psi:F\rar\bC^\times$.
\end{enumerate}
\end{thmD}
By \cite[Theorem 1.2]{Hen85}, Theorem D shows that our construction indeed equals the usual local Langlands correspondence for $\GL_n$.
We now discuss the proof of Theorem D. Compatibility with local class field theory follows from earlier work, as our use of $1$-dimensional formal $\cO$-modules recovers the Lubin--Tate description of local class field theory when $n=1$. Compatibility with twists of characters follows by embedding into the global setting, using Theorem B, and applying the strong multiplicity one theorem \cite[Theorem 3.3.(b)]{BR17}. We prove compatibility with central characters via similar global means---this time, we reduce to the situation of induced representations by Brauer induction, and then we prove some cases of non-Galois automorphic induction in order to pass to the automorphic side. From here, Henniart's trick of twisting by highly ramified characters \cite[Lemma 4.2]{Hen86} implies compatibility with $L$-functions and $\eps$-factors, and then compatibility with duals follows from the decomposition of $L$-functions of pairs in terms of $L$-functions of characters.
\subsection*{Outline} In \S\ref{s:deformationspaces}, we introduce local shtukas, the geometry of their deformation spaces, and the test function $f_{\tau,h}$. Next, in \S\ref{s:firstinductivelemma}, we recall the Bernstein--Zelevinsky classification, the Bernstein center, and Schneider--Zink tempered types. We use this to prove Lemma \ref{ss:firstinductivelemma}, which serves as the framework for proving Theorem A. The primary goal of \S\ref{s:parabolicinduction}--\S\ref{s:localglobal} is to prove the ingredients needed for running Lemma \ref{ss:firstinductivelemma}. In \S\ref{s:parabolicinduction}, we use the geometry of deformation spaces along with the Bernstein--Zelevinsky induction-restriction formula to prove compatibility with parabolic induction. In \S\ref{s:lubintatetower}, we compare our deformation spaces with the Lubin--Tate tower. With the exception of Lemma \ref{ss:nearbycyclesalgebraization}, the entirety of \S\ref{s:deformationspaces}--\S\ref{s:lubintatetower} is local.
At this point, we switch to a global setup. In \S\ref{s:modulispaces}, we introduce $\sD$-elliptic sheaves, their moduli spaces, and their relationship with local shtukas. In \S\ref{s:nearbycycles}, we present an explicit calculation of nearby cycle sheaves on the special fibers of these moduli spaces at bad reduction. In \S\ref{s:langlandskottwitz}, we employ a variant of the Langlands--Kottwitz method, using the Serre--Tate theorem as introduced in \ref{ss:extendedserretate} to relate the cohomology of the moduli space of $\sD$-elliptic sheaves with the local test function $f_{\tau,h}$. In \S\ref{s:localglobal}, we deduce Theorem B from \S\ref{s:langlandskottwitz}, and then we embed our local situation into our global one to finish the proof of Theorem A.
We now turn to the proofs of Theorem C and Theorem D. In \S\ref{s:secondinductivelemma}, we recall cyclic base change, and we combine this with local-global embedding theorems to prove Theorem C. In \S\ref{s:llc}, we use Theorem B (as well as compatibility with automorphic induction) to prove some cases of non-Galois automorphic induction. We tie these results together to prove Theorem D, concluding our proof of the local Langlands correspondence for $\GL_n$ over $F$.
\subsection*{Notation}
Let $\ell\neq p$ be a prime number. We fix an identification $\ov\bQ_\ell=\bC$ of fields. Unless otherwise specified, all representations are over $\bC$.
Throughout \S\ref{s:deformationspaces}--\S\ref{s:lubintatetower}, let $F$ be a local field of positive characteristic. Write $\cO$ for its ring of integers, fix a uniformizer $\vpi$ of $\cO$, and write $\ka$ for the residue field $\cO/\vpi$. This choice of $\vpi$ yields an identification $\cO=\ka\llb\vpi$. We denote $\#\ka$ using $q$. We write $v$ for the normalized discrete valuation on $F$, and we denote the normalized valuation on $F$ using $\abs-$. We fix a separable closure $F^\sep$ in which our separable extensions of $F$ lie, and we write $\ov\ka$ for the residue field of $F^\sep$, which is a separable closure of $\ka$. We write $\bC_\vpi$ for the completion of $F^\sep$, and we view completions of separable extensions of $F$ as closed subfields of $\bC_\vpi$.
We write $W_F$ for the Weil group of $F$ with respect to this choice of $F^\sep$, and we view Weil groups of separable extensions of $F$ as living inside $W_F$. We denote the inertia subgroup of $F$ using $I_F$. We have a canonical short exact sequence
\begin{align*}
1\rar I_F\rar W_F\rar^v\bZ\rar1,
\end{align*}
and we identify $1$ in $\bZ$ with the geometric $q$-Frobenius in $\Gal(\ov\ka/\ka)$. We write $\Art:F^\times\rar^\sim W_F^\ab$ for the local Artin isomorphism normalized by sending uniformizers to geometric $q$-Frobenii.
\subsection*{Acknowledgments} The author is tremendously indebted to Sophie Morel for her advice on this project, suggesting this topic of research, and her encouragement. The author extends his gratitude to Richard Taylor for his careful reading of an earlier iteration of this paper as well as clarifying some misconceptions. Many of the ideas in this paper are due to Peter Scholze, and the author thanks him for answering some questions about \cite{Sch13}. The author would also like to thank Thomas Haines for some helpful conversations on nonarchimedean harmonic analysis, and to thank Alain Genestier for directing him to Boyer's thesis \cite{Boy99}.
\section{Deformation spaces of local shtukas}\label{s:deformationspaces}
In this section, we introduce \emph{local shtukas}, which are the equi-characteristic analogue of isocrystals. We then narrow our scope to \emph{effective minuscule} local shtukas, which are the analogue of Dieudonn\'e modules. A similar Dieudonn\'e equivalence relates $1$-dimensional connected local shtukas to $1$-dimensional formal $\cO$-modules, which allows us to use results of Drinfeld to study their deformation spaces. From here, we deduce finitude properties about the cohomology of these deformation spaces from a global algebraization result, and we conclude by defining the test functions $\phi_{\tau,h}$ and $f_{\tau,h}$ using these cohomology groups.
\subsection{}
Given a scheme $S$ over $\Spec\cO$, we write $\ze$ for the image of $\vpi$ in $\sO_S$. Note that requiring $\ze$ to be locally nilpotent is the same as requiring $S\rar\Spec\cO$ to factor as $S\rar\Spf\cO\rar\Spec\cO$. We consider $\sO_S\llb{\vpi}$ and $\sO_S\llp{\vpi}$ as sheaves of formal power and Laurent series on $\abs{S}$, respectively \cite[p.~4]{HS16}.
\begin{defn*}
Let $S$ be a scheme over $\Spf\cO$. An \emph{local shtuka} over $S$ is a pair $(\sM,\sF)$, where
\begin{enumerate}[$\bullet$]
\item $\sM$ is a locally free $\sO_S\llb\vpi$-module of finite rank,
\item $\sF:\sg^*\sM[\frac1\vpi]\rar\sM[\textstyle\frac1\vpi]$ is an $\sO_S\llp\vpi$-module isomorphism,
\end{enumerate}
where $\sg$ denotes the absolute $q$-Frobenius on $S$ and its canonical lifts to $\sO_S\llb\vpi$ and $\sO_S\llp\vpi$. A \emph{morphism} of local shtukas is a morphism $f:\sM\rar\sM'$ of locally free $\sO_S\llb\vpi$-modules satisfying $f[\frac1\vpi]\circ\sF = \sF'\circ f[\frac1\vpi]$. A \emph{quasi-isogeny} of local shtukas is an invertible element in $\Hom(\sM,\sM')[\frac1\vpi]$. When such an element exists, we say $\sM$ and $\sM'$ are \emph{isogenous}.
\end{defn*}
\subsection{}\label{ss:localshtukaconjugacy}For any field $\la$ over $\ka$, we see that isomorphism classes of local shtukas over $\Spec\la$ of rank $n$ correspond to $\GL_n(\la\llb\vpi)$-$\sg$-conjugacy classes in $\GL_n(\la\llp\vpi)$ via sending $\de$ in $\GL_n(\la\llp\vpi)$ to $(\la\llb\vpi^{ n},\de\circ\sg^{\oplus n})$, where $\de$ acts via left multiplication on column vectors. In further analogy with isocrystals, we also have a \emph{Dieudonn\'e--Manin classification} \cite[(2.4.5)]{Lau96} for isogeny classes of local shtukas over algebraically closed fields.
\begin{defn}
Let $\sM$ be a local shtuka over $S$. We say that $\sM$ is \emph{effective} if $\sF$ is the localization of an $\sO_S\llb\vpi$-module morphism $\sg^*\sM\rar\sM$, which we henceforth refer to as $\sF$ by abuse of notation. For an effective local shtuka $\sM$, one can show that $\coker\sF$ is a locally free $\sO_S$-module of finite rank \cite[Lemma 2.3]{HS16}, and we say the \emph{dimension} of $\sM$ is the rank of $\coker\sF$. We say $\sM$ is \emph{\'etale} if its dimension equals zero.
We say $\sM$ is \emph{effective minuscule} if $\vpi-\ze$ annihilates $\coker\sF$. For an effective minuscule local shtuka $\sM$, we say $\sM$ is \emph{connected} if $\sF$ is $\vpi$-adically nilpotent.
\end{defn}
\subsection{}\label{ss:effectiveminusculelocalshtukaconjugacy}
In our above description \ref{ss:localshtukaconjugacy} of isomorphism classes of local shtukas over $\Spec\la$, we see that the local shtuka corresponding to $\de$ is effective if and only if $\de$ lies in $\M_n(\la\llb\vpi)$. By the Cartan decomposition, it is effective minuscule if and only if $\de$ lies in
\begin{align*}
\coprod_{d=0}^n\GL_n(\la\llb\vpi)\diag(\underbrace{\vpi,\dotsc,\vpi}_{d\,\text{times}},1,\dotsc,1)\GL_n(\la\llb\vpi),
\end{align*}
where $d$ corresponds to the dimension of $(\la\llb\vpi^{n},\de\circ\sg^{\oplus n})$.
\subsection{}
Just as with Dieudonn\'e modules, effective minuscule local shtukas have a good theory of
\begin{enumerate}[$\bullet$]
\item truncated variants, called \emph{finite $\ka$-shtukas} \cite[Definition 2.6]{HS16}, which we remark can be defined over any scheme $S$ over $\Spec\cO$,
\item a contravariant \emph{Dieudonn\'e equivalence} \cite[Theorem 5.2]{HS16} between finite $\ka$-shtukas and certain finite module schemes called \emph{strict $\ka$-modules} \cite[Definition 4.8]{HS16} that extends to an anti-equivalence \cite[Theorem 8.3]{HS16} between effective minuscule local shtukas and certain module sheaves called \emph{$\vpi$-divisible local Anderson modules} \cite[Definition 7.1]{HS16},
\item a \emph{connected-\'etale} short exact sequence over local Artinian rings that splits over perfect fields \cite[Proposition 2.9]{HS16}.
\end{enumerate}
\subsection{}\label{ss:localshtukaalgebraicclosure}
Let $\br{F}$ be the completion of the maximal unramified extension of $F$, and write $\br\cO$ for its ring of integers. Our choice of $\vpi$ yields an identification $\br\cO=\ov\ka\llb\vpi$.
We begin by examining the case $S=\Spec\ov\ka$. Let $\de$ be an element of $\GL_n(\br\cO)\diag(\vpi,1,\dotsc,1)\GL_n(\br\cO)$, and write $\br{H}_\de$ for the associated effective minuscule local shtuka over $\Spec\ov\ka$ of rank $n$ and dimension $1$. The connected-\'etale sequence yields a decomposition
\begin{align*}
\br{H}_\de = \br{H}_\de^\circ\oplus\br{H}_\de^\et,
\end{align*}
where $\br{H}_\de^\circ$ and $\br{H}_\de^\et$ are the connected and \'etale parts of $\br{H}_\de$, respectively. Write $k$ for the rank of $\br{H}_\de^\circ$.
As $\br{H}_\de^\circ$ is the connected part, it must have dimension $1$, so under the Dieudonn\'e equivalence it corresponds to the unique formal $\cO$-module of height $k$ and dimension $1$, i.e. the Lubin--Tate module \cite[Proposition 1.7.1)]{Dri74}. By applying the Dieudonn\'e equivalence to $\br{H}_\de^\et$ and using the algebraic closedness of $\Spec\ov\ka$, we also see that $\br{H}_\de^\et$ is the unique \'etale effective minuscule local shtuka over $\Spec\ov\ka$ of rank $n-k$, i.e. the constant $\vpi$-divisible local Anderson module $\ul{(F/\cO)}^{n-k}$. Thus under \ref{ss:localshtukaconjugacy}, we see that $\br{H}_\de^\et$ corresponds to the $\GL_{n-k}(\br\cO)$-$\sg$-conjugacy class of $1$, and $H_\de^\circ$ corresponds to a basic element $\de^\circ$ in $\GL_k(\br\cO)\diag(\vpi,1,\dotsc,1)\GL_k(\br\cO)$, up to $\GL_k(\br\cO)$-$\sg$-conjugation.
\subsection{}\label{ss:localshtukaunramifiedextension}
Let $F_r$ be the $r$-th degree unramified extension of $F$, write $\cO_r$ for its ring of integers, and write $\ka_r$ for $\cO_r/\vpi$, which is the $r$-degree extension of $\ka$. Our choice of $\vpi$ yields an identification $\cO_r=\ka_r\llb\vpi$.
We now turn to the case of $S=\Spec\ka_r$. Let $\de$ be an element of $\GL_n(\cO_r)\diag(\vpi,1,\dotsc,1)\GL_n(\cO_r)$, and write $H_\de$ for the associated effective minuscule local shtuka over $\Spec\ka_r$ of rank $n$ and dimension $1$. The connected-\'etale sequence yields a decomposition
\begin{align*}
H_\de = H_\de^\circ\oplus H_\de^\et,
\end{align*}
where $H_\de^\circ$ and $H^\et_\de$ are the connected and \'etale parts of $H_\de$, respectively. Write $k$ for the rank of $H_\de^\circ$. Under \ref{ss:localshtukaconjugacy}, we see that $H_\de^\et$ corresponds to some $\de^\et$ in $\GL_{n-k}(\cO_r)$, up to $\GL_{n-k}(\cO_r)$-$\sg$-conjugation, and one can show that $H_\de^\circ$ corresponds to some $\de^\circ$ in $\GL_k(\cO_r)\diag(\vpi,1,\dotsc,1)\GL_k(\cO_r)$, up to $\GL_k(\cO_r)$-$\sg$-conjugation, whose norm
\begin{align*}
\N(\de^\circ)\deq\de^\circ\sg(\de^\circ)\dotsm\sg^{r-1}(\de^\circ)
\end{align*}
is $\GL_k(F_r)$-conjugate to an elliptic element in $\GL_k(F)$. We denote the set of such $\de$ by $B_{n,k}$, and we write $B_k$ for $B_{k,k}$. Note that the pullback of $H_\de$ to $\Spec\ov\ka$ is isomorphic to $\br{H}_\de$.
\subsection{}
To define our deformation spaces of local shtukas (with additional data), we need to introduce a notion of \emph{Drinfeld level structures} for effective local shtukas. For any effective local shtuka $\sM$, its $m$-th level truncation is $\sM/\vpi^m$. For any finite $\ka$-shtuka $M$, we write $\Dr(M)$ for the corresponding finite $\ka$-strict module \cite[p.~16]{HS16}, which is an $\cO/\vpi^m$-module scheme over $S$.
\begin{defn*}
Let $\sM$ be an effective minuscule local shtuka over $S$ of constant rank $n$. We say a \emph{Drinfeld level-$m$ structure} on $\sM$ is a Drinfeld level-$m$ structure on $\Dr(\sM/\vpi^m)$, that is, an $\cO/\vpi^m$-module morphism
\begin{align*}
\al:(\vpi^{-m}\cO/\cO)^n\rar\Dr(\sM/\vpi^m)(S)
\end{align*}
such that the collection of all $\al(x)$ for $x$ in $(\vpi^{-m}\cO/\cO)^n$ forms a full set of sections of $\Dr(\sM/\vpi^m)$ as in \cite[(1.8.2)]{KM85}. For any Drinfeld level-$m$ structure $\al$, its restriction to $(\vpi^{-m'}\cO/\cO)^n$ is a Drinfeld level-$m'$ structure, which we denote by $\res\al_{m'}$. Furthermore, if $S$ is the spectrum of a local Artinian ring, then the restriction of $\al$ to $\ker\al$ is a Drinfeld level-$m$ structure on the connected part $\sM^\circ$ of $\sM$. We denote this by $\al^\circ$.
\end{defn*}
\subsection{}
We now initiate our study of deformation spaces. Write $\wh\cC$ for the opposite category of the category whose
\begin{enumerate}[$\bullet$]
\item objects are complete Noetherian local $\br\cO$-algebras $A$ with residue field $\ov\ka$,
\item morphisms are local $\br\cO$-algebra morphisms,
\end{enumerate}
and write $\cC$ for the full subcategory of $\wh\cC$ consisting of Artinian rings. We identify $\wh\cC$ with a full subcategory of formal schemes over $\Spf\br\cO$. Note that for $A$ in $\cC$, we have $\Spf{A}=\Spec{A}$.
For any contravariant functor $E:\cC\rar(\text{Set})$, we write $\wh{E}:\wh\cC\rar(\text{Set})$ for the extension of $E$ to $\wh\cC$ given by sending
\begin{align*}
A\mapsto\invlim_i E(A/\fm^i_A),
\end{align*}
where $\fm_A$ is the maximal ideal of $A$. We say $E$ \emph{has a deformation space} if $\wh{E}$ is representable by a finite disjoint union $\fX$ of formal schemes in $\wh\cC$. By Yoneda's lemma, such an $\fX$ is unique up to isomorphism.
\subsection{}\label{ss:connecteddeformationalgebraicclosure}
Return to the situation in \ref{ss:localshtukaalgebraicclosure}, and let $\al$ be a Drinfeld level-$m$ structure on $\br{H}_\de$. Write $\br{E}_{\de,\al}:\cC\rar(\text{Set})$ for the functor sending
\begin{align*}
A\mapsto\{\mbox{triples }(H',\al',\io')\}/\sim,
\end{align*}
where $H'$ is a local shtuka over $\Spec{A}$, $\al'$ is a Drinfeld level-$m$ structure on $H'$, and $\io'$ is an isomorphism $\br{H}_\de\rar^\sim H'_{\ov\ka}$ such that $(\al')_{\ov\ka} = \io'\circ\al$. In other words, $\br{E}_{\de,\al}$ parametrizes deformations of $(\br{H}_\de,\al)$. Note that $\br{E}_{\de,\al}$ has a right action of $\GL_n(\cO/\vpi^m)$ given by sending $(H',\al',\io')$ to $(H',\al'\circ\ga,\io')$ for any $\ga$ in $\GL_n(\cO/\vpi^m)$.
By applying the Dieudonn\'e equivalence to rephrase the problem in terms of $\vpi$-divisible local Anderson modules, we immediately deduce the following from classical results of Drinfeld.
\begin{prop*}[{\cite[Proposition 4.2]{Dri74}, \cite[Proposition 4.3]{Dri74}, \cite[2.1.2.(ii)]{Str08}}]\hfill
\begin{enumerate}[(i)]
\item The functor $\br{E}_{\de,\al}$ has a deformation space, which we denote using $\br\fX_{\de,\al}=\Spf\br{R}_{\de,\al}$. This $\br{R}_{\de,\al}$ is a regular complete local Noetherian $\br\cO$-algebra.
\item Write $\br\fX_\de=\Spf\br{R}_\de$ for the deformation space with Drinfeld level-$0$ structure, that is, no Drinfeld level structure. Choosing a coordinate on $\br{H}_\de^\circ$ and an isomorphism $\br{H}_\de\rar^\sim\br{H}_\de^\circ\oplus\ul{(F/\cO)}^{n-k}$ induces an isomorphism from $\br{R}_\de$ to the formal power series ring $\br\cO\llb{s_1,\dotsc,s_{n-1}}$.
\item The same choice as in (ii) induces an isomorphism from $\br{R}_{\de,\al}$ to the formal power series ring $\br{R}_{\de^\circ,\al^\circ}\llb{t_1,\dotsc,t_{n-k}}$, where the $\br{R}_{\de^\circ,\al^\circ}$-algebra structure is given by the restriction morphism $\br{R}_{\de^\circ,\al^\circ}\rar\br{R}_{\de,\al}$.
\item For all non-negative integers $m'\leq m$, the restriction morphism $\br{R}_{\de,\res\al_{m'}}\rar\br{R}_{\de,\al}$ is finite flat. Its generic fiber $\br{R}_{\de,\res\al_{m'}}[\frac1\vpi]\rar\br{R}_{\de,\al}[\textstyle\frac1\vpi]$ is a Galois extension of rings, and the left action of
\begin{align*}
\ker\left(\GL_n(\cO/\vpi^m)\rar\GL_n(\cO/\vpi^{m'})\right)
\end{align*}
on $\br{R}_{\de,\res\al_{m'}}\rar\br{R}_{\de,\al}$ yields the Galois action on $\br{R}_{\de,\res\al_{m'}}[\frac1\vpi]\rar\br{R}_{\de,\al}[\textstyle\frac1\vpi]$.
\end{enumerate}
\end{prop*}
In particular, parts (ii) and (iv) imply that $\br\fX_{\de,\al}$ has dimension $n-1$ over $\Spf\br\cO$.
\subsection{}\label{ss:drinfeldparameters}
Let $e_1,\dotsc,e_k$ be an $\cO/\vpi^m$-basis of $\ker\al$. After choosing a coordinate on $\br{H}_{\de^\circ}$, we may identify its $A$-points with $\fm_A$. Classical results of Drinfeld yield certain local parameters of $\br{R}_{\de,\al}$ which satisfy the following relationship with Drinfeld level-$m$ structures.
\begin{prop*}[{\cite[Proposition 4.3.2)]{Dri74}}]
There exist local parameters $x_1,\dotsc,x_k$ of the regular local ring $\br{R}_{\de^\circ,\al^\circ}$ such that, for any morphism $f:\br{R}_{\de^\circ,\al^\circ}\rar A$ in $\wh\cC$, the image of $x_i$ under $f$ equals the element of $\fm_A$ corresponding to $\al'(e_i)$, where $(H',\al',\io')$ is the deformation of $(\br{H}^\circ_\de,\al^\circ)$ corresponding to $f$.
\end{prop*}
\subsection{}\label{ss:deformationalgebraicclosure}
Next, we allow the Drinfeld level-$m$ structure to vary. Write $\br{E}_{\de,m}:\cC\rar(\text{Set})$ for the functor sending
\begin{align*}
A\mapsto\{\mbox{triples }(H',\al',\io')\}/\sim,
\end{align*}
where $H'$ is a local shtuka over $\Spec{A}$, $\al'$ is a Drinfeld level-$m$ structure on $H'$, and $\io'$ is an isomorphism $\br{H}_\de\rar^\sim H'_{\ov\ka}$. In other words, $\br{E}_{\de,m}$ parametrizes deformations of $\br{H}_\de$ along with a Drinfeld level-$m$ structure. As in \ref{ss:connecteddeformationalgebraicclosure}, our functor $\br{E}_{\de,m}$ has a right action of $\GL_n(\cO/\vpi^m)$.
Because every $\al'$ yields a Drinfeld level-$m$ structure $\al'_{\ov\ka}$ of $\br{H}_\de$, we have
\begin{align*}
\br{E}_{\de,m} = \coprod_\al\br{E}_{\de,\al},
\end{align*}
where $\al$ ranges over all Drinfeld level-$m$ structures on $\br{H}_\de$. This disjoint union respects both restriction to the connected component as well as restriction to level-$m'$, where $m'\leq m$. By decomposing Drinfeld level-$m$ structures in terms of the connected-\'etale sequence, we deduce the following result from Proposition \ref{ss:connecteddeformationalgebraicclosure}.
\begin{prop*}
The functor $\br{E}_{\de,m}$ has a deformation space, which we denote using $\br\fX_{\de,m}$. We have an identification
\begin{align*}
\br\fX_{\de,m} = \coprod_\al\br\fX_{\de,\al} = \coprod_V\coprod_{\ker\al=V}\Spf\br{R}_{\de^\circ,\al^\circ}\llb{t_1,\dotsc,t_{n-k}},
\end{align*}
where $V$ ranges over all $\cO/\vpi^m$-linear direct summands of $(\vpi^{-m}\cO/\cO)^n$ with rank $k$. Under the first identification, the restriction morphisms $\br\fX_{\de,m}\rar\br\fX_{\de,m'}$ equal the disjoint union of the restriction morphisms $\br\fX_{\de,\al}\rar\br\fX_{\de,\res\al_{m'}}$.
\end{prop*}
Fix such a $V$. Since Drinfeld level-$m$ structures on \'etale group schemes are precisely group isomorphisms \cite[(1.8.3)]{KM85}, the set of all $\al$ satisfying $\ker\al=V$ is a right principal homogeneous space for $\GL((\vpi^{-m}\cO/\cO)^n/V)$.
\subsection{}\label{ss:deformationunramifiedextension}
Now suppose that $\de$ lies in $\GL_n(\cO_r)\diag(\vpi,1,\dotsc,1)\GL_n(\cO_r)$. This will allow us to descend $\br\fX_{\de,m}$ to a formal scheme $\fX_{\de,m}$ over $\Spf\cO_r$ as follows. Let $\sg$ be any element of $\Gal(\ov\ka/\ka_r)$, which we identify with its canonical lift to $\Aut(\br\cO/\cO_r)$, and write $\sg$ for $\Spec\sg$ by abuse of notation. The automorphism of $\br{H}_\de = H_{\de,\ov\ka}$ given by $\id_{H_\de}\times_{\ka_r}\sg$ lies over $\sg$ and hence induces an isomorphism $\sg^*\br{H}_\de\rar^\sim\br{H}_\de$ over $\Spec\ov\ka$, which we denote using $f_\sg$.
Note that $\sg^*\br{H}_\de$ is isomorphic to $\br{H}_{\sg(\de)}$ and that $\sg^*\br\fX_{\de,m}$ is isomorphic to $\br\fX_{\sg(\de),m}$. We obtain an isomorphism $\vp_\sg:\br\fX_{\de,m}\rar^\sim\sg^*\br\fX_{\de,m}$ by sending $(H',\al',\io')$ to $(H',\al',\io'\circ f_\sg)$. Since the morphisms $f_\sg$ satisfy the cocycle condition, the morphisms $\vp_\sg$ do as well (albeit contravariantly), and thus the $\vp_\sg$ provide Weil descent datum for $\br\fX_{\de,m}$ as in \cite[Definition (3.5)]{RZ96}\footnote{The definition here is stated for mixed characteristic, but it adapts to equal characteristic by using formal power series instead of Witt vectors.} One can show this Weil descent datum is effective, and we write $\fX_{\de,m}$ for the resulting formal scheme over $\Spf\cO_r$. The right action of $\GL_n(\cO/\vpi^m)$ on $\br\fX_{\de,m}$ commutes with $\vp_\sg$ and hence also descends to a right action on $\fX_{\de,m}$. The restriction morphisms similarly descend to morphisms $\fX_{\de,m}\rar\fX_{\de,m'}$.
\subsection{}\label{ss:nearbycyclesalgebraization}
Let $\fX$ be a special formal scheme over $\Spf{A}$ as in \cite[p.~370]{Ber96} for a complete valuation ring $A$. For any complete nonarchimedean field $K$ containing $A$, we write $\fX_K$ for the adic pullback of $\fX$ to $K$ \cite[p.~370]{Ber96}. Writing $\la$ for the residue field of $K$, we denote the adic pullback\footnote{These are referred to in \cite{Ber96} as the generic and special fibers, respectively. Our terminology stems from interpreting them as pullbacks in the category of adic spaces.} of $\fX$ to $\la$ by $\fX_\la$ \cite[p.~370]{Ber96}. These constructions correspond to generic and special fibers, respectively, in the setting of nonarchimedean analytic geometry.
We shall now introduce the cohomology of our deformation spaces. Write $R^i\psi_{\de,m}$ for the $\ov\bQ_\ell$-vector space $H^i(\fX_{\de,m,\bC_\vpi},\ov\bQ_\ell)$. The zero-dimensionality of $\fX_{\de,m,\ov\ka}$ and the nearby cycles spectral sequence \cite[Corollary 2.5]{Ber96} show that
\begin{align*}
R^i\psi_{\de,m} = H^0(\fX_{\de,m,\ov\ka},R^i\Psi_{\fX_{\de,m,\bC_\vpi}}\ov\bQ_\ell),
\end{align*}
where $R^i\Psi_{\fX_{\de,m,\bC_\vpi}}$ denotes the $i$-th nearby cycles functor\footnote{This is referred to in \cite{Ber96} and \cite{HT01} as the vanishing cycles functor.} \cite[p.~373]{Ber96} on $\fX_{\de,m,\cO_{\bC_\vpi}}$. This equality explains our choice of notation for $R^i\psi_{\de,m}$. To prove the finite-dimensionality of $R^i\psi_{\de,m}$ as well as eventually prove the admissibility of a certain $\GL_n(\cO)$-action, we take recourse to the following algebraization result.
\begin{lem*}\label{lem:alglol}
There exists a projective scheme $\cM$ over $\Spec\br\cO$, a zero-dimensional closed subscheme $\cZ$ of $\cM_{\ov\ka}$, and a right action of $\GL_n(\cO/\vpi^m)$ on the pair $(\cM,\cZ)$ such that the completion of $\cM$ at $\cZ$ is isomorphic to $\br\fX_{\de,m}$ with its right action of $\GL_n(\cO/\vpi^m)$.
\end{lem*}
The desired algebraization comes from a moduli space of \emph{$\sD$-elliptic sheaves with bad reduction}, but we won't introduce these moduli spaces until \S\ref{s:modulispaces}, and we won't explain how to relate them to local shtukas until \S\ref{s:langlandskottwitz}. Although we record the proof here, one can safely take this algebraization result as a black box.
\begin{proof}[Proof of Lemma \ref{lem:alglol}]
Let $C=\bP^1_\ka$ be our curve of interest, and let $\infty$ and $o$ be distinct $\ka$-points of $C$. Let $D$ be any central division algebra over $\ka(C)$ of dimension $n^2$ that splits at $o$ and $\infty$, and let $\sD$ be a maximal order of $D$, which can be constructed using \ref{ss:orders} because division algebras split at cofinitely many places. Proposition \ref{ss:dellipticsheavesisomorphismclasses} allows us to find a $\sD$-elliptic sheaf $(\sE_i,t_i,j_i)_i$ over $\ov\ka$ whose local shtuka $\sM_o'$ at $o$ as in \ref{ss:dellipticsheavestolocalshtukazero} is isomorphic to $\br{H}_\de$.
Denote the corresponding $\ov\ka$-point of $\cM_{\varnothing,\br\cO}$ using $z$. Write $o^m$ for the finite closed subscheme of $C$ supported on $o$ with multiplicity $m$, and write $\pi:\cM_{o^m,\br\cO}\rar\cM_{\varnothing,\br\cO}$ for the restriction morphism. Then \ref{ss:extendedserretate} says that $\cM=\cM_{o^m,\br\cO}$ and $\cZ=\pi^{-1}(z)$ yield the desired algebraization of $\br\fX_{\de,m}$.
\end{proof}
\subsection{}\label{ss:fundamentalrepresentation}
With Lemma \ref{ss:nearbycyclesalgebraization} in hand, Berkovich's nearby cycles comparison theorem \cite[Theorem 3.1]{Ber96} implies that $R^i\psi_{\de,m}$ is finite-dimensional over $\ov\bQ_\ell$ and vanishes for $i>n-1$ by Proposition \ref{ss:connecteddeformationalgebraicclosure}. Furthermore, it has commuting continuous left actions of $W_{F_r}$ and $\GL_n(\cO/\vpi^m)$. Write $R^i\psi_\de$ for the direct limit
\begin{align*}
R^i\psi_\de \deq \dirlim_mR^i\psi_{\de,m},
\end{align*}
where the transition maps $R^i\psi_{\de,m'}\rar R^i\psi_{\de,m}$ are induced by the restriction morphisms $\fX_{\de,m}\rar\fX_{\de,m'}$ for $m'\leq m$. Now $R^i\psi_\de$ has commuting left actions of $W_{F_r}$ and $\GL_n(\cO)$, and Proposition \ref{ss:connecteddeformationalgebraicclosure}.(iv) shows that
\begin{align*}
(R^i\psi_\de)^{1+\vpi^m\!\M_n(\cO)} = R^i\psi_{\de,m}.
\end{align*}
Therefore $R^i\psi_\de$ is a $\GL_n(\cO)\times I_F$-admissible/continuous representation as in \cite[p.~24]{HT01} of $\GL_n(\cO)\times W_{F_r}$ over $\ov\bQ_\ell$. Write $[R\psi_\de]$ for the virtual representation
\begin{align*}
[R\psi_\de] \deq \sum_{i=0}^\infty(-1)^iR^i\psi_\de.
\end{align*}
\subsection{}\label{ss:sigmaconjugationisopen}
At this point, we can finally begin to define the test function $f_{\tau,h}$ mentioned in Theorem A. We shall start by defining $\phi_{\tau,h}$, which will end up being a transfer of $f_{\tau,h}$. Let $\tau$ be an element of $W_F$ with positive $v(\tau)$, write $r=v(\tau)$, and let $h$ be a function in $C^\infty_c(\GL_n(\cO))$. We write $\phi_{\tau,h}:\GL_n(F_r)\rar\bC$ for the function
\begin{align*}
\de\mapsto
\begin{cases}
\tr(\tau\times h|[R\psi_\de]) & \mbox{if }\de\mbox{ is in }\GL_n(\cO_r)\diag(\vpi,1,\dotsc,1)\GL_n(\cO_r),\\
0 & \mbox{otherwise,}
\end{cases}
\end{align*}
where we have identified $\ov\bQ_\ell$ with $\bC$.
\begin{lem*}
The map $\GL_n(\br{F})\rar\GL_n(\br{F})$ given by $g\mapsto g^{-1}\de\sg(g)$ is open, where $\sg$ denotes the lift of $q$-Frobenius.
\end{lem*}
\begin{proof}
This follows from reducing to the Lie algebra situation, using the Dieudonn\'e--Manin classification to casework on $\de$, and applying the nonarchimedean Banach open mapping theorem, c.f. \cite[Lemma 4.4]{Sch13c}.
\end{proof}
Lemma \ref{ss:sigmaconjugationisopen} implies that the subsets $B_{n,k}$ of $\GL_n(\cO_r)\diag(\vpi,1,\dotsc,1)\GL_n(\cO_r)$ are open. Because
\begin{align*}
\GL_n(\cO_r)\diag(\vpi,1,\dotsc,1)\GL_n(\cO_r) = \coprod_{k=1}^nB_{n,k},
\end{align*}
we see that the subsets $B_{n,k}$ are also closed.
One use Lemma \ref{ss:sigmaconjugationisopen} and proceed as in \cite[Proposition 4.3]{Sch13c} to show that $\phi_{\tau,h}$ is locally constant. As $\phi_{\tau,h}$ is supported on the compact subset $\GL_n(\cO_r)\diag(\vpi,1,\dotsc,1)\GL_n(\cO_r)$ by definition, we see that $\phi_{\tau,h}$ is in $C^\infty_c(\GL_n(\cO_r))$.
\subsection{}\label{ss:ftauhdefinition}
Let $f_{\tau,h}$ be a transfer of $\phi_{\tau,h}$, i.e. a function in $C^\infty_c(\GL_n(F))$ such that $f_{\tau,h}$ and $\phi_{\tau,h}$ have matching twisted orbital integrals \cite[(I.2.5, prop.)]{HL11}. We remark that while $f_{\tau,h}$ is not uniquely determined as a function in $C^\infty_c(\GL_n(F))$, its orbital integrals are well-defined by the fact that they equal the corresponding twisted orbital integrals of $\phi_{\tau,h}$. Hence the Weyl integration formula \cite[(II.2.10, formula (1)]{HL11} implies that the trace of $f_{\tau,h}$ on admissible representations of $\GL_n(F)$ is also well-defined.
Since we have now defined $f_{\tau,h}$, the statement of Theorem A makes sense.
\section{The first inductive lemma: trace identities and parabolic induction}\label{s:firstinductivelemma}
In this section, we begin by recalling the \emph{Bernstein--Zelevinsky classification} of irreducible smooth representations of $\GL_n(F)$, which we use to prove a lemma on checking equalities of traces. Then, we use Schneider--Zink's theory of \emph{tempered types} for $\GL_n(F)$, which is a version of Bushnell--Kutzko types for tempered representations, to construct test functions that pick out certain representations.
With these preliminaries, we proceed to the main result of this section: a lemma which allows us to prove Theorem A by inducting on $n$, provided that we verify certain conditions. The proof of our lemma uses theory of the \emph{Bernstein center}, which describes the center of the category of smooth representations of $\GL_n(F)$. In subsequent sections, we will focus on verifying the necessary conditions for our inductive lemma.
\subsection{}\label{ss:essentiallyl2}
First, we recall Bernstein--Zelevinsky's description of irreducible essentially $L^2$ representations of $\GL_n(F)$. Let $\De$ be a segment
\begin{align*}
\De = \{\pi_0[\textstyle\frac{1-m}2], \pi_0[\frac{3-m}2],\dotsc,\pi_0[\frac{m-1}2]\}
\end{align*}
as in \cite[3.1]{Zel80}, where $m$ is a positive divisor of $n$, $\pi_0$ is an irreducible cuspidal representation of $\GL_{n/m}(F)$, and $[s]$ denotes twisting by the unramified character $\abs\det^s$. Consider the normalized parabolic induction
\begin{align*}
\nInd_{P(F)}^{\GL_n(F)}(\pi_0[\textstyle\frac{1-m}2]\otimes\pi_0[\frac{3-m}2]\otimes\dotsb\otimes\pi_0[\frac{m-1}2]),
\end{align*}
where $P$ is the standard parabolic subgroup of $\GL_n$ with block sizes $(m,\dotsc,m)$. This representation of $\GL_n(F)$ has a unique irreducible quotient \cite[9.1]{Zel80}, which we denote by $Q(\De)$.
Recall that the irreducible essentially $L^2$ representations of $\GL_n(F)$ are those isomorphic to $Q(\De)$ \cite[9.3]{Zel80}. We see that $\De$ is the cuspidal support of $Q(\De)$, and note that $Q(\De)$ has unitary central character if and only if $\pi_0$ does.
\subsection{}\label{ss:bernsteinzelevinsky}
Next, we proceed to arbitrary irreducible smooth representations of $\GL_n(F)$. Let $\{\De_1,\dotsc,\De_t\}$ be a collection of segments such that $\De_i$ does not precede $\De_j$ as in \cite[4.1]{Zel80} for $i<j$. Each $Q(\De_i)$ is an irreducible essentially $L^2$ representation of $\GL_{n_i}(F)$, and we may form the normalized parabolic induction
\begin{align*}
\nInd_{P(F)}^{\GL_n(F)}\left(Q(\De_1)\otimes\dotsb\otimes Q(\De_t)\right),
\end{align*}
where $n=n_1+\dotsb+n_t$, and $P$ is the standard parabolic subgroup of $\GL_n$ with block sizes $(n_1,\dotsc,n_t)$. When none of the $\De_i$ are linked, this representation of $\GL_n(F)$ is irreducible \cite[9.7.(a)]{Zel80}. For general $\De_i$, it has a unique irreducible quotient, which we denote by $Q(\De_1,\dotsc,\De_t)$. Furthermore, the induced representation above is irreducible precisely when $Q(\De_1,\dotsc,\De_t)$ is a generic representation of $\GL_n(F)$ \cite[9.7.(b)]{Zel80}.
The isomorphism class of $Q(\De_1,\dotsc,\De_t)$ does not depend on the ordering of the $\{\De_1,\dotsc,\De_t\}$, as long as it still satisfies the condition that $\De_i$ does not precede $\De_j$ for $i<j$. Recall that every irreducible smooth representation of $\GL_n(F)$ is isomorphic to $Q(\De_1,\dotsc,\De_t)$ for a unique choice of $\{\De_1,\dotsc,\De_t\}$ \cite[9.7.(b)]{Zel80}, up to reordering. We see that the multiset $\De_1\coprod\dotsb\coprod\De_t$ is the cuspidal support of $Q(\De_1,\dotsc,\De_t)$. Furthermore, recall that $Q(\De_1,\dotsc,\De_t)$ is tempered if and only if every $Q(\De_i)$ is $L^2$ \cite{Jac77}.
\subsection{}\label{ss:generalizedsteinbergspeh}
The following terminology generalizes the Steinberg and trivial representations, respectively.
\begin{defn*}
Let $t$ be a positive divisor of $n$, and let $\pi_0$ be an irreducible cuspidal representation of $\GL_{n/t}(F)$. We write $\St_t(\pi_0)$ for the representation
\begin{align*}
\St_t(\pi_0)\deq Q\left(\{\pi_0[\textstyle\frac{1-t}2], \pi_0[\frac{3-t}2],\dotsc,\pi_0[\frac{t-1}2]\}\right),
\end{align*}
and we write $\Sp_t(\pi_0)$ for the representation
\begin{align*}
\Sp_t(\pi_0) \deq Q\left(\{\pi_0[\textstyle\frac{t-1}2]\},\{\pi_0[\frac{t-3}2]\},\dotsc,\{\pi_0[\frac{1-t}2]\}\right).
\end{align*}
We say that a representation of $\GL_n(F)$ is a \emph{Steinberg module} if it is isomorphic to some $\Sp_t(\pi_0)$ for some $\pi_0$ with unitary central character, and we say it is a \emph{Speh module}\footnote{What we call a Steinberg module is often called a generalized Steinberg representation. However, what we call a Speh module is actually less general than the usual definition of Speh representations.} if it is isomorphic to $\Sp_t(\pi_0)$ for some $\pi_0$ with unitary central character. Note that \ref{ss:essentiallyl2} indicates that being a Steinberg module means the same thing as being irreducible $L^2$.
\end{defn*}
We recover the usual Steinberg and trivial representations by taking $t=n$ and letting $\pi_0$ be the trivial representation.
\subsection{}\label{ss:kazhdanvariant}
We now turn to the following lemma on checking equalities of traces.
\begin{lem*}
Let $f$ be a function in $C^\infty_c(\GL_n(F))$, and suppose that $\tr(f|\pi)=0$ whenever $\pi$ is an irreducible smooth representation of $\GL_n(F)$ that is
\begin{enumerate}[(a)]
\item tempered but not $L^2$,
\item a Speh module.
\end{enumerate}
Then $\tr(f|\pi)=0$ for all irreducible smooth representations $\pi$ of $\GL_n(F)$.
\end{lem*}
\begin{proof}
Kazhdan's density theorem \cite[Theorem 0]{Kaz86}\footnote{The proof given here is stated for $p$-adic fields, but it only uses the Langlands classification and hence carries over to any nonarchimedean local field.} indicates that it suffices to check $\tr(f|\pi)=0$ for all irreducible tempered $\pi$, and (a) above further reduces it to checking $\tr(f|\pi)=0$ whenever $\pi$ is irreducible $L^2$. Thus we can write $\pi$ as $\St_t(\pi_0)$, where $t$ is some positive divisor of $n$, we set $d\deq n/t$, and $\pi_0$ is an irreducible cuspidal representation of $\GL_d(F)$ with unitary central character.
If $t=1$, then $\pi=\pi_0=\Sp_1(\pi_0)$, so (b) shows that $\tr(f|\pi)=0$. Next, suppose that $t\geq2$. A routine calculation \cite[Lemma I.3.2]{HT01} using the graph-theoretic description of the Jordan--H\"older factors of
\begin{align*}
\nInd_{P(F)}^{\GL_n(F)}(\pi_0[\textstyle\frac{1-t}2]\otimes\pi_0[\frac{3-t}2]\otimes\dotsb\otimes\pi_0[\frac{t-1}2]),
\end{align*}
where $P$ is the standard parabolic subgroup of $\GL_n$ with block sizes $(d,\dotsb,d)$, yields the following equality of virtual smooth representations of $\GL_n(F)$:
\begin{align*}
\Sp_t(\pi_0)+(-1)^t\pi = \sum_{j=1}^{t-1}(-1)^{j-1}\nInd_{P_j(L)}^{\GL_n(L)}\left(\Sp_{t-j}(\pi_0[\textstyle\frac{t-1+j}2])\otimes \St_j(\pi_0[\frac{j-1}2])\right),
\end{align*}
where $P_j$ is the standard parabolic subgroup of $\GL_n$ with block sizes $(d(t-j),dj)$. Note that $\Sp_t(\pi_0)$ is a Speh module. Therefore once we prove the following lemma, taking traces and using (b) yields the desired result.
\end{proof}
\begin{lem}\label{lem:tempguys}
Let $f$ be a function in $C^\infty_c(\GL_n(F))$, and suppose that $\tr(f|\pi)=0$ whenever $\pi$ is irreducible tempered but not $L^2$. Then for all partitions $n=n_1+n_2$ and irreducible smooth representations $\pi_i$ of $\GL_{n_i}(F)$, we have
\begin{align*}
\tr\left(f|\nInd_{P(F)}^{\GL_n(F)}(\pi_1\otimes\pi_2)\right)=0,
\end{align*}
where $P$ is the standard parabolic subgroup of $\GL_n$ with block sizes $(n_1,n_2)$.
\end{lem}
For any function $f$ in $C_c^\infty(\GL_n(F))$, we write $f^{P,\GL_n(\cO)}$ for its normalized $\GL_n(\cO)$-invariant constant term along a parabolic subgroup $P$ as in \cite[p.~80]{Lem16}. It is a function in $C_c^\infty(M(F))$, where $M$ is the standard Levi subgroup of $P$.
\begin{proof}[Proof of Lemma \ref{lem:tempguys}]
Van Dijk's formula \cite[Theorem 5.9]{Lem16} implies that
\begin{align*}
\tr\left(f|\nInd_{P(F)}^{\GL_n(F)}(\pi_1\otimes\pi_2)\right) = \tr(f^{P,\GL_n(\cO)}|\pi_1\otimes\pi_2).
\end{align*}
Recall from \ref{ss:bernsteinzelevinsky} that, if $\pi_1$ and $\pi_2$ are tempered, then
\begin{align*}
\nInd_{P(F)}^{\GL_n(F)}(\pi_1\otimes\pi_2)
\end{align*}
is irreducible tempered but not $L^2$. In this situation, the left-hand side of the above equation vanishes by assumption. As $\pi_1$ and $\pi_2$ run over tempered representations of $\GL_{n_1}(F)$ and $\GL_{n_2}(F)$, respectively, $\pi_1\otimes\pi_2$ runs over irreducible tempered representations of $M=\GL_{n_1}(F)\times\GL_{n_2}(F)$. Thus applying Kazhdan's density theorem \cite[Theorem 0]{Kaz86} to $f^{P,\GL_n(\cO)}$ indicates that the right-hand and hence left-hand side of the above equation vanishes for all $\pi_1$ and $\pi_2$, as desired.
\end{proof}
\subsection{}\label{ss:schneiderzinktestfunctions}
By using Schneider--Zink's tempered types, we can construct the following test functions that pick out Speh modules.
\begin{lem*}
Let $\pi=\Sp_t(\pi_0)$ be a Speh module, where $t$ is a positive divisor of $n$, and $\pi_0$ is an irreducible cuspidal representation of $\GL_{n/t}(F)$ with unitary central character. Then there exists a function $h$ in $C^\infty_c(\GL_n(\cO))$ such that
\begin{enumerate}[(i)]
\item $\tr(h|\pi)=1$,
\item if $\pi'$ is an irreducible tempered representation of $\GL_n(F)$ that is not isomorphic to $Q\left(\{\pi_0[s_1]\},\dotsc,\{\pi_0[s_t]\}\right)$ for some purely imaginary $s_1,\dotsc,s_t$, then $\tr(h|\pi')=0$.
\end{enumerate}
\end{lem*}
\begin{proof}
We reduce this to tempered type theory as follows. Suppose we could find an irreducible smooth representation $\la$ of $\GL_n(\cO)$ such that
\begin{enumerate}[(i)]
\item $\pi$ contains $\la$ with multiplicity $1$,
\item if $\pi'$ is an irreducible tempered representation of $\GL_n(F)$ that is not isomorphic to $Q\left(\{\pi_0[s_1]\},\dotsc,\{\pi_0[s_t]\}\right)$ for some purely imaginary $s_1,\dotsc,s_t$, then $\pi'$ does not contain $\la$.
\end{enumerate}
Then Schur orthogonality shows that the function $g\mapsto\ov{\tr(g|\la)}$ yields the desired $h$. As for finding such a $\la$, we use \cite{SZ99}. More precisely, let $\la$ be the representation $\sg_\cP(\la)$ as in \cite[p.~30]{SZ99} for $\cP=t\cdot\De(\pi_0,1)$, where we adopt the notation of \cite{SZ99}. Note that $\pi$ lies in $\im Q_\cP$, so (i) follows from \cite[Prop. 11, i.]{SZ99}. Furthermore, we see that $\cP$ is a maximal partition-valued function on $\cC$, so (ii) follows from \cite[Prop. 11, iii.]{SZ99}.
\end{proof}
\subsection{}\label{ss:firstinductivelemma}
Finally, we can introduce the main result of this section: our inductive lemma.
\begin{lemfirst}
Assume that the following conditions hold for all admissible representations $\pi$ of $\GL_n(F)$:
\begin{enumerate}[(a)]
\item Theorem A is true for all $n'<n$,
\item if $\pi$ is isomorphic to the normalized parabolic induction
\begin{align*}
\nInd^{\GL_n(F)}_{P(F)}(\pi_1\otimes\dotsb\otimes\pi_t),
\end{align*}
where $t\geq2$, the $\pi_i$ are irreducible smooth representations of $\GL_{n_i}(F)$, we have $n=n_1+\dotsb+n_t$, and $P$ is the standard parabolic subgroup of $\GL_n$ with block sizes $(n_1,\dotsc,n_t)$, then we have an equality of traces
\begin{align*}
\tr(f_{\tau,h}|\pi) = \tr\left(\tau|\rho(\pi_1)(\textstyle\frac{n-n_1}2)\oplus\dotsb\oplus\rho(\pi_t)(\frac{n-n_t}2)\right)\tr(h|\pi)
\end{align*}
for all $\tau$ in $W_F$ with $v(\tau)>0$ and $h$ in $C^\infty_c(\GL_n(\cO))$,
\item if $\pi$ is a Speh module or irreducible essentially $L^2$, there exists an $n$-dimensional $\bQ_{\geq0}$-virtual continuous representation $\rho(\pi)$ of $W_F$ satisfying the trace condition
\begin{align*}
\tr(f_{\tau,h}|\pi) = \tr\left(\tau|\rho(\pi)\right)\tr(h|\pi)
\end{align*}
for all $\tau$ in $W_F$ with $v(\tau)>0$ and $h$ in $C^\infty_c(\GL_n(\cO))$,
\item if $\pi$ is irreducible cuspidal, then the representation $\rho(\pi)$ from condition (c) is actually a $\bZ$-virtual continuous representation of $W_F$.
\end{enumerate}
Then Theorem A is true for $n$.
\end{lemfirst}
Since semisimple representations of $W_F$ are determined by their traces, any $\rho(\pi)$ satisfying the equality of traces in Theorem A.(i) is unique up to isomorphism. In particular, the $\rho(\pi)$ whose existence is posited by Theorem A.(i) is isomorphic to the $\rho(\pi)$ whose existence is provided by condition (c).
We now proceed to prove the first inductive lemma. To this end, for the remainder of this section we shall assume that conditions (a)--(d) hold. Later, we shall verify these conditions in \S\ref{s:parabolicinduction}, \S\ref{s:lubintatetower}, and \S\ref{s:localglobal}.
\subsection{}\label{ss:cuspidalrho}
We begin with the following observation. Suppose that $\pi$ is irreducible cuspidal. Then condition (c) provides an $n$-dimensional $\bQ$-virtual continuous representation $\rho(\pi)$ of $W_F$ such that, when expanded in terms of the $\bQ$-basis of irreducible continuous representations of $W_F$, every coefficient is non-negative. Condition (d) implies that these coefficients are integers, so $\rho(\pi)$ corresponds to an actual $n$-dimensional semisimple continuous representation of $W_F$.
\subsection{}\label{ss:bernsteincenter}
Now let $\pi$ be any irreducible smooth representation of $\GL_n(F)$. Before proceeding, we gather a few recollections on the Bernstein center. For any $f$ in the Bernstein center $\wh\cZ$ of $\GL_n(F)$, we write $\tr(f|\pi)$ for the scalar by which $f$ acts on $\pi$. We identify $\wh\cZ$ with the product of the rings of regular functions on the Bernstein components of $\GL_n(F)$ \cite[2.10]{Ber84}.
Write $\pi_1,\dotsc,\pi_t$ for the cuspidal support of $\pi$, where the $\pi_i$ are irreducible cuspidal representations of $\GL_{n_i}(F)$ such that $n=n_1+\dotsb+n_t$. Write $P$ for the standard parabolic subgroup of $\GL_n$ with block sizes $(n_1,\dotsc,n_t)$. For any function $h$ in $C^\infty_c(\GL_n(F))$, van Dijk's formula \cite[Theorem 5.9]{Lem16} implies that the assignment
\begin{align*}
\pi\mapsto\tr\left(h|\nInd_{P(F)}^{\GL_n(F)}(\pi_1\otimes\dotsb\otimes\pi_t)\right)
\end{align*}
yields an element of $\wh\cZ$, which we shall denote using $\wh{h}$. When $\nInd_{P(F)}^{\GL_n(F)}(\pi_1\otimes\dotsb\otimes\pi_t)$ is irreducible and hence isomorphic to $\pi$, we have $\tr(\wh{h}|\pi)=\tr(h|\pi)$. However, we stress that this is false for general $\pi$.
\subsection{}\label{ss:functiondesire}
We construct an element $f_\tau$ of $\wh\cZ$ as follows. Write $r(\pi)$ for the representation
\begin{align*}
r(\pi)\deq\rho(\pi_1)(\textstyle\frac{1-n_1}2)\oplus\dotsb\oplus\rho(\pi_t)(\frac{1-n_t}2),
\end{align*}
which is well-defined for $t=1$ by \ref{ss:cuspidalrho} and for $t\geq2$ by Theorem A.(i) for the $\pi_i$. Note that $r(\pi)$ is semisimple. We shall see in Proposition \ref{ss:convolutiontraceequality} that $r(\pi)$ is always isomorphic to $\rec(\pi)$, where the latter is defined as in Theorem A.
\begin{prop*}
The function $f_\tau$ whose action on $\pi$ is given by $\tr\left(\tau|\sg(\pi)(\frac{n-1}2)\right)$ is an element of $\wh\cZ$.
\end{prop*}
This fulfills the desire to have a function-like object (in this case, an element of the Bernstein center, which is a distribution) associated with $\tau$ whose trace equals the action of $\tau$ on $r(\pi)$ and hence on $\rec(\pi)$.
\begin{proof}
It suffices to prove that $\pi\mapsto\tr\left(\tau|r(\pi)(\frac{n-1}2)\right)$ is a regular function on every Bernstein component, so we may focus on one particular $\pi$. First, use Schur orthogonality to obtain a function $h_1$ in $C^\infty_c(\GL_n(\cO))$ that satisfies
\begin{align*}
\tr\left(h_1|\nInd_{P(F)}^{\GL_n(F)}(\pi_1\otimes\dotsb\otimes\pi_t)\right)=1,
\end{align*}
where $P$ is the standard parabolic subgroup of $\GL_n$ with block sizes $(n_1,\dotsc,n_t)$. Now any other irreducible representation $\pi'$ in the Bernstein component of $\pi$ is a subquotient of
\begin{align*}
\nInd_{P(F)}^{\GL_n(F)}(\pi_1[s_1]\otimes\dotsb\otimes\pi_t[s_t]),
\end{align*}
where the $s_i$ are complex numbers. The Iwasawa decomposition implies that $h_1^{P,\GL_n(\cO)}$ is supported on $M(\cO)$, where $M$ is the standard Levi subgroup of $P$, so van Dijk's formula \cite[Theorem 5.9]{Lem16} shows that
\begin{align*}
\tr\left(h_1|\nInd_{P(F)}^{\GL_n(F)}(\pi_1[s_1]\otimes\dotsb\otimes\pi_t[s_t])\right)= \tr\left(h_1|\nInd_{P(F)}^{\GL_n(F)}(\pi_1\otimes\dotsb\otimes\pi_t)\right)=1.
\end{align*}
Because $\pi_1[s_1],\dotsc,\pi_t[s_t]$ is the cuspidal support of $\pi'$, we obtain
\begin{align*}
\tr\left(\tau|r(\pi')(\textstyle\frac{n-1}2)\right) &= \tr\left(\tau|\rho(\pi_1[s_1])(\textstyle\frac{n-n_1}2)\oplus\dotsb\oplus\rho(\pi_t[s_t])(\frac{n-n_t}2)\right) \\
&= \tr\left(\tau|\rho(\pi_1[s_1])(\textstyle\frac{n-n_1}2)\oplus\dotsb\oplus\rho(\pi_t[s_t])(\frac{n-n_t}2)\right)\tr\left(h_1|\nInd_{P(F)}^{\GL_n(F)}(\pi_1[s_1]\otimes\dotsb\otimes\pi_t[s_t])\right).
\end{align*}
This equals
\begin{align*}
\tr\left(f_{\tau,h_1}|\nInd_{P(F)}^{\GL_n(F)}(\pi_1[s_1]\otimes\dotsb\otimes\pi_t[s_t])\right)
\end{align*}
for $t=1$ by condition (c) and for $t\geq2$ by condition (b). Altogether, we see that $f_\tau$ acts on the Bernstein component of $\pi$ via $\wh{f_{\tau,h_1}}$, so it is a regular function.
\end{proof}
\subsection{}\label{ss:convolutiontraceequality}
We can relate the distribution $f_\tau$ to the test function $f_{\tau,h}$ as follows. Recall that the convolution product $f_\tau*h$ is a function in $C^\infty_c(\GL_n(F))$.
\begin{prop*}
We have $\tr(f_{\tau,h}|\pi) = \tr(f_\tau*h|\pi)$.
\end{prop*}
Note that, since $f_\tau$ lies in the Bernstein center of $\GL_n(F)$, we have
\begin{align*}
\tr(f_\tau*h|\pi) = \tr(f_\tau|\pi)\tr(h|\pi) = \tr\left(\tau|r(\pi)(\textstyle\frac{n-1}2)\right)\tr(h|\pi).
\end{align*}
By taking $\rho(\pi)=r(\pi)(\textstyle\frac{n-1}2)$, we see that this proposition gives the existence part of Theorem A.(i) for $n$.
\begin{proof}[Proof of Proposition \ref{ss:convolutiontraceequality}]
By Lemma \ref{ss:kazhdanvariant}, it suffices to consider $\pi$ that are either Speh modules or irreducible tempered but not $L^2$. In the latter case, the descriptions given in \ref{ss:bernsteinzelevinsky} imply that $\pi$ is isomorphic to a representation of the form considered in condition (b). Therefore condition (b) yields the desired result in this case.
Turning to the case where $\pi$ is a Speh module, suppose that $\pi$ is isomorphic to $\Sp_t(\pi_0)$ for some positive divisor $t$ of $n$ and irreducible cuspidal representation $\pi_0$ of $\GL_{n/t}(F)$ with unitary central character. If $t=1$, then $\pi=\pi_0$ is cuspidal, and condition (c) gives the desired equality. Therefore suppose that $t\geq2$, and let $h_2$ be a function in $C^\infty_c(\GL_n(\cO))$ as in Lemma \ref{ss:schneiderzinktestfunctions}. We start by proving the case where $h=h_2$.
\begin{lem}\label{lem:nestedkazhdan}
For all irreducible smooth representations $\pi'$ of $\GL_n(F)$, we have $\tr\left(f_{\tau,h_2}|\pi'\right) = \tr\left(f_\tau*h_2|\pi'\right)$.
\end{lem}
\begin{proof}
Kazhdan's density theorem \cite[Theorem 0]{Kaz86} indicates that it suffices to check this for irreducible tempered $\pi'$. When $\pi'$ is not $L^2$, this identity is a case of Proposition \ref{ss:convolutiontraceequality} we proved above, so we now tackle the case when $\pi'$ is $L^2$. Since $t\geq2$, the descriptions given in \ref{ss:essentiallyl2} and \ref{ss:bernsteinzelevinsky} indicate that $\pi'$ cannot be of the form $Q\left(\{\pi_0[s_1]\},\dotsc,\{\pi_0[s_t]\}\right)$ for purely imaginary $s_1,\dotsc,s_t$. Thus condition (c) and Lemma \ref{ss:schneiderzinktestfunctions}.(ii) imply that
\begin{align*}
\tr\left(f_{\tau,h_2}|\pi'\right) = \tr\left(\tau|\rho(\pi')\right)\tr(h_2|\pi') = 0 = \tr\left(\tau|r(\pi)(\textstyle\frac{n-1}2)\right)\tr(h_2|\pi) = \tr\left(f_\tau*h_2|\pi'\right),
\end{align*}
which concludes the proof of Lemma \ref{lem:nestedkazhdan}.
\end{proof}
Return to the proof of Proposition \ref{ss:convolutiontraceequality}. By plugging $\pi'=\pi$ into Lemma \ref{lem:nestedkazhdan} and using condition (c) and Lemma \ref{ss:schneiderzinktestfunctions}.(i), we see that
\begin{align*}
\tr\left(\tau|\rho(\pi)\right) = \tr\left(f_{\tau,h_2}|\pi\right) = \tr\left(f_\tau*h_2|\pi\right) = \tr\left(\tau|r(\pi)(\textstyle\frac{n-1}2)\right).
\end{align*}
Because semisimple representations of $W_F$ are determined by their traces, this indicates that $\rho(\pi)$ is isomorphic to $r(\pi)(\textstyle\frac{n-1}2)$. In particular, applying condition (c) again verifies the desired equality when $\pi$ is a Speh module.
\end{proof}
We conclude this section by finishing the proof of Lemma \ref{ss:firstinductivelemma}, that is, by proving that Theorem A holds for $n$.
\begin{proof}[Proof of Lemma \ref{ss:firstinductivelemma}]
We have already noted that Proposition \ref{ss:convolutiontraceequality} gives the existence part of Theorem A.(i) for $n$. Since semisimple representations of $W_F$ are determined by their traces, any such $\rho(\pi)$ satisfying the equality of traces in Theorem A.(i) is unique. As for Theorem A.(ii), it immediately follows from the associativity of normalized parabolic induction as well as the definition of $\sg(\pi)$ in terms of the cuspidal support of $\pi$.
\end{proof}
The next several sections will focus on verifying that conditions (b)--(d) hold, sometimes while assuming the inductive condition (a). More precisely, we shall prove condition (b) in Proposition \ref{ss:conditionb} under the assumption that the inductive condition (a) holds, we will prove condition (d) in Proposition \ref{ss:conditiond} while black-boxing condition (c), and we shall prove condition (c) itself in Proposition \ref{s:conditionc}. We will also prove the $n=1$ case of Theorem A in Proposition \ref{ss:theoremabasecase}, which serves as the base case for applying Lemma \ref{ss:firstinductivelemma}.
\section{Geometry of parabolic induction}\label{s:parabolicinduction}
Our goal in this section is to prove that condition (b) in Lemma \ref{ss:firstinductivelemma} holds, under the assumption that the inductive condition (a) holds. To do this, we start by using \'etale local shtukas to prove the existence of transfers for $\GL_n(\cO)$-conjugation-invariant functions in $C^\infty_c(\GL_n(\cO))$. We then use work from this proof, along with our description of the deformation spaces $\br\fX_{\de,m}$ from \S\ref{s:deformationspaces}, to prove an identity relating $\tr(f_{\tau,h}|\pi)$ with the traces of $\phi_{\tau,h'}$ on certain Jacquet restrictions of $\pi$ (where $h'$ now ranges over functions in $C^\infty_c(M(\cO))$ for certain Levi blocks $M$ in $\GL_n$). The proof of this identity uses \emph{Casselman's theorem} on characters of Jacquet modules. We finish the proof that condition (b) holds, under the assumption that the inductive condition (a) holds, by using the Bernstein--Zelevinsky \emph{induction-restriction formula} to relate this aforementioned trace identity to a reformulated version of condition (b).
\subsection{}\label{ss:normbijection}
We begin by using two descriptions of isomorphism classes of \'etale local shtukas over $\Spec\ka_r$ to prove the following bijection. In the end, this amounts to an application of completed unramified descent.
\begin{lem*}
The norm map $\de\mapsto\de\sg(\de)\dotsm\sg^{r-1}(\de)$ induces a bijection
\begin{align*}
\N:\{\GL_n(\cO_r)\mbox{-}\sg\mbox{-conjugacy classes in }\GL_n(\cO_r)\}\rar^\sim\{\GL_n(\cO)\mbox{-conjugacy classes in }\GL_n(\cO)\}.
\end{align*}
\end{lem*}
\begin{proof}
We shall construct bijections between both the left-hand as well as right-hand sides and the set
\begin{align*}
\{\mbox{isomorphism classes of \'etale local shtukas over }\Spec\ka_r\mbox{ of rank }n\}
\end{align*}
such that the composed bijection is equal to $\N$. On the left-hand side, \ref{ss:effectiveminusculelocalshtukaconjugacy} yields the desired bijection.
We now turn to the right-hand side. Let $\sM$ be an \'etale local shtuka over $\Spec\ka_r$. Completed unramified descent implies that the isomorphism class of $\sM$ is determined by the isomorphism class of $\br\sM\deq\sM_{\ov\ka}$ along with a descent isomorphism $f:\sg^{-r,*}\br\sM\rar^\sim\br\sM$ corresponding to the action of the geometric $q^r$-Frobenius map $\sg^{-r}$. As $\br\sM$ must be isomorphic to $(\ov\ka\llb\vpi^n,\sg^{\oplus n})$ by \ref{ss:localshtukaalgebraicclosure}, any such isomorphism identifies $\Aut(\br\sM)$ with $\GL_n(\cO)$. Write $\br\sF:\sg^*\br\sM\rar\br\sM$ for the Frobenius of $\br\sM$, which is an isomorphism because $\br\sM$ is \'etale, and write $\ga$ for the automorphism
\begin{align*}
\ga\deq\br\sF\circ\dotsb\circ\sg^{r-1,*}\br\sF\circ\sg^{r,*}f
\end{align*}
of $\br\sM$, viewed as an element of $\GL_n(\cO)$. Since any other choice of isomorphism $\br\sM\rar^\sim (\ov\ka\llb\vpi^n,\sg^{\oplus n})$ preserves the $\GL_n(\cO)$-conjugacy class of $\ga$, this yields a bijection $\ga\leftrightarrow\sM$ between $\GL_n(\cO)$-conjugacy classes in $\GL_n(\cO)$ and isomorphism classes of \'etale local shtukas over $\Spec\ka_r$ of rank $n$.
Recall that $\br\sF$ corresponds to $\de\circ\sg^{\oplus n}$ under \ref{ss:effectiveminusculelocalshtukaconjugacy}. Since $f$ corresponds to $(\sg^{\oplus n})^{-r}$, we see that $\ga$ corresponds to
\begin{align*}
\underbrace{(\de\circ\sg^{\oplus n})\circ\dotsb\circ(\de\circ\sg^{\oplus n})}_{r\,\text{times}}\circ (\sg^{\oplus n})^{-r} = \left(\de\sg(\de)\dotsm\sg^{r-1}(\de)\right)\circ(\sg^{\oplus n})^r\circ (\sg^{\oplus n})^{-r} = \N\de,
\end{align*}
as desired.
\end{proof}
\subsection{}\label{ss:etalelocalshtukatransfer}
We now construct certain functions with matching twisted orbital integrals, that is, certain transfers of functions. Let $h$ be a $\GL_n(\cO)$-conjugation-invariant function in $C^\infty_c(\GL_n(\cO))$, and form the function $\phi:\GL_n(\cO_r)\rar\bC$ by sending $\de\mapsto h(\N\de)$. One can show that $\phi$ is locally constant \cite[Corollary 2.3]{Sch13b}\footnote{The proof given here is stated for mixed characteristic, but it works verbatim for any nonarchimedean field.}.
\begin{lem*}
The functions $h$ and $\phi$ have matching twisted orbital integrals.
\end{lem*}
\begin{proof}
We need to prove that $\O_{\N\de}(h)=\TO_{\de,\sg}(\phi)$ for all regular $\de$ in $\GL_n(\cO_r)$, where $\O_{\N\de}$ and $\TO_{\de,\sg}$ denote the orbital integral on $\N\de$ and twisted orbital integral on $\de$, respectively, with respect to compatible Haar measures. We can prove this as follows: write $\sM$ for the \'etale local shtuka corresponding to $\de$ under \ref{ss:effectiveminusculelocalshtukaconjugacy}, and write $X$ for the set of isomorphism classes of quasi-isogenies $\be:\sM\dasharrow\sM'$ between \'etale local shtukas. Note that the group $\Ga$ of self-quasi-isogenies $\sM\dasharrow\sM$ has a left action on $X$ given by sending $\be$ to $\be\circ g^{-1}$ for any $g$ in $\Ga$.
One can equate both $\O_{\N\de}(h)$ as well as $\TO_{\de,\sg}(\phi)$ to the sum
\begin{align*}
\sum_{(\sM',\be)}\wt{h}(\sM',\be),
\end{align*}
where $(\sM',\be)$ ranges over elements in $\Ga\bs X$, and $\wt{h}$ is the function sending $(\sM',\be)$ to $h(\de')$, where $\de'$ corresponds to $\sM'$ under \ref{ss:effectiveminusculelocalshtukaconjugacy}. In a line of reasoning similar to the proof of Lemma \ref{ss:normbijection}, we relate $\TO_{\de,\sg}(\phi)$ to the above sum by using \ref{ss:effectiveminusculelocalshtukaconjugacy}, and we relate $\O_{\N\de}(h)$ to the above sum by using completed unramified descent, c.f. \cite[Proposition 4.3]{Sch13}
\end{proof}
\subsection{}\label{ss:deformationtestfunction}
Next, we recast our results from \S\ref{s:deformationspaces} in terms of our test functions. For this, recall that for any $\de$ in
\begin{align*}
\GL_n(\cO_r)\diag(\vpi,1,\dotsc,1)\GL_n(\cO_r),
\end{align*}
we constructed an associated effective minuscule local shtuka $H_\de$ over $\Spec\ka_r$ of dimension $1$. Its connected--\'etale decomposition \cite[Proposition 2.9]{HS16} is $H_{\de^\circ}\oplus H_{\de^\et}$, where $H_{\de^\circ}$ has rank $k$, and $B_{n,k}$ is the set of $\de$ such that $H_\de$ decomposes in this way. By considering $\de$ as an element of $\GL_n(\br\cO)\diag(\vpi,1,\dotsc,1)\GL_n(\br\cO)$, we obtain the pullback $\br{H}_\de$ of $H_\de$ to $\Spec\ov\ka$, and the space $\br\fX_{\de,m}$ parametrizes deformations of $\br{H}_\de$ along with a Drinfeld level-$m$ structure. We defined a virtual representation $[R\psi_\de]$ in terms of the cohomology of $\br\fX_{\de,m}$ as $m$ varies, and we defined $\phi_{\tau,h}(\de)$ as a trace on $[R\psi_\de]$.
\begin{prop*}
We have an equality
\begin{align*}
\phi_{\tau,h}(\de) = \tr\left((\tau\times h)|\Ind_{P_k(\cO)}^{\GL_n(\cO)}\left([R\psi_{\de^\circ}]\otimes C^\infty_c(\GL_{n-k}(\cO))\right)\right),
\end{align*}
where $P_k$ is the standard parabolic subgroup of $\GL_n$ with block sizes $(k,n-k)$, $\GL_{n-k}(\cO)$ acts on $C^\infty_c(\GL_{n-k}(\cO))$ by left inverse multiplication, and $W_{F_r}$ acts on $C^\infty_c(\GL_{n-k}(\cO))$ via the unramified action sending $\tau$ to the action of right multiplication by $\N\de^\et$.
\end{prop*}
\begin{proof}
Let $f_{\sg^{-r}}:\sg^{-r,*}\br{H}_\de\rar^\sim\br{H}_\de$ and $\vp_{\sg^{-r}}:\br\fX_{\de,m}\rar\sg^{-r,*}\br\fX_{\de,m}$ be as in \ref{ss:deformationunramifiedextension}. Using the Dieudonn\'e equivalence \cite[Theorem 8.3]{HS16} to pass to $\vpi$-divisible local Anderson modules, the proof of Lemma \ref{ss:normbijection} shows that the triple $(H',\al',\io'\circ f_{\sg^{-r}})$ is isomorphic to the triple $(H',\N\de^\et\circ\al',\io')$. Therefore $\vp_{\sg^{-r}}$ sends $\br\fX_{\de,\al}$ to $\sg^{-r,^*}\br\fX_{\de,\N\de^\et\circ\al}$. Taking cohomology of both sides in Proposition \ref{ss:deformationalgebraicclosure} and using \cite[Lemma I.5.6]{HT01} give us
\begin{align*}
R\psi^i_{\de,m} = \Ind_{P_k(\cO/\vpi^m)}^{\GL_n(\cO/\vpi^m)}\left(R\psi^i_{\de^\circ,m}\otimes\ov\bQ_\ell[\GL_{n-k}(\cO/\vpi^m)]\right),
\end{align*}
where $W_{F_r}$ acts on $\ov\bQ_\ell[\GL_{n-k}(\cO)]$ via the unramified action sending $\sg^{-r}I_F$ and hence $\tau$ to the action of right multiplication by $N\de^\et$. Taking the direct limit over $m$ and forming the alternating sum over $i$ gives an equality
\begin{align*}
[R\psi_\de] = \Ind_{P_k(\cO)}^{\GL_n(\cO)}\left([R\psi_{\de^\circ}]\otimes C^\infty_c(\GL_{n-k}(\cO))\right)
\end{align*}
of virtual representations, where we have identified $\ov\bQ_\ell$ with $\bC$. Taking the trace of $\tau\times h$ yields the desired result.
\end{proof}
\subsection{}\label{ss:twistedcharacters}
Before proceeding, we gather a few facts about twisted characters and automorphic base change. Recall that the norm map $\de\mapsto\de\sg(\de)\dotsm\sg^{r-1}(\de)$ induces an injection
\begin{align*}
\N:\{\GL_n(F_r)\mbox{-}\sg\mbox{-conjugacy classes in }\GL_n(F_r)\}\inj{}\{\GL_n(F_r)\mbox{-conjugacy classes in }\GL_n(F_r)\}
\end{align*}
\cite[Lemma 4.2]{Lan80}. For any finite length smooth representation $\pi$ of $\GL_n(F)$, its character distribution $\tr(-|\pi)$ is represented by a locally integrable conjugation-invariant function $\Te_\pi:\GL_n(F)\rar\bC$ \cite[Theorem (7.1)]{Lem04}, which we call the \emph{character} of $\pi$. Naturally, taking characters is additive. The function $\Te_{\pi/F_r}^{\sg}:\GL_n(F_r)\rar\bC$ that sends $\de\mapsto\Te_\pi(\N\de)$ is locally integrable \cite[(II.2.8)]{HL11}, and we denote the associated distribution by $\tr\left((-,\sg)|\pi/F_r\right):C^\infty_c(\GL_n(F_r))\rar\bC$.
The twisted Weyl integration formula \cite[(II.2.10, formula (1)]{HL11} implies that, if $f$ and $\phi$ are functions in $C^\infty_c(\GL(F))$ and $C^\infty_c(\GL_n(F_r))$, respectively, such that $f$ and $\phi$ have matching twisted orbital integrals, then we have
\begin{align*}
\tr\left((\phi,\sg)|\pi/F_r\right) = \tr(f|\pi).
\end{align*}
As suggested by our notation, if $\pi$ is generic, then the twisted character of its base change lift $\pi_{F_r}$ to $\GL_n(F_r)$ equals $\Te_{\pi/F_r}^{\sg}$, and we have $\tr((\phi,\sg)|\pi_{F_r})=\tr\left((\phi,\sg)|\pi/F_r\right)$ \cite[(II.2.8)]{HL11}.
\subsection{}\label{ss:unramifiedtwistsofrho}
We use \ref{ss:twistedcharacters} to obtain the following compatibility result for $\rho$ and unramified twists.
\begin{prop*}
Let $\pi$ be an irreducible smooth representation of $\GL_n(F)$, and suppose that Theorem A.(i) holds for $\pi$. Then for any complex number $s$, Theorem A.(i) holds for $\pi[s]$. That is, there exists a unique $n$-dimensional semisimple continuous representation $\rho(\pi[s])$ of $W_F$ satisfying the following property:
\begin{align*}
\mbox{for all }\tau\mbox{ in }W_F\mbox{ with }v(\tau)>0\mbox{ and }h\mbox{ in }C^\infty_c(\GL_n(\cO))\mbox{, we have }\tr\left(f_{\tau,h}|\pi[s]\right) = \tr\left(\tau|\rho(\pi[s])\right)\tr\left(h|\pi[s]\right).
\end{align*}
This $\rho(\pi[s])$ is given by $\rho(\pi[s])=\rho(\pi)(s)$.
\end{prop*}
\begin{proof}
We immediately have $\tr\left(f_{\tau,h}|\pi[s]\right) = \tr\left((\phi_{\tau,h},\sg)|\pi[s]/F_r\right)$ by \ref{ss:twistedcharacters}. Unpacking definitions shows that the right hand side equals
\begin{align*}
\int_{\GL_n(\cO_r)\diag(\vpi,1,\dotsc,1)\GL_n(\cO_r)}\!\dif\de\,\tr\left(\tau\times h|[R\psi_\de]\right)\Te_{\pi[s]}(\N\de).
\end{align*}
Since $\pi[s]$ is the twist of $\pi$ by the character $\abs\det^s$, we see that $\Te_{\pi[s]}(\N\de) = \Te_\pi(\N\de)\abs{\det\N\de}^s$. As $\de$ lies in
\begin{align*}
\GL_n(\cO_r)\diag(\vpi,1,\dotsc,1)\GL_n(\cO_r),
\end{align*}
its determinant has valuation $1$, which makes $\det\N\de$ have valuation $r$. Therefore the above integral becomes
\begin{align*}
\frac1{q^{rs}}\int_{\GL_n(\cO_r)\diag(\vpi,1,\dotsc,1)\GL_n(\cO_r)}\!\dif\de\,\tr\left(\tau\times h|[R\psi_\de]\right)\Te_\pi(\N\de) = \frac1{q^{rs}}\tr\left(\tau|\rho(\pi)\right)\tr\left(h|\pi\right)
\end{align*}
by applying Theorem A.(i) for $\pi$. Because $\tau$ acts through $(s)$ via $1/q^{rs}$, the desired result follows.
\end{proof}
\subsection{}\label{ss:peterweyl}
In order to state the trace identity involving $f_{\tau,h}$ and Jacquet restriction, we first need to recall the Peter--Weyl theorem for profinite groups $G$. One version of it says that we have a canonical isomorphism of smooth representations of $G\times G$
\begin{align*}
C^\infty_c(G)&\rar^\sim\bigoplus_\la\End_\bC(\la)=\bigoplus_\la\la^\vee\otimes\la\\
f&\longmapsto\bigoplus_\la\la(f),
\end{align*}
where $\la$ ranges over isomorphism classes of irreducible smooth representations of $G$, and $G\times G$ acts on $C^\infty_c(G)$ via letting $(g_1,g_2)$ send $f$ to the function $x\mapsto f(g_2^{-1}xg_1)$. This implies, along with Schur orthogonality, that the trace map induces a $\bC$-linear bijection
\begin{align*}
\tr:\{\bC\mbox{-virtual admissible representations of }G\}\rar^\sim\{\mbox{conjugation-invariant distributions on }G\}.
\end{align*}
We shall use this bijection to view conjugation-invariant distributions as $\bC$-virtual admissible representations.
\subsection{}\label{ss:jacquetmoduletraceidentity}
We may now introduce an identity that relates the trace of $f_{\tau,h}$ on $\pi$ with traces on certain Jacquet restrictions of $\pi$. The proof shall critically use Proposition \ref{ss:deformationtestfunction}, which itself is derived from the geometry of $\br\fX_{\de,m}$ as described in \S\ref{s:deformationspaces}.
For all integers $1\leq k \leq n$, write $N_k$ for the unipotent radical of $P_k$. Write $\pi_{N_k}$ for the unnormalized Jacquet restriction of $\pi$, and suppose that we have an equality of virtual representations
\begin{align*}
\pi_{N_k} = \sum_{i=1}^{t_k}\pi_{1,i}^k\otimes\pi_{2,i}^k,
\end{align*}
where $t_k$ is a non-negative integer, and the $\pi_{1,i}^k$ and $\pi_{2,i}^k$ are finite length smooth representations of $\GL_k(F)$ and $\GL_{n-k}(F)$, respectively. Note that we do not require the $\pi_{1,i}^k$ and $\pi_{2,i}^k$ to be irreducible. For any subset $X$, we denote the indicator function on that subset using $\bf1_X$.
\begin{prop*}
We have an equality of traces
\begin{align*}
\tr(f_{\tau,h}|\pi) = \sum_{k=1}^nq^{(n-k)r}\sum_{i=1}^{t_k}\tr\left(h|\Ind_{P_k(\cO)}^{\GL_n(\cO)}\left(\tr((\bf1_{B_k}\cdot\phi_{\tau,-},\sg)|\pi_{1,i}^k/F_r)\otimes\pi_{2,i}^k\right)\right),
\end{align*}
where we view the conjugation-invariant distribution $\tr((\bf1_{B_k}\cdot\phi_{\tau,-},\sg)|\pi_{1,i}^k/F_r)$ as a $\bC$-virtual admissible representation of $\GL_k(\cO)$ via \ref{ss:peterweyl}.
\end{prop*}
In the proof, we take our Haar measures compatibly whenever possible.
\begin{proof}
We immediately have $\tr(f_{\tau,h}|\pi)=\tr\left((\phi_{\tau,h},\sg)|\pi/F_r\right)$. Because $\phi_{\tau,h}$ is supported on
\begin{align*}
\GL_n(\cO_r)\diag(\vpi,1,\dotsc,1)\GL_n(\cO_r) = \coprod_{k=1}^nB_{n,k},
\end{align*}
it suffices to show that
\begin{align*}
\tr\left((\bf1_{B_{n,k}}\cdot\phi_{\tau,h},\sg)|\pi/F_r\right) = q^{(n-k)r}\sum_{i=1}^{t_k}\tr\left(h|\Ind_{P_k(\cO)}^{\GL_n(\cO)}\left(\tr((\bf1_{B_k}\cdot\phi_{\tau,-},\sg)|\pi_{1,i}^k/F_r)\otimes\pi_{2,i}^k\right)\right)
\end{align*}
for all $k$. We begin with left-hand side. Applying the twisted Weyl integration formula \cite[(II.2.10, formula (1)]{HL11} to the function $\bf1_{B_{n,k}}\cdot\phi_{\tau,h}$ in $C^\infty_c(\GL_n(F_r))$ gives
\begin{align*}
\tr\left((\bf1_{B_{n,k}}\cdot\phi_{\tau,h},\sg)|\pi/F_r\right) = \sum_T\frac1{\#W(T,\GL_n)}\int_{T(F_r)^{\sg-1}\bs T(F_r)}\!\dif\de\,\abs{D_{\GL_n}(\N\de)}\TO_{\de,\sg}(\bf1_{B_{n,k}}\cdot\phi_{\tau,h})\Te_{\pi/F_r}^{\sg}(\de),
\end{align*}
where $T$ runs over conjugacy classes of maximal tori of $\GL_n$ over $F$, and $D_{\GL_n}:\GL_n(F)\rar F$ is the regular function defined by the equation
\begin{align*}
\det(t-\ad\ga+1|\fg\fl_n(F))\equiv D_{\GL_n}(\ga)t^n\pmod{t^{n+1}}.
\end{align*}
The decomposition $\de=\de^\circ\oplus\de^\et$, along with the description of $\de^\circ$ and $\de^\et$ in \ref{ss:localshtukaunramifiedextension}, shows that the only terms remaining in the above sum are
\begin{align*}
&\sum_{T_k}\frac1{\#W(T_k,\GL_k)}\sum_{T_{n-k}}\frac1{\#W(T_{n-k},\GL_{n-k})}\\
&\cdot\int_{(T_k(F_r)^{\sg-1}\bs T_k(F_r)_1)\times (T_{n-k}(F_r)^{\sg-1}\bs T_{n-k}(\cO_r))}\!\dif\de\,\abs{D_{\GL_n}(\N\de)}\TO_{\de,\sg}(\bf1_{B_{n,k}}\cdot\phi_{\tau,h})\Te_\pi(\N\de),
\end{align*}
where $T_k$ runs over conjugacy classes of anisotropic mod center maximal tori of $\GL_k$, $T_{n-k}$ runs over conjugacy classes of maximal tori of $\GL_{n-k}$, and we have $T=T_k\times T_{n-k}$.
With visible product decompositions beginning to form, let us make the change of variables $\de=(\de_k,\de_{n-k})$. Our goal is to break up the integrand in terms of $\GL_k$ and $\GL_{n-k}$. By using block matrices to expand $\abs{D_{\GL_n}(\N\de)}$, we see that
\begin{align*}
\abs{D_{\GL_n}(\N\de)} = q^{(n-k)r}\abs{D_{\GL_k}(\N\de_k)}\cdot\abs{D_{\GL_{n-k}}(\N\de_{n-k})}.
\end{align*}
To obtain a similar decomposition of $\TO_{\de,\sg}(\bf1_{B_{b,k}}\cdot\phi_{\tau,h})$, we shall use the following result.
\begin{lem}
We have an equality of twisted orbital integrals
\begin{align*}
\TO_{\de,\sg}(\bf1_{B_{n,k}}\cdot\phi_{\tau,h}) = \TO_{(\de_k,\de_{n-k}),\sg}((\bf1_{B_{n,k}}\cdot\phi_{\tau,h})|_{M_k(F_r)}),
\end{align*}
where $M_k$ is the standard Levi subgroup of $P_k$, $(\bf1_{B_{n,k}}\cdot\phi_{\tau,h})|_{M_k(F_r)}$ is the restriction of $\bf1_{B_{n,k}}\cdot\phi_{\tau,h}$ to $M_k(F_r)$, and the right-hand side is the twisted orbital integral of $(\bf1_{B_{n,k}}\cdot\phi_{\tau,h})|_{M_k(F_r)}$ on $(\de_k,\de_{n-k})$.
\end{lem}
\begin{proof}
Using Nakayama's lemma over $\cO$ to reduce to the residue field, we can explicitly calculate the normalized $\GL_n(\cO_r)$-$\sg$-invariant constant term $(\bf1_{B_{n,k}}\cdot\phi_{\tau,h})^{P_k,\sg,\GL_n(\cO_r)}$ of $\bf1_{B_{n,k}}\cdot\phi_{\tau,h}$ along $P_k$ to be
\begin{align*}
(\bf1_{B_{n,k}}\cdot\phi_{\tau,h})^{P_k,\sg,\GL_n(\cO_r)} = q^{\frac12(n-k)r}(\bf1_{B_{n,k}}\cdot\phi_{\tau,h})|_{M_k(F_r)}
\end{align*}
\cite[Lemma 6.6]{Sch13b}. The desired equality follows by applying a standard result \cite[(4.4.9)]{Lau96} relating the twisted orbital integrals of $\bf1_{B_{n,k}}\cdot\phi_{\tau,h}$ and $(\bf1_{B_{n,k}}\cdot\phi_{\tau,h})^{P_k,\sg,\GL_n(\cO_r)}$.
\end{proof}
Return to the proof of Proposition \ref{ss:jacquetmoduletraceidentity}. Casselman's theorem \cite[5.2]{Cas77} shows that
\begin{align*}
\Te_\pi(\N\de) = \Te_{\pi_{N_k}}(\N\de) = \sum_{i=1}^{t_k}\Te_{\pi_{1,i}^k}(\N\de_k)\Te_{\pi_{2,i}^k}(\N\de_{n-k}),
\end{align*}
and by putting all of this together, we can rewrite our sum of integrals as
\begin{align*}
&q^{(n-k)r}\sum_{i=1}^{t_k}\sum_{T_k}\frac1{\#W(T_k,\GL_k)}\int_{T_k(F_r)^{\sg-1}\bs T_k(F_r)_1}\!\dif\de_k\,\abs{D_{\GL_k}(\N\de_k)}\Te_{\pi_{1,i}^k}(\N\de_k)\sum_{T_{n-k}}\frac1{\#W(T_{n-k},\GL_{n-k})}\\
&\cdot\int_{T_{n-k}(F_r)^{\sg-1}\bs T_{n-k}(\cO_r)}\!\dif\de_{n-k}\,\abs{D_{\GL_{n-k}}(\N\de_{n-k})}\TO_{(\de_k,\de_{n-k}),\sg}(\phi_{\tau,h}|_{M_k(F_r)})\Te_{\pi_{2,i}^k}(\N\de_{n-k}),
\end{align*}
where we drop $\bf1_{B_{n,k}}$ since our $\de$ all lie in $B_{n,k}$.
Before proceeding further, we apply Proposition \ref{ss:deformationtestfunction} as follows. Write $\wt{f}_{\tau,h}(\de_k,-):\GL_{n-k}(\cO)\rar\bC$ for the function that sends
\begin{align*}
\ga\mapsto\tr\left((\ga,h)|\Ind_{P_k(\cO)}^{\GL_n(\cO)}\left(\phi_{\tau,-}(\de_k)\otimes C^\infty_c(\GL_{n-k}(\cO))\right)\right),
\end{align*}
where we view the conjugation-invariant distribution $\phi_{\tau,-}(\de_k)$ as a $\bC$-virtual admissible representation of $\GL_k(\cO)$ via \ref{ss:peterweyl}, $\ga$ acts trivially on $\phi_{\tau,-}(\de_k)$, and $\ga$ acts on $C^\infty_c(\GL_{n-k}(\cO))$ via right multiplication. Note that, with Proposition \ref{ss:deformationtestfunction} in mind, $\wt{f}_{\tau,h}$ heavily resembles $\phi_{\tau,h}$.
\begin{lem}\label{lem:ftilde}
The function $\wt{f}_{\tau,h}(\de_k,-)$ is in $C^\infty_c(\GL_{n-k}(\cO))$, and it satisfies the equalities
\begin{align*}
\tr((\phi_{\tau,h}(\de_k,-),\sg)|\pi_{2,i}^k/F_r) = \tr(\wt{f}_{\tau,h}(\de_k,-)|\pi_{2,i}^k) = \tr\left(h|\Ind_{P_k(\cO)}^{\GL_n(\cO)}(\phi_{\tau,-}(\de_k)\otimes\pi_{2,i}^k)\right).
\end{align*}
\begin{proof}
Proposition \ref{ss:deformationtestfunction} indicates that $\phi_{\tau,h}(\de_k,\de)=\wt{f}_{\tau,h}(\de_k,\N\de)$. Thus \ref{ss:etalelocalshtukatransfer} reduces the first claim to the statement that $\phi_{\tau,h}$ is in $C^\infty_c(\GL_n(\cO_r))$, which holds by \ref{ss:sigmaconjugationisopen}. For the second claim, note that Lemma \ref{ss:etalelocalshtukatransfer} implies $\wt{f}_{\tau,h}(\de_k,-)$ and $\phi_{\tau,h}(\de_k,-)$ have matching twisted orbital integrals. Therefore \ref{ss:twistedcharacters} yields $\tr((\phi_{\tau,h}(\de_k,-),\sg)|\pi_{2,i}^k/F_r) = \tr(\wt{f}_{\tau,h}(\de_k,-)|\pi_{2,i}^k)$.
From here, applying the Peter--Weyl theorem and Schur orthogonality to the definition of $\wt{f}_{\tau,h}(\de_k,-)$ show that $\tr(\wt{f}_{\tau,h}(\de_k,-)|\pi_{2,i}^k)=\tr\left(h|\Ind_{P_k(\cO)}^{\GL_n(\cO)}(\phi_{\tau,-}(\de_k)\otimes\pi_{2,i}^k)\right)$.
\end{proof}
\end{lem}
Return to the proof of Proposition \ref{ss:jacquetmoduletraceidentity}. Applying the twisted Weyl integration formula \cite[(II.2.10, formula (1)]{HL11} to the function $\TO_{\de_k,\sg}(\phi_{\tau,h}(-,-))$ in $C^\infty_c(\GL_{n-k}(F_r))$ allows us to collapse the sum over $T_{n-k}$ and simplify our expression into
\begin{align*}
q^{(n-k)r}\sum_{i=1}^{t_k}\sum_{T_k}\frac1{\#W(T_k,\GL_k)}\int_{T_k(F_r)^{\sg-1}\bs T_k(F_r)_1}\!\dif\de_k\,\abs{D_{\GL_k}(\N\de_k)}\Te_{\pi_{1,i}^k}(\N\de_k)\TO_{\de_k,\sg}(\tr(\wt{f}_{\tau,h}(-,-)|\pi_{2,i}^k)),
\end{align*}
where we first used $\tr((\TO_{\de_k,\sg}(\phi_{\tau,h}(-,-)),\sg)|\pi_{2,i}^k/F_r)=\TO_{\de_k,\sg}(\tr((\phi_{\tau,h}(-,-),\sg)|\pi_{2,i}^k/F_r))$ and then the first equality in Lemma \ref{lem:ftilde}. Applying the twisted Weyl integration formula to the function $\bf1_{B_k}\cdot\tr(\wt{f}_{\tau,h}(-,-)|\pi_{2,i}^k)$ in $C^\infty_c(\GL_k(F_r))$ lets us collapse the sum over $T_k$, and we get
\begin{align*}
q^{(n-k)r}\sum_{i=1}^{t_k}\tr\left((\bf1_{B_k}\cdot\tr(\wt{f}_{\tau,h}(-,-)|\pi_{2,i}^k),\sg)|\pi_{1,i}^k/F_r)\right).
\end{align*}
Now the second equality in Lemma \ref{lem:ftilde} turns this into
\begin{align*}
q^{(n-k)r}\sum_{i=1}^{t_k}\tr\left(h|\Ind_{P_k(\cO)}^{\GL_n(\cO)}\left(\tr((\bf1_{B_k}\cdot\phi_{\tau,-},\sg)|\pi_{1,i}^k/F_r)\otimes\pi_{2,i}^k\right)\right),
\end{align*}
which completes the proof of Proposition \ref{ss:jacquetmoduletraceidentity}.
\end{proof}
\subsection{}\label{ss:properlyinducedsupport}
Before moving on to the proof of condition (b) in Lemma \ref{ss:firstinductivelemma}, we record a useful consequence of van Djik's formula. Let $P$ be a proper standard parabolic subgroup of $\GL_n$, write $M$ for the standard Levi subgroup of $P$, let $\pi'$ be an irreducible smooth representation of $M(F)$, and write $\pi$ for the induced representation
\begin{align*}
\pi\deq\nInd_{M(F)}^{\GL_n(F)}\pi'.
\end{align*}
As $\pi'$ has finite length, so does $\pi$.
\begin{lem*}
Let $\phi$ be a function in $C^\infty_c(\GL_n(F_r))$ that is supported in elements of $\GL_n(F_r)$ whose norm is elliptic. Then we have $\tr((\phi,\sg)|\pi/F_r)=0$.
\end{lem*}
\begin{proof}
Let $f$ be a transfer of $\phi$, i.e. a function in $C^\infty_c(\GL_n(F))$ such that $f$ and $\phi$ have matching twisted orbital integrals. Then $f$ is supported in elliptic elements of $\GL_n(F)$, and no such element is contained in $P(F)$ because $P$ is a proper parabolic subgroup. Therefore the normalized $\GL_n(\cO)$-invariant constant term $f^{P,\GL_n(\cO)}$ of $f$ along $P$ is identically zero on $M(F)$. Our remark from \ref{ss:twistedcharacters} and van Djik's formula \cite[Theorem 5.9]{Lem16} together yield
\begin{align*}
\tr((\phi,\sg)|\pi/F_r) = \tr(f|\pi) =\tr(f^{P,\GL_n(\cO)}|\pi')=0,
\end{align*}
as desired.
\end{proof}
We will apply the above lemma to $\bf1_{B_k}\cdot\phi_{\tau,h}$ in the proof of Proposition \ref{ss:conditionb}.
\subsection{}\label{ss:conditionb}
Finally, we use Proposition \ref{ss:jacquetmoduletraceidentity} to prove condition (b), assuming condition (a).
\begin{prop*}
Assume that the inductive condition (a) in Lemma \ref{ss:firstinductivelemma} holds, that is, Theorem A is true for $n'<n$. Let $\pi$ be a smooth representation of $\GL_n(F)$, and suppose it is of the form
\begin{align*}
\pi=\nInd_{P(F)}^{\GL_n(F)}(\pi_1\otimes\dotsb\otimes\pi_t),
\end{align*}
where $t\geq2$, the $\pi_i$ are some irreducible smooth representations of $\GL_{n_i}(F)$ with $n=n_1+\dotsb+n_t$, and $P$ is the standard parabolic subgroup of $\GL_n$ with block sizes $(n_1,\dotsc,n_t)$. Then we have an equality of traces
\begin{align*}
\tr(f_{\tau,h}|\pi) = \tr\left(\tau|\rho(\pi_1)(\textstyle\frac{n-n_1}2)\oplus\dotsb\oplus\rho(\pi_t)(\frac{n-n_t}2)\right)\tr(h|\pi)
\end{align*}
for all $\tau$ in $W_F$ with $v(\tau)>0$ and $h$ in $C^\infty_c(\GL_n(\cO))$.
\end{prop*}
In the proof of Proposition \ref{ss:conditionb}, we shall frequently pass between normalized and un-normalized induction. For this, one immediately checks that
\begin{align*}
\nInd_{Q(F)}^{\GL_m(F)}(\la_1\otimes\la_2) = \Ind_{Q(F)}^{\GL_m(F)}(\la_1[\textstyle\frac{m_2}2]\otimes\la_2[-\frac{m_1}2]),
\end{align*}
where the $\la_i$ are irreducible smooth representations of $\GL_{m_i}(F)$ with $m=m_1+m_2$, and $Q$ is the standard parabolic subgroup of $\GL_m$ with block sizes $(m_1,m_2)$.
We also recall that the two representations
\begin{align*}
\nInd_{Q(F)}^{\GL_m(F)}(\la_1\otimes\la_2)\mbox{ and }\nInd_{Q'(F)}^{\GL_m(F)}(\la_2\otimes\la_1),
\end{align*}
where $Q'$ denotes the standard parabolic subgroup of $\GL_n$ with block sizes $(m_2,m_1)$, have the same Jorder--H\"older series \cite[2.9]{BZ77}. This enables us to identify them as virtual representations of $\GL_n(F)$.
\begin{proof}[Proof of Proposition \ref{ss:conditionb}]
We start with some immediate reductions. By using the transitivity of parabolic induction \cite[1.7]{Zel80}, it suffices to prove the proposition for $t=2$. Furthermore, because swapping $\pi_1$ and $\pi_2$ leaves the virtual representation $\pi$ unchanged, we may assume that $n_1\leq n_2$. Finally, by taking the trace of $h$ afterwards, Proposition \ref{ss:jacquetmoduletraceidentity} shows that it suffices to find, for all integers $1\leq k\leq n$, a decomposition of virtual representations
\begin{align*}
\pi_{N_k} = \sum_{i=1}^{t_k}\pi_{1,i}^k\otimes\pi_{2,i}^k
\end{align*}
of $\GL_k(F)\times\GL_{n-k}(F)$, where the $\pi_{1,i}^k$ and $\pi_{2,i}^k$ have finite length, such that we have an equality of $\bC$-virtual representations
\begin{align}\label{eq:goal}
\sum_{k=1}^nq^{(n-k)r}\sum_{i=1}^{t_k}\Ind_{P_k(\cO)}^{\GL_n(\cO)}\left(\tr((\bf1_{B_k}\cdot\phi_{\tau,-},\sg)|\pi_{1,i}^k/F_r)\otimes\pi_{2,i}^k\right) = \tr\left(\tau|\rho(\pi_1)(\textstyle\frac{n-n_1}2)\oplus\rho(\pi_2)(\frac{n-n_2}2)\right)\pi\tag{$\star$}
\end{align}
of $\GL_n(\cO)$. We will choose these $\pi_{1,i}^k$ and $\pi_{2,i}^k$ in a way that relates them to $\pi_1$ and $\pi_2$. To do so, we shall use the following \emph{induction-restriction formula} of Bernstein--Zelevinsky.
\subsection{}\label{ss:bernsteinzelevinskyinductionrestriction}
Write $T$ for the standard maximal torus of $\GL_n$, and write $B$ for the standard Borel subgroup of $\GL_n$. Let $P$ and $Q$ be standard parabolic subgroups of $\GL_n$, and denote their standard Levi subgroups using $M$ and $N$, respectively. We identify the Weyl group $W(T,\GL_n)$ with the symmetric group $\fS_n$ acting by permuting entries. Write $W_{M,N}$ for the subset
\begin{align*}
W_{M,N}\deq\{w\in W(T,\GL_n)|w(M\cap B)\subseteq B\mbox{ and }w^{-1}(N\cap B)\subseteq B\}
\end{align*}
of $W(T,\GL_n)$.
\begin{lem*}[{\cite[2.12]{BZ77}}]
Let $\pi$ be an irreducible smooth representation of $\GL_n(F)$. Then we have an equality of virtual representations
\begin{align*}
\nRes_{Q(F)}^{\GL_n(F)}(\nInd_{P(F)}^{\GL_n(F)}\pi) = \sum_w\nInd_{Q'(F)}^{N(F)}(w(\nRes_{P'(F)}^{M(F)}\pi))
\end{align*}
of $N(L)$, where $\nRes$ denotes normalized Jacquet restriction, $w$ runs over all elements of $W_{M,N}$, $P'$ denotes the standard parabolic subgroup of $M$ whose standard Levi subgroup is $M\cap w^{-1}(N)$, and $Q'$ denotes the standard parabolic subgroup of $N$ whose standard Levi subgroup is $w(M)\cap N$.
\end{lem*}
Return to the proof of Proposition \ref{ss:conditionb}. Our goal will be to apply $\tr((\bf1_{B_k}\cdot\phi_{\tau,-},\sg)|-/F_r)$ to the $\pi_{1,i}^k$. At this point, Lemma \ref{ss:properlyinducedsupport} indicates that the terms corresponding to properly parabolically induced $\pi_{1,i}^k$ will vanish. Therefore, when using the induction-restriction formula for $M=\GL_{n_1}\times\GL_{n_2}$ and $N=\GL_k\times\GL_{n-k}$, we will only be interested in the terms for which $w(M)\cap N$ contains $\GL_k$.
With these choices of $M$ and $N$, now $W_{M,N}$ corresponds precisely to the set of permutations $w$ in $\fS_n$ for which
\begin{itemize}
\item $w$ is order-preserving on $\{1,\dotsc,n_1\}$ and $\{n_1+1,\dotsc,n\}$,
\item $w^{-1}$ is order-preserving on $\{1,\dotsc,k\}$ and $\{k+1,\dotsc,n\}$.
\end{itemize}
Furthermore, the $w(M)\cap N\supseteq\GL_k$ condition is equivalent to asking that
\begin{align*}
w(\{1,\dotsc,n_1\})\supseteq\{1,\dotsc,k\}\mbox{ or }w(\{n_1+1,\dotsc,n\})\supseteq\{1,\dotsc,k\}.
\end{align*}
If $k\leq n_1$, then the only such $w$ in $W_{M,N}$ are the identity permutation and the permutation $\te$ that sends $\{1,\dotsc,n_1\}$ to $\{n_2+1,\dotsc,n\}$ and $\{n_1+1,\dotsc,n\}$ to $\{1,\dotsc,n_2\}$ while preserving their internal orders. If instead $n_1< k\leq n_2$, then the identity no longer satisfies $w(M)\cap N\supseteq\GL_k$, although $\te$ continues to do so. Finally, for $n_2<k$, no element of $W_{M,N}$ has $w(M)\cap N\supseteq\GL_k$. Altogether, Lemma \ref{ss:bernsteinzelevinskyinductionrestriction} yields an equality
\begin{align}\label{eq:jacquetrestriction}
\pi_{N_k} = &\Ind_{P_{n_1}(F)\cap(\GL_k(F)\times\GL_{n-k}(F))}^{\GL_k(F)\times\GL_{n-k}(F)}(\pi'_{N_{k_1}}\otimes\pi'')\nonumber\\
&+\Ind_{P_{n_2}(F)\cap(\GL_k(F)\times\GL_{n-k}(F))}^{\GL_k(F)\times\GL_{n-k}(F)}(\pi''_{N_{k_2}}[n_1]\otimes\pi'[-n_2])+\sum_{j=1}^{u_k}\tau_{1,j}^k\otimes\tau_{2,j}^k
\end{align}
of virtual representations of $\GL_k(F)\times\GL_{n-k}(F)$, where $\pi'$ denotes $\pi_1[\frac{n_2}2]$, $\pi''$ denotes $\pi_2[-\frac{n_1}2]$, $N_{k_1}$ denotes the unipotent radical of $P_{n_1}\cap\GL_k$, $N_{k_2}$ denotes the unipotent radical of $P_{n_2}\cap\GL_k$, and the $\tau_{1,1}^k,\dotsc,\tau_{1,u_k}^k$ (respectively $\tau_{2,1}^k,\dotsc,\tau_{2,u_k}^k$) are smooth representations of $\GL_k(F)$ (respectively $\GL_{n-k}(F)$) such that the $\tau_{1,j}^k$ are properly induced representations of $\GL_k(F)$. As remarked above, if $k>n_1$ or $k>n_2$, then we can absorb the first or second term on the right hand side into the third summation term, respectively.
So suppose that $k\leq n_1$ and $k\leq n_2$. We shall begin by studying the first term on the right hand side of Equation (\ref{eq:jacquetrestriction}). Because the unnormalized Jacquet restriction $\pi'_{N_{k_1}}$ of $\pi'$ is a smooth representation of $\GL_k(F)\times\GL_{n_1-k}(F)$ of finite length, we can write its virtual representation as
\begin{align*}
\pi'_{N_{k_1}} = \sum_{i=1}^{t_k'}\pi'^k_{1,i}\otimes\pi'^k_{2,i},
\end{align*}
where $t_k'$ is a non-negative integer, and the $\pi'^k_{1,i}$ and $\pi'^k_{2,i}$ are irreducible smooth representations of $\GL_k(F)$ and $\GL_{n_1-k}(F)$, respectively. Write $P_{k,n_1}$ for the standard parabolic subgroup of $\GL_n$ with block sizes $(k,n_1-k,n_2)$. By applying $\Ind_{P_{n_1}(F)\cap(\GL_k(F)\times\GL_{n-k}(F))}^{\GL_k(F)\times\GL_{n-k}(F)}(-\otimes\pi'')$ to the above decomposition of $\pi'_{N_{k_1}}$, we obtain the equality of virtual representations
\begin{align}\label{eq:virtual1}
\Ind_{P_{n_1}(F)\cap(\GL_k(F)\times\GL_{n-k}(F))}^{\GL_k(F)\times\GL_{n-k}(F)}(\pi'_{N_{k_1}}\otimes\pi'') = \sum_{i=1}^{t_k'}\pi_{1,i}'^k\otimes\Ind_{P_{k,n_1}(F)\cap\GL_{n-k}(F)}^{\GL_{n-k}(F)}(\pi_{2,i}'^k\otimes\pi''),
\end{align}
giving an alternate description of the first term on the right hand side of Equation (\ref{eq:jacquetrestriction}).
Now Proposition \ref{ss:jacquetmoduletraceidentity} for $\pi'$ indicates that
\begin{align*}
\sum_{k=1}^{n_1}q^{(n_1-k)r}\sum_{i=1}^{t_k'}\tr\left(h_1|\Ind_{P_k(\cO)\cap\GL_{n_1}(\cO)}^{\GL_{n_1}(\cO)}\left(\tr((\bf1_{B_k}\cdot\phi_{\tau,-},\sg)|\BC(\pi_{1,i}'^k))\otimes\pi_{2,i}'^k\right)\right) = \tr(f_{\tau,h_1}|\pi')
\end{align*}
for all functions $h_1$ in $C^\infty_c(\GL_{n_1}(\cO))$. Theorem A.(i) for $\pi'$ implies that $\tr(f_{\tau,h_1}|\pi') = \tr\left(\tau|\rho(\pi')\right)\tr(h_1|\pi')$, so \ref{ss:peterweyl} yields an equality of $\bC$-virtual representations
\begin{align*}
\sum_{k=1}^{n_1}q^{(n_1-k)r}\sum_{i=1}^{t_k'}\Ind_{P_k(\cO)\cap\GL_{n_1}(\cO)}^{\GL_{n_1}(\cO)}\left(\tr((\bf1_{B_k}\cdot\phi_{\tau,-},\sg)|\BC(\pi_{1,i}'^k))\otimes\pi_{2,i}'^k\right) = \tr\left(\tau|\rho(\pi')\right)\pi'
\end{align*}
of $\GL_{n_1}(\cO)$. Since we defined $\pi'$ to be $\pi_1[\textstyle\frac{n_2}2]$, taking $\Ind_{P_{n_1}(F)}^{\GL_n(F)}(-\otimes\pi'')$ on both sides shows that
\begin{align}\label{eq:finaleq1}
\sum_{k=1}^{n_1}q^{(n-k)r}\sum_{i=1}^{t_k'}\Ind_{P_{k,n_1}(\cO)}^{\GL_n(\cO)}\left(\tr((\bf1_{B_k}\cdot\phi_{\tau,-},\sg)|\BC(\pi_{1,i}'^k))\otimes\pi_{2,i}'^k\otimes\pi''\right) = \tr\left(\tau|\rho(\pi_1)(\textstyle\frac{n_2}2)\right)\pi
\end{align}
as $\bC$-virtual representations of $\GL_n(\cO)$, where we used Proposition \ref{ss:unramifiedtwistsofrho} to identify $\rho(\pi')$ with $\rho(\pi_1)(\frac{n_2}2)$.
Next, we turn to the second term on the right hand side of Equation (\ref{eq:jacquetrestriction}). As with the first term, we have an equality of virtual representations
\begin{align*}
\pi''_{N_{k_2}} = \sum_{i=1}^{t_k''}\pi''^k_{1,i}\otimes\pi''^k_{2,i},
\end{align*}
where $t_k''$ is a non-negative integer, and the $\pi''^k_{1,i}$ and $\pi''^k_{2,i}$ are irreducible smooth representations of $\GL_k(F)$ and $\GL_{n_2-k}(F)$, respectively. Write $P_{k,n_2}$ for the standard parabolic subgroup of $\GL_n$ with block sizes $(k,n_2-k,n_1)$. By applying $\Ind_{P_{n_2}(F)\cap(\GL_k(F)\times\GL_{n-k}(F))}^{\GL_k(F)\times\GL_{n-k}(F)}((-)[n_1]\otimes\pi'[-n_2])$ to the above decomposition of $\pi''_{N_{k_2}}$, we obtain the equality of virtual representations
\begin{align}\label{eq:virtual2}
\Ind_{P_{n_2}(F)\cap(\GL_k(F)\times\GL_{n-k}(F))}^{\GL_k(F)\times\GL_{n-k}(F)}(\pi''_{N_{k_2}}[n_1]\otimes\pi'[-n_2]) = \sum_{i=1}^{t_k''}\pi_{1,i}''^k[n_1]\otimes\Ind_{P_{k,n_2}(F)\cap\GL_{n-k}(F)}^{\GL_{n-k}(F)}(\pi_{2,i}''^k[n_1]\otimes\pi'[-n_2]),
\end{align}
giving an alternate description of the second term on the right hand side of Equation (\ref{eq:jacquetrestriction}).
Now Proposition \ref{ss:jacquetmoduletraceidentity} for $\pi''$ indicates that
\begin{align*}
\sum_{k=1}^{n_2}q^{(n_2-k)r}\sum_{i=1}^{t_k''}\tr\left(h_2|\Ind_{P_k(\cO)\cap\GL_{n_2}(\cO)}^{\GL_{n_2}(\cO)}\left(\tr((\bf1_{B_k}\cdot\phi_{\tau,-},\sg)|\BC(\pi_{1,i}''^k))\otimes\pi_{2,i}''^k\right)\right) = \tr(f_{\tau,h_2}|\pi'')
\end{align*}
for all functions $h_2$ in $C^\infty_c(\GL_{n_2}(\cO))$. Theorem A.(i) for $\pi''$ implies that $\tr(f_{\tau,h_2}|\pi'')=\tr\left(\tau|\rho(\pi'')\right)\tr(h_2|\pi'')$, so \ref{ss:peterweyl} yields an equality of $\bC$-virtual representations
\begin{align*}
\sum_{k=1}^{n_2}q^{(n_2-k)r}\sum_{i=1}^{t_k''}\Ind_{P_k(\cO)\cap\GL_{n_2}(\cO)}^{\GL_{n_2}(\cO)}\left(\tr((\bf1_{B_k}\cdot\phi_{\tau,-},\sg)|\BC(\pi_{1,i}''^k))\otimes\pi_{2,i}''^k\right) = \tr\left(\tau|\rho(\pi'')\right)\pi''
\end{align*}
of $\GL_{n_2}(\cO)$. Since we defined $\pi''$ to be $\pi_2[-\textstyle\frac{n_1}2]$, taking $\Ind_{P_{n_2}(F)}^{\GL_n(F)}((-)[n_1]\otimes\pi'[-n_2])$ on both sides shows that
\begin{align}\label{eq:finaleq2}
\sum_{k=1}^{n_2}q^{(n_2-k)r}\sum_{i=1}^{t_k''}\Ind_{P_{k,n_2}(\cO)}^{\GL_n(\cO)}\left(\tr((\bf1_{B_k}\cdot\phi_{\tau,-},\sg)|\BC(\pi_{1,i}''^k[n_1]))\otimes\pi_{2,i}''^k[n_1]\otimes\pi'[-n_2]\right) = \tr\left(\tau|\rho(\pi_2)(\textstyle\frac{n_1}2)\right)\pi
\end{align}
as $\bC$-virtual representations of $\GL_n(\cO)$, where we used Proposition \ref{ss:unramifiedtwistsofrho} to identify $\rho(\pi'')$ with $\rho(\pi_2)(-\frac{n_2}2)$.
We piece together the above work as follows. Equation (\ref{eq:jacquetrestriction}), Equation (\ref{eq:virtual1}), and Equation (\ref{eq:virtual2}) indicate that
\begin{align*}
\pi_{N_k} &= \sum_{i=1}^{t_k'}\pi_{1,i}'^k\otimes\Ind_{P_{k,n_1}(F)\cap\GL_{n-k}(F)}^{\GL_{n-k}(F)}(\pi_{2,i}'^k\otimes\pi'')\\
&+\sum_{i=1}^{t_k''}\pi_{1,i}''^k[n_1]\otimes\Ind_{P_{k,n_2}(F)\cap\GL_{n-k}(F)}^{\GL_{n-k}(F)}(\pi_{2,i}''^k[n_1]\otimes\pi'[-n_2])+\sum_{j=1}^{u_k}\tau_{1,j}^k\otimes\tau_{2,j}^k,
\end{align*}
and we take this decomposition for our $\pi^k_{1,i}$ and $\pi^k_{2,i}$. Because the $\tau_{1,j}^k$ are properly induced, Lemma \ref{ss:properlyinducedsupport} shows that the $\tr((\bf1_{B_k}\cdot\phi_{\tau,-},\sg)|\BC(\tau_{1,j}^k))$ terms vanish. Therefore the sum of Equation (\ref{eq:finaleq1}) and Equation (\ref{eq:finaleq2}) is precisely Equation (\ref{eq:goal}), and this concludes the proof of Proposition \ref{ss:conditionb}.
\end{proof}
\section{The Lubin--Tate tower}\label{s:lubintatetower}
In this section, our goal is to prove that condition (d) in Lemma \ref{ss:firstinductivelemma} holds, under the assumption that condition (c) holds. Although we postpone the proof of condition (c) for later, our proof of condition (d) does not use anything from the construction of condition (c)---we may safely black box the latter. Along the way, we also prove the $n=1$ case of Theorem A, which serves as the base case for applying Lemma \ref{ss:firstinductivelemma}.
We begin by using connected local shtukas to prove, in the same vein as Lemma \ref{ss:normbijection} and Lemma \ref{ss:etalelocalshtukatransfer}, the existence of transfers for certain conjugation-invariant functions on division algebras over $F$. We then introduce the \emph{Lubin--Tate tower}, which is closely related to our deformation spaces from \S\ref{s:deformationspaces}. By studying this relation, along with the fact that it gives the Lubin--Tate proof of local class field theory, we prove the $n=1$ case of Theorem A. We also use this relation to compare the cohomologies of the Lubin--Tate tower and $\br\fX_{\de,m}$. This comparison is stated in terms of the \emph{local Jacquet--Langlands correspondence}, which relates representations of $\GL_n(F)$ with representations of division algebras. Finally, we use results from Jacquet--Piatetski-Shapiro--Shalika's theory of \emph{new-vectors}, which are certain elements in irreducible generic representations of $\GL_n(F)$, to conclude the proof of condition (d).
\subsection{}
We start by using two descriptions of isomorphism classes of connected local shtukas over $\Spec\ka_r$ to prove the following proposition. Write $B$ for the central division algebra over $F$ of Hasse invariant $\frac1n$, and write $\cO_B$ for its ring of integers. By abuse of notation, we denote the normalized valuation on $B$ using $v$. Write $B_r$ for the subset of $b$ in $B$ satisfying $v(b)=r$.
\begin{lem*}\label{ss:noncommutativenormbijection}
There exists a bijection
\begin{align*}
\N:\{\GL_n(\cO_r)\mbox{-}\sg\mbox{-conjugacy classes in }B_n\}\rar^\sim\{\cO_B^\times\mbox{-conjugacy classes in }B_r\}
\end{align*}
such that the characteristic polynomial of $\N\de$ equals that of $\de\sg(\de)\dotsm\sg^{r-1}(\de)$. By abuse of notation, we call this the \emph{norm map}.
\end{lem*}
This result and its proof are entirely analogous to those of Lemma \ref{ss:normbijection}.
\begin{proof}
We shall construct bijections between both the left-hand as well as right-hand sides and the set
\begin{align*}
\{\mbox{isomorphism classes of connected local shtukas over }\Spec\ka_r\mbox{ of rank }n\},
\end{align*}
and we denote the composed bijection using $\N$. On the left-hand side, \ref{ss:effectiveminusculelocalshtukaconjugacy} yields the desired bijection.
We now turn to the right-hand side. Let $\sM$ be a connected local shtuka over $\Spec\ka_r$. Completed unramified descent implies that the isomorphism class of $\sM$ is determined by the isomorphism class of $\br\sM\deq\sM_{\ov\ka}$ along with a descent isomorphism $f:\sg^{-r,*}\br\sM\rar^\sim\br\sM$ corresponding to the action of the geometric $q^r$-Frobenius map $\sg^{-r}$. As $\br\sM$ must be isomorphic to the local shtuka associated with the Lubin--Tate module, after choosing such an isomorphism we may identify $\End(\br\sM)$ with $\cO_B$. Write $\br\sF:\sg^*\br\sM\rar\br\sM$ for the Frobenius of $\br\sM$, which corresponds to an element of valuation $1$ in $\cO_B$, and write $\ga$ for the endomorphism
\begin{align*}
\ga\deq\br\sF\circ\dotsb\circ\sg^{r-1,*}\br\sF\circ\sg^{r,*}f
\end{align*}
of $\br\sM$, viewed as an element of $\cO_B$. As $\sg^{r,*}f$ is an isomorphism and hence corresponds to an element of valuation $0$ in $\cO_B$, this shows that $v(\ga)=r$. Since any other choice of isomorphism between $\br\sM$ and the local shtuka associated with the Lubin--Tate module preserves the $\cO_B^\times$-conjugacy class of $\ga$, this yields a bijection $\ga\leftrightarrow\sM$ between $\cO_B^\times$-conjugacy classes in $B_r$ and isomorphism classes of connected local shtukas over $\Spec\ka_r$ of rank $n$.
Recall that $\br\sF$ corresponds to $\de\circ\sg^{\oplus n}$ under \ref{ss:effectiveminusculelocalshtukaconjugacy}. Since $f$ corresponds to $(\sg^{\oplus n})^{-r}$, we see that $\ga$ corresponds to
\begin{align*}
\underbrace{(\de\circ\sg^{\oplus n})\circ\dotsb\circ(\de\circ\sg^{\oplus n})}_{r\,\text{times}}\circ (\sg^{\oplus n})^{-r} = \left(\de\sg(\de)\dotsm\sg^{r-1}(\de)\right)\circ(\sg^{\oplus n})^r\circ (\sg^{\oplus n})^{-r} = \de\sg(\de)\dotsm\sg^{r-1}(\de),
\end{align*}
as desired.
\end{proof}
\subsection{}\label{ss:connectedlocalshtukatransfer}
We will now construct certain functions with matching twisted orbital integrals, that is, certain transfers of functions. Let $h$ be an $\cO_B^\times$-conjugation-invariant function in $C^\infty_c(B_r)$, and form the function $\phi:B_n\rar\bC$ by sending $\de\mapsto h(\N\de)$. One can show that $\phi$ is locally constant \cite[Corollary 2.3]{Sch13b}\footnote{The proof given here is stated for mixed characteristic and $\GL_n$, but it only uses Lang's lemma and the relationship between inner twists of $\GL_n$ and central division algebras. In particular, it adapts to equal characteristic and any inner twist of $\GL_n$.}.
\begin{lem*}
The functions $h$ and $\phi$ have matching twisted orbital integrals.
\end{lem*}
This result and its proof are entirely analogous to those of Lemma \ref{ss:etalelocalshtukatransfer}.
\begin{proof}
We need to prove that $\O_{\N\de}(h)=\TO_{\de,\sg}(\phi)$ for all regular $\de$ in $B_n$, where $\O_{\N\de}$ and $\TO_{\de,\sg}$ denote the orbital integral on $\N\de$ and twisted orbital integral on $\de$, respectively, with respect to compatible Haar measures. We can prove this as follows: write $\sM$ for the connected local shtuka corresponding to $\de$ under \ref{ss:effectiveminusculelocalshtukaconjugacy}, and write $X$ for the set of isomorphism classes of quasi-isogenies $\be:\sM\dasharrow\sM'$ between connected local shtukas. Note that the group $\Ga$ of self-quasi-isogenies $\sM\dasharrow\sM$ has a left action on $X$ given by sending $\be$ to $\be\circ g^{-1}$ for any $g$ in $\Ga$.
One can equate both $\O_{\N\de}(h)$ as well as $\TO_{\de,\sg}(\phi)$ to the sum
\begin{align*}
\sum_{(\sM',\be)}\wt{h}(\sM',\be),
\end{align*}
where $(\sM',\be)$ ranges over all elements in $\Ga\bs X$, and $\wt{h}$ is the function sending $(\sM',\be)$ to $h(\de')$, where $\de'$ corresponds to $\sM'$ under \ref{ss:effectiveminusculelocalshtukaconjugacy}. In a line of reasoning similar to the proof of Lemma \ref{ss:noncommutativenormbijection}, we relate $\TO_{\de,\sg}(\phi)$ to the above sum by using \ref{ss:effectiveminusculelocalshtukaconjugacy}, and we relate $\O_{\N\de}(h)$ to the above sum by using completed unramified descent, c.f. \cite[Proposition 4.7]{Sch13}.
\end{proof}
\subsection{}
At this point, we introduce the \emph{Lubin--Tate tower}. Let $\de$ be in $B_n$. By the Dieudonn\'e--Manin classification \cite[(2.4.5)]{Lau96}, the isomorphism class of $\br{H}_\de$ is independent of our choice of $\de$, and we may take $\de$ to lie in $\GL_n(F)$. We write $\br{H}$ for $\br{H}_\de$. The inverse system $(\br\fX_{\de,m})_m$ does not depend on $\de$ either, so we shall rewrite it as
\begin{align*}
\dotsm\rar\br\fX_m\rar\dotsm\rar\br\fX_1\rar\br\fX_0.
\end{align*}
This is the \emph{Lubin--Tate tower}. Of course, all our statements from \S\ref{s:deformationspaces} concerning $\br{H}_\de$ also apply to the Lubin--Tate tower:
\begin{enumerate}[$\bullet$]
\item this inverse system has a right action of $\GL_n(\cO)$,
\item we can form the cohomology group $R^i\psi_m\deq H^i(\br\fX_{m,\bC_\vpi},\ov\bQ_\ell)$, which is isomorphic to $H^0(\br\fX_{m,\ov\ka},R^i\Psi_{\br\fX_{m,\bC_\vpi}},\ov\bQ_\ell)$,
\item we can form the direct limit $R^i\psi\deq\dirlim_mR^i\psi_m$,
\item we can take the alternating sum $[R\psi]\deq\sum_{i=0}^\infty(-1)^iR^i\psi$.
\end{enumerate}
We shall now explain how the left action of $I_F$ on $R^i\psi_m$ actually extends to an action of $(B^\times\times W_F)_0$, where $(B^\times\times W_F)_0$ is the subgroup
\begin{align*}
(B^\times\times W_F)_0\deq\{b\times\tau\in B^\times\times W_F|v(b)+v(\tau)=0\}
\end{align*}
of $B^\times\times W_F$. We start by describing the action for $b\times \tau$ with non-negative $v(\tau)=-v(b)$. Let $r=v(\tau)=v(b^{-1})$, and note that $b^{-1}$ lies in $\cO_B$. By abuse of notation, write $b^{-1}:\br{H}\rar\br{H}$ for the corresponding endomorphism of $\br{H}$, write $\br{F}:\sg^*\br{H}\rar\br{H}$ for the Frobenius of $\br{H}$, and write $f_\tau:\sg^{-r,*}\br{H}\rar^\sim\br{H}$ for the isomorphism from \ref{ss:deformationunramifiedextension}. Using the isomorphism $\sg^*\br{H}\cong\br{H}$ obtained from the fact that $\de$ lies in $\GL_n(F)$, the map $\br{F}$ corresponds to an element of valuation $1$ in $\cO_B$. Therefore the quasi-isogeny
\begin{align*}
b\times\tau\deq f_\tau\circ(\sg^{-r,*}\br{F}^{-1}\circ\dotsb\circ\sg^{-1,*}\br{F}^{-1})\circ b^{-1}
\end{align*}
corresponds to an element of valuation $0$ and hence equals an automorphism of $\br{H}$.
As for $b\times\tau$ with negative $v(\tau)$, we define $b\times\tau$ as the inverse of $d^{-1}\times\tau^{-1}$. Since we view $\cO_B$ as acting from the right on $\br{H}$, we see that in all cases $(b\times\tau)\circ(b'\times\tau') = bb'\times\tau\tau'$.
By sending the triple $(H',\al',\io')$ to $(H',\al',\io'\circ(d\times\tau))$, we obtain a right action on $\br\fX_m$ and hence its base change $\br\fX_{m,\cO_{\bC_\vpi}}$. Taking cohomology gives us the desired left action on $R^i\psi_m$. Forming the direct limit over all $m$ yields a $(\GL_n(\cO)\times\cO_B^\times)\times I_F$-admissible/continuous representation as in \cite[p.~24]{HT01} of $\GL_n(\cO)\times(B^\times\times W_F)_0$ over $\ov\bQ_\ell$.
\subsection{}\label{ss:lubintatecomparison}
Now return to the situation of an arbitrary $\de$ in $B_n$. We shall begin studying the relationship between $\br\fX_{\de,m}$ and the Lubin--Tate tower by considering the virtual representations $[R\psi_\de]$ and $[R\psi]$.
\begin{prop*}
We have an equality of traces
\begin{align*}
\tr(\tau\times h|[R\psi_\de]) = \tr(h\times(\N\de)^{-1}\times\tau|[R\psi]).
\end{align*}
\end{prop*}
\begin{proof}
Under our $\GL_n(\cO)$-equivariant isomorphism $\br{H}_\de\cong\br{H}$, we see that $\br{F}$ corresponds to $\de\circ\sg^{\oplus n}$. Therefore work from the proof of Lemma \ref{ss:noncommutativenormbijection} shows that
\begin{align*}
(\N\de,\tau) = f_\tau\circ(\sg^{-r,*}\br{F}^{-1}\circ\dotsb\circ\sg^{-1,*}\br{F}^{-1})\circ\left((\br{F}\circ\dotsb\circ\sg^{r-1,*}\br{F}\circ\sg^{r,*}f)^{-1}\right)^{-1} = f_\tau\circ\sg^{r,*}f,
\end{align*}
where $f:\sg^{-r,*}\br{H}\rar^\sim\br{H}$ corresponds to the action of $\sg^{-r}$. The description of $\fX_{\de,m}$ in \ref{ss:deformationunramifiedextension} shows that precomposing with $f_\tau\circ\sg^{r,*}f$ induces the action of $\tau$ on $R^i\psi_{\de,m}$. Thus the action of $\ga\times(\N\de)^{-1}\times\tau$ on $R^i\psi_m$ corresponds to the action of $\tau\times\ga$ on $R^i\psi_{\de,m}$ for all $\ga$ in $\GL_n(\cO)$. Taking the direct limit over all $m$, forming the alternating sum over all $i$, and taking traces yields the desired result.
\end{proof}
\subsection{}\label{ss:lubintatelocalclassfieldtheory}
Before turning to the proof of the $n=1$ case of Theorem A, we take a brief interlude to describe the cohomology of the Lubin--Tate tower in terms of the Lubin--Tate proof of local class field theory. Note that in the $n=1$ case, $\GL_1(\cO)=\cO^\times$ and $(B^\times\times W_F)_0=(F^\times\times W_F)_0$. Recall that $\Art:F^\times\rar^\sim W^\ab_F$ denotes the local reciprocity isomorphism that sends uniformizers to geometric $q$-Frobenii. We denote the maximal unramified extension of $F$ using $F^\text{nr}$, and we denote the extension of $F^\text{nr}$ corresponding to $(\cO/\vpi^m)^\times$ using $F^{\text{nr},\,m}$.
In our calculations for the $n=1$ case, we take $\de=\vpi$.
\begin{prop*}
In the $n=1$ case, we have an equality of representations
\begin{align*}
[R\psi] = C^\infty_c(\cO^\times),
\end{align*}
where $\cO^\times$ acts by inverse left multiplication, and $(F^\times\times W_F)_0$ acts via right multiplication by $\Art^{-1}(\tau)^{-1}b$.
\end{prop*}
We only distinguish between left and right multiplication to maintain the situation for general $n\geq1$---there is no difference between them in the $n=1$ case, because all groups involved only act through their abelianizations.
\begin{proof}[Proof of Proposition \ref{ss:lubintatelocalclassfieldtheory}]
Recall from \S\ref{s:deformationspaces} that $\br\fX_m=\Spf\br{R}_m$ for some regular complete local Noetherian $\br\cO$-algebra $\br{R}_m$. The Lubin--Tate proof of local class field theory shows that $\br{R}_m$ is isomorphic to the completion of $\cO^{\text{nr},\,m}$, where $\cO^{\text{nr},\,m}$ is the ring of integers of $F^{\text{nr},\,m}$. Therefore $\br\fX_{m,\bC_\vpi}=\Spf\br{R}_m\wh\otimes_{\br\cO}\bC_\vpi$ consists of one point for each element of $\Gal(\br{R}_m[\frac1\vpi]/\br{F})=(\cO/\vpi^m)^\times$.
Under this identification, the right action of $\cO^\times$ on $\br\fX_{m,\bC_\vpi}$ is given by sending $g$ to $\ga^{-1}g$ for any $\ga$ in $\cO^\times$, and the Lubin--Tate proof of local class field theory implies that the right action of $(F^\times\times W_F)_0$ is given by sending $g$ to $g\Art^{-1}(\tau)b^{-1}$ for any $\tau\times b$ in $(F^\times\times W_F)_0$. Taking cohomology shows that
\begin{align*}
R^0\psi_m = \ov\bQ_\ell[(\cO/\vpi^m)^\times],
\end{align*}
where $\cO^\times$ acts by inverse left multiplication, and $(F^\times\times W_F)_0$ acts via right multiplication by $\Art^{-1}(\tau)^{-1}b$. Finally, as $R^i\psi_m$ vanishes for $i>n-1=0$, we have $[R\psi]=R^0\psi$. Forming the direct limit over all $m$ yields the desired result, where we have identified $\ov\bQ_\ell$ and $\bC$.
\end{proof}
\subsection{}\label{ss:theoremabasecase}
With Proposition \ref{ss:lubintatecomparison} and Proposition \ref{ss:lubintatelocalclassfieldtheory} in hand, we can now prove the $n=1$ case of Theorem A. This amounts to rewriting the Lubin--Tate tower in terms of the Lubin--Tate proof of local class field theory.
\begin{prop*}
Let $\chi$ be an irreducible smooth representation of $\GL_1(F)$, that is, a smooth character $\chi:F^\times\rar\bC^\times$. Then Theorem A holds for $\chi$, that is, there exists a unique continuous character $\rho(\chi):W_F\rar\bC$ satisfying the following property:
\begin{align*}
\mbox{for all }\tau\mbox{ in }W_F\mbox{ with }v(\tau)>0\mbox{ and }h\mbox{ in }C^\infty_c(\GL_n(\cO))\mbox{, we have }\tr(f_{\tau,h}|\chi) = \tr\left(\tau|\rho(\chi)\right)\tr(h|\chi).
\end{align*}
This $\rho(\chi)$ is given by $\rho(\chi)=\chi\circ\Art^{-1}$.
\end{prop*}
Note that we need not consider Theorem A.(ii) for $n=1$, since $\GL_1$ has no proper parabolic subgroups. As before, we always take compatible Haar measures, but we generally omit their precise description.
\begin{proof}[Proof of Proposition \ref{ss:theoremabasecase}]
We first use \ref{ss:twistedcharacters} to show that
\begin{align*}
\tr(f_{\tau,h}|\chi) = \tr((\phi_{\tau,h},\sg)|\BC(\chi)) = \int_{\vpi\cO_r^\times}\!\dif\de\,\tr(\tau\times h|[R\psi_\de])\chi(\N\de),
\end{align*}
and then Proposition \ref{ss:lubintatecomparison} implies that the above integral is equal to
\begin{align*}
\int_{\vpi\cO^\times_r}\!\dif\de\,\tr(h\times(\N\de)^{-1}\times\tau|[R\psi])\chi(\N\de).
\end{align*}
Since $\tr(h\times(-)^{-1}\times\tau|[R\psi])$ and $\tr(h\times(\N-)^{-1}\times\tau|[R\psi])$ have matching twisted orbital integrals by Lemma \ref{ss:connectedlocalshtukatransfer}, making the change of variables $b=\N\de$ gives us
\begin{align*}
\int_{\vpi^r\cO^\times}\!\dif b\,\tr(h\times b^{-1}\times\tau|[R\psi])\chi(b).
\end{align*}
Using Proposition \ref{ss:lubintatelocalclassfieldtheory} and applying the Peter--Weyl theorem to $\cO^\times$ (which is really just Pontryagin duality, since $\cO^\times$ is commutative) shows that the above integral is equal to
\begin{align*}
\tr(h|\chi)\tr(\Art^{-1}(\tau)|\chi) = \tr(h|\chi)\tr(\tau|\chi\circ\Art^{-1}).
\end{align*}
Therefore $\rho(\chi)=\chi\circ\Art^{-1}$ satisfies the desired property. Since $\chi$ is a character, it equals its trace, so it is also characterized uniquely by this property.
\end{proof}
\subsection{}
Now return to the arbitrary $n\geq1$ case. We introduce some notation on the multiplicity of $\la$ in $R^i\psi$, where $\la$ is an irreducible smooth representation of $B^\times$. Write $R^i\psi(\la)$ for the vector space
\begin{align*}
R^i\psi(\la)\deq\Hom_{\cO_B^\times}(\res\la_{\cO_B^\times},R^i\psi).
\end{align*}
This has a left action of $\GL_n(\cO)\times W_F$ as follows. For any $f$ in $R^i\psi(\la)$ and $(\ga,\tau)$ in $\GL_n(\cO)\times W_F$, we set
\begin{align*}
((\ga\times\tau)f)(v) = (\ga\times b\times\tau)f(b^{-1}v)
\end{align*}
for all $v$ in $\la$, where $b$ is any element of $B^\times$ satisfying $v(b)+v(\tau)=0$. Since $f$ commutes with $\cO_B^\times$, this action is independent of our choice of $b$. We see that $R^i\psi(\la)$ is a $\GL_n(\cO)\times I_F$-admissible/continuous representation of $\GL_n(\cO)\times W_F$ over $\ov\bQ_\ell$. Write $[R\psi](\la)$ for the virtual representation $\sum_{i=0}^\infty(-1)^iR^i\psi(\la)$.
\subsection{}\label{ss:localjacquetlanglands}
In order to state our next result, we recall the \emph{local Jacquet--Langlands correspondence}. Let $B$ be a central division algebra over $F$ of dimension $n^2$. Since every irreducible smooth representation $\la$ of $B^\times$ is finite-dimensional, its character distribution is represented by the function $\Te_\la:B^\times\rar\bC$ that sends $b\mapsto\tr(b|\la)$. This finite-dimensionality also implies that every such representation is essentially $L^2$.
There exists a unique bijection \cite[Th. 1.1]{Bad00}
\begin{align*}
\JL:\left\{
\begin{tabular}{c}
isomorphism classes of irreducible essentially \\
$L^2$ representations of $\GL_n(F)$
\end{tabular}\right\}\rar^\sim \left\{\begin{tabular}{c}
isomorphism classes of irreducible \\
representations of $B^\times$
\end{tabular}\right\}
\end{align*}
such that, for all regular elliptic elements $\ga$ in $\GL_n(F)$ and $b$ in $B^\times$ whose characteristic polynomials are the same, we have the equality
\begin{align*}
\Te_\pi(\ga) = (-1)^{n-1}\Te_{\JL(\pi)}(b).
\end{align*}
Furthermore, the central characters of $\pi$ and $\JL(\pi)$ are equal.
\begin{prop}\label{prop:lubintatecohomology}
Assume that condition (c) in Lemma \ref{ss:firstinductivelemma} holds, and suppose that $\pi$ is an irreducible cuspidal representation of $\GL_n(F)$. Then we have an equality of virtual $\GL_n(\cO)\times I_F$-admissible/continuous representations
\begin{align*}
(-1)^{n-1}\res\pi_{\GL_n(\cO)}\otimes\rho(\pi) = [R\psi](\JL(\pi)).
\end{align*}
\end{prop}
\begin{proof}
We begin by using condition (c) in Lemma \ref{ss:firstinductivelemma} and \ref{ss:twistedcharacters} to see that
\begin{align*}
\tr(h\times\tau|\res\pi_{\GL_n(\cO)}\otimes\rho(\pi)) = \tr(h|\pi)\tr(\tau|\rho(\pi)) = \tr(f_{\tau,h}|\pi) = \tr((\phi_{\tau,h},\sg)|\BC(\pi)).
\end{align*}
Then, Proposition \ref{ss:lubintatecomparison} shows that this twisted trace is equal to the integral
\begin{align*}
\int_{\GL_n(\cO_r)\diag(\vpi,1,\dotsc,1)\GL_n(\cO_r)}\!\dif\de\,\tr(h\times(\N\de)^{-1}\times\tau|[R\psi])\Te_\pi^{\sg}(\de).
\end{align*}
To narrow down our domain of integration, we shall use the following lemma.
\begin{lem}\label{lem:cuspidaltwistedcharacter}
The twisted character $\Te_\pi^{\sg}$ vanishes on $\GL_n(\cO_r)\diag(\vpi,1,\dotsc,1)\GL_n(\cO_r)\ssm B_n$.
\end{lem}
\begin{proof}
Recall that we have the decomposition
\begin{align*}
\GL_n(\cO_r)\diag(\vpi,1,\dotsc,1)\GL_n(\cO_r) = \coprod_{k=1}^nB_{n,k},
\end{align*}
and recall the description of $B_{n,k}$ from \ref{ss:localshtukaunramifiedextension} in terms of $\de=\de^\circ\oplus\de^\et$. Write $P_\de$ for the parabolic subgroup of $\GL_n$ over $F_r$ associated with $\de$. This shows that $P_\de$ is a proper subgroup if and only if $\de$ does not lie in $B_n$, and because $\pi$ is cuspidal, Casselman's theorem \cite[5.2]{Cas77} yields the desired result.
\end{proof}
Return to the proof of Proposition \ref{prop:lubintatecohomology}. Lemma \ref{lem:cuspidaltwistedcharacter} implies that our integral becomes
\begin{align*}
\int_{B_n}\!\dif\de\,\tr(h\times(\N\de)^{-1}\times\tau|[R\psi])\Te_\pi(\N\de).
\end{align*}
As $\tr(h\times(-)^{-1}\times\tau|[R\psi])$ and $\tr(h\times(\N-)^{-1}\times\tau|[R\psi])$ have matching twisted orbital integrals by Lemma \ref{ss:connectedlocalshtukatransfer}, making the change of variables $b=\N\de$ and applying the local Jacquet--Langlands correspondence indicate that the above expression equals
\begin{align*}
(-1)^{n-1}\int_{B_r}\!\dif b\,\tr(h\times b^{-1}\times\tau|[R\psi])\Te_{\JL(\pi)}(b) = (-1)^{n-1}\tr(h\times\tau|[R\psi](\JL(\pi))).
\end{align*}
Since virtual $\GL_n(\cO)\times I_F$-admissible/continuous representations of $\GL_n(\cO)\times W_F$ are determined by their traces, this yields the desired result.
\end{proof}
\subsection{}\label{ss:conditiond}
We conclude this section by verifying that, under the assumption that condition (c) from Lemma \ref{ss:firstinductivelemma} holds, condition (d) holds as well.
\begin{prop*}
Assume that condition (c) in Lemma \ref{ss:firstinductivelemma} holds, and suppose that $\pi$ is an irreducible cuspidal representation of $\GL_n(F)$. Then the $\bQ_{\geq0}$-virtual continuous representation $\rho(\pi)$ is actually a $\bZ$-virtual continuous representation of $W_F$.
\end{prop*}
\begin{proof}
Proposition \ref{prop:lubintatecohomology} shows that the $\bZ$-virtual representation $[R\psi](\JL(\pi))$ equals $\res\pi_{\GL_n(\cO)}\otimes\rho(\pi)$ as virtual $\GL_n(\cO)\times I_F$-admissible/continuous representations of $\GL_n(\cO)\times W_F$ over $\ov\bQ_\ell$. Thus its $\la$-isotypic component is also a $\bZ$-virtual representation for any irreducible smooth representation $\la$ of $\GL_n(\cO)$. If we could find such a $\la$ that is contained in $\pi$ with multiplicity $1$, then the $\la$-isotypic component of $[R\psi](\JL(\pi))$ would equal $\rho(\pi)$. Therefore, once we prove the following lemma, the result follows.
\end{proof}
\begin{lem}
Let $\pi$ be an irreducible cuspidal representation of $\GL_n(F)$. Then there exists an irreducible smooth representation $\la$ of $\GL_n(\cO)$ such that $\pi$ contains $\la$ with multiplicity $1$.
\end{lem}
\begin{proof}
Since $\pi$ is cuspidal and hence generic, the theory of new-vectors \cite[Theorem (5.1).(ii)]{JPSS81} shows that there exists a non-negative number $a$ such that $\dim\pi^{K(a)}=1$, where $K(a)$ is the subgroup
\begin{align*}
K(a)\deq\left\{\matr{a}{b}{c}{d}\in\GL_n(\cO)\middle|
a\in\GL_{n-1}(\cO),\,c\equiv0\pmod{\vpi^a},\,d\equiv1\pmod{\vpi^a}\right\}
\end{align*}
for positive $a$, and $K(a)\deq\GL_n(\cO)$ for $a=0$. In other words, $\dim\Hom_{K(a)}(\bC,\res\pi_{K(a)})=1$, where $\bC$ denotes the trivial representation. Frobenius reciprocity yields
\begin{align*}
\dim\Hom_{\GL_n(\cO)}(\cInd_{K(a)}^{\GL_n(\cO)}\bC,\res\pi_{\GL_n(\cO)}) = \dim\Hom_{K(a)}(\bC,\res\pi_{K(a)})=1.
\end{align*}
The Peter--Weyl theorem then implies that any irreducible smooth subrepresentation $\la$ of $\cInd_{K(a)}^{\GL_n(\cO)}\bC$ is contained in $\pi$ with multiplicity $1$, as desired.
\end{proof}
\section{Moduli spaces of $\sD$-elliptic sheaves}\label{s:modulispaces}
At this point, we shift our focus from local considerations to global ones. In this section, we begin by introducing \emph{$\sD$-elliptic sheaves}, which are the equi-characteristic analog of abelian varieties equipped with certain endomorphism structures. Afterwards, we introduce moduli spaces of $\sD$-elliptic sheaves, which therefore correspond to certain Shimura varieties and their integral models. The cohomology of these moduli spaces plays an important role in the Langlands correspondence. We can also obtain local shtukas from $\sD$-elliptic sheaves, and this is the equi-characteristic version of taking the $p$-divisible group of an abelian variety. We conclude this section by introducing a \emph{Serre--Tate theorem}, which relates deformations of $\sD$-elliptic sheaves to deformations of their associated local shtukas. This will eventually allow us to prove condition (c) in Lemma \ref{ss:firstinductivelemma}.
\subsection{}
We start by switching our notation to a global context. In these next few sections, let $\ka$ be a finite field of characteristic $p$ and cardinality $q$, and fix a separable closure $\ov\ka$ of $\ka$. We view all separable extensions of $\ka$ as lying in $\ov\ka$. Let $C$ be a geometrically connected proper smooth curve over $\ka$, and write $\bf{F}$ for its field of rational functions. Denote the adele ring of $C$ using $\bA$, denote the ring of integers of $\bA$ using $\bO$, and for any finite closed subscheme $I$ of $C$, write $\bK_I$ for the ideal of $\bO$ corresponding to $I$.
For any place $x$ of $C$, write $\cO_x$ for the local ring given by the completion of $C$ at $x$, and write $\bf{F}_x$ for the fraction field of $\cO_x$. Then $\bf{F}_x$ is a completion of $\bf{F}$ at $x$. Choose a uniformizer $\vpi_x$ of $x$ in $\bf{F}$, write $\ka_x$ for $\cO_x/\vpi_x$, write $q_x$ for $\#\ka_x$, and write $\deg{x}$ for $[\ka_x:\ka]$. By abuse of notation, we write $x$ for the normalized valuation corresponding to $x$. We also write $\deg:\bA^\times\rar\bZ$ for the function given by sending
\begin{align*}
(a_x)_x\mapsto-\sum_xx(a_x)\deg{x}.
\end{align*}
We fix a closed point $\infty$ in $C$, which shall serve as a replacement for the archimedean places at infinity.
\subsection{}\label{ss:divisionalgebras}
Since division algebras are fundamental to our constructions, we recall some facts about them here. For any central division algebra $D$ over $\bf{F}$, write $D_x$ for $D\otimes_\bf{F}\bf{F}_x$, and write $\inv_x(D)$ for the Hasse invariant of $D_x$. Recall that the Brauer group of $\bf{F}$ fits into an exact sequence
\begin{align*}
0\rar\Br(\bf{F})\rar\bigoplus_x\Br(\bf{F}_x)=\bigoplus_x\bQ/\bZ\rar\bQ/\bZ\rar0,
\end{align*}
where $x$ runs over all places of $C$, the first map sends $D\mapsto(D_x)_x$, and the second map sums the Hasse invariants of any element in $\bigoplus_x\Br(\bf{F}_x)$. In particular, we have $\inv_x(D)=0$ for cofinitely many $x$, that is, $D$ is split at $x$ for cofinitely many $x$. For any central division algebra $D$ in $\Br(\bf{F})$, its dimension equals the square of the least common denominator of $\inv_x(D)$ as $x$ runs over all places of $\bf{F}$.
We now fix a central division algebra $D$ over $\bf{F}$ of dimension $n^2$ such that $\inv_\infty(D)=0$, and we write $\text{Bad}$ for the set of places $x$ of $C$ where $\inv_x(D)\neq0$, that is, where $D$ ramifies. Then $\text{Bad}$ is finite, and $\infty$ does not lie in $\text{Bad}$. We write $C'$ for the open subscheme $C\ssm\text{Bad}$ of $C$.
\subsection{}
Next, we introduce \emph{orders} of $D$, which shall provide some sort of integrality structure on $D$.
\begin{defn*}
Let $\sD$ be a locally free (not necessarily commutative) $\sO_C$-algebra of rank $n^2$. We say $\sD$ is an \emph{order} of $D$ if its generic fiber $\sD_\bf{F}$ is isomorphic to $D$. For an order $\sD$ of $D$, we say $\sD$ is \emph{maximal} if it is maximal with respect to injective $\sO_C$-algebra morphisms $\sD'\inj{}\sD$ between orders of $D$.
\end{defn*}
We denote the completion of $\sD$ at $x$ using $\sD_x$, which is a free $\cO_x$-algebra of rank $n^2$, and we identify its generic fiber $\sD_x[\frac1{\vpi_x}]$ with $D_x$.
\subsection{}\label{ss:orders}
Let $\sD$ be an order of $D$. Let $U$ be an affine open subset of $C$, and let $M$ be an $\bf{F}$-basis of $D$ contained in $\sD(U)$. Because $D$ splits at cofinitely many places, checking valuations shows that $\cO_x\cdot M=\sD_x$ for cofinitely many $x$. Conversely, let $(\sD_x)_x$ be a collection of $\cO_x$-orders (where $x$ ranges over all places of $C$) such that there exists an $\bf{F}$-basis $M$ of $D$ for which $\cO_x\cdot M=\sD_x$ for cofinitely many $x$. One can use the Riemann--Roch theorem to show that the Zariski sheaf
\begin{align*}
U\mapsto\bigcap_{x'}\sD_{x'}\cap D,
\end{align*}
where $x'$ runs over all closed points in $U$, is an order of $D$.
This construction yields a bijection between isomorphism classes of orders of $D$ and collections $(\sM_x)_x$ of $\cO_x$-orders such that there exists an $\bf{F}$-basis $M$ of $D$ satisfying $\cO_x\cdot M=\sM_x$ for cofinitely many $x$. Note that $\sD$ is maximal if and only if every $\sM_x$ is maximal.
\subsection{}
From now on, fix a maximal order $\sD$ of $D$. We may now introduce $\sD$-elliptic sheaves. For any scheme $S$ over $\ka$, write $\sg$ for the absolute $q$-Frobenius on $S$, and write ${}^\sg$ for $(\id_C\times_\ka\sg)^*$.
\begin{defn*}
Let $S$ be a scheme over $\ka$. A \emph{$\sD$-elliptic sheaf} over $S$ is a commutative diagram of sheaves on $C\times_\ka S$
\begin{align*}
\xymatrixcolsep{5pc}
\xymatrixrowsep{3pc}
\xymatrix{\dotsm\ar[r]^-{j_{i-2}}& \sE_{i-1}\ar[r]^-{j_{i-1}} &\sE_i\ar[r]^-{j_i} &\sE_{i+1}\ar[r]^-{j_{i+1}} &\dotsm\\
\dotsm\ar[r]^-{{}^\sg\!j_{i-2}} \ar[ur]^-{t_{i-2}}& {}^\sg\!\sE_{i-1}\ar[r]^-{{}^\sg\!j_{i-1}} \ar[ur]^-{t_{i-1}}&{}^\sg\!\sE_i\ar[r]^-{{}^\sg\!j_i} \ar[ur]^-{t_i}&{}^\sg\!\sE_{i+1}\ar[r]^-{{}^\sg\!j_{i+1}} \ar[ur]^-{t_{i+1}}&\dotsm
}
\end{align*}
where the $\sE_i$ are locally free right $\sD\boxtimes\sO_S$-modules of rank $1$ (and hence vector bundles over $S$ of rank $n^2$), and the $t_i$ and $j_i$ are injective morphisms of right $\sD\boxtimes\sO_S$-modules satisfying the following conditions:
\begin{enumerate}[$\bullet$]
\item for all $i$, $\sE_{i+n\deg\infty}$ is isomorphic to $\sE_i(\infty)$, and this isomorphism identifies $\sE_i\inj{}\sE_{i+n\deg\infty}$ with the canonical injection $\sE_i\inj{}\sE_i(\infty)$,
\item there exists a morphism $i_\infty:S\rar\infty$ such that $\coker j_1$ is supported on the image of its graph $\Ga_\infty:S\rar\infty\times_\ka S$,
\item there exists a morphism $i_o:S\rar C'\ssm\infty$ such that $\coker t_1$ is supported on the image of its graph $\Ga_o:S\rar C\times_\ka S$,
\item when viewed as $\sO_S$-modules, the $\coker{j_i}$ and $\coker{t_i}$ are locally free of rank $n$.
\end{enumerate}
We denote $\sD$-elliptic sheaves by $(\sE_i,t_i,j_i)_i$. We say that $i_\infty$ is the \emph{pole} of $(\sE_i,t_i,j_i)_i$, and we say that $i_o$ is the \emph{zero} of $(\sE_i,t_i,j_i)_i$.
\end{defn*}
Since the image of $i_o$ is disjoint from $\infty$, we see $t_i$ induces an isomorphism ${}^\sg\!(\sE_i/\sE_{i-1})\rar^\sim\sE_{i+1}/\sE_i$. Thus $\coker j_{i+1}$ is supported on the image of the graph of $i_\infty\circ\sg^i$. And as the image of $i_o$ is disjoint from $\infty$, we see that $\coker t_i$ is supported on the image of $\Ga_o$ for all $i$.
\subsection{}\label{ss:modulispacedellipticsheavesnolevel}
We proceed to describe the \emph{moduli space} of $\sD$-elliptic sheaves. For any scheme $S$ over $\ka$, write $\cE\ell\ell_\sD(S)$ for the category whose objects are $\sD$-elliptic sheaves and whose morphisms are isomorphisms of $\sD$-elliptic sheaves. Then $\cE\ell\ell_\sD$ forms an fppf stack over $\ka$ \cite[(2.4)]{LRS93}, and the assignment sending a $\sD$-elliptic sheaf over $S$ to its zero $i_o:S\rar C'\ssm\infty$ yields a morphism $\cE\ell\ell_\sD\rar C'\ssm\infty$. We have the following result of Laumon--Rapoport--Stuhler.
\begin{prop*}[{\cite[(4.1)]{LRS93}}] Our $\cE\ell\ell_\sD$ is Deligne--Mumford stack, and the morphism $\cE\ell\ell_\sD\rar C'\ssm\infty$ is smooth of relative dimension $n-1$.
\end{prop*}
This morphism $\cE\ell\ell_\sD\rar C'\ssm\infty$ is the equi-characteristic analog of the structure morphism from an integral model of a Shimura variety to $\Spec\bZ$. Given that $\sD$-elliptic sheaves $(\sE_i,t_i,j_i)_i$ correspond to abelian schemes $A\rar S$ in our analogy, the zero of $(\sE_i,t_i,j_i)_i$ therefore corresponds to the characteristic of $S$.
\subsection{}
Just as with abelian varieties, we have a notion of \emph{level structure} for $\sD$-elliptic sheaves. The situation is more complicated when the characteristic divides the level, so we begin by excluding this case. Let $I$ be a finite closed subscheme of $C\ssm\infty$.
\begin{defn*}
Let $(\sE_i,t_i,j_i)_i$ be $\sD$-elliptic sheaf over $S$ such that the image of $i_o$ does not meet $I$. A \emph{level-$I$ structure} on $(\sE_i,t_i,j_i)_i$ is an isomorphism of right $\res\sD_I\boxtimes\sO_S$-modules
\begin{align*}
\io:\res\sD_I\boxtimes\sO_S\rar^\sim\res\sE_{I\times_\ka S}
\end{align*}
such that the diagram
\begin{align*}
\xymatrixcolsep{3pc}
\xymatrix{
{}^\sg\!(\res\sD_I\boxtimes\sO_S)\ar[r]^-{f_\sg}\ar[d]^-{{}^\sg\!\io} & \res\sD_I\boxtimes\sO_S\ar[d]^-\io\\
{}^\sg\!\res\sE_{I\times_\ka S}\ar[r]^-{\res{j}_{I\times_\ka S}} & \res\sE_{I\times_\ka S}
}
\end{align*}
commutes, where $f_\sg$ denotes the linearization of $\id\times_\ka\sg:\res\sD_I\boxtimes\sO_S\rar\res\sD_I\boxtimes\sO_S$. For any closed subscheme $I'$ of $I$ and level-$I$ structure $\io$, its restriction $\res\io_{I'}$ is a level-$I'$ structure.
\end{defn*}
\subsection{}\label{ss:modulispacedellipticsheaveslevel}
We now incorporate the data of level-$I$ structures in order to introduce more moduli spaces of $\sD$-elliptic sheaves. Write $\cE\ell\ell_{\sD,I}(S)$ for the category whose
\begin{enumerate}[$\bullet$]
\item objects are pairs $((\sE_i,t_i,j_i)_i,\io)$, where $(\sE_i,t_i,j_i)_i$ is a $\sD$-elliptic sheaf over $S$, and $\io$ is a level-$I$ structure on $(\sE_i,t_i,j_i)_i$,
\item morphisms are isomorphisms of pairs.
\end{enumerate}
Then $\cE\ell\ell_{\sD,I}$ forms a Deligne--Mumford stack over $\ka$ \cite[(4.1)]{LRS93}. We naturally identify $\cE\ell\ell_{\sD,\varnothing}=\cE\ell\ell_\sD$, and as in \ref{ss:modulispacedellipticsheavesnolevel}, we have a morphism $\cE\ell\ell_{\sD,I}\rar C'\ssm(I\cup\infty)$ given by sending any $\sD$-elliptic sheaf with level-$I$ structure to its zero.
By abuse of notation, we shall write $\sD\otimes\bO/\bK_I$ for the product ring $\prod_x\sD_x/\bK_{I,x}\sD_x$, where $x$ runs over all places in $C$. Since $\bK_{I,x}=\cO_x$ for cofinitely many $x$, we see that $\sD\otimes\bO/\bK_I$ is finite. As $I$ varies, the stacks $\cE\ell\ell_{\sD,I}$ are related as follows.
\begin{prop*}[{\cite[(4.1)]{LRS93}}] For any closed subscheme $I'$ of $I$, the restriction morphism $\cE\ell\ell_{\sD,I}\rar\res{\cE\ell\ell_{\sD,I'}}_{C'\ssm(I\cup\infty)}$ is finite representable Galois, and the right action of
\begin{align*}
\ker\left((\sD\otimes\bO/\bK_I)^\times\rar(\sD\otimes\bO/\bK_{I'})^\times\right)
\end{align*}
on $\cE\ell\ell_{\sD,I}\rar\res{\cE\ell\ell_{\sD,I'}}_{C'\ssm(I\cup\infty)}$ given by multiplication on the level-$I$ structure yields the Galois action.
\end{prop*}
By combining this with Proposition \ref{ss:modulispacedellipticsheavesnolevel}, we see that $\cE\ell\ell_{\sD,I}\rar C'\ssm(I\cup\infty)$ is smooth of relative dimension $n-1$.
\subsection{}\label{ss:dellipticsheavestwistingbybundle}
In anticipation for studying the cohomology of $\cE\ell\ell_{\sD,I}$, we present some operations that one can perform on $\cE\ell\ell_{\sD,I}$. First, write $\Pic_I(C)$ for the group
\begin{align*}
\Pic_I(C)\deq\{\mbox{pairs }(\sL,\be)\}/\!\sim,
\end{align*}
where $\sL$ is a line bundle on $C$, and $\be$ is an isomorphism $\be:\sO_I\rar^\sim\res\sL_I$ of $\sO_I$-modules, under tensor product. As $\sD$-elliptic sheaves consist of vector bundles, we can twist them by line bundles. That is, for any $\sD$-elliptic sheaf $(\sE_i,t_i,j_i)_i$ over $S$ with level-$I$ structure $\io$, we obtain another $\sD$-elliptic sheaf over $S$ with level-$I$ structure as follows:
\begin{align*}
(\sL,\be)\cdot((\sE_i,t_i,j_i)_i,\io) \deq ((\sE_i\otimes(\sL\boxtimes\sO_S),t_i\otimes\id,j_i\otimes\id)_i,\io\otimes\be).
\end{align*}
This yields an action of $\Pic_I(C)$ on $\cE\ell\ell_{\sD,I}$ over $C'\ssm(I\cup\infty)$.
\subsection{}
As $I$ varies, the moduli spaces of $\sD$-elliptic sheaves with level-$I$ structure form a system of stacks $(\cE\ell\ell_{\sD,I})_I$, which corresponds to how Shimura varieties form systems of schemes. We package this system into one object as follows. First, let $T$ be a finite set of places of $\bf{F}$. Write $\bA^T$ for the ring of adeles away from $T$, and more generally for any subset $X$ of $\bA$, write $X^T$ for the projection of $X$ to $\bA^T$. Similarly, write $\bA_T$ for the ring of adeles at $T$, and write $X_T$ for the projection of $X$ to $\bA_T$. We write $\cE\ell\ell_\sD^\infty$ for the inverse limit
\begin{align*}
\cE\ell\ell_\sD^\infty \deq \invlim_I\cE\ell\ell_{\sD,I},
\end{align*}
where $I$ ranges over all finite closed subschemes of $C$ that do not meet $\infty$. This inverse limit yields a morphism $\cE\ell\ell_\sD^\infty\rar\Spec{\bf{F}}$. By abuse of notation, write $\bO^\infty$ for the $\sO_C$-algebra given by
\begin{align*}
U\mapsto\prod_{x'}\cO_{x'},
\end{align*}
where $x'$ runs over all closed points in $U\ssm\infty$. We see that $\cE\ell\ell_\sD^\infty(S)$ is the category whose \cite[(7.1)]{LRS93}
\begin{enumerate}[$\bullet$]
\item objects are pairs $((\sE_i,t_i,j_i)_i,\io^\infty)$, where $(\sE_i,t_i,j_i)_i$ is a $\sD$-elliptic sheaf over $S$ and $\io^\infty$ is a $(\sD\otimes_{\sO_C}\bO^\infty)\boxtimes\sO_S$-linear isomorphism
\begin{align*}
\io^\infty:(\sD\otimes_{\sO_C}\bO^\infty)\boxtimes\sO_S\rar^\sim\sE\otimes(\bO^\infty\boxtimes\sO_S)
\end{align*}
such that the diagram
\begin{align*}
\xymatrixcolsep{3pc}
\xymatrix{
{}^\sg\!((\sD\otimes_{\sO_C}\bO^\infty)\boxtimes\sO_S)\ar[r]^-{f_\sg}\ar[d]^-{{}^\sg\!\io^\infty} & (\sD\otimes_{\sO_C}\bO^\infty)\boxtimes\sO_S\ar[d]^-{\io^\infty}\\
{}^\sg\!\sE\otimes(\bO^\infty\boxtimes\sO_S)\ar[r]^-{j\boxtimes\id} & \sE\otimes(\bO^\infty\boxtimes\sO_S)
}
\end{align*}
commutes, where $f_\sg$ denotes the linearization of $\id\times_\ka\sg:(\sD\otimes_{\sO_C}\bO^\infty)\boxtimes\sO_S\rar(\sD\otimes_{\sO_C}\bO^\infty)\boxtimes\sO_S$,
\item morphisms are isomorphisms of pairs.
\end{enumerate}
\subsection{}
The action of twisting by line bundles is compatible with the inverse limit, so $\cE\ell\ell_\sD^\infty$ has an action of
\begin{align*}
\invlim_I\Pic_I(C) = \bf{F}^\times\bs\bA^\times/\bf{F}_\infty^\times = \bf{F}^{\infty,\times}\bs\bA^{\infty,\times}
\end{align*}
over $\bf{F}$, where we identify the idele $(a_x)_x$ with the line bundle $\sO_C(-\sum_xx(a_x))$ along with the trivialization induced via multiplication by $(a_x)_x$. In particular, $\cE\ell\ell_\sD^\infty$ has an action of $\bA^{\infty,\times}$. We see that $\cE\ell\ell_\sD^\infty$ also has a right action of
\begin{align*}
\invlim_I(\sD\otimes\bO/\bK_I)^\times = (\sD\otimes\bO^\infty)^\times
\end{align*}
over $\bf{F}$ given by multiplication on the level structure, and both of these actions actually arise from a right action of $(D\otimes\bA^\infty)^\times$ on $\cE\ell\ell_\sD^\infty$ \cite[(7.3)]{LRS93}. This is the \emph{Hecke} action, analogous to the Hecke action on Shimura varieties. Writing $\sK_I$ for the subgroup
\begin{align*}
\sK_I\deq\ker\left((\sD\otimes\bO)^\times\rar(\sD\otimes\cO/\bK_I)^\times\right),
\end{align*}
we see that Proposition \ref{ss:modulispacedellipticsheaveslevel} implies $\cE\ell\ell_\sD^\infty/\sK_I^\infty=\cE\ell\ell_{\sD,I}$.
\subsection{}
Since $\sD$-elliptic sheaves are essentially vector bundles, they have a notion of \emph{degree}.
\begin{defn*}
Let $(\sE_i,t_i,j_i)_i$ be a $\sD$-elliptic sheaf. We say its \emph{degree} is the locally constant function
\begin{align*}
\frac{\deg\sE_1-\deg\sD}n
\end{align*}
on the base scheme $S$, where we use $1$ instead of $0$ to avoid confusion with $o$. One can show that the degree is integer-valued \cite[p.~49]{Laf97}.
\end{defn*}
By partitioning the base scheme $S$ by the degree of $(\sE_i,t_i,j_i)_i$, we can write $\cE\ell\ell_{\sD,I}$ as a disjoint union
\begin{align*}
\cE\ell\ell_{\sD,I} = \coprod_d\cE\ell\ell_{\sD,I,d},
\end{align*}
where $d$ ranges over all integers, and $\cE\ell\ell_{\sD,I,d}$ is the open substack of $\sD$-elliptic sheaves with level-$I$ structure whose degree equals $d$. Note that the restriction morphisms $\cE\ell\ell_{\sD,I}\rar\cE\ell\ell_{\sD,I'}$ preserve degree, so we obtain a similar decomposition
\begin{align*}
\cE\ell\ell_\sD^\infty = \coprod_d\cE\ell\ell_{\sD,d}^\infty,
\end{align*}
where $\cE\ell\ell_{\sD,d}^\infty=\invlim_I\cE\ell\ell_{\sD,I,d}$ for $I$ running over all finite closed subschemes of $C\ssm\infty$.
\subsection{}\label{ss:dellipticsheavestranslation}
We shall define an action of $\bZ$ on $\cE\ell\ell_{\sD,I}$ over $C'\ssm(I\cup\infty)$ as follows. For any integer $l$ and $\sD$-elliptic sheaf $(\sE_i,t_i,j_i)_i$, write
\begin{align*}
[l](\sE_i,t_i,j_i)_i\deq(\sE_{i+l},t_{i+l},j_{i+l})_i,
\end{align*}
that is, $[l]$ acts via translation by $l$ on the index $i$. Then $[l](\sE_i,t_i,j_i)_i$ also forms a $\sD$-elliptic sheaf, so this yields an action of $\bZ$ on $\cE\ell\ell_{\sD,I}$. The definition of a $\sD$-elliptic sheaf implies that $\deg\sE_{i+l}=\deg\sE_i+nl$, so the degree of $[l](\cE_i,t_i,j_i)_i$ is $l$ plus that of $(\cE_i,t_i,j_i)_i$. This allows us to identify the quotient stack $\cE\ell\ell_{\sD,I}/\bZ$ with
\begin{align*}
\cE\ell\ell_{\sD,I}/\bZ = \cE\ell\ell_{\sD,I,0}
\end{align*}
as stacks over $\ka$. The translation action of $\bZ$ commutes with restriction morphisms as well as the Hecke action, so we obtain a similar description of $\cE\ell\ell_\sD^\infty/\bZ$. The Hecke action also descends to $\cE\ell\ell_{\sD,I}/\bZ$. By passing to this quotient stack (or equivalently, by restricting the degree), we obtain the following finite-type representability result.
\begin{prop*}[{\cite[(6.2)]{LRS93}}]
Suppose $I$ is nonempty. Then $\cE\ell\ell_{\sD,I}/\bZ$ is actually a projective scheme over $C'\ssm(I\cup\infty)$.
\end{prop*}
Thus $\cE\ell\ell_\sD^\infty/\bZ=\invlim_I\cE\ell\ell_{\sD,I}/\bZ$ is also representable by a scheme.
\subsection{}
Before we proceed to the cohomology of our moduli spaces, we introduce a covering of $\cE\ell\ell_{\sD,I}$ which we shall use to construct our coefficient sheaves. Write $\ov{B}$ for the central division algebra over $\bf{F}_\infty$ of Hasse invariant $-\frac1n$.
\begin{prop*}[{\cite[(8.11)]{LRS93}}]
There exists a pro-Galois covering
\begin{align*}
\wt{\cE\ell\ell}_{\sD,I}\rar\cE\ell\ell_{\sD,I},
\end{align*}
where $\wt{\cE\ell\ell}_{\sD,I}$ is a scheme, whose Galois group is given by a right action of $\ov{B}^\times/\vpi_\infty^\bZ$. This covering is compatible with restriction morphisms $\cE\ell\ell_{\sD,I}\rar\cE\ell\ell_{\sD,I'}$, so taking the inverse limit yields an analogous pro-Galois covering
\begin{align*}
\wt{\cE\ell\ell}_\sD^\infty\rar\cE\ell\ell_\sD^\infty
\end{align*}
whose Galois group is given by a right action of $\ov{B}^\times/\vpi_\infty^\bZ$.
\end{prop*}
Briefly, the covering $\wt{\cE\ell\ell}_{\sD,I}\rar\cE\ell\ell_{\sD,I}$ is given as follows. Given a $\sD$-elliptic sheaf $(\sE_i,t_i,j_i)_i$, we can construct an object resembling a local shtuka at the place $\infty$ (in a fashion similar to the construction to be given in \ref{ss:dellipticsheavestolocalshtuka}). This ``local shtuka at $\infty$'' will always be isomorphic to a fixed object analogous to the local shtuka of slope $-\frac1n$ under the Dieudonn\'e--Manin classification \cite[(2.4.5)]{Lau96}, and the covering $\wt{\cE\ell\ell}_{\sD,I}\rar\cE\ell\ell_{\sD,I}$ parametrizes isomorphisms between our ``local shtuka at $\infty$'' and this fixed object. The Galois action is given by composition with this isomorphism, as the endomorphism ring of a local shtuka of slope $-\frac1n$ at $\infty$ is $\ov{B}$.
The stack $\wt{\cE\ell\ell}_{\sD,I}$ also has an action of $\bZ$ by translation, and it is preserved under the morphism $\wt{\cE\ell\ell}_{\sD,I}\rar\cE\ell\ell_{\sD,I}$. Thus we obtain a pro-Galois morphism $\wt{\cE\ell\ell}_{\sD,I}/\bZ\rar \cE\ell\ell_{\sD,I}/\bZ$ whose Galois group is given by a right action of $\ov{B}^\times/\vpi_\infty^\bZ$. This yields a similar pro-Galois morphism $\wt{\cE\ell\ell}_\sD^\infty/\bZ\rar \cE\ell\ell_\sD^\infty/\bZ$ as well.
The space $\wt{\cE\ell\ell}_\sD^\infty$ is the equi-characteristic analog of the Hermitian symmetric space covering a Shimura variety.
\subsection{}\label{ss:dellipticsheavescohomology}
At this point, we can finally introduce the cohomology of our moduli spaces. Fix a separable closure $\bf{F}^\sep$ of $\bf{F}$, and view all separable extensions of $\bf{F}$ as lying in $\bf{F}^\sep$. For every place $x$ of $C$, choose a separable closure $\bf{F}^\sep_x$ of $\bf{F}_x$, fix an embedding $\bf{F}^\sep\inj{}\bf{F}^\sep_x$, and form the absolute Galois groups $G_\bf{F}$ and $G_x$ of $\bf{F}$ and $\bf{F}_x$, respectively, with respect to these separable closures. Write $\bC_x$ for the completion of $\bf{F}^\sep_x$. We view $G_x$ as a subgroup of $G_\bf{F}$ via our embedding, and we denote the arithmetic $q_x$-Frobenius at $x$ using $\sg_x$.
Fix an irreducible smooth representation
\begin{align*}
\xi:\ov{B}^\times/\vpi_\infty^\bZ\rar\GL_N(\ov\bQ_\ell).
\end{align*}
Since $\ov{B}^\times/\vpi_\infty^\bZ$ is compact, we see $\xi$ has finite image and is therefore defined over a finite extension of $\bQ_\ell$ in $\ov\bQ_\ell$. Write $\cL_{\xi,I}$ for the $\ell$-adic sheaf on $\cE\ell\ell_{\sD,I}/\bZ$ induced from $\xi$ and the pro-Galois morphism $\wt{\cE\ell\ell}_{\sD,I}/\bZ\rar \cE\ell\ell_{\sD,I}/\bZ$ via monodromy, and write $\cL_{\xi}$ for the analogous $\ell$-adic sheaf on $\cE\ell\ell^\infty_\sD/\bZ$. Form the $\ov\bQ_\ell$-vector space
\begin{align*}
H^i_{\xi,\eta} \deq H^i(\cE\ell\ell^\infty_{\sD,\bf{F}^\sep}/\bZ,\cL_{\xi}) = \dirlim_IH^i(\cE\ell\ell_{\sD,I,\bf{F}^\sep}/\bZ,\cL_{\xi,I,\bf{F}^\sep}),
\end{align*}
where $I$ runs over all finite closed subschemes of $C\ssm\infty$. This has a left action of $(D\otimes\bA^\infty)^\times\times G_\bf{F}$, and it vanishes for sufficiently large $i$. Proposition \ref{ss:modulispacedellipticsheaveslevel} implies that
\begin{align*}
(H^i_{\xi,\eta})^{\sK_I^\infty} = H^i(\cE\ell\ell_{\sD,I,\bf{F}^\sep}/\bZ,\cL_{\xi,I,\bf{F}^\sep}),
\end{align*}
which shows that $H^i_{\xi,\eta}$ is an admissible/continuous representation as in \cite[p.~24]{HT01} of $(D\otimes\bA^\infty)^\times\times G_\bf{F}$ over $\ov\bQ_\ell$. Furthermore, for any $g$ in $(D\otimes\bA^\infty)^\times$, the action of
\begin{align*}
\frac1{\vol(\sK_I^\infty)}\bf1_{\sK_I^\infty g\sK_I^\infty}\in C^\infty_c((D\otimes\bA^\infty)^\times)
\end{align*}
on $H^i_{\xi,\eta}$ is induced by a correspondence \cite[(7.5)]{LRS93}
\begin{align*}
\xymatrix{& \ar[ld]_-{c_{1,\bf{F}}}\cE\ell\ell_{\sD,\bf{F}}^\infty/(\bZ\times(\sK_I^\infty\cap g^{-1}\sK_I^\infty g))\ar[rd]^-{c_{2,\bf{F}}} & \\
\cE\ell\ell_{\sD,\bf{F}}^\infty/(\bZ\times\sK_I^\infty) & & \cE\ell\ell_{\sD,\bf{F}}^\infty/(\bZ\times\sK_I^\infty)
}
\end{align*}
over $\bf{F}$, where $c_{1,\bf{F}}$ is induced by further quotienting by $\sK_I^\infty$, and $c_{2,\bf{F}}$ is induced by the right action of $g^{-1}$ followed by quotienting by $\sK_I^\infty$. This is the equi-characteristic analog of \emph{Hecke correspondences}.
Proposition \ref{ss:modulispacedellipticsheaveslevel} implies that $c_{1,\bf{F}}$ and $c_{2,\bf{F}}$ are finite \'etale. We write $c_\bf{F}:\cE\ell\ell_{\sD,I,\bf{F}}/\bZ\dashrightarrow\cE\ell\ell_{\sD,I,\bf{F}}/\bZ$ for the correspondence formed by $c_{1,\bf{F}}$ and $c_{2,\bf{F}}$, and we write $[H_\xi]$ for the virtual representation $[H_\xi]\deq\sum_{i=0}^\infty(-1)^iH^i_{\xi,\eta}$.
\subsection{}\label{ss:dellipticsheavestolocalshtukazero}
Now we introduce a construction that takes any $\sD$-elliptic sheaf over a certain base and yields an object resembling a local shtuka. Let $o$ be a closed point in $C'\ssm\infty$, let $S$ be a scheme over $\Spec\cO_o$, and let $(\sE_i,t_i,j_i)_i$ be a $\sD$-elliptic sheaf over $S$.
Let $\Ga_o$ be the image of the graph of $i_o:S\rar C\times_\ka S$, and denote completions along $\Ga_o$ using $(-)^\wedge_{\Ga_o}$. Then $(\sE_1)^\wedge_{\Ga_o}$ is a vector bundle on $(C\times_\ka S)^\wedge_{\Ga_o}$ of rank $n^2$ with a right action of $\sD_o=\M_n(\cO_o)$, and $(j_1)^\wedge_{\Ga_o}$ is an isomorphism because $\Ga_o$ lies away from $\infty$. Thus we may form the composed morphism
\begin{align*}
(j_1)^{\wedge,-1}_{\Ga_o}\circ(t_1)^\wedge_{\Ga_o}:({}^\sg\!\sE_1)^\wedge_{\Ga_o}\rar(\sE_1)^\wedge_{\Ga_o}.
\end{align*}
We see that the adic pullback of $(j_1)^{\wedge,-1}_{\Ga_o}\circ(t_1)^\wedge_{\Ga_o}$ as in \cite[p.~370]{Ber96} to $\bf{F}_o$ is an isomorphism, and its cokernel is a vector bundle on $S$ of rank $n$.
We may identify $(C\times_\ka S)^\wedge_{\Ga_o}$ with the locally ringed space whose support is $\abs{S}$ and whose structure sheaf is $\sO_S\llb{\vpi_o}$ \cite[Lemma 5.3]{EH14}. From this point of view, the pair
\begin{align*}
((\sE_1)^\wedge_{\Ga_o},(j_1)^{\wedge,-1}_{\Ga_o}\circ(t_1)^\wedge_{\Ga_o})
\end{align*}
corresponds to a pair $(\sM_o,\sF_o)$, where $\sM_o$ is a locally free $\sO_S\llb{\vpi_o}$-module of rank $n^2$ with a right $\sO_S\llb{\vpi_o}$-linear action of $\M_n(\cO_o)$, and $\sF_o:\sg^*_o\sM_o\rar\sM_o$ is an $\sO_S\llb{\vpi_o}$-linear morphism such that $\coker\sF_o$ is a locally free $\sO_S$-module of rank $n$. Note that $\sF_o[\frac1{\vpi_o}]$ is an isomorphism.
Now $\sM_o$ is a free right $\M_n(\cO_o)$-module of rank $1$. Applying Morita equivalence to the right $\M_n(\cO_o)$-action shows that there exists pair $(\sM_o',\sF_o')$ satisfying
\begin{align*}
(\sM_o,\sF_o) = (\sM_o'^{\oplus n},\sF_o'^{\oplus n}),
\end{align*}
where $\sM_o'$ is a locally free $\sO_S\llb{\vpi_o}$-module of rank $n$, and $\sF_o':\sg^*_o\sM_o'\rar\sM_o'$ is an $\sO_S\llb{\vpi_o}$-linear morphism whose cokernel is a locally free $\sO_S$-module of rank $1$ and whose generic fiber is an isomorphism. Note that if $S$ is actually a scheme over $\Spf\cO_o$, then $(\sM_o',\sF_o')$ actually forms an effective minuscule local shtuka over $S$ of rank $n$ and dimension $1$.
\subsection{}
With the above construction, we can treat level structures in the case when characteristics divide the level. As with abelian varieties, this requires a notion of \emph{Drinfeld level structure}.
Begin by recalling some material from \S\ref{s:deformationspaces}. We have a notion of finite $\ka_o$-shtukas, which are truncated versions of local shtukas, and they correspond to certain module schemes called strict $\ka_o$-modules under a Dieudonn\'e equivalence. Now the quotient module $\sM'_o/\vpi_o^m$ is a finite $\ka_o$-shtuka, and we denote the corresponding finite $\ka_o$-strict module using $\Dr(\sM'_o/\vpi_o^m)$. As with local shtukas, we define Drinfeld level structures via using $\Dr(-)$ to pass to module schemes.
\begin{defn*}
Let $S$ be a scheme over $\Spec\cO_o$, and let $(\sE_i,t_i,j_i)_i$ be a $\sD$-elliptic sheaf over $S$. We say a \emph{Drinfeld level-$m$ structure} on $(\sE_i,t_i,j_i)_i$ is a Drinfeld level-$m$ structure on $\Dr(\sM'_o/\vpi_o^m)$, that is, an $\cO_o/\vpi_o^m$-module morphism
\begin{align*}
\al:(\vpi^{-m}\cO_o/\cO_o)^n\rar\Dr(\sM'_o/\vpi^m_o)(S)
\end{align*}
such that the collection of all $\al(x)$ for $x$ in $(\vpi^{-m}\cO_o/\cO_o)^n$ forms a full set of sections of $\Dr(\sM'_o/\vpi^m_o)$ as in \cite[(1.8.2)]{KM85}. For any non-negative integer $m'\leq m$ and Drinfeld level-$m$ structure $\al$, its restriction to $(\vpi^{-m'}_o\cO_o/\cO_o)^n$ is a Drinfeld level-$m'$ structure.
\end{defn*}
\subsection{}
We may finally define moduli spaces of $\sD$-elliptic sheaves at bad reduction. For any scheme $S$ over $\Spec\cO_o$, write $\cM_{I}^*(S)$ for the category whose
\begin{enumerate}[$\bullet$]
\item objects are triples $((\sE_i,t_i,j_i)_i,\io,\al)$, where $(\sE_i,t_i,j_i)_i$ is a $\sD$-elliptic sheaf over $S$, $\io$ is a level $(I\ssm o)$-structure on $(\sE_i,t_i,j_i)_i$, and $\al$ is a Drinfeld level-$m$ structure on $(\sE_i,t_i,j_i)_i$, where $m$ is the multiplicity of $o$ in $I$,
\item morphisms are isomorphisms of pairs.
\end{enumerate}
Then $\cM_{I}^*$ forms an fppf stack over $\ka$, and when $I$ does not contain $o$, we see that $\cM_{I}^*$ is simply the pullback of $\cE\ell\ell_{\sD,I}$ to $\Spec\cO_o$.
By sending a $\sD$-elliptic sheaf to its zero, we obtain a morphism $\cM_{I}^*\rar\Spec\cO_o$. For any closed subscheme $I'$ of $I$, restriction of level structures gives us a morphism $\cM_{I}^*\rar\cM_{I'}^*$ over $\Spec\cO_o$, and partitioning by degrees allows us to obtain a decomposition
\begin{align*}
\cM_{I}^* = \coprod_d\cM_{I,d}^*
\end{align*}
that respects restriction morphisms, where $d$ runs over all integers.
We see that the translation action of $\bZ$ from \ref{ss:dellipticsheavestranslation} naturally extends to an action on $\cM_I^*$, and we write $\cM_{I}$ for the quotient stack $\cM_{I}\deq\cM_I/\bZ$. We have the following representability result, which follows from work of Katz--Mazur.
\begin{prop*}[{\cite[(1.10.13)]{KM85}}]
The restriction morphism $\cM_{I}^*\rar\cM_{I'}^*$ is finite and representable.
\end{prop*}
By combining this with Proposition \ref{ss:dellipticsheavestranslation}, we see that $\cM_{I}$ is a projective scheme of dimension $n-1$ over $\Spec\cO_o$ whenever $I\ssm o$ is nonempty. The scheme $\cM_{I}$ shall be our focus in the next few sections.
\subsection{}\label{ss:badreductionheckecorrespondence}
In this subsection, we extend the Hecke action to $\cM_{I}$. We use $o^m$ to denote the finite closed subscheme of $C$ supported on $o$ with multiplicity $m$. Since Drinfeld level-$m$ structures over $\bf{F}_o$ are equivalent to level-$o^m$ structures over $\bf{F}_o$ \cite[Proposition 7.1.3]{Boy99}, the generic fiber $\cM_{I,\bf{F}_o}$ is isomorphic to the pullback $\cE\ell\ell_{\sD,I,\bf{F}_o}/\bZ$. Under this identification, the right action of $(D\otimes\bA^\infty)^\times$ on $\cE\ell\ell_{\sD,\bf{F}_o}^\infty/a^\bZ$ extends \cite[p.~599]{Boy99} to a right action on the inverse limit
\begin{align*}
\cM^\infty \deq \invlim_I\cM_{I},
\end{align*}
where $I$ runs over all finite closed subschemes of $C\ssm\infty$. This action satisfies $\cM^\infty/\sK_I^\infty=\cM_{I}$. Furthermore, for any $g$ in $D\otimes\bA^\infty$, the correspondence from \ref{ss:dellipticsheavescohomology} extends to a correspondence \cite[p.~600]{Boy99}
\begin{align*}
\xymatrix{& \ar[ld]_-{c_{1,\cO_o}}\cM^\infty/(\sK_I^\infty\cap g^{-1}\sK_I^\infty g)\ar[rd]^-{c_{2,\cO_o}} & \\
\cM^\infty/\sK_I^\infty & & \cM^\infty/\sK_I^\infty
}
\end{align*}
where $c_{1,\cO_o}$ and $c_{2,\cO_o}$ are finite morphisms over $\Spec\cO_o$. We write $c_{\cO_o}:\cM_{I}\dashrightarrow\cM_{I}^\infty$ for the correspondence formed by $c_{1,\cO_o}$ and $c_{2,\cO_o}$.
\subsection{}\label{ss:boyerserretate}
Next, we present the \emph{Serre--Tate theorem} for $\sD$-elliptic sheaves, which relates deformations of $\sD$-elliptic sheaves with deformations of their associated local shtukas. Let $m$ be the multiplicity of $o$ in $I$, let $z$ be a $\ov\ka$-point of $\cM_{I}$, and let $(\sE_i,t_i,j_i)_i$ be the corresponding $\sD$-elliptic sheaf over $\ov\ka$ with level $(I\ssm o)$-structure $\io$ and Drinfeld level-$m$ structure $\al$. Then $\al$ yields a Drinfeld level-$m$ structure for the local shtuka $\sM'_o$ formed in \ref{ss:dellipticsheavestolocalshtukazero}. Recall from \ref{ss:effectiveminusculelocalshtukaconjugacy} that $\sM'_o$ corresponds to some $\de_o$ in
\begin{align*}
\GL_n(\br\cO_o)\diag(\vpi_o,1,\dotsc,1)\GL_n(\br\cO_o)
\end{align*}
uniquely up to $\GL_n(\br\cO_o)$-$\sg_o$-conjugacy, and recall that the deformation space of $(\sM'_o,\al)$ is $\br\fX_{\de_o,\al}$. Write $(\cM_{I,\br\cO_o})_z^\wedge$ for the completion of $\cM_{I,\br\cO_o}$ at $z$. Boyer proves the following comparison result for deformation spaces.
\begin{prop*}[{\cite[Theorem 7.4.4]{Boy99}}]
Our $((\sE_i,t_i,j_i)_i,\io,\al)\mapsto(\sM'_o,\al)$ induces an isomorphism $(\cM_{I,\br\cO_o})_z^\wedge\rar^\sim\br\fX_{\de_o,\al}$ over $\Spf\br\cO_o$ that preserves the right action of $\GL_n(\cO_o)$ as well as the Weil action.
\end{prop*}
\subsection{}\label{ss:extendedserretate}
We extend the above results to $\br\fX_{\de_o,m}$, where $\br\fX_{\de_o,m}$ is the deformation space of $\sM'_o$ along with a Drinfeld level-$m$ structure as in \ref{ss:deformationalgebraicclosure}. Write $\pi:\cM_{I,\br\cO_o}\rar\cM_{I\ssm o,\br\cO_o}$ for the restriction morphism. We see that $\pi^{-1}(\pi(z))$ corresponds to all possible Drinfeld level-$m$ structures on $(\sE_i,t_i,j_i)_i$. Therefore the isomorphism in Proposition \ref{ss:boyerserretate} also induces an isomorphism $(\cM_{I,\br\cO_o})^\wedge_{\pi^{-1}(\pi(z))}\rar^\sim\br\fX_{\de_o,m}$ over $\Spf\br\cO_o$, where $(\cM_{I,\br\cO_o})^\wedge_{\pi^{-1}(\pi(z))}$ is the completion of $\cM_{I,\br\cO_o}$ along $\pi^{-1}(\pi(z))$, and this isomorphism preserves the right action of $\GL_n(\cO_o)$ as well as the Weil action.
\subsection{}\label{ss:dellipticsheavestolocalshtuka}
We conclude this section by discussing a variant of the construction from \ref{ss:dellipticsheavestolocalshtukazero}. Let $S$ be a scheme over $\Spec\cO_o$, let $(\sE_i,t_i,j_i)_i$ be a $\sD$-elliptic sheaf over $S$, let $x$ be a closed point in $C\ssm o$, and let $f:S\rar C$ be a morphism over $\ka$ whose image is $x$.
Write $\Ga_f$ for the image of the graph of $f$ in $C\times_\ka S$, and denote completions along $\Ga_f$ using $(-)^\wedge_{\Ga_f}$. Then $(\sE_1)_{\Ga_f}^\wedge$ is a vector bundle on $(C\times_\ka S)_{\Ga_f}^\wedge$ of rank $n^2$ with a right action of $\sD_x$. The adic generic fibers $(j_1)^\wedge_{\Ga_f,\bf{F}_x}$ and $(t_1)^\wedge_{\Ga_f,\bf{F}_x}$ as in \cite[p.~370]{Ber96} are isomorphisms, so the composed morphism
\begin{align*}
(j_1)^{\wedge,-1}_{\Ga_f,\bf{F}_x}\circ(t_1)^\wedge_{\Ga_f,\bf{F}_x}:({}^\sg\!\sE_1)^\wedge_{\Ga_f,\bf{F}_x}\rar^\sim(\sE_1)^\wedge_{\Ga_f,\bf{F}_x}
\end{align*}
is also an isomorphism. Similarly to \ref{ss:dellipticsheavestolocalshtukazero}, we may identify $(C\times_\ka S)^\wedge_{\Ga_f}$ with the locally ringed space whose support is $\abs{S}$ and whose structure sheaf is $\sO_S\llb{\vpi_x}$ \cite[Lemma 5.3]{EH14}. From this point of view, the pair
\begin{align*}
((\sE_1)_{\Ga_f}^\wedge,(j_1)^{\wedge,-1}_{\Ga_f,\bf{F}_x}\circ(t_1)^\wedge_{\Ga_f,\bf{F}_x})
\end{align*}
corresponds to a pair $(\sM_x,\sF_x)$, where $\sM_x$ is a locally free $\sO_S\llb{\vpi_x}$-module of rank $n^2$ with a right $\sO_S\llb{\vpi_x}$-linear action of $\sD_x$, and $\sF_x:\sg^*_x\sM_x[\frac1{\vpi_x}]\rar^\sim\sM_x[\textstyle\frac1{\vpi_x}]$ is an $\sO_S\llp{\vpi_x}$-linear isomorphism that commutes with the right $\sD_x$-action. Note that $\sM_x$ is a free right $\sD_x$-module of rank $1$.
When $x$ does not equal $\infty$, the graph $\Ga_f$ lies away from both $o\times_\ka S$ and $\infty\times_\ka S$, so the completions $(t_1)_{\Ga_f}^\wedge$ and $(j_1)_{\Ga_f}^\wedge$ are isomorphisms. In this case, the morphism $\sF_x$ is the localization of an $\sO_S\llb{\vpi_x}$-module isomorphism $\sg^*_x\sM_x\rar^\sim\sM_x$, which we also refer to using $\sF_x$ by abuse of notation.
Finally, in the case when $x$ equals $\infty$, we have $\sD_\infty=\M_n(\cO_\infty)$ because $D$ splits at $\infty$. Applying Morita equivalence to the right $\M_n(\cO_\infty)$-action on $\sM_\infty$ provides a pair $(\sM_\infty',\sF_\infty')$ that satisfies
\begin{align*}
(\sM_\infty,\sF_\infty) = (\sM_\infty'^{\oplus n},\sF_\infty'^{\oplus n}),
\end{align*}
where $\sM_\infty'$ is a locally free $\sO_S\llb{\vpi_\infty}$-module of rank $n$, and $\sF_\infty':\sg_\infty^*\sM_\infty'[\frac1{\vpi_\infty}]\rar^\sim\sM_\infty'[\textstyle\frac1{\vpi_\infty}]$ is an $\sO_S\llp{\vpi_\infty}$-linear isomorphism. Furthermore, if $S$ is the spectrum of $\ov\ka$, then the isogeny class of $\sM_\infty'$ has slope $-\frac1n$ \cite[(9.8).(i)]{LRS93} under the Dieudonn\'e--Manin classification \cite[(2.4.5)]{Lau96}.
\section{A nearby cycles calculation and semisimple trace}\label{s:nearbycycles}
Write $\bf{F}_{o,r}$ for the $r$-th degree unramified extension of $\bf{F}_o$, and recall our test function $\phi_{\tau,h}$ in $C^\infty_c(\GL_n(\bf{F}_{o,r}))$ as in \S\ref{s:deformationspaces}. Our goal in this section is to express the integral
\begin{align*}
\int_{\sg_o^{-r}I_{\bf{F}_o}}\!\dif\tau\,\phi_{\tau,h}
\end{align*}
in terms of representations of $\GL_n(\cO_o)$. We shall use this description in \S\ref{s:secondinductivelemma} to show that the preimage of any unramified representation under $\pi\mapsto\rec\pi$ remains unramified, which plays an important role in our proof that $\rec$ is bijective.
To begin, we present a geometric calculation of the inertia invariants of nearby cycles, due to Scholze. This calculation relies on a case of \emph{Grothendieck's purity conjecture} as proved by Thomason, which describes relative cohomology in the \'etale setting. By using results from \S\ref{s:deformationspaces}, we verify that the hypotheses of this calculation apply to $\cM_I$ as well as explicitly compute the inertia invariants in terms of representations of $\GL_n(\cO_o/\vpi_o^m)$, where $m$ denotes the multiplicity of $o$ in $I$. We then relate nearby cycles to \emph{semisimple trace}, which is a better-behaved variant of Frobenius traces for ramified representations of $W_{\bf{F}_o}$. Finally, we conclude by applying results from \S\ref{s:deformationspaces} once more to describe our integral in terms of semisimple traces.
\subsection{}\label{ss:inertiainvariantspullback}
We start by introducing notation for the calculation of nearby cycles. Let $\cX$ be a scheme over $\cO_o$ of finite type, and write
\begin{align*}
\ov\j:\cX_{\br{\bf{F}}_o}\rar\cX_{\br\cO_o}\mbox{ and }\ov\i:\cX_{\ov\ka}\rar\cX_{\br\cO_o}
\end{align*}
for the inclusion morphisms of the generic and special fibers of $\cX_{\br\cO_o}$, respectively. Denote the derived $I_{\bf{F}_o}$-invariants functor by $R_{I_{\bf{F}_o}}$, and denote the derived nearby cycles functor on $\cX_{\br\cO_o}$ by $R\Psi_{\cX_{\br{\bf{F}}_o}}$. The Galois description of \'etale sheaves implies that
\begin{align*}
(R_{I_{\bf{F}_o}}R\Psi_{\cX_{\br{\bf{F}}_o}})(-) = \ov\i^*R\ov\j_*((-)_{\br{\bf{F}}_o})
\end{align*}
as functors from $D^b_c(\cX_{\bf{F}_o},\ov\bQ_\ell)$ to $D^b_c(\cX_{\ka_o},\ov\bQ_\ell)$, since the non-derived versions of both sides are equal.
\subsection{}\label{ss:stratificationcomplex}
Next, we use the combinatorics of the geometry of $\cX$ to construct a certain family of $\ov\bQ_\ell$-vector spaces involved in the calculation of nearby cycles. Assume that $\cX$ is regular, and suppose that the morphism $\cX\rar\Spec\cO_o$ is flat of relative dimension $d$. Suppose we have a \emph{stratification} of $\cX_{\ka_o}$, i.e. assume that $\cX_{\ka_o}$ can be written as
\begin{align*}
\cX_{\ka_o}=\bigcup_j\mathring\cZ_j,
\end{align*}
where $j$ ranges over some finite indexing set $\bJ$, and the $\mathring\cZ_j$ are disjoint irreducible locally closed subsets of $\cX_{\ka_o}$ whose closures $\cZ_j\deq\ov{\mathring\cZ_j}$ are regular and equal to unions of $\mathring\cZ_k$ for some $k$. For any $j$ in $\bJ$, write $c(j)$ for the codimension of $\cZ_j$ in $\cX$, which is positive because $\cZ_j$ lies in $\cX_{\ka_o}$. We write $k\succ j$ if $\cZ_k$ strictly contains $\cZ_j$.
By inducting on $c(j)$, we shall assign a finite-dimensional $\ov\bQ_\ell$-vector space $W_j$ to any $j$ in $\bJ$ as follows. In the $c(j)=1$ case, set $W_j\deq\ov\bQ_\ell$. In the $c(j)=2$ case, we let
\begin{align*}
W_j\deq \ker\left(\bigoplus_kW_k\rar\ov\bQ_\ell\right),
\end{align*}
where $k$ runs over elements of $\bJ$ satisfying $k\succ j$ and $c(k)=1$, and the maps in the direct sum are the identity morphisms. Thus $W_j$ is just the kernel of the summation map. Note that we have a morphism $W_j\rar W_k$ given by
\begin{align*}
W_j\inj{}\bigoplus_kW_k\surj{}W_k
\end{align*}
for all $k\succ j$ with $c(k)=1$. Finally, in the $c(j)\geq3$ case, set
\begin{align*}
W_j\deq \ker\left(\bigoplus_kW_k\rar\bigoplus_lW_l\right),
\end{align*}
where now $k$ runs over all elements of $\bJ$ satisfying $k\succ j$ and $c(k)=c(j)-1$, $l$ runs over all elements of $\bJ$ satisfying $l\succ j$ and $c(l)=c(j)-2$, and the maps in the direct sum are of the form $W_k\rar W_l$ for $l\succ k$. Note that we have a morphism $W_j\rar W_k$ as before via inclusion and projection, so we may indeed inductively continue this construction.
Let $G$ be a group, and suppose it acts on $\cX_{\ka_o}$ in a manner preserving the stratification. Then the stabilizer of $\cZ_j$ in $G$ acts from the left on $W_j$ via permuting the $\cZ_k$ and hence $W_k$ for which $k\succ j$.
\subsection{}\label{ss:scholzenearbycycles}
We now present Scholze's calculation of the inertia invariants of nearby cycles. Let $j$ be in $\bJ$. Consider the chain complex
\begin{align*}
0\rar W_j\rar\bigoplus_{k_1} W_{k_1}\rar\bigoplus_{k_2}W_{k_2}\rar\dotsb\rar\bigoplus_{k_{c(j)-1}}W_{k_{c(j)-1}}\rar\ov\bQ_\ell\rar0,
\end{align*}
where $k_s$ runs over elements of $\bJ$ satisfying $k_s\succ j$ and $c(k_s)=c(j)-s$, and the maps in the direct sum are of the form $W_{k_s}\rar W_{k_{s+1}}$ for $k_{s+1}\succ k_s$. Note that this complex is exact at $W_j$ by construction. For any $\ka_o$-point $z$ of $\cX$, write $\ov{z}$ for the corresponding $\ov\ka$-point of $\cX$.
\begin{prop*}[{\cite[Theorem 5.3]{Sch13b}}]
If the above chain complex is exact for all $j$ in $\bJ$, then we have a canonical isomorphism
\begin{align*}
(\ov\i^*R^i\ov\j_*\ov\bQ_\ell)_{\ov{z}}\rar^\sim\bigoplus_kW_k(-i)
\end{align*}
for all $\ka_o$-points $z$ of $\cX$ and non-negative integers $i$, where $k$ ranges over all $k$ in $\bJ$ such that $z$ lies in $\cZ_k$ and $c(k)=i$.
\end{prop*}
The proof of this proposition uses a case of \emph{Grothendieck's purity conjecture} as proved by Thomason \cite[Corollary 3.9]{Tho84}, which says that relative \'etale cohomology is concentrated in the expected degree according to codimension.
\subsection{}
In order to apply Proposition \ref{ss:scholzenearbycycles}, we must first describe a stratification of $\cM_{I,\ka_o}$ as in \ref{ss:stratificationcomplex}. This stratification will be defined in terms of Drinfeld level-$m$ structures, and it is called the \emph{Newton stratification}.
Let $V$ be an $\cO_o/\vpi_o^m$-linear direct summand of $(\vpi_o^{-m}\cO_o/\cO_o)^n$, and write $\cM_I^V$ for the subfunctor of $\cM_I$ consisting of triples $((\sE_i,t_i,j_i)_i,\io,\al)$ that satisfy
\begin{align*}
\sum_v[\al(v)] = \#V\cdot[0],
\end{align*}
where $v$ runs over all elements of $V$, $[0]$ denotes the zero section of $\Dr(\sM'_o/\vpi_o^m)$, the sum is taken as closed subschemes of $\Dr(\sM'_o/\vpi_o^m)$, and $\#V\cdot[0]$ denotes $\#V$-fold sum of $[0]$ as a closed subscheme of $\Dr(\sM'_o/\vpi_o^m)$. Here, $\sM'_o$ denotes the local shtuka of $(\sE_i,t_i,j_i)_i$ at $o$ as in \ref{ss:dellipticsheavestolocalshtukazero}, and $\Dr(\sM'_o/\vpi^m_o)$ denotes the strict $\ka_o$-module associated with the finite $\ka_o$-shtuka $\sM'_o/\vpi^m_o$ under the Dieudonn\'e equivalence \cite[Theorem 5.2]{HS16}.
\begin{prop}\label{ss:initialstratification}\hfill
\begin{enumerate}[(i)]
\item The subfunctor $\cM_I^V$ is a regular closed subscheme of $\cM_I$ that is equidimensional of dimension $n-\rk_{\cO_o/\vpi^m_o}(V)$.
\item The special fiber $\cM_{I,\ka_o}$ equals $\bigcup_V\cM_I^V$, where $V$ ranges over all nonzero $\cO_o/\vpi_o^m$-linear direct summands of $(\vpi_o^{-m}\cO_o/\cO_o)^n$.
\item For any $\cO_o/\vpi^m_o$-linear direct summand $V'$ of $(\vpi^{-m}_o\cO_o/\cO_o)^n$, if any irreducible component of $\cM^{V'}_I$ lies in $\cM^{V}_I$, then $V'$ contains $V$. Conversely, if $V'$ contains $V$, then $\cM^{V'}_I$ lies in $\cM^{V}_I$.
\end{enumerate}
\end{prop}
Hence we obtain a decomposition
\begin{align*}
\cM_{I,\ka_o} = \bigcup_V\mathring\cM_I^V,
\end{align*}
where $V$ runs over all nonzero $\cO_o/\vpi_o^m$-linear direct summands of $(\vpi_o^{-m}\cO_o/\cO_o)^n$, and the $\mathring\cM_I^V$ are the disjoint locally closed subschemes
\begin{align*}
\mathring\cM_I^V \deq \cM^V_I\ssm\bigcup_{V'}\cM^{V'}_I,
\end{align*}
where $V'$ runs over all $\cO_o/\vpi_o^m$-linear direct summands of $(\vpi_o^{-m}\cO_o/\cO_o)^n$ that strictly contain $V$. Note that the only condition in \ref{ss:stratificationcomplex} that this decomposition of $\cM_{I,\ka_o}$ does not satisfy is that the $\cM_I^V$ might not be connected. Furthermore, we see that the right action of $\GL_n(\cO_o/\vpi_o^m)$ on $\cM_{I,\ka_o}$ given by composition with Drinfeld level-$m$ structures preserves this decomposition.
\begin{proof}[Proof of Proposition \ref{ss:initialstratification}]
The second part of (iii) is immediate. And when $V$ is zero, $\cM^V_I$ equals all of $\cM_I$, so the whole proposition for this $V$ follows immediately. Now suppose $V$ is nonzero. Let $\ov{z}$ be a geometric point of $\cM_I^V$, and write $((\sE_i,t_i,j_i)_i,\io,\al)$ for the corresponding triple. If $\ov{z}$ lay inside the generic fiber $\cM_{I,\bf{F}_o}$, then $\Dr(\sM'_o/\vpi_o^m)$ would be \'etale \cite[Proposition 7.1.3]{Boy99} and hence could not possibly satisfy the relation required for $\ov{z}$ to lie in $\cM_I^V$. Therefore $\cM_I^V$ is contained in $\cM_{I,\ka_o}$. As the $\cM_I^V$ cover $\cM_I$ as $V$ varies, this proves part (ii).
We can check parts (i) and (iii) by passing to the completion of closed points, so now assume that $\ov{z}$ is a $\ov\ka$-point. Write $\de_o$ for the element of
\begin{align*}
\GL_n(\br\cO_o)\diag(\vpi_o,1,\dotsc,1)\GL_n(\br\cO_o),
\end{align*}
unique up to $\GL_n(\br\cO_o)$-$\sg_o$-conjugacy, which corresponds to $\sM'_o$. Then the completion of $\cM_I$ at $\ov{z}$ is isomorphic to $\br\fX_{\de_o,\al}$ by Proposition \ref{ss:boyerserretate}. Recall from Proposition \ref{ss:connecteddeformationalgebraicclosure} that the restriction morphism induces an isomorphism $\br{R}_{\de_o^\circ,\al^\circ}\llb{T_1,\dotsc,T_{n-k}}\rar^\sim\br{R}_{\de_o,\al}$, where $k$ is the height of the connected part $\sM_o'^\circ$ of $\sM_o'$, $\de_o^\circ$ is an element of $\GL_k(\br\cO_o)\diag(\vpi_o,1,\dotsc,1)\GL_k(\br\cO_o)$ corresponding to $\sM_o'^\circ$,\footnote{We apologize for the notation.} and $\al^\circ$ is the restriction of $\al$ to $\ker\al$. The condition defining $\cM_I^V$ depends only on $\sM_o'^\circ$, so we may further narrow our focus to the local deformation ring $\br{R}_{\de_o^\circ,\al^\circ}$.
Let $e_1,\dotsc,e_s$ be an $\cO_o/\vpi_o^m$-basis of $V$. If $\ov{z}$ lies in $\cM_I^V$, then $\ker\al$ contains $V$, so here we may extend this to an $\cO_o/\vpi_o^m$-basis $e_1,\dotsc,e_k$ of $\ker\al$. For any local Artinian $\br\cO_o$-algebra $A$ with residue field $\ov\ka$ and local $\br\cO_o$-algebra morphism $f:\br{R}_{\de_o^\circ,\al^\circ}\rar A$, write $(H',\al',\io')$ for the corresponding triple. Recall from Proposition \ref{ss:drinfeldparameters} that $\br{R}_{\de_o^\circ,\al^\circ}$ has a choice of local parameters $X_1,\dots,X_k$ such that, for all such $A$ and $f$, the image of $X_i$ under $f$ equals the element of $\fm_A$ corresponding to $\al'(e_i)$. Now the condition defining $\cM_I^V$ is equivalent to
\begin{align*}
\prod_v(T-\al'(v)) = (T)^{\#V}
\end{align*}
as ideals in the formal power series ring $A\llb{T}$, where $v$ runs over all elements in $V$. This is equivalent to the condition that $X_1=\dotsb=X_j=0$. This proves the first part of (iii), and as $X_1,\dotsc,X_k$ is a regular sequence, this also proves part (i).
\end{proof}
\subsection{}
The decomposition of $\cM_{I,\ka_o}$ from Proposition \ref{ss:initialstratification} consists of closed subschemes $\cM_I^V$ which might not be connected. We can immediately rectify this issue (which is necessary for applying Proposition \ref{ss:scholzenearbycycles}) by refining this decomposition into its irreducible components. To simplify the exposition, we proceed as follows.
Let $z$ be a $\ka_o$-point of $\cM_I$, and write $\ov{z}$ for the corresponding $\ov\ka$-point of $\cM_I$. Then $z$ corresponds to a $\sD$-elliptic sheaf $(\sE_i,t_i,j_i)_i$ with Drinfeld level-$m$ structure $\al$. Note that, for any $\cO_o/\vpi_o^m$-linear direct summand $V$ of $(\vpi_o^{-m}\cO_o/\cO_o)^n$, the closed subscheme $\cM^V_I$ contains $z$ if and only if $V$ lies in $\ker\al$.
Given such a $V$, write $\mathring\cM^{V,0}_I,\dotsc,\mathring\cM^{V,l_V}_I$ for the irreducible components of $\mathring\cM^V_I$, and write $\cM^{V,0}_I,\dotsc,\cM^{V,l_V}_I$ for their closures, respectively. These are the irreducible components of $\cM_I^V$, and regularity implies that they are disjoint. We label them such that $\cM^{V,0}_I$ is the one containing $z$. For any $\cO_o/\vpi_o^m$-linear direct summand $V'$ lying in $V$, the disjointness of the $\cM^{V',0}_I,\dotsc,\cM^{V',l_{V'}}_I$ implies that precisely one of them contains $\cM_I^{V,a}$, where $a$ is any integer $0\leq a\leq l_V$. Therefore
\begin{align*}
\cM_{I,\ka_o}^z\deq\bigcup_{(V,a)}\mathring\cM_I^{V,a},
\end{align*}
where $(V,a)$ runs over all pairs for which
\begin{enumerate}[$\bullet$]
\item $V$ is a nonzero $\cO_o/\vpi^m_o$-linear direct summand of $(\vpi_o^{-m}\cO_o/\cO_o)^n$ such that $\cM_I^V$ contains $z$,
\item $a$ is an integer $0\leq a\leq l_V$,
\end{enumerate}
is an open subscheme of $\cM_{I,\ka_o}$. Furthermore, the $\mathring\cM_I^{V,a}$ now form a stratification of $\cM_{I,\ka_o}^z$ as in \ref{ss:stratificationcomplex}, where the indexing set equals the collection of such pairs $(V,a)$. Proposition \ref{ss:initialstratification} shows that $c(V,a)=\rk_{\cO_o/\vpi_o^m}(V)$. Note that the right action of $\GL_n(\cO_o/\vpi_o^m)$ on $\cM_{I,\ka_o}$ restricts to an action on $\cM_{I,\ka_o}^z$ that preserves this stratification.
\subsection{}\label{ss:modulispacecomplex}
At this point, we specialize the general constructions from \ref{ss:stratificationcomplex} to our specific situation of $\cM_{I,\ka_o}^z$. Let $W_{V,a}$ be the $\ov\bQ_\ell$-vector space associated with $\cM_I^{V,a}$ as in \ref{ss:stratificationcomplex}, and make a change of basis in $(\vpi_o^{-m}\cO_o/\cO_o)^n$ to identify $V$ with $(\vpi_o^{-m}\cO_o/\cO_o)^k$, where $k\leq n$ is rank of $V$ as a free $\cO_o/\vpi_o^m$-module. Then the stabilizer of $\cM_I^{V,a}$ under the $\GL_n(\cO_o/\vpi_o^m)$-action is $P(\cO_o/\vpi_o^m)$, where $P$ denotes the standard parabolic subgroup of $\GL_n$ with block sizes $(k,n-k)$, so $W_{V,a}$ has a left action by the finite group $P(\cO_o/\vpi_o^m)$.
Proposition \ref{ss:initialstratification}.(iii) implies that the $\cM^{V',a'}_I$ containing $\cM^{V,a}_I$ are precisely those for which $V'$ lies in $V$. Furthermore, after choosing such a $V'$, the integer $a'$ is uniquely determined by this containment condition. By making another change of basis, we may identify $V'$ with $(\vpi_o^{-m}\cO_o/\cO_o)^{k'}$, where $k'\leq k$ is the rank of $V'$ as a free $\cO_o/\vpi_o^m$-module. Then $Q(\cO_o/\vpi_o^m)$ is the stabilizer of $\cM_I^{V',a'}$ under the action of $\GL_n(\cO_o/\vpi_o^m)$, where $Q$ denotes the standard parabolic subgroup of $\GL_n$ with block sizes $(k',n-k')$. Note that $Q$ contains the unipotent radical of $P$ as well as the Levi factor $\GL_{n-k}$ of $P$, so we see that $P(\cO_o/\vpi_o^m)$ acts on $W_{V,a}$ through the quotient $\GL_k(\cO_o/\vpi_o^m)$.
\subsection{}\label{ss:finitesteinberg}
Now \ref{ss:modulispacecomplex} shows that $\cM^{V,a}_I$ is contained in a unique irreducible component of $\cM_I^{V'}$ whenever $V'$ lies in $V$. Therefore, by inducting on $k$ and examining the construction of $W_{V,a}$, we see that $W_{V,a}$ is isomorphic to the Steinberg representation
\begin{align*}
\St_k^{\cO_o/\vpi_o^m}\deq\ker\left(\Ind_{B(\cO_o/\vpi_o^m)}^{\GL_k(\cO_o/\vpi_o^m)}\ov\bQ_\ell\rar\bigoplus_P\Ind_{P(\cO_o/\vpi_o^m)}^{\GL_k(\cO_o/\vpi_o^m)}\ov\bQ_\ell\right)
\end{align*}
of $\GL_k(\cO_o/\vpi_o^m)$, where $B$ denotes the standard Borel subgroup of $\GL_k$ of upper triangular matrices, $\ov\bQ_\ell$ denotes the trivial representation, and $P$ ranges over all standard parabolic subgroups of $\GL_k$ that do not equal $B$. Under this identification, the chain complex from Proposition \ref{ss:scholzenearbycycles} formed from the $\cM_I^{V,a}$ becomes
\begin{align*}
0\rar\St_k^{\cO_o/\vpi_o^m}\rar&\Ind_{P_{k-1}(\cO_o/\vpi_o^m)}^{\GL_k(\cO_o/\vpi_o^m)}\left(\St_{k-1}^{\cO_o/\vpi_o^m}\otimes\,\ov\bQ_\ell\right)\rar \Ind_{P_{k-2}(\cO_o/\vpi_o^m)}^{\GL_k(\cO_o/\vpi_o^m)}\left(\St_{k-2}^{\cO_o/\vpi_o^m}\otimes\,\ov\bQ_\ell\right)\rar\dotsb\\
\dotsb&\rar\Ind_{P_1(\cO_o/\vpi_o^m)}^{\GL_k(\cO_o/\vpi_o^m)}\left(\ov\bQ_\ell\otimes\ov\bQ_\ell\right)\rar\ov\bQ_\ell\rar0,
\end{align*}
where $P_s$ denotes the standard parabolic subgroup of $\GL_k$ with block sizes $(s,k-s)$.
\subsection{}\label{ss:exacthypothesis}
We proceed to verify that the chain complex in \ref{ss:finitesteinberg} is exact, that is, the hypotheses of Proposition \ref{ss:scholzenearbycycles} hold for the stratification $\cM_{I,\ka_o}^z=\bigcup_{(V,a)}\mathring\cM_I^{V,a}$.
\begin{lem*}
For all positive integers $k$, the chain complex
\begin{align*}
0\rar\St_k^{\cO_o/\vpi_o^m}\rar&\Ind_{P_{k-1}(\cO_o/\vpi_o^m)}^{\GL_k(\cO_o/\vpi_o^m)}\left(\St_{k-1}^{\cO_o/\vpi_o^m}\otimes\,\ov\bQ_\ell\right)\rar \Ind_{P_{k-2}(\cO_o/\vpi_o^m)}^{\GL_k(\cO_o/\vpi_o^m)}\left(\St_{k-2}^{\cO_o/\vpi_o^m}\otimes\,\ov\bQ_\ell\right)\rar\dotsb\\
\dotsb&\rar\Ind_{P_1(\cO_o/\vpi_o^m)}^{\GL_k(\cO_o/\vpi_o^m)}\left(\ov\bQ_\ell\otimes\ov\bQ_\ell\right)\rar\ov\bQ_\ell\rar0,
\end{align*}
is exact.
\end{lem*}
In the proof, we will frequently pass between induced representations of $\GL_k(\bf{F}_o)$ and $\GL_k(\cO_o)$. For this, one can use the fact that $P(\cO_o)\bs\GL_k(\cO_o)$ equals $P(\bf{F}_o)\bs\GL_k(\bf{F}_o)$ to show that
\begin{align*}
\Ind_{P(\bf{F}_o)}^{\GL_k(\bf{F}_o)}\la = \Ind_{P(\cO_o)}^{\GL_k(\cO_o)}\la
\end{align*}
as representations of $\GL_k(\cO_o)$, where $P$ is a parabolic subgroup of $\GL_k$ over $\cO_o$, and $\la$ is a smooth representation of $P(\bf{F}_o)$.
\begin{proof}[Proof of Lemma \ref{ss:exacthypothesis}]
We reduce this to a fact about representations of $\GL_k(\bf{F}_o)$ as follows. Suppose we could show that the chain complex
\begin{align*}
0\rar\St_k\rar&\nInd_{P_{k-1}(\bf{F}_o)}^{\GL_k(\bf{F}_o)}\left(\St_{k-1}\otimes\,\ov\bQ_\ell\right)\rar \nInd_{P_{k-2}(\bf{F}_o)}^{\GL_k(\bf{F}_o)}\left(\St_{k-2}\otimes\,\ov\bQ_\ell\right)\rar\dotsb\\
\dotsb&\rar\nInd_{P_1(\bf{F}_o)}^{\GL_k(\bf{F}_o)}\left(\ov\bQ_\ell\otimes\ov\bQ_\ell\right)\rar\ov\bQ_\ell\rar0,
\end{align*}
is exact, where $\St_s$ is the Steinberg representation of $\GL_s(\bf{F}_o)$, that is, the representation $\St_s(\ov\bQ_\ell)$ in the terminology of Definition \ref{ss:generalizedsteinbergspeh}. As the modulus character $\de_{P_s}$ of $P_s(\bf{F}_o)$ vanishes on $P_s(\cO_o)$, we see that the above chain complex is isomorphic as representations of $\GL_n(\cO_o)$ to
\begin{align*}
0\rar\St_k\rar&\Ind_{P_{k-1}(\cO_o)}^{\GL_k(\cO_o)}\left(\St_{k-1}\otimes\,\ov\bQ_\ell\right)\rar \Ind_{P_{k-2}(\cO_o)}^{\GL_k(\cO_o)}\left(\St_{k-2}\otimes\,\ov\bQ_\ell\right)\rar\dotsb\\
\dotsb&\rar\Ind_{P_1(\cO_o)}^{\GL_k(\cO_o)}\left(\ov\bQ_\ell\otimes\ov\bQ_\ell\right)\rar\ov\bQ_\ell\rar0,
\end{align*}
which is therefore also exact.
Subgroups of $\cO_o$-points are compact, so modulus characters vanish on them. Therefore we have
\begin{align*}
\St_s=\ker\left(\nInd_{B(\bf{F}_o)}^{\GL_s(\bf{F}_o)}\de_B^{1/2}\rar\bigoplus_P\nInd_{P(\bf{F}_o)}^{\GL_s(\bf{F}_o)}\de_P^{1/2}\right) = \ker\left(\Ind_{B(\cO_o)}^{\GL_s(\cO_o)}\ov\bQ_\ell\rar\bigoplus_P\Ind_{P(\cO_o)}^{\GL_s(\cO_o)}\ov\bQ_\ell\right)
\end{align*}
as representations of $\GL_s(\cO_o)$, where $B$ denotes the standard Borel subgroup of $\GL_s$, $P$ runs over all standard parabolic subgroups of $\GL_b$ that do not equal $B$, and $\de_B$ and $\de_P$ denote the modulus characters, where we have identified $\ov\bQ_\ell$ with $\bC$. Thus taking $(1+\vpi_o^m\M_s(\cO_o/\vpi_o^m))$-invariants of $\St_s$ yields $\St_s^{\cO_o/\vpi_o^m}$. Because the Peter--Weyl theorem implies that taking $(1+\vpi_o^m\M_k(\cO_o))$-invariants preserves exactness, we conclude that
\begin{align*}
0\rar\St_k^{\cO_o/\vpi_o^m}\rar&\Ind_{P_{k-1}(\cO_o/\vpi_o^m)}^{\GL_k(\cO_o/\vpi_o^m)}\left(\St_{k-1}^{\cO_o/\vpi_o^m}\otimes\,\ov\bQ_\ell\right)\rar \Ind_{P_{k-2}(\cO_o/\vpi_o^m)}^{\GL_k(\cO_o/\vpi_o^m)}\left(\St_{k-2}^{\cO_o/\vpi_o^m}\otimes\,\ov\bQ_\ell\right)\rar\dotsb\\
\dotsb&\rar\Ind_{P_1(\cO_o/\vpi_o^m)}^{\GL_k(\cO_o/\vpi_o^m)}\left(\ov\bQ_\ell\otimes\ov\bQ_\ell\right)\rar\ov\bQ_\ell\rar0
\end{align*}
is exact, as desired.
Therefore it suffices to show that our chain complex of representations of $\GL_k(\bf{F}_o)$ is exact. A routine calculation \cite[Lemma I.3.2]{HT01} using the graph-theoretic description of the Jordan--H\"older factors of
\begin{align*}
\Ind_{P_s(\bf{F}_o)}^{\GL_k(\bf{F}_o)}\left(\St_s\otimes\,\ov\bQ_\ell\right)
\end{align*}
shows that there exists irreducible smooth representations $\pi_0,\dotsc,\pi_{k-1}$ of $\GL_k(\bf{F}_o)$ such that the above representation has $\pi_s$ as a subrepresentation and $\pi_{s-1}$ as a quotient. Because the maps in our chain complex are nonzero, this fact indicates that our chain complex of representations of $\GL_n(\bf{F}_o)$ is exact, which concludes the proof.
\end{proof}
\subsection{}\label{ss:modulinearbycycles}
With Lemma \ref{ss:exacthypothesis} in hand, we may apply Proposition \ref{ss:scholzenearbycycles} to our situation. This finally allows us to compute the inertia invariants of nearby cycles on $\cM_I$.
\begin{cor*}
Write $k$ for the rank of $\ker\al$ as a free $\cO_o/\vpi_o^m$-module. We have a canonical isomorphism
\begin{align*}
(R_{I_{\bf{F}_o}}^iR\Psi_{\cM_{I,\br{\bf{F}}_o}}\ov\bQ_\ell)_{\ov{z}} \rar^\sim
\begin{cases}
\Ind_{P_i(\cO_o/\vpi_o^m)}^{\GL_k(\cO_o/\vpi_o^m)}\left(\St_i^{\cO_o/\vpi_o^m}\otimes\,\ov\bQ_\ell\right)(-i) & \mbox{ if }0\leq i \leq k,\\
0 & \mbox{otherwise},
\end{cases}
\end{align*}
as representations of $\GL_k(\cO_o/\vpi_o^m)\times W_{\bf{F}_o}$, where $P_i$ denotes the standard parabolic subgroup of $\GL_k$ with block sizes $(i,k-i)$.
\end{cor*}
\begin{proof}
Because $\cM_{I,\ka_o}^z$ is an open subscheme of $\cM_{I,\ka_o}$ and stalks are local, we may replace $\cM_I$ with its open subscheme $\cM_{I,\ka_o}^z\cup\cM_{I,\bf{F}_o}$. In \ref{ss:inertiainvariantspullback}, we saw that
\begin{align*}
R_{I_{\bf{F}_o}}^iR\Psi_{\cM_{I,\br{\bf{F}}_o}}\ov\bQ_\ell = \ov\i^*R^i\ov\j_*\ov\bQ_\ell,
\end{align*}
and Lemma \ref{ss:exacthypothesis} implies that we may compute $\ov\i^*R^i\ov\j_*\ov\bQ_\ell$ by means of Proposition \ref{ss:scholzenearbycycles}. Recall that $z$ lies in $\cM_I^V$ if and only if $V$ lies in $\ker\al$. Therefore the same argument as in \ref{ss:finitesteinberg} shows that the right hand side of Proposition \ref{ss:scholzenearbycycles} yields
\begin{align*}
\Ind_{P_i(\cO_o/\vpi_o^m)}^{\GL_k(\cO_o/\vpi_o^m)}\left(\St_i^{\cO_o/\vpi_o^m}\otimes\,\ov\bQ_\ell\right)(-i),
\end{align*}
if $0\leq i\leq k$, as desired, while we get zero otherwise because $\ker\al$ contains no $\cO_o/\vpi_o^m$-linear direct summands of rank $i>k$.
\end{proof}
\subsection{}
At this point, we turn to \emph{semisimple traces}. Let $H$ be a finite group, and let $Y$ be a finite-dimensional continuous representation of $H\times W_{\bf{F}_{o,r}}$ over $\ov\bQ_\ell$. Because taking $I_{\bf{F}_o}$-invariants is not exact, the operation $Y\mapsto\tr(\sg_o^{-r}|Y^{I_{\bf{F}_o}})$ is not additive. One way of rectifying this starts by passing to certain filtrations of $Y$.
\begin{defn*}
An exhaustive filtration
\begin{align*}
0=Y_0\subseteq Y_1\subseteq\dotsb\subseteq Y_d=Y
\end{align*}
of $H\times W_{\bf{F}_{o,r}}$-subrepresentations is \emph{admissible} if $I_{\bf{F}_o}$ acts through a finite quotient on the associated graded representation
\begin{align*}
\gr Y_\bullet\deq\bigoplus_{i=1}^dY_i/Y_{i-1}.
\end{align*}
\end{defn*}
Note that refinements of admissible filtrations remain admissible, as do their sums and intersections.
\subsection{}\label{ss:ladicmonodromy}
To ensure that admissible filtrations of $Y$ exist, we shall use the following version of Grothendieck's $\ell$-adic monodromy theorem.
\begin{lem*}
There exists an admissible filtration of $Y$.
\end{lem*}
\begin{proof}
One can prove this precisely as in \cite[Lemma 7.2]{Sch11}: the argument proceeds as in the usual proof of Grothendieck's $\ell$-adic monodromy theorem, except we use $W_{\bf{F}_{o,r}}$ in place of $G_o$ and carry around the extra commuting action of $H$ throughout.
\end{proof}
\begin{defn}
Let $r$ be a non-negative integer, and let $h$ be in $H$. The \emph{semisimple trace} of $h\times \sg^{-r}_o$ on $Y$ is
\begin{align*}
\tr^\text{ss}\left(h\times \sg^{-r}_o|Y\right)\deq\tr\left(h\times \sg^{-r}_o|(\gr Y_\bullet)^{I_{\bf{F}_o}}\right),
\end{align*}
where $Y_\bullet$ is any admissible filtration of $Y$.
\end{defn}
Note that the common refinement of two admissible filtrations remains admissible. Taking $I_{\bf{F}_o}$-invariants on the associated graded representation is exact because $I_{\bf{F}_o}$ acts through a finite quotient, so we see that $\tr^\text{ss}(h\times \sg^{-r}_o|Y)$ is independent of the choice of $Y_\bullet$. This same observation shows that $\tr^\text{ss}(h\times \sg^{-r}_o|-)$ itself is additive. Therefore semisimple trace descends to the derived category of finite-dimensional continuous representations of $H\times W_{\bf{F}_o,r}$ over $\ov\bQ_\ell$, and it is additive on exact triangles.
\subsection{}\label{ss:inertiainvariantssemisimpletrace}
Taking derived $I_{\bf{F}_o}$-invariants is also additive on exact triangles, so we can consider the additive function $\tr(h\times\sg_o^{-r}|-)\circ R_{I_{\bf{F}_o}}$ as well. It has the following relationship with semisimple trace.
\begin{lem*}
We have
\begin{align*}
\tr\left(h\times\sg_o^{-r}|-\right)\circ R_{I_{\bf{F}_o}} = (1-q_o^r)\tr^\text{ss}\left(h\times\sg_o^{-r}|-\right)
\end{align*}
as functions on the derived category of finite-dimensional continuous representations of $H\times W_{\bf{F}_{o,r}}$ over $\ov\bQ_\ell$.
\end{lem*}
\begin{proof}
One can prove this exactly as in \cite[Lemma 7.5]{Sch11}: first, reduce to the case of a complex concentrated in one degree, and take wild inertia invariants. Since wild inertia is a pro-$p$ group, while the modules in question are $\ell$-torsion, Maschke's theorem indicates that this is exact. Next, we take tame inertia invariants: as tame inertia is pro-cyclic, we can take the standard resolution for pro-cyclic groups to conclude the proof.
\end{proof}
\subsection{}\label{ss:semisimpletraceintegral}
Semisimple traces have the following interpretation in terms of integrals of traces.
\begin{lem*}
We have an equality
\begin{align*}
\tr^\text{ss}\left(h\times\sg_o^{-r}|Y\right) = \int_{\sg_o^{-r}I_{\bf{F}_o}}\!\dif\tau\,\tr\left(h\times\tau|Y\right),
\end{align*}
where $\dif\tau$ is the Haar measure on $W_{\bf{F}_o}$ that gives $I_{\bf{F}_o}$ volume $1$.
\end{lem*}
\begin{proof}
Because both sides are additive in $Y$, it suffices to prove this when $Y$ is irreducible. Lemma \ref{ss:ladicmonodromy} implies that $I_{\bf{F}_o}$ acts on $Y$ through a finite quotient, so we obtain
\begin{align*}
\tr^\text{ss}\left(h\times\sg_o^{-r}|Y\right) = \tr\left(h\times\sg_o^{-r}|Y^{I_{\bf{F}_o}}\right).
\end{align*}
Furthermore, $Y$ is a smooth representation of $W_{\bf{F}_{o,r}}$, and the action of $h\times\sg_o^{-r}$ on $Y^{I_{\bf{F}_o}}$ is given by the action of the function
\begin{align*}
\bf1_{h\times\sg_o^{-r}I_{\bf{F}_o}}\in C^\infty_c(H\times W_{\bf{F}_{o,r}})
\end{align*}
on $Y$, where $\bf1_{h\times\sg_o^{-r}I_{\bf{F}_o}}$ is the indicator function on $h\times\sg_p^{-r}I_{\bf{F}_o}$. The trace of this function is given by the above integral, so we obtain the desired equality.
\end{proof}
\subsection{}\label{ss:sectionsixfinalresult}
Let $h$ be a function in $C^\infty_c(\GL_n(\cO_o))$. We conclude by combining the results in this section to write the integral of $\phi_{\tau,h}$, as $\tau$ ranges over elements of $\sg_o^{-r}I_{\bf{F}_o}$, in terms of representations of $\GL_n(\cO_o)$. Write $\cO_{o,r}$ for the ring of integers of $\bf{F}_{o,r}$, and write $\ka_{o,r}$ for the residue field of $\cO_{o,r}$. Let $\de_o$ be an element of $\GL_n(\bf{F}_{o,r})$, and if $\de_o$ lies in $\GL_n(\cO_{o,r})\diag(\vpi_o,1,\dotsc,1)\GL_n(\cO_{o,r})$, write $k$ for the rank of the connected part of the associated effective minuscule local shtuka $H_{\de_o}$ over $\Spec\ka_{o,r}$ as in \S\ref{s:deformationspaces}.
\begin{cor*}
The integral
\begin{align*}
\int_{\sg_o^{-r}I_{\bf{F}_o}}\!\dif\tau\,\phi_{\tau,h}(\de_o)
\end{align*}
vanishes if $\de_o$ does not lie in $\GL_n(\cO_{o,r})\diag(\vpi_o,1,\dotsc,1)\GL_n(\cO_{o,r})$. On the other hand, if $\de_o$ lies in this double coset, then the above integral equals
\begin{align*}
\fC_k\deq\frac1{1-q_o^r}\sum_{i=0}^k(-1)^iq_o^{ir}\tr\left(h|\Ind_{P_{i,k}(\cO_o)}^{\GL_n(\cO_o)}\left(\St_i\otimes\,\bC\right)\right),
\end{align*}
where $P_{i,k}$ is the standard parabolic subgroup of $\GL_n$ with block sizes $(i,k-i,n-k)$, and $\bC$ denotes the trivial representation of $\GL_{k-i}(\cO_o)\times\GL_{n-k}(\cO_o)$.
\end{cor*}
\begin{proof}
Because $\phi_{\tau,h}(\de_o)$ vanishes for $\de_o$ not in $\GL_n(\cO_{o,r})\diag(\vpi_o,1,\dotsc,1)\GL_n(\cO_{o,r})$ by definition, it suffices to prove this when $\de_o$ actually does lie in this double coset. Recalling the definition of $\phi_{\tau,h}(\de_o)$ for such $\de_o$ yields
\begin{align*}
\int_{\sg_o^{-r}I_{\bf{F}_o}}\!\dif\tau\,\phi_{\tau,h}(\de_o) = \int_{\sg_o^{-r}I_{\bf{F}_o}}\!\dif\tau\,\tr\left(\tau\times h|[R\psi_{\de_o}]\right),
\end{align*}
where $[R\psi_{\de_o}]$ is the virtual $\GL_n(\cO_o)\times I_{\bf{F}_o}$-admissible/continuous representation as in \cite[p.~24]{HT01} of $\GL_n(\cO_o)\times W_{\bf{F}_o}$ over $\ov\bQ_\ell$ from \ref{ss:fundamentalrepresentation}. Let $m$ be a positive integer for which $h$ descends to a function in $\ov\bQ_\ell[\GL_n(\cO_o/\vpi_o^m)]$, where we have identified $\ov\bQ_\ell$ with $\bC$. We denote this element in $\ov\bQ_\ell[\GL_n(\cO_o/\vpi_o^m)]$ using $h$ by abuse of notation. Then this integral of traces becomes
\begin{align*}
\sum_{i=0}^\infty(-1)^i\int_{\sg_o^{-r}I_{\bf{F}_o}}\!\dif\tau\,\tr\left(\tau\times h|R^i\psi_{\de_o,m}\right),
\end{align*}
where $R^i\psi_{\de_o,m}$ is the finite-dimensional continuous representation of $\GL_n(\cO_o/\vpi_o^m)\times W_{\bf{F}_o}$ from \ref{ss:nearbycyclesalgebraization}, since
\begin{align*}
\sum_{i=0}^\infty(-1)^iR^i\psi_{\de_o,m}
\end{align*}
are the $(1+\vpi_o^m\M_n(\cO_o))$-invariants of $[R\psi_{\de_o}]$. Lemma \ref{ss:semisimpletraceintegral} shows that our above sum of integrals equals
\begin{align*}
\sum_{i=0}^\infty(-1)^i\tr^\text{ss}\left(\sg_o^{-r}\times h|R^i\psi_{\de_o,m}\right).
\end{align*}
Next, Lemma \ref{ss:inertiainvariantssemisimpletrace} implies that this sum of traces equals
\begin{align*}
\frac1{1-q_o^r}\sum_{i=0}^\infty(-1)^i\tr\left(\sg_o^{-r}\times h|R_{I_{\bf{F}_o}}R^i\psi_{\de_o,m}\right).
\end{align*}
Using $\cM_{o^m,\br\cO_o}$ as an algebraization of $\br\fX_{\de_o,m}$ as in the proof of Lemma \ref{ss:nearbycyclesalgebraization}, write $z$ for the point of $\cM_{\varnothing,\br\cO_o}$ corresponding to $\de_o$, and write $\pi:\cM_{o^m,\br\cO_o}\rar\cM_{\varnothing,\br\cO_o}$ for the restriction morphism. Then Berkovich's nearby cycles comparison theorem \cite[Theorem 3.1]{Ber96} shows that the above sum becomes
\begin{align*}
\frac1{1-q_o^r}\sum_{i=0}^\infty(-1)^i\tr\left(\sg_o^{-r}\times h|R^i_{I_{\bf{F}_o}}\res{R\Psi_{\cM_{o^m,\br{\bf{F}}_o}}\ov\bQ_\ell}_{\pi^{-1}(z)}\right),
\end{align*}
where now we use the alternating product to expand $R_{I_{\bf{F}_o}}$ instead of $R\Psi_{\cM_{o^m,\br{\bf{F}}_o}}$. Applying the decomposition of $\br\fX_{\de_o,m}$ in Proposition \ref{ss:deformationalgebraicclosure} alongside the calculation of inertia invariants in Corollary \ref{ss:modulinearbycycles} yields the desired expression, where we have identified $\ov\bQ_\ell$ with $\bC$.
\end{proof}
\section{Langlands--Kottwitz counting and the Serre--Tate trick}\label{s:langlandskottwitz}
In this section, all representations shall be taken over $\ov\bQ_\ell$, and we view $\bC$-valued functions as $\ov\bQ_\ell$-valued ones via our fixed identification $\ov\bQ_\ell=\bC$. Let $f_{\tau,h}$ be our test function in $C^\infty_c(\GL_n(\bf{F}_o))$ from \S\ref{s:deformationspaces}, and let $f^{\infty,o}$ be any function in $C^\infty((D\otimes\bA^{\infty,o})^\times)$. Our goal is to relate the trace of $f^{\infty,o}\times h\times\tau$ to the trace of $f^{\infty,o}\times f_{\tau,h}$ on the virtual representation $[H_\xi]$. We begin by using the nearby cycles spectral sequence to pass from the generic fiber to the special fiber. Then, we use \emph{Deligne's conjecture} as proven by Fujiwara, which describes this trace in terms of manageable local terms, to convert this into a sum of terms indexed by geometric points in the special fiber. These points correspond to isomorphism classes of $\sD$-elliptic sheaves with extra structure.
We express this sum in terms of orbital integrals by using the equi-characteristic analog of \emph{Langlands--Kottwitz counting}, which describes isomorphism classes of $\sD$-elliptic sheaves in terms of the algebraic group-theoretic data associated with their local shtukas. Here, the Serre--Tate theorem from \S\ref{s:modulispaces} allows us to convert contributions from $h\times\tau$ into contributions from $f_{\tau,h}$. We then use the \emph{Selberg trace formula} to rewrite this expression in terms of automorphic representations. Finally, results of Laumon--Rapoport--Stuhler on the cohomology of $\cE\ell\ell_{\sD,I}$ allow us to conclude.
\subsection{}\label{ss:galoistohecke}
The following identity is the main result of this section. Write $v_o:W_{\bf{F}_o}\rar\bZ$ for the canonical surjection that sends geometric $q_o$-Frobenius at $o$ to $1$ in $\bZ$.
\begin{prop*}
Let $\tau$ be an element $W_{\bf{F}_o}$ with $v_o(\tau)>0$, let $h$ be a function in $C^\infty_c(\GL_n(\cO_o))$, and let $f^{\infty,o}$ be a function in $C^\infty_c((D\otimes\bA^{\infty,o})^\times)$. Then we have an equality of traces
\begin{align*}
\tr(f^{\infty,o}\times h\times\tau|[H_\xi]) = \textstyle\frac1n\tr(f^{\infty,o}\times f_{\tau,h}|[H_\xi]),
\end{align*}
where $f_{\tau,h}$ in $C^\infty_c(\GL_n(\bf{F}_o))$ is the function defined as in \ref{ss:ftauhdefinition}.
\end{prop*}
\begin{proof}
We start with some immediate reductions. Note that both sides are $\ov\bQ_\ell$-linear with respect to the function $f^{\infty,o}\times h$ in $C^\infty_c((D\otimes\bA^\infty)^\times)$. Therefore it suffices to prove this equality for
\begin{align*}
f^{\infty,o} = \frac1{\vol(\sK_I^{\infty,o})}\bf1_{\sK^{\infty,o}_Ig^o\sK^{\infty,o}_I}\mbox{ and }h = \frac1{\vol(\sK_{I,o}^\infty)}\bf1_{\sK_{I,o}^\infty g_o\sK_{I,o}^\infty},
\end{align*}
where $I$ can be any sufficiently large finite closed subscheme of $C\ssm\infty$, and $g$ is an element of $(D\otimes\bA^\infty)^\times$ for which $g_o$ lies in $\GL_n(\cO_o)$. With these choices of $f^{\infty,o}$ and $h$, our trace becomes
\begin{align*}
\tr(f^{\infty,o}\times h\times\tau|[H_\xi]) = \tr(c_{\bf{F}^\sep}\times\tau|[H_\xi]),
\end{align*}
where $c_{\bf{F}^\sep}=(c_{1,\bf{F}^\sep},c_{2,\bf{F}^\sep})$ is the pullback \cite[1.1.8]{Var07} to $\bf{F}^\sep$ of the correspondence from \ref{ss:dellipticsheavescohomology} that induces the action of $f^{\infty,o}\times h$. Next, we pass to $\bC_o$ and apply nearby cycles, which shows that this trace equals
\begin{align*}
\sum_{i=0}^\infty(-1)^i\tr\left(c_{\ov\ka}\times\tau|H^i(\cM_{I,\ov\ka},R\Psi_{\cM_{I,\bC_o}}\cL_{\xi,I,\bC_o})\right),
\end{align*}
where $R\Psi_{\cM_{I,\bC_o}}$ denotes the derived nearby cycles functor \cite[XIII (1.3.2.3)]{DK73} on $\cM_{I,\cO_{\bC_o}}$, and $c_{\ov\ka}=(c_{1,\ov\ka},c_{2,\ov\ka})$ is the pullback to $\ov\ka$ of the correspondence $c_{\cO_o}$ from \ref{ss:badreductionheckecorrespondence} that extends $c_\bf{F}$. Writing $\pi:\cM_{I}\rar\cM_{I\ssm o}$ for the restriction morphism, we see that the above sum equals
\begin{align*}
\sum_{i=0}^\infty(-1)^i\tr\left(\pi_{\ov\ka,*}c_{\ov\ka}\times\tau|H^i(\cM_{I\ssm o,\ov\ka},\pi_{\ov\ka,*}R\Psi_{\cM_{I,\bC_o}}\cL_{\xi,I,\bC_o})\right),
\end{align*}
where $\pi_{\ov\ka,*}c_{\ov\ka}=(\pi_{\ov\ka,*}c_{1,\ov\ka},\pi_{\ov\ka,*}c_{1,\ov\ka})$ is the pushforward correspondence \cite[1.1.6]{Var07} of $c_{\ov\ka}$. Next, write $r=v_o(\tau)$. The independence of characters implies that it suffices to prove our desired equalities for sufficiently large $r$. With this reduction in hand, we may now apply Deligne's conjecture as proven by Fujiwara \cite[Corollary 5.4.5]{Fuj97} to obtain the sum
\begin{align*}
\sum_y\tr\left((\pi_{\ov\ka,*}c_{\ov\ka}\times\tau)_y|(\pi_{\ov\ka,*}R\Psi_{\cM_{I,\bC_o}}\cL_{\xi,I,\bC_o})_{c_{2,\ov\ka}(y)}\right),
\end{align*}
where $y$ runs over all $\ov\ka$-points of $\Fix(\sg_o^r\circ\pi_{\ov\ka,*}c_{\ov\ka})$, and $\Fix(\sg_o^r\circ\pi_{\ov\ka,*}c_{\ov\ka})$ in turn denotes the fiber product
\begin{align*}
\xymatrix{\Fix(\sg_o^r\circ c_{\ov\ka})\ar[r]\ar[d] & \cM^\infty_{\ov\ka}/(\sK_I^{\infty,o}\cap g^{o,-1}\sK_I^{\infty,o} g^o)\ar[d]^-{(\sg_o^r\circ\pi_{\ov\ka,*}c_{1,\ov\ka},\pi_{\ov\ka,*}c_{2,\ov\ka})}\\
\cM_{I\ssm o,\ov\ka}\ar[r]^-{\De_{\cM_{I\ssm o},\ov\ka}} & \cM_{I\ssm o,\ov\ka}\times_{\ov\ka}\cM_{I\ssm o,\ov\ka}}
\end{align*}
At this point, we shall use the following lemma to decompose the $\ell$-adic sheaf whose cohomology we are studying.
\begin{lem}\label{lem:derivediso}
We have an isomorphism in the derived category of constructible $\ell$-adic sheaves
\begin{align*}
(\pi_{\ov\ka,*}R\Psi_{\cM_{I,\bC_o}}\ov\bQ_\ell)\otimes\cL_{I\ssm o,\xi,\ov\ka}\rar^\sim\pi_{\ov\ka,*}R\Psi_{\cM_{I,\bC_o}}\cL_{\xi,I,\bC_o}
\end{align*}
that preserves the actions of $I_{\bf{F}_o}$ and $(D\otimes\bA^\infty)^\times$.
\end{lem}
For the proof of this lemma, we will need the following notation. Write
\begin{align*}
\ov\j:\cM_{I\ssm o,\bC_o}\rar\cM_{I\ssm o,\cO_{\bC_o}}\mbox{ and }\ov\i:\cM_{I\ssm o,\ov\ka}\rar\cM_{I\ssm o,\cO_{\bC_o}}
\end{align*}
for the inclusion morphisms of the generic and special fibers of $\cM_{I\ssm o,\cO_{\bC_o}}$, respectively. In the proof of this lemma, we shall freely use the fact that
\begin{align*}
R\Psi_{\cM_{I\ssm o,\bC_o}}\pi_{\bC_o,*} = \pi_{\ov\ka,*}R\Psi_{\cM_{I,\bC_o}},
\end{align*}
which follows \cite[XIII (1.3.6)]{DK73} from finite base change.
\begin{proof}[Proof of Lemma \ref{lem:derivediso}]
We start by identifying
\begin{align*}
(\pi_{\ov\ka,*}R\Psi_{\cM_{I,\bC_o}}\ov\bQ_\ell)\otimes\cL_{I\ssm o,\xi,\ov\ka} = (R\Psi_{\cM_{I\ssm o,\bC_o}}\pi_{\bC_o,*}\ov\bQ_\ell)\otimes\ov\i^*\cL_{I\ssm o,\xi,\cO_{\bC_o}}.
\end{align*}
Next, applying the projection formula to canonical adjunction morphisms yields a morphism \cite[(0.2)]{Var07}
\begin{align}\label{eq:morphism1}
(\pi_{\ov\ka,*}R\Psi_{\cM_{I,\bC_o}}\ov\bQ_\ell)\otimes\cL_{I\ssm o,\xi,\ov\ka}\rar\ov\i^*(R\ov\j_*\pi_{\bC_o,*}\ov\bQ_\ell\otimes R\ov\j_*\ov\j^*\cL_{I\ssm o,\xi,\cO_{\bC_o}}),\tag{$\star$}
\end{align}
and taking $\ov\i^*$ of a similar morphism \cite[(0.1)]{Var07}
\begin{align*}
R\ov\j_*\pi_{\bC_o,*}\ov\bQ_\ell\otimes R\ov\j_*\ov\j^*\cL_{I\ssm o,\xi,\cO_{\bC_o}}\rar R\ov\j_*(\pi_{\bC_o,*}\ov\bQ_\ell\otimes\ov\j^*\cL_{I\ssm o,\xi,\cO_{\bC_o}})
\end{align*}
provides another morphism
\begin{align}\label{eq:morphism2}
\ov\i^*(R\ov\j_*\pi_{\bC_o,*}\ov\bQ_\ell\otimes R\ov\j_*\ov\j^*\cL_{I\ssm o,\xi,\cO_{\bC_o}})\rar R\Psi_{\cM_{I\ssm o,\bC_o}}(\pi_{\bC_o,*}\ov\bQ_\ell\otimes\cL_{I\ssm o,\xi,\bC_o}).\tag{$\star\star$}
\end{align}
By reducing to finite coefficients and then checking on an \'etale trivialization, we see that the composition of Equation (\ref{eq:morphism1}) and Equation (\ref{eq:morphism2}) is an isomorphism. Applying the projection formula \cite[XVII (5.2.9)]{AGV73} to $\ov\bQ_\ell$ and $\cL_{I\ssm o,\xi,\bC_o}$ implies that
\begin{align*}
R\Psi_{\cM_{I\ssm o,\bC_o}}(\pi_{\bC_o,*}\ov\bQ_\ell\otimes\cL_{I\ssm o,\xi,\bC_o}) =R\Psi_{\cM_{I\ssm o,\bC_o}}\pi_{\bC_o,*}\pi^*_{\bC_o}\cL_{I\ssm o,\xi,\bC_o} = \pi_{\ov\ka,*}R\Psi_{\cM_{I,\bC_o}}\cL_{\xi,I,\bC_o},
\end{align*}
as desired.
\end{proof}
Return to the proof of Proposition \ref{ss:galoistohecke}. With this tensor product decomposition, our sum of traces becomes
\begin{align*}
\sum_y\tr\left((\pi_{\ov\ka,*}c_{\ov\ka}\times\tau)_y|(\pi_{\ov\ka,*}R\Psi_{\cM_{I,\bC_o}}\ov\bQ_\ell)_{c_{2,\ov\ka}(y)}\right)\tr\left((\pi_{\ov\ka,*}c_{\ov\ka}\times\tau)_y|(\cL_{I\ssm o,\xi,\ov\ka})_{c_{2,\ov\ka}(y)}\right).
\end{align*}
Write $z$ for $c_{2,\ov\ka}(y)$, write $(\sE_i,t_i,j_i)_i$ for the $\sD$-elliptic sheaf over $\ov\ka$ corresponding to $z$, write $\sM'_o$ for its local shtuka at $o$ as in \ref{ss:dellipticsheavestolocalshtukazero}, and write $\de_o$ for the corresponding element of the double coset
\begin{align*}
\ang{\vpi_o}\deq\GL_n(\br\cO_o)\diag(\vpi_o,1,\dotsc,1)\GL_n(\br\cO_o),
\end{align*}
which is unique up to $\GL_n(\br\cO_o)$-$\sg_o$-conjugacy. Because $y$ lies in $\Fix(\sg_o^r\circ\pi_{\ov\ka,*}c_{\ov\ka})$, we see $z$ is defined over $\ka_r$, so $\de_o$ can naturally be chosen to lie in $\ang{\vpi_o}\cap\GL_n(\bf{F}_{o,r})$.
Writing $m$ for the multiplicity of $o$ in $I$, we see that finite base change implies that the above sum equals
\begin{align*}
\sum_y\tr\left(c_{\ov\ka}\times\tau|H^i(\pi^{-1}(z),\res{R\Psi_{\cM_{I,\bC_o}}\ov\bQ_\ell}_{\pi^{-1}(z)})\right)\tr\left((\pi_{\ov\ka,*}c_{\ov\ka}\times\tau)_y|(\cL_{I\ssm o,\xi,\ov\ka})_{z}\right).
\end{align*}
Because $\br\fX_{\de_o,m}$ is the completion of $\cM_{I,\br\cO_o}$ along $\pi^{-1}(z)$ by \ref{ss:extendedserretate}, Berkovich's nearby cycles comparison theorem \cite[Theorem 3.1]{Ber96} indicates that this equals
\begin{align*}
&\sum_y\sum_{i=0}^\infty\tr\left(g_o\times\tau|H^i(\fX_{\de_o,m,\ov\ka},R^i\Psi_{\fX_{\de_o,m,\bC_o}}\ov\bQ_\ell)\right)\tr\left((\pi_{\ov\ka,*}c_{\ov\ka}\times\tau)_y|(\cL_{I\ssm o,\xi,\ov\ka})_{z}\right)\\
=&\sum_y\tr(\tau\times h|[R\psi_{\de_o}])\tr\left((\pi_{\ov\ka,*}c_{\ov\ka}\times\tau)_y|(\cL_{I\ssm o,\xi,\ov\ka})_{z}\right) = \sum_y\phi_{\tau,h}(\de_o)\tr\left((\pi_{\ov\ka,*}c_{\ov\ka}\times\tau)_y|(\cL_{I\ssm o,\xi,\ov\ka})_{z}\right),
\end{align*}
where $[R\psi_{\de_o}]$ is the virtual $\GL_n(\cO_o)\times I_{\bf{F}_o}$-admissible/continuous representation of $\GL_n(\cO_o)\times W_{\bf{F}_o}$ defined in \ref{ss:fundamentalrepresentation}, and $\phi_{\tau,h}$ in $C_c^\infty(\GL_n(\bf{F}_{o,r}))$ is the function defined in \ref{ss:sigmaconjugationisopen}.
\subsection{}
In order to express the above sum in terms of orbital integrals, we now introduce \emph{Langlands--Kottwitz counting}. This technique counts isomorphism classes of $\sD$-elliptic sheaves over $\ov\ka$ whose zero lies over $o$ by
\begin{enumerate}[(1)]
\item first describing the \emph{isogeny classes} of $\sD$-elliptic sheaves in terms of conjugacy classes in certain groups,
\item then counting the number of isomorphism classes in each isogeny class by taking orbital integrals.
\end{enumerate}
Since $\sD$-elliptic sheaves consist of vector bundles, it is quite natural to reinterpret them in terms of linear algebraic groups. This incarnation of the Langlands--Kottwitz method is heavily based on work of Drinfeld and Laumon.
\subsection{}
To define isogeny classes of $\sD$-elliptic sheaves, we first introduce \emph{$\vp$-spaces}.
\begin{defn*}
A \emph{$\vp$-space} is a pair $(V,\vp)$, where
\begin{enumerate}[$\bullet$]
\item $V$ is a finite-dimensional $\bf{F}\otimes_\ka\ov\ka$-vector space,
\item $\vp:{}^\sg\!V\rar^\sim V$ is an $\bf{F}\otimes_\ka\ov\ka$-linear isomorphism,
\end{enumerate}
where ${}^\sg$ denotes $(\id_\bf{F}\otimes_\ka\sg)^*$.
\end{defn*}
Note that $\vp$-spaces resemble local shtukas and $\sD$-elliptic sheaves, with an important difference: $\vp$-spaces lie over the generic fiber of the curve $C$, rather than completions of $C$ at closed points in the case of local shtukas or (a specific open subset of) the entire curve $C$ in the case of $\sD$-elliptic sheaves.
\subsection{}\label{ss:vpguys}
For any $\sD$-elliptic sheaf $(\sE_i,t_i,j_i)_i$ over $\ov\ka$, the generic fiber $\sE_{1,\bf{F}\otimes_\ka\ov\ka}$ is a finite-dimensional $\bf{F}\otimes_\ka\ov\ka$-vector space, and the map $(j_{1,\bf{F}\otimes_\ka\ov\ka})^{-1}\circ t_{1,\bf{F}\otimes_\ka\ov\ka}$ is a $\bf{F}\otimes_\ka\ov\ka$-linear isomorphism. Therefore the pair
\begin{align*}
(V,\vp)\deq\left(\sE_{1,\bf{F}\otimes_\ka\ov\ka},(j_{1,\bf{F}\otimes_\ka\ov\ka})^{-1}\circ t_{1,\bf{F}\otimes_\ka\ov\ka}\right)
\end{align*}
forms a $\vp$-space, and it has a right action of $D$ for which $V$ is a free right $D$-module of rank $1$. We call this the \emph{generic fiber} of $(\sE_i,t_i,f_i)_i$, and we say two $\sD$-elliptic sheaves over $\ov\ka$ are \emph{isogenous} if they have isomorphic generic fibers. Thus isogeny classes of $\sD$-elliptic sheaves correspond to certain isomorphism classes of $\vp$-spaces.
Suppose that the zero of $(\sE_i,t_i,j_i)_i$ lies over $o$. For every place $x$ of $\bf{F}$, fix a map $\ka_x\rar\ov\ka$ over $\ka$, and write $\sM_x$ for the resulting local shtuka of $(\sE_i,t_i,j_i)_i$ at $x$ as constructed in \ref{ss:dellipticsheavestolocalshtukazero} for $x=o$ or \ref{ss:dellipticsheavestolocalshtuka} for $x\neq o$. Since the set of all maps $\ka_x\rar\ov\ka$ over $\ka$ are cyclically permuted via composition with $\sg$, we can view $\sM_x$ as a free module over
\begin{align*}
\cO_x\wh\otimes_\ka\ov\ka = \prod_\io\cO_x\wh\otimes_{\ka_x,\io}\ov\ka
\end{align*}
equipped with a $\sg$-semilinear automorphism after inverting $\vpi_x$, where the $\io$ runs over all maps $\ka_x\rar\ov\ka$ over $\ka$ \cite[p.~33]{Dri88}. Under this perspective, we may naturally identify $(\sM_x[\frac1{\vpi_x}],\sF_x)$ with $(V\wh\otimes_\bf{F}\bf{F}_x,\vp\wh\otimes_\bf{F}\id_{\bf{F}_x})$.
\subsection{}
Next, we introduce \emph{$\vp$-pairs}, which will provide an alternative way of describing $\vp$-spaces.
\begin{defn*}
A \emph{$\vp$-pair} is a pair $(\wt{\bf{F}},\wt\Pi)$, where
\begin{enumerate}[$\bullet$]
\item $\wt{\bf{F}}$ is a finite separable (ring) extension of $\bf{F}$,
\item $\wt\Pi$ is an element of $\wt{\bf{F}}^\times\otimes_\bZ\bQ$ that is not contained in $\bf{F}'^\times\otimes_\bZ\bQ$ for any proper $\bf{F}$-subalgebra $\bf{F}'$ of $\wt{\bf{F}}$.
\end{enumerate}
Note that $\wt{\bf{F}}$ is a product of field extensions of $\bf{F}$, so it has a well-defined notion of places. For any place $\wt{x}$ of $\wt{\bf{F}}$, the $\bQ$-valued valuation $\wt{x}(\wt\Pi)$ is well-defined, and we write $d(\wt\Pi)$ for the least common denominator of $\wt{x}(\wt\Pi)\deg\wt{x}$ as $\wt{x}$ ranges over all places of $\wt{\bf{F}}$, where $\deg\wt{x}$ is taken with respect to $\ka$.
\end{defn*}
Therefore $\vp$-pairs roughly correspond to elements in extensions (up to roots of unity) that cannot be obtained from subextensions by taking rational powers.
\subsection{}\label{ss:phispacesphipairs}
The notions of a $\vp$-pair and a $\vp$-space are related as follows.
\begin{prop*}[{\cite[Proposition 2.1]{Dri88}}]
We have a canonical map
\begin{align*}
\{\mbox{isomorphism classes of }\vp\mbox{-spaces}\}&\rar\{\mbox{isomorphism classes of }\vp\mbox{-pairs}\}\\
(V,\vp)&\longmapsto(\wt{\bf{F}}_{(V,\vp)},\wt\Pi_{(V,\vp)})
\end{align*}
that induces a bijection from isomorphism classes of irreducible $\vp$-spaces to isomorphism classes of $\vp$-pairs for which $\wt{\bf{F}}$ is a field. In that situation, the dimension of $V$ over $\bf{F}\otimes_\ka\ov\ka$ equals $[\wt{\bf{F}}_{(V,\vp)}:\bf{F}]d(\wt\Pi_{(V,\vp)})$, and the endomorphism ring of $(V,\vp)$ is isomorphic to the central division algebra $\wt{D}$ over $\wt{\bf{F}}_{(V,\vp)}$ with Hasse invariants
\begin{align*}
\inv_{\wt{x}}(\wt{D}) = -\wt{x}(\wt\Pi_{(V,\vp)})\deg\wt{x}.
\end{align*}
\end{prop*}
The above map is constructed by replacing $\ov\ka$ with a finite extension of $\ka$ of degree $a$ and then considering the $\bf{F}$-algebra generated by $\vp^a$, where taking $a$ to be sufficiently divisible ensures that the resulting construction is independent of the choice of $a$ \cite[p.~31]{Dri88}. This is the equi-characteristic analogue of Honda--Tate theory.
\subsection{}\label{ss:dellipticsheavesphipairs}
We can describe which $\vp$-spaces occur as generic fibers of $\sD$-elliptic sheaves in terms of their associated $\vp$-pairs. This completes our description of the isogeny classes of $\sD$-elliptic sheaves.
\begin{prop*}[{\cite[(9.13)]{LRS93}}]
The isomorphism classes of $\vp$-spaces that arise from $\sD$-elliptic sheaves over $\ov\ka$ with zero lying over $o$ map precisely to $\vp$-pairs $(\wt{\bf{F}},\wt\Pi)$ for which
\begin{enumerate}[$\bullet$]
\item $\wt{\bf{F}}$ is a field, and $[\wt{\bf{F}}:\bf{F}]$ divides $n$,
\item there exists only one place $\wt\infty$ of $\wt{\bf{F}}$ lying over $\infty$, and it satisfies $\wt\infty(\wt\Pi)\deg\wt\infty=-[\wt{\bf{F}}:\bf{F}]/n$,
\item there exists only one other place $\wt{o}\neq\wt\infty$ of $\wt{\bf{F}}$ satisfying $\wt{o}(\wt\Pi)\neq0$, and it lies over $o$.
\end{enumerate}
For any $\sD$-elliptic sheaf $(\sE_i,t_i,j_i)_i$ lying in the isogeny class corresponding to $(\wt{\bf{F}},\wt\Pi)$, the height of the connected part of $\sM_o'$ equals $n[\wt{\bf{F}}_{\wt{o}}:\bf{F}_o]/[\wt{\bf{F}}:\bf{F}]$. Furthermore, if we write $(W,\psi)$ for the irreducible $\vp$-space corresponding to $(\wt{\bf{F}},\wt\Pi)$, then $(V,\vp)$ is isomorphic to $(W,\psi)^{\oplus n}$.
\end{prop*}
By checking Hasse invariants of the endomorphism ring $\wt{D}$ of $(W,\psi)$ and applying Proposition \ref{ss:phispacesphipairs}, we also deduce that $\wt{o}(\wt\Pi)\deg\wt{o}=[\wt{\bf{F}}:\bf{F}]/n$.
Return to the proof of Proposition \ref{ss:galoistohecke}. By gathering terms in the same isogeny class, we can rewrite our sum as
\begin{align*}
\sum_{(\wt{\bf{F}},\wt\Pi)}\sum_y\phi_{\tau,h}(\de_o)\tr\left((\pi_{\ov\ka,*}c_{\ov\ka}\times\tau)_y|(\cL_{I\ssm o,\xi,\ov\ka})_{z}\right),
\end{align*}
where $(\wt{\bf{F}},\wt\Pi)$ runs over all $\vp$-pairs satisfying the conditions in Proposition \ref{ss:dellipticsheavesphipairs}, and $y$ runs over $\ov\ka$-points of $\Fix(\sg^r_o\circ c_{\ov\ka})$ that lie in the isogeny class corresponding to $(\wt{\bf{F}},\wt\Pi)$.
\subsection{}\label{ss:dellipticsheavesisomorphismclasses}
At this point, we initiate the second part of the Langlands--Kottwitz method: counting the number of isomorphism classes in each isogeny class. We begin by describing \emph{all} isomorphism classes---we shall refine our description by isogeny class afterwards.
\begin{prop*}[{\cite[(9.4)]{LRS93}}]
The map $(\sE_i,t_i,j_i)_i\mapsto((V,\vp),(\sM_x)_x)$, where
\begin{enumerate}[$\bullet$]
\item $(V,\vp)$ is the generic fiber of $(\sE_i,t_i,j_i)_i$,
\item $x$ runs through all places of $\bf{F}$, and $\sM_x$ is as in \ref{ss:vpguys},
\end{enumerate}
yields a bijection from isomorphism classes of $\sD$-elliptic sheaves over $\ov\ka$ with zero lying over $o$ to isomorphism classes of pairs $((V,\vp),(\sM_x)_x)$, where
\begin{enumerate}[$\bullet$]
\item $(V,\vp)$ is a $\vp$-space with a right action of $D$ such that $V$ is a free right $D\otimes_\ka\ov\ka$-module of rank $1$,
\item $x$ runs over all places in $\bf{F}$, and $\sM_x$ is a local shtuka over $\ov\ka$ corresponding to a right $\sD_x\wh\otimes_\ka\ov\ka$-submodule of rank $1$ in $V\wh\otimes_\bf{F}\bf{F}_x$ that is stable under $\vp\wh\otimes_\bf{F}\bf{F}_x$,
\end{enumerate}
that satisfy the following conditions:
\begin{enumerate}[(i)]
\item the $\vp$-pair corresponding to $(V,\vp)$ satisfies the conditions in Proposition \ref{ss:dellipticsheavesphipairs},
\item the Morita reduction $\sM'_o$ of $\sM_o$ is effective minuscule of dimension $1$ and rank $n$,
\item the isogeny class of the Morita reduction $\sM'_\infty$ of $\sM_\infty$ has slope $-\frac1n$ under the Dieudonn\'e--Manin classification,
\item $\sM_x$ is \'etale for all closed points $x$ in $C\ssm\{o,\infty\}$,
\item there exists a generator $b$ of $V$ as a free right $D\otimes_\ka\ov\ka$-module such that, for cofinitely many $x$, we have
\begin{align*}
\sM_x = b(\sD_x\wh\otimes_\ka\ov\ka)
\end{align*}
as right $\sD_x$-modules equipped with a $\sg$-semilinear automorphism, where we view $\sM_x$ as the corresponding free $\sD_x\wh\otimes_\ka\ov\ka$-submodule of $V\wh\otimes_\bf{F}\bf{F}_x$ of rank $1$ equipped with a $\sg$-semilinear automorphism.
\end{enumerate}
\end{prop*}
\subsection{}\label{ss:dellipticsheavesisogenyclass}
Next, we use Proposition \ref{ss:dellipticsheavesisomorphismclasses} to obtain a description of the isomorphism classes in a given isogeny class. Fix a $\vp$-pair $(\wt{\bf{F}},\wt\Pi)$ satisfying the conditions of Proposition \ref{ss:dellipticsheavesphipairs}, and write $\wt{D}$ for the endomorphism ring of $(V_{(\wt{\bf{F}},\wt\Pi)},\vp_{(\wt{\bf{F}},\wt\Pi)})$. Write $\cM_I(\ov\ka)_{(\wt{\bf{F}},\wt\Pi)}$ for the set of points in $\cM_I(\ov\ka)$ lying in the isogeny class corresponding to $(\wt{\bf{F}},\wt\Pi)$. Our goal is to describe $\cM_I(\ov\ka)_{(\wt{\bf{F}},\wt\Pi)}$ using algebraic group-theoretic data.
Let $k=n[\wt{\bf{F}}_{\wt{o}}:\bf{F}_o]/[\wt{\bf{F}}:\bf{F}]$, write $\ve_o^{\wt{o}}$ for the identity matrix in $\GL_{n-k}(\bf{F}_o)$, write $\ve_{\wt{o}}$ for the $k$-by-$k$ matrix
\begin{align*}
\ve_{\wt{o}}\deq
\begin{bmatrix}
0 & 1 & & \\
&\ddots & \ddots & \\
& & \ddots& 1 \\
\vpi_o & & &0
\end{bmatrix}\in\GL_k(\bf{F}_o),
\end{align*}
and write $\ve_o$ for the block matrix $\ve_o^{\wt{o}}\oplus\ve_{\wt{o}}$ in $\GL_n(\bf{F}_o)$. Note that, for any $\sD$-elliptic sheaf in $\cM_I(\ov\ka)_{(\wt{\bf{F}},\wt\Pi)}$, Proposition \ref{ss:dellipticsheavesisomorphismclasses} says that its local shtuka at $o$ corresponds to $\ve_o$ via \ref{ss:localshtukaconjugacy}, by the Dieudonn\'e--Manin classification.
Write $\sY_o$ for the subset
\begin{align*}
\sY_o\deq\{h_o\in\GL_n(\br{\bf{F}}_o)|h_o^{-1}\ve_o\sg_o(h_o)\in\ang{\vpi_o}\},
\end{align*}
and write $\br\sK_{o,m}$ for the subgroup
\begin{align*}
\br\sK_{o,m}\deq\ker\left(\GL_n(\br\cO_o)\rar\GL_n(\br\cO_o/\vpi_o^m)\right).
\end{align*}
\begin{prop*}
We have a bijection
\begin{align*}
\cM_I(\ov\ka)_{(\wt{\bf{F}},\wt\Pi)}\rar^\sim \wt{D}^\times\bs\left((D\otimes\bA^{\infty,o})^\times/\sK^{\infty,o}_I\times \sY_o/\br\sK_{o,m}\right)
\end{align*}
for which $g^{\infty,o}$ acts on the right-hand side via left multiplication on $(D\otimes\bA^{\infty,o})^\times$, and $\sg_o^{-1}$ acts on the right-hand side via sending $h_o\mapsto\ve_o\sg_o(h)$ on $\sY_o$.
\end{prop*}
\begin{proof}
We use the description of $\cM_I(\ov\ka)_{(\wt{\bf{F}},\wt\Pi)}$ given in Proposition \ref{ss:dellipticsheavesisomorphismclasses}. Under this description, the translation action of $\bZ$ sends $((V,\vp),(\sM_x)_x)$ to $((V,\vp),(\vp^l(\sM_x))_x)$ for all integers $l$. The characterization of $\sM_\infty'$ given in Proposition \ref{ss:dellipticsheavesisomorphismclasses}.(iii) implies that we may fix the position of $\sM'_\infty$ in $V\wh\otimes_\bf{F}\bf{F}_\infty$ via translating by $\bZ$.
Next, fix a generator $b$ of $V$ as a free right $D\otimes_\ka\ov\ka$-module. For all closed points $x$ in $C\ssm\infty$, let $h_x$ be an element of $D_x$ such that $bh_x$ generates $\sM_x$ over $\sD_x\wh\otimes_\ka\ov\ka$. Parts (iv) and (v) of Proposition \ref{ss:dellipticsheavesisomorphismclasses} show that $(h_x)_x$ is an element of $(D\otimes\bA^{\infty,o})^\times$ as $x$ runs over all closed points in $C\ssm\{o,\infty\}$, and the level-$I$ structures shows that $(h_x)_x$ is well-defined up to right multiplication by $\sK_I^{\infty,o}$. Similarly, $h_o$ yields a well-defined element of $\sY_o/\br\sK_{o,m}$.
Finally, the left action of $\wt{D}^\times$ on $(V,\vp)$ yields an embedding
\begin{align*}
\wt{D}^\times\inj{}(D\otimes\bA^{\infty,o})^\times\times\sY_o
\end{align*}
that preserves the action of $\wt{D}^\times$ on $(V,\vp)$, and taking the quotient of
\begin{align*}
(D\otimes\bA^{\infty,o})^\times/\sK^{\infty,o}_I\times \sY_o/\br\sK_{o,m}
\end{align*}
with respect to this action concludes the proof, by Proposition \ref{ss:dellipticsheavesisomorphismclasses}.
\end{proof}
\subsection{}\label{ss:dellipticsheavesadelicisogenyclass}
With Proposition \ref{ss:dellipticsheavesisogenyclass} in hand, we shall give an analogous description of $\Fix(\sg_o^r\circ\pi_*c_{\ov\ka})$. Write $\Fix(\sg_o^r\circ\pi_*c_{\ov\ka})_{(\wt{\bf{F}},\wt\Pi)}$ for the set of $y$ in $\Fix(\sg_o^r\circ\pi_*c_{\ov\ka})$ lying in the isogeny class corresponding to $(\wt{\bf{F}},\wt\Pi)$. Our goal is to describe $\Fix(\sg_o^r\circ\pi_*c_{\ov\ka})_{(\wt{\bf{F}},\wt\Pi)}$ in terms of the bijection from Proposition \ref{ss:dellipticsheavesisogenyclass}.
Write $\sK_{r,m}$ for the subgroup
\begin{align*}
\sK_{r,m}\deq\ker\left(\GL_n(\cO_{o,r})\rar\GL_n(\cO_{o,r}/\vpi_o^m)\right).
\end{align*}
For any element $\wt{d}$ in $\wt{D}^\times$, we say $\wt{d}$ is \emph{$r$-admissible} if there exists some $h_{\wt{d}}$ in $\GL_n(\br{\bf{F}}_o)$ such that
\begin{align*}
h^{-1}_{\wt{d}}\cdot\wt{d}^{-1}\cdot\N_o\ve_o\cdot\sg_o^r(h_{\wt{d}})=1,
\end{align*}
where we view $\wt{d}^{-1}$ as an element of $(\wt{D}\otimes_{\wt{\bf{F}}}\br{\bf{F}}_o)^\times=\GL_n(\br{\bf{F}}_o)$, and $\N_o$ denotes the norm map. Note that $r$-admissibility descends to a property on $\wt{D}^\times$-conjugacy classes in $\wt{D}^\times$.
For any group $G$ and element $g$ in $G$, we denote the centralizer of $g$ in $G$ by $G_g$. We write $\sY_{\wt{d}}$ for the double quotient space
\begin{align*}
\sY_{\wt{d}}\deq\wt{D}^\times_{\wt{d}}\bs\left((D\otimes\bA^{\infty,o})^\times/\sK_I^{\infty,o}\times\GL_n(\bf{F}_{o,r})/\sK_{r,m}\right).
\end{align*}
\begin{prop*}\label{prop:bijdisjoint}
We have a bijection from $\Fix(\sg_o^r\circ\pi_*c_{\ov\ka})_{(\wt{\bf{F}},\wt\Pi)}$ to the disjoint union
\begin{align*}
\coprod_{\wt{d}}\left\{\wt{D}^\times_{\wt{d}}\left(h^{\infty,o}\sK_I^{\infty,o},h_{\wt{d}}h_r\sK_{r,m}\right)\in\sY_{\wt{d}}\middle|(h^{\infty,o})^{-1}\wt{d}h^{\infty,o}\in\sK^{\infty,o}_Ig^o\sK_I^{\infty,o}\mbox{ and }h_r^{-1}\wt\ga\sg_o(h_r)\in\ang{\vpi_o}\cap\GL_n(\cO_{o,r})\right\},
\end{align*}
where $\wt{d}$ runs over all $r$-admissible $\wt{D}^\times$-conjugacy classes in $\wt{D}^\times$, and $\wt\ga$ is defined to be $h_{\wt{d}}^{-1}\ve_o\sg_o^r(h_{\wt{d}})$.
\end{prop*}
Note that $\wt\ga$ is fixed by $\sg_o^r$ and hence lies in $\GL_n(\bf{F}_{o,r})$.
\begin{proof}[Proof of Proposition \ref{prop:bijdisjoint}]
Begin by applying Proposition \ref{ss:dellipticsheavesisomorphismclasses} to $g^{o,-1}\sK_I^{\infty,o}g^o$ instead of $\sK_I^{\infty,o}$ to obtain an adelic description of $\Fix(\sg_o^r\circ\pi_*c_{\ov\ka})_{(\wt{\bf{F}},\wt\Pi)}$as in \cite[p.~53]{Lau96}. Next, simplify the resulting description by using our largeness hypothesis on $I$: this ensures that $\sK_I^{\infty,o}$ is small enough to apply the same argument as in \cite[(3.2.6)]{Lau96}. Conclude as in \cite[(3.2.7)]{Lau96}.
\end{proof}
Return to the proof of Proposition \ref{ss:galoistohecke}. Since $\de_o$ corresponds to $h_r^{-1}\wt\ga\sg_o(h_r)$ under Proposition \ref{ss:dellipticsheavesadelicisogenyclass}, our sum equals
\begin{align*}
\sum_{(\wt{\bf{F}},\wt\Pi)}\sum_{\wt{d}}\vol\left(\wt{D}^\times_{\wt{d}}\bs\left((D\otimes\bA^{\infty,o})^\times_{\wt{d}}\times\GL_{n,\wt\ga}^{\sg_o}(\bf{F}_o)\right)\right)\O_{\wt{d}}(f^{\infty,o})\TO_{\wt\ga,\sg_o}(\phi_{\tau,h})\tr\left((\pi_{\ov\ka,*}c_{\ov\ka}\times\tau)_y|(\cL_{I\ssm o,\xi,\ov\ka})_{z}\right),
\end{align*}
where $\wt{d}$ runs over all $r$-admissible $\wt{D}^\times$-conjugacy classes in $\wt{D}^\times$, and $\GL_{n,\wt\ga}^{\sg_o}$ denotes the $\sg_o$-centralizer of $\wt\ga$ in $\GL_n(\bf{F}_{o,r})$. As before, we do not explicate our Haar measures, but we choose them compatibly whenever possible. We may rewrite the volume factors in the above sum via the following lemma.
\begin{lem}[{\cite[(11.7)]{LRS93}}]\label{lem:volumelemma}
The embeddings
\begin{align*}
(\wt{D}\otimes\bA^{\infty,o})^\times_{\wt{d}}\inj{}(D\otimes\bA^{\infty,o})^\times_{\wt{d}}\mbox{ and }(\wt{D}\otimes \bf{F}_o)^\times\inj{}\GL_{n,\wt\ga}^{\sg_o}(\bf{F}_o)
\end{align*}
induced by the left action of $\wt{D}^\times$ as in the proof of Proposition \ref{ss:dellipticsheavesisogenyclass} are isomorphisms.
\end{lem}
Return to the proof of Proposition \ref{ss:galoistohecke}. Lemma \ref{lem:volumelemma} indicates that we can modify our volume terms to make our expression of interest equal
\begin{align*}
\sum_{(\wt{\bf{F}},\wt\Pi)}\sum_{\wt{d}}\vol\left(\wt{D}^\times_{\wt{d}}\bs(\wt{D}\otimes\bA^\infty)^\times_{\wt{d}}\right)\O_{\wt{d}}(f^{\infty,o})\TO_{\wt\ga,\sg_o}(\phi_{\tau,h})\tr\left((\pi_{\ov\ka,*}c_{\ov\ka}\times\tau)_y|(\cL_{I\ssm o,\xi,\ov\ka})_{z}\right).
\end{align*}
At this point, we have two goals: to rewrite this as a sum over certain conjugacy classes of $D^\times$ instead of $\wt{D}^\times$, and to rewrite the trace on $(\cL_{I\ssm o,\xi})_z$ in terms of algebraic group-theoretic data. For this, we shall use the following lemma, which transfers both conjugacy classes as well as volumes of their stabilizers.
\subsection{}\label{ss:dtildetodtransfer}
Write $\rn:D\rar \bf{F}$ for the reduced norm of $D$, and let $\ga$ be an element of $D^\times$. Write $\bf{F}'$ for the finite extension $\bf{F}[\ga]$ of $\bf{F}$. We say that $\ga$ is \emph{$r$-admissible} if $o(\rn\ga)=r$ and there exists a place $o'$ of $\bf{F}'$ above $o$ such that, for all other places $x'\neq o'$ of $\bf{F}'$ lying above $o$, we have $x'(\ga)=0$. Note that $r$-admissibility descends to a property on $D^\times$-conjugacy classes in $D^\times$.
\begin{lem*}[{\cite[(11.9)]{LRS93}, \cite[(3.5.4)]{Lau96}}]
We have a bijection
\begin{align*}
&\{D^\times\mbox{-conjugacy classes in }D^\times\mbox{ that are }r\mbox{-admissible and elliptic in }D_\infty^\times\}\\
&\rar^\sim\coprod_{(\wt{\bf{F}},\wt\Pi)}\{\wt{D}^\times\mbox{-conjugacy classes in }\wt{D}^\times\mbox{ that are }r\mbox{-admissible}\},
\end{align*}
where $(\wt{\bf{F}},\wt\Pi)$ ranges over all $\vp$-pairs satisfying the conditions of Proposition \ref{ss:dellipticsheavesphipairs}. If we denote this bijection by $\ga\mapsto\wt{d}$, then the $D^\times_o=\GL_n(\bf{F}_o)$-conjugacy class of $\N_o\wt\ga$ equals that of $\ga$, and we have an equality of traces
\begin{align*}
\tr\left((\pi_{\ov\ka,*}c_{\ov\ka}\times\tau)_y|(\cL_{I\ssm o,\xi,\ov\ka})_{z}\right) = \Te_\xi(\ov\ga),
\end{align*}
where $\ov\ga$ is any elliptic element of $\ov{B}^\times$ with the same characteristic polynomial as $\ga$, and $\Te_\xi$ is the character of $\xi$. Furthermore, we have an equality of volumes
\begin{align*}
\vol\left(\wt{D}^\times_{\wt{d}}\bs(\wt{D}\otimes\bA^\infty)^\times_{\wt{d}}\right) = a(\ga)\frac1{n\cdot\vol(\vpi_\infty^\bZ\bs\wt{D}_{\wt\infty}^\times)},
\end{align*}
where $a(\ga)$ is another volume factor
\begin{align*}
a(\ga) \deq \vol\left(\vpi_\infty^\bZ D^\times_\ga\bs(D\otimes\bA)^\times_\ga\right).
\end{align*}
\end{lem*}
Return to the proof of Proposition \ref{ss:galoistohecke}. The above lemma implies that our sum becomes
\begin{align*}
\sum_\ga a(\ga)\frac{\ve(\ga)}{n\cdot\vol(\vpi_\infty^\bZ\bs\wt{D}_{\wt\infty}^\times)}\O_\ga(f^{\infty,o})\ve(\ga)\TO_{\wt\ga,\sg_o}(\phi_{\tau,h})\Te_\xi(\ov\ga),
\end{align*}
where $\ga$ runs over all $D^\times$-conjugacy classes in $D^\times$ that are $r$-admissible and elliptic in $D^\times_\infty$, and $\ve(\ga)$ is the Kottwitz sign $\ve(\ga)\deq(-1)^{n/[\bf{F}_\infty[\ga]:\bf{F}_\infty]-1}$. Since $\phi_{\tau,h}$ and $f_{\tau,h}$ have matching orbital integrals, applying Lemma \ref{ss:dtildetodtransfer} further changes our sum to
\begin{align*}
\sum_\ga a(\ga)\frac{\ve(\ga)}{n\cdot\vol(\vpi_\infty^\bZ\bs\wt{D}_{\wt\infty}^\times)}\O_\ga(f^{\infty,o})\O_\ga(f_{\tau,h})\Te_\xi(\ov\ga) = \sum_\ga a(\ga)\frac{\ve(\ga)}{n\cdot\vol(\vpi_\infty^\bZ\bs\wt{D}_{\wt\infty}^\times)}\Te_\xi(\ov\ga)\O_\ga(f^{\infty,o}\times f_{\tau,h}).
\end{align*}
We now want to absorb the $\ve(\ga)\vol(\vpi_\infty^\bZ\bs\wt{D}_{\wt\infty}^\times)^{-1}\Te_\xi(\ov\ga)$ term into our orbital integral. In order to do so, we shall introduce the following special function on $D^\times_\infty=\GL_n(\bf{F}_\infty)$.
\subsection{}
We now introduce \emph{pseudo-coefficients}, which are certain functions in $C^\infty_c(\GL_n(\bf{F}_\infty))$. Recall that $\vpi_\infty$ acts trivially under $\xi$, which implies that $\xi$ has unitary central character. Therefore the local Jacquet--Langlands correspondence as in \ref{ss:localjacquetlanglands} yields an irreducible $L^2$ representation $\JL^{-1}(\xi)$ of $\GL_n(\bf{F}_\infty)$. Let $f_{\xi,\infty}$ in $C^\infty_c(\GL_n(\bf{F}_\infty))$ be the corresponding pseudo-coefficient of $\JL^{-1}(\xi)$ as in \cite[Section 5]{BR17}.
\begin{lem*}[{\cite[p.~2203]{BR17}, \cite[(13.8)]{LRS93}}]
Let $\ga$ be an element of $\GL_n(\bf{F}_\infty)$. Then the function $f_{\xi,\infty}$ satisfies
\begin{align*}
\O_\ga(f_{\xi,\infty}) =
\begin{cases}
\displaystyle\frac{\ve(\ga)}{\vol(\vpi_\infty^\bZ\bs\wt{D}_{\wt\infty}^\times)}\Te_\xi(\ov\ga) &\mbox{if }\ga\mbox{ is elliptic,}\\
0 & \mbox{otherwise.}
\end{cases}
\end{align*}
\end{lem*}
Return to the proof of Proposition \ref{ss:galoistohecke}. The pseudo-coefficient $f_{\xi,\infty}$ allows us to rewrite our sum as
\begin{align*}
\frac1n\sum_\ga a(\ga)\O_\ga(f_{\xi,\infty}\times f^{\infty,o}\times f_{\tau,h}).
\end{align*}
\subsection{}
Before using the Selberg trace formula, we first need some notation regarding automorphic representations. Write $\cA(\vpi_\infty^\bZ D^\times\bs(D\otimes\bA)^\times)$ for the $\ov\bQ_\ell$-vector space
\begin{align*}
\cA(\vpi_\infty^\bZ D^\times\bs(D\otimes\bA)^\times)\deq\{f:\vpi_\infty^\bZ D^\times\bs(D\otimes\bA)^\times\rar\ov\bQ_\ell\mid f\mbox{ is locally constant}\}.
\end{align*}
Then $\cA(\vpi_\infty^\bZ D^\times\bs(D\otimes\bA)^\times)$ has a left action of $\vpi_\infty^\bZ\bs(D\otimes\bA)^\times$ via right multiplication, and we see that this is a smooth representation of $\vpi_\infty^\bZ\bs(D\otimes\bA)^\times$. Now $\vpi_\infty^\bZ D^\times\bs(D\otimes\bA)^\times$ is compact because $D$ is a division algebra, so we obtain a decomposition
\begin{align*}
\cA(\vpi_\infty^\bZ D^\times\bs(D\otimes\bA)^\times) = \bigoplus_{\wt\Pi}\wt\Pi^{\oplus m(\wt\Pi)},
\end{align*}
where $\wt\Pi$ ranges over all irreducible admissible representations of $\vpi^\bZ_\infty\bs(D\otimes\bA)^\times$, and $m(\wt\Pi)$ is a non-negative integer. If $m(\wt\Pi)$ is nonzero, we say $\wt\Pi$ is \emph{automorphic}. The weak multiplicity one theorem \cite[Theorem 3.3.(a)]{BR17} indicates that $m(\wt\Pi)$ is at most $1$.
Return to the proof of Proposition \ref{ss:galoistohecke}. Applying the Selberg trace formula to $\cA(\vpi_\infty^\bZ D^\times\bs(D\otimes\bA)^\times)$ shows that our sum of orbital integrals equals the sum of traces
\begin{align*}
\frac1n\sum_{\wt\Pi} m(\wt\Pi)\tr(f_{\xi,\infty}\times f^{\infty,o}\times f_{\tau,h}|\wt\Pi),
\end{align*}
where $\wt\Pi$ runs over all irreducible automorphic representations of $\vpi_\infty^\bZ\bs(D\otimes\bA)^\times$. Therefore the following proposition of Laumon--Rapoport--Stuhler concludes the proof of Proposition \ref{ss:galoistohecke}.
\end{proof}
\begin{prop}
We have an equality of traces
\begin{align*}
\sum_{\wt\Pi} m(\wt\Pi)\tr(f_{\xi,\infty}\times f^{\infty,o}\times f_{\tau,h}|\wt\Pi) = \tr(f^{\infty,o}\times f_{\tau,h}|[H_\xi]),
\end{align*}
where $\wt\Pi$ ranges over all irreducible automorphic representations of $\vpi_\infty^\bZ\bs(D\otimes\bA)^\times$.
\end{prop}
\begin{proof}
This is the generalization of \cite[(13.6)]{LRS93} as described in \cite[(13.8)]{LRS93}. This generalization relies on the existence of the pseudo-coefficient $f_{\xi,\infty}$, which we have already seen exists as in \cite[Section 5]{BR17}.
\end{proof}
We remark that the proof of this proposition uses similar point-counting methods as we do, except that Laumon--Rapoport--Stuhler immediately pass to the level of $(D\otimes\bA^\infty)^\times$, rather than remaining at the level of $\GL_n(\bf{F}_{o,r})$ at the place $o$. The latter is necessary for incorporating the contributions of $\phi_{\tau,h}$ and hence $f_{\tau,h}$.
\section{Local-global compatibility}\label{s:localglobal}
Our goal in this section is to prove that condition (c) in Lemma \ref{ss:firstinductivelemma} holds, which completes our proof of Theorem A. Along the way, we shall prove Theorem B. Returning to the local notation of \S\ref{s:deformationspaces}--\S\ref{s:lubintatetower} for a moment, we shall construct the desired virtual representation $\rho(\pi)$ of $W_F$ from our cohomology representation $[H_\xi]$.
Now revert back to global notation conventions. We start by recalling results of Laumon--Rapoport--Stuhler that specify the automorphic representations of $(D\otimes\bA^\infty)^\times$ occurring in $[H_\xi]$. Next, we use the \emph{strong multiplicity one theorem}, which says that automorphic representations of $(D\otimes\bA)^\times$ are determined by their local components at cofinitely many places, to convert Proposition \ref{ss:galoistohecke} into a global analog of condition (c) in Lemma \ref{ss:firstinductivelemma}. Finally, we introduce the \emph{global Jacquet--Langlands correspondence}, which enables us to pass between automorphic representations of $(D\otimes\bA)^\times$ and $\GL_n(\bA)$. This finishes the proof of Theorem B, our titular local-global compatibility result.
At this point, we want to use Theorem B to prove condition (c) in Lemma \ref{ss:firstinductivelemma}. To do so, we need to embed local representations of $\GL_n$ into global representations. We start by presenting such an embedding for $L^2$ representations, which uses the \emph{Deligne--Kazhdan simple trace formula}. Trace formula methods also allow us to embed cuspidal representations globally. From here, we use M\oe glin--Waldspurger's description of the discrete automorphic spectrum of $\GL_n$ in order to realize Speh modules in a global setting, and this enables us to complete the proof of condition (c) in Lemma \ref{ss:firstinductivelemma}.
\subsection{}
First, we introduce some notation on multiplicity spaces. Let $\wt\Pi^\infty$ be an irreducible admissible representation of $(D\otimes\bA^\infty)^\times$, and write $H_{\xi,\eta}^i(\wt\Pi^\infty)$ for the $\ov\bQ_\ell$-vector space
\begin{align*}
H^i_{\xi,\eta}(\wt\Pi^\infty) \deq \Hom_{(D\otimes\bA^\infty)^\times}(\wt\Pi^\infty,H^{i,\text{ss}}_{\xi,\eta}),
\end{align*}
that is, the multiplicity of $\wt\Pi^\infty$ in $H^{i,\text{ss}}_{\xi,\eta}$, where $H^{i,\text{ss}}_{\xi,\eta}$ denotes the semisimplification of $H^{i}_{\xi,\eta}$. We see that $H^{i}_{\xi,\eta}(\wt\Pi^\infty)$ is a continuous finite-dimensional representation of $G_\bf{F}$ over $\ov\bQ_\ell$, where $G_\bf{F}$ denotes the Galois group of the global field $\bf{F}$. Write $[H_\xi(\wt\Pi^\infty)]$ for the virtual representation $\sum_{i=0}^\infty(-1)^i H^i_{\xi,\eta}(\wt\Pi^\infty)$, which we see equals the multiplicity of $\wt\Pi^\infty$ in $[H_\xi]$. Therefore $\wt\Pi^\infty$ occurs in $[H_\xi]$ if and only if $[H_\xi(\wt\Pi^\infty)]$ is nonzero.
\subsection{}\label{ss:laumonrapoportstuhlerisotypiccomponents}
The question of whether $\wt\Pi^\infty$ occurs in $[H_\xi]$ is connected to automorphic representations of $\vpi_\infty^\bZ\bs(D\otimes\bA)^\times$ as well as the representation $\xi$ of $\ov{B}^\times$. Since $\vpi_\infty$ acts trivially under $\xi$, we see $\xi$ has unitary central character. Therefore the representation $\JL^{-1}(\xi)$ as in \ref{ss:localjacquetlanglands} is an irreducible $L^2$ representation of $\GL_n(\bf{F}_\infty)$ and hence isomorphic to a Steinberg module
\begin{align*}
\JL^{-1}(\xi) = \St_t(\pi_{0,\xi})
\end{align*}
as in \ref{ss:generalizedsteinbergspeh}, where $t$ is a positive divisor of $n$, $d=\frac{n}t$, and $\pi_{0,\xi}$ is an irreducible cuspidal representation of $\GL_d(\bf{F}_\infty)$ with unitary central character. Recall that we can also form the Speh module $\Sp_t(\pi_{0,\xi})$ as in \ref{ss:generalizedsteinbergspeh}, which is an irreducible smooth representation of $\GL_n(\bf{F}_\infty)$ The following result of Laumon--Rapoport--Stuhler determines precisely when $\wt\Pi^\infty$ occurs in $[H_\xi]$.
\begin{prop*}\hfill
\begin{enumerate}[(i)]
\item The virtual representation $[H_\xi(\wt\Pi^\infty)]$ of $G_\bf{F}$ is nonzero if and only if either $\wt\Pi^\infty\otimes\St_t(\pi_{0,\xi})$ or $\wt\Pi^\infty\otimes\Sp_t(\pi_{0,\xi})$ is an irreducible automorphic representation of $\vpi_\infty^\bZ\bs(D\otimes\bA)^\times$. Furthermore, we have $\dim[H_\xi(\wt\Pi^\infty)]=n$.
\item In addition, we have $H^i_{\xi,\eta}(\wt\Pi^\infty)=0$ either for all odd $i$ or all even $i$.
\end{enumerate}
\end{prop*}
\begin{proof}
Part (ii) and the first statement of part (i) comprise the generalization of \cite[(14.7)]{LRS93} as described in \cite[(14.21)]{LRS93}. This generalization follows from the global Jacquet--Langlands correspondence between $D^\times$ and $\GL_n(\bf{F})$ (which is now known \cite[Theorem 3.2]{BR17}), by the remark made in \cite[(14.24)]{LRS93}. As for the dimension statement in part (i), this follows from Deligne's purity theorem, c.f. \cite[(14.11)(ii)]{LRS93}.
\end{proof}
\subsection{}\label{ss:globalconditionc}
Next, we use Proposition \ref{ss:galoistohecke} to obtain the following result, which resembles condition (c) in Lemma \ref{ss:firstinductivelemma}.
\begin{prop*}
Let $\tau$ be an element of $W_{\bf{F}_o}$ with $v_o(\tau)>0$, and let $h$ be a function in $C^\infty_c(\GL_n(\cO_o))$. Let $\wt\Pi^\infty$ be an irreducible admissible representation of $(D\otimes\bA^\infty)^\times$ for which $[H_\xi(\wt\Pi^\infty)]$ is nonzero. Then we have an equality of traces
\begin{align*}
\tr(f_{\tau,h}|\wt\Pi_o^\infty) = \tr(\tau|[H_\xi(\wt\Pi^\infty)])\tr(h|\wt\Pi_o^\infty),
\end{align*}
where $\wt\Pi_o^\infty$ denotes the component of $\wt\Pi^\infty$ at $o$.
\end{prop*}
For any irreducible admissible representation $\wt\Pi'^\infty$ of $(D\otimes\bA^\infty)^\times$, we write $\wt\Pi'^{\infty,o}$ for the component of $\wt\Pi'^\infty$ away from $o$.
\begin{proof}
Let $I$ be a sufficiently large finite closed subscheme of $C\ssm\infty$ such that $\wt\Pi^{\infty,o}$ has nonzero $\sK_I^{\infty,o}$-invariants and $h$ is invariant under $\sK_{I,o}^\infty$. Because $[H_\xi]$ is admissible, we see that $[H_\xi]^{\sK_I^\infty}$ is finite-dimensional, so $[H_\xi]$ can only contain finitely many non-isomorphic irreducible admissible representations $\wt\Pi'^\infty$ of $(D\otimes\bA^\infty)^\times$ with nonzero $\sK_I^\infty$-invariants. By applying the Chinese remainder theorem to these left $C^\infty_c((D\otimes\bA^{\infty,o})^\times\!/\!/\sK_I^{\infty,o})$-modules $\wt\Pi'^\infty$, we obtain a function $f^{\infty,o}$ in $C^\infty_c((D\otimes\bA^{\infty,o})^\times)$ satisfying the following properties:
\begin{enumerate}[(i)]
\item $\tr(f^{\infty,o}|\wt\Pi^{\infty,o})=1$,
\item if $\tr(f^{\infty,o}|\wt\Pi'^\infty)\neq0$ for one of our aforementioned $\wt\Pi'^\infty$, then $\wt\Pi'^{\infty,o}$ is isomorphic to $\wt\Pi^{\infty,o}$.
\end{enumerate}
In order to strengthen property (ii) to show that $\wt\Pi'^\infty$ is isomorphic to $\wt\Pi^\infty$ (hence extending our isomorphism to the component at $o$), we use the \emph{strong multiplicity one theorem}. We recall its statement below.
\subsection{}\label{ss:strongmultiplicityone}
Let $T$ be a finite set of places of $\bf{F}$. For any irreducible admissible representation $\wt\Pi$ of $\vpi^\bZ_\infty\bs(D\otimes\bA)^\times$, we write $\wt\Pi^T$ for the component of $\wt\Pi$ away from $T$, and we write $\wt\Pi_T$ for the component of $\wt\Pi$ at $T$.
\begin{thm*}[{\cite[Theorem 3.3.(b)]{BR17}}]
Let $\wt\Pi$ and $\wt\Pi'$ be two irreducible automorphic representations of $\vpi_\infty^\bZ\bs(D\otimes\bA)^\times$, and let $T$ be a finite set of places of $\bf{F}$. If $\wt\Pi^T$ is isomorphic to $\wt\Pi'^T$, then $\wt\Pi$ is isomorphic to $\wt\Pi'$.
\end{thm*}
We remark that \cite{BR17} deduces the strong multiplicity one theorem by using the global Jacquet--Langlands correspondence \cite[Theorem 3.2]{BR17} to reduce to the case of $\GL_n(\bf{F})$.
Return to the proof of Proposition \ref{ss:globalconditionc}. The strong multiplicity one theorem and Proposition \ref{ss:laumonrapoportstuhlerisotypiccomponents}.(i) imply that, if the hypothesis of property (ii) holds, then $\wt\Pi'^\infty$ is actually isomorphic to $\wt\Pi^\infty$. Therefore plugging $f^{\infty,o}$ into Proposition \ref{ss:galoistohecke} shows that
\begin{align*}
\tr(h|\wt\Pi_o^\infty)\tr(\tau|[H_\xi(\wt\Pi^\infty)]) = \tr(f^{\infty,o}\times h\times\tau|[H_\xi]) = \textstyle\frac1n\tr(f^{\infty,o}\times f_{\tau,h}|[H_\xi]) = \frac1n\dim[H_\xi(\wt\Pi^\infty)]\tr(f_{\tau,h}|\wt\Pi_o^\infty),
\end{align*}
and applying the second statement in Proposition \ref{ss:laumonrapoportstuhlerisotypiccomponents}.(i) concludes the proof of Proposition \ref{ss:globalconditionc}.
\end{proof}
\subsection{}\label{ss:globaljacquetlanglands}
To convert automorphic representations of $\GL_n(\bA)$ into those of $(D\otimes\bA)^\times$, we use the \emph{global Jacquet--Langlands correspondence}. Recall from \ref{ss:divisionalgebras} that $\text{Bad}$ denotes the set of places where $D$ ramifies.
From this point onwards, assume that $D_x$ is a division algebra for all places $x$ in $\text{Bad}$. We say that an irreducible discrete automorphic representation $\Pi$ of $\GL_n(\bA)$ is \emph{$D$-admissible} if, for all places $x$ in $\text{Bad}$, the local component $\Pi_x$ is a Steinberg module or Speh module, as a representation of $\GL_n(\bf{F}_x)$.
There exists a unique bijection \cite[Theorem 3.2]{BR17}
\begin{align*}
\JL:\left\{
\begin{tabular}{c}
isomorphism classes of irreducible\\
$D$-admissible representations of $\GL_n(\bA)$
\end{tabular}\right\}\rar^\sim \left\{\begin{tabular}{c}
isomorphism classes of irreducible\\
automorphic representations of $(D\otimes\bA)^\times$
\end{tabular}\right\}
\end{align*}
such that, for all places $x$ of $\bf{F}$,
\begin{enumerate}[(i)]
\item if $x$ does not lie in $\text{Bad}$, then $\JL(\Pi)_x$ is isomorphic to $\Pi_x$,
\item if $x$ lies in $\text{Bad}$, and $\Pi_x$ is isomorphic to either $\St_t(\pi_0)$ or $\Sp_t(\pi_0)$ as in Definition \ref{ss:generalizedsteinbergspeh}, then $\JL(\Pi)_x$ is isomorphic to $\JL(\St_t(\pi_0))$, where the latter $\JL$ denotes the local Jacquet--Langlands correspondence from \ref{ss:localjacquetlanglands}.
\end{enumerate}
\subsection{}\label{ss:choiceofdivisionalgebra}
Before using the global Jacquet--Langlands correspondence to deduce Theorem B, let us fix our choice of division algebra. Fix three distinct places $x_1$, $x_2$, and $\infty$ of $\bf{F}$, and let $D$ be the central division algebra over $\bf{F}$ of dimension $n^2$ defined by
\begin{align*}
\inv_x(D) =
\begin{cases}
\frac1n &\mbox{ if }x=x_1,\,\\
-\frac1n &\mbox{ if }x=x_2,\,\\
0 & \mbox{otherwise,}
\end{cases}
\end{align*}
as in \ref{ss:divisionalgebras}. Let $\sD$ be a maximal order of $D$, which can be constructed using \ref{ss:orders} because division algebras split at cofinitely many places.
\subsection{}\label{prop:localglobalcompatibility}
We now proceed to prove Theorem B, using Proposition \ref{ss:globalconditionc} and the global Jacquet--Langlands correspondence.
\begin{prop*}
Let $\Pi$ be an irreducible discrete automorphic representation of $\GL_n(\bA)$ whose components at $x_1$, $x_2$, and $\infty$ are either irreducible $L^2$ representations or Speh modules of $\GL_n(\bf{F}_{x_1})$, $\GL_n(\bf{F}_{x_2})$, and $\GL_n(\bf{F}_\infty)$, respectively. Then there exists a unique $n$-dimensional semisimple continuous representation $R(\Pi)$ of $G_\bf{F}$ over $\ov\bQ_\ell$ such that, for all places $o$ of $\bf{F}$ not lying in $\{x_1,x_2,\infty\}$, the restriction of $R(\Pi)$ to $W_{\bf{F}_o}$ satisfies
\begin{align*}
\res{R(\Pi)}_{W_{\bf{F}_o}} = \rho(\Pi_o),
\end{align*}
where we identify $\ov\bQ_\ell$ with $\bC$.
\end{prop*}
\begin{proof}
The Chebotarev density theorem implies that, as $o$ varies over all places of $\bf{F}$ not lying in $\{x_1,x_2,\infty\}$, the conjugacy classes of geometric $q_o$-Frobenius elements at $o$ in $G_\bf{F}$ are dense. Now $n$-dimensional continuous semisimple representations of $G_\bf{F}$ are determined, up to isomorphism, by their characteristic polynomials, and said polynomials are continuous in $G_\bf{F}$, so we see that the above condition determines $R(\Pi)$ uniquely.
We turn to the existence of $R(\Pi)$. Our hypotheses indicate that $\Pi$ is $D$-admissible, so we can form the automorphic representation $\wt\Pi\deq\JL(\Pi)$ of $(D\otimes\bA)^\times$. Now $\wt\Pi_\infty=\Pi_\infty$ is either a Steinberg module or a Speh module, so Proposition \ref{ss:laumonrapoportstuhlerisotypiccomponents}.(i) ensures that there exists an irreducible smooth representation of $\ov{B}^\times/\vpi_\infty^\bZ$ such that $[H_\xi(\wt\Pi^\infty)]$ has dimension $n$. Finally, Proposition \ref{ss:globalconditionc} indicates that $\res{[H_\xi(\wt\Pi^\infty)]}_{W_{\bf{F}_o}}$ satisfies the defining condition Theorem A.(i) of $\rho(\Pi_o)$, so taking $R(\Pi)=[H_\xi(\wt\Pi^\infty)]$ yields the desired result.
\end{proof}
\subsection{}
Briefly return to the local notation of \S\ref{s:deformationspaces}--\S\ref{s:lubintatetower}. In order to convert Theorem B into a proof of condition (c) in Lemma \ref{ss:firstinductivelemma}, we must find some global automorphic representation $\wt\Pi$ of $\GL_n(\bA)$ such that our local representation $\pi$ is isomorphic to $\wt\Pi_o$ as a representation of $\GL_n(\bf{F}_o)$. That is, we must embed local representations into global ones. Recall that the two classes of possibilities for $\pi$ from condition (c) of Lemma \ref{ss:firstinductivelemma} that we must consider are
\begin{enumerate}[(1)]
\item essentially $L^2$ representations, which we will reduce to the case of $L^2$ representations via Proposition \ref{ss:unramifiedtwistsofrho},
\item Speh modules, which we reduce to the case of cuspidal representations by M\oe glin--Waldspurger's classification of the discrete automorphic spectrum of $\GL_n$.
\end{enumerate}
\subsection{}\label{ss:embedl2}
Return to our global notation. We start by embedding local $L^2$ representations into global representations.
\begin{lem*}
Let $\pi$ be an irreducible $L^2$ representation of $\GL_n(\bf{F}_o)$. Then there exists an irreducible automorphic representation $\Pi$ of $\GL_n(\bA)$ whose
\begin{enumerate}[(i)]
\item component at $o$ is isomorphic to $\pi$,
\item components at $x_1$, $x_2$, and $\infty$ are irreducible $L^2$ representations of $\GL_n(\bf{F}_{x_1})$, $\GL_n(\bf{F}_{x_2})$, and $\GL_n(\bf{F}_\infty)$, respectively, with unitary central characters.
\end{enumerate}
\end{lem*}
\begin{proof}
This is proven precisely as in \cite[Corollary VI.2.5]{HT01} using the simple trace formula \cite[A.1.d Theorem]{DKV84}.
\end{proof}
\subsection{}\label{ss:embedcuspidal}
As a first step towards embedding Speh modules into a global setting, we first realize certain local cuspidal representations in a global context.
\begin{lem*}
Let $\pi$ be an irreducible cuspidal representation of $\GL_n(\bf{F}_o)$ with unitary central character. Then there exists an irreducible cuspidal automorphic representation $\Pi$ of $\GL_n(\bA)$ whose
\begin{enumerate}[(i)]
\item component at $o$ is isomorphic to $\pi$,
\item components at $x_1$, $x_2$, and $\infty$ are irreducible cuspidal representations of $\GL_n(\bf{F}_{x_1})$, $\GL_n(\bf{F}_{x_2})$, and $\GL_n(\bf{F}_\infty)$, respectively, with unitary central characters.
\end{enumerate}
\begin{proof}
This follows from the proof of \cite[(15.2)]{LRS93}, where we use $\infty$ instead of $x_3$ and $o$ instead of $x_0$.
\end{proof}
\end{lem*}
\subsection{}\label{ss:moeglinwaldspurger}
We want to use M\oe glin--Waldspurger's classification of the discrete automorphic spectrum of $\GL_n$. To present this classification, let us recall some notation from \ref{ss:bernsteinzelevinsky} on local representations. Briefly return to the local notation of \S\ref{s:deformationspaces}--\S\ref{s:lubintatetower}, and let $\{\De_1,\dotsc,\De_t\}$ be a collection of segments as in \cite[3.1]{Zel80} such that $\De_i$ does not precede $\De_j$ as in \cite[4.1]{Zel80} for $i<j$. Then each associated $Q(\De_i)$ is an irreducible essentially $L^2$ representation of $\GL_{n_i}(F)$, and recall that $Q(\De_1,\dotsc,\De_t)$ denotes the unique irreducible quotient of
\begin{align*}
\nInd_{P(F)}^{\GL_n(F)}\left(Q(\De_1)\otimes\dotsb\otimes Q(\De_t)\right),
\end{align*}
where $n=n_1+\dotsb+n_t$, and $P$ is the standard parabolic subgroup of $\GL_n$ with block sizes $(n_1,\dotsc,n_t)$.
Now return to the global notation. Work of M\oe glin--Waldspurger implies the following.
\begin{lem*}[{\cite[Theorem]{MW89}}]
Let $d$ be a positive divisor of $n$, write $d\deq\frac{n}t$, and let $\Pi_0$ be an irreducible automorphic cuspidal representation of $\GL_d(\bA)$. Then the restricted tensor product
\begin{align*}
\Pi\deq\sideset{}{'}\bigotimes_xQ\left(\{\Pi_{0,x}[\textstyle\frac{t-1}2]\},\dotsc,\{\Pi_{0,x}[\frac{1-t}2]\}\right),
\end{align*}
where $x$ ranges over all places of $\bf{F}$, is an irreducible discrete automorphic representation of $\GL_n(\bA)$.
\end{lem*}
\subsection{}\label{ss:embedspeh}
We apply Lemma \ref{ss:moeglinwaldspurger} to embed Speh modules into global representations.
\begin{lem*}
Let $\pi$ be a Speh module, as a representation of $\GL_n(\bf{F}_o)$. Then there exists an irreducible discrete automorphic representation $\Pi$ of $\GL_n(\bA)$ whose
\begin{enumerate}[(i)]
\item component at $o$ is isomorphic to $\pi$,
\item components at $x_1$, $x_2$, and $\infty$ are Speh modules, as representations of $\GL_n(\bf{F}_{x_1})$, $\GL_n(\bf{F}_{x_2})$, and $\GL_n(\bf{F}_\infty)$, respectively.
\end{enumerate}
\end{lem*}
\begin{proof}
Write $\pi$ as $\pi=\Sp_t(\pi_0)$, where $t$ is some positive divisor of $n$, and $\pi_0$ is an irreducible cuspidal representation of $\GL_{n/t}(\bf{F}_o)$ with unitary central character. Now Lemma \ref{ss:embedcuspidal} provides an irreducible cuspidal automorphic representation $\wt\Pi_0$ of $\GL_{n/t}(\bA)$ whose component at $o$ is isomorphic to $\pi_0$ and whose components at $x_1$, $x_2$, and $\infty$ are irreducible cuspidal representations of $\GL_n(\bf{F}_{x_1})$, $\GL_n(\bf{F}_{x_2})$, and $\GL_n(\bf{F}_\infty)$, respectively, with unitary central characters. Then restricted tensor product $\Pi$ associated with $\Pi_0$ as in Lemma \ref{ss:moeglinwaldspurger} has components
\begin{align*}
\Pi_o = \Sp_t(\pi_0) = \pi,\, \Pi_{x_1}= \Sp_t(\Pi_{0,x_1}),\, \Pi_{x_2}= \Sp_t(\Pi_{0,x_2}),\mbox{ and }\Pi_\infty = \Sp_t(\Pi_{0,\infty}),
\end{align*}
as desired.
\end{proof}
\subsection{}\label{s:conditionc}
Return to the local notation of \S\ref{s:deformationspaces}--\S\ref{s:lubintatetower}. Thus $F$ is a local field of positive characteristic, $\cO$ denotes its ring of integers, $\vpi$ is a fixed uniformizer, and $\ka$ denotes $\cO/\vpi$. We will finally complete our proof of condition (c) in Lemma \ref{ss:firstinductivelemma} by explicitly embedding our local situation into the global situation of $\sD$-elliptic sheaves.
\begin{prop*}
Let $\pi$ be a Speh module or an irreducible essentially $L^2$ representation of $\GL_n(F)$. Then there exists an $n$-dimensional $\bQ_{\geq0}$-virtual continuous representation $\rho(\pi)$ of $W_F$ satisfying the trace condition
\begin{align*}
\tr(f_{\tau,h}|\pi) = \tr\left(\tau|\rho(\pi)\right)\tr(h|\pi)
\end{align*}
for all $\tau$ in $W_F$ with $v(\tau)>0$ and $h$ in $C^\infty_c(\GL_n(\cO))$.
\end{prop*}
\begin{proof}
Our goal is to apply Theorem B. Let $C=\bP^1_\ka$ be our curve of interest, choose $o$ in \ref{ss:choiceofdivisionalgebra} to be a $\ka$-point of $C$, and choose $x_1$, $x_2$, and $\infty$ in \ref{ss:choiceofdivisionalgebra} to be distinct closed points in $C\ssm o$. Note that $\bf{F}_o$ is isomorphic to $F$.
Let us first consider the case when $\pi$ has unitary central character, that is, $\pi$ is either a Speh module or $L^2$. Then Lemma \ref{ss:embedl2} or Lemma \ref{ss:embedspeh}, respectively, yields an irreducible discrete automorphic representation $\Pi$ of $\GL_n(\bA)$ satisfying the hypotheses of Theorem B such that $\Pi_o$ is isomorphic to $\pi$. Proceeding to apply Theorem B to $\Pi$ shows that we may take $\rho(\pi)=\res{R(\Pi)}_{W_{\bf{F}_o}}$.
In the last remaining case when $\pi$ is an arbitrary irreducible essentially $L^2$ representation $\pi$ of $\GL_n(F)$, note that $\pi$ is isomorphic to an unramified twist of an $L^2$ representation. Thus the above work, along with Proposition \ref{ss:unramifiedtwistsofrho}, concludes the proof in this case.
\end{proof}
By using Proposition \ref{ss:conditionb}, Proposition \ref{s:conditionc}, and Proposition \ref{ss:conditiond} to verify that conditions (b)--(d) of Lemma \ref{ss:firstinductivelemma} hold and using Proposition \ref{ss:theoremabasecase} to check the $n=1$ base case, this concludes our proof by induction of Theorem A.
\section{The second inductive lemma: bijectivity of the correspondence}\label{s:secondinductivelemma}
Retain the local notation of \S\ref{s:deformationspaces}--\S\ref{s:lubintatetower}. The purpose of this section is to prove Theorem C, i.e. that $\pi\mapsto\rec\pi$ yields a bijection from isomorphism classes of irreducible cuspidal representations of $\GL_n(F)$ to isomorphism classes of $n$-dimensional irreducible continuous representations of $W_F$. We start by collecting some facts on restricting representations of $W_F$ to representations of $W_E$ for cyclic extensions $E/F$. Next, we use these facts to motivate \emph{automorphic base change}, an analogous operation that turns representations of $\GL_n(F)$ into representations of $\GL_n(E)$. Automorphic base change yields section's first main result: a lemma which allows us to prove Theorem C by inducting on $n$, provided that we verify certain conditions.
We then proceed to verify the conditions needed for this inductive lemma, thus completing the proof of Theorem C. The main ingredients are Theorem A, Theorem B, and our nearby cycles calculation from \S\ref{s:nearbycycles}. More precisely, we apply results from \S\ref{s:nearbycycles} to show that the preimage of any unramified representation under $\rec$ remained unramified, and we use Theorem B to show that $\rec$ is compatible with automorphic base change as well as twisting by characters.
\subsection{}\label{ss:galoissidebasechange}
We begin by establishing some notation on field extensions. Let $E$ be a finite extension of $F$ inside $F^\sep$. For any finite-dimensional semisimple continuous representation $\rho'$ of $W_E$ and any map $\al$ in $G_F$, write $\rho'^\al$ for the finite-dimensional semisimple continuous representation of $W_{\al^{-1}(E)}$ given by $\tau\mapsto\rho'(\al\circ\tau\circ\al^{-1})$ for all $\tau$ in $W_{\al^{-1}(E)}$.
Now assume that $E/F$ is a cyclic extension. Hence $\al^{-1}(E)=E$. Write $r$ for the degree of $E/F$, and fix a generator $\sg$ of $\Gal(E/F)=W_F/W_E$, which identifies it with $\bZ/r\bZ$. Write $\fK(E/F)$ for the set of group homomorphisms $\Gal(E/F)\rar\bC^\times$, and interpret $\fK(E/F)$ as a set of characters of $W_F$. Note that $\fK(E/F)$ acts on the set of isomorphism classes of irreducible continuous finite-dimensional representations of $W_F$ via twisting.
When $r$ is prime, one can use Frobenius reciprocity and the Mackey formula to verify the following assertions:
\begin{enumerate}[(i)]
\item Let $\rho'$ be an irreducible continuous finite-dimensional representation of $W_E$. Then $\rho'$ extends to a representation $\rho$ of $W_F$ if and only if $\rho'=\rho'^\sg$. Furthermore, if this is the case, then there are $r$ isomorphism classes of such $\rho'$, and any two differ by a twist of a character in $\fK(E/F)$.
\item Let $\rho$ be an irreducible continuous finite-dimensional representation of $W_F$. Then $\res\rho_{W_E}$ is reducible if and only if the stabilizer of $\rho$ in $\fK(E/F)$ is nontrivial (and hence all of $\fK(E/F)$). Furthermore, if this is the case, then
\begin{align*}
\res\rho_{W_E} = \rho'\oplus\dotsb\oplus\rho'^{\sg^{r-1}}
\end{align*}
for some irreducible continuous finite-dimensional representation $\rho'$ of $W_E$ satisfying $\rho'\neq\rho'^\sg$, and $\rho$ is the unique finite-dimensional semisimple continuous representation of $W_F$ with this property.
\end{enumerate}
We remark that the same argument works in the global setting, for which we shall use entirely analogous notation.
\subsection{}\label{ss:basechangeliftcuspidal}
Next, we recall \emph{automorphic base change} in the local setting. For any finite extension $E$ of $F$ inside $F^\sep$, any irreducible smooth representation $\pi'$ of $\GL_n(E)$, and any map $\al$ in $G_F$, write $\pi'^\al$ for the irreducible smooth representation of $\GL_n(\al^{-1}(E))$ given by $g\mapsto\pi'(\al(g))$ for all $g$ in $\GL_n(\al^{-1}(E))$. Note that $\Te_{\pi'^\al}=\Te_{\pi'}\circ\al$.
As in \ref{ss:galoissidebasechange}, specialize to cyclic extensions $E$ of $F$, and adopt the notation of \ref{ss:galoissidebasechange} as well. Since $\Art$ induces an isomorphism $F^\times/\Nm_{E/F}(E^\times)\rar^\sim\Gal(E/F)$, we can view $\fK(E/F)$ as a set of characters of $\GL_n(F)$ via precomposition with $\det\circ\Art$. Thus $\fK(E/F)$ acts on the set of isomorphism classes of irreducible smooth representations of $\GL_n(F)$ by twisting.
First, let $\pi$ be an irreducible cuspidal representation of $\GL_n(F)$. Write $u$ for the cardinality of the stabilizer of $\pi$ in $\fK(E/F)$. Then $u$ divides $n$, and there exists an irreducible cuspidal representation $\pi'$ of $\GL_{n/u}(E)$ such that
\begin{align*}
Q\left(\{\pi'\},\dotsc,\{\pi'^{\sg^{u-1}}\}\right)
\end{align*}
is an irreducible generic representation of $\GL_n(E)$ and is isomorphic to the base change lift of $\pi$ to $\GL_n(E)$ \cite[(II.4.12, prop.)]{HL11}. In particular, the description of genericity from \ref{ss:bernsteinzelevinsky} indicates that $\pi'$ is isomorphic to $\pi'^\sg$ if and only if $u=1$. Note that this is the automorphic analogue of \ref{ss:galoissidebasechange}.
\subsection{}\label{ss:basechangeliftlocal}
We now turn to local automorphic base change for generic representations. For any irreducible cuspidal representation $\tau$ of $\GL_d(F)$, write $\De(\tau,m)$ for the segment $\{\tau,\dotsc,\tau[m-1]\}$ as in \cite[3.1]{Zel80}. Recall from \ref{ss:bernsteinzelevinsky} that every irreducible generic representation $\pi$ of $\GL_n(F)$ is of the form
\begin{align*}
\nInd_{P(F)}^{\GL_n(F)}\left(Q(\De(\pi_1,m_1))\otimes\dotsb\otimes Q(\De(\pi_t,m_t))\right),
\end{align*}
where $n=d_1m_1+\dotsb+d_tm_t$, the $\pi_i$ are irreducible cuspidal representations of $\GL_{d_i}(F)$, and $P$ is the standard parabolic subgroup of $\GL_n$ with block sizes $(d_1m_1,\dotsc,d_tm_t)$. Furthermore, none of the $\De(\pi_i,m_i)$ are linked.
At this point, suppose that $\pi$ also has $\fK(E/F)$-regular segments as in \cite[(II.3.4)]{HL11}. Write $u_i$ for the divisor of $d_i$ and $\pi'_i$ for the irreducible cuspidal representation of $\GL_{d_i/u_i}(E)$ associated with $\pi_i$ as in \ref{ss:basechangeliftcuspidal}. Then the base change lift $\pi_E$ of $\pi$ to $\GL_n(E)$ is isomorphic to \cite[(II.4.4)]{HL11}, \cite[(II.4.12, cor.)]{HL11}
\begin{align*}
Q\left(\De(\pi'_1,m_1),\dotsc,\De(\pi_1'^{\sg^{u_1-1}},m_1),\dotsc,\De(\pi'_t,m_t),\dotsc,\De(\pi_t'^{\sg^{u_t-1}},m_t)\right).
\end{align*}
Note that $\pi_E$ is an irreducible generic representation of $\GL_n(E)$.
Next, let $\xi$ be any irreducible generic representation of $\GL_n(F)$ with $\fK(E/F)$-regular segments such that the base change lift of $\xi$ to $\GL_n(E)$ is isomorphic to $\pi_E$. Then $\xi$ must be isomorphic to
\begin{align*}
\nInd_{P(F)}^{\GL_n(F)}\left(Q(\De(\chi_1\cdot\pi_1,m_1))\otimes\dotsb\otimes Q(\De(\chi_t\cdot\pi_t,m_t))\right)
\end{align*}
for some $\chi_1,\dotsc,\chi_t$ in $\fK(E/F)$ \cite[(II.4.13, lem.)]{HL11}. Finally, every $\sg$-stable irreducible generic representation $\pi'$ of $\GL_n(E)$ is the base change lift of some irreducible generic representation of $\GL_n(F)$ with $\fK(E/F)$-regular segments \cite[(II.1.4)]{HL11}.
\subsection{}\label{ss:basechangeliftglobal}
We will also need automorphic base change in the global setting, for which we briefly return to our global notation. Hence $\bf{F}$ is a global function field, and $\bA$ denotes its ring of adeles. Let $\bf{E}$ be a cyclic extension of $\bf{F}$, write $\bA_\bf{E}$ for the ring of adeles of $\bf{E}$, and fix a generator $\sg$ of $\Gal(\bf{E}/\bf{F})=W_{\bf{F}}/W_{\bf{E}}$. Then $\sg$ acts on the set of isomorphism classes of irreducible discrete automorphic representations of $\GL_n(\bA_\bf{E})$ via precomposition.
Let $\Pi$ be an irreducible cuspidal automorphic representation of $\GL_n(\bA)$. Then there exists a unique irreducible discrete automorphic representation $\Pi_{\bf{E}}$ of $\GL_n(\bA_{\bf{E}})$ such that, for all places $o$ of $\bf{F}$ and places $o'$ of $\bf{E}$ lying above $o$, the base change lift of $\Pi_o$ to $\GL_n(\bf{E}_{o'})$ is isomorphic to $\Pi_{\bf{E},o'}$ \cite[(IV.1.3)]{HL11}. Next, let $\Xi$ be any irreducible cuspidal automorphic representation of $\GL_n(\bA)$ such that $\Xi_\bf{E}$ is isomorphic to $\Pi_\bf{E}$. Then $\Xi$ must be isomorphic to $\Pi\otimes(X\circ\det)$ for some character $X:\bA^\times/(\bf{F}^\times\Nm_{\bf{E}/\bf{F}}(\bA_\bf{E}^\times))\rar\bC^\times$ \cite[Theorem III.3.1]{AC89}.\footnote{The proof given here is stated for number fields, but it only uses the relationship between unramified local $L$-functions and Satake parameters, as well as the fact that $L(\Pi\times\Pi',s)$ has a pole at $s=1$ if and only if $\Pi\cong\Pi'^\vee$. In particular, it carries over to arbitrary global fields.} Finally, every $\sg$-stable irreducible cuspidal representation $\Pi'$ of $\GL_n(\bA_\bf{E})$ is isomorphic to $\Pi_{\bf{E}}$ for some irreducible cuspidal automorphic representation $\Pi$ of $\GL_n(\bA)$.
\subsection{}\label{ss:automorphismscompatibility}
Return to the local notation of \S\ref{s:deformationspaces}--\S\ref{s:lubintatetower}. We first verify that $\rec$ is compatible with automorphisms.
\begin{lem*}
For any $\al$ in $G_F$ and any irreducible smooth representation $\pi'$ of $\GL_n(E)$, we have an isomorphism
\begin{align*}
\rec(\pi'^\al) = \rec(\pi')^\al.
\end{align*}
\end{lem*}
\begin{proof}
We immediately see $\al$ preserves the Tate twist $(\textstyle\frac{1-n}2)$, so it suffices to show that $\rho(\pi'^\al) = \rho(\pi')^\al$. We will do so by verifying that $\rho(\pi')^\al$ satisfies the defining property of $\rho(\pi'^\al)$ as in Theorem A.(i). Write $\cO_E$ for the ring of integers of $E$. Let $\tau$ be an element of $W_{\al^{-1}(E)}$ satisfying $r\deq v(\tau)>0$, and let $h$ be a function in $C^\infty_c(\GL_n(\al^{-1}(\cO_E)))$. Now \ref{ss:twistedcharacters} yields
\begin{align*}
\tr\left(f_{\tau,h}|\pi'^\al\right) = \tr\left((\phi_{\tau,h},\sg)|\pi'^\al/E_r\right) = \int_{\GL_n(\al^{-1}(E))}\!\dif\de'\,\phi_{\tau,h}(\de')\Te_{\pi'^\al}(\N\de'),
\end{align*}
where $\phi_{\tau,h}$ is our test function in $C^\infty_c(\GL_n(\al^{-1}(E)_r))$ from \S\ref{s:deformationspaces}. Fixing a uniformizer $\vpi_E$ of $E$ and expanding the definition of $\phi_{\tau,h}$, we see that this integral equals
\begin{align*}
\int_{\GL_n(\al^{-1}(\cO_E))\diag(\al^{-1}(\vpi_E),1,\dotsc,1)\GL_n(\al^{-1}(\cO_E))}\!\dif\de'\,\tr(\tau\times h|[R\psi_{\de'}])\Te_{\pi'}(\al(\N\de')).
\end{align*}
Upon making the change of variables $\de=\al(\de')$, our integral becomes
\begin{align*}
&\int_{\GL_n(\cO_E)\diag(\vpi_E,1,\dotsc,1)\GL_n(\cO_E)}\!\dif\de\,\tr(\tau\times h|[R\psi_{\al^{-1}(\de)}])\Te_{\pi'}(\N(\al(\de')))\\
=\,&\int_{\GL_n(\cO_E)\diag(\vpi_E,1,\dotsc,1)\GL_n(\cO_E)}\!\dif\de\,\tr((\al\circ\tau\circ\al^{-1})\times h\circ\al^{-1}|[R\psi_\de])\Te_{\pi'}(\N\de)\\
=\,&\int_{\GL_n(E)}\!\dif\de\,\phi_{\al\circ\tau\circ\al^{-1},h\circ\al^{-1}}(\de)\Te_{\pi'}(\N\de) = \tr\left((\phi_{\al\circ\tau\circ\al^{-1},h\circ\al^{-1}},\sg)|\pi'/E_r\right).
\end{align*}
Applying \ref{ss:twistedcharacters} once more indicates that this equals $\tr\left(f_{\al\circ\tau\circ\al^{-1},h\circ\al^{-1}}|\pi'\right)$. From here, Theorem A.(i) shows that
\begin{align*}
\tr\left(f_{\al\circ\tau\circ\al^{-1},h\circ\al^{-1}}|\pi'\right) = \tr\big(\al^{-1}\circ\tau\circ\al|\rho(\pi')\big)\tr\big(h\circ\al^{-1}|\pi'\big) = \tr\left(\tau|\rho(\pi')^\al\right)\tr\big(h\circ\al^{-1}|\pi'\big).
\end{align*}
Thus all that remains is to prove that $\tr\big(h\circ\al^{-1}|\pi'\big)=\tr\big(h|\pi'^\al\big)$. Now $\tr\big(h|\pi'^\al\big)$ is the trace of the operator
\begin{align*}
v\mapsto\int_{\GL_n(\al^{-1}(E))}\!\dif g'\,h(g')\pi'(\al(g'))v = \int_{\GL_n(E)}\!\dif g\,h(\al^{-1}(g))\pi'(g)v,
\end{align*}
where we have made the change of variables $g=\al(g')$. This shows that $\tr(h|\pi'^\al) = \tr(h\circ\al^{-1}|\pi')$, concluding our proof of Lemma \ref{ss:automorphismscompatibility}.
\end{proof}
\subsection{}\label{ss:secondinductivelemma}
Finally, we can introduce our inductive lemma.
\begin{lemsecond}
Assume that the following conditions hold for all irreducible smooth representations $\pi$ of $\GL_n(F)$:
\begin{enumerate}[(a)]
\item Theorem C is true for $n'<n$,
\item if $n=1$, then $\rec\pi$ is isomorphic to $\pi\circ\Art^{-1}$,
\item if $\pi$ is isomorphic to a subquotient of the normalized parabolic induction of
\begin{align*}
\pi_1\otimes\dotsb\otimes\pi_t,
\end{align*}
where the $\pi_i$ are irreducible smooth representations of $\GL_{n_i}(F)$ for which $n_1+\dotsb+n_t=n$. Then
\begin{align*}
\rec(\pi) = \rec(\pi_1)\oplus\dotsb\oplus\rec(\pi_t),
\end{align*}
\item for all smooth characters $\chi:F^\times\rar\bC^\times$, we have
\begin{align*}
\rec(\pi\otimes(\chi\circ\det)) = \rec(\pi)\otimes\rec(\chi),
\end{align*}
\item if $\pi$ is generic, then for all cyclic extensions $E/F$ of prime degree, we have
\begin{align*}
\rec(\pi_E) = \res{\rec(\pi)}_{W_E},
\end{align*}
where $\pi_E$ denotes the base change lift of $\pi$ to $\GL_n(E)$,
\item if $\rec(\pi)$ is an unramified representation of $W_F$, then $\pi$ is an unramified representation of $\GL_n(F)$.
\end{enumerate}
Then Theorem C is true for $n$.
\end{lemsecond}
In the first part of this section, our goal is to prove the second inductive lemma. To this end, starting from this point, assume that conditions (a)--(f) hold. We have already proved condition (b) in Proposition \ref{ss:theoremabasecase} and condition (c) in Theorem A. After proving Lemma \ref{ss:secondinductivelemma}, we shall prove conditions (d)--(f) in the remainder of this section, using results from \S\ref{s:nearbycycles} and \S\ref{s:localglobal}.
\subsection{}\label{ss:solvablebasechange}
We begin by immediately upgrading condition (e) to finite solvable extensions $E/F$.
\begin{prop*}
Let $\pi$ be generic. Then, for all finite solvable extensions $E/F$, we have
\begin{align*}
\rec(\pi_E) = \res{\rec(\pi)}_{W_E},
\end{align*}
where $\pi_E$ denotes the base change lift of $\pi$ to $\GL_n(E)$.
\end{prop*}
\begin{proof}
We induct on the degree of $E$ over $F$. The result is immediate for $E=F$, and condition (e) ensures that it holds for prime $[E:F]$. In general, the solvability of $E/F$ yields a Galois subextension $E\supset E'\supseteq F$ such that $E/E'$ is cyclic of prime degree and $E'/F$ is solvable. Writing $\pi_{E'}$ for the base change lift of $\pi$ to $\GL_n(E')$, we see that $\rec(\pi_{E'}) = \res{\rec(\pi)}_{W_{E'}}$ by the inductive hypothesis. Transitivity of base change indicates that $\pi_E$ is also the base change lift of $\pi_{E'}$ to $\GL_n(E)$, so condition (e) gives us
\begin{gather*}
\rec(\pi_E) = \res{\rec(\pi_{E'})}_{W_E} = \res{\rec(\pi)}_{W_E}.\qedhere
\end{gather*}
\end{proof}
\subsection{}\label{ss:unramifiedsolvablebasechange}
We make the following observation. Let $\rho$ be a finite-dimensional semisimple continuous representation of $W_F$. Then $\rho$ is smooth, so $\res\rho_{I_E}$ is trivial for some finite Galois extension $E$ of $F$. As $F$ is a nonarchimedean local field, $E/F$ is solvable. Altogether, $\rho$ becomes unramified after passing to a solvable extension.
In particular, by letting $\rho=\rec\pi$ for any irreducible generic representation $\pi$ of $\GL_n(F)$, Proposition \ref{ss:solvablebasechange} and condition (f) imply that the base change lift $\pi_E$ of $\pi$ to $\GL_n(E)$ is unramified.
\subsection{}\label{ss:cuspidaltoirreducible}
We now check that $\rec$ sends irreducible cuspidal representations to irreducible ones.
\begin{prop*}
Suppose that $\pi$ is irreducible cuspidal. Then $\rec\pi$ is an irreducible representation of $W_F$.
\end{prop*}
\begin{proof}
This is immediate for $n=1$, so suppose that $n\geq2$. Now \ref{ss:unramifiedsolvablebasechange} yields a finite solvable extension $E/F$ for which the base change lift $\pi_E$ of $\pi$ to $\GL_n(E)$ is unramified. As $n\geq2$, we see that $\pi_E$ cannot be cuspidal. Therefore, by replacing $E/F$ with a subextension if necessary, we can find a Galois subextension $E\supset E'\supseteq F$ satisfying the following properties:
\begin{enumerate}[$\bullet$]
\item $E/E'$ is cyclic of prime degree $r$,
\item $E'/F$ is solvable,
\item the base change lift $\pi_{E'}$ of $\pi$ to $\GL_n(E')$ remains cuspidal.
\end{enumerate}
Since $\pi_E$ is also the base change lift of $\pi_{E'}$ to $\GL_n(E)$, we can apply \ref{ss:basechangeliftcuspidal} to obtain a positive integer $u$ and an irreducible cuspidal representation $\pi'$ of $\GL_{n/u}(E)$ such that $\pi_E$ is isomorphic to
\begin{align*}
Q\left(\{\pi'\},\dotsc,\{\pi'^{\sg^{u-1}}\}\right),
\end{align*}
where $\sg$ is a generator of $\Gal(E/E')$. Thus condition (e) and condition (c) indicate that $\rec(\pi_{E'})$ restricts to
\begin{align*}
\res{\rec(\pi_{E'})}_{W_E} = \rec(\pi_E) = \rec(\pi')\oplus\dotsb\oplus\rec(\pi')^{\sg^{r-1}}.
\end{align*}
Now $u$ is the cardinality of a subgroup of $\fK(E/E')$, so $u$ divides $r$. But $\pi_E$ is not cuspidal, which forces $u=r$. Hence $\pi'$ is not isomorphic to $\pi'^\sg$. From here, Theorem C for $\pi'$ and Lemma \ref{ss:automorphismscompatibility} show that $\rec(\pi')$ is not isomorphic to $\rec(\pi'^\sg) = \rec(\pi')^\sg$, so \ref{ss:galoissidebasechange}.(ii) indicates that $\rec(\pi_{E'})$ is irreducible.
Finally, because Proposition \ref{ss:solvablebasechange} shows that $\rec(\pi_{E'}) = \res{\rec(\pi)}_{W_{E'}}$, we see that $\rec(\pi)$ is irreducible as well.
\end{proof}
\subsection{}\label{ss:lemmas=1step}
Next, we establish a few lemmas on the bijectivity of $\rec$ in special cases.
\begin{lem*}
Let $\rho$ be an $n$-dimensional irreducible continuous representation of $W_F$, and let $E/F$ be a cyclic extension of prime degree. If $\res\rho_{W_E}$ is reducible, then there exists a unique irreducible cuspidal representation $\pi$ of $\GL_n(E)$ such that $\rec\pi=\rho$.
\end{lem*}
\begin{proof}
Write $r$ for the degree of $E/F$, and fix a generator $\sg$ of $\Gal(E/F)$. Then \ref{ss:galoissidebasechange}.(ii) yields an irreducible continuous finite-dimensional representation $\rho'$ of $W_E$ satisfying $\res\rho_{W_E}=\rho'\oplus\dots\oplus\rho'^{\sg^{r-1}}$ and $\rho'\neq\rho'^\sg$. Now Theorem C for $\rho'$ provides a unique irreducible cuspidal representation $\pi'$ of $\GL_{n/r}(E)$ such that $\rec(\pi')$ is isomorphic to $\rho'$.
Define $\wt\pi$ to be the irreducible smooth representation
\begin{align*}
\wt\pi\deq Q\left(\{\pi'\},\dotsc,\{\pi'^{\sg^{r-1}}\}\right)
\end{align*}
of $\GL_n(E)$. We see that $\wt\pi$ is generic, and van Dijk's formula \cite[Theorem 5.9]{Lem16} implies that $\wt\pi$ is $\sg$-stable. Therefore \ref{ss:basechangeliftlocal} shows that $\wt\pi$ is the base change lift of some irreducible generic representation $\pi$ of $\GL_n(F)$. Furthermore, because the segments $\{\pi'\},\dotsc,\{\pi'^{\sg^{r-1}}\}$ in $\wt\pi$ have length $1$, our description of possibilities for $\pi$ in \ref{ss:basechangeliftlocal} indicates $\pi$ is cuspidal.
Condition (e), condition (c), and Lemma \ref{ss:automorphismscompatibility} yield
\begin{align*}
\res{\rec(\pi)}_{W_E} = \rec(\wt\pi) = \rec(\pi')\oplus\dotsb\oplus\rec(\pi'^{\sg^{r-1}}) = \rho'\oplus\dotsb\oplus\rho'^{\sg^{r-1}}.
\end{align*}
The uniqueness of $\rho$ from \ref{ss:galoissidebasechange}.(ii) therefore indicates that $\rec\pi=\rho$. This takes care of the existence of $\pi$.
We now tackle uniqueness. Suppose $\xi$ is any irreducible cuspidal representation of $\GL_n(F)$ with $\rec(\xi)=\rho$, and write $\xi_E$ for the base change lift of $\xi$ to $\GL_n(E)$. Condition (e) and condition (c) show that $\rec(\xi_E)=\res\rho_{W_E}$ is reducible, so Proposition \ref{ss:cuspidaltoirreducible} implies that $\xi_E$ is not cuspidal. Therefore our description of base change in \ref{ss:basechangeliftcuspidal} gives
\begin{align*}
\xi_E = Q\left(\{\xi'\},\dotsc,\{\xi'^{\sg^{r-1}}\}\right)
\end{align*}
for some irreducible cuspidal representation $\xi'$ of $\GL_{n/r}(E)$. Applying condition (e), condition (c), and Lemma \ref{ss:automorphismscompatibility} once more yields
\begin{align*}
\rho'\oplus\dotsb\oplus\rho'^{\sg^{r-1}} = \res{\rec(\pi)}_{W_E} = \rec(\xi_E) = \rec(\xi')\oplus\dotsb\oplus\rec(\xi'^{\sg^{r-1}}) = \rec(\xi')\oplus\dotsb\oplus\rec(\xi')^{\sg^{r-1}}.
\end{align*}
Theorem C for $\xi'$ tells us that $\rec(\xi')$ is irreducible. As $\rho'$ is irreducible too, we see that $\rho'$ is isomorphic to $\rec(\xi')^{\sg^i}$ for some $0\leq i\leq r-1$. After replacing $\xi'$ with $\xi'^{\sg^i}$, we get $\rho' = \rec(\xi')$. Similarly replacing $\pi'$ with $\pi'^{\sg^i}$ for some $0\leq i\leq r-1$ yields $\rho'=\rec(\pi')$, and hence $\pi'=\xi'$ via Theorem C for $\rho'$. This in turn yields $\wt\pi=\xi_E$. Since $\wt\pi$ is the base change lift of $\pi$ to $\GL_n(E)$, we see from \ref{ss:basechangeliftlocal} that $\xi$ must be isomorphic to $\chi\cdot\pi$ for some $\chi$ in $\fK(E/F)$. But the stabilizer of $\pi$ in $\fK(E/F)$ has cardinality $r$ by \ref{ss:basechangeliftcuspidal}. Thus we finally obtain $\xi=\chi\cdot\pi=\pi$, as desired.
\end{proof}
\begin{lem}\label{lem:lemmas=2step}
Let $\rho$ be an $n$-dimensional irreducible continuous representation of $W_F$, and let $E/F$ be a cyclic extension of prime degree. Suppose that there exists a unique irreducible cuspidal representation $\wt\pi$ of $\GL_n(E)$ such that $\rec\wt\pi = \res\rho_{W_E}$. If $\res\rho_{W_E}$ is irreducible, then there also exists a unique irreducible cuspidal representation $\pi$ of $\GL_n(F)$ such that $\rec\pi=\rho$.
\end{lem}
\begin{proof}
Write $r$ for the degree of $E/F$, and fix a generator $\sg$ of $\Gal(E/F)$. As \ref{ss:galoissidebasechange}.(i) indicates that
\begin{align*}
(\res\rho_{W_E})^\sg = \res\rho_{W_E},
\end{align*}
our uniqueness hypothesis on $\wt\pi$ and \ref{ss:automorphismscompatibility} yield $\wt\pi^\sg = \wt\pi$. Thus $\wt\pi$ is $\sg$-stable, so \ref{ss:basechangeliftlocal} shows that it is the base change lift of some irreducible generic representation $\pi$ of $\GL_n(F)$. Furthermore, because $\wt\pi$ is cuspidal, our description of possibilities for $\pi$ implies that $\pi$ is also cuspidal. Consequently, Proposition \ref{ss:cuspidaltoirreducible} indicates that $\rec\pi$ is irreducible. Now condition (e) yields
\begin{align*}
\res{\rec(\pi)}_{W_E} = \rec(\wt\pi) = \res\rho_{W_E},
\end{align*}
so \ref{ss:galoissidebasechange}.(i) shows that $\rec(\pi)$ is isomorphic to $\chi\cdot\rho$ for some $\chi$ in $\fK(E/F)$. After replacing $\pi$ with $\chi^{-1}\cdot\pi$, condition (d) tells us that $\rec\pi=\rho$. This takes care of the existence of $\pi$.
As for uniqueness, suppose $\xi$ is any irreducible cuspidal representation of $\GL_n(F)$ satisfying $\rec(\xi)=\rho$. Writing $\xi_E$ for the base change lift of $\xi$ to $\GL_n(E)$, we see that condition (e) gives us
\begin{align*}
\rec(\xi_E) = \res\rho_{W_E} = \rec(\wt\pi).
\end{align*}
Condition (c) and the irreducibility of $\res\rho_{W_E}$ imply that $\xi_E$ is cuspidal, so our uniqueness hypothesis on $\wt\pi$ indicates that $\xi_E = \wt\pi$. As $\xi_E$ is the base change lift of $\xi$ to $\GL_n(E)$, we see from \ref{ss:basechangeliftlocal} that $\xi$ must be isomorphic to $\chi\cdot\pi$ for some $\chi$ in $\fK(E/F)$. From here, condition (d) yields
\begin{align*}
\rho = \rec\xi = \rec(\chi\cdot\pi) = \chi\cdot\rec\pi = \chi\cdot\rho,
\end{align*}
and \ref{ss:galoissidebasechange} along with the irreducibility of $\res\rho_{W_E}$ imply that $\chi=1$. Hence $\xi=\pi$, as desired.
\end{proof}
We shall finally wrap up the proof of Lemma \ref{ss:secondinductivelemma} itself, that is, prove Theorem C holds for $n$.
\begin{proof}[Proof of Lemma \ref{ss:secondinductivelemma}] Proposition \ref{ss:cuspidaltoirreducible} shows that $\pi\mapsto\rec\pi$ indeed yields a map from isomorphism classes of irreducible cuspidal representations of $\GL_n(F)$ to isomorphism classes of $n$-dimensional irreducible continuous representations of $W_F$.
Next, we proceed towards bijectivity. Let $\rho$ be an $n$-dimensional irreducible continuous representation of $W_F$. Bijectivity is immediate for $n=1$, so suppose that $n\geq2$. Now \ref{ss:unramifiedsolvablebasechange} yields a finite solvable extension $E/F$ for which $\res\rho_{W_E}$ is unramified. As $n\geq2$, this implies that $\res\rho_{W_E}$ cannot be irreducible. Thus, by replacing $E/F$ with a subextension if necessary, we obtain a tower of field extensions $E=E_s\supset\dotsb\supset E_0 = F$ such that
\begin{enumerate}[$\bullet$]
\item $E_{j+1}/E_j$ is cyclic of prime degree for all $0\leq j\leq s-1$,
\item $\res\rho_{E_{s-1}}$ is irreducible,
\item $\res\rho_{E_s}$ is reducible.
\end{enumerate}
Lemma \ref{ss:lemmas=1step} provides a unique irreducible cuspidal representation $\pi_{s-1}$ of $\GL_n(E_{s-1})$ satisfying $\rec\pi_{s-1}=\res\rho_{E_{s-1}}$. From here, repeated applications of Lemma \ref{lem:lemmas=2step} yield unique irreducible cuspidal representations $\pi_j$ of $\GL_n(E_j)$ such that $\rec\pi_j=\res\rho_{E_j}$, and the $j=0$ case is precisely the desired result.
\end{proof}
Now that we have Lemma \ref{ss:secondinductivelemma}, we turn our attention towards verifying that conditions (b)--(f) hold. Recall that we already proved condition (b) in Proposition \ref{ss:theoremabasecase} and condition (c) in Theorem A.
\subsection{}\label{lem:unramifiedlocallanglands}
To prove condition (d) and condition (e), we use the following explicit description of $\rec\pi$ for unramified $\pi$.
\begin{lem*}
Let $\pi$ be isomorphic to $Q\left(\{\chi_1\},\dotsc,\{\chi_n\}\right)$, where the $\chi_i:F^\times\rar\bC^\times$ are unramified characters sending $\vpi$ to $z_i$ in $\bC^\times$. Then $\rec\pi$ is isomorphic to the $n$-dimensional unramified representation of $W_F$ where geometric $q$-Frobenius acts via $\diag(z_1,\dotsc,z_n)$. In particular, for all irreducible unramified representations $\pi$ and $\pi'$ of $\GL_n(F)$, we have $L(\pi\times\pi',s)=L(\rec(\pi)\otimes\rec(\pi'),s)$.
\end{lem*}
\begin{proof}
Theorem A shows that $\rec\pi$ is isomorphic to $\rec(\chi_1)\oplus\dotsb\oplus\rec(\chi_n)$, and Proposition \ref{ss:theoremabasecase} allows us to conclude $\rec\pi$ has the desired form. Next, any irreducible unramified representation $\pi$ of $\GL_n(F)$ is isomorphic to $Q\left(\{\chi_1\},\dotsc,\{\chi_n\}\right)$ for some unramified characters $\chi_i:F^\times\rar\bC^\times$ \cite{Cas80}. Write $z_i$ for $\chi_i(\vpi)$, and form the analogous $\chi'$ and $z_i'$ for $\pi'$ too. Then we have
\begin{gather*}
L(\pi\times\pi',s) = \prod_{i,j=1}^nL(\chi_i\times\chi'_j,s) = \prod_{i,j=1}^n(1-z_iz_jq^{-s})^{-1} = L(\rec(\pi)\otimes\rec(\pi'),s).\qedhere
\end{gather*}
\end{proof}
\subsection{}
From here, we shall prove condition (d) and condition (e) via embedding into the global situation. More precisely, we will use Theorem B along with the Chebotarev density theorem. Let us recall some global notation: $C$ denotes a geometrically connected proper smooth curve over $\ka$, $\bf{F}$ denotes its field of rational functions, $\bA$ denotes its ring of adeles, and $W_{\bf{F}}$ denotes the Weil group of $\bf{F}$. By abuse of notation, we write $\Art:\bA^\times/\bf{F}^\times\rar^\sim W_{\bf{F}}^\ab$ for the global Artin isomorphism normalized by sending uniformizers to geometric Frobenii. Fix distinct places $x_1$, $x_2$, and $\infty$ of $\bf{F}$.
\begin{prop}\label{prop:compatwithtwisting}
Let $\pi$ be an irreducible smooth representation of $\GL_n(F)$. For all smooth characters $\chi:F^\times\rar\bC^\times$, we have
\begin{align*}
\rec(\pi\otimes(\chi\circ\det)) = \rec(\pi)\otimes\rec(\chi).
\end{align*}
\end{prop}
\begin{proof}
We begin with some reductions. Theorem A indicates that both sides are compatible with parabolic induction, so it suffices to prove this for cuspidal $\pi$. As $\rec(\pi)=\rho(\pi)(\frac{1-n}2)$ and $\rec(\chi)=\chi\circ\Art^{-1}$, this is equivalent to
\begin{align*}
\rho(\pi\otimes(\chi\circ\det)) = \rho(\pi)\otimes(\chi\circ\Art^{-1}).
\end{align*}
Finally, note that $\chi$ is the product of a finite order character with $\abs\cdot^s$ for some complex number $s$. The $\chi=\abs\cdot^s$ case follows immediately from Proposition \ref{ss:unramifiedtwistsofrho}, so we need only consider the case when $\chi$ has finite order.
Assume this is the case, and let $C=\bP^1_\ka$ be our curve of interest. Choose $o$ to be a $\ka$-point of $C$, and choose $x_1$, $x_2$, and $\infty$ in \ref{ss:choiceofdivisionalgebra} to be distinct closed points in $C\ssm o$. Note that $\bf{F}_o$ is isomorphic to $F$. Since $\chi$ has finite order, there exists a finite order smooth character $X:\bA^\times/\bf{F}^\times\rar\bC^\times$ such that $\res{X}_{\bf{F}_o^\times}=\chi$ \cite[Theorem 5 of Chapter X]{AT09}. Furthermore, as $\pi$ is cuspidal, Lemma \ref{ss:embedcuspidal} provides an irreducible cuspidal representation $\Pi$ of $\GL_n(\bA)$ whose component at $o$ is isomorphic to $\pi$ and whose components at $x_1$, $x_2$, and $\infty$ are irreducible cuspidal representations of $\GL_n(\bf{F}_{x_1})$, $\GL_n(\bf{F}_{x_2})$, and $\GL_n(\bf{F}_\infty)$, respectively, with unitary central characters.
Because the smooth character $X\circ\Art^{-1}:W_{\bf{F}}\rar\bC^\times$ has finite image, it extends to a continuous character $\Xi:G_{\bf{F}}\rar\ov\bQ_\ell^\times$, where we identify $\ov\bQ_\ell$ with $\bC$ \cite[(IV.2.2)]{HL11}. Note also that $\Pi\otimes(X\circ\det)$ is an irreducible cuspidal representation of $\GL_n(\bA)$. Therefore we may apply Theorem B to obtain the $n$-dimensional semisimple continuous representations
\begin{align*}
R_1\deq R(\Pi\otimes(X\circ\det))\mbox{ and } R_2\deq R(\Pi)\otimes \Xi
\end{align*}
of $G_{\bf{F}}$ over $\ov\bQ_\ell$. Write $T$ for the set of places $x$ of $\bf{F}$ such that
\begin{enumerate}[$\bullet$]
\item $\res{X}_{\bf{F}_x^\times}$ is unramified,
\item $\Pi_x$ is unramified,
\item $x$ does not lie in $\{x_1,x_2,\infty\}$,
\end{enumerate}
and note that $T$ is cofinite. For $x$ in $T$, Lemma \ref{lem:unramifiedlocallanglands} indicates that $\res{R_1}_{W_{\bf{F}_x}}$ and $\res{R_2}_{W_{\bf{F}_x}}$ are isomorphic.
The Chebotarev density theorem implies that, as $x$ varies over all places of $\bf{F}$ not lying in $T$, the conjugacy classes in $G_{\bf{F}}$ of arithmetic $q_x$-Frobenius elements at $x$ are dense. Because $n$-dimensional continuous semisimple representations of $G_{\bf{F}}$ are determined, up to isomorphism, by their characteristic polynomials (and said polynomials are continuous in $G_{\bf{F}}$), we see that $R_1=R_2$ as representations of $G_{\bf{F}}$. Restricting both sides to $W_{\bf{F}_o}$ and applying Theorem B again yields the desired result.
\end{proof}
\begin{prop}\label{prop:compatwithbasechange}
Let $\pi$ be an irreducible generic representation of $\GL_n(F)$. For all cyclic extensions $E/F$ of prime degree, we have
\begin{align*}
\rec(\pi_E) = \res{\rec(\pi)}_{W_E},
\end{align*}
where $\pi_E$ denotes the base change lift of $\pi$ to $\GL_n(E)$.
\end{prop}
\begin{proof}
Applying Theorem A to the decomposition of $\pi_E$ from \ref{ss:basechangeliftlocal} shows that it suffices to consider cuspidal $\pi$. With this reduction in hand, first set $C=\bP^1_\ka$ as our curve of interest, and choose a $\ka$-point $o$ of $C$. We may identify $\bf{F}_o$ with $F$. By writing $E=F[t]/f(t)$ and approximating $f(t)$ using a polynomial with entries in $\bf{F}$, Krasner's lemma yields a separable extension $\bf{E}/\bf{F}$ such that $o$ is inert in $\bf{E}$, and $\bf{E}_{o'}/\bf{F}_o$ can be identified with $E/F$, where $o'$ is the unique place of $\bf{E}$ dividing $o$. By replacing $\bf{E}$ with its Galois closure and replacing $\bf{F}$ (and changing $C$ accordingly) with the subfield corresponding to the decomposition group of $o'$, we may assume that $\bf{E}/\bf{F}$ is Galois and hence cyclic as well.
Since $\bf{E}$ is a global function field, it corresponds to a geometrically connected proper smooth curve $C'$ over a finite field $\ka'$. Thus we can apply the results of \S\ref{s:modulispaces}--\S\ref{s:localglobal} to $\bf{E}$ and $C'$. By the Chebotarev density theorem, we may choose $x_1$, $x_2$, and $\infty$ in \ref{ss:choiceofdivisionalgebra} for $\bf{F}$ to be distinct closed points in $C\ssm o$ such that they split completely in $C'$. Choose $x_1'$, $x_2'$, and $\infty'$ in \ref{ss:choiceofdivisionalgebra} for $\bf{E}$ lying above $x_1$, $x_2$, and $\infty$, respectively. This allows us to identify $\bf{E}_{x_1'}$ with $\bf{F}_{x_1}$, $\bf{E}_{x_2'}$ with $\bf{F}_{x_2}$, and $\bf{E}_{\infty'}$ with $\bf{F}_\infty$.
Because $\pi$ is cuspidal, Lemma \ref{ss:embedcuspidal} gives an irreducible cuspidal representation $\Pi$ of $\GL_n(\bA)$ whose component at $o$ is isomorphic to $\pi$ and whose components at $x_1$, $x_2$, and $\infty$ are irreducible cuspidal representations of $\GL_n(\bf{F}_{x_1})$, $\GL_n(\bf{F}_{x_2})$, and $\GL_n(\bf{F}_\infty)$, respectively, with unitary central characters. As $x_1$, $x_2$, and $\infty$ split in $\bf{E}$, the irreducible discrete automorphic representation $\Pi_{\bf{E}}$ formed in \ref{ss:basechangeliftglobal} has components
\begin{align*}
\Pi_{\bf{E},x_1'}=\Pi_{x_1},\,\Pi_{\bf{E},x_2'}=\Pi_{x_2},\mbox{ and }\Pi_{\bf{E},\infty'}=\Pi_\infty.
\end{align*}
In particular, these components are irreducible cuspidal with unitary central characters. Hence we can apply Theorem B to $\Pi_{\bf{E}}$ to form the $n$-dimensional semisimple continuous representations
\begin{align*}
R_1\deq R(\Pi_{\bf{E}})\mbox{ and }R_2\deq \res{R(\Pi)}_{G_{\bf{E}}}
\end{align*}
of $G_{\bf{E}}$ over $\ov\bQ_\ell$.
Write $T$ for the set of places $x$ of $\bf{F}$ such that $\Pi_x$ is unramified and $x$ does not lie in $\{x_1,x_2,\infty\}$, which is a cofinite set of places. For all $x$ in $T$, Lemma \ref{lem:unramifiedlocallanglands} and the description of base change lifts in \ref{ss:basechangeliftcuspidal} and \ref{ss:basechangeliftglobal} indicate that $\res{R_1}_{W_{\bf{F}_x}}$ is isomorphic to $\res{R_2}_{W_{\bf{F}_x}}$. From here, we conclude using the Chebotarev density theorem and Theorem B, as in the proof of Proposition \ref{prop:compatwithtwisting}.
\end{proof}
\subsection{}\label{ss:compatwithunramified}
Finally, we prove condition (f) using our results from \S\ref{s:nearbycycles}.
\begin{prop*}
Let $\pi$ be an irreducible smooth representation of $\GL_n(F)$. If $\rec(\pi)$ is an unramified representation of $W_F$, then $\pi$ is an unramified representation of $\GL_n(F)$.
\end{prop*}
\begin{proof}
Write $\pi_1,\dotsc,\pi_t$ for the cuspidal support of $\pi$, where the $\pi_i$ are irreducible cuspidal representations of $\GL_{n_i}(F)$ such that $n=n_1+\dotsb+n_t$. Now Theorem A.(ii) yields
\begin{align*}
\rec(\pi) = \rec(\pi_1)\oplus\dotsb\oplus\rec(\pi_t).
\end{align*}
If $\pi$ is not unramified, then there exists some $i$ such that either
\begin{enumerate}[(1)]
\item $n_i=1$ and $\pi_i:F^\times\rar\bC^\times$ is not an unramified character,
\item $n_i\geq2$.
\end{enumerate}
In case (1), Proposition \ref{ss:theoremabasecase} shows that $\rec(\pi_i)$ and hence $\rec(\pi)$ is not unramified, so we need only tackle case (2). By replacing $\pi$ with $\pi_i$, it suffices to assume that $\pi$ is cuspidal and $n\geq2$. Finally, because $\rec(\pi)$ is an unramified twist of $\rho(\pi)$, we need only show that $\rho(\pi)$ is not unramified.
We begin by using Schur orthogonality to obtain a function $h$ in $C^\infty_c(\GL_n(\cO))$ that satisfies $\tr(h|\pi)=1$. Then we have
\begin{align*}
\tr\left(\sg^{-r}|\rho(\pi)^{I_F}\right) = \tr\left(\bf1_{\sg^{-r}I_F}|\rho(\pi)\right)\tr(h|\pi) = \int_{\sg^{-r}I_F}\!\dif\tau\,\tr\left(\tau|\rho(\pi)\right)\tr(h|\pi),
\end{align*}
where $\bf1_{\sg^{-r}I_F}$ is the indicator function on $\sg^{-r}I_F$. Applying Theorem A.(i), we obtain
\begin{align*}
\int_{\sg^{-r}I_F}\!\dif\tau\,\tr\left(f_{\tau,h}|\pi\right),
\end{align*}
and \ref{ss:twistedcharacters} indicates that this integral becomes
\begin{align*}
\int_{\sg^{-r}I_F}\!\dif\tau\,\tr\left((\phi_{\tau,h},\sg)|\pi/F_r\right) = \int_{\sg^{-r}I_F}\!\dif\tau\,\int_{\GL_n(\cO_r)\diag(\vpi,1,\dotsc,1)\GL_n(\cO_r)}\!\dif\de\,\phi_{\tau,h}(\de)\Te_\pi^\sg(\de).
\end{align*}
Recall the subset $B_n\subseteq\GL_n(\cO_r)\diag(\vpi,1,\dotsc,1)\GL_n(\cO_r)$, which was defined in \ref{ss:localshtukaunramifiedextension} using the decomposition $\de=\de^\circ\oplus\de^\et$. As $\pi$ is cuspidal, we can use Lemma \ref{lem:cuspidaltwistedcharacter} to convert our integral into
\begin{align*}
\int_{\sg^{-r}I_F}\!\dif\tau\,\int_{B_n}\!\dif\de\,\phi_{\tau,h}(\de)\Te_\pi^\sg(\de).
\end{align*}
Now let $C=\bP^1_\ka$ be our curve of interest, and choose a $\ka$-point $o$ of $C$. Since we may identify $\bf{F}_o$ with $F$, the results of \S\ref{s:nearbycycles} apply to our situation. Namely, Fubini's theorem and Corollary \ref{ss:sectionsixfinalresult} allow us to rewrite this integral as
\begin{align*}
\int_{\sg^{-r}I_F}\!\dif\tau\,\phi_{\tau,h}(\de)\int_{B_n}\!\dif\de\,\Te_\pi^\sg(\de) = \fC_n\int_{B_n}\!\dif\de\,\Te_\pi^\sg(\de),
\end{align*}
where $\fC_n$ is as in Corollary \ref{ss:sectionsixfinalresult}. The definition of $\Te_\pi^\sg$ turns the above expression into
\begin{align*}
\fC_n\int_{B_n}\!\dif\de\,\Te_\pi(\N\de),
\end{align*}
and from here Lemma \ref{ss:noncommutativenormbijection} and the local Jacquet--Langlands correspondence yield
\begin{align*}
\fC_n(-1)^{n-1}\int_{B_r}\!\dif b\,\Te_{\JL(\pi)}(b) = \fC_n(-1)^{n-1}\tr\left(\bf1_{B_r}|\JL(\pi)\right) = \fC_n(-1)^{n-1}\tr\left(\bf1_{B_r}|\JL(\pi)^{\cO_B^\times}\right),
\end{align*}
where $B$ is the central division algebra over $F$ of Hasse invariant $\frac1n$, and $B_r$ is the subset of valuation $r$ elements in $B$. Because $\pi$ has no $\GL_n(\cO)$-invariants, its image under the local Jacquet--Langlands correspondence has no $\cO_B^\times$-invariants. Hence the above trace vanishes.
Altogether, we have shown that $\tr\left(\sg^{-r}|\rho(\pi)^{I_F}\right)$ vanishes for any positive integer $r$. As semisimple representations are determined by their traces, we have $\rho(\pi)^{I_F}=0$. In particular, $\rho(\pi)$ is not unramified.
\end{proof}
By using Theorem A, Proposition \ref{prop:compatwithtwisting}, Proposition \ref{prop:compatwithbasechange}, and Proposition \ref{ss:compatwithunramified} to verify that conditions (b)--(f) of Lemma \ref{ss:secondinductivelemma} hold and using Proposition \ref{ss:theoremabasecase} to check the $n=1$ base case, this concludes our proof by induction of Theorem C.
\section{Duals, $L$-functions, and $\eps$-factors}\label{s:llc}
In this section, our goal is to prove Theorem D, i.e. that $\pi\mapsto\rec\pi$ satisfies Henniart's properties \cite[Theorem 1.2]{Hen85} characterizing the local Langlands correspondence for $\GL_n$ over $F$. We begin by collecting facts on inducing representations of $W_\bf{E}$ to representations of $W_\bf{F}$ for separable extensions $\bf{E}/\bf{F}$. Similar statements hold for extensions $E/F$ of local fields. We use these facts to motivate \emph{automorphic induction}, an analogous operation that turns representations of $\GL_{n}(\bA_\bf{E})$ (respectively $\GL_{n}(E)$) into representations of $\GL_{n[\bf{E}:\bf{F}]}(\bA)$ (respectively $\GL_{n[E:F]}(F)$) for cyclic extensions. Combining this with Theorem B allows us to prove automorphic induction for some non-Galois extensions $\bf{E}/\bf{F}$.
We use this \emph{non-Galois automorphic induction} to show that $\rec$ is compatible with central characters. More precisely, we apply Brauer induction to reduce to the case of induced representations, embed into the global setting, and then invoke our non-Galois automorphic induction. Afterwards, we use compatibility with central characters to prove that $\rec$ preserves $L$-functions and $\eps$-factors, by twisting with highly ramified characters. Finally, compatibility with duals follows from the decomposition of $L$-functions of pairs in terms of $L$-functions of characters. This concludes our proof of the local Langlands correspondence for $\GL_n$ over $F$.
\subsection{}\label{ss:inductionongroups}
Let us recall some global notation: $C$ denotes a geometrically connected proper smooth curve over $\ka$, $\bf{F}$ denotes its field of rational functions, $\bA$ denotes its ring of adeles, and $W_{\bf{F}}$ denotes the Weil group of $\bf{F}$.
Let $\bf{E}$ be a finite extension of $\bf{F}$ inside $\bf{F}^\sep$. Now write $\wt{\bf{E}}$ for the Galois closure of $\bf{E}$ over $\bf{F}$, and let $R:W_\bf{E}\rar\bC^\times$ be a smooth character. One can use Frobenius reciprocity and the Mackey formula to show that $\Ind_{W_\bf{E}}^{W_\bf{F}}R$ is irreducible if and only if the stabilizer of $\res{R}_{W_{\wt{\bf{E}}}}$ in $\Gal(\wt{\bf{E}}/\bf{F})$ equals $\Gal(\wt{\bf{E}}/\bf{E})$. The same argument works in the local setting: let $E$ be a finite extension of $F$ in $F^\sep$, and let $\rho:W_E\rar\bC^\times$ be a smooth character. Writing $\wt{E}$ for the Galois closure of $E$ over $F$, we see that $\Ind_{W_E}^{W_F}\rho$ is irreducible if and only if the stabilizer of $\res\rho_{W_{\wt{E}}}$ in $\Gal(\wt{E}/F)$ equals $\Gal(\wt{E}/E)$.
\subsection{}\label{ss:automorphicinductionlocal}
We begin by recalling \emph{automorphic induction} in the local case. Let $E$ be a cyclic $F$-algebra, and write $\fK(E/F)$ for the set of group homomorphisms $\Gal(E/F)\rar\bC^\times$. By reducing to the case when $E$ is a field, one can show that $\Art$ induces an isomorphism $F^\times/\Nm_{E/F}(E^\times)\rar^\sim\Gal(E/F)$, so $\fK(E/F)$ acts on isomorphism classes of smooth representations of $\GL_{n[E:F]}(F)$ as in \ref{ss:basechangeliftcuspidal}.
Let $\pi$ be an irreducible tempered representation of $\GL_n(E)$. Then there exists a unique irreducible tempered representation $I_{E/F}(\pi)$ of $\GL_{n[E:F]}(F)$ that is fixed by $\fK(E/F)$ and satisfies a certain character identity involving $\Te_\pi$ \cite[Theorem 1.3]{HH95}. This representation satisfies $L(I_{E/F}(\pi),s) = L(\pi,s)$ \cite[Theorem 1.4.(a)]{HH95}. Furthermore, $I_{E/F}(\pi)$ is cuspidal if and only if the stabilizer of $\pi$ in $\Gal(E/F)$ is trivial \cite[Proposition 5.5]{HH95}. Note that this is the automorphic analog of \ref{ss:inductionongroups}.
\subsection{}\label{ss:automorphicinductionglobal}
Next, we introduce automorphic induction in the global setting. Let $\bf{E}/\bf{F}$ be a finite cyclic extension, and write $\bA_\bf{E}$ for the ring of adeles of $\bf{E}$. Then for any irreducible cuspidal automorphic representation $\Pi$ of $\GL_n(\bA_\bf{E})$, there exists a unique irreducible automorphic representation $I_{\bf{E}/\bf{F}}(\Pi)$ of $\GL_{n[\bf{E}:\bf{F}]}(\bA)$ such that, for cofinitely many places $o$ of $\bf{F}$ for which $\Pi$ is unramified at every place $o'$ of $\bf{E}$ above $o$, we have $L(I_{\bf{E}/\bf{F}}(\Pi)_o,s)=\prod_{o'\mid o}L(\Pi_{o'},s)$ \cite[(IV.1.8)]{HL11}. In addition, for any place $o$ of $\bf{F}$ where the irreducible smooth representation $\Pi_o$ of $\GL_n(\bf{E}\otimes_{\bf{F}}\bf{F}_o)$ is tempered, we have $I_{\bf{E}/\bf{F}}(\Pi)_o=I_{\bf{E}\otimes_{\bf{F}}\bf{F}_o/\bf{F}_o}(\Pi_o)$ \cite[(IV.1.9)]{HL11}.
\subsection{}\label{defn:associatedreps}
We establish some terminology that reflects the connection between automorphic and Galois representations.
\begin{defn*}
Let $R$ be an $n$-dimensional semisimple continuous representation of $G_\bf{F}$ over $\ov\bQ_\ell$, and let $\Pi$ be an irreducible automorphic representation of $\GL_n(\bA)$. We say $R$ and $\Pi$ are \emph{associated} if there exists a finite set $T$ of places of $\bf{F}$ such that, for all places $x$ of $\bf{F}$ not in $T$,
\begin{enumerate}[$(a)$]
\item $\res{R}_{W_{\bf{F}_x}}$ and $\Pi_x$ are unramified,
\item $\rec\Pi_x$ is isomorphic to $\res{R}_{W_{\bf{F}_x}}$, where we identify $\ov\bQ_\ell$ with $\bC$.
\end{enumerate}
\end{defn*}
Note that either one of $R$ or $\Pi$ determines the other, by the strong multiplicity one theorem and the Chebotarev density theorem. Furthermore, when there exist three places $\{x_1,x_2,\infty\}$ of $\bf{F}$ where $\Pi$ is irreducible $L^2$ or a Speh module, Theorem B shows that $R$ is isomorphic to $R(\Pi)(\frac{1-n}2)$ and that condition (b) is true for all $x$ not in $\{x_1,x_2,\infty\}$, even if $\res{R}_{W_{\bf{F}_x}}$ or $\Pi_x$ are ramified.
\subsection{}\label{ss:associatedreps}
We view automorphic induction within the framework of Definition \ref{defn:associatedreps} as follows. Let $\bf{E}/\bf{F}$ be a finite cyclic extension. Let $R'$ be an $n$-dimensional continuous semisimple continuous representation of $G_\bf{E}$ over $\ov\bQ_\ell$, and let $\Pi'$ be an irreducible automorphic representation of $\GL_n(\bA_\bf{E})$. If $R'$ and $\Pi'$ are associated, then Lemma \ref{lem:unramifiedlocallanglands}, \ref{ss:automorphicinductionglobal}, and \ref{ss:automorphicinductionlocal} show that $\Ind_{G_{\bf{E}}}^{G_\bf{F}}R'$ and $I_{\bf{E}/\bf{F}}(\Pi')$ are associated.
Next, let $X:\bA^\times/\bf{F}^\times\rar\bC^\times$ be a finite order smooth character. The smooth character $X\circ\Art^{-1}:W_{\bf{F}}\rar\bC^\times$ has finite image, so it extends uniquely to a continuous character $\Xi:G_{\bf{F}}\rar\bC^\times$ \cite[(IV.2.2)]{HL11}. Proposition \ref{ss:theoremabasecase} and the local-global compatibility of the Artin map imply that $\Xi$ and $X$ are associated.
Finally, let $R$ be an $n$-dimensional semisimple continuous representation of $G_\bf{F}$ over $\ov\bQ_\ell$, and let $\Pi$ be an irreducible automorphic representation of $\GL_n(\bA_\bf{F})$ such that $R$ and $\Pi$ are associated. Since any irreducible unramified representation of $\GL_n(\bf{F}_x)$ is isomorphic to $Q\left(\{\chi_1\},\dotsc,\{\chi_n\}\right)$ for some unramified characters $\chi_i:\bf{F}^\times_x\rar\bC^\times$ \cite{Cas80}, Lemma \ref{ss:compatwithunramified} shows that $\rec\om_{\Pi,x}=\det\res{R}_{W_{\bf{F}_x}}$ for cofinitely many places $x$ of $\bf{F}$. Then the Chebotarev density theorem, Proposition \ref{ss:theoremabasecase}, and the local-global compatibility of the Artin map indicate that $\det{R}$ is the unique extension of $\om_\Pi\circ\Art^{-1}$ to a continuous character $G_\bf{F}\rar\ov\bQ_\ell^\times$, where we identify $\ov\bQ_\ell^\times$ with $\bC$.
\subsection{}\label{ss:nongaloisinduction} We apply Theorem B via \ref{defn:associatedreps} to prove automorphic induction in a non-Galois setting.
\begin{prop*}
Let $\bf{E}/\bf{F}$ be a finite separable extension, and let $X:\bA_\bf{E}^\times/\bf{E}^\times\rar\bC^\times$ be a finite order smooth character. Assume that
\begin{enumerate}[(a)]
\item the Galois closure $\wt{\bf{E}}$ of $\bf{E}/\bf{F}$ is solvable,
\item there exist three places $\{x_1,x_2,\infty\}$ of $\bf{F}$ inert in $\wt{\bf{E}}$ such that, for all $x$ in $\{x_1,x_2,\infty\}$, the stabilizer of $\res{(X\circ\Art^{-1})}_{W_{\wt{\bf{E}}_{\wt{e}}}}$ in $\Gal(\wt{\bf{E}}_{\wt{e}}/\bf{F}_x)$ equals $\Gal(\wt{\bf{E}}_{\wt{e}}/\bf{E}_e)$, where $\wt{e}$ (respectively $e$) is the unique place of $\wt{\bf{E}}$ (respectively $\bf{E}$) lying above $x$.
\end{enumerate}
Then there exists an irreducible cuspidal automorphic representation $I_{\bf{E}}^{\bf{F}}(X)$ of $\GL_{[\bf{E}:\bf{F}]}(\bA)$ associated with $\Ind_{G_{\bf{E}}}^{G_{\bf{F}}}(\wh{X})$, where $\Xi:G_{\bf{E}}\rar\bC^\times$ is the character associated with $X$ as in \ref{ss:associatedreps}. Furthermore, the components of $I^\bf{F}_\bf{E}(X)$ at $x_1$, $x_2$, and $\infty$ are irreducible cuspidal representations with unitary central characters.
\end{prop*}
\begin{proof}
We induct on the degree of $\bf{E}$ over $\bf{F}$, where the result is immediate for $\bf{E}=\bf{F}$. In general, the solvability of $\bf{E}/\bf{F}$ yields a Galois subextension $\wt{\bf{E}}\supseteq\bf{K}\supset\bf{F}$ such that $\bf{K}/\bf{F}$ is cyclic of prime degree. Write $k$ for the place of $\bf{K}$ above $x$, and write $k'$ for the place of $\bf{K}\bf{E}$ above $x$. Then the stabilizer of
\begin{align*}
\res{(X\circ\Nm_{\bf{K}\bf{E}/\bf{E}}\circ\Art^{-1})}_{W_{\wt{\bf{E}}_{\wt{e}}}} = \res{\Big(\res{(X\circ\Art^{-1})}_{W_{\bf{K}\bf{E}}}\Big)}_{W_{\wt{\bf{E}}_{\wt{e}}}} = \res{(X\circ\Art^{-1})}_{W_{\wt{\bf{E}}_{\wt{e}}}}
\end{align*}
in $\Gal(\wt{\bf{E}}_{\wt{e}}/\bf{K}_k)$ equals $\Gal(\wt{\bf{E}}_{\wt{e}}/\bf{E}_e)\cap\Gal(\wt{\bf{E}}_{\wt{e}}/\bf{K}_k)=\Gal(\wt{\bf{E}}_{\wt{e}}/\bf{K}\bf{E}_{k'})$. Also, note that the unique extension of $X\circ\Nm_{\bf{K}\bf{E}/\bf{E}}\circ\Art^{-1}$ to a continuous character of $G_{\bf{K}\bf{E}}$ is $\res{\Xi}_{G_{\bf{K}\bf{E}}}$. Finally, as $[\wt{\bf{E}}:\bf{K}]<[\wt{\bf{E}}:\bf{F}]$, we can apply the inductive hypothesis to obtain an irreducible cuspidal automorphic representation $I^\bf{K}_{\bf{K}\bf{E}}(X\circ\Nm_{\bf{K}\bf{E}/\bf{E}})$ of $\GL_{[\bf{K}\bf{E}:\bf{K}]}(\bA_\bf{K})$ associated with $\Ind_{G_{\bf{K}\bf{E}}}^{G_\bf{K}}(\res{\Xi}_{G_{\bf{K}\bf{E}}})$. Furthermore, $I^\bf{K}_{\bf{K}\bf{E}}(X\circ\Nm_{\bf{K}\bf{E}/\bf{E}})$ is cuspidal with unitary central character at $k$. We now casework:
\begin{enumerate}[(1)]
\item Suppose that $\bf{E}$ contains $\bf{K}$. Then $\bf{K}\bf{E}=\bf{E}$, and we define $I^\bf{F}_\bf{E}(X)$ to be $I_{\bf{K}/\bf{F}}(I^\bf{K}_{\bf{E}}(X))$. Now \ref{ss:associatedreps} indicates that $I^\bf{F}_\bf{E}(X)$ and $\Ind^{G_{\bf{F}}}_{G_{\bf{K}}}\Ind_{G_{\bf{E}}}^{G_\bf{K}}(\Xi)=\Ind_{G_\bf{E}}^{G_\bf{F}}(\Xi)$ are associated. Because $x$ is inert in $\bf{K}$ and $I^\bf{K}_\bf{E}(X)$ is cuspidal at $k$, we see from \ref{ss:automorphicinductionglobal} that $I^\bf{F}_\bf{E}(X)_x = I_{\bf{K}_k/\bf{F}_x}(I^\bf{K}_\bf{E}(X)_k)$. Note that the stabilizer of $I^\bf{K}_\bf{E}(X)_k$ in $\Gal(\bf{K}_k/\bf{F}_x)$ equals the image of $\Gal(\wt{\bf{E}}_{\wt{e}}/\bf{E}_e)$. Since $\bf{E}_e$ contains $\bf{K}_k$, this is trivial, and hence \ref{ss:automorphicinductionlocal} shows that $I_{\bf{K}_k/\bf{F}_x}(I^\bf{K}_\bf{E}(X)_k)$ is cuspidal.
\item Suppose that $\bf{E}$ does not contain $\bf{K}$. Then $\bf{K}\cap\bf{E} = \bf{F}$ and $[\bf{E}:\bf{F}]=[\bf{K}\bf{E}:\bf{K}]$, so $\Ind^{G_\bf{K}}_{G_{\bf{K}\bf{E}}}(\res{\Xi}_{G_{\bf{K}\bf{E}}})$ is isomorphic to $\res{\Ind^{G_{\bf{F}}}_{G_{\bf{E}}}(\Xi)}_{G_{\bf{K}}}$. Next, fix a generator $\sg$ of $\Gal(\bf{K}/\bf{F})$. As $I^\bf{K}_{\bf{K}\bf{E}}(X\circ\Nm_{\bf{K}\bf{E}/\bf{E}})$ and $\res{\Ind^{G_{\bf{F}}}_{G_{\bf{E}}}(\Xi)}_{G_{\bf{K}}}$ are associated, there exist cofinitely many places $v$ of $\bf{K}$ where both are unramified and
\begin{align*}
\rec(I^\bf{K}_{\bf{K}\bf{E}}(X\circ\Nm_{\bf{K}\bf{E}/\bf{E}})_v^\sg) = \rec(I^\bf{K}_{\bf{K}\bf{E}}(X\circ\Nm_{\bf{K}\bf{E}/\bf{E}})_v)^\sg = \Big(\res{\Ind^{G_{\bf{F}}}_{G_{\bf{E}}}(\Xi)}_{W_{\bf{K}_v}}\Big)^\sg = \res{\Ind^{G_{\bf{F}}}_{G_{\bf{E}}}(\Xi)}_{W_{\bf{K}_v}},
\end{align*}
by Lemma \ref{ss:automorphismscompatibility} and \ref{ss:galoissidebasechange}.(i). Then Lemma \ref{lem:unramifiedlocallanglands} indicates that the local representations $I^\bf{K}_{\bf{K}\bf{E}}(X\circ\Nm_{\bf{K}\bf{E}/\bf{E}})_v^\sg$ and $I^\bf{K}_{\bf{K}\bf{E}}(X\circ\Nm_{\bf{K}\bf{E}/\bf{E}})_v$ are isomorphic. Hence the strong multiplicity one theorem shows that the global representations $I^\bf{K}_{\bf{K}\bf{E}}(X\circ\Nm_{\bf{K}\bf{E}/\bf{E}})^\sg$ and $I^\bf{K}_{\bf{K}\bf{E}}(X\circ\Nm_{\bf{K}\bf{E}/\bf{E}})$ are isomorphic, so \ref{ss:basechangeliftglobal} yields an irreducible discrete automorphic representation $I^\bf{F}_\bf{E}(X)$ of $\GL_{[\bf{E}:\bf{F}]}(\bA)$ such that $I^\bf{F}_\bf{E}(X)_\bf{K}$ is isomorphic to $I^\bf{K}_{\bf{K}\bf{E}}(X\circ\Nm_{\bf{K}\bf{E}/\bf{E}})$. Because $\Gal(\bf{K}_{k_\infty}/\bf{F}_\infty)=\Gal(\bf{K}/\bf{F})$, where $k_\infty$ is the place of $\bf{K}$ above $\infty$, Proposition \ref{prop:compatwithbasechange} and \ref{ss:galoissidebasechange}.(i) show that we may choose $I^\bf{F}_\bf{E}(X)$ such that $\rec I^\bf{F}_\bf{E}(X)_\infty=\res{\Ind_{G_\bf{E}}^{G_\bf{F}}(\Xi)}_{W_{\bf{F}_\infty}}$.
Now the base change lift of $I^\bf{F}_\bf{E}(X)_x$ to $\GL_{[\bf{E}:\bf{F}]}(\bf{K}_k)$ is cuspidal with unitary central character, so \ref{ss:basechangeliftlocal} indicates $I^\bf{F}_\bf{E}(X)_x$ is also cuspidal with unitary central character. Thus we may apply Theorem B to obtain a semisimple continuous representation $\Sg$ of $G_\bf{F}$ over $\ov\bQ_\ell$ associated with $I^\bf{F}_\bf{E}(X)$. Proposition \ref{prop:compatwithbasechange} implies that $\res{\Sg}_{G_\bf{K}}$ is isomorphic to $\res{\Ind^{G_{\bf{F}}}_{G_{\bf{E}}}(\Xi)}_{G_{\bf{K}}}$, so \ref{ss:galoissidebasechange}.(i) shows that $\Sg$ is isomorphic to $\Ind^{G_{\bf{F}}}_{G_{\bf{E}}}(\Xi)$ tensored with a character $\Gal(\bf{K}/\bf{F})\rar\bC^\times$. By restricting to $W_{\bf{F}_\infty}$, we see that this character is trivial. Altogether, $I_\bf{E}^\bf{F}(X)$ is associated with $\Sg=\Ind^{G_\bf{F}}_{G_\bf{E}}(\Xi)$.
\end{enumerate}
Finally, as $I^\bf{F}_\bf{E}(X)$ is cuspidal at one place, we see that $I^\bf{F}_\bf{E}(X)$ itself is cuspidal.
\end{proof}
\subsection{}\label{ss:galoisgrothendieckgroup}
We now introduce some notation for $\bZ$-virtual representations of $W_F$. Write $\sG_F$ for the Grothendieck group of finite-dimensional continuous representations of $W_F$. Then $\sG_F$ is free over $\bZ$, with a $\bZ$-basis given by isomorphism classes of finite-dimensional irreducible continuous representations $\rho$ of $W_F$. As every such $\rho$ is of the form $\sg(s)$ for some complex number $s$ and continuous representation $\sg$ of $G_F$, Brauer induction shows that $\sG_F$ is $\bZ$-spanned by elements of the form $\Ind_{W_E}^{W_F}\chi$, where $E$ runs over finite separable extensions of $F$, and $\chi:W_E\rar\bC^\times$ runs over smooth characters \cite[4.10]{Del73}.
The assignments $\rho\mapsto\det\rho$ and $\rho\mapsto\rho^\vee$ for finite-dimensional continuous representations $\rho$ of $W_F$ are additive. Therefore they extend to homomorphisms
\begin{align*}
\det:\sG_F\rar\Hom_{\text{cts}}(W_F,\bC^\times)\mbox{ and }(-)^\vee:\sG_F\rar\sG_F.
\end{align*}
Write $\fM$ for the field of meromorphic functions on $\bC$, and fix a nontrivial continuous homomorphism $\psi:F\rar\bC^\times$. We also define $\bZ$-bilinear maps
\begin{align*}
L(-\otimes-,s):\sG_F\times\sG_F\rar\fM^\times\mbox{ and }\eps(-\otimes-,\psi,s):\sG_F\times\sG_F\rar\fM^\times
\end{align*}
by extending via $\bZ$-linearity from their values on pairs $(\rho,\rho')$ of isomorphism classes of finite-dimensional irreducible continuous representations of $W_F$.
\subsection{}\label{ss:automorphicgrothendieckgroup}
The following forms an automorphic analogue of $\sG_F$. Write $\sA_F$ for the free $\bZ$-basis with generators given by isomorphism classes of irreducible cuspidal representations $\pi$ of $\GL_n(F)$, where $n$ ranges over all positive integers. The assignment $\pi\mapsto\rec(\pi)$ extends to a map $\rec:\sA_F\rar\sG_F$, and Theorem C implies that this is an isomorphism.
Write $\om_\pi:F^\times\rar\bC^\times$ for the central character of $\pi$. We define $\bZ$-linear maps
\begin{gather*}
\om_\pi:\sA_F\rar\Hom_\text{cts}(F^\times,\bC^\times)\mbox{ and }(-)^\vee:\sA_F\rar\sA_F,\\
L(-\times-,s):\sA_F\otimes_\bZ\sA_F\rar\fM^\times\mbox{ and }\eps(-\times-,\psi,s):\sA_F\otimes_\bZ\sA_F\rar\fM^\times.
\end{gather*}
by extending via $\bZ$-linearity from their values on isomorphism classes of irreducible cuspidal representations of $\GL_n(F)$, or pairs thereof.
\subsection{}
We now prove Theorem D, by using Proposition \ref{ss:nongaloisinduction} and embedding into the global situation.
\begin{prop*}\hfill
\begin{enumerate}[(i)]
\item For any smooth character $\chi:F^\times\rar\bC^\times$, we have $\rec(\chi) = \chi\circ\Art^{-1}$.
\item For any irreducible cuspidal representation $\pi$ of $\GL_n(F)$ and smooth character $\chi:F^\times\rar\bC^\times$, we have
\begin{align*}
\rec(\pi\otimes(\chi\circ\det)) = \rec(\pi)\otimes\rec(\chi).
\end{align*}
\item For any irreducible cuspidal representation $\pi$ of $\GL_n(F)$, we have
\begin{align*}
\rec(\om_\pi) = \det\circ\rec(\pi)\mbox{ and }\rec(\pi^\vee) = \rec(\pi)^\vee.
\end{align*}
\item For any irreducible cuspidal representations $\pi$ of $\GL_n(F)$ and $\pi'$ of $\GL_{n'}(F)$, we have
\begin{align*}
L(\pi\times\pi',s) = L(\rec(\pi)\otimes\rec(\pi'),s)\mbox{ and } \eps(\pi\times\pi',\psi,s) = \eps(\rec(\pi)\otimes\rec(\pi'),\psi,s).
\end{align*}
\end{enumerate}
\end{prop*}
\begin{proof}
We have already proved (i) in Proposition \ref{ss:theoremabasecase} and (ii) in Proposition \ref{prop:compatwithtwisting}. Now both sides of (iii) are $\bZ$-linear in $\pi$, and both sides of (iv) are $\bZ$-bilinear in $(\pi,\pi')$. Therefore \ref{ss:galoisgrothendieckgroup} and \ref{ss:automorphicgrothendieckgroup} show that (iii) and (iv) follow from considering the same equations for $\pi=\rec^{-1}(\Ind_{W_E}^{W_F}\rho)$ and $\pi'=\rec^{-1}(\Ind_{W_{E'}}^{W_F}\rho')$ in $\sA_F$, where $E$ and $E'$ are separable extensions of $F$, and $\rho:W_E\rar\bC^\times$ and $\rho':W_{E'}\rar\bC^\times$ are smooth characters.
Note that $\rho$ is the product of a finite order character with a Tate twist $(s)$ for some complex number $s$. The projection formula and (ii) indicate that twisting $\rho$ by $(-s)$ results in twisting $\pi$ by $(-s)$. Now (ii) implies that both sides of (iii) and (iv) are compatible with Tate twists, so we need only consider the case when $\rho$ has finite order. In the same way, we may assume $\rho'$ also has finite order.
With this reduction in hand, write $\wt{E}$ for the Galois closure of $E$. The proof of Proposition \ref{prop:compatwithbasechange} yields a Galois extension $\wt{\bf{E}}/\bf{F}$ of global function fields and a place $z$ of $\wt{\bf{E}}$ such that $z$ is inert over $\bf{F}$ and $\wt{\bf{E}}_z/\bf{F}_o$ can be identified with $\wt{E}/F$, where $o$ is the place of $\bf{F}$ below $z$. Note then that $\Gal(\wt{\bf{E}}/\bf{F})$ is identified with $\Gal(\wt{E}/F)$. Writing $\bf{E}$ for the subfield of $\wt{\bf{E}}$ corresponding to $\Gal(\wt{E}/E)$, we see that this identifies $\bf{E}_y$ with $E$, where $y$ is the place of $\bf{E}$ below $z$. We apply this to $E'$ to similarly obtain extensions $\wt{\bf{E}}'\supseteq\bf{E}'\supseteq\bf{F}'$ with places $z'$, $y'$, and $o'$.
Because $\wt{\bf{E}}/\bf{F}$ is inert at one place $o$, the Chebotarev density theorem provides three more places $\{x_1,x_2,\infty\}$ of $\bf{F}$ that are inert in $\wt{\bf{E}}$. For all $x$ in $\{x_1,x_2,\infty\}$, there exists a finite order smooth character $\chi_x:\bf{E}_e^\times\rar\bC^\times$ such that the stabilizer of $\chi_x\circ\Nm_{\wt{\bf{E}}_{\wt{e}}/\bf{E}_e}$ in $\Gal(\wt{\bf{E}}_{\wt{e}}/\bf{F}_x)$ equals $\Gal(\wt{\bf{E}}_{\wt{e}}/\bf{E}_e)$, where $\wt{e}$ (respectively $e$) is the unique place of $\wt{\bf{E}}$ (respectively $\bf{E}$) lying above $x$ \cite[Lemma 4.7]{Har98}. As $\rho\circ\Art$ also has finite order, there exists a finite order smooth character $X:\bA_{\bf{E}}^\times/\bf{E}^\times\rar\bC^\times$ such that $\res{X}_{\bf{E}_y^\times}=\rho\circ\Art$ and $\res{X}_{\bf{E}_x^\times}=\chi_x$ for all $x$ in $\{x_1,x_2,\infty\}$ \cite[Theorem 5 of Chapter X]{AT09}. The same discussion yields analogous places $\{x_1',x_2',\infty'\}$ of $\bf{F}'$ and an analogous finite order smooth character $X':\bA^\times_{\bf{E}'}/\bf{E}'^\times\rar\bC^\times$.
Since $F$ is a nonarchimedean local field, $\wt{E}/F$ and hence $\wt{\bf{E}}/\bf{F}$ are solvable. Therefore we may apply Proposition \ref{ss:nongaloisinduction} to obtain an irreducible cuspidal automorphic representation $I^\bf{F}_\bf{E}(X)$ of $\GL_{[\bf{E}:\bf{F}]}(\bA)$ associated with $\Ind_{G_\bf{E}}^{G_\bf{F}}(\Xi)$, where $\Xi:G_{\bf{E}}\rar\bC^\times$ is the character associated with $X$ as in \ref{ss:associatedreps}. Furthermore, the components of $I^\bf{F}_\bf{E}(X)$ at $x_1$, $x_2$, and $\infty$ are irreducible cuspidal representations, so \ref{defn:associatedreps} shows that
\begin{align*}
\rec(I^\bf{F}_\bf{E}(X)_v) = \res{\Ind_{G_\bf{E}}^{G_\bf{F}}(\Xi)}_{W_{\bf{F}_v}}
\end{align*}
for all places $v$ of $\bf{F}$ not in $\{x_1,x_2,\infty\}$. Taking $v=o$ yields
\begin{align*}
\rec(I^\bf{F}_\bf{E}(X)_o) = \res{\Ind_{G_\bf{E}}^{G_\bf{F}}(\Xi)}_{W_{\bf{F}_o}} = \Ind^{W_{\bf{F}_o}}_{W_{\bf{E}_y}}(\res{X}_{\bf{E}^\times_y}\circ\Art^{-1}) = \Ind^{W_F}_{W_E}\rho,
\end{align*}
so $\pi$ is the component of $I^\bf{F}_\bf{E}(X)$ at $o$. We apply this to $\bf{E}'/\bf{F}'$ and $X'$ to similarly obtain an irreducible cuspidal automorphic representation $I^{\bf{F}'}_{\bf{E}'}(X')$ associated with $\Ind^{G_{\bf{F}'}}_{G_{\bf{E}'}}(\Xi')$.
Now \ref{ss:associatedreps} implies that $\om_{I^\bf{F}_\bf{E}(X)}$ equals $\big(\det\Ind_{G_\bf{E}}^{G_\bf{F}}(\Xi)\big)\circ\Art$, so restricting to $o$ tells us that
\begin{align*}
\om_\pi = \om_{I^\bf{F}_\bf{E}(X)_o} = \det\res{\Ind^{G_\bf{F}}_{G_\bf{E}}(\Xi)}_{W_{\bf{F}_o}} = \det\Ind^{W_F}_{W_E}\rho = \det\rec\pi.
\end{align*}
This completes the first part of (iii). Additionally, because $\big(\det\Ind_{G_\bf{E}}^{G_\bf{F}}(\Xi)\big)\circ\Art$ has finite image, $\om_{I^\bf{F}_\bf{E}(X)}$ does as well. And \ref{ss:inductionongroups} shows that the restriction of $\Ind_{G_\bf{E}}^{G_\bf{F}}(\Xi)$ to $W_{\bf{F}_\infty}$ is irreducible, so $\Ind_{G_\bf{E}}^{G_\bf{F}}(\Xi)$ itself is irreducible. The same discussion holds for $I^{\bf{F}'}_{\bf{E}'}(X')$ and $\Ind^{G_{\bf{F}'}}_{G_{\bf{E}'}}(\Xi')$, so we can apply \cite[Theorem 2.4]{Hen00} to conclude that (iv) holds.
Finally, return to the situation where $\pi$ and $\pi'$ are irreducible cuspidal representations. Then
\begin{align*}
L(\pi\times\pi',s) = \prod_\chi L(\chi,s),
\end{align*}
where $\chi$ ranges over unramified characters $F^\times\rar\bC^\times$ such that $\pi'^\vee\otimes(\chi\circ\det)$ is isomorphic to $\pi$. Since $L(\chi,s)$ has a pole at $s=1$ if and only if $\chi$ is trivial, we see that $L(\pi\times\pi',s)$ has a pole at $s=1$ if and only if $\pi'^\vee=\pi$. We have an analogous decomposition of $L(\rec(\pi)\otimes\rec(\pi'),s)$, and the same argument indicates that $L(\rec(\pi)\otimes\rec(\pi'),s)$ has a pole at $s=1$ if and only if $\rec(\pi')^\vee=\rec(\pi)$. Applying (iv) completes the second part of (iii).
\end{proof}
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\section{Introduction} {\label{section_introduction}}
Hydrophobic force causes attraction between non-polar solutes in liquid water and is known to be an important driving force in micellar aggregation, cell membrane formation, large protein structures \citep{BallChemRev2008,*PrivalovAdvProtChem1988,*Tanford1997}. Although hydrophobic force is widely known to be pervading distinct physical, chemical and biological phenomena, its occurence as a solvent induced interaction was first suggested only in 1945, by Frank and Evans through calorimetric studies on hydrocarbons \citep{Frank1945} and later elucidated in biological context by Kauzmann \citep{Kauzmann1959}. The former study noted that transferring a small hydrophobe into water was accompanied by unfavorable free energy change \citep{Butler1937}, dominated by entropy reduction due to reorganization of vicinal water molecules \citep{Frank1945}. Hence, two hydrophobes show tendency to coalesce in order to minimize the unfavorable free energy. In the following years the applicability of this viewpoint for different sizes of hydrophobes such as alkanes, proteins has been discussed \citep{KlotzScience1958,*ScheregaJCP1962,*Engberts1993}. Theoretical and simulational investigations were limited to small solutes and interaction between them on scales of few Angstroms \citep{PrattJCP1977,*BerneJCP1979,*StillingerJCP1979,*TanakaJCP1987,*Herman1993}. In early 1980s the first direct measurement of an attractive force between hydrophobic surfaces has been carried out using surface force apparatus \citep{IsrNature1982}. Large hydrophobic surfaces made up of chemisorbed mica were employed in a crossed cylinder geometry and the measured force between them was related to interaction free energy per unit surface area using Derjaguin approximation \citep{Isrbook1992}. The force was seen to be influential upto hundreds of Angstroms and stronger than inter-surface van der Waal forces \citep{IsrNature1982}. The qualitative aspect of the result i.e., long range and monotonic nature of the force withstood the test of time \citep{Claesson2001,MeyerPNAS2006,Hammer2010}. To understand the same, various physical mechanisms were proposed, some generic to any fluid such as metastability of confined fluid \citep{ClaessonScience1988,*LuzarPRL2003}, dewetting-induced cavitation under liquid-vapor coexistence conditions \citep{LumJPC1999}, fluid structuring effects \citep{MarceljaCPL1976,*Eriksson1989,*Besseling1997} and some, dependent on surface details like correlated dipolar fluctuations \citep{AttardJPC1989,*PodgornikJCP1989,*Tsao1993}, charged bilayer patches \citep{MeyerPNAS2005}, nanobubbles \citep{AttardLangmuir1996,*TyrellPRL2001,*AttardPhysicaA2002}. The above phenomenological descriptions, however, are either envisaged in narrow range of fluid conditions or specifically depend on hydrophobization patterns on surfaces. Besides, they were unsuccessful in reproducing generic features of hydrophobic force seen in experiments \citep{Claesson2001}. We attempt to understand in a model study the nature of hydrophobic force by analyzing correlations of hydrogen bond fluctuations in liquid water and the effect of modification caused by presence of surfaces on these fluctuations. The analysis is carried out at generic fluid conditions within the model and applies to generic surfaces.
We discuss some essential aspects in modelling water and past attempts in this direction. Hydrogen bonding is an orientation-dependent attraction between two water molecules \citep{StillingerScience1980}. In order to analyze hydrogen bond fluctuations it is therefore essential to envisage both density and orientational degrees of freedom of each water molecule. There are models galore proposed and analyzed to reproduce anomalous thermodynamic properties of water \citep{BellJoP1972,*BlumJCP1985,*JaglaJCP1999,*TruskettJCP1999,*StanleyJSP2003,Guillot2002,*Vega2009}. However, theoretical attempts to envisage fluctuations in water are limited to Ornstein-Zernike-like phenomenological approaches, wherein integral equations only in terms of density correlation were heuristically proposed and are numerically solved using different closure approximations \citep{HansenSimpleLiquids,*Gray1984,BlumJCP1990,*RichardiJCP1999}. Wertheim's theory of associating fluids envisages similar density correlations to be solved in compliance with steric constraints imposed by formation of associated n-mers \citep{WertheimJStatPhy1984Vol35,*Wertheim2JStatPhy1984Vol35,*WertheimJStatPhy1986Vol42}. The success of these descriptions crucially depends on approximation schemes employed \citep{HansenSimpleLiquids,*Gray1984,KolafaMolPhys1989,*EvansJCP2003,*SciortinoJCP2008,*DillJCP2009}. Other approaches specific to molecular fluids such as RISM were seen to be less predictive in case of associating fluids \citep{ChandlerARPC1978}. Also, perturbation theories based on smallness of interaction strength found little success in reproducing liquid phase properties of water-like fluids \citep{AndersenJCP1973Vol59}.
Quantum mechanical calculations on water dimer in gas phase and diffraction study of ice forms provide sufficient evidence for specific nature of hydrogen bonding \citep{Eisenberg1969}, i.e., positively polarized hydrogen atom covalently bonded to an oxygen interacts only with negatively polarized lone-pair of neighboring oxygen. The specificity necessitates the density of hydrogen bonds and dangling bonds (hydrogens or lone-pairs which are not hydrogen bonded) to be commensurate with molecular density, stated as a \textit{sum rule} \citep{KanthPhysica2011}. Consequently, density and orientational fluctuations (the latter being inherently connected to bond fluctuations) are not totally independent; their long wavelength fluctuations, especially, are to be consistent with the sum rule. Effective interaction models for water designed for numerical simulations (molecular dynamics) \citep{RahmanJCP1971Vol55,*StillingerJCP1974Vol60,Guillot2002,*Vega2009} provide successful instances of (implictly) envisaging most essential features of hydrogen bond fluctuations consistent with the sum rule. A water molecule is often modelled as a polar molecule with charges corresponding to hydrogens and lone-pairs placed at vertices of a tetrahedron \citep{RahmanJCP1971Vol55,*StillingerJCP1974Vol60,MahoneyJCP2000Vol112}. A complete description of molecular correlations in such models can be achieved by defining a set of orthonormal vectors in terms of atomic coordinates and defining correlations among them. Large scale molecular dynamics (MD) simulations at ambient conditions reveal that density correlations are short ranged and extend no further than $12$ \AA{}; whereas, dipolar orientations of water molecule, which are receptive to bond fluctuations in the neighborhood, are correlated over large distances, atleast upto $75$ \AA{} \citep{KanthPRE2010}. Two correlation lengths of order $5.2$ \AA{} and $24$ \AA{} were inferred. Coulomb interactions, surprisingly, have little effect on asymptotic behavior of these correlations \citep{KanthPRE2010}. It is also suggested that mesoscopic hydrophobic solutes when coupled to dipolar fluctuations in water medium experience a long range exponential force, which is attractive in nature and dependent on shape and mutual orientation of solute surfaces \citep{KanthPRE2010}. The exponential decay bears a striking consistency with that seen in experiments measuring hydrophobic force between large surfaces \citep{IsrNature1982,Claesson2001}. For the case of large hydrophobic surfaces, correlations in confined water need to be ascertained. To simulate such a system the surfaces need to be several times larger than the longest correlation length in the system in order to obtain proper equilibrium conditions. This requires huge system size that would render the simulation prohibitively resource intensive. In addition, the accompanying free energy change could be very small due to weak nature of correlations at large distances. Instead, we take analytic route to describe hydrogen bond fluctuations in water and the effect of spatial confinement on them.
Hydrophobes are known to interact unfavorably with water molecules in contrast to strong and cohesive water-water interactions. In addition, large surfaces substantially disrupt the hydrogen bond network whose fluctuations are suppressed at surface boundaries. The setting is ideally suited for fluctuation-induced force between the surfaces driven by thermal energy in confined water. Forces of this nature are generically called \textit{Casimir forces} as they were first discussed by Casimir in the case of electromagnetic fluctuations confined between conducting plates \citep{Casimir1948} which was later studied in detail by Lifshitz \citep{Lifshitz1956}. Such forces are now envisaged in widely different contexts \citep{KardarRMP1999}. Fisher and de Gennes argued that when a binary liquid mixture is confined between surfaces which have specific affinity towards one of the fluid components, Casimir-like density fluctuations in the liquid give rise to an effective force between surfaces \citep{Fisher1978}. Origin of the force is entropic in nature; in that, the free energy is increased due to restriction imposed on fluctuations by the boundaries, thereby system tends to minimize the separation in order to reduce the free energy cost. We investigate hydrophobic force to be a manifestation of Casimir-like force, here, due to density and orientational fluctuations in liquid water.
When water is confined between hydrophobic surfaces the inherent field fluctuations vanish on surfaces. Furthermore, water molecules at interface with each surface have restricted orientational entropy owing to repulsive hydrophobe-water interactions. This effect gives rise to modified fluctuations at each interface. We study the collective consequences of these effects on the overall free energy of the system in a model study. We define a simple water model wherein density and orientations of a water molecule are envisaged. The specific nature of hydrogen bonding between molecules is incorporated and associated constraints on the bond network are taken care of in the analysis. Partition function is solved about a mean field which is consistently deduced at arbitrary densities within the model. Fluctuation properties are also deduced. Large correlation length is seen for orientational fluctuations. Two macroscopic surfaces are envisaged as boundaries in a spatial dimension. The change in free energy in the system due to the presence of surfaces is calculated and is seen to be composed of three important contributions : (i) Casimir part, which arises solely from discretization of fluctuation modes between boundaries and is generic to all surfaces; (ii) Interfacial tension, which is free energy change due to modified fluctuations at hydrophobe-water interface. It is dependent on nature of surface-water interaction and to a small extent, also on separation distance between the surfaces; (iii) Interfacial fluctuations-induced free energy, which is due to correlations of modified fluctuations at both interfaces. It depends on type of both surfaces and their interaction with water. The results are discussed for different types of surfaces such as hydrophobic and hydrophilic. We find that the Casimir part is leading contribution and is an inverse power-law function of separation distance. However, numerically the magnitude of Casimir part is significant only upto four times the longest correlation length in the model. Interfacial tension also varies with separation distance, but its variation is numerically insignificant. Interfacial fluctuations-induced contribution is seen to be exponentially decaying with distance, analogous to the force form deduced for mesoscopic surfaces \citep{KanthPRE2010}. Furthermore, we find that all the contributions are of comparable order of magnitude consistent with experimental values. The dependence of the force on fluid conditions like temperature, average hydrogen bonds is also discussed. Our results indicate that hydrophobic force qualitatively imitates Casimir-like force behavior within our model study. It is desirable to emulate the computation within more realistic models of water possibly with the help of MD simulations. We also looked at transverse density profile for confined water and show that an increase in density occurs near interfaces.
\section{Model for water} {\label{section_watermodel}}
We envisage our water model in the background of a lattice to exactly account for hard-sphere repulsion between atoms at short distances. Also, constraints of hydrogen bond network are explicitly taken care of in lattice background. We developed molecular mean field (MMF) technique in an earlier paper \citep{KanthPhysica2011} to address the same model in the infinite volume limit. We present here a brief summary of the model and MMF technique and then, address the case of confined water.
On a three dimensional hypercubic lattice we define occupation field $W(r) = \lbrace 0, 1 \rbrace$ corresponding to water being absent or present, respectively, at the site $r = (x,y,z)$. At each \textit{occupied} site we define bond arm field $H_{\alpha}(r)$ which resides on the links around the site $r$. $H_{\alpha}(r) = \lbrace 0, \pm 1 \rbrace$, where $\alpha = \lbrace \pm 1, \pm 2, \pm 3 \rbrace$ correspond to six directions around the site. $H_{\alpha}(r) = 1$ refers to hydrogen arm on the corresponding link, $-1$ to lone pair arm and $0$ for no arm. The constraints between $W(r)$ and $H_{\alpha}(r)$ being,
\begin{align}
\sum_{\alpha} H_{\alpha}^{2}(r) \ &= \ 4 W(r) \label{H2constraint}
\\ \sum_{\alpha} H_{\alpha}(r) \ &= \ 0 \label{Hconstraint}
\end{align}
which imply that every water molecule has two hydrogen and two lone-pair arms only. A hydrogen bond is realized when two water molecules two lattice units apart have one of each's hydrogen and lone-pair arms meet at a site, as shown in Fig.(\ref{fig_allowedconfigs}). When two molecules are on near-neighbor sites they are disallowed to have any non-zero bond arm on the link between them. The constraint is given by :
\begin{equation}
W(r) \left( \sum_{\alpha} H^{2}_{\alpha}(r + e_{\alpha}) \right) = 0 \label{unoccupied_Hconstraint}
\end{equation}
We write a general interaction Hamiltonian in terms of $H_{\alpha}$ field as below :
\begin{equation}
{\cal H} \ = \ \frac{\tilde{\lambda}}{2} \sum_{r} \sum_{\alpha, \alpha^{'}} H_{\alpha}(r - e_{\alpha}) H_{\alpha^{'}}(r - e_{\alpha^{'}})
\end{equation}
where, $\tilde{\lambda}$ is interaction strength and $\alpha$, $\alpha^{'}$ denote directions around site $r$. There are additional restrictions on $H_{\alpha}$ field, namely,
\\ (i) at any site no more than two bond arms meet i.e.,
\begin{equation}
0 \leq \sum_{\alpha} H^{2}_{\alpha}(r + e_{\alpha}) \leq 2 \label{unoccupied_H2values}
\end{equation}
(ii) two non-zero bond arms are disallowed from meeting at a site i.e., anti-bonds are disallowed,
\begin{equation}
-1 \leq \sum_{\alpha} H_{\alpha}(r + e_{\alpha}) \leq 1 \label{unoccupied_Hvalues}
\end{equation}
The grand canonical partition function for the system at a finite chemical potential $\tilde{\mu}$ for water and inverse temperature $\beta$ is given by :
\begin{equation}
Z = \prod_{r} \sum_{\displaystyle W(r), H_{\alpha}(r)}^{'} \exp \left[ - \beta \sum_{r} \left( {\cal H} - \tilde{\mu} W(r) \right) \right] \label{partitionfunction}
\end{equation}
where, the prime over summation indicates that the $W(r)$ and $H_{\alpha}(r)$ sum have to be carried out in compliance with Eqs.(\ref{H2constraint},\ref{Hconstraint},\ref{unoccupied_Hconstraint},\ref{unoccupied_H2values},\ref{unoccupied_Hvalues}). Evaluating $Z$ amounts to enumerating all possible bond configurations that satisfy above constraints and calculating the exponential in Eq.(\ref{partitionfunction}) for those configurations over allowed range of $W$ and $H_{\alpha}$ at fixed values of $\tilde{\mu}$, $\beta$ and $V$ the volume of the system.
The restrictions represented by Eqs.(\ref{unoccupied_Hconstraint},\ref{unoccupied_H2values},\ref{unoccupied_Hvalues}) are at sites where there is no water. These are shown in Fig.(\ref{fig_disallowedconfigs}). To implement them in our analysis it is useful to define two discrete integer fields $b(r)$, $q(r)$ :
\begin{subequations}
\label{bqdefinition}
\begin{align}
b(r) \ & = \ \sum_{\alpha} H_{\alpha}^{2}(r+{e}_{\alpha})
\\ q(r) \ & = \ \sum_{\alpha} H_{\alpha}(r+{e}_{\alpha})
\end{align}
\end{subequations}
The discrete field $b(r)$ counts the number of non-zero arms in the neighborhood of site $r$, while $q(r)$ measures the charge i.e, difference between number of hydrogen arms and lone-pair arms meeting at site $r$. By construction, $b(r)$ varies between $0$ and $6$ on a three dimensional hyper-cubic lattice and $q(r)$ in turn varies between $-b(r)$ to $b(r)$. By imposing the condition that $b(r) \leq 2$ in our analysis we ensured that no more than two arms can meet at a site. Furthermore, for $b(r) = 2$ we demand $q(r) = 0$ to disallow anti-bond configurations. In terms of these variables, Eqs.(\ref{unoccupied_Hconstraint},\ref{unoccupied_H2values},\ref{unoccupied_Hvalues}) can be rewritten as :
\begin{align}
W(r) b(r) \ & = \ 0 \label{bconstraint}
\\ (b(r), q(r)) \ & = \ \left\lbrace (0,0), (1,1), (1,-1), (2,0) \right\rbrace \label{bqvalues}
\end{align}
where, $b(r)$ and $q(r)$ values are restricted only to the above set of mutually exclusive pairs. We now rewrite the partition function as below :
\begin{equation}
Z \ = \ \prod_{r} \sum_{\displaystyle { W(r), H_{\alpha}(r)} \atop \displaystyle { b(r), q(r) } }^{'} \exp \left[ - \beta \sum_{r} \left( {\cal H} - \tilde{\nu} q^{2}(r) - \tilde{\mu} W(r) \right) \right]
\end{equation}
where, we have additionally introduced a chemical potential $\tilde{\nu}$ for dangling bond configuration i.e., $(b,q) = (1, \pm 1)$. The fields $b$, $q$ are summed over allowed range given in Eq.(\ref{bqvalues}) and the prime over summation indicates that Eqs.(\ref{H2constraint},\ref{Hconstraint},\ref{bqdefinition},\ref{bconstraint}) act as constraints in the evaluation. Note that, since only hydrogen bond interaction is envisaged in the model, the Hamiltonian ${\cal H}$ can be rewritten as a simple expression :
\begin{equation}
{\cal H} \ = \ - \tilde{\lambda} \sum_{r} \ \delta({b(r),2}) \label{Hamiltonian_redefinition}
\end{equation}
where, Kronecker delta function denoted here as $\delta({p,q})$ is defined as $\delta({p,q}) = 1$ for $p = q$ and $0$ otherwise. All the possible hydrogen bond configurations are implied from solving the non-local constraints Eq.(\ref{bqdefinition}). These constraints are enforced in the partition function by introducing auxiliary fields, as given below :
\begin{subequations}
\label{auxiliaryfields}
\begin{align}
& \delta\left( b(r), \sum_{\alpha} H^{2}_{\alpha}(r + {e}_{\alpha}) \right) = \frac{1}{2N+1} \sum_{\displaystyle \eta(r)} \exp\left[ -i \frac{\pi}{N} \eta(r) \left( b(r) - \sum_{\alpha} H^{2}_{\alpha}(r + {e}_{\alpha}) \right) \right]
\\ & \delta\left( q(r), \sum_{\alpha} H_{\alpha}(r + {e}_{\alpha}) \right) = \frac{1}{2N+1} \sum_{\displaystyle \phi(r)} \exp\left[ - i \frac{\pi}{N} \phi(r) \left( q(r) - \sum_{\alpha} H_{\alpha}(r + {e}_{\alpha}) \right) \right]
\end{align}
\end{subequations}
where, $\eta(r)$ and $\phi(r)$ act as dual fields to density and charge of bond arms in a local neighborhood. The discrete $\eta$ and $\phi$ fields take integer values in the range $[-N, N]$ at every site, where $N$ is any suitably large integer (greater than $8$).
The partition function can be rewritten in terms of new variables and auxiliary fields as :
\begin{align}
\nonumber & Z \ = \ \left[ \prod_{r} \sum^{'}_{\displaystyle W(r), H_{\alpha}(r) \atop \displaystyle b(r), q(r)} \frac{1}{(2N+1)^{2}} \sum_{\displaystyle \eta(r), \phi(r)} \right] \exp \sum_{r} \left[ \vphantom{\sum_{\alpha}} -\beta ( {\cal H} - \tilde{\nu} q^{2}(r) - \tilde{\mu} W(r) ) \right.
\\ & + \left. {i \frac{\pi}{N}} {\displaystyle \eta(r) \left( \sum_{\alpha} H^{2}_{\alpha}(r + {e}_{\alpha}) - b(r) \right) } + {i \frac{\pi}{N}} \phi(r) \left( \sum_{\alpha} H_{\alpha}(r + {e}_{\alpha}) - q(r) \right) \right] \label{Zequation}
\end{align}
Here, prime over summation refers to sum being restricted to local on-site constraints Eqs.(\ref{H2constraint},\ref{Hconstraint},\ref{bconstraint}) only. The introduction of auxiliary fields $\eta(r)$ and $\phi(r)$ allows summation over other discrete fields $W(r), H_{\alpha}(r), b(r), q(r)$ within their respective allowed range at each site \textit{without} any restriction from the neighborhood configurations i.e., as if a single site functional $Z_{site}$, as given below :
\begin{equation}
Z \ = \ \int [{\cal D} \eta] [{\cal D}\phi] \prod_{r} Z_{site}(\eta(r), \phi(r), \nabla_{\alpha}\eta, \nabla_{\alpha}\phi) \label{Zequation_v2}
\end{equation}
where, the summation over $\eta, \phi$ fields is transformed into an integral in the limit of $N \rightarrow \infty$ and $Z_{site}$ is given as sum of weights corresponding to each allowed state i.e., void, dangling bond, hydrogen bond and water. It is given by,
\begin{align}
Z_{site} \ & = \ 1 + 2 \nu e^{-i \eta(r)} \cos(\phi(r)) + \lambda e^{-2 i \eta(r)} + \mu C(\eta, \phi, \nabla_{\alpha}\eta, \nabla_{\alpha}\phi) ) \label{zsite}
\\ C(\eta, \phi, \nabla_{\alpha}\eta, \nabla_{\alpha}\phi) & = \sum^{'}_{\displaystyle \stackrel{\displaystyle H_{\alpha} = 0,\pm 1}{\alpha = \pm 1, \pm 2, \pm 3}} \exp\left[ i \sum_{\alpha} ( H^{2}_{\alpha}(r) {\eta}(r+e_{\alpha}) + H_{\alpha}(r) {\phi}(r+e_{\alpha}) ) \right] \label{Cweight}
\end{align}
where, $\nu \equiv \exp(\beta \tilde{\nu})$, $\lambda \equiv \exp(\beta \tilde{\lambda})$ and $\mu \equiv \exp(\beta \tilde{\mu})$ are fugacities of dangling bond, hydrogen bond and water states, respectively. The orientational degrees of freedom of water yields $C(\eta, \phi, \nabla_{\alpha}\eta, \nabla_{\alpha}\phi)$ given by Eq.(\ref{Cweight}), where the summation is over orientations at site $r$. The prime over summation indicates $H_{\alpha}$'s of each orientation satisfy Eqs.(\ref{H2constraint},\ref{Hconstraint}). The exponential corresponds to an orientation and it is a function of dual fields at near-neighbor sites towards which non-zero bond arms of the orientation are directed. The densities of dangling bond (DB), hydrogen bond (HB) and water ($\rho$) are calculated from partial derivative of partition function with respect to $\beta \tilde{\nu}$, $\beta \tilde{\lambda}$, $\beta \tilde{\mu}$, respectively.
\subsection{MMF theory} {\label{section_MMF}}
The partition function has a unique maximum at isotropic and homogeneous field configuration $\eta = \phi = 0$. $Z_{site}$ at the maximum is given by $Z_{o}$ :
\begin{equation}
Z_{o} = (1 + 2 \nu + \lambda + 90 \mu)
\end{equation}
The extremization condition $\sum_{x}({\delta}/{\delta \eta}) Z = 0$ implies the sum rule of the system i.e., $\textnormal{DB} + 2 \textnormal{HB} = 4 \rho$ exactly \citep{KanthPhysica2011}, while extremization with respect to $\phi$ field is trivially satisfied. To the zeroth order, partition function is $Z = (Z_{o})^{V}$ and sum rule translates as :
\begin{equation}
2 \nu + 2 \lambda \ = \ 4(90 \mu)
\end{equation}
Using this relation, the densities of dangling bond, hydrogen bond and water are given upto zeroth order as :
\begin{subequations}
\label{densities_zerothorder}
\begin{align}
\textnormal{DB} \ & \equiv \ \nu \frac{\partial }{\partial \nu}(\ln Z) \ = \ \frac{ 4\nu }{ 2 + 5\nu + 3 \lambda }
\\ \textnormal{HB} \ & \equiv \ \lambda \frac{\partial }{\partial \lambda}(\ln Z) \ = \ \frac{ 2 \lambda }{ 2 + 5 \nu + 3\lambda }
\\ \rho \ & \equiv \ \mu \frac{\partial }{\partial \mu}(\ln Z) \ = \ \frac{ \nu + \lambda }{ 2 + 5 \nu + 3\lambda }
\end{align}
\end{subequations}
Eliminating $\lambda$ from equations for DB, HB we obtain :
\begin{equation}
\textnormal{HB} \ = \ 2 \rho - \frac{ \nu }{ \nu + 1 } (1 - 3 \rho)
\label{equationofnetwork}
\end{equation}
We call Eq.(\ref{equationofnetwork}) the \textit{equation of network}. It is a manifestation of sum rule in terms of model parameters. We choose dangling bond energy parameter to be zero i.e. $\nu = 1$ and measure temperature ($\beta^{-1}$) in units of hydrogen bond strength ($\tilde{\lambda}$). To zeroth order the theory is now parameter-free and all densities can be obtained as a function of temperature only. The equation of network can also be written in terms of average hydrogen bonds per molecule $h \equiv \frac{2 \textnormal{HB}}{\rho}$ as :
\begin{equation}
h = 7 - \frac{1}{\rho} \label{equationofnetwork_h_rho}
\end{equation}
From zeroth order partition function the mean field free energy $G_{m}$ per unit volume can be given in terms of densities as :
\begin{equation}
\beta G_{m} = \ln( 1 - 5 \rho + HB )
\label{equationofstate}
\end{equation}
Eq.(\ref{equationofstate}) is analogous to equation of state for a system of hard spheres. It correctly predicts the density saturation in the model at $\rho = \frac{1}{3}$, $\textnormal{HB} = \frac{2}{3}$ or $h = 4$. Thus, equation of network is a manifestation and density saturation effect is a direct consequence of the sum rule.
Using Eq.(\ref{zsite}) for $Z_{site}$ we expand dual fields upto quadratic order about their maximum, perform Fourier transform on their fluctuations in a large cubic box using periodic boundary conditions, then integrate the resulting Gaussian functional in Eq.(\ref{Zequation_v2}) over all field configurations. This yields total free energy per unit volume $G$ upto one-loop correction which includes leading contributions due to fluctuations in density and orientations :
\begin{equation}
\beta G \ = \ \beta G_{m} + \frac{1}{2} \int\limits^{\pi}_{-\pi} \frac{d^{3}k}{(2 \pi)^{3}} \ln \left(P_{\eta \eta}(\vec{k}) P_{\phi \phi}(\vec{k}) \right) \label{oneloop}
\end{equation}
where,
\begin{subequations}
\label{propagators}
\begin{align}
P_{\eta \eta}(\vec{k}) & = \displaystyle 64 \rho \left(\frac{9}{10} - \rho \right)
\left[ \left(\Delta - \frac{9}{20 (\frac{9}{10} - \rho)} \right)^{2} + \frac{3 (\frac{9}{25} - \rho)}{8 (\frac{9}{10} - \rho)^{2}} \right]
\\ P_{\phi \phi}(\vec{k}) & = \displaystyle \frac{96 \rho}{5} \left[ \Delta (1 -\Delta) + \frac{5 \text{(DB)}}{96 \rho} \right]
\end{align}
\end{subequations}
and $\Delta = \frac{1}{6} \sum_{i=1}^{3} (1 - \text{cos}(k_{i}))$; $k_{i}$ are vector components of $\vec{k}$. $P_{\eta \eta}$, $P_{\phi \phi}$ are fluctuation propagators of dual fields. The above expressions for propagators are simplified to the leading order using sum rule. Precise expressions in terms of original fugacities are given in Appendix (\ref{app_correlationfunctions}). The term involving propagators in the free energy expression (Eq.\ref{oneloop}) is the entropy contribution about the mean field.
The correlation functions in the system in the momentum space are given by propagators $P_{\eta \eta}$ and $P_{\phi \phi}$. Density correlations are dominated by $\eta(r)$ correlations. They display coordination peaks in position space reminiscent of radial distribution function of fluids \citep{HansenSimpleLiquids} and do not have any long distance behavior (Fig.\ref{fig_h_beta}). Orientational correlations are dominated by $\phi(r)$ correlations at large distances. They display a correlation length of upto $4$ lattice units in liquid phase (Fig.\ref{fig_h_beta}).
We also envisaged Coulomb interaction between bond arm charges via a new dual field that couples to the charges. We find that Coulomb interactions have little effect on the asymptotic behavior of orientational fluctuations and also on MMF results like equation of network and equation of state \citep{KanthPhysica2011}.
\section{Water confined between macroscopic surfaces} {\label{section_confinedwater}}
We now study the case of water confined between two macroscopic hydrophobic surfaces. As a result of confinement the structure of fluctuations in the system is restricted by the boundaries, thereby causing entropy to be a function of separation distance between surfaces. In analogy with Casimir interaction, the distance-dependent entropy component of free energy of confined water leads to an effective interaction between hydrophobic surfaces. In addition, due to hydrophobe-water interactions, orientational fluctuations are modified at interface of each surface. The modified fluctuations and their correlations lead to interfacial tension proportional to area of surface and an induced interaction between the surfaces. The net effect is an interaction force that acts over distances longer than typical hydration structure of water. We utilize MMF framework to analyze these effects in a unified fashion within the proposed water model.
We envisage surfaces in the $(x,y)$ plane of rectangular coordinate system; one present at $z = 0$ and other at $z = L$ (Fig.\ref{fig_confinedwater}). Each surface excludes water from its region of occupation. Hence, $W = 0$ on surface sites. On the immediate layer, i.e., at $z = 1$ or $z = L-1$ called \textit{interface} layer, water can be present and can take various orientations. For a hydrophobic surface if a non-zero bond arm of interface water is directed towards the surface, there would be a dangling bond on surface site; else a void state occurs. There can never be a hydrogen bond on surface i.e., $b \neq 2$ on surface. We will take care of these possibilities explicitly in our analysis. Consequently, we need not introduce $\eta$ and $\phi$ integrals (Eq.\ref{auxiliaryfields}) on the surface. Alternatively, we set $\eta = \phi = 0$ on surfaces.
The calculation of partition function begins with formulating the site functional $Z_{site}$ at each site, which comprises weights corresponding to each allowed state in the model. The site functional for all the sites in bulk region is of same form as given by Eq.(\ref{zsite}). On interface sites, weights corresponding to void state, dangling bond and hydrogen bond states remain unaltered. When a water molecule is present on a interface site its bond arms can orient in all possible ways. Only if one of the arms is towards the surface we assign a weight $\exp(\beta \tilde{\nu}_{S})$ to the corresponding orientation. For an ideal hydrophobic surface i.e., which is indifferent to bond arms of vicinal water, $\tilde{\nu}_{S} = 0$ (in general, $\tilde{\nu}_{S}$ can be positive or negative). Consequently, orientational weights for a water state on interface (with surface in $e_3$ direction) are given by :
\begin{align}
\nonumber & \left. C(\eta, \phi) \right|_{\textnormal{interface}} = \left( \sum^{'}_{\displaystyle \stackrel{\displaystyle \alpha \neq 3 \ H_{\alpha} = 0,\pm 1}{H_{3} = 0}} + \ \exp(\beta \tilde{\nu_{S}}) \sum^{'}_{\displaystyle \stackrel{\displaystyle \alpha \neq 3 \ H_{\alpha} = 0,\pm 1}{H_{3} = \pm 1}} \right)
\\ \nonumber & \qquad \qquad \qquad \qquad \qquad \qquad \exp\left[ i \sum_{\alpha} ( H^{2}_{\alpha}(r) {\eta}(r+e_{\alpha}) + H_{\alpha}(r) {\phi}(r+e_{\alpha}) ) \right]
\\ \nonumber & \ \ = \ C(\eta, \phi) + \nu_{S} \sum^{'}_{\displaystyle \stackrel{\displaystyle \alpha \neq 3 \ H_{\alpha} = 0,\pm 1}{H_{3} = \pm 1}} \exp\left[ i \sum_{\alpha} ( H^{2}_{\alpha}(r) {\eta}(r+e_{\alpha}) + H_{\alpha}(r) {\phi}(r+e_{\alpha}) ) \right]
\\ & \ \ \equiv \ C(\eta, \phi) + \nu_{S} C^{'}(\eta, \phi) \label{Caffected}
\end{align}
where, prime over $H_{\alpha}$ sum implies constraints Eqs.(\ref{H2constraint},\ref{Hconstraint}), $C^{'}(\eta, \phi)$ corresponds to affected orientations only i.e., those with $H_{3} = \pm 1$ and, $\nu_{S} \equiv \exp(\beta \tilde{\nu_{S}}) - 1$ is a function of surface-water interaction strength. The site functional $Z_{I}$ for any interfacial site can be arranged as :
\begin{equation}
Z_{I} \ = \ Z_{site} + \nu_{S} \mu C^{'}(\eta, \phi)
\label{interfaceaction}
\end{equation}
By definition, $\nu_{S}$ ranges from $-1$ to $\infty$. We remark that for a perfect hydrophobic surface, $\nu_{S} = 0$.
The modified site functional at interfacial sites can be recast in the expression for full partition function, such that the following decomposition is deduced:
\begin{align}
\nonumber Z_{\left| \right|} & \ = \ \int [{\cal D} \eta] [{\cal D}\phi] \prod_{r} Z_{site} \prod_{r_1 \in I_1} \left( 1 + \Gamma(r_1) \right) \prod_{r_2 \in I_2} \left( 1 + \Gamma(r_2) ) \right)
\\ & \ = \ Z \left\langle \exp\left( \sum_{r_1 \in I_1} \ln( 1 + \Gamma(r_1) ) + \sum_{r_2 \in I_2} \ln( 1 + \Gamma(r_2) ) \right) \right\rangle
\label{Zinterfaces}
\end{align}
where, $Z_{\left| \right|}$ is partition function for the system with surfaces; $Z$ is for corresponding unperturbed case ($\nu_{S} = 0$) with $\eta = \phi = 0$ on surfaces and $\Gamma(r)$ is defined only on interfacial sites. It is relative orientational weight of affected orientations with respect to $Z_{site}$, i.e.,
\begin{equation}
\Gamma(r) \ = \ \frac{ \nu_{S} \mu C^{'}(\eta, \phi) }{ Z_{site}(r) }
\label{affectedorientations}
\end{equation}
The partition function for unperturbed case $Z$ can be evaluated using MMF technique. The leading mean field energy is obtained from the maximum of $Z_{site}$ at each site and fluctuations in $\eta$, $\phi$ fields are analyzed subject to vanishing boundary conditions on the surfaces. The interfaces-dependent part in $Z_{\left| \right|}$ is evaluated using cluster technique and the corresponding free energy is obtained. The resulting form of total free energy $G_{tot}$ per unit lattice area is organized to be:
\begin{equation}
G_{tot} = G_{o} + G_{C} + \gamma_{S_1} + \gamma_{S_2} + G_{\Gamma}
\label{Gfparts}
\end{equation}
where, $G_{o} + G_{C}$ is the free energy obtained from evaluation of $Z$, analogous to Eq.(\ref{oneloop}). $G_{o}$ includes leading terms proportional to $L$ and constants obtained in large $L$ limit. They contribute only to bulk pressure of the system. $G_{C}$ is the remaining $L$-dependent part. $\gamma_{S_1}$, $\gamma_{S_2}$ are free energy contributions due to surface-water interaction and evaluated only on respective interface sites $I_1$ and $I_2$ respectively. $G_{\Gamma}$ constitutes terms which involve sites of both interfaces. Expression for each of the terms is deduced in the remaining section and their relevance to hydrophobic interaction is elucidated.
We first evaluate $Z$ using the MMF technique described in the previous section. We identify the maximum of the functional to be at $\eta = \phi = 0$. It yields mean field free energy, which to the leading order is given by $L G_{m}$ (Eq.\ref{equationofstate}). The dual fields are then expanded upto quadratic order about their maximum and the resulting Gaussian functional is integrated over all field configurations to give one-loop contribution to free energy. In the process, the following Fourier transform is employed which satisfies the boundary conditions :
\begin{equation}
\eta(\vec{r}) \ = \ \frac{2}{L} \ \sum_{n = 1}^{L-1} \ \int\limits_{-\pi}^{\pi} \frac{(dk_1) (dk_2) }{(2 \pi)^{2}} \ \tilde{\eta}(\vec{k}) \exp(i k_1 x + i k_2 y) \sin\left( {n \pi z}/{L} \right)
\label{FTcbc}
\end{equation}
where $\vec{r} = (x,y,z)$ is position vector and $\vec{k} = (k_1,k_2,k_3 = \frac{n \pi}{L})$ denote modes in momentum space. Similarly for $\phi$ field.
The entropy contribution to free energy for unperturbed system is discrete analog of that of bulk water (Eq.\ref{oneloop}), in that the integral over wavevector in $z$-direction is replaced by a summation over a restricted number of wavevectors i.e., $k_3 = \frac{\pi}{L}, \frac{2 \pi}{L}, \ldots, \frac{\pi(L-1)}{L}$. To analyze $L$-dependence, we define entropy contribution per unit area in each mode in $z$-direction as :
\begin{equation}
S(k_{3}) \ = \ \frac{1}{2} \int\limits_{-\pi}^{\pi} \frac{ (dk_1) (dk_2) }{(2 \pi)^{2}} \ln\left(P_{\eta \eta}(\vec{k}) P_{\phi \phi}(\vec{k}) \right)
\label{entropy_per_mode}
\end{equation}
where, the propagators $P_{\eta \eta}$, $P_{\phi \phi}$ are the same as in the case of bulk water. Total entropy contribution to free energy of confined water is $S(k_{3})$ summed over allowed values of $k_{3}$. Its large-$L$ behavior can be enumerated using Euler-Maclaurin series expansion \citep{Boas1971} :
\begin{equation}
\sum_{k_{3} = \frac{\pi}{L}}^{\frac{\pi}{L}(L-1)} S(k_{3}) \ = \ L \int\limits_{0}^{\pi} \frac{d{k_{3}}}{\pi} S(k_{3}) - \frac{1}{2} \left( S(0) + S(\pi) \right) + \beta G_{C}
\label{EMseries}
\end{equation}
On right hand side of Eq.(\ref{EMseries}), first term is total entropy contribution in the same volume of bulk water. $S(0)$ and $S(\pi)$ are free energy densities in modes $k_{3} = 0$ and $k_{3} = \pi$ respectively. They are independent of $L$. From Eq.(\ref{EMseries}) we infer $G_{C}$ to be the net difference in entropy contribution per unit area between confined water and bulk water in the same volume. $G_{C}$ can be calculated as a series expansion in $\frac{1}{L}$, the leading term being :
\begin{equation}
\beta G_{C} \ \simeq \ \frac{\pi}{B_{2} L} \left[ \left. \frac{\partial}{\partial k_{3}} S(k_{3}) \right|_{k_{3} = \pi} - \left. \frac{\partial}{\partial k_{3}} S(k_{3}) \right|_{k_{3} = 0} \right] \quad \textnormal{for large L}
\label{Gcasimir}
\end{equation}
where, $B_{2} = 2$ is first Bernoulli constant. $G_{C}$ is analogous to the Casimir interaction energy derived for the case of conducting plates confining electromagnetic fluctuations \citep{Casimir1948}. Hence, we call $G_{C}$ the \textit{Casimir part} of free energy. It falls-off asymptotically as $\frac{1}{L}$ for large $L$.
In the expression for partition function (Eq.\ref{Zinterfaces}) average over interface terms is now pursued. At each interfacial site, $\ln(1 + \Gamma(r)) \simeq \Gamma(r)$ is the leading order term. This is justified because in Eq.(\ref{affectedorientations}) for $\Gamma(r)$, we note that $\frac{\mu C^{'}(\eta,\phi)}{Z_{site}} \simeq \frac{\rho C^{'}(\eta,\phi)}{90}$ whose maximum value is always less than $1$, since $\rho < \frac{1}{3}$ and $\left| \frac{ C^{'}(\eta, \phi) }{90} \right| < \frac{2}{3}$, $C^{'}(0,0) = 60$. From Eq.(\ref{Zinterfaces}) the leading order contribution from interface terms is given by :
\begin{equation}
\frac{Z_{\left| \right|}}{Z} \ = \ \left\langle \exp\left( \sum_{r_{1} \in I_{1}} \Gamma(r_{1}) + \sum_{r_{2} \in I_{2}} \Gamma(r_{2}) \right) \right\rangle
\end{equation}
The average can be evaluated using cluster technique\footnote{If $A$ and $B$ are functions of a random variable whose probability distribution is known, the average $<\exp(A + B)>$ over the probability distribution is given by : $ \left\langle \exp(A + B) \right\rangle = \exp\left[ <A> + <B> + \frac{1}{2} (<A^{2}> - <A>^{2} + <B^{2}> - <B>^{2}) + <AB> - \right.$ $ \left. <A> <B> + \ldots \right] $ }. Terms that involve sites of same interface and those involving sites of both interfaces are segregated. $\gamma_{S}$ is defined to constitute terms corresponding to sites on same interface. Each of them is proportional to $\nu_{S}$ or its higher order. $\gamma_{S}$ is given to the leading order as :
\begin{equation}
{- \beta \gamma_{S} A}{} \ = \ \left[ \left\langle \sum_{r \in I} \Gamma(r) \right\rangle + \left\langle \sum_{\stackrel{r_{1}, r_{2} \in I}{r_{1} \neq r_{2}}} \Gamma(r_{1}) \Gamma(r_{2}) \right\rangle - \left\langle \sum_{r \in I} \Gamma(r) \right\rangle^{2} \right] \quad \label{interfacialtension}
\end{equation}
where, $A$ is area of the surface. $\gamma_{S}$ arises due to surface-water interaction and consequent effect on orientational fluctuations in the interfacial region.
Each of the averages in Eq.(\ref{interfacialtension}) can be evaluated using a functional integration relation\footnote{ If $\phi$ is a random field whose action is known and when a constant external field $J$ couples to $\phi$ such that their interaction is $i J \phi(r)$, then $< \exp(i J (\phi(r_{1}) + \phi(r_{2})) ) >$ $= \exp[- \frac{1}{2} J^{2} ( < \phi(r_{1}) \phi(r_{1}) > + <\phi(r_{2}) \phi(r_{2}) + 2 < \phi(r_{1}) \phi(r_{2}) > ) + \ldots ]$. If two-point correlation is the leading order, then the subsequent terms of higher order denoted by $(\ldots)$ can be ignored}. For an interface site with surface in $e_{3}$ direction, using Eqs.(\ref{Caffected},\ref{affectedorientations}) $\left\langle \Gamma(r) \right\rangle $ is given to the leading order as :
\begin{align}
\nonumber \left\langle \Gamma(r) \right\rangle \ = \ \left( \frac{\nu_{S} \rho}{90} \right) \sum^{'}_{\displaystyle \stackrel{\displaystyle \alpha \neq 3 \ H_{\alpha} = 0,\pm 1}{H_{3} = \pm 1}} \exp & \left[ \sum_{\alpha, \alpha^{'}} \left( H^{2}_{\alpha}(r) H^{2}_{\alpha^{'}}(r) {\cal G}_{\eta}(r+e_{\alpha}, r+e_{\alpha^{'}}) \right. \right.
\\ & \left. \left. + \ H_{\alpha}(r) H_{\alpha^{'}}(r) {\cal G}_{\phi}(r+e_{\alpha}, r+e_{\alpha^{'}}) \vphantom{H^{2}_{\alpha^{'}}} \right) \vphantom{\sum_{\alpha}} \right] \label{average_Gamma}
\end{align}
where, the $H_{\alpha}$ summation is over affected orientations at site $r$. The prime over summation indicates $H_{\alpha}$'s of each orientation satisfy Eqs.(\ref{H2constraint},\ref{Hconstraint}). The exponential in Eq.(\ref{average_Gamma}) corresponds to one such orientation. $H_{\alpha}$, $H_{\alpha^{'}}$ are bond arms of the same orientation; $r + e_{\alpha}$, $r + e_{\alpha^{'}}$ are the bond arm locations. The average $\left\langle \Gamma(r_{1}) \Gamma(r_{2}) \right\rangle$ is given to leading order as :
\begin{align}
\nonumber & \left\langle \Gamma(r_{1}) \Gamma(r_{2}) \right\rangle \ = \ \displaystyle \left( \frac{\nu_{S} \rho}{90} \right)^{2}
\sum^{'}_{\displaystyle \stackrel{\displaystyle \alpha \neq 3 \ H_{\alpha} = 0,\pm 1}{H_{3} = \pm 1}} \ \ \sum^{'}_{\displaystyle \stackrel{\displaystyle \kappa \neq 3 \ H_{\kappa} = 0,\pm 1}{H_{3} = \pm 1}} \exp \left[ \vphantom{\sum_{\alpha^{'}}} \right.
\\ \nonumber & \left. \displaystyle \sum_{\alpha, \alpha^{'}} \left( H^{2}_{\alpha}(r_{1}) H^{2}_{\alpha^{'}}(r_{1}) {\cal G}_{\eta}(r_{1} + e_{\alpha}, r_{1} + e_{\alpha^{'}}) + H_{\alpha}(r_{1}) H_{\alpha^{'}}(r_{1}) {\cal G}_{\phi}(r_{1} + e_{\alpha}, r_{1} + e_{\alpha^{'}}) \right) \right.
\\ \nonumber & \left. \displaystyle + \sum_{\kappa, \kappa^{'}} \left( H^{2}_{\kappa}(r_{2}) H^{2}_{\kappa^{'}}(r_{2}) {\cal G}_{\eta}(r_{2} + e_{\kappa}, r_{2} + e_{\kappa^{'}}) + H_{\kappa}(r_{2}) H_{\kappa^{'}}(r_{2}) {\cal G}_{\phi}(r_{2} + e_{\kappa}, r_{2} + e_{\kappa^{'}}) \right) \right.
\\ & \left. \displaystyle + \sum_{\alpha, \kappa} \left( H^{2}_{\alpha}(r_{1}) H^{2}_{\kappa}(r_{2}) {\cal G}_{\eta}(r_{1} + e_{\alpha}, r_{2} + e_{\kappa}) + H_{\alpha}(r_{1}) H_{\kappa}(r_{2}) {\cal G}_{\phi}(r_{1} + e_{\alpha}, r_{2} + e_{\kappa}) \right) \vphantom{\sum_{\alpha^{'}}} \right] \label{average_GammaGamma}
\end{align}
where, $H_{\alpha}$, $H_{\alpha^{'}}$ are bond arms of an affected orientation at site $r_{1}$ and $H_{\kappa}$, $H_{\kappa^{'}}$ are those of an orientation at site $r_{2}$. The exponential corresponds to product of the two orientations and the summation is over all possible products. The two-point Green's function ${\cal G}_{\eta}(r_{1},r_{2})$ for $\eta$-field fluctuations between any two arbitrary sites $r_{1} = (x_{1},y_{1},z_{1})$ and $r_{2} = (x_{2},y_{2},z_{2})$ is given by,
\begin{align}
{\cal G}_{\eta}(r_{1}, r_{2}) \ = \ \frac{2}{L} \ \sum_{n = 1}^{L-1} \ \int\limits_{-\pi}^{\pi} \frac{ (dk_{1})(dk_{2}) }{(2 \pi)^{2}} \exp\left(i k_{1} (x_{1} - x_{2}) + i k_{2} (y_{1} - y_{2}) \right) \frac{\displaystyle \sin\left( {n \pi z_{1}}/{L}\right) \sin\left( {n \pi z_{2}}/{L}\right) }{ P_{\eta \eta}(\vec{k}) } \label{Greensfunction}
\end{align}
Similarly, ${\cal G}_{\phi}(r_{1}, r_{2})$ for $\phi$ field can be defined using the propagator $P_{\phi \phi}(\vec{k})$.
The expression for $\gamma_{S}$ indicates that it varies with separation distance, owing to the $L$-dependent Green's functions. The asymptotic value of $\gamma_{S}$ is the interfacial tension for hydrophobic surface in contact with water. The leading correction term is proportional to $\frac{1}{L}$ for large-$L$ and contributes to force between the surfaces.
From the cluster expansion of partition function, terms that involve sites of both interfaces are grouped as $G_{\Gamma}$. It is given to the leading order as :
\begin{equation}
{- \beta G_{\Gamma} A } \ = \ \left[ \left\langle \sum_{r_{1} \in I_{1}} \Gamma(r_{1}) \sum_{r_{2} \in I_{2}} \Gamma(r_{2}) \right\rangle - \left\langle \sum_{r_{1} \in I_{1}} \Gamma(r_{1}) \right\rangle \left\langle \sum_{r_{2} \in I_{2}} \Gamma(r_{2}) \right\rangle \right] \label{IIFE}
\end{equation}
Effectively, $G_{\Gamma}$ is connected correlation between orientational fluctuations of both interfaces. Hence, we call this contribution \textit{interfacial fluctuations-induced part} of free energy. The averages in Eq.(\ref{IIFE}) can be evaluated using Eq.(\ref{average_Gamma}) with $\nu_{S}$ corresponding to each interface and using Eq.(\ref{average_GammaGamma}) with proportionality factor $(\nu_{S_{1}} \nu_{S_{2}})$ instead of $(\nu_{S})^{2}$. The identity of sites is as per given in the expression for $G_{\Gamma}$ (Eq.\ref{IIFE}).
The long distance behavior of $G_{\Gamma}$ is dominated by $\phi(r)$ correlations, $\eta(r)$ being short ranged. Between two hydrophobic surfaces, to the leading order $G_{\Gamma}$ is proportional to square of orientational correlations i.e., $({\cal G}_{\phi}(r))^{2}$, where ${\cal G}_{\phi}(r)$ is an exponentially falling-off function for large $r$ (Appendix \ref{app_correlationfunctions}).
For the case of mesoscopic surfaces hydrophobic force is suggested to arise from orientational correlations between water molecules at both interfaces \citep{KanthPRE2010}. The force is seen to decay exponentially with separation distance, asymptotically. $G_{\Gamma}$ is thus analogous to hydrophobic interaction free energy of mesoscopic surfaces. However for macroscopic surfaces, in addition to $G_{\Gamma}$, hydrophobic force obtains contributions from Casimir part and interfacial tension. This distinguishes hydrophobic interaction between large surfaces from that of between small surfaces both qualitatively and quantitatively. The non-additive nature of hydrophobic interaction with increasing size of surfaces has attracted considerable attention \citep{ChandlerNature2005,*AshbaughRMP2006} and our work provides a direction to elucidate the size dependence in terms of hydrogen bond fluctuations in water.
\subsection{Hydrophilic surfaces} {\label{section_hydrophilicsurfaces}}
We can envisage surfaces of generic heterogeniety in our calculation. The heterogeniety could be in terms of space-dependent $\nu_{S}$ and/or charge on surface. One of the simplest cases is a homogeneous hydrophilic surface with a fixed charge on each site. We first consider the case of a positively charged hydrophilic surface. On its interface, the site functional comprises weights corresponding to all states. When a water molecule is present on interface, its hydrogen arm is restricted from pointing in surface direction. We assign an energetic penalty to such orientations and the site functional can be arranged, analogous to the case of a hydrophobic surface, as :
\begin{equation*}
Z_{I} \ = \ Z_{site} + \nu_{S} \mu C^{'}(\eta, \phi)
\end{equation*}
Here, $\nu_{S} \in (-1,0)$ (ideally, $\nu_{S} = -1$) and the orientational weights corresponding to affected orientations $C^{'}(\eta, \phi)$ are given by :
\begin{equation}
C^{'}_{+}(\eta, \phi) = \sum^{'}_{\displaystyle \stackrel{\displaystyle \alpha \neq 3 \ H_{\alpha} = 0,\pm 1}{H_{3} = 1}} \exp\left[ i \sum_{\alpha} ( H^{2}_{\alpha}(r) {\eta}(r+e_{\alpha}) + H_{\alpha}(r) {\phi}(r+e_{\alpha}) ) \right]
\label{Caffected_pos_hydrophilic}
\end{equation}
The above expression is for an interface site with surface in $e_{3}$ direction. A negatively charged hydrophilic surface can also be envisaged such that for interface water orientations with lone-pair arm in surface direction are energetically penalized. Here, the weights for affected orientations are :
\begin{equation}
C^{'}_{-}(\eta, \phi) = \sum^{'}_{\displaystyle \stackrel{\displaystyle \alpha \neq 3 \ H_{\alpha} = 0,\pm 1}{H_{3} = -1}} \exp\left[ i \sum_{\alpha} ( H^{2}_{\alpha}(r) {\eta}(r+e_{\alpha}) + H_{\alpha}(r) {\phi}(r+e_{\alpha}) ) \right]
\label{Caffected_neg_hydrophilic}
\end{equation}
We now compute the free energy components $G_{C}$, $\gamma_{S_{1}}$, $\gamma_{S_{2}}$, $G_{\Gamma}$ using their respective expressions for different types of surfaces. $\nu_{S}$ is an arbitrary parameter in the calculation. It is chosen close to its ideal value for each surface type. The properties of water enter the computation via Green's functions ${\cal G}_{\eta}$, ${\cal G}_{\phi}$. These are computed within the model using Eq.(\ref{Greensfunction}). Due to $L$-dependent modes in the confined direction, all the free energy components that depend on fluctuations are expected to vary with separation distance $L$.
\section{Results : Hydrophobic force, interfacial tension} {\label{section_hydrophobicforce}}
We first mention that this computation is totally parameter-free on lattice. Hence, the best way to interpret results is in terms of physically observable quantities such as $\rho$ and hydrogen bond density. Indeed because of equation of network (Eq.\ref{equationofnetwork}) only one of them is independent. We find that it is best to describe in terms of $h = \frac{2 \textnormal{HB}}{\rho}$, the average number of hydrogen bonds per molecule. Temperature is conjugate to HB (total hydrogen bond density) and hence it is also implicitly fixed self-consistently due to equation of network, as shown in Fig.(\ref{fig_h_beta}). The relation between $\rho$ and $h$ is simple at zeroth order in MMF theory (Eq.\ref{equationofnetwork_h_rho}), but it becomes non-linear at one-loop level. (Zeroth order is still a reasonable approximation \citep{KanthPhysica2011}.) Hence, all densities DB, HB, $\rho$ and the free energy components given by Eqs.(\ref{EMseries},\ref{interfacialtension},\ref{IIFE}) are evaluated from partition function upto one-loop order using the corresponding expressions for propagators (Appendix \ref{app_correlationfunctions}).
In our model MMF theory describes liquid for $h > 3$ reasonably consistently. For $2 < h < 3$ MMF approximation is not seen to be good, namely, one-loop order terms are either comparable or exceed zero-loop term. So, we choose to present our results for $h > 3$. All the potentials and energies are computed in the units of hydrogen bond strength $\tilde{\lambda}$ taken to be unity. The lattice constant in the model is arbitrary. By computing physical lengthscales such as correlation length it can be fixed. Correlation lengths for density ($\xi_{\eta}$) and orientational fluctuations ($\xi_{\phi}$) to the leading order are simple expressions given in Appendix (\ref{app_correlationfunctions}), but a precise expression to one-loop order is implicitly given. In Fig.(\ref{fig_h_beta}) we plot correlation lengths as a function of $h$. $\xi_{\eta}$ is only about one lattice unit in liquid phase and does not vary considerably with $h$, while $\xi_{\phi}$ increases with $h$. In MD simulation density correlation length is not seen; this is consistent with MMF result since $\xi_{\eta}$ is equal to the minimum length possible in the model and also independent of $h$. Orientational correlation lengths inferred from MD simulation are $5.2$ \AA{} and $24$ \AA{}, of which the latter is weaker in strength (one-tenth) relative to the shorter one \citep{KanthPRE2010}. In our water model we have only one orientational correlation length $\xi_{\phi}$ which we relate to $5.2$ \AA{}. For liquid water $h$ value is suggested to be about $3.6$ \citep{MahoneyJCP2000Vol112}. From Fig.(\ref{fig_h_beta}) $h = 3.58$ corresponds to $\xi_{\phi} \simeq 3.3$ lattice units. Consequently, we infer that $1$ lattice unit $\simeq \frac{5.2}{3.3} = 1.57$ \AA{}.
In Fig.(\ref{fig_Gall_zz}) various contributions to interaction free energy and their relative magnitudes are plotted as a function of separation distance $L$ between surfaces. The plot is presented for $h = 3.58$. Casimir part $G_{C}$ gives the most attractive force, followed by $G_{\Gamma}$, while the interfacial term $\gamma_{S}$ is repulsive, albeit very small. $G_{C}$, $\gamma_{S}$ fall-off as $\frac{1}{L}$ for large $L$ from our analytic calculations. Numerically, beyond $15$ lattice units they are insignificant. All the plots are presented for lattice distance $L \geq 5$. For smaller $L$ the results are predominantly influenced by surface effects. In the model, for $L = 4$ there is only one layer which can have free orientations (besides two interface layers), while for $L \geq 5$ there are two or more such layers.
Force is computed as discrete derivative of total free energy with respect to $L$ and plotted in Fig.(\ref{fig_Force_zz}) for various $h$. The curves effectively show that the force can manifest upto a length of about $15$ lattice units which translates to about four times the orientational correlation length in the model. All the free energy components and force obtain major contributions from orientational fluctuations.
Figs.(\ref{fig_Gcasimir},\ref{fig_Git_z},\ref{fig_Gf_zz}) display the $h$-dependence of $G_{C}$, $\gamma_{S}$ and $G_{\Gamma}$ functions. The Casimir part $G_{C}$ monotonically increases in magnitude with $h$. The interfacial tension contribution $\gamma_{S}$ decreases and is always slightly repulsive. Interfacial fluctuations-induced part $G_{\Gamma}$ increases with $h$ for $L \geq 6$. At shorter distances it decreases with increasing $h$. This indicates that the adhesion strength of $G_{\Gamma}$ component is higher for higher temperatures. This behavior is qualitatively similar to the temperature dependence of interaction free energy for mesoscopic hydrophobic surfaces \citep{Frank1945,WidomPCCP2003}. This reaffirms our interpretation that $G_{\Gamma}$ component is analogous to hydrophobic interaction free energy for mesoscopic surfaces.
Fig.(\ref{fig_Gf_pp_pn}) is the plot for $G_{\Gamma}$ contribution between two hydrophilic surfaces, both of same type (hydrogen donor/acceptor) and of dissimilar type. $G_{\Gamma}$ in this case is proportional to ${\cal G}_{\phi}$ and hence, correlation length is twice as longer in range than in the case of hydrophobic surfaces (where $G_{\Gamma}$ is proportional to $({\cal G}_{\phi})^{2}$). At short distances it is seen to be attractive for both combinations. However, for large distances it is weakly repulsive between like-charged surfaces, in contrast to attraction between oppositely charged surfaces. Fig.(\ref{fig_Force_pp_pn}) depicts the force between hydrophilic surfaces for both similar and dissimilar combinations. As expected, the dissimilar pair of surfaces have marginally larger attraction than that of similar surfaces. It is interesting to note that like-charged hydrophilic surfaces also have a net attraction. This is due to dominance of Casimir part $G_{C}$ which is indifferent to surface charge.
Fig.(\ref{fig_Force_pz}) displays force between a hydrophobic and hydrophilic surface. It bears similar profile as in the case of two hydrophobic surfaces. This is expected because essentially $G_{\Gamma}$ is qualitatively same for both cases i.e., proportional to $({\cal G}_{\phi})^{2}$. For all surface combinations the force is seen to increase in magnitude with $h$, dominantly due to indifference of Casimir part to surface types. This is a consequence of the fact that the entropy induced forces are largely charge neutral.
Next, we make an attempt to relate our computational results to those of experiments. The free energy values presented in the graphs are in the units where hydrogen bond strength is unity. Generally, dimensionful quantities in lattice models and those in corresponding continuum models are not the same. So it is best to compare dimensionless quantities. In our instance, for $h = 3.58$ and $L = 6$ lattice units which translates to $6 \times 1.57 \textnormal{\AA{}} \simeq 9.5 \textnormal{\AA{}}$, $\displaystyle \frac{ |G_{tot}(6) - G_{tot}(\infty) | }{ \gamma_{S}(\infty) } \simeq \frac{9 \times 10^{-5}}{8.5 \times 10^{-3}} \simeq 10^{-2}$. From experiments interaction free energy estimate when two hydrophobic plates are about $10$ \AA{} apart is about $1 \ \textnormal{mJ} \ \textnormal{m}^{-2}$ \citep{Hammer2010}, while interfacial tension is in the range $50 - 100 \ \textnormal{mJ} \ \textnormal{m}^{-2}$ \citep{Goebel1997}; their ratio agrees with our computation. In experiments the free energy values are also measured for larger distances all the way up to $100$ \AA{}. Unfortunately our model is not good for these distances. This discrepancy was already noticed when our results were compared with MD simulation. The simple water model has only one orientational correlation length, while there are more than one in both MD simulations \citep{KanthPhysica2011} and surface force apparatus experiments \citep{Claesson2001}. We conclude that while order of magnitude estimate of the strength of hydrophobic force is in agreement with Casimir-like energies envisaged here, a few more important details are perhaps missing in our simple model of water.
\section{Transverse density profile} {\label{section_densityprofile}}
We also deduce expression for water density profile along the confinement direction $z$. $\rho(z)$ is obtained by assuming chemical potential of water $\tilde{\mu}$ to be $z$-dependent and then, a partial derivative of $\ln(Z_{||})$ is taken with respect to $\beta \tilde{\mu}(z)$. At both interfaces i.e., $z = 1$ and $z = L-1$, the modified fugacity provides additional correction to average density. The expression for density profile is given by :
\begin{equation}
\rho(z) \ \equiv \ \frac{ \partial (\ln (Z_{||})) }{ \partial (\beta \tilde{\mu}(z)) }
\ = \ \rho_{C}(z) + \frac{1}{A} \left\langle \frac{ \partial }{ \partial (\beta \tilde{\mu}(z)) } \left( \sum_{r_{1} \in I_{1}} \Gamma(r_{1}) + \sum_{r_{2} \in I_{2}} \Gamma(r_{2}) \right) \right\rangle + \ldots
\label{rhoz}
\end{equation}
$\rho_{C}$ is obtained from differentiating $Z$ in Eq.(\ref{Zinterfaces}). It is the density profile between ideal hydrophobic surfaces ($\nu_{S} = 0$) and is the dominant contribution at all positions. The explicit expression for $\rho_{C}(z)$ upto one-loop order is given in the Appendix (\ref{app_densityprofile}). The interfaces-dependent term in Eq.(\ref{rhoz}) can be analyzed using Eq.(\ref{average_Gamma}). This contribution is only at $z =1$ and $z = L-1$.
The transverse density profile is shown in Fig.(\ref{fig_densityprofile}) after scaling $\rho(z)$ with respect to bulk density value. At both interfaces there is a characterstic rise in density. From expressions of $\rho_{C}(z)$ (Appendix \ref{app_densityprofile}) and interface terms (Eq.\ref{average_Gamma}) it is evident that net contribution of $\phi$ field correlations is numerically small since density is charge-neutral quantity and linear $\phi$-dependent terms tend to cancel each other. Hence, away from interfaces density reaches bulk density value rapidly within a distance $\xi_{\eta}$. Many a model simulations in the past computed the transverse density profile for water confined between model hydrophobic surfaces. The short distance density increase is generically observed \citep{Paschek2001,*PradeepPRE2005,GardePNAS2009}. At ambient conditions the magnitude of interfacial density is seen to be typically $1.3$ times bulk density value near surfaces with alkane headgroups \citep{GardePNAS2009} and independent of $L$. In our model study we see an $L$-independent increase of magnitude $1.2$ for an ideal hydrophobic surface. The under-estimation could possibly be due to discrete orientational freedom envisaged in our model. Also, alkane head-groups in simulations may have an extra entropy due to fluctuating short length polymer chains.
The rise in interfacial density is also seen for water in the vicinity of hydrophilic surfaces \citep{GardePNAS2009,BerkowitzJCP2006}. In our model study $\rho(z)$ between hydrophilic surfaces also displays qualitatively similar profile and a lower magnitude of interfacial density compared to that near an ideal hydrophobic surface. In all cases the phenomenon is seen to be a consequence of the fact that water density has to vanish on the surface. This is compensated by an increase at the interface and the system comes back to its bulk equilibrium density within a distance $\xi_{\eta}$ from the interface.
We also calculate density correlations within the interfacial plane and between sites on interface and away from interface. Density correlations between any two sites $r$ and $r^{'}$ can be calculated from :
\begin{equation}
\langle W(r) W(r^{'}) \rangle \ = \ \left\langle \frac{\mu \lbrace \ldots \rbrace}{Z_{site}(r)} \frac{\mu \lbrace \ldots \rbrace}{Z_{site}(r^{'})} \right\rangle
\end{equation}
where, $Z_{site}(r)$ is site functional at $r$. To compute density correlations on same interface, the site functional at both sites is given by Eq.(\ref{interfaceaction}). For density correlations between a site on interface and another, away from interface, the site functionals are given by Eqs.(\ref{interfaceaction}, \ref{zsite}) respectively. $\mu \lbrace \ldots \rbrace$ refers to the term proportional to $\mu$ in the respective site functional. The connected part of the correlation is given by $<W(r) W(r^{'})>_{c} \ \equiv ( <W(r) W(r^{'})> - <W(r)> <W(r^{'})> )$. The explicit expression in each context is deduced upto one-loop order in terms of ${\cal G}_{\eta}$, ${\cal G}_{\phi}$ and are given in Appendix (\ref{app_interface_bulk_correlations}).
Density correlations scaled appropriately with respect to bulk density value are plotted in Fig.(\ref{fig_ww}). The plot corresponds to $h = 3.58$. The figure essentially indicates density correlations do not extend beyond few molecular diameters from the interface. Also, there is no significant difference between correlations within an interface and that of between interface and non-interface sites.
Similarly, orientational correlations can also be analyzed using the expressions for orientational weights given in Appendix (\ref{app_orientationalweight}). Their effect persists upto longer distance away from interface, proportional to the long correlation length of $\phi$ field.
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\section{Introduction}
\label{sec:introduction}
A special class of finite-dimensional modules of the derived subalgebra of the Drinfel'd--Jimbo quantum group $U_q'(\asl_n)$ called Kirillov--Reshetikhin (KR) modules have received significant attention over the past 20 years. KR modules have many remarkable properties and deep connections with mathematical physics. For example, KR modules arise in the study of certain solvable lattice models~\cite{BBB16, JM95, KP84}. Their characters (resp. $q$-characters~\cite{FM01, FR99}) satisfy the Q-system (resp. T-system) relations, which come from a certain cluster algebra~\cite{dFK09, Hernandez10, Nakajima03II}. This gives a fermionic formula interpretation and a relation to the string hypothesis in the Bethe ansatz for solving Heisenberg spin chains. The graded characters of (resp. Demazure submodules of) tensor products of certain KR modules, the fundamental representations, are also (resp. nonsymmetric) Macdonald polynomials at $t = 0$~\cite{LNSSS14, LNSSS14II} (resp.~\cite{LNSSS15}).
In the seminal papers~\cite{K90, K91}, Kashiwara defined the crystal basis of a representation of a quantum group, which is a basis that is well-behaved in the $q \to 0$ limit and affords a combinatorial description. Furthermore, he showed every irreducible highest weight representation admits a crystal basis $B(\lambda)$. While KR modules are cyclic modules, they are not highest weight modules. Yet, KR modules for $U_q'(\asl_n)$ admit crystal bases~\cite{KKMMNN92} (conjecturally for all affine types~\cite{HKOTY99,HKOTT02}, which is known for non-exceptional types~\cite{OS12} and some other special cases~\cite{JS10, KMOY07, Yamane98}), which are known as Kirillov--Reshetikhin (KR) crystals, and contain even further connections to mathematical physics. For example, KR crystals are in bijection with combinatorial objects that arise naturally from the Bethe ansatz called rigged configurations~\cite{KKR86, KR86, KSS02, DS06}. KR crystals $B^{1,s}$ can be used to model the Takahashi--Satsuma box-ball system~\cite{TS90}, where rigged configurations are invariants called action-angle variables~\cite{KOSTY06, Takagi05}. They are also perfect crystals~\cite{FOS10}, and therefore, they can be used to construct the Kyoto path model~\cite{KKMMNN91,KKMMNN92,OSS03IV}, which came from the study of 2D solvable lattice models and Baxter's corner transfer matrix~\cite{B89}.
Despite intense study, relatively little is understood about KR crystals. In particular, there is currently not a combinatorial model for KR crystals where all crystal operators are given by the same rules, the model is valid for general $B^{r,s}$, and the model is given uniformly across all affine types. By using the decomposition into $U_q(\fsl_n)$-crystals and the Dynkin diagram automorphism, we can lift the tableaux model of~\cite{KN94} to a model for KR crystals for $U_q'(\asl_n)$~\cite{Shimozono02}. However, this process obscures the affine crystal operators as it uses the promotion operator of Sch\"utzenberger~\cite{Sch72}, and so it is desirable to have a model where all of the crystal operators are given by the same rules. A similar procedure was utilized in~\cite{FOS09, JS10}, but using type-dependent information and it fails for type $E_8^{(1)}$ due to the Dynkin diagram not admitting any non-trivial automorphisms.
Partial progress has been made on this problem. Naito and Sagaki constructed a model uniform across all types for tensor products of $B^{r,1}$ by using the usual crystal structure on Lakshmibai--Seshadri (LS) paths for level-zero representations and projecting onto the classical weight space~\cite{NS03, NS05, NS06, NS06II, NS08II}. There is another description of these paths called quantum LS paths~\cite{LNSSS14, LNSSS16}. Lenart and Lubovsky performed a similar construction using a discrete version of quantum LS paths called the quantum alcove path model~\cite{LL15}. Yet, it is not known how to extend these models for general $B^{r,s}$. On the other side, models for $B^{r,s}$ were constructed in~\cite{Kus13, Kus16, Kwon13} for type $A_n^{(1)}$, but these are not known to extend (uniformly) to other affine types.
There is a $t$-analog of $q$-characters (or $q,t$-characters for short) that was studied by Nakajima~\cite{Nakajima01, Nakajima03II, Nakajima03, Nakajima04, Nakajima10}. From this study, Nakajima gave a $U_q(\fsl_n)$-crystal structure on the monomials that appear in the $q$-character~\cite{Nakajima03}. Based on this model, Kashiwara~\cite{K03II} independently constructed a different crystal structure on the $q$-character monomials. These two crystal structures were later simultaneously generalized by Sam and Tingley~\cite{ST14}, where a connection to quiver varieties was also made. This is known as the Nakajima monomial model. We note that Kashiwara's crystal structure works for $U_q(\asl_n)$-crystals $B(\lambda)$, but Nakajima's is only valid when $n$ is odd.
For an extremal level-zero crystal $B(\lambda)$ (so $\lambda$ is a level-zero weight), there exists an automorphism $\kappa$ such that $B(\lambda) / \kappa \cong \bigotimes_{i=1}^N B^{r_i, 1}$ as $U_q'(\asl_n)$-crystals, where $\lambda = \sum_{i=1}^N \Lambda_{r_i}$. This was the construction of Naito and Sagaki previously mentioned, where the description was given explicitly in terms of LS paths. A similar construction was given for Nakajima monomials by Hernandez and Nakajima~\cite{HN06}.
Nakajima's $q,t$-characters have also been well-studied using a variety of techniques. While their definition is combinatorial, Nakajima used quiver varieties to show their existence in simply-laced types~\cite{Nakajima04}. Hernandez reformulated the definition to be purely algebraic by using a $t$-analog of screening operators~\cite{Hernandez04}. Nakajima also showed that $q,t$-characters can be used to determine the change of basis from standard to simple $U_q(\asl_n)$-modules in the Grothendieck group~\cite{Nakajima01}. From this, Kodera and Naoi showed that the graded decomposition of a standard modules in terms of simple modules categorifies the graded decomposition into $U_q(\fsl_n)$-modules of a tensor product of fundamental representations~\cite{KN12}. The graded decomposition polynomials in this case are also Kostka polynomials~\cite{KSS02} and are a specialization of Macdonald polynomials at $t=0$. The $q,t$-characters are also related to the natural Jordan filtration of $\ell$-weight spaces by the Heisenberg subalgebra of $U_q(\asl_n)$~\cite{Zegers15}. Furthermore, Qin used a slight variation of $q,t$-characters to realize the generic basis, the dual PBW basis, and the canonical basis of a quantum cluster algebra~\cite{Qin14}.
Cluster algebras~\cite{FZ02} also have strong connections to characters of KR crystals and Nakajima monomials. Hernandez and Leclerc gave an algorithm to compute $q$-characters as cluster variables of a cluster algebra from a certain semi-infinite quiver~\cite{HL16}. For a double Bruhat cell $G^{u,v} = BuB \cap B_- v B_-$, the coordinate ring $\mathbb{C}[G^{u,v}]$ is an upper cluster algebra whose generalized minors are the cluster variables~\cite{BFZ05}. Kanakubo and Nakashima showed that the generalized minors of $G^{u,e}$ can be expressed as the sum over the Nakajima monomials in a Demazure subcrystal~\cite{KN15}. A further connection between cluster variables in $\mathbb{C}[G^{u,u^{-1}}]$, with $u$ a Coxeter element, and representation theory was given by Rupel, Stella, and Williams in~\cite{RSW16}. In particular, they show the regular cluster variables in the coordinate ring of the loop group of $SL_n$ are restrictions of generalized minors of level-zero representations. The specialization of nonsymmetric Macdonald polynomials at $t = \infty$ can also be described as a Demazure submodule of a tensor product of fundamental representations~\cite{NNS16}, which satisfy the quantum Q-system relations of~\cite{dFK15, dFK16}.
All of this suggests that there should exist a natural description of tensor products of KR crystals in terms of Nakajima monomials. Indeed, $q,t$-characters are given as the sum over Nakajima monomials graded by energy, which admit a classical crystal structure. Therefore, it is evidence that the entire $q,t$-character could be constructed by adding $0$-arrows to get a tensor product of KR crystals.
The main result of this paper is a model for the KR crystal $B^{1,s}$ in type $A_n^{(1)}$ using Nakajima monomials. From this construction, we are able to describe the tensor product of KR crystals $\bigotimes_{i=1}^N B^{1,s_i}$ using only Nakajima monomials ({\it i.e.}, no tensor products). Furthermore, we are able to recover the Kyoto path model. From this construction, we are able to relate the models of~\cite{ST14, Tingley08} with the Kyoto path model. We also extend our construction to give a Nakajima monomial description of the coherent limit $B_{\infty}$, where we recover the analog of the Kyoto path model for $B(\infty)$ and the characterization of $B(\infty)$ given in~\cite[Thm.~5.1]{KKS07}.
The results of our paper suggest a crystal interpretation for the fusion construction of~\cite{KKMMNN91, KKMMNN92}. Indeed, the kernel of the $R$-matrix is approximately given by a commutator relation on the elements of $B^{1,1}$, which are uniquely determined by the variables $X_{i,k}$. By considering the tensor product as multiplication, we can relate our construction with the kernel of the $R$-matrix. Furthermore, our construction gives an explanation of the link between the Nakajima monomial model, the abacus model, multipartition model, and quiver varieties that was explored in~\cite{ST14, Tingley08, Tingley10}. While our model does not naturally extend to general $B^{r,s}$ or to other affine types, there is evidence that our construction can be modified to the general case.
We now give one potential application of our results. To do so, we recall that geometric crystals were introduced by Berenstein and Kazhdan~\cite{BK00, BK07}, where the $U_q(\fsl_n)$-crystal structure of $B(\infty)$ is lifted to actions on algebraic varieties. This was generalized to a lifting of certain $U_q(\asl_n)$-crystals using Schubert varieties~\cite{Nakashima05}. Nakashima has lifted Nakajima monomials to describe the decoration function of geometric crystals (specifically as generalized minors) in~\cite{Nakashima14} and made a connection to the polyhedral model in~\cite{Nakashima13}. There is also a lifting of the $U_q'(\asl_n)$-crystal $B_{\infty}$ to the geometric setting given in~\cite{KNO08} and the coherent limits of $\{ B^{r,s} \}_{s=1}^{\infty}$ in~\cite{MN16}. The geometric $R$-matrix has also been studied~\cite{KNO10, Yamada01}, which was then used to relate a quotient of the liftings of $B_{\infty}$ to the unipotent loop group in~\cite{LP11}.
We expect that our results can be lifted to a statement on geometric crystals, connecting the results of Nakashima with the work on geometric analogs of $B_{\infty}$. Furthermore, we believe our results could be used to construct a geometric lifting of the path model embeddings and give a geometric lifting of highest weight $U_q(\asl_n)$-crystals. We also believe that our results could give a connection to the cluster algebra (geometric) $R$-matrices that were recently introduced in~\cite{ILP16}.
We also consider our results as evidence of a deep connection between cluster algebras, $q,t$-characters, and KR crystals. Indeed, KR modules could be considered as loop group representations and are closely related to level-zero representations as mentioned above. Furthermore, after removing certain $0$-arrows, KR crystals are Demazure subcrystals of affine highest weight crystals~\cite{ST12}. Therefore, we believe that the Nakajima monomials appearing in a realization of general KR crystals to give a connection between the work of~\cite{HL16, KN15, Nakashima05, Nakashima14, RSW16}.
This paper is organized as follows. In Section~\ref{sec:background}, we give a background on crystals, KR crystals, Nakajima monomials, and the Kyoto path model. In Section~\ref{sec:monomial_model}, we construct a model for $B^{1,s}$ using Nakajima monomials. In Section~\ref{sec:Kyoto_path}, we give a method to construct tensor products of KR crystals as a map on Nakajima monomials and relate our construction to the Kyoto path model. In Section~\ref{sec:coherent_limit}, we describe the crystal corresponding to the coherent limit of $\{B^{1,s}\}_{s=1}^{\infty}$. In Section~\ref{sec:extensions}, we describe the relationship between our model and other models for highest weight $U_q(\asl_n)$-crystals, generalizations of our model to $B^{r,1}$, and possible extensions to other types.
\subsection*{Acknowledgements}
The authors would like to thank Peter Tingley for valuable discussions. The authors would like to thank Masato Okado, Ben Salisbury, and Anne Schilling for comments on earlier drafts of this paper. TS would like to thank Rinat Kedem and Bolor Turmunkh for valuable discussions. This work benefited from computations using {\sc SageMath}~\cite{sage, combinat}.
\section{Background}
\label{sec:background}
In this section, we provide the necessary background.
\subsection{Crystals}
Let $\asl_n$ be the affine Kac--Moody Lie algebra of type $A_{n-1}^{(1)}$ with index set $I = \{0, 1, \dotsc, n-1\}$, Cartan matrix $(a_{ij})_{i,j \in I}$, simple roots $\{\alpha_i\}_{i \in I}$, simple coroots $\{h_i\}_{i \in I}$, fundamental weights $\{\Lambda_i\}_{i \in I}$, weight lattice $P = \operatorname{span}_{\mathbb{Z}} \{\Lambda_i \mid i \in I\}$, dual weight lattice $P^{\vee}$, canonical pairing $\langle\ ,\ \rangle \colon P^{\vee} \times P \to \mathbb{Z}$ given by $\inner{h_i}{\alpha_j} = a_{ij}$, and quantum group $U_q(\asl_n)$.
See Figure~\ref{fig:dynkin_diagram} for the Dynkin diagram of $\asl_n$.
Let $P^+ = \operatorname{span}_{\mathbb{Z}_{\geq 0}} \{\Lambda_i \mid i \in I\}$ denote the dominant integral weights.
Note that $\fsl_n$ is the canonical simple Lie algebra given by the index set $I_0 = I \setminus \{0\}$. Let $\clfw_i$ denote the natural projection of $\Lambda_i$ onto the weight lattice $\cl{P}$ of $\fsl_n$.
Let $c = h_0 + h_1 + \cdots + h_{n-1}$ denote the canonical central element of $\asl_n$. We define the level of a weight $\lambda$ as $\inner{c}{\lambda}$. Let $P_s^+ := \{ \lambda \in P^+ \mid \inner{c}{\lambda} = s \}$ denote the set of level $s$ weights.
\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=0.6]
\draw (6 cm, -1 cm) -- (3 cm,-1 cm) -- (1 cm,0) -- (7.5 cm, 1 cm) -- (14 cm, 0) -- (12 cm, -1 cm) -- (9 cm, -1 cm);
\draw[style=dashed] (6 cm,-1 cm) -- (9 cm,-1 cm);
\draw[fill=white] (7.5 cm, 1 cm) circle (.25cm) node[above=4pt]{$0$};
\draw[fill=white] (1 cm, 0 cm) circle (.25cm) node[left=4pt]{$1$};
\draw[fill=white] (3 cm, -1 cm) circle (.25cm) node[below=4pt]{$2$};
\draw[fill=white] (6 cm, -1 cm) circle (.25cm) node[below=4pt]{$3$};
\draw[fill=white] (9 cm, -1 cm) circle (.25cm) node[below=4pt]{$n-3$};
\draw[fill=white] (12 cm, -1 cm) circle (.25cm) node[below=4pt]{$n-2$};
\draw[fill=white] (14 cm, 0 cm) circle (.25cm) node[right=4pt]{$n-1$};
\end{tikzpicture}
\end{center}
\caption{Dynkin diagram of $\asl_n$.}
\label{fig:dynkin_diagram}
\end{figure}
We write $U_q'(\asl_n) = U_q([\asl_n, \asl_n])$, and let $\delta = \alpha_0 + \alpha_1 \cdots + \alpha_{n-1}$ denote the null root. Note that the $U_q'(\asl_n)$ fundamental weights and simple roots are also given by $\{\Lambda_i\}_{i \in I}$ and $\{\alpha_i\}_{i \in I}$, respectively, but are considered in the weight lattice $P / \mathbb{Z} \delta$.
An \defn{abstract $U_q(\asl_n)$-crystal} is a set $B$ with \defn{crystal operators} $e_i, f_i \colon B \to B \sqcup \{0\}$ for $i \in I$, statistics $\varepsilon_i, \varphi_i \colon B \to \mathbb{Z} \sqcup \{-\infty\}$, and \defn{weight function} $\wt \colon B \to P$ that satisfy the following conditions for all $i \in I$:
\begin{enumerate}
\item[(1)] $\varphi_i(b) = \varepsilon_i(b) + \inner{h_i}{\wt(b)}$ for all $b \in B$ and $i \in I$;
\item[(2)] if $e_i b \neq 0$ for $b \in B$, then
\begin{enumerate}
\item $\varepsilon_i(f_i b) = \varepsilon_i(b) - 1$,
\item $\varphi_i(f_i b) = \varphi_i(b) + 1$,
\item $\wt(f_i b) = \wt(b) + \alpha_i$;
\end{enumerate}
\item[(3)] if $f_i b \neq 0$ for $b \in B$, then
\begin{enumerate}
\item $\varepsilon_i(f_i b) = \varepsilon_i(b) + 1$,
\item $\varphi_i(f_i b) = \varphi_i(b) - 1$,
\item $\wt(f_i b) = \wt(b) - \alpha_i$;
\end{enumerate}
\item[(4)] $f_i b = b'$ if and only if $b = e_i b'$ for $b, b' \in B$ and $i \in I$;
\item[(5)] if $\varphi_i(b) = -\infty$ for $b \in B$, then $e_i b = f_i b = 0$.
\end{enumerate}
Define
\[
\varepsilon(b) = \sum_{i \in I} \varepsilon_i(b) \Lambda_i,
\qquad\qquad
\varphi(b) = \sum_{i \in I} \varphi_i(b) \Lambda_i,
\]
We say an element $b \in B$ is \defn{highest weight} if $e_i b = 0$ for all $i \in I$. If $B$ is a $U_q'(\asl_n)$-crystal, then we say $b \in B$ is \defn{classically highest weight} if $e_i b = 0$ for all $i \in I_0$.
We say an abstract $U_q(\asl_n)$-crystal is \defn{regular} if
\[
\varepsilon_i(b) = \max \{ k \mid e_i^k b \neq 0 \},
\qquad \qquad \varphi_i(b) = \max \{ k \mid f_i^k b \neq 0 \}.
\]
\begin{remark}
The term regular is sometimes called \defn{seminormal} in the literature.
\end{remark}
We call an abstract $U_q(\asl_n)$-crystal $B$ a \defn{$U_q(\asl_n)$-crystal} if $B$ is the crystal basis of some $U_q(\asl_n)$-module.
Kashiwara showed in~\cite{K91} that the irreducible highest weight $U_q(\mathfrak{g})$-module $V(\lambda)$ admits a crystal basis, where $\mathfrak{g}$ is a symmetrizable Kac--Moody Lie algebra and $\lambda$ is a dominant integrable weight. We denote this crystal basis by $B(\lambda)$, and let $u_{\lambda} \in B(\lambda)$ denote the unique highest weight element, which is the unique element of weight $\lambda$.
There is also an analog of the crystal corresponding to the lower half of $U_q(\mathfrak{g})$ denoted by $B(\infty)$, and let $u_{\infty}$ denote the highest weight element of $B(\infty)$.
We define the \defn{tensor product} of abstract $U_q(\asl_n)$-crystals $B_1$ and $B_2$ as the crystal $B_2 \otimes B_1$ that is the Cartesian product $B_2 \times B_1$ with the crystal structure
\begin{align*}
e_i(b_2 \otimes b_1) & = \begin{cases}
e_i b_2 \otimes b_1 & \text{if } \varepsilon_i(b_2) > \varphi_i(b_1), \\
b_2 \otimes e_i b_1 & \text{if } \varepsilon_i(b_2) \leq \varphi_i(b_1),
\end{cases}
\\ f_i(b_2 \otimes b_1) & = \begin{cases}
f_i b_2 \otimes b_1 & \text{if } \varepsilon_i(b_2) \geq \varphi_i(b_1), \\
b_2 \otimes f_i b_1 & \text{if } \varepsilon_i(b_2) < \varphi_i(b_1),
\end{cases}
\\ \varepsilon_i(b_2 \otimes b_1) & = \max(\varepsilon_i(b_1), \varepsilon_i(b_2) - \inner{h_i}{\wt(b_1)}),
\\ \varphi_i(b_2 \otimes b_1) & = \max(\varphi_i(b_2), \varphi_i(b_1) + \inner{h_i}{\wt(b_2)}),
\\ \wt(b_2 \otimes b_1) & = \wt(b_2) + \wt(b_1).
\end{align*}
\begin{remark}
Our tensor product convention is opposite of Kashiwara~\cite{K91}.
\end{remark}
Let $B_1$ and $B_2$ be two abstract $U_q(\mathfrak{g})$-crystals. A \defn{crystal morphism} $\psi \colon B_1 \to B_2$ is a map $B_1 \sqcup \{0\} \to B_2 \sqcup \{0\}$ with $\psi(0) = 0$ such that the following properties hold for all $b \in B_1$:
\begin{itemize}
\item[(1)] If $\psi(b) \in B_2$, then $\wt\bigl(\psi(b)\bigr) = \wt(b)$, $\varepsilon_i\bigl(\psi(b)\bigr) = \varepsilon_i(b)$, and $\varphi_i\bigl(\psi(b)\bigr) = \varphi_i(b)$.
\item[(2)] We have $\psi(e_i b) = e_i \psi(b)$ if $\psi(e_i b) \neq 0$ and $e_i \psi(b) \neq 0$.
\item[(3)] We have $\psi(f_i b) = f_i \psi(b)$ if $\psi(f_i b) \neq 0$ and $f_i \psi(b) \neq 0$.
\end{itemize}
An \defn{embedding} and \defn{isomorphism} is a crystal morphism such that the induced map $B_1 \sqcup \{0\} \to B_2 \sqcup \{0\}$ is an embedding and bijection respectively. A crystal morphism is \defn{strict} if it commutes with all crystal operators.
\subsection{Nakajima monomials}
Next, we recall the Nakajima monomial realization of crystals following~\cite{ST14}.
Let $\mathcal{M}$ denote the set of Laurent monomials in the commuting variables $\{Y_{i,k}\}_{i \in I, k \in \mathbb{Z}}$. Fix an integer $K$, and then fix integers $c_{i,i+1}$ and $c_{i+1,i}$ such that $K = c_{i,i+1} + c_{i+1,i}$ for all $i \in I$, where all indices are taken mod $n$. For a monomial $m = \prod_{i \in I} \prod_{k \in \mathbb{Z}} Y_{i,k}^{y_{i,k}}$, define
\begin{align*}
\varepsilon_i(m) & = -\min_{k \in \mathbb{Z}} \sum_{s > k} y_{i,s},
& k_e(m) & = \max \left\{ k \hspace{5pt}\vline\hspace{5pt} \varepsilon_i(m) = - \sum_{s > k} y_{i,s} \right\},
\\ \varphi_i(m) & = \max_{k \in \mathbb{Z}} \sum_{s \leq k} y_{i,s},
& k_f(m) & = \min \left\{ k \hspace{5pt}\vline\hspace{5pt} \varphi_i(m) = \sum_{s \leq k} y_{i,s} \right\},
\end{align*}
\[
\wt(m) = \sum_{\substack{i \in I \\ k \in \mathbb{Z}}} y_{i,k} \Lambda_i
\]
Define the crystal operators $e_i,f_i \colon \mathcal{M} \to \mathcal{M} \sqcup \{ 0 \}$ by
\begin{align*}
e_i(m) & = \begin{cases} 0 & \text{if } \varepsilon_i(m) = 0, \\ m A_{i, k_e(m) - K} & \text{if } \varepsilon_i(m) > 0, \end{cases}
\\ f_i(m) & = \begin{cases} 0 & \text{if } \varphi_i(m) = 0, \\ m A_{i, k_f(m) - K}^{-1} & \text{if } \varphi_i(m) > 0, \end{cases}
\end{align*}
where
\[
A_{i,k} = Y_{i,k} Y_{i,k+K} Y_{i-1,k+c_{i,i-1}}^{-1} Y_{i+1,k+c_{i,i+1}}^{-1}.
\]
We note that the crystal structure of Kashiwara~\cite{K03II} is when $K = 1$, and that of Nakajima~\cite{Nakajima03} is when $c_{i,i+1} = c_{i+1,i} = 1$ for all $i \in I$. Note that in the case of Nakajima, we prohibit odd length cycles in the Dynkin diagram; that is, we can only consider types $\asl_{2k+1}$.
Let $\mathcal{M}(m)$ denote the closure of $m$ under the crystal operators $e_i$ and $f_i$ for all $i \in I$.
\begin{thm}[{\cite{ST14}}]
\label{thm:highest_weight_monomials}
Let $\lambda \in P^+$, then
\[
\mathcal{M}\left( \prod_{i \in I} Y_{i,k_i}^{\inner{h_i}{\lambda}} \right) \cong B(\lambda),
\]
for any $(k_i)_{i \in I}$.
\end{thm}
For convenience, we define
\[
Y_{\lambda} := \prod_{i \in I} Y_{i, 0}^{\inner{h_i}{\lambda}},
\]
and we denote $\mathbf{1} = Y_0$, where $0 \in P^+$. We also define $\mathcal{M}(\lambda) := \mathcal{M}(Y_{\lambda})$.
We modify the definition of $k_f(m)$ to only be the \emph{finite} partial sums $0 \leq s \leq k$:
\[
k_f^{\dagger}(m) = \min \left\{ k \hspace{5pt}\vline\hspace{5pt} \varphi_i(m) = \sum_{0 \leq s \leq k} y_{i,s} \right\}.
\]
Next, we define the modified crystal operator
\[
f_i^{\dagger}(m) = m A_{i,k_f^{\dagger}(m) - K}^{-1},
\]
and then define $\mathcal{M}(\infty)$ as the closure of $\mathbf{1}$ under $f_i^{\dagger}$.
\begin{thm}[{\cite{KKS07}}]
\label{thm:monomial_infinity}
Suppose $c_{ij} \in \mathbb{Z}_{\geq 0}$ and $K = 1$. We have
\[
\mathcal{M}(\infty) \cong B(\infty).
\]
\end{thm}
There is also another set of variables
\begin{equation}
\label{eq:X_variables}
X_{i,k} := Y_{i-1,k+1}^{-1} Y_{i,k}
\end{equation}
introduced in~\cite{KKS07}, which was used in their description of $\mathcal{M}(\infty)$ in type $A_n^{(1)}$.
Unless otherwise stated, we consider $c_{ij} = 1$ if $(i, j) = (n, 0)$ or $i < j$ when $(i, j) \neq (0, n)$ and $c_{ij} = 0$ otherwise (hence $K = 1$). Note that this corresponds to orienting the Dynkin diagram into an ordered cycle, where we draw an arrow $i \to i+1$ implying $c_{i,i+1} = 1$ and the other values $c_{i,j}$ for $j \neq i \pm 1$ do not affect the crystal structure.
\subsection{Kirillov--Reshetikhin crystals}
\label{sec:KR_crystals}
A \defn{Kirillov--Reshetikhin (KR) module} $W^{r,s}$, where $r \in I_0$ and $s \in \mathbb{Z}_{> 0}$, is a particular irreducible finite-dimensional $U_q'(\asl_n)$-module that has many remarkable properties. KR modules are classified by their Drinfel'd polynomials, and $W^{r,s}$ is the minimal affinizations of the highest weight $U_q(\fsl_n)$-representation $V(s \clfw_r)$~\cite{CP95, CP98}. In particular, it was shown in~\cite{KKMMNN92} that the KR module $W^{r,s}$ admits a crystal basis $B^{r,s}$ called a \defn{Kirillov--Reshetikhin (KR) crystal}.
Another property is that the KR crystal $B^{r,s}$ is a \defn{perfect crystal of level $s$}, which means it satisfies the following conditions:
\begin{enumerate}
\item $B^{r,s} \otimes B^{r,s}$ is connected.
\item $\cl{\wt}(b) \in s \clfw_r + \sum_{i \in I_0} \mathbb{Z}_{\leq 0} \alpha_i$ for all $b \in B^{r,s}$.
\item $\inner{c}{\varepsilon(b)} \geq s$ for all $b \in B^{r,s}$.
\item For all $\lambda \in P_s^+$, there exists unique elements $b_{\lambda}, b^{\lambda} \in B^{r,s}$ such that
\[
\varepsilon(b_{\lambda}) = \lambda = \varphi(b^{\lambda}).
\]
\end{enumerate}
Additionally, we have $B^{r,s} \cong B(s\clfw_r)$ as $U_q(\fsl_n)$-crystals. Next, recall that the Dynkin diagram automorphism $i \mapsto i+1 \mod n$ induces a (twisted) $U_q(\asl_n)$-crystal isomorphism $\pr \colon B^{r,s} \to B^{r,s}$ called the \defn{promotion isomorphism}~\cite{Shimozono02}. On semistandard tableaux~\cite{KN94}, the map $\pr$ is the (weak) promotion operator of Sch\"utzenberger~\cite{Sch72}. Hence, $B^{r,s}$ is a regular crystal, and we define the remaining crystal structure on $B^{r,s}$ by
\begin{align*}
e_0(b) & = \pr^{-1} \circ e_1 \circ \pr,
\\ f_0(b) & = \pr^{-1} \circ f_1 \circ \pr,
\\ \wt(b) & = \cl{\wt}(b) + k_0 \Lambda_0,
\end{align*}
where $k_0$ is such that $\inner{\wt(b)}{c} = 0$ ({\it i.e.}, it is a level 0 weight).
We will be focusing on the KR crystal $B^{1,s}$, which is the crystal that naturally corresponds to the \defn{vector representation} of $W^{1,s}$. We have
\[
B^{1,s} = \left\{ (x_1, \dotsc, x_n) \hspace{5pt}\vline\hspace{5pt} x_1, \dotsc, x_n \in \mathbb{Z}_{\geq 0},\, \sum_{i=1}^n x_i = s \right\}
\]
with the crystal structure
\begin{subequations}
\label{eq:vector_crystal}
\begin{align}
e_i(x_1, \dotsc, x_n) & = \begin{cases} 0 & \text{if } x_{i+1} = 0, \\ (x_1, \dotsc, x_i + 1, x_{i+1} - 1, \dotsc, x_n) & \text{if } x_{i+1} > 0, \end{cases}
\\ f_i(x_1, \dotsc, x_n) & = \begin{cases} 0 & \text{if } x_i = 0, \\ (x_1, \dotsc, x_i - 1, x_{i+1} + 1, \dotsc, x_n) & \text{if } x_i > 0, \end{cases}
\\ \varepsilon_i(x_1, \dotsc, x_n) & = x_{i+1},
\\ \varphi_i(x_1, \dotsc, x_n) & = x_i,
\\ \wt(x_1, \dotsc, x_n) & = \sum_{i \in I} (x_i - x_{i+1}) \Lambda_i,
\end{align}
\end{subequations}
where all indices are understood mod $n$. Note that $B^{1,s}$ is a regular crystal.
We will also need the \defn{affinization} of a $U_q'(\asl_n)$-crystal $B$, which is defined as follows.
The affinization of $B$ is the $U_q(\asl_n)$-crystal $\widehat{B} =\{ b(k) \mid b \in B, k \in \mathbb{Z} \}$, whose crystal structure is given by
\begin{align*}
e_i\bigl(b(k)\bigr) & = \begin{cases} (e_0 b)(k+1) & \text{if } i = 0, \\ (e_i b)(k) & \text{if } i \neq 0, \end{cases}
\\ f_i\bigl(b(k)\bigr) & = \begin{cases} (f_0 b)(k-1) & \text{if } i = 0, \\ (f_i b)(k) & \text{if } i \neq 0, \end{cases}
\\ \varepsilon_i\bigl(b(k)\bigr) & = \varepsilon_i(b),
\\ \varphi_i\bigl(b(k)\bigr) & = \varphi_i(b),
\\ \wt\bigl(b(k)\bigr) & = \wt(b) + k \delta.
\end{align*}
We can construct the coherent limit of the family $\{ B^{1,s} \}_{s=1}^{\infty}$ as follows. The coherent limit is given by
\[
B_{\infty} = \left\{ (x_1, \dotsc, x_n) \hspace{5pt}\vline\hspace{5pt} x_1, \dotsc, x_n \in \mathbb{Z},\, \sum_i x_i = 0 \right\}
\]
with the same crystal structure as in Equation~\eqref{eq:vector_crystal} except
\begin{align*}
e_i(x_1, \dotsc, x_n) & = (x_1, \dotsc, x_i + 1, x_{i+1} - 1, \dotsc, x_n),
\\ f_i(x_1, \dotsc, x_n) & = (x_1, \dotsc, x_i - 1, x_{i+1} + 1, \dotsc, x_n).
\end{align*}
We note that $B_{\infty}$ is generated by $b_{\infty} = (0, 0, \dotsc, 0)$, but it is not a regular crystal.
\subsection{Kyoto path model}
We recall some of the results of~\cite{KKMMNN91,KKMMNN92,OSS03IV}, which gives a model for highest weight $U_q(\asl_n)$-crystals using KR crystals.
\begin{thm}
\label{thm:kyoto_embedding}
Let $\lambda$ be a level $s$ weight. Let $B$ be a perfect crystal of level $s$. Let $b^{\lambda} \in B$ be the unique element such that $\varphi(b^{\lambda}) = \lambda$. Let $\mu = \varepsilon(b^{\lambda})$. The morphism
\[
\widehat{\Psi} \colon B(\lambda) \to \widehat{B} \otimes B(\mu)
\]
defined by $u_{\lambda} \mapsto b^{\lambda}(0) \otimes u_{\mu}$ is a $U_q(\asl_n)$-crystal isomorphism.
\end{thm}
By iterating $\widehat{\Psi}$, we obtain the \defn{Kyoto path model}. For a level $s$ weight $\lambda$, we can construct a model for $B(\lambda)$ by
\begin{equation}
\label{eq:kyoto_path_model}
\widehat{\Psi}^{(+\infty)} \colon B(\lambda) \to \widehat{B}^{1,s} \otimes \widehat{B}^{1,s} \otimes \cdots
\end{equation}
since $B^{1,s}$ is a perfect crystal of level $s$.
Note that to define $e_i$ for Equation~\eqref{eq:kyoto_path_model}, we consider $e_i b = 0$ if it would be otherwise undefined.
Furthermore, $\widehat{\Psi}^{(+\infty)}(u_{\lambda})$ is eventually cyclic and, for any $b \in B(\lambda)$, the element $\widehat{\Psi}^{(+\infty)}(b)$ only differs from $\widehat{\Psi}^{(+\infty)}(u_{\lambda})$ in a finite number of factors. Therefore, for any element $b$ can consider Theorem~\ref{thm:kyoto_embedding} iterated $N \gg 1$ times (that depends on $b$) $\widehat{\Psi}^{(N)}(b)$ to define the crystal structure on the Kyoto path model using only the KR crystal $B^{1,s}$.
We note that there are analogous results for $U_q'(\asl_n)$-crystals by considering the branching rule from $U_q(\asl_n)$ to $U_q'(\asl_n)$. In particular, there exists a $U_q'(\asl_n)$-crystal isomorphism
\begin{equation}
\label{eq:kyoto_prime}
\Psi \colon B(\lambda) \to B \otimes B(\mu)
\end{equation}
defined by $u_{\lambda} \mapsto b^{\lambda} \otimes u_{\mu}$.
There is an analog of the Kyoto path model for $B(\infty)$ given by Kang, Kashiwara, and Misra~\cite{KKM94} by iterating the following isomorphism.
\begin{thm}
\label{thm:Binf_Kyoto_path}
Let $B_{\infty}$ denote the coherent limit of $\{B_s\}_{s=1}^{\infty}$, where $B_s$ is a perfect crystal of level $s$. Let $b_{\infty} \in B_{\infty}$ denote the unique element of weight $0$. Then the morphism
\[
\Upsilon \colon B(\infty) \to B_{\infty} \otimes B(\infty)
\]
defined by $u_{\infty} \mapsto b_{\infty} \otimes u_{\infty}$ is a $U_q'(\asl_n)$-crystal isomorphism.
\end{thm}
\section{Monomial realization of $B^{1,s}$}
\label{sec:monomial_model}
In this section, we describe the construction of $B^{1,s}$ using Kashiwara's crystal structure of Nakajima monomials.
Define
\[
\mathcal{M}^{1,s} := \left\{ \prod_{i=1}^n Y_{i-1,1}^{-x_i} Y_{i,0}^{x_i} \hspace{5pt}\vline\hspace{5pt} x_1, \dotsc, x_n \in \mathbb{Z}_{\geq 0}, \sum_{i=1}^n x_i = s \right\}.
\]
\begin{thm}
\label{thm:single_row_isomorphism}
We have
\[
\mathcal{M}^{1,s} \cong B^{1,s}
\]
as $U_q'(\asl_n)$-crystals.
\end{thm}
\begin{proof}
Let $\Phi \colon B^{1,s} \to \mathcal{M}^{1,s}$ be the map
\[
(x_1, \dotsc, x_n) \mapsto \prod_{i=1}^n Y_{i-1,1}^{-x_i} Y_{i,0}^{x_i},
\]
and it is clear that $\Phi$ is a bijection. Thus it remains to show that $\Phi$ commutes with the crystal operators.
We restrict to the variables only containing $i$. It is sufficient to consider
\[
m = Y_{i,1}^{k_1}\, Y_{i,0}^{k_2}
\]
with $k_1\leq 0 \leq k_2$. Then
\[
\varphi_i(m) = \max\left\{\sum_{s\leq k} y_{i,s} \hspace{5pt}\vline\hspace{5pt} k \in \mathbb{Z} \right\} = \max\{k_2, k_2+k_1\} = k_2 \geq 0
\]
and
\[
\varepsilon_i(m) = \max\left\{-\sum_{k < s} y_{i,s} \hspace{5pt}\vline\hspace{5pt} k \in \mathbb{Z} \right\} = \max\{-k_2-k_1,-k_1\} = -k_1 \geq 0
\]
since $k_1 \leq 0 \leq k_2$.
Hence
\begin{align*}
k_f(m) & = \min\left\{k \hspace{5pt}\vline\hspace{5pt} \varphi_i(m) = \sum_{s\leq k} y_{i,s}\right\}
= \min\left\{k \hspace{5pt}\vline\hspace{5pt} k_2 = \sum_{s \leq k} y_{i,s}\right\} = 0,
\\ k_e(m) & = \max\left\{k \hspace{5pt}\vline\hspace{5pt} \varphi_i(m) = \sum_{s\leq k} y_{i,s}\right\}
= \max\left\{k \hspace{5pt}\vline\hspace{5pt} k_2 = \sum_{s \leq k} y_{i,s}\right\} = 0,
\end{align*}
and note that we have used an alternative form of $k_e$ (see, {\it e.g.},~\cite{K03II}).
Therefore, we have
\begin{align*}
A_{i,k_f(m)} = A_{i,k_e(m)} = A_{i,0} &= Y_{i,0} \, Y_{i,1} \prod_{j\neq i} Y_{j,c_{ji}}^{\langle h_j,\alpha_i\rangle}
\\ &=Y_{i,0} \, Y_{i,1} \, Y_{i-1,c_{i-1,i}}^{-1} \, Y_{i+1,c_{i+1,i}}^{-1}
\\ &=Y_{i,0} \, Y_{i,1} \, Y_{i-1,1}^{-1} \, Y_{i+1,0}^{-1}.
\end{align*}
Since $\varphi_i(m) > 0$ and $\varepsilon_i(m) > 0$, we have
\begin{align*}
f_i(m) & =
A_{i,k_f}^{-1} m =
Y_{i,0}^{k_2-1} \, Y_{i,1}^{k_1-1} \, Y_{i-1,1}^{} \, Y_{i+1,0}^{},
\\ e_i(m) & =
A_{i,k_e} m =
Y_{i,0}^{k_2+1} \, Y_{i,1}^{k_1+1} \, Y_{i-1,1}^{-1} \, Y_{i+1,0}^{-1}.
\end{align*}
For $i \in I$, we have
\begin{align*}
f_i \bigl( \Phi (x_1, \dotsc, x_n) \bigr) &= f_i \left( \prod_{k=1}^n Y_{k-1,1}^{-x_k} Y_{k,0}^{x_k} \right)
\\ &= f_i\left(Y_{i-1,1}^{-x_i} Y_{i,0}^{x_i} \,\, Y_{i,1}^{-x_{i+1}} Y_{i+1,0}^{x_{i+1}} \right) \prod_{k\neq i,i+1} Y_{k-1,1}^{-x_k} Y_{k,0}^{x_k}
\\ &= f_i\left(Y_{i,1}^{-x_{i+1}} Y_{i,0}^{x_i} \right) \,\, Y_{i-1,1}^{-x_i} Y_{i+1,0}^{x_{i+1}} \prod_{k\neq i,i+1} Y_{k-1,1}^{-x_k} Y_{k,0}^{x_k}
\\ &= \left(Y_{i,1}^{-x_{i+1}-1} Y_{i,0}^{x_i-1} \,\,\,\, Y_{i-1,1} Y_{i+1,0} \right)
\,\, Y_{i-1,1}^{-x_i} Y_{i+1,0}^{x_{i+1}} \prod_{k\neq i,i+1} Y_{k-1,1}^{-x_k} Y_{k,0}^{x_k}
\\ &= Y_{i-1,1}^{-x_i+1} \, Y_{i,0}^{x_i-1} \, Y_{i,1}^{-x_{i+1}-1} \, Y_{i+1,0}^{x_{i+1}+1} \prod_{k\neq i,i+1} Y_{k-1,1}^{-x_k} Y_{k,0}^{x_k}
\\ &= \Phi(\dotsc, x_i - 1, x_{i+1} + 1, \ldots)
\\ &= \Phi\bigl( f_i (x_1, \dotsc, x_n) \bigr).
\end{align*}
where all indices are taken mod $n$. Similarly, we have $e_i\bigl( \Phi (x_1, \dotsc, x_n) \bigr) = \Phi\bigl( e_i (x_1, \dotsc, x_n) \bigr)$.
\end{proof}
Note that we can define $\Phi(x_1, \dotsc, x_n) = X_{1,0}^{x_1} X_{2,0}^{x_2} \dotsm X_{n,0}^{x_n}$, where the variables $X_{i,k}$ are given by Equation~\eqref{eq:X_variables}.
\begin{ex}
The crystal $\mathcal{M}^{1,3}$ for $\asl_3$ is given by Figure~\ref{fig:asl3_M13_ex}.
\end{ex}
\begin{figure}
\[
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\node (node_8) at (38.0bp,314.0bp) [draw,draw=none] {$Y_{0,0} Y_{0,1}^{-2} Y_{1,0}^{2} Y_{2,1}^{-1} $};
\node (node_7) at (258.0bp,238.0bp) [draw,draw=none] {$Y_{1,1}^{-3} Y_{2,0}^{3} $};
\node (node_6) at (115.0bp,86.0bp) [draw,draw=none] {$Y_{0,0}^{2} Y_{1,1}^{-1} Y_{2,0} Y_{2,1}^{-2} $};
\node (node_5) at (113.0bp,390.0bp) [draw,draw=none] {$Y_{0,1}^{-2} Y_{1,0}^{2} Y_{1,1}^{-1} Y_{2,0} $};
\node (node_4) at (65.0bp,466.0bp) [draw,draw=none] {$Y_{0,1}^{-3} Y_{1,0}^{3} $};
\node (node_3) at (195.0bp,162.0bp) [draw,draw=none] {$Y_{0,0} Y_{1,1}^{-2} Y_{2,0}^{2} Y_{2,1}^{-1} $};
\node (node_2) at (40.0bp,162.0bp) [draw,draw=none] {$Y_{0,0}^{2} Y_{0,1}^{-1} Y_{1,0} Y_{2,1}^{-2} $};
\node (node_1) at (109.0bp,238.0bp) [draw,draw=none] {$Y_{0,0} Y_{0,1}^{-1} Y_{1,0} Y_{1,1}^{-1} Y_{2,0} Y_{2,1}^{-1} $};
\node (node_0) at (67.0bp,10.0bp) [draw,draw=none] {$Y_{0,0}^{3} Y_{2,1}^{-3} $};
\draw [black,<-] (node_4) ..controls (55.089bp,433.79bp) and (50.04bp,416.0bp) .. (47.0bp,400.0bp) .. controls (41.735bp,372.3bp) and (39.328bp,338.9bp) .. (node_8);
\definecolor{strokecol}{rgb}{0.0,0.0,0.0};
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\draw (35.0bp,238.0bp) node {$0$};
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\draw (242.0bp,200.0bp) node {$2$};
\draw [blue,->] (node_2) ..controls (55.611bp,142.7bp) and (69.761bp,126.72bp) .. (83.0bp,114.0bp) .. controls (87.089bp,110.07bp) and (91.657bp,106.03bp) .. (node_6);
\draw (92.0bp,124.0bp) node {$1$};
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\end{tikzpicture}
\]
\caption{The crystal $\mathcal{M}^{1,3}$ for $\asl_3$.}
\label{fig:asl3_M13_ex}
\end{figure}
\section{Relation to Kyoto path model}
\label{sec:Kyoto_path}
To define $\mathcal{M}^{1,s}$, we considered the crystal generated from $Y_{0,1}^{-s} Y_{1,0}^s$. However, by shifting the monomials, we can construct an isomorphism with the tensor product. Indeed, let $\tau_j$ be the map given by $Y_{i,k} \mapsto Y_{i,k+j}$ for all $i \in I$ and $k \in \mathbb{Z}$. Let $\mathcal{M} \cdot \mathcal{M}' = \{ m \cdot m' \mid m \in \mathcal{M}, m' \in \mathcal{M}' \}$, where $\cdot$ denotes the usual multiplication in $\mathcal{M}$, under the usual crystal operators.
\begin{thm}
\label{thm:nakajima_tensor_product}
Let $j_1, j_2, \dotsc, j_N \in \mathbb{Z}_{\geq 0}$ be pairwise distinct. We have
\[
\prod_{k=1}^N \tau_{j_k}(\mathcal{M}^{1,s_k}) \cong \bigotimes_{k=1}^N B^{1,s_k}.
\]
\end{thm}
\begin{proof}
Let $\Phi \colon \bigotimes_{k=1}^N B^{1,s_k} \to \prod_{k=1}^N \tau_{j_k}(\mathcal{M}^{1,s_k}) $ be the map
\[
\Phi(b_1 \otimes \cdots \otimes b_N) = \prod_{k=1}^N \tau_{j_k}\bigl(\Phi_{s_k}(b_k)\bigr),
\]
where $\Phi_{s_k} \colon B^{1,s_k} \to \mathcal{M}^{1,s_k}$ is the isomorphism given by Theorem~\ref{thm:single_row_isomorphism}.
From the combinatorial $R$-matrix and the commutativity of the $Y_{i,k}$, we can assume without loss of generality that $j_1 < j_2 < \cdots < j_N$.
We show the claim holds by induction on $N$. Theorem~\ref{thm:single_row_isomorphism} says this holds when $N = 1$. Thus assume the claim holds for $N-1$.
Consider the tensor product $B \otimes B^{1,s_N}$, where $B = \bigotimes_{k=1}^{N-1} B^{1,s_k}$, and $j_N > j_{N-1}$. Recall from the proof of Theorem~\ref{thm:single_row_isomorphism} that $-\varepsilon_i(m)$ and $\varphi_i(m)$ are equal to the powers of $Y_{i,1}$ and $Y_{i,0}$, respectively, appearing in $m \in \mathcal{M}^{1,s_k}$. Let $\psi \colon \Phi(B) \otimes \tau_{j_N}(\mathcal{M}^{1,s_N}) \to \Phi(B) \cdot \tau_{j_N}(\mathcal{M}^{1,s_N})$.
Consider
\begin{align*}
m & = \prod_{i \in I} \prod_{k \in \mathbb{Z}} Y_{i,k}^{y_{i,k}} \in \Phi(B),
\\ m' & = \prod_{i \in I} \prod_{k \in \mathbb{Z}} Y_{i,k}^{y'_{i,k}} \in \tau_{j_N}(\mathcal{M}^{1,s_N}).
\end{align*}
Note that $\psi(m \otimes m') = m \cdot m'$. Also, for all $i \in I$, we have $y_{i,p} = 0$ for all $p > j_N$, $y'_{i,p} = 0$ for all $p < j_N$ or $p > j_N + 1$.
Thus, we have
\begin{subequations}
\begin{align}
\label{eq:product_epsilon}
\sum_{p > k} y_{i,p} + y'_{i,p} & =
\begin{cases}
y'_{i,j_N + 1} + y'_{i, j_N} + \sum_{p > k} y_{i,p} & \text{if } k < j_N, \\
y'_{i,j_N + 1} & \text{if } k = j_N, \\
0 & \text{if } k > j_N,
\end{cases}
\\ \label{eq:product_phi}
\sum_{p \leq k} y_{i,p} + y'_{i,p} & =
\begin{cases}
\sum_{p \leq k} y_{i,p} & \text{if } k < j_N, \\
y'_{i,j_N} + \sum_{p \leq j_N} y_{i,p} & \text{if } k = j_N, \\
y'_{i,j_N + 1} + y'_{i, j_N} + \sum_{p \leq j_N} y_{i,p} & \text{if } k > j_N.
\end{cases}
\end{align}
\end{subequations}
Note that
\[
\inner{h_i}{\wt(m')} = \varphi_i(m') - \varepsilon_i(m') = y'_{i,j_N} + y'_{i,j_N+1}.
\]
From Equation~\eqref{eq:product_epsilon}, either $y'_{i,j_N+1}$ obtains the minimum in $\varepsilon_i(m \cdot m')$ or
$\varepsilon_i(m \cdot m') = \varepsilon_i(m) - \inner{h_i}{\wt(m')}$.
Hence, we have
\[
\varepsilon_i(m \otimes m') = \max\bigl(\varepsilon_i(m'), \varepsilon_i(m) - \inner{h_i}{\wt(m')}\bigr) = \varepsilon_i(m \cdot m').
\]
Next, from Equation~\eqref{eq:product_phi}, either $\varphi_i(m)$ obtains the maximum in $\varphi_i(m \cdot m')$ or
$\varphi_i(m \cdot m') = \varphi_i(m') + \inner{h_i}{\wt(m)}$
as $y'_{i,j_N+1} \leq 0$ and $\sum_{s \leq j_N} y_{i,s} = \inner{h_i}{\wt(m)}$.
Hence, we have
\[
\varphi_i(m \otimes m') = \max\bigl(\varphi_i(m), \varphi_i(m') + \inner{h_i}{\wt(m)}\bigr) = \varphi_i(m \cdot m').
\]
It is clear that $\wt(m \otimes m') = \wt(m) + \wt(m') = \wt(m \cdot m')$.
Thus it remains to show $\psi$ commutes with the crystal operators.
Suppose $k_e(m \cdot m') < j_N$. Thus, we must have
\[
-\varepsilon_i(m') = y'_{i,j_N} < y'_{i,j_N} + y'_{i,j_N+1} + \sum_{p > k} y_{i,p} = -\varepsilon_i(m) + \varphi_i(m') - \varepsilon_i(m')
\]
for some $k$ from Equation~\eqref{eq:product_epsilon} as $k_e(m \cdot m')$ is the maximum index. Hence, we have $\varphi_i(m') > \varepsilon_i(m)$.
Similarly, if $k_e(m \cdot m') = j_N$, then we have
\[
y'_{i, j_N+1} = -\varepsilon_i(m \cdot m') = -\varepsilon_i(m') \geq -\varepsilon_i(m) + \varphi_i(m') - \varepsilon_i(m')
\]
from Equation~\eqref{eq:product_epsilon}.
Thus we have $\varphi_i(m') \leq \varepsilon_i(m)$. Therefore, we have
\[
e_i\bigl(\psi(m \otimes m')\bigr) = \psi\bigl(e_i(m \otimes m')\bigr) = \psi(e_i m \otimes m').
\]
Suppose $k_f(m \cdot m') < j_N$. Thus we must have
\[
\varphi_i(m') + \varphi_i(m) - \varepsilon_i(m) = y'_{i, j_N} + \inner{h_i}{\wt(m)} \leq \varphi_i(m \cdot m')
\]
from Equation~\eqref{eq:product_phi} and that $k_f(m \cdot m')$ is the minimal index.
Thus, we have $\varphi_i(m') \leq \varepsilon_i(m)$. Similarly, if $k_f(m \cdot m') = j_N$, then we have
\[
\varphi_i(m \cdot m') = \varphi_i(m') + \varphi_i(m) - \varepsilon_i(m) > \varphi_i(m)
\]
as this is the minimal index such that $\varphi_i(m \cdot m')$ is achieved from Equation~\eqref{eq:product_phi}. Hence, we have $\varphi_i(m') > \varepsilon_i(m)$. Therefore, we have
\[
f_i\bigl(\psi(m \otimes m')\bigr) = \psi\bigl(f_i(m \otimes m')\bigr) = \psi(f_i m \otimes m').
\]
\end{proof}
\begin{thm}
\label{thm:monomial_kyoto_path}
Let $\lambda \in P^+$ be a level $s$ weight. Let $\Xi \colon B(\lambda) \to \mathcal{M}(\lambda)$ be the canonical isomorphism. Let $\Psi \colon B(\lambda) \to B^{1,s} \otimes B(\mu)$ be the isomorphism from Equation~\eqref{eq:kyoto_prime}. Let $\Phi \colon B^{1,s} \otimes B(\mu) \to \mathcal{M}(\lambda)$ denote the map given by
\[
\Phi(b \otimes b') = \Phi_s(b) \cdot \tau_1\bigl(\Phi_{\mu}(b')\bigr),
\]
where $\Phi_s \colon B^{1,s} \to \mathcal{M}^{1,s}$ and $\Phi_{\mu} \colon B(\mu) \to \mathcal{M}(\mu)$ be the isomorphisms from Theorem~\ref{thm:single_row_isomorphism} and Theorem~\ref{thm:highest_weight_monomials}, respectively.
Then the diagram
\[
\xymatrixrowsep{3.5pc}
\xymatrixcolsep{4.5pc}
\xymatrix{B(\lambda) \ar[r]^-{\Psi} \ar[dr]_{\Xi} & B^{1,s} \otimes B(\mu) \ar[d]^{\Phi} \\ & \mathcal{M}(\lambda)}
\]
commutes.
\end{thm}
\begin{proof}
Note that the crystal operators imply that any $m \in \mathcal{M}(\mu)$ does not contain any $Y_{i,k}$ with $k \in \mathbb{Z}_{<0}$.
To show the diagram commutes, it is sufficient to show that for $b = b^{\lambda} \otimes u_{\mu}$, we have
\[
\Phi(b) = \Phi_s(b^{\lambda}) \cdot \tau_1\bigl(\Phi_{\mu}(u_{\mu})\bigr) = Y_{\lambda}
\]
as the proof of Theorem~\ref{thm:nakajima_tensor_product} implies the crystal operators commute with $\Phi$.
From the definition of $\Psi$, we have $\varphi(b^{\lambda}) = \lambda$
and $\varepsilon(b^{\lambda}) = \mu$.
Next, note that we have
\begin{align*}
\tau_1\bigl(\Phi_{\mu}(u_{\mu})\bigr) & = \tau_1(Y_{\mu}) = \prod_{i \in I} Y_{i,1}^{\inner{h_i}{\mu}} = \prod_{i \in I} Y_{i,1}^{\varepsilon_i(b^{\lambda})},
\\ \Phi_s(b^{\lambda}) & = \prod_{i \in I} Y_{i,1}^{-\varepsilon_i(b^{\lambda})} Y_{i,0}^{\varphi_i(b^{\lambda})}.
\end{align*}
Hence, we have
\[
\Phi(b) = \prod_{i \in I} Y_{i,1}^{\varepsilon_i(b^{\lambda})} \prod_{i \in I} Y_{i, 1}^{-\varepsilon_i(b^{\lambda})} Y_{i,0}^{\varphi_i(b^{\lambda})} = Y_{\lambda}.
\]
\end{proof}
\begin{remark}
Note that Theorem~\ref{thm:monomial_kyoto_path} implies that there exists an isomorphism $\mathcal{M}(\lambda) \cong \mathcal{M}^{1,s} \otimes \mathcal{M}(\mu)$. This is also implied by the proof of Theorem~\ref{thm:monomial_kyoto_path}, which gives an alternative proof of the existence of the isomorphism given by Equation~\eqref{eq:kyoto_prime}. Moreover, this also recovers Theorem~\ref{thm:kyoto_embedding}.
\end{remark}
\begin{ex}
Consider type $U_q'(\asl_5)$. The ground-state path is given by
\[
\young(5) \otimes \young(4) \otimes \young(3) \otimes \young(2) \otimes \young(1) \otimes \young(5) \otimes \young(4) \otimes \young(3) \otimes \young(2) \otimes \young(1) \otimes \cdots.
\]
Therefore, by iterating the isomorphism $\Phi$ from Theorem~\ref{thm:monomial_kyoto_path}, we have
\begin{align*}
\Phi(u_{\Lambda_0}) & = Y_{0,0} Y_{4,1}^{-1} \tau_1(Y_{4,0} Y_{3,1}^{-1}) \tau_2(Y_{3,0} Y_{2,1}^{-1}) \tau_3(Y_{2,0} Y_{1,1}^{-1}) \tau_4(Y_{1,0} Y_{0,1}^{-1}) \tau_5(Y_{0,0} Y_{4,1}^{-1}) \cdots
\\ & = Y_{0,0} Y_{4,1}^{-1} Y_{4,1} Y_{3,2}^{-1} Y_{3,2} Y_{2,3}^{-1} Y_{2,3} Y_{1,4}^{-1} Y_{1,4} Y_{0,5}^{-1} Y_{0,5} Y_{4,6}^{-1} \cdots
\\ & = Y_{0,0}
\end{align*}
\end{ex}
By restricting the tensor product given above to a finite number of factors and not including the highest weight crystal, we have
\[
\mathcal{M}(Y_{-m,m}^{-s} Y_{0,0}^s) \cong \bigl(B^{1,s})^{\otimes m}.
\]
\section{Coherent limit and $B(\infty)$}
\label{sec:coherent_limit}
We describe the coherent limit $B_{\infty}$ in terms of Nakajima monomials.
Let $\mathcal{M}_{\infty}$ denote the coherent limit of $\mathcal{M}^{1,s}$. We explicitly give the crystal structure on $\mathcal{M}_{\infty}$ as follows. We define
\begin{align*}
k'_e(m) & = \begin{cases} 0 & \text{if $k_e(m)$ is undefined}, \\ k_e(m) & \text{otherwise}, \end{cases}
\\ k'_f(m) & = \begin{cases} 0 & \text{if $k_f(m)$ is undefined}, \\ k_f(m) & \text{otherwise}, \end{cases}
\end{align*}
and consider $\mathcal{M}_{\infty}$ as the closure of $\mathbf{1}$ under the modified crystal operators
\[
e_i'(m) = m A_{i, k'_e(m) - 1},
\hspace{50pt}
f_i'(m) = m A_{i, k'_f(m) - 1}^{-1}.
\]
\begin{thm}
\label{thm:monomial_coherent_limit}
The map $\Psi \colon B_{\infty} \to \mathcal{M}_{\infty}$ given by
\[
(x_1, \dotsc, x_n) \mapsto X_{1,0}^{x_1} \cdots X_{n,0}^{x_n}
\]
is a $U_q'(\asl_n)$-crystal isomophism.
\end{thm}
\begin{proof}
This is similar to the proof of Theorem~\ref{thm:single_row_isomorphism}.
\end{proof}
\begin{thm}
\label{thm:Binf_monomial}
The map
\[
\Theta \colon \mathcal{M}_{\infty} \otimes \mathcal{M}(\infty) \to \mathcal{M}(\infty)
\]
given by $m \otimes m' \mapsto m \cdot \tau_1(m')$ is a $U_q'(\asl_n)$-crystal isomorphism.
\end{thm}
\begin{proof}
Let $\mathbf{1}^{\infty}$ denote the highest weight element of $\mathcal{M}(\infty)$ (which is the constant monomial $Y_0$).
Note that $\tau_1(\mathbf{1}^{\infty}) = \mathbf{1}^{\infty}$. The proof is similar to the proof of Theorem~\ref{thm:nakajima_tensor_product} and Theorem~\ref{thm:monomial_kyoto_path}.
\end{proof}
Note that Theorem~\ref{thm:Binf_monomial} is the Nakajima monomial version of Theorem~\ref{thm:Binf_Kyoto_path}. Hence, we recover the path model for $B(\infty)$ and~\cite[Thm.~5.1]{KKS07}.
\section{Extensions and Connections}
\label{sec:extensions}
In this section, we give some connections of our results to other crystal models. We also give (potential) extensions of our results to $B^{r,1}$ and other types.
\subsection{Single columns}
We can only do $B^{1,s}$ in type $A_n^{(1)}$ using Kashiwara's crystal structure without a quotient precisely because the Dynkin diagram contains a directed cycle. We never repeat an $f_i$ along any path from $b \in B^{1,s}$ back to itself which matches that of the path in the Dynkin diagram. So we never increase our indices. This is false in all other orientations of the Dynkin diagram. However, we are able to construct $B^{r,1}$ using a modification of Nakajima monomials.
We note that elements of $B^{r,1}$ can be described by $(x_1, \dotsc, x_n)$ such that $0 \leq x_i \leq 1$ and $\sum_{i=1}^n x_i = r$ with the crystal structure the same as the vector representation given in Section~\ref{sec:KR_crystals}. Hence, we have the following fact, which does not appear in the literature as far as we are aware, but is likely known to experts.
\begin{prop}
\label{prop:column_embedding}
There exists a crystal embedding $B^{r,1} \to B^{1,r}$.
\end{prop}
Therefore, we can quotient our monomials by $X_{i,k}^2$ and obtain a description for $B^{r,1}$ in terms of Nakajima monomials. Explicitly, given a monomial $m$, define modified crystal operator $\overline{f}_i$ by $\overline{f}_i(m) = f_i(m)$ if $f_i(m)$ does not contain an $X_{i,k}^2$ for some $(i, k) \in I \times \mathbb{Z}$ and $\overline{f}_i(m) = 0$ otherwise. The definition of $\overline{e}_i$ is defined similarly by replacing $f_i$ with $e_i$. Let $\overline{\mathcal{M}}(m)$ denote the closure of $m$ under $\overline{e}_i$ and $\overline{f}_i$.
\begin{prop}
\label{prop:monomial_columns}
For any $k \in \mathbb{Z}$, we have
\[
B^{r,1} \cong \overline{\mathcal{M}}\left( \prod_{i=1}^r X_{i,k} \right)
\]
as $U_q'(\asl_n)$-crystals.
\end{prop}
Recall that the highest weight $U_q(\fsl_n)$-crystal $B(\clfw_r)$ corresponding to $B^{r,1}$ can be constructed from the exterior product of the basic representation $B(\clfw_1)$ of $U_q'(\fsl_n)$. Therefore, it is more natural to consider the variables $X_{i,k}$ as anticommuting variables to describe $B(\clfw_r)$ using Nakajima monomials (up to a sign).
\subsection{Relations to other models}
Recall the abacus model from~\cite{Tingley08}, where we model $B(\lambda)$ by $\ell$ strings, with $\ell$ being the level of $\lambda$, with beads and the crystal operators act by moving a bead. Fix some abacus configuration $\psi$. Let $\psi_i^k$ denote the number of beads such that it is the $k$-th bead in a string in position $i$ modulo $n$. Define
\[
\Phi(\psi) = \prod_{i \in I} \prod_{k=0}^{\infty} X_{i,k}^{\psi_i^k},
\]
and we note that this is just a translation of the map $J$ described in~\cite[Thm.~5.1]{Tingley08} in terms of our Nakajima monomial model. Therefore, $\Phi$ is a $U_q'(\asl_n)$-crystal isomorphism.
Additionally, a bijection between Nakajima monomials and quiver varieties was constructed in~\cite[Thm.8.5]{ST14}. Indeed, we realize $\widehat{B}^{1,1}$ by the quiver variety represented by an infinite string of arrows
\[
[b] =
\begin{array}{ccccccc}
\cdots & \longrightarrow & (b-2) & \longrightarrow & (b-1) & \longrightarrow & b
\end{array}.
\]
where $f_i [b]$ adds an arrow $b \to b+1$ if $b \equiv i \mod n$ and is $0$ otherwise. We can consider this as an infinite number of bins with a single ball, where the crystal operator $f_i$ moves the ball if it is in the $b$-th bin, where $b \equiv i \mod n$. Thus, we can model $B^{1,1}$ by considering the bins in a cycle of length $n$. Furthermore, from the projection of $\widehat{B}^{1,1}$ onto $B^{1,1}$, we expect a description of $B^{1,1}$ using cyclic quiver varieties.
\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=0.5]
\foreach \x in {0, 72, 144, 216, 288}
{
\draw (\x:5)+(-0.5,0.5) -- +(-0.5,-0.5) -- +(0.5, -0.5) -- +(0.5, 0.5);
\draw[->] (\x+10:5) to[in=\x-30,out=\x+100] (\x+60:5);
\pgfmathtruncatemacro{\label}{\x / 72};
\draw (\x:5)+(1,0) node {\label};
}
\draw[fill=blue!50] (0:5) circle[radius=0.35];
\draw[fill=red!50] (144:5) circle[radius=0.35];
\draw[fill=yellow!50] (216:5) circle[radius=0.35];
\draw[fill=green!50] (246:5) circle[radius=0.35];
\draw (0:5) node {\small $1$};
\draw (144:5) node {\small $2$};
\draw (216:5) node {\small $3$};
\draw (246:5) node {\small $4$};
\draw[->, thick, dashed] (226:5.8) to[in=180,out=312] node[midway,below] {$f_3$} (274:5.8);
\end{tikzpicture}
\end{center}
\caption{The model of $B^{1,4}$ for $U_q(\asl_5)$ using balls in bins.}
\label{fig:ball_representation}
\end{figure}
We can extend this ball-bin model into $B^{1,s}$ by using $s$-colored balls, where $f_i$ moves the largest colored ball in bin $i$. See Figure~\ref{fig:ball_representation} for an example. Similarly, we can construct $\widehat{B}^{1,s}$ by starting with an $s$-fold stack of $[0]$ and $f_i$ adding an arrow to the bottom string (if possible).
\begin{ex}
Consider $\widehat{B}^{1,3}$ for $U_q'(\asl_4)$. We can represent the element $(1, 0, 2, 0)(3)$ by the infinite strings
\[
[13, 15, 15] =
\begin{array}{ccccccccccc}
\cdots & \longrightarrow & 11 & \longrightarrow & 12 & \longrightarrow & 13
\\ \cdots & \longrightarrow & 11 & \longrightarrow & 12 & \longrightarrow & 13 & \longrightarrow & 14 & \longrightarrow & 15
\\ \cdots & \longrightarrow & 11 & \longrightarrow & 12 & \longrightarrow & 13 & \longrightarrow & 14 & \longrightarrow & 15
\end{array}.
\]
Then, we have
\begin{align*}
f_1 [13, 15, 15] & = [14, 15, 15] =
\begin{array}{ccccccccccc}
\cdots & \longrightarrow & 11 & \longrightarrow & 12 & \longrightarrow & 13 & \longrightarrow & 14
\\ \cdots & \longrightarrow & 11 & \longrightarrow & 12 & \longrightarrow & 13 & \longrightarrow & 14 & \longrightarrow & 15
\\ \cdots & \longrightarrow & 11 & \longrightarrow & 12 & \longrightarrow & 13 & \longrightarrow & 14 & \longrightarrow & 15
\end{array},
\\ f_3 [13, 15, 15] & = [13, 15, 16] =
\begin{array}{ccccccccccccc}
\cdots & \longrightarrow & 11 & \longrightarrow & 12 & \longrightarrow & 13 & \longrightarrow & 14
\\ \cdots & \longrightarrow & 11 & \longrightarrow & 12 & \longrightarrow & 13 & \longrightarrow & 14 & \longrightarrow & 15
\\ \cdots & \longrightarrow & 11 & \longrightarrow & 12 & \longrightarrow & 13 & \longrightarrow & 14 & \longrightarrow & 15 & \longrightarrow 16
\end{array}.
\\
\end{align*}
\end{ex}
We can also realize $B^{r,1}$ by considering the $n$ bins placed on a cycle, but now each bin can hold at most one ball. The crystal operators $f_i$ acts by moving a ball from bin $i$ to bin $i+1$ if possible, and otherwise is defined as $0$.
We recall that bijections between the partition model, the abacus model, and cylindric plane partition model for $B(\lambda)$ are given in~\cite{Tingley08}. A bijection $\Xi$ between the Nakajima monomial model and the partition model for $B(\Lambda_i)$ was given in~\cite{Tingley10}. Thus, our bijection $\Phi$ connects these two results, and moreover, generalizes the bijection $\Xi$ to higher levels.
\subsection{$R$-matrix kernel and other types}
From Theorem~\ref{thm:nakajima_tensor_product}, we can construct the tensor product $B^{1,1} \otimes B^{1,1}$ by considering the product $\mathcal{M}^{1,1}$ with its shifted version $\tau_1(\mathcal{M}^{1,1})$. However, if we want to avoid the shift, we can still construct $\mathcal{M}^{1,1} \otimes \mathcal{M}^{1,1}$ by multiplication but having multiplication twisted by a generic parameter $t$. This is a special case of the results of~\cite{Nakajima03, Nakajima03II, Nakajima04, Hernandez04, KN12} expressed in terms of Kashiwara's variation of Nakajima monomials. The kernel of the appropriate parameterized $R$-matrix is generated by
\[
K = \{m \otimes m' - t m' \otimes m \mid m \neq m' \in \mathcal{M}^{1,1}\}.
\]
Note that we can construct the elements of $K$ by taking all twisted commutators for the $q,t$-character, where we consider $\otimes$ as multiplication.
We also recall there is a statistic called \defn{energy} on tensor products of KR crystals~\cite{KKMMNN91,KKMMNN92}. We denote the energy of $b$ by $E(b)$. If we take the quotient of $\mathcal{M}^{1,1} \otimes \mathcal{M}^{1,1}$ where we consider elements as $t^{E(b)} b$, the quotient by $K$ results in $\mathcal{M}^{1,2}$. This can also be extended to $\mathcal{M}^{1,s}$.
\begin{figure}
\[
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\node (node_7) at (166.5bp,228.5bp) [draw,draw=none] {$Y_{0,1}^{-1} Y_{1,0} \otimes Y_{1,1}^{-1} Y_{2,0} $};
\node (node_6) at (205.5bp,82.5bp) [draw,draw=none] {$Y_{1,1}^{-1} Y_{2,0} \otimes Y_{0,0} Y_{2,1}^{-1} $};
\node (node_5) at (356.5bp,301.5bp) [draw,draw=none] {$Y_{1,1}^{-1} Y_{2,0} \otimes Y_{0,1}^{-1} Y_{1,0} $};
\node (node_4) at (258.5bp,155.5bp) [draw,draw=none] {$Y_{0,0} Y_{2,1}^{-1} \otimes Y_{1,1}^{-1} Y_{2,0} $};
\node (node_3) at (291.5bp,9.5bp) [draw,draw=none] {$Y_{0,0} Y_{2,1}^{-1} \otimes Y_{0,0} Y_{2,1}^{-1} $};
\node (node_2) at (288.5bp,228.5bp) [draw,draw=none] {$Y_{0,0} Y_{2,1}^{-1} \otimes Y_{0,1}^{-1} Y_{1,0} $};
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\node (node_0) at (70.5bp,125.5bp) [draw,draw=none] {$Y_{1,1}^{-1} Y_{2,0} \otimes Y_{1,1}^{-1} Y_{2,0} $};
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\end{tikzpicture}
\quad
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\node (node_5) at (99.025bp,9.5bp) [draw,draw=none] {$Y_{0,0}^{2} Y_{2,1}^{-2} $};
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\node (node_3) at (50.025bp,82.5bp) [draw,draw=none] {$Y_{0,0} Y_{1,1}^{-1} Y_{2,0} Y_{2,1}^{-1} $};
\node (node_2) at (99.025bp,301.5bp) [draw,draw=none] {$Y_{0,1}^{-2} Y_{1,0}^{2} $};
\node (node_1) at (127.03bp,155.5bp) [draw,draw=none] {$Y_{0,0} Y_{0,1}^{-1} Y_{1,0} Y_{2,1}^{-1} $};
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\end{tikzpicture}
\]
\caption{The $U_q'(\asl_3)$-crystal $\mathcal{M}^{1,1} \otimes \mathcal{M}^{1,1}$ (left) and $\mathcal{M}^{1,2}$ (right).}
\label{fig:tensor_product}
\end{figure}
\begin{ex}
Consider the $U_q'(\asl_3)$-crystals $\mathcal{M}^{1,1} \otimes \mathcal{M}^{1,1}$ and $\mathcal{M}^{1,2}$ (see Figure~\ref{fig:tensor_product}). The graded $q$-character of $\mathcal{T} = \mathcal{M}^{1,1} \otimes \mathcal{M}^{1,1}$ is
\begin{align*}
\sum_{m \otimes m' \in \mathcal{T}} t^{E(m \otimes m')} m \cdot m' & =
Y_{0,1}^{-2} Y_{1,0}^{2} + \left(t + 1\right) Y_{0,0} Y_{1,1}^{-1} Y_{2,0} Y_{2,1}^{-1} + \left(t + 1\right) Y_{0,0} Y_{0,1}^{-1} Y_{1,0} Y_{2,1}^{-1}
\\ & \hspace{10pt} + \left(t + 1\right) Y_{0,1}^{-1} Y_{1,0} Y_{1,1}^{-1} Y_{2,0} + Y_{1,1}^{-2} Y_{2,0}^{2} + Y_{0,0}^{2} Y_{2,1}^{-2}.
\end{align*}
By considering the (graded) decomposition into $U_q(\fsl_3)$-crystals, we get the same decomposition (after $t \mapsto t^2$) as computed by~\cite{Nakajima03, Nakajima03II, Nakajima04, Hernandez04} (note that we are using different Nakajima monomials). This correspondence is an example of the results of~\cite{KN12}. However, if we take the quotient of the graded $q$-character generated by $K$ (with replacing $\otimes$ by $\cdot$), we obtain
\[
Y_{1,1}^{-2} Y_{2,0}^{2} + Y_{0,0} Y_{1,1}^{-1} Y_{2,0} Y_{2,1}^{-1} + Y_{0,0} Y_{0,1}^{-1} Y_{1,0} Y_{2,1}^{-1} + Y_{0,1}^{-2} Y_{1,0}^{2} + Y_{0,0}^{2} Y_{2,1}^{-2} + Y_{0,1}^{-1} Y_{1,0} Y_{1,1}^{-1} Y_{2,0},
\]
which is the graded $q$-character of $\mathcal{M}^{1,2}$.
\end{ex}
\begin{figure}
\begin{center}
\begin{tikzpicture}[>=latex,line join=bevel,xscale=1.1, yscale=0.9, every node/.style={scale=0.9},baseline=0]
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\end{tikzpicture}
\hspace{60pt}
\begin{tikzpicture}[>=latex,line join=bevel,scale=0.9, every node/.style={scale=0.9}]
\node (node_2) at (104.0bp,155.5bp) [draw,draw=none] {$Y_{0,2} Y_{1,2}^{-1} $};
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\draw (112.5bp,338.0bp) node {$1$};
\draw [black,<-] (node_1) ..controls (178.218bp,340.0bp) and (178.218bp,190.0bp) .. (node_2);
\draw (175.5bp,265.0bp) node {$0$};
\end{tikzpicture}
\end{center}
\caption{A portion of the level-zero crystal $\mathcal{M}(Y_{0,1}^{-1} Y_{1,0})$ (left) and $\mathcal{M}^{1,1}$ constructed from $\mathcal{M}(Y_{0,1}^{-1} Y_{1,0}) / \kappa$ (right) in type $C_2^{(1)}$.}
\label{fig:type_C2}
\end{figure}
Our construction works because $B^{1,1}$ in type $A_n^{(1)}$ ``follows'' our orientation of the Dynkin diagram. More generally, in order to construct $\mathcal{M}^{1,1}$, we need to take a quotient of a level-zero crystal $\mathcal{M}(m)$, where $m$ is some monomial whose weight is of level $0$. For example, in type $C_2^{(1)}$ with $c_{ij} = 1$ if $i < j$ and $0$ otherwise, we can construct $\mathcal{M}^{1,1}$ as a quotient of $\mathcal{M}(Y_{0,1}^{-1} Y_{1,0})$ by an automorphism $\kappa$ given by $Y_{i,k}^{-1} Y_{i',k'} \mapsto Y_{i,k-2}^{-1} Y_{i',k'-2}^{-1}$ as in Figure~\ref{fig:type_C2}. However, $\mathcal{M}^{1,1} \cdot \mathcal{M}^{1,1}$ is not isomorphic to $B^{1,2}$ as the former has 10 elements and the latter has 11. This also agrees with instead taking $\mathcal{M}(Y_{0,1}^{-2} Y_{1,0}^2)$ and then taking the quotient by $\kappa$.
Additionally, the quotient of the kernel of the parameterized $R$-matrix corresponding to $B^{1,1} \otimes B^{1,1}$ gives rise to a twisted commutator in type $A_n^{(1)}$. Therefore, we expect that by considering the variables in $\mathcal{M}^{1,1}$ as non-commuting variables and then taking an appropriate quotient by the kernel of the $R$-matrix, we could construct general $B^{r,s}$ for all affine types. Furthermore, by relating the extra parameter in the kernel to energy, we expect to recover the results of~\cite{KN12} that relate the $q,t$-characters of standard modules to those of simple modules. Indeed, for the type $C_2^{(1)}$ case considered above, if we add the grading by energy in $\mathcal{M}^{1,1} \cdot \mathcal{M}^{1,1}$, then we obtain 14 elements. However, if we additionally quotient by the kernel of the $R$-matrix, we can construct a Nakajima monomial realization of $B^{1,2}$.
|
train/arxiv
|
BkiUaWbxK3YB9ohkJdFH
| 5 | 1 |
\section{Introduction}
Wall slip is a widespread phenomenon in the flow of various liquids. In Newtonian liquids, interest in slip has revived due to its relevance for flow in nano-porous media, microfluidic devices and along superhydrophobic surfaces [\cite{Barrat1999, Zhu2001, BocquetSoftMat07_slip, NetoRepProPhys_Newtslip}]. The so-called slip-length $l_s$, the distance to the wall at which the velocity profile
extrapolates to zero, can reach many molecular diameters, depending on
wettability, and strongly affects flow when it is comparable to the
system dimensions. Polymer slip has also received considerable
attention, e.g. [\cite{Hatzikiriakos1991,Hatzikiriakos1992,BrochardGennesLangmuir92_boundaryslip,Westover1966,Leger1997, Mhetar1998}]. Here changes in the chain relaxation dynamics near the wall govern slip: typically a transition from weak slip at small flow
rate to strong slip ($l_s$ exceeding hundreds of micrometers [\cite
{Mhetar1998}]) for large flow rate is seen, driven by chain
disentanglement near the wall. In surfactant solutions, recent work [\cite
{SalmonPRE03,BecuPRL04_bandingdynamics,Manneville2004,LettingaPRL09_WLM_slip,Becu2007}
] has shown more complex behavior, where slip and wall interactions are
coupled to the shear-banding in these systems.
The most prominent examples of slip in industrial and daily applications
occur in flow of complex, multi-phase fluids [\cite
{Yoshimura1988,Barnes1995,Larson1999}]. Meaningful characterization of the
bulk flow properties of these systems requires proper insight into boundary effects [\cite{BuscallJRheo2010_slip}]. Over the last
decades, many studies of slip in these systems have appeared, e.g. in
particulate (colloidal) suspensions [\cite
{Yilmazer1989,Aral1994,HartmanKok2004,Kalyon2005,Jana1995,Cohen2006,Isa2007,Soltani,Persello}
], colloidal gels [\cite{Buscall1993, Russel2000,
WallsJRheo03_slip,VaradanSolomonJRheo2003_gelflowconfocal,GibaudMannevillePRL08_slipyield,Wassenius2005}
] and emulsions and foams [\cite{Bertola2003,Princen1985,
Meeker2004a,Meeker2004b,Salmon2003,DenkovColSurfSciA05_foamfriction,Katgert2008}].
Despite this large body of work, it is challenging to gain microscopic
insight into the nature of slip and understand its dependence on material
composition, wall properties, and flow rate. Broadly speaking, slip results
from depletion of the dispersed phase near a smooth wall, giving a low
viscosity, high shear boundary layer which reduces the \textit{apparent}
bulk viscosity \footnote{
We refer here to `intrinsic' depletion under uniform stress, as opposed to
depletion of dispersed phase due to shear migration as may arise under
non-uniform stress.}. However, both the structure and the origin of this
``layer'' vary greatly between different systems and the interplay with
(non)linear bulk rheology can give rise to slip being pronounced either at
large or small flow rate.
Slip at low stress or applied shear rate can occur in dispersions where
caging, aggregation or `jamming' leads to a solid-like microstructure and
mechanical behavior. The system then exhibits a yield stress, $\sigma_y$,
below which the micro-structure remains intact but, depending on wall
interactions and roughness, \textit{apparent} flow of the material can
still be measured. This was recently studied for concentrated emulsions and other soft
particle pastes [\cite{Meeker2004a,Seth2008}] in presence of smooth walls.
There, elasto-hydrodynamic lubrication, associated with the particles'
deformability, causes a lubrication layer between the compressed packing and
the wall with velocity dependent thickness. This gives a nonlinear relation
between the slip stress $\sigma$ and the slip velocity $v_s$, but this
mechanism only occurs for very weakly or non-repulsive particle-wall
interactions [\cite{Meeker2004b,Seth2008}]. Note that nonlinear lubrication
is of strong interest to applications of many other soft solids, e.g.
hydrogels, where combined wall-network repulsion and adsorption determines
the friction properties [\cite{Gong2010}].
For suspensions of `hard' particles, most studies of slip have been performed
for \textit{non-Brownian} systems, i.e. at large P\'{e}clet number $\mathrm{
Pe}= \dot\gamma \tau_B >> 100$, where $\dot\gamma$ is the true bulk flow
rate and $\tau_B$ the Brownian relaxation time. There, slip occurs both in solid- [\cite
{Yilmazer1989, Kalyon2005}] and liquid-like [\cite{Jana1995}] suspensions
(albeit without consensus on the quantitative behavior), but little attention has been given to the effect of
Brownian motion on slip in colloidal systems. \cite{HartmanKok2004} found
that Brownian motion prevents depletion at low shear rates in dilute
colloids, so that slip becomes apparent only for $\mathrm{Pe} \gtrsim 10$.
However, for more concentrated colloids, `crowding' competes with Brownian
relaxation. This causes structural arrest, a glass transition [\cite
{Pusey1986,Megen1998}], which for hard-sphere (HS) colloids occurs at a
volume fraction of $\phi_g \simeq 0.58$. The associated change from liquid
to solid-like rheology [Petekidis et al. (2004)] can significantly
affect the slip response.
The slip and flow behavior in these concentrated HS colloidal suspensions is
the subject of this paper. In [\cite{Ballesta2008}] we have given a short
account of some of the results, here we present a more extensive study of
the behavior. Using rheo-microscopy [\cite{Besseling2009}], we show that for
$\phi> \phi_g$, slip becomes dominant in the rheology for smooth, non-stick
walls. We address the dependence of slip on colloid-wall (van der Waals)
interactions and concentration and quantitatively describe the transition
from slip to yielding for different combinations of confining walls via a
phenomenological model. Below the yield stress, slip causes full plug flow,
qualitatively similar to jammed emulsions and microgels [\cite{Meeker2004a,
Seth2008}], but quantitatively different; above a threshold stress, the slip
stress increases linearly with the slip velocity (Bingham slip response) due
to a sub-colloidal solvent layer with velocity-independent thickness. The
concentration dependence of the Bingham slip parameters shows a direct
relation to the osmotic pressure and yield stress of the suspensions, which
can be understood on a semi-quantitative basis but still lacks a full
theoretical description. We further show that deep in the glass regime,
shear localization effects [\cite{Besseling2010}] accompany the slip-shear
transition and discuss residual slip in concentrated liquid suspensions ($
\phi< \phi_g$), a feature not uncovered previously~[\cite{Ballesta2008}] due
to its limited effect on the bulk rheology.
The paper is structured as follows. After a description of the measurement
setup and suspensions in Sec.~\ref{sec_methods}, we present rheology and
local velocimetry results for various hard sphere suspensions, colloid-wall interactions
and confining walls conditions in Sec.~\ref{sec_results}. In Sec.~\ref
{Sec_model}, we describe the phenomenological model for slip and yielding,
generalizing the results in [\cite{Ballesta2008}] to account for geometries
with various confining walls, and in Sec.~\ref{sec_compare} we compare the
predictions with the experimental results. The physical origin of the
Bingham slip parameters is analyzed in Sec.~\ref{Sec_slipparam}, along with
the effect of van der Waals interactions. Sect.~\ref{sec_localization} and
\ref{section_slip_liq} deal with shear localization for $\phi > \phi_g$ and
residual slip in liquid suspensions for $\phi<\phi_g$, respectively, and we conclude in Sect.~\ref{disc_concl}.
\section{Samples and methods}
\label{sec_methods}
\subsection{Colloidal suspensions}
\label{subsec_colloids}
We used polymethylmethacrylate (PMMA) colloids of various sizes (radii $
a=138 $~nm, $150$~nm, $302$~nm and a fluorescent batch with $a=652$~nm,
measured by light scattering, polydispersity $\sim 15\%$), sterically
stabilized with a poly-12-hydroxystearic acid (PHSA) layer [\cite{Barret74}] and dispersed in a refractive index (RI) matching solvent mixture of decalin and tetralin (viscosity $\eta_s=2.2$~mPas). A few measurements were
also conducted using pure decalin as solvent. Non-fluorescent (RI-matched) samples were
seeded with $\sim 0.5\%$ of the fluorescent particles (labeled with
nitrobenzoxadiazole, NBD), which served as tracers during confocal imaging
(Fig.~\ref{fig_setup}(b)). In these solvents the colloids interact almost
like perfect HSs [\cite{Bryant2002}]. In the decalin-tetralin mixture the refractive index of the colloids is $\sim 1.50-1.51$, somewhat larger than the bulk PMMA value ($1.49$), due to solvent absorbtion of the colloids and slight swelling, estimated to be $\sim 10~\%$.
Batches of different volume fractions $\phi$ were prepared by diluting
samples centrifuged to a random close packed sediment, with volume fraction $
\phi_{sed}=\phi_{rcp}$. One traditional method to determine $\phi$ for HS
suspensions employs the crystal-fluid coexistence boundaries [\cite{Segre1995,Pusey1986}].
Because of polydispersity our suspensions do not crystallize, and $\phi$ has
to be determined differently [\cite{Poon2011}]. We have chosen the following method. We first
measure the mass density of the solvent $\rho_{s}$ (densitometer: Anton
Paar DMA 4500). We then measure the density $\rho_{rcp}$ of the close packed sediment
after centrifugation by taking a small sample, diluting it by a factor $
f=10$ by weight in solvent and measuring the density $\rho^*$ of this
diluted suspension. The density of the sediment is then deduced from $
\rho_{rcp}=\rho^*\rho_{s}/[(1+f)\rho_{s}-f\rho^*]$. Samples at a
given $\phi/\phi_{rcp} < 1$ for the main experiments are then prepared by
adding a solvent mass $M_s$ to the known weight $M_t$ of the stock
sediment (giving $\phi/\phi_{rcp}=[1+\frac{M_s}{M_t}\frac{\rho_{rcp}}{\rho_s}]^{-1}$) and homogenizing the sample rigorously. The densitometry results are given in table~\ref{tab0}; $\rho_{s}$ slightly increases with increasing particle
size, probably due to a slight increase in RI for larger particles
and thus a slight change in the RI-matching composition of the
solvent mixture. We emphasize that, without knowledge of the colloid mass density $\rho_c$, this method does not provide $\phi_{rcp}$ but only $\phi/\phi_{rcp}$. When mentioning absolute volume fractions, we have
assumed $\phi_{rcp}=0.67$, as found in simulations of spheres with a polydispersity of $\sim 15\%$ [\cite{fixschaert}] (monodisperse
spheres give $\phi_{rcp}=0.64$). The value of $\phi_{rcp}=0.67$ may be used to extract the colloid density via
$\phi_{rcp}=(\rho_{rcp}-\rho_s)/(\rho_c-\rho_s)$. The resulting densities $\rho_c \sim 1.12-1.13~$g/ml are reduced compared to the bulk
PMMA value of $\sim 1.2~$g/ml, but reasonable match what is expected from the above mentioned $\sim 10\%$ swelling of the particles
\footnote{We do not separately take into account the reduction of $\rho_c$ arising from the PHS-layer. The density of swollen colloids is then simply $\rho_{c}=(\rho_{PMMA}s^{-3})+[1-s^{-3}]\rho_s$ with $s \simeq 1.1$ the swelling ratio. As the swelling is not exactly known, we have used bare particle radii, measured in decalin, throughout the paper.}. We note that the effect of polydispersity and compaction rate on the value of $\phi_{rcp}$ is still under study [\cite{hermesdijkstra_epl10}] and not without controversy, see also [\cite{Poon2011}]. Yet, in relative terms, our approach is quite accurate, the main uncertainty being a variation $\delta\phi_{rcp}/\phi_{rcp} \simeq 0.005$ found between
different centrifugation runs via drying and weighting. The value of this uncertainty is
close to the maximum variation in the `jamming' fraction for our
polydispersity, as found in simulations of HSs for different compression rates [\cite{hermesdijkstra_epl10}], but our $\delta\phi_{rcp}$ (in two tests we performed, using a single batch) was obtained under similar centrifugation conditions. Moreover, the sedimentation Peclet number
${\rm Pe}_s = 4 \pi a^4 \Delta\rho g_c / 3k_B T$ (with $\Delta\rho=\rho_c-\rho_s$ and
$g_c$ the acceleration), lies in the range ${\rm Pe}_s\sim 0.005-1$ from the smallest to the largest particles. Thus, `jamming' effects (which may cause $\phi_{sed}<\phi_{rcp}$) are strongly limited, except possibly for the $a=652~$nm sample. Most important, for the analysis of the $\phi$ dependence of rheological and slip parameters (Sect.~\ref{Sec_slipparam}), we employ the directly measured {\it reduced} volume fraction $\phi/\phi_{rcp}$, which suffers the least from possible ambiguities [\cite{Poon2011}].
\begin{table}[tbp]
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
sample name & asm340 & asm247 & asm209 & asm195 \\ \hline
$a$ (nm) & 138 & 150 & 300 & 652 \\ \hline
$\rho_{s}$ (g.ml$^{-1}$) & 0.9153 & 0.915 & 0.92023 & 0.92543 \\ \hline
$\rho_{RCP}$ (g.ml$^{-1}$) & 1.0596 & 1.0594 & 1.07 & 1.0644 \\ \hline
\end{tabular}
\end{center}
\caption{Particle size $a$, density of the index-matched solvent $\rho_{s}$ and density of the random close packed sediment $\rho_{RCP}$ for the different samples.}
\label{tab0}
\end{table}
\subsection{Measurement system}
\label{subsec_instr}
We use a stress-controlled rheometer (AR2000, TA Instruments) with a
cone-plate geometry (radius $r_c=20$~mm, cone angle $\theta=1^{\circ}$
unless mentioned otherwise) and a glass slide (radius $25$~mm, thickness $\sim 180$~$\mu$m) as the bottom plate. A solvent trap saturates the
atmosphere around the sample minimizing evaporation. We image the flow with
a confocal scanner (VT-Eye, Visitech International) through the glass plate
via a piezo-mounted objective (oil immersion, magnification $\geq 60\times$
), and optics mounted on an adjustable arm [\cite{Besseling2009}]. We take
2D movies in the flow-vorticity ($x$--$y$) plane at a frame rate $\leq 90$
~Hz at various distances $r$ from the cone center and at equally spaced depths $z \leq h$, with $z$ the velocity-gradient direction and $h$ the local gapsize (Fig.~\ref{fig_setup}(a)), from which we extract,
via image analysis [\cite{Besseling2009}], the velocity profiles $v(z)$ at
different $r$. The typical error bar in the reduced velocity $v(z)/v(h)$ is $\lesssim 5\%$, achieved by imaging sufficiently large displacements, i.e. over timescales of at least a few times $1/\dot\gamma_a$ (see also [\cite{Besseling2010}]). The observation window is
$\sim 50\times 50$~$\mu$m$^2$, similar to the gap sizes where imaging is performed ($r=2$--$10$~mm, $h=35$--$175$~$\mu$m). Over the
observation window, the variation in $h$ ($ \delta h \simeq 0.9$~$\mu$m) is
negligible, $\delta h /h \ll 1$, and the geometry locally mimics
parallel plates separated by $h=r\tan (\theta)$. Experiments
were performed at controlled applied shear rate $\dot\gamma_a$ (using the rheometer's fast feedback), going from
high to low rates, unless stated otherwise. Stress controlled measurements
gave the same results; in particular, we do not find differences between `static' (measured for increasing stress) and dynamic stress thresholds for flow or slip (the latter measured on reducing $\sigma$ or $\dot\gamma_a$), except for possible initial hysteresis just after loading of very concentrated samples (Sect.~\ref{Sec_slipparam}).
\begin{figure}[tbp]
\center\scalebox{1}{\includegraphics{fig1.eps}}
\caption{(a) Cone-plate rheometer with transparent plate and optics
connected via an adjustable arm to the confocal scanner. The cone angle $
\protect\theta$ and radius $r_c$ are shown. (b) Confocal image of an
RI-matched suspension with the $a=652$~nm fluorescent tracers showing as
bright spots. Scale bar: $10~\protect\mu$m. (c) AFM image of the sintered layer of $a=652$~nm colloids on the glass slide. The color scale marks height variations of $\sim 500~$nm. Scale bar: $2.5~\protect\mu$m. Inset: confocal image of a coated slide on a larger scale, $50\times50~\protect\mu$m.}
\label{fig_setup}
\end{figure}
\subsection{Wall properties}
\label{subsec_wallprops}
We have used the following preparations of the surfaces of the cone-plate
geometry: (\textit{i}) To prevent slip, the cone, the glass slide or both
can be made rough on a scale similar to or larger than the particle radius by
spin-coating a $\phi \sim 0.3$ suspension of $a=652~$nm radius particles and
sintering the resulting dense disordered colloidal monolayer for one hour at
$\sim 120^{\circ}~$C in a vacuum oven. This sintering causes adhesion of the colloids to the glass (or metal cone), probably accompanied by local redistribution of the PHS stabilizer at these temperatures. Nevertheless, the sintering leaves the corrugated nature of the disordered monolayer intact, giving a wall roughness $\sim 500~$nm, see Fig.~\ref{fig_setup}(c) and the inset. The suppression of slip associated with this rough coating has been evidenced in [\cite{Besseling2009}] (Fig. 16 in that paper) and is also shown in Sect.~\ref{sec_localization}. (\textit{ii}) Use of the glass slides
without coating gives a surface which is very smooth on the colloidal scale
(local roughness $<1~$nm measured by AFM ). The slides were used either
untreated or cleaned with ethanol, methanol or a Piranha solution ($98\%$ H$
_2$SO$_4$:$30\%$H$_2$O$_2$ aqueous solution, ratio $7:3$ by mass ), but these
cleaning methods did not give systematic differences in slip behavior. We
also performed temperature-controlled rheological measurements (without
imaging) using the smooth glass slides as bottom plate, achieved by
anchoring the slides on the rheometer Peltier plate using thermal paste. The
stainless steel surface of the cone (cleaned with acetone) is also smooth on
the colloidal scale, but interacts differently with the colloids (Sect.~\ref{sec_wallinteraction}).
\section{Main experimental results}
\label{sec_results}
Rheological studies of concentrated HS suspensions have shown the
emergence of a finite dynamic yield stress for $\phi \gtrsim 0.57-0.58$ [\cite{Petekidis2004, Pham2006}]. For $\phi<\phi_g$, the flow curve, i.e. a plot
of the measured stress ($\sigma_m$) versus applied shear rate ($\dot\gamma_a
), exhibits Newtonian behavior at low $\dot\gamma_a$ followed by strong
shear thinning at higher $\dot\gamma_a$, while for $\phi>\phi_g$ suspensions are
glassy and have a yield stress $\sigma_y$ below which no flow occurs. In this case the flow curves exhibit a Herschel-Bulkley behavior
described by $\sigma_m=\sigma_y+\alpha\dot\gamma_a^n$. This behavior is
geometry independent [\cite{Pham2008}], at least for $\phi \lesssim 0.6$,
indicating that $\dot\gamma_a$ and the bulk flow rate of the material,
\dot\gamma$, are the same. For larger $\phi$, different geometries may
cause small differences for $\sigma \simeq\sigma_y$ due to shear-banding
\cite{Besseling2010}], but the bulk $\sigma(\dot\gamma)$ behavior is
nevertheless consistent with a HB form.
\begin{figure}
\center\scalebox{1}{\includegraphics{fig2.eps}}
\caption{Measured shear stress $\protect\sigma_m$ versus applied shear rate
\dot\protect\gamma_a$ for $a=300$~nm colloids at $\protect\phi \sim 0.64$ in
RI-matching solvent ($\square$) and decalin ($\bullet$), using a smooth
glass slide and coated cone. Full line: fit of the low $\dot\protect\gamma_a$
branch to Eq.~(\protect\ref{eq:binghameta}), giving $\protect\eta_{\rm eff}=38.3~ $Pa$\cdot$s, $\protect\sigma_s=2.94~$Pa. The data for decalin are
multiplied by a factor $1.5$ for comparison, the difference with the
RI-matching solvent is due to slightly different $\protect\phi$.}
\label{fig_im_nim}
\end{figure}
We first illustrate how the rheology for $\phi > \phi_g$ is affected when
the colloid-wall interaction is changed. Figure~\ref{fig_im_nim} shows the
stress measured with the rheometer, $\sigma_m$, versus the \textit{applied}
shear rate $\dot\gamma_a$ for a suspension in pure decalin as well as in the
RI-matching decalin-tetralin mixture, at similar volume fractions. For the
RI-mismatched sample, the flow curve clearly shows a yield stress and an
overall Herschel-Bulkley response. However, in the RI-matched sample, the
flow curve at low $\dot\gamma_a$ exhibits a branch with Bingham-like
behavior:
\begin{eqnarray}
\sigma_m=\sigma_s+\eta_{\mathrm{eff}} \dot\gamma_a. \label{eq:binghameta}
\end{eqnarray}
Here $\sigma_s$ is a threshold stress below which the (apparent) flow
completely stops and $\eta_{\mathrm{eff}}$ is an effective viscosity
characterizing the stress increase in this branch. For large $\dot\gamma_a$
the stress attains the same nonlinear behavior as the sample in decalin. As shown below,
the small $\dot\gamma_a$ behavior marks full slip along the glass slide (due to the
strongly suppressed colloid-wall vdW attraction) with a vanishing shear rate $\dot\gamma$ in the bulk of the sample.
\begin{figure}
\scalebox{1}{\includegraphics{fig3.eps}}
\caption{(a,b) Measured stress versus $\dot\protect\gamma_a$ for coated cone
and plate ($\circ$) and with (a) uncoated plate ($\square$) and $\protect\ph
=0.59$ and $a=138$~nm, and (b) uncoated cone and uncoated plate ($\square$)
with $\protect\phi=0.59$ and $a=150$~nm. Regime $\mathrm{I}$ in (a) and
\mathrm{I_a}$ in (b) represent full slip along one boundary; regime $\mathrm
I_b}$ in (b) represents full slip along two boundaries; and regime $\mathrm{II}$ in
(a),(b) mark slip plus bulk flow. In (a),(b) the dash-dot curves are
Hershel-Bulkley fits with $n=0.5$, Eq.~(\protect\ref{eq_HB}), giving
\protect\alpha=6.1~$Pa$\cdot$s$^{1/2}$, $\protect\sigma_y=5.5~$Pa in (a) and
\protect\alpha=8~$Pa$\cdot$s$^{1/2}$, $\protect\sigma_y=14~$Pa in (b); dashed
curves are fits to the Bingham form Eq.~\protect\ref{eq:binghameta}. In
regime $\mathrm{I}$ in (a), where Eq.~(\protect\ref{eq_sigetafI}) applies,
this gives $\protect\beta=8.2 \cdot 10^4~$Pa s/m, $\protect\sigma_s=1.8~$Pa.
In regime $\mathrm{I_a}$ or $\mathrm{I_b}$ in (b), where Eq.~(\protect\re
{eq_sigetafI}) and Eq.~(\protect\ref{eq_sigetafIb}) apply, the parameters
are $\protect\beta=8.7.10^4~$Pa$\cdot$s$\cdot$m$^{-1}$, $\protect\sigma_1=5.2~$Pa, $\protec
\sigma_2=3.4~$Pa; full lines in regime II are the global flow curves from
Eq.~\protect\ref{eq_sigmamtot} in Appendix~\protect\ref{appsec_slipone} using
the above parameters. (c) Normalized velocity profiles $v(z)/v_{cone}$ for
the suspension in (a) at $r=3$~mm with coated surfaces (filled symbols) and
at $r=2.5$~mm with uncoated plate (open symbols) for various $\dot\protec
\gamma_a$. (d) $v(z)/v_{cone}$ for the data in (b) for uncoated cone and plate, at $r=5.5$~mm and
various $\dot\protect\gamma_a$. Full lines: linear fits. Dotted lines:
behavior without slip.}
\label{fig_globalrheo}
\end{figure}
We next discuss results for different combinations of smooth and colloid-coated surfaces. Figure~\ref{fig_globalrheo} shows flow curves
for RI-matched samples with both walls or one wall coated, Fig.~\re
{fig_globalrheo}(a), and with two uncoated walls (smooth glass plate and
stainless steel cone), Fig.~\ref{fig_globalrheo}(b). With both surfaces
coated (Fig.~\ref{fig_globalrheo}(a), open circles), a Herschel-Bulkley
behavior is recovered, confirming that coating prevents slip, as we also
directly observed via microscopy [\cite{Besseling2007,Ballesta2008}]. If only
one surface is coated (squares in Fig.~\ref{fig_globalrheo}(a)), we observe
a similar slip response as described above: below a critical applied shear
rate $\dot\gamma_{a,c}$ bulk shear vanishes in the entire gap (see
below and the appendix) and we have a Bingham response; we define this as
regime I. For $\dot\gamma_a>\dot\gamma_{a,c}$ the flow curve deviates from
this Bingham regime and approaches the flow curve obtained with coated geometry, this regime is noted regime
II. If both walls are uncoated, squares in Fig.~\ref{fig_globalrheo}(b), the
flow curve exhibits two successive Bingham regimes at low shear rate:
\mathrm{I_a}$ marking slip at the plate, $\mathrm{I_b}$ marking slip at both
surfaces (see Sec.~\ref{Sec_model}); the second Bingham regime, $\mathrm{I_
}$, has a higher slip stress and an effective viscosity $\eta_{\mathrm{eff}}$
half the value of that in regime $\mathrm{I_a}$. Eventually, at high shear
rate, the curve again tends toward the HB flow curve.
We now turn to the velocimetry results obtained by simultaneous confocal
imaging of the flow, Fig.~\ref{fig_globalrheo}(c,d). In Fig.~\re
{fig_globalrheo}(c) the filled symbols show flow profiles with both walls
coated. These profiles are nearly linear and $v(z)$ reaches zero at the
glass plate and the applied velocity at the cone, showing that the coating
provides a no-slip condition. The open symbols in Fig.~\ref{fig_globalrheo
(c),(d) show $v(z)$ for one uncoated surface (c) or two uncoated surfaces
(d). In both cases $v(z)$ is essentially linear but exhibits slip at the
non-coated walls. The profiles can be fitted by $v=v_s+\dot\gamma z$, with
v_s$ the slip velocity at the plate and $\dot\gamma$ the bulk shear rate. At
the smallest $\dot\gamma_a$, $\dot\gamma$ is zero; the suspension sticks to
the coated cone and rotates as a solid body, slipping over the smooth glass.
This causes the apparent flow below the yield stress in regime $\mathrm{I}$
in Fig.~\ref{fig_globalrheo}(a) or $\mathrm{I_a}$ in (b). Figure~\re
{fig_globalrheo}(c) shows that plug flow can persist in regime II for
r=2.5~ $mm and $\dot\gamma_a=0.5~$s$^{-1}$, as explained later. With two
uncoated surfaces, Fig.~\ref{fig_globalrheo}(d), the velocity profiles for
intermediate $\dot\gamma_a$ show slip at both walls, with solid body
rotation at a fraction of the cone velocity. This occurs in regime $\mathrm
I_b}$ and, as shown later, also in regime $\mathrm{II}$
at small $r$. Eventually, for applied rates $\dot\gamma_a \geq 5~$s$^{-1}$,
both in Fig.~\ref{fig_globalrheo}(c) and (d) the bulk of the sample starts
to yield, $\dot\gamma>0$. The difference between $\dot \gamma$ and
\dot\gamma_a$ decreases on increasing $\dot\gamma_a$ and the flow curve
approaches bulk HB behavior.
\begin{figure}
\scalebox{1}{\includegraphics{fig4.eps}}
\caption{(a) Flow curves for two uncoated surfaces, $a=150$~nm, and various
\protect\phi$. The lower branches of the full lines are fits to Eqs.~
\protect\ref{eq_sigetafI},\protect\ref{eq_sigetafIb}), the upper branch is
Eq.~\protect\ref{eq_sigmamtot} for slip at two surfaces. The parameters are
\protect\sigma_{1,2}=[7,5.7]~$Pa, $\protect\beta=19.0 \cdot 10^4~$Pa$\cdot$s$\cdot$m$^{-1}$,
\protect\sigma_y=23.5~$Pa, $\protect\alpha=18.0~$Pa$\cdot$s$^{1/2}$ for $\protec
\phi=0.61$ and $\protect\sigma_{1,2}=[28,16.6]~$Pa, $\protect\beta=26.3\cdot
10^4~$Pa$\cdot$s$\cdot$m$^{-1}$, $\protect\sigma_y=82~$Pa, $\protect\alpha=50.6~$Pa$\cdot$s$^{1/2}$
for $\protect\phi=0.63$ (the $\phi=0.59$ parameters are as in Fig.~\ref{fig_globalrheo}b). (b) Ratio of the slip threshold stresses at the
bottom and top surface, $\protect\sigma_2/\protect\sigma_1$, versus $\protec
\phi$, the dotted line is the average value.}
\label{fig_op2slip}
\end{figure}
The typical behavior described above, with the transition from a Bingham to
a HB flow regime, is representative for all RI-matched suspensions with
\phi > 0.57$ and at least one smooth wall. We illustrate this in Fig.~\re
{fig_op2slip} and Fig.~\ref{powerlawone}. The former shows the two-step
Bingham behavior for two uncoated walls, as in Fig.~\ref{fig_globalrheo}(b)
but now including two other concentrations. Both the slip threshold stress (
\sigma_1 $ for the cone, $\sigma_2$ for the plate) and $\sigma_y$ increase
strongly with $\phi$. Figure.~\ref{fig_op2slip}(b) shows that the ratio
between the slip threshold stress for the two surfaces is essentially
independent of $\phi$. In Fig.~\ref{powerlawone} we show
(\sigma-\sigma_s)/\eta_{\mathrm{eff}}$ versus applied shear rate for
suspensions with $a=138$~nm at various $\phi$, using a coated cone and
smooth glass plate. Plotted this way, all curves at small $\dot\gamma_a$
(regime I) are linear and overlap, following Eq.~(\ref{eq:binghameta}). In
addition to the threshold stress, also the fitted values of the Bingham
viscosity show a strong $\phi$-dependence, both of which are discussed in Sec.~\re
{Sec_slipparam}.
\begin{figure}
\center\scalebox{1}{\includegraphics{fig5.eps}}
\caption{Shear stress minus slip stress $\protect\sigma_s(\protect\phi)$
divided by the Bingham viscosity $\protect\eta_{\mathrm{eff}}(\protect\phi)$
for $a=138$~nm and $\protect\phi=0.585$--$0.65$. $\protect\sigma_s$ and
\protect\eta_{\mathrm{eff}}$ were extracted from fits of the small $\do
\protect\gamma_a$ behavior to Eq.~(\protect\ref{eq:binghameta}). Dashed
line: $\protect\sigma_m-\protect\sigma_s=\dot\protect\gamma_a \protect\eta_
\mathrm{eff}}$.}
\label{powerlawone}
\end{figure}
Figure~\ref{fig_localrdep}(a) shows velocity profiles from the experiment in
Fig.~\ref{fig_globalrheo}(a) ($\phi=0.59$, coated cone, uncoated plate), but
at a constant applied shear rate $\dot\gamma_a=1.1$~s$^{-1}$, at various
distances $r$. For this applied rate, $\sigma_m$ is close to its value
measured in the absence of slip (see Fig.~\ref{fig_globalrheo}(a)), but
slip is nevertheless visible. For large $r$, slip and bulk flow are
simultaneously present, the mechanism governing slip in regime I thus
persists when bulk shear is present. The slip gets more pronounced for
smaller $r$, which is shown directly in Fig.~\ref{fig_localrdep}(b) via the
r$-dependence of the local shear rate $\dot\gamma(r)$ extracted from Fig.
\ref{fig_localrdep}(a): $\dot\gamma(r)$ decreases strongly with $r$ and
eventually vanishes completely for $r< r_y \simeq 2.5~$mm, with $r_y$ a
``yielding radius'' inside of which no shear is present. These data demonstrate
directly that, in presence of slip, the stress in a cone-plate, while
essentially uniform across the gap, is radially non-uniform: for small $r$,
\sigma(r) < \sigma_y$, while for larger $r$, $\sigma(r)>\sigma_y$.
\begin{figure}
\scalebox{1}{\includegraphics{fig6.eps}}
\caption{(a) Velocity profiles for fixed shear rate $\dot\protect\gamm
_a=1.1 $~s$^{-1}$, $\protect\phi=0.59$ and $a=138$~nm with uncoated glass
and coated cone, at selected distances $r$ from the center. (b)
Corresponding local shear rate versus $r$. Full line: Eq.~(\protect\re
{eq_1u_dg_dga}) with $h=r \tan(\protect\theta)$, using Eq.~(\protect\re
{eq_param1slip}) and rheological parameters as in Fig.~\protect\re
{fig_globalrheo}(a). Inset: corresponding $r$-dependence of the normalized slip velocity $\tilde{v}_s=v_2/(\dot\gamma r \tan{\theta})$, along
with the prediction using Eqs.~(\protect\ref{eq_ratevslip}),(\protect\ref{eq_1u_dg_dga}) and (\protect\ref{eq_param1slip}) with $v_1=0$
(full line).}
\label{fig_localrdep}
\end{figure}
The velocity profiles at small $\dot\gamma_a$ in Figs.~\ref{fig_globalrheo} suggest that
in regime $\mathrm{I}$ the suspension moves as a solid body, without bulk
shear, down to the glass plate. To get direct evidence for this and exclude
locally non-uniform flow as seen in colloidal crystals [\cit
{Derks2004,Cohen2006}], we imaged the first layers of a suspension of fully
fluorescent particles ($a=652$~nm) in regime I in the RI-matching solvent.
As reported in Fig. 2a of [\cite{Ballesta2008}], in regime $\mathrm{I}$
below the critical applied shear rate $\dot\gamma_{a,c}$ mentioned earlier,
we observed plug flow with $v_s=v_{cone}$ down to the first layer of
particles. There is thus no highly sheared layer of colloids and the Bingham
slip response, Eq.~\ref{eq:binghameta}, reflects the behavior of a
lubrication layer between the first particles and the wall. Due to the
smooth wall, the colloids do exhibit some layering at the surface, but this
vanishes completely beyond $\sim 4$ particle diameters (Fig. 2a of [\cite{Ballesta2008
]), where the structure is that of a fully disordered colloidal glass. Even near the wall, no significant ordering was observed within the imaging plane.
\section{Model for the slip and yield behavior}
\label{Sec_model}
The local and global rheology in the previous section can be described via a
model based on the same assumption used in [\cite{Yoshimura1988, Russel2000,
Kalyon2005}], i.e. that the stress at the wall due to slip matches the
stress in the bulk. As suggested by the phenomenology of the smooth wall
flow curves, Fig.~\ref{fig_globalrheo}(a),(b) and the Bingham form Eq.~\re
{eq:binghameta} for the slip branch, the local stress $\sigma_{\mathrm{slip
} $ and the slip velocity $v_i$ of the colloids along a wall, labeled by $i
, can be related via:
\begin{eqnarray}
\sigma_{\mathrm{slip}}=\sigma_i + \beta_i v_i. \label{eq_slipstress}
\end{eqnarray}
Here $\beta_i v_i$ is a hydrodynamic term reflecting the lubrication between
the first layer of colloids and the wall and the threshold stress $\sigma_i$
is similar to a Coulomb friction term. Unless particles strongly interact
with one of the smooth surfaces the mean distance of a particle to the
surface will mostly be determined by the available free space, which only
depends on $\phi$; hence we assume that $\beta_1=\beta_2=\beta(\phi)$. On
the other hand we expect $\sigma_i$ to depend on the wall interaction which
can differ between the surfaces. A more detailed description of these terms
will be given later. Below we use the convention $i=1$ for the top
surface, $i=2$ for the bottom plate and also assume that $\sigma_1\geq \sigma_2$.
For the bulk flow, we use the HB form, taking for simplicity an exponent $n=0.5$:
\begin{eqnarray}
\sigma_{\mathrm{bulk}}=\sigma_y + \alpha \dot\gamma^{0.5}. \label{eq_HB}
\end{eqnarray}
Although the measured $n$ can vary ($0.35 \lesssim n \lesssim 0.6$) for $1
\gtrsim \phi/\phi_{rcp} \gtrsim 0.85$, as long as $n<1$ the results below
remain qualitatively correct. The relative contribution of slip and flow is
then evaluated via $\sigma_{\mathrm{bulk}}=\sigma_{\mathrm{slip}}$ along
with the relation between $\dot\gamma$ and $v_{1,2}$:
\begin{eqnarray}
h\dot\gamma+v_1+v_2=h \dot\gamma_a. \label{eq_ratevslip}
\end{eqnarray}
Due to the dependence on the gap size $h$, the results are geometry
dependent. We illustrate this for infinite parallel plates and the
cone-plate geometry.
\subsection{Infinite parallel plates}
This idealized situation serves as the basis for subsequent
analysis of the cone-plate geometry. Various cases are possible depending on
the relative values of $\sigma_s$, $\sigma$ and $\sigma_y$. We first
consider the case $\sigma <\sigma_{y}$. In this regime bulk shear is absent,
$\dot\gamma=0$. For $\sigma_2\leq \sigma <\sigma_1, \sigma_y$ (regime $\mathrm{I}$
or $\mathrm{I_a}$) slip is localized at one surface, leading to
\sigma=\sigma_2 +\beta h\dot\gamma_a$. For $\sigma_1,\sigma_2 \leq
\sigma<\sigma_{y}$ (regime $\mathrm{I_b}$) the sample slips at both surfaces
and we have $\sigma = (\sigma_1+\sigma_2)/2+\beta h\dot\gamma_a/2$.
Whenever $\sigma_2<\sigma_y$, yielding sets in at a slip velocity
v_2=(\sigma_y-\sigma_2)/\beta$. This defines the transition from regime I or
$\mathrm{I_b}$ to regime II, where bulk flow and slip coexists. The relation
$\dot\gamma(\dot\gamma_a)$ is obtained by equating Eqs.~(\ref{eq_slipstress}) and ({\ref{eq_HB}), using Eq.~(\ref{eq_ratevslip}). For slip at one wall (
v_1=0 $, $\sigma_2\leq\sigma_y\leq \sigma <\sigma_1$), this gives:
\begin{eqnarray} \label{eq_1u_dg_dga}
\dot\gamma =\dot\gamma_a -\dot\gamma_y +\dot\gamma_0\left(1- \sqrt{1+\frac{
}{\dot\gamma_0}\left(\dot\gamma_a -\dot\gamma_y \right)}\right),
\end{eqnarray}
\begin{eqnarray} \label{eq_param1slip}
\Delta\sigma =\sigma_y -\sigma_2 \mbox{, } \dot\gamma_y=\frac{\Delta\sigma}
\beta h} \mbox{, and } \dot\gamma_0=\frac{\alpha^2}{2h^2\beta^2}.
\end{eqnarray}
For $\sigma_{1,2}<\sigma_y<\sigma$, the sample yields and slips at both
surfaces. The relation $\dot\gamma(\dot\gamma_a)$ is still given by Eq.~(\re
{eq_1u_dg_dga}), with:
\begin{eqnarray} \label{eq_param2slip}
\Delta\sigma =\sigma_y -\frac{\sigma_1 +\sigma_2}{2} \mbox{, } \dot\gamma_y
\frac{2\Delta\sigma}{\beta h} \mbox{, and } \dot\gamma_0=\frac{2\alpha^2}
h^2\beta^2}.
\end{eqnarray}
Equation~(\ref{eq_1u_dg_dga}) can be written in dimensionless form, giving
the following master curve for the local shear rate versus applied rate in
regime II:
\begin{eqnarray}
\dot\Gamma=1+\Omega-\sqrt{1+2\Omega}, \label{eq_mastercurve}
\end{eqnarray}
with $\dot\Gamma=\dot\gamma/\dot\gamma_0$ and $\Omega=(\dot\gamma_a-\do
\gamma_y)/\dot\gamma_0$, in which the $h$ dependence is absorbed. This
form describes the vanishing bulk shear rate for $\dot\gamma_a \rightarrow
\dot\gamma_y$ and the approach towards the HB curve (
\dot\gamma\rightarrow \dot\gamma_a$) for large rate. This can also be
described in terms of the slip length $l_s$, see App.~\ref{subannexe_ls};
entering regime II, $l_s$ decreases and approaches $l_s \sim
\dot\gamma^{-1/2}$ for large $\dot\gamma_a$.
\subsection{Cone and plate}
\label{sec_coneplate}
To analyze the cone-plate geometry, the variation in gap size needs to be
accounted for. Approximating the geometry at a distance $r$ by parallel
plates, the local stress is deduced as before, with
\sigma(r)=\sigma(h/\tan(\theta),\dot\gamma_a)$. The measured stress
\sigma_m $ is found by integrating $\sigma(r)$ over the entire geometry:
\begin{equation}
\sigma_m=\frac{1}{\pi r_c^2}\int_0^{r_c}\sigma(r)2\pi rdr.
\label{eq_stressint}
\end{equation}
We again first consider the case $\sigma(r) <\sigma_{y}$, i.e. solid body
rotation of the entire sample ($\dot\gamma=0$ for all $r$). In regime
\mathrm{I}$ or $\mathrm{I_a}$, where slip occurs at the plate ($\sigma_2\leq
\sigma <\sigma_1,\sigma_y$) we have:
\begin{equation}
\sigma_m^{\mathrm{I}}=\sigma_2+2\beta \tan(\theta) r_c \dot\gamma_a /3.
\label{eq_sigetafI}
\end{equation}
which is the Bingham form Eq.~({\ref{eq:binghameta}) with $\eta_{\mathrm{eff
}=2\beta \tan(\theta) r_c/3$. For complete slip at two surfaces
\sigma_1,\sigma_2\leq \sigma <\sigma_{y}$, regime $\mathrm{I_b}$, the solid body
rotation rate of the sample is reduced, see Eq.~(\ref{eq_omega_Ib}) in App.
\ref{appsec_sliptwo}. The stress is obtained from a balance of the total
stress on the bottom plate and the cone (App.~\ref{appsec_sliptwo}):
\begin{equation}
\sigma_m^{\mathrm{I_b}}=(\sigma_1+\sigma_2)/2+\beta \tan(\theta) r_c
\dot\gamma_a /3. \label{eq_sigetafIb}
\end{equation}
The transition I$\mathrm{_a} \rightarrow$ I$\mathrm{_b}$ occurs when
\sigma_m^{\mathrm{I_a}}=\sigma_m^{\mathrm{I_b}}$ giving Eq.~(\re
{eq_omega_Iab}) in App.~\ref{appsec_sliptwo}. }
\begin{figure}
\center\scalebox{1}{\includegraphics{fig7.eps}}
\caption{(a) Full line: HB flow curve with $n=0.5$, $\protect\alpha=6.1$~Pa.s
$^{-0.5}$, and $\protect\sigma_y=5.5$~Pa. Other curves: calculations for
different geometries with slip at the bottom plate using $\protect\bet
=82000 $~Pa.s$/$m, $\protect\sigma_s=1.8$~Pa: cone-plate with $\protect\thet
=1^{\circ}$, $r_c=20$~mm (dashed line, Eqs.~(\protect\ref{eq_sigetafI}
\protect\ref{eq_sigmamtot}); parallel plate with $h=2\tan(\protect\thet
)r_c/3$ (dotted) and $h=\tan(\protect\theta)r_c/3$ (dash-dot),
using Eqs.~(\protect\ref{eq_HB},\protect\ref{eq_1u_dg_dga},\protect\re
{eq_param1slip}). (b) Zoom in on (a).}
\label{fig_compare}
\end{figure}
Due to the non-uniform slip velocity and associated nonuniform
stress, the yielding transition and flow in regime II differ from that
for parallel plates. For slip at the bottom plate ($\sigma_1=\infty$, $v_1=0
), when increasing $\dot\gamma_a$, yielding starts when $\sigma_
\mathrm{slip}}=\sigma_y$, i.e. at a radius $r_y=(\sigma_y-\sigma_2)/(\bet
\tan(\theta) \dot\gamma_a)$. The applied rate where regime II starts follows
from $r_y=r_c$:
\begin{equation} \label{eq_dotgamac}
\dot\gamma_{a,c}= (\sigma_y-\sigma_2)/\beta r_c \tan(\theta)
\end{equation}
For $\dot\gamma_a>\dot\gamma_{a,c}$, the boundary
between solid body rotation ($r<r_y$) and slip and shear ($r>r_y$) moves
inward. The measured stress in regime II follows from Eq.~\ref{eq_stressint}
integrated over these two regions. The result, Eq.~(\ref{eq_sigmamtot}) in
App.~\ref{appsec_slipone} is shown for specific parameters in Fig.~\re
{fig_compare}, along with that for parallel plates for two different gaps.
The transition I-II for the cone-plate is smoother than for parallel
plates due to the 'mixed' nature of the transition in the former.
For slip at two surfaces, the transition to regime II is somewhat more
complicated due to a small difference in $\sigma(r)$ between bottom plate
and cone in regime I. Yet, with the approximation $\sigma_1-\sigma_2 \ll
\sigma_y$ the analysis is essentially the same as for slip at one surface,
see App.~\ref{appsec_sliptwo}: the transition to regime II is described by
Eq.~(\ref{eq_dotgamac}) with the substitution $\sigma_2 \rightarrow
(\sigma_1+\sigma_2)/2$ and $\beta \rightarrow \beta/2$.
\section{Comparison with experiment}
\label{sec_compare}
\subsection{Global rheology}
\label{checkglobal}
The uniform lubrication layer and solid body rotation in regime I lead to an
effective viscosity $\eta_{\mathrm{eff}} \sim \beta r_c \tan (\theta)$,
which can be verified by comparing slip branches for different cones. Figure
\ref{fig_geo}(a) shows results for coated cones with different $r_c$ and
\theta$ and uncoated glass plate for the same sample ($a=138$~nm, $\phi=0.63
). The data for $r_c=20~$mm, $\theta=2^{\circ}$ and $r_c=10~$mm,
\theta=4^{\circ}$ indeed superimpose for all $\dot\gamma_a$, as all geometry
dependence of $\sigma_m$ in regime I and II (Eq.~(\ref{eq_sigmamtot}))
enters via $r_c\tan(\theta)$. When plotted versus the velocity at the edge
of the geometry, $\dot\gamma_a r_c \tan (\theta)$, Fig.~\ref{fig_geo}(b),
all slip branches superimpose.
\begin{figure}
\center\scalebox{1}{\includegraphics{fig8.eps}}
\caption{(a) Flow curves for an RI-matched suspension with $a=138$~nm,
\protect\phi=0.63$, using different coated cones and smooth plates. (b)
\protect\sigma_m$ versus velocity at the geometry edge; symbols as in (a).}
\label{fig_geo}
\end{figure}
The transition from Bingham slip to HB bulk flow behavior in the global flow
curves is well described by Eqs.~(\ref{eq_sigetafI}, \ref{eq_sigetafIb}) and
(\ref{eq_sigmamtot}). The parameters $\alpha$, $\sigma_y$ and $\beta$ and
\sigma_{1,2}$ entering Eq.~(\ref{eq_sigmamtot}) (for regime II) follow from
fits of the global rheology with and without coating. Examples of the
predicted full flow curves are shown in Fig.~\ref{fig_globalrheo}(a)(b) and
Fig.~\ref{fig_op2slip}. In Fig.~\ref{fig_globalrheo}(a) the measured
transition from regime I to II for one smooth wall is well captured. In Fig.
\ref{fig_globalrheo}(b) and ~\ref{fig_op2slip}, where both walls are
uncoated, the transition from slip at one surface to two surfaces and the
subsequent yielding are also well described. The small discrepancy for
\phi=0.63$ can be attributed to both a slight reduction of the HB exponent
n<0.5$ for the largest $\phi$ and the presence of shear localization
near the yield stress in this $\phi$ regime, see Sec.~\re
{sec_localization}.
\begin{figure}
\scalebox{1}{\includegraphics{fig9.eps}}
\caption{Velocimetry results corresponding to the data in Fig.~\protect\re
{fig_globalrheo}. (a),(b) Measured normalized local shear rate versus $\do
\protect\gamma_a$ for (a) $r=2.5$~mm ($\square$) and $r=4$~mm ($\circ$), and
(b) $r=2.5$~mm ($\square$) and $r=5.5$~mm ($\circ$). (c),(d) Normalized slip
velocity $v_s/v_{cone}$ at the glass plate corresponding to data in (a),(b).
In (e) the normalized slip velocity at the cone is shown. Data in (a),(c)
are for $a=138$~nm, coated cone and uncoated plate. Data in (b),(d) and (e)
are for $a=150$~nm, uncoated cone and uncoated plate. Full lines in (a),(b)
for $\dot\protect\gamma>0$ are given by Eqs.~(\protect\ref{eq_1u_dg_dga}
\protect\ref{eq_param1slip}) and Eqs.~(\protect\ref{eq_1u_dg_dga},\protect
\ref{eq_param2slip}), respectively, with parameters given in the
caption of Fig.~\protect\ref{fig_globalrheo}. In (d),(e), the curves in
regime I$\mathrm{_b}$ are given by Eq.~(\protect\ref{eq_omega_Ib}); the
transition I$\mathrm{_a} \rightarrow$ I$\mathrm{_b}$ occurs at $\dot\protec
\gamma_a^*$ given in Eq.~(\protect\ref{eq_omega_Iab}). The bulk (slip)
velocity at the second $v_{1,2}$ plateau (corresponding to $r<r_y$ in regime
II) is given by Eq.~(\protect\ref{eq_rotvelunshear}); In (c) and (d),(e) the
curves for the largest $\dot\protect\gamma_a$ (where $r>r_y$) follow from
those in (a),(b) via Eq.~(\protect\ref{eq_ratevslip}).}
\label{fig_localrheo}
\end{figure}
\subsection{Local shear rate}
\label{checklocal}
The model can be checked further by comparing flow profiles to
the predictions for local shear rate and slip velocities in Sec.~\re
{Sec_model}. To obtain these quantities, $v(z)$ is fitted to
$v=\dot\gamma z + v_s$. Figure~\ref{fig_localrheo}(a),(c) shows the
reduced bulk rate $\dot\gamma/\dot\gamma_a$ and slip velocity $v_s$ at the
plate from $v(z)$ taken at $r=2.5$~mm (shown in Fig.~\re
{fig_globalrheo}(c)) and $r=4~$mm. As expected from Eq.~(\ref{eq_1u_dg_dga
), bulk shear starts at smaller $\dot \gamma_a$ for larger $r$, hence for
r<r_c$ the solid body rotation extends beyond the transition rate
\dot\gamma_{a,c} \simeq 0.135~$s$^{-1}$ given by Eq.~(\ref{eq_dotgamac}).
Similarly Fig.~\ref{fig_localrheo}(b-d-e) presents the local bulk rate and
slip velocities $v_1(\dot\gamma_a)$, $v_2(\dot\gamma_a)$ for uncoated cone
and plate taken at $r=5.5$~mm (from Fig.~\ref{fig_globalrheo
(d)) and at $r=2.5~$mm. Both the transition from regime I$\mathrm{_a}$ to I$\mathrm{_b}$ and that to regime II are well described. Due to the small difference between the slip stresses $\sigma_1$ and $\sigma_2$ for bottom
and top, the extent of regime $\mathrm{I_b}$ is limited. Note that $v_1$ and
$v_2$ are essentially constant between $\dot\gamma_{a,c}$ and
(2\sigma_y-\sigma_1-\sigma_2)/\beta r\tan(\theta)\simeq 4-7~$s$^{-1}$, which
motivated the simplification of the model for two uncoated plates in App.
\ref{appsec_sliptwo}. For $\dot \gamma_a \gtrsim 4-7~$s$^{-1}$, the bulk
shear rate $\dot\gamma$ and $v_{1,2}$ also match the predictions.
\begin{figure}
\scalebox{1}{\includegraphics{fig10.eps}}
\caption{(a) Local shear rate $\dot\protect\gamma$ extracted from the
measured $v(z)$, versus $\dot\protect\gamma_a$ for different $\protect\phi$,
$a$ and $r$, see symbols in (b), for slip at the bottom plate only ($a=138~
nm data) and at both surfaces ($a=150~$nm data). The full lines represent
Eq.~(\protect\ref{eq_1u_dg_dga}) with the rheological parameters entering
via Eq.~(\protect\ref{eq_param1slip}) or Eq.~(\protect\ref{eq_param2slip})
with $h=r\tan(\protect\theta)$. Dotted line: $\dot\protect\gamm
=\dot\protect\gamma_a$. (b) Normalized local shear rate $\dot\Gamma$ versus
normalized applied rate $\Omega$ for various $r$, $\protect\phi$ and two
particle sizes. The full line shows Eq.~(\protect\ref{eq_mastercurve}).}
\label{fig_mcurve}
\end{figure}
The radial variation of $\dot \gamma$ over the geometry in regime II,
shown in Fig.~\ref{fig_localrdep}(b), is also well explained by the model.
The line in Fig.~\ref{fig_localrdep}(b) shows that
both the onset of yielding at $r_y\simeq (\sigma_y-\sigma_s)/(\beta
\theta\dot\gamma_a)=2.35~$mm and the $r$-dependence of the bulk shear rate
directly follow from the measured rheological parameters. Finally, in Fig.
\ref{fig_mcurve}(a) we show $\dot\gamma$ versus $\dot\gamma_a$ for different
$\phi$ and particle sizes along with the model predictions (see
caption). Using the normalization described below Eq.~\ref{eq_mastercurve},
\dot\Gamma=\dot\gamma/\dot\gamma_0$ and $\Omega=(\dot\gamma_a-\dot\gamma_y)
\dot\gamma_0$, all data indeed collapse on the master curve
\dot\Gamma=1+\Omega-\sqrt{1+2\Omega}$, Fig.~\ref{fig_mcurve}(b). Overall,
both macroscopic and microscopic observations thus validate the
phenomenological model.
\section{Analysis of bulk flow and slip parameters}
Before discussing the nature of the slip behavior, we first present the bulk rheological parameters, measured in detail for the $a=138~$nm and $a=150~$nm samples using rough, coated walls. Figure~\ref{fig_bulkrheovsphi}(a) shows that $\sigma_y$, plotted as function of the normalized distance to random close packing, strongly increases towards rcp and is reasonably described by $\sigma_y a^3/k_B T \simeq 0.01 (1-\phi/\phi_{rcp})^{-3}$. This increase of $\sigma_y$ is due to tightening of particle cages and strongly increasing entropic barriers. The HB exponent $n$, Fig.~\ref{fig_bulkrheovsphi}(b), is in the range $0.4-0.55$ with a slightly decreasing trend for $\phi \rightarrow \phi_{rcp}$ \footnote{These exponents do not necessarily reflect terminal slopes of $\log(\sigma_m)$ vs $\log(\dot\gamma_a)$. Somewhat larger values ($n\lesssim 0.65$) have also been observed in other PMMA HS suspensions [\cite{Koumakis2012}]}. Further, the HB parameter $\alpha$ also strongly increases with $\phi$, Fig.~\ref{fig_bulkrheovsphi}(c), following approximately $\alpha \propto (1-\phi/\phi_{rcp})^{-2.5} \simeq (1-\phi/\phi_{rcp})^{n-p}$. This implies the following scaling of the bulk HB rheology:
\begin{equation}
\sigma=\sigma_y+\alpha \dot\gamma^n=\sigma_y(\phi)[1+ A(1-\phi/\phi_{rcp})^n (\dot\gamma \tau_B)^n]=\sigma_y(\phi)[1+A(\dot\gamma \tau_m(\phi))^n], \label{eq:HB scaling}
\end{equation}
\begin{figure}[!h]
\scalebox{1.4}{\includegraphics{extrascaling.eps}}
\caption{Bulk HB parameters for $a=138$~nm and $a=150~$nm versus $1-(\phi/\phi_{rcp})$. (a) Normalized yield stress. Line: $\sigma_y=\sigma_0[1-(\phi/\phi_{rcp})]^{-p}$ with $p=3$ and $\sigma_0=0.01 k_BT/a^3$. (b) the HB epxonent $n$. (c) The HB parameter $\alpha$. Lines: $\alpha=A\sigma_0 \tau_B^n [1-(\phi/\phi_{rcp})]^{n-p}$ with $A=10$, $n=0.45$, $p=3$ and $\tau_B=0.0285~$s and $\tau_B=0.036~$s the Brownian times.}
\label{fig_bulkrheovsphi}
\end{figure}
with $\tau_m=\tau_B(1-\phi/\phi_{rcp})$ a characteristic microscopic $\phi$ dependent (in-cage relaxation) time scale in the glass, $\tau_B$ the Brownian time and $A$ given by $A=\alpha/[\sigma_y \tau_B^n(1-\phi/\phi_{rcp})^n] $. While this phenomenological behavior has no solid theoretical basis, it is physically plausible and has been used succesfully in [\cite{Besseling2010}]. We do not further pursue its interpretation here, but instead turn to the behavior of the slip parameters.
\label{Sec_slipparam}
\subsection{$\protect\phi$ dependence for non-stick walls}
As mentioned, for smooth, non-stick walls the Bingham slip parameters $\beta$ and $\sigma_s$ in Eq.~(\re
{eq_slipstress}) show a strong increase with $\phi$. We show
the parameters extracted from the rheology in Fig.~\ref{fig_numphidep
(a),(b), where in (a) the lubrication parameter has been normalized by the
solvent viscosity and particle size, while in (b) the stress is normalized
by $k_BT/a^3$. We observe a characteristic increase of the parameters for $\phi/\phi_{rcp} \rightarrow 0$
for the different particle sizes, except for deviations for the $a=652~
nm particles \footnote{The latter might be partly due to a somewhat reduced sediment volume fraction $\phi_{sed}<\phi_{rcp}$ from centrifugation of this batch (as discussed in Sect.~\ref{subsec_colloids}), thus slightly overestimating of $\phi/\phi_{rcp}$.}.
\begin{figure}
\scalebox{0.8}{\includegraphics{fig11.eps}}
\caption{(a) Normalized lubrication parameter $\protect\beta a/\protect\et
_s $ versus $1-\protect\phi/\protect\phi_{rcp}$. Symbols: data for different
particle size (see (b)); full line: result based on Eq.~(\protect\re
{eq_empiricbeta}). Dotted line: result based on Eq.~(\protect\re
{eq:slipexpr}) and the explanation in the text. Inset: ($\bullet$) $\protec
\lambda$ versus $1-\protect\phi/\protect\phi_{rcp}$ from a numerical
evaluation of Eq.~(\protect\ref{eq:lambda}). The full line represent
\protect\lambda \propto \Pi/\Pi_0$. (b) Normalized slip stress $\protec
\sigma_s a^3/k_BT$ versus $1-\protect\phi/\protect\phi_{rcp}$, the dotted
line is $\protect\sigma_s = 0.45 \Pi$, the full line represents Eq.~(\protect
\ref{eq_empiricsigmas}) with $A=0.005$ and $m=2.5$.}
\label{fig_numphidep}
\end{figure}
We now attempt to rationalize this behavior in terms of the physical
properties of the suspension and the (hard) interaction with the wall. We
first discuss the lubrication parameter $\beta$. The linear increase of
stress with velocity, Eq.~(\ref{eq_slipstress}), implies that $\beta$ is
governed by hydrodynamic friction between the first layer of particles and
the wall, with an effective lubrication layer thickness independent of
velocity. For a single particle of radius $a$, centered at a distance
a+\delta$ from the wall, moving with constant velocity $v$ in a solvent of
viscosity $\eta_s$, the drag force it experiences is $\eta_s v a
f(\delta)$, where $f(\delta/a) \simeq 18.1-10 \ln(\delta/a)$ for $\delta \ll
a$. $f(\delta)$ reflects the wall induced hydrodynamic reduction of the
particle mobility [\cite{Goldman1967}] for no-slip boundary conditions.
For a distribution of particles with concentration $n(\delta)$ moving with
respect to the wall, the mean stress on the wall follows from integrating
over $\delta$. The lubrication parameter $\beta=\sigma/ v$ may thus be
written in normalized form as:
\begin{equation}
\beta a / \eta_s = \sigma a/ \eta_s v = \int_0^{\infty} a^2 f(\delta )
n(\delta) d\delta. \label{eq:slipexpr}
\end{equation}
We assume that beyond the first layer the fluid and colloids move together,
hence the integral can be cut off at $\delta=a$. With the further assumption
that solvent flow in the lubrication layer does not change the distribution
n(\delta)$ and that colloid interactions can be ignored, $\beta$ may be
evaluated from the ``equilibrium'' distribution $n_e(\delta)$. It was
demonstrated in [\cite{Henderson1984}] that the contact value $n_e(0)$
follows $n_e(0)=\Pi/k_BT$ with $\Pi$ the osmotic pressure of the suspension
at rest. While $\Pi(\phi)$ is uncertain for $\phi > \phi_g$ ([\cit
{hermesdijkstra_epl10},\cite{Phan1996}, \cite{Tokuyama2007}]), a widely used
form is [\cite{Brady1995}]:
\begin{equation}
\Pi = 2.9\Pi_0/(1-\phi/\phi_{\mathrm{rcp}}), ~~~~~~\mathrm{with}~~~~~\Pi_0 =
3 \phi k_B T/(4\pi a^3).
\end{equation}
Simulations for $\phi<0.5$ in [\cite{Henderson1984}] further showed that in
the first layer $n_e$ decreases as $n(\delta)=n_e(0) \exp(-3\lambda
\delta(1+\delta/a+\delta^2/3a^2)/a)$, with $\lambda$ a $\phi$-dependent
parameter. Employing this form also for the colloidal glass, $\lambda(\phi)$
can be extracted by equating $\phi$ in the first layer to the bulk value:
\begin{equation}
\frac{2\pi a^2}{3}\int_0^a n(\delta) d\delta=\phi \sim a^2\int
n_e(0)\exp(-\lambda \delta/a) d\delta \sim n_e(0)a^3/\lambda
\label{eq:lambda}
\end{equation}
Numerical solution of the l.h.s. of Eq.~\ref{eq:lambda} gives $\lambda(\phi)
\propto \Pi(\phi)/(k_B T/a^3)$ (inset to Fig.~\ref{fig_numphidep}(a)), as
also confirmed by the scaling in the r.h.s. of Eq.~\ref{eq:lambda}. Using
\lambda(\phi)$ we then calculate $\beta$ from Eq.~\ref{eq:slipexpr}. The
result, shown by the dashed line in Fig.~\ref{fig_numphidep}(a), is finite
for all $\phi$ and diverges as $\phi \rightarrow \phi_{rcp}$, approximately
as $\beta a /\eta_s \propto - \ln(1-\phi/\phi_{rcp})$. While this qualitatively
accounts for the data, and Eq.~\ref{eq:slipexpr} matches the experimental $\beta\propto \eta a^{-1}$ scaling
(Figs.~\ref{Tdep} and \ref{fig_numphidep}(a)), the predicted $\beta(\phi)$ clearly
does not properly describe the experimental results.
A possible explanation for the discrepancy is that the chosen form for
f(\delta)$ only applies for a single particle. However, while colloid
interactions are known to limit the wall induced reduction of diffusion in
concentrated colloidal liquids [\cite{Michailidou2009}], recent simulations
\cite{Swan2010}] show that for small $\delta$ the logarithmic $f(\delta)$
nevertheless holds. An alternative explanation is that the ``equilibrium''
form $n_e$ does not correctly represent the near wall particle distribution.
This can be due to the fact that already at rest the structure is out of
equilibrium (since $\phi > \phi_g$), possibly combined with the layering or
a change in concentration in the wall layer as observed in [\cite{Dullens}].
Equally likely is that $n(\delta)$ differs from $n_e$ due to the actual flow present
in the slip layer, similar to dilute systems [\cite{Polverari1995}]. This
non-equilibrium effect can be quantified by comparing the shear rate in the
lubrication layer with the inverse timescale $1/\tau_m$ for ``cage''
exploration (discussed at the start of this section), via a ``wall'' P\'{e}clet number $\mathrm{Pe}_w=\tau_m v_s /
\xi = \tau_m (\sigma-\sigma_s)/\eta_s$ ($\xi$ is a mean layer thickness
discussed below). Here $\tau_m$ is estimated using the mean free particle
space (also described below) and the short time diffusion coefficient [\cit
{BradyCOColIntSci96_rheodifoverview}] as $\tau_m =
3\tau_B[1-\phi/\phi_{rcp}]$, in line with $\tau_m$ inferred from the $\phi$-scaling of the bulk rheology at the start of this section. Using the Brownian time $\tau_B \gtrsim 30~$ms
and $\sigma-\sigma_s \gtrsim 0.2~$Pa, we have $\mathrm{Pe}_w = O(1)$ and
larger for our data in the slip regime. Thus, during slip $n(\delta)$ is
indeed expected to differ considerably from $n_e$ [\cite{BradyJCP93}], but
further measurements are required to confirm this.
Empirically, the bulk of the experimental data are well described by the
form:
\begin{equation}
\beta a /\eta_s \equiv a/\xi \simeq 0.9/(1-\phi/\phi_{rcp}) \simeq
\Pi/3.2\Pi_0, \label{eq_empiricbeta}
\end{equation}
where the l.h.s. defines the mean lubrication layer thickness $\xi \propto 1/\Pi$. In fact, the mean spacing $\langle
s \rangle$ between colloids in the bulk is estimated as $\langle s \rangle
=a((\phi_{rcp}/\phi)^{1/3}-1)\simeq (a/3)(1-(\phi/\phi_{rcp}))$, hence the
empirical form implies $\xi \simeq 3.4 \langle s \rangle$. Using $n_e$ to
calculate $\xi$ gives similarly $\xi = a^2 \int_0^a \delta~ n(\delta) d
\delta \simeq 2 a \Pi_0/\Pi \simeq 2 \langle s \rangle$, but this form can
also result from other distributions, different from the 'equilibrium' one.
Overall, the analysis suggests that the non-equilibrium behavior in the
lubrication layer underlies the behavior of $\beta(\phi)$. Note
that our $\beta(\phi)$ matches the predicted $\phi$ dependence of the high
frequency viscosity or inverse short time diffusion constant in concentrated
suspensions [\cite{BradyCOColIntSci96_rheodifoverview}], for which, to our
knowledge, no experimental verification yet exist for $\phi \rightarrow
\phi_{rcp}$.
Next we discuss the behavior of the slip threshold $\sigma_s$. As seen in
Fig.~\ref{fig_numphidep}(b), $\sigma_s \propto 1/a^3$, suggesting a relation
to the osmotic pressure. In this context, $\sigma_s$ and $\Pi$ may be
naively linked via a phenomenological Coulomb friction mechanism using
\sigma_s = \mu \Pi$ with $\mu$ the friction coefficient. Comparing this with
the data, we see that a value $\mu \simeq 0.45$ can describe $\sigma_s$ at
intermediate $1-\phi/\phi_{rcp}$. Furthermore, recalling the experiments
with two uncoated surfaces in Fig.~\ref{fig_op2slip}, we found that the
ratio between $\sigma_s$ at the bottom plate and the cone is insensitive to
\phi$, Fig.~\ref{fig_op2slip}(b), which might support the Coulomb friction
scenario. However, the overall $\phi$ dependence of the data in Fig.~\re
{fig_numphidep}(b) is inconsistent with $\sigma_s \propto \Pi$ for constant
\mu$. The data are considerably more scattered than for $\beta(\phi)$ and
also exhibit unsystematic variation of $\sigma_s$ within a factor $2$ for
different glass cleaning methods. While preventing a precise description
of the divergence, the data are reasonably described by:
\begin{equation}
\sigma_s a^3 /k_B T \simeq A (1-\phi/\phi_{rcp})^{-m},
\label{eq_empiricsigmas}
\end{equation}
with $A \simeq 0.005$ and $m \simeq 2.5$. This dependence mimics the
behavior of the yield stress, $\sigma_y a^3/k_B T \simeq
0.01 (1-\phi/\phi_{rcp})^{-3}$. Thus, $\sigma_s/\sigma_y=O(0.1)$ and decreases weakly with $\phi$ as
\sigma_s/\sigma_y \simeq 0.5(1-\phi/\phi_{rcp})^{0.5}$. Therefore $
\sigma_s(\phi)$ may represent the stress required for the presumed
reorganization of the particle distribution when slip sets in and the
similarity between $\sigma_s(\phi)$ and $\sigma_y(\phi)$ could be connected
to a similar change of the particle distribution due to 'cage' breaking for
\sigma=\sigma_y$.
\begin{figure}
\scalebox{1}{\includegraphics{fig12.eps}}
\caption{(a) Measured stress $\protect\sigma_m$ versus $\dot\protect\gamma_a$
for a suspension with $a=138$~nm, $\protect\phi=0.61$ in RI-matching
solvent, immediately after loading, going from small to large $\dot\gamma_a$ ($\triangle $), followed by large to small $\dot\gamma_a$ ($\bigtriangledown
), again small to large($\circ $) and finally large to small $\dot\gamma_a$ (
\diamond $). (b) The power $m$ and (c) $\protect\sigma_s$, both obtained from a fit of the low $\dot\protec
\gamma_a$ branch to $\protect\sigma_m-\protect\sigma_s \propto \dot\protec
\gamma_a^m$, versus run number.}
\label{fig_disc}
\end{figure}
A definite interpretation of $\sigma_s$ is thus still lacking. We conclude
by showing that nevertheless normal stresses in the system do seem to affect
$\sigma_s$. For concentrated samples, the rheology of the system may show a
transient behavior after initial sample loading, associated with local shear
thickening behavior during loading. This is shown in Fig.~\ref{fig_disc}
where a sample is submitted to repeated low-to-high-to-low shear rate cycles
after loading. Fitting the low shear part of the first flow curve
\Blue{$\triangle$}) to $\sigma_m-\sigma_s \propto \dot\gamma_a^m$, we find
m \simeq 0.65$, i.e. in between the exponent $1$ for Bingham slip and $0.5$
for bulk flow, indicating a mixture of shear and slip. For
the first run $\sigma_s$ is significantly larger than for the next cycle(s),
where both $\sigma_s$ and $m$ reach a constant value $\sigma_s\simeq 2$~Pa
and $m\simeq 1$. Assuming that local shear thickened regions lead to
relatively large local normal stresses, increasing $\sigma_s$, the reduced
\sigma_s$ in repeat runs is consistent with flow induced relaxation of these
local normal stresses.
\subsection{Effect of wall interaction}
\label{sec_wallinteraction}
\begin{figure}[tbp]
\center\scalebox{0.5}{\includegraphics{fig13.eps}}
\caption{Mean squared displacement of $a=652~$nm particles versus time in
dilute suspensions in a decalin-tetralin mixture far from the glass (
\Black{$\blacksquare$}), close to the glass ($\bullet$) and close to the
glass in decalin ($\blacktriangle$). The lines represent diffusive behavior
\Delta x^2=2 D t$ with $D=0.139~\protect\mu$m$^2/$s, $D=0.062~\protect\mu$m
^2/$s and $D=0.014~\protect\mu$m$^2/$s from top to bottom. Inset:
trajectories at $z\sim a$ of particles in decalin (the stuck particle and
short trajectory) and a single long particle trajectory in decalin-tetralin
$\bullet$).}
\label{fig_diff}
\end{figure}
As discussed, the slip response for HS glasses strongly depends on the colloid-wall vdW
interaction \footnote{Note that, generally, attractive colloid-wall forces alone do not guarantee elimination of slip. For e.g. flocculated gels, with a heterogeneous network of aggregates, a smooth, attractive (or even particle coated) wall can act as an easy `fracture' plane resulting effectively in slip [\cite{Buscall1993}]. In these cases larger scale wall roughness - of the order of the aggregate size - is needed to prevent slip.}. As shown in Fig.~\re
{fig_im_nim}, for sufficient attraction slip vanishes, implying that the
slip threshold stress $\sigma _{s}>\sigma _{y}$. To characterize directly
the attraction, we imaged dilute suspensions of the $a=652$~nm particles and
analyzed the near-wall motion both in RI-matching and RI-mismatching
solvents, without flow. In Fig.~\ref{fig_diff} we show the mean squared
displacements (MSD) for the two cases, for particles imaged at the surface
(i.e. $z\simeq a$ within the microscope $z-$ resolution) and the MSD away
from the surface. As expected, [\cite{Goldman1967, Sharma2008,
Michailidou2009}] the MSD is smaller close to the wall than in the bulk.
Moreover, with RI-matching the near wall MSD is five times larger than in
the RI-mismatching solvent; in the latter case particles are (temporarily) stuck to the surface, evidencing the vdW attractions. With
RI-matching the mobility is enhanced, vdW forces are reduced and
insufficient to stick particles to the surface.
\begin{figure}
\scalebox{1}{\includegraphics{fig14.eps}}
\caption{Temperature dependence of (a) the normalized Bingham slip stress
and (b) Bingham viscosity $\protect\eta_{\mathrm{eff}}(T)$ normalized by the
temperature dependent solvent viscosity $\protect\eta_{\mathrm{s}}(T)$. Data
are for $\protect\phi=0.59$, $a=138$~nm using a coated cone and smooth
glass. The inset to (b) shows the un-renormalized data $\protect\eta_
\mathrm{eff}}(T)$.}
\label{Tdep}
\end{figure}
Even though the vdW interaction is strongly reduced by RI-matching, it
cannot be completely suppressed. To study the role of remaining vdW forces
on the slip stress $\sigma _{s}$ (see Eq.~\ref{eq:binghameta}), we have
measured flow curves on the same sample, with smooth glass and coated cone,
at various temperatures $T$. Changing $T$ changes the RIs (mainly of the
solvent) and might thus be observable in the dependence of $\sigma _{s}$ on
T$. For all temperatures measured ($5-45$~C$^{\circ }$) the flow curves
exhibited the Bingham slip branch (data not shown), from which we extracted
\sigma _{s}(T)$ and the effective viscosity $\eta _{\mathrm{eff}}(T)$ via
Eq.~\ref{eq:binghameta}. Figure~\ref{Tdep}(a) shows that the normalized slip
stress, $\sigma _{s}a^{3}/k_{B}T$, is nearly temperature independent, but
exhibits a small drop of $\sim 10~\%$ for $T=25^{o}$C. For decalin $\partial
_{T}n_R=-4.4\cdot 10^{-4}$~K$^{-1}$, introducing a temperature dependent
solvent refractive index $n_R$ in the calculations presented in App.~\ref{B1} can be used to obtain a $T$-dependent particle-wall
interaction. For
what interests us here, there is a temperature interval where $n_{\mathrm
PMMA}}<n_{\mathrm{R,solvent}}<n_{\mathrm{R,glass}}$ and thus colloid-wall
interactions may become slightly repulsive (see Table 3 below). The observed
minimum of $\sigma _{s}$ can qualitatively be associated with such slight
repulsion, although a calculation suggests that this should occur
around 40$^{\circ }$C rather than 25$^{\circ }$C as observed experimentally
(figure \ref{Tdep}(a)). We attribute the difference to the approximations
made in the calculation of the vdW interactions (App.~\ref{B1}). Overall,
we conclude that vdW forces do affect $\sigma_{s}$ but, in the range of
RI-mismatch considered here, have only a modest effect. The data for
the Bingham viscosity $\eta _{\mathrm{eff}}(T)$, inset to Fig.~\ref{Tdep
(b), also provide useful information. While $\eta _{\mathrm{eff}}(T)$
exhibits a clear temperature variation, when normalized by the temperature
dependent solvent viscosity, $\eta _{\mathrm{s}}(T)$ no $T$ dependence is
detected (Fig.~\ref{Tdep}(b)) suggesting that the viscous slip is due to a
lubrication layer of pure solvent between the colloids and the wall.
Rheology experiments along with near-wall motion measurements evidenced that sufficiently strong vdW attraction between colloids and walls
suppresses slip in the HS glasses even with smooth walls. Changing the index matching influences both the vdW interaction between two particles and the particle-wall interaction. We calculated the interaction $V_{pw}$ between
the wall and a colloid of radius $a$ separated by a distance
\delta$, and the interaction energy $V_{pp}$ between two colloids separated by a distance $2\delta $ using the formulas in
App.~\ref{B1}. The parameters for each solvent are given in
table ~\ref{tab_ind}. The resulting particle-particle and particle-wall
interactions for the various $a$ and the different solvents are shown in
table~\ref{tab3} using $\delta =10$~nm as thickness of the steric
layer. In decalin, particle-particle vdW attractions are weaker than
k_{B}T/10$ and thus can be neglected. However, the particle-wall (glass) vdW
attraction is of the order of $k_{B}T$ for the larger particles explaining
the significant tendency of these particles to stick to the surface,
although such attractions are clearly weaker than $k_{B}T/10$ for smaller
particles. Finally, the vdW attraction between colloids and the metallic
cone is stronger than $k_{B}T$ for all particle sizes suggesting that a
layer of stuck particles at the cone should be expected. In decalin-tetralin
mixture all attractions are at least an order of magnitude smaller than
k_{B}T$. Hence in index matching solvent particle-particle and particle-wall
interaction are reduced to their hard sphere counterparts.
\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Component & decalin & decaline-tetraline & PMMA & swollen PMMA & glass \\
\hline
$n_R$ & 1.47 & 1.51 & 1.49 & 1.51 & 1.523 \\ \hline
$\epsilon$ & 2.43 & 2.63 & 2.6 & 2.6 & 3.4 \\ \hline
\end{tabular
\end{center}
\caption{Dielectric permittivity $\protect\epsilon $ and index of refraction
$n_R$\ for the solvents, glass, and PMMA.}
\label{tab_ind}
\end{table}
Still, even with a good index matching, some residual van der Waals forces
are present (see table~\ref{tab3}). Moreover, in decalin-tetralin mixtures,
the particle-glass surface interaction is positive, which denotes a repulsive
force. This is due to the fact that in this case $\epsilon _{1}<\epsilon
_{3}<\epsilon _{2}$ (see table~\ref{tab_ind}). Even if such a repulsive
interaction should enhance slip, the small energies involved allow us to
mostly neglect this effect. Two other factors that can affect the slip
parameter by modifying the friction coefficient are the variations of the
index-matching solvent composition and the polymer stabilizing layer between
different batches and particles, and the glass plates that are replaced in each
measurement. These factors may introduce some experimental uncertainty
resulting in a variation of $\sigma _{s}$ by up to a factor of 2 between two
different experiments. Thus, the friction coefficient may vary both for
different particle sizes and between different experiments with the same
sample.
\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
sample name & asm340 & asm247 & asm209 & asm195 \\ \hline
$a$ (nm) & 138 & 150 & 300 & 652 \\ \hline
$V_{pp}^{dec}$ ($k_{B}T$) & -0.011 & -0.012 & -0.032 & -0.082 \\ \hline
$V_{pw}^{dec}$ ($k_{B}T$) & -0.099 & -0.11 & -0.24 & -0.55 \\ \hline
$V_{pp}^{mix}$ ($k_{B}T$) & -67 e-7 & -76e-7 & -20 e-6 & -51 e-6 \\ \hline
$V_{pw}^{mix}$ ($k_{B}T$) & 0.00097 & 0.0011 & 0.0024 & 0.0054 \\ \hline
$V_{p-steel}^{dec}$ ($k_{B}T$) & -4.58 & -5.05 & -11.1 & -25.4 \\ \hline
$V_{p-steel}^{mix}$ ($k_{B}T$) & 0.0076 & 0.0084 & 0.018 & 0.042 \\ \hline
\end{tabular
\end{center}
\caption{van der Waals interactions for different particle sizes $a$, surfaces
and solvents: particle-particle interaction in decalin $V_{pp}^{dec}$,
particle-wall interaction in decalin $V_{pw}^{dec}$, particle-particle
interaction in decalin-tetralin $V_{pp}^{mix}$, particle-wall
interaction in decalin-tetralin $V_{pw}^{mix}$, as well as the particle-steel cone interaction for decalin ($V_{p-steel}^{dec}$) and for decalin-tetralin ($V_{p-steel}^{mix}
).}
\label{tab3}
\end{table}
\section{Shear localization}
\label{sec_localization}
So far we have focused on results for which, at a given $r$ (i.e. gap size)
the shear rate $\dot{\gamma}(z)$ is essentially uniform over the gap when
the suspensions start to yield. However, in experiments where slip is suppressed by coating both
cone and plate, we have observed shear
banding for small shear rates (near $\sigma _{y}$) and large $\phi $. An
example is shown in Fig.~\ref{fig_localisation}(a). For large rates, $v(z)$
is approximately linear, but for $\dot{\gamma}\leq 3~$s$^{-1}$ the profiles
become strongly nonlinear, with shear localization detected near the walls
and a vanishing shear rate in the bulk for the smallest $\gamma_a$. In
\cite{Besseling2010}], we have shown that this behavior, with a \textit
continuous} variation of $\dot{\gamma}(z)$ over the gap, can be explained by
very small concentration gradients ($\delta \phi \lesssim 0.003$), caused by
a dilation-like flow instability due to shear concentration coupling (SCC)
\cite{Schmitt1995}]. This is qualitatively different from other soft glasses
[\cite{Besseling2010}] and cannot be explained using earlier models for
heterogeneous glassy flow, e.g. involving specific wall rheology [\cit
{Bocquet2009}]. This instability and the associated nonlinearity in $v(z)$
sets in below a typical rate $\dot{\gamma}_{c}(\phi )$ which becomes
appreciable only for large $\phi $.
While localization is most easily observed for coated walls, where the
average bulk shear rate $\langle\dot\gamma\rangle=[v(z_g)-v(z=0)]/z_g$
equals $\dot\gamma_a$, we have also detected nonlinear profiles for smooth
walls near the slip to yield transition, $\langle\dot\gamma\rangle \to 0$,
for large $\phi$, Fig.~\ref{fig_localisation}(b). For
\dot\gamma_a=3-5~$s$^{-1}$ the suspension has started to yield, i.e.
\langle\dot\gamma\rangle>0$, but the $\dot\gamma(z)$ is
strongly reduced (enhanced) near the smooth (rough) wall compared to
\langle\dot\gamma\rangle$. Indeed, for $\dot\gamma_a=5~$s$^{-1}$ the mean
shear rate is $\langle\dot\gamma\rangle \sim 0.2\dot\gamma_a=1~$s$^{-1}$,
similar to the value of $\dot\gamma_a$ below which banding becomes
significant for rough walls, Fig.~\ref{fig_localisation}(a). Due to such nonlinear flow near yielding, it is difficult to determine $\sigma_{y}$ very precisely for $\phi/\phi_{rcp}\gtrsim
0.94$. The slightly reduced concentration $\phi (z)<\phi $ in the fluidized
bands allows flow for $\sigma$ (very) slightly below the average $\sigma_y$. Thus
the determined $\sigma_{y}$ for large $\phi $ may be slightly
underestimated, preventing exact calculation of $\dot{\gamma}(r)$ for large
\phi $ where $\dot{\gamma}_{c}$ is large. Therefore, we have only analyzed
the slip to yield transition for $[v(z_{g})-v(z=0)]/z_{g}>\dot{\gamma}_{c}$,
i.e. where the induced bulk flow is essentially linear.
\begin{figure}[tbp]
\scalebox{1}{\includegraphics{fig15.eps}}
\caption{(a) Velocity profiles for $\protect\phi=0.62$, $r=5$~mm and both
surfaces coated, for various $\dot\protect\gamma_a$. (b) Same as (a) but for
coated cone, smooth plate and $r=5.5$~mm.}
\label{fig_localisation}
\end{figure}
\section{Slip below the glass transition}
\label{section_slip_liq}
In our earlier work~[\cite{Ballesta2008}], we reported that for
concentrations well below the glass transition, $\phi <\phi _{g}$, the flow
curves and velocity profiles showed no indication of slip or shear-banding
independent on the surface roughness. Figure \ref{fig_liquid} illustrates
this for $\phi =0.52$ where the flow curves for coated and smooth surfaces
are essentially identical and the flow profiles (Fig.\ref{fig_liquid}~(b))
are very close to linear for all $\dot{\gamma}_{a}$. However, recent
velocimetry data for suspensions closer to the glass transition clearly
reveal slip at the lowest applied shear rate, although the effect is not
detectable in the rheology of these samples.
\begin{figure}
\scalebox{1}{\includegraphics{fig16.eps}}
\caption{(a) Flow curve for an RI-matched suspension with $a=138$~nm,
\protect\phi=0.52$, with smooth ($\square$) and rough walls (full
line). (b) $v(z)$ for smooth walls at $r=5.5~$mm and for
various $\dot\gamma_a$.}
\label{fig_liquid}
\end{figure}
Figure~\ref{fig_liquidslip} shows the flow curve and velocity profiles for a
$\phi =0.535$ suspension with coated cone and smooth glass surface. In the
flow curve, inset to Fig.~\ref{fig_liquidslip}(a), the low $\dot\gamma_a$
Newtonian behavior could not be resolved; only the shear thinning
behavior could be detected.
However, the velocity profiles in Fig.~\ref{fig_liquidslip}(a) clearly show
slip, but with a finite bulk shear rate $\dot\gamma>0$ (no plug flow),
similar to what is observed in glassy samples for $\sigma>\sigma_y$. We extracted the dependence of $\dot\gamma$ on $v_s$ for various gap sizes. The results in Fig.~\re
{fig_liquidslip}(b) show that $\dot{\gamma}=C
v_{s}^{2}$. This is consistent with the model presented earlier, but taking
into account the fact that in the concentrated liquid regime (here $\phi
=0.535$) the yield stress is absent. Using
\sigma _{s}=0$, we obtain $\sigma \simeq \beta v_{s}$ and $\sigma=\alpha \dot
\gamma}^{0.5}$ giving $C=(\beta/\alpha)^{2}$. From the imaging data in Fig.~\re
{fig_liquidslip}(b) we find $C=0.13 \cdot 10^{12}~$s/m$^2$ at this $\phi$. To compare this with the rheology, a fit of the flow curve gives $\alpha
=0.177$~Pa.s$^{0.5}$, while $\beta$ follows from extrapolation of the relation $\beta =0.9\eta
_{s}a^{-1}(\phi _{rcp}-\phi )^{-1}$ in the glassy state. This yields $\beta \simeq 5.8\cdot 10^{4}$~Pa.s$/$m at this $\phi$, giving $(\beta/\alpha)^2=0.11 \cdot 10^{12}~$s/m$^2$, in very reasonable agreement with the data. The model can thus also describe residual slip of liquids, assuming $\sigma_{s,y}=0$.
\begin{figure}
\scalebox{1}{\includegraphics{fig17.eps}}
\caption{(a) Normalized flow profiles at $r=12.5~$mm for $\protect\phi=0.535$
at various $\dot\protect\gamma_a$ using a smooth cone and smooth glass.
Inset: measured flow curve. (b) Local shear rate $\dot\protect\gamma$ versus
slip velocity $v_s$ at various $r$. Full line: fit to a power law $\do
\protect\gamma\propto v_s^{\protect\nu}$ with $\protect\nu=1.92\pm0.3$.
Dotted line: $\dot\protect\gamma= C v_s^2$ with $C=0.13 \cdot 10^{12}~$s/m
^2 $. Magenta discontinued line: $\dot\protect\gamma= C v_s^2$ with $C=
\protect\beta/\protect\alpha)^2$ (see text).}
\label{fig_liquidslip}
\end{figure}
We can compare the slip behavior of glasses and liquids further as follows. The rheology of
liquid-like suspensions at low $\dot \gamma$ is characterized by a Newtonian flow (
\sigma =\eta _{0}\dot{\gamma}$ with $\eta
_{0}=\eta_0(\phi)$ the zero shear viscosity), which evolves towards nonlinear shear thinning behavior at higher shear rates ( $\sigma \propto \dot{\gamma}^{n}$). As a result of the low shear rate Newtonian behavior, any applied stress
results in shear at all $r^{\prime }$s for any such small $\dot{\gamm
}_{a}$. At small $\dot\gamma$, the reduced
apparent viscosity is $\eta _{m}=\eta _{0}/(1+\eta _{0}/\beta h)$ (in plate-plate approximation\footnote{In cone-plate, setting $h=r \tan(\theta)$ and integrating as in Eq.~\ref{eq_stressint}, the refined expression for the apparent viscosity of a Newtonian liquid with slip is $\eta_m/\eta_0=1-2x_c[1-x_c\ln(1+x_c^{-1})]$ with $x_c=\eta_0/(\beta\theta r_c)$.}). A
rough estimate of the difference between slip and no slip measurements can be obtained by using the phenomenological
form for the divergence $\eta
_{0}(\phi)$ on approaching the glass transition (e.g. [\cite{Meeker1997}]): $\eta _{0}=\eta
_{s}(1-\phi \phi _{g}^{-1})^{-2}$ and the earlier mentioned extrapolation of $\beta =0.9\eta
_{s}a^{-1}(\phi _{rcp}-\phi )^{-1}$ to $\phi< \phi_g$.
For $\phi =0.52$, $a=138$~nm and $h=50$~$\mu $m this leads to $\eta
_{m}=0.9588\eta _{0}$. A $\sim 4\%$ difference is roughly within the
experimental uncertainty, which explains why we did not note slip at low $\phi$ in [\cit
{Ballesta2008}]. However, as $\phi $ approaches $\phi _{g}$, $\eta _{0}$
diverges while $\eta _{m}$ tends towards $\beta h$ and slip becomes apparent. For intermediate and large $\dot\gamma$, where the rheology can be approximated by $\sigma \propto \dot{\gamma
^{n}$ with $n\simeq 0.5$ (as stated above, and in agreement with a semi-empirical expression in
\cite{Krieger1959}]), a calculation of the flow curves with and without slip shows that in all cases the relative difference between $\eta _{m}$ and
\eta $ increases with $\phi$ and decreases as $\dot{\gamma}_{a}$ increases,
similarly to the glassy state. Since in the low-shear Newtonian regime the difference is already small (except very close to $\phi_g$), the presence of slip is even more difficult to detect in the nonlinear rheology of liquids at larger rate. However, for $\phi>\phi_g$ the presence of a yield stress and plug flow leads
to a stress difference $\sim \sigma _{y}$ regardless of the geometry, which explains why slip is easily detected in the rheology of HS glasses both for low and intermediate shear rates.
\section{Discussion and conclusions}
\label{disc_concl}
The rheology and velocimetry results and the modeling we presented clearly show that local particle-wall interactions, the character of the boundaries, the geometry and $\phi$ dependent nonlinear bulk rheology strongly affect the measured rheology of concentrated HS suspensions. In general for yield stress fluids, if
a flow curve exhibits a `kink' and a stress drop at low $\dot\gamma_a$ (with a power
law $\dot{\gamma}_{a}^{m}$) that is absent when rough surfaces are used, one can conclude that ($i$) the sample slips at one or both surfaces and ($ii$) locally the transmitted stress is proportional to $v_{s}^{m}$, independent of the geometry. The detailed phenomenological model we presented shows that, from a well characterized slip law $\sigma(v_s)$ and bulk rheology, the overall flow curve and local flow profiles can be accurately predicted (if more complex behavior such as shearbanding and possible non-stationary behavior can be ignored).
For our HS glasses with smooth non-stick walls, a velocity independent, but $\phi$-dependent, lubrication layer forms, leading to $m=1$, but vdW attractions, in non-RI-matched suspensions, easily suppress slip, leading to a slip stress $\sigma_s \geq \sigma_y$. Standard (non-imaging) rheology experiments for HSs are therefore unlikely to be affected by slip. Yet, with many recent studies of nonlinear colloidal flow focusing on microscopic properties [\cite{Cohen2006,Besseling2007}] via microscopy on RI matched suspensions, slip is an important ingredient, and we have shown here that the slip response can be related semi-quantitatively to a bulk property of the suspension (osmotic pressure). For other yield stress fluids, in particular jammed emulsions, the slip behavior may follow a different powerlaw, i.e. $m\simeq 0.5$ has been observed in [\cite{Meeker2004b,Seth2008}] due to elastohydrodynamic lubrication for non-repulsive smooth walls. {\it With} repulsion, $m\simeq 1$ is recovered, implying that our model of the slip-yield transition may carry over to emulsions. Further, emulsions with $\phi_g \lesssim \phi \lesssim \phi_{rcp}$ exhibit HS like (Brownian) glassy behavior [\cite{Gang1999}], for which we thus expect similar Bingham slip behavior as for the HSs.
In non-Brownian suspensions [\cit
{Jana1995,Soltani,Kalyon2005}], slip is also characterized by $m\simeq 1$, i.e. $\sigma \propto v_s$, i.e. a lubrication layer with a thickness independent of $v_s$ [\cite{Kalyon2005,Yilmazer1989}]. Here, contrary to colloids, no
slip stress is observed, as expected from the (near) absence of osmotic or wall-interaction effects. However, the detailed mechanism for slip in non-Brownian systems is still unclear as shown by the different phenomenological
relations found: $\delta /a\simeq 0.125$ for concentrated but Newtonian suspensions ([\cite{Jana1995}],$\phi$-independent) , $\delta
/a\simeq 0.06-0.15$ (for pastes of polydisperse spheres [\cite{Soltani}]), and $\delta/a \simeq 2/[1-(\phi/\phi _{rcp})]$ ([\cite{Kalyon2005}], for systems including polydisperse and non-spherical particles). Interestingly, the latter is similar to our Eq.~\ref{eq_empiricbeta} for the slip of colloidal glasses, but lacks a theoretical basis for non-Bronwian systems. Moreover, in the latter two cases, slip was measured for non-uniform stress, such that shear induced migration may affect the interpretation.
We can also compare the results with those for (depletion) flocculated colloidal gels in [\cite{Buscall1993}]. There, a linear slip response has also been observed, with $\delta$ decreasing from $\sim ~1\mu$m to $\sim 10~$nm from $\phi\sim 0.2$ to $\phi\sim 0.55$, without siginificant dependence on particle size or colloid attraction strength. Here the nature of the slip layer is likely determined by the $\phi$ dependent aggregate lengthscale and structure, rather than the particle size. It is worth noting again that in these systems, even significant colloid-wall attraction is generally unable to suppress slip; unless the wall roughness is very large, the boundary typically acts as a weak `fracture' plane and a slip response is induced. Insight in the nature of this behavior and a theoretical understanding are still lacking.
In conclusion, we have shown the existence of Bingham-type slip response in colloidal HS glasses near smooth non-stick walls. A phenomenological model quantitatively accounts for the global rheology and local flow profiles. Slip in HS glasses is effectively caused by Brownian motion, creating a lubrication layer and slip response governed by the suspensions osmotic pressure, evidenced by the particle size dependence and divergence for $\phi \rightarrow \phi_{rcp}$ of the slip stress and slip viscosity. For HSs, slip is suppressed by colloidal scale wall roughness or sufficient vdW wall attraction. Slip can also occur in concentrated liquid-like suspensions, but is {\it partial} ($\dot\gamma_a > \dot\gamma \neq 0$) due to absence of a yield stress. This is also described within the phenomenological model, but has only limited effect on the bulk rheology. Our study of HSs and the study of [\cite{Meeker2004b,Seth2008}] for emulsions, together with future similar studies for other yield stress fluids, should provide improved predictability of yield stress fluid flows in industrial processing and applications.
\subsubsection{Acknowledgements}
We thank K.N. Pham, J. Arlt and N. Pham for advice and help with the
experiments, A.B. Schofield for particle synthesis and sizing and M.E.
Cates, A. Morozov and D. Marenduzzo for useful discussions. R.B. and W.P.
acknowledge funding through EP/D067650 and EP/D071070/1. L.I. was funded by the EU network
MRTN-CT-2003-504712. G. P. and P. B. acknowledge EU funding from ToK
\textquotedblleft Cosines\textquotedblright\ (MTCDCT-2005-029944), NMP Small
\textquotedblleft Nanodirect\textquotedblright\ (CPFP7- 213948-2) and NoE
"SoftComp".
|
train/arxiv
|
BkiUdXE4c3aisLf0PvQi
| 5 | 1 |
\section{Introduction}
Blazars dominate the scene of extragalactic gamma ray astronomy as space borne
missions and \v{C}erenkov atmospheric telescopes have shown.
BL Lac objects ( BL Lacs or BLLs)
are a main sub-class of \textit{blazars} that by definition exhibit featureless
spectra or very weak lines, most probably because of the relativistic
enhancement of the continuum.
Surely, the recent launch of AGILE and GLAST gamma-ray observatories
and the upgrading of the existing ground-based \v{C}erenkov telescopes will
significantly increase the interest in the BL Lac objects.
The determination of the redshift is mandatory in order to characterize these
sources (e.g. \ to determine nuclear and host galaxy luminosity of the
sources), but it is arduous, in most of the objects, to detect the weak
spectral features over the continuum. Indeed, a redshift determination exists
for only about half of the known BLLs. The detection of the weak spectral
lines necessarily requires the use of $8-10$ meters telescopes.
In addition to the issue of redshift, high S/N and spectral resolution
are also of importance for detecting the host galaxy independently of
imaging, since its emission may appear superposed to the non-thermal
continuum of the nucleus.
Absorption lines may be related to the host galaxy itself, to the
intergalactic medium and to the interstellar medium of our galaxy and its
halo. The emission lines are the most direct probe to the physical conditions
around the nucleus.
We have an ongoing program for BL Lac's spectroscopy at the European Southern
Observatory (ESO) $8-$ meter Very Large Telescope (VLT),
which utilizes the telescope in service mode under non-optimal seeing
conditions.
The results of the first three runs (2003 and 2004) referred to 35 BLLs.
We measured new redshifts for 17 sources, while for the rest of the
objects we have given upper limits using a technique specifically designed
for this project. Details are given in \citet[][ S05 and S06 in the
following]{sbarufatti05_vlt1, sbarufatti06_vlt2}, together with the criteria
on the sample selection.
In this paper we present the spectra of 15 objects observed in 2006
(Guest Observer run: ESO 077.B-0045).
Data reduction and analysis procedures are described in section 2. Results are
reported in section 3 along with specific comments about each source. Summary
and conclusion are given in section 4. Throughout this paper we assume the
following cosmological parameters: H$_0$=70 km s$^{-1}$ Mpc$^{-1}$,
$\Omega_\Lambda$=0.7, $\Omega_{\rm m}$=0.3.
\section{Observations and data analysis}
The observations (Table \ref{tab:journal})
were performed between 2006 March through August in Service Mode at
the ESO VLT UT2 (Kueyen) telescope, equipped with the FOcal Reducer and low
dispersion Spectrograph (FORS1), using the 300V+I grism combined with a
2\arcsec{ } slit, yielding a dispersion of 112 \AA~mm$^{-1}$ (corresponding to
2.64 \AA~pixel$^{-1}$) and a spectral resolution of 15 \AA~covering the
3800--8000 \AA~range.
The seeing during observations was in the range 0.5-2.5\arcsec, with an
average of 1\arcsec.
We performed data reduction using IRAF\footnote{IRAF (Image Reduction and
Analysis Facility) is distributed by the National Optical Astronomy
Observatories, which are operated by the Association of Universities for
Research in Astronomy, Inc., under cooperative agreement with the National
Science Foundation.} \citep{tody86,tody93}, following the standard procedures
for spectral analysis. This includes bias subtraction, flat fielding, and
removal of bad pixels. For each target, we obtained three spectra for an
optimal correction of the cosmic rays and to check for the reality of
weak spectral features. The individual frames were then combined into
a single average image. Wavelength calibration was performed using the spectra
of a He/Ne/Ar lamp,
resulting in an accuracy of $\sim$3 \AA \ (rms). From these calibrated final
images, we extracted the one-dimensional spectra inside a
2\arcsec$\times$6\arcsec aperture, adopting an optimal extraction algorithm
\citep{horne86} to improve the Signal to Noise ratio (S/N).
As a part of a fill-in program, our observations did not require optimal
photometric conditions. However, the sky was clear for most of the
observations. This enabled us to perform a spectrophotometric calibration of
the data using standard stars \citep{oke90}. We estimate an uncertainty
of the order of 10\% in the flux calibration because of the not optimal sky
condition. Flux losses due to the slit not being oriented along the parallactic
angle are negligible with respect to the flux calibration uncertainties.
All the spectra were dereddened following the extinction law by
\citet{cardelli89} and assuming the E$_{\textrm{B-V}}$ values computed by
\citet{schlegel98}.
\section{Results}\label{sec:results}
In Figure \ref{fig:spectra} we give the optical spectrum for each source. In
order to make apparent the shape of the continuum and the faint spectral
features, we report both the flux calibrated and the normalized spectra.
Intrinsic and intervening spectral features are identified
with atomic species. Absorptions caused by the Galactic
interstellar medium are indicated with ISM for simple atomic species, and with
DIB (Diffuse Interstellar Band) for complex molecules. The Earth symbol is used
to mark telluric absorptions. All spectra can be retrieved in electronic form
at \texttt{http://www.oapd.inaf.it/zbllac/}, where all the results of our program are
archived.
\subsection{Continuum emission and host galaxy contribution}\label{sec:cont}
The optical spectrum of a BL Lac can be
described, in a first approximation, as the superposition of two components.
The first one is a non-thermal continuum emitted by the active nucleus.
The second is the contribution from the stars and the ISM of the
BLL host galaxy. Extensive studies in the past \citep[e.g.][]{urry00}
have shown that these galaxies are usually giant ellipticals
the optical magnitudes of which follow a narrow Gaussian distribution centered
around M$_{\rm R}^{\rm host}=-22.9\pm0.5$ \citep{sbarufatti05_hst}. In most
cases the host galaxy signature was not detected in our spectrum because it
was too faint with respect to the nuclear component. For these objects, we
performed a fit of the optical continuum with a simple power-law model
($F_\lambda \propto \lambda^{-\alpha}$). In two cases however (RBS 1752 and RBS
1915) the host galaxy spectral features were detected, and we performed a fit
to the spectrum using a two component model (see Fig. \ref{fig:specdec}): a
power-law plus an elliptical galaxy spectrum as in the template given in
\citet[][see also S06 for details on the fitting procedure]{kinney96}.
The results from our fits are reported in Table \ref{tab:fits}, where for each
object we give the power-law index, the object class \citep[High
Energy Peaked BLL, HBL; Low Energy Peaked BLL, LBL, see][for a definition]
{padovani95b} the apparent magnitude, and the extinction
coefficient. For RBS 1752 and RBS 1915 we report also the absolute magnitudes
of the host galaxies corrected for the aperture effect which were
M$_{\rm R}= -23.3$ and M$_{\rm R}$= -22.4 respectively, in agreement with the
expected distribution.
\subsection{Line detection and redshift determination}
Since emission and absorption lines in a BLL spectrum can be very faint, their
detection can be a difficult task. Using the same technique presented in S06,
we estimated the minimum detectable equivalent width EW$_{\rm min}$~ for each spectrum,
and considered all features with equivalent width (EW) above this threshold as
line candidates which were then carefully inspected for identification or
rejection.
EW$_{\rm min}$ ~ values for each spectrum are given in table 1, while line
identifications, Full Width Half Maximum (FWHM) and EW are reported in table
\ref{tab:lines}. The continuum and line properties confirmed the BL Lac
classification for 12 objects. In section \ref{sec:notes} we report notes on
the individual sources.
\subsubsection{Redshift lower limits}
Most of the confirmed BLL in our sample show featureless spectra, despite the
high S/N reached using VLT. As extensively discussed in S06
(4.2.4), it is possible to estimate a lower limit to the redshift of such
sources, knowing the EW$_{\rm min}$ ~ and the nucleus apparent magnitude and exploiting
the assumption that BL Lac hosts can be considered as candles
\citep{sbarufatti05_hst}.
We remark here that Equation 1 in S06 contains a typographical error, which was
noted also by \citet{finke08}. The correct expression is:
\begin{equation}\label{eq:1}
\mathrm{EW}_{\rm obs}=\frac{(1+z) \times \mathrm{EW}_0}{1+\rho / A(z)}
\end{equation}
where EW$_{\rm obs}$ is the observed equivalent width, EW$_{0}$ is the
equivalent width of the feature in the host galaxy template \citep{kinney96},
$\rho$ is the nucleus-to-host flux ratio, $z$ is the redshift and $A(z)$ is
the aperture correction.
We applied this procedure to all featureless spectra in the sample, obtaining
the lower limits reported in section \ref{sec:notes}.
\subsection{Notes on individual objects.}\label{sec:notes}
\paragraph{PKS 0019+058}
This radio selected source \citep{condon74} was first classified as a BLL
by \citet{fricke83}, based on a featureless optical spectrum. We observed this
object in two epochs separated by about a month, noticing an optical
variability of 0.7 mag ($R$ band) and an evolution of the spectral index from
0.65 to 0.76. The reality of the optical variability is fully confirmed by
the $R$-band images exposed prior to the spectra which enable direct
photometry. The optical variability was known, both in magnitude
\citep[$V= 19.2$ and $V>21$ in][ respectively]{fricke83, abraham91}
and in spectral index \citep[$\alpha_{\nu}$=0.8 and 0.94 in ][]{fricke83,
chen05}. We find no intrinsic feature with EW $>$ EW$_{\rm min}$~ in any of the two
spectra. The NaI$\lambda$5891 absorption from the Galaxy ISM is detected in
both spectra with EW $=0.43$ \AA~ (July 12) and 0.66 \AA~ (August 8).
A marginally significant excess around 5600 \AA~ is present in both
observations, but it is most probably spurious because of a residuals left
after the subtraction of a nearby atmospheric emission line.
The most stringent redshift lower limits, obtained from the August 2006
spectrum is $z>$0.64.
\paragraph{GC 0109+224}
This source was discovered in radio observations by \citet{davis71}, and
subsequently classified as a BL Lac by \citet{owen77}. Strong flux and
polarization variability was reported both in the radio and optical band
\citep[e.g.][]{katajainen00, ciprini03, ciprini04}. Optical imaging by
\citet{falomo96} and \citet{nilsson03} failed to detect the host
galaxy, while the detection reported by \citet{wright98} is dubious,
since it has not been confirmed by any subsequent observation.
The lower limit on the redshift proposed by \citet{falomo96} is $z>$ 0.4.
Previous optical spectroscopy by \citet{wills79,falomo94,sbarufatti06_eso36}
performed with 2--4 m class telescopes showed featureless spectra,
while \citet{healey08} report z=0.265 for this object based on an
unpublished optical spectrum.
The lower limit to the redshift as deduced by \citet{sbarufatti06_eso36}
was based on an EW$_{\rm min}$$= 0.43$ \AA~ that yielded $z > 0.18$. The considerably
higher S/N spectrum we obtained with VLT has EW$_{\rm min}$$=0.09$ \AA, which in turn
implies $z>$ 0.25, which improves the previous spectroscopical limit, but
it is still less stringent than the imaging limit.
\paragraph{RBS 0231}
This X-ray-selected object \citep{voges99} was classified as a BLL by
\citet{schwope00}, and as a HBL by \citet{brinkmann00}. No previous
optical spectroscopy was published. The VLT spectrum is featureless,
with EW$_{\rm min}$$=1.27$ \AA, which implies $z>0.41$.
\paragraph{PKS 0823$-$223}
This radio selected BL Lac \citep{allen82} is characterized by a number of
absorption lines in the UV and optical band \citep[e.g.][]{rao06,
meiring07, falomo90, veron90, falomo94}. These features are consistent with
the presence of a sub-Damped Lyman $\alpha$ Absorber at $z = 0.91$. In our VLT
spectrum we detect the absorption features of
FeII $\lambda\lambda$2373.7, 2383.2, 2585.9, 2599.4, MgII
$\lambda$2798 and MgI $\lambda$2852 at the same redshift.
\paragraph{PKS 1057$-$79}
This radio selected object \citep{shimmins81} was proposed as a
counterpart for the $\gamma$-ray source 2EGS 1050$-$7650 by
\citet{tornikoski02}. No previous optical spectroscopy was published.
Our VLT spectrum shows several emission lines ([OIII]
$\lambda\lambda$4959,5007, [NeIII] $\lambda$3868, MgII $\lambda$2798), at
$z=0.581$. Since the FWHM of the MgII line (see table \ref{tab:lines})
is in excess of $1000~km~s^{-1}$ we propose to classify this object as a
broad line AGN.
\paragraph{PKS 1145$-$676}
This radio source was classified as a quasar due to its point-like appearance
by \citet{white87}. The flat radio spectrum and optical variability
\citep{beasley97, costa02} prompted a \textit{blazar} classification. We
detect several emission lines ([OII] $\lambda$3727, H$\beta~ \lambda$4861,
[OIII] $\lambda\lambda$4959,5007, H$\alpha~ \lambda$6563, [NII] $\lambda$6585)
at $z= 0.210$. The observed EW are in the range 4--25 \AA, pointing towards a
FSRQ classification. The FWHM of the H$\beta~$ line could indicate
that this source is a narrow lines object, but the fact that we are unable
to measure the H$\alpha~$ FWHM due to the blending with the [NII] lines
permits only a type $\sim1.9$ classification.
\paragraph{OM 280}
This radio selected source \citep{colla72} was classified as a BLL due
to its featureless spectrum by \citet{strittmatter74}. Subsequent optical
spectroscopy by \citet{rector01} also failed to discover any intrinsic
spectral feature. The host galaxy was not detected in deep HST imaging
\citep{urry00}, implying $z>0.63$ \citep{sbarufatti05_hst}. The VLT
spectrum is featureless, with EW$_{\rm min}$$=0.35$ \AA, which implies a redshift
lower limit of 0.20, less stringent than the one from imaging.
\paragraph{PMN J1323$-$3652}
This radio selected source was classified as a candidate BL Lac by the WGA
catalog \citep{wga2000} and the Deep X-ray Radio Blazar survey
\citep[DXRBS,][]{landt01}, that also provided a featureless
optical spectrum. However, our VLT optical spectrum clearly shows the
absorption lines and the characteristic shape of a Galactic F-type star. This
indicates a mis-identification of the optical counterpart of this source.
\paragraph{OQ 012}
This radio selected BLL \citep{weiler80} showed a featureless optical spectrum
in observations by \citet{falomo94}. \citet{richards04} gave a photometric
redshift estimate $z=0.475$ based on data from the Sloan
Digital Sky Survey (SDSS). Our VLT spectrum shows an absorption line by
Galactic ISM (NaI $\lambda$5891), but no intrinsic features. The EW$_{\rm min}$~ is
0.31 \AA, which implies $z>0.65$, inconsistent with the photometric estimate.
\paragraph{PMNJ 1539$-$0658}
This radio source \citep{griffith95} was classified as a BL Lac in the DXRBS
\citep{landt01}, which also provided a featureless optical spectrum. In the
VLT spectrum we detect the NaI $\lambda$5891 absorption line from the Galaxy,
but no intrinsic features. The minimum detectable EW is 0.61 \AA, which
implies $z>0.80$.
\paragraph{PKS 1830-589}
This radio-selected BL Lac \citep{griffith95, landt01} showed a featureless
optical spectrum when observed in the DXRBS. Our VLT spectrum is also
featureless (NaI$\lambda$5891 is marginally detected with EW = 0.4 \AA). The
minimum detectable equivalent width is EW$_{\rm min}$$=0.46$ \AA, which gives
a lower limit $z>0.45$.
\paragraph{RBS 1752}
This X-ray-selected BLL \citep{voges99} had a tentative redshift $z=0.449$
proposed in the Sedentary Survey \citep{giommi05, piranomonte07}, based on
the possible detection of the host galaxy spectral features in an ESO 3.6 m
optical spectrum. The high S/N obtained using VLT allowed us to detect
some weak host galaxy lines (CaII $\lambda\lambda$3934,3968, G band
$\lambda$4305, and MgI $\lambda$5175) at $z=0.449$. However, the G band is
possibly contaminated by the [OI] atmospheric line at 6300 \AA, and the
MgI line is very close to the telluric O$_2$ A band. therefore, while the
absorption features reported by \citet{piranomonte07} are confirmed, the
redshift remains tentative because of the lack of other firm absorption
features.
The fit to the spectrum with a power-law plus elliptical galaxy model gives
M$_{\rm R}^{\rm host}=-23.3$, in good agreement with the expected distribution
M$_{\rm R}^{\rm host}=-22.9\pm0.5$.
\paragraph{RBS 1915}
This X-ray selected object \citep{voges99} was classified as a BLL by
\citet{schwope00}. The optical spectrum reported by \citet{chavushyan00} was
featureless. Our VLT spectrum shows faint absorption lines from the BLL host
galaxy (CaII $\lambda\lambda$3934,3968, G band $\lambda$4305, and MgI
$\lambda$5175) at $z=0.243$. We performed a fit to the spectrum using a
power-law plus elliptical galaxy model, obtaining M$_R^{host}$=-22.4, consistent
with the expected distribution (M$_R^{host}$=-22.9$\pm$0.5).
\paragraph{TXS 2346+052}
This radio selected source \citep{large81} was classified as a BL Lac because
of its flat radio spectrum \citep{gorshkov00} and the featureless optical
spectrum \citep{chavushyan00}. Our VLT spectrum shows several emission lines
(MgII $\lambda$2798, [OII] $\lambda$3727, [NeIII] $\lambda$3868, [OIII]
$\lambda\lambda$4959,5007) at $z=0.419$. The observed EW of the MgII
and [OIII] lines (exceeding 5 \AA), rules out a BL Lac classification and
suggests a FSRQ nature for this source. The EW ratio between the [OII] and
[OIII] lines are untypical for an AGN, possibly indicating an ongoing star
formation \citep[as seen in PKS 2005-489 by ][]{bressan06}.
A measurement of the equivalent width of the H$\alpha$+[NII] line system
(which is out of the observed spectral range) could help to clarify this issue.
\paragraph{1RXS J235730.1$-$171801}
This X-ray selected object \citep{voges99} was classified as a BLL by
\citet{schwope00}. Our previous VLT observations (S06)
gave a limit $z>$ 0.85. The spectrum presented here has a slightly lower S/N
(110, to be compared with 150 of the earlier observation, because
of the different seeing conditions between the observations), that gives
EW$_{\rm min}$$=0.22$ \AA~ and $z>0.60$. CaII$\lambda$3934 and NaI$\lambda$5891
absorptions from the Galaxy ISM are marginally detected (they were also
detected in the S06 spectrum, along with several DIBs).
No significant flux variations were detected between two different observation
periods.
\section{Summary and conclusions}
Of 15 observed objects, we confirm the BL Lac classification for 11
sources, and the detection of a sub-DLA system in PKS 0823$-223$
($z\geq0.911$). PKS 1145$-$676 and TXS2346+052 are reclassified as
FSRQ ($z=0.210$~ and $z=0.419$ respectively), while PMN J1323$-$3652 is a
F-type star. For 4 BLLs we are able to give a new determination of the redshift
(PKS 1057$-$79 $z=0.569$; RBS 1752 $z=0.448$; RBS 1915, $z=0.243$).
For the remaining 8 BLLs, we give redshift lower limits based on the
minimum detectable equivalent width of their featureless spectra.
On the whole, our BL Lac spectroscopy database now contains 45 confirmed BL
Lacs observed with VLT, with 20 redshifts determinated by detection of faint
lines, and 25 redshift lower limits.
In those cases where even VLT+FORS observations are inconclusive,
a further increase in the S/N ratio is required, for example through the use of
adaptive optics, Very Large Telescope Interferometry, the Large Binocular
Telescope, observations in the near infrared region, where the nucleus-to-host
ratio is smaller than in the optical range, or, in the future, even the use of
Extremely Large Telescopes. Alternatively, it would be possible to
observe these sources when the active nucleus goes into a low state, since the
decrease in the N/H ratio would make easier to detect the features of the host
galaxy.
\acknowledgments
Based on observations collected at the European Organisation for Astronomical
Research in the Southern Hemisphere, Chile. Observing proposal:
ESO 077.B-0045 (PI: S. Ciprini).
SC acknowledges the funding by the European Community's Human Potential
Programme under contract HPRN-CT-2002-00321.
|
train/arxiv
|
BkiUeKXxaJiQn7ar4fvF
| 5 | 1 |
\section{Introduction}
\begin{figure*}
\centering
\subfigure[ \label{fig:snapshot_add} ]{ \includegraphics[scale=0.3]{IL_addition} }
\subfigure[ \label{fig:snapshot_sub} ]{ \includegraphics[scale=0.3]{IL_substitution} }
\caption{Snapshots from the simulation showing (a)~the \emph{addition} of IL to the
binary polymer electrolyte {$\text{PEO}_{20}\text{LiTFSI}$}\ and (b)~the \emph{substitution} of PEO
chains by IL molecules from the same electrolyte.
PEO chains are shown in red, lithium ions in green, whereas all other
ions (PYR$_\mathrm{13}^+$ and TFSI$^-$) are invisible. }
\label{fig:snapshots}
\end{figure*}
Polymer electrolytes -- typically consisting of an amorphous polymer matrix and a salt dissolved in it~\cite{Wright} --
are promising candidates for many technological devices such as batteries or fuel cells~\cite{Armand,Gray,BruceJCSFaraday1993}.
Here, a commonly used polymer is poly(ethylene oxide) (PEO), whereas lithium-bis\-(tri\-fluoro\-meth\-ane)sulfon\-imide (LiTFSI)
is often employed as conducting salt, as the large TFSI anion reduces ion aggregation.
However, at ambient temperatures, the conductivity is still too low for an efficient technological use.
Several attempts have been made in the past to overcome this deficiency.
Common modifications of the classical polymer-salt systems are the
addition of nanoparticles~\cite{ScrosatiSSI1992} or organic solvent
molecules~\cite{BorghiniElectrochimActa1996,BandaraElectrochimActa1998,KimSSI2002}.
However, in case of the latter, the resulting electrolytes
suffer from the high volatility and thus flammability of
the solvent as well as from its reaction with lithium metal
electrodes~\cite{KimSSI2002}.
Here, Passerini~{\it et al.\,}~\cite{PasseriniElectrochemCommun2003,PasseriniElectrochemActa2005,PasseriniJPowerSources2006,PasseriniJoost}
demonstrated that the use of an ionic liquid (IL) instead of a conventional
solvent has several advantages:
The ILs are non-volatile, non-flammable~\cite{AdamNature2000}
and exhibit a wide electrochemical stability window~\cite{MacFarlaneNature1999,MacFarlaneJPCB1999,PasseriniJPCB2008}.
In this way, improved polymer electrolytes can be designed,
which show an increased conductivity combined with inherent
stability~\cite{PasseriniElectrochemCommun2003,PasseriniJoost},
and are thus ideal to create light-weighted but powerful
batteries~\cite{PasseriniElectrochemActa2005,PasseriniJPowerSources2006}.
In a recent molecular dynamics (MD) simulation study~\cite{DiddensMacroLett2013},
we investigated a ternary polymer electrolyte
consisting of {$\text{PEO}_{20}\text{LiTFSI}$}\ and a variable mole fraction $x$ of the IL
{\it N}-methyl-{\it N}-propyl\-pyr\-rolid\-inium bis(trifluoromethane)sulfonimide (PYR$_\mathrm{13}$TFSI).
As observed experimentally by Passerini~{\it et al.\,}~\cite{PasseriniElectrochemCommun2003,PasseriniJoost}, we also
found a clear increase of the lithium diffusion coefficient with $x$, and were able to show that the main
reason for this enhancement can be attributed to the plasticization of the PEO backbone by the IL.
That is, the presence of the IL enhances the PEO dynamics, and consequently, the lithium ions
coordinated to the PEO backbone also become faster.
In this contribution, we will significantly expand the scope of our previous analysis~\cite{DiddensMacroLett2013},
and investigate the lithium transport in ternary polymer electrolytes in a more general way.
This is not only due to the fact that we present additional insights on the ternary {$\text{PEO}_{20}\text{LiTFSI}\cdot\,x\text{~PYR$_\mathrm{13}$TFSI}$}\ mixtures (in which IL is \emph{added}
to binary {$\text{PEO}_{20}\text{LiTFSI}$}), but also follow a conceptually different approach how to create ternary polymer electrolytes, that is, we also
\emph{substitute} PEO chains in {$\text{PEO}_{20}\text{LiTFSI}$}\ by PYR$_\mathrm{13}$TFSI molecules.
This procedure is carried out in such a way that the overall lithium volume concentration is held constant.
Figure~\ref{fig:snapshots} shows snapshots from the simulations for the two different scenarios.
The motivation for our extended study is twofold:
First, although Passerini~{\it et al.\,}\ found in their recent work~\cite{PasseriniJoost} that the lithium diffusion coefficient
increases with the IL fraction for an ether oxygen-to-lithium ion ($\text{EO}:\text{Li}$) ratio of $20:1$, they simultaneously
observed no significant changes for lower ratios of $10:1$ or $5:1$.
This observation becomes even more puzzling by additional insights from Raman measurements~\cite{PasseriniJoost}, which revealed
that with increasing $x$, the lithium-TFSI coordination increases at the expense of the lithium-PEO coordination (especially for
low $\text{EO}:\text{Li}$ ratios).
Although these findings rather suggest a lithium transport mechanism which is at least partly decoupled from the PEO chains, it is
obvious from the approximately constant lithium diffusion coefficients that the lithium motion cannot be similar to that in pure IL/LiTFSI
electrolytes, which would be significantly faster~\cite{PasseriniJPCB2005,PasseriniJPCC2011}.
Nevertheless, the transport mechanism has to change at some point when PEO is successively replaced by PYR$_\mathrm{13}$TFSI
molecules, which gives us a second, theoretical motivation for our extended study.
Thus, the concentration series of the PEO substitution serves as a model electrolyte in a double sense, as both the
$\text{EO}:\text{Li}$ ratio decreases with increasing IL fraction, while at the same time a crossover from a PEO-based lithium
transport mechanism to a TFSI-based transport mechanism is enforced.
In addition to the MD simulations, we also employ an analytical ion transport model~\cite{MaitraHeuerPRL2007} to clarify the issues
outlined above.
This model expresses the macroscopic lithium diffusion in PEO-based polymer electrolytes via three different microscopic mechanisms,
each characterized by a specific time scale (see sketch in Figure~\ref{fig:sketch_transport_processes}):
1.~Diffusion along the PEO chain, which can be interpreted as an effective one-dimensional random walk along
the curvilinear path of the polymer chain.
This motion can be characterized by the time scale $\tau_1$ the ion needs to explore the entire PEO chain.
2.~Segmental motion of the PEO chain, which can be separated into the center-of-mass motion and the internal
dynamics.
For non-entangled chains, the internal dynamics can be described by the Rouse model~\cite{RouseJCP1953} via an effective Rouse time
$\tau_2$, characterizing the motion of the bound PEO segments.
3.~Intersegmental transfer of the cation from one PEO chain to another, which can be quantified by the average
residence time $\tau_3$ at a given chain.
Previous simulations~\cite{MaitraHeuerPRL2007} have shown that this mechanism can be viewed as a renewal
process within the framework of the Dynamic Bond Percolation (DBP) model~\cite{rev_DBP}, since the dynamics of
a given lithium ion becomes independent of its past after being transferred to another PEO chain.
One particular advantage of our model is that it allows one to extrapolate the lithium diffusion coefficients
derived from the numerical data to the experimentally relevant long-chain limit.
\begin{figure}
\centering
\includegraphics[scale=0.45]{jumps}
\caption{Sketch depicting the three different cation
transport mechanisms in PEO-salt electrolytes.
Each mechanism is characterized by a specific
time scale. }
\label{fig:sketch_transport_processes}
\end{figure}
However, one should keep in mind that the ion transport model has originally been developed for the archetypal, binary
polymer electrolytes.
Although our previous study~\cite{DiddensMacroLett2013} showed that the lithium transport mechanism qualitatively
remains the same for the case of IL addition, it is a priori unclear if this also holds for the PEO substitution,
as the transport mechanism will certainly change at some point when PEO is removed from the electrolyte.
In fact, it will turn out that the IL-mediated lithium transport may substantially contribute to the overall
lithium migration, and we will propose a revised model accounting for this additional, fourth transport mechanism.
Further advancements compared to our original study~\cite{DiddensMacroLett2013} are comprehensive analyses related
to the microscopic factors contributing to the individual transport mechanisms.
For example, we will demonstrate that the plasticization by the IL is only one important requirement to yield an
enhanced segmental mobility and thus a faster lithium diffusion.
Moreover, the effect of the IL on the polymer dynamics is also interesting from a fundamental point of view, as it
gives rise to hydrodynamic interactions among distinct PEO segments, reflected by a Zimm-like scaling of the Rouse-mode
relaxation times.
Concerning the other mechanisms, we will show that \mbox{e.\,g.\,}\ the renewal process (characterized by $\tau_3$) can be understood
in terms of a few structural and dynamical observables.
These additional insights could ultimately contribute to a more directed improvement of battery materials.
For all ternary electrolytes, {$\text{PEO}_{20}\text{LiTFSI}$}\ will be used as a reference.
In what follows, the electrolytes with a constant $\text{EO}:\text{Li}$ ratio, \mbox{i.\,e.\,}\ $20:1$, and a variable amount $x$ of PYR$_\mathrm{13}$TFSI
will be denoted as {$\text{PEO}_{20}\text{LiTFSI}\cdot\,x\text{~PYR$_\mathrm{13}$TFSI}$}.
The new electrolyte types, devised to study the differences in the lithium transport mechanisms between
binary PEO/LiTFSI and PYR$_\mathrm{13}$TFSI/LiTFSI, will be denoted as {$\text{PEO}_{20-\alpha x}\text{LiTFSI}\cdot\,x\text{~PYR$_\mathrm{13}$TFSI}$}.
Here, $\alpha$ is the ratio of the respective partial molar volumes of a PYR$_\mathrm{13}$TFSI ion pair and
a PEO monomer (see below).
For reasons of simplicity, we also abbreviate PEO as \lq P\rq\ (\mbox{i.\,e.\,}\ polymer or PEO) and LiTFSI
as \lq S\rq\ (\mbox{i.\,e.\,}\ salt) in the following, leading to the short-hand notations {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}\ and {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\
for the two distinct classes of ternary electrolytes.
This article is organized as follows:
Section~\ref{sec:md_details} describes the details of the MD simulation procedure and the setup of the systems.
In Section~\ref{sec:structure}, we discuss the lithium ion coordination sphere and the structure of the PEO chains.
Section~\ref{sec:lithium_dynamics} deals with the two transport processes involving changes in the lithium coordination
sphere (intersegmental transfer and diffusion along the chain), followed by a detailed analysis of the polymer dynamics
(Section~\ref{sec:polymer_dynamics}).
Having characterized all three transport mechanisms, we compare the predictions of our model with the experimental data from
Passerini~{\it et al.\,}~\cite{PasseriniJoost} (Section~\ref{sec:cmp_exp}).
Finally, we conclude (Section~\ref{sec:conclusion}).
\section{MD Simulation Methodology}
\label{sec:md_details}
The MD simulations were performed with a modified version of the sander module of the AMBER
package~\cite{AMBER10}, allowing us to use a force field specifically designed for
PEO/LiTFSI~\cite{BorodinJPCB2006_PEO,BorodinJPCB2006_PEOLiTFSI} and PYR$_\mathrm{13}$TFSI~\cite{BorodinJPCB2006_IL}
(see appendix, Section~\ref{sec:validation} for a validation of the modified AMBER code and Section~\ref{sec:benchmark}
for a benchmark assessing its performance).
The initial configurations were created by randomly placing the individual molecules on a simple
cubic lattice with a lattice constant of $36\text{~\AA}$ irrespective of their type, mixing the
system in this way.
In case of PEO, the chains already had coiled conformations as under melt conditions, thereby
circumventing an expensive equilibration.
The binary PEO/LiTFSI electrolyte contained $10$~PEO chains with $N=54$ monomers each as well as
$27$~LiTFSI molecules.
In case of the addition of IL, \mbox{i.\,e.\,}\ {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}, the simulation cell also contained $18$ ($x=0.66$) or $87$
($x=3.24$) PYR$_\mathrm{13}$TFSI molecules.
The latter system is shown in Figure~\ref{fig:snapshot_add}.
In case of the substitution of PEO ({$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}) and thus a constant lithium volume concentration, the
partial molar volumes of PEO and PYR$_\mathrm{13}$TFSI must be known.
Through several short equilibration runs of the respective binary electrolytes, namely $\text{PEO}_n\text{LiTFSI}$
($n=30$, $20$ and $10$) and $\text{PYR}_\mathrm{13}\text{TFSI}\cdot x\text{~LiTFSI}$
($x=0.20$, $0.24$ and $0.26$) in the $NpT$ ensemble, we found that the molar volumes of both PEO and PYR$_\mathrm{13}$TFSI
are independent of the LiTFSI fraction for the investigated concentration ranges.
Their ratio reveals that approximately $\alpha=7.14$ PEO monomers occupy the same volume as one PYR$_\mathrm{13}$TFSI
ion pair.
Based on this value, we created the following systems: $8$~PEO chains with $15$~PYR$_\mathrm{13}$TFSI molecules
({$\text{P}_{16}\text{S}\cdot0.556\text{~IL}$}), $6$~PEO chains with $30$~PYR$_\mathrm{13}$TFSI molecules ({$\text{P}_{12}\text{S}\cdot1.111\text{~IL}$}), and $4$~PEO chains with
$45$~PYR$_\mathrm{13}$TFSI molecules ({$\text{P}_{8}\text{S}\cdot1.667\text{~IL}$}, see Figure~\ref{fig:snapshot_sub}), all of them containing, as above, $27$~LiTFSI ion pairs.
Neat PEO ($16$ chains with $N=54$) and the pure PYR$_\mathrm{13}$TFSI/LiTFSI electrolyte ($76$~PYR$_\mathrm{13}$TFSI
molecules and $27$~LiTFSI ion pairs, {$\text{PYR}_\mathrm{13}\text{TFSI}\cdot0.262\text{~LiTFSI}$}) serve as reference within this context.
The systems were equilibrated in the $NpT$ ensemble for $70-80\text{~ns}$ using the PME technique~\cite{DardenJCP2000}.
Afterwards, production runs with a total length of $200\text{~ns}$ were performed in the $NVT$ ensemble.
A time step of $1\text{~fs}$ was used in all simulations to propagate the systems.
The temperature was maintained by the Berendsen thermostat~\cite{BerendsenJCP1984} with a reference temperature of
$423\text{~K}$.
All bonds involving hydrogen were constrained by the SHAKE algorithm~\cite{SHAKE}.
The induceable point dipoles were integrated by a Car-Parrinello-like
scheme~\cite{vanBelleMolPhys1992}, and the charge-dipole interactions between atoms
separated by three bonds ($1$-$4$ interactions) were scaled by a factor of $0.2$.
For all other nonbonded contributions, the full $1$-$4$ interaction was taken
into account.
Dipole-dipole interactions were damped using a Thole screening~\cite{TholeChemPhys1981} with a dimensionless damping
parameter~\cite{BorodinJPCB2006_PEO} of $a_\mathrm{T}=0.4$.
\begin{table*}
\centering
\footnotesize
\begin{tabular}{l c c r c c}
\hline
\hline
system & $r_1$ & $r_2$ & $b_0^2$ [$\text{\AA$^2$}$] & $\langle{\bf R}_\mathrm{e}^2\rangle$ [$\text{\AA$^2$}$] & $\langle{\bf R}_\mathrm{g}^2\rangle$ [$\text{\AA$^2$}$] \\
\hline
PEO & - & - & $10.4$ & $1904\pm43$ & $306\pm35$ \\
\hline
{$\text{P}_{20}\text{S}$}\ & $0.47$ & $0.53$ & $9.7$ & $1573\pm112$ & $260\pm42$ \\
\hline
\multicolumn{6}{c}{ {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}\ } \\
\hline
{$\text{P}_{20}\text{S}\cdot0.66\text{~IL}$}\ & $0.53$ & $0.47$ & $9.7$ & $1654\pm121$ & $272\pm38$ \\
{$\text{P}_{20}\text{S}\cdot3.24\text{~IL}$}\ & $0.76$ & $0.24$ & $9.6$ & $1498\pm57$ & $249\pm37$ \\
\hline
\multicolumn{6}{c}{ {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\ } \\
\hline
{$\text{P}_{16}\text{S}\cdot0.556\text{~IL}$}\ & $0.55$ & $0.45$ & $9.5$ & $1767\pm271$ & $267\pm50$ \\
{$\text{P}_{12}\text{S}\cdot1.111\text{~IL}$}\ & $0.64$ & $0.35$ & $9.3$ & $1349\pm150$ & $227\pm43$ \\
{$\text{P}_{8}\text{S}\cdot1.667\text{~IL}$}\ & $0.84$ & $0.14$ & $9.1$ & $1840\pm430$ & $280\pm80$ \\
\hline
\hline
\end{tabular}
\normalsize
\caption{Fraction of lithium ions coordinating to one ($r_1$) or two ($r_2$) PEO chains,
mean squared chemical bond length $b_0^2$, mean squared end-to-end
vector $\langle{\bf R}_\mathrm{e}^2\rangle$ and mean squared radius
of gyration $\langle{\bf R}_\mathrm{g}^2\rangle$ of the PEO chains. }
\label{tab:structure}
\end{table*}
Table~\ref{tab:boxlength} (appendix) shows the average values of the box
lengths as determined from the $NpT$ runs and subsequently used in the $NVT$ simulations.
One indeed observes that box length is nearly the same for all {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\ systems.
Slight deviations for {$\text{PYR}_\mathrm{13}\text{TFSI}\cdot0.262\text{~LiTFSI}$}\ may result from rounded molecule numbers in the simulation.
\section{Structural Properties}
\label{sec:structure}
\subsection{Lithium Coordination}
In order to quantify the local structure around the lithium ions, radial distribution functions
(denoted as $g(r)$ in the following) have been computed for the atom pairs {$\text{Li}^\text{+}$}$-\text{O}_\text{PEO}$
and {$\text{Li}^\text{+}$}$-\text{O}_\text{TFSI}$ (see \ref{ssec:gofr}).
Both coordination types exhibit a sharp peak around $2\text{~\AA}$, corresponding to the first
coordination shell, which is in good agreement with neutron diffraction experiments~\cite{MaoPRL2000}
and quantum chemistry calculations~\cite{JohanssonPolym1999,BaboulJAmChemSoc1999}.
At larger distances no significant structural arrangement can be found.
When successively adding the IL, one observes that the peak positions of both {$\text{Li}^\text{+}-\text{O}_\text{PEO}$}\
and {$\text{Li}^\text{+}-\text{O}_\text{TFSI}$}\ remain the same for all electrolytes.
Thus, the same criterion to define temporary lithium bonds will be used for the subsequent analysis.
We consider a EO and a {$\text{Li}^\text{+}$}\ as bound if their distance is not larger than $3.0\text{~\AA}$.
In analogy, we consider a {$\text{Li}^\text{+}$}\ and a TFSI oxygen as bound if their distance is not larger than $2.7\text{~\AA}$.
The probability distribution functions to find a lithium ion with a certain number of EOs
or TFSI oxygens in its first coordination shell is also shown in the appendix (Figure~\ref{fig:coordbins}).
To summarize, we find that the predominant lithium coordination consists of $4-5$ EOs for
all electrolytes, which is again in good agreement with the experimental data~\cite{MaoPRL2000} and
the quantum-chemical results~\cite{JohanssonPolym1999,BaboulJAmChemSoc1999}.
In those complexes where the $4-5$ EOs originate from a single PEO molecule, the polymer
chain wraps helically around the cation.
For complexes involving two PEO chains, typically $2-3$ EOs from each chain coordinate to
the ion.
Coordinations by TFSI oxygens are mostly rare, and often the anion coordinates only briefly
to the lithium ion (cf. Figure~\ref{fig:snapshots}, where all ions are in the vicinity of a PEO chain).
The only exceptions are {$\text{P}_{8}\text{S}\cdot1.667\text{~IL}$}\ and, though less pronounced, {$\text{P}_{12}\text{S}\cdot1.111\text{~IL}$}.
Here, the lithium ions more likely coordinate to $1-2$ TFSI oxygens in contrast to all other systems,
and some are even coordinated to TFSI only ($0.4\text{~\%}$ for {$\text{P}_{12}\text{S}\cdot1.111\text{~IL}$}\ and $2.5\text{~\%}$ for {$\text{P}_{8}\text{S}\cdot1.667\text{~IL}$}).
As in pure IL/LiTFSI, the prevalent coordination number in the latter scenario is about $3-4$ TFSI oxygens,
originating mostly from different anions.
This specific coordination has also been observed previously in MD simulations~\cite{BorodinJPCB2006_ILLiTFSI},
although experimental work emphasizes that also the $\text{Li}(\text{TFSI})_2$ complex is important~\cite{LasseguesPCCP2006}.
Table~\ref{tab:structure} summarizes the fraction of lithium ions coordinating to one or two PEO chains
(denoted as $r_1$ and $r_2$, respectively).
Coordinations to three PEO chains were rarely observed and had a very brief life time of a few picoseconds only.
Therefore, these events were neglected for the subsequent analysis.
For the pure polymer electrolyte, the fractions of complexes involving one and two chains are nearly equal.
Similar binding energies for both coordination types have also been found in quantum chemistry
calculations~\cite{BaboulJAmChemSoc1999}.
The fraction of lithium ions coordinating to one PEO molecule increases with the IL content for both types of electrolytes.
In case of {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}, this is a consequence of the reduced PEO concentration, as it becomes less likely that a lithium ion encounters a
second chain in the semidilute case. This is also confirmed by the observation that the fraction of {$\text{Li}^\text{+}$}\ coordinating to one PEO
chain increases linearly with $x$, showing that the changes in the coordination sphere are purely statistical.
For {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}, however, these trends are stronger than linear, which can be attributed to the fact that the PEO chains are
successively removed from the system, and the remaining PEO chains become more crowded by {$\text{Li}^\text{+}$}\ (see below).
Thus, the different coordination shell can not solely be explained as a simple dilution effect.
\subsection{Statical Polymer Structure}
Due to the helical coordination structure of the PEO backbone, the local polymer
structure changes, and the conformational phase space of the chain is reduced.
Moreover, in case of the ternary electrolytes, the additional IL molecules
dilute the PEO chains, thus inducing a crossover from a polymer melt
to a semidilute solution, which may also alter the equilibrium conformation
of the polymer chains~\cite{DoiEdwards}.
Table~\ref{tab:structure} summarizes the mean squared distance $b_0^2$
between two chemical monomers and the mean squared end-to-end vector
$\langle{\bf R}_\mathrm{e}^2\rangle$.
Due to the crown-ether-like coordination of the PEO backbone, $b_0^2$ is
smaller in all lithium-containing systems.
For {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}, this trend becomes more pronounced with increasing IL concentration.
A contraction of the polymer chain can also be observed for {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}\ from
$\langle{\bf R}_\mathrm{e}^2\rangle$, which also decreases, whereas for
{$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\ no clear predictions can be made within the statistical uncertainties.
When determining the ratio of $\langle{\bf R}_\mathrm{e}^2\rangle$ and
the radius of gyration $\langle{\bf R}_\mathrm{g}^2\rangle$ (Table~\ref{tab:structure}), one finds
values close to the ideal ratio of six for a Gaussian chain~\cite{DoiEdwards}
for all electrolytes, again indicating that on a global scale the chains are approximately Gaussian.
\begin{figure}
\centering
\includegraphics[scale=0.3]{Xp2}
\caption{Rouse mode amplitudes $\langle{\bf X}_p^2\rangle$ in dependence of
the mode number $p$ calculated from the expression of the discrete
Rouse model, \mbox{i.\,e.\,}\ ${\bf X}_p(t) = (1/N)\sum_{n=1}^N \cos((p\pi (n-1/2)/N)){\bf R}_n(t)$.
The curves of the {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\ electrolytes have been shifted by one order of magnitude. }
\label{fig:Xp2}
\end{figure}
Also for the scaling of the Rouse mode amplitudes $\langle{\bf X}_p^2\rangle$,
we observe only slight deviations from the respective curve for pure PEO (Figure~\ref{fig:Xp2}).
Here, the modes ${\bf X}_p$ were determined from the simulation data using the respective
expression of the discrete Rouse model~\cite{VerdierJCP1966},
\mbox{i.\,e.\,}\ ${\bf X}_p(t) = (1/N)\sum_{n=1}^N \cos(p\pi(n-1/2)/N)\,{\bf R}_n(t)$, where the ${\bf R}_n$
denote the position vectors of the individual monomers.
In the limit of low mode numbers $p$, we find the expected Rousean scaling
$\langle{\bf X}_p^2\rangle\propto p^{-2}$, again demonstrating that the
chain structure remains relatively ideal upon the addition of IL.
Thus, no significant swelling of the chains can be observed, and the
structural properties are similar as in neat PEO.
On a local scale however, the PEO chains become rather contracted due to
the helical coordination sphere of the lithium ions.
\section{Lithium Dynamics}
\label{sec:lithium_dynamics}
\begin{figure}
\centering
\includegraphics[scale=0.3]{msd_Li_all}
\caption{MSDs of the lithium ions in the individual electrolytes.
The curves of the {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\ electrolytes have been shifted by one order of magnitude.
The curve for the pure IL-salt solution {$\text{PYR}_\mathrm{13}\text{TFSI}\cdot0.262\text{~LiTFSI}$}\ is also shown for comparison. }
\label{fig:msd_Li}
\end{figure}
The MSDs of the lithium ions in the individual electrolytes are shown in Figure~\ref{fig:msd_Li}.
For {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}, one indeed observes an increase of the lithium mobility upon the addition of IL.
In particular, this increase becomes visible at $1-10\text{~ns}$, whereas the onset to
diffusive behavior occurs on the same time scale for all systems.
Nevertheless, the MSDs are much smaller than for pure {$\text{PYR}_\mathrm{13}\text{TFSI}\cdot0.262\text{~LiTFSI}$}, and the crossover to diffusion
occurs much later.
This is consistent with the observation that nearly all cations coordinate to the PEO chains,
and that the lithium transport in the ternary electrolytes changes not too drastically as compared
to {$\text{P}_{20}\text{S}$}.
Interestingly, for {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}, no significant difference in the lithium MSD can be found.
In the following, we will employ the lithium ion transport model in order to rationalize these observations.
First, we will focus on the intersegmental transfer and the diffusion along the chain, followed by an in-detail
analysis of the polymer dynamics.
Once all ingredients are combined, we will use the transport model to compute the macroscopic lithium diffusion
coefficients from the numerical data, and compare these values to the experimental data from Passerini~{\it et al.\,}~\cite{PasseriniJoost}.
\subsection{Lithium Ion Transfer Mechanism}
First, we will study the lithium ion transfer between two different polymer chains, which is vital for the long-range cation transport and thus any
macroscopically measurable lithium diffusion.
That is, when only considering the diffusion along the chain and the PEO dynamics as possible transport mechanisms, the ion remains
confined to a finite volume characterized by the radius of gyration of the PEO chains.
Moreover, the diffusion of the center of mass of the polymer chain becomes irrelevant in the experimentally relevant long-chain limit.
Previous MD simulations focusing on this mechanism have shown that the dynamics of the lithium ion is independent from its past
after such a transfer processes~\cite{MaitraHeuerPRL2007}, leading to Markovian behavior for time scales larger than $\tau_3$, and that
these processes can thus be regarded as renewal events in the spirit of the DBP model~\cite{rev_DBP}.
We start with the investigation of the detailed mechanism of the cation transfer process.
During most transfers, the ion is coordinated to two PEO chains simultaneously (\mbox{i.\,e.\,}\ the
leaving chain and the entering chain) as a transition state.
In some events, the cation is also temporarily coordinated by TFSI anions only and migrates
to another PEO chain in this way, which will be discussed below.
\begin{figure}
\centering
\includegraphics[scale=0.3]{logtaubin_rescale_all}
\caption{Histogram of the life times $\tau$ of the intermediate complexes involving two PEO chains
which facilitate the cationic transfer. Solid: all events, dashed: successful events only
(see text). }
\label{fig:tau_trans}
\end{figure}
Figure~\ref{fig:tau_trans} shows a histogram of the life times $\tau$ of the intermediate in which a
{$\text{Li}^\text{+}$}\ is coordinated by two PEO chains (solid lines).
One clearly observes that the resulting distribution is very broad.
Whereas most complexes exist only briefly (\mbox{i.\,e.\,}\ a few picoseconds), some exist over nearly the
whole simulation length of $200\text{~ns}$.
For all electrolytes, no significant difference can be found for $p(\tau)$, suggesting a rather
universal mechanism.
This is also reasonable when keeping in mind that the lithium coordination sphere containing two
PEO chains is relatively compact, and that the surrounding molecules thus hardly affect the
transfer process once the complex has formed.
Of course, it is questionable if also the brief coordinations in Figure~\ref{fig:tau_trans} can result
in a successful cation transfer, since the separation of the leaving chain and the simultaneous
formation of a new, stable coordination sphere on the entering chain will need a certain time.
For this purpose, Figure~\ref{fig:tau_trans} also shows $p(\tau)$ for complexes which resulted in a
successful transfer.
The criterion for a successful transfer was that the lithium ion remained detached from the old
chain for at least $100\text{~ps}$, which was motivated by a more detailed analysis revealing that non-Markovian,
short-time backjumps to the old chain occur up to about $100\text{~ps}$ (not shown).
From Figure~\ref{fig:tau_trans} one indeed observes that the probability for short $\tau$ is lower
for real renewal events (dashed lines).
Again, the curves are nearly identical for all electrolytes.
In order to study the transfer mechanism in more detail, we monitored the progress of the cation
transfer as follows: When a second PEO chain enters the coordination sphere of a cation that is
already coordinated to a PEO molecule, the EO coordination number $n$ at the first chain naturally
decreases.
For brief coordinations of the second chain, the crown-ether-like structure of the first chain
will hardly be perturbed.
On the other hand, for longer times, one would expect a rather symmetric coordination by both PEO
chains, leading to smaller $n$.
In case of a successful cation transfer, the EO coordination number decreases even further to zero.
The minimum coordination number $n_\mathrm{min}$ that is reached during $\tau$ can therefore be
used to monitor the progress and the success of the cation transfer.
\begin{figure}
\centering
\includegraphics[scale=0.3]{p_n_p_n_min_all}
\caption{Probability $p(n)$ to find a specific EO coordination number $n$ for a cation coordinated
to two PEO chains (solid) as well as the probability $p(n_\mathrm{min})$ to find a minimum
EO coordination number $n_\mathrm{min}$ on the first PEO chain during the life time of the
complex (dashed). }
\label{fig:p_n_min}
\end{figure}
Figure~\ref{fig:p_n_min} shows the probability $p(n_\mathrm{min})$ to find a specific minimum coordination
number $n_\mathrm{min}$ as well as the probability to find any coordination number $n$ on the first
PEO chain during $\tau$.
One clearly observes that on average $2-3$~EOs of each chain coordinate to the {$\text{Li}^\text{+}$} (solid lines).
This coordination is independent from the IL concentration as expected from the dense packing of the
coordination sphere.
For $p(n_\mathrm{min})$, one notices that the brief coordinations corresponding to $n_\mathrm{min}=4-5$
become less likely with increasing IL concentration for {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}.
As a consequence, the relative probability that a complex with $n_\mathrm{min}=2$ is formed increases.
For {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}, no such effect can be observed, and $p(n_\mathrm{min})$ remains basically unaltered by the
presence of the IL.
This indicates that the increased probability of $n_\mathrm{min}=2$ for {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\ is not related to a
simple concentration effect, which would also be present for {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}, but rather to the fact that the PEO molecules
are successively substituted by IL.
Due to this substitution, each PEO chain coordinates to more cations, resulting in an enhanced rigidity of the PEO backbone,
which impedes the formation of the helical structure with $n=4-5$ and thus leads to a larger value for $p(n_\mathrm{min}=2)$.
In total, a complete cation transfer with $n_\mathrm{min}=0$ has the same relative probability of about
$10\text{~\%}$ for all IL concentrations and $\text{EO}:\text{Li}$ ratios.
Obviously, the critical step is the encounter of a second PEO chain, or, more precisely, a free PEO segment.
Once the complex has formed, it decays according to a universal distribution of life times (Figure~\ref{fig:tau_trans}).
A remarkable feature, however, is the fact that although $p(n_\mathrm{min}=2)$ (which corresponds to a symmetric coordination
by both PEO chains) for {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\ increases with $x$, the complete transfer with $p(n_\mathrm{min}=0)$ does not become more
likely, indicating that other, less trivial factors affect the outcome of a PEO-{$\text{Li}^\text{+}$}-PEO encounter.
It is also worth noting that, in contrast to earlier findings from MD simulations of a PEO/LiBF$_4$
electrolyte~\cite{MaitraHeuerPRL2007}, no significant influence of the anion on the PEO-to-PEO transfer
could be observed.
Most likely, the minor importance of the TFSI anions for the transfer arises from the suppressed tendency
to form ion pairs or higher-order ionic clusters as compared to BF$_4^-$.
\subsection{Renewal Time}
In order to determine $\tau_3$, we counted the number of transfer processes
$N_\mathrm{trans}$ from the simulations.
As above, brief transfers and successive backjumps after less than $100\text{~ps}$
were excluded, while on the other hand transfer processes via the anions were also
counted, since they serve as a renewal process in the same way.
Subsequently, the $\tau_3$-values were determined according to
$\tau_3=t_\mathrm{max}N_{\mathrm{Li}^+}/N_\mathrm{trans}$, where $t_\mathrm{max}=200\text{~ns}$
is the simulation length and $N_{\mathrm{Li}^+}=27$ is the number of lithium ions in the
simulation box.
\begin{table*}
\centering
\footnotesize
\begin{tabular}{l c c r r r r r}
\hline
\hline
system & $\tau_3$ [ns] & $V_\mathrm{PEO}/V$ & $p_\mathrm{EO}^\mathrm{free}$ [\%] & $p_\mathrm{IL}$ [\%] & $p_\mathrm{trans,IL}$ [\%] & $\langle\tau_\mathrm{Li}\rangle_\mathrm{IL}$ [ns] & $\langle\Delta{\bf R}^2_\mathrm{Li}\rangle_\mathrm{IL}$ [$\text{\AA$^2$}$] \\
\hline
{$\text{P}_{20}\text{S}$}\ & $17.1\pm1.3$ & $0.87$ & $72.3$ & $0.02$ & $2.5$ & $0.25 \pm 0.08$ & $31 \pm 8$ \\
\hline
\multicolumn{8}{c}{ {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}\ } \\
\hline
{$\text{P}_{20}\text{S}\cdot0.66\text{~IL}$}\ & $18.4\pm1.4$ & $0.72$ & $71.9$ & $0.03$ & $1.0$ & $0.58 \pm 0.31$ & $34 \pm 13$ \\
{$\text{P}_{20}\text{S}\cdot3.24\text{~IL}$}\ & $24.1\pm1.3$ & $0.43$ & $73.1$ & $0.36$ & $8.5$ & $0.99 \pm 0.22$ & $102 \pm 26$ \\
\hline
\multicolumn{8}{c}{ {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\ } \\
\hline
{$\text{P}_{16}\text{S}\cdot0.556\text{~IL}$}\ & $25.0\pm2.0$ & $0.70$ & $65.2$ & $0.07$ & $4.6$ & $0.38 \pm 0.14$ & $34 \pm 10$ \\
{$\text{P}_{12}\text{S}\cdot1.111\text{~IL}$}\ & $28.4\pm2.2$ & $0.52$ & $54.7$ & $0.43$ & $13.7$ & $0.88 \pm 0.20$ & $79 \pm 17$ \\
{$\text{P}_{8}\text{S}\cdot1.667\text{~IL}$}\ & $32.7\pm2.5$ & $0.35$ & $36.7$ & $2.50$ & $38.2$ & $2.10 \pm 0.31$ & $226 \pm 51$ \\
\hline
\hline
\end{tabular}
\normalsize
\caption{Average residence times $\tau_3$ of a lithium ion at a given PEO chain.
Here, $p_\mathrm{trans,IL}$ is the probability that the lithium ion is transferred
by TFSI anions, $\langle\tau_\mathrm{Li}\rangle_\mathrm{IL}$ corresponds to the average
duration the {$\text{Li}^\text{+}$}\ is coordinated to TFSI only in such a transfer, and
$\langle\Delta{\bf R}^2_\mathrm{Li}\rangle_\mathrm{IL}$ is the average squared
distance the ion covers during this time. }
\label{tab:tau3}
\end{table*}
The individual values for $\tau_3$ are summarized in Table~\ref{tab:tau3}.
One observes for both the {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}\ and the {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\ electrolytes that $\tau_3$ increases with increasing IL concentration.
Since the PEO molecules become more and more diluted, this can at least partly be explained as a concentration effect.
However, the stronger increase for {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\ indicates that not only the reduced PEO concentration, but also the reduced
number of free EOs, which are necessary for a successful transfer, leads to an increase of $\tau_3$.
To understand these trends in more detail, we will figure out the individual contributions to the $\tau_3$-values in the following.
When the lithium ion moves through the electrolyte, it has to encounter a free PEO segment in order to be transferred (naturally,
above the glass-transition temperature, the segments themselves also show significant motion).
The probability for such an event can be estimated on the basis a few structural parameters:
First, the volume element into which the {$\text{Li}^\text{+}$}\ moves has to be occupied by PEO.
Here, the probability to encounter any PEO segment (free or bound) can be obtained from the volume fraction of PEO in
the simulation cell, \mbox{i.\,e.\,}\ $V_\mathrm{PEO}/V$.
Second, the PEO segment has to be free, which is necessary to form the helical coordination sphere.
Here, we defined a free segment as a block of at least four free EOs, which occurs with probability $p_\mathrm{EO}^\mathrm{free}$.
The overall probability to encounter a free PEO segment (at least four EOs) is then given by the product of $V_\mathrm{PEO}/V$
and $p_\mathrm{EO}^\mathrm{free}$ (both quantities are shown in Table~\ref{tab:tau3}).
Apart from these structural ingredients, one also needs dynamical information how fast the PEO segments (and the attached lithium ions) move.
That is, the faster both the lithium ions and the PEO segments rearrange, the more opportunities the ions have to find a free PEO segment
and to be transferred.
Although the effect of the IL on the polymer dynamics will be discussed in Section~\ref{sec:polymer_dynamics}, we nevertheless use the parameter
$C_\mathrm{R}$ (Table~\ref{tab:tau12}), which is the prefactor of the Rouse-like $\text{MSD}\propto t^{1/2}$ for the average EOs, as an anticipatory
measure for the polymer dynamics.
Since $C_\mathrm{R}$ quantifies the motion of the average EOs, it implicitly contains the dynamics of both the free PEO segments and the lithium
ions (due to the cooperative motion with the bound EOs, which are also contained in $C_\mathrm{R}$; see below).
When plotting the product $(V_\mathrm{PEO}/V)\,p_\mathrm{EO}^\mathrm{free}\,C_\mathrm{R}$ versus the observed $\tau_3$, we find a
nearly perfect linear relationship (correlation coefficient $-0.99$) within the statistical error of $\tau_3$.
This demonstrates that $\tau_3$ is essentially determined by three factors: the volume fraction of PEO and the fraction of free PEO segments
(or, alternatively, the volume concentration of free PEO segments) on one hand, and the polymer dynamics on the other hand.
Table~\ref{tab:tau3} also shows the probability $p_\mathrm{trans,IL}$ that a TFSI-supported transfer takes place
($p_\mathrm{trans,IL}=N_\mathrm{trans,IL}/N_\mathrm{trans}$; \mbox{i.\,e.\,}\ the number of transfers in which the {$\text{Li}^\text{+}$}\ is intermediately
coordinated to TFSI only relative to the total number of transfers).
For high IL concentrations, it becomes more likely for a lithium ion to migrate through the IL-rich regions of the electrolyte.
Since this scenario is less likely for {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}, one can conclude that the transfer via TFSI in {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\ is mainly caused by the
crowded PEO chains and the lower fraction of free EOs.
Also shown in Table~\ref{tab:tau3} is the mean time $\langle\tau_\mathrm{Li}\rangle_\mathrm{IL}$ the {$\text{Li}^\text{+}$}\ is coordinated to TFSI only
in a PEO-TFSI-PEO transfer, and the average squared distance $\langle\Delta{\bf R}^2_\mathrm{Li}\rangle_\mathrm{IL}$ the lithium
ion covers during this period.
The values for $\langle\tau_\mathrm{Li}\rangle_\mathrm{IL}$ and $\langle\Delta{\bf R}^2_\mathrm{Li}\rangle_\mathrm{IL}$ show
similar trends as $p_\mathrm{trans,IL}$: With increasing IL fraction and decreasing number of free EOs ({$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\ electrolytes),
both $\langle\tau_\mathrm{Li}\rangle_\mathrm{IL}$ and $\langle\Delta{\bf R}^2_\mathrm{Li}\rangle_\mathrm{IL}$ show a significant increase.
From the Einstein relation $D_\mathrm{Li,IL}=\langle\Delta{\bf R}^2_\mathrm{Li}\rangle_\mathrm{IL}/(6\,\langle\tau_\mathrm{Li}\rangle_\mathrm{IL})$,
we find values close to the lithium diffusion coefficient in {$\text{PYR}_\mathrm{13}\text{TFSI}\cdot0.262\text{~LiTFSI}$}\ ($D_\mathrm{Li,IL}=14.014\text{~\AA$^2$}\text{ns$^{-1}$}$).
Thus, for {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}, a change of the entire lithium transport mechanism becomes visible, and the PEO-TFSI-PEO transitions may
substantially contribute to the overall lithium MSD for sufficiently large values of $p_\mathrm{trans,IL}$ and $\langle\Delta{\bf R}^2_\mathrm{Li}\rangle_\mathrm{IL}$.
On the other hand, from the probability $p_\mathrm{IL}$ to find a lithium ion coordinated to TFSI only at \emph{any} time (Table~\ref{tab:tau3}),
one can conclude that the impact of the IL-mediated transport is negligible for most investigated electrolytes (see discussion in Section~\ref{sec:cmp_exp}).
\subsection{Diffusion along the Chain}
\label{sec:diff_chain}
In order to quantify the quasi-one-dimensional random walk of the lithium ions along the PEO chains,
all EOs have been numbered consecutively, and the mean squared change $\langle\Delta n^2(t)\rangle$
of the average EO index $n$ has been computed (Figure~\ref{fig:delta_n_msd}).
\begin{figure}
\centering
\includegraphics[scale=0.3]{delta_n_msd}
\caption{Mean square change of the average EO index $\langle\Delta n^2(t)\rangle$ for
the {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}\ and {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\ electrolytes.
The curves of the {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\ electrolytes have been shifted by one order of magnitude. }
\label{fig:delta_n_msd}
\end{figure}
Starting from about $100\text{~ps}$, the dynamics crosses over to a regime that is only slightly
subdiffusive (\mbox{i.\,e.\,}\ $\langle\Delta n^2(t)\rangle\propto t^{0.8}$).
Qualitatively, this behavior is found for all compositions.
For {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}, no significant change of $\langle\Delta n^2(t)\rangle$ can be observed when
varying the IL concentration, indicating that the motion along the chain is basically
determined by the characteristic motion of the PEO backbone, and that the surrounding
molecules (\mbox{i.\,e.\,}\ PEO chains or IL molecules) have virtually no influence on this mechanism.
Interestingly, the magnitude of $\langle\Delta n^2(t)\rangle$ is essentially the same for
lithium ions bound to one or to two PEO chains (not shown).
Switching to {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}, one observes that the motion along the chain becomes slower with
increasing IL concentration, especially for {$\text{P}_{12}\text{S}\cdot1.111\text{~IL}$}\ and {$\text{P}_{8}\text{S}\cdot1.667\text{~IL}$}\ (Figure~\ref{fig:delta_n_msd}),
which can be attributed to the lower fraction of free EOs in the system.
Interestingly, the value for the exponent of $\langle\Delta n^2(t)\rangle$ remains basically
the same in this case, and only the magnitude of the functional form $\langle\Delta n^2(t)\rangle\propto t^{0.8}$
becomes slightly lower.
In principle, this type of motion could be quantified by the respective diffusion coefficient
$D_1$ extracted from Figure~\ref{fig:delta_n_msd} according to the Einstein relation
\begin{equation}
D_1=\lim_{t\rightarrow\infty}\frac{\langle\Delta n^2(t)\rangle}{2\,t}\,\text{.}
\end{equation}
Unfortunately, the $\langle\Delta n^2(t)\rangle$-curves in Figure~\ref{fig:delta_n_msd} are not
purely diffusive, whereas on larger time scales the statistics are too bad, since the ions
are additionally transferred between distinct PEO chains, thus making it impossible to track
the motion along the chain any further.
In order to estimate the net effect of this mechanism, \mbox{i.\,e.\,}\ the number of traversed monomers
during $\tau_3$, $D_1$ was approximately estimated as
\begin{equation}
\label{eq:D1}
D_1(\tau_3)=\frac{\langle\Delta n^2(\tau_3)\rangle}{2\,\tau_3}\,\text{.}
\end{equation}
Since the maximum observation time in Figure~\ref{fig:delta_n_msd} is lower than the $\tau_3$-values,
we assumed that the scaling $\langle\Delta n^2(t)\rangle\propto t^{0.8}$ persists until $t=\tau_3$,
and simply extrapolated the curves in Figure~\ref{fig:delta_n_msd}.
Subsequently, the time scale $\tau_1$ was calculated by the respective expression from the
transport model~\cite{MaitraHeuerPRL2007}:
\begin{equation}
\tau_1=\frac{\left(N-1\right)^2}{\pi^2\,D_1}
\end{equation}
\begin{table*}
\centering
\small
\begin{tabular}{l c c c c c c}
\hline
\hline
system & $\tau_1$ [ns] & $\langle{\bf R}_\mathrm{e}^2\rangle_\mathrm{eff}$ [$\text{\AA$^2$}$] & $\tau_\mathrm{R}$ [ns] & $\tau_2$ [ns] & $C_\mathrm{R}$ [$\text{\AA$^2$ ns$^{-1/2}$}$] & $C_2$ [$\text{\AA$^2$ ns$^{-1/2}$}$] \\
\hline
PEO & - & $1979$ & $22$ & - & $151.5$ & - \\
\hline
{$\text{P}_{20}\text{S}$}\ & $147$ & $1662$ & $45$ & $167$ & $89.0$ & $46.2$ \\
\hline
\multicolumn{7}{c}{ {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}\ } \\
\hline
{$\text{P}_{20}\text{S}\cdot0.66\text{~IL}$}\ & $140$ & $1570$ & $37$ & $89$ & $92.7$ & $59.8$ \\
{$\text{P}_{20}\text{S}\cdot3.24\text{~IL}$}\ & $127$ & $1571$ & $24$ & $68$ & $115.2$ & $68.4$ \\
\hline
\multicolumn{7}{c}{ {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\ } \\
\hline
{$\text{P}_{16}\text{S}\cdot0.556\text{~IL}$}\ & $181$ & $1479$ & $43$ & $145$ & $81.0$ & $44.1$ \\
{$\text{P}_{12}\text{S}\cdot1.111\text{~IL}$}\ & $208$ & $1359$ & $35$ & $145$ & $82.5$ & $40.5$ \\
{$\text{P}_{8}\text{S}\cdot1.667\text{~IL}$}\ & $301$ & $1151$ & $28$ & $104$ & $78.1$ & $40.5$ \\
\hline
\hline
\end{tabular}
\normalsize
\caption{Parameters characterizing the two intramolecular transport mechanisms (see text for further explanation).
The values for pure PEO are also shown for comparison. }
\label{tab:tau12}
\end{table*}
Table~\ref{tab:tau12} summarizes the resulting $\tau_1$-values.
For the {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\ electrolytes, one clearly observes an increase of $\tau_1$ with
increasing IL concentration, since more and more EOs become involved in complexes with
lithium, and the PEO backbone becomes less mobile.
Contrarily, for the {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}\ electrolytes, $\tau_1$ decreases only slightly, which can mainly be
attributed to the dependence of $\tau_1$ on $\tau_3$ (Equation~\ref{eq:D1}).
In total, the motion along the chain basically remains unaltered in these electrolytes,
as also suggested by Figure~\ref{fig:delta_n_msd}.
\section{Polymer Dynamics}
\label{sec:polymer_dynamics}
\subsection{Segmental Dynamics}
\label{sec:segmental_dynamcis}
\begin{figure*}
\centering
\subfigure[ \label{fig:EO_Li_msd_x} ]{ \includegraphics[scale=0.3]{EO_Li_msd_add} }
\subfigure[ \label{fig:EO_Li_msd_minusx} ]{ \includegraphics[scale=0.3]{EO_Li_msd_sub} }
\caption{MSDs of the average EOs (circles), bound EOs (diamonds), lithium ions
bound to these EOs ($\left|\Delta n\left(t\right)\right|\leq 1$, solid lines in the inset of Figure~\ref{fig:EO_Li_msd_x}).
The inset of Figure~\ref{fig:EO_Li_msd_minusx} shows the MSD of the central EOs of a PEO segment consisting of at least seven free EOs.
All MSDs have been computed in the center-of-mass frame of the PEO chain.
The dashed and the dotted lines show the respective Rouse fits. }
\label{fig:EO_Li_msd}
\end{figure*}
In order to quantify the segmental polymer motion, the MSDs $\langle\Delta{\bf R}^2(t)\rangle$
of the EOs relative to the center of mass of the PEO chain have been calculated
(Figure~\ref{fig:EO_Li_msd}).
This quantity has been computed for all EOs (\mbox{i.\,e.\,}\ irrespective of the presence of an ion),
for EOs bound to a lithium ion and for the respective attached ions.
The criterion to consider a cation or EO as bound was that the average EO
index of the ion did not change more than one, \mbox{i.\,e.\,}\ $\left|\Delta n(t)\right|\leq 1$
for all time frames during $t$.
For the bound EOs, no further distinction between additional coordinations
of the lithium ion to \mbox{e.\,g.\,}\ another PEO chain or a TFSI molecule was made.
Thus, these effects are already implicitly contained in the curves in Figure~\ref{fig:EO_Li_msd}.
Of course, one might wonder if cations bound to two PEO molecules show the same dynamical characteristics as
ions bound to single polymer chain, as the $\text{Li}(\text{PEO})_2$ complexes could be viewed as temporary crosslinks.
A more detailed analysis (published in the appendix of our previous study~\cite{DiddensMacroLett2013}) indeed revealed that
there is a conceptual difference between these two coordinations.
However, the temporary crosslinks do not affect the polymer dynamics, but rather the motion along the chain:
Whereas lithium ions coordinated to one chain are transported by both intramolecular mechanisms,
ions bound to two chains experience the polymer dynamics only.
This is due to the fact that the PEO chains move reptation-like along their own contour past the cation, which results in
a non-zero $\langle\Delta n^2(t)\rangle$ (Figure~\ref{fig:delta_n_msd}), but does not contribute to the overall lithium transport.
Since there is no qualitative effect of the coordination on the polymer dynamics (possibly also due to the fact that the
exchange between both coordination types is sufficiently fast), we proceed to characterize the average dynamics of the bound
PEO segments with a single $\tau_2$ only.
The average EOs (circles) show typical Rouse-like motion with the
characteristic relaxation time $\tau_\mathrm{R}$.
The dynamics of the bound EOs (diamonds) is qualitatively the same
but protracted.
Therefore, it is possible to characterize the intramolecular dynamics
of the bound EOs by using a larger, effective Rouse time $\tau_2$.
The lithium ions attached to these EOs (inset, solid lines) closely
follow the bound EOs, which gives evidence for their cooperative motion.
On short time scales, the MSD of the EOs is larger than the lithium MSD
due to the additional internal degrees of freedom of the PEO backbone,
but the MSD of the bound cations catches up on longer time scales
(approximately $1\text{~ns}$).
Due to the collective motion, $\tau_2$ also characterizes the dynamics of the attached lithium ions.
Figure~\ref{fig:EO_Li_msd} also shows the Rouse fits
\begin{equation}
\label{eq:seg_msd}
\langle\Delta{\bf R}^2(t)\rangle = \frac{2\langle{\bf R}_\mathrm{e}^2\rangle}{\pi^2}\sum_{p=1}^\infty \frac{\left[1-\exp{\left(-\frac{tp^2}{\tau_\mathrm{R}}\right)}\right]}{p^2}
\end{equation}
for the average (dashed lines) and the bound EOs (dotted lines).
It is important to mention that the precise value of $\tau_2$ obtained from these
fits also depends on the value of $\langle{\bf R}_\mathrm{e}^2\rangle$ entering
the prefactor of Equation~\ref{eq:seg_msd}.
In order to obtain a fit consistent with the plateau value at large $t$, the MSDs
of the average EOs were fitted using two parameters, \mbox{i.\,e.\,}\ $\tau_\mathrm{R}$ and
$\langle{\bf R}_\mathrm{e}^2\rangle_\mathrm{eff}$.
In this way, the $\langle{\bf R}_\mathrm{e}^2\rangle$-values in Table~\ref{tab:structure} are replaced by an effective
mean squared end-to-end vector $\langle{\bf R}_\mathrm{e}^2\rangle_\mathrm{eff}$,
characterizing the maximum accessible intramolecular distance for the cations.
Once $\langle{\bf R}_\mathrm{e}^2\rangle_\mathrm{eff}$ was determined from the MSDs
of the average EOs, the MSDs of the bound EOs were fitted using this value
in combination with a single fit parameter $\tau_2$ only.
The fitting parameters $\langle{\bf R}_\mathrm{e}^2\rangle_\mathrm{eff}$, $\tau_\mathrm{R}$
and $\tau_2$ are summarized in Table~\ref{tab:tau12} (deviations from our previous study~\cite{DiddensMamol2010} on
{$\text{P}_{20}\text{S}$}\ arise from the shorter simulation length of about $27\text{~ns}$ and the slightly modified fitting procedure).
In case of {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}, $\langle{\bf R}_\mathrm{e}^2\rangle_\mathrm{eff}$ decreases
only slightly when IL is added.
The $\langle{\bf R}_\mathrm{e}^2\rangle_\mathrm{eff}$-values are in reasonable agreement with the respective
$\langle{\bf R}_\mathrm{e}^2\rangle$-values in Table~\ref{tab:structure}.
Compared to pure PEO, the $\langle{\bf R}_\mathrm{e}^2\rangle_\mathrm{eff}$
are about $16-20\text{~\%}$ lower in these systems and show good agreement
with the respective values in Table~\ref{tab:structure}.
This demonstrates that the PEO chains in these electrolytes behave relatively ideal,
thus additionally validating our analysis in terms of the Rouse model.
Both $\tau_\mathrm{R}$ and $\tau_2$ decrease significantly, clearly indicating that
the dynamics of the PEO segments becomes faster with increasing IL concentration.
Therefore, the IL can be regarded as plasticizer.
Due to the enhanced polymer mobility, the lithium ions also move faster while they
are coordinated to a specific chain, and the overall lithium MSD increases as observed
in Figure~\ref{fig:msd_Li}.
A similar observation has been made experimentally for other plasticizers like
ethylene/propylene carbonate~\cite{BandaraElectrochimActa1998,KimSSI2002} or short PEO
chains embedded in a high-molecular weight matrix~\cite{BorghiniElectrochimActa1996,KimSSI2002}.
Naturally, this effect has also been observed for ILs in the
experiments~\cite{PasseriniJoost} that motivated the present work.
For {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}, $\langle{\bf R}_\mathrm{e}^2\rangle_\mathrm{eff}$ clearly decreases when the PEO
chains are successively substituted by IL molecules.
The mismatch with the $\langle{\bf R}_\mathrm{e}^2\rangle$-values in Table~\ref{tab:structure} results
from the different averaging procedure:
Whereas $\langle{\bf R}_\mathrm{e}^2\rangle$ is determined by the outer monomers only and thus
rather characterizes the global polymer structure, $\langle{\bf R}_\mathrm{e}^2\rangle_\mathrm{eff}$
contains contributions from all monomers, making it more susceptible to structural features on short
and intermediate length scales.
Note that the latter quantity is also statistically much better defined, since the average is taken over all
EOs instead of the outermost EOs only.
The decrease of $\langle{\bf R}_\mathrm{e}^2\rangle_\mathrm{eff}$ can be rationalized by the
fact that more and more lithium ions coordinate to a specific PEO chain, thus diminishing the
\emph{average} intramolecular distances, and, consequently, the maximum accessible displacement
for most ions.
At the same time, it is still possible that the PEO chains are stretched on a global scale, which might
arise from \mbox{e.\,g.\,}\ the electrostatic repulsion between distinct ions at a given chain (although for {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$},
one cannot make clear predictions within the uncertainties of $\langle{\bf R}_\mathrm{e}^2\rangle$ in
Table~\ref{tab:structure}).
In combination with the decline of $\langle{\bf R}_\mathrm{e}^2\rangle_\mathrm{eff}$,
the values for $\tau_\mathrm{R}$ and $\tau_2$ also decrease when the IL concentration
is raised (Table~\ref{tab:tau12}).
However, it must be pointed out that the $\tau_\mathrm{R}$- and the $\tau_2$-values
of different electrolytes can only be quantitatively compared in conjunction with
the precise value of $\langle{\bf R}_\mathrm{e}^2\rangle_\mathrm{eff}$.
Therefore, a direct comparison of $\tau_\mathrm{R}$ and $\tau_2$ is only valid for similar
$\langle{\bf R}_\mathrm{e}^2\rangle_\mathrm{eff}$, as in the case of {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}.
This is also highlighted by the observations from Figure~\ref{fig:EO_Li_msd_minusx}, where
the EOs (both types) rather become slower with the IL fraction.
It is convenient to express $\langle{\bf R}_\mathrm{e}^2\rangle_\mathrm{eff}$ and $\tau_\mathrm{R}$
(or $\tau_2$) within a single meaningful number.
To this purpose, the prefactor of the characteristic Rouse-like motion for intermediate
time scales (\mbox{i.\,e.\,}\ $\tau_\mathrm{R}/N^2\ll t \ll \tau_\mathrm{R}$),
$\langle\Delta{\bf R}^2(t)\rangle=(2\langle{\bf R}_\mathrm{e}^2\rangle/\pi^{3/2})\sqrt{t/\tau_\mathrm{R}}$,
can be calculated, making it possible to directly measure the magnitude of the MSDs.
Table~\ref{tab:tau12} displays the constants $C_\mathrm{R}=2\langle{\bf R}_\mathrm{e}^2\rangle_\mathrm{eff}/(\pi^3 \tau_\mathrm{R})^{1/2}$
and $C_2=2\langle{\bf R}_\mathrm{e}^2\rangle_\mathrm{eff}/(\pi^3 \tau_2)^{1/2}$ for the individual systems,
thereby quantifying the mobilities of the average and the bound EOs, respectively.
Here, one indeed observes the same trends as from the direct comparison of the MSDs:
For {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}, the segmental mobility expressed by $C_\mathrm{R}$ and $C_2$ clearly increases
with the IL concentration, whereas it slightly decreases in case of the {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\
electrolytes, meaning that also the lithium dynamics decreases.
This behavior can again be explained by the reduced flexibility of the PEO backbone,
since the fraction of bound EOs increases when the PEO chains are gradually substituted by IL.
Nevertheless, the plasticizing is also present for {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}.
This can be seen from the inset of Figure~\ref{fig:EO_Li_msd_minusx}, which shows the MSDs of the central EOs
of all those PEO segments consisting of at least seven free EOs.
Again, one observes an increase in the polymer dynamics with increasing IL fraction.
However, since the fraction of free EOs and the probability to encounter an entirely free PEO segment
(cf. $p_\mathrm{EO}^\mathrm{free}$ in Table~\ref{tab:tau3}) is significantly lower, the plasticizing is barely
visible in {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}.
Rather, for the average EOs, the progressive coordination of lithium ions to the PEO chains screens the
plasticizing effect of the IL.
Finally, it is worth mentioning that for both investigated types of electrolytes, the center-of-mass dynamics of the PEO
chains becomes faster with increasing IL concentration.
(The impact of this increase on the lithium diffusion coefficient will be discussed in Section~\ref{sec:cmp_exp}).
However, this observation can be easily rationalized by the fact that motion of the individual chains is less hindered by
other chains when the PEO concentration is reduced.
\subsection{Hydrodynamic Interactions}
An especially interesting effect of the IL on the polymer dynamics can be observed
from the scaling of the Rouse mode relaxation times.
The correlators $\langle{\bf X}_p(0){\bf X}_p(t)\rangle$ display a stretched exponential
relaxation (not shown), and a Kohlrausch-Williams-Watts fit,
$\langle{\bf X}_p(0){\bf X}_p(t)\rangle/\langle{\bf X}_p^2\rangle = \exp(-({t}/{\hat{\tau}_p})^\beta)$,
was used to extract the parameters $\hat{\tau}_p$ and $\beta$
(here, the hats have been used to avoid confusion with the time scales $\tau_1$, $\tau_2$ and $\tau_3$
of the lithium ion transport model).
The mean relaxation time $\langle\hat{\tau}_p\rangle$ is then given by
$\langle\hat{\tau}_p\rangle=\hat{\tau}_p\beta^{-1}\Gamma(\beta^{-1})$, where $\Gamma$
is the gamma function.
The resulting scaling of $\langle\hat{\tau}_p\rangle$ is shown in Figure~\ref{fig:taup}.
Deviations from the values for $\tau_\mathrm{R}\equiv \langle\hat{\tau}_1\rangle$ in Table~\ref{tab:tau12}
result from the fact that the segmental MSD reflects an average over all modes, whereas
$\langle\hat{\tau}_1\rangle$ is determined from the first mode only.
\begin{figure}
\centering
\includegraphics[scale=0.3]{taup}
\caption{Average Rouse mode relaxation times $\langle\hat{\tau}_p\rangle$ in dependence of the mode number $p$.
The curves of the {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\ electrolytes have been shifted by one order of magnitude. }
\label{fig:taup}
\end{figure}
For low mode numbers, both pure PEO as well as {$\text{P}_{20}\text{S}$}\ exhibit the same scaling
of approximately $\langle\hat{\tau}_p\rangle\propto p^{-2}$, thereby indicating that the
global modes are Rousean.
The absolute values of the $\langle\hat{\tau}_p\rangle$ are systematically higher for
{$\text{P}_{20}\text{S}$}\ due to the decelerating effect of the lithium ions.
In contrast to this, one observes a clear decrease of the relaxation times at low $p$
for {$\text{P}_{20}\text{S}\cdot3.24\text{~IL}$}\ as compared to {$\text{P}_{20}\text{S}$}.
As already observed from the fits of the segmental MSDs (Figure~\ref{fig:EO_Li_msd}), {$\text{P}_{20}\text{S}\cdot3.24\text{~IL}$}\
approximately has the same value for $\tau_\mathrm{R}$ as pure PEO, whereas the local
$\langle\hat{\tau}_p\rangle$ ($p\gg 1$) are basically the same as for the binary {$\text{P}_{20}\text{S}$}.
However, a faster global relaxation combined with a similar local relaxation can only be
realized by a less steep dependence of $\langle\hat{\tau}_p\rangle$ on $p$.
These observations are compatible with a scenario in which hydrodynamic interactions, arising
from the presence of the IL, contribute to the overall polymer motion.
From an analytical point of view, these interactions can readily be taken into account via the Zimm model~\cite{ZimmJCP1956,DoiEdwards}.
In its simplest form, \mbox{i.\,e.\,}\ when no swelling of the chains is present (cf. $\langle{\bf R}_\mathrm{e}^2\rangle$
in Table~\ref{tab:structure} and $\langle{\bf X}_p^2\rangle$ in Figure~\ref{fig:Xp2}), the equilibrium
distribution of the chain conformation is Gaussian as in the Rouse model.
Therefore, the static polymer properties remain unaffected, and only the dynamics is
altered, leading to a scaling of the relaxation times of ${\hat\tau}_p\propto p^{-3/2}$.
Interestingly, for {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}, the scaling of the $\langle\hat{\tau}_p\rangle$ for
intermediate $p$ does not change significantly.
However, the uncertainties of these curves increase when the PEO chains are successively
replaced by IL (up to $60\text{~\%}$ in the worst case, \mbox{i.\,e.\,}\ $p=1$ for {$\text{P}_{8}\text{S}\cdot1.667\text{~IL}$}).
For the local modes, the $\langle\hat{\tau}_p\rangle$ increase with the IL concentration, again
highlighting the reduced flexibility of the PEO backbone.
Therefore, in total, the replacement of PEO by IL rather leads to a decrease of the polymer
dynamics, and the plasticizing effect of the IL is not sufficient to compensate for the
slowing down of the PEO segments caused by the attached lithium ions.
Although the scaling of the $\langle\hat{\tau}_p\rangle$ is rather Zimm-like~\cite{ZimmJCP1956}
(\mbox{i.\,e.\,}\ $\langle\hat{\tau}_p\rangle\propto p^{-3/2}$) for {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}, especially for {$\text{P}_{20}\text{S}\cdot3.24\text{~IL}$}, the segmental
MSD nevertheless shows the characteristic Rousean proportionality $\text{MSD}\propto t^{1/2}$
for $t\geq 1\text{~ns}$ (Figure~\ref{fig:EO_Li_msd}) instead of the respective Zimm scaling~\cite{DoiEdwards}
$\text{MSD}\propto t^{2/3}$.
Only on time scales up to about $1\text{~ns}$, the MSD of the average EOs is lower and slightly steeper
than the Rouse predictions for {$\text{P}_{20}\text{S}\cdot3.24\text{~IL}$}.
This is in clear contrast to {$\text{P}_{20}\text{S}$}, for which the PEO curve is larger than the Rouse prediction
due to the additional internal degrees of freedom of the PEO backbone.
Thus, this feature reflects another minor imprint of the hydrodynamic interactions.
On larger length scales however, the hydrodynamic interactions are screened, since the monomer
concentration in the PEO-IL solutions is still relatively high.
Consequently, the chains interpenetrate strongly as in the pure polymer melt, and the momentum
transfer via the low-molecular solvent acts only on short length scales.
A similar dynamical behavior has also been reported previously for semidilute solutions of model
polymers~\cite{DuenwegPRE2001} simulated by a combination of stochastic Langevin dynamics and
Lattice Boltzmann techniques~\cite{DuenwegJCP1999}.
Since the motion on larger scales is still Rousean, we also left Equation~\ref{eq:seg_msd} unchanged in
order to extract $\tau_2$ for the {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}\ electrolytes.
\section{Comparison with Experiments}
\label{sec:cmp_exp}
Finally, we compare the predictions of the lithium transport model to the experimental data
taken from ref.~\citenum{PasseriniJoost}.
Of course, a direct comparison of $D_\mathrm{Li}$ as extracted from Figure~\ref{fig:msd_Li} is impossible,
since the center-of-mass motion of the short PEO chains ($N=54$) in our simulations significantly contributes to
the lithium diffusion, whereas most experimental studies focus on the long-chain limit
(\mbox{e.\,g.\,}\ $M_\mathrm{w}=4.000.000\text{~g/mol}$ in ref.~\citenum{PasseriniJoost}).
However, due the explicit $N$-dependence of the three time scales~\cite{MaitraHeuerPRL2007},
\mbox{i.\,e.\,}\ $\tau_1\propto N^2$, $\tau_2\propto N^2$ and $\tau_3\propto N^0$, and using the fact that the
center-of-mass motion vanishes for $N\rightarrow\infty$, this problem can be easily circumvented.
For the scaling of $\tau_2$, entanglement effects may become relevant, which would manifest
itself by a steeper dependence of the polymer relaxation time on $N$ for chain lengths larger than
the entanglement length $N_\mathrm{e}$.
However, if $\tau_3<\tau_\mathrm{e}$, meaning that the lithium ion leaves the chain before the
latter begins to reptate, the dynamical contribution to the lithium ions is still Rousean~\cite{DoiEdwards}.
In the case of PEO, experiments revealed that the entanglement regime sets in from about
$N\approx 75$~\cite{AppelMamol1993,ShiSSI1993}.
Based on these observations, one can estimate $\tau_\mathrm{e}$ according to
$\tau_\mathrm{e}=\tau_\mathrm{R}(N=75)=\tau_\mathrm{R}(75/54)^2$.
For {$\text{P}_{20}\text{S}$}, this leads to $\tau_\mathrm{e}\approx 87\text{~ns}$, which is substantially
larger than $\tau_3$.
Also in case of the highly plasticized {$\text{P}_{20}\text{S}\cdot3.24\text{~IL}$}\ one finds $\tau_\mathrm{e}\approx 46\text{~ns}>\tau_3$.
Therefore, the lithium ion leaves the PEO chain before the tube constraints become noticeable,
and the transport model can also be applied in the limit of long chains.
Due to the renewal events (\mbox{i.\,e.\,}\ the interchain transfer), $D_\mathrm{Li}$ can be expressed as
\begin{equation}
\label{eq:D_renewal}
D_\mathrm{Li}(N)=\frac{\langle g_{12}(N,\tau_3)\rangle_\mathrm{M3}}{6\tau_3}+D_\mathrm{PEO}(N)\,\text{,}
\end{equation}
where $\langle g_{12}(\tau_3)\rangle_\mathrm{M3}$ corresponds to an averaged, elementary
step length between two successive renewal events (abbreviated as M3), and is characterized
by the intramolecular transport mechanisms (\mbox{i.\,e.\,}\ the diffusion along the chain and the polymer dynamics).
The center-of-mass motion of the PEO chains with length $N$ is characterized by the diffusion
coefficient $D_\mathrm{PEO}$.
As shown in earlier work~\cite{MaitraHeuerPRL2007}, $g_{12}$ can be described by a Rouse-like expression
\begin{equation}
\label{eq:g12model}
g_{12}(t) = \frac{2\langle{\bf R}_\mathrm{e}^2\rangle}{\pi^2}\sum_{p=1}^\infty\frac{\left[1-\exp{\left(-\frac{tp^2}{\tau_{12}}\right)}\right]}{p^2}\,\text{,}
\end{equation}
where $1/\tau_{12}=1/\tau_1+1/\tau_2$ is a combined relaxation rate due to both mechanisms.
As argued in Section~\ref{sec:polymer_dynamics}, lithium ions coordinated to two PEO chains experience
no effective transport due to the diffusion along the chain (see also supporting information of ref.~\citenum{DiddensMacroLett2013}).
However, with respect to Equation~\ref{eq:g12model}, this effect can be easily captured.
For a sufficiently fast exchange between coordinations to one and to two PEO chains, the average intramolecular
lithium MSD $\bar{g}_{12}$ is simply determined by the structural quantities $r_1$ and $r_2$ (Table~\ref{tab:structure}):
\begin{equation}
\label{eq:g12_g2model}
\bar{g}_{12}(t) = r_\mathrm{1}\,g_{12}(t) + r_2\,g_{2}(t)
\end{equation}
That is, for the fraction $r_1$ of cations bound to one PEO chain, Equation~\ref{eq:g12model} remains valid, whereas for
ions bound to two chains with $\tau_{1,\mathrm{\,2\,PEO}}\rightarrow\infty$, only $\tau_2$ is important.
For exponentially distributed residence times at the distinct PEO chains (which is fulfilled to a good approximation
in our simulations except for shortest residence times lower than $300\text{~ps}$, which account only for a few percent),
one can write down an analytical expression for the average MSD at a given chain:
\begin{equation}
\begin{aligned}
\langle g_{12}(\tau_3)\rangle_\mathrm{M3} &= \frac{1}{\tau_3}\int_{0}^\infty dt\exp{\left(-\frac{t}{\tau_3}\right)}\,g_{12}(t) \\
&= \frac{2\langle{\bf R}_\mathrm{e}^2\rangle}{\pi^2}\sum_{p=1}^\infty\frac{1}{p^2}\left[1-\frac{1}{p^2\frac{\tau_3}{\tau_{12}}+1}\right]
\end{aligned}
\label{eq:g12_expdist}
\end{equation}
Solving this expression numerically, values for $D_\mathrm{Li}$ were obtained for $N=54$ ($D_\mathrm{Li}^\mathrm{sim}$) and
$N\rightarrow\infty$ ($D_\mathrm{Li}^{\infty}$), which are summarized in Table~\ref{tab:D_Li} (here, a correction similar to
Equation~\ref{eq:g12_g2model} has been applied).
For the calculation of $D_\mathrm{Li}^\mathrm{sim}$, $D_\mathrm{PEO}$ was simply extracted from the simulation.
When comparing $D_\mathrm{Li}^\mathrm{sim}$ with the lithium diffusion coefficients directly extracted from Figure~\ref{fig:msd_Li},
we find reasonable agreement between both values (although the diffusion coefficients from the MSDs are about $5-20\text{~\%}$
higher, which is due to the fact that the lithium motion has not yet become fully diffusive and the statistical uncertainties
are relatively large in the long-time regime).
However, when using the transport model to calculate the lithium MSDs as in our previous work~\cite{DiddensMacroLett2013}
(which was done by simulating a random-walk motion whose step length was given by Equation~\ref{eq:g12model}, and in which random reorientations
at given time steps sampled from a Poisson process modeled the renewal events), we find quantitative agreement within the error bars for
all systems, showing that the predictions of our model are consistent with the numerical data.
\begin{table*}
\centering
\footnotesize
\begin{tabular}{l c c c l}
\hline
\hline
system & $D_\mathrm{Li}^\mathrm{sim}$ [$\text{\AA$^2$}\text{ns}^{-1}$] & $D_\mathrm{Li}^{\infty}$ [$\text{\AA$^2$}\text{ns}^{-1}$] & \multicolumn{2}{c}{$D_\mathrm{Li}^\mathrm{exp}$ [$\text{\AA$^2$}\text{ns}^{-1}$] (ref.~\citenum{PasseriniJoost})} \\
\hline
{$\text{P}_{20}\text{S}$}\ & $2.945$ ($2.947$) & $1.947$ ($1.949$) & $0.052$ & ({$\text{P}_{20}\text{S}\cdot1.0\text{~IL}$}) \\
\hline
\multicolumn{5}{c}{ {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}\ } \\
\hline
{$\text{P}_{20}\text{S}\cdot0.66\text{~IL}$}\ & $3.542$ ($3.545$) & $2.309$ ($2.313$) & $0.118$ & ({$\text{P}_{20}\text{S}\cdot2.0\text{~IL}$}) \\
{$\text{P}_{20}\text{S}\cdot3.24\text{~IL}$}\ & $4.257$ ($4.292$) & $2.392$ ($2.434$) & $0.126$ & ({$\text{P}_{20}\text{S}\cdot4.0\text{~IL}$}) \\
\hline
\multicolumn{5}{c}{ {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\ } \\
\hline
{$\text{P}_{16}\text{S}\cdot0.556\text{~IL}$}\ & $2.573$ ($2.581$) & $1.511$ ($1.520$) & $0.051$ & ({$\text{P}_{10}\text{S}\cdot1.0\text{~IL}$}) \\
{$\text{P}_{12}\text{S}\cdot1.111\text{~IL}$}\ & $2.716$ ($2.765$) & $1.311$ ($1.366$) & $0.063$ & ({$\text{P}_{10}\text{S}\cdot2.0\text{~IL}$}) \\
{$\text{P}_{8}\text{S}\cdot1.667\text{~IL}$}\ & $2.812$ ($3.092$) & $1.166$ ($1.487$) & $0.046$ & ({$\text{P}_{5}\text{S}\cdot1.0\text{~IL}$}) \\
\hline
\hline
\end{tabular}
\normalsize
\caption{Lithium diffusion coefficient $D_\mathrm{Li}$ calculated by the lithium transport model (in particular Equations~\ref{eq:D_renewal}
and \ref{eq:g12_expdist}) for $N=54$ ($D_\mathrm{Li}^\mathrm{sim}$) and $N\rightarrow\infty$ ($D_\mathrm{Li}^{\infty}$).
The values in parentheses for $D_\mathrm{Li}^\mathrm{sim}$ and $D_\mathrm{Li}^{\infty}$ are corrected values, for which
the lithium migration through IL-rich regions has been taken into account.
Experimental values~\cite{PasseriniJoost} for similar compositions at $T=323\text{~K}$ are also shown. }
\label{tab:D_Li}
\end{table*}
In the following, we will compare our results with the experimental data ($D_\mathrm{Li}^\mathrm{exp}$)
at $T=323\text{~K}$ taken from ref.~\citenum{PasseriniJoost}.
Despite the temperature gap between simulation and experiment, we observe identical trends:
For a similar dilution series, in which the IL is {\it added} to {$\text{P}_{20}\text{S}$}, $D_\mathrm{Li}^\mathrm{exp}$
also increases significantly with increasing IL content.
In contrast to this, $D_\mathrm{Li}^\mathrm{exp}$ rather decreases when the PEO chains are {\it substituted}
by IL, and, consequently, the $\text{EO}:\text{Li}$ ratio decreases.
As already discussed in context with the segmental PEO motion (Section~\ref{sec:segmental_dynamcis}), these observations can be understood as follows:
In case of {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}, the IL acts as a plasticizer, and the enhanced polymer dynamics leads to a faster lithium ion dynamics.
On the other hand, when the $\text{EO}:\text{Li}$ ratio simultaneously decreases as for {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}, the remaining PEO chains
become more and more rigid, which approximately cancels the plasticizing effect of the IL.
Thus, two opposing trends related to the segmental mobility can be found in ternary polymer
electrolyte-ionic liquid mixtures: the slowdown of the polymer motion with decreasing $\text{EO}:\text{Li}$
ratio (also encountered in the binary systems) and the plasticizing due to the IL, leading to an
enhancement of the polymer dynamics (\mbox{i.\,e.\,}\ both the segmental and the center-of-mass motion).
Naturally, for the technologically relevant long-chain limit, the enhancement of the center-of-mass
motion of the PEO chains is irrelevant.
For these reasons, one should keep in mind that especially the polymer segments remain mobile enough when
designing new electrolyte materials.
It should be pointed out that the $D_\mathrm{Li}$-values in Table~\ref{tab:D_Li} do not contain the contribution from those lithium ions
which are coordinated by IL molecules only and thus move faster.
In particular for the electrolytes {$\text{P}_{12}\text{S}\cdot1.111\text{~IL}$}\ and {$\text{P}_{8}\text{S}\cdot1.667\text{~IL}$}, for which this additional, fourth mechanism becomes relevant (Table~\ref{tab:tau3}),
this would lead to noticeably higher $D_\mathrm{Li}$-values in Table~\ref{tab:D_Li}.
Here, a correction to $D_\mathrm{Li}$ can easily be estimated:
At a given time, the fraction of lithium ions coordinated by TFSI only is given by $p_\mathrm{IL}$ (Table~\ref{tab:tau3}).
Since we found that dynamics of these ions is similar to that in {$\text{PYR}_\mathrm{13}\text{TFSI}\cdot0.262\text{~LiTFSI}$}, one can use the respective lithium diffusion coefficient
($D_\mathrm{Li,IL}=14.014\text{~\AA$^2$}\text{ns$^{-1}$}$) to estimate their contribution to the overall diffusion coefficient,
whereas the relative contribution of the remaining lithium ions is still described by Equation~\ref{eq:D_renewal}:
\begin{equation}
\label{eq:D_renewal_corr}
D_\mathrm{Li}^\mathrm{corr}(N) = (1-p_\mathrm{IL})\,D_\mathrm{Li}(N) + p_\mathrm{IL}\,D_\mathrm{Li,IL}
\end{equation}
The respective corrected values for $D_\mathrm{Li}^\mathrm{sim}$ and $D_\mathrm{Li}^{\infty}$ are given in parentheses in Table~\ref{tab:D_Li}.
As already expected from the values of $p_\mathrm{IL}$, significant contributions arise only for {$\text{P}_{12}\text{S}\cdot1.111\text{~IL}$}\ and {$\text{P}_{8}\text{S}\cdot1.667\text{~IL}$}\
($p_\mathrm{IL}=0.4\text{~\%}$ and $2.5\text{~\%}$, respectively), whereas $D_\mathrm{Li}$ for all other electrolytes remains
basically unaltered.
Thus, for most electrolytes, the value of $D_\mathrm{Li}$ is determined by essentially three contributions (Equation~\ref{eq:D_renewal}):
the intrachain distance $\langle g_{12}(\tau_3)\rangle_\mathrm{M3}$ the ion covers while
connected to the same chain, the center-of-mass motion of the PEO molecules characterized by $D_\mathrm{PEO}$
and the renewal events measured by $\tau_3$, which facilitate the diffusive motion of the ions in the long-time
limit for $N\rightarrow\infty$.
Moreover, these trends may also oppose each other as in case of {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}.
For these reasons, it is desirable to quantify in how far the modified intrachain transport
and the altered renewal rates contribute to the overall change of $D_\mathrm{Li}$ relative to
the pure polymer electrolyte {$\text{P}_{20}\text{S}$}.
\begin{table}
\centering
\footnotesize
\subtable[$N=54$]{
\begin{tabular}{l r r r r}
\hline
\hline
system & $\langle g_{12}(\tau_3)\rangle_\mathrm{M3}$ & $\tau_3^{-1}$ & $D_\mathrm{PEO}$ & $D_\mathrm{Li}$ \\
\hline
{$\text{P}_{20}\text{S}$}\ & $\pm0$ & $\pm0$ & $\pm0$ & $\pm0$ \\
\hline
\multicolumn{5}{c}{ {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}\ } \\
\hline
{$\text{P}_{20}\text{S}\cdot0.66\text{~IL}$}\ & $+22$ & $-7$ & $+30$ & $+20$ \\
{$\text{P}_{20}\text{S}\cdot3.24\text{~IL}$}\ & $+54$ & $-29$ & $+92$ & $+45$ \\
\hline
\multicolumn{5}{c}{ {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\ } \\
\hline
{$\text{P}_{16}\text{S}\cdot0.556\text{~IL}$}\ & $+9$ & $-32$ & $+4$ & $-13$ \\
{$\text{P}_{12}\text{S}\cdot1.111\text{~IL}$}\ & $+6$ & $-40$ & $+30$ & $-8$ \\
{$\text{P}_{8}\text{S}\cdot1.667\text{~IL}$}\ & $+5$ & $-48$ & $+50$ & $-5$ \\
\hline
\hline
\end{tabular}
}
\subtable[$N\rightarrow\infty$]{
\begin{tabular}{l r r r r}
\hline
\hline
system & $\langle g_{12}(\tau_3)\rangle_\mathrm{M3}$ & $\tau_3^{-1}$ & $D_\mathrm{PEO}$ & $D_\mathrm{Li}$ \\
\hline
{$\text{P}_{20}\text{S}$}\ & $\pm0$ & $\pm0$ & - & $\pm0$ \\
\hline
\multicolumn{5}{c}{ {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}\ } \\
\hline
{$\text{P}_{20}\text{S}\cdot0.66\text{~IL}$}\ & $+28$ & $-7$ & - & $+19$ \\
{$\text{P}_{20}\text{S}\cdot3.24\text{~IL}$}\ & $+73$ & $-29$ & - & $+23$ \\
\hline
\multicolumn{5}{c}{ {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\ } \\
\hline
{$\text{P}_{16}\text{S}\cdot0.556\text{~IL}$}\ & $+14$ & $-32$ & - & $-22$ \\
{$\text{P}_{12}\text{S}\cdot1.111\text{~IL}$}\ & $+12$ & $-40$ & - & $-33$ \\
{$\text{P}_{8}\text{S}\cdot1.667\text{~IL}$}\ & $+15$ & $-48$ & - & $-40$ \\
\hline
\hline
\end{tabular}
}
\normalsize
\caption{Percental changes of the intramolecular contribution $\langle g_{12}(\tau_3)\rangle_\mathrm{M3}$,
the renewal rates $\tau_3^{-1}$, the center-of-mass motion $D_\mathrm{PEO}$ and the resulting
$D_\mathrm{Li}$-value relative to {$\text{P}_{20}\text{S}$}\ for (a)~$N=54$ and (b)~$N\rightarrow\infty$. }
\label{tab:M12_M3_contr}
\end{table}
Table~\ref{tab:M12_M3_contr} shows the percentage by which the intramolecular
contributions ($\langle g_{12}(\tau_3)\rangle_\mathrm{M3}$),
the center-of-mass motion of the PEO chains ($D_\mathrm{PEO}$)
and the renewal rates ($\tau_3^{-1}$) as well as the overall $D_\mathrm{Li}$-values
change relative to the pure polymer electrolyte {$\text{P}_{20}\text{S}$}\ for $N=54$ and $N\rightarrow\infty$.
For {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}\ one clearly observes that the intramolecular contribution increases
with the IL concentration, mainly as a result of the increased segmental
mobility.
For finite PEO chains, the enhanced $D_\mathrm{PEO}$-value additionally increases
the lithium motion.
In contrast to this, the contribution of the renewal processes decreases for
these electrolytes.
However, the plasticizing effect overcompensates this decrease, and $D_\mathrm{Li}$
increases for both $N=54$ and $N\rightarrow\infty$.
Remarkably, the relative intramolecular contribution also increases for
{$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}, although the enhancement is weaker than for {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}.
At first glance, this may contradict the findings presented in Section~\ref{sec:segmental_dynamcis}, where it
was observed that both the diffusion along the PEO chain and the segmental
motion become slower when IL is added to the system.
However, this mismatch can easily be resolved by the following considerations:
First, the fraction of lithium ions coordinating to one PEO chain only increases
when going from {$\text{P}_{20}\text{S}$}\ to {$\text{P}_{8}\text{S}\cdot1.667\text{~IL}$}\ (Table~\ref{tab:structure}).
Since lithium ions coordinated to two polymer chains experience no effective transport via this mechanism
(supporting information in ref.~\citenum{DiddensMacroLett2013}), the diffusion along the PEO backbone becomes more efficient in this way.
The second, more important effect leading to the observations from Table~\ref{tab:M12_M3_contr}
is the fact that $\langle g_{12}(\tau_3)\rangle_\mathrm{M3}$ is evaluated for different
$\tau_3$ (Equation~\ref{eq:g12_expdist}).
Thus, with increasing $\tau_3$ also the distance $\langle g_{12}(\tau_3)\rangle_\mathrm{M3}$
increases, although this dependence is weaker than linear, and the overall
$D_\mathrm{Li}$ diminishes in this case.
Consistently, calculating the individual values for $\langle g_{12}(\tau_3)\rangle_\mathrm{M3}$
with a constant value of $\tau_3=17.1\text{~ns}$ as observed for {$\text{P}_{20}\text{S}$}\ (Table~\ref{tab:tau3}), the intramolecular
contribution indeed decreases for {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}.
The only contribution which displays a clear increase is $D_\mathrm{PEO}$.
Thus, for $N=54$, $D_\mathrm{Li}$ diminishes only slightly, whereas a significant decrease
can be found in case of $N\rightarrow\infty$.
So far, we focused only on the high-temperature limit which we can address in our simulations.
Interestingly, the relative increase of $D_\mathrm{Li}^\mathrm{exp}$ upon the addition of IL becomes much more
pronounced in the low-temperature regime~\cite{PasseriniJoost}.
Although simulations at low temperature would be too costly, we are nevertheless able to make some statements
about this regime.
For example, in case of the addition of IL, it was reported that the glass-transition temperature $T_\mathrm{g}$
significantly decreases (up to $35\text{~K}$) with increasing IL concentration~\cite{PasseriniJoost}, which
is a first indication that also at ambient temperatures the enhanced polymer dynamics contributes to a larger
$D_\mathrm{Li}$.
In contrast to this, the experimentally observed decrease of $T_\mathrm{g}$ is significantly weaker for lower
$\text{EO}:\text{Li}$ ratios~\cite{PasseriniJoost} (which resembles the case of IL substitution).
Simultaneously, $D_\mathrm{Li}^\mathrm{exp}$ approximately remains constant at $323\text{~K}$.
A detailed analysis of the low-temperature regime is already underway.
\section{Conclusion}
\label{sec:conclusion}
In this work, we presented an exhaustive MD simulation study on ternary polymer electrolytes consisting of
PEO/LiTFSI and the ionic liquid PYR$_\mathrm{13}$TFSI.
In total, we investigated two different concentration series, namely (a)~the \emph{addition} of the ionic liquid
to {$\text{PEO}_{20}\text{LiTFSI}$}, and (b)~the \emph{substitution} of the PEO chains in {$\text{PEO}_{20}\text{LiTFSI}$}\ by the ionic liquid, subject to the
constraint of a constant lithium volume concentration.
The latter electrolytes not only reflected an interpolation between {$\text{PEO}_{20}\text{LiTFSI}$}\ and {$\text{PYR}_\mathrm{13}\text{TFSI}\cdot0.262\text{~LiTFSI}$}, but also served as
model systems for electrolytes with lower $\text{EO}:\text{Li}$ ratios.
In addition to the simulations, we also utilized a lithium ion transport model~\cite{MaitraHeuerPRL2007}
to express the effect of the ionic liquid on the lithium diffusion via a few microscopic parameters.
Based on the combined insights from the simulations and the transport model, we were able to formulate general
statements in how far the lithium ion transport mechanism changes when the composition of the electrolyte is
varied.
In case of sufficiently high $\text{EO}:\text{Li}$ ratios (in particular $20:1$), we find that the lithium
transport almost exclusively takes place at the PEO chains, and that the transport mechanism therefore is
qualitatively the same as in binary {$\text{PEO}_{20}\text{LiTFSI}$}.
For lower ratios, however, more and more lithium ions coordinate to a specific PEO chain, with the result
that a progressive coordination by TFSI anions (both partially and totally) becomes visible.
Accordingly, a completely new, fourth transport mechanism, in which the lithium ions are decoupled from the
PEO chains, emerges due to the lack of free ether oxygens capable to bind the lithium ions.
This mechanism may significantly contribute to the macroscopic lithium diffusion or even dominate it.
From a theoretical perspective, we incorporated this mechanism into our transport model.
For all electrolytes, we find reasonable agreement of the model predictions with the experimentally observed
diffusion coefficients~\cite{PasseriniJoost}.
Apart from these qualitative differences, the lithium transport is also altered on a quantitative level.
For instance, the renewal rate $\tau_3^{-1}$, characterizing the transfer between distinct PEO chains,
decreases with increasing ionic-liquid concentration and with decreasing fraction of free ether oxygens.
Consequently, $\tau_3$ is especially large for diluted electrolytes with crowded PEO chains (\mbox{i.\,e.\,}\ the
{$\text{PEO}_{20-\alpha x}\text{LiTFSI}\cdot\,x\text{~PYR$_\mathrm{13}$TFSI}$}\ systems).
Concerning the polymer dynamics, it turned out that the increase of the lithium diffusivity in {$\text{PEO}_{20}\text{LiTFSI}\cdot\,x\text{~PYR$_\mathrm{13}$TFSI}$}\ can
be attributed to the enhanced segmental mobility of the PEO chains, and that the ionic liquid thus serves
as a plasticizer.
As a result of the faster PEO motion, also the lithium ions coordinating to the PEO chains become faster.
Interestingly, the plasticization is also visible in the scaling of the Rouse-mode relaxation times, which
exhibit a slightly Zimm-like scaling characteristic for semidilute polymer solutions.
An opposing effect on the PEO dynamics, however, is the coordination to the lithium ions, which becomes
especially pronounced for low $\text{EO}:\text{Li}$ ratios.
Accordingly, the average PEO dynamics in {$\text{PEO}_{20-\alpha x}\text{LiTFSI}\cdot\,x\text{~PYR$_\mathrm{13}$TFSI}$}\ is slowed down by the progressive coordination of
lithium ions, and the plasticizing effect of the ionic liquid becomes visible only for larger, entirely
uncoordinated polymer segments.
In summary, one therefore faces the situation that the segmental mobility of the PEO chains plays a decisive
role in ternary polymer electrolytes:
Whereas the lithium ions slow down the PEO chains, the ionic liquid accelerates the polymer motion.
Thus, for the design of new battery materials, one should attempt to render the latter effect the
dominating one.
\begin{acknowledgments}
The authors would like to thank Oleg Borodin, Nicolaas A. Stolwijk, Stefano Passerini
and Mario Joost for helpful discussions and for providing the experimental data.
Financial support from the NRW Graduate School of Chemistry is also greatly appreciated.
\end{acknowledgments}
\begin{appendix}
\section{Part~A: Further Analyses}
\subsection{Simulation Box Size}
\begin{table}
\centering
\begin{tabular}{l c}
\hline
\hline
system & $L_\mathrm{box}$ [\AA] \\
\hline
PEO & $40.11$ \\
\hline
{$\text{P}_{20}\text{S}$}\ & $35.96$ \\
\hline
\multicolumn{2}{c}{ {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}\ } \\
\hline
{$\text{P}_{20}\text{S}\cdot0.66\text{~IL}$}\ & $38.27$ \\
{$\text{P}_{20}\text{S}\cdot3.24\text{~IL}$}\ & $45.29$ \\
\hline
\multicolumn{2}{c}{ {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\ } \\
\hline
{$\text{P}_{16}\text{S}\cdot0.556\text{~IL}$}\ & $35.93$ \\
{$\text{P}_{12}\text{S}\cdot1.111\text{~IL}$}\ & $35.96$ \\
{$\text{P}_{8}\text{S}\cdot1.667\text{~IL}$}\ & $36.00$ \\
\hline
{$\text{IL}\cdot0.262\text{~S}$}\ & $36.33$ \\
\hline
\hline
\end{tabular}
\caption{Length of the simulation boxes for the different electrolytes. }
\label{tab:boxlength}
\end{table}
The sizes of the individual simulation boxes are summarized in Table~\ref{tab:boxlength}.
One clearly observes that while for the {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}\ systems the volume of the simulation box increases,
$L_\mathrm{box}$ for the {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\ electrolytes is nearly identical for each system, meaning
that the lithium volume concentration ($27$ lithium ions in each box) remains constant.
\subsection{Radial Distribution Functions}
\label{ssec:gofr}
\begin{figure*}
\centering
\subfigure[ \label{fig:gofr_LiO_PEO_x} ]{ \includegraphics[scale=0.3]{gofr_PEO_O-Li_add} }
\subfigure[ \label{fig:gofr_LiO_TFSI_x} ]{ \includegraphics[scale=0.3]{gofr_TFSI_O-Li_add} }
\subfigure[ \label{fig:gofr_LiO_PEO_xminus} ]{ \includegraphics[scale=0.3]{gofr_PEO_O-Li_sub} }
\subfigure[ \label{fig:gofr_LiO_TFSI_xminus} ]{ \includegraphics[scale=0.3]{gofr_TFSI_O-Li_sub} }
\caption{Upper: radial distribution functions in {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}\ for the atom pairs (a)~{$\text{Li}^\text{+}-\text{O}_\text{PEO}$}\ and (b)~{$\text{Li}^\text{+}-\text{O}_\text{TFSI}$}.
Lower: radial distribution functions in {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}\ for the atom pairs (c)~{$\text{Li}^\text{+}-\text{O}_\text{PEO}$}\ and (d)~{$\text{Li}^\text{+}-\text{O}_\text{TFSI}$}. }
\label{fig:gofr_Li}
\end{figure*}
Figure~\ref{fig:gofr_Li} shows the radial distribution functions $g(r)$ for the atom pairs {$\text{Li}^\text{+}$}$-\text{O}_\text{PEO}$ and {$\text{Li}^\text{+}$}$-\text{O}_\text{TFSI}$.
Both coordination types exhibit a sharp peak around $2\text{~\AA}$ corresponding to the first coordination shell.
For {$\text{Li}^\text{+}-\text{O}_\text{PEO}$}, the first coordination sphere directly crosses over into a second small peak around $3.3\text{~\AA}$.
At larger distances no significant structural arrangement can be found.
For the coordination of {$\text{Li}^\text{+}-\text{O}_\text{TFSI}$}, peaks become noticeable also at larger distances around $6\text{~\AA}$ and $8\text{~\AA}$, thus demonstrating
that long-ranged correlations are present as also observed for a PEO$_{20}$LiI electrolyte~\cite{MaitraJPCB2008}.
The same observation can be made for the PEO-free electrolyte {$\text{PYR}_\mathrm{13}\text{TFSI}\cdot0.262\text{~LiTFSI}$}, as also reported in previous MD studies~\cite{BorodinJPCB2006_ILLiTFSI}.
The pair correlation function of the PYR$_\mathrm{13}$ cations and TFSI anions (\mbox{i.\,e.\,}\ {$\text{N}_\text{PYR$_\mathrm{13}$}-\text{O}_\text{TFSI}$}) exhibits only a weak first coordination peak
between $3.6$ and $5.0\text{~\AA}$.
When successively adding the IL, one observes that the peak positions of both {$\text{Li}^\text{+}-\text{O}_\text{PEO}$}\ and {$\text{Li}^\text{+}-\text{O}_\text{TFSI}$}\ remain the same for all electrolytes.
For {$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}, the EO coordination numbers extracted from the integral over the first shell increase slightly, partly as a result of the lower
fraction of lithium ions coordinating to two PEO chains (Table~\ref{tab:structure} in the main article), for which the EO coordination number is
lower due to steric effects.
In {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}, the absolute EO coordination number (as determined from the integral over $g(r)$) decreases with increasing IL content.
This trend can be explained by the decreasing number of possible coordination sites, since the PEO concentration reduces.
Naturally, the coordination number of the TFSI oxygens increases with increasing IL content in both types of electrolytes.
\subsection{Lithium Coordination Shell}
\begin{figure*}
\centering
\subfigure[ \label{fig:EO_bin_x} ]{ \includegraphics[scale=0.3]{PILbin_add} }
\subfigure[ \label{fig:EO_bin_xminus} ]{ \includegraphics[scale=0.3]{PILbin_sub} }
\caption{Probability $p(n)$ to find a certain coordination number $n$ of EOs
(irrespective if the ion is tied to one or two PEO chains) or TFSI oxygens
in (a)~{$\text{P}_{20}\text{S}\cdot\,x\text{~IL}$}\ and (b)~{$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}. }
\label{fig:coordbins}
\end{figure*}
Figure~\ref{fig:coordbins} shows the probability distribution functions $p(n)$ to find a lithium ion with $n$ EOs or TFSI oxygens in its first coordination shell.
One observes that for {$\text{PEO}_{20}\text{LiTFSI}\cdot\,x\text{~PYR$_\mathrm{13}$TFSI}$}, the coordination numbers are very similar in all systems and only change slightly with the IL concentration.
In contrast to this, for {$\text{P}_{20-\alpha x}\text{S}\cdot\,x\text{~IL}$}, a coordination number of $n=2$ EOs becomes noticeable, and simultaneously the probability for one or two coordinating
anions increases.
\section{Part~B: Force Field Validation}
\label{sec:validation}
In order to validate the results from the AMBER simulations, comparative simulations with Lucretius~\cite{Lucretius} (the original code for which the force
field used in this study has been developed) have been performed.
We focus on the pure polymer electrolyte {$\text{PEO}_{20}\text{LiTFSI}$}.
For the Lucretius simulation, the same techniques and parameters as in the original study~\cite{BorodinMamol2006} have been used.
Both MD codes yielded the same density when simulating an $NpT$ ensemble, thus giving a first indication
that the AMBER results are reasonable.
In the following, $NVT$ simulations of the respective codes will be compared
(the results are the same for $NpT$ simulations).
Since the Lucretius version used in this comparison (\mbox{i.\,e.\,}\ the same as in ref.~\citenum{BorodinMamol2006}) does not allow parallel calculations, the total
simulation length was restricted to $50\text{~ns}$ as compared to the $200\text{~ns}$ of the AMBER simulations (note that in the meantime a newer, parallelized
Lucretius version is also available).
\subsection{Dihedral Distributions}
\begin{figure*}
\centering
\subfigure[ \label{fig:dihedral_OCCO} ]{ \includegraphics[scale=0.3]{dihedral_O-C-C-O_bin_val} }
\subfigure[ \label{fig:dihedral_COCC} ]{ \includegraphics[scale=0.3]{dihedral_C-O-C-C_bin_val} }
\caption{Probability distribution function of two dihedral types of the PEO backbone for {$\text{PEO}_{20}\text{LiTFSI}$}\ at
$T=423\text{~K}$. (a) O-C-C-O and (b) C-O-C-C. }
\label{fig:dihedral_bin}
\end{figure*}
Figure~\ref{fig:dihedral_bin} shows the probability distribution function $p(\phi)$ of the dihedral angles
$\phi$ describing the rotation around the carbon-carbon ($\text{O-C-C-O}$) and the carbon-oxygen ($\text{C-O-C-C}$)
bonds of the PEO backbone.
For the $\text{O-C-C-O}$-dihedral, one observes perfect agreement between the two codes, whereas for the conformations
of the $\text{C-O-C-C}$-dihedral slight differences can be observed.
These deviations can be rationalized by the fact that the bond length constraining (SHAKE~\cite{SHAKE}) for non-hydrogen
bonds is not possible in parallel AMBER simulations.
Since these bonds can be stretched in these simulations, the $1$-$4$ repulsions between the outer carbon atoms of the
$\text{C-O-C-C}$-dihedral is mitigated, and the {\it gauche}-conformation ($\phi\approx 75^\circ$) becomes more populated
compared to the {\it trans}-conformation ($\phi=180^\circ$).
Indeed, the same observation can be made for a Lucretius simulation (length $25\text{~ns}$) without bond length constraining.
However, these slight deviations have no influence on the dynamics of the PEO backbone and the lithium motion (see discussion below).
\subsection{Radial Distribution Functions}
\begin{figure*}
\centering
\subfigure[ \label{fig:gofr_validation_PEO_Li} ]{ \includegraphics[scale=0.3]{gofr_PEO-O_Li_val} }
\subfigure[ \label{fig:gofr_validation_TFSI_Li} ]{ \includegraphics[scale=0.3]{gofr_TFSI-O_Li_val} }
\caption{Radial distribution functions $g(r)$ of the
(a)~{$\text{Li}^\text{+}$}$-\text{O}_\text{PEO}$ and the
(b)~{$\text{Li}^\text{+}$}$-\text{O}_\text{TFSI}$ interaction. }
\label{fig:gofr_validation}
\end{figure*}
Figure~\ref{fig:gofr_validation} shows the radial distribution functions $g(r)$ for the {$\text{Li}^\text{+}$}$-\text{O}_\text{PEO}$ and the
{$\text{Li}^\text{+}$}$-\text{O}_\text{TFSI}$ interaction. For both coordination types one observes reasonable agreement.
Again, the slight deviations for the second coordination peak are related to the fact that the molecules in AMBER are more
flexible due to the unconstrained bonds.
\subsection{Mean Square Displacements}
\begin{figure}
\centering
\includegraphics[scale=0.3]{msd_all_val}
\caption{MSDs for the individual molecule types in {$\text{PEO}_{20}\text{LiTFSI}$}\ simulated with Lucretius and AMBER. }
\label{fig:msd_validation}
\end{figure}
Figure~\ref{fig:msd_validation} shows the MSDs for the individual molecular species in {$\text{PEO}_{20}\text{LiTFSI}$}.
Since the AMBER simulation extends over $200\text{~ns}$ and the MSDs thus have much smaller error bars, only the error
bars of the Lucretius curves are shown for clarity.
For PEO, an additional distinction was made between the segmental dynamics (EOs) and center-of-mass motion.
The agreement for all molecule types is quantitative.
Deviations for larger time scale of several ten nanoseconds are within the error bars of the shorter Lucretius simulation.
In total, the agreement between the two MD codes is convincing.
\section{Part~C: Benchmark Calculations}
\label{sec:benchmark}
Benchmark calculations have been performed for the modified AMBER code.
Again, we focus on the electrolyte {$\text{PEO}_{20}\text{LiTFSI}$}\ ($4772$ atoms) at $T=423\text{~K}$ in the $NVT$ ensemble
using various numbers of CPUs (Intel Xeon Nehalem X5550 with $2.67\text{~GHz}$ each, PALMA cluster~\cite{PALMA}).
The simulations were carried out using various treatments of the polarization, in particular without polarization, iterative solution
of the inducible dipoles, as well as the Car-Parrinello scheme described in ref.~\citenum{vanBelleMolPhys1992}.
\begin{figure*}
\centering
\subfigure[ \label{fig:hour_per_ns} ]{ \includegraphics[scale=0.3]{time_per_ns_hours} }
\subfigure[ \label{fig:ns_per_day} ]{ \includegraphics[scale=0.3]{nanoseconds_per_day} }
\caption{Computation time for {$\text{PEO}_{20}\text{LiTFSI}$}\ ($4772$ atoms) in dependence of the number of CPUs for various polarization treatments.
For convenience, the accessible MD time per day is also shown. }
\label{fig:benchmark}
\end{figure*}
The computation time is shown in Figure~\ref{fig:benchmark}.
For convenience, the resulting MD time per day is also shown in Figure~\ref{fig:benchmark}.
For small processor numbers, one observes an ideal linear increase of the computation speed.
From about eight CPUs this scaling drops below linear scaling, and finally saturates at a maximum computation speed for $32\text{~CPUs}$.
A more detailed analysis of the computation times of the distinct contributions (\mbox{i.\,e.\,}\ bonded, van-der-Waals or electrostatic interactions)
reveals that the saturation as well as the subsequent decrease in Figure~\ref{fig:benchmark} arises from the evaluation of the reciprocal
part of the electrostatic energies and forces.
For this step, all spatial coordinates of the system are involved in the calculation, requiring the excessive communication between the
individual CPUs.
Consequently, the speed-up gained by the larger number of CPUs cannot compensate for this massive broadcast, so that the real computation
time saturates and finally decreases again.
The incorporation of polarization leads to a decrease of the simulation speed.
Here, the Car-Parrinello-like treatment of the dipoles~\cite{vanBelleMolPhys1992} is significantly faster than the iterative solution of
the inducible dipoles.
For many CPUs, their relative difference becomes even larger.
As discussed above, this effect is related to the enormous broadcast and the exchange of the inducible dipoles between the individual CPUs.
This slowdown becomes especially heavy for the iterative scheme, as the dipoles have to be gathered and redistributed on the nodes in each
iteration step.
\end{appendix}
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\chapter{ \sffamily{{\bf T}he Standard Model and beyond...}}\label{SM}
\section{{\bf T}he Standard Model}\label{SM-intro}
The quest for explaining diverse physical phenomena with a single ``supreme''
theory is perhaps deeply embedded in the human mind. The journey was started
long ago with {\it{Michael Faraday}} and later with {\it{James Clerk Maxwell}}
with the unification of the electric and the magnetic forces
as the electromagnetic force. The inspiring successful past
has finally led us to the Standard Model (SM)
(see reviews \cite{c1Abers-Lee,c1Beg-Sirlin} and
\cite{c1Halzen-Martin,c1Cheng-Li,c1Griffiths,c1Burgess-Moore})
of elementary Particle Physics. In the SM three of the four fundamental
interactions, namely electromagnetic, weak and strong interactions
are framed together. The first stride towards
the SM was taken by {\it{Sheldon Glashow}}
\cite{c1Glashow-1961} by unifying the theories of
electromagnetic and weak interactions as the electroweak theory.
Finally, with pioneering contributions from {\it{Steven Weinberg}}
\cite{c1Weinberg-1967} and {\it{Abdus Salam}} \cite{c1Salam-1968}
and including the third fundamental interaction of nature, namely
the strong interaction the Standard Model of particle physics
emerged in its modern form. Ever since, the SM has successfully
explained host of experimental results and precisely predicted a
wide variety of phenomena. Over time and through many experiments by
many physicists, the Standard Model has become established as a well-tested
physics theory.
\begin{flushleft}
{\it{$\maltese$ The quarks and leptons}}
\end{flushleft}
The SM contains elementary particles which are the basic ingredients
of all the matter surrounding us. These particles are divided
into two broad classes, namely, {\it{quarks}} and {\it{leptons}}.
These particles are called {\it{fermions}} since they are spin $\frac{1}{2}$
particles. Each group of quarks and leptons consists of six members, which
are ``paired up'' or appear in {\it{generations}}. The lightest
and most stable particles make up the first generation, whereas
the heavier and less stable particles belong to the second and
third generations. The six quarks are paired in the three
generations, namely the `up quark $(u)$' and the `down quark $(d)$'
form the first generation, followed by the second generation containing the
`charm quark $(c)$' and `strange quark $(s)$', and finally the
`top quark $(t)$' and `bottom quark $(b)$' of the third generation.
The leptons are similarly arranged in three generations, namely the
`electron $(e)$' and the `electron-neutrino $(\nu_e)$',
the `muon $(\mu)$' and the `muon-neutrino $(\nu_\mu)$',
and the `tau $(\tau)$' and the `tau-neutrino $(\nu_\tau)$'.
\begin{flushleft}
{\it{$\maltese$ There are gauge bosons too}}
\end{flushleft}
Apart from the quarks and leptons the SM also contains
different types of {\it{spin-1}} bosons, responsible
for mediation of the electromagnetic, weak and
the strong interaction. These force mediators essentially
emerge as a natural consequence
of the theoretical fabrication of the SM, which relies on the principle
of local gauge invariance with the gauge group ${\rm SU}(3)_C \times
{\rm SU}(2)_L \times {\rm U}(1)_Y$. The force mediator gauge bosons
are $\bf{n^2-1}$ in number for an ${\rm SU}(n)$ group and
belong to the adjoint representation of the group.
The group ${\rm SU}(3)_C$ is
associated with the {\it{colour}} symmetry in the quark sector and
under this group one obtains the so-called colour triplets.
Each quark $(q)$ can carry a {\it{colour charge}} under the ${\rm SU}(3)_C$
group\footnote{The colour quantum number was introduced
for quarks \cite{c1Greenberg:1964pe} to save the {\it{Fermi}} statistics.
These are some hypothetical charges having no connection with the
real life colour of light.} (very similar to electric
charges under ${\rm U}(1)_{em}$
symmetry). Each quark carries one of the three
fundamental colours (${\bf{3}}$ representation), namely,
red $(R)$, green $(G)$ and blue $(B)$. In a similar fashion
an anti-quark $(\bar{q})$ has the complementary colours
(${\bf{\bar 3}}$ representation), cyan $(\overline R)$,
magenta $(\overline G)$ and yellow $(\overline B)$. The accompanying eight
force mediators are known as gluons $(G^a_\mu)$.
The gluons belong to the adjoint representation of ${\rm SU}(3)_C$.
However, all of the hadrons (bound states of quarks) are colour singlet.
Three weak bosons $(W^a_\mu)$ are the force mediators
for ${\rm SU}(2)_L$ group, under which left-chiral quark and lepton
fields transform as doublets. The remaining gauge group ${\rm U}(1)_Y$
provides hypercharge quantum number $(Y)$ to all the SM particles
and the corresponding gauge boson is denoted by $B_\mu$. In describing
different gauge bosons the index `$\mu$' ($=1,..,4$) has been used to
denote {\it{Lorentz}} index. The index `$a$' appears for the
non-Abelian gauge groups\footnote{Yang and Mills \cite{c1Yang-Mills}.}
and they take values $1,..,8$ for ${\rm SU}(3)_C$
and $1,2,3$ for ${\rm SU}(2)_L$.
Different transformations for the SM fermions and gauge bosons under
the gauge group ${\rm SU}(3)_C \times {\rm SU}(2)_L \times {\rm U}(1)_Y$
are shown below\footnote{We
choose $Q=T_3 + \frac{Y}{2}$, where $Q$ is the electric charge,
$T_3$ is the third component of the weak isospin ($\pm\frac{1}{2}$
for an ${\rm SU(2)}$ doublet) and $Y$ is the weak hypercharge.}
\begin{eqnarray}
&&L_{i_L} = \left(\begin{array}{c}
\nu_{\ell_i} \\
\ell_i
\end{array}\right)_L \sim ({\bf{1,2,-1}}),
~~\ell_{i_R} \sim ({\bf{1,1,-2}}
,\nonumber \\
&&Q_{i_L} = \left(\begin{array}{c}
u_i \\
d_i
\end{array}\right)_L \sim ({\bf{3,2,{\frac{1}{3}}}}),~~u_{i_R} \sim
({\bf{3,1,{\frac{4}{3}}}}),~~d_{i_R} \sim ({\bf{3,1,-{\frac{2}{3}}}}),\nonumber \\
&& G^a_\mu \sim ({\bf{8,0,0}}),~~W^a_\mu \sim ({\bf{1,3,0}}),
~~B_\mu \sim ({\bf{1,1,0}}),
\label{SM-gauge-group}
\end{eqnarray}
where $\ell_i=e,\mu,\tau$, $u_i=u,c,t$ and $d_i=d,s,b$. The
singlet representation is given by ${\bf{1}}$.
\begin{flushleft}
{\it{$\maltese$ Massive particles in the SM~?}}
\end{flushleft}
Principle of gauge invariance demands for massless
gauge bosons which act as the force mediators. In addition,
all of the SM fermions (quarks and leptons) are supposed
to be exactly massless, as a consequence of the gauge invariance.
But these are in clear contradiction to observational facts. In
reality one encounters with massive fermions. Also,
the short range nature of the weak interaction indicates towards
some massive mediators. This apparent contradiction
between gauge invariance and massive gauge boson
was resolved by the celebrated method
of {\it{spontaneous breaking of gauge symmetry
}} \cite{c1Anderson-1958,c1Englert-1964,c1Higgs1-1964,c1Higgs-1964,
c1Guralnik-1964}. The initial SM gauge group
after spontaneous symmetry breaking (SSB) reduces to ${\rm SU}(3)_C
\times {\rm U}(1)_{em}$, leaving the colour and electric charges
to be conserved in nature. Consequently, the corresponding gauge
bosons, gluons and photon, respectively remain massless ensuing gauge invariance,
whereas the weak force mediators ($W^{\pm}$ and $Z$ bosons) become massive.
Symbolically,
\begin{eqnarray}
{\rm SU}(3)_C \times {\rm SU}(2)_L \times {\rm U}(1)_Y
~~~\underrightarrow{SSB}~~~
{\rm SU}(3)_C \times {\rm U}(1)_{em}.
\label{SSB}
\end{eqnarray}
Since ${\rm SU}(3)_C$ is unbroken in nature, all the particles
existing freely in nature are forced to be colour neutral. In a similar
fashion unbroken ${\rm U}(1)_{em}$ implies that any charged particles
having free existence in nature must have their charges as integral
multiple of that of a electron or its antiparticle. It is interesting
to note that quarks have fractional charges but they are not free
in nature since ${\rm SU}(3)_C$ is unbroken.
\begin{list}{}{}
\item
\begin{flushleft}
{{$\blacklozenge$ Spontaneous symmetry breaking}}
\end{flushleft}
Let us consider a Hamiltonian $H_0$ which is invariant under some
symmetry transformation. If this symmetry of $H_0$ is
not realized by the particle spectrum, the symmetry is
spontaneously broken. A more illustrative example
is shown in figure \ref{SSB-pic}. Here the minima
of the potential lie on a circle (white dashed)
rather than being a specific point. Each of these points are
equally eligible for being the minimum and whenever the
red ball chooses a specific minimum, the symmetry of the
ground state (the state of minimum energy) is spontaneously broken.
In other words,
when the symmetry of $H_0$ is not respected by the ground
state, the symmetry is spontaneously broken. It turns
out that the degeneracy in the ground state is essential
for spontaneous symmetry breaking.
\begin{figure}[ht]
\centering
\includegraphics[width=5.45cm,keepaspectratio]{FigsPDF/SSB-pic.pdf}
\caption{Spontaneous breaking of symmetry through
the choice of a specific degenerate ground state.}
\label{SSB-pic}
\end{figure}
\end{list}
Everything seems to work fine with the massive gauge bosons.
But the demon lies within the method of spontaneous symmetry
breaking itself. The {\it{spontaneous breakdown}} of a {\it{continuous symmetry}}
implies the existence of {\it{massless, spinless particles}} as
suggested by {\it{Goldstone}} theorem.\footnote{Initially by Nambu
\cite{c1Nambu-1960}, Nambu and Jona-Lasino. \cite{c1Nambu1-1961,c1Nambu2-1961}.
General proof by Goldstone \cite{c1Goldstone1-1961,c1Goldstone2-1962}.}
They are known as {\it{Nambu-Goldstone}} or simply {\it{Goldstone}}
bosons. So the SSB apart from generating gauge boson masses also
produces massless scalars which are not yet experimentally detected.
This is the crisis point when the celebrated
``Higgs-mechanism''\footnote{The actual name
should read as Brout-Englert-Higgs-Guralnik-Hagen-Kibble mechanism
after all the contributors. Brout and Englert \cite{c1Englert-1964},
Higgs \cite{c1Higgs1-1964,c1Higgs-1964}, Guralnik, Hagen and Kibble \cite{c1Guralnik-1964}.}
resolves the crisis situation.
The unwanted
massless scalars are now eaten up by the gauge boson fields
and they turn out to be the badly needed longitudinal polarization
mode for the ``massive'' gauge bosons.
So this is essentially the reappearance of three degrees of freedom
associated with three massless scalars in the form of three
longitudinal polarization modes for the massive gauge bosons.
This entire mechanism happens
without breaking the gauge invariance of the theory explicitly.
This mechanism for generating gauge boson masses is also consistent
with the renormalizability of a theory with massive gauge
bosons.\footnote{Veltman and 't Hooft,
\cite{c1'tHooft-1971,c1'tHooft-1972}.} The fermion masses also
emerge as a consequence of Higgs mechanism.
\begin{flushleft}
{\it{$\maltese$ Higgs sector of the SM and mass generation}}
\end{flushleft}
So the only scalar (spin-$0$) in the SM is the Higgs boson.
Higgs mechanism is incorporated in the SM through a complex scalar doublet
$\Phi$ with the following transformation properties under the SM
gauge group.
\begin{eqnarray}
\Phi = \left(\begin{array}{c}
\phi^+ \\
\phi^0
\end{array}\right) \sim ({\bf{1,2,1}}).
\label{SM-Higgs}
\end{eqnarray}
The potential for $\Phi$ is written as
\begin{eqnarray}
V(\Phi) = \mu^2 \Phi^{\dagger} \Phi + \lambda (\Phi^{\dagger} \Phi)^2,
\label{SM-Higgs-pot}
\end{eqnarray}
with $\mu^2 < 0$ and $\lambda > 0$ (so that the potential is bounded
from below). Only a colour and charge (electric) neutral component
can acquire a vacuum expectation value (VEV), since
even after SSB the theory remains invariant under
${\rm SU(3)}_C \times {\rm U(1)}_{em}$
(see eqn.(\ref{SSB})). Now with a suitable choice of gauge
(``unitary gauge''), so that the Goldstone bosons disappear
one ends up with
\begin{eqnarray}
\Phi = \frac{1}{\sqrt{2}}\left(\begin{array}{c}
0 \\
v + h^0
\end{array}\right),
\label{SM-Higgs-VEV}
\end{eqnarray}
where $h^0$ is the physical Higgs field and `$v$' is the
VEV for $\mathcal{R}e(\phi^0)$ (all other fields
acquire zero VEVs) with
$v^2 = -\frac{\mu^2}{\lambda}$. At this moment it is apparent
that eqn.(\ref{SSB}) can be recasted as
\begin{eqnarray}
{\rm SU}(2)_L \times {\rm U}(1)_Y
~~~\underrightarrow{SSB}~~~
{\rm U}(1)_{em},
\label{EWSB}
\end{eqnarray}
which is essentially the breaking of the electroweak symmetry
since the ${\rm{SU(3)}}_C$ sector remains unaffected.
Thus the phenomena of SSB in the context of the SM is identical
with the electroweak symmetry breaking (EWSB). The weak bosons,
$W^a_\mu$ and $\rm U(1)_Y$ gauge boson $B_\mu$ now mix together and
finally yield three massive vector bosons $(W^\pm_\mu, Z^0_\mu)$
and one massless photon $(A^0_\mu)$:
\begin{eqnarray}
&&W^\pm_\mu = \frac{W^1_\mu \mp i W^2_\mu}{\sqrt{2}},\nonumber\\
&&Z^0_\mu = \rm{cos}\theta_W W^3_\mu - \rm{sin}\theta_W B_\mu,\nonumber\\
&&A^0_\mu = \rm{sin}\theta_W W^3_\mu + \rm{cos}\theta_W B_\mu,
\label{WZboson-Photon}
\end{eqnarray}
where $\theta_W$ is the {\it{Weinberg}} angle or weak mixing angle.\footnote{
At present $\rm{sin}^2\theta_W = 0.231$ (evaluated at $M_Z$ with
renormalization scheme $\overline{MS}$) \cite{c1Nakamura-c2}.} In
terms of the $\rm SU(2)_L$ and $\rm U(1)_Y$ gauge couplings $(g_2, g_1)$
one can write
\begin{eqnarray}
g_2 ~{\rm{sin}}\theta_W = g_1 ~{\rm{cos}}\theta_W.
\label{Weak-angle}
\end{eqnarray}
The $W^\pm_\mu, Z^0_\mu$ boson masses are given by
\begin{eqnarray}
M_{W} = \frac{g_2 v}{2},
~~~~M_{Z} = \frac{v}{2} \sqrt{g^2_1 + g^2_2},
\label{WZ-mass}
\end{eqnarray}
with $v^2 = -\frac{\mu^2}{\lambda}$. The mass of physical
Higgs boson $(h^0)$ is given by $m^2_{h^0} = 2 v^2 \lambda$.
Note that $m_{h^0} > 0$ since $\mu^2 < 0$.
Interestingly, ratio of the quantities $M^2_W$ and
$M^2_Z \cos^2\theta_W$ is equal to one at the tree level
(see eqns. (\ref{Weak-angle}) and (\ref{WZ-mass})). This
ratio is defined as the $\rho$-parameter, which is an
important parameter for electroweak precision test:
\begin{eqnarray}
\rho = \frac{M^2_W}{M^2_Z {\rm cos}^2\theta_W} = 1.
\label{rho-param}
\end{eqnarray}
There exists an alternative realization of the $\rho$-parameter. The
$\rho$-parameter specifies the relative strength of the neutral current
(mediated through $Z$-bosons) to the charged current (mediated through $W^\pm$-boson)
weak interactions.
For the purpose of fermion mass generation consider the
Lagrangian containing interactions between Higgs field
and matter fermions.
\begin{eqnarray}
-\mathcal{L}_{\text{Yukawa}} = y_{\ell_i} \overline{L_i} \Phi e_i
+ y_{d_i} \overline{Q_i} \Phi d_i + y_{u_i} \overline{Q_i} \tilde{\Phi} u_i
+ \text{Hermitian conjugate},
\label{fermion-mass}
\end{eqnarray}
where $y_{\ell_i,u_i,d_i}$ are the {\it{Yukawa}} couplings for the
charged leptons, up-type quarks and down-type quarks,
respectively. The $\rm SU(2)_L$ doublet and singlet
quark and lepton fields are shown in eqn.(\ref{SM-gauge-group}).
The field $\tilde{\Phi}$ is used to generate masses for
the up-type quarks and it is given by
\begin{eqnarray}
\tilde\Phi = - i \sigma_2 \Phi^* =
i \left(\begin{array}{cc}
0 & -i \\
i & 0
\end{array}\right)\left(\begin{array}{c}
\phi^- \\
{\phi^0}^*
\end{array}\right) =
\left(\begin{array}{c}
-{\phi^0}^* \\
\phi^-
\end{array}\right).
\label{conju-Higgs}
\end{eqnarray}
The fermion masses and their interactions with Higgs field
emerge after the EWSB using eqn.(\ref{fermion-mass}). For example
considering the {\it{electron}} these terms are as follows
\begin{eqnarray}
\mathcal{L}^{\text{electron}}_{\text{Yukawa}} = - \frac{y_e (v + h^0)}{\sqrt{2}}
(\overline {e_L} e_R + \overline {e_R} e_L),
\label{fermion-mass-Higgs}
\end{eqnarray}
where $e_{L} = P_L L_e $ (see eqn.(\ref{SM-gauge-group})).
So with four component spinor $e$ as
$\left(\begin{array}{c}
e_L \\
e_R
\end{array}\right)$, eqn.(\ref{fermion-mass-Higgs})
can be rewritten as
\begin{eqnarray}
\mathcal{L}^{\text{electron}}_{\text{Yukawa}} = - m_e \overline{e} e
- \frac{m_e}{v} \overline{e} e h^0,
\label{fermion-mass-Higgs-2}
\end{eqnarray}
with $m_e = \frac{Y_e v}{\sqrt{2}}$ as mass of the
electron. The particle spectrum of the SM can be
written in a tabular form as shown in table \ref{SM-spectrum}.
\begin{table}[t]
\footnotesize
\centering
\begin{tabular}{ c c c c c}
\hline \hline
Particle & mass in GeV & Spin & Electric Charge & Colour charge \\ \hline \hline
electron $(e)$ & 5.109$\times10^{-4}$ & $\frac{1}{2}$ & -1 & 0 \\
\vspace*{0.1cm}
muon $(\mu)$ & 0.105 & $\frac{1}{2}$ & -1 & 0 \\
\vspace*{0.1cm}
tau $(\tau)$ & 1.776 & $\frac{1}{2}$ & -1 & 0 \\
\vspace*{0.1cm}
neutrinos $(\nu_{e,\mu,\tau})$ & 0 & $\frac{1}{2}$ & 0 & 0 \\
\vspace*{0.1cm}
up-quark $(u)$ & 2.49$\times10^{-3}$ & $\frac{1}{2}$ & $\frac{2}{3}$ & yes \\
\vspace*{0.1cm}
down-quark $(d)$ & 5.05$\times10^{-3}$ & $\frac{1}{2}$ & $-\frac{1}{3}$ & yes \\
\vspace*{0.1cm}
charm-quark $(c)$ & 1.27 & $\frac{1}{2}$ & $\frac{2}{3}$ & yes \\
\vspace*{0.1cm}
strange-quark $(s)$ & 0.101 & $\frac{1}{2}$ & $-\frac{1}{3}$ & yes \\
\vspace*{0.1cm}
top-quark $(t)$ & 172.0 & $\frac{1}{2}$ & $\frac{2}{3}$ & yes \\
\vspace*{0.1cm}
bottom-quark $(b)$ & 4.19 & $\frac{1}{2}$ & $-\frac{1}{3}$ & yes \\
\vspace*{0.1cm}
W-boson $(W^\pm)$ & 80.399 & 1 & $\pm 1$ & 0 \\
\vspace*{0.1cm}
Z-boson $(Z^0)$ & 91.187 & 1 & 0 & 0 \\
\vspace*{0.1cm}
photon $(\gamma)$ & 0 & 1 & 0 & 0 \\
\vspace*{0.1cm}
gluon $(g)$ & 0 & 1 & 0 & yes \\
\vspace*{0.1cm}
Higgs $(h^0)$ & {\bf{?}} & 0 & 0 & 0 \\ \hline \hline
\end{tabular}
\caption{\label{SM-spectrum}
The particle spectrum of the SM \cite{c1Nakamura-c2}.
Each of the charged particles are accompanied
by charge conjugate states of equal mass.
The charge neutral particles act as their own
antiparticles with all charge like quantum numbers
as opposite to that of the corresponding particles.
Evidence for Higgs boson is yet experimentally
missing and thus Higgs mass is denoted as `$?$'.
The neutrinos are presented with zero masses since
we are considering the SM only (see section \ref{suc-prob}).
}
\end{table}
\begin{flushleft}
{\it{$\maltese$ SM interactions}}
\end{flushleft}
Based on the discussion above, the complete Lagrangian for the SM
can be written as
\begin{eqnarray}
\mathcal{L}_{SM} = \mathcal{L}_{1} + \mathcal{L}_{2}
+ \mathcal{L}_{3} + \mathcal{L}_{4},
\label{SM-Lagrangian}
\end{eqnarray}
where
\begin{enumerate}
\item
$\mathcal{L}_{1}$ is the part of the Lagrangian which
contains kinetic energy terms and self-interaction terms
for the gauge bosons. After the EWSB these gauge bosons are
known as $W^\pm, Z^0$, gluons and photon. So we have
\begin{eqnarray}
\mathcal{L}_{1} = \sum_{\mu,\nu=1}^{4} \left[
-\frac{1}{4} \sum_{a=1}^{8} G^a_{\mu\nu} G^{\mu\nu}_a
- \frac{1}{4} \sum_{i=1}^{3} W^i_{\mu\nu} W^{\mu\nu}_i
- \frac{1}{4} B_{\mu\nu} B^{\mu\nu}\right],
\label{Lagrangian-gauge}
\end{eqnarray}
where
\begin{eqnarray}
&&G^a_{\mu\nu} = \partial_\mu G^a_\nu - \partial_\nu G^a_\mu
-g_3 f_{abc} G^b_\mu G^c_\nu,\nonumber\\
&&W^i_{\mu\nu} = \partial_\mu W^i_\nu - \partial_\nu W^i_\mu
-g_2 \epsilon_{ijk} W^j_\mu W^k_\nu,\nonumber\\
&&B_{\mu\nu} = \partial_\mu B_\nu - \partial_\nu B_\mu,
\label{non-Abelian-gauge}
\end{eqnarray}
with $f_{abc}$ and $\epsilon_{ijk}$ as the structure constants of the
respective non-Abelian groups. $g_3$ is the coupling constant for
$\rm {SU(3)_C}$ group.
\item
Kinetic energy terms for quarks and leptons belong to
$\mathcal{L}_2$. This part of the Lagrangian also contains
the interaction terms between the elementary fermions and gauge
bosons. Symbolically,
\begin{eqnarray}
\mathcal{L}_{2} = i \overline{\chi_L} \not\!\!{D} \chi_L +
i \overline{\chi_R} \not\!\!{D} \chi_R,
\label{Lagrangian-fermion}
\end{eqnarray}
where $\not\!\!{D} = \gamma^\mu D_\mu$ with $D_\mu$ as the covariant
derivative.\footnote{Replacement of ordinary derivative $(\partial_\mu)$
by $D_\mu$ is essential for a gauge transformation, so that $D_\mu \psi$
transforms covariantly under gauge transformation, similar to
the matter field, $\psi$.} The quantity $\chi_L$ stands for lepton and
quark $\rm{SU(2)_L}$ doublets whereas $\chi_R$ denotes $\rm{SU(2)_L}$
singlet fields (see eqn.(\ref{SM-gauge-group})). The
covariant derivative $D_\mu$ for different fermion
fields are written as (using eqn.(\ref{SM-gauge-group}))
\begin{eqnarray}
D_\mu Q_i &=& \left[\partial_\mu + i g_1 \frac{1}{6} B_\mu
+ i \sum_{i=1}^{3} g_2 \frac{1}{2} \sigma_i.W^i_\mu \right] Q_i,\nonumber\\
D_\mu u_i &=& \left[\partial_\mu + i g_1 \frac{2}{3} B_\mu \right] u_i,\nonumber\\
D_\mu d_i &=& \left[\partial_\mu - i g_1 \frac{1}{3} B_\mu \right] d_i,\nonumber\\
D_\mu L_i &=& \left[\partial_\mu - i g_1 \frac{1}{2} B_\mu
+ i \sum_{i=1}^{3} g_2 \frac{1}{2} \sigma_i.W^i_\mu \right] L_i,\nonumber\\
D_\mu e_i &=& \left[\partial_\mu - i g_1 B_\mu \right] e_i.\nonumber\\
\label{Dmu-fermions}
\end{eqnarray}
But these are the information for $\rm{SU(2)_L}\times\rm{U(1)_Y}$
only. What happens to the $\rm{SU(3)_C}$ part$?$ Obviously, for the
leptons there will be no problem since they are $\rm{SU(3)_C}$ singlet
after all (see eqn.(\ref{SM-gauge-group})). For the quarks the
$\rm{SU(3)_C}$ part can be taken care of in the following fashion,
\begin{eqnarray}
D_\mu
\left(\begin{array}{c}
{q_i}_R \\
{q_i}_G \\
{q_i}_B
\end{array}\right)
&=& \left[\partial_\mu
+ i \sum_{a=1}^{8} g_3 \frac{1}{2} \lambda_a.G^a_\mu \right]
\left(\begin{array}{c}
{q_i}_R \\
{q_i}_G \\
{q_i}_B
\end{array}\right),
\label{Dmu-QCD-quarks}
\end{eqnarray}
where $R,G$ and $B$ are the three types of colour charge and
$\lambda_a$'s are eight Gell-Mann matrices. $q_i$ is
triplet under $\rm{SU(3)_C}$, where `$i$' stands for different
types of left handed or right handed
(under ${\rm{SU(2)_L}}$) quark flavours, namely
$u,d,c,s,t$ and $b$.
\item
The terms representing physical Higgs mass and Higgs self-interactions
along with interaction terms between Higgs and the gauge bosons
are inhoused in $\mathcal{L}_3$
\begin{eqnarray}
\mathcal{L}_{3} = (D^\mu \Phi)^{\dagger} (D_\mu \Phi)
- V(\Phi).
\label{Lagrangian-Higgs}
\end{eqnarray}
The expressions for $\Phi$ and $V(\Phi)$ are given in
eqns.(\ref{SM-Higgs}) and (\ref{SM-Higgs-pot}), respectively.
For $\Phi$ the covariant derivative $D_\mu$ is given by
\begin{eqnarray}
D_\mu \Phi &=& \left[\partial_\mu + i g_1 \frac{1}{2} B_\mu
+ i \sum_{i=1}^{3} g_2 \frac{1}{2} \sigma_i.W^i_\mu \right] \Phi.
\label{Dmu-Higgs}
\end{eqnarray}
\item
The remaining Lagrangian $\mathcal{L}_4$ contains lepton
and quark mass terms and their interaction terms with Higgs
field $(h^0)$ (after EWSB). The expression for $\mathcal{L}_4$
is shown in eqn.(\ref{fermion-mass}). The elementary fermions get
their masses through respective Yukawa couplings, which are
{\it{free parameters}} of the theory. It turns out that in the SM the flavour
states are not necessarily the mass eigenstates, and it is
possible to relate them through an {\it{unitary transformation}}.
In case of the quarks this matrix is known as the
CKM (Cabibbo-Kobayashi-Maskawa) \cite{c1Cabibbo-1963,c1Kobayashi-1973}
matrix. This $3\times3$
unitary matrix contains three mixing angles and one phase.
The massless neutrinos in the SM make the corresponding
leptonic mixing matrix a trivial one (Identity matrix).
All possible interactions of the SM are shown in figure \ref{SM-int}.
The loops represent self-interactions like $h^0h^0h^0$, $h^0h^0h^0h^0$
(from the choice of potential, see eqn. (\ref{SM-Higgs-pot}))
$W^\pm W^\pm W^\mp W^\mp$, $ggg$ or $gggg$ (due to non-Abelian interactions)
and also interactions like $W^\pm W^\mp Z Z$, $W^\pm W^\mp \gamma \gamma$,
$W^\pm W^\mp Z$, $W^\pm W^\mp \gamma$ etc.
\begin{figure}[ht]
\centering
\includegraphics[width=8.95cm,keepaspectratio]{FigsPDF/SM-int.pdf}
\caption{Interactions of the Standard Model. See text for more details.}
\label{SM-int}
\end{figure}
\end{enumerate}
\section{{\bf A}pparent successes and the dark sides}\label{suc-prob}
The SM is an extremely successful theory to explain a host of
elementary particle interactions. Masses of the $W^\pm$ and $Z$
bosons as predicted by the SM theory are very close to their
experimentally measured values. The SM also predicted the existence
of the charm quark from the requirement to suppress flavour
changing neutral current (FCNC)\footnote{Glashow,
Iliopoulos and Maiani \cite{c1Glashow-1970}.} before it was actually
discovered in 1974. In a similar fashion the SM also predicted
the mass of the heavy top quark in the right region
before its discovery. Besides, all of the SM particles except Higgs boson
have been discovered already and their masses are also measured very precisely
\cite{c1Nakamura-c2}. Indeed, apart from Higgs sector, rest of the SM has
been already analysed for higher order processes and their spectacular
accuracy as revealed by a host of experiments has firmly established the
success of the SM.
Unfortunately, the so-called glorious success of the SM suffers serious
threat from various theoretical and experimental perspective. One of
the main stumbling blocks is definitely the Higgs boson, yet to
be observed in an experiment and its mass. Some of these shortcomings
are listed below.
\begin{enumerate}
\item
The SM has a large number of free parameters ({\bf{19}}).
The parameters are {\bf{9}} Yukawa couplings (or elementary
fermion masses) + {\bf{3}} angles and {\bf{one}} phase
of CKM matrix + {\bf {3}} gauge couplings $g_1,g_2,g_3$\footnote{An
alternate set could be $g_3$, $e$ (the unit of electric charge) and
the Weinberg angle $\theta_W$.} + {\bf {2}} parameters
$(\mu, \lambda)$ from scalar potential (see eqn.(\ref{SM-Higgs-pot}))
+ {\bf{one}} vacuum angle for quantum chromodynamics (QCD).
The number of free parameters is rather large for a fundamental theory.
\item
There are no theoretical explanation why there exist only
three generations of quarks and leptons. Also the huge mass
hierarchy between different generations (from first to third),
that is to say why mass of the top quark $(m_t) \gg$ mass of
the up-quark $(m_u)$ (see table \ref{SM-spectrum}), is unexplained.
\item
The single phase of CKM matrix accounts for
many Charge-Parity (CP) violating processes. However, one needs
additional source of CP-violation to account for
the large matter-anti matter asymmetry of the universe.
\item
The most familiar force in our everyday lives, gravity, is not a part of
the SM. Since the effect of gravity dominates near the ``{\it{Planck Scale $(M_P)$}}'',
$(\sim 10^{19}$ GeV) the SM still works fine despite its reluctant exclusion of the
gravitational interaction. In conclusion, the Standard Model cannot be a theory which is
valid for all energy scales.
\item
There is no room for a cold Dark Matter candidate inside
the SM framework, which has been firmly established by now from the
observed astrophysical and cosmological evidences.
\item
Neutrinos are exactly massless in the Standard Model as a
consequence of the particle content, gauge invariance, renormalizability
and Lorentz invariance. However, the
experimental results from atmospheric, solar and reactor neutrino experiments
suggest that the neutrinos do have non-zero masses with non-trivial mixing
among different neutrino flavours \cite{c1Schwetz:2008er,c1GonzalezGarcia:2010er}.
In order to generate masses and mixing for
the neutrinos, one must extend the SM framework by introducing additional
symmetries or particles or both.
But in reality the consequence of a massive neutrino is far serious
than asking for an extension of the SM. As written earlier, the
massive neutrinos trigger a non-trivial mixing in the charged lepton
sector just like the CKM matrix\footnote{Known as the PMNS matrix, will
be addressed in chapter \ref{neut} in more details.}, but with large
off-diagonal entries. It remains to explain why the structure of the
mixing matrix for the leptons are so different from the quarks?
\item
Perhaps the severe most of all the drawbacks is associated
with Higgs boson mass. In the Standard Model,
Higgs boson mass is totally unprotected by any
symmetry argument. In other words putting $m_{h^0}=0$, does not
enhance any symmetry of the theory.\footnote{Note
that putting zero for fermion or
gauge boson mass however enhances the symmetry of the Lagrangian.
In this case the chiral and gauge symmetry, respectively.}
Higgs mass can be as large as the ``Grand Unified Theory (GUT)"
scale ($10^{16} ~\rm{GeV}$) or the ``{\it{Planck Scale}}" ($10^{19} ~\rm{GeV}$)
when radiative corrections are included. This is the so called
{\it{gauge hierarchy problem}}. However, from several theoretical
arguments \cite{c1Cornwall-1973,c1Cornwall-1974,c1Smith-1973,c1Lee-1977,
c1Dashen:1983ts,c1Lindner:1985uk,c1Stevenson:1985zy,c1Hasenfratz:1987tk,
c1Luscher:1987ay,c1Hasenfratz:1987eh,c1Luscher:1987ek,c1Kuti:1987nr,c1Sher:1988mj,
c1Lindner:1988ww,c1Callaway:1988ya,c1Ford:1992mv,c1Sher:1993mf,
c1Choudhury:1997bn} and various experimental searches
\cite{c1Abbiendi:1998rd,c1Barate:2003sz,c1Nakamura-c2} Higgs boson mass is
expected to be in the range of a few hundreds of GeV,
which requires unnatural fine tuning of parameters
($\sim$ one part in $10^{38}$) for all orders in perturbation
theory. Different one-loop
diagrams contributing to the radiative correction to
Higgs boson mass are shown in figure \ref{Higgs-loop}.
\begin{figure}[ht]
\centering
\includegraphics[width=10.95cm]{FigsPDF/Higgs-loop.pdf}
\caption{One-loop radiative corrections to the Higgs boson mass from (a) and (b)
self-interactions (c) and (d) interactions with gauge bosons
and (e) interactions with fermions (f).}
\label{Higgs-loop}
\end{figure}
It is clear from figure \ref{Higgs-loop}, the contribution from the
fermion loop is proportional to the squared Yukawa couplings $(y^2_f)$.
As a corollary these contributions are negligible
except when heavy quarks are running in the loop. Contributions from the
diagrams (b) and (c) are {\it{logarithmically divergent}} which
is well under control due to the behaviour of {\it{log}} function. The contributions
from diagrams (a), (d) and (e) are {\it{quadratically divergent}},
which are the sources of the hierarchy problem.
\begin{list}{}{}
\item
\begin{flushleft}
{{$\blacklozenge$ Loop correction and divergences}}
\end{flushleft}
Consider the diagram (e) of figure \ref{Higgs-loop}, which
represents the fermionic loop contribution to the scalar
two point function. Assuming the loop momentum to be `$k$'
and the momentum for the external leg to be `$p$'
this contribution can be written as
\begin{eqnarray}
\Pi^f_{h^0h^0}(p^2=0) &=& (-1) \int \frac{d^4k}{(2\pi)^4}
(\frac{-i y_f}{\sqrt{2}})^2{\rm{Tr}}\left[ \frac{i}{\not\!\!{k}-m_f}
\frac{i}{\not\!\!{k}-m_f} \right],\nonumber\\
&=& (-\frac{y^2_f}{2}) \int \frac{d^4k}{(2\pi)^4}
{\rm{Tr}}\left[\frac{(\not\!\!{k}+m_f)(\not\!\!{k}+m_f)}{(k^2-m^2_f)^2}\right],\nonumber\\
&=& (-2 y^2_f) \int \frac{d^4k}{(2\pi)^4}
\left[\frac{(k^2 + m^2_f)}{(k^2-m^2_f)^2}\right],\nonumber\\
&=& {-2 y^2_f} \int \frac{d^4k}{(2\pi)^4}
\left[\frac{1}{(k^2-m^2_f)}
+ \frac{2 m^2_f}{(k^2-m^2_f)^2} \right],\nonumber\\
\label{Higgs-fermion-loop}
\end{eqnarray}
where the $(-1)$ factor appears for closed fermion loop and
`$i$' comes from the {\it{Feynman}} rules (see eqn.(\ref{fermion-mass})).
Fermion propagator is written as $i/(\not\!\!{k}-m_f)$. Here some of
the properties of Dirac Gamma matrices have been used.
Now in eqn.(\ref{Higgs-fermion-loop}) Higgs mass appears
nowhere which justifies the fact that setting $m_{h^0}=0$ does
not increase any symmetry of the Lagrangian. From naive
power counting argument the second term of
eqn.(\ref{Higgs-fermion-loop}) is logarithmically divergent
whereas the first term is quadratically divergent. Suppose the
theory of the SM is valid upto Planck scale and the cut off
scale $\Lambda$ (scale upto which a certain theory is valid)
lies there, then the correction to the
Higgs boson mass goes as (using eqn.(\ref{Higgs-fermion-loop})),
\begin{eqnarray}
\delta m^2_{h^0} \approx - \frac{y^2_f}{8 \pi^2} \Lambda^2 +
{\text{logarithmic terms}}.
\label{Higgs-fermion-approx}
\end{eqnarray}
The renormalized Higgs mass squared is then given by
\begin{eqnarray}
\tilde{m}^2_{h^0} = m^2_{h^0,{\text{bare}}} + \delta m^2_{h^0},
\label{Higgs-masssq-renorm}
\end{eqnarray}
and looking at eqn.(\ref{Higgs-fermion-approx}) the requirement of
fine tuning for a TeV scale Higgs mass is apparent.
Note that mass generation for all of the SM particles
solely depend on Higgs. So in a sense the entire mass
spectrum of the SM will be driven towards a high scale
with the radiative correction in Higgs boson mass.
\end{list}
\end{enumerate}
The list of drawbacks keep on increasing with
issues like unification of gauge couplings at a high scale and
a few more. To summarize, all of these unanswered questions
have opened up an entire new area of physics, popularly known as
``Beyond the Standard Model (BSM)'' physics. Some of the well-known candidates are
{\it{supersymmetry}}\footnote{First proposed in the context of hadronic physics,
by Hironari Miyazawa (1966).} \cite{c1Gervais:1971ji,c1Golfand-1971,c1Volkov-1973,
c1Wess:1974tw,c1Wess:1973kz}, {\it{theories with extra spatial dimensions}}
\cite{c1ArkaniHamed1-1998,c1ArkaniHamed2-1998,c1Randall-1999} and many others.
In this proposed thesis we plan to study some of the problems mentioned
earlier in the context of a supersymmetric theory and look for signatures
of such a theory at the ongoing Large Hadron Collider (LHC) experiment.
\chapter{ \sffamily{{\bf
}}\label{appenG}
\section{Feynman diagrams for the tree level $\ntrl1$ decay}\label{LN-decay}
Possible two-body and three-body final states (at the tree level) arising from the $R_p$-violating
decays of a lightest neutralino, $\ntrl1$ are shown here
\begin{figure}[ht]
\centering
\vspace*{0.5cm}
\includegraphics[height=2.00cm]{FigsPDF/LSP-decay2.pdf}
\caption{Feynman diagrams for the possible two body decays of the lightest neutralino.
$h^0$ is the lightest Higgs boson of the MSSM which is similar to the SM Higgs boson.}
\label{LSP-decay2}
\end{figure}
\begin{figure}[ht]
\centering
\vspace*{0.5cm}
\includegraphics[height=15.80cm,keepaspectratio]{FigsPDF/LSP-decay3.pdf}
\caption{Feynman diagrams for the possible three body decays of the lightest neutralino.
$S^0,P^0,S^\pm$ represent the scalar, the pseudoscalar and the charged scalar states, respectively.}
\label{LSP-decay3}
\end{figure}
\chapter*{\sffamily{{\bf A}cknowledgment}}
I am grateful to the Council of Scientific and Industrial Research,
Government of India for providing me the financial assistance for the completion
of my thesis work (Award No. 09/080(0539)/2007-EMR-I (Date 12.03.2007)). I am
also thankful to my home institute for a junior research fellowship that I
had enjoyed from August, 2006 to January, 2007.
\vspace{0.1cm}
I have no words to express my gratitude to the members of theoretical
high energy physics group of the Department of Theoretical Physics
of my home institute, particularly Dr. Utpal Chattopadhyay
and Dr. Sourov Roy. Their gratuitous infinite patience for my academic and personal problems,
unconditional affection to me, heart-rending analysis of my performance,
cordial pray for my success, spontaneous and constant motivations for
facing new challenges were beyond the conventional teacher-student relation.
I am also grateful to Dr. Dilip Kumar Ghosh, Dr. Pushan Majumdar,
Dr. Koushik Ray, Prof. Siddhartha Sen and Prof. Soumitra SenGupta of the same group for their
encouragement, spontaneous affection, crucial guidance and of course criticism,
in academics and life beyond it. It is also my pleasure to thank
Dr. Shudhanshu Sekhar Mandal and Dr. Krishnendu Sengupta of the
condensed matter group. It is an honour for me to express my respect
to Prof. Jayanta Kumar Bhattacharjee not only for his marvelous teaching,
but also for explaining me a different meaning of academics.
\vspace{0.1cm}
I sincerely acknowledge the hard efforts and sincere
commitments of my collaborators Dr. Priyotosh Bandopadhyay and Dr. Paramita Dey
to the research projects. I am thankful to them for their level of tolerance
to my infinite curiosity in spite of their extreme busy schedules. I
learned several new techniques and some rare insights
of the subjects from them. In the course of scientific collaboration I
have been privileged to work with Prof. Biswarup Mukhopadhyaya,
who never allowed me to realize the two decades of age difference between us.
Apart from his precious scientific guidance (I was also fortunate enough to attend
his teaching), his affection and inspiration for me has earned an eternal mark
in my memory just like his signature smile.
\vspace{0.1cm}
It is my duty to express my sincere gratitude to all of my teachers, starting
from the very basic level till date. It was their kind and hard efforts which help
me to reach here.
I am especially grateful to Ms. Anuradha SenSarma and Mr. Malay Ghosh
for their enthusiastic efforts and selfless sacrifices during my school days.
I have no words to express my respect to Prof. Anirban Kundu and Prof. Amitava
Raychaudhuri for their precious guidance and unconventional teaching during
my post-graduate studies. I am really fortunate enough to receive their
affection and guidance till date.
In this connection I express my modest gratitude to some
of the renowned experts
of the community for their valuable advise and encouragement.
They were always very generous to answer even some of my stupid questions,
in spite of their extremely busy professional schedules. I am particularly
grateful to Dr. Satyaki Bhattacharya, Prof. Debajyoti Choudhury, Dr. Anindya Datta,
Dr. Aseshkrishna Datta, Dr. Manas Maity, Prof. Bruce Mellado,
Dr. Sujoy Poddar, Dr. Subhendu Rakshit and Prof. Sreerup
Raychaudhuri for many useful suggestions and very helpful discussions.
\vspace{0.1cm}
I also express my humble thanks to my home institute,
Indian Association for the Cultivation of Science, for providing all the facilities
like high-performance personal desktop, constant and affluent access to high-speed
internet, a homely atmosphere and definitely a world class library. I am also
thankful to all the non-teaching staff members of my department
(Mr. Subrata Balti, Mr. Bikash Darji, Mr. Bhudeb Ghosh, Mr. Tapan Moulik and Mr. Suresh Mondal)
who were always there to assist us. It is my honour to thank the Director of my home institute,
Prof. Kankan Bhattacharyya for the encouragement I received from him.
\vspace{0.1cm}
It is a pleasure to express my thanks to my colleagues and friends who were
always there to cheer me up when things were not so smooth either in
academics or in personal life.
My cordial and special thanks to
Dr. Naba Kumar Bera, Dr. Debottam Das, Sudipto Paul Chowdhury, Dwipesh Majumder
and Joydip Mitra who were not just my colleagues but were, are and always
will be my brothers. I am really thankful to them and also to
Dr. Shyamal Biswas, Amit Chakraborty, Dr. Dipanjan Chakrabarti, Manimala Chakraborty,
Sabyasachi Chakraborty, Anirban Datta, Ashmita Das, Sanjib Ghosh,
Dr. R. S. Hundi, Dr. Ratna Koley, Dr. Debaprasad Maity,
Sourav Mondal, Subhadeep Mondal, Sanhita Modak, Shreyoshi Mondal,
Dr. Soumya Prasad Mukherjee, Sutirtha Mukherjee, Tapan Naskar, Dr. Himadri Sekhar Samanta,
Kush Saha, Ipsita Saha and Ankur Sensharma for making my office my second home.
\vspace{0.1cm}
It is definitely the worst injustice to acknowledge the support of my family as without
them I believe it is just like getting lost in crowd.
\vspace{0.1cm}
I cannot resist myself to show my humble tribute to three personalities, who by the
philosophy of their lives and works have influenced diverse aspects of my life.
The scientist who was born long before his time, Richard P. Feynman, the writer
who showed that field of expertise is not really a constraint, Narayan Sanyal
and my old friend, Mark.
\begin{flushright}
\vspace{1.5cm}
Pradipta Ghosh
\end{flushright}
\chapter{ \sffamily{{\bf
}}\label{appenE}
In this appendix we give the detail expressions for the renormalized self energy functions
${\widetilde \Sigma}^V_{ij}$ and ${\widetilde \Pi}^V_{ij}$. Different $(O^{ff^{\prime}b})$\footnote{$f$
is a neutralino, $f'$ is either a neutralino or a chargino or a quark and $b$ is either a scalar
(CP-even or CP-odd or charged or squark) or a vector boson $(W^\pm,Z)$.} couplings are given in
appendix \ref{appenD}.
\section{The ${\widetilde \Sigma}^V_{ij}$ function}\label{self-energy-sigma}
The regularized function ${\widetilde \Sigma}^V_{ij}$ is given as
\begin{eqnarray}\label{Sigma-part}
-\frac{1}{16 \pi^2}&&\left[ \frac{{\widetilde g}^2}{2}
\sum_{r=1}^{8} \sum_{k=1}^{10} \left(O^{nnh}_{Lkir} O^{nnh}_{Rjkr}
+ O^{nnh}_{Ljkr} O^{nnh}_{Rkir}\right) B_1(p^2,m^2_{{\widetilde \chi}^0_k},m^2_{S^0_r})
\right. \nonumber\\
- && \frac{{\widetilde g}^2}{2} \sum_{r=1}^{7} \sum_{k=1}^{10} \left(O^{nna}_{Lkir} O^{nna}_{Rjkr}
+ O^{nna}_{Ljkr} O^{nna}_{Rkir}\right) B_1(p^2,m^2_{{\widetilde \chi}^0_k},m^2_{P^0_r})\nonumber\\
+ && {g^2_2} \sum_{k=1}^{10} \left( O^{nnz}_{Lki} O^{nnz}_{Ljk}
+ O^{nnz}_{Rki} O^{nnz}_{Rjk} \right) {B_1(p^2,m^2_{{\widetilde \chi}^0_k},m^2_{Z^0_{\mu}})}\nonumber\\
+ && {2 g^2_2} \sum_{k=1}^{5} \left( O^{cnw}_{Lki} O^{ncw}_{Ljk}
+ O^{cnw}_{Rki} O^{ncw}_{Rjk} \right) {B_1(p^2,m^2_{{\widetilde\chi}^{\mp}_k},m^2_{W^{\pm}_{\mu}})}\nonumber\\
+ && {{\widetilde g}^2} \sum_{r=1}^{7} \sum_{k=1}^{5} \left(O^{cns}_{Lkir} O^{ncs}_{Rjkr}
+ O^{ncs}_{Ljkr} O^{cns}_{Rkir}\right) {B_1(p^2,m^2_{{\widetilde\chi}^{\mp}_k},m^2_{S^{\pm}_r})}\nonumber\\
+ && 3 {{\widetilde g}^2} \sum_{r=1}^{6} \sum_{k=1}^{3} \left(O^{un\widetilde u}_{Lkir} O^{nu\widetilde u}_{Rjkr}
+ O^{nu\widetilde u}_{Ljkr} O^{un\widetilde u}_{Rkir}\right) {B_1(p^2,m^2_{u_k},m^2_{\widetilde u_r})}\nonumber\\
+ && \left.
3{{\widetilde g}^2} \sum_{r=1}^{6} \sum_{k=1}^{3} \left(O^{dn\widetilde d}_{Lkir} O^{nd\widetilde d}_{Rjkr}
+ O^{nd\widetilde d}_{Ljkr} O^{dn\widetilde d}_{Rkir}\right) {B_1(p^2,m^2_{d_k},m^2_{\widetilde d_r})} \right].\nonumber\\
\end{eqnarray}
\section{The ${\widetilde \Pi}^V_{ij}$ function}\label{self-energy-pi}
In similar fashion ${\widetilde \Pi}^V_{ij}$ looks like
\begin{eqnarray}\label{Pi-part}
\nonumber \\
-\frac{1}{16 \pi^2}&&\left[ {{\widetilde g}^2} \sum_{r=1}^{8}
\sum_{k=1}^{10} \frac{m_{{\widetilde \chi}^0_k}}{2} \left(O^{nnh}_{Lkir} O^{nnh}_{Ljkr}
+ O^{nnh}_{Rkir} O^{nnh}_{Rjkr}\right) B_0(p^2,m^2_{{\widetilde \chi}^0_k},m^2_{S^0_r})
\right. \nonumber\\
- &&{{\widetilde g}^2 } \sum_{r=1}^{7} \sum_{k=1}^{10} \frac{m_{{\widetilde \chi}^0_k}}{2} \left(O^{nna}_{Lkir} O^{nna}_{Ljkr}
+ O^{nna}_{Rkir} O^{nna}_{Rjkr}\right) B_0(p^2,m^2_{{\widetilde \chi}^0_k},m^2_{P^0_r})\nonumber\\
- && {2 g^2_2} \sum_{k=1}^{10} {m_{{\widetilde \chi}^0_k}} \left(O^{nnz}_{Lki} O^{nnz}_{Rjk}
+ O^{nnz}_{Ljk} O^{nnz}_{Rki}\right) {B_0(p^2,m^2_{{\widetilde \chi}^0_k},m^2_{Z^0_{\mu}})}\nonumber\\
- && {4 g^2_2} \sum_{k=1}^{5} {m_{\widetilde{\chi}^{\pm}_k}} \left(O^{cnw}_{Lki} O^{ncw}_{Rjk}
+ O^{cnw}_{Rki} O^{ncw}_{Ljk}\right) {B_0(p^2,m^2_{{\widetilde\chi}^{\mp}_k},m^2_{W^{\pm}_{\mu}})}\nonumber\\
+ && {{\widetilde g}^2} \sum_{r=1}^{7} \sum_{k=1}^{5} {m_{\widetilde{\chi}^{\pm}_k}}
\left(O^{cns}_{Lkir} O^{ncs}_{Ljkr} + O^{ncs}_{Rjkr} O^{cns}_{Rkir}\right)
{B_0(p^2,m^2_{{\widetilde\chi}^{\mp}_k},m^2_{S^{\pm}_r})}\nonumber\\
+ && 3{{\widetilde g}^2} \sum_{r=1}^{6} \sum_{k=1}^{3} {m_{u_k}} \left(O^{un\widetilde u}_{Lkir} O^{nu\widetilde u}_{Ljkr}
+ O^{un\widetilde u}_{Rkir} O^{nu\widetilde u}_{Rjkr}\right) {B_0(p^2,m^2_{u_k},m^2_{\widetilde u_r})}\nonumber\\
+ && \left.
3{{\widetilde g}^2} \sum_{r=1}^{6} \sum_{k=1}^{3} {m_{d_k}} \left(O^{dn\widetilde d}_{Lkir} O^{nd\widetilde d}_{Ljkr}
+ O^{dn\widetilde d}_{Rkir} O^{nd\widetilde d}_{Rjkr}\right) {B_0(p^2,m^2_{d_k},m^2_{\widetilde d_r})} \right].\nonumber\\
\end{eqnarray}
Note that the quark - squark loops (second row, right most diagram of figure \ref{one-loop-diagrams})
receive an extra enhancement by a factor of $3$ from {\it{three}} different quark colours.
The Passarino-Veltman functions $(B_0,B_1)$ are given in appendix \ref{appenF}.
\chapter{ \sffamily{{\bf
}}\label{appenI}
\section{Three body decays of the $\ntrl1$ LSP}\label{LSP-3body-decay}
In this appendix we write down the spin-averaged (sum over spins
of the final state particles and average over the spin of initial particle) matrix
element square ($\overline{|\mathbb{M}|^2}$) for possible three body decays of a neutralino LSP
$\ntrl1$. The possible decays are given by eqn. (\ref{2-3-body-decays}). Since neutralinos
are fermion by nature, an average over the initial particle spin will yield a factor
of $\frac{1}{2}$, that is, mathematically,
\begin{equation}
\overline{|\mathbb{M}|^2}=\frac{N_c X_1 X_2}{(2.{\frac{1}{2}}+1)}
\left[\sum_i M^\dagger_i M_i + 2 \Re \left(\sum_{i \neq j} M^\dagger_i M_j\right)\right],
\label{mat-ele}
\end{equation}
where we put spin of the neutralino, $S_{\widetilde{\chi}^0_i}=\frac{1}{2}$ in the factor
$\frac{1}{(2.S_{\widetilde{\chi}^0_i}+1)}$. The second terms of eqn.(\ref{mat-ele}) represent
interference terms in case multiple Feynman diagrams exist for a given process.
$M_i$ represents amplitude of the $i$-th Feynman Diagram of a given process.
$N_c$ is the colour factor which is $3(1)$ for processes involving quarks(leptons).
The quantities $X_{1,2}$ are associated with two vertices of a three body decay
process (see figure \ref{LSP-qqnu} for example). These factors are $2$ for a $\ntrl1-\nu
-{\rm{neutral~boson}}$ vertex\footnote{Also true for $\nu-\nu -{\rm{neutral~boson}}$ vertex,
appears in $\ntrl1 \to \nu \nu \bar{\nu}$ process.} since $\ntrl1,\nu$ are Majorana particles
\cite{AppHaber:1984rc} and equal to $1$ for all other vertices. All processes are calculated
using 't-Hooft-Feynman gauge.
\section{Process $\ntrl1\to q \bar{q} \nu$}\label{LSP-qqnu-decay}
We start with the processes involving down type quarks $(q=d,s,b)$ first and later
for $q=u,c$. We represent different down and up type quarks generically by $d$ and
$u$, respectively. We write down all possible $M^\dagger_iM_j$ for the five diagrams
shown in figure \ref{LSP-qqnu}. The four-momentum assignments are as follows
\begin{equation}
\ntrl1(P) \to q(k)+\bar{q}(k')+\nu_i(p),
\label{momentum-qqnu}
\end{equation}
where $i$ stands for $i$-th neutrino flavour. $i=1,2,3$ or $e,\mu,\tau$.
$\ntrl1$ is the lightest of the seven heavy neutralino states (see
eqn.(\ref{neutralino_mass_eigenstate_matrixform})).
\begin{eqnarray}
\bullet~ M^\dagger_1M_1 (\ntrl1\to q \bar{q}\sum \nu_i) &=&
\frac{4 g^4_2}{cos^2_{\theta_W}\left[((k+k')^2-m^2_Z)^2+m^2_Z\Gamma^2_Z\right]} \nonumber\\
&&\sum_i
\left[(P.k)(p.k')A^{qq\nu_i}_{11}+(P.k')(p.k)B^{qq\nu_i}_{11}+m^2_q(P.p)C^{qq\nu_i}_{11}\right],\nonumber\\
\label{mat11-qqnu}
\end{eqnarray}
where $q=d(u)$, $\Gamma_Z$ is the $Z$-boson decay width and
\begin{eqnarray}
A^{qq\nu_i}_{11} &=& \left(O^{{1iZ}^*}_{Li1} O^{{1iZ}}_{Li1}
+ O^{{1iZ}^*}_{Ri1} O^{{1iZ}}_{Ri1}\right)\left(\frac{1(4)}{9}sin^4{\theta_W}
-\frac{1(2)}{6}sin^2{\theta_W}+\frac{1}{8}\right)\nonumber\\
&+& \left(O^{{1iZ}^*}_{Li1} O^{{1iZ}}_{Li1}
- O^{{1iZ}^*}_{Ri1} O^{{1iZ}}_{Ri1}\right)
\left(\frac{1(2)}{6}sin^2{\theta_W}-\frac{1}{8}\right),\nonumber\\
B^{qq\nu_i}_{11} &=& \left(O^{{1iZ}^*}_{Li1} O^{{1iZ}}_{Li1}
+ O^{{1iZ}^*}_{Ri1} O^{{1iZ}}_{Ri1}\right)\left(\frac{1(4)}{9}sin^4{\theta_W}
-\frac{1(2)}{6}sin^2{\theta_W}+\frac{1}{8}\right)\nonumber\\
&-& \left(O^{{1iZ}^*}_{Li1} O^{{1iZ}}_{Li1}
- O^{{1iZ}^*}_{Ri1} O^{{1iZ}}_{Ri1}\right)
\left(\frac{1}{6}sin^2{\theta_W}-\frac{1}{8}\right),\nonumber\\
C^{qq\nu_i}_{11} &=& \left(O^{{1iZ}^*}_{Li1} O^{{1iZ}}_{Li1}
+ O^{{1iZ}^*}_{Ri1} O^{{1iZ}}_{Ri1}\right)\left(\frac{1(4)}{9}sin^4{\theta_W}
-\frac{1(2)}{6}sin^2{\theta_W}\right).
\label{mat11-qqnu-det}
\end{eqnarray}
\begin{eqnarray}
\bullet~ M^\dagger_2M_2 (\ntrl1\to q \bar{q}\sum \nu_i) &=& \sum^2_{r,s=1}
\frac{4 \widetilde {g}^4}{\left[((k'+p)^2-m^2_{\widetilde {q}_r})
((k'+p)^2-m^2_{\widetilde {q}_s})\right]} \nonumber\\
&&\sum_i
\left[(P.k)(p.k')A^{qq\nu_i}_{22}+ m_q m_{\ntrl1} (p.k')B^{qq\nu_i}_{22}\right],
\label{mat22-qqnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{qq\nu_i}_{22} &=& \left(O^{q1\widetilde q}_{Lq1\widetilde q_s}O^{{q1\widetilde q}^*}_{Lq1\widetilde q_r}
+ O^{q1\widetilde q}_{Rq1\widetilde q_s}O^{{q1\widetilde q}^*}_{Rq1\widetilde q_r}\right)
\left(O^{iq\widetilde q}_{Liq\widetilde q_s}O^{{iq\widetilde q}^*}_{Liq\widetilde q_r}
+ O^{iq\widetilde q}_{Riq\widetilde q_s}O^{{iq\widetilde q}^*}_{Riq\widetilde q_r}\right),\nonumber\\
B^{qq\nu_i}_{22} &=& \left(O^{q1\widetilde q}_{Lq1\widetilde q_s}O^{{q1\widetilde q}^*}_{Rq1\widetilde q_r}
+ O^{q1\widetilde q}_{Rq1\widetilde q_s}O^{{q1\widetilde q}^*}_{Lq1\widetilde q_r}\right)
\left(O^{iq\widetilde q}_{Liq\widetilde q_s}O^{{iq\widetilde q}^*}_{Liq\widetilde q_r}
+ O^{iq\widetilde q}_{Riq\widetilde q_s}O^{{iq\widetilde q}^*}_{Riq\widetilde q_r}\right).
\label{mat22-qqnu-det}
\end{eqnarray}
\begin{eqnarray}
\bullet~ M^\dagger_3M_3 (\ntrl1\to q \bar{q}\sum \nu_i) &=& \sum^2_{r,s=1}
\frac{4 \widetilde {g}^4}{\left[((k+p)^2-m^2_{\widetilde {q}_r})
((k+p)^2-m^2_{\widetilde {q}_s})\right]} \nonumber\\
&&\sum_i
\left[(P.k')(p.k)A^{qq\nu_i}_{33}+ m_q m_{\ntrl1} (p.k)B^{qq\nu_i}_{33}\right],
\label{mat33-qqnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{qq\nu_i}_{33} &=& \left(O^{1q\widetilde q}_{L1q\widetilde q_s}O^{{1q\widetilde q}^*}_{L1q\widetilde q_r}
+ O^{1q\widetilde q}_{R1q\widetilde q_s}O^{{1q\widetilde q}^*}_{R1q\widetilde q_r}\right)
\left(O^{qi\widetilde q}_{Lqi\widetilde q_s}O^{{qi\widetilde q}^*}_{Lqi\widetilde q_r}
+ O^{qi\widetilde q}_{Rqi\widetilde q_s}O^{{qi\widetilde q}^*}_{Rqi\widetilde q_r}\right),\nonumber\\
B^{qq\nu_i}_{33} &=& \left(O^{1q\widetilde q}_{L1q\widetilde q_s}O^{{1q\widetilde q}^*}_{R1q\widetilde q_r}
+ O^{1q\widetilde q}_{R1q\widetilde q_s}O^{{1q\widetilde q}^*}_{L1q\widetilde q_r}\right)
\left(O^{qi\widetilde q}_{Lqi\widetilde q_s}O^{{qi\widetilde q}^*}_{Lqi\widetilde q_r}
+ O^{qi\widetilde q}_{Rqi\widetilde q_s}O^{{qi\widetilde q}^*}_{Rqi\widetilde q_r}\right).
\label{mat33-qqnu-det}
\end{eqnarray}
\begin{eqnarray}
\bullet~ M^\dagger_4M_4 (\ntrl1\to q \bar{q}\sum \nu_i) &=& \sum^8_{k,l=1}
\frac{2 \widetilde {g}^4}{\left[((k+k')^2-m^2_{S^0_k})
((k+k')^2-m^2_{S^0_l})\right]} \nonumber\\
&&\sum_i
\left[(P.p)(k.k')A^{qq\nu_i}_{44}- m^2_q (P.p)B^{qq\nu_i}_{44}\right],
\label{mat44-qqnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{qq\nu_i}_{44} &=& \left(O^{{qqh}^*}_{Lqqk} O^{qqh}_{Lqql}
+ O^{{qqh}^*}_{Rqqk} O^{qqh}_{Rqql} \right)
\left(O^{{i1h}^*}_{Li1k} O^{i1h}_{Li1l}
+ O^{{i1h}^*}_{Ri1k} O^{i1h}_{Ri1l} \right),\nonumber\\
B^{qq\nu_i}_{44} &=& \left(O^{{qqh}^*}_{Lqqk} O^{qqh}_{Rqql}
+ O^{{qqh}^*}_{Rqqk} O^{qqh}_{Lqql} \right)
\left(O^{{i1h}^*}_{Li1k} O^{i1h}_{Li1l}
+ O^{{i1h}^*}_{Ri1k} O^{i1h}_{Ri1l} \right).
\label{mat44-qqnu-det}
\end{eqnarray}
\begin{eqnarray}
\bullet~ M^\dagger_5M_5 (\ntrl1\to q \bar{q}\sum \nu_i) &=& \sum^8_{k,l=1}
\frac{2 \widetilde {g}^4}{\left[((k+k')^2-m^2_{P^0_k})
((k+k')^2-m^2_{P^0_l})\right]} \nonumber\\
&&\sum_i
\left[(P.p)(k.k')A^{qq\nu_i}_{55}- m^2_q (P.p)B^{qq\nu_i}_{55}\right],
\label{mat55-qqnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{qq\nu_i}_{55} &=& \left(O^{{qqa}^*}_{Lqqk} O^{qqa}_{Lqql}
+ O^{{qqa}^*}_{Rqqk} O^{qqa}_{Rqql} \right)
\left(O^{{i1a}^*}_{Li1k} O^{i1a}_{Li1l}
+ O^{{i1a}^*}_{Ri1k} O^{i1a}_{Ri1l} \right),\nonumber\\
B^{qq\nu_i}_{55} &=& \left(O^{{qqa}^*}_{Lqqk} O^{qqa}_{Rqql}
+ O^{{qqa}^*}_{Rqqk} O^{qqa}_{Lqql} \right)
\left(O^{{i1a}^*}_{Li1k} O^{i1a}_{Li1l}
+ O^{{i1a}^*}_{Ri1k} O^{i1a}_{Ri1l} \right).
\label{mat55-qqnu-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_1M_2 (\ntrl1\to q \bar{q}\sum \nu_i) = -(+)\sum^2_{r=1}
\frac{2 g^2_2 \widetilde {g}^2 sec\theta_W}{\left[((k+k')^2-m^2_Z-im_Z\Gamma_Z)
((p+k')^2-m^2_{\widetilde q_r})\right]} \nonumber\\
&&\sum_i
\left[2(p.k')(P.k)(A^{qq\nu_i}_{12}C^{qq\nu_i}_{12}+B^{qq\nu_i}_{12}D^{qq\nu_i}_{12})
+ m_q m_{\ntrl1}(p.k)(A^{qq\nu_i}_{12}D^{qq\nu_i}_{12}+B^{qq\nu_i}_{12}C^{qq\nu_i}_{12})
\right.\nonumber\\
&&+ \left.
2m_q m_{\ntrl1}(p.k')(B^{qq\nu_i}_{12}E^{qq\nu_i}_{12}+A^{qq\nu_i}_{12}F^{qq\nu_i}_{12})
+m^2_q(P.p)(A^{qq\nu_i}_{12}E^{qq\nu_i}_{12}+B^{qq\nu_i}_{12}F^{qq\nu_i}_{12})
\right],\nonumber\\
\label{mat12-qqnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{qq\nu_i}_{12} &=& O^{{1iZ}^*}_{Li1} O^{iq\widetilde q}_{Riq\widetilde q_r},
~~B^{qq\nu_i}_{12} = O^{{1iZ}^*}_{Ri1} O^{iq\widetilde q}_{Liq\widetilde q_r},
~~C^{qq\nu_i}_{12} = \frac{1(2)}{3}sin^2\theta_W O^{q1\widetilde q}_{Lq1\widetilde q_r},\nonumber\\
D^{qq\nu_i}_{12} &=& \frac{1(2)}{3}sin^2\theta_W O^{q1\widetilde q}_{Rq1\widetilde q_r}
-\frac{1}{2} O^{q1\widetilde q}_{Rq1\widetilde q_r},
~~E^{qq\nu_i}_{12} = \frac{1(2)}{3}sin^2\theta_W O^{q1\widetilde q}_{Lq1\widetilde q_r}
-\frac{1}{2} O^{q1\widetilde q}_{Lq1\widetilde q_r},\nonumber\\
F^{qq\nu_i}_{12} &=& \frac{1(2)}{3}sin^2\theta_W O^{q1\widetilde q}_{Rq1\widetilde q_r}.
\label{mat12-qqnu-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_1M_3 (\ntrl1\to q \bar{q}\sum \nu_i) = (-)\sum^2_{r=1}
\frac{2 g^2_2 \widetilde {g}^2 sec\theta_W}{\left[((k+k')^2-m^2_Z-im_Z\Gamma_Z)
((p+k)^2-m^2_{\widetilde q_r})\right]} \nonumber\\
&&\sum_i
\left[2(p.k)(P.k')(A^{qq\nu_i}_{13}C^{qq\nu_i}_{13}+B^{qq\nu_i}_{13}D^{qq\nu_i}_{13})
+ m_q m_{\ntrl1}(p.k')(A^{qq\nu_i}_{13}D^{qq\nu_i}_{13}+B^{qq\nu_i}_{13}C^{qq\nu_i}_{13})
\right.\nonumber\\
&&+ \left.
2m_q m_{\ntrl1}(p.k)(B^{qq\nu_i}_{13}E^{qq\nu_i}_{13}+A^{qq\nu_i}_{13}F^{qq\nu_i}_{13})
+m^2_q(P.p)(A^{qq\nu_i}_{13}E^{qq\nu_i}_{13}+B^{qq\nu_i}_{13}F^{qq\nu_i}_{13})
\right],\nonumber\\
\label{mat13-qqnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{qq\nu_i}_{13} &=& O^{{1iZ}^*}_{Ri1} O^{qi\widetilde q}_{Lqi\widetilde q_r},
~~B^{qq\nu_i}_{13} = O^{{1iZ}^*}_{Li1} O^{qi\widetilde q}_{Rqi\widetilde q_r},
~~C^{qq\nu_i}_{13} = \frac{1(2)}{3}sin^2\theta_W O^{1q\widetilde q}_{R1q\widetilde q_r},\nonumber\\
D^{qq\nu_i}_{13} &=& \frac{1(2)}{3}sin^2\theta_W O^{1q\widetilde q}_{L1q\widetilde q_r}
-\frac{1}{2} O^{1q\widetilde q}_{L1q\widetilde q_r},
~~E^{qq\nu_i}_{13} = \frac{1(2)}{3}sin^2\theta_W O^{1q\widetilde q}_{R1q\widetilde q_r}
-\frac{1}{2} O^{1q\widetilde q}_{R1q\widetilde q_r},\nonumber\\
F^{qq\nu_i}_{13} &=& \frac{1(2)}{3}sin^2\theta_W O^{1q\widetilde q}_{L1q\widetilde q_r}.
\label{mat13-qqnu-det}
\end{eqnarray}
\begin{eqnarray}
\bullet~ M^\dagger_1M_4 (\ntrl1\to q \bar{q}\sum \nu_i) &=& -(+)\sum^8_{k=1}
\frac{\sqrt{2} g^2_2 \widetilde {g}^2 sec\theta_W}{\left[((k+k')^2-m^2_Z-im_Z\Gamma_Z)
((k+k')^2-m^2_{S^0_k})\right]} \nonumber\\
&&\sum_i
\left[m_q m_{\ntrl1} (p.k') A^{qq\nu_i}_{14} - m_q m_{\ntrl1} (p.k) B^{qq\nu_i}_{14}
\right],
\label{mat14-qqnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{qq\nu_i}_{14} &=& \left(O^{{1iZ}^*}_{Ri1} O^{i1h}_{Li1k}
+ O^{{1iZ}^*}_{Li1} O^{i1h}_{Ri1k}\right)\nonumber\\
&\times&\left\{\left(\frac{1(2)}{3}sin^2\theta_W-\frac{1}{2}\right)O^{qqh}_{Lbbk}
+ \frac{1(2)}{3}sin^2\theta_W O^{qqh}_{Rbbk}\right\},\nonumber\\
B^{qq\nu_i}_{14} &=& \left(O^{{1iZ}^*}_{Ri1} O^{i1h}_{Li1k}
+ O^{{1iZ}^*}_{Li1} O^{i1h}_{Ri1k}\right)\nonumber\\
&\times&\left\{\left(\frac{1(2)}{3}sin^2\theta_W-\frac{1}{2}\right)O^{qqh}_{Rbbk}
+ \frac{1(2)}{3}sin^2\theta_W O^{qqh}_{Lbbk}\right\}.
\label{mat14-qqnu-det}
\end{eqnarray}
\begin{figure}[ht]
\centering
\vspace*{0.5cm}
\includegraphics[height=6.00cm]{FigsPDF/LSP-qqnu.pdf}
\caption{Feynman diagrams for the possible three body decays of the lightest supersymmetric
particle into $q\bar{q}\nu$ final states, with $q\neq t$. $\widetilde{q}_{1,2}$ are the squarks
in the mass eigenbasis (see eqn.(\ref{squark-mass-basis})).
$S^0_m,P^0_m$ are the neutral scalar and pseudoscalar states of the $\mu\nu$SSM as
shown by eqns.(\ref{scalar-mass-basis}), (\ref{pseudoscalar-mass-basis}).}
\label{LSP-qqnu}
\end{figure}
\begin{eqnarray}
\bullet~ M^\dagger_1M_5 (\ntrl1\to q \bar{q}\sum \nu_i) &=& (-)\sum^8_{k=1}
\frac{\sqrt{2} g^2_2 \widetilde {g}^2 sec\theta_W}{\left[((k+k')^2-m^2_Z-im_Z\Gamma_Z)
((k+k')^2-m^2_{P^0_k})\right]} \nonumber\\
&&\sum_i
\left[m_q m_{\ntrl1} (p.k') A^{qq\nu_i}_{15} - m_q m_{\ntrl1} (p.k) B^{qq\nu_i}_{15}
\right],
\label{mat15-qqnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{qq\nu_i}_{15} &=& \left(O^{{1iZ}^*}_{Ri1} O^{i1a}_{Li1k}
+ O^{{1iZ}^*}_{Li1} O^{i1a}_{Ri1k}\right)\nonumber\\
&\times&\left\{\left(\frac{1(2)}{3}sin^2\theta_W-\frac{1}{2}\right)O^{qqa}_{Lbbk}
+ \frac{1(2)}{3}sin^2\theta_W O^{qqa}_{Rbbk}\right\},\nonumber\\
B^{qq\nu_i}_{15} &=& \left(O^{{1iZ}^*}_{Ri1} O^{i1a}_{Li1k}
+ O^{{1iZ}^*}_{Li1} O^{i1a}_{Ri1k}\right)\nonumber\\
&\times&\left\{\left(\frac{1(2)}{3}sin^2\theta_W-\frac{1}{2}\right)O^{qqa}_{Rbbk}
+ \frac{1(2)}{3}sin^2\theta_W O^{qqa}_{Lbbk}\right\}.
\label{mat15-qqnu-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_2M_3 (\ntrl1\to q \bar{q}\sum \nu_i) = -\sum^2_{r,s=1}
\frac{2 \widetilde {g}^4}{\left[((p+k')^2-m^2_{\widetilde q_r})
((p+k)^2-m^2_{\widetilde q_s})\right]} \nonumber\\
&&\sum_i
\left[\{(P.k)(p.k')-(P.p)(k.k')+(P.k')(p.k)\} (A^{qq\nu_i}_{23} O^{{q1\widetilde q}^*}_{Lq1\widetilde q_r}
+B^{qq\nu_i}_{23} O^{{q1\widetilde q}^*}_{Rq1\widetilde q_r})
\right.\nonumber\\
&+& m_q m_{\ntrl1} (p.k') (A^{qq\nu_i}_{23} O^{{q1\widetilde q}^*}_{Rq1\widetilde q_r}
+B^{qq\nu_i}_{23} O^{{q1\widetilde q}^*}_{Lq1\widetilde q_r})
+ m_q m_{\ntrl1} (p.k) (C^{qq\nu_i}_{23} O^{{q1\widetilde q}^*}_{Rq1\widetilde q_r}
+D^{qq\nu_i}_{23} O^{{q1\widetilde q}^*}_{Lq1\widetilde q_r})\nonumber\\
&+& \left.
m^2_q(P.p) (C^{qq\nu_i}_{23} O^{{q1\widetilde q}^*}_{Lq1\widetilde q_r}
+D^{qq\nu_i}_{23} O^{{q1\widetilde q}^*}_{Rq1\widetilde q_r})
\right],
\label{mat23-qqnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{qq\nu_i}_{23} &=& O^{qi\widetilde q}_{Lqi\widetilde q_s} O^{{iq\widetilde q}^*}_{Liq\widetilde q_r} O^{1q\widetilde q}_{L1q\widetilde q_s},
~~B^{qq\nu_i}_{23} = O^{qi\widetilde q}_{Rqi\widetilde q_s} O^{{iq\widetilde q}^*}_{Riq\widetilde q_r} O^{1q\widetilde q}_{R1q\widetilde q_s},\nonumber\\
C^{qq\nu_i}_{23} &=& O^{qi\widetilde q}_{Rqi\widetilde q_s} O^{{iq\widetilde q}^*}_{Riq\widetilde q_r} O^{1q\widetilde q}_{L1q\widetilde q_s},
~~D^{qq\nu_i}_{23} = O^{qi\widetilde q}_{Lqi\widetilde q_s} O^{{iq\widetilde q}^*}_{Liq\widetilde q_r} O^{1q\widetilde q}_{R1q\widetilde q_s}.
\label{mat23-qqnu-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_2M_4 (\ntrl1\to q \bar{q}\sum \nu_i) = \sum^2_{r=1}\sum^8_{k=1}
\frac{\sqrt{2} \widetilde {g}^4}{\left[((p+k')^2-m^2_{\widetilde q_r})
((k+k')^2-m^2_{S^0_k})\right]} \nonumber\\
&&\sum_i
\left[\{(P.p)(k.k')-(P.k')(p.k)+(p.k')(P.k)\} (A^{qq\nu_i}_{24} O^{{q1\widetilde q}^*}_{Rq1\widetilde q_r}
+B^{qq\nu_i}_{24} O^{{q1\widetilde q}^*}_{Lq1\widetilde q_r})
\right.\nonumber\\
&-& i \epsilon_{\mu\nu\rho\sigma} p^\mu P^\nu k^\rho {k'}^\sigma
(A^{qq\nu_i}_{24} O^{{q1\widetilde q}^*}_{Rq1\widetilde q_r}
-B^{qq\nu_i}_{24} O^{{q1\widetilde q}^*}_{Lq1\widetilde q_r})\nonumber\\
&+& m_q m_{\ntrl1} (p.k') (A^{qq\nu_i}_{24} O^{{q1\widetilde q}^*}_{Lq1\widetilde q_r}
+B^{qq\nu_i}_{24} O^{{q1\widetilde q}^*}_{Rq1\widetilde q_r})
- m_q m_{\ntrl1} (p.k) (C^{qq\nu_i}_{24} O^{{q1\widetilde q}^*}_{Lq1\widetilde q_r}
+D^{qq\nu_i}_{24} O^{{q1\widetilde q}^*}_{Rq1\widetilde q_r})\nonumber\\
&-& \left.
m^2_q(P.p) (C^{qq\nu_i}_{24} O^{{q1\widetilde q}^*}_{Rq1\widetilde q_r}
+D^{qq\nu_i}_{24} O^{{q1\widetilde q}^*}_{Lq1\widetilde q_r})
\right],
\label{mat24-qqnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{qq\nu_i}_{24} &=& O^{qqh}_{Rqqk} O^{{iq\widetilde q}^*}_{Riq\widetilde q_r} O^{i1h}_{Ri1k},
~~B^{qq\nu_i}_{24} = O^{qqh}_{Lqqk} O^{{iq\widetilde q}^*}_{Liq\widetilde q_r} O^{i1h}_{Li1k},\nonumber\\
C^{qq\nu_i}_{24} &=& O^{qqh}_{Lqqk} O^{{iq\widetilde q}^*}_{Riq\widetilde q_r} O^{i1h}_{Ri1k},
~~D^{qq\nu_i}_{24} = O^{qqh}_{Rqqk} O^{{iq\widetilde q}^*}_{Liq\widetilde q_r} O^{i1h}_{Li1k}.
\label{mat24-qqnu-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_2M_5 (\ntrl1\to q \bar{q}\sum \nu_i) = -\sum^2_{r=1}\sum^8_{k=1}
\frac{\sqrt{2} \widetilde {g}^4}{\left[((p+k')^2-m^2_{\widetilde q_r})
((k+k')^2-m^2_{P^0_k})\right]} \nonumber\\
&&\sum_i
\left[\{(P.p)(k.k')-(P.k')(p.k)+(p.k')(P.k)\} (A^{qq\nu_i}_{25} O^{{q1\widetilde q}^*}_{Rq1\widetilde q_r}
+B^{qq\nu_i}_{25} O^{{q1\widetilde q}^*}_{Lq1\widetilde q_r})
\right.\nonumber\\
&-& i \epsilon_{\mu\nu\rho\sigma} p^\mu P^\nu k^\rho {k'}^\sigma
(A^{qq\nu_i}_{25} O^{{q1\widetilde q}^*}_{Rq1\widetilde q_r}
-B^{qq\nu_i}_{25} O^{{q1\widetilde q}^*}_{Lq1\widetilde q_r})\nonumber\\
&+& m_q m_{\ntrl1} (p.k') (A^{qq\nu_i}_{25} O^{{q1\widetilde q}^*}_{Lq1\widetilde q_r}
+B^{qq\nu_i}_{25} O^{{q1\widetilde q}^*}_{Rq1\widetilde q_r})
- m_q m_{\ntrl1} (p.k) (C^{qq\nu_i}_{25} O^{{q1\widetilde q}^*}_{Lq1\widetilde q_r}
+D^{qq\nu_i}_{25} O^{{q1\widetilde q}^*}_{Rq1\widetilde q_r})\nonumber\\
&-& \left.
m^2_q(P.p) (C^{qq\nu_i}_{25} O^{{q1\widetilde q}^*}_{Rq1\widetilde q_r}
+D^{qq\nu_i}_{25} O^{{q1\widetilde q}^*}_{Lq1\widetilde q_r})
\right],
\label{mat25-qqnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{qq\nu_i}_{25} &=& O^{qqa}_{Rqqk} O^{{iq\widetilde q}^*}_{Riq\widetilde q_r} O^{i1a}_{Ri1k},
~~B^{qq\nu_i}_{25} = O^{qqa}_{Lqqk} O^{{iq\widetilde q}^*}_{Liq\widetilde q_r} O^{i1a}_{Li1k},\nonumber\\
C^{qq\nu_i}_{25} &=& O^{qqa}_{Lqqk} O^{{iq\widetilde q}^*}_{Riq\widetilde q_r} O^{i1a}_{Ri1k},
~~D^{qq\nu_i}_{25} = O^{qqa}_{Rqqk} O^{{iq\widetilde q}^*}_{Liq\widetilde q_r} O^{i1a}_{Li1k}.
\label{mat25-qqnu-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_3M_4 (\ntrl1\to q \bar{q}\sum \nu_i) = \sum^2_{r=1}\sum^8_{k=1}
\frac{\sqrt{2} \widetilde {g}^4}{\left[((p+k)^2-m^2_{\widetilde q_r})
((k+k')^2-m^2_{S^0_k})\right]} \nonumber\\
&&\sum_i
\left[\{(P.k')(p.k)-(P.k)(p.k')+(P.p)(k.k')\} (A^{qq\nu_i}_{34} O^{{1q\widetilde q}^*}_{L1q\widetilde q_r}
+B^{qq\nu_i}_{34} O^{{1q\widetilde q}^*}_{R1q\widetilde q_r})
\right.\nonumber\\
&+& m_q m_{\ntrl1} (p.k) (A^{qq\nu_i}_{34} O^{{1q\widetilde q}^*}_{R1q\widetilde q_r}
+B^{qq\nu_i}_{34} O^{{1q\widetilde q}^*}_{L1q\widetilde q_r})
- m_q m_{\ntrl1} (p.k') (C^{qq\nu_i}_{34} O^{{1q\widetilde q}^*}_{R1q\widetilde q_r}
+D^{qq\nu_i}_{34} O^{{1q\widetilde q}^*}_{L1q\widetilde q_r})\nonumber\\
&-& \left.
m^2_q(P.p) (C^{qq\nu_i}_{34} O^{{1q\widetilde q}^*}_{L1q\widetilde q_r}
+D^{qq\nu_i}_{34} O^{{1q\widetilde q}^*}_{R1q\widetilde q_r})
\right],
\label{mat34-qqnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{qq\nu_i}_{34} &=& O^{qqh}_{Lqqk} O^{{qi\widetilde q}^*}_{Lqi\widetilde q_r} O^{i1h}_{Li1k},
~~B^{qq\nu_i}_{34} = O^{qqh}_{Rqqk} O^{{qi\widetilde q}^*}_{Rqi\widetilde q_r} O^{i1h}_{Ri1k},\nonumber\\
C^{qq\nu_i}_{34} &=& O^{qqh}_{Rqqk} O^{{qi\widetilde q}^*}_{Lqi\widetilde q_r} O^{i1h}_{Li1k},
~~D^{qq\nu_i}_{34} = O^{qqh}_{Lqqk} O^{{qi\widetilde q}^*}_{Rqi\widetilde q_r} O^{i1h}_{Ri1k}.
\label{mat34-qqnu-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_3M_5 (\ntrl1\to q \bar{q}\sum \nu_i) = -\sum^2_{r=1}\sum^8_{k=1}
\frac{\sqrt{2} \widetilde {g}^4}{\left[((p+k)^2-m^2_{\widetilde q_r})
((k+k')^2-m^2_{P^0_k})\right]} \nonumber\\
&&\sum_i
\left[\{(P.k')(p.k)-(P.k)(p.k')+(P.p)(k.k')\} (A^{qq\nu_i}_{35} O^{{1q\widetilde q}^*}_{L1q\widetilde q_r}
+B^{qq\nu_i}_{35} O^{{1q\widetilde q}^*}_{R1q\widetilde q_r})
\right.\nonumber\\
&+& m_q m_{\ntrl1} (p.k) (A^{qq\nu_i}_{35} O^{{1q\widetilde q}^*}_{R1q\widetilde q_r}
+B^{qq\nu_i}_{35} O^{{1q\widetilde q}^*}_{L1q\widetilde q_r})
- m_q m_{\ntrl1} (p.k') (C^{qq\nu_i}_{35} O^{{1q\widetilde q}^*}_{R1q\widetilde q_r}
+D^{qq\nu_i}_{35} O^{{1q\widetilde q}^*}_{L1q\widetilde q_r})\nonumber\\
&-& \left.
m^2_q(P.p) (C^{qq\nu_i}_{35} O^{{1q\widetilde q}^*}_{L1q\widetilde q_r}
+D^{qq\nu_i}_{35} O^{{1q\widetilde q}^*}_{R1q\widetilde q_r})
\right],
\label{mat35-qqnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{qq\nu_i}_{35} &=& O^{qqa}_{Lqqk} O^{{qi\widetilde q}^*}_{Lqi\widetilde q_r} O^{i1a}_{Li1k},
~~B^{qq\nu_i}_{35} = O^{qqa}_{Rqqk} O^{{qi\widetilde q}^*}_{Rqi\widetilde q_r} O^{i1a}_{Ri1k},\nonumber\\
C^{qq\nu_i}_{35} &=& O^{qqa}_{Rqqk} O^{{qi\widetilde q}^*}_{Lqi\widetilde q_r} O^{i1a}_{Li1k},
~~D^{qq\nu_i}_{35} = O^{qqa}_{Lqqk} O^{{qi\widetilde q}^*}_{Rqi\widetilde q_r} O^{i1a}_{Ri1k}.
\label{mat35-qqnu-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_4M_5 (\ntrl1\to q \bar{q}\sum \nu_i) = -\sum^8_{k,l=1}
\frac{2 \widetilde {g}^4}{\left[((k+k')^2-m^2_{S^0_k})
((k+k')^2-m^2_{P^0_l})\right]} \nonumber\\
&&\sum_i
\left[(P.p)(k.k')\left(O^{{i1h}^*}_{Li1k} O^{i1a}_{Li1l} + O^{{i1h}^*}_{Ri1k} O^{i1a}_{Ri1l} \right)
\left(O^{{qqh}^*}_{Lqqk} O^{qqa}_{Lqql} + O^{{qqh}^*}_{Rqqk} O^{qqa}_{Rqql} \right)
\right.\nonumber\\
&&-\left.
m^2_q(P.p)\left(O^{{i1h}^*}_{Li1k} O^{i1a}_{Li1l} + O^{{i1h}^*}_{Ri1k} O^{i1a}_{Ri1l} \right)
\left(O^{{qqh}^*}_{Rqqk} O^{qqa}_{Lqql} + O^{{qqh}^*}_{Lqqk} O^{qqa}_{Rqql} \right)
\right].
\label{mat45-qqnu}
\end{eqnarray}
Values for Weinberg angle $\theta_W$ and $\Gamma_Z$ are given in ref. \cite{AppNakamura-c2}.
Quark masses are also taken from ref. \cite{AppNakamura-c2}. All the relevant couplings
are given in appendices \ref{appenD} and \ref{appenH}.
\section{Process $\ntrl1\to \ell^+_i \ell^-_j \nu_k$}\label{LSP-llnu-decay}
We represent different leptons $(e,\mu,\tau)$ generically by $\ell$.
We write down all possible $M^\dagger_iM_j$ for the seven diagrams
shown in figure \ref{LSP-llnu}. We treat the charge conjugate leptons
as charginos (see eqn.(\ref{neutralino-chargino}))
as shown in eqn.(\ref{chargino_mass-basis_weak-basis_relations}).
The four-momentum assignments are as follows
\begin{equation}
\ntrl1(P) \to \ell^+_i(k)+\ell^-_j(k')+\nu_k(p),
\label{momentum-llnu}
\end{equation}
where $i,j,k$ stand for various lepton flavours. $i,j,k=1,2,3$ or $e,\mu,\tau$.
\begin{figure}[ht]
\centering
\vspace*{0.5cm}
\includegraphics[height=9.00cm]{FigsPDF/LSP-llnu.pdf}
\caption{Feynman diagrams for the possible three body decays of the lightest supersymmetric
particle into $\ell^+_i+\ell^-_j+\nu_k$ final states.
$S^\pm_r,S^0_r,P^0_r$ are the charged scalar, neutral scalar and pseudoscalar states of
the $\mu\nu$SSM as shown by eqns.(\ref{charged-scalar-mass-basis}), (\ref{scalar-mass-basis}),
(\ref{pseudoscalar-mass-basis}).}
\label{LSP-llnu}
\end{figure}
\begin{eqnarray}
&&\bullet~ M^\dagger_1M_1 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) =
\frac{8{g}^4_2}{\left[((p+k')^2-m^2_{W})^2+ m^2_W \Gamma^2_W\right]} \nonumber\\
&&\sum_{i,j,k}
\left[2(P.k')(p.k) A^{\ell^+_i \ell^-_j \nu_k}_{11}
+ 2(P.p)(k.k') B^{\ell^+_i \ell^-_j \nu_k}_{11}
-m_{\ell_i}m_{\ntrl1}(p.k') C^{\ell^+_i \ell^-_j \nu_k}_{11}
\right],\nonumber\\
\label{mat11-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{11} &=& \left(O^{{cnw}*}_{Li1} O^{cnw}_{Li1}
O^{{ncw}*}_{Lkj} O^{ncw}_{Lkj} + O^{{cnw}*}_{Ri1} O^{cnw}_{Ri1}
O^{{ncw}*}_{Rkj} O^{ncw}_{Rkj}\right),\nonumber\\
B^{\ell^+_i \ell^-_j \nu_k}_{11} &=& \left(O^{{cnw}*}_{Li1} O^{cnw}_{Li1}
O^{{ncw}*}_{Rkj} O^{ncw}_{Rkj} + O^{{cnw}*}_{Ri1} O^{cnw}_{Ri1}
O^{{ncw}*}_{Lkj} O^{ncw}_{Lkj}\right),\nonumber\\
C^{\ell^+_i \ell^-_j \nu_k}_{11} &=& \left(O^{{cnw}*}_{Ri1} O^{cnw}_{Li1}
+ O^{{cnw}*}_{Li1} O^{cnw}_{Ri1}\right)
\left(O^{{ncw}*}_{Lkj} O^{ncw}_{Lkj}+ O^{{ncw}*}_{Rkj} O^{ncw}_{Rkj}\right).
\label{mat11-llnu-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_2M_2 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) =
\frac{8{g}^4_2}{\left[((p+k)^2-m^2_{W})^2+ m^2_W \Gamma^2_W\right]} \nonumber\\
&&\sum_{i,j,k}
\left[2(P.k)(p.k') A^{\ell^+_i \ell^-_j \nu_k}_{22}
+ 2(P.p)(k.k') B^{\ell^+_i \ell^-_j \nu_k}_{22}
-m_{\ell_j}m_{\ntrl1}(p.k) C^{\ell^+_i \ell^-_j \nu_k}_{22}
\right],\nonumber\\
\label{mat22-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{22} &=& \left(O^{{ncw}*}_{L1j} O^{ncw}_{L1j}
O^{{cnw}*}_{Lik} O^{cnw}_{Lik} + O^{{ncw}*}_{R1j} O^{ncw}_{R1j}
O^{{cnw}*}_{Rik} O^{cnw}_{Rik}\right),\nonumber\\
B^{\ell^+_i \ell^-_j \nu_k}_{22} &=& \left(O^{{ncw}*}_{L1j} O^{ncw}_{L1j}
O^{{cnw}*}_{Rik} O^{cnw}_{Rik} + O^{{ncw}*}_{R1j} O^{ncw}_{R1j}
O^{{cnw}*}_{Lik} O^{cnw}_{Lik}\right),\nonumber\\
C^{\ell^+_i \ell^-_j \nu_k}_{22} &=& \left(O^{{ncw}*}_{R1j} O^{ncw}_{L1j}
+ O^{{ncw}*}_{L1j} O^{ncw}_{R1j}\right)
\left(O^{{cnw}*}_{Lik} O^{cnw}_{Lik}+ O^{{cnw}*}_{Rik} O^{cnw}_{Rik}\right).
\label{mat22-llnu-det}
\end{eqnarray}
\begin{eqnarray}
\bullet~ M^\dagger_3M_3 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) &=& \sum^8_{r,l=1}
\frac{4\widetilde{g}^4}{\left[((p+k')^2-m^2_{S^\pm_r})((p+k')^2-m^2_{S^\pm_l})\right]} \nonumber\\
&&\sum_{i,j,k}
\left[(P.k)(p.k') A^{\ell^+_i \ell^-_j \nu_k}_{33}
+m_{\ell_i}m_{\ntrl1}(p.k') B^{\ell^+_i \ell^-_j \nu_k}_{33}
\right],\nonumber\\
\label{mat33-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{33} &=& \left(O^{{cns}*}_{Li1r} O^{cns}_{Li1l}
+ O^{{cns}*}_{Ri1r} O^{cns}_{Ri1l}\right)
\left(O^{{ncs}*}_{Lkjr} O^{ncs}_{Lkjl}+ O^{{ncs}*}_{Rkjr} O^{ncs}_{Rkjl}\right),\nonumber\\
B^{\ell^+_i \ell^-_j \nu_k}_{33} &=& \left(O^{{cns}*}_{Ri1r} O^{cns}_{Li1l}
+ O^{{cns}*}_{Li1r} O^{cns}_{Ri1l}\right)
\left(O^{{ncs}*}_{Lkjr} O^{ncs}_{Lkjl}+ O^{{ncs}*}_{Rkjr} O^{ncs}_{Rkjl}\right).
\label{mat33-llnu-det}
\end{eqnarray}
\begin{eqnarray}
\bullet~ M^\dagger_4M_4 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) &=& \sum^8_{r,l=1}
\frac{4\widetilde{g}^4}{\left[((p+k)^2-m^2_{S^\pm_r})((p+k)^2-m^2_{S^\pm_l})\right]} \nonumber\\
&&\sum_{i,j,k}
\left[(P.k')(p.k) A^{\ell^+_i \ell^-_j \nu_k}_{44}
+m_{\ell_j}m_{\ntrl1}(p.k) B^{\ell^+_i \ell^-_j \nu_k}_{44}
\right],\nonumber\\
\label{mat44-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{44} &=& \left(O^{{ncs}*}_{L1jr} O^{ncs}_{L1jl}
+ O^{{ncs}*}_{R1jr} O^{ncs}_{R1jl}\right)
\left(O^{{cns}*}_{Likr} O^{cns}_{Likl}+ O^{{cns}*}_{Rikr} O^{cns}_{Rikl}\right),\nonumber\\
B^{\ell^+_i \ell^-_j \nu_k}_{44} &=& \left(O^{{ncs}*}_{R1jr} O^{ncs}_{L1jl}
+ O^{{ncs}*}_{L1jr} O^{ncs}_{R1jl}\right)
\left(O^{{cns}*}_{Likr} O^{cns}_{Likl}+ O^{{cns}*}_{Rikr} O^{cns}_{Rikl}\right).
\label{mat44-llnu-det}
\end{eqnarray}
\begin{eqnarray}
\bullet~ M^\dagger_5M_5 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) &=& \sum^8_{r,l=1}
\frac{\widetilde{g}^4}{\left[((k+k')^2-m^2_{S^0_r})((k+k')^2-m^2_{S^0_l})\right]} \nonumber\\
&&\sum_{i,j,k}
\left[(P.p)(k.k') A^{\ell^+_i \ell^-_j \nu_k}_{55}
-m_{\ell_i}m_{\ell_j}(P.p) B^{\ell^+_i \ell^-_j \nu_k}_{55}
\right],\nonumber\\
\label{mat55-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{55} &=& \left(O^{{nnh}*}_{Lk1r} O^{nnh}_{Lk1l}
+ O^{{nnh}*}_{Rk1r} O^{nnh}_{Rk1l}\right)
\left(O^{{cch}*}_{Lijr} O^{cch}_{Lijl}+ O^{{cch}*}_{Rijr} O^{cch}_{Rijl}\right),\nonumber\\
B^{\ell^+_i \ell^-_j \nu_k}_{55} &=& \left(O^{{nnh}*}_{Lk1r} O^{nnh}_{Lk1l}
+ O^{{nnh}*}_{Rk1r} O^{nnh}_{Rk1l}\right)
\left(O^{{cch}*}_{Rijr} O^{cch}_{Lijl}+ O^{{cch}*}_{Lijr} O^{cch}_{Rijl}\right).
\label{mat55-llnu-det}
\end{eqnarray}
\begin{eqnarray}
\bullet~ M^\dagger_6M_6 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) &=& \sum^8_{r,l=1}
\frac{\widetilde{g}^4}{\left[((k+k')^2-m^2_{P^0_r})((k+k')^2-m^2_{P^0_l})\right]} \nonumber\\
&&\sum_{i,j,k}
\left[(P.p)(k.k') A^{\ell^+_i \ell^-_j \nu_k}_{66}
-m_{\ell_i}m_{\ell_j}(P.p) B^{\ell^+_i \ell^-_j \nu_k}_{66}
\right],\nonumber\\
\label{mat66-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{66} &=& \left(O^{{nna}*}_{Lk1r} O^{nna}_{Lk1l}
+ O^{{nna}*}_{Rk1r} O^{nna}_{Rk1l}\right)
\left(O^{{cca}*}_{Lijr} O^{cca}_{Lijl}+ O^{{cca}*}_{Rijr} O^{cca}_{Rijl}\right),\nonumber\\
B^{\ell^+_i \ell^-_j \nu_k}_{66} &=& \left(O^{{nna}*}_{Lk1r} O^{nna}_{Lk1l}
+ O^{{nna}*}_{Rk1r} O^{nna}_{Rk1l}\right)
\left(O^{{cca}*}_{Rijr} O^{cca}_{Lijl}+ O^{{cca}*}_{Lijr} O^{cca}_{Rijl}\right).
\label{mat66-llnu-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_7M_7 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) =
\frac{2{g}^4_2}{cos^2\theta_W\left[((k+k')^2-m^2_{Z})^2 + m^2_Z \Gamma^2_Z\right]} \nonumber\\
&&\sum_{i,j,k}
\left[2(P.k')(p.k) A^{\ell^+_i \ell^-_j \nu_k}_{77}
+ 2(P.k)(p.k')B^{\ell^+_i \ell^-_j \nu_k}_{77}
-m_{\ell_i}m_{\ell_j}(P.p) C^{\ell^+_i \ell^-_j \nu_k}_{77}
\right],\nonumber\\
\label{mat77-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{77} &=& \left(O^{{nnz}^*}_{Lk1} O^{nnz}_{Lk1}
O^{{ccz}^*}_{Lij} O^{ccz}_{Lij} + O^{{nnz}^*}_{Rk1} O^{nnz}_{Rk1}
O^{{ccz}^*}_{Rij} O^{ccz}_{Rij}\right),\nonumber\\
B^{\ell^+_i \ell^-_j \nu_k}_{77} &=& \left(O^{{nnz}^*}_{Lk1} O^{nnz}_{Lk1}
O^{{ccz}^*}_{Rij} O^{ccz}_{Rij} + O^{{nnz}^*}_{Rk1} O^{nnz}_{Rk1}
O^{{ccz}^*}_{Lij} O^{ccz}_{Lij}\right),\nonumber\\
C^{\ell^+_i \ell^-_j \nu_k}_{77} &=& \left(O^{{nnz}^*}_{Lk1} O^{nnz}_{Lk1}
+ O^{{nnz}^*}_{Rk1} O^{nnz}_{Rk1}\right)\left(
O^{{ccz}^*}_{Rij} O^{ccz}_{Lij} + O^{{ccz}^*}_{Lij} O^{ccz}_{Rij}\right).
\label{mat77-llnu-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_1M_2 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) = \nonumber\\
&&\frac{8{g}^4_2}{\left[((p+k')^2-m^2_{W}) -i m_W \Gamma_W\right]
\left[((p+k)^2-m^2_{W})^2 +i m_W \Gamma_W\right]}
\nonumber\\
&&\sum_{i,j,k}
\left[-2(P.p)(k.k') (A^{\ell^+_i \ell^-_j \nu_k}_{12} O^{{cnw}^*}_{Li1}
+ B^{\ell^+_i \ell^-_j \nu_k}_{12} O^{{cnw}^*}_{Ri1})
\right.\nonumber\\
&&+ m_{\ell_i} m_{\ntrl1} (p.k') (A^{\ell^+_i \ell^-_j \nu_k}_{12} O^{{cnw}^*}_{Ri1}
+ B^{\ell^+_i \ell^-_j \nu_k}_{12} O^{{cnw}^*}_{Li1}) \nonumber\\
&&+m_{\ell_j} m_{\ntrl1} (p.k) (C^{\ell^+_i \ell^-_j \nu_k}_{12} O^{{cnw}^*}_{Ri1}
+ D^{\ell^+_i \ell^-_j \nu_k}_{12} O^{{cnw}^*}_{Li1})\nonumber\\
&&+\left.
m_{\ell_i} m_{\ell_j} (P.p) (C^{\ell^+_i \ell^-_j \nu_k}_{12} O^{{cnw}^*}_{Li1}
+ D^{\ell^+_i \ell^-_j \nu_k}_{12} O^{{cnw}^*}_{Ri1})
\right],\nonumber\\
\label{mat12-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{12} &=& O^{{ncw}^*}_{Rkj} O^{ncw}_{L1j} O^{cnw}_{Lik},
~~B^{\ell^+_i \ell^-_j \nu_k}_{12} = O^{{ncw}^*}_{Lkj} O^{ncw}_{R1j} O^{cnw}_{Rik},\nonumber\\
C^{\ell^+_i \ell^-_j \nu_k}_{12} &=& O^{{ncw}^*}_{Lkj} O^{ncw}_{L1j} O^{cnw}_{Rik},
~~D^{\ell^+_i \ell^-_j \nu_k}_{12} = O^{{ncw}^*}_{Rkj} O^{ncw}_{R1j} O^{cnw}_{Lik},\nonumber\\
\label{mat12-llnu-det}
\end{eqnarray}
\begin{eqnarray}
\bullet~ M^\dagger_1M_3 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) &=& \sum^8_{l=1}
\frac{4g^2_2\widetilde{g}^2}{\left[(((p+k')^2-m^2_{W}) -i m_W \Gamma_W)
((p+k')^2-m^2_{S^\pm_l})\right]}
\nonumber\\
&&\sum_{i,j,k}
\left[ m_{\ell_j}m_{\ntrl1} (p.k) A^{\ell^+_i \ell^-_j \nu_k}_{13}
+ m_{\ell_i} m_{\ell_j} (P.p) B^{\ell^+_i \ell^-_j \nu_k}_{13} \right],\nonumber\\
\label{mat13-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{13} &=& \left(O^{{cnw}^*}_{Ri1} O^{cns}_{Li1l}
+ O^{{cnw}^*}_{Li1} O^{cns}_{Ri1l}\right)
\left(O^{{ncw}^*}_{Rkj} O^{ncs}_{Lkjl}
+ O^{{ncw}^*}_{Lkj} O^{ncs}_{Rkjl}\right),\nonumber\\
B^{\ell^+_i \ell^-_j \nu_k}_{13} &=& \left(O^{{cnw}^*}_{Li1} O^{cns}_{Li1l}
+ O^{{cnw}^*}_{Ri1} O^{cns}_{Ri1l}\right)
\left(O^{{ncw}^*}_{Rkj} O^{ncs}_{Lkjl}
+ O^{{ncw}^*}_{Lkj} O^{ncs}_{Rkjl}\right).
\label{mat13-llnu-det}
\end{eqnarray}
\begin{eqnarray}
\bullet~ M^\dagger_1M_4 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) &=& - \sum^8_{r=1}
\frac{4g^2_2\widetilde{g}^2}{\left[(((p+k')^2-m^2_{W}) -i m_W \Gamma_W)
((p+k)^2-m^2_{S^\pm_r})\right]}
\nonumber\\
&&\sum_{i,j,k}
\left[2(P.k')(p.k) (A^{\ell^+_i \ell^-_j \nu_k}_{14} O^{{cnw}^*}_{Li1}
+ B^{\ell^+_i \ell^-_j \nu_k}_{14} O^{{cnw}^*}_{Ri1})
\right.\nonumber\\
&&-m_{\ell_i} m_{\ntrl1} (p.k') (A^{\ell^+_i \ell^-_j \nu_k}_{14} O^{{cnw}^*}_{Ri1}
+ B^{\ell^+_i \ell^-_j \nu_k}_{14} O^{{cnw}^*}_{Li1})\nonumber\\
&&+2 m_{\ell_j} m_{\ntrl1} (p.k) (C^{\ell^+_i \ell^-_j \nu_k}_{14} O^{{cnw}^*}_{Ri1}
+ D^{\ell^+_i \ell^-_j \nu_k}_{14} O^{{cnw}^*}_{Li1})\nonumber\\
&&-\left.
m_{\ell_i}m_{\ell_j} (P.p) (C^{\ell^+_i \ell^-_j \nu_k}_{14} O^{{cnw}^*}_{Li1}
+ D^{\ell^+_i \ell^-_j \nu_k}_{14} O^{{cnw}^*}_{Ri1})\right],
\label{mat14-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{14} &=& O^{{ncw}^*}_{Lkj} O^{ncs}_{L1jl} O^{cns}_{Rikl},
~~B^{\ell^+_i \ell^-_j \nu_k}_{14} = O^{{ncw}^*}_{Rkj} O^{ncs}_{R1jl} O^{cns}_{Likl},\nonumber\\
C^{\ell^+_i \ell^-_j \nu_k}_{14} &=& O^{{ncw}^*}_{Rkj} O^{ncs}_{L1jl} O^{cns}_{Likl},
~~D^{\ell^+_i \ell^-_j \nu_k}_{14} = O^{{ncw}^*}_{Lkj} O^{ncs}_{R1jl} O^{cns}_{Rikl}.
\label{mat14-llnu-det}
\end{eqnarray}
\begin{eqnarray}
\bullet~ M^\dagger_1M_5 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) &=& - \sum^8_{r=1}
\frac{2g^2_2\widetilde{g}^2}{\left[(((p+k')^2-m^2_{W}) -i m_W \Gamma_W)
((k+k')^2-m^2_{S^0_r})\right]}
\nonumber\\
&&\sum_{i,j,k}
\left[2(P.p)(k.k') (A^{\ell^+_i \ell^-_j \nu_k}_{15} O^{{cnw}^*}_{Li1}
+ B^{\ell^+_i \ell^-_j \nu_k}_{15} O^{{cnw}^*}_{Ri1})
\right.\nonumber\\
&&-m_{\ell_i} m_{\ntrl1} (p.k') (A^{\ell^+_i \ell^-_j \nu_k}_{15} O^{{cnw}^*}_{Ri1}
+ B^{\ell^+_i \ell^-_j \nu_k}_{15} O^{{cnw}^*}_{Li1})\nonumber\\
&&+m_{\ell_j} m_{\ntrl1} (p.k) (C^{\ell^+_i \ell^-_j \nu_k}_{15} O^{{cnw}^*}_{Ri1}
+ D^{\ell^+_i \ell^-_j \nu_k}_{15} O^{{cnw}^*}_{Li1})\nonumber\\
&&-\left.
2m_{\ell_i}m_{\ell_j} (P.p) (C^{\ell^+_i \ell^-_j \nu_k}_{15} O^{{cnw}^*}_{Li1}
+ D^{\ell^+_i \ell^-_j \nu_k}_{15} O^{{cnw}^*}_{Ri1})\right],
\label{mat15-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{15} &=& O^{{ncw}^*}_{Rkj} O^{nnh}_{Lk1l} O^{cch}_{Rijl},
~~B^{\ell^+_i \ell^-_j \nu_k}_{15} = O^{{ncw}^*}_{Lkj} O^{nnh}_{Rk1l} O^{cch}_{Lijl},\nonumber\\
C^{\ell^+_i \ell^-_j \nu_k}_{15} &=& O^{{ncw}^*}_{Rkj} O^{nnh}_{Lk1l} O^{cch}_{Lijl},
~~D^{\ell^+_i \ell^-_j \nu_k}_{15} = O^{{ncw}^*}_{Lkj} O^{nnh}_{Rk1l} O^{cch}_{Rijl}.
\label{mat15-llnu-det}
\end{eqnarray}
\begin{eqnarray}
\bullet~ M^\dagger_1M_6 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) &=& \sum^8_{r=1}
\frac{2g^2_2\widetilde{g}^2}{\left[(((p+k')^2-m^2_{W}) -i m_W \Gamma_W)
((k+k')^2-m^2_{P^0_r})\right]}
\nonumber\\
&&\sum_{i,j,k}
\left[2(P.p)(k.k') (A^{\ell^+_i \ell^-_j \nu_k}_{16} O^{{cnw}^*}_{Li1}
+ B^{\ell^+_i \ell^-_j \nu_k}_{16} O^{{cnw}^*}_{Ri1})
\right.\nonumber\\
&&-m_{\ell_i} m_{\ntrl1} (p.k') (A^{\ell^+_i \ell^-_j \nu_k}_{16} O^{{cnw}^*}_{Ri1}
+ B^{\ell^+_i \ell^-_j \nu_k}_{16} O^{{cnw}^*}_{Li1})\nonumber\\
&&+m_{\ell_j} m_{\ntrl1} (p.k) (C^{\ell^+_i \ell^-_j \nu_k}_{16} O^{{cnw}^*}_{Ri1}
+ D^{\ell^+_i \ell^-_j \nu_k}_{16} O^{{cnw}^*}_{Li1})\nonumber\\
&&-\left.
2m_{\ell_i}m_{\ell_j} (P.p) (C^{\ell^+_i \ell^-_j \nu_k}_{16} O^{{cnw}^*}_{Li1}
+ D^{\ell^+_i \ell^-_j \nu_k}_{16} O^{{cnw}^*}_{Ri1})\right],\nonumber\\
\label{mat16-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{16} &=& O^{{ncw}^*}_{Rkj} O^{nna}_{Lk1l} O^{cca}_{Rijl},
~~B^{\ell^+_i \ell^-_j \nu_k}_{16} = O^{{ncw}^*}_{Lkj} O^{nna}_{Rk1l} O^{cca}_{Lijl},\nonumber\\
C^{\ell^+_i \ell^-_j \nu_k}_{16} &=& O^{{ncw}^*}_{Rkj} O^{nna}_{Lk1l} O^{cca}_{Lijl},
~~D^{\ell^+_i \ell^-_j \nu_k}_{16} = O^{{ncw}^*}_{Lkj} O^{nna}_{Rk1l} O^{cca}_{Rijl}.
\label{mat16-llnu-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_1M_7 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) =\nonumber\\
&&-\frac{4g^4_2see\theta_W}{\left[(((p+k')^2-m^2_{W}) -i m_W \Gamma_W)\right]
\left[(((k+k')^2-m^2_{Z}) + i m_Z \Gamma_Z)\right]}
\nonumber\\
&&\sum_{i,j,k}
\left[2(P.k')(p.k) (A^{\ell^+_i \ell^-_j \nu_k}_{17} O^{{cnw}^*}_{Li1}
+ B^{\ell^+_i \ell^-_j \nu_k}_{17} O^{{cnw}^*}_{Ri1})
\right.\nonumber\\
&&-m_{\ell_i} m_{\ntrl1} (p.k') (A^{\ell^+_i \ell^-_j \nu_k}_{17} O^{{cnw}^*}_{Ri1}
+ B^{\ell^+_i \ell^-_j \nu_k}_{17} O^{{cnw}^*}_{Li1})\nonumber\\
&&+m_{\ell_j} m_{\ntrl1} (p.k) (C^{\ell^+_i \ell^-_j \nu_k}_{17} O^{{cnw}^*}_{Ri1}
+ D^{\ell^+_i \ell^-_j \nu_k}_{17} O^{{cnw}^*}_{Li1})\nonumber\\
&&+\left.
m_{\ell_i}m_{\ell_j} (P.p) (C^{\ell^+_i \ell^-_j \nu_k}_{17} O^{{cnw}^*}_{Li1}
+ D^{\ell^+_i \ell^-_j \nu_k}_{17} O^{{cnw}^*}_{Ri1})\right],\nonumber\\
\label{mat17-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{17} &=& O^{{ncw}^*}_{Lkj} O^{nnz}_{Lk1} O^{ccz}_{Lij},
~~B^{\ell^+_i \ell^-_j \nu_k}_{17} = O^{{ncw}^*}_{Rkj} O^{nnz}_{Rk1} O^{ccz}_{Rij},\nonumber\\
C^{\ell^+_i \ell^-_j \nu_k}_{17} &=& O^{{ncw}^*}_{Lkj} O^{nnz}_{Lk1} O^{ccz}_{Rij},
~~D^{\ell^+_i \ell^-_j \nu_k}_{17} = O^{{ncw}^*}_{Rkj} O^{nnz}_{Rk1} O^{ccz}_{Lij}.
\label{mat17-llnu-det}
\end{eqnarray}
\begin{eqnarray}
\bullet~ M^\dagger_2M_3 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) &=& \sum^8_{r=1}
\frac{4g^2_2 \widetilde{g}^2}{\left[(((p+k)^2-m^2_{W}) -i m_W \Gamma_W)
((p+k')^2-m^2_{S^\pm_r})\right]}
\nonumber\\
&&\sum_{i,j,k}
\left[2(P.k)(p.k') (A^{\ell^+_i \ell^-_j \nu_k}_{23} O^{{ncw}^*}_{L1j}
+ B^{\ell^+_i \ell^-_j \nu_k}_{23} O^{{ncw}^*}_{R1j})
\right.\nonumber\\
&&-m_{\ell_j} m_{\ntrl1} (p.k) (A^{\ell^+_i \ell^-_j \nu_k}_{23} O^{{ncw}^*}_{R1j}
+ B^{\ell^+_i \ell^-_j \nu_k}_{23} O^{{ncw}^*}_{L1j})\nonumber\\
&&+2m_{\ell_i} m_{\ntrl1} (p.k') (C^{\ell^+_i \ell^-_j \nu_k}_{23} O^{{ncw}^*}_{R1j}
+ D^{\ell^+_i \ell^-_j \nu_k}_{23} O^{{ncw}^*}_{L1j})\nonumber\\
&&-\left.
m_{\ell_i}m_{\ell_j} (P.p) (C^{\ell^+_i \ell^-_j \nu_k}_{23} O^{{ncw}^*}_{L1j}
+ D^{\ell^+_i \ell^-_j \nu_k}_{23} O^{{ncw}^*}_{R1j})\right],\nonumber\\
\label{mat23-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{23} &=& O^{{cnw}^*}_{Rik} O^{cns}_{Li1r} O^{ncs}_{Rkjr},
~~B^{\ell^+_i \ell^-_j \nu_k}_{23} = O^{{cnw}^*}_{Lik} O^{cns}_{Ri1r} O^{ncs}_{Lkjr},\nonumber\\
C^{\ell^+_i \ell^-_j \nu_k}_{23} &=& O^{{cnw}^*}_{Lik} O^{cns}_{Li1r} O^{ncs}_{Lkjr},
~~D^{\ell^+_i \ell^-_j \nu_k}_{23} = O^{{cnw}^*}_{Rik} O^{cns}_{Ri1r} O^{ncs}_{Rkjr}.
\label{mat23-llnu-det}
\end{eqnarray}
\begin{eqnarray}
\bullet~ M^\dagger_2M_4 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) &=& -\sum^8_{r=1}
\frac{4g^2_2 \widetilde{g}^2}{\left[(((p+k)^2-m^2_{W}) -i m_W \Gamma_W)
((p+k)^2-m^2_{S^\pm_r})\right]}
\nonumber\\
&&\sum_{i,j,k}
\left[m_{\ell_i} m_{\ntrl1} (p.k') A^{\ell^+_i \ell^-_j \nu_k}_{24}
+ m_{\ell_i} m_{\ell_j} (P.p) B^{\ell^+_i \ell^-_j \nu_k}_{24} \right],\nonumber\\
\label{mat24-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{24} &=& \left(O^{{ncw}^*}_{R1j} O^{ncs}_{L1jr}
+ O^{{ncw}^*}_{L1j} O^{ncs}_{R1jr}\right)
\left(O^{{cnw}^*}_{Lik} O^{cns}_{Likr}
+ O^{{cnw}^*}_{Rik} O^{cns}_{Rikr}\right),\nonumber\\
B^{\ell^+_i \ell^-_j \nu_k}_{24} &=& \left(O^{{ncw}^*}_{L1j} O^{ncs}_{L1jr}
+ O^{{ncw}^*}_{R1j} O^{ncs}_{R1jr}\right)
\left(O^{{cnw}^*}_{Lik} O^{cns}_{Likr}
+ O^{{cnw}^*}_{Rik} O^{cns}_{Rikr}\right).
\label{mat24-llnu-det}
\end{eqnarray}
\begin{eqnarray}
\bullet~ M^\dagger_2M_5 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) &=& \sum^8_{r=1}
\frac{2g^2_2 \widetilde{g}^2}{\left[(((p+k)^2-m^2_{W}) -i m_W \Gamma_W)
((k+k')^2-m^2_{S^0_r})\right]}
\nonumber\\
&&\sum_{i,j,k}
\left[2(P.p)(k.k') (A^{\ell^+_i \ell^-_j \nu_k}_{25} O^{{ncw}^*}_{L1j}
+ B^{\ell^+_i \ell^-_j \nu_k}_{25} O^{{ncw}^*}_{R1j})
\right.\nonumber\\
&&-m_{\ell_j} m_{\ntrl1} (p.k) (A^{\ell^+_i \ell^-_j \nu_k}_{25} O^{{ncw}^*}_{R1j}
+ B^{\ell^+_i \ell^-_j \nu_k}_{25} O^{{ncw}^*}_{L1j})\nonumber\\
&&+m_{\ell_i} m_{\ntrl1} (p.k') (C^{\ell^+_i \ell^-_j \nu_k}_{25} O^{{ncw}^*}_{R1j}
+ D^{\ell^+_i \ell^-_j \nu_k}_{25} O^{{ncw}^*}_{L1j})\nonumber\\
&&-\left.
2m_{\ell_i}m_{\ell_j} (P.p) (C^{\ell^+_i \ell^-_j \nu_k}_{25} O^{{ncw}^*}_{L1j}
+ D^{\ell^+_i \ell^-_j \nu_k}_{25} O^{{ncw}^*}_{R1j})\right],\nonumber\\
\label{mat25-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{25} &=& O^{{cnw}^*}_{Lik} O^{nnh}_{Lk1r} O^{cch}_{Rijr},
~~B^{\ell^+_i \ell^-_j \nu_k}_{25} = O^{{cnw}^*}_{Rik} O^{nnh}_{Rk1r} O^{cch}_{Lijr},\nonumber\\
C^{\ell^+_i \ell^-_j \nu_k}_{25} &=& O^{{cnw}^*}_{Lik} O^{nnh}_{Lk1r} O^{cch}_{Lijr},
~~D^{\ell^+_i \ell^-_j \nu_k}_{25} = O^{{cnw}^*}_{Rik} O^{nnh}_{Rk1r} O^{cch}_{Rijr}.
\label{mat25-llnu-det}
\end{eqnarray}
\begin{eqnarray}
\bullet~ M^\dagger_2M_6 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) &=& -\sum^8_{r=1}
\frac{2g^2_2 \widetilde{g}^2}{\left[(((p+k)^2-m^2_{W}) -i m_W \Gamma_W)
((k+k')^2-m^2_{P^0_r})\right]}
\nonumber\\
&&\sum_{i,j,k}
\left[2(P.p)(k.k') (A^{\ell^+_i \ell^-_j \nu_k}_{26} O^{{ncw}^*}_{L1j}
+ B^{\ell^+_i \ell^-_j \nu_k}_{26} O^{{ncw}^*}_{R1j})
\right.\nonumber\\
&&-m_{\ell_j} m_{\ntrl1} (p.k) (A^{\ell^+_i \ell^-_j \nu_k}_{26} O^{{ncw}^*}_{R1j}
+ B^{\ell^+_i \ell^-_j \nu_k}_{26} O^{{ncw}^*}_{L1j})\nonumber\\
&&+m_{\ell_i} m_{\ntrl1} (p.k') (C^{\ell^+_i \ell^-_j \nu_k}_{26} O^{{ncw}^*}_{R1j}
+ D^{\ell^+_i \ell^-_j \nu_k}_{26} O^{{ncw}^*}_{L1j})\nonumber\\
&&-\left.
2m_{\ell_i}m_{\ell_j} (P.p) (C^{\ell^+_i \ell^-_j \nu_k}_{26} O^{{ncw}^*}_{L1j}
+ D^{\ell^+_i \ell^-_j \nu_k}_{26} O^{{ncw}^*}_{R1j})\right],\nonumber\\
\label{mat26-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{26} &=& O^{{cnw}^*}_{Lik} O^{nna}_{Lk1r} O^{cca}_{Rijr},
~~B^{\ell^+_i \ell^-_j \nu_k}_{26} = O^{{cnw}^*}_{Rik} O^{nna}_{Rk1r} O^{cca}_{Lijr},\nonumber\\
C^{\ell^+_i \ell^-_j \nu_k}_{26} &=& O^{{cnw}^*}_{Lik} O^{nna}_{Lk1r} O^{cca}_{Lijr},
~~D^{\ell^+_i \ell^-_j \nu_k}_{26} = O^{{cnw}^*}_{Rik} O^{nna}_{Rk1r} O^{cca}_{Rijr}.
\label{mat26-llnu-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_2M_7 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) = \nonumber\\
&&-\frac{4g^4_2sec\theta_W}{\left[(((p+k)^2-m^2_{W}) -i m_W \Gamma_W)\right]
\left[(((k+k')^2-m^2_{Z}) + i m_Z \Gamma_Z)\right]}
\nonumber\\
&&\sum_{i,j,k}
\left[2(P.k)(p.k') (A^{\ell^+_i \ell^-_j \nu_k}_{27} O^{{ncw}^*}_{L1j}
+ B^{\ell^+_i \ell^-_j \nu_k}_{27} O^{{ncw}^*}_{R1j})
\right.\nonumber\\
&&-m_{\ell_j} m_{\ntrl1} (p.k) (A^{\ell^+_i \ell^-_j \nu_k}_{27} O^{{ncw}^*}_{R1j}
+ B^{\ell^+_i \ell^-_j \nu_k}_{27} O^{{ncw}^*}_{L1j})\nonumber\\
&&+m_{\ell_i} m_{\ntrl1} (p.k') (C^{\ell^+_i \ell^-_j \nu_k}_{27} O^{{ncw}^*}_{R1j}
+ D^{\ell^+_i \ell^-_j \nu_k}_{27} O^{{ncw}^*}_{L1j})\nonumber\\
&&+\left.
m_{\ell_i}m_{\ell_j} (P.p) (C^{\ell^+_i \ell^-_j \nu_k}_{27} O^{{ncw}^*}_{L1j}
+ D^{\ell^+_i \ell^-_j \nu_k}_{27} O^{{ncw}^*}_{R1j})\right],\nonumber\\
\label{mat27-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{27} &=& O^{{cnw}^*}_{Rik} O^{nnz}_{Lk1} O^{ccz}_{Rij},
~~B^{\ell^+_i \ell^-_j \nu_k}_{27} = O^{{cnw}^*}_{Lik} O^{nnz}_{Rk1} O^{ccz}_{Lij},\nonumber\\
C^{\ell^+_i \ell^-_j \nu_k}_{27} &=& O^{{cnw}^*}_{Rik} O^{nnz}_{Lk1} O^{ccz}_{Lij},
~~D^{\ell^+_i \ell^-_j \nu_k}_{27} = O^{{cnw}^*}_{Lik} O^{nnz}_{Rk1} O^{ccz}_{Rij}.
\label{mat27-llnu-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_3M_4 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) = -\sum^8_{r,l=1}
\frac{2\widetilde{g}^4}{\left[((p+k')^2-m^2_{S^\pm_r})((p+k)^2-m^2_{S^\pm_l})\right]}
\nonumber\\
&&\sum_{i,j,k}
\left[\left\{(P.k)(p.k')-(P.p)(k.k')+(P.k')(p.k)\right\}
(A^{\ell^+_i \ell^-_j \nu_k}_{34} O^{{cns}^*}_{Li1r}
+ B^{\ell^+_i \ell^-_j \nu_k}_{34} O^{{cns}^*}_{Ri1r})
\right.\nonumber\\
&&+m_{\ell_i} m_{\ntrl1} (p.k) (A^{\ell^+_i \ell^-_j \nu_k}_{34} O^{{cns}^*}_{Ri1r}
+ B^{\ell^+_i \ell^-_j \nu_k}_{34} O^{{cns}^*}_{Li1r})\nonumber\\
&&+m_{\ell_j} m_{\ntrl1} (p.k') (C^{\ell^+_i \ell^-_j \nu_k}_{34} O^{{cns}^*}_{Ri1r}
+ D^{\ell^+_i \ell^-_j \nu_k}_{34} O^{{cns}^*}_{Li1r})\nonumber\\
&&+\left.
m_{\ell_i}m_{\ell_j} (P.p) (C^{\ell^+_i \ell^-_j \nu_k}_{34} O^{{cns}^*}_{Li1r}
+ D^{\ell^+_i \ell^-_j \nu_k}_{34} O^{{cns}^*}_{Ri1r})\right],
\label{mat34-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{34} &=& O^{{ncs}^*}_{Lkjr} O^{ncs}_{L1jl} O^{cns}_{Likl},
~~B^{\ell^+_i \ell^-_j \nu_k}_{34} = O^{{ncs}^*}_{Rkjr} O^{ncs}_{R1jl} O^{cns}_{Rikl},\nonumber\\
C^{\ell^+_i \ell^-_j \nu_k}_{34} &=& O^{{ncs}^*}_{Rkjr} O^{ncs}_{L1jl} O^{cns}_{Rikl},
~~D^{\ell^+_i \ell^-_j \nu_k}_{34} = O^{{ncs}^*}_{Lkjr} O^{ncs}_{R1jl} O^{cns}_{Likl}.
\label{mat34-llnu-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_3M_5 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) = \sum^8_{r,l=1}
\frac{\widetilde{g}^4}{\left[((p+k')^2-m^2_{S^\pm_r})((k+k')^2-m^2_{S^0_l})\right]}
\nonumber\\
&&\sum_{i,j,k}
\left[\left\{(P.k)(p.k')-(P.k')(p.k)+(P.p)(k.k')\right\}
(A^{\ell^+_i \ell^-_j \nu_k}_{35} O^{{cns}^*}_{Li1r}
+ B^{\ell^+_i \ell^-_j \nu_k}_{35} O^{{cns}^*}_{Ri1r})
\right.\nonumber\\
&&+m_{\ell_i} m_{\ntrl1} (p.k') (A^{\ell^+_i \ell^-_j \nu_k}_{35} O^{{cns}^*}_{Ri1r}
+ B^{\ell^+_i \ell^-_j \nu_k}_{35} O^{{cns}^*}_{Li1r})\nonumber\\
&&-m_{\ell_j} m_{\ntrl1} (p.k) (C^{\ell^+_i \ell^-_j \nu_k}_{35} O^{{cns}^*}_{Ri1r}
+ D^{\ell^+_i \ell^-_j \nu_k}_{35} O^{{cns}^*}_{Li1r})\nonumber\\
&&-\left.
m_{\ell_i}m_{\ell_j} (P.p) (C^{\ell^+_i \ell^-_j \nu_k}_{35} O^{{cns}^*}_{Li1r}
+ D^{\ell^+_i \ell^-_j \nu_k}_{35} O^{{cns}^*}_{Ri1r})\right],
\label{mat35-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{35} &=& O^{{ncs}^*}_{Lkjr} O^{nnh}_{Lk1l} O^{cch}_{Lijl},
~~B^{\ell^+_i \ell^-_j \nu_k}_{35} = O^{{ncs}^*}_{Rkjr} O^{nnh}_{Rk1l} O^{cch}_{Rijl},\nonumber\\
C^{\ell^+_i \ell^-_j \nu_k}_{35} &=& O^{{ncs}^*}_{Lkjr} O^{nnh}_{Lk1l} O^{cch}_{Rijl},
~~D^{\ell^+_i \ell^-_j \nu_k}_{35} = O^{{ncs}^*}_{Rkjr} O^{nnh}_{Rk1l} O^{cch}_{Lijl}.
\label{mat35-llnu-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_3M_6 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) = -\sum^8_{r,l=1}
\frac{\widetilde{g}^4}{\left[((p+k')^2-m^2_{S^\pm_r})((k+k')^2-m^2_{P^0_l})\right]}
\nonumber\\
&&\sum_{i,j,k}
\left[\left\{(P.k)(p.k')-(P.k')(p.k)+(P.p)(k.k')\right\}
(A^{\ell^+_i \ell^-_j \nu_k}_{36} O^{{cns}^*}_{Li1r}
+ B^{\ell^+_i \ell^-_j \nu_k}_{36} O^{{cns}^*}_{Ri1r})
\right.\nonumber\\
&&+m_{\ell_i} m_{\ntrl1} (p.k') (A^{\ell^+_i \ell^-_j \nu_k}_{36} O^{{cns}^*}_{Ri1r}
+ B^{\ell^+_i \ell^-_j \nu_k}_{36} O^{{cns}^*}_{Li1r})\nonumber\\
&&-m_{\ell_j} m_{\ntrl1} (p.k) (C^{\ell^+_i \ell^-_j \nu_k}_{36} O^{{cns}^*}_{Ri1r}
+ D^{\ell^+_i \ell^-_j \nu_k}_{36} O^{{cns}^*}_{Li1r})\nonumber\\
&&-\left.
m_{\ell_i}m_{\ell_j} (P.p) (C^{\ell^+_i \ell^-_j \nu_k}_{36} O^{{cns}^*}_{Li1r}
+ D^{\ell^+_i \ell^-_j \nu_k}_{36} O^{{cns}^*}_{Ri1r})\right],
\label{mat36-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{36} &=& O^{{ncs}^*}_{Lkjr} O^{nna}_{Lk1l} O^{cca}_{Lijl},
~~B^{\ell^+_i \ell^-_j \nu_k}_{36} = O^{{ncs}^*}_{Rkjr} O^{nna}_{Rk1l} O^{cca}_{Rijl},\nonumber\\
C^{\ell^+_i \ell^-_j \nu_k}_{36} &=& O^{{ncs}^*}_{Lkjr} O^{nna}_{Lk1l} O^{cca}_{Rijl},
~~D^{\ell^+_i \ell^-_j \nu_k}_{36} = O^{{ncs}^*}_{Rkjr} O^{nna}_{Rk1l} O^{cca}_{Lijl}.
\label{mat36-llnu-det}
\end{eqnarray}
\begin{eqnarray}
\bullet~ M^\dagger_3M_7 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) &=& -\sum^8_{r=1}
\frac{2g^2_2\widetilde{g}^2}{\left[((p+k')^2-m^2_{S^\pm_r})((k+k')^2-m^2_{Z}+im_Z\Gamma_Z)\right]}
\nonumber\\
&&\sum_{i,j,k}
\left[2(P.k)(p.k') (A^{\ell^+_i \ell^-_j \nu_k}_{37} O^{{ccz}}_{Lij}
+ B^{\ell^+_i \ell^-_j \nu_k}_{37} O^{{ccz}}_{Rij})
\right.\nonumber\\
&&+m_{\ell_i} m_{\ell_j} (P.p) (A^{\ell^+_i \ell^-_j \nu_k}_{37} O^{{ccz}}_{Rij}
+ B^{\ell^+_i \ell^-_j \nu_k}_{37} O^{{ccz}}_{Lij})\nonumber\\
&&+2m_{\ell_i} m_{\ntrl1} (p.k') (C^{\ell^+_i \ell^-_j \nu_k}_{37} O^{{ccz}}_{Rij}
+ D^{\ell^+_i \ell^-_j \nu_k}_{37} O^{{ccz}}_{Lij})\nonumber\\
&&+\left.
m_{\ell_j}m_{\ntrl1} (p.k) (C^{\ell^+_i \ell^-_j \nu_k}_{37} O^{{ccz}}_{Lij}
+ D^{\ell^+_i \ell^-_j \nu_k}_{37} O^{{ccz}}_{Rij})\right],
\label{mat37-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{37} &=& O^{{ncs}^*}_{Lkjr} O^{nnz}_{Rk1} O^{{cns}^*}_{Ri1r},
~~B^{\ell^+_i \ell^-_j \nu_k}_{37} = O^{{ncs}^*}_{Rkjr} O^{nnz}_{Lk1} O^{{cns}^*}_{Li1r},\nonumber\\
C^{\ell^+_i \ell^-_j \nu_k}_{37} &=& O^{{ncs}^*}_{Rkjr} O^{nnz}_{Lk1} O^{{cns}^*}_{Ri1r},
~~D^{\ell^+_i \ell^-_j \nu_k}_{37} = O^{{ncs}^*}_{Lkjr} O^{nnz}_{Rk1} O^{{cns}^*}_{Li1r}.
\label{mat37-llnu-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_4M_5 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) = \sum^8_{r,l=1}
\frac{\widetilde{g}^4}{\left[((p+k)^2-m^2_{S^\pm_r})((k+k')^2-m^2_{S^0_l})\right]}
\nonumber\\
&&\sum_{i,j,k}
\left[\left\{(P.k')(p.k)-(P.k)(p.k')+(P.p)(k.k')\right\}
(A^{\ell^+_i \ell^-_j \nu_k}_{45} O^{{ncs}^*}_{L1jr}
+ B^{\ell^+_i \ell^-_j \nu_k}_{45} O^{{ncs}^*}_{R1jr})
\right.\nonumber\\
&&+m_{\ell_j} m_{\ntrl1} (p.k) (A^{\ell^+_i \ell^-_j \nu_k}_{45} O^{{ncs}^*}_{R1jr}
+ B^{\ell^+_i \ell^-_j \nu_k}_{45} O^{{ncs}^*}_{L1jr})\nonumber\\
&&-m_{\ell_i} m_{\ntrl1} (p.k') (C^{\ell^+_i \ell^-_j \nu_k}_{45} O^{{ncs}^*}_{R1jr}
+ D^{\ell^+_i \ell^-_j \nu_k}_{45} O^{{ncs}^*}_{L1jr})\nonumber\\
&&-\left.
m_{\ell_i}m_{\ell_j} (P.p) (C^{\ell^+_i \ell^-_j \nu_k}_{45} O^{{ncs}^*}_{L1jr}
+ D^{\ell^+_i \ell^-_j \nu_k}_{45} O^{{ncs}^*}_{R1jr})\right],
\label{mat45-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{45} &=& O^{{cns}^*}_{Likr} O^{nnh}_{Lk1l} O^{cch}_{Lijl},
~~B^{\ell^+_i \ell^-_j \nu_k}_{45} = O^{{cns}^*}_{Rikr} O^{nnh}_{Rk1l} O^{cch}_{Rijl},\nonumber\\
C^{\ell^+_i \ell^-_j \nu_k}_{45} &=& O^{{cns}^*}_{Likr} O^{nnh}_{Lk1l} O^{cch}_{Rijl},
~~D^{\ell^+_i \ell^-_j \nu_k}_{45} = O^{{cns}^*}_{Rikr} O^{nnh}_{Rk1l} O^{cch}_{Lijl}.
\label{mat45-llnu-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_4M_6 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) = -\sum^8_{r,l=1}
\frac{\widetilde{g}^4}{\left[((p+k)^2-m^2_{S^\pm_r})((k+k')^2-m^2_{P^0_l})\right]}
\nonumber\\
&&\sum_{i,j,k}
\left[\left\{(P.k')(p.k)-(P.k)(p.k')+(P.p)(k.k')\right\}
(A^{\ell^+_i \ell^-_j \nu_k}_{46} O^{{ncs}^*}_{L1jr}
+ B^{\ell^+_i \ell^-_j \nu_k}_{46} O^{{ncs}^*}_{R1jr})
\right.\nonumber\\
&&+m_{\ell_j} m_{\ntrl1} (p.k) (A^{\ell^+_i \ell^-_j \nu_k}_{46} O^{{ncs}^*}_{R1jr}
+ B^{\ell^+_i \ell^-_j \nu_k}_{46} O^{{ncs}^*}_{L1jr})\nonumber\\
&&-m_{\ell_i} m_{\ntrl1} (p.k') (C^{\ell^+_i \ell^-_j \nu_k}_{46} O^{{ncs}^*}_{R1jr}
+ D^{\ell^+_i \ell^-_j \nu_k}_{46} O^{{ncs}^*}_{L1jr})\nonumber\\
&&-\left.
m_{\ell_i}m_{\ell_j} (P.p) (C^{\ell^+_i \ell^-_j \nu_k}_{46} O^{{ncs}^*}_{L1jr}
+ D^{\ell^+_i \ell^-_j \nu_k}_{46} O^{{ncs}^*}_{R1jr})\right],
\label{mat46-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{46} &=& O^{{cns}^*}_{Likr} O^{nna}_{Lk1l} O^{cca}_{Lijl},
~~B^{\ell^+_i \ell^-_j \nu_k}_{46} = O^{{cns}^*}_{Rikr} O^{nna}_{Rk1l} O^{cca}_{Rijl},\nonumber\\
C^{\ell^+_i \ell^-_j \nu_k}_{46} &=& O^{{cns}^*}_{Likr} O^{nna}_{Lk1l} O^{cca}_{Rijl},
~~D^{\ell^+_i \ell^-_j \nu_k}_{46} = O^{{cns}^*}_{Rikr} O^{nna}_{Rk1l} O^{cca}_{Lijl}.
\label{mat46-llnu-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_4M_7 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) = \sum^8_{r=1}
\frac{2g^2_2\widetilde{g}^2}{\left[((p+k)^2-m^2_{S^\pm_r})((k+k')^2-m^2_{Z}+im_Z\Gamma_Z)\right]}
\nonumber\\
&&\sum_{i,j,k}
\left[2(P.k')(p.k)(A^{\ell^+_i \ell^-_j \nu_k}_{47} O^{ccz}_{Rij}
+ B^{\ell^+_i \ell^-_j \nu_k}_{47} O^{ccz}_{Lij})
\right.\nonumber\\
&&+m_{\ell_i} m_{\ell_j} (P.p) (A^{\ell^+_i \ell^-_j \nu_k}_{47} O^{ccz}_{Lij}
+ B^{\ell^+_i \ell^-_j \nu_k}_{47} O^{ccz}_{Rij})\nonumber\\
&&+2m_{\ell_j} m_{\ntrl1} (p.k) (C^{\ell^+_i \ell^-_j \nu_k}_{47} O^{ccz}_{Lij}
+ D^{\ell^+_i \ell^-_j \nu_k}_{47} O^{ccz}_{Rij})\nonumber\\
&&+\left.
m_{\ell_i}m_{\ntrl1} (p.k') (C^{\ell^+_i \ell^-_j \nu_k}_{47} O^{ccz}_{Rij}
+ D^{\ell^+_i \ell^-_j \nu_k}_{47} O^{ccz}_{Lij})\right],
\label{mat47-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{47} &=& O^{{cns}^*}_{Likr} O^{nnz}_{Rk1} O^{{ncs}^*}_{R1jr},
~~B^{\ell^+_i \ell^-_j \nu_k}_{47} = O^{{cns}^*}_{Rikr} O^{nnz}_{Lk1} O^{{ncs}^*}_{L1jr},\nonumber\\
C^{\ell^+_i \ell^-_j \nu_k}_{47} &=& O^{{cns}^*}_{Rikr} O^{nnz}_{Lk1} O^{{ncs}^*}_{R1jr},
~~D^{\ell^+_i \ell^-_j \nu_k}_{47} = O^{{cns}^*}_{Likr} O^{nnz}_{Rk1} O^{{ncs}^*}_{L1jr}.
\label{mat47-llnu-det}
\end{eqnarray}
\begin{eqnarray}
\bullet~ M^\dagger_5M_6 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) &=& -\sum^8_{r,l=1}
\frac{\widetilde{g}^4}{\left[((k+k')^2-m^2_{S^0_r})((k+k')^2-m^2_{P^0_l})\right]}
\nonumber\\
&&\sum_{i,j,k}
\left[(P.p)(k.k')A^{\ell^+_i \ell^-_j \nu_k}_{56}
-m_{\ell_i}m_{\ell_j}(P.p)B^{\ell^+_i \ell^-_j \nu_k}_{56}\right],\nonumber\\
\label{mat56-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{56} &=& \left(O^{{cch}^*}_{Lijr} O^{cca}_{Lijl}
+O^{{cch}^*}_{Rijr} O^{cca}_{Rijl}\right)
\left(O^{{nnh}^*}_{Lk1r} O^{nna}_{Lk1l}
+O^{{nnh}^*}_{Rk1r} O^{nna}_{Rk1l}\right),\nonumber\\
B^{\ell^+_i \ell^-_j \nu_k}_{56} &=& \left(O^{{cch}^*}_{Rijr} O^{cca}_{Lijl}
+O^{{cch}^*}_{Lijr} O^{cca}_{Rijl}\right)
\left(O^{{nnh}^*}_{Lk1r} O^{nna}_{Lk1l}
+O^{{nnh}^*}_{Rk1r} O^{nna}_{Rk1l}\right).
\label{mat56-llnu-det}
\end{eqnarray}
\begin{eqnarray}
\bullet~ M^\dagger_5M_7 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) &=& -\sum^8_{r=1}
\frac{g^2_2\widetilde{g}^2see\theta_W}{\left[((k+k')^2-m^2_{S^0_r})((k+k')^2-m^2_{Z}+im_Z\Gamma_Z)
\right]} \nonumber\\
&&\sum_{i,j,k}
\left[m_{\ell_i}m_{\ntrl1}(p.k')A^{\ell^+_i \ell^-_j \nu_k}_{57}
-m_{\ell_j}m_{\ntrl1}(p.k)B^{\ell^+_i \ell^-_j \nu_k}_{57}\right],\nonumber\\
\label{mat57-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{57} &=& \left(O^{{cch}^*}_{Rijr} O^{ccz}_{Rij}
+O^{{cch}^*}_{Lijr} O^{ccz}_{Lij}\right)
\left(O^{{nnh}^*}_{Rk1r} O^{nnz}_{Lk1}
+O^{{nnh}^*}_{Lk1r} O^{nnz}_{Rk1}\right),\nonumber\\
B^{\ell^+_i \ell^-_j \nu_k}_{57} &=& \left(O^{{cch}^*}_{Rijr} O^{ccz}_{Lij}
+O^{{cch}^*}_{Lijr} O^{ccz}_{Rij}\right)
\left(O^{{nnh}^*}_{Rk1r} O^{nnz}_{Lk1}
+O^{{nnh}^*}_{Lk1r} O^{nnz}_{Rk1}\right).
\label{mat57-llnu-det}
\end{eqnarray}
\begin{eqnarray}
\bullet~ M^\dagger_6M_7 (\ntrl1\to \sum \ell^+_i \ell^-_j \nu_k) &=& \sum^8_{r=1}
\frac{g^2_2\widetilde{g}^2see\theta_W}{\left[((k+k')^2-m^2_{P^0_r})((k+k')^2-m^2_{Z}+im_Z\Gamma_Z)
\right]} \nonumber\\
&&\sum_{i,j,k}
\left[m_{\ell_i}m_{\ntrl1}(p.k')A^{\ell^+_i \ell^-_j \nu_k}_{67}
-m_{\ell_j}m_{\ntrl1}(p.k)B^{\ell^+_i \ell^-_j \nu_k}_{67}\right],\nonumber\\
\label{mat67-llnu}
\end{eqnarray}
where
\begin{eqnarray}
A^{\ell^+_i \ell^-_j \nu_k}_{67} &=& \left(O^{{cca}^*}_{Rijr} O^{ccz}_{Rij}
+O^{{cca}^*}_{Lijr} O^{ccz}_{Lij}\right)
\left(O^{{nna}^*}_{Rk1r} O^{nnz}_{Lk1}
+O^{{nna}^*}_{Lk1r} O^{nnz}_{Rk1}\right),\nonumber\\
B^{\ell^+_i \ell^-_j \nu_k}_{67} &=& \left(O^{{cca}^*}_{Rijr} O^{ccz}_{Lij}
+O^{{cca}^*}_{Lijr} O^{ccz}_{Rij}\right)
\left(O^{{nna}^*}_{Rk1r} O^{nnz}_{Lk1}
+O^{{nna}^*}_{Lk1r} O^{nnz}_{Rk1}\right).
\label{mat67-llnu-det}
\end{eqnarray}
$\Gamma_W$ and $\Gamma_Z$ are the decay width for $W$ and $Z$-boson, respectively
and their values are given in ref. \cite{AppNakamura-c2}. All the lepton
masses are also taken from ref. \cite{AppNakamura-c2}.
\section{Process $\ntrl1\to \nu_i \overline{\nu}_j \nu_k$}\label{LSP-3nu-decay}
We represent different lepton flavours $(e,\mu,\tau)$ by $i,j,k$.
We write down all possible $M^\dagger_iM_j$ for the three diagrams
shown in figure \ref{LSP-3nu}. The four-momentum assignments are as follows
\begin{equation}
\ntrl1(P) \to \nu_i(p)+\overline{\nu_j}(k)+\nu_k(k').
\label{momentum-3nu}
\end{equation}
\begin{figure}[ht]
\centering
\vspace*{0.5cm}
\includegraphics[height=3.00cm]{FigsPDF/LSP-3nu.pdf}
\caption{Feynman diagrams for the possible three body decays of the lightest supersymmetric
particle into $\nu_i\overline{\nu_j}\nu_k$ final states.
$S^0_r,P^0_r$ are the neutral scalar and pseudoscalar states of the $\mu\nu$SSM as
shown by eqns.(\ref{scalar-mass-basis}), (\ref{pseudoscalar-mass-basis}).}
\label{LSP-3nu}
\end{figure}
\begin{eqnarray}
&&\bullet~ M^\dagger_1M_1 (\ntrl1\to \sum \nu_i \overline{\nu_j} \nu_k) = \sum^8_{r,l=1}
\frac{\widetilde {g}^4}{\left[((k+k')^2-m^2_{S^0_r})
((k+k')^2-m^2_{S^0_l})\right]} \nonumber\\
&&\times\sum_{i,j,k}(P.p)(k.k')
\left(O^{{nnh}^*}_{Li1r}O^{nnh}_{Li1l}+O^{{nnh}^*}_{Ri1r}O^{nnh}_{Ri1l}\right)
\left(O^{{nnh}^*}_{Lkjr}O^{nnh}_{Lkjl}+O^{{nnh}^*}_{Rkjr}O^{nnh}_{Rkjl}\right).\nonumber\\
\label{mat11-3nu}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_2M_2 (\ntrl1\to \sum \nu_i \overline{\nu_j} \nu_k) = \sum^8_{r,l=1}
\frac{\widetilde {g}^4}{\left[((k+k')^2-m^2_{P^0_r})
((k+k')^2-m^2_{P^0_l})\right]} \nonumber\\
&&\times\sum_{i,j,k}(P.p)(k.k')
\left(O^{{nna}^*}_{Li1r}O^{nna}_{Li1l}+O^{{nna}^*}_{Ri1r}O^{nna}_{Ri1l}\right)
\left(O^{{nna}^*}_{Lkjr}O^{nna}_{Lkjl}+O^{{nna}^*}_{Rkjr}O^{nna}_{Rkjl}\right).\nonumber\\
\label{mat22-3nu}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_3M_3 (\ntrl1\to \sum \nu_i \overline{\nu_j} \nu_k) =
\frac{{g}^4}{\left[((k+k')^2-m^2_{Z})^2 + m^2_Z \Gamma^2_Z\right]} \nonumber\\
&&\times\sum_{i,j,k}
\left[(P.k)(p.k')\left(O^{{nnz}^*}_{Li1}O^{nnz}_{Li1}O^{{nnz}^*}_{Lkj}O^{nnz}_{Lkj}
+O^{{nnz}^*}_{Ri1}O^{nnz}_{Ri1}O^{{nnz}^*}_{Rkj}O^{nnz}_{Rkj}\right)
\right.\nonumber\\
&&+\left.
(P.k')(p.k)\left(O^{{nnz}^*}_{Li1}O^{nnz}_{Li1}O^{{nnz}^*}_{Rkj}O^{nnz}_{Rkj}
+O^{{nnz}^*}_{Ri1}O^{nnz}_{Ri1}O^{{nnz}^*}_{Lkj}O^{nnz}_{Lkj}\right)\right].
\label{mat33-3nu}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_1M_2 (\ntrl1\to \sum \nu_i \overline{\nu_j} \nu_k) = -\sum^8_{r,l=1}
\frac{\widetilde {g}^4}{\left[((k+k')^2-m^2_{S^0_r})
((k+k')^2-m^2_{P^0_l})\right]} \nonumber\\
&&\times\sum_{i,j,k}(P.p)(k.k')
\left(O^{{nnh}^*}_{Li1r}O^{nna}_{Li1l}+O^{{nnh}^*}_{Ri1r}O^{nna}_{Ri1l}\right)
\left(O^{{nnh}^*}_{Lkjr}O^{nna}_{Lkjl}+O^{{nnh}^*}_{Rkjr}O^{nna}_{Rkjl}\right).\nonumber\\
\label{mat12-3nu}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_1M_3 (\ntrl1\to \sum \nu_i \overline{\nu_j} \nu_k) = 0.
\label{mat13-3nu}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_2M_3 (\ntrl1\to \sum \nu_i \overline{\nu_j} \nu_k) = 0.
\label{mat23-3nu}
\end{eqnarray}
\section{Process $\ntrl1\to \bar{u}_i d_j \ell^+_k$}\label{LSP-qql1-decay}
We represent different lepton flavours $(e,\mu,\tau)$ by $k$. $u_i(d_j)$ stands
for different up-type and down-type quarks $(u,c(d,s,b))$, except the top.
We write down all possible $M^\dagger_iM_j$ for the four diagrams
shown in figure \ref{LSP-qql1}.
Required couplings are given in appendices \ref{appenD} and \ref{appenH}.
The four-momentum assignments are as follows
\begin{equation}
\ntrl1(P) \to \ell^+_k(p)+\bar{u}_i(k)+d_j(k').
\label{momentum-qql1}
\end{equation}
\begin{figure}[ht]
\centering
\vspace*{0.5cm}
\includegraphics[height=6.00cm]{FigsPDF/LSP-qql1.pdf}
\caption{Feynman diagrams for the possible three body decays of the lightest supersymmetric
particle into $\bar{u}_i d_j \ell^+_k$ final states.
$S^-_r$ are the charged scalar states of the $\mu\nu$SSM as
shown by eqn.(\ref{charged-scalar-mass-basis}). $\widetilde u(\widetilde d)$ are the up and down-type
squarks as shown by eqn.(\ref{squark-mass-basis}) corresponding
to $\bar{u}_i$ and $d_j$.}
\label{LSP-qql1}
\end{figure}
\begin{eqnarray}
&&\bullet~ M^\dagger_1M_1 (\ntrl1\to \sum \bar{u}_i d_j \ell^+_k) = \sum_{i,j,k}
\frac{4 {g}^4 |V^{CKM}_{ij}|^2}{\left[(((k+k')^2-m^2_{W})^2+m^2_W\Gamma^2_W)\right]} \nonumber\\
&&\left[2(P.k)(p.k')O^{{cnw}^*}_{Lk1} O^{cnw}_{Lk1}
+ 2(P.k')(p.k) O^{{cnw}^*}_{Rk1} O^{cnw}_{Rk1}
\right.\nonumber\\
&&-\left.
m_{\ell_k} m_{\ntrl1} (k.k') \left(O^{{cnw}^*}_{Rk1} O^{cnw}_{Lk1}
+ O^{{cnw}^*}_{Lk1} O^{cnw}_{Rk1}\right)\right].
\label{mat11-qql1}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_2M_2 (\ntrl1\to \sum \bar{u}_i d_j \ell^+_k) = \sum_{i,j,k}
\sum^8_{r,l=1}\frac{4 \widetilde{g}^4}{\left[((k+k')^2-m^2_{S^\pm_r})((k+k')^2-m^2_{S^\pm_l})
\right]} \nonumber\\
&&\left[(P.p)\left(O^{{cns}^*}_{Lk1r} O^{cns}_{Lk1l}
+ O^{{cns}^*}_{Rk1r} O^{cns}_{Rk1l}\right)+
m_{\ell_k} m_{\ntrl1} \left(O^{{cns}^*}_{Lk1r} O^{cns}_{Rk1l}
+ O^{{cns}^*}_{Rk1r} O^{cns}_{Lk1l}\right)\right]\nonumber\\
&&\times \left[(k.k')\left(O^{{uds}^*}_{Lijl} O^{uds}_{Lijr}
+ O^{{uds}^*}_{Rijl} O^{uds}_{Rijr}\right)-
m_{u_i}m_{d_j} \left(O^{{uds}^*}_{Rijl} O^{uds}_{Lijr}
+ O^{{uds}^*}_{Lijl} O^{uds}_{Rijr}\right)\right].\nonumber\\
\label{mat22-qql1}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_3M_3 (\ntrl1\to \sum \bar{u}_i d_j \ell^+_k) = \sum_{i,j,k}
\sum^2_{r,l=1}\frac{4 \widetilde{g}^4}{\left[((p+k)^2-m^2_{\widetilde d_r})((p+k)^2-m^2_{\widetilde d_l})
\right]} \nonumber\\
&&\left[(P.k')\left(O^{{dn\widetilde d}^*}_{Lj1r} O^{dn\widetilde d}_{Lj1l}
+ O^{{dn\widetilde d}^*}_{Rj1r} O^{dn\widetilde d}_{Rj1l}\right)+
m_{d_j} m_{\ntrl1} \left(O^{{dn\widetilde d}^*}_{Rj1r} O^{dn\widetilde d}_{Lj1l}
+ O^{{dn\widetilde d}^*}_{Lj1r} O^{dn\widetilde d}_{Rj1l}\right)\right]\nonumber\\
&&\times \left[(p.k)\left(O^{{ucd}^*}_{Likl} O^{ucd}_{Likr}
+ O^{{ucd}^*}_{Rikl} O^{ucd}_{Rikr}\right)-
m_{u_i}m_{\ell_k} \left(O^{{ucd}^*}_{Rikl} O^{ucd}_{Likr}
+ O^{{ucd}^*}_{Likl} O^{ucd}_{Rikr}\right)\right].\nonumber\\
\label{mat33-qql1}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_4M_4 (\ntrl1\to \sum \bar{u}_i d_j \ell^+_k) = \sum_{i,j,k}
\sum^2_{r,l=1}\frac{4 \widetilde{g}^4}{\left[((p+k')^2-m^2_{\widetilde u_r})((p+k')^2-m^2_{\widetilde u_l})
\right]} \nonumber\\
&&\left[(P.k)\left(O^{{nu\widetilde u}^*}_{L1ir} O^{nu\widetilde u}_{L1il}
+ O^{{nu\widetilde u}^*}_{R1ir} O^{nu\widetilde u}_{R1il}\right)+
m_{u_i} m_{\ntrl1} \left(O^{{nu\widetilde u}^*}_{R1ir} O^{nu\widetilde u}_{L1il}
+ O^{{nu\widetilde u}^*}_{L1ir} O^{nu\widetilde u}_{R1il}\right)\right]\nonumber\\
&&\times \left[(p.k')\left(O^{{cdu}^*}_{Lkjl} O^{cdu}_{Lkjr}
+ O^{{cdu}^*}_{Rkjl} O^{cdu}_{Rkjr}\right)-
m_{d_j}m_{\ell_k} \left(O^{{cdu}^*}_{Rkjl} O^{cdu}_{Lkjr}
+ O^{{cdu}^*}_{Lkjl} O^{cdu}_{Rkjr}\right)\right].\nonumber\\
\label{mat44-qql1}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_1M_2 (\ntrl1\to \sum \bar{u}_i d_j \ell^+_k) = \nonumber\\
&&\sum_{i,j,k}
\sum^8_{r=1}\frac{2\sqrt{2} g^2_2\widetilde{g}^2 V^{CKM}_{ij}}
{\left[((k+k')^2-m^2_{W}-im_W\Gamma_W)((k+k')^2-m^2_{S^\pm_r})
\right]} \nonumber\\
&&\left[m_{u_i}m_{\ntrl1}(p.k')O^{{uds}^*}_{Lijr}
A^{\bar{u}_i d_j \ell^+_k}_{12}
+m_{\ell_k} m_{u_i}(P.k') O^{{uds}^*}_{Lijr}
B^{\bar{u}_i d_j \ell^+_k}_{12}
\right.\nonumber\\
&&-\left.
m_{d_j}m_{\ntrl1}(p.k) O^{{uds}^*}_{Rijr}
A^{\bar{u}_i d_j \ell^+_k}_{12}
-m_{d_j}m_{\ell_k}(P.k) O^{{uds}^*}_{Rijr}
B^{\bar{u}_i d_j \ell^+_k}_{12}\right].\nonumber\\
\label{mat12-qql1}
\end{eqnarray}
where
\begin{eqnarray}
A^{\bar{u}_i d_j \ell^+_k}_{12} &=& \left(O^{{cnw}^*}_{Lk1} O^{cns}_{Rk1r}
+O^{{cnw}^*}_{Rk1} O^{cns}_{Lk1r}\right),
~~B^{\bar{u}_i d_j \ell^+_k}_{12} = \left(O^{{cnw}^*}_{Lk1} O^{cns}_{Lk1r}
+O^{{cnw}^*}_{Rk1} O^{cns}_{Rk1r}\right).\nonumber\\
\label{mat12-qql1-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_1M_3 (\ntrl1\to \sum \bar{u}_i d_j \ell^+_k) = \nonumber\\
&&-\sum_{i,j,k}
\sum^2_{r=1}\frac{2\sqrt{2} g^2_2\widetilde{g}^2 V^{CKM}_{ij}}
{\left[((k+k')^2-m^2_{W}-im_W\Gamma_W)((p+k)^2-m^2_{\widetilde d_r})
\right]} \nonumber\\
&&\left[2(P.k')(p.k)A^{\bar{u}_i d_j \ell^+_k}_{13}
-m_{\ell_k}m_{\ntrl1} (k.k') B^{\bar{u}_i d_j \ell^+_k}_{13}
+m_{u_i}m_{\ntrl1}(p.k')C^{\bar{u}_i d_j \ell^+_k}_{13}
\right.\nonumber\\
&&-2m_{u_i}m_{\ell_k} (P.k') D^{\bar{u}_i d_j \ell^+_k}_{13}
+2m_{d_j}m_{\ntrl1}(p.k)E^{\bar{u}_i d_j \ell^+_k}_{13}
-m_{d_j}m_{\ell_k}(P.k)F^{\bar{u}_i d_j \ell^+_k}_{13}\nonumber\\
&&+\left.
m_{u_i}m_{d_j}(P.p)G^{\bar{u}_i d_j \ell^+_k}_{13}
-2m_{u_i}m_{d_j}m_{\ell_k}m_{\ntrl1}H^{\bar{u}_i d_j \ell^+_k}_{13}\right],
\label{mat13-qql1}
\end{eqnarray}
where
\begin{eqnarray}
A^{\bar{u}_i d_j \ell^+_k}_{13} &=& O^{{cnw}^*}_{Rk1} O^{{ucd}^*}_{Rikr}
O^{dn\widetilde d}_{Rj1r},
~~B^{\bar{u}_i d_j \ell^+_k}_{13} = O^{{cnw}^*}_{Lk1} O^{{ucd}^*}_{Rikr}
O^{dn\widetilde d}_{Rj1r},\nonumber\\
C^{\bar{u}_i d_j \ell^+_k}_{13} &=& O^{{cnw}^*}_{Lk1} O^{{ucd}^*}_{Likr}
O^{dn\widetilde d}_{Rj1r},
~~D^{\bar{u}_i d_j \ell^+_k}_{13} = O^{{cnw}^*}_{Rk1} O^{{ucd}^*}_{Likr}
O^{dn\widetilde d}_{Rj1r},\nonumber\\
E^{\bar{u}_i d_j \ell^+_k}_{13} &=& O^{{cnw}^*}_{Rk1} O^{{ucd}^*}_{Rikr}
O^{dn\widetilde d}_{Lj1r},
~~F^{\bar{u}_i d_j \ell^+_k}_{13} = O^{{cnw}^*}_{Lk1} O^{{ucd}^*}_{Rikr}
O^{dn\widetilde d}_{Lj1r},\nonumber\\
G^{\bar{u}_i d_j \ell^+_k}_{13} &=& O^{{cnw}^*}_{Lk1} O^{{ucd}^*}_{Likr}
O^{dn\widetilde d}_{Lj1r},
~~H^{\bar{u}_i d_j \ell^+_k}_{13} = O^{{cnw}^*}_{Rk1} O^{{ucd}^*}_{Likr}
O^{dn\widetilde d}_{Lj1r}.
\label{mat13-qql1-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_1M_4 (\ntrl1\to \sum \bar{u}_i d_j \ell^+_k) = \nonumber\\
&&\sum_{i,j,k}
\sum^2_{r=1}\frac{2\sqrt{2} g^2_2\widetilde{g}^2 V^{CKM}_{ij}}
{\left[((k+k')^2-m^2_{W}-im_W\Gamma_W)((p+k')^2-m^2_{\widetilde u_r})
\right]} \nonumber\\
&&\left[2(P.k)(p.k')A^{\bar{u}_i d_j \ell^+_k}_{14}
-m_{\ell_k}m_{\ntrl1} (k.k') B^{\bar{u}_i d_j \ell^+_k}_{14}
+m_{d_j}m_{\ntrl1}(p.k)C^{\bar{u}_i d_j \ell^+_k}_{14}
\right.\nonumber\\
&&-2m_{d_j}m_{\ell_k} (P.k) D^{\bar{u}_i d_j \ell^+_k}_{14}
+2m_{u_i}m_{\ntrl1}(p.k')E^{\bar{u}_i d_j \ell^+_k}_{14}
-m_{u_i}m_{\ell_k}(P.k')F^{\bar{u}_i d_j \ell^+_k}_{14}\nonumber\\
&&+\left.
m_{u_i}m_{d_j}(P.p)G^{\bar{u}_i d_j \ell^+_k}_{14}
-2m_{u_i}m_{d_j}m_{\ell_k}m_{\ntrl1}H^{\bar{u}_i d_j \ell^+_k}_{14}\right],
\label{mat14-qql1}
\end{eqnarray}
where
\begin{eqnarray}
A^{\bar{u}_i d_j \ell^+_k}_{14} &=& O^{{cnw}^*}_{Lk1} O^{{cdu}^*}_{Lkjr}
O^{nu\widetilde u}_{L1ir},
~~B^{\bar{u}_i d_j \ell^+_k}_{14} = O^{{cnw}^*}_{Rk1} O^{{cdu}^*}_{Lkjr}
O^{nu\widetilde u}_{L1ir},\nonumber\\
C^{\bar{u}_i d_j \ell^+_k}_{14} &=& O^{{cnw}^*}_{Rk1} O^{{cdu}^*}_{Rkjr}
O^{nu\widetilde u}_{L1ir},
~~D^{\bar{u}_i d_j \ell^+_k}_{14} = O^{{cnw}^*}_{Lk1} O^{{cdu}^*}_{Rkjr}
O^{nu\widetilde u}_{L1ir},\nonumber\\
E^{\bar{u}_i d_j \ell^+_k}_{14} &=& O^{{cnw}^*}_{Lk1} O^{{cdu}^*}_{Lkjr}
O^{nu\widetilde u}_{R1ir},
~~F^{\bar{u}_i d_j \ell^+_k}_{14} = O^{{cnw}^*}_{Rk1} O^{{cdu}^*}_{Lkjr}
O^{nu\widetilde u}_{R1ir},\nonumber\\
G^{\bar{u}_i d_j \ell^+_k}_{14} &=& O^{{cnw}^*}_{Rk1} O^{{cdu}^*}_{Rkjr}
O^{nu\widetilde u}_{R1ir},
~~H^{\bar{u}_i d_j \ell^+_k}_{14} = O^{{cnw}^*}_{Lk1} O^{{cdu}^*}_{Rkjr}
O^{nu\widetilde u}_{R1ir}.
\label{mat14-qql1-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_2M_3 (\ntrl1\to \sum \bar{u}_i d_j \ell^+_k) = \sum_{i,j,k}
\sum^8_{r=1}\sum^2_{l=1}\frac{2\widetilde{g}^4}
{\left[((k+k')^2-m^2_{S^\pm_r})((p+k)^2-m^2_{\widetilde d_l})
\right]} \nonumber\\
&&\left[\{(P.p)(k.k')-(P.k)(p.k')+(P.k')(p.k)\}
\left(A^{\bar{u}_i d_j \ell^+_k}_{23}O^{{cns}^*}_{Lk1r}
+B^{\bar{u}_i d_j \ell^+_k}_{23}O^{{cns}^*}_{Rk1r}\right)
\right.\nonumber\\
&&+m_{\ell_k}m_{\ntrl1}(k.k')\left(A^{\bar{u}_i d_j \ell^+_k}_{23}O^{{cns}^*}_{Rk1r}
+B^{\bar{u}_i d_j \ell^+_k}_{23}O^{{cns}^*}_{Lk1r}\right)\nonumber\\
&&-m_{u_i}m_{\ntrl1}(p.k')\left(C^{\bar{u}_i d_j \ell^+_k}_{23}O^{{cns}^*}_{Rk1r}
+D^{\bar{u}_i d_j \ell^+_k}_{23}O^{{cns}^*}_{Lk1r}\right)\nonumber\\
&&-m_{u_i}m_{\ell_k}(P.k')\left(C^{\bar{u}_i d_j \ell^+_k}_{23}O^{{cns}^*}_{Lk1r}
+D^{\bar{u}_i d_j \ell^+_k}_{23}O^{{cns}^*}_{Rk1r}\right)\nonumber\\
&&+m_{d_j}m_{\ntrl1}(p.k)\left(E^{\bar{u}_i d_j \ell^+_k}_{23}O^{{cns}^*}_{Rk1r}
+F^{\bar{u}_i d_j \ell^+_k}_{23}O^{{cns}^*}_{Lk1r}\right)\nonumber\\
&&+m_{d_j}m_{\ell_k}(P.k)\left(E^{\bar{u}_i d_j \ell^+_k}_{23}O^{{cns}^*}_{Lk1r}
+F^{\bar{u}_i d_j \ell^+_k}_{23}O^{{cns}^*}_{Rk1r}\right)\nonumber\\
&&-m_{u_i}m_{d_j}(P.p)\left(G^{\bar{u}_i d_j \ell^+_k}_{23}O^{{cns}^*}_{Lk1r}
+H^{\bar{u}_i d_j \ell^+_k}_{23}O^{{cns}^*}_{Rk1r}\right)\nonumber\\
&&-\left.
m_{u_i}m_{d_j}m_{\ell_k}m_{\ntrl1}\left(G^{\bar{u}_i d_j \ell^+_k}_{23}O^{{cns}^*}_{Rk1r}
+H^{\bar{u}_i d_j \ell^+_k}_{23}O^{{cns}^*}_{Lk1r}\right)\right],
\label{mat23-qql1}
\end{eqnarray}
where
\begin{eqnarray}
A^{\bar{u}_i d_j \ell^+_k}_{23} &=& O^{{ucd}^*}_{Rikl} O^{uds}_{Rijr}
O^{dn\widetilde d}_{Lj1l},
~~B^{\bar{u}_i d_j \ell^+_k}_{23} = O^{{ucd}^*}_{Likl} O^{uds}_{Lijr}
O^{dn\widetilde d}_{Rj1l},\nonumber\\
C^{\bar{u}_i d_j \ell^+_k}_{23} &=& O^{{ucd}^*}_{Likl} O^{uds}_{Rijr}
O^{dn\widetilde d}_{Lj1l},
~~D^{\bar{u}_i d_j \ell^+_k}_{23} = O^{{ucd}^*}_{Rikl} O^{uds}_{Lijr}
O^{dn\widetilde d}_{Rj1l},\nonumber\\
E^{\bar{u}_i d_j \ell^+_k}_{23} &=& O^{{ucd}^*}_{Likl} O^{uds}_{Lijr}
O^{dn\widetilde d}_{Lj1l},
~~F^{\bar{u}_i d_j \ell^+_k}_{23} = O^{{ucd}^*}_{Rikl} O^{uds}_{Rijr}
O^{dn\widetilde d}_{Rj1l},\nonumber\\
G^{\bar{u}_i d_j \ell^+_k}_{23} &=& O^{{ucd}^*}_{Rikl} O^{uds}_{Lijr}
O^{dn\widetilde d}_{Lj1l},
~~H^{\bar{u}_i d_j \ell^+_k}_{23} = O^{{ucd}^*}_{Likl} O^{uds}_{Rijr}
O^{dn\widetilde d}_{Rj1l}.
\label{mat23-qql1-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_2M_4 (\ntrl1\to \sum \bar{u}_i d_j \ell^+_k) = \sum_{i,j,k}
\sum^8_{r=1}\sum^2_{l=1}\frac{2\widetilde{g}^4}
{\left[((k+k')^2-m^2_{S^\pm_r})((p+k')^2-m^2_{\widetilde u_l})
\right]} \nonumber\\
&&\left[\{(P.p)(k.k')-(P.k')(p.k)+(P.k)(p.k')\}
\left(A^{\bar{u}_i d_j \ell^+_k}_{24}O^{{cns}^*}_{Lk1r}
+B^{\bar{u}_i d_j \ell^+_k}_{24}O^{{cns}^*}_{Rk1r}\right)
\right.\nonumber\\
&&+m_{\ell_k}m_{\ntrl1}(k.k')\left(A^{\bar{u}_i d_j \ell^+_k}_{24}O^{{cns}^*}_{Rk1r}
+B^{\bar{u}_i d_j \ell^+_k}_{24}O^{{cns}^*}_{Lk1r}\right)\nonumber\\
&&-m_{d_j}m_{\ntrl1}(p.k)\left(C^{\bar{u}_i d_j \ell^+_k}_{24}O^{{cns}^*}_{Rk1r}
+D^{\bar{u}_i d_j \ell^+_k}_{24}O^{{cns}^*}_{Lk1r}\right)\nonumber\\
&&-m_{d_j}m_{\ell_k}(P.k)\left(C^{\bar{u}_i d_j \ell^+_k}_{24}O^{{cns}^*}_{Lk1r}
+D^{\bar{u}_i d_j \ell^+_k}_{24}O^{{cns}^*}_{Rk1r}\right)\nonumber\\
&&+m_{u_i}m_{\ntrl1}(p.k')\left(E^{\bar{u}_i d_j \ell^+_k}_{24}O^{{cns}^*}_{Rk1r}
+F^{\bar{u}_i d_j \ell^+_k}_{24}O^{{cns}^*}_{Lk1r}\right)\nonumber\\
&&+m_{u_i}m_{\ell_k}(P.k')\left(E^{\bar{u}_i d_j \ell^+_k}_{24}O^{{cns}^*}_{Lk1r}
+F^{\bar{u}_i d_j \ell^+_k}_{24}O^{{cns}^*}_{Rk1r}\right)\nonumber\\
&&-m_{u_i}m_{d_j}(P.p)\left(G^{\bar{u}_i d_j \ell^+_k}_{24}O^{{cns}^*}_{Lk1r}
+H^{\bar{u}_i d_j \ell^+_k}_{24}O^{{cns}^*}_{Rk1r}\right)\nonumber\\
&&-\left.
m_{u_i}m_{d_j}m_{\ell_k}m_{\ntrl1}\left(G^{\bar{u}_i d_j \ell^+_k}_{24}O^{{cns}^*}_{Rk1r}
+H^{\bar{u}_i d_j \ell^+_k}_{24}O^{{cns}^*}_{Lk1r}\right)\right],
\label{mat24-qql1}
\end{eqnarray}
where
\begin{eqnarray}
A^{\bar{u}_i d_j \ell^+_k}_{24} &=& O^{{cdu}^*}_{Rkjl} O^{uds}_{Rijr}
O^{nu\widetilde u}_{L1il},
~~B^{\bar{u}_i d_j \ell^+_k}_{24} = O^{{cdu}^*}_{Lkjl} O^{uds}_{Lijr}
O^{nu\widetilde u}_{R1il},\nonumber\\
C^{\bar{u}_i d_j \ell^+_k}_{24} &=& O^{{cdu}^*}_{Lkjl} O^{uds}_{Rijr}
O^{nu\widetilde u}_{L1il},
~~D^{\bar{u}_i d_j \ell^+_k}_{24} = O^{{cdu}^*}_{Rkjl} O^{uds}_{Lijr}
O^{nu\widetilde u}_{R1il},\nonumber\\
E^{\bar{u}_i d_j \ell^+_k}_{24} &=& O^{{cdu}^*}_{Lkjl} O^{uds}_{Lijr}
O^{nu\widetilde u}_{L1il},
~~F^{\bar{u}_i d_j \ell^+_k}_{24} = O^{{cdu}^*}_{Rkjl} O^{uds}_{Rijr}
O^{nu\widetilde u}_{R1il},\nonumber\\
G^{\bar{u}_i d_j \ell^+_k}_{24} &=& O^{{cdu}^*}_{Rkjl} O^{uds}_{Lijr}
O^{nu\widetilde u}_{L1il},
~~H^{\bar{u}_i d_j \ell^+_k}_{24} = O^{{cdu}^*}_{Lkjl} O^{uds}_{Rijr}
O^{nu\widetilde u}_{R1il}.
\label{mat24-qql1-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_3M_4 (\ntrl1\to \sum \bar{u}_i d_j \ell^+_k) = -\sum_{i,j,k}
\sum^2_{r,l=1}\frac{2\widetilde{g}^4}
{\left[((p+k)^2-m^2_{\widetilde d_r})((p+k')^2-m^2_{\widetilde u_l})
\right]} \nonumber\\
&&\left[\{(P.k')(p.k)-(P.p)(k.k')+(P.k)(p.k')\}
\left(A^{\bar{u}_i d_j \ell^+_k}_{34}O^{{dn\widetilde d}^*}_{Lj1r}
+B^{\bar{u}_i d_j \ell^+_k}_{34}O^{{dn\widetilde d}^*}_{Rj1r}\right)
\right.\nonumber\\
&&+m_{d_j}m_{\ntrl1}(p.k)\left(A^{\bar{u}_i d_j \ell^+_k}_{34}O^{{dn\widetilde d}^*}_{Rj1r}
+B^{\bar{u}_i d_j \ell^+_k}_{34}O^{{dn\widetilde d}^*}_{Lj1r}\right)\nonumber\\
&&-m_{\ell_k}m_{\ntrl1}(k.k')\left(C^{\bar{u}_i d_j \ell^+_k}_{34}O^{{dn\widetilde d}^*}_{Rj1r}
+D^{\bar{u}_i d_j \ell^+_k}_{34}O^{{dn\widetilde d}^*}_{Lj1r}\right)\nonumber\\
&&-m_{d_j}m_{\ell_k}(P.k)\left(C^{\bar{u}_i d_j \ell^+_k}_{34}O^{{dn\widetilde d}^*}_{Lj1r}
+D^{\bar{u}_i d_j \ell^+_k}_{34}O^{{dn\widetilde d}^*}_{Rj1r}\right)\nonumber\\
&&+m_{u_i}m_{\ntrl1}(p.k')\left(E^{\bar{u}_i d_j \ell^+_k}_{34}O^{{dn\widetilde d}^*}_{Rj1r}
+F^{\bar{u}_i d_j \ell^+_k}_{34}O^{{dn\widetilde d}^*}_{Lj1r}\right)\nonumber\\
&&+m_{u_i}m_{d_j}(P.p)\left(E^{\bar{u}_i d_j \ell^+_k}_{34}O^{{dn\widetilde d}^*}_{Lj1r}
+F^{\bar{u}_i d_j \ell^+_k}_{34}O^{{dn\widetilde d}^*}_{Rj1r}\right)\nonumber\\
&&-m_{u_i}m_{\ell_k}(P.k')\left(G^{\bar{u}_i d_j \ell^+_k}_{34}O^{{dn\widetilde d}^*}_{Lj1r}
+H^{\bar{u}_i d_j \ell^+_k}_{34}O^{{dn\widetilde d}^*}_{Rj1r}\right)\nonumber\\
&&-\left.
m_{u_i}m_{d_j}m_{\ell_k}m_{\ntrl1}\left(G^{\bar{u}_i d_j \ell^+_k}_{34}O^{{dn\widetilde d}^*}_{Rj1r}
+H^{\bar{u}_i d_j \ell^+_k}_{34}O^{{dn\widetilde d}^*}_{Lj1r}\right)\right],
\label{mat34-qql1}
\end{eqnarray}
where
\begin{eqnarray}
A^{\bar{u}_i d_j \ell^+_k}_{34} &=& O^{{cdu}^*}_{Rkjl} O^{ucd}_{Rikr}
O^{nu\widetilde u}_{L1il},
~~B^{\bar{u}_i d_j \ell^+_k}_{34} = O^{{cdu}^*}_{Lkjl} O^{ucd}_{Likr}
O^{nu\widetilde u}_{R1il},\nonumber\\
C^{\bar{u}_i d_j \ell^+_k}_{34} &=& O^{{cdu}^*}_{Lkjl} O^{ucd}_{Rikr}
O^{nu\widetilde u}_{L1il},
~~D^{\bar{u}_i d_j \ell^+_k}_{34} = O^{{cdu}^*}_{Rkjl} O^{ucd}_{Likr}
O^{nu\widetilde u}_{R1il},\nonumber\\
E^{\bar{u}_i d_j \ell^+_k}_{34} &=& O^{{cdu}^*}_{Lkjl} O^{ucd}_{Likr}
O^{nu\widetilde u}_{L1il},
~~F^{\bar{u}_i d_j \ell^+_k}_{34} = O^{{cdu}^*}_{Rkjl} O^{ucd}_{Rikr}
O^{nu\widetilde u}_{R1il},\nonumber\\
G^{\bar{u}_i d_j \ell^+_k}_{34} &=& O^{{cdu}^*}_{Rkjl} O^{ucd}_{Likr}
O^{nu\widetilde u}_{L1il},
~~H^{\bar{u}_i d_j \ell^+_k}_{34} = O^{{cdu}^*}_{Lkjl} O^{ucd}_{Rikr}
O^{nu\widetilde u}_{R1il}.
\label{mat34-qql1-det}
\end{eqnarray}
\section{Process $\ntrl1\to u_i \bar{d}_j \ell^-_k$}\label{LSP-qql2-decay}
We represent different lepton flavours $(e,\mu,\tau)$ by $k$. $u_i(d_j)$ stands
for different up-type and down-type quarks $(u,c(d,s,b))$, except the top.
We write down all possible $M^\dagger_iM_j$ for the four diagrams
shown in figure \ref{LSP-qql2}.
Required couplings are given in appendices \ref{appenD} and \ref{appenH}.
The four-momentum assignments are as follows
\begin{equation}
\ntrl1(P) \to \ell^-_k(p)+u_i(k)+\bar{d}_j(k').
\label{momentum-qql2}
\end{equation}
\begin{figure}[ht]
\centering
\vspace*{0.5cm}
\includegraphics[height=6.00cm]{FigsPDF/LSP-qql2.pdf}
\caption{Feynman diagrams for the possible three body decays of the lightest supersymmetric
particle into $u_i \bar{d}_j \ell^-_k$ final states.
$S^+_r$ are the charged scalar states of the $\mu\nu$SSM as
shown by eqn.(\ref{charged-scalar-mass-basis}). $\widetilde u(\widetilde d)$ are the up and down-type
squarks as shown by eqn.(\ref{squark-mass-basis}) corresponding
to $\bar{u}_i$ and $d_j$.}
\label{LSP-qql2}
\end{figure}
\begin{eqnarray}
&&\bullet~ M^\dagger_1M_1 (\ntrl1\to \sum u_i \bar{d}_j \ell^-_k) =
M^\dagger_1M_1 (\ntrl1\to \sum \bar{u}_i d_j \ell^+_k)
\label{mat11-qql2}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_2M_2 (\ntrl1\to \sum u_i \bar{d}_j \ell^-_k) =
M^\dagger_2M_2 (\ntrl1\to \sum \bar{u}_i d_j \ell^+_k)
\label{mat22-qql2}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_3M_3 (\ntrl1\to \sum u_i \bar{d}_j \ell^-_k) =
M^\dagger_3M_3 (\ntrl1\to \sum \bar{u}_i d_j \ell^+_k)
\label{mat33-qql2}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_4M_4 (\ntrl1\to \sum u_i \bar{d}_j \ell^-_k) =
M^\dagger_4M_4 (\ntrl1\to \sum \bar{u}_i d_j \ell^+_k)
\label{mat44-qql2}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_1M_2 (\ntrl1\to \sum u_i \bar{d}_j \ell^-_k) = \nonumber\\
&&\sum_{i,j,k}
\sum^8_{r=1}\frac{2\sqrt{2} g^2_2\widetilde{g}^2 V^{{CKM}^*}_{ij}}
{\left[((k+k')^2-m^2_{W}-im_W\Gamma_W)((k+k')^2-m^2_{S^\pm_r})
\right]} \nonumber\\
&&\left[m_{d_j}m_{\ntrl1}(p.k)O^{{uds}}_{Rijr}
A^{u_i \bar{d}_j \ell^-_k}_{12}
+m_{\ell_k} m_{d_j}(P.k) O^{{uds}}_{Rijr}
B^{u_i \bar{d}_j \ell^-_k}_{12}
\right.\nonumber\\
&&-\left.
m_{u_i}m_{\ntrl1}(p.k') O^{{uds}}_{Lijr}
A^{u_i \bar{d}_j \ell^-_k}_{12}
-m_{u_i}m_{\ell_k}(P.k') O^{{uds}}_{Lijr}
B^{u_i \bar{d}_j \ell^-_k}_{12}\right],\nonumber\\
\label{mat12-qql2}
\end{eqnarray}
where
\begin{eqnarray}
A^{u_i \bar{d}_j \ell^-_k}_{12} &=& \left(O^{{cnw}}_{Rk1} O^{{cns}^*}_{Rk1r}
+O^{{cnw}}_{Lk1} O^{{cns}^*}_{Lk1r}\right),
~~B^{u_i \bar{d}_j \ell^-_k}_{12} = \left(O^{{cnw}}_{Lk1} O^{{cns}^*}_{Rk1r}
+O^{{cnw}}_{Rk1} O^{{cns}^*}_{Lk1r}\right).\nonumber\\
\label{mat12-qql2-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_1M_3 (\ntrl1\to \sum u_i \bar{d}_j \ell^-_k) = \nonumber\\
&&\sum_{i,j,k}
\sum^2_{r=1}\frac{2\sqrt{2} g^2_2\widetilde{g}^2 V^{{CKM}^*}_{ij}}
{\left[((k+k')^2-m^2_{W}-im_W\Gamma_W)((p+k)^2-m^2_{\widetilde d_r})
\right]} \nonumber\\
&&\left[2(P.k)(p.k')A^{u_i \bar{d}_j \ell^-_k}_{13}
-m_{\ell_k}m_{\ntrl1} (k.k') B^{u_i \bar{d}_j \ell^-_k}_{13}
+m_{u_i}m_{\ntrl1}(p.k')C^{u_i \bar{d}_j \ell^-_k}_{13}
\right.\nonumber\\
&&-2m_{u_i}m_{\ell_k} (P.k') D^{u_i \bar{d}_j \ell^-_k}_{13}
+2m_{d_j}m_{\ntrl1}(p.k)E^{u_i \bar{d}_j \ell^-_k}_{13}
-m_{d_j}m_{\ell_k}(P.k)F^{u_i \bar{d}_j \ell^-_k}_{13}\nonumber\\
&&+\left.
m_{u_i}m_{d_j}(P.p)G^{u_i \bar{d}_j \ell^-_k}_{13}
-2m_{u_i}m_{d_j}m_{\ell_k}m_{\ntrl1}H^{u_i \bar{d}_j \ell^-_k}_{13}\right],
\label{mat13-qql2}
\end{eqnarray}
where
\begin{eqnarray}
A^{u_i \bar{d}_j \ell^-_k}_{13} &=& O^{{cnw}}_{Lk1} O^{{ucd}}_{Rikr}
O^{{dn\widetilde d}^*}_{Rj1r},
~~B^{u_i \bar{d}_j \ell^-_k}_{13} = O^{{cnw}}_{Rk1} O^{{ucd}}_{Rikr}
O^{{dn\widetilde d}^*}_{Rj1r},\nonumber\\
C^{u_i \bar{d}_j \ell^-_k}_{13} &=& O^{{cnw}}_{Rk1} O^{{ucd}}_{Likr}
O^{{dn\widetilde d}^*}_{Rj1r},
~~D^{u_i \bar{d}_j \ell^-_k}_{13} = O^{{cnw}}_{Lk1} O^{{ucd}}_{Likr}
O^{{dn\widetilde d}^*}_{Rj1r},\nonumber\\
E^{u_i \bar{d}_j \ell^-_k}_{13} &=& O^{{cnw}}_{Lk1} O^{{ucd}}_{Rikr}
O^{{dn\widetilde d}^*}_{Lj1r},
~~F^{u_i \bar{d}_j \ell^-_k}_{13} = O^{{cnw}}_{Rk1} O^{{ucd}}_{Rikr}
O^{{dn\widetilde d}^*}_{Lj1r},\nonumber\\
G^{u_i \bar{d}_j \ell^-_k}_{13} &=& O^{{cnw}}_{Rk1} O^{{ucd}}_{Likr}
O^{{dn\widetilde d}^*}_{Lj1r},
~~H^{u_i \bar{d}_j \ell^-_k}_{13} = O^{{cnw}}_{Lk1} O^{{ucd}}_{Likr}
O^{{dn\widetilde d}^*}_{Lj1r}.
\label{mat13-qql2-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_1M_4 (\ntrl1\to \sum u_i \bar{d}_j \ell^-_k) = \nonumber\\
&&-\sum_{i,j,k}
\sum^2_{r=1}\frac{2\sqrt{2} g^2_2\widetilde{g}^2 V^{{CKM}^*}_{ij}}
{\left[((k+k')^2-m^2_{W}-im_W\Gamma_W)((p+k')^2-m^2_{\widetilde u_r})
\right]} \nonumber\\
&&\left[2(P.k)(p.k')A^{u_i \bar{d}_j \ell^-_k}_{14}
-m_{\ell_k}m_{\ntrl1} (k.k') B^{u_i \bar{d}_j \ell^-_k}_{14}
+m_{d_j}m_{\ntrl1}(p.k)C^{u_i \bar{d}_j \ell^-_k}_{14}
\right.\nonumber\\
&&-2m_{d_j}m_{\ell_k} (P.k) D^{u_i \bar{d}_j \ell^-_k}_{14}
+2m_{u_i}m_{\ntrl1}(p.k')E^{u_i \bar{d}_j \ell^-_k}_{14}
-m_{u_i}m_{\ell_k}(P.k')F^{u_i \bar{d}_j \ell^-_k}_{14}\nonumber\\
&&+\left.
m_{u_i}m_{d_j}(P.p)G^{u_i \bar{d}_j \ell^-_k}_{14}
-2m_{u_i}m_{d_j}m_{\ell_k}m_{\ntrl1}H^{u_i \bar{d}_j \ell^-_k}_{14}\right],
\label{mat14-qql2}
\end{eqnarray}
where
\begin{eqnarray}
A^{u_i \bar{d}_j \ell^-_k}_{14} &=& O^{{cnw}}_{Rk1} O^{{cdu}}_{Lkjr}
O^{{nu\widetilde u}^*}_{L1ir},
~~B^{u_i \bar{d}_j \ell^-_k}_{14} = O^{{cnw}}_{Lk1} O^{{cdu}}_{Lkjr}
O^{{nu\widetilde u}^*}_{L1ir},\nonumber\\
C^{u_i \bar{d}_j \ell^-_k}_{14} &=& O^{{cnw}}_{Lk1} O^{{cdu}}_{Rkjr}
O^{{nu\widetilde u}^*}_{L1ir},
~~D^{u_i \bar{d}_j \ell^-_k}_{14} = O^{{cnw}}_{Rk1} O^{{cdu}}_{Rkjr}
O^{{nu\widetilde u}^*}_{L1ir},\nonumber\\
E^{u_i \bar{d}_j \ell^-_k}_{14} &=& O^{{cnw}}_{Rk1} O^{{cdu}}_{Lkjr}
O^{{nu\widetilde u}^*}_{R1ir},
~~F^{u_i \bar{d}_j \ell^-_k}_{14} = O^{{cnw}}_{Lk1} O^{{cdu}}_{Lkjr}
O^{{nu\widetilde u}^*}_{R1ir},\nonumber\\
G^{u_i \bar{d}_j \ell^-_k}_{14} &=& O^{{cnw}}_{Lk1} O^{{cdu}}_{Rkjr}
O^{{nu\widetilde u}^*}_{R1ir},
~~H^{u_i \bar{d}_j \ell^-_k}_{14} = O^{{cnw}}_{Rk1} O^{{cdu}}_{Rkjr}
O^{{nu\widetilde u}^*}_{R1ir}.
\label{mat14-qql2-det}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_2M_3 (\ntrl1\to \sum u_i \bar{d}_j \ell^-_k)
\underbrace{=}_{L\Longleftrightarrow R}
M^\dagger_2M_3 (\ntrl1\to \sum \bar{u}_i d_j \ell^+_k)^*.
\label{mat23-qql2}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_2M_4 (\ntrl1\to \sum u_i \bar{d}_j \ell^-_k)
\underbrace{=}_{L\Longleftrightarrow R}
M^\dagger_2M_4 (\ntrl1\to \sum \bar{u}_i d_j \ell^+_k)^*.
\label{mat24-qql2}
\end{eqnarray}
\begin{eqnarray}
&&\bullet~ M^\dagger_3M_4 (\ntrl1\to \sum u_i \bar{d}_j \ell^-_k)
\underbrace{=}_{L\Longleftrightarrow R}
M^\dagger_3M_4 (\ntrl1\to \sum \bar{u}_i d_j \ell^+_k)^*.
\label{mat34-qql2}
\end{eqnarray}
$V^{CKM}_{ij}$ are the entries of the CKM matrix and their values are given
in ref.\cite{AppNakamura-c2}.
\chapter{ \sffamily{{\bf
}}\label{appenH}
\section{Feynman rules}\label{Feynman rules-2}
The relevant Feynman rules required for the calculation of the
possible two-decays of the scalar and pseudoscalar states are shown in this
appendix. The factors $\eta_{i}$,$\eta_{j}$ and
$\epsilon_{i}$, $\epsilon_{j}$ are the
proper signs of neutralino and chargino masses \cite{AppGunion:1984yn}.
\subsection*{$\bigstar$~Chargino-chargino-neutral scalar}\label{cch}
The Lagrangian using four component spinor notation can be written as
\begin{equation}
\mathcal{L}^{cch}= - \frac{\widetilde g}{\sqrt{2}} \overline{{\widetilde \chi}_i}
(O^{cch}_{Lijk} P_L + O^{cch}_{Rijk} P_R) {{\widetilde \chi}_j} S^0_k,
\label{chargino-chargino-neutral scalar}
\end{equation}
where
\begin{eqnarray}
{\widetilde g}O^{cch}_{Lijk} = & & \epsilon_j \left[\mathbf R^{S^0}_{k1} \left(
Y^{mn}_e \mathbf U^*_{i,m+2} \mathbf V^*_{j,n+2} + g_2 \mathbf U^*_{i2} \mathbf V^*_{j1} \right)
\right. \nonumber\\
& &+ g_2 \mathbf R^{S^0}_{k2} \mathbf U^*_{i1} \mathbf V^*_{j2} \nonumber\\
& &+ \mathbf R^{S^0}_{k,m+2} \left(\lambda^m \mathbf U^*_{i2} \mathbf V^*_{j2}
-Y^{mn}_\nu \mathbf U^*_{i,n+2} \mathbf V^*_{j2} \right) \nonumber\\
& &+ \left.
\mathbf R^{S^0}_{k,m+5} \left(g_2 \mathbf U^*_{i,m+2} \mathbf V^*_{j1}
-Y^{mn}_e \mathbf U^*_{i2} \mathbf V^*_{j,n+2} \right) \right],
\label{cch-couplings}
\end{eqnarray}
and
\begin{equation}
\widetilde g O^{cch}_{Rijk} =(\widetilde g O^{cch}_{Ljik})^*.
\label{cch-R-couplings}
\end{equation}
\subsection*{$\bigstar$~Chargino-chargino-neutral pseudoscalar}\label{cca}
The Lagrangian using four component spinor notation can be written as
\begin{equation}
\mathcal{L}^{cch}= - i\frac{\widetilde g}{\sqrt{2}} \overline{{\widetilde \chi}_i}
(O^{cca}_{Lijk} P_L + O^{cca}_{Rijk} P_R) {{\widetilde \chi}_j} P^0_k,
\label{chargino-chargino-neutral pseudoscalar}
\end{equation}
where
\begin{eqnarray}
{\widetilde g}O^{cca}_{Lijk} = & & \epsilon_j \left[\mathbf R^{P^0}_{k1} \left(
Y^{mn}_e \mathbf U^*_{i,m+2} \mathbf V^*_{j,n+2} - g_2 \mathbf U^*_{i2} \mathbf V^*_{j1} \right)
\right. \nonumber\\
& &- g_2 \mathbf R^{P^0}_{k2} \mathbf U^*_{i1} \mathbf V^*_{j2} \nonumber\\
& &+ \mathbf R^{P^0}_{k,m+2} \left(\lambda^m \mathbf U^*_{i2} \mathbf V^*_{j2}
-Y^{mn}_\nu \mathbf U^*_{i,n+2} \mathbf V^*_{j2} \right) \nonumber\\
& &- \left.
\mathbf R^{P^0}_{k,m+5} \left(g_2 \mathbf U^*_{i,m+2} \mathbf V^*_{j1}
+ Y^{mn}_e \mathbf U^*_{i2} \mathbf V^*_{j,n+2} \right) \right],
\label{cca-couplings}
\end{eqnarray}
and
\begin{equation}
\widetilde g O^{cca}_{Rijk} =-(\widetilde g O^{cca}_{Ljik})^*.
\label{cca-R-couplings}
\end{equation}
\subsection*{$\bigstar$~Down-quark-down-quark-neutral scalar}\label{ddh}
The Lagrangian using four component spinor notation can be written as
\begin{equation}
\mathcal{L}^{ddh}= - {\widetilde g} \overline{d_i}
(O^{ddh}_{Lijk} P_L + O^{ddh}_{Rijk} P_R) {d_j} S^0_k ,
\label{down-down-neutral scalar}
\end{equation}
where
\begin{eqnarray}
{\widetilde g}O^{ddh}_{Lijk} = & & \frac{1}{\sqrt{2}}
Y^{mn}_d \mathbf R^{S^0}_{k1} \mathbf R^{d}_{L_{im}} \mathbf R^{d}_{L_{jn}},\nonumber\\
{\widetilde g}O^{ddh}_{Rijk} = & & ({\widetilde g} O^{ddh}_{Ljik})^*.
\label{ddh-couplings}
\end{eqnarray}
\subsection*{$\bigstar$~Down-quark-down-quark-neutral pseudoscalar}\label{dda}
The Lagrangian using four component spinor notation can be written as
\begin{equation}
\mathcal{L}^{dda}= - i{\widetilde g} \overline{d_i}
(O^{dda}_{Lijk} P_L + O^{dda}_{Rijk} P_R) {d_j} P^0_k ,
\label{down-down-neutral pseudoscalar}
\end{equation}
where
\begin{eqnarray}
{\widetilde g}O^{dda}_{Lijk} = & & \frac{1}{\sqrt{2}}
Y^{mn}_d \mathbf R^{P^0}_{k1} \mathbf R^{d}_{L_{im}} \mathbf R^{d}_{L_{jn}},\nonumber\\
{\widetilde g}O^{dda}_{Rijk} = & & -({\widetilde g} O^{dda}_{Ljik})^*.
\label{dda-couplings}
\end{eqnarray}
\subsection*{$\bigstar$~Up-quark-up-quark-neutral scalar}\label{uuh}
The Lagrangian using four component spinor notation can be written as
\begin{equation}
\mathcal{L}^{uuh}= - {\widetilde g} \overline{u_i}
(O^{uuh}_{Lijk} P_L + O^{uuh}_{Rijk} P_R) {u_j} S^0_k ,
\label{up-up-neutral scalar}
\end{equation}
where
\begin{eqnarray}
{\widetilde g}O^{uuh}_{Lijk} = & & \frac{1}{\sqrt{2}}
Y^{mn}_u \mathbf R^{S^0}_{k2} \mathbf R^{u}_{L_{im}} \mathbf R^{u}_{L_{jn}},\nonumber\\
{\widetilde g}O^{uuh}_{Rijk} = & & ({\widetilde g} O^{uuh}_{Ljik})^*.
\label{uuh-couplings}
\end{eqnarray}
\subsection*{$\bigstar$~Up-quark-up-quark-neutral pseudoscalar}\label{uua}
The Lagrangian using four component spinor notation can be written as
\begin{equation}
\mathcal{L}^{uua}= - i{\widetilde g} \overline{u_i}
(O^{uua}_{Lijk} P_L + O^{uua}_{Rijk} P_R) {u_j} P^0_k ,
\label{up-up-neutral pseudoscalar}
\end{equation}
where
\begin{eqnarray}
{\widetilde g}O^{uua}_{Lijk} = & & \frac{1}{\sqrt{2}}
Y^{mn}_u \mathbf R^{P^0}_{k2} \mathbf R^{u}_{L_{im}} \mathbf R^{u}_{L_{jn}},\nonumber\\
{\widetilde g}O^{uua}_{Rijk} = & & -({\widetilde g} O^{uua}_{Ljik})^*.
\label{uua-couplings}
\end{eqnarray}
\subsection*{$\bigstar$~Quark-squark-chargino}\label{qsqc}
The Lagrangian using four component spinor notation can be written as
\begin{equation}
\mathcal{L}^{q\widetilde q c}= -\widetilde{g} \overline{\widetilde {\chi}^c_i}
\left(O^{cdu}_{Lijk} P_L + O^{cdu}_{Rijk} P_R\right) d_j \widetilde u^*_k
-\widetilde{g} \overline{u_i} \left(O^{ucd}_{Lijk} P_L + O^{ucd}_{Rijk} P_R\right)
\widetilde{\chi}_j \widetilde d_k + h.c,
\label{q-sq-chargino}
\end{equation}
where
\begin{eqnarray}
\widetilde{g} O^{cdu}_{Lijk} = & & -Y^{mn}_u \mathbf V^*_{i2} \mathbf R^{d}_{L_{jm}} \mathbf R^{\widetilde u}_{k,n+3}
+g_2 \mathbf V^*_{i1} \mathbf R^{d}_{L_{jm}} \mathbf R^{\widetilde u}_{km},\nonumber\\
\widetilde{g} O^{cdu}_{Rijk} = & & -Y^{{mn}^*}_d \mathbf U_{i2} {\mathbf R^{d}}^*_{R_{jn}} \mathbf R^{\widetilde u}_{km},\nonumber\\
\widetilde{g} O^{ucd}_{Lijk} = & & -Y^{mn}_u \mathbf V^*_{j2} \mathbf R^{u}_{R_{in}} {\mathbf R^{\widetilde d}}^*_{km},\nonumber\\
\widetilde{g} O^{ucd}_{Rijk} = & & -Y^{{mn}^*}_d \mathbf U_{j2} {\mathbf R^{u}}^*_{L_{im}} {\mathbf R^{\widetilde d}}^*_{k,n+3}
+g_2 \mathbf U_{j1} {\mathbf R^{u}}^*_{L_{im}} {\mathbf R^{\widetilde d}}^*_{km}.
\label{qsqc-couplings}
\end{eqnarray}
The charge conjugated chargino spinor $\widetilde{\chi}^c$ is defined
by eqn.(\ref{neutralino-chargino}).
\subsection*{$\bigstar$~Quark-quark-charged scalar}\label{qqcs}
The Lagrangian is written as
\begin{equation}
\mathcal{L}^{qqs}= -\widetilde{g} \overline{u_i}
\left(O^{uds}_{Lijk} P_L + O^{uds}_{Rijk} P_R\right) d_j S^+_k
+ h.c,
\label{q-q-charged-scalar}
\end{equation}
where
\begin{eqnarray}
\widetilde{g} O^{uds}_{Lijk} = & & -Y^{mn}_u \mathbf R^{u}_{R_{in}} \mathbf R^{d}_{L_{jm}} \mathbf R^{S^{\pm}}_{k2},\nonumber\\
\widetilde{g} O^{uds}_{Rijk} = & & -Y^{{mn}^*}_d {\mathbf R^{u}}^*_{L_{im}} {\mathbf R^{d}}^*_{R_{jn}} \mathbf R^{S^{\pm}}_{k1}.
\label{qqs-couplings}
\end{eqnarray}
\section{Squared matrix elements for $h^0\to \widetilde{\chi}^0_i \widetilde{\chi}^0_j, b \bar{b}
$}\label{h-mat-ele}
\begin{eqnarray}
\bullet~ |M|^2 (h^0\to \widetilde{\chi}^0_i \widetilde{\chi}^0_j) &&=
2\widetilde{g}^2(m^2_{h^0}-(m^2_{\widetilde{\chi}^0_i}+m^2_{\widetilde{\chi}^0_j}))\left(
O^{{nnh}^*}_{Lij4}O^{nnh}_{Lij4} + O^{{nnh}^*}_{Rij4}O^{nnh}_{Rij4}\right)\nonumber\\
&&-4 \widetilde{g}^4
m_{\widetilde{\chi}^0_i} m_{\widetilde{\chi}^0_j}\left(
O^{{nnh}^*}_{Rij4}O^{nnh}_{Lij4} + O^{{nnh}^*}_{Lij4}O^{nnh}_{Rij4}\right),
\label{math-n1n1}
\end{eqnarray}
where we have used the favour of an extra $2$ factor for $\widetilde{\chi}^0_i-\widetilde{\chi}^0_j-h^0$
vertex \cite{AppHaber:1984rc} (also see appendix \ref{appenD}).
\begin{eqnarray}
\bullet~ |M|^2 (h^0\to b \bar{b}) &&=
3\widetilde{g}^2(m^2_{h^0}-2m^2_b)\left(
O^{{ddh}^*}_{L334}O^{ddh}_{L334} + O^{{ddh}^*}_{R334}O^{ddh}_{R334}\right)\nonumber\\
&&-6 \widetilde{g}^4
m^2_{b} \left(
O^{{ddh}^*}_{R334}O^{ddh}_{L334} + O^{{ddh}^*}_{L334}O^{ddh}_{R334}\right),
\label{math-bb}
\end{eqnarray}
where we have used relations from appendix \ref{appenD}, section \ref{Feynman rules-2}
and put $3$ for the colour factor.
\chapter{ \sffamily{{\bf
}}\label{appenD}
\section{Feynman rules}\label{Feynman rules-1}
The relevant Feynman rules required for the calculation of the
one-loop contributions to the neutralino masses (see figure \ref{one-loop-diagrams},
section \ref{loop-neut-calc}) are shown here \cite{AppGhosh:2010zi}.
Some of these Feynman rules have been derived also in ref.\cite{AppGhosh:2008yh}
for calculating two body decays of the lightest neutralino, $\ntrl1$.
Feynman rules for MSSM are given in references \cite{AppHaber:1984rc,AppRosiek:1989rs,AppRosiek:1995kg}
and in references \cite{AppGunion:1984yn,AppGunion:1986nh,AppFranke:1995tf,AppFranke:1995tc} for MSSM
with singlet superfields. Feynman rules for $R_p$-violating MSSM are studied in
references \cite{AppHempfling:1995wj,AppHirsch:2000ef,AppDiaz:2003as}.
The required Feynman rules are (using relations of
form {\it{neutralino-fermion-scalar/gauge boson}} and they are listed below.
\subsection*{$\bigstar$~Neutralino-neutralino-neutral scalar}\label{nnh}
The Lagrangian using four component spinor notation can be written as
\begin{equation}
\mathcal{L}^{nnh}= - \frac{\widetilde g}{\sqrt{2}} \overline{{\widetilde \chi}^0_i}
(O^{nnh}_{Lijk} P_L + O^{nnh}_{Rijk} P_R) {{\widetilde \chi}^0_j} S^0_k,
\label{Neutralino-neutralino-neutral-scalar}
\end{equation}
where ${\widetilde g} O^{nnh}_{Lijk}$ is given by
\begin{eqnarray}
&& \eta_{j} \frac{1}{2} \left[\mathbf R^{S^0}_{k1}
\left( \frac{g_2}{\sqrt{2}} {\mathbf N}^*_{i2} \mathbf N^*_{j3} - \frac{g_1}{\sqrt{2}} \mathbf N^*_{i1} \mathbf N^*_{j3}
- \lambda^m \mathbf N^*_{i4} \mathbf N^*_{j,m+4}\right)
\right. \nonumber \\
&&- \mathbf R^{S^0}_{k2} \left(\frac{g_2}{\sqrt{2}} {\mathbf N}^*_{i2} \mathbf N^*_{j4}
-\frac{g_1}{\sqrt{2}} \mathbf N^*_{i1} \mathbf N^*_{j4} + \lambda^m \mathbf N^*_{i3} \mathbf N^*_{j,m+4}
- Y^{mn}_{\nu} \mathbf N^*_{i,n+4} \mathbf N^*_{j,m+7}\right)\nonumber \\
&&+ \mathbf R^{S^0}_{k,m+2} \left( Y^{mn}_{\nu} \mathbf N^*_{i4} \mathbf N^*_{j,n+7}
- \lambda^m \mathbf N^*_{i3} \mathbf N^*_{j4} + \kappa^{mnp} \mathbf N^*_{i,n+4} \mathbf N^*_{j,p+4}\right)\nonumber \\
&&+ \left.
\mathbf R^{S^0}_{k,m+5} \left( \frac{g_2}{\sqrt{2}} {\mathbf N}^*_{i2} \mathbf N^*_{j,m+7}
- \frac{g_1}{\sqrt{2}} \mathbf N^*_{i1} \mathbf N^*_{j,m+7} + Y^{mn}_{\nu} \mathbf N^*_{i4} \mathbf N^*_{j,n+4}\right)\right] \nonumber\\
&&+
(i\leftrightarrow j),\nonumber\\
\label{nnh-L-couplings}
\end{eqnarray}
and\footnote{A typos \cite{AppGhosh:2010zi} in the expression of $O^{nnh}_{Rijk}$ has been corrected.}
\begin{equation}
O^{nnh}_{Rijk} =(O^{nnh}_{Ljik})^*.
\label{nnh-R-couplings}
\end{equation}
\subsection*{$\bigstar$~Neutralino-neutralino-neutral pseudoscalar}\label{nna}
The Lagrangian using four component spinor notation can be written as
\begin{equation}
\mathcal{L}^{nna}= - i\frac{\widetilde g}{\sqrt{2}} \overline{{\widetilde \chi}^0_i}
(O^{nna}_{Lijk} P_L + O^{nna}_{Rijk} P_R) {{\widetilde \chi}^0_j} P^0_k,
\label{Neutralino-neutralino-neutral-pseudoscalar}
\end{equation}
where ${\widetilde g} O^{nna}_{Lijk}$ is given as
\begin{eqnarray}
&&\eta_{j} \frac{1}{2} \left[\mathbf R^{P^0}_{k1}
\left( -\frac{g_2}{\sqrt{2}} {\mathbf N}^*_{i2} \mathbf N^*_{j3} + \frac{g_1}{\sqrt{2}} \mathbf N^*_{i1} \mathbf N^*_{j3}
- \lambda^m \mathbf N^*_{i4} \mathbf N^*_{j,m+4}\right)
\right. \nonumber \\
&&+ \mathbf R^{P^0}_{k2} \left( \frac{g_2}{\sqrt{2}} {\mathbf N}^*_{i2} \mathbf N^*_{j4}
- \frac{g_1}{\sqrt{2}} \mathbf N^*_{i1} \mathbf N^*_{j4} - \lambda^m \mathbf N^*_{i3} \mathbf N^*_{j,m+4}
+ Y^{mn}_{\nu} \mathbf N^*_{i,n+4} \mathbf N^*_{j,m+7}\right)\nonumber \\
&&+ \mathbf R^{P^0}_{k,m+2} \left( Y^{mn}_{\nu} \mathbf N^*_{i4} \mathbf N^*_{j,n+7}
- \lambda^m \mathbf N^*_{i3} \mathbf N^*_{j4} + \kappa^{mnp} \mathbf N^*_{i,n+4} \mathbf N^*_{j,p+4}\right)\nonumber \\
&&+ \left.
\mathbf R^{P^0}_{k,m+5} \left( -\frac{g_2}{\sqrt{2}} {\mathbf N}^*_{i2} \mathbf N^*_{j,m+7}
+ \frac{g_1}{\sqrt{2}} \mathbf N^*_{i1} \mathbf N^*_{j,m+7} + Y^{mn}_{\nu} \mathbf N^*_{i4} \mathbf N^*_{j,n+4}\right)\right] \nonumber \\
&&+(i\leftrightarrow j),\nonumber\\
\label{nna-L-couplings}
\end{eqnarray}
and\footnote{A typos \cite{AppGhosh:2010zi} in the expression of $O^{nna}_{Rijk}$ has been corrected.}
\begin{equation}
O^{nna}_{Rijk} =-(O^{nna}_{Ljik})^*.
\label{nna-R-couplings}
\end{equation}
\subsection*{$\bigstar$~Neutralino-neutralino-$Z^0$}\label{nnz}
The Lagrangian using four component spinor notation can be written as
\begin{equation}
\mathcal{L}^{nnz}= - \frac{g_2}{2} \overline{{\widetilde \chi}^0_i} \gamma^{\mu}
(O^{nnz}_{Lij} P_L + O^{nnz}_{Rij} P_R) {{\widetilde \chi}^0_j} Z^0_{\mu},
\label{Neutralino-neutralino-neutral-Z-boson}
\end{equation}
where
\begin{eqnarray}
O^{nnz}_{Lij} = & & \eta_{i} \eta_{j} \frac{1}{2 \cos_{\theta_W}} \left(\mathbf N_{i3} \mathbf N^*_{j3}
- \mathbf N_{i4} \mathbf N^*_{j4} + \mathbf N_{i,m+7} \mathbf N^*_{j,m+7}\right), \nonumber \\
O^{nnz}_{Rij} = & & \frac{1}{2 \cos_{\theta_W}} \left(-\mathbf N^*_{i3} \mathbf N_{j3} + \mathbf N^*_{i4} \mathbf N_{j4}
- \mathbf N^*_{i,m+7} \mathbf N_{j,m+7}\right).
\label{nnz-L-R-couplings}
\end{eqnarray}
\subsection*{$\bigstar$~Neutralino-chargino-charged scalar}\label{ncs}
The Lagrangian using four component spinor notation can be written as
\begin{equation}
\mathcal{L}^{ncs}= - {\widetilde g} \overline{{\widetilde \chi}_i} (O^{cns}_{Lijk} P_L
+ O^{cns}_{Rijk} P_R) {{\widetilde \chi}^0_j} S^+_k
- {\widetilde g} \overline{{\widetilde \chi}^0_i} (O^{ncs}_{Lijk} P_L
+ O^{ncs}_{Rijk} P_R) {{\widetilde \chi}_j} S^-_k,
\label{Neutralino-chargino-chargedscalar}
\end{equation}
where
\begin{eqnarray}
{\widetilde g} O^{cns}_{Lijk} = & &\eta_{j} \left[\mathbf R^{S^{\pm}}_{k1}
\left( -\frac{g_2}{\sqrt{2}} {\mathbf U}^*_{i2} \mathbf N^*_{j2} - \frac{g_1}{\sqrt{2}} \mathbf U^*_{i2} \mathbf N^*_{j1}
+ {g_2} \mathbf U^*_{i1} \mathbf N^*_{j3}\right)
\right. \nonumber \\
& &+ \mathbf R^{S^{\pm}}_{k2} \left( \lambda^m \mathbf U^*_{i2} \mathbf N^*_{j,m+4} - Y^{mn}_{\nu} \mathbf U^*_{i,m+2} \mathbf N^*_{j,n+4}\right) \nonumber \\
& &+ \mathbf R^{S^{\pm}}_{k,m+2} \left(Y^{mn}_e \mathbf U^*_{i,n+2} \mathbf N^*_{j3} - Y^{mn}_e \mathbf U^*_{i2} \mathbf N^*_{j,n+7} \right)\nonumber \\
& &+ \left.
\mathbf R^{S^{\pm}}_{k,m+5} \left( {g_2} {\mathbf U}^*_{i1} \mathbf N^*_{j,m+7} - \frac{g_2}{\sqrt{2}} \mathbf U^*_{i,m+2} \mathbf N^*_{j2}
- \frac{g_1}{\sqrt{2}} \mathbf U^*_{i,m+2} \mathbf N^*_{j1}\right)\right],\nonumber \\
\nonumber\\
\label{cns-L-couplings}
{\widetilde g} O^{cns}_{Rijk} = & &\epsilon_{i} \left[\mathbf R^{S^{\pm}}_{k1}
\left( \lambda^m \mathbf V_{i2} \mathbf N_{j,m+4} -Y^{mn}_e \mathbf V_{i,n+2} \mathbf N_{j,m+7}\right)
\right. \nonumber \\
& &+ \mathbf R^{S^{\pm}}_{k2} \left( \frac{g_2}{\sqrt{2}} {\mathbf V}_{i2} \mathbf N_{j2}
+ \frac{g_1}{\sqrt{2}} \mathbf V_{i2} \mathbf N_{j1} + {g_2} \mathbf V_{i1} \mathbf N_{j4}\right) \nonumber \\
& &+ \sqrt{2} {g_1} \mathbf R^{S^{\pm}}_{k,m+2} \mathbf V_{i,m+2} \mathbf N_{j1}\nonumber \\
& &+ \left.
\mathbf R^{S^{\pm}}_{k,m+5} \left(Y^{mn}_e \mathbf V_{i,n+2} \mathbf N_{j3}
- Y^{mn}_{\nu} \mathbf V_{i2} \mathbf N_{j,n+4} \right)\right],\nonumber \\
\label{cns-R-couplings}
\end{eqnarray}
and
\begin{equation}
O^{ncs}_{Lijk} =(O^{cns}_{Rjik})^*, \quad O^{ncs}_{Rijk} =(O^{cns}_{Ljik})^*.
\label{ncs-L-R-couplings}
\end{equation}
\subsection*{$\bigstar$~Neutralino-chargino-$W$}\label{ncw}
The Lagrangian using four component spinor notation can be written as
\begin{equation}
\mathcal{L}^{ncw}= - {g_2} \overline{{\widetilde \chi}_i} \gamma^{\mu}
(O^{cnw}_{Lij} P_L + O^{cnw}_{Rij} P_R) {{\widetilde \chi}^0_j}
W^+_{\mu} - {g_2} \overline{{\widetilde \chi}^0_i} \gamma^{\mu} (O^{ncw}_{Lij} P_L
+ O^{ncw}_{Rij} P_R) {{\widetilde \chi}_j} W^-_{\mu}.
\label{Neutralino-chargino-W-boson}
\end{equation}
where
\begin{eqnarray}{\label{ncw-L-R-couplings}}
O^{cnw}_{Lij} = & &- \epsilon_{i} \eta_{j} \left(\mathbf V_{i1} \mathbf N^*_{j2}
- \frac{1}{\sqrt{2}} \mathbf V_{i2} \mathbf N^*_{j4}\right), \nonumber \\
O^{cnw}_{Rij} = & &- \mathbf U^*_{i1} \mathbf N_{j2} - \frac{1}{\sqrt{2}} \mathbf U^*_{i2} \mathbf N_{j3}
- \frac{1}{\sqrt{2}} \mathbf U^*_{i,n+2} \mathbf N_{j,n+7},
\end{eqnarray}
and
\begin{equation}
O^{ncw}_{Lij} =(O^{cnw}_{Lji})^*, \quad O^{ncw}_{Rij} =(O^{cnw}_{Rji})^*.
\label{cnw-L-R-couplings}
\end{equation}
The factors $\eta_{j}$ and $\epsilon_{i}$ are the proper signs of neutralino
and chargino masses \cite{AppGunion:1984yn}. They have values $\pm{1}$.
\subsection*{$\bigstar$~Neutralino-quark-squark}\label{nqsq}
The Lagrangian using four component spinor notation can be written as
\begin{equation}
\mathcal{L}^{nq\widetilde q}= - {\widetilde g} \overline{q_i} (O^{qn\widetilde q}_{Lijk} P_L
+ O^{qn\widetilde q}_{Rijk} P_R) {{\widetilde \chi}^0_j} \widetilde q_k
- {\widetilde g} \overline{{\widetilde \chi}^0_i} (O^{nq\widetilde q}_{Lijk} P_L
+ O^{nq\widetilde q}_{Rijk} P_R) q_j \widetilde q^*_k.
\label{Neutralino-quark-squark}
\end{equation}
where
\begin{equation}
O^{qn\widetilde q}_{Lijk} =(O^{nq\widetilde q}_{Rjik})^*, \quad O^{qn\widetilde q}_{Rijk} =(O^{nq\widetilde q}_{Ljik})^*,
\label{q-n-squark-L-R-couplings}
\end{equation}
and
\begin{eqnarray}
{\widetilde g} O^{nu\widetilde u}_{Lijk} = & & \mathbf R^{\widetilde u}_{km} \left(\frac{g_2}{\sqrt{2}} \mathbf N^*_{i2} \mathbf R^{u}_{L_{jm}}
+ \frac{g_1}{3\sqrt{2}} \mathbf N^*_{i1} \mathbf R^{u}_{L_{jm}}\right)
+ Y^{nm}_u \mathbf R^{\widetilde u}_{k,m+3} \mathbf N^*_{i4} \mathbf R^{u}_{L_{jn}} , \nonumber \\
{\widetilde g} O^{nu\widetilde u}_{Rijk} = & & Y^{{mn}^*}_u \mathbf R^{\widetilde u}_{km} \mathbf N_{i4} R^{u^*}_{R_{jn}}
- \frac{4{g_1}}{3 \sqrt{2}} \mathbf R^{\widetilde u}_{k,m+3} \mathbf N_{i1} R^{u^*}_{R_{jm}},\nonumber \\
{\widetilde g} O^{nd\widetilde d}_{Lijk} = & & \mathbf R^{\widetilde d}_{km} \left(-\frac{g_2}{\sqrt{2}} \mathbf N^*_{i2} \mathbf R^{d}_{L_{jm}}
+ \frac{g_1}{3\sqrt{2}} \mathbf N^*_{i1} \mathbf R^{d}_{L_{jm}}\right)
+ Y^{nm}_d \mathbf R^{\widetilde d}_{k,m+3} \mathbf N^*_{i3} \mathbf R^{d}_{L_{jn}} , \nonumber \\
{\widetilde g} O^{nd\widetilde d}_{Rijk} = & & Y^{{mn}^*}_d \mathbf R^{\widetilde d}_{km} \mathbf N_{i3} R^{d^*}_{R_{jn}}
+ \frac{2{g_1}}{3 \sqrt{2}} \mathbf R^{\widetilde d}_{k,m+3} \mathbf N_{i1} R^{d^*}_{R_{jm}}.
\label{n-q-squark-L-R-couplings}
\end{eqnarray}
Note that for couplings of the type $\widetilde \chi^0 \widetilde \chi^0 \mathbf B$,
with $\mathbf B$ as either a scalar (CP-even, CP-odd) or a vector boson ($Z$)
the associated Feynman rules must be multiplied by a $2$ factor in calculations. This feature
is a special property of a Majorana fermion since a Majorana field, being self conjugate
(eqn.(\ref{neut-mass-a2})) contains both creation and annihilation operators \cite{AppHaber:1984rc}.
We have extensively used a set of relations between weak or flavour eigenbasis and mass eigenbasis,
both for the scalars and fermions to derive all these Feynman rules.
For the scalars (CP-even scalar, CP-odd scalar, charged
scalar and scalar quarks) these relations are given by eqns.(\ref{scalar-mass-basis}),
(\ref{pseudoscalar-mass-basis}), (\ref{charged-scalar-mass-basis}) and (\ref{squark-mass-basis}).
Similar relations for the four component neutralinos and charginos (eqn.(\ref{neutralino-chargino}))
are given below.
\vspace{0.1cm}
\noindent
$\blacksquare$~{\it{Neutralinos}}
\begin{eqnarray}\label{neutralino_mass-basis_weak-basis_relations}
& &{P_L}{\widetilde{B}}^0 = P_L N^*_{i1} \widetilde{\chi}^0_i,
\quad\ {P_L}{\widetilde{W}}^0_3 = P_L N^*_{i2} \widetilde{\chi}^0_i,
\quad\ {P_L}{\widetilde{H}}_j = P_L N^*_{i,j+2} \widetilde{\chi}^0_i,\nonumber \\
& &{P_L}{\nu}_k = P_L N^*_{i,k+7} \widetilde{\chi}^0_i,
\quad\ {P_L}{\nu}^{c}_k = P_L N^*_{i,k+4} \widetilde{\chi}^0_i,\nonumber \\
& &{P_R}{\widetilde{B}}^0 = P_R N_{i1} \widetilde{\chi}^0_i,
\quad\ {P_R}{\widetilde{W}}^0_3 = P_R N_{i2} \widetilde{\chi}^0_i,
\quad\ {P_R}{\widetilde{H}}_j = P_R N_{i,j+2},\quad \nonumber \\
& &{P_R}{\nu}_k = P_R N_{i,k+7} \widetilde{\chi}^0_i,
\quad\ {P_R}{\nu}^{c}_k = P_R N_{i,k+4} \widetilde{\chi}^0_i, \nonumber \\
& & \text{where} \quad j = 1,2 \quad k = 1,2,3, \quad i = 1,2,...,10
\end{eqnarray}
and
\begin{equation}\label{P-L-P-R}
P_{L}=\left(\frac{1-{\gamma^5}}{2}\right), \quad
P_{R}=\left(\frac{1+{\gamma^5}}{2}\right).
\end{equation}
\vspace{0.1cm}
\noindent
$\blacksquare$~{\it{Charginos}}
\begin{eqnarray}\label{chargino_mass-basis_weak-basis_relations}
& &{P_L}{\widetilde{W}} = P_L V^*_{i1} \widetilde{\chi}_i,
\quad\ {P_L}{\widetilde{H}} = P_L V^*_{i2} \widetilde{\chi}_i,
\quad\ {P_L}{l_k} = P_L U^*_{i,k+2} \widetilde{\chi}^c_i,\nonumber \\
& &{P_R}{\widetilde{W}} = P_R U_{i1} \widetilde{\chi}_i,
\quad\ {P_R}{\widetilde{H}} = P_R U_{i2} \widetilde{\chi}_i,
\quad\ {P_R}{l_k} = P_R V_{i,k+2} \widetilde{\chi}^c_i,\nonumber \\
& &{P_L}{\widetilde{W}^c} = P_L U^*_{i1} \widetilde{\chi}^c_i,
\quad\ {P_L}{\widetilde{H}^c} = P_L U^*_{i2} \widetilde{\chi}^c_i,
\quad\ {P_L}{l^c_k} = P_L V^*_{i,k+2} \widetilde{\chi}_i,\nonumber \\
& &{P_R}{\widetilde{W}^c} = P_R V_{i1} \widetilde{\chi}^c_i,
\quad\ {P_R}{\widetilde{H}^c} = P_R V_{i2} \widetilde{\chi}^c_i,
\quad\ {P_R}{l^c_k} = P_R U_{i,k+2} \widetilde{\chi}_i,\nonumber \\
\end{eqnarray}
where $k = 1,2,3$, and $i$ varies from $1$ to $5$.
\chapter*{\sffamily{{\bf M}otivation and plan of the thesis}}\label{motiv}
The standard model of the particle physics is extremely successful in
explaining the elementary particle interactions, as has been firmly
established by a host of experiments. However, unfortunately
there exist certain issues where the standard model is an apparent
failure, like unnatural fine tuning associated with the mass of the
hitherto unseen Higgs boson or explaining massive neutrinos, as
confirmed by neutrino oscillation experiments. A collective approach to
address these shortcomings requires extension beyond the standard model
framework. The weak scale supersymmetry has been a very favourite
choice to explain physics beyond the standard model where by virtue
of the construction, the mass of Higgs boson is apparently free
from fine-tuning problem. On the other hand, violation
of a discrete symmetry called $R$-parity is an intrinsically
supersymmetric way of accommodating massive neutrinos.
But, in spite of all these successes supersymmetric theories
are also not free from drawbacks and that results in a wide variety
of models. Besides, not a single supersymmetric particle has
been experimentally discovered yet. Nevertheless, possibility
of discovering weak scale supersymmetric particles
as well as Higgs boson are highly envisaged with the initiation of
the large hadron collider experiment at CERN.
In this thesis we plan to study a few phenomenological aspects
of a particular variant of $R$-parity violating supersymmetric model, popularly known as
the $\mu\nu$SSM. This model offers a solution for the $\mu$-problem
of the minimal supersymmetric standard model and simultaneously accommodate massive
neutrinos with the use of a common set of right-handed neutrino superfields.
Initially, we aimed to accommodate massive neutrinos in this model
consistent with the three flavour global neutrino data with tree level
analysis for different schemes of light neutrino masses. Besides,
as the lightest supersymmetric particle is unstable
due to $R$-parity violation, we also tried to explore the possible correlations
between light neutrino mixing angles with the branching ratios of the
decay modes of the lightest supersymmetric particle
(which is usually the lightest neutralino for an appreciable region of the parameter space)
as a possible check of this model
in a collider experiment. Later on we looked forward to re-investigate
the tree level analysis with the inclusion of one-loop radiative corrections.
We were also keen to study the sensitivity of our one-loop corrected results to the
light neutrino mass hierarchy. Finally, we proposed an unconventional background free
signal for Higgs boson in $\mu\nu$SSM which can concurrently act as a probe to the
seesaw scale. A signal of this kind not only can lead to an early discovery,
but also act as an unique collider signature of $\mu\nu$SSM.
This thesis is organized as follows, we start with a brief introduction of the
standard model in chapter \ref{SM}, discuss the very basics of mathematical
formulations and address the apparent successes and shortcomings. We start our
discussion in chapter \ref{susy} by studying how the quadratic
divergences in the standard model Higgs boson mass can be handled in a supersymmetric theory.
We also discuss the relevant mathematical formulations,
address the successes and drawbacks of the minimal supersymmetric standard model
with special attentions on the $\mu$-problem and the $R$-parity. A small discussion
on the next-to-minimal supersymmetric standard model has also been addressed. We
devote chapter \ref{neut} for neutrinos. The issues of neutrino mass generation
both in supersymmetric and non-supersymmetric models have been addressed for
tree level as well as for one-loop level analysis. Besides, implications of neutrino
physics in a collider analysis has been discussed. Light neutrino masses
and mixing in $\mu\nu$SSM both for tree level and one-loop level analysis are given in
chapter \ref{munuSSM-neut}. The $\mu\nu$SSM model has been discussed more extensively
in this chapter. We present the results of correlation study between the neutrino
mixing angles and the branching ratios of the decay modes of the lightest neutralino
in $\mu\nu$SSM in chapter \ref{munuSSM-LSP}. Our results are given for different natures of the
lightest neutralino with different hierarchies in light neutrino masses. Finally,
in chapter \ref{munuSSM-Higgs} we present an unusual background free signal for
Higgs boson in $\mu\nu$SSM, which can lead to early discovery. We list our conclusions in
chapter \ref{con-sum}. Various technical details, like different mass matrices, couplings,
matrix element squares of the three-body decays of the lightest supersymmetric particle,
Feynman diagrams etc. are relegated to the appendices.
\chapter{ \sffamily{{\bf
}}\label{appenC}
\section{Details of expansion matrix $\xi$}\label{Details-xi}
In this appendix the entries of the expansion matrix $\xi$ are given in details
\begin{eqnarray}\label{expansion-parameter-terms}
& &\xi_{i1} \approx \frac{\sqrt{2} g_1 \mu m^2_{\nu^c} M_2 A}{12 D}b_i, \nonumber \\
& &\xi_{i2} \approx -\frac{\sqrt{2} g_2 \mu m^2_{\nu^c} M_1 A}{12 D}b_i, \nonumber \\
& &\xi_{i3} \approx -\frac{m^2_{\nu^c} M^{\prime}}{2 D} \left\{
{\left(\lambda v_2 v^2 - 4 \mu A {\frac{M}{v_2}}\right)}a_i + {m_{\nu^c} v_2 v^c}b_i\right.\nonumber\\
& &-\left.
{3\lambda \left(\lambda v_1 v^2-2 m_{\nu^c} v^c v_2\right) }c_i\right\}, \nonumber \\
& &\xi_{i4} \approx -\frac{m^2_{\nu^c} M^{\prime}}{2 D} \left\{ {\lambda v_1 v^2}a_i
+ {m_{\nu^c} v_1 v^c}b_i+{3\lambda^2 v_2 v^2}c_i\right\}, \nonumber \\
& &\xi_{i,4+i} \approx \frac{m_{\nu^c} M^{\prime}}{2 D}\left\{ 2 \lambda \left( \lambda v^4 (1-\frac{1}{2}
{\rm{sin}^2{2\beta}}) + \frac{m_{\nu^c}}{2} v^c v^2 {\rm{sin}{2\beta}}
\right.\right.\nonumber\\
& &+\left.\left.
A v^2 {\rm{sin}{2\beta}} - 4 \mu M A\right) a_i
- {\mu m_{\nu^c} v^2 {\rm{cos}{2\beta}}}b_i\right\}, \nonumber \\
& &\xi_{16} \approx \xi_{17} \approx -\frac{m_{\nu^c} M^{\prime}}{2 D}\left\{ {\lambda \left
( \lambda {v^4} - 4 \mu M A\right)}a_1 + \frac{\mu m_{\nu^c} v^2}{3}b_1
- {2 \lambda \mu m_{\nu^c} v^2_2}c_1\right\}, \nonumber \\
& &\xi_{25} \approx \xi_{27} \approx -\frac{m_{\nu^c} M^{\prime}}{2 D}\left\{ {\lambda \left( \lambda {v^4}
- 4 \mu M A\right)}a_2 + \frac{\mu m_{\nu^c} v^2}{3}b_2 - {2 \lambda \mu m_{\nu^c} v^2_2}c_2\right\}, \nonumber \\
& &\xi_{35} \approx \xi_{36} \approx -\frac{m_{\nu^c} M^{\prime}}{2 D}\left\{ {\lambda \left( \lambda {v^4}
- 4 \mu M A\right)}a_3 + \frac{\mu m_{\nu^c} v^2}{3}b_3 - {2 \lambda \mu m_{\nu^c} v^2}c_3\right\}, \nonumber\\
\end{eqnarray}
where using eqn.(\ref{assumption1})
\begin{eqnarray}
& &a_i = Y_{\nu}^{ii} v_2, ~b_i = (Y_{\nu}^{ii} v_1 + 3 {\lambda} {v'_i}),~c_i = {v'_i}, \nonumber \\
& &m_{\nu^c} = 2 \kappa v^c, ~\mu = 3 \lambda v^c,~A = ({\kappa}{v^c}^2 + {\lambda} v_1 v_2),\nonumber \\
& &v_2 = v {\rm{sin}{\beta}},~v_1 = v {\rm{cos}{\beta}}, ~D = Det\left[M_{7\times7}\right],\nonumber \\
& &\frac{1}{M} = \frac{g^2_1}{M_1} +\frac{g^2_2}{M_2}, ~M^{\prime} = \frac{M_1 M_2}{M},
\label{specifications-2}
\end{eqnarray}
with ${i} = {e,\mu,\tau} ~\equiv{1,2,3}$.
\section{Tree level analysis with perturbative calculation}\label{tree-perturbed}
In the unperturbed basis ${\mathcal{B}} b_ib_j$ with ${\mathcal{B}}
={\frac{2}{3}}{\frac{A {v^c}}{\Delta}}$ the eigenvalues and eigenvectors are
given by
\begin{eqnarray}
& & 0, 0, {\mathcal{B}}{(b^2_e + b^2_\mu + b^2_\tau)}, \nonumber\\
& & \left(\begin{array}{ccc}
-\frac{b_{\tau}}{b_e} & 0 & 1
\end{array}\right)^{T},
\left(\begin{array}{ccc}
-\frac{b_{\mu}}{b_e} & 1 & 0
\end{array}\right)^{T},
\left(\begin{array}{ccc}
\frac{b_e}{b_{\tau}} & \frac{b_{\mu}}{b_{\tau}} & 1
\end{array}\right)^{T},
\label{unperturbed}
\end{eqnarray}
where $b_i$s are given by eqn.(\ref{specifications}). We choose the co-efficient of
$a_ia_j$ term to be ${\mathcal{A}} (= {\frac{1}{6 {\kappa}{v^c}}})$.
The set of orthonormal eigenvectors are obtained using Gram-Schmidt orthonormalization
procedure. The set of orthonormal eigenvectors obtained in this case are
\begin{eqnarray}
& &\text{y}_1 =
{\frac{b_e}{\sqrt{b^2_e + b^2_{\tau}}}}\left(\begin{array}{c}
-\frac{b_{\tau}}{b_e} \\
0 \\
1
\end{array}\right),\nonumber \\
& &\text{y}_2 =
{\frac{\sqrt{b^2_e + b^2_{\tau}}}{{{\Omega}_b}}}
\left(\begin{array}{c}
-\frac{b_e b_{\mu}}{b^2_e + b^2_{\tau}} \\
1 \\
-\frac{b_{\mu} b_{\tau}}{b^2_e + b^2_{\tau}}
\end{array}\right),\nonumber \\
& &\text{y}_3 =
{\frac{b_{\tau}}{{{\Omega}_b}}}\left(\begin{array}{c}
\frac{b_e}{b_{\tau}} \\
\frac{b_{\mu}}{b_{\tau}} \\
1
\end{array}\right),
\label{eigenvectors}
\end{eqnarray}
where
\begin{equation}\label{omegab}
{\Omega}_b={\sqrt{b^2_e + b^2_{\mu} + b^2_{\tau}}}.
\end{equation}
Using degenerate perturbation theory for this set of orthonormal eigenvectors,
the modified eigenvalues $m^\prime_{\pm}$ and $m^\prime_3$ are obtained as
\begin{eqnarray}
m^\prime_{\pm} &=& -{\frac{\mathcal{A}}{{\Omega}_b^2}} \left\{{\Pi_{ab}}
\pm {\sqrt{\left[-3 {{\Omega}^2_b}{\left({\Sigma_{ab}}\right)^2} +
\left({\Pi_{ab}}\right)^2 \right]}} \right\},
\nonumber \\
m^\prime_3 &=& {\mathcal{B}}{{\Omega}^2_b} -
{\frac{2{\mathcal{A}}}{{\Omega}_b^2}} \left\{(\sum_{i}{a_i}{b_i})^2 -
3 {\Lambda}_{ab}\right\},
\label{eigenvalues}
\end{eqnarray}
where
\begin{eqnarray}
{\Lambda}_{ab}={{\sum_{i < j}} {a_i} {a_j} {b_i} {b_j}},~~
{\Pi_{ab}}={\sum_{i < j}}(a_i b_j+ a_j b_i)^2-{{\Lambda}_{ab}},
~~{\Sigma_{ab}}&=&{\sum_{i \neq j \neq k}}{a_i}{a_j}{b_k}.\nonumber\\
\label{eigenvalues-details}
\end{eqnarray}
As one can see from eqn.(\ref{eigenvalues}), the correction to the eigenvalues
are proportional to the coefficient $\mathcal{A}$ appearing in
ordinary seesaw (eqn.(\ref{ordinaryseesaw})). This is
a well expected result since we treat the ordinary seesaw terms
as the perturbation. Let us note in passing that this effect is absent if only one
generation of left chiral neutrino is considered, whereas for two and three
generations of left chiral neutrino the ordinary seesaw effect exists.
This can be understood from the most general calculation involving
n-generations of left chiral neutrinos, where the coefficients of
$\mathcal{A}$ pick up an extra factor $(n-1)$ (see section \ref{n-gen}).
With the set of orthonormal eigenvectors in
eqn. (\ref{eigenvectors}) and the eigenvalues in eqn.(\ref{eigenvalues}), it is
possible to write down the eigenvectors of matrix given by eqn.(\ref{mnuij-compact1})
in the following form
\begin{eqnarray}
(\mathcal{Y}_1)_{3\times1} = {\alpha_1}{y_1} + {\alpha_2}{y_2},
~~(\mathcal{Y}_2)_{3\times1} = {{\alpha}^\prime}_{1}{y_1} + {{\alpha}^\prime}_{2}
{y_2},
~~(\mathcal{Y}_3)_{3\times1} = {y_3},
\label{evec_anal}
\end{eqnarray}
where ${\alpha_1}$, ${\alpha_2}$, $\alpha^\prime_1$,
$\alpha^\prime_2$, are calculated using degenerate perturbation
theory and their analytical expressions are given by
\begin{eqnarray}
&&\alpha_1 = \pm \left( \frac {h_{12}} {\sqrt{h^2_{12} + (h_{11} -
m^\prime_+)^2}}\right),
~~\alpha_2 = \mp \left( \frac {h_{11} - m^\prime_+} {\sqrt{h^2_{12} + (h_{11}
- m^\prime_+)^2}}\right),\nonumber\\
&&\alpha^\prime_1 = \pm \left( \frac {h_{12}} {\sqrt{h^2_{12} + (h_{11}
- m^\prime_-)^2}}\right),
~~\alpha^\prime_2 = \mp \left( \frac {h_{11} - m^\prime_-}
{\sqrt{h^2_{12} + (h_{11} - m^\prime_-)^2}}\right).\nonumber\\
\label{alphas}
\end{eqnarray}
Here $m^\prime_+$, $m^\prime_-$ are given by eqn.(\ref{eigenvalues}) and
$h_{11}$, $h_{12}$ are given by
\begin{eqnarray}
&&h_{11} = -\frac {2{\mathcal{A}} \left({a^2_{\tau}} {b^2_e} +
{a_e}{a_{\tau}}{b_e}{b_{\tau}} + {a^2_e} {b^2_{\tau}}\right)}{b^2_+},\nonumber\\
&&h_{12} = \frac{{\mathcal{A}}\left[a_{\mu}(a_{\tau} b_e - a_e b_{\tau}){b^2_+}
- {b_{\mu}}\left(2 b_e b_{\tau}{a^2_-}+ a_e
a_{\tau}{b^2_-}\right)\right]}{{\Omega_b}{b^2_+}},
\label{h1112}
\end{eqnarray}
where
\begin{eqnarray}\label{h12_specifications}
b^2_{\pm} = (b^2_e \pm b^2_{\tau}),~~a^2_- = (a^2_e - a^2_{\tau}),
\end{eqnarray}
and $\Omega_b$ has been defined in eqn.({\ref{omegab}}).
The light neutrino mixing matrix or PMNS matrix $U$ (eqn.(\ref{PMNS1})) can be constructed using
the eigenvectors given in eqn.(\ref{evec_anal}) and it looks like
\begin{eqnarray}
{U}&=&
\left(\begin{array}{ccc}
{\mathcal{Y}_1} & {\mathcal{Y}_2} & {\mathcal{Y}_3}
\end{array}\right)_{3\times3}.
\label{mneutrino_mixing_numerical}
\end{eqnarray}
\section{See-saw masses with n generations}\label{n-gen}
For the sake of completeness we mention the neutrino mass generation
in $\mu\nu$SSM with $n$ generations of lepton family.
The most general form of effective neutrino mass matrix is given by
\begin{eqnarray}
({M^{seesaw}_{\nu}})_{ij} &=& {\frac{1}{2 n \kappa {v^c}}} {a_{i}} {a_{j}}
(1-n\delta_{ij}) + {\frac{2 A {v^c}}{n \Delta}} {b_{i}} {b_{j}}.
\nonumber \\
\label{mnuij-compact-general}
\end{eqnarray}
In this situation eqn.(\ref{specifications}), eqn.(\ref{omegab}) and eqn.(\ref{eigenvalues})
are modified as follows
\begin{eqnarray}\label{omegab-general}
{\Omega}_b&=&{\sum_m} b^2_m \nonumber \\
\text{where} ~~b_m~&=&~(Y^{mm}_\nu v_1 + n \lambda {\nu}_m)~~m=1,..,n,
\end{eqnarray}
\begin{eqnarray}
m^\prime_r &=& -{\frac{(n-1)\mathcal{A}}{2{\Omega}^2_b}} \left\{{\Pi_{ab}} -~(-1)^{n-r}
{\sqrt{\left[-3 {{\Omega}^2_b}{\left({\Sigma_{ab}}\right)^2} +\left({\Pi_{ab}}\right)^2 \right]}}
\right\},
\nonumber \\
m^\prime_n &=&{\mathcal{B}}{{\Omega}^2_b} - {\frac{(n-1){\mathcal{A}}}{{\Omega}^2_b}}
\left\{(\sum_{i}{a^2_i}{b^2_i})^2 - 3(n-2) {\Lambda}_{ab}\right\},
\label{eigenvalues-general}
\end{eqnarray}
where ${\mathcal{A}} = {\frac{1}{2~n {\kappa}{v^c}}} $, ${\mathcal{B}}
={\frac{2}{n}}{\frac{A {v^c}}{\Delta}}$, $\mu ~= n \lambda v^c$, $r~=~1,...,(n-1)$ and
\begin{eqnarray}
{\Lambda}_{ab}&=&{{\sum_{i < j}} {a_i} {a_j} {b_i} {b_j}},\nonumber \\
{\Pi_{ab}}&=&{\sum_{i < j}}{(a_i b_j + a_j b_i)}^2-(n-2){{\Lambda}_{ab}},\nonumber \\
{\Sigma_{ab}}&=&{\sum_{i \neq j \neq k}}{a_i}{a_j}{b_k}\nonumber \\
\text{where}~ i,~j,~k ~&=&~1,.....,n.
\label{notation-general}
\end{eqnarray}
\chapter{ \sffamily{{\bf
}}\label{appenB}
\section{Scalar mass squared matrices in $\mu\nu$SSM}\label{munuSSM-matsq-scalar}
Decomposition of various neutral scalar fields of $\mu\nu$SSM in
real ($\Re$) and imaginary (${\Im}$) parts are as follows
\begin{eqnarray}
{H^0_d}&=&\Re{H^0_d}+\Im{H^0_d}=
{H^0_{d{\mathcal R}}}+i{H^0_{d{\mathcal I}}},\nonumber \\
{H^0_u}&=&\Re{H^0_u}+\Im{H^0_u}=
{H^0_{u{\mathcal R}}}+i{H^0_{u{\mathcal I}}}, \nonumber \\
{\widetilde \nu}^c_k&=&\Re{\widetilde \nu^c_k}+\Im{\widetilde \nu^c_k}=
{\widetilde \nu}^c_{k{\mathcal R}}
+i{\widetilde \nu}^c_{k{\mathcal I}},\nonumber \\
{\widetilde \nu}_k&=&\Re{\widetilde \nu_k}+\Im{\widetilde \nu_k}=
{\widetilde \nu}_{k{\mathcal R}}+i{\widetilde \nu}_{k{\mathcal I}}.
\label{decomposition}
\end{eqnarray}
Only the real components get VEVs as indicated in eqn.(\ref{munuSSM-vevs}).
The entries of the scalar and pseudoscalar mass-squared matrices are defined as
\begin{equation}
({M^2_S})^{\alpha \beta}=\langle\frac{1}{2}\frac{\partial^2{V_{neutral}}}
{\partial{\phi^{\alpha}_{\mathcal R}}\partial{\phi^{\beta}_{\mathcal R}}}
\rangle,
~~({M^2_P})^{\alpha \beta}=\langle\frac{1}{2}\frac{\partial^2{V_{neutral}}}
{\partial{\phi^{\alpha}_{\mathcal I}}\partial{\phi^{\beta}_{\mathcal I}}}
\rangle,
\label{mass_square_matrix_working_formula}
\end{equation}
where
\begin{eqnarray}
{\phi^{\alpha}_{\mathcal R}}&=&{H^0_{d{\mathcal R}}}, {H^0_{u{\mathcal R}}},
{\widetilde \nu}^c_{k{\mathcal R}}, {\widetilde \nu}_{k{\mathcal R}},\nonumber \\
{\phi^{\alpha}_{\mathcal I}}&=&{H^0_{d{\mathcal I}}}, {H^0_{u{\mathcal I}}},
{\widetilde \nu}^c_{k{\mathcal I}}, {\widetilde \nu}_{k{\mathcal I}}.
\end{eqnarray}
Note that the Greek indices $\alpha, \beta$ are used to refer
various scalar and pseudoscalar Higgs and both $\rm{SU(2)_L}$ doublet and singlet
sneutrinos, that is $H^0_d, H^0_u, {\widetilde \nu}^c_k, {\widetilde \nu}_k$,
whereas k is used as a subscript to specify various flavours of doublet and
singlet sneutrinos i.e., $k=e, {\mu}, {\tau}$ in the flavour
(weak interaction) basis.
\begin{flushleft}
{\it{$\maltese$ Neutral scalar}} $\blacktriangleright$
\end{flushleft}
In the flavour basis or weak interaction basis $\Phi^T_{S}=({H^0_{d{\mathcal R}}},{H^0_{u{\mathcal R}}},
{{\widetilde{\nu}}^c_{n{\mathcal R}}},{{\widetilde{\nu}_{n{\mathcal R}}}})$,\footnote{In refs.
\cite{AppGhosh:2008yh,AppGhosh:2010zi} ${H^0_{1{\mathcal R}}},{H^0_{2{\mathcal R}}}$ was used
in lieu of ${H^0_{d{\mathcal R}}},{H^0_{u{\mathcal R}}}$.} the scalar mass term in
the Lagrangian is of the form
\begin{equation}\label{scalar_Lagrangian}
{\mathcal{L}_{scalar}^{mass}} = {\Phi_{S}^T} {M}^2_{S} {\Phi_{S}},
\end{equation}
where ${M}^2_{S}$ is an $8\times8$ symmetric matrix. The mass eigenvectors are
\begin{equation}
S^0_\alpha = \mathbf R^{S^0}_{\alpha \beta} \Phi_{S_\beta},
\label{scalar-mass-basis}
\end{equation}
with the diagonal mass matrix
\begin{equation}
(\mathcal{M}^{diag}_{S})^2_{\alpha \beta} = \mathbf R^{S^0}_{\alpha \gamma} {M}^2_{S_{\gamma \delta}}
\mathbf R^{S^0}_{\beta \delta}.
\label{scalar-diagonal-mass-matrix}
\end{equation}
\begin{flushleft}
{\it{$\maltese$ Neutral pseudoscalar}} $\blacktriangleright$
\end{flushleft}
In the weak interaction basis $\Phi^T_{P}=(H^0_{d{\mathcal I}},H^0_{u{\mathcal I}},
{\widetilde \nu}^c_{n{\mathcal I}},{\widetilde \nu}_{n{\mathcal I}})$, the pseudoscalar
mass term in the Lagrangian is of the form
\begin{equation}\label{pseudoscalar_Lagrangian}
{\mathcal{L}_{pseudoscalar}^{mass}} = {\Phi_{P}^T} {M}^2_{P} {\Phi_{P}},
\end{equation}
where ${M}^2_{P}$ is an $8\times8$ symmetric matrix. The mass eigenvectors are defined as
\begin{eqnarray}
P^0_\alpha = \mathbf R^{P^0}_{\alpha \beta} \Phi_{P_\beta},
\label{pseudoscalar-mass-basis}
\end{eqnarray}
with the diagonal mass matrix
\begin{eqnarray}
(\mathcal{M}^{diag}_{P})^2_{\alpha \beta} = \mathbf R^{P^0}_{\alpha \gamma}
{M}^2_{P_{\gamma \delta}} \mathbf R^{P^0}_{\beta \delta}.
\label{pseudoscalar-diagonal-mass-matrix}
\end{eqnarray}
\begin{flushleft}
{\it{$\maltese$ Charged scalar}} $\blacktriangleright$
\end{flushleft}
In the weak basis $\Phi^{+^T}_{C}=({H^+_{d}}, {H^+_{u}},
{{\widetilde{e}}^+_{Rn}} ,{{\widetilde{e}}^+_{Ln}})$\footnote{In refs.\cite{AppGhosh:2008yh,AppGhosh:2010zi}
$\Phi^{+^T}_{C}=({H^+_{1}}, {H^+_{2}}, {{\widetilde{e}}^+_{Rn}} ,{{\widetilde{e}}^+_{Ln}})$
basis was used.} the charged scalar mass term in
the Lagrangian is of the form
\begin{equation}\label{charged-scalar_Lagrangian}
{\mathcal{L}_{charged~scalar}^{mass}} = {\Phi_{C}^{-^T}} {M}^2_{C^{\pm}}
{\Phi_{C}^{+}},
\end{equation}
where ${M}^2_{C^{\pm}}$ is an $8\times8$ symmetric matrix. The mass eigenvectors are
\begin{eqnarray}
S^{\pm}_\alpha = \mathbf R^{S^{\pm}}_{\alpha \beta} \Phi^{\pm}_{C_\beta},
\label{charged-scalar-mass-basis}
\end{eqnarray}
with the diagonal mass matrix
\begin{eqnarray}
(\mathcal{M}^{diag}_{C^{\pm}})^2_{\alpha \beta} = \mathbf R^{S^{\pm}}_{\alpha \gamma}
{M}^2_{C^{\pm}_{\gamma \delta}} \mathbf R^{S^{\pm}}_{\beta \delta}.
\label{charged-scalar-diagonal-mass-matrix}
\end{eqnarray}
The independent entries of the $8\times8$ symmetric matrix ${M}^2_{S}$
(eqn. (\ref{scalar_Lagrangian})) using
eqn. (\ref{Minim-munuSSM}) and eqn. (\ref{Abbrevations}) \footnote{A typo in
$(M^2_S)^{H^0_{d{\mathcal R}} H^0_{u{\mathcal R}}}$ in ref.\cite{AppGhosh:2008yh}
has been corrected.}
are given by
\begin{eqnarray}
(M^2_S)^{H^0_{d{\mathcal R}} H^0_{d{\mathcal R}}}&=&{\frac{1}{v_1}}\left[{\sum_j}{\lambda^j}{v_2}
\left({\sum_{ik}} {\kappa^{ijk}}{v^c_i}{v^c_k}\right)+{\sum_j}{\lambda^j}
{r^j}{v^2_2}+{\mu} {\sum_j}{r^j_c}{v'_j}\right. \nonumber\\
&+& \left.
{\sum_i}(A_{\lambda}{\lambda})^{i}{v^c_i}{v_2}\right] +2{\gamma_g}{v^2_1}, \nonumber \\
(M^2_S)^{H^0_{d{\mathcal R}} H^0_{u{\mathcal R}}}&=&-2{\sum_j}{\lambda^j}{\rho^j}{v_2}-{\sum_{i,j,k}}{\lambda^j}
{\kappa^{ijk}}{v^c_i}{v^c_k}-2{\gamma_g}{v_1}{v_2}-{\sum_i}{({A_{\lambda}}
{\lambda})^i}{v^c_i}, \nonumber\\
(M^2_S)^{H^0_{u{\mathcal R}} H^0_{u{\mathcal R}}}&=&{\frac{1}{v_2}}\left[-{\sum_{j}}{\rho^{j}}\left({\sum_{l,k}}
\kappa^{ljk}{v^c_l}{v^c_k}\right)-{\sum_{i,j}}(A_{\nu}{Y_{\nu}})^{ij}
{v'_i}{v^c_j}\right. \nonumber\\
&+& \left.
{\sum_i}(A_\lambda {\lambda})^{i} {v^c_i}{v_1}\right]+2{\gamma_g}{v^2_2},\nonumber \\
(M^2_S)^{H^0_{d{\mathcal R}} {\widetilde{\nu}^c_{m{\mathcal R}}}}&=&-2{\sum_j}{\lambda^j}{u^{mj}_c}{v_2}+2{\mu}
{v_1}{\lambda^m}-{\lambda^m}{\sum_i}{r^i_c}{v'_i}-{\mu}{r^m}-
{({A_{\lambda}}{\lambda})^m}{v_2}, \nonumber\\
(M^2_S)^{H^0_{d{\mathcal R}} {\widetilde{\nu}_{m{\mathcal R}}}}&=&-{\sum_j}{\lambda^j}{Y^{mj}_{\nu}}{v^2_2}-
{\mu}{r^m_c}+2{\gamma_g}{v'_m}{v_1},\nonumber \\
(M^2_S)^{H^0_{u{\mathcal R}} {\widetilde{\nu}^c_{m{\mathcal R}}}}&=&2{\sum_j}{u^{mj}_c}{\rho^j}+2{\lambda^m}
{\mu}{v_2}+2{\sum_i}{Y^{im}_{\nu}}{r^i_c}{v_2} + {\sum_i}{(A_{\nu}{Y_{\nu}})^{im}}{v'_i}\nonumber\\
&-&{(A_{\lambda}{\lambda})^{m}}{v_1},\nonumber \\
(M^2_S)^{H^0_{u{\mathcal R}} {\widetilde{\nu}_{m{\mathcal R}}}}&=&2{\sum_j}{Y^{mj}_{\nu}}{\rho^j}{v_2}
+{\sum_{i,j,k}}{Y^{mj}_{\nu}}{\kappa^{ijk}}{v^c_i}{v^c_k}-2{\gamma_g}{v'_m}{v_2}
+{\sum_j}{(A_{\nu}{Y_{\nu}})^{mj}}{v^c_j},\nonumber \\
(M^2_S)^{{\widetilde{\nu}^c_{n{\mathcal R}}} {\widetilde{\nu}^c_{m{\mathcal R}}}}&=&2{\sum_j}{\kappa^{jnm}}{\zeta^j}
+4{\sum_j}{u^{mj}_c}{u^{nj}_c}+{\rho^m}{\rho^n}+{h^{nm}}{v^2_2}\nonumber \\
&+&{(m^2_{\widetilde{\nu}^c})^{mn}}+2{\sum_i}{(A_{\kappa}{\kappa})^{imn}}{v^c_i},
\nonumber \\
(M^2_S)^{{\widetilde{\nu}^c_{n{\mathcal R}}} {\widetilde{\nu}_{m{\mathcal R}}}}&=&2{\sum_j}{Y^{nj}_{\nu}}{u^{mj}_c}{v_2}
+{Y^{nm}_{\nu}}{\sum_i}{r^i_c}{v'_i}+{r^n_c}{r^m}-{\mu}{v_1}{Y^{nm}_{\nu}}\nonumber \\
&-&{\lambda^m}{r^n_c}{v_1}+{(A_{\nu}{Y_{\nu}})^{nm}}{v_2},\nonumber \\
(M^2_S)^{{\widetilde{\nu}_{n{\mathcal R}}} {\widetilde{\nu}_{m{\mathcal R}}}}&=&{\sum_j}{Y^{nj}_{\nu}}{Y^{mj}_{\nu}}{v^2_2}
+{r^m_c}{r^n_c}+{\gamma_g}{\xi_{\upsilon}}{\delta_{nm}}+2{\gamma_g}{v'_n}
{v'_m}+{(m^2_{\widetilde{L}})^{mn}}.
\label{element_of_scalar_mass_matrix}
\end{eqnarray}
Similarly independent elements of $8\times8$ symmetric matrix $\mathcal{M}^2_{P}$
(eqn. (\ref{pseudoscalar_Lagrangian}))
using eqn. (\ref{Minim-munuSSM}) and eqn. (\ref{Abbrevations}) are given by
\begin{eqnarray}
(M^2_P)^{H^0_{d{\mathcal I}} H^0_{d{\mathcal I}}}&=&{\frac{1}{v_1}}
\left[{\sum_j}{\lambda^j}{v_2} \left({\sum_{ik}} {\kappa^{ijk}}{v^c_i}
{v^c_k}\right)+{\sum_j} {\lambda^j}{r^j}{v^2_2}+ {\mu}{\sum_j}{r^j_c}{v'_j}
\right.\nonumber\\
&+&\left.
{\sum_i} (A_{\lambda}{\lambda})^{i}{v^c_i}{v_2} \right], \nonumber \\
(M^2_P)^{H^0_{d{\mathcal I}} H^0_{u{\mathcal I}}}&=&{\sum_{i,j,k}}{\lambda^j}
{\kappa^{ijk}} {v^c_i}{v^c_k} + \sum_i (A_{\lambda} {\lambda})^{i}{v^c_i},
\nonumber \\
(M^2_P)^{H^0_{u{\mathcal I}} H^0_{u{\mathcal I}}}&=&{\frac{1}{v_2}}
\left[-{\sum_{j}}{\rho^{j}} \left({\sum_{l,k}} \kappa^{ljk}{v^c_l}
{\nu^c_k}\right)-{\sum_{i,j}} (A_{\nu}{Y_{\nu}})^{ij} {v'_i}{v^c_j}\right.\nonumber\\
&+&\left.
{\sum_i}(A_\lambda {\lambda})^{i} {v^c_i}{v_1}\right],
\nonumber \\
(M^2_P)^{H^0_{d{\mathcal I}} {\widetilde \nu}^c_{m{\mathcal I}}}&=&-2 {\sum_j}
{\lambda^j}{u^{mj}_c}{v_2} - {\mu}{r^m} + {\lambda^m}{\sum_i}{r^i_c}{v'_i}
+ (A_{\lambda}{\lambda})^m {v_2}, \nonumber \\
(M^2_P)^{H^0_{d{\mathcal I}} {\widetilde \nu}_{m{\mathcal I}}}&=&-{\sum_j}
{\lambda^j}{Y^{mj}_{\nu}} {v^2_2} - {\mu}{r^m_c},\nonumber \\
(M^2_P)^{H^0_{u{\mathcal I}} {\widetilde \nu}^c_{m{\mathcal I}}}&=&2{\sum_j}
{u^{mj}_c}{\rho^j} - {\sum_i}(A_{\nu} Y_{\nu})^{im}{v'_i} +
(A_{\lambda}{\lambda})^m {v_1}, \nonumber \\
(M^2_P)^{H^0_{u{\mathcal I}} {\widetilde \nu}_{m{\mathcal I}}}&=&-{\sum_{i,j,k}}
{Y^{mj}_{\nu}} {\kappa^{ijk}} {v^c_i}{v^c_k}-{\sum_j}
({A_{\nu}}{Y_{\nu}})^{mj}{v^c_j},
\nonumber \\
(M^2_P)^{{\widetilde \nu}^c_{n{\mathcal I}} {\widetilde \nu}^c_{m{\mathcal I}}}&=&
-2{\sum_j}{\kappa^{jnm}} {\zeta^j}+4{\sum_j} {u^{mj}_c}{u^{nj}_c}+{\rho^m}
{\rho^n}+{h^{nm}}{v^2_2} \nonumber\\
&+&(m^2_{\widetilde{\nu}^c})^{nm} -2{\sum_{i}} ({A_{\kappa}}{\kappa})^{inm} {v^c_i},\nonumber \\
(M^2_P)^{{\widetilde \nu}^c_{n{\mathcal I}} {\widetilde \nu}_{m{\mathcal I}}}&=&
2{\sum_j}{u^{mj}_c} {Y^{nj}_{\nu}}{v_2}-{Y^{nm}_{\nu}}{\sum_i}{r^i_c}{v'_i}
+{r^n_c}{r^m} +{\mu}{v_1}{Y^{nm}_{\nu}}\nonumber\\
&-&{\lambda^m}{r^n_c}{v_1} - {({A_{\nu}}{Y_{\nu}})^{nm}} {v_2},\nonumber \\
(M^2_P)^{{\widetilde \nu}_{n{\mathcal I}} {\widetilde \nu}_{m{\mathcal I}}}&=&
{\sum_j}{Y^{mj}_{\nu}} {Y^{nj}_{\nu}}{v^2_2} +{r^m_c}{r^n_c}
+(m^2_{\widetilde{L}})^{nm}+{\gamma_g} {\xi_{\upsilon}} {\delta_{mn}}.
\label{element_of_pseudoscalar_mass_matrix}
\end{eqnarray}
In eqns.(\ref{element_of_scalar_mass_matrix}), (\ref{element_of_pseudoscalar_mass_matrix})
$h^{nm}={\lambda^n}{\lambda^m}+{\sum}{Y_{\nu}^{in}} {Y_{\nu}^{im}}$ has been used.
One eigenvalue of $\mathcal{M}^2_{P}$ matrix is zero which corresponds
to the neutral Goldstone boson.
Finally, the independent entries of $\mathcal{M}^2_{C}$
using eqn. (\ref{Minim-munuSSM}) and eqn. (\ref{Abbrevations}) are given by
\begin{eqnarray}
(M^2_C)^{H_{d} H_{d}}&=&{\frac{1}{v_1}}\left[{\sum_j} \lambda^j \zeta^j v_2
+ \mu {\sum_j} r^j_c v'_j + {\sum_i} ( A_{\lambda} \lambda )^i v^c_i v_2
\right] \nonumber\\
&+& {\sum_{i,j,k}} Y^{ij}_e Y^{kj}_e v'_i v'_k - \frac{{g_2}^2}{2}
({\sum_i} v'^2_i - v^2_2 ), \nonumber \\
(M^2_C)^{H_{d} H_{u}}&=& -{\sum_j} \lambda^{j^2} v_1 v_2 + {\sum_j} \lambda^j r^j v_2
+ {\sum_j} \lambda^j u^{ij}_c v^c_i + \frac{{g_2}^2}{2} v_1 v_2 \nonumber\\
&+& {\sum_i} (A_{\lambda} \lambda )^i v^c_i, \nonumber\\
(M^2_C)^{H_{u} H_{u}}&=&{\frac{1}{v_2}}\left[ - {\sum_j} \rho^j \zeta^j
- {\sum_{i,j}} ( A_\nu Y_\nu )^{ij} v'_i v^c_j + {\sum_i} ( A_\lambda
\lambda )^i v^c_i v_1\right] \nonumber\\
&+& \frac{{g_2}^2}{2} ( {\sum_i} v'^2_i + v^2_1 ),\nonumber \\
(M^2_C)^{H_{d} {{\widetilde{e}}_{Rm}}}&=& - {\sum_i} r^i_c Y^{im}_e v_2 - {\sum_i}
( A_e Y_e )^{im} v'_i, \nonumber\\
(M^2_C)^{H_{d} {{\widetilde{e}}_{Lm}}}&=& - \mu r^m_c - {\sum_{i,j}} Y^{mj}_e Y^{ij}_e
v'_i v_1 + \frac{g_2^2}{2} v'_m v_1,\nonumber \\
(M^2_C)^{H_{u} {{\widetilde{e}}_{Rm}}}&=& - \mu {\sum_i} Y^{mi}_e v'_i - {\sum_i}
Y^{im}_e r^i_c v_1,\nonumber \\
(M^2_C)^{H_{u} {{\widetilde{e}}_{Lm}}}&=& - {\sum_j} Y^{mj}_{\nu} \zeta^j +
\frac{g_2^2}{2} v'_m v_2 - {\sum_i} (A_{\nu} Y_{\nu})^{mi} v^c_i,\nonumber \\
(M^2_C)^{{{\widetilde{e}}_{Rn}} {{\widetilde{e}}_{Rm}}}&=& {\sum_{i,j}} Y^{im}_e Y^{jn}_e v'_i v'_j
+ {\sum_i} Y^{im}_e Y^{in}_e v^2_1 + (m^2_{\widetilde{e}^c})^{mn} -
\frac{g_1^2}{2} {\xi_\upsilon} \delta_{mn},\nonumber \\
(M^2_C)^{{{\widetilde{e}}_{Rn}} {{\widetilde{e}}_{Lm}}}&=& - \mu Y^{mn}_e v_2 + (A_e Y_e)^{nm} v_1 ,\nonumber \\
(M^2_C)^{{{\widetilde{e}}_{Ln}} {{\widetilde{e}}_{Lm}}}&=& r^m_c r^n_c + {\sum_j} Y^{mj}_e Y^{nj}_e
v^2_1 + \gamma_g {\xi_\upsilon} \delta_{mn} -\frac{g_2^2}{2}{\xi_\upsilon}
\delta_{mn} \nonumber \\
&+& \frac{g_2^2}{2} v'_m v'_n + (m^2_{\widetilde{L}})^{mn}.
\label{element_of_charged-scalar_mass_matrix}
\end{eqnarray}
For the charged scalar mass-squared matrix, seven out of eight eigenvalues are
positive and the remaining one is a massless charged Goldstone boson.
Note that in eqns. (\ref{element_of_scalar_mass_matrix}),
(\ref{element_of_pseudoscalar_mass_matrix}), (\ref{element_of_charged-scalar_mass_matrix})
we have used $v^c_i$ and $v'_i$ to represent VEV of $i$-th right handed and left handed sneutrino,
respectively. In ref. \cite{AppGhosh:2008yh} these were represented by $\nu^c_i$ and $\nu_i$,
respectively.
\begin{flushleft}
{\it{$\maltese$ Squark mass matrices}} $\blacktriangleright$
\end{flushleft}
In the weak basis,
$\widetilde{u'}_i=(\widetilde{u}_{L_i},\widetilde{u}^*_{R_i})$ and
$\widetilde{d'}_i=(\widetilde{d}_{L_i},\widetilde{d}^*_{R_i} )$, we get
\begin{equation}
\mathcal{L}^{mass}_{squark}=
\frac{1}{2}\widetilde{u'_i}^\dag M_{\widetilde{u_{ij}}}^2\ \widetilde{u'_j}
+\frac{1}{2}\widetilde{d'_i}^{\dag} M_{\widetilde{d_{ij}}}^2\ \widetilde{d'_j}\ ,
\label{squark-mass-lagrangian}
\end{equation}
where $\widetilde{q}=(\widetilde{u'},\widetilde{d'})$. Explicitly for up and down type
squarks $(\widetilde{u},\widetilde{d})$, using eqn.(\ref{Abbrevations}) the entries are
\begin{eqnarray}
(M^2_{\widetilde{u}})^{L_iL_j}&=&
(m^2_{\widetilde{Q}})^{ij} + \frac{1}{6}(\frac{3g^2_2}{2}
- \frac{g_1^2}{2}){\xi_{\upsilon}} \delta^{ij}+
\sum_n {Y_u^{in} Y_u^{jn} v_2^2}\ , \nonumber\\
(M^2_{\widetilde{u}})^{R_iR_j}&=&
(m^2_{\widetilde{u}^c})^{ij}+ \frac{g^2_1}{3}{\xi_{\upsilon}}\delta^{ij}
+ \sum_n {Y_u^{ni} Y_u^{nj}v_2^2}\ , \nonumber\\
(M^2_{\widetilde{u}})^{L_iR_j}&=&
(A_u Y_u)^{ij} v_2 -Y_u^{ij} v_1 \mu
+ Y_u^{ij} \sum_l{r^l_c v'_l}\ ,\nonumber\\
(M^2_{\widetilde{u}})^{R_iL_j}&=&
(M^2_{\widetilde{u}})^{L_jR_i}\ ,
\label{entries-of-usquark-mass-matrix}
\end{eqnarray}
and
\begin{eqnarray}
(M^2_{\widetilde{d}})^{L_iL_j}&=&
(m^2_{\widetilde{Q}})^{ij} - \frac{1}{6}(\frac{3g^2_2}{2}
+ \frac{g_1^2}{2}){\xi_{\upsilon}}\delta^{ij}+
\sum_n {Y_d^{in} Y_d^{jn} v_1^2}\ , \nonumber\\
(M^2_{\widetilde{d}})^{R_iR_j}&=&
(m^2_{\widetilde{d}^c})^{ij} - \frac{g^2_1}{6}{\xi_{\upsilon}}\delta^{ij}
+ \sum_n {Y_d^{ni} Y_d^{nj}v_1^2}\ , \nonumber\\
(M^2_{\widetilde{d}})^{L_iR_j}&=&
(A_d Y_d)^{ij} v_1 -Y_d^{ij} v_2 \mu \ ,\nonumber\\
(M^2_{\widetilde{d}})^{R_iL_j}&=&
(M^2_{\widetilde{d}})^{L_jR_i}\ .
\label{entries-of-dsquark-mass-matrix}
\end{eqnarray}
For the mass eigenstate $\widetilde{\mathbf{q}}_i$ we have
\begin{equation}
\widetilde{\mathbf{q}}_i = \mathbf R^{\widetilde q}_{ij} \widetilde{q}_j\ ,
\label{squark-mass-basis}
\end{equation}
with the diagonal mass matrix
\begin{equation}
(\mathcal{M}^{\text{diag}}_{\widetilde{q}})^2_{ij}
= \mathbf R^{\widetilde q}_{il} M^2_{\widetilde{q}_{lk}} \mathbf R^{\widetilde q}_{jk} .
\label{squark-diagonal-mass-matrix}
\end{equation}
\section{Quark mass matrices in $\mu\nu$SSM}
\label{munuSSM-matsq-fermion}
The mass matrices for up and down quarks are $3\times3$ and they are diagonalized using bi-unitary
transformation. Entries of up and down quark mass matrices $m^u_{3\times3}~\rm{and}~m^d_{3\times3}$ are
same as the MSSM and are given below
\begin{eqnarray}
(m^u_{3\times3})_{ij} &=& Y_u^{ij} v_2, \nonumber \\
(m^d_{3\times3})_{ij} &=& Y_d^{ij} v_1.
\label{quark-mass-matrix}
\end{eqnarray}
The quark mass matrices are diagonalized as follows
\begin{eqnarray}\label{quark_mass_eigenstate_matrixform}
{\mathbf R^{u}_L}^* m^u_{3\times3} {\mathbf R^{u}_R}^{-1} &=& \mathcal{M}^{diag}_U, \nonumber \\
{\mathbf R^{d}_L}^* m^d_{3\times3} {\mathbf R^{d}_R}^{-1} &=& \mathcal{M}^{diag}_D.
\end{eqnarray}
\chapter{ \sffamily{{\bf
}}\label{appenA}
\section{Scalar mass squared matrices in MSSM}\label{MSSM-matsq-scalar}
\begin{flushleft}
{\it{$\maltese$ Neutral scalar,
$(\mathcal{M}^2_{MSSM-scalar})_{2\times2}$}}
in the basis $(\Re H^0_d, \Re H^0_u) \Rrightarrow$
\end{flushleft}
\begin{eqnarray}
\left(\begin{array}{c c}
B_\mu tan\beta -\frac{\mu}{v_1} \varepsilon^\alpha v'_\alpha + 2\gamma_g v^2_1
& -2 \gamma_g v_1 v_2 + B_\mu \\
-2 \gamma_g v_1 v_2 + B_\mu &
B_\mu cot\beta + B_{\varepsilon_\alpha} \frac{v'_\alpha}{v_2} + 2\gamma_g v^2_2
\end{array}\right).
\label{MSSM-cp-even-scalar}
\end{eqnarray}
\begin{flushleft}
{\it{$\maltese$ Neutral pseudoscalar
$(\mathcal{M}^2_{MSSM-pseudoscalar})_{2\times2}$}}
in the basis $(\Im H^0_d, \Im H^0_u) \Rrightarrow$
\end{flushleft}
\begin{eqnarray}
\left(\begin{array}{c c}
B_\mu tan\beta -\mu \varepsilon^\alpha \frac{v'_\alpha}{v_1}
& B_\mu \\
B_\mu &
B_\mu cot\beta + B_{\varepsilon_\alpha} \frac{v'_\alpha}{v_2}
\end{array}\right).
\label{MSSM-cp-odd-scalar}
\end{eqnarray}
\begin{flushleft}
{\it{$\maltese$ Charged scalar
$(\mathcal{M}^2_{MSSM-charged})_{2\times2}$}}
in the basis $(H^+_d, H^+_u) \Rrightarrow$
\end{flushleft}
\begin{eqnarray}
(\mathcal{M}^2_{MSSM-charged})_{2\times2} = \left(\begin{array}{c c}
C^2_{11} & C^2_{12}\\
C^2_{21} & C^2_{22}
\end{array}\right),
\label{MSSM-charged-scalar1}
\end{eqnarray}
where
\begin{eqnarray}
C^2_{11} &=& B_\mu tan\beta -\mu\varepsilon^\alpha \frac{v'_\alpha}{v_1}
+ Y^{\alpha\rho}_e Y^{\beta\rho}_e v'_\alpha v'_\beta
-\frac{g^2_2}{2}\{v'^2_\alpha-v^2_2\}, \nonumber \\
C^2_{12} &=& B_\mu + \frac{g^2_2}{2}v_1v_2, ~C^2_{21} = C^2_{12}, \nonumber \\
C^2_{22} &=& B_\mu cot\beta + B_{\varepsilon_\alpha} \frac{v'_\alpha}{v_2}
+ \frac{g^2_2}{2}\{v'^2_\alpha+v^2_1\}.
\label{MSSM-charged-scalar2}
\end{eqnarray}
In these derivations minimization equations for $H_u,H_d$ has been
used, which are given by
\begin{eqnarray}
{\varepsilon^2_\alpha} v_2- B_{\varepsilon_\alpha}{v'_\alpha}-B_\mu v_1 + ({m^2_{H_u}}+{\mu^2}) v_2
-{\gamma_g}{\xi_{\upsilon}} {v_2}
= 0,\nonumber \\
\mu \varepsilon^\alpha v'_\alpha -B_\mu v_2 + ({m^2_{H_d}}+\mu^2)v_1
+{\gamma_g}{\xi_{\upsilon}} {v_1} = 0,
\label{MSSM-Higgs-minima}
\end{eqnarray}
with $\gamma_{g} = \frac{1}{4}({g_1^2 + g_2^2})$ and
$\xi_{\upsilon} ={\sum {v'^2_\alpha} + v_1^2 -v_2^2}$.
\section{Fermionic mass matrices in MSSM}
\label{MSSM-matsq-fermion}
\begin{flushleft}
{\it{$\maltese$ Chargino mass matrix
$(M^{chargino}_{MSSM})_{2\times2}$}}
in the basis $-i \widetilde {\lambda}^{+}_{2}, \widetilde{H}^+_u$ (column)
and $-i \widetilde {\lambda}^{-}_{2}, \widetilde{H}^-_d$ (row)
$\Rrightarrow$
\end{flushleft}
\begin{eqnarray}
(M^{chargino}_{MSSM})_{2\times2} =
\left(\begin{array}{c c}
M_2 & g_2v_2\\
g_2v_1& \mu
\end{array}\right).
\label{MSSM-chargino}
\end{eqnarray}
\begin{flushleft}
{\it{$\maltese$ Neutralino mass matrix
$(M^{neutralino}_{MSSM})_{4\times4}$}}
in the basis $-i\widetilde B^0, -i\widetilde W_3^0, \widetilde H_d^0,
\widetilde H_u^0\Rrightarrow$
\end{flushleft}
\begin{eqnarray}
(M^{neutralino}_{MSSM})_{4\times4} =
\left(\begin{array}{c c c c}
M_1 & 0 & -\frac{g_1}{\sqrt{2}}v_1 & \frac{g_1}{\sqrt{2}}v_2 \\
0 & M_2 & \frac{g_2}{\sqrt{2}}v_1 & -\frac{g_2}{\sqrt{2}}v_2 \\
-\frac{g_1}{\sqrt{2}}v_1 & \frac{g_2}{\sqrt{2}}v_1 & 0 & -{\mu} \\
\frac{g_1}{\sqrt{2}}v_2 & -\frac{g_2}{\sqrt{2}}v_2 & -{\mu} & 0
\end{array}\right).
\label{MSSM-neutralino}
\end{eqnarray}
\chapter{ \sffamily{{\bf N}eutrinos
}}\label{neut}
Long back in 1930, a new particle was suggested by {\it{Pauli}}
to preserve the conservation of energy, conservation of momentum, and
conservation of angular momentum in beta decay \cite{c3Fermi:1934sk,
c3Fermi:1934hr}\footnote{To be specific, this was an electron neutrino.
$\nu_\mu$ and $\nu_\tau$ were hypothesized later in 1940 and 1970, respectively.}.
The name {\it{neutrino}} was coined by {\it{Fermi}} in 1934. The much
desired experimental evidence for neutrinos (actually $\nu_e$) was finally
achieved in 1956 \cite{c3Cowan:1992xc}. In 1962, muon neutrino was discovered
\cite{c3Danby:1962nd}. However, it took a long time till 2000 to discover
$\nu_\tau$ \cite{c3Kodama:2000mp}.
Neutrino sources are everywhere, however, they are broadly classifiable in
two major classes, namely, (1) natural sources and (2) man made neutrinos.
Natural neutrino sources are nuclear $\beta$ decay ($\nu_e$), solar neutrinos ($\nu_e$),
atmospheric neutrinos ($\nu_e,\nu_\mu$ and their anti-neutrinos) and supernovae neutrinos
(all flavours) mainly. Man made neutrinos are produced by the particle accelerators and
neutrinos coming out of nuclear reactors.
Neutrino physics has been seeking huge attention for the last few decades.
Different aspects of neutrino physics have been discussed in references
\cite{c3Frampton:1982qi,c3Boehm:1987fc,c3Bilenky:1987ty,c3Bahcall:1989ks,c3Kayser:1989iu,
c3Mohapatra:1991ng,c3Gelmini:1994az,c3Barbieri:1998mq,c3Bilenky:1998dt,c3Fisher:1999fb,
c3Kayser:2000pe,c3Langacker:2000fp,c3Caldwell:2001pc,c3GonzalezGarcia:2002dz,c3Pakvasa:2003zv,
c3Barger:2003qi,c3Grossman:2003eb,c3King:2003jb,c3Altarelli:2003ph,c3Altarelli:2004za,c3Giunti:2004yg,
c3Kayser:2005cd,c3Mohapatra:2005wg,c3Mohapatra:2006gs,c3Strumia:2006db,c3Valle:2006vb,
c3GonzalezGarcia:2007ib}.
\section{{\bf N}eutrinos in the Standard Model}\label{neut-SM}
The neutrinos as discussed in chapter \ref{SM}, appear to be a part of the SM.
Confining our attention within the SM, it is worth listing the information about
neutrinos, that ``lies within the SM''
\begin{enumerate}
\item
They are spin $\frac{1}{2}$ objects and thus follow Fermi-Dirac statistics
\cite{c3Fermi,c3Dirac:1926jz}.
\item
Neutrinos are electrically neutral fundamental particles, belonging to
the lepton family. The SM contains three neutrinos, corresponding
to three charged leptons.
\item
They are a part of the weak isospin ($\rm{SU(2)_L}$) doublet.
Being charge and colour neutral neutrinos are sensitive to weak interaction only.
\item
There exist two kinds of neutrino interactions in nature,
(1) neutral and (2) charge current interactions (see figure \ref{CC-NC}).
\begin{figure}[ht]
\centering
\includegraphics[width=7.45cm]{FigsPDF/CC-NC.pdf}
\caption{Feynman diagram for the neutral and charged
current interactions. $\nu_i$ stand for different neutrino flavours
like $\nu_e,\nu_\mu,\nu_\tau$. The charged leptons $(e,\mu,\tau)$
are represented by $l_i$s.}
\label{CC-NC}
\end{figure}
\item
There are only left-chiral \cite{c3Lee:1956qn,c3Wu:1957my}
(spin anti-parallel to the momentum direction)
neutrinos in nature, without any right-handed counter part. But there
exists anti-neutrinos of right chirality (spin parallel to momentum).
\item
Neutrinos are exactly mass less in the SM.
\item
Since the neutrinos are massless within the framework of
the SM, the mass basis and the weak interaction basis are same for the charged leptons.
In other words, there exists no leptonic analogue
of the CKM matrix (see ref.\cite{c3Cabibbo-1963,c3Kobayashi-1973})
with vanishing neutrino mass.
\end{enumerate}
The massless neutrinos seem to work fine with the SM until the
first hint of neutrino oscillation appeared in 1969 \cite{c3Davis:1968cp},
which requires massive neutrinos!\footnote{The first idea of neutrino
oscillation was given by Bruno Pontecorvo
\cite{c3Pontecorvo:1957cp,c3Pontecorvo:1957qd}.}. However, maintaining
the gauge invariance, Lorentz invariance and renormalizability, there is
absolutely no room for massive neutrinos in the SM (see reviews
\cite{c3Akhmedov:1999uz,c3Mohapatra:2005wg}). It is then apparent
that to explain neutrino oscillation the SM framework requires extension.
We leave these modifications for time being until section \ref{neut-mass}.
It is rather more important to know the phenomenon of neutrino oscillation.
Besides, it is also important to know if neutrinos posses non-zero mass,
what will be the possible experimental impressions?
\section{{\bf N}eutrino oscillation}\label{neut-osc}
\begin{flushleft}
{\it{$\maltese$ Evidences of neutrino oscillation}}
\end{flushleft}
\noindent
{\bf{I. Atmospheric neutrino problem $\blacktriangleright$}}
Consider the atmospheric neutrinos, which are coming from the interaction
of cosmic rays with the earth's atmosphere. The charged pion $(\pi^\pm)$
produced in the interaction, has the following decay chain
\begin{eqnarray}
\pi^\pm \to \mu^\pm + \nu_\mu(\overline{\nu_\mu}),
\label{neut-atm1}
\end{eqnarray}
followed by
\begin{eqnarray}
\mu^\pm \to e^\pm + \nu_e(\overline{\nu_e}) + \overline{\nu_\mu}({\nu_\mu}).
\label{neut-atm2}
\end{eqnarray}
These neutrinos(anti-neutrinos) take part in charge current interaction
(see figure \ref{CC-NC}) and produce detectable charged leptons. Looking
at eqns. (\ref{neut-atm1}, \ref{neut-atm2}) one would naively expect
number wise\footnote{This number is actually not exactly $2$, because of
various uncertainties like, geometry of cosmic ray flux and neutrino flux,
solar activities,uncertainty in cross section measurements, etc.},
\begin{equation}
R_{\frac{\mu}{e}}=\frac{\nu_\mu(\overline{\nu_\mu})}{\nu_e(\overline{\nu_e})}
= 2.
\label{neut-atm3}
\end{equation}
However, in reality $R_{\frac{\mu}{e}}$ is much smaller $(\sim 0.6)$, as observed by
experiments like Kamiokande \cite{c3Hirata:1992ku,
c3Fukuda:1994mc}, NUSEX \cite{c3Aglietta:1988be}, IMB \cite{c3Casper:1990ac,c3BeckerSzendy:1992hq},
Soudan-2 \cite{c3Allison:1996yb}, MACRO \cite{c3Ambrosio:1998wu,c3Ronga:1998ei},
Super-Kamoikande \cite{c3Fukuda:1998mi,c3Fukuda:1998ah}. The diminution in $R_{\frac{\mu}{e}}$
as observed by a host of experiments indicates a deficit of muon (anti)neutrino flux.
This apparent discrepancy between predicted and observed neutrino flux defines the
atmospheric neutrino problem.
\vspace{0.2cm}
\noindent
{\bf{II. Solar neutrino problem $\blacktriangleright$}}
The Sun gets huge energy by fusing hydrogen $(^1_1{\rm H})$ to helium $(^4_2{\rm He})$
in thermonuclear reactions. There exist a few viable candidates for this reaction
chain, like proton-proton $(pp)$ cycle, carbon-nitrogen-oxygen (CNO) cycle
\cite{c3Bethe:1939bt,c3Bahcall:2000xc} etc,
although the $pp$ cycle appears to be the dominant one. The sun is a major
source of electron neutrinos (see also ref.\cite{c3Bahcall:1995gw,c3Bahcall:2000kh})
following the process
\begin{equation}
4p \to {^4_2{\rm He}} + 2 e^+ + 2 \nu_e,
\label{neut-sol1}
\end{equation}
where $e^+$ is a positron. There exist a host of literature concerning
the standard solar model \cite{c3Bahcall:1987jc,c3Bahcall:1989ks,c3TurckChieze:1988tj,
c3TurckChieze:1993dw,c3Bahcall:1992hn,c3Bahcall:1998wm}, which account for the
number of solar neutrinos expected to be detected in an earth based detector.
However, only one-third of the expected solar neutrino flux has been detected
by experiments like Homestake \cite{c3Davis:1968cp,c3Davis:1994jw,c3Cleveland:1998nv},
SAGE \cite{c3Abdurashitov:1996dp,c3Abdurashitov:1999zd,
c3Abdurashitov:2009tn}, GALLEX \cite{c3Anselmann:1992um,
c3Hampel:1998xg}, GNO \cite{c3Altmann:2000ft},
Kamiokande\cite{c3Hirata:1990xa}, Super-Kamiokande \cite{c3Fukuda:2001nj,
c3Cravens:2008zn}, SNO \cite{c3Boger:1999bb,c3Ahmad:2002jz,
c3Aharmim:2005gt} etc. The disappearance of a large fraction of solar neutrinos
defines the solar neutrino problem.
There were numerous attempts to explain the discrepancy between the measured
and the predicted neutrino flux for the solar and the atmospheric neutrinos.
In fact, these neutrino deficits lead to the proposal of various theoretical
models\footnote{A discussion of these models is beyond the scope of this thesis.
See ref.\cite{c3Vicente:2011pf} for further discussions.}.
However, with the idea of Bruno Pontecorvo \cite{c3Pontecorvo:1957cp,
c3Pontecorvo:1957qd,c3Pontecorvo:1967fh}, it seems more logical to think
about some sort of {\it{conversion}} among neutrino flavours while they
propagate through vacuum or matter, which can lead to diminution of
a specific type of flavor as reported by experiments.
\begin{flushleft}
{\it{$\maltese$ Theory of neutrino oscillation}}
\end{flushleft}
In order to explain the Solar and atmospheric neutrino
deficits, as discussed earlier it is expected that a neutrino of a
specific flavour, say $a$, during propagation can alter its flavour to
some other one, say $b$, at a later stage of time. Now from our knowledge of
quantum mechanics it is evident that,
\begin{enumerate}
\item
The set of linearly independent mass eigenstates form a complete
basis.
\item
Any arbitrary state can be expressed as a linear combination of
the linearly independent mass eigenstates.
\end{enumerate}
So, if neutrinos oscillate,
\cite{c3Kayser:1981ye,c3Giunti:1991ca,c3Rich:1993wu,c3Grossman:1996eh}
the flavour eigenstates, $\nu_e,\nu_\mu,\nu_\tau$ must
differ from the physical or mass eigenstates and it is possible to express
them as a linear combination of the neutrino mass eigenstates,
$\nu_1,\nu_2,\nu_3$\footnote{Assuming three active light neutrino flavour
\cite{c3Nakamura-c2}. There are controversies
concerning more than three light neutrino flavours
\cite{c3Athanassopoulos:1996jb,c3Athanassopoulos:1997pv,
c3AguilarArevalo:2007it,c3AguilarArevalo:2010wv}.
This may be the so-called sterile neutrino which
mixes with three light neutrinos, but is phobic to weak interactions, so that
invisible decay width of $Z$-boson remains sacred. Nevertheless, there
exists literature \cite{c3Ma:1995gf,c3Goswami:1995yq,c3Sarkar:1995dd,c3Gaur:1998uk}
which deals with more than three neutrino species.}. Thus, we define
\begin{eqnarray}
|\nu'_a\rangle = U^*_{a i} |\nu_i\rangle,
\label{neut-osc-1}
\end{eqnarray}
where $\nu'_a,\nu_i$ are flavour and mass eigenstates for neutrinos,
respectively and $U^*_{a i}$ are the coefficients, carrying information of
``conversion''. So if at time, $t=0$ we have a flavour state $\nu_a$,
then the probability for transforming to another flavour state $\nu_b$
at a later time $t$ is given by (using eqn.(\ref{neut-osc-1})),
\begin{eqnarray}
P(\nu_a\to \nu_b;t) = \sum_{j} |U_{bj}e^{-i E_j t} U^*_{aj}|^2.
\label{neut-osc-2}
\end{eqnarray}
Eqn.(\ref{neut-osc-2}) is the key equation for neutrino oscillation
and the underlying physics can be explained in three pieces,
\vspace{0.1cm}
\noindent
I. $U^*_{aj}$ is the amplitude of transformation of a flavour state $\nu_a$
into some mass eigenstate $\nu_j$.
\vspace{0.1cm}
\noindent
II. Immediately after that, the factor $e^{-i E_j t}$ governs the evolution
of mass eigenstate $\nu_j$ with time.
\vspace{0.1cm}
\noindent
III. Finally, $U_{bj}$ is the amplitude of transformation of a time evolved
mass eigenstate $\nu_j$ into some other flavor state $\nu_b$.
A bit of algebraic trick for relativistic neutrinos of momentum $p$,
($E_j \simeq p + \frac{m^2_j}{2E}$) yields
\begin{eqnarray}
P(\nu_a\to \nu_b;t) = \sum_{j,k} U^*_{bk} U_{ak} U_{bj} U^*_{aj}
e^{-i \Delta m^2_{jk} \frac{L}{2E}},
\label{neut-osc-3}
\end{eqnarray}
where $\Delta m^2_{jk} = m^2_{j} - m^2_{k}$. $m_i$ is the mass of $\nu_i$
state and $L\simeq t$ (L is the distance traversed by a neutrino in time $t$
to change its flavour) using natural unit system for relativistic neutrinos.
It is clear from eqn.(\ref{neut-osc-3}) that oscillation probability
depends on the squared mass differences rather than individual masses,
thus it is impossible to probe the absolute mass scale for neutrinos
with oscillation data.
It is important to note from eqn.(\ref{neut-osc-3}), one can define
the survival probability for a flavour $\nu_a$ as
\begin{eqnarray}
P(\nu_a\to \nu_a;t) = 1- \sum_{j,k} U^*_{ak} U_{ak} U_{aj} U^*_{aj}
e^{-i \Delta m^2_{jk} \frac{L}{2E}}.
\label{neut-osc-4}
\end{eqnarray}
With the aid of eqn.(\ref{neut-osc-4}), deficit of a particular flavour
in the solar and the atmospheric neutrino flux can be explained.
However, even using eqn.(\ref{neut-osc-4}) it is hardly possible to
account for the solar neutrino problem. This
was an apparent puzzle until the matter effects in the enhancement of neutrino
oscillation were understood. Eqn.(\ref{neut-osc-4}) works only for
oscillations in vacuum\cite{c3Gribov:1968kq}. The much desired modification
for explaining matter effect induced enhanced oscillations to accommodate
the solar neutrino deficit was given by Mikheyev, Smirnov and Wolfenstein
\cite{c3Wolfenstein:1977ue,c3Mikheev:1986gs,c3Mikheev:1986wj}.
This is popularly known as the MSW effect.
\begin{flushleft}
{\it{$\maltese$ What do we know about oscillations?}}
\end{flushleft}
It has been argued already that the theory of neutrino oscillation
is sensitive to squared mass differences. It is also confirmed by this
time that, it is indeed possible to explain oscillation phenomena
with two massive neutrinos, consequently, two squared mass differences
are enough. We define them as $\Delta m^2_{solar}$ and $\Delta m^2_{atm}$,
where the word atmospheric is abbreviated as {\it{atm}}. From the observational
fact $\Delta m^2_{atm}$($\sim$ $10^{-3} {\rm eV}^2$) $\gg$
$\Delta m^2_{solar}$($\sim$ $10^{-5} {\rm eV}^2$). The sign of $\Delta m^2_{solar}$
has been affirmed experimentally to be positive, but $\Delta m^2_{atm}$
can be either positive or negative. With this sign ambiguity, two types
of light neutrino mass spectrum are possible, namely {\it{normal}} and
{\it{inverted}}. Mathematically, (i) normal hierarchy:
$m_1 < m_2 \sim \sqrt{\Delta m^2_{solar}}$ ,~$m_3 \sim
\sqrt{|\Delta m^2_{atm}|}$, (ii) inverted hierarchy: $m_1 \approx m_2 \sim
\sqrt{|\Delta m^2_{atm}|}$, $m_3 \ll \sqrt{|\Delta m^2_{atm}|}$, where
$m_1,m_2,m_3$ are light neutrino masses\footnote{It is useful to note that
$m_2>m_1$ irrespective of mass hierarchy, since $\Delta m^2_{solar}>0$ always.
However, $\Delta m^2_{atm}>0$ for normal hierarchy whereas $<0$ for the
inverted one.}. There exists a third possibility
of light neutrino mass ordering, where $m_1 \approx m_2 \approx m_3 \gg
\sqrt{|\Delta m^2_{atm}|}$ with finely splitted $m_i$s in
order to satisfy oscillation data. This is known as the quasi-degenerate
spectrum. Note that, it is impossible to
accommodate quasi-degenerate spectrum unless all three neutrinos are massive
whereas for the normal or inverted hierarchical scheme of light neutrino mass
at least two neutrinos must be massive \cite{c3Fukuda:1998mi,c3Ahmad:2002jz,c3Eguchi:2002dm}.
Probability of flavour oscillation also contains the elements of
conversion matrix, $U$ (eqn.(\ref{neut-osc-3})). The matrix $U$
acts as the bridge between flavour and mass eigenstates, having
one index from both the basis. This matrix is the leptonic analogue
of the CKM matrix (see chapter \ref{SM}) and is known as the
Pontecorvo-Maki-Nakagawa-Sakata or the PMNS matrix
\cite{c3Pontecorvo:1957cp,c3Pontecorvo:1957qd,c3Maki:1962mu,c3Pontecorvo:1967fh}.
In three flavour model this can be parametrized as
\cite{c3Schechter:1980gr,c3Chau:1984fp,c3Eidelman:2004wy}
\begin{equation}
U =
\left(\begin{array}{ccc}
{c_{12}}{c_{13}} & {s_{12}}{c_{13}} & {s_{13}} e^{-i\delta} \\ \\
-{s_{12}}{c_{23}}-{c_{12}}{s_{23}}{s_{13}}e^{i\delta} & {c_{12}}{c_{23}}
-{s_{12}}{s_{23}}{s_{13}}e^{i\delta} & {s_{23}}{c_{13}}\\ \\
{s_{12}}{s_{23}}-{c_{12}}{c_{23}}{s_{13}}e^{i\delta} & -{c_{12}}{s_{23}}
-{s_{12}}{c_{23}}{s_{13}}e^{-i\delta} & {c_{23}}{c_{13}}
\end{array}\right).U'(\alpha),
\label{PMNS1}
\end{equation}
where $c_{ij} = \cos{\theta_{ij}}$, $s_{ij} = \sin{\theta_{ij}}$ and
$U'(\alpha) = diag(e^{-i\alpha_1/2},1,e^{-i\alpha_2/2})$.
Here $\alpha_1,\alpha_2,\delta$ are complex phases. The phases $\alpha_1,\alpha_2$
can be non-zero only if neutrinos are Majorana particle in nature
(will be addressed later). Neutrino oscillation is insensitive to Majorana phases.
The phase $\delta$ is a Dirac CP-violating phase and can appear in oscillation
dynamics. We stick to Charge-Parity(CP)-preserving case (zero phases)
throughout this thesis.
It is interesting to note that unlike the CKM matrix of quark sector,
the PMNS matrix has a rather non-trivial structure. Present
experimental evidence favours a tri-bimaximal mixing in
light neutrino sector \cite{c3Harrison:2002er}, though there exist other alternatives
\cite{c3Harrison:1994iv,c3Altarelli:1998sr,c3Barger:1998ta,c3Baltz:1998ey,c3Barger:1998ed
,c3Bjorken:2005rm} (see also \cite{c3Mohapatra:1998ka} and references
therein)\footnote{Some of these proposals are now experimentally ruled out.}.
The atmospheric mixing angle $(\theta_{23})$ is close to maximal $(\sim \pi/4)$ and the
solar mixing angle $(\theta_{12})$ is also large $(\sim 35^\circ)$. The third
and the remaining mixing angle, the reactor angle $(\theta_{13})$ is
the most difficult one to measure. There exist a host of literature,
both on theoretical prediction and experimental observation
for the value and the measurement of the angle $\theta_{13}$.
(see ref. \cite{c3Ardellier:2006mn,c3Guo:2007ug,c3Fogli:2008jx,c3Schwetz:2008er,
c3Maltoni:2008ka,c3GonzalezGarcia:2010er} for recent updates. Also see ref.
\cite{c3Mezzetto:2009cr}). At the zeroth-order approximation
\begin{equation}
\theta_{23} = \frac{\pi}{4},~\theta_{12} \simeq 35^\circ,~\theta_{13} =0.
\label{PMNS2}
\end{equation}
Recently, non-zero value for the angle $\theta_{13}$ has been
reported by the T2K collaboration \cite{Abe:2011sj} both
for the normal and inverted hierarchy in the light neutrino masses.
For normal (inverted) hierarchy and at $90\%$ C.L. this value
is reported to be
\begin{equation}
0.03(0.04)< {\rm sin}^2{2\theta_{13}} < 0.28(0.34) .
\label{T2K-13}
\end{equation}
The oscillation parameters $(\Delta m^2_{sol},\Delta m^2_{atm},\theta_{23},
\theta_{12},\theta_{13})$ are highly constrained by experiments.
In table \ref{osc-para} best-fit values of these parameters for the global
three-flavor neutrino oscillation data are given \cite{c3Schwetz:2008er}.
\begin{table}[ht]
\centering
\begin{tabular}{c || c || c}
\hline \hline
Parameter & Best fit & 3$\sigma$ limit\\ \hline \hline
$\Delta m^2_{sol} \times 10^5$ ${\rm eV}^2$ &
$7.65^{+0.23}_{-0.20}$ & $7.05 - 8.34$\\
$|\Delta m^2_{atm}| \times 10^3$ ${\rm eV}^2$
& $2.40^{+0.12}_{-0.11}$ & $2.07 - 2.75$\\
${\rm sin}^2\theta_{23}$
& $0.50^{+0.07}_{-0.06}$ & $0.25 - 0.37$ \\
${\rm sin}^2\theta_{12}$
& $0.304^{+0.022}_{-0.016}$ & $0.36 - 0.67$ \\
${\rm sin}^2\theta_{13}$
& $0.01^{+0.016}_{-0.011}$ & $\leqslant 0.056$ \\ \hline \hline
\end{tabular}
\caption{\label{osc-para}
Best fit values and $3\sigma$ ranges of oscillation parameters from
three flavour global data \cite{c3Schwetz:2008er}.
}
\end{table}
The experiments like Borexino \cite{c3Arpesella:2008mt},
CHOOZ \cite{c3Apollonio:1999ae,c3Apollonio:2002gd},
Double Chooz \cite{c3Ardellier:2004ui,c3Ardellier:2006mn},
KamLAND \cite{c3:2008ee,c3Gando:2010aa}, Kamiokande \cite{c3Wendell:2010md},
Super-Kamiokande \cite{c3Cravens:2008zn,c3Abe:2010hy},
K2K \cite{c3Ahn:2006zza}, MINOS \cite{c3Adamson:2008zt,c3Adamson:2011ig},
GNO \cite{c3Kirsten:2003ev,c3Altmann:2005ix}, SNO \cite{c3Aharmim:2009gd}
and others are now in the era of precision measurements. More, sophisticated experiments
like RENO \cite{c3Ahn:2010vy}, OPERA \cite{c3Acquafredda:2006ki,c3Acquafredda:2009zz} etc.
have already been initiated and an extremely precise global fit
is well anticipated in near future. One can go through ref. \cite{c3Schwetz:2011qt}
for a recent analysis of the precision results.
\vspace*{0.5cm}
\begin{flushleft}
{\it{$\maltese$ Searching for a neutrino mass}}
\end{flushleft}
Theory of neutrino oscillation depends on squared mass differences, which
are shown in table \ref{osc-para}. It is then, natural to ask that what is
the absolute scale for a neutrino mass. Is it small, $\sim$ a few eV so that
small squared mass differences (see table \ref{osc-para}) seem natural or
the absolute masses are much larger and have unnatural fine splittings
among them.
Possible evidences of absolute neutrino mass scale can come from different
experiments which are discussed below.
\vspace{0.1cm}
\noindent
I. {\bf{Tritrium beta decay}} $\blacktriangleright$ There are a host of experimental
collaboration (Mainz \cite{c3Bonn:2001tw,c3Kraus:2004zw},
Troitsk \cite{c3Belesev:1995sb,c3Lobashev:2001uu},
KARTIN \cite{c3Osipowicz:2001sq,c3Lobashev:2003kt}) looking for the
modification in the beta spectrum in the process $^3{\rm H}\to {^3{\rm {He}}}+e^-+\overline{\nu_e}$
in the presence of non-zero $m_i$. Indeed, the Kurie plot \cite{c3PhysRev.49.368}
shows deviation near the endpoint with $m_i\neq0$ (see figure \ref{Kurie}).
\begin{figure}[ht]
\centering
\includegraphics[width=7.05cm]{FigsPDF/Kurie.pdf}
\caption{Kurie function, $K(E_e)$ versus energy $(E_e)$ of $\beta$-particle $(e^-)$ plot
for neutrino mass, $m_\nu=0$ (solid line) and $m_\nu \neq 0$ (dashed line). $E_0$ is the
energy release.}
\label{Kurie}
\end{figure}
The experiments, however in reality
measure an effective neutrino mass $m_\beta = \sqrt{\sum |U_{ei}|^2 m^2_i}$
($U$ is the PMNS matrix). The present bound is \cite{c3Nakamura-c2}
\begin{eqnarray}
m_\beta < 2.0 ~{\rm eV}.
\label{neut-nonosc-1}
\end{eqnarray}
\vspace{0.1cm}
\noindent
II. {\bf{Neutrinoless double beta decay}} $\blacktriangleright$ Consider two beta decays,
($n\to p^++e^-+\nu_e$ or $d\to u+e^-+\nu_e$, in the quark level of proton (neutron)
simultaneously, such that (anti)neutrino emitted in one decay is
somehow absorbed in the other. This leads to the neutrinoless double beta decay,
$0\nu\beta\beta$ (figure \ref{0nubb1}).
\begin{figure}[ht]
\centering
\includegraphics[width=3.15cm]{FigsPDF/0nubb1.pdf}
\caption{Diagram for neutrinoless double beta decay in the SM.
Subscript $L$ signifies the left handed nature of weak interaction.}
\label{0nubb1}
\end{figure}
However, it is clear from figure \ref{0nubb1}, this breaks lepton
number conservation. The quantity $m_L$ represents Majorana mass
(will be addressed soon) of a left-handed neutrino, which is responsible
for this lepton number violation $(\not\!\!{L})$.
Not only the lepton number is broken in this interaction, but
through Majorana mass term $m_L$ process like this also breaks
chirality conservation \cite{c3Beshtoev:2009zz}. The measurable quantity is defined as
$m_{\beta\beta} = \sum U_{ei}^2 m_i$. Since
$m_{\beta\beta} \propto U_{ei}^2$, rather than $|U_{ei}|^2$, information
about the $\delta$-phase is not lost until one asks for CP-preservation
in the lepton sector. Experimental reporting of neutrinoless double beta decay
is controversial. The result obtained
by Heidelberg-Moscow experiment \cite{c3Baudis:1999xd,
c3KlapdorKleingrothaus:2004wj,
c3KlapdorKleingrothaus:2006ff}, CUORICINO \cite{c3Arnaboldi:2008ds} suggests
\begin{eqnarray}
m_{\beta\beta} < 0.2-0.6 ~{\rm eV}.
\label{neut-nonosc-2}
\end{eqnarray}
However, there are experiments like CUORE \cite{c3Arnaboldi:2008ds},
EXO \cite{c3Gornea:2010zz}, GERDA \cite{c3Smolnikov:2008fu}, MAJORANA and
a few others, which are expected to shed light on this
claim in near future. One important point about $0\nu\beta\beta$ is
that unless a neutrino possesses a Majorana mass term,
$m_{\beta\beta}=0$. This is true for different models
and has been confirmed by model independent analysis \cite{c3Schechter:1981bd}.
\begin{figure}[ht]
\centering
\includegraphics[width=3.15cm]{FigsPDF/0nubb2-mod.pdf}
\caption{Diagram for neutrinoless double beta decay in $R_p$-violating
supersymmetric models. See text for further details.}
\label{0nubb2}
\end{figure}
It must be emphasized here that, actually the presence
of Majorana mass term is a sufficient condition for non-zero $m_{\beta\beta}$,
but not necessary, for example $0\nu\beta\beta$ in $R_p$-violating
(section \ref{R-parity}) supersymmetric model (see figure \ref{0nubb2})
can occur without a neutrino Majorana mass term (see ref. \cite{c3Hirsch:1995ek}).
In figure \ref{0nubb2},
$\tilde g$ represents a gluino, superpartner of a gluon and $\tilde u$ is a
up-type squark (see figure \ref{MSSM-particle}).
\vspace{0.1cm}
\noindent
III. {\bf{Cosmology}} $\blacktriangleright$ Neutrino masses are also constrained by
the standard big-bang cosmology. However, in this case the bound exists
on sum of neutrino masses. There were earlier works \cite{c3Gershtein:1966gg,
c3Cowsik:1972gh,c3Szalay:1976ef} in this connection, where, a bound on the sum
of neutrino mass was obtained from the bound on the
total density of the universe. However, the present bound as obtained from
sophisticated experiment like WMAP \cite{c3Komatsu:2008hk,c3Dunkley:2008ie,
c3Hinshaw:2008kr,c3Komatsu:2010fb,c3Larson:2010gs,c3Jarosik:2010iu} is much
stringent and is given by
\begin{eqnarray}
\sum_{i=1}^{3} m_{i} \leq 0.58 ~{\rm eV}(95\%~{\rm C.L.}).
\label{neut-nonosc-3}
\end{eqnarray}
Note that only for the case of quasi-degenerate
light neutrino masses individual masses can be much larger
compared to the squared mass differences (see table \ref{osc-para}).
Thus quasi-degenerate
neutrinos masses are highly constrained by eqn.(\ref{neut-nonosc-3}).
So far we have addressed the features of oscillation and non-oscillation
experiments to constrain the neutrino physics. It is the time to
demonstrate the origin of neutrino mass. But even before that, it
is important to discuss the nature of neutrino masses, that is
whether they are Dirac or Majorana particles \cite{c3Fiorini:2006cg,
c3Kayser:2009zz}.
\section{{\bf M}odels of neutrino mass}\label{neut-mass}
\begin{flushleft}
{\it{$\maltese$ Nature of neutrino mass, Dirac versus Majorana}}
\end{flushleft}
It is well-known that the charge conjugation operator
$\hat C$ is defined as
\begin{eqnarray}
\hat C: \psi \to \psi^c = C {\bar{\psi}}^T,
\label{neut-mass-a1}
\end{eqnarray}
where $C$ is the matrix representation for $\hat C$, $T$ denotes transpose
and $\psi$ is a four component spinor. It is then apparent that for
a charge neutral fermion
\begin{eqnarray}
\psi^c = \psi.
\label{neut-mass-a2}
\end{eqnarray}
Any $\psi$ which obeys eqn.(\ref{neut-mass-a2}) is known as a
Majorana fermion\footnote{A free Majorana field is an eigenstate of charge conjugation operator.}.
On the contrary, the so-called Dirac fermions are
known to follow $\psi\neq\psi^c$. Now, since the neutrinos are
the only charge neutral particle in the SM there is a possibility,
that a neutrino is a Majorana particle,
\cite{c3Majorana:1937vz} rather than a Dirac particle \cite{c3Dirac:1928hu}.
\vspace{0.1cm}
\noindent
I. {\bf{Dirac Mass}} $\blacktriangleright$ If there were right-handed neutrinos
$(\nu_R)$ in the SM, a non-zero Dirac mass $(m_D)$ is well expected. The
mass term using eqn.(\ref{fermion-mass}) can be written as
\begin{eqnarray}
y_\nu \overline{L} \tilde{\Phi} \nu_R~+~h.c = \frac{y_\nu .v}{\sqrt{2}} \overline{\nu_L} \nu_R
~+~h.c= m_D \overline{\nu_L} \nu_R~+~h.c,
\label{neut-mass-a3}
\end{eqnarray}
where $\nu_{L(R)}$ is a left(right) handed neutrino field and $y_\nu$
is the neutrino Yukawa coupling. Demanding a neutrino mass $\sim$
1 eV one gets $y_\nu$ $\sim$ $10^{-11}$. But immediately then, it
is legitimate to ask why $m_D$ is extremely small compared to other
masses as shown in table \ref{SM-spectrum} or alternatively why
$y_\nu$ is much smaller, compared to say electron Yukawa coupling, $Y_e
$ $\sim$ $10^{-6}$.
The Dirac mass terms respect the total lepton number $L$, but not the
family lepton number, $L_e,L_\mu,L_\tau$. If $m_D\neq0$, a neutrino
is different from an anti-neutrino.
\vspace{0.1cm}
\noindent
II. {\bf{Majorana Mass}} $\blacktriangleright$ A Majorana mass term not only violate
the family lepton number, but also the total lepton number. In
general they are given by
\begin{eqnarray}
m_{ii} \overline{\nu_i} \nu^c_i,
\label{neut-mass-a4}
\end{eqnarray}
where, $i=L,R$. $\nu^c_{L(R)}$ represents a CP conjugated state.
A Majorana spinor has only two degrees of freedom because of
eqn.(\ref{neut-mass-a2}), whereas a Dirac spinor has four degrees of freedom.
Thus, two degenerate Majorana neutrinos of opposite CP, when maximally
mixed form a Dirac neutrino. A Majorana neutrino is believed to
be one of the main reasons for non-vanishing amplitude in $0\nu\beta\beta$
(see section \ref{neut-osc}). However, just like the Dirac case
it is also important to explain how a neutrino gets a tiny Majorana
mass? The answer will be given in the subsequent paragraph.
In the most general occasion, a neutrino can posses a ``Dirac + Majorana'' mass
term. A term of this kind can lead to extremely interesting
neutrino-anti neutrino oscillation which violates total lepton
number (see ref. \cite{c3Bilenky:1998dt} for detailed discussion).
It is also important to mention that since neutrino oscillation
does not violate total lepton number, it is hardly possible to
differentiate between a Dirac and a Majorana nature from the
theory of neutrino oscillation. The $0\nu\beta\beta$ is definitely
an evidence for Majorana nature. Besides, one can use the favour
of electric and magnetic dipole moment measurement, to discriminate
these scenarios \cite{c3Bernstein:1963qh,c3Okun:1986hi,c3Schechter:1981hw,
c3Nieves:1981zt,c3Kayser:1982br,c3Shrock:1982sc,c3Babu:1987be,c3Raffelt:1999gv}.
\subsection{{\bf M}ass models I}\label{neut-mass-I}
It is apparent by now that we need to extend either the particle
content of the SM or need to enlarge the gauge group to accommodate neutrino
mass. In this subsection we discuss the models for neutrino
mass generation without introducing supersymmetry. Some of these models
generate masses at the tree level and the remaining through loop
processes.
\vspace*{0.02cm}
\begin{center}
{\bf{1. Renormalizable interaction with triplet Higgs}}.
\end{center}
\vspace*{0.02cm}
Consider a term in the Lagrangian as
\begin{eqnarray}
f_{\Delta} (\ell^T C i \sigma_2 {\bm{\sigma}} \ell) {\Delta} ~+h.c,
\label{neut-mass-I1}
\end{eqnarray}
where $C$ is the charge conjugation matrix, ${\bm{\sigma}}$'s are Pauli
matrices and $\Delta$ is a {\it{triplet}} Higgs field with hypercharge,
$Y=2$. If further we assume that ${\Delta}$ has $L=-2$, then Lagrangian
given by eqn.(\ref{neut-mass-I1}) conserves lepton number. The mass term
for neutrinos will be then
\begin{eqnarray}
m_\nu \approx {f_\Delta v_\Delta},
\label{neut-mass-I2}
\end{eqnarray}
where $v_\Delta$ is the VEV of ${\Delta}$ field. But this will also
produce massless triplet Majoron \cite{c3Gelmini:1980re,c3Chikashige:1980ui,
c3Georgi:1981pg} since lepton number is spontaneously
broken by the VEV of $\rm{SU(2)_L}$ triplet ${\Delta}$ field. Missing
experimental evidence has ruled out this model.
One alternative could be to put $L=0$ for ${\Delta}$, which breaks the lepton number
explicitly. This situation, though free of triplet Majoron are highly
constrained from $\rho$ parameter measurement (eqn.(\ref{rho-param})), which requires
$v_\Delta<8$ GeV. Once again for $m_\nu \sim$ 1 eV, $f_\Delta \sim$
$10^{-10}$.
\vspace*{0.05cm}
\begin{center}
{\bf{2. Non-renormalizable interactions}}.
\end{center}
If we want to build a model for tiny neutrino mass with the SM
degrees of freedom, the immediate possibility that emerges is the
so-called {\it{dimension five}} Weinberg operator \cite{c3Weinberg:1979sa,c3Weinberg:1980bf}
\begin{eqnarray}
Y_{ij}\frac{(\ell_i \Phi)(\Phi \ell_j)}{M},
\label{neut-mass-I3}
\end{eqnarray}
where $\ell$ are the SM ${\rm{SU(2)}_L}$ lepton doublets (eqn.(\ref{SM-gauge-group}))
and $\Phi$ is the SM Higgs doublet (eqn.(\ref{SM-Higgs})) with VEV $\frac{v}{\sqrt{2}}$.
$Y_{ij}$ stands for some dimensionless coupling. $M$ is some high scale in the theory,
and is the messenger of some new physics. Thus the small Majorana neutrino mass
coming from this $\Delta L = 2$ operator is
\begin{eqnarray}
(m_\nu)_{ij}=\frac{Y_{ij} v^2}{2 M}.
\label{neut-mass-I4}
\end{eqnarray}
Note that if $M$ is large enough ($\sim$ $10^{14}$ GeV) the coupling,
$Y_{ij} \sim$ 1 (close to perturbative cutoff), for right order of
magnitude in the neutrino mass $(m_\nu)$ $\sim 0.1$ eV. However,
this is a rather challenging scenario, since it is hardly possible
to probe $M$ ($\sim$ $10^{14}$ GeV) in a collider experiment.
One viable alternative is a TeV scale $M$,
which is possible to explore in a collider experiment.
Note that for such a choice of $M$, $Y_{ij}$ is much smaller.
Maintaining the gauge invariance and renormalizability, the effective
operator can arise from three possible sources.
\vspace{0.1cm}
\noindent
I. {\bf{Fermion singlet}} $\blacktriangleright$ The intermediate massive particle is
a SM gauge singlet particle, $(S)$. This is the example of Type-I seesaw
mechanism \cite{c3Minkowski:1977sc,c3Yanagida:1979as,c3Glashow:1979nm,c3Mohapatra:1979ia,
c3GellMann:1980vs,c3Schechter:1980gr,c3Schechter:1981cv} (seesaw mechanism will be
discussed later in more details).
The light neutrino mass is given by
\begin{eqnarray}
m_\nu=\frac{f^2_s v^2}{2 {M_S}},
\label{neut-mass-I5}
\end{eqnarray}
where $M_S$ is the mass of the singlet fermion and $f_s$ is the
$\ell \Phi S$ coupling. (see figure \ref{seesaw} (a) and (b)). It
is important to note that the $\Delta L=2$ effect can arise either
using a singlet with non-zero lepton number (right handed neutrino,
$\nu_R$) (figure \ref{seesaw} (a)) or through a singlet, $S$ without
lepton number (figure \ref{seesaw} (b)).
\begin{figure}[ht]
\centering
\includegraphics[width=9.95cm]{FigsPDF/seesaw.pdf}
\caption{Different types of seesaw mechanism. The cross on the fermionic propagator
signifies a Majorana mass term for the corresponding fermion.}
\label{seesaw}
\end{figure}
\vspace{0.1cm}
\noindent
II. {\bf{Scalar triplet}} $\blacktriangleright$ The intermediate massive particle is
a scalar triplet $(\Delta)$ under the group ${\rm{SU(2)}_L}$. It is singlet under
${\rm{SU(3)}_C}$ and has hypercharge, $Y = 2$. This is the so-called Type-II
seesaw mechanism \cite{c3Konetschny:1977bn,c3Magg:1980ut,c3Lazarides:1980nt,c3Cheng:1980qt,
c3Mohapatra:1980yp}. The light neutrino mass is given by
\begin{eqnarray}
m_\nu=\frac{f_\Delta s_\Delta v^2}{2 {M^2_{\Delta}}},
\label{neut-mass-I6}
\end{eqnarray}
where $M_\Delta$ is the mass of the scalar triplet. $f_\Delta$
and $s_\Delta$ are the $LL\Delta$ and $\Phi\Phi\Delta$ coupling,
respectively (see figure \ref{seesaw} (c)).
\vspace{0.1cm}
\noindent
III. {\bf{Fermion triplet}} $\blacktriangleright$ A triplet fermion $(\Sigma)$ acts
as the mediator and this is an illustration of the Type-III seesaw
mechanism \cite{c3Foot:1988aq,c3Ma:1998dn}.
This $\Sigma$ is a triplet under ${\rm{SU(2)}_L}$ but singlet under
${\rm{SU(3)}_C}$. Hypercharge for $\Sigma$ is zero.
The light neutrino mass is given by
\begin{eqnarray}
m_\nu=\frac{f^2_\Sigma v^2}{2 {M_{\Sigma}}},
\label{neut-mass-I7}
\end{eqnarray}
where $M_\Sigma$ is the mass of the fermion triplet. $f_\Sigma$
is the $L\Phi\Sigma$ coupling (see figure \ref{seesaw} (d)).
A very important aspect of these seesaw models and the associated
Majorana nature is that they can produce same sign dilepton at a
collider experiment \cite{c3Keung:1983uu}
apart from a non-zero amplitude for the $0\nu\beta\beta$ process.
The collider phenomenology for Type-II or III seesaw models
are more attractive compared to the Type-I scenario, due to
the involvement of a SM gauge singlet fermion in the latter case. Also a seesaw generated
neutrino mass can have implications in flavour violating processes,
\cite{c3Hall:1983id,c3Lee:1984kr,c3Lee:1984tn,c3Dawson:1985vr,c3Dimopoulos:1988jw}
leptogenesis \cite{c3Fukugita:1986hr,c3Luty:1992un,c3Buchmuller:2004nz,
c3Davidson:2008bu}. However, none of these issues are
addressed here.
There exist other interesting seesaw models like (a) Inverse seesaw
\cite{c3Mohapatra:1986bd} (requires an extra SM singlet $S$ apart from $\nu_R$),
(b) Linear seesaw \cite{c3Malinsky:2005bi}, (c) Double seesaw
\cite{c3Mohapatra:1986aw,c3Mohapatra:1986bd}, (d) Hybrid seesaw
\cite{c3Wetterich:1981bx,c3Wetterich:1998vh,c3Antusch:2004xd,c3Chen:2005jm}
etc. Some of these models have
definite group theoretical motivation. Also neutrino masses can arise
in the left-right symmetric model \cite{c3Pati:1974yy,c3Mohapatra:1974gc,
c3Senjanovic:1975rk,c3Senjanovic:1978ev}. It is important to
note that the Weinberg operator can also give rise to neutrino
mass via loop effects \cite{c3Zee:1980ai,c3Zee:1985id,c3Babu:1988ki,c3Branco:1988ex,
c3Ma:2006km,c3Ma:2008uza}. Some of the very early attempts
in this connection have been addressed in references \cite{c3Georgi:1972hy,c3Cheng:1977ir}.
However, any more involved discussion of these
topics are beyond the scope of this thesis.
A comprehensive information about various neutrino mass models is given
in ref. \cite{c3Vicente:2011pf}.
\begin{flushleft}
{\it{$\maltese$ The seesaw mechanism}}
\end{flushleft}
It has been already argued that the seesaw mechanism (Type-I,II,III and others)
is perhaps the most convenient way to generate small Majorana masses for neutrinos.
But what is actually a seesaw mechanism and how does it lead to small Majorana mass?
It is true that a Majorana mass term violates lepton number by {\it{two units}},
but this could happen either through a pair of $\Delta L=1$ effects
(see figure \ref{seesaw} (b),(d)) or by a $\Delta L=2$ vertex
(see figure \ref{seesaw} (a),(c)). We will discuss the canonical seesaw mechanism
(Type-I, however this analysis is also applicable for Type-III) using a simple model
containing left-handed neutrino, $\nu_L$ and some fermion $f$, either
a SM gauge singlet (Type-I seesaw) or an ${\rm{SU(2)}_L}$ triplet (Type-III seesaw).
Further we assume that to start with Majorana mass for $\nu_L$ is absent.
Majorana mass for $f$ is given by $M_f$ and the co-efficient of the
mixing term $(\nu_Lf)$ is written as $m_m$. The mass matrix in the $\nu_L,f$
basis is given by
\begin{eqnarray}
\mathcal{M} = \left(\begin{array}{cc}
0 & m_m \\
m_m & M_f
\end{array}\right).
\label{seesaw-mat}
\end{eqnarray}
If $M_f\gg m_m$, the eigenvalues are given as
\begin{eqnarray}
m_{light} \simeq -\frac{m^2_m}{M_f},{\rm{~and}}~m_{heavy} \simeq M_f.
\label{seesaw-eigval1}
\end{eqnarray}
If $\chi_1,\chi_2$ form a new basis where
$\mathcal{M}\to~diag(m_{light},m_{heavy})$, then mixing between
$\chi_1,\chi_2$ and $\nu_L,f$ basis is parametrized by an angle
$\theta$ with
\begin{eqnarray}
{\rm tan}2\theta = \frac{2 m_m}{M_f}.
\label{seesaw-mixing}
\end{eqnarray}
Eqn.(\ref{seesaw-eigval1}) is the celebrated seesaw formula
for neutrino mass generation. Now the left neutrino possesses
a non-zero Majorana mass term which was {\it{zero}} to start with. Also
the mass $m_{light}$, is suppressed by a factor $\frac{m_m}{M_f}$,
and thus always is small as long as $M_f\gg m_m$.
Considering three generations of light neutrinos,
the unitary matrix (orthogonal in the absence of complex phases) $U'$ which rotates
the off-diagonal basis ($m_{light}$ is a $3\times3$ matrix for
the three generation case) to the
diagonal one is known as the PMNS matrix (eqn.(\ref{PMNS1})).
Mathematically,
\begin{eqnarray}
U^{'^T} m_{light} U' = diag(m_{\nu_i}),~~i=1,2,3.
\label{seesaw-PMNS}
\end{eqnarray}
In the Type-I and Type-III seesaw process, the
effective leptonic mixing matrix or PMNS matrix looses its unitarity
\cite{c3Langacker:1988up,c3Antusch:2006vwa,c3Ma:2009du} $\sim \frac{m_m}{M_f}$.
The unitary nature is restored when $M_f\to\infty$. This feature is however
absent in Type-II seesaw mechanism. A discussion on the phenomenological implications
of this non-unitarity is beyond the scope of this thesis.
It is essential to note that when $f$ is a right handed neutrino, $\nu_R$,
then $m_m\equiv m_D$, the Dirac mass term. Further replacing $M_f$ by
$M_R$, in the limit $m_D\gg M_R$, we get from eqn.(\ref{seesaw-eigval1})
\begin{eqnarray}
m_{light} \simeq m_D-\frac{M_R}{2},~{\rm{and}}~|m_{heavy}| \simeq m_D+\frac{M_R}{2}.
\label{seesaw-pseudo-Dirac}
\end{eqnarray}
This pair is known to behave as Dirac neutrino in various
aspects and is named as {\it{quasi-Dirac neutrinos}}
\cite{c3Wolfenstein:1981kw,c3Bilenky:1987ty}.
\subsection{{\bf M}ass models II}\label{neut-mass-II}
In this subsection we try to address the issues of neutrino
mass generation in a supersymmetric theory \cite{c3Hirsch:2004he}, which is
one of the prime themes of this thesis. $\not\!\!{R_p}$ through bilinear terms ($\varepsilon_i$,
see eqn.(\ref{MSSM-superpotential-2})) is the simplest extension of the MSSM
\cite{c3Diaz:1997xc}, which provides a framework for neutrino masses and mixing angles
consistent with experiments. It is important to clarify
that there are various sources for light neutrino mass
generation in supersymmetry without $\not\!\!{R_p}$ (section \ref{R-parity}),
for example see refs. \cite{c3Borzumati:1986qx,c3Hisano:1995cp,c3Hisano:1995nq,
c3Hirsch:1997vz,c3Grossman:1997is,c3Hirsch:1997dm,c3Davidson:1998bi,
c3Aulakh:1999cd,c3Casas:1999ac,c3ArkaniHamed:2000bq,c3Das:2010wp,c3Aoki:2010ib,c3Abada:2010ym}.
But we stick to a very special case where the origin of neutrino mass
generation is entirely supersymmetric, namely through $R_p$-violation.
An introduction to $\not\!\!{R_p}$ was given in section \ref{R-parity} and
here we will concentrate only on the effect of $\not\!\!{R_p}$ in neutrino mass
generation.
The effect of $\not\!\!{R_p}$ and neutrino masses in a supersymmetric theory
has received immense interest for a long time and there exist a host
of analyses to shed light on different phenomenological
aspects of broken $R_p$ (see section \ref{R-parity} and references therein).
We quote a few of these references having connections with the theme of this
thesis, namely (a) neutrino mass generation either through explicit
$\not\!\!{R_p}$ \cite{c3Joshipura:1994ib,c3de-Campos:1995av,c3Banks:1995by,
c3Nowakowski:1995dx,c3Diaz:1997xc,c3Akeroyd:1997iq,c3Faessler:1997db,c3Hirsch:1998kc,
c3Chang:1999nu,c3Datta:1999yd,c3Diaz:1999is,c3Aulakh:2000sn,c3Restrepo:2001me,
c3Abada:2006qn,c3Abada:2006gh,c3Hall:1983id,
c3Lee:1984kr,c3Lee:1984tn,c3Ellis:1984gi,c3Dimopoulos:1988jw,
c3Barbieri:1990qj,c3Enqvist:1992ef,c3Hempfling:1995wj,c3de-Carlos:1996du,
c3Borzumati:1996hd,c3Nilles:1996ij,c3Nardi:1996iy,c3Roy:1996bua,c3Drees:1997id,
c3Chun:1998gp,c3Bednyakov:1998cx,c3Ferrandis:1998ii,c3Grossman:1998py,c3Chun:1998ub,
c3Rakshit:1998kd,c3Kaplan:1999ds,c3Ma:1999ni,c3Joshipura:1999hr,c3Choi:1999tq,
c3Grossman:1999hc,c3Romao:1999up,c3Abada:1999ai,c3Haug:1999kr,c3Chun:1999bq,
c3Takayama:1999pc,c3Davidson:1999mc,c3Adhikari:1999pa,c3Hirsch:2000jt,c3Hirsch:2000ef,
c3Davidson:2000uc,c3Grossman:2000ex,
c3Abada:2000xr,c3Mira:2000gg,c3Davidson:2000ne,c3Porod:2000hv,c3Joshipura:2001mq,c3Barger:2001xe,
c3Abada:2001zh,c3Joshipura:2002fc,c3Chun:2002rh,c3Borzumati:2002bf,c3Abada:2002ju,c3Chun:2002vp,c3Diaz:2003as,
c3Grossman:2003gq,c3Rakshit:2004rj,c3Jung:2004rd,c3Chun:2004mu,c3Chemtob:2006ur,
c3Dedes:2006ni,c3Mukhopadhyaya:2006is,c3Chun:2006ss,c3Dedes:2007ef,c3Dey:2008ht,c3Hundi:2009yf,c3JeanLouis:2009du}
or through spontaneous $\not\!\!{R_p}$ \cite{c3Romao:1991ex,
c3Nogueira:1990wz,c3Romao:1991tp,c3Giudice:1992jg,c3Kitano:1999qb,c3Frank:2007un,c3Hirsch:2008ur,
c3Mitra:2009jj,c3Ross:1984yg,c3Dawson:1985vr,
c3Santamaria:1987uq,c3Umemura:1993wc} (tree and(or) loop corrections)
and (b) neutrino mass generation and(or) collider phenomenology
\cite{c3Ellis:1984gi,c3Barger:1989rk,c3Dawson:1985vr,c3Dimopoulos:1988jw,
c3Barbieri:1989vb,c3Nogueira:1990wz,c3GonzalezGarcia:1991ap,c3Romao:1991tp,
c3Dreiner:1991pe,c3Enqvist:1992ef,c3Godbole:1992fb,c3Butterworth:1992tc,
c3de-Campos:1995av,c3Bhattacharyya:1995pr,c3Bhattacharyya:1995bw,c3Nowakowski:1995dx,
c3Adhikari:1996bm,c3Choudhury:1996ia,c3Nilles:1996ij,c3Roy:1996bua,c3Kim:1997rr,c3Choudhury:1997dt,
c3Akeroyd:1997iq,c3Mukhopadhyaya:1998xj,c3Diaz:1998wq,c3Chun:1998ub,c3Bisset:1998hy,
c3Mukhopadhyaya:1999gy,c3Choi:1999tq,c3Ghosh:1999ix,c3Allanach:1999bf,c3Romao:1999up,
c3Takayama:1999pc,c3Davidson:1999mc,c3Adhikari:1999pa,c3Porod:2000hv,c3Datta:2000ci,
c3Restrepo:2001me,c3Barger:2001xe,c3Saha:2002kt,c3Chun:2002rh,c3Hirsch:2003fe,c3Jung:2004rd,
c3Datta:2006ak,c3Hirsch:2008ur,c3Arhrib:2009mz,c3Bandyopadhyay:2009xa,c3Perez:2009mu,c3Nath:2010zj,
c3DeCampos:2010yu}.
We start with a brief discussion of spontaneous $\not\!\!{R_p}$ and later
we will address the issues of neutrino mass generation with explicit
breaking of $R_p$.
\vspace*{0.02cm}
\begin{center}
{\bf{I. Spontaneously broken $R$-parity}}.
\end{center}
\vspace*{0.02cm}
The idea of spontaneous $\not\!\!{R_p}$ was first implemented in ref.\cite{c3Aulakh:1982yn}
through spontaneous violation of the lepton number. The lepton number
violation occurs through the left sneutrino VEVs. It was revealed in ref.
\cite{c3Nieves:1983kn} that if supersymmetry breaking terms include
trilinear scalar couplings and gaugino Majorana masses, only one
neutrino mass would be generated at the tree level. Remaining two small
masses are generated at the one-loop level \cite{c3Aulakh:1982yn}. Different
phenomenological implications for such a model
were addressed in references \cite{c3Ellis:1984gi,c3Ross:1984yg,
c3Santamaria:1987uq,c3Santamaria:1988ic,c3Santamaria:1988zm}. A consequence of
spontaneous $\not\!\!{L}$ appears in the form of a massless Nambu-Goldstone boson
called Majoron \cite{c3Chikashige:1980ui,c3Gelmini:1980re}. Unfortunately,
a Majoron, arising from the breaking of gauge non-singlet fields
(in this case a doublet Majoron from the left sneutrino VEVs which is a
member of ${\rm{SU(2)}_L}$ family) is strongly disfavored by
electroweak precision measurements ($Z$-boson decay width)
\cite{c3Adeva:1989mn,c3Decamp:1989tu,c3Akrawy:1989pi,c3Aarnio:1989tv}
and astrophysical constraints \cite{c3Georgi:1981pg,c3Fukugita:1982ep,c3Raffelt:1996wa,
c3Kachelriess:2000qc}. Thus this doublet-Majoron model is ruled out
\cite{c3GonzalezGarcia:1989zh,c3Romao:1989yh,c3Nogueira:1990wz}.
The possible shortcomings of a doublet Majoron model are removable by
using the VEV of a gauge-singlet field as suggested in ref.\cite{c3Masiero:1990uj}.
Most of these models break the lepton number spontaneously by giving VEV to a
singlet field carrying one unit of lepton number \cite{c3Masiero:1990uj,c3Romao:1991ex,
c3Romao:1992vu}. However, there exists model where the singlet field carries
two unit of lepton number \cite{c3Giudice:1992jg}. This singlet Majoron model
\cite{c3Chikashige:1980ui} is not ruled out by LEP data. More phenomenological
implications of this class of models are addressed in refs. \cite{c3Romao:1991tp,
c3Hirsch:2004rw,c3Hirsch:2005wd,c3Hirsch:2006di,c3Hirsch:2008ur,c3Hirsch:2009ee,
c3Bhattacharyya:2010kr}.
We just briefly mentioned the basics of spontaneous $\not\!\!{R_p}$ for the sake
of completeness. These issues are not a part of this thesis work and hence we
do not elaborate further. A dedicated discussion on the spontaneous violation of
$R_p$ has been given in ref.\cite{c3Vicente:2011pf}.
\vspace*{0.02cm}
\begin{center}
{\bf{II. Explicit breaking of $R$-parity}}.
\end{center}
\vspace*{0.02cm}
The MSSM superpotential with $R_p$ violating terms was given by
eqns. (\ref{MSSM-superpotential-1}) and (\ref{MSSM-superpotential-2}).
Since we aim to generate Majorana masses for the light neutrinos,
we consider violation of the lepton number only and thus the
baryon number violating terms
($\frac{1}{2} \lambda^{''}_{ijk} \hat u^c_i \hat d^c_j \hat d^c_k$)
are dropped for the rest of the discussions. It is perhaps, best
to start with the simple most example of $\not\!\!{R_p}$, namely bilinear
$R_p$-violation ($bR_pV$) and continue the discussion with the trilinear
terms ($tR_pV$) later.
\begin{flushleft}
{\it{$\maltese$ Bilinear $R$-parity violation}}
\end{flushleft}
The superpotential and soft terms are given by
(see eqns. (\ref{MSSM-superpotential-1}), (\ref{Lsoft-MSSM})
and (\ref{MSSM-superpotential-2}))
\begin{eqnarray}
W^{bR_pV} &=& W^{MSSM} - \epsilon_{ab} \varepsilon^i \hat L^a_i\hat H^b_u, \nonumber \\
-\mathcal{L}^{bR_pV}_{\text{soft}}
&=& -\mathcal{L}^{MSSM}_{\text{soft}}
- \epsilon_{ab} B_{\varepsilon_i} \hat L^a_i\hat H^b_u.
\label{MSSM-brpv}
\end{eqnarray}
Now what are the implications of eqn.(\ref{MSSM-brpv})?
\vspace{0.1cm}
\noindent
1. $R_p$ is violated through lepton number violation by odd unit,
$\Delta L = 1$. This is an explicit breaking and so
there is no possibility for
an experimentally disfavored doublet Majoron emission.
\vspace{0.3cm}
\noindent
2. Similar to eqn.(\ref{MSSM-scalar-pot}) one can construct the
neutral scalar potential, $V^{bR_pV}_{neutral~scalar}$. Interestingly
now one get non-zero VEVs for the left sneutrino fields using the
suitable minimization condition
\begin{equation}
\sum_{j} (m^2_{\widetilde{L}})^{ji} {v'_j}- B_{\varepsilon_i} v_2
+{\gamma_g}{\xi_{\upsilon}}{v'_i}
+{\varepsilon_i} {\eta} =0,
\label{Minim-Lsneut}
\end{equation}
where
\begin{eqnarray}
\eta &=&\sum_{i} {\varepsilon^i}{v'_i} + {\mu} v_1 ,
~\gamma_{g} = \frac{1}{4}({g_1^2 + g_2^2}),
~\xi_{\upsilon} ={\sum_{i} {v'^2_i} + v_1^2 -v_2^2}.
\label{Abbrevations-1}
\end{eqnarray}
$v_1,v_2$ are VEVs for down and up-type Higgs fields, respectively.
$v'_i$ is the VEV acquired by `$i$'-th sneutrino
field. The soft masses $(m^2_{\widetilde{L}})^{ji}$ are assumed to
be symmetric in `$i$' and `$j$' indices.
The masses for $W,Z$ bosons now should be given by
\begin{eqnarray}
M_W = \frac{g_2 v_{new}}{\sqrt{2}} ,
~~M_Z = \frac{v_{new}}{\sqrt{2}} \sqrt{g^2_1 + g^2_2},
\label{wz-mod-mass}
\end{eqnarray}
where $v^2_{new} = {\sum {v'^2_i} + v_1^2 + v_2^2}$.
It is apparent from eqn.(\ref{wz-mod-mass}) that to
maintain the electroweak precision, $\sum {v'^2_i} \ll v^2_1,v^2_2$,
so that ${\sum {v'^2_i} + v_1^2 + v_2^2} \simeq v_1^2 + v_2^2$ to
a very good approximation.
\vspace{0.3cm}
\noindent
3. Significance of the lepton number is lost, indeed without
a designated lepton number there is no difference between a
lepton superfield $(\hat L_i)$ and a down-type Higgs superfield,
$\hat H_d$. As a consequence now the neutral sleptons
(left sneutrinos $(\widetilde{\nu})$ in this
case) can mix with CP-odd (pseudoscalar) and even (scalar)
neutral Higgs bosons. Similar mixing is allowed between charged Higgs
and the charged sleptons. These enlarged scalar and
pseudoscalar mass squared matrices in the
basis $(\Re{H^0_d},\Re{H^0_u},\Re{\widetilde{\nu}_\alpha})$
and $(\Im{H^0_d},\Im{H^0_u},\Im{\widetilde{\nu}_\alpha})$
respectively, are given by
\begin{eqnarray}
(a)~(M^2_{scalar})_{5\times5} = \left(\begin{array}{c c}
(\mathcal{M}^2_{MSSM-scalar})_{2\times2} &
(\mathcal{S}^2_{\widetilde{\nu}_\alpha H^0_i})_{2\times3}\\
(\mathcal{S}^2_{\widetilde{\nu}_\alpha H^0_i})^T
& (\mathcal{S}^2_{\widetilde{\nu}_\alpha \widetilde{\nu}_\beta})_{3\times3}
\end{array}\right),
\label{brpMSSM-cp-even-1}
\end{eqnarray}
where $i=\left(\begin{array}{c}
d\\
u
\end{array}\right)$ with $\alpha,\beta=1,2,3$ or $e,\mu,\tau$ and
\begin{eqnarray}
(\mathcal{S}^2_{\widetilde{\nu}_\alpha H^0_d}) &=&
(\mu \varepsilon_\alpha + 2\gamma_{g} v'_\alpha v_1),
~(\mathcal{S}^2_{\widetilde{\nu}_\alpha H^0_u}) =
(- B_{\varepsilon_\alpha} - 2\gamma_{g} v'_\alpha v_2), \nonumber \\
(\mathcal{S}^2_{\widetilde{\nu}_\alpha \widetilde{\nu}_\beta}) &=&
{\varepsilon_\alpha}{\varepsilon_\beta}
+{\gamma_g}{\xi_{\upsilon}}{\delta_{\alpha\beta}}
+2{\gamma_g}{v'_\alpha}{v'_\beta}+{(m^2_{\tilde{L}})_{\alpha\beta}},
\label{brpMSSM-cp-even-2}
\end{eqnarray}
and $(b)$
\begin{eqnarray}
(M^2_{pseudoscalar})_{5\times5} = \left(\begin{array}{c c}
(\mathcal{M}^2_{MSSM-pseudoscalar})_{2\times2} &
(\mathcal{P}^2_{\widetilde{\nu}_\alpha H^0_i})_{2\times3}\\
(\mathcal{P}^2_{\widetilde{\nu}_\alpha H^0_i})^T
& (\mathcal{P}^2_{\widetilde{\nu}_\alpha \widetilde{\nu}_\beta})_{3\times3}
\end{array}\right),
\label{brpMSSM-cp-odd-1}
\end{eqnarray}
with
\begin{eqnarray}
(\mathcal{P}^2_{\widetilde{\nu}_\alpha H^0_d}) &=&
(-\mu \varepsilon_\alpha),
~(\mathcal{P}^2_{\widetilde{\nu}_\alpha H^0_u}) =
(B_{\varepsilon_\alpha}), \nonumber \\
(\mathcal{P}^2_{\widetilde{\nu}_\alpha \widetilde{\nu}_\beta}) &=&
{\varepsilon_\alpha}{\varepsilon_\beta}
+{\gamma_g}{\xi_{\upsilon}}{\delta_{\alpha\beta}}
+{(m^2_{\tilde{L}})_{\alpha\beta}}.
\label{brpMSSM-cp-odd-2}
\end{eqnarray}
Here `$\Re$' and `$\Im$' correspond to the real and imaginary part
of a neutral scalar field.
The charged scalar mass squared matrix with the basis choice
$({H^+_d},{H^+_u},{\widetilde{\ell}^+_{\alpha_R}},
{\widetilde{\ell}^+_{\alpha_L}})$ looks like
\begin{eqnarray}
(M^2_{charged~scalar})_{8\times8} = \left(\begin{array}{c c}
(\mathcal{M}^2_{MSSM-charged})_{2\times2} &
(\mathcal{C}^2_{\widetilde{\ell}_{\alpha_X} H_i})_{2\times6}\\
(\mathcal{C}^2_{\widetilde{\ell}_{\alpha_X} H_i})^T
& (\mathcal{C}^2_{\widetilde{\ell}_{\alpha_X} \widetilde{\ell}_{\beta_Y}})_{6\times6}
\end{array}\right),
\label{brpMSSM-charged-1}
\end{eqnarray}
where $X,Y=L,R$ and
\begin{eqnarray}
(\mathcal{C}^2_{\widetilde{\ell}_{\alpha_R} H_d})_{1\times3} &=&
(Y^{\alpha\beta}_e \varepsilon_\beta v_2 - (A_e Y_e )^{\alpha\beta} v'_\beta),\nonumber \\
(\mathcal{C}^2_{\widetilde{\ell}_{\alpha_L} H_d})_{1\times3} &=&
(\mu \varepsilon_\alpha - Y^{\alpha a}_e Y^{\beta a}_e
v'_\beta v_1 + \frac{g_2^2}{2} v'_\alpha v_1),\nonumber \\
(\mathcal{C}^2_{\widetilde{\ell}_{\alpha_R} H_u})_{1\times3} &=&
(- \mu Y^{\beta \alpha}_e v'_\beta + Y^{\beta \alpha}_e \varepsilon_\beta v_1),
~(\mathcal{C}^2_{\widetilde{\ell}_{\alpha_L} H_u})_{1\times3} =
(\frac{g_2^2}{2} v'_\alpha v_2 + B_{\varepsilon_\alpha}),\nonumber \\
(\mathcal{C}^2_{\widetilde{\ell}_{\alpha_L} \widetilde{\ell}_{\beta_L}})_{3\times3} &=&
(\varepsilon_\alpha \varepsilon_\beta + Y^{\alpha \rho}_e Y^{\beta \rho}_e
v^2_1 + \gamma_g {\xi_\upsilon} \delta_{\alpha\beta} -
\frac{g_2^2}{2} \mathcal{D}_{\alpha\beta} + (m^2_{\tilde{L}})^{\alpha\beta}), \nonumber \\
(\mathcal{C}^2_{\widetilde{\ell}_{\alpha_R} \widetilde{\ell}_{\beta_R}})_{3\times3} &=&
(Y^{\rho\alpha}_e Y^{\sigma\beta}_e v'_\rho v'_\sigma
+ Y^{\rho\alpha}_e Y^{\rho\beta}_e v^2_1 + (m^2_{\tilde{e}^c})^{\alpha\beta} -
\frac{g_1^2}{2} {\xi_\upsilon} \delta_{\alpha\beta}), \nonumber \\
(\mathcal{C}^2_{\widetilde{\ell}_{\alpha_L} \widetilde{\ell}_{\beta_R}})_{3\times3} &=&
(- \mu Y^{\alpha\beta}_e v_2 + (A_e Y_e)^{\alpha\beta} v_1),
\label{brpMSSM-charged-2}
\end{eqnarray}
where $\mathcal{D}_{\alpha\beta} = \{{\xi_\upsilon}
\delta_{\alpha\beta} - v'_\alpha v'_\beta\}$.
The soft-terms are assumed to be symmetric. The ${2\times2}$ MSSM scalar,
pseudoscalar and charged scalar mass squared matrices are given in
appendix \ref{appenA}.
\vspace{0.3cm}
\noindent
4. In a similar fashion charged leptons $(\ell_\alpha \equiv e, \mu, \tau)$
mix with charged gauginos
as well as with charged higgsinos
and yield an enhanced chargino mass matrix.
In the basis, $-i \tilde {\lambda}^{+}_{2}, \tilde{H}^+_u, \ell^+_{\alpha_R}$ (column)
and $-i \tilde {\lambda}^{-}_{2}, \tilde{H}^-_d, \ell^-_{\beta_L}$ (row)
\begin{eqnarray}
(M_{chargino})_{5\times5} =
\left(\begin{array}{c c}
(M^{chargino}_{MSSM})_{2\times2} &
\left(\begin{array}{c}
0\\
Y^{\rho\alpha}v'_{\rho}
\end{array}\right)_{2\times3}\\
\left(\begin{array}{c c}
g_2 v'_\alpha & \varepsilon_\alpha
\end{array}\right)_{3\times2}
& (Y^{\beta\alpha}v_1)_{3\times3}
\end{array}\right).
\label{brpMSSM-chargino}
\end{eqnarray}
With this enhancement eqn.(\ref{MSSM-charginos}) looks like
\begin{eqnarray}
&&\chi^+_i = V_{i1} \widetilde W^+ + V_{i2} \widetilde H^+_u + V_{i,\alpha+2} \ell^+_{\alpha_R}, \nonumber \\
&&\chi^-_i = U_{i1} \widetilde W^- + U_{i2} \widetilde H^-_d + U_{i,\alpha+2} \ell^-_{\alpha_L}.
\label{brpMSSM-charginos-reln}
\end{eqnarray}
The neutral fermions also behave in a similar manner. The neutralino mass
matrix now can accommodate three light neutrinos $(\nu \equiv \nu_L)$ apart from
the four MSSM neutralinos. The extended neutralino mass matrix in the basis
$\tilde B^0, \tilde W_3^0, \tilde H_d^0,\tilde H_u^0,{\nu_\alpha}$ is written as
\begin{eqnarray}
(M_{neutralino})_{7\times7} =
\left(\begin{array}{c c}
(M^{neutralino}_{MSSM})_{4\times4} &
((m)_{3\times4})^T\\
(m)_{3\times4}
& (0)_{3\times3}
\end{array}\right),
\label{brpMSSM-neutralino}
\end{eqnarray}
with
\begin{eqnarray}
(m)_{3\times4} =
\left(\begin{array}{c c c c}
-\frac{g_1}{\sqrt{2}}{v'_\alpha} & \frac{g_2}{\sqrt{2}}{v'_\alpha}
& 0 & -{\epsilon_\alpha}
\end{array}\right).
\label{brpMSSM-neutralino2}
\end{eqnarray}
Just like the charginos, for the neutralinos one can rewrite
eqn.(\ref{MSSM-neutralinos}) in modified form as
\begin{equation}
\chi^0_i = N_{i1} \widetilde B^0 + N_{i2} \widetilde W^0_3 + N_{i3} \widetilde H^0_d
+ N_{i4} \widetilde H^0_u + N_{i,\alpha+4} \nu_\alpha.
\label{MSSM-neutralinos-reln}
\end{equation}
Chargino and neutralino mass matrices for MSSM are given
in appendix \ref{appenA}.
\vspace{0.3cm}
\noindent
5. In eqn.(\ref{brpMSSM-neutralino}) entries of the $4\times4$ MSSM
block are $\sim$ TeV scale, which are $\gg$ entries of $(m)_{3\times4}$.
Besides, the $3\times3$ null matrix $(0)_{3\times3}$ signifies the
absence of Majorana mass terms for the left-handed neutrinos. This matrix has
a form similar to that of eqn.(\ref{seesaw-mat}), thus the effective
light neutrino mass matrix is given by (using eqn.(\ref{seesaw-eigval1}))
\begin{equation}
m_{seesaw} = -(m)_{3\times4}\{(M^{neutralino}_{MSSM})_{4\times4}\}^{-1}
\{(m)_{3\times4}\}^T,
\label{seesaw-brpv}
\end{equation}
or in component form
\begin{equation}
(m_{seesaw})_{\alpha\beta} = \frac{g^2_1M_2+g^2_2M_1}
{2 ~Det[(M^{neutralino}_{MSSM})_{4\times4}]}
(\mu v'_\alpha - \varepsilon_\alpha v_1)(\mu v'_\beta- \varepsilon_\beta v_1).
\label{seesaw-brpv-component}
\end{equation}
Assuming $M_1,M_2,\mu,v_1,v_2$ are $\sim \widetilde{m}$, a generic mass scale
(say EWSB scale or the scale of the soft supersymmetry breaking terms)
and $g_1,g_2\sim$ $\cal{O}$ $(1)$ we get from
eqn.(\ref{seesaw-brpv-component})
\begin{equation}
(m_{seesaw})_{\alpha\beta} \approx \underbrace{\frac{v'_\alpha v'_\beta}{\widetilde{m}}}
_{I} + \overbrace{\frac{\varepsilon_\alpha \varepsilon_\beta}{\widetilde{m}}}^{II}
-\underbrace{\frac{(\varepsilon_\alpha v'_\beta + \alpha\leftrightarrow\beta)}{\widetilde{m}}}_{III}.
\label{seesaw-brpv-component-2}
\end{equation}
The first term of eqn.(\ref{seesaw-brpv-component-2}) is coming from the
{\it{gaugino seesaw}} effect, which is
originating though the mixing of light neutrinos with either a bino $(\widetilde B^0)$
or a neutral wino $(\widetilde W^0_3)$. This is also another example for a
Type-I (bino) $+$ Type-III (wino) seesaw (see figure \ref{brpv-seesaw} (a),(b)).
\begin{figure}[ht]
\centering
\includegraphics[width=9.95cm]{FigsPDF/brpv-seesaw.pdf}
\caption{Different types of tree level contributions to the neutrino mass in a
$bR_pV$ supersymmetric model. The cross on the neutralino propagator signifies
a Majorana mass term for the neutralino.
}
\label{brpv-seesaw}
\end{figure}
The second and third contributions are represented by $(c)$ and $(d)$
of figure \ref{brpv-seesaw}. There is one extremely important point to note
about this analysis, that is if $\varepsilon_\alpha=0$ but $B_{\varepsilon_\alpha}$ are not,
even then $v'_\alpha\neq0$ (see eqn.(\ref{Minim-Lsneut})). Thus even if
$R_p$ violation is rotated away from the superpotential, effect of $\not\!\!{R_p}$
in the soft terms can still trigger non-zero neutrino mass as shown by $(a,b)$
of figure \ref{brpv-seesaw}. However, this analysis is strictly valid
if $B_{\varepsilon_\alpha} \not\propto \varepsilon_\alpha$.
If we define $\mu'_\alpha = (\mu v'_\alpha - \varepsilon_\alpha v_1)$, then using the following
set of relations, namely, $g^2_1/(g^2_1+g^2_2) = {\rm sin}^2\theta_W$,
$g^2_2/(g^2_1+g^2_2) = {\rm cos}^2\theta_W$, $M^2_Z = (1/2)(g^2_1+g^2_2)(v^2_1+v^2_2)$
and the fact $Det[(M^{neutralino}_{MSSM})_{4\times4}] =
(g^2_1M_2+g^2_2M_1)v_1v_2\mu - M_1M_2\mu^2$, we get an alternative expression
of eqn.(\ref{seesaw-brpv-component-2})
\begin{equation}
(m_{seesaw})_{\alpha\beta} \approx
\frac{\mu'_\alpha \mu'_\beta}{\widetilde{m}}{\rm cos}^2\beta,
\label{seesaw-brpv-component-3}
\end{equation}
where $tan\beta = v_2/v_1$ holds good with $v'_\alpha \ll v_1,v_2$.
The problem with this tree level effective light neutrino mass matrix
is that, it gives only one {\it{non-zero}} eigenvalue, given by
\begin{equation}
m_{neut} =
\frac{|\mu'_\alpha|^2}{\widetilde{m}}{\rm cos}^2\beta.
\label{seesaw-brpv-component-4}
\end{equation}
The {\it{only non-zero}} neutrino mass at the tree level of a $bR_pV$
model is suppressed by squared $R_p$-violating parameter and also
by ${\rm tan}^{-2}\beta$ for ${\rm tan}\beta\gg1$. With $\varepsilon_\alpha\sim$ $10^{-4}$
GeV and $\widetilde m\sim$ $1$ TeV one gets $m_{neut}\sim$ $10^{-{11}}$ GeV,
which is the scale for the atmospheric neutrinos\footnote{Assuming normal hierarchy
in light neutrino masses.}. But to accommodate three flavour global data
\cite{c3Schwetz:2008er,c3GonzalezGarcia:2010er}
one requires at least two massive neutrinos!
\begin{flushleft}
{\it{$\blacklozenge$ Loop corrections in bilinear $R$-parity violation}}
\end{flushleft}
The remedy to this problem can come from the one-loop
contributions to the light neutrino masses. The dominant diagrams are
shown in figure \ref{brpv-loops}. Before discussing these diagrams
and their contributions further it is
worthy to explain the meaning of symbols used in figure \ref{brpv-loops}.
The quantity $B'_\alpha$ denotes mixing between a left handed
sneutrino $\widetilde \nu_\alpha$ (see eqns.(\ref{brpMSSM-cp-even-2}), (\ref{brpMSSM-cp-odd-2}))
and physical MSSM Higgs bosons (eqn.(\ref{Higgs-mass-gauge})).
$\widetilde{\mu}_\alpha$ is either $\varepsilon_\alpha$ ($\nu_\alpha \widetilde H_u$
mixing, see eqn.(\ref{brpMSSM-neutralino2})) or $g_1 v'_\alpha,g_2 v'_\alpha$
($\nu_\alpha \widetilde B^0,\nu_\alpha \widetilde W^0_3$ mixing, see eqn.(\ref{brpMSSM-neutralino2}))
(figure $(a)$ and $(b)$). In figure $(c)$ a {\it{blob}} on the scalar line
indicates a mixing between left and right handed up-type squarks, which exists
if one has either gauginos $(\widetilde B^0,\widetilde W^0_3)$ or up-type higgsino $(\widetilde H^0_u)$
on both the sides. However, if one puts gauginos on one side and higgsino
on the other, then this left-right mixing is absent. This situation is represented by
a void circle on the scalar line around the blob. In figure $(d)$ $g_\alpha,g_\beta$
represents neutrino-gaugino mixing (eqn.(\ref{brpMSSM-neutralino2})). $f$ denotes a
{\it{down-type}} fermion, that is either a charged lepton, $(\ell_k=e,\mu,\tau)$
or a down quark, $(d_k=d,s,b)$. There also exist more complicated diagrams
for down-type fermion loops as shown in figure $(e,f)$. $\eta_\chi$ represents
mixing of a down-type higgsino with neutral gauginos and up-type higgsino
(see eqn.(\ref{MSSM-neutralino})). The last two diagrams $(g,h)$ arise from
chargino-charged scalar contribution to neutrino mass. A {\it{cross}} on the
fermion line represents a mass insertion, responsible for a chirality flip.
In all of these diagrams $\Delta L=2$ effect is coming from a pair of $\Delta L=1$
contributions. For diagrams $(g,h)$ the blobs and the cross represent mixing only
without any chirality flip (see eqns.(\ref{brpMSSM-charged-2}),(\ref{brpMSSM-chargino})).
These diagrams are shown for a general
basis where both of the bilinear $R_p$-violating parameters $(\varepsilon_\alpha)$
and sneutrino VEVs $(v'_\alpha)$ are non-vanishing. When $v'_\alpha=0$, using the
minimization condition for left sneutrinos (eqn. (\ref{Minim-Lsneut})),
diagram $(a)$ of figure \ref{brpv-loops} reduces to the well-known
$BB$-loop \cite{c3Grossman:1997is,c3Davidson:2000uc,c3Davidson:2000ne,c3Grossman:2003gq}.
\begin{figure}[ht]
\centering
\includegraphics[width=12.05cm]{FigsPDF/brpv-loops.pdf}
\caption{Neutrino mass generation through loops in a
model with $bR_pV$. For details of used symbols see text.}
\label{brpv-loops}
\end{figure}
This $BB$ loop can either give mass to one more neutrino state
(not to that one which was already massive at the tree level
so long $B_{\varepsilon_\alpha} \not\propto \varepsilon_\alpha$) when sneutrino
masses are degenerate or can contribute to all three
light neutrino masses with non-degenerate sneutrinos. Assuming all
the scalar and neutralino masses $\sim \widetilde m$, an approximate
expression for this loop contribution to light neutrino masses
with degenerate sneutrinos is given by \cite{c3Grossman:2003gq,c3Rakshit:2004rj}.
\begin{equation}
m^{BB}_{\alpha\beta} \simeq
\frac{g^2_2}{64 \pi^2 \cos^2\beta}
\frac{B'_\alpha B'_\beta}{\widetilde m^3}.
\label{BB-loop}
\end{equation}
It is important to mention that in order to generate
solar neutrino mass square difference
using loop corrections one
should naively expect $B'\sim$ $(0.1-1)$ ${\rm {GeV^2}}$, with
the assumption of normal hierarchical structure in light neutrino masses.
In a similar fashion the loop shown by diagram $(b)$ of figure \ref{brpv-loops}
is an example of the $\mu B$-type loop at $v'_\alpha=0$ \cite{c3Davidson:2000uc,
c3Davidson:2000ne,c3Grossman:2003gq,c3Rakshit:2004rj}. This loop involves
neutrino-gaugino or neutrino-higgsino mixing (collectively labeled as $\widetilde \mu_\alpha$,
see eqn.(\ref{brpMSSM-neutralino2})) together with sneutrino-Higgs mixing ($B'_\beta$,
see eqns.(\ref{brpMSSM-cp-even-2}), (\ref{brpMSSM-cp-odd-2}), (\ref{Higgs-mass-gauge})).
Assuming all the masses (Higgs, sneutrino,
neutralino) are at the weak scale $\widetilde m$, an approximate contribution is
given by \cite{c3Davidson:2000uc,c3Davidson:2000ne,c3Grossman:2003gq,c3Rakshit:2004rj}
\begin{equation}
m^{\mu B}_{\alpha\beta} \simeq
\frac{g^2_2}{64 \pi^2 \cos\beta}
\frac{\widetilde \mu_\alpha B'_\beta + \widetilde \mu_\beta B'_\alpha}{\widetilde m^2}.
\label{muB-loop}
\end{equation}
It is evident from the structure of right hand side of eqn.(\ref{muB-loop}) that
the $\mu B$ loop contributes to more than one neutrino masses. However, presence
of $\widetilde\mu_\alpha$ makes this loop contribution sub-leading to neutrino
masses compared to the $BB$ loop \cite{c3Chun:2002vp,c3Grossman:2003gq,c3Rakshit:2004rj}.
For large values of $\tan\beta$
$(\tan\beta \gg 1)$ the $BB$-loop and the $\mu B$-loop are enhanced by
$\tan^2\beta$ and $\tan\beta$, respectively.
Contributions to neutrino masses from quark-squark loops are given
by diagrams $(c,d,e,f)$ of figure \ref{brpv-loops}. Diagram $(c)$ represents
an up-type quark-squark loops. This diagram can yield large contribution
to neutrino mass particularly when it is a top-stop $(t-\tilde {t})$ loop,
because of the large top Yukawa
coupling, $Y_t$. This loop contribution is proportional
to $\widetilde \mu_\alpha \widetilde \mu_\beta$, which is exactly same as the tree level
one (see eqn.(\ref{seesaw-brpv-component-3})), thus this entire
effect eventually gives a correction to a neutrino mass which is already
massive at the tree level \cite{c3Mukhopadhyaya:2006is}. An approximate expression
for this loop is given by
\begin{equation}
m^{u_k {\widetilde {u}_k}}_{\alpha\beta}(no~blob) \simeq
\frac{N_c f^2_{u \widetilde u}}{16 \pi^2}
\frac{m_{u_k}\widetilde \mu_\alpha \widetilde \mu_\beta}{\widetilde m^2},
~~m^{u_k {\widetilde {u}_k}}_{\alpha\beta}(blob) \simeq
\frac{N_c f^2_{u \widetilde u}}{16 \pi^2}
\frac{m^2_{u_k}\widetilde \mu_\alpha \widetilde \mu_\beta}{\widetilde m^3},
\label{upsup-loop}
\end{equation}
where $m_{u_k}$ is the mass of up-quark of type $k$. The coupling factor
$f^2_{u \widetilde u}$ is either $g_ig_j$ or $g_iY_{u_k}$\footnote{$u_k
\equiv u,c,t$.} with $i=1,2$. $N_c$ is the colour factor which is $3$ for
quarks. For the case of left right sfermion mixing we use the relation
\begin{equation}
m^{2^{LR}}_{\widetilde f_k} \approx m_{f_k} \widetilde m.
\label{fermion-sfermion-relation}
\end{equation}
In a similar fashion for a down type fermion-sfermion,
$f_k-\widetilde f_k$ (charged lepton-slepton or down quark-squark) (see diagram
$(d)$ of figure \ref{brpv-loops}) an approximate expression is given by
(using eqn.(\ref{fermion-sfermion-relation}))
\begin{equation}
m^{f_k {\widetilde {f}_k}}_{\alpha\beta} \simeq
\frac{N_c f^2_{f \widetilde f}}{16 \pi^2}
\frac{m^2_{f_k}g_\alpha g_\beta}{\widetilde m^3},
\label{fdsfd-loop}
\end{equation}
where $m_{f_k}$ is the mass of down-type fermion of type $k$\footnote{
$f=d_k \equiv d,s,b$ or $f=\ell_k\equiv e,\mu,\tau$.}. $N_c=3$
for quarks but $=1$ for leptons. The coupling factor
$f^2_{f \widetilde f}$ is $g_ig_j$ with $i=1,2$. The quantity $g_\alpha$ denotes mixing
between a neutrino and a gaugino. However,
for down-type fermion-sfermion there exist other complicated loop diagrams
like $(e,f)$ \cite{c3Mukhopadhyaya:2006is} of figure
\ref{brpv-loops}. These loops give contribution of the approximate form
\begin{equation}
m'^{f_k {\widetilde {f}_k}}_{\alpha\beta} \simeq
\frac{N_c f'^2_{f \widetilde f} \eta_\chi}{16 \pi^2}
\frac{m_{f_k}(\widetilde \mu_\alpha g_\beta + \alpha\leftrightarrow\beta)}{\widetilde m^3},
\label{fdsfd-loop-2}
\end{equation}
for diagram $(e)$ and
\begin{equation}
m''^{f_k {\widetilde {f}_k}}_{\alpha\beta} \simeq
\frac{N_c f''^2_{f \widetilde f} \eta^2_\chi}{16 \pi^2}
\frac{m^2_{f_k}\widetilde \mu_\alpha \widetilde \mu_\beta}{\widetilde m^5},
\label{fdsfd-loop-3}
\end{equation}
for diagram $(f)$, respectively. The quantity $f'^2_{f \widetilde f}$ is $g_i Y_{f_k}$
whereas $f''^2_{f \widetilde f}$ represents $Y^2_{f_k}$ with $i=1,2$
and $Y_{f_k}$ being either charged lepton or down quark Yukawa couplings.
It is apparent that eqns.(\ref{fdsfd-loop}), (\ref{fdsfd-loop-3})
once again contribute to the ``{\it{same neutrino}}'' which is already massive
at the tree level. However, eqn.(\ref{fdsfd-loop-2}) will contribute
to more than one neutrino masses. Note that since
contributions of these set of diagrams are proportional to the fermion mass,
$m_{f_k}$, they are important only for bottom quark and tau-lepton along with the corresponding
scalar states running in the loop.
Diagrams $(g,h)$ are the chargino-charged scalar loop contributions
to light neutrino mass \cite{c3Dedes:2006ni}. An approximate form for these loops
are given by
\begin{equation}
m^{(g)}_{\alpha\beta} \simeq
\frac{g^2_2 Y_{\ell_k}}{16 \pi^2}
\frac{v'_\alpha B''_\beta}{\widetilde m},
~~
m^{(h)}_{\alpha\beta} \simeq
\frac{Y^3_{\ell_k}}{16 \pi^2}
\frac{v'_\alpha B''_\beta}{\widetilde m},
\label{csc-loop}
\end{equation}
where $Y_{\ell_k}$ are the charged lepton Yukawa couplings and
$B''_\beta$ $(\sim B')$ represents a generic charged slepton-charged Higgs mixing
(see eqn.(\ref{brpMSSM-charged-2})). These contributions vanishes identically
when $v'_\alpha=0$. These contributions
being proportional to small parameters like $v',Y_\ell$, are {\it{much smaller}}
compared to the other types of loops. Various couplings needed here
can be found in references like \cite{c3Haber:1984rc,c3Gunion:1984yn,c3Rosiek:1989rs,c3Rosiek:1995kg,
c3Hempfling:1995wj,c3Hirsch:2000ef,c3Diaz:2003as}.
\begin{flushleft}
{\it{$\blacklozenge$ Trilinear $R$-parity violation and loop corrections}}
\end{flushleft}
The so-called trilinear couplings,
contribute to light neutrino mass through loops only \cite{c3Grossman:1997is,
c3Grossman:1998py,c3Grossman:2003gq,c3Davidson:2000uc}. Possible
diagrams are shown in figure \ref{trpv-loops}.
\begin{figure}[ht]
\centering
\includegraphics[width=12.35cm]{FigsPDF/trpv-loops.pdf}
\caption{Neutrino mass generation through loops in a
model with $tR_pV$.}
\label{trpv-loops}
\end{figure}
Using eqn.(\ref{fermion-sfermion-relation})
these contributions can be written as
\begin{equation}
m^{\lambda\lam}_{\alpha\beta} \simeq
\frac{N_c\lambda_{\alpha\sigma\rho}\lambda_{\beta\sigma\rho}}{8 \pi^2}
\frac{m_{\ell_\sigma} m_{\ell_\rho}}{\widetilde m},
~~
m^{\lambda'\lambda'}_{\alpha\beta} \simeq
\frac{N_c\lambda'_{\alpha\sigma\rho}\lambda'_{\beta\sigma\rho}}{8 \pi^2}
\frac{m_{d_\sigma} m_{d_\rho}}{\widetilde m},
\label{ll-lplp-loop}
\end{equation}
where $N_c$ is $1(3)$ for $\lambda\lam(\lambda'\lambda')$ loop. Contributions
of these diagrams are suppressed by squared $R_p$ violating couplings
$\lambda^2,\lambda'^2$ and squared charged lepton, down-type quark masses
apart from usual loop suppression factor. Thus usually these loop
contributions are quiet small \cite{c3Grossman:1998py}.
\begin{figure}[ht]
\centering
\includegraphics[width=12.35cm]{FigsPDF/brpv-trpv.pdf}
\caption{Neutrino mass generation through loops in a
model with both $bR_pV$ and $tR_pV$. $\widetilde{f}_\sigma$ is either
a charged slepton with $f_{\beta\sigma\sigma}=\lambda_{\beta\sigma\sigma}$
or a down-type squark with $f_{\beta\sigma\sigma}=\lambda'_{\beta\sigma\sigma}$.
$\widetilde\mu_\alpha,\eta_\chi,g_\alpha$ are same as explained in figure \ref{trpv-loops}.
The cross have similar explanation as discussed
in figure \ref{brpv-loops}.}
\label{brpv-trpv-loops}
\end{figure}
\begin{flushleft}
{\it{$\blacklozenge$ Loop corrections in $bR_pV+tR_pV$}}
\end{flushleft}
There also exist a class of one-loop diagrams which involve
both bilinear and trilinear $R_p$ violating couplings, as
shown figure \ref{brpv-trpv-loops} \cite{c3Davidson:2000uc,c3Davidson:2000ne
,c3Grossman:2003gq,c3Rakshit:2004rj}. One can write down
these loop contributions approximately as
\begin{equation}
(i)~~~m^{\mu f}_{\alpha\beta} \simeq
\frac{N_c \widetilde \mu_\alpha \eta_\chi Y_{f_\sigma} f_{\beta\sigma\sigma}}{16 \pi^2}
\frac{m^2_{f_\sigma} }{\widetilde m^3} + \alpha \leftrightarrow \beta,
\label{mu-ll-lplp-loop-1}
\end{equation}
for diagram $(a)$ where $N_c=1(3)$ for charged lepton (down-type quark),
$Y_{f_\sigma}$ is either a charged lepton or a down-type Yukawa coupling and
\begin{equation}
(ii)~~~m^{\mu f}_{\alpha\beta} \simeq
\frac{N_c g_\alpha g_i f_{\beta\sigma\sigma}}{16 \pi^2}
\frac{m_{f_\sigma} }{\widetilde m} + \alpha \leftrightarrow \beta,
\label{mu-ll-lplp-loop-2}
\end{equation}
for diagram $(b)$.
The quantity $f_{\alpha\sigma\sigma}$ represents either $\lambda_{\alpha\sigma\sigma}$
or $\lambda'_{\alpha\sigma\sigma}$ couplings. $g_\alpha$
represents a neutrino-gaugino mixing (see eqn.(\ref{brpMSSM-neutralino2})).
$i=1,2$. These contributions are suppressed by a pair of $R_p$-violating
couplings $(\mu\lambda/\mu\lambda')$ or product of sneutrino VEVs and trilinear
$R_p$-violating couplings $(v'\lambda/v'\lambda')$, a loop factor and at least by a
fermion mass ($\propto$ Yukawa coupling) \cite{c3Grossman:2003gq,c3Rakshit:2004rj}.
Also contributions of diagram $(a)$ is negligible
compared to that of $(b)$ by a factor of squared Yukawa coupling. Contributions
of these loops are second order in the above mentioned suppression factors
(similar to that of $\mu B$ loop) once the tree level effect is taken into account.
There are literature
where these loop contributions are analysed in a basis independent formalism
\cite{c3Davidson:2000uc,c3Davidson:2000ne,
c3Rakshit:2004rj} (also see refs.\cite{c3Davidson:1996cc,
c3Davidson:1997mc,c3Davidson:1998yy,c3Ferrandis:1998ii,c3Grossman:2000ex} for basis
independent parameterizations of $\not\!\!{R_p}$). For this discussion we
stick to the ``mass insertion approximation'' but alternatively it is also
possible to perform these entire analysis in physical or mass basis
\cite{c3Hempfling:1995wj,c3Hirsch:2000ef,c3Dedes:2006ni}. The mass
insertion approximation works well since the effect of $R_p$-violating parameters
are expected to be small in order to account for neutrino data.
All of these calculations are performed assuming no
flavour mixing for the sfermions.
\vspace{0.2cm}
\begin{flushleft}
{\it{$\blacklozenge$ A comparative study of different loop contribution}}
\end{flushleft}
Usually the trilinear loops $(\lambda\lam, \lambda'\lambda')$ are doubly Yukawa
suppressed (through fermion masses) and they yield rather small contributions.
The $\mu B$-type, $\mu \lambda, \mu \lambda'$ loop contributions to the light neutrino
masses are second order in suppression factors. The $\mu \lambda, \mu \lambda'$
loop contributions are also suppressed by single Yukawa coupling. The Yukawa
couplings (either double or single) are also present in the quark-squark or
charged lepton-slepton loops. However, in most of the occasions they give
corrections to the tree level neutrino mass, though other contributions
can also exist (see eqn.(\ref{fdsfd-loop-2})). These loops are sometimes
dominant \cite{c3Kaplan:1999ds,c3Hirsch:2000ef,c3Diaz:2003as} provided the
$BB$-type loop suffers large cancellation among
different Higgs contributions. In general the second neutrino receives
major contribution from the $BB$ loop.
In the situation when $\tan\beta$ is large, the tree
level contribution (see eqn.(\ref{seesaw-brpv-component-3})) can be smaller
compared to the loop contributions. In this situation, the tree level result
usually account for the solar neutrino mass scale whereas the loop corrections
generate the atmospheric mass scale. In conventional scenario when tree level
effect is leading, it is easy to fit the normal hierarchical spectrum of
neutrino mass in an $R_p$-violating theory.
\section{{\bf T}esting neutrino oscillation at Collider}\label{corre}
We have already spent enough time to discuss the issue of light
neutrino mass generation. It is then legitimate to ask what
are the possible experimental implications of a massive neutrino?
It was first advocated in ref.\cite{c3Mukhopadhyaya:1998xj} that
in a simple supersymmetric model with only $bR_pV$ it is possible
to get some kind of relation between the neutrino sector and the decays
of the LSP.
This kind of model predicts comparable numbers of muons and taus, produced
together with the $W$-boson, in decays of the lightest neutralino. Usually
for an appreciable region of parameter space the lightest neutralino is the LSP.
Additionally, the appearance of a measurable ``displaced vertex''
was also addressed in ref.\cite{c3Mukhopadhyaya:1998xj} which is
extremely useful for a collider related study to efface undesired backgrounds.
This novel feature also has been addressed in refs.\cite{c3Chun:1998ub,c3Choi:1999tq,
c3Porod:2000hv,c3DeCampos:2010yu}. See also refs.\cite{c3Bartl:2000yh,c3Restrepo:2001me,
c3Hirsch:2002ys,c3Bartl:2003uq,c3Hirsch:2003fe,c3Bartl:2003sr} for tests of neutrino
properties at accelerator experiments.
The correlation between a LSP decay and neutrino physics is apparent
for supersymmetric models with bilinear $\not\!\!{R_p}$, since the same parameter $\varepsilon_\alpha$
is involved in both the analysis. For example, if the neutralino LSP, $\ntrl1$
decays into a charged lepton and $W$-boson \cite{c3Mukhopadhyaya:1998xj} then
following \cite{c3Romao:1999up,c3Porod:2000hv} one can get approximately
\begin{equation}
\frac{(\ntrl1\to\mu^\pm W^\mp)}{(\ntrl1\to\tau^\pm W^\mp)}
\simeq \left(\frac{\mu'_\mu}{\mu'_\tau}\right)^2= \tan^2\theta_{23},
\label{mu-tau-23}
\end{equation}
where $\mu_\alpha = \mu v'_\alpha - \varepsilon_\alpha v_1$ with $\alpha=e,\mu,\tau$
and $\tan^2\theta_{23}$ is the atmospheric mixing angle.
Similar correlations with trilinear $\not\!\!{R_p}$ parameters are lost \cite{c3Choi:1999tq}
since the model became less predictive with a larger set of parameters.
A rigorous discussion of these correlations has
been given in ref.\cite{c3Nath:2010zj}.
We note in passing that when the LSP is no longer stable (due to $\not\!\!{R_p}$) it is
not necessary for them to be charge or colour neutral \cite{c3Wolfram:1978gp,c3Dover:1979sn,
c3Ellis:1983ew}. With broken $R_p$ any sparticle (charginos \cite{c3Feng:1999fu},
squarks, gluinos \cite{c3Raby:1997bpa,c3Baer:1998pg,c3Raby:1998xr}, sneutrinos
\cite{c3Hagelin:1984wv}, (see also ref.\cite{c3Ellis:1983ew}))
can be the LSP. It was pointed out in ref.\cite{c3Hirsch:2003fe} that whatever be the LSP,
measurements of branching ratios at future accelerators will provide a definite test of
bilinear $R_p$ breaking as the model of neutrino mass.
\chapter{ \sffamily{{\bf S}upersymmetry
}}\label{susy}
\section{{\bf W}aking up to the idea}\label{intro}
The effect of radiative correction drives the
``natural'' Higgs mass, and therefore the entire SM particle spectra
to some ultimate cutoff of the theory, namely, the Planck scale.
A solution to this hierarchy problem could
be that, either the Higgs boson is some sort of composite
particle rather than being a fundamental particle
or the SM is an effective theory valid upto a certain energy
scale so that the cutoff scale to the theory lies far below the Planck scale.
It is also a viable alternative that there exists no Higgs boson at
all and we need some alternative mechanism
to generate masses for the SM particles\footnote{These issues are well
studied in the literature and beyond the theme of this thesis.}.
However, it is also possible that even in the presence of
quadratic divergences the Higgs boson mass can be in the
range of a few hundreds of GeV to a TeV provided different sources of
radiative corrections cancel the quadratic divergent pieces.
It is indeed possible to cancel the total one-loop
quadratic divergences (shown in chapter \ref{SM}, section \ref{suc-prob}) by
explicitly canceling contributions between bosonic
and fermionic loop with some postulated relation
between their masses. However, this cancellation is not motivated
by any symmetry argument and thus a rather accidental
cancellation of this kind fails for higher order loops.
Driven by this simple argument let us assume that there are
two additional complex scalar fields $\widetilde f_L$ and
$\widetilde f_R$ corresponding to a fermion $f$ which couples
to field $\Phi$ (see eqn.(\ref{SM-Higgs})) in the following manner
\begin{eqnarray}
\mathcal{L}_{\widetilde f \widetilde f h^0}
&=& \widetilde {\lambda_f} |\Phi|^2
(|\widetilde f_L|^2 + |\widetilde f_R|^2), \nonumber \\
&\underrightarrow{EWSB}& \frac{1}{2} \widetilde {\lambda_f} h^{0^2}
(|\widetilde f_L|^2 + |\widetilde f_R|^2) +
v \widetilde {\lambda_f} h^{0}
(|\widetilde f_L|^2 + |\widetilde f_R|^2) +..,
\label{c2-susy-motivation}
\end{eqnarray}
where $h^0$ is the physical Higgs field (see eqn.(\ref{SM-Higgs-VEV})).
A Lagrangian of the form of eqn.(\ref{c2-susy-motivation}) will yield
additional one-loop contributions to Higgs mass. Note that in order
to get a potential bounded from below, $\widetilde {\lambda_f}<0$.
\begin{figure}[ht]
\centering
\includegraphics[width=10.95cm]{FigsPDF/Higgs-loop-c2.pdf}
\caption{New diagrams contributing to Higgs mass correction
from Lagrangian $\mathcal{L}_{\widetilde f \widetilde f h^0}$
(eqn.(\ref{c2-susy-motivation})).}
\label{Higgs-loop-c2}
\end{figure}
The additional contributions to the two point function for Higgs mass
via the loops (figure \ref{Higgs-loop-c2}) can be written as
\begin{eqnarray}
\Pi^{\widetilde f}_{h^0h^0}(p^2=0) &=&
-\widetilde {\lambda_f} \int \frac{d^4k}{(2\pi)^4}
\left(\frac{1}{k^2-m^2_{\widetilde f_L}}
+ \frac{1}{k^2-m^2_{\widetilde f_R}} \right) \nonumber \\
&+& (v \widetilde {\lambda_f})^2 \int \frac{d^4k}{(2\pi)^4}
\left(\frac{1}{(k^2-m^2_{\widetilde f_L})^2}
+ \frac{1}{(k^2-m^2_{\widetilde f_R})^2} \right).
\label{c2-susy-loop1}
\end{eqnarray}
Eqn.(\ref{c2-susy-loop1}) contains two types of divergences,
(a) the first line which is quadratically divergent and (b)
second line, which is logarithmically divergent. Following
similar procedure to that of deriving eqn.(\ref{Higgs-fermion-approx}),
one can see that the total two point function $\Pi^{\widetilde f}_{h^0h^0}(p^2=0)$
$ +~\Pi^{f}_{h^0h^0}(p^2=0)$ (see eqn.(\ref{Higgs-fermion-loop}))
is {\it{completely}} free from quadratic divergences, provided
\begin{eqnarray}
\widetilde \lambda_f = - y^2_f.
\label{coup-equality}
\end{eqnarray}
It is extremely important to note that eqn.(\ref{coup-equality})
is {\it{independent}} of mass of $f,\widetilde f_L$ and $\widetilde f_R$,
namely $m_f, m_{\widetilde f_L}$ and $m_{\widetilde f_R}$ respectively.
The remaining part of $\Pi^{\widetilde f}_{h^0h^0}(p^2=0)$
$ +~\Pi^{f}_{h^0h^0}(p^2=0)$, containing logarithmic divergences
can be explicitly written as (using eqn.(\ref{coup-equality})
and dropping $p^2$)
\begin{eqnarray}
\Pi^{\widetilde f}_{h^0h^0}(0) +
\Pi^{f}_{h^0h^0}(0) &=&
\frac{i y^2_f}{16 \pi^2}
\left[-2m^2_f(1-ln \frac{m^2_f}{\mu^2_R})
+ 4 m^2_f ln \frac{m^2_f}{\mu^2_R}\right] \nonumber \\
&+& \frac{i y^2_f}{16 \pi^2}
\left[+2m^2_{\widetilde f}(1-ln \frac{m^2_{\widetilde f}}{\mu^2_R})
- 4 m^2_{\widetilde f} ln \frac{m^2_{\widetilde f}}{\mu^2_R}\right],
\label{c2-susy-loop-log}
\end{eqnarray}
with $m_{\widetilde f_L} = m_{\widetilde f_R} = m_{\widetilde f}$. $\mu_R$
is the scale of renormalization. If further one considers
{\bf{$m_{\widetilde f} = m_{f}$}} then from eqn.(\ref{c2-susy-loop-log}),
$\Pi^{\widetilde f}_{h^0h^0}(0) + \Pi^{f}_{h^0h^0}(0) = 0$, i.e. sum of the
two point functions via the loop vanishes! This theory
is absolutely free from hierarchy problem. However, in order
to achieve a theory free from quadratic divergences, such cancellation between
fermionic and bosonic contributions must persists for all higher orders also.
This is indeed a unavoidable feature of a theory, if there exists a symmetry
relating fermion and boson masses and couplings\footnote{The hint of such a
symmetry is evident from $m_{\widetilde f} = m_{f}$.}.
\section{{\bf B}asics of supersymmetry algebra}\label{susy-basics}
A symmetry which transforms a fermionic state into a bosonic one
is known as supersymmetry (SUSY)
\cite{c2Gervais:1971ji,c2Neveu:1971rx,c2Ramond:1971gb,c2Aharonov:1971kf,c2Wess:1974tw,c2Wess:1973kz,
c2Ferrara:1974ac,c2Iwasaki:1973pw,c2Fayet:1976cr,c2Nilles-1983,c2Gates-1983,c2Haber:1984rc,
c2Sohnius-1985,c2Drees-1996,c2Lykken-1996,c2Dine-1996,c2Martin-1997,c2Bilal-2001,c2Wess-1992,
c2Bailin:1994qt,
c2Weinberg-2000,c2Drees-2004,c2Baer-2006} (also see references of \cite{c2Martin-1997}).
The generator $(Q)$ of SUSY thus satisfies
\begin{eqnarray}
Q|Boson\rangle = |Fermion\rangle,
~~Q|Fermion\rangle = |Boson\rangle.
\label{susy-generator}
\end{eqnarray}
In eqn.(\ref{susy-generator}) spin of the left and right hand
side differs by {\it{half-integral}} number and thus
$Q$ must be a {\it{spinorial}} object in nature and hence follows anti-commutation
relation. Corresponding Hermitian conjugate $(\overline{Q})$ is also
another viable generator since spinors are complex objects.
It is absolutely important to study the space-time property
of $Q$, because they change the spin (and hence statistics also)
of a particle and spin is related to the behaviour under {\it{spatial
rotations}}.
Let us think about an unitary operator $\mathcal{U}$, representing
a rotation by $360^\circ$ about some axis in configuration space,
then
\begin{eqnarray}
&&\mathcal{U}Q|Boson\rangle =
\mathcal{U}Q\mathcal{U}^{-1}\mathcal{U}|Boson\rangle
= \mathcal{U}|Fermion\rangle,\nonumber \\
&&\mathcal{U}Q|Fermion\rangle =
\mathcal{U}Q\mathcal{U}^{-1}\mathcal{U}|Fermion\rangle
= \mathcal{U}|Boson\rangle.
\label{susy-gen-prop}
\end{eqnarray}
However, under a rotation by $360^\circ$ (see ref.\cite{c2Feynman-1987})
\begin{eqnarray}
\mathcal{U}|Boson\rangle = |Boson\rangle,
~~\mathcal{U}|Fermion\rangle = -|Fermion\rangle.
\label{susy-gen-prop-2}
\end{eqnarray}
Combining eqns.(\ref{susy-gen-prop}),
(\ref{susy-gen-prop-2}) one ends up with
\begin{eqnarray}
\mathcal{U}Q\mathcal{U}^{-1} = -Q,
~~\Rrightarrow \{Q,\mathcal{U}\}=0.
\label{susy-gen-prop-3}
\end{eqnarray}
Extending this analysis for any Lorentz transformations it is possible
to show that $Q$ does not commute with the generators of Lorentz transformation.
On the contrary, under space-time translation,
\begin{eqnarray}
P_\mu|Boson\rangle = |Boson\rangle,
~~P_\mu|Fermion\rangle = |Fermion\rangle.
\label{susy-gen-prop-3a}
\end{eqnarray}
Eqns.(\ref{susy-gen-prop-3a}) and (\ref{susy-generator}) together
imply that $Q$ (also $\bar{Q}$) is invariant under space-time translations.
that is
\begin{equation}
[Q,P^\mu ]=[\bar{Q},P^\mu ]=0.
\label{susy-gen-prop-4}
\end{equation}
It is obvious from eqns.(\ref{susy-gen-prop-3}) and (\ref{susy-gen-prop-4}),
that supersymmetry is indeed a space-time symmetry. In fact now the
largest possible space-time symmetry is no longer {\it{Poincar\'{e}}}
symmetry but the supersymmetry itself with larger number of generators,\footnote{This
statement is consistent with the statement of {\it{Coleman-Mandula}}
theorem \cite{c2Coleman:1967ad} and {\it{Haag-Lopuszanski-Sohnius}} theorem \cite{c2Haag:1974qh}.}
$M^{\mu\nu}$ (Lorentz transformation $\Rrightarrow$ spatial rotations and boosts),
$P^{\mu}$ (Poincar\'{e} transformation $\Rrightarrow$ translations) and
$Q,\bar{Q}$ (SUSY transformations). It has been argued earlier that the SUSY
generators ${Q,\bar{Q}}$ are {\it{anti-commuting}} rather than
being commutative. So what is $\{Q,\bar{Q}\}$? Since $Q,\bar{Q}$
are spinorial in nature, then expression for $\{Q,\bar{Q}\}$ must
be bosonic in nature and definitely has to be another symmetry
generator of the larger group.
In general, one can expect that $\{Q,\bar{Q}\}$ should be a combination of $P^\mu$
and $M^{\mu\nu}$ (with appropriate index contraction), However, after
a brief calculation one gets
\begin{equation}
\{Q,\bar{Q}\}\propto P^\mu.
\label{susy-gen-prop-5}
\end{equation}
Eqn.(\ref{susy-gen-prop-5}) is the basic of the
SUSY algebra which contains generators of the SUSY transformations
$(Q,\bar{Q})$ on the left hand side and generator for space-time
translations, $P^\mu$ on the other side. This suggests that
successive operation of two finite SUSY transformations will
induce a space-time translation on the states under operation.
The quantity $\{Q,\bar{Q}\}$ is a Hermitian operator with
positive definite eigenvalue, that is
\begin{equation}
\langle...|\{Q,\bar{Q}\}|...\rangle
=|Q|...\rangle|^2 + |\bar{Q}|...\rangle|^2
\ge 0.
\label{susy-gen-prop-6}
\end{equation}
Summing over all the SUSY generators and using eqns.(\ref{susy-gen-prop-5})
and (\ref{susy-gen-prop-6}) one gets
\begin{equation}
\sum_{Q}\{Q,\bar{Q}\}\propto P^0,
\label{susy-gen-prop-7}
\end{equation}
where $P^0$ is the total energy of the system or the eigenvalue
of the Hamiltonian, thus
Hamiltonian of supersymmetric theory contains no negative eigenvalues.
If $|0\rangle$ denotes the {\it{vacuum}} or the lowest energy state
of any supersymmetric theory then
following eqns.(\ref{susy-gen-prop-6}) and (\ref{susy-gen-prop-7}) one
obtains $P^0|0\rangle = 0$. This is again true if $Q|0\rangle = 0$ and
$\bar{Q}|0\rangle = 0$ for all $Q,\bar{Q}$. This implies that any one-particle
state with non-zero energy cannot be invariant under SUSY transformations.
So there must be one or more supersymmetric partners ({\it{superpartners}})
$Q|1\rangle$ or $\bar{Q}|1\rangle$ for every one-particle state $|1\rangle$.
Spin of superpartner state differs by $\frac{1}{2}$ unit from that of $|1\rangle$.
The state $|1\rangle$ together with its superpartner state said to form a {\it{supermultiplet}}.
In a supermultiplet different states are connected in between through one or more
SUSY transformations. Inside a supermultiplet the number of fermionic
degrees of freedom $(n_F)$ must be equal to that for bosonic one
$(n_B)$. A supermultiplet must contain at least one boson and
one fermion state. This simple most supermultiplet is known as the {\it{chiral}}
supermultiplet which contains a Weyl spinor (two degrees of freedom) and one
complex scalar (two degrees of freedom). It is important to note that the translational
invariance of SUSY generators (see eqn.(\ref{susy-gen-prop-4})) imply
{\it{All states in a supermultiplet must have same mass}}\footnote{It
is interesting to note that supercharge $Q$ satisfies
$[Q,P^2] = 0$ but $[Q,W^2] \neq 0$, where $W^\mu( = \frac{1}{2}\epsilon^{\mu\nu\rho\sigma}
M_{\nu\rho}P_\sigma)$ is the {\it{Pauli-Lubanski}} vector. Note that eigenvalue
of $W^2\propto s(s+1)$ where $s$ is spin of a particle. Thus in general
members of a supermultiplet should have same mass but different spins, which is
the virtue of supersymmetry.}.
It must be emphasized here that throughout the calculation
indices for $Q$ and $\bar{Q}$ have been suppressed. In reality
$Q\equiv Q^i_a$ where `$i=1,2,...N$' is the number of supercharges
and `$a$' is the spinor index. To be specific one should explicitly
write (for $i=1$), $Q_\alpha,\bar{Q}_{\dot{\alpha}}$, where $\alpha,\dot{\alpha}$
are spinorial indices belonging to two different representations
of the Lorentz group. We stick to $i=1$ for this thesis.
Details of SUSY algebra is given in
refs.\cite{c2MullerKirsten-1986,c2Simonsen-1995}.
\section{{\bf C}onstructing a supersymmetric Lagrangian} \label{susy-Lagrangian}
Consider a supersymmetric Lagrangian with a single Weyl fermion,
$\psi$ (contains two helicity states, $\Rrightarrow n_F=2$) and a
complex scalar, $\phi$ ($\Rrightarrow n_B=2$) without any
interaction terms. This two component Weyl spinor
and the associated complex scalar are said to form a chiral supermultiplet.
The free Lagrangian, which contains only kinetic terms is written as
\begin{equation}
\mathcal{L}^{susy} = -\partial_\mu \phi^* \partial^\mu \phi
+ i \psi^\dagger \overline{\sigma}^\mu \partial_\mu \psi,
\label{susy-Lag-1}
\end{equation}
where $\overline{\sigma}^\mu = \bf{1},-\sigma_i$.
Eqn.(\ref{susy-Lag-1}) represents a massless, non-interacting supersymmetric
model known as {\it{Wess-Zumino}} model \cite{c2Wess:1974tw}. The action
$\mathcal{S}^{susy} (=\int d^4x \mathcal{L}^{susy})$
is invariant under the set of transformations, given as
\begin{eqnarray}
&& \delta \phi = \epsilon_\alpha \psi^\alpha \equiv \epsilon \psi,
~~\Rrightarrow \delta \phi^* = \epsilon^\dagger \psi^\dagger,\nonumber \\
&& \delta \psi_\alpha = -i(\sigma^\mu \epsilon^\dagger)_\alpha \partial \phi,
~~\Rrightarrow \delta \psi^\dagger_{\dot{\alpha}}
= i(\epsilon \sigma^\mu)_{\dot{\alpha}} \partial \phi^*,
\label{susy-Lag-2}
\end{eqnarray}
where $\epsilon^\alpha$ parametrizes infinitesimal SUSY transformation.
It is clear from eqn.(\ref{susy-Lag-2}), on the basis of dimensional
argument that $\epsilon^\alpha$ must be spinorial object
and hence anti-commuting in nature. They have mass dimension
$[M]^{-\frac{1}{2}}$. It is important to note that
$\partial_\mu \epsilon^\alpha = 0$ for global SUSY transformation.
\begin{flushleft}
{\it{$\maltese$ Is supersymmetry algebra closed?}}
\end{flushleft}
It has already been stated that $\mathcal{S}^{susy}$ is
invariant under SUSY transformations (eqn.(\ref{susy-Lag-2})).
But does it also indicate that the SUSY algebra is closed?
In other words, is it true that two successive SUSY transformations
(parametrized by $\epsilon_1,\epsilon_2$) is indeed another symmetry of the
theory? In reality one finds
\begin{eqnarray}
[\delta_{\ep2},\delta_{\ep1}] X
&=& -i(\epsilon_1\sigma_\mu\epsilon^\dagger_2 - \epsilon_2\sigma_\mu\epsilon^\dagger_1)
\partial^\mu X,
\label{susy-Lag-3}
\end{eqnarray}
where $X=\phi,\psi_\alpha$, which means that
commutator of two successive supersymmetry transformations
is equivalent to the space-time translation of the respective fields.
This is absolutely consistent with our realization of
eqn.(\ref{susy-gen-prop-5}). But there is a {\it{flaw}}
in the above statement. In order to obtain eqn.(\ref{susy-Lag-3})
one has to use the equation of motion for the massless fermions
and therefore the SUSY algebra closes only in
on-shell limit. So how to close SUSY algebra even
in off-shell. A more elucidate statement for this problem
should read as how to match the bosonic degrees of freedom
to that of a fermionic one in off-shell? The remedy of this
problem can come from adding some auxiliary field, $F$
(with mass dimension $2$) in the theory which can provide
the required extra bosonic degrees of freedom.
Being auxiliary, $F$ cannot posses a kinetic term ($\mathcal{L}_{auxiliary} = F^*F$,
Euler-Lagrange equation is $F = F^* = 0$). So the modified set of
transformations read as
\begin{eqnarray}
&& \delta \phi = \epsilon \psi,
~~\Rrightarrow \delta \phi^* = \epsilon^\dagger \psi^\dagger,\nonumber \\
&& \delta \psi_\alpha = -i(\sigma^\mu \epsilon^\dagger)_\alpha \partial \phi
+ \epsilon_\alpha F,
~~\Rrightarrow \delta \psi^\dagger_{\dot{\alpha}}
= i(\epsilon \sigma^\mu)_{\dot{\alpha}} \partial \phi^*
+ \epsilon^\dagger_{\dot{\alpha}} F^*\nonumber \\
&& \delta F = -i \epsilon^\dagger \bar{\sigma}^\mu \partial_\mu \psi
~~\Rrightarrow \delta F^*
= i \partial_\mu \psi^\dagger \bar{\sigma}^\mu \epsilon.
\label{susy-Lag-4}
\end{eqnarray}
Eqn.(\ref{susy-Lag-1}) also receives modification and for `$i$' number
of chiral supermultiplets is given by
\begin{equation}
\mathcal{L}^{chiral} =
-\underbrace{\partial_\mu \phi^{i^*} \partial^\mu \phi_i}_{\mathcal{L}_{scalar}}
+\underbrace{i \psi^{i^\dagger} \overline{\sigma}^\mu \partial_\mu \psi_i}_{\mathcal{L}_{fermion}}
+\underbrace{F^{i^*} F_i}_{\mathcal{L}_{auxiliary}}.
\label{susy-Lag-5}
\end{equation}
\begin{flushleft}
{\it{$\maltese$ Gauge bosons}}
\end{flushleft}
Theory of the SM also contains different types of gauge bosons. So in
order to supersymmetrize the SM one must consider some ``fermionic counterparts''
also to complete the set. The massless spin one gauge boson $(A^a_\mu)$
and the accompanying spin $\frac{1}{2}$ supersymmetric partner
(two component Weyl spinor, called {\it{gauginos ($\lambda^a$)}}) also belong
to the same multiplet, known as the gauge supermultiplet. The index `$a$'
runs over adjoint representation of the associated $\rm{SU(N)}$ group. It is
interesting to note that since gauge bosons belong to the adjoint representation,
hence a gauge supermultiplet is a real representation. Just like the case
of chiral supermultiplet one has to rely on some auxiliary fields
$D^a$ to close off-shell SUSY algebra. The corresponding Lagrangian
is written as
\begin{equation}
\mathcal{L}^{gauge} =
-\overbrace{\frac{1}{4}F^a_{\mu\nu}F^{\mu\nu}_a}^{F^a_{\mu\nu}
= \partial_\mu A^a_\nu -\partial_\nu A^a_\mu + g f^{abc} A^b_{\mu} A^c_{\nu}}
+\underbrace{i \lambda^{a\dagger} \bar{\sigma}^\mu D_\mu \lambda^a}_{D_\mu
\lambda^a = \partial_\mu \lambda^a + g f^{abc} A^b_\mu \lambda^c}
+~ \frac{1}{2} D^a D^a,
\label{susy-Lag-6}
\end{equation}
where $F^a_{\mu\nu}$ is the Yang-Mills field strength
and $D_\mu \lambda^a$ is the covariant derivative for gaugino
field, $\lambda^a$. The set of SUSY transformations which leave the
action $\mathcal{S}^{gauge} (=\int d^4x \mathcal{L}^{gauge})$
invariant are written as
\begin{eqnarray}
&& \delta A^a_\mu = -\frac{1}{\sqrt{2}} (\epsilon^\dagger \bar{\sigma}_\mu \lambda^a
+ \lambda^{a^\dagger} \bar{\sigma}_\mu \epsilon),\nonumber \\
&& \delta \lambda^a_\alpha = \frac{i}{2\sqrt{2}} (\sigma^\mu \bar{\sigma}^\nu \epsilon)_\alpha
F^a_{\mu\nu} + \frac{1}{\sqrt{2}} \epsilon_\alpha D^a,\nonumber \\
&& \delta D^a = -\frac{i}{\sqrt{2}} (\epsilon^\dagger \bar{\sigma}_\mu D_\mu \lambda^a
- D_\mu \lambda^{a^\dagger} \bar{\sigma}_\mu \epsilon).
\label{susy-Lag-7}
\end{eqnarray}
\begin{flushleft}
{\it{$\maltese$ Interactions in a supersymmetric theory}}
\end{flushleft}
A supersymmetrize version of the SM should include an interaction
Lagrangian invariant under SUSY transformations. From the
argument of renormalizability and naive power counting
the most general interaction Lagrangian (without gauge interaction)
appears to be
\begin{equation}
\mathcal{L}^{int} = \underbrace{-\frac{1}{2}W^{ij}\psi_i\psi_j}^{[W^{ij}] = [mass]^1}
+ \overbrace{W^{i}F_i}^{[W^{i}] = [mass]^2}
+ \overbrace{x^{ij} F_i F_j}^{[x^{ij}] = [mass]^0}
+~ c.c - \underbrace{U}_{[U] = [mass]^4},
\label{susy-Lag-8}
\end{equation}
where $x^{ij},W^{ij},W^i,U$ all are polynomials of
$\phi,\phi^*$ (scalar fields) with degrees $0,1,2,4$. However, invariance under
SUSY transformations restricts the form of eqn. (\ref{susy-Lag-8})
as
\begin{equation}
\mathcal{L}^{int} = (-\frac{1}{2} W^{ij}\psi_i\psi_j
+ W^{i}F_i) + c.c.
\label{susy-Lag-9}
\end{equation}
It turns out that in order to maintain the interaction Lagrangian
invariant under supersymmetry transformations,
the quantity $W^{ij}$ must to be analytic function of $\phi_i$
and thus cannot contain a $\phi^*_i$. It is convenient to define
a quantity $W$ such that $W^{ij} = \frac{\partial W}{\partial \phi_i \partial \phi_j}$
and $W^{i} = \frac{\partial W}{\partial \phi_i}$. The entity $W$ in
most general form looks like
\begin{equation}
W = h_i \phi_i + \frac{1}{2} M^{ij} \phi_i \phi_j
+ \frac{1}{3!} f^{ijk} \phi_i \phi_j \phi_k.
\label{susy-Lag-10}
\end{equation}
First term of eqn.(\ref{susy-Lag-10}) vanishes for the supersymmetric
version of the SM as $h^i = 0$ in the absence of a
gauge singlet scalar field. It is important to note that in an equivalent language,
the quantity $W$ is said to be a function of the chiral superfields
\cite{c2Ferrara:1974ac,c2Salam:1974yz}. A superfield is a single object
that contains as components all of the bosonic, fermionic, and auxiliary
fields within the corresponding supermultiplet. That is
\begin{eqnarray}
&&{\bf \Phi} \supset (\phi,\psi,F),~{\rm{or}},\nonumber\\
&&{\bf \Phi}(y^\mu,\theta) = \phi(y^\mu) + \theta \psi(y^\mu)
+ \theta\theta F(y^\mu)~{\rm{and}},\nonumber\\
&&{\bf \Phi}^\dagger(\bar{y}^\mu,\bar{\theta}) = \phi^*(\bar{y}^\mu)
+ \bar{\theta} \bar{\psi}(\bar{y}^\mu)
+ \bar{\theta}\bar{\theta} F(\bar{y}^\mu),
\label{superfield-defn}
\end{eqnarray}
where $y^\mu~(=x^\mu-i\theta \sigma^\mu \bar{\theta})$ and
$\bar{y}^\mu~(=\bar{x}^\mu+i\theta \sigma^\mu \bar{\theta})$ represent left and
right chiral superspace coordinates, respectively. It is important
to note that in case of the $(3+1)$ dimensional field theory $x^\mu$ represents
the set of coordinates. However, for implementation
of SUSY with $(3+1)$ dimensional field theory one needs to consider
{\it{superspace}} with supercoordinate $(x^\mu,\theta^\alpha,\bar{\theta}_{\dot{\alpha}})$.
$\theta^\alpha,\bar{\theta}_{\dot{\alpha}}$ are spinorial coordinates spanning the
fermionic subspace of the superspace. Any superfield, which is
a function of $y$ and $\theta$ ($\bar{y}$ and $\bar{\theta}$) only,
would be known as a left(right) chiral superfield. Alternatively,
if one defines chiral covariant derivatives $\bf{\mathcal{D}}_A$ and
$\bf{\mathcal{\bar{D}}}_{\bar A}$ as
\begin{eqnarray}
{\bf{\mathcal{D}}_A} \bar{y}^\mu = 0,~{\bf{\mathcal{\bar{D}}}_{\bar A}} y^\mu=0,
\label{chiral-covariant-defn}
\end{eqnarray}
then a left and a right chiral superfield is defined as
\begin{eqnarray}
\bf{\mathcal{D}}_A {\bf \Phi} = 0
~{\rm{and}}
~\bf{\mathcal{\bar{D}}}_{\bar A} {\bf \Phi^\dagger}=0,
\label{chiral-field-defn2}
\end{eqnarray}
The gauge quantum numbers and the mass dimension of a chiral superfield are the
same as that of its scalar component, thus in the superfield
formulation, eqn.(\ref{susy-Lag-10}) can be recasted as
\begin{equation}
W = h_i {\bf \Phi}_i + \frac{1}{2} M^{ij} {\bf \Phi}_i {\bf \Phi}_j
+ \frac{1}{3!} f^{ijk} {\bf \Phi}_i {\bf \Phi}_j {\bf \Phi}_k.
\label{susy-Lag-10-superfield}
\end{equation}
The quantity $W$ is now called a superpotential. The superpotential $W$
now not only determines the scalar interactions of the theory,
but also determines fermion masses as well as different Yukawa
couplings. Note that $W(W^\dagger)$ is an
analytical function of the left(right) chiral superfield.
Coming back to interaction Lagrangian, using the equation of motion for $F$ and $F^*$
finally one ends up with
\begin{equation}
\mathcal{L}^{int} = -\frac{1}{2}(W^{ij}\psi_i\psi_j
+ W^*_{ij}\psi^{\dagger^i}\psi^{\dagger^j}) - 2 W^i W^*_i.
\label{susy-Lag-11}
\end{equation}
The last and remaining interactions are coming from
the interaction between gauge and chiral supermultiplets.
In presence of the gauge interactions SUSY transformations
of eqn.(\ref{susy-Lag-4}) suffer the following modification,
$\Rrightarrow \partial_\mu \rightarrow D_\mu$. It is also interesting
to know that in presence of interactions, Euler-Lagrange equations
for $D^a$ modify as $D^a = -g(\phi^{i^*} T^a \phi^i)$ with $T^a$
as the generator of the group.
So finally with the help of eqns.(\ref{susy-Lag-5}), (\ref{susy-Lag-6}),
(\ref{susy-Lag-11}) and including the effect of gauge interactions
the complete supersymmetric Lagrangian looks like
\begin{eqnarray}
\mathcal{L}^{total} &=& - \partial_\mu \phi^{i^*} \partial^\mu \phi_i
+ i \psi^{i^\dagger} \overline{\sigma}^\mu \partial_\mu \psi_i
-\frac{1}{4}F^a_{\mu\nu}F^{\mu\nu}_a
+i \lambda^{a\dagger} \bar{\sigma}^\mu D_\mu \lambda^a \nonumber \\
-&&\left[\{\frac{1}{2}(W^{ij}\psi_i\psi_j
+\sqrt{2} g (\phi^*_iT^a_{ij}\psi_j)\lambda^a \} + h.c\right]
\nonumber \\
-&&V(\phi,\phi^*) \left\{\Rrightarrow {
\underbrace{W^*_i W^i}_{F^*_i F^i} +
\overbrace{\frac{1}{2}\sum_{a} g^2_a
(\phi^{i^*} T^a \phi_i)^2}^{\sum \frac{1}{2}D^aD^a}}\right\}.
\label{susy-Lag-12}
\end{eqnarray}
In eqn.(\ref{susy-Lag-12}) index `$a$' runs over three of the SM
gauge group, ${\rm SU}(3)_C \times {\rm SU}(2)_L \times {\rm U}(1)_Y$.
Potential $V(\phi,\phi^*)$, by definition (see eqn.(\ref{susy-Lag-12}))
is bounded from below with minima at the origin.
\section{{\bf S}USY breaking} \label{susy-breaking}
In a supersymmetric theory fermion and boson belonging to the
same supermultiplet must have equal mass.
This statement can be re-framed in
a different way. Consider the supersymmetric partner
of electron (called selectron, $\widetilde e$), then SUSY invariance
demands, $m_e = m_{\widetilde e} = 5.109\times10^{-4} {\rm{GeV}}$ (see table \ref{SM-spectrum}),
where $m_{\widetilde e}$ is mass of the selectron. But till date there exists no experimental
evidence (see ref.\cite{c2Nakamura-c2}) for a selectron. That simply indicates that
supersymmetry is a broken symmetry in nature.
The immediate question arises then what is the pattern of SUSY
breaking? Is it a spontaneous or an explicit breaking? With the
successful implementation of massive gauge bosons in the SM, it
is naturally tempting to consider a spontaneous SUSY breaking first.
\begin{flushleft}
{\it{$\blacklozenge$ Spontaneous breaking of SUSY}}
\end{flushleft}
In the case of spontaneous SUSY breaking the supersymmetric Lagrangian
remains unchanged, however, vacuum of the theory is no longer symmetric
under SUSY transformations. This will in turn cause
splitting in masses between fermionic and bosonic states within the
same multiplet connected by supersymmetry transformation.
From the argument given in section \ref{susy-basics}
it is evident that the spontaneous breaking of supersymmetry
occurs when the supercharges $Q,\bar{Q}$ (the SUSY generators)
fail to annihilate the vacuum of the theory. In other words if supersymmetry
is broken spontaneously (see figure \ref{SSB-susy}), the vacuum must have
positive energy, i.e. $\langle0|\mathcal{H}^{susy}|0\rangle \equiv
\langle\mathcal{H}^{susy}\rangle> 0$ (see eqn.(\ref{susy-gen-prop-6})).
$\mathcal{H}^{susy}$ is the SUSY Hamiltonian. Neglecting the space-time
effects one gets
\begin{equation}
\langle0|\mathcal{H}^{susy}|0\rangle = \langle0|\mathcal{V}^{susy}|0\rangle,
\label{susy-Lag-13}
\end{equation}
where $\mathcal{V}^{susy}$ is given by $V(\phi,\phi^*)$ (see eqn.(\ref{susy-Lag-12})).
Therefore spontaneous breaking of SUSY implies
\begin{equation}
\underbrace{\langle F \rangle \neq 0}_{\tt{F-type~breaking}}
~~{\rm{or}}~~
\overbrace{\langle D \rangle \neq 0}^{\tt{D-type~breaking}}.
\label{susy-break-1}
\end{equation}
It is interesting to note that eqn.(\ref{susy-break-1}) does not
contain $D^a$ because if the theory is gauge invariant
then $\langle D \rangle = 0$ holds for Abelian vector superfield
only. It is informative to note that the spontaneous breaking of
a supersymmetric theory through $F$-term is known as
{\it{O'raifeartaigh}} mechanism \cite{c2O'Raifeartaigh:1975pr} and
the one from $D$-term as {\it{Fayet-Iliopoulos}}
mechanism \cite{c2Fayet:1974jb,c2Fayet:1975yh}. In the case of
global \footnote{The infinitesimal SUSY transformation parameter
$\epsilon_\alpha$ is a space-time independent quantity.} SUSY breaking,
the broken generator is $Q$, and hence the Nambu-Goldstone particle must
be a massless neutral spin $\frac{1}{2}$ Weyl fermion (known as {\it{goldstino}}).
The goldstino in not the supersymmetric partner of Goldstone boson,
but a Goldstone fermion itself.
\begin{figure}[ht]
\centering
\includegraphics[width=8.95cm,keepaspectratio]{FigsPDF/SSB-susy.pdf}
\caption{Vacua of a supersymmetric theory. (i) exactly supersymmetric
and (ii) SUSY is spontaneously broken.}
\label{SSB-susy}
\end{figure}
But there are drawbacks with this simple approach.
The supersymmetric particle spectrum is known to follow
certain sum rules, known as the supertrace
sum rules which must vanish.
The supertrace of the tree-level squared-mass eigenvalues is
defined with a weighted sum over all particles with spin $j$ as
STr$(m^2) \equiv \sum (-1)^j (2j+1) Tr(m^2_j) = 0$
\cite{c2Ferrara:1979iz,c2Ferrara:1979wa}. This theorem holds
for sets of states having same quantum numbers.
But, a vanishing supertrace indicates
that some of the supersymmetric particles must be lighter compared
to that of the SM, which is of course not observed experimentally so far.
However, this relation holds true at the tree level and for renormalizable
theories. So supersymmetry can be spontaneously broken in some
``hidden sector'' which only couples to the ``visible'' or ``observable''
SM sector through loop mediated or through non-renormalizable interactions.
These intermediate states which appear in loops or are integrated out
to produce non-renormalizable interactions are known as the ``messengers''
or ``mediators''.
Some of the well-motivated
communication schemes are {\it{supergravity, anomaly mediation, gauge mediation,
gaugino mediation}} and many others
(see review \cite{Kolda:1997wt,c2Intriligator:2007cp}).
In all of these scenario SUSY is
spontaneously broken at some hidden or secluded
sector, containing fields singlet under the SM gauge group at
some distinct energy scale and the information of breaking
is communicated to the observable minimal sector via
some messenger interaction. A discussion on these issues
is beyond the scope of this thesis.
\begin{flushleft}
{\it{$\blacklozenge$ Explicit SUSY breaking and soft-terms}}
\end{flushleft}
It is now well understood that with the minimal field content
SUSY has to be broken explicitly.
But what happens to Higgs mass hierarchy if SUSY is broken in nature?
It turns out that in order to have a theory free from quadratic divergence
as well as to have the desired convergent behaviour of supersymmetric
theories at high energies along with the nonrenormalization of its
superpotential couplings, the explicit SUSY breaking terms must be
soft \cite{c2Girardello:1981wz,c2Dimopoulos:1981zb,c2Sakai:1981gr,c2Hall:1990ac}.
The word soft essentially implies that all field operators occurring
in explicit SUSY breaking Lagrangian must have a mass dimension
less than four.
The possible most general \cite{c2Fayet:1976cr,c2Girardello:1981wz}
soft supersymmetry breaking terms inhoused in
$\mathcal{L}_{soft}$ are\footnote{It is interesting to note that
terms like $-\frac{1}{2}c^{jk}_i \phi^{i^*} \phi_j \phi_k + c.c$
are also viable candidates for $\mathcal{L}_{soft}$, however they can
generate quadratic divergence from the loop in the presence of gauge singlet
chiral superfields. A term like this becomes soft \cite{c2Hall:1990ac}
in the absence of singlet superfields.
One more important lesson is that the mass dimension of any coupling
in $\mathcal{L}_{soft}$ has to be less than four is a necessary but not sufficient
condition for the softness of any operator.}
\begin{eqnarray}
\mathcal{L}_{soft} &&= -\left(\frac{1}{2}M_a \lambda^a \lambda^a
+ \frac{1}{3!} a^{ijk} \phi_i \phi_j \phi_k + \frac{1}{2} b^{ij} \phi_i \phi_j
+ t^i \phi_i\right) + c.c \nonumber \\
&&- (m^2)^i_j \phi^{j^*} \phi_i.
\label{susy-soft-1}
\end{eqnarray}
In eqn.(\ref{susy-soft-1}) terms like $t^i \phi_i$ are possible only if
there exist gauge singlet superfields and thus these terms are
absent from the minimal supersymmetric version of the SM. $M_a$'s are
the gaugino soft mass terms, $(m^2)^j_i$ are the coefficients for
scalar squared mass terms and $b^{ij},a^{ijk}$ are the couplings for
quadratic and cubic scalar interactions.
\begin{flushleft}
{\it{$\blacklozenge$ Higgs mass hierarchy and $\mathcal{L}_{soft}$}}
\end{flushleft}
\begin{list}{}{}
\item
The form of eqn.(\ref{susy-soft-1}) indicates modification of
Lagrangian shown in eqn.(\ref{c2-susy-motivation}). Adding a possible
interaction term of the form
$\frac{\lambda_f A_{\widetilde f}}{\sqrt{2}} \widetilde{f_L}\widetilde{f^*_R}h^0 + h.c$
(scalar cubic interaction) in eqn.(\ref{c2-susy-motivation})
in turn modifies the two-point function via the loop
(see eqn.(\ref{c2-susy-loop-log})) as
\begin{eqnarray}
\Pi^{\widetilde f}_{h^0h^0}(0) +
\Pi^{f}_{h^0h^0}(0) &=&
-\frac{i y^2_f}{16 \pi^2}
\left[4 \delta^2 + (2 \delta^2 + |A_{\widetilde f}|^2) ln \frac{m^2_f}{\mu^2_R}\right]\nonumber\\
&+& {\text{higher orders}},
\label{c2-susy-loop-soft-mod}
\end{eqnarray}
where $\delta^2 = m^2_{\widetilde f} - m^2_f$ and we
assume $|\delta|,|A_{\widetilde f}|\ll m_f$. The most important observation about
eqn.(\ref{c2-susy-loop-soft-mod}) is that, in the exact supersymmetric
limit
\begin{equation}
m^2_{\widetilde f} = m^2_f,~~A_{\widetilde f} = 0,
\label{c2-susy-limit}
\end{equation}
that is, entire one loop renormalization of the Higgs self
energy vanishes \footnote{Actually this condition is true for
all orders of perturbation theory and is a consequence of the
nonrenormalization theorem \cite{c2Wess:1973kz,c2Iliopoulos:1974zv,
c2Ferrara:1974fv,c2Zumino:1974bg,c2Grisaru:1979wc}.}. It is also clear
from eqn.(\ref{c2-susy-loop-soft-mod}) that Higgs self energy
is linearly proportional to the SUSY breaking parameters
$(\delta^2,|A_{\widetilde f}|^2)$. Thus supersymmetric theories are
free from quadratic divergences, unless $m^2_{\widetilde f} \gg m^2_f$.
This is an extremely important relation, which indicates that
in order to have a TeV scale Higgs boson mass (theoretical limit)
the soft terms $(A_{\widetilde f})$ and the sparticle masses $(m_{\widetilde f})$
must lie in the same energy scale (reason why we are dreaming to
discover SUSY at the large hadron collider experiment).
\end{list}
\section{{\bf M}inimal Supersymmetric Standard Model} \label{susy-MSSM}
We are now well equipped to study the Minimal Supersymmetric Standard
Model or MSSM (see reviews \cite{c2Nilles-1983,c2Sohnius-1985,c2Martin-1997}).
It is always illuminating to start with a description of
the particle content. Each of the SM fermions have their bosonic counterparts,
known as sfermions. Fermionic counterpart for a gauge boson is known as
a gaugino. Higgsino is the fermionic counter part for a Higgs boson.
It is important to re-emphasize that since a superpotential
is invariant under supersymmetry transformation it cannot involve an chiral and a anti-chiral
superfield at the same time. In other words a superpotential $(W)$ is
an analytical functions of chiral superfields only ($W^\dagger$
contains anti-chiral superfields only) and thus two Higgs doublets are essential for MSSM.
In addition, the condition for
anomaly cancellation in the higgsino sector, which is a requirement of renormalizability
also asks for two Higgs doublets, $H_u$ and $H_d$. It must be remembered that
each of the supersymmetric particle ({\it{sparticle}}) has same set of gauge
quantum numbers under the SM gauge group as their SM counterpart,
as shown in eqn.(\ref{SM-gauge-group}). The Higgs doublet $H_u$ behaves like
eqn.(\ref{SM-Higgs}), whereas the other doublet
$H_d$ under ${\rm SU}(3)_C \times {\rm SU}(2)_L \times {\rm U}(1)_Y$ transforms as,
\begin{eqnarray}
H_d = \left(\begin{array}{c}
H^0_d \\
H^-_d
\end{array}\right) \sim ({\bf{1,2,-1}}).
\label{Hd-Higgs}
\end{eqnarray}
The particle content of the MSSM is shown in figure \ref{MSSM-particle}.
\begin{figure}[ht]
\centering
\includegraphics[width=10.75cm,keepaspectratio]{FigsPDF/MSSM.pdf}
\caption{particle content of the MSSM.}
\label{MSSM-particle}
\end{figure}
Every lepton $(\ell)$ and quark $(q)$ of the SM (spin $\frac{1}{2}$)
is accompanied by a {\it{slepton $(\widetilde{\ell})$}} and
{\it{squark $(\widetilde{q})$}} (spin $0$). Corresponding to two Higgs fields
$H_u$ and $H_d$ (denoted as $H$ in figure \ref{MSSM-particle})
there exist two {\it{Higgsino}} fields ($\widetilde{H_U},\widetilde{H_d}$) as well
(denoted as $\widetilde H$ in figure \ref{MSSM-particle}). The electroweak gauge bosons
$W,Z$, gluons $(g)$ and photon $(\gamma)$ are associated with their
superpartner states, namely, {\it{wino $(\widetilde W)$}}, {\it{zino $(\widetilde Z)$}},
{\it{gluino $(\widetilde g)$}} and {\it{photino $(\widetilde \gamma)$}}\footnote{Another
alternative set in lieu of $Z,\gamma$
could be $B,W^3$, where $B$ and $W^3$ are the $\rm{U(1)_Y}$ and $\rm{SU(2)_L}$
gauge bosons, respectively. Correspondingly on the right hand side
one should have $\widetilde{W}_3,\widetilde{B}\Longleftrightarrow \widetilde \gamma, \widetilde Z$.}.
Without further clarification we will concentrate first on the MSSM superpotential
and then on the soft terms. We will not talk about the
kinetic terms i.e, the free Lagrangian and the gauge interactions
(see ref.\cite{c2Drees-2004} for an extensive discussions).
\begin{flushleft}
{\it{$\maltese$ MSSM superpotential and soft terms}}
\end{flushleft}
The superpotential for the MSSM is written as
\begin{eqnarray}
W^{MSSM} &=& \epsilon_{ab}(Y^{ij}_u\hat H^b_u\hat Q^a_i\hat u^c_j +
Y^{ij}_d\hat H^a_d \hat Q^b_i\hat d^c_j + Y^{ij}_e\hat H^a_d\hat
L^b_i\hat e^c_j - \mu \hat H^a_d\hat H^b_u ),\nonumber \\
\label{MSSM-superpotential-1}
\end{eqnarray}
where $\hat H_d$ and $\hat H_u$ are the down-type and up-type Higgs
superfields, respectively. The $\hat Q_i$ are ${SU(2)}_L$ doublet quark superfields,
${\hat u}^c_j$ [${\hat d}^c_j$] are $\rm{SU(2)}_L$ singlet up-type [down-type] quark
superfields. The $\hat L_i$ are the doublet lepton superfields, and the
${\hat e}^c_j$ are the singlet charged lepton superfields. Here $a,b$ are SU(2)
indices, and $\epsilon_{12}$ = --$\epsilon_{21}$ = 1. Note that
$u^c_i,d^c_i,e^c_i \equiv u^*_{i_R},d^*_{i_R},\ell^*_{i_R}$
(see eqn.(\ref{SM-gauge-group})). The only coupling of the superpotential $W$,
that has a positive mass dimension is the $\mu$-parameter.
The corresponding soft SUSY breaking Lagrangian can be
written as
\begin{eqnarray}
-\mathcal{L}^{MSSM}_{\text{soft}} &=&
(m_{\tilde{Q}}^2)^{ij} {\tilde Q^{a^*}_i} \tilde{Q^a_j}
+(m_{\tilde u^c}^{2})^{ij}
{\tilde u^{c^*}_i} \tilde u^c_j
+(m_{\tilde d^c}^2)^{ij}{\tilde d^{c^*}_i}\tilde d^c_j
- \epsilon_{ab} B_\mu \hat H^a_d\hat H^b_u \nonumber \\
&+&(m_{\tilde{L}}^2)^{ij} {\tilde L^{a^*}_i}\tilde{L^a_j}
+ (m_{\tilde e^c}^2)^{ij}{\tilde e^{c^*}_i}\tilde e^c_j
+ m_{H_d}^2 {H^{a^*}_d} H^a_d + m_{H_u}^2 {H^{a^*}_u} H^a_u \nonumber \\
&+& \left[ \epsilon_{ab} \left\{
(A_uY_u)^{ij} H_u^b\tilde Q^a_i \tilde u_j^c +
(A_dY_d)^{ij} H_d^a \tilde Q^b_i \tilde d_j^c \right. \right.\nonumber \\
&+& \left. \left.
(A_eY_e)^{ij} H_d^a \tilde L^b_i \tilde e_j^c
\right\} - \frac{1}{2} \sum_{i=1}^3 M_i \widetilde {\lambda_i}+ \text{h.c.} \right].
\label{Lsoft-MSSM}
\end{eqnarray}
In eqn.(\ref{Lsoft-MSSM}), the first two lines consist of squared-mass terms
of squarks, sleptons and Higgses along with a bilinear
term $(\epsilon_{ab} B_\mu \hat H^a_d\hat H^b_u)$ in two Higgs superfields.
The next line contains the trilinear scalar
couplings. Finally, in the last line $M_3, M_2$, and $M_1$ are
Majorana masses corresponding to $\rm{SU(3)}_c$, $\rm{SU(2)}_L$
and $\rm{U(1)}_Y$ gauginos $\widetilde{\lambda}_3, \widetilde{\lambda}_2$,
and $\widetilde{\lambda}_1$, respectively.
The tree level scalar potential is given by (see eqn.(\ref{susy-Lag-12}))
\begin{eqnarray}
V^{MSSM}_{scalar} &=& V^{MSSM}_{soft} + \frac{1}{2} D^a D^a
+ \left|\frac{\partial W^{MSSM}}{\partial \phi^{MSSM}}
\right|^2,
\label{MSSM-scalar-pot}
\end{eqnarray}
where $V^{MSSM}_{soft}$ contains only the scalar couplings of
eqn.(\ref{Lsoft-MSSM}) and $\Phi^{MSSM}$ represents scalar
component of any of the MSSM chiral superfields. Only the neutral scalar
fields develop vacuum expectation values while minimizing
the scalar potential $V^{MSSM}_{scalar}$ as follows
\begin{eqnarray}
\langle H_d^0 \rangle = v_1,~\langle H_u^0 \rangle = v_2.
\label{MSSM-VEV}
\end{eqnarray}
It is evident from eqns.(\ref{MSSM-superpotential-1}) and
(\ref{Lsoft-MSSM}) that the MSSM has a very rich particle
spectra. Note that the matrices associated with bilinear terms in fields
(particles or sparticles) are often appear with off-diagonal
entries after EWSB (see chapter \ref{SM}). Clearly entries of these matrices cannot
represent any physical masses. So in general these off-diagonal matrices
of the gauge or flavour basis can be rotated in a diagonal basis
using suitable unitary or bi-unitary transformations.
All the scalar mass squared matrices are inhoused in $V^{MSSM}_{scalar}$.
\begin{flushleft}
{\it{$\maltese$ Gauge versus Mass eigen-basis}}
\end{flushleft}
\begin{itemize}
\item
The squarks $(\widetilde q)$ and sleptons $(\widetilde l)$
The squark and slepton mass square matrices in the flavour basis
are bilinears in $\widetilde {f_L}^* \widetilde {f_L}$, $\widetilde {f_R}^* \widetilde {f_R}$
and $\widetilde {f_L}^* \widetilde {f_R} + \text{c.c}$ where $\widetilde f \equiv \widetilde l
/ \widetilde q$. It is always possible to rotate them into another basis $\widetilde f_1,\widetilde f_2$
where only combination like $\widetilde {f_1}^* \widetilde {f_1}, \widetilde {f_2}^* \widetilde {f_2}$
exists. The basis $\widetilde f_{1,2}$ is known as the mass basis for squarks
and sleptons. The orthogonal mixing matrix relating $\widetilde f_{L,R}$ and
$\widetilde f_{1,2}$ contains an angle `$\theta$'
which depends on the ratio of the off-diagonal entry in $\widetilde f_{L,R}$ basis
and the difference in diagonal entries in the same basis. It can be shown
(see for example ref. \cite{c2Drees-2004}) that
for the first two generations of squark and charged slepton the effect of
off diagonal mixing is negligible and to a very good approximation
$\widetilde f_{L,R}$ can be treated as the mass basis. So we conclude that
\begin{table}[ht]
\footnotesize
\centering
\begin{tabular}{c c}
\hline \hline
Gauge or flavour basis & Mass basis \\ \hline \hline
$\widetilde e_L,\widetilde e_R,\widetilde \mu_L,\widetilde \mu_R$ &
$\widetilde e_L,\widetilde e_R,\widetilde \mu_L,\widetilde \mu_R$ \\
\vspace*{0.1cm}
$\widetilde u_L,\widetilde u_R,\widetilde d_L,\widetilde d_R$
& $\widetilde u_L,\widetilde u_R,\widetilde d_L,\widetilde d_R$ \\
\vspace*{0.1cm}
$\widetilde c_L,\widetilde c_R,\widetilde s_L,\widetilde s_R$
& $\widetilde c_L,\widetilde c_R,\widetilde s_L,\widetilde s_R$ \\ \hline \hline
\end{tabular}
\end{table}
However, this simple minded approach fails for the
third family of slepton and squark due to relatively large Yukawa
coupling. This is because, it is the effect of Yukawa coupling
which controls the size of the off-diagonal term. Summing up,
for the third family
\begin{table}[ht]
\footnotesize
\centering
\begin{tabular}{c c}
\hline \hline
Gauge or flavour basis & Mass basis \\ \hline \hline
$\widetilde \tau_L,\widetilde \tau_R$ &
$\widetilde \tau_1,\widetilde \tau_2$ \\
\vspace*{0.1cm}
$\widetilde b_L,\widetilde b_R,\widetilde t_L,\widetilde t_R$
& $\widetilde b_1,\widetilde b_2,\widetilde t_1,\widetilde t_2$ \\ \hline \hline
\end{tabular}
\end{table}
It remains to talk about the left sneutrinos which do not
have any right handed counter part. The degenerate squared mass for
all three generations of left sneutrino is given by
\begin{eqnarray}
M^2_{\widetilde \nu} = m^2_{\widetilde L} + \frac{1}{2} M^2_Z {\rm cos} 2\beta,
\label{left-sneutrinos}
\end{eqnarray}
where $\rm{tan}\beta=\frac{v_2}{v_1}$ is the ratio of two Higgs VEVs and
$M_Z$ is the $Z$ boson mass given by $M^2_Z = \frac{1}{2}(g^2_1
+g^2_2)(v^2_1+v^2_2)$. Mass for the $W^\pm$-bosons are given
by $M^2_W = \frac{g^2_2}{2}(v^2_1+v^2_2)$.
\item
The neutral and charged supersymmetric fermions
The neutral supersymmetric fermions $(-i\widetilde B^0,-i\widetilde W^0_3,\widetilde H^0_d,
\widetilde H^0_u)$ are known to form a $4\times4$ symmetric matrix
in the flavour basis. This symmetric matrix is diagonalizable
using a single unitary matrix $N$ and the corresponding four mass
eigenstates are known as neutralinos, $\chi^0_i$ (two-component spinor).
Mathematically,
\begin{equation}
\chi^0_i = N_{i1} \widetilde B^0 + N_{i2} \widetilde W^0_3 + N_{i3} \widetilde H^0_d
+ N_{i4} \widetilde H^0_u,
\label{MSSM-neutralinos}
\end{equation}
with $N_{ij}$ as the elements of the matrix $N$.
The charged fermions $\psi^+=-i\widetilde W^+,\widetilde H^+_u$ and
$\psi^-=-i\widetilde W^-,\widetilde H^-_d$ on the other hand form a
$4\times4$ mass matrix in the Lagrangian as follows
\begin{eqnarray}
\mathcal{L}^{chargino}_{MSSM} =
-\frac{1}{2}
\left(\begin{array}{c c}
\psi^+ & \psi^-
\end{array}\right)
\left(\begin{array}{c c}
0 & (M^{chargino}_{MSSM})^T_{2\times2}\\
(M^{chargino}_{MSSM})_{2\times2} & 0
\end{array}\right)
\left(\begin{array}{c}
\psi^+ \\
\psi^-
\end{array}\right) + h.c.\nonumber\\
\label{MSSM-charginos-Lagrangian}
\end{eqnarray}
The $2\times2$ non-symmetric matrix $(M^{chargino}_{MSSM})_{2\times2}$
(see appendix \ref{appenA}) requires a bi-unitary transformation to
go the two-component physical charged fermion
eigenstates, known as charginos, $\chi^\pm_i$. If $U,V$ are the two required
transformation matrices, then
\begin{eqnarray}
&&\chi^+_i = V_{i1} \widetilde W^+ + V_{i2} \widetilde H^+_u, \nonumber \\
&&\chi^-_i = U_{i1} \widetilde W^- + U_{i2} \widetilde H^-_d.
\label{MSSM-charginos}
\end{eqnarray}
It is important to re-emphasize that all the charged and neutral
spinors considered here are two-component Weyl spinors. They
can be used further to construct the corresponding four-component
spinors. The neutralino and chargino mass matrices for MSSM are given
in appendix \ref{appenA}.
\item
The neutral and the charged leptons and the
quarks are treated in MSSM similar to that of the SM.
\item
The gauge bosons are also treated in similar fashion.
\item
Higgs bosons in MSSM
Let us write down two Higgs doublet of the MSSM
in the real $(\Re)$ and imaginary $(\Im)$ parts as follows
\begin{eqnarray}
&&H_d = \left(\begin{array}{c}
H^0_d \\
H^-_d
\end{array}\right) =
\left(\begin{array}{c}
\Re{H^0_d} + i \Im{H^0_d} \\
\Re{H^-_d} + i \Im{H^-_d}
\end{array}\right) =
\left(\begin{array}{c}
h_1 + i h_2 \\
h_3 + i h_4
\end{array}\right),\nonumber\\
&&H_u = \left(\begin{array}{c}
H^+_u \\
H^0_u
\end{array}\right) =
\left(\begin{array}{c}
\Re{H^+_u} + i \Im{H^+_u} \\
\Re{H^0_u} + i \Im{H^0_u}
\end{array}\right) =
\left(\begin{array}{c}
h_5 + i h_6 \\
h_7 + i h_8
\end{array}\right)
\label{MSSM-Higgs-doublets}
\end{eqnarray}
Out of this eight Higgs field $(h_i)$, only the neutral real
fields can develope a {\it{non-zero}} VEV which are
(recasting eqn.(\ref{MSSM-VEV}))
\begin{eqnarray}
\langle \Re{H_d^0} \rangle = v_1,~\langle \Re{H_u^0} \rangle = v_2.
\label{MSSM-VEV-2}
\end{eqnarray}
These eight Higgs fields are further classifiable into three groups,
namely (1) CP-even $(h_1,h_7)$, (2) CP-odd $(h_2,h_8)$
and (3) charged $(h_{3-6})$. In the mass basis these
give five physical Higgs states, $h^0,H^0,A^0,H^\pm$ and
three Goldstone bosons $(G^0,G^\pm)$. In terms of mathematical relations,
\begin{eqnarray}
&&H^0 = \sqrt{2}\left((\Re{H^0_d}-v_1){\rm cos}\alpha+(\Re{H^0_u}-v_2){\rm sin}\alpha\right),\nonumber \\
&&h^0 = \sqrt{2}\left(-(\Re{H^0_d}-v_1){\rm sin}\alpha+(\Re{H^0_u}-v_2){\rm cos}\alpha\right),\nonumber \\
&&H^- = \left((\Re{H^-_d} + i \Im{H^-_d}){\rm sin}\beta
+ (\Re{H^+_u} + i \Im{H^+_u})^\dagger {\rm cos}\beta\right),\nonumber\\
&&A^0 = \sqrt{2} \left(-\Im{H^0_d}{\rm sin}\beta + \Im{H^0_u}{\rm cos}\beta\right),\nonumber\\
&&G^0 = \sqrt{2} \left(\Im{H^0_d}{\rm cos}\beta - \Im{H^0_u}{\rm sin}\beta\right),\nonumber\\
&&G^- = \left((\Re{H^-_d} + i \Im{H^-_d}){\rm cos}\beta
- (\Re{H^+_u} + i \Im{H^+_u})^\dagger {\rm sin}\beta\right),\nonumber\\
&& H^+ = (H^-)^\dagger,~~G^+ = (G^-)^\dagger,
\label{Higgs-mass-gauge}
\end{eqnarray}
where $\alpha$ is a mixing angle relating the gauge and mass basis of
CP-even Higgs fields. Scalar (CP-even), pseudoscalar (CP-odd)
and charged scalar mass squared matrices in the flavour basis for
MSSM Higgs fields are given in appendix \ref{appenA}.
Physical Higgs boson squared masses are given by
(using eqns.(\ref{MSSM-superpotential-1}),(\ref{Lsoft-MSSM}))
\begin{eqnarray}
&&m^2_{A^0} = \frac{2 B_\mu} {{\rm sin}{2\beta}} ,\nonumber \\
&&m^2_{H^0} = \frac{1}{2}\left[m^2_{A^0} + M^2_Z
+ \sqrt{(m^2_{A^0} + M^2_Z)^2-4 m^2_{A^0} M^2_Z {\rm cos}^22\beta}\right],\nonumber \\
&&m^2_{h^0} = \frac{1}{2}\left[m^2_{A^0} + M^2_Z
- \sqrt{(m^2_{A^0} + M^2_Z)^2-4 m^2_{A^0} M^2_Z {\rm cos}^22\beta}\right],\nonumber \\
&&m^2_{H^\pm} = m^2_{A^0} + M^2_W.
\label{Higgs-mass-MSSM}
\end{eqnarray}
From eqn.(\ref{Higgs-mass-MSSM}) one can achieve a theoretical upper
limit of the lightest Higgs boson mass \cite{c2Casas:1994us,c2Casas:1994qy},
$(m_{h^0})$ at the tree level after a bit of algebraic
exercise as \cite{c2Gunion:1989we,c2Gunion:1992hs,c2Gunion:1984yn},
\begin{equation}
m_{h^0} \le M_Z |{\rm cos}2\beta|.
\label{Higgs-mass-MSSM-lightest}
\end{equation}
The lightest Higgs mass can however, receives significant radiative
corrections from higher order processes, which are capable of
altering the lightest Higgs mass bound drastically. Note that
the value for angle $\beta$ is between $0$ to $\frac{\pi}{2}$. Thus
it is easy to conclude that $m_{h^0}$ at the tree level can be
at most of the order of the $Z$-boson mass. But this is already
ruled out by the LEP experiment
\cite{c2Barate:2003sz,c2Nakamura-c2}. So it is evident that inclusion of
loop correction \cite{c2Haber:1990aw,c2Okada:1990vk,c2Okada:1990gg,
c2Ellis:1990nz,c2Ellis:1991zd,c2Espinosa:1991fc} (see also ref.\cite{c2Chankowski:1992er} and
references therein) to lightest Higgs boson mass in MSSM is
extremely important. The dominant contribution arises from top-stop
loop and assuming masses for sparticles below $1$ TeV we get $m_{h^0} \le 135$
GeV\footnote{This limit can be further relaxed to $m_{h^0} \le 150$
GeV, assuming all couplings in the theory remain perturbative up
to the unification scale \cite{c2Kane:1992kq,c2Espinosa:1992hp}.}.
The conditions for the tree level
Higgs potential to be bounded from below (in the direction $v_1 = v_2$)
as well as the condition for EWSB are
\begin{eqnarray}
&&m_{H_d}^2 + m_{H_u}^2 + 2 |\mu|^2 \ge 2 |B_\mu|,\nonumber \\
&&(m_{H_d}^2 + |\mu|^2)(m_{H_u}^2 + |\mu|^2)< B^2_\mu.
\label{Higgs-pot-cond}
\end{eqnarray}
It is extremely important to note that if $B_\mu,m_{H_d}^2,m_{H_u}^2$
all are zero, i.e. there exist no soft SUSY breaking terms, the EWSB
turns out to be impossible. So in a sense SUSY breaking is somehow
related to the EWSB.
We conclude the description of the MSSM with a note on the corresponding set
of Feynman rules. The number of vertices are extremely large for
a supersymmetric theory even in the minimal version, and consequently
there exist a huge number of Feynman rules. The rules are far more
complicated compared to the SM because of the presence of Majorana
particles (particles, that are antiparticles of their own, neutralinos for
example). For a complete set of Feynman rules
for the MSSM see references \cite{c2Haber:1984rc,c2Gunion:1984yn,c2Gunion:1989we,
c2Rosiek:1989rs,c2Rosiek:1995kg}. A detailed analysis for the Higgs boson in supersymmetry
and related phenomenology are addressed in a series of references
\cite{c2Gunion:1984yn,c2Gunion:1986nh,c2Gunion:1988yc,c2Gunion:1992tq}.
\end{itemize}
\section{{\bf T}he $R$-parity} \label{R-parity}
The superpotential for MSSM was shown in eqn.(\ref{MSSM-superpotential-1}).
This superpotential is gauge (the SM gauge group) invariant, Lorentz invariant
and maintains renormalizability. However, it is natural to ask that
what is preventing the following terms to appear in $W^{MSSM}$, which
are also gauge and Lorentz invariant and definitely renormalizable:
\begin{eqnarray}
W^{extra} &=& \epsilon_{ab}(- \varepsilon^i \hat L^a_i\hat H^b_u
+ \frac{1}{2} \lambda_{ijk}\hat L^a_i\hat L^b_j\hat e^c_k
+ \lambda^{'}_{ijk} \hat L^a_i \hat Q^b_j\hat d^c_k
+ \frac{1}{2} \lambda^{''}_{ijk} \hat u^c_i \hat d^c_j \hat d^c_k).\nonumber\\
\label{MSSM-superpotential-2}
\end{eqnarray}
Of course, all of these terms violate either lepton $(L)$ \cite{c2Weinberg:1981wj,
c2Hall:1983id} or baryon $(B)$ \cite{c2Weinberg:1981wj,
c2Zwirner:1984is} number by odd units. The second and the third terms of
eqn.(\ref{MSSM-superpotential-2}) violate lepton number by one unit whereas
the fourth term violates baryon number by one unit.
Now it is well known that in the SM, lepton and baryon numbers
are conserved at the perturbative level. In the SM, $L$ and $B$ are the accidental
symmetry of the Lagrangian, that is to say that these are not symmetries
imposed on the Lagrangian, rather they are consequence of the gauge
and Lorentz invariance, renormalizability and, of course, particle content
of the SM. Moreover, these numbers are no way related to any fundamental
symmetries of nature, since they are known to be violated by non-perturbative
electroweak effects \cite{c2'tHooft:1976up}. So it is rather difficult to drop these terms
from a general MSSM superpotential unless one assumes $B,L$ conservation
as a postulate for the MSSM. However, in the presence of these terms
there exists new contribution to the proton decay process
$(p\to\ell^+\pi^0$ with $\ell^+=e^+,\mu^+$) as shown
in figure \ref{proton-decay}.
\begin{figure}[ht]
\centering
\includegraphics[width=7.45cm]{FigsPDF/proton-decay.pdf}
\caption{Feynman diagrams for the process $p \to \ell^+ \pi^0$ with
$\ell^+=e^+,\mu^+$.}
\label{proton-decay}
\end{figure}
This process (see figure \ref{proton-decay}) will yield a
proton life time $\approx 10^{-9}$ sec,
assuming $\lambda',\lambda^{''}\sim \cal {O}$ $(10^{-1})$ and TeV scale
squark masses. However, the known experimental bound for proton
lifetime is $>10^{32}$ years \cite{c2NH:2009gd,c2Nakamura-c2}. So in order to
explain proton stability either these new couplings $(\lambda,\lambda^{'},\lambda^{''})$
are extremely small (which again requires explanation) or their products
(appear in the decay width for the process $p\to\ell^+\pi^0$) are
very small or these
terms are somehow forbidden from the MSSM superpotential. In fact,
to avoid very fast proton decay mediated through squarks of masses
of the order of the electroweak scale, simultaneous presence of
$\lambda',\lambda''$ type couplings must be forbidden unless the product
$\lambda'\lambda''$ is severely constrained (see figure \ref{proton-decay}).
The $\lambda$ type of operators are not so stringently suppressed, and therefore
still a lot of freedom remains (see ref.\cite{c2Dreiner:2006gu} and references therein).
It turns out that since these new terms (see eqn.(\ref{MSSM-superpotential-2}))
violate either lepton or baryon number by odd units it is possible
to restrict them from appearing in $W^{MSSM}$ by imposing a discrete
symmetry called $R$-parity $(R_p)$,\footnote{See also matter parity
\cite{c2Dimopoulos:1981zb,c2Weinberg:1981wj,c2Sakai:1981pk,c2Dimopoulos:1981dw}.}
\cite{c2Farrar:1978xj,c2Weinberg:1981wj,c2Sakai:1981pk,c2Dimopoulos:1981dw} defined as,
\begin{equation}
R_p = (-1)^{3(B-L)+2s},
\label{R-parity-defn}
\end{equation}
where $s$ is the spin of the particle. Since $L$ is an integer, an
alternative expression for $R_p$ is also given by
\begin{equation}
R_p = (-1)^{3B+L+2s}.
\label{R-parity-defn-2}
\end{equation}
It is interesting to
note that since different states within a supermultiplet have
different spins, they must have different $R_p$. It turns out
that by construction all the SM particles have $R_p = +1$
and for all superpartners, $R_p = -1$. This is a discrete $Z_2$
symmetry and multiplicative in nature. It is important to note that
$R_p$ conservation would require (1) even number of
sparticles at each interaction vertex, and (2) the lightest
supersymmetric particle (LSP) has no lighter $R_p=-1$ states to decay
and thus it is absolutely stable (see figure \ref{Rpc-Rpv}).
Thus the LSP for a supersymmetric model with conserved $R_p$ can
act as a natural dark matter candidate. It
must be remembered that the soft supersymmetry breaking Lagrangian
will also contain $R_p$ violating terms \cite{c2Allanach:2003eb,c2de-Carlos:1996du}.
\begin{figure}[ht]
\centering
\includegraphics[height=3.10cm]{FigsPDF/RpC-RpV.pdf}
\caption{With $R_p$ conservation the LSP is
forced to be stable due to unavailability of any lighter sparticle
states ({\it{left}}), whereas for the $R_p$-violating scenario the LSP can
decay into SM particles ({\it{right}}).}
\label{Rpc-Rpv}
\end{figure}
Looking at eqn.(\ref{MSSM-superpotential-2}) it is clear that
sources for $R_p$ violation $(\not\!\!{R_p})$ (see references
\cite{c2Ellis:1984gi,c2Ross:1984yg,c2Dawson:1985vr,c2Barbieri:1985ty,c2Barger:1989rk,
c2Ibanez:1991pr,c2Bhattacharyya:1996nj,c2Dreiner:1997uz,c2Bhattacharyya:1997vv,
c2Barbier:1998fe,
c2Allanach:1999bf,c2Barbier:2004ez,c2Chemtob:2004xr}) are either bilinear $(\epsilon)$
\cite{c2Joshipura:1994wm,c2Joshipura:1994ib,c2de-Campos:1995av,c2Smirnov:1995ey,
c2Nowakowski:1995dx,c2Hempfling:1995wj,c2Nilles:1996ij,c2deCarlos:1996yh,c2Roy:1996bua,
c2Diaz:1997xc,c2Akeroyd:1997iq,c2Hirsch:2000ef,c2RestrepoQuintero:2001pk} or trilinear
$(\lambda,\lambda^{'},\lambda^{''})$ \cite{c2Barger:1989rk,
c2Enqvist:1992ef,c2de-Carlos:1996du,c2deCarlos:1996yh,c2Dreiner:1997uz,
c2Choudhury:1997dt,c2Rakshit:1998kd,c2Raychaudhuri:1999zg} in nature. The simple
most example of $\not\!\!{R_p}$ turns out to be bilinear. It is interesting to
note that these bilinear terms are removable from superpotential by using field
redefinitions, however they reappear as trilinear couplings both in superpotential
and in soft SUSY breaking Lagrangian \cite{c2Hall:1983id,c2Lee:1984kr,c2Lee:1984tn}
along with the original bilinear $R_p$-violating terms, that were in
the soft SUSY breaking Lagrangian to start with.
The effect of rotating away $L_iH_u$ term from the superpotential
by a redefinition of the lepton and Higgs superfields are bound to
show up via the scalar potential \cite{c2de-Campos:1995av}.
Also even if bilinear terms are rotated away at one energy scale, they reappear
in some other energy scale as the couplings evolve radiatively \cite{c2Barger:1995qe}.
The trilinear couplings can also give rise to bilinear terms in one-loops
(see figure \ref{t-b}) \cite{c2de-Carlos:1996du}.
Note that $\not\!\!{R_p}$ can be either explicit
(like eqn.(\ref{MSSM-superpotential-2})) \cite{c2Hall:1983id,c2Lee:1984kr,
c2Lee:1984tn,c2Ellis:1984gi} or spontaneous \cite{c2Aulakh:1982yn,c2Ellis:1984gi,c2Ross:1984yg,
c2Santamaria:1987uq,c2Santamaria:1988zm,c2Santamaria:1988ic,
c2Masiero:1990uj,c2Nogueira:1990wz,c2Romao:1992vu}.
\begin{figure}[ht]
\centering
\includegraphics[height=3.10cm]{FigsPDF/t-b.pdf}
\caption{One loop diagrams contributing to bilinear terms like $L_iH_u,L_iH_d$
using the trilinear couplings $\lambda,\lambda'$.}
\label{t-b}
\end{figure}
Here as a digression it should be mentioned that $R_p$ can
be embedded into a larger continuous group (see, for example, ref.\cite{c2Joshipura:2000sn}
and references therein) which is finally abandoned
for phenomenological reasons\footnote{A continuous symmetry
would prefer massless gauginos, which is already ruled out
by experiments.}. However, its $Z_2$ subgroup could still be retained,
which is the $R_p$.
To summarize, it seems that $R_p$ violation is a natural
feature for supersymmetric theories, since $R_p$-violating terms
(see eqn.(\ref{MSSM-superpotential-2})) are not forbidden
to appear in the MSSM superpotential by the arguments of
gauge and Lorentz invariance or renormalizability.
On the contrary, assumption of $R_p$-conservation to prevent proton decay
appears to be an ad hoc one. Besides,
models with $R_p$-violation are also phenomenologically very rich. Of course,
it is natural to ask about the fate of the proton. But considering
either lepton or baryon number violation at a time proton stability can be achieved.
It is true that with$~\not\!\!{R_p}$ the LSP is no longer stable and can decay
into the SM particles. The stable LSP (in case it is colour and charge neutral) can be a
natural candidate for the Dark matter \cite{c2Goldberg:1983nd,c2Ellis:1983ew}.
However, their exist other viable dark matter candidates even for a theory with$~\not\!\!{R_p}$,
namely, gravitino \cite{c2Borgani:1996ag,c2Takayama:2000uz,c2Hirsch:2005ag},
axion \cite{c2Kim:1986ax,c2Raffelt:1996wa} and
axino \cite{c2Chun:1999cq,c2Chun:2006ss} (supersymmetric partner of axion).
It is important to note that a decaying LSP has very different
and enriched implications in a collider study. Unlike models
with $R_p$ conservation, which yield large missing energy signature
at the end of any supersymmetric process, effect of $\not\!\!{R_p}$ can often
produce interesting visible final states detectable in a collider
experiments. Models with bilinear $\not\!\!{R_p}$ are especially interesting
concerning collider studies \cite{Guchait:1996xw,Huitu:1997qu,
BarShalom:1998av,Choudhury:1998ta,c2Mukhopadhyaya:1998xj,c2Choi:1999tq,
Ghosh:1999ix,Hikasa:1999wy,Chiappetta:1999cd,Lebedev:1999ze,Moreau:2000bs,
Ghosh:2000gx,Choudhury:2002av,c2Hirsch:2005ag,Li:2006he}, as they admit direct
mixing between neutrino and neutralinos.
Finally, it remains to be mentioned the most important
aspect of $R_p$ violation, namely, generation of the neutrino mass.
It is impossible to generate neutrino masses in a supersymmetric
model with $R_p$ conservation along with minimal field content
(see eqn.(\ref{MSSM-superpotential-1})). It is
rather important to clarify the importance of $\not\!\!{R_p}$ in neutrino
mass generation. There are other ways to generate
light neutrino masses, both in supersymmetric or non-supersymmetric
models like adding extra particles or enhancing the gauge group
(left-right symmetric models \cite{c2Pati:1974yy} for example) and many others.
But generating massive neutrinos with $\not\!\!{R_p}$ is a pure
supersymmetric phenomenon without any SM analog.
More on the issue of light neutrino mass generation
and $\not\!\!{R_p}$ will be addressed in the next chapter.
To complete the discussion, it is important to mention that
these $\not\!\!{R_p}$ couplings are highly constrained by experimental
limits on different physical processes, like neutron-anti neutron
scattering \cite{c2BasteroGil:1996ps,c2BasteroGil:1996vc,c2Brahmachari:1994wd,
c2Goity:1994dq}, neutrinoless double beta decay \cite{c2Enqvist:1992ef,c2Hirsch:1995zi,
c2Hirsch:1995ek,c2Faessler:1996ph,c2Faessler:1997db,c2Hirsch:1998kc},
precision measurements of $Z$ decay \cite{c2Bhattacharyya:1995bw,
c2Lebedev:1999ze,c2Takeuchi:2000dc}, proton decay \cite{c2Smirnov:1996bg,
c2Bhattacharyya:1998dt,c2Bhattacharyya:1998bx}, Majorana masses for
neutrinos \cite{c2Dimopoulos:1988jw,c2Godbole:1992fb,c2Drees:1997id,
c2Rakshit:1998kd,c2Borzumati:2002bf,c2Bhattacharyya:1999tv} etc.
Discussion on different supersymmetric models with and without $R_p$
conservation, proposed
in the literature is given in a recent review \cite{c2Munoz:2007fa}.
\section{{\bf S}uccesses of supersymmetry} \label{MSSM-success}
So far, we tried to formulate the theory of MSSM step
by step starting from the very basics. It is perhaps the appropriate
place to discuss the success of the supersymmetric
theories over most of the shortcomings of the SM (see section \ref{suc-prob}).
We are about to discuss all the seven points made in section \ref{suc-prob}
but in reverse order.
\begin{enumerate}
\item
The last point deals with Higgs mass hierarchy in the SM. It has
been shown earlier that how a supersymmetric theory can predict
a finite Higgs mass without any {\it{quadratic divergences}} even
though SUSY is broken in nature.
\item
It is true that MSSM with $R_p$ conservation predicts massless neutrinos
similar to the SM. However, as argued in the earlier section,
supersymmetric theories are capable of accommodating massive neutrinos
if $R_p$ is broken. Just for the sake of completeness, let us mention that there
exist also certain non-minimal supersymmetric models, which can account for the
neutrino masses with seesaw mechanism. Such models include e.g. right-
handed neutrinos or other very heavy particles. In the next chapter
these possibilities will be explored in detail.
\item
The SM hardly offers any room for a suitable dark matter candidate.
But as described in
section \ref{R-parity} the lightest supersymmetric particle is a
good candidate for the dark matter in a supersymmetric model with
$R_p$ conservation. Nevertheless, as stated in section \ref{R-parity},
there exist other viable dark matter candidates (gravitino, axion etc.)
even for an $R_p$-violating supersymmetric theory.
\item
The apparent exclusion of gravitational interaction from the SM
is still maintained in supersymmetric theories, so long one considers
global supersymmetry.
A locally supersymmetric theory together with the theory of general relativity
can incorporate gravitational interaction in SUSY. This theory is popularly
known as supergravity theory.
\item
Concerning point no.(3) of section \ref{suc-prob}, there are other
sources of CP-violation in the MSSM itself, which can account for
the large matter-anti matter asymmetry of the universe. In general,
apart from one CKM phase there exist many different phases in the MSSM,
particularly in the soft supersymmetry breaking sector. However, some of
these are subjected to strong phenomenological constraints.
\item
It is true that the number of free parameters in a general
MSSM theory is larger $(> 100)$ \cite{c2Dimopoulos:1995ju,c2Giudice:1998bp}
compared to that of the SM. However, there are models where most of these parameters can
be achieved through evolution of a fewer number of parameters at a higher
scale. For example in minimal supergravity \cite{c2Nath:1983fp,c2Nilles-1983}
theory the number of free parameters is just five.
\end{enumerate}
It has to be emphasized here, that this will be a rather incorrect
statement that supersymmetric theories are free from any drawbacks. It is
definitely true that they provide explanations to
some of the shortcomings of the SM in a few
occasions, but not always. As an example supersymmetric
theories are more prone to FCNC through the sparticle mediated
processes \cite{c2Hall:1985dx,c2Dine:1993np,c2Dine:1993su,c2Hagelin:1992tc,
c2Dimopoulos:1995ju,c2Sutter:1995kp,c2Haber:1997if}. This
problem can be removed using clever tricks, but
a related discussion is beyond the scope of this thesis. Another,
well known problem of MSSM, the $\mu$-problem will be addressed
in the following section.
The main stumbling block for any supersymmetric theory is that
there are no experimental evidence for supersymmetry till date.
All the experimental bounds on different phenomenological processes
with supersymmetric effects are basically exclusion limit.
\section{{\bf T}he $\mu$-problem} \label{mu-prob}
The $\mu$-parameter, associated with the bilinear term in Higgs
superfields (see eqn.(\ref{MSSM-superpotential-1})) is the only
coupling in the MSSM superpotential having a non-zero positive mass
dimension. The problem appears when
one consider the EWSB condition, which is given by
\begin{equation}
{\frac{1}{2}} M^2_Z =
\frac{m^2_{H_d}-m^2_{H_u} \tan^2{\beta}}{{\tan^2{\beta}}-1}-|\mu^2|,
\label{mu-problem}
\end{equation}
where $m^2_{H_d},m^2_{H_u}$ are given by eqn.(\ref{Lsoft-MSSM}),
$\rm{tan}\beta = \frac{v_2}{v_1}$ and $M_Z$ is the $Z$ boson mass.
The $Z$ boson mass is very preciously measured to be $91.187$ GeV
(see table \ref{SM-spectrum}). So it is expected that all the entries
of the right hand side of eqn.(\ref{mu-problem}) (without any fine cancellation)
should have the same order of magnitudes. But how could this happen?
$m^2_{H_d},m^2_{H_u}$ are coming from the soft supersymmetric breaking
sector with entries at the TeV scale. On the other hand, $\mu$ belongs to SUSY invariant
$W^{MSSM}$ (eqn.(\ref{MSSM-superpotential-1})), which naturally can be as large as the Planck
scale.
So why these two scales appear to be of the same order of magnitude
without having any a priori connection in between? This defines
the $\mu$-problem \cite{c2Kim:1983dt}. An alternative statement could be
why $\mu^2\sim m^2_{soft}$ and not $\sim M^2_{Planck}$.
It seems easy to solve this problem by starting with $\mu=0$ at $W^{MSSM}$
and then use the favour of radiative corrections to generate a non-zero $\mu$
term. But there are some phenomenological problems of this approach and
moreover $\mu=0$ will give zero VEV for $H_d$ along with the presence
of unwanted Weinberg-Wilczek axion \cite{c2Weinberg:1977ma,c2Wilczek:1977pj}.
So it is apparent that one needs to consider either $\mu\neq0$ or
require extra fields.
The requirement of additional fields often lead to other problems
and consequently do not predict satisfactory models \cite{c2Polchinski:1982an,
c2Nilles:1982mp,c2Lahanas:1982bk,c2Ferrara:1982ke,c2AlvarezGaume:1983gj}.
There exist indeed various solutions to the $\mu$-problem where in most
of the occasions the $\mu$-term is absent at the tree level
and a TeV or electroweak scale $\mu$-term arises from
the VEV(s) of new fields. These VEVs are obtained by minimizing the
potential which also involves soft SUSY breaking terms. Thus, the fact
$\mu^2\sim m^2_{soft}$ turns out to be rather natural. Different
solutions to the $\mu$-problem have been addressed in references
\cite{c2Giudice:1988yz,c2Kim:1991mv,c2Chun:1991xm,c2Casas:1992mk,
c2Giudice:1993rc,c2Kim:1994eu,c2Yanagida:1997yf,c2Dimopoulos:1997je,
c2Kim:1999fg,c2Langacker:1999hs,c2Choi:1999yaa,c2Mafi:2000kg}.
Some of these mechanisms are operational at very high energies and
thus are hardly testable experimentally.
Perhaps the simple most dynamical solution to the $\mu$-problem is
offered by next-to minimal supersymmetric standard model or
NMSSM (see review \cite{c2Ellwanger:2009dp} and references therein).
In NMSSM the bilinear term $\epsilon_{ab} \hat H^a_d\hat H^b_u$
gets replaced by $\epsilon_{ab} \lambda\hat S \hat H^a_d\hat H^b_u$.
The superfield $\hat S$ is singlet \cite{c2Fayet-NMSSM,c2Kaul-NMSSM,c2Barbieri-NMSSM,
c2Nilles-NMSSM,c2Frere-NMSSM,c2Derendinger-NMSSM} under the SM gauge group. After
the EWSB an effective $\mu$ term is given by
\begin{equation}
\mu = \lambda v_s,
\label{mu-gen-NMSSM}
\end{equation}
where $v_s= \langle S \rangle$, is the VEV acquired by the
scalar component of the superfield $\hat{S}$. The VEV calculation
invokes the soft SUSY breaking terms and hence in general the VEVs
are at the TeV scale. It is now clear that the $\mu$-term
of eqn.(\ref{mu-gen-NMSSM}) is of the right order of magnitude and
it is indeed connected to $m^2_{soft}$. The NMSSM superpotential
assumes a $Z_3$ symmetry which forbids any bilinear term in superpotential.
It is important to note that any term in the superpotential
with a non-zero positive mass dimension suffers the similar fate.
In fact the bilinear $\not\!\!{R_p}$ terms (see eqn.(\ref{MSSM-superpotential-2}))
are also associated with similar kind of problem known as the
$\epsilon$-problem \cite{c2Nilles:1996ij}. A common origin for the $\varepsilon_i$ (to
account for the neutrino oscillation data), and the $\mu$-term
can be achieved using a horizontal family symmetry as suggested in
ref.\cite{c2Mira:2000gg}.
\section{{\bf N}ext-to-Minimal Supersymmetric Standard Model} \label{NMSSM}
It is perhaps logical and consistent with the theme of this thesis
to give a brief introduction of the NMSSM. The NMSSM superpotential,
is given by (see review \cite{c2Maniatis:2009re,c2Ellwanger:2009dp})
\begin{equation}
W^{NMSSM} = W^{'^{MSSM}} - \epsilon_{ab} \lambda\hat S \hat H^a_d\hat H^b_u
+ \frac{1}{3} \kappa \hat{S}^3,
\label{NMSSM-superpotential}
\end{equation}
where $W^{'^{MSSM}}$ is the MSSM superpotential (eqn.(\ref{MSSM-superpotential-1}))
without the $\mu$-term. In a similar fashion if $\mathcal{L}^{'^{MSSM}}_{soft}$
denotes $\mathcal{L}^{MSSM}_{soft}$ without the $B_\mu$ term (see eqn.(\ref{Lsoft-MSSM})),
then
\begin{equation}
-\mathcal{L}^{NMSSM}_{soft} = -\mathcal{L}^{'^{MSSM}}_{soft}
+ (m^2_{\widetilde S}) \widetilde S \widetilde S
- \epsilon_{ab} (A_{\lambda}\lambda) \tilde S H_d^a H_u^b
+ \frac{1}{3} (A_{\kappa}\kappa) \tilde S^3 + h.c.
\label{Lsoft-NMSSM}
\end{equation}
However, even in NMSSM, if $R_p$ is conserved, light neutrinos
are exactly massless. NMSSM models of neutrino mass generation
will be discussed in the next chapter.
Particle spectrum for NMSSM will be enlarged over that of the MSSM
due to extra particle content. However, $\hat S$ being SM gauge
singlet only the neutralino sector and the neutral Higgs sector receives
modifications. The neutralino mass matrix is now a $5\times5$
symmetric matrix and there will be one more CP-odd and CP-even neutral
scalar states, compared to that of the MSSM. The phenomenology of NMSSM is definitely much
enriched compared to the MSSM. This is essentially due to the admixture of new singlet
states with MSSM fields. For example, theoretical lower bound on the lightest Higgs mass is
now given by \cite{c2Drees:1988fc} (For Higgs sector in NMSSM also see references
\cite{c2Ellis:1988er,c2Haber:1990aw,c2Binetruy:1991mk,c2Espinosa:1991gr,c2Pietroni:1992in,
c2Moroi:1992zk,c2Elliott:1993uc,c2Elliott:1993bs,c2Pandita:1993tg,c2Pandita:1993hx,c2Wu:1994eb,
c2Franke:1995xn,c2King:1995ys,c2Franke:1995tc,c2Choudhury:1995bx,c2Ham:1996mi,c2Daikoku:2000eq,c2Ellwanger:2000fh,
c2Panagiotakopoulos:2000wp,c2Davies:2001uv,c2Ellwanger:2001iw,c2Ellwanger:2004gz,c2Ham:2004nv,
c2Ellwanger:2004xm,c2Ellwanger:2005fh,c2Ellwanger:2006rm,c2Ellwanger:2006rn,c2Cavicchia:2008fn,
c2Degrassi:2009yq})
\begin{equation}
m^{'^2}_{h^0} \le M^2_Z \left[{\rm cos}^22\beta
+ \frac{2\lambda^2{\rm cos}^2_{\theta_W}}{g^2_2}~{\rm sin}^22\beta\right].
\label{Higgs-mass-NMSSM-lightest}
\end{equation}
which is clearly different from eqn.(\ref{Higgs-mass-MSSM-lightest}).
It is interesting to note that the lower limit of tree level
lightest Higgs boson mass in NMSSM depends on $\lambda$ and hence
it is in general difficult to put some upper bound on $m'_{h^0}$
without constraining $\lambda$.
The last term in eqn.(\ref{NMSSM-superpotential}) with coefficient $\kappa$ is
included in order to avoid an unacceptable axion associated to the breaking of a
global ${\rm{U(1)}}$ symmetry \cite{c2Ellis:1988er}. This term is perfectly allowed
by all symmetry. However, the discrete $Z_3$ symmetry of the NMSSM superpotential
(see section \ref{mu-prob}) when spontaneously broken leads to three degenerate
vacua. Casually disconnected parts of the Universe then would have randomly chosen
one of these three equivalent minima leading to the formation of the dangerous
cosmological domain wall \cite{c2Ellis:1986mq,c2Rai:1992xw,c2Abel:1995wk}. However,
solutions to this problem exist
\cite{c2Abel:1996cr,c2Panagiotakopoulos:1998yw,c2Panagiotakopoulos:1999ah},
but these issues are beyond the scope of this thesis\footnote{One
solution of this problem is to put $\kappa=0$ in the NMSSM superpotential by some
symmetry argument. This simplified version is known as Minimally NMSSM or MNMSSM.}.
Another major problem of the NMSSM
theories are associated with the stability of gauge hierarchy arising from the
{\it{tadpole}} contribution of the singlet field.
Diverse phenomenological aspects of NMSSM models are discussed in
references \cite{c2Franke:1994hj,c2Ananthanarayan:1995xq,c2King:1995vk,c2Franke:1995tf,c2Ananthanarayan:1995zr,
c2Franke:1995tc,
c2Ananthanarayan:1996zv,c2Kim:1996jh,c2Asatrian:1996dg,c2Ellwanger:1998vi,c2Ellwanger:1998jk,c2Demir:1999mm,
c2BasteroGil:2000bw,c2Nevzorov:2000uv,c2Nevzorov:2001vj,c2Menon:2004wv,c2Hiller:2004ii,c2Cerdeno:2004xw,
c2Ellwanger:2005uu,c2Funakubo:2005pu,c2Ellwanger:2005uu,c2Ellwanger:2005fh,c2Belanger:2005kh,c2Kraml:2005nx,
c2Ellwanger:2005dv,c2Gunion:2005rw,c2Dermisek:2005gg,c2Schuster:2005py,c2Arhrib:2006sx,c2Huber:2006wf,
c2Moretti:2006hq,c2Ferrer:2006hy,c2Dermisek:2006wr,c2Ma:2006te,c2Dermisek:2006py,c2Cerdeno:2007sn,
c2Mangano:2007gi,c2Balazs:2007pf,c2Dermisek:2007yt,c2Hugonie:2007vd,c2Dermisek:2007ah,c2Domingo:2007dx,
c2Barbieri:2007tu,c2Forshaw:2007ra,c2Akeroyd:2007yj,c2Heng:2008rc,c2Djouadi:2008uw,c2Djouadi:2008yj,
c2Chun:2008pg,c2Ellwanger:2008py,c2He:2008zw,c2Belyaev:2008gj,c2Domingo:2008bb,c2Chang:2008np,
c2Gogoladze:2008wz,c2Cao:2008un,c2Domingo:2008rr,c2Kraml:2008zr,c2Djouadi:2008uj,c2Belanger:2008nt,
c2Dermisek:2008uu,c2Cao:2009ad,c2Cerdeno:2009dv,c2Dermisek:2010mg,c2Das:2010ww,c2Staub:2010ty,c2Abada:2010ym}.
\vspace{1cm}
The prime focus of this thesis remains the issues of neutrino masses and mixing
in supersymmetric theories. It has been already argued that massive neutrinos
can be accommodated in supersymmetric theories either through $\not\!\!{R_p}$ or using
seesaw mechanism with non-minimal field contents.
Besides, mass generation is possible both with the tree level and loop
level analysis. However, even before it is important to
note the evidences as well as the basics of the neutrino oscillation.
It is also interesting to note the implications of the massive neutrinos
in an accelerator experiment. We aim to discuss these issues in details
in the next chapter along with other phenomenological implications.
\chapter{ \sffamily{{\bf $\mu$}$\nu$SSM: neutrino masses and mixing
}}\label{munuSSM-neut}
\section{{\bf I}ntroducing $\mu\nu$SSM}
As discussed earlier, the minimal supersymmetric
standard model (MSSM) is not free from drawbacks. We have addressed
these issues in the context of the $\mu$-problem \cite{c4Kim:1983dt}
and light neutrino mass generation. Supersymmetric theories
can accommodate massive neutrinos either through $\not\!\!{R_p}$ or using
seesaw mechanism. Regarding the $\mu$-problem, as discussed in section \ref{NMSSM},
a simple solution is given by the NMSSM. There exist a host of NMSSM models
where the superpotential contains either explicit $R_p$-violating
couplings \cite{c4Abada:2006qn,c4Abada:2006gh,c4Hundi:2009yf,c4JeanLouis:2009du,
c4Pandita:1999jd,c4Pandita:2001cv,c4Chemtob:2006ur}
or use spontaneous violation of $R_p$ \cite{c4Kitano:1999qb} to accommodate
light neutrino masses apart from offering a solution to the $\mu$-problem.
Unfortunately, NMSSM models of neutrino mass generation with $bR_pV$ suffer from the
$\epsilon$-problem \cite{c4Nilles:1996ij}. Besides, with bilinear $\not\!\!{R_p}$ not
all the neutrino masses are generated at
the tree level. Thus loop corrections are unavoidable to account for the three flavour
oscillation data. Loop effects are compulsory
for models with $tR_pV$ where all of the neutrino masses appear at loop level.
Certainly, larger number
of trilinear couplings reduce the predictability of these models. An elegant alternative
is given by NMSSM models with spontaneous $\not\!\!{R_p}$ where apart from a singlet superfield, $\hat S$
(to solve the $\mu$-problem) one requires a right-handed neutrino superfield,
$\hat \nu^c$ to accommodate massive neutrinos. The
issues of light neutrino mass generation together with a solution to the $\mu$-problem in
$R_p$-conserving NMSSM models have been addressed in references
\cite{c4Frank:2007un,c4Das:2010wp,c4Abada:2010ym}.
So, in a nutshell, the well-known NMSSM models of neutrino mass generation either suffer
from the naturalness problem or are less predictive due to the presence of
either large number of couplings
or additional superfields. Now following the structure of the SM it seems rather
natural to add right-handed neutrino superfields with the MSSM superfields in order
to generate neutrino masses. Also being a SM gauge singlet, a right-handed neutrino superfield,
$\hat \nu^c$ can act as a viable alternative for the singlet field $(\hat S)$ of NMSSM
used to solve the $\mu$-problem.
The novel idea of solving the $\mu$-problem and light neutrino mass generation
simultaneously in a supersymmetric model using {\it{only}} right-handed neutrino
superfields, $\hat \nu^c_i$ was advocated in ref.\cite{c4LopezFogliani:2005yw}.
This model is popularly known as the {\it{``$\mu$ from $\nu$'' supersymmetric standard
model}} or $\mu\nu$SSM \cite{c4LopezFogliani:2005yw}. Details of this model
will be provided in the next sub-section.
In this chapter we plan to discuss the $\mu\nu$SSM model first with necessary details
like neutral scalar potential, minimization conditions, scalar sector, fermionic sector etc.
and later we aim to discuss the issues of light neutrino masses and mixing in the $\mu\nu$SSM
at the tree level as well as with one-loop radiative corrections.
\section{{\bf T}he model} \label{munuSSM-model}
In this section we introduce the model along the lines of ref.\cite{c4LopezFogliani:2005yw},
discuss its basic features and set our notations. Throughout this thesis we consider
three generations of right-handed neutrino superfield $(\hat \nu^c_i)$ apart from the
MSSM superfields as proposed in ref.\cite{c4LopezFogliani:2005yw}. We start with
the model superpotential and the soft terms and continue our discussion with the minimization
conditions later.
\begin{flushleft}
{\it{$\maltese$ Superpotential}}
\end{flushleft}
The $\mu\nu$SSM superpotential is given by
\begin{equation}
W^{\mu\nu SSM} = W^{'^{MSSM}} + \epsilon_{ab} Y^{ij}_\nu \hat H^b_u\hat L^a_i\hat \nu^c_j
-\underbrace{\epsilon_{ab} \lambda^i\hat \nu^c_i\hat H^a_d\hat H^b_u}_{\Delta L=1}
+\overbrace{\frac{1}{3}\kappa^{ijk}\hat \nu^c_i\hat \nu^c_j\hat \nu^c_k}^{\Delta L=3},
\label{munuSSM-superpotential}
\end{equation}
where $W^{'^{MSSM}}$ is the MSSM superpotential (see eqns.(\ref{MSSM-superpotential-1}),
(\ref{NMSSM-superpotential}))
but without the $\epsilon_{ab}\mu \hat H^a_d\hat H^b_u$-term. The superfields
$\hat H_d,\hat H_u,\hat L_i$ are usual MSSM down-type Higgs, up-type Higgs
and $\rm{SU(2)_L}$ doublet lepton superfields. Since right-handed neutrinos carry
a non-zero lepton number, the third and fourth terms of eqn.(\ref{munuSSM-superpotential})
violate lepton number by odd unit(s) ({\it{one}} and {\it{three}}, respectively).
Violation of lepton number by odd units is the source of $\not\!\!{R_p}$ (eqn.(\ref{R-parity-defn-2}))
is $\mu\nu$SSM.
It is important to mention the implications of different terms of
eqn.(\ref{munuSSM-superpotential}) at this stage.
\vspace*{0.1cm}
\noindent
$\bullet$ The second term $\epsilon_{ab} Y^{ij}_\nu \hat H^b_u\hat L^a_i\hat \nu^c_j$
respects lepton number conservation to start with. However, after EWSB these
terms give rise to effective bilinear $R_p$-violating terms as $\varepsilon^i L_i H_u$
with $\varepsilon^i=Y^{ij}_\nu v^c_j$. $v^c_j$ denotes the VEV acquired
by $j$-th right-handed sneutrino. Besides, a term of this kind also give rise
to Dirac neutrino mass matrix with entries as $m_{D_{ij}} = Y^{ij}_\nu v_2$.
\vspace*{0.1cm}
\noindent
$\bullet$ The third term $\epsilon_{ab} \lambda^i\hat \nu^c_i\hat H^a_d\hat H^b_u$
after EWSB generates an effective $\mu$-term as $\mu=\sum \lambda_i v^c_i$. This term
violates lepton number by one unit.
\vspace*{0.1cm}
\noindent
$\bullet$ The last term $\frac{1}{3}\kappa^{ijk}\hat \nu^c_i\hat \nu^c_j\hat \nu^c_k$
violates lepton number by three units. Note that this term is allowed
by all possible symmetry arguments. Now if $\kappa^{ijk}=0$ to start with then
the Lagrangian has a global $\rm{U(1)}$
symmetry which is broken spontaneously by the VEVs of the scalar
fields and leads to unacceptable massless axion.
In order to avoid axions non-zero values for $\kappa^{ijk}$
are essential \cite{c4Ellis:1988er}. Besides, after EWSB the last term of
eqn.(\ref{munuSSM-superpotential}) produces entries for the right-handed neutrino
Majorana mass matrix as $m_{\nu^c_{ij}}=2\kappa^{ijk} v^c_k$.
\vspace*{0.1cm}
\noindent
$\bullet$ As already mentioned, a $Z_3$ symmetry is imposed on the $\mu\nu$SSM superpotential
(eqn.(\ref{munuSSM-superpotential})) to forbid appearance of any bilinear term.
This feature is similar to the NMSSM models as stated in section \ref{NMSSM}. Thus
similar to the NMSSM, the $\mu\nu$SSM also suffers from the problem of
cosmological domain wall formation \cite{c4Ellis:1986mq,c4Rai:1992xw,c4Abel:1995wk}.
However, the problem can be ameliorated through well known methods \cite{c4Abel:1996cr,
c4Panagiotakopoulos:1998yw,c4Panagiotakopoulos:1999ah}.
\vspace*{0.1cm}
\noindent
$\bullet$ The conventional trilinear couplings $\lambda^{ijk},\lambda'^{ijk}$ (see eqn.(\ref{MSSM-superpotential-2}))
can be generated in $\mu\nu$SSM at one-loop level as shown in figure \ref{munuSSM-llp}
\cite{c4Escudero:2008jg}.
\begin{figure}[ht]
\centering
\includegraphics[width=7.45cm]{FigsPDF/munuSSM-llp.pdf}
\caption{One-loop generation of the $\lambda^{ijk},\lambda'^{ijk}$ terms in the superpotential. These terms are
proportional to product of two Yukawa couplings and $\lambda$. Product of two Yukawa couplings
assures smallness of the $\lambda^{ijk},\lambda'^{ijk}$ couplings.}
\label{munuSSM-llp}
\end{figure}
A term of the type
$Y_\nu L H_u \nu^c$ has been considered also in ref.
\cite{c4Mukhopadhyaya:2006is} in the context of light
neutrino mass generation, but without offering any attempts to solve the $\mu$-problem.
In ref.\cite{c4Farzan:2005ez} couplings
of the form $Y_\nu L H_u \nu^c$ and $\lambda H_d H_u \nu^c$ were considered
along with Majorana mass terms $\frac{1}{2}M^{ij}\nu^c_i\nu^c_j$ for right-handed neutrinos.
However, in this case
the contribution of $\lambda H_d H_u \nu^c$ for generating the $\mu$-term is negligible
because the right-handed sneutrinos $(\widetilde \nu^c_i)$ , being super heavy, do not acquire
sizable vacuum expectation values. In reference \cite{c4Farzan:2005ez} the term
$\lambda H_d H_u \nu^c$ has been utilized for the purpose of thermal seesaw leptogenesis.
\begin{flushleft}
{\it{$\maltese$ Soft terms}}
\end{flushleft}
Confining ourselves in the framework of supergravity mediated supersymmetry breaking, the
Lagrangian $\mathcal{L}^{\mu\nu SSM}_{soft}$, containing the soft-supersymmetry-breaking
terms is given by
\begin{eqnarray}
-\mathcal{L}^{\mu\nu SSM}_{soft} &=& -\mathcal{L}^{'^{MSSM}}_{soft}
+ (m^2_{\widetilde \nu^c})^{ij} \widetilde \nu^{c^*}_i \widetilde \nu^c_j \nonumber \\
&+& \{ \epsilon_{ab} (A_{\nu}Y_\nu)^{ij} H_u^b \widetilde L^a_i \widetilde \nu^c_j
- \epsilon_{ab} (A_{\lambda}\lambda)^i \widetilde \nu^c_i H_d^a H_u^b \nonumber \\
&+& \frac{1}{3} (A_{\kappa}\kappa)^{ijk} \widetilde \nu^c_i \widetilde \nu^c_j \widetilde \nu^c_k + h.c\},
\label{munuSSM-soft}
\end{eqnarray}
where $\mathcal{L}^{'^{MSSM}}_{soft}$ denotes $\mathcal{L}^{MSSM}_{soft}$
without the $B_\mu$ term (see eqn.(\ref{Lsoft-MSSM})). $(m^2_{\widetilde \nu^c})^{ij}$
denote soft square masses for right-handed sneutrinos.
\vspace{1cm}
\begin{flushleft}
{\it{$\maltese$ Scalar potential and minimization}}
\end{flushleft}
The tree-level scalar potential receives the usual D and F term
(see eqn.(\ref{MSSM-scalar-pot}), where $\left|\frac{\partial W^{\mu\nu SSM}}
{\partial \phi^{\mu\nu SSM}} \right|^2 \equiv F^*F$
with $\phi$ as any superfields of the $\mu\nu$SSM)
contributions, in addition to the terms from $\mathcal{L}^{\mu\nu SSM}_{soft}$.
We adhere to the $CP$-preserving case, so that only the real parts of
the neutral scalar fields develop, in general, the following VEVs,
\begin{eqnarray}
\langle H_d^0 \rangle = v_1 \, , \quad \langle H_u^0 \rangle = v_2 \,
, \quad \langle \widetilde \nu_i \rangle = v'_i \, , \quad
\langle \widetilde \nu_i^c \rangle = v^c_i.
\label{munuSSM-vevs}
\end{eqnarray}
In eqn.(\ref{munuSSM-vevs}) $i=1,2,3 \equiv e, \mu, \tau$.
The tree level neutral scalar potential looks like \cite{c4LopezFogliani:2005yw,
c4Escudero:2008jg,c4Ghosh:2008yh,c4Bartl:2009an,c4Ghosh:2010zi}
\begin{eqnarray}
\langle V_{\text{neutral}}\rangle &=&
\left|\sum_{i,j}Y^{ij}_{\nu}{v^{\prime}_i}{v^c_j}
- \sum_{i}\lambda^i{v^c_i} v_1\right|^2\ \nonumber \\
&+& \sum_{j}\left|\sum_{i} Y^{ij}_{\nu}{v^{\prime}_i}v_2
-\lambda^jv_1v_2+\sum_{i,k} \kappa^{ijk}{v^c_i}{v^c_k}\right|^2
+ \left|\sum_{i}\lambda^i{v^c_i}v_2\right|^2 \nonumber \\
&+& \sum_{i}\left|\sum_{j}Y^{ij}_{\nu}v_2 {v^c_j}\right|^2
+(\frac{g_1^2+g_2^2}{8}) \left[\sum_{i}|v'_i|^2+|v_1|^2 -
|v_2|^2\right]^2 \nonumber \\
&+& \left [\sum_{i,j}(A_\nu Y_\nu )^{ij} {v'_i} {v^c_j}v_2 -\sum_{i}
(A_\lambda \lambda )^{i} {v^c_i}v_1v_2 + {\rm H.c.} \right]\nonumber \\
&+& \left [\sum_{i,j,k}\frac{1}{3}(A_\kappa
\kappa )^{ijk}v^c_iv^c_jv^c_k + {\rm H.c.} \right]
+ \sum_{i,j} (m_{\widetilde{L}}^2)^{ij}{v'_i}^* {v'_j}\nonumber \\
&+& \sum_{i,j}(m_{\widetilde{\nu}^c}^2)^{ij} {v^{c^*}_i}{v^c_j} +
m_{H_u}^2|v_2|^2 + m_{H_d}^2|v_1|^2 .
\label{Vneut-munuSSM}
\end{eqnarray}
One important thing is to notice that the potential is bounded from below
because the coefficient of the fourth power of all the eight superfields are
positive (see eqn.(\ref{Vneut-munuSSM})). We shall further assume that all the parameters
present in the scalar potential are real. From eqn.(\ref{Vneut-munuSSM}), the minimization
conditions with respect to $v^c_i,~v'_i,v_2,~v_1$ are
\begin{eqnarray}
&&2{\sum_{j}} {{u^{ij}_c}} {\zeta^{j}} + \sum_{k} Y^{ki}_{\nu} {r^{k}_c} {v_2^2}
+\sum_{j} (m^2_{\widetilde{\nu}^c})^{ji}
{v^c_j}+ {\rho^i \eta} + {\mu} {\lambda^i v_2^2}+(A_x x)^{i} =0,\nonumber\\
&&{\sum_{j}} {Y_{\nu}}^{ij} {v_2} {\zeta^{j}} +\sum_{j} (m^2_{\widetilde{L}})^{ji} {v'_j}
+\sum_{j} (A_{\nu} Y_{\nu})^{ij} {v^c_j} v_2 +{\gamma_g}{\xi_{\upsilon}}{v'_i}
+{r^i_c} {\eta} =0,\nonumber\\
&&{\sum_{j}}{\rho^{j}}{\zeta^{j}} + {{\sum_{i}} {r^i_c}^2 v_2}+ {\sum_{i}}({A_{\nu}Y_{\nu}})^{ij}
{v'_i} {v^c_j}-\sum_{i}({A_{\lambda} {\lambda}})^i {v^c_i} v_1 + X^u v_2 = 0,\nonumber \\
&&-{\sum_{j}}{\lambda^j}v_2 {\zeta^{j}}- {\mu} {\sum_{j}} {r^j_c} {v'_j} -\sum_{i}({A_{\lambda}
{\lambda}})^i {v^c_i} v_2 + X^d {v_1} = 0,
\label{Minim-munuSSM}
\end{eqnarray}
with $X^u = {m^2_{H_u}} + {\mu^2} -{\gamma_g}{\xi_{\upsilon}}$,
$X^d = {m^2_{H_d}} + {\mu^2} +{\gamma_g}{\xi_{\upsilon}}$ and
\begin{eqnarray}
(A_x x)^{i} &=& \sum_{j} (A_{\nu} Y_{\nu})^{ji} {v'_j} v_2 + \sum_{j,k}
({A_\kappa} {\kappa})^{ijk} {v^c_j} {v^c_k}
- (A_{\lambda} {\lambda})^i v_1 v_2, \nonumber \\
{r^{i}_c} &=& \varepsilon^i = {\sum_{j}Y^{ij}_{\nu} {v^c_{j}}},
~~{r^{i}} = {\sum_{j}Y^{ij}_{\nu} {v'_{j}}},
~~{u^{ij}_c} = {\sum_{k}}\kappa^{ijk}{v^c_k}, \nonumber \\
\zeta^{j} &=& \sum_{i}
{u^{ij}_c} {v^c_i} + {r^j} v_2 -
{\lambda}^j v_1 v_2 , ~~\mu = \sum_{i}\lambda^{i} {v^c_i},
~~\rho^{i} = {r^i} - \lambda^i v_1,\nonumber \\
\eta &=&\sum_{i} {r^i_c}{v'_i} - {\mu} v_1 ,
~~\gamma_{g} = \frac{1}{4}({g_1^2 + g_2^2}),
~~\xi_{\upsilon} ={\sum_{i} {v'^2_i} + v_1^2 -v_2^2}. \nonumber \\
\label{Abbrevations}
\end{eqnarray}
In deriving the above equations, it has been assumed that
${\kappa}^{ijk}$, $({{A_\kappa} {\kappa}})^{ijk}$, $Y^{ij}_{\nu}$,
$(A_{\nu} Y_{\nu})^{ij}$, $(m^2_{\widetilde{\nu}^c})^{ij}$,
$(m^2_{\widetilde{L}})^{ij}$ are all symmetric in their indices.
It is important to know that now the Majorana masses for right-handed
neutrinos $(2\kappa^{ijk}v^c_k)$ are at the TeV scale with $\kappa\sim$
$\cal{O}$ $(1)$ and TeV scale $v^c_i$ (see first one of eqn.(\ref{Minim-munuSSM})).
For neutrino Dirac masses $(Y_{\nu}^{ij} v_2)$ $\sim$ $10^{-4}$ GeV
the neutrino Yukawa couplings $(Y^{ij}_\nu)$ must also be very small
$\sim \cal{O}$ $(10^{-7})$, in order to get
correct neutrino mass scale using a seesaw mechanism involving TeV scale
right-handed neutrino. This immediately
tells us that in the limit $Y_{\nu}^{ij} \rightarrow 0$,
(see second one of eqn.(\ref{Minim-munuSSM}))
$v'_{i} \rightarrow 0$. So in order to get appropriate neutrino
mass scale both $Y_{\nu}^{ij}$ and $v'_{i}$ have to be small.
Ignoring the terms of the second order in $Y_{\nu}^{ij}$
and considering $({v'^2_i}+v^2_1-v^2_2) \approx (v^2_1-v^2_2)$
(which is a good approximation), and
\textbf{$(m^2_{\tilde{L}})^{ij} = (m^2_{\tilde{L}}) \delta^{ij}$}, we can
easily solve second one of eqn.(\ref{Minim-munuSSM}) as (using eqn. (\ref{Abbrevations}))
\begin{equation}
{v'_i}\approx - \left\{{\frac{{Y_{\nu}}^{ik}{u^{kj}_c} {v_2} - {\mu}{v_1}
{Y^{ij}_{\nu}} +(A_{\nu} Y_{\nu})^{ij} v_2}{{\gamma_{g}} ({v_1^2 -v_2^2})+
(m^2_{\tilde{L}})}}\right\}{v^c_{j}} +
\left\{{\frac{{Y_{\nu}}^{ij}{\lambda}^j v_1 v^2_2}{{\gamma_{g}}
({v_1^2 -v_2^2})+ (m^2_{\tilde{L}})}}\right\}.
\label{sneutrino_VEV_simplified}
\end{equation}
Note from eqn.(\ref{sneutrino_VEV_simplified}), that the left-handed sneutrinos
can acquire, in general, non-vanishing, non-degenerate VEVs even in the limit
of zero vacuum expectation values of the gauge singlet sneutrinos \cite{c4Ghosh:2008yh}.
However, zero VEVs of all the three gauge singlet sneutrinos is not an acceptable
solution since in that case no $\mu$-term $(\sum \lambda_i v^c_i)$ will be generated.
It is essential to ensure that the extremum value of the potential corresponds to the
minimum of the potential, by studying the second derivatives.
The neutral scalar potential and the minimization conditions
in $\mu\nu$SSM but for complex VEVs, have been discussed in ref.\cite{c4Fidalgo:2009dm} in the
context of spontaneous $CP$ violation and its implications in neutrino physics.
\section{{\bf S}calar sector of $\mu\nu$SSM}\label{munuSSM-scalar}
It is evident from eqns.(\ref{munuSSM-superpotential}) and (\ref{munuSSM-soft})
that lepton number $(L)$ is no longer conserved in $\mu\nu$SSM.
In this situation states having zero lepton number can mix with states having $L\neq0$. These
lepton number violating mixings in turn result in larger $(8\times8)$ mass squared matrices
for $CP$-even neutral scalar, $CP$-odd neutral pseudoscalar
and charged scalar states. This is a consequence of the fact that
in $\mu\nu$SSM three generations of left and right-handed sneutrinos
can mix with neutral Higgs bosons. In a similar fashion charged sleptons
mix with the charged Higgs bosons. The enhancement
over the $2\times2$ MSSM structure (see appendix \ref{appenA}) is phenomenologically
very rich. Detailed structures for neutral scalar, pseudoscalar and the charged scalar mass
squared matrices are given in appendix \ref{appenB}. In our numerical analysis we
confirm the existence of two charged and one neutral Goldstone boson(s) in
the charged scalar and pseudoscalar sector. In addition, we have checked
that all the eigenvalues of the scalar, pseudoscalar, and charged scalar mass-squared
matrices (apart from the Goldstone bosons) appear to be positive ({\it{non-tachyonic}})
for a minima. These matrices are addressed in refs.\cite{c4Escudero:2008jg,c4Ghosh:2008yh,
c4Bartl:2009an}. In appendix \ref{appenB} squark mass squared matrices are also addressed
\cite{c4Escudero:2008jg,c4Ghosh:2008yh}.
Before discussing the scalar sector of this model further, it is important
to point out the approximation and simplification used for involved numerical
analysis.
For numerical calculations we assume all soft-masses, $\lambda^i,\kappa^{ijk}$ and the
corresponding soft parameters $(A_\lambda\lam)^i,(A_\kappa\kappa)^{ijk}$ to be flavour
diagonal as well as flavour blind. However the neutrino Yukawa couplings $(Y^{ij}_\nu)$
and the respective soft parameters $(A_\nu Y_\nu)^{ij}$ are chosen to be flavour
diagonal. For simplicity all three right sneutrino VEVs are assumed to be
degenerate $(v^c)$. Mathematically,
\begin{eqnarray}
&&\kappa^{ijk} = \kappa \delta^{ij} \delta^{jk},
~~(A_\kappa\kappa)^{ijk} = (A_\kappa\kappa) \delta^{ij} \delta^{jk},\nonumber\\
&& Y^{ij}_\nu = Y^{ii}_\nu \delta^{ij},~~(A_\nu Y_\nu)^{ij} = (A_\nu Y_\nu)^{ii} \delta^{ij},\nonumber\\
&& \lambda^i = \lambda,~~(A_\lambda\lam)^i=(A_\lambda\lam),~~v^c_i=v^c,\nonumber\\
&& (m^2_{\tilde{L}})^{ij}=(m^2_{\tilde{L}})\delta^{ij},
~~(m^2_{\tilde{\nu}^c})^{ij}=(m^2_{\tilde{\nu}^c})\delta^{ij}.
\label{assumption1}
\end{eqnarray}
Coming back to the scalar sector of the $\mu\nu$SSM, apart from excluding
the corner of parameter space responsible for tachyons,
additional constraints on the parameter space can come from the existence
of false minima as well as from the perturbativity of the model parameters
(free from Landau pole).
A detailed discussion on this issue has been presented in
ref. \cite{c4Escudero:2008jg} and the regions excluded by the existence of false
minima have been shown. One can check from these figures that mostly the lower
part of the region allowed by the absence of tachyons, are excluded by the
existence of false minima. In our analysis, we have chosen the parameter
points in such a way that they should be well above the regions disallowed by
the existence of false minima. Nevertheless, in the case of gauge-singlet
neutrino ($\nu^c$) dominated lightest neutralino (to be discussed in the next chapter),
the value of $\kappa$ that we have chosen is $0.07$ with two different values of
$\lambda$, namely, $0.1$ and $0.29$. In this case, there is a possibility that
these points might fall into the regions disallowed by the existence of false
minima. However, we have checked that even if we take the value of $\kappa$
to be higher ($0.2$ or so), with appropriately chosen $\lambda$, our conclusions
do not change much. For such a point in the parameter space, it is likely that
the existence of false minima can be avoided.
Let us also mention here that the sign of the $\mu$-term is controlled by
the sign of the VEV $v^c$ (assuming a positive $\lambda$), which is
controlled by the signs of $A_\lambda \lambda$ and $A_\kappa \kappa$.
If $A_\lambda \lambda$ is negative and $A_\kappa \kappa$ is positive then the
sign of the $\mu$ parameter is negative whereas for opposite signs of the
above quantities, we get a positive sign for the $\mu$ parameter.
The eigenvalues of the scalar mass-squared matrices and the right-handed sneutrino
VEVs $(v^c)$ are not very sensitive to the change in neutrino
Yukawa couplings ($Y_\nu$ $\sim \cal {O}$ $(10^{-7})$) and the corresponding
soft parameter $A_\nu Y_\nu$ ($\sim \cal {O}$ $(10^{-4})$ GeV). On the other
hand, the values of $\tan\beta$ and the coefficients $\lambda$ and $\kappa$ are
very important in order to satisfy various constraints on the scalar sector
mentioned earlier. In figure \ref{scalar_sector_for_tan_beta_values}, we have
plotted the allowed regions in the ($\lambda$--$\kappa$) plane for
$\tan\beta=10$ \cite{c4Ghosh:2008yh}. Relevant parameters are given in
table \ref{lkplot-parameters}.
\begin{figure}[ht]
\centering
\includegraphics[width=6.45cm]{FigsPDF/tb10lk.pdf}
\caption{Allowed regions in ($\lambda$--$\kappa$) plane which satisfy
various constraints on the scalar sector, for tan$\beta =10$. $\lambda$ and
$\kappa$ were allowed to vary from $0.005$ to $0.50$ and $0.005$ to $0.70$, respectively.
Corresponding set of other parameters are given in table \ref{lkplot-parameters}.}
\label{scalar_sector_for_tan_beta_values}
\end{figure}
\begin{table}[ht]
\centering
\begin{tabular}{c | c || c | c}
\hline \hline
Parameter & Chosen Value & Parameter & Chosen Value \\ \hline \hline
$(A_\lambda \lambda)$ & $1000\times\lambda$ GeV& $(A_\kappa \kappa)$ & $1000\times\kappa$ GeV\\
$Y^{11}_\nu$ & $5.0\times10^{-7}$ & $(A_\nu Y_\nu)^{11}$ & $5.0\times10^{-4}$ GeV\\
$Y^{22}_\nu$ & $4.0\times10^{-7}$ & $(A_\nu Y_\nu)^{22}$ & $4.0\times10^{-4}$ GeV\\
$Y^{33}_\nu$ & $3.0\times10^{-7}$ & $(A_\nu Y_\nu)^{33}$ & $3.0\times10^{-4}$ GeV\\
$m^2_{\widetilde L}$ & $400^2$ GeV$^2$ & $m^2_{\widetilde \nu^c}$ & $300^2$ GeV$^2$\\
\hline \hline
\end{tabular}
\caption{\label{lkplot-parameters}
Relevant parameter choices for figure \ref{scalar_sector_for_tan_beta_values}
consistent with the EWSB conditions and non-tachyonic nature for squared scalar masses.
Eqn.(\ref{assumption1}) has been used and we choose $\tan\beta=10$.}
\end{table}
The upper limit of the value
of $\kappa$ is taken to be $\sim$ 0.7 because of the constraints coming from
the existence of Landau pole \cite{c4Escudero:2008jg}.
With the values of different parameters satisfying the constraints in the
scalar sector (see figure \ref{scalar_sector_for_tan_beta_values}), we will go on to
calculate the neutrino masses and the mixing patterns in the next few sections.
It is also important to discuss the bounds on the lightest Higgs boson mass
in $\mu\nu$SSM. Neglecting small neutrino Yukawa couplings $Y^{ij}_\nu$,
the tree level upper bound on the lightest neutral
Higgs mass \cite{c4Drees:1988fc,c4Ellis:1988er,c4Binetruy:1991mk,c4Espinosa:1991gr,
c4Espinosa:1992hp} is given by (see eqn.(\ref{Higgs-mass-NMSSM-lightest}))
\begin{equation}
m^{'^2}_{h^0} \lesssim M^2_Z \left[\cos^22\beta
+ 3.62~\lambda^i\lambda_i~\sin^22\beta\right].
\label{Higgs-mass-NMSSM-lightest-2}
\end{equation}
Apparently, one can optimize this bound by choosing small $\tan\beta$ and
large $\lambda^i\lambda_i$ values simultaneously.
Similar to the NMSSM \cite{c4Espinosa:1998re,c4Daikoku:2000eq,c4Ellwanger:2006rm}
the upper bound for the lightest ${\rm{SU(2)_L}}$ doublet-like
Higgs boson mass in the $\mu\nu$SSM is $\sim 140$ GeV
for $\tan\beta \sim2$ \cite{c4Escudero:2008jg}. Such a conclusion strictly demands small mixing
among the MSSM Higgs and the right-handed sneutrinos $\widetilde \nu^c_i$ (see eqns.
(\ref{element_of_scalar_mass_matrix}), (\ref{element_of_pseudoscalar_mass_matrix})).
It should be mentioned at this point that the radiative corrections to the
lightest Higgs boson mass, can be significant in some regions of the parameter space
as discussed in ref.\cite{c4Escudero:2008jg}. It has been shown that the light
Higgs mass larger than the LEP lower limit of 114 GeV can be obtained with the
value of $A_t$ (trilinear coupling in the scalar sector for the stop) within
1-2.4 TeV and when the mixing of the light Higgs with the right-handed
sneutrino is small. The latter requirement is fulfilled in most of the cases
that we have considered and in some cases the mixing is slightly larger.
However, there is always the freedom of choosing the value of $A_t$
appropriately. Hence, it would be fair to say that the experimental limits on
the light Higgs boson mass can be satisfied in our analysis.
Before starting the next section we want to emphasize that the
parameters chosen for our numerical analysis are just for illustrative purpose.
These are not some particular and specific choices in some sacred corner of the
model space. Since we have a large parameter space, it is always possible
to choose a different parameter point with the same characteristic features
satisfying all the experimental constraints.
\section{{\bf F}ermions in $\mu\nu$SSM}\label{munuSSM-fermion}
Effect of $\not\!\!{R_p}$ in the superpotential and in the soft terms (eqns.(\ref{munuSSM-superpotential}),
(\ref{munuSSM-soft})) is responsible for enrichment in the scalar sector. In an identical
fashion, the neutral and the charged fermion mass matrices also receive enhancement
through lepton number violating couplings.
\begin{flushleft}
{\it{$\maltese$ Neutralino mass matrix}}
\end{flushleft}
The neutral fermions of the MSSM $(\widetilde B^0,\widetilde W^0_3,\widetilde H^0_d,\widetilde H^0_u)$,
through second, third and fourth terms of $\mu\nu$SSM superpotential
(eqn.(\ref{munuSSM-superpotential})), can mix with
three generations of left and right-handed neutrinos, $\nu_i$ and $\nu^c_i$
respectively. The neutralino mass matrix for $\mu\nu$SSM is thus a $10\times10$ symmetric
matrix \cite{c4LopezFogliani:2005yw,c4Escudero:2008jg,c4Ghosh:2008yh,c4Bartl:2009an,c4Ghosh:2010zi}.
In the weak interaction basis defined by
\begin{eqnarray}
{\Psi^0}^T = \left(\tilde B^0, \tilde W_3^0, \tilde H_d^0,
\tilde H_u^0,{\nu^c_\alpha},{\nu_{\alpha}} \right),
\label{neutralino_basis}
\end{eqnarray}
where $\alpha=1,2,3\equiv e, \mu, \tau$.
The neutral fermion mass term in the Lagrangian is of the form
\begin{eqnarray}
{\mathcal{L}_{neutral}^{mass}} = -\frac{1}{2}{{\Psi^0}^T} \mathcal{M}_n
{\Psi^0} + \text{H.c.},
\label{weak-basis-Lagrangian-neutralino}
\end{eqnarray}
The massless neutrinos now can acquire masses due to their mixing with the MSSM neutralinos
and the gauge singlet right-handed
neutrinos. The three lightest eigenvalues of this $10\times10$ neutralino mass
matrix correspond to the three light physical neutrinos, which are expected
to be very small in order to satisfy the experimental data on massive
neutrinos (see table \ref{osc-para}). The matrix $\mathcal{M}_n$ can be written
in the following fashion
\begin{equation}
\mathcal{M}_n =
\left(\begin{array}{cc}
M_{7\times 7} & m_{3\times 7}^T \\
m_{3\times 7} & 0_{3\times 3}
\end{array}\right),
\label{neutralino-seesaw}
\end{equation}
where using eqn.(\ref{Abbrevations})
\begin{equation}
M_{7\times7} =
\left(\begin{array}{ccccccc}
M_1 & 0 & -\frac{g_1}{\sqrt{2}}v_1 & \frac{g_1}{\sqrt{2}}v_2 & 0 & 0 & 0 \\ \\
0 & M_2 & \frac{g_2}{\sqrt{2}}v_1 & -\frac{g_2}{\sqrt{2}}v_2 & 0 & 0 & 0 \\ \\
-\frac{g_1}{\sqrt{2}}v_1 & \frac{g_2}{\sqrt{2}}v_1 & 0 & -{\mu} &
-{\lambda^e}v_2 & -{\lambda^{\mu}}v_2 & -{\lambda^{\tau}}v_2 \\ \\
\frac{g_1}{\sqrt{2}}v_2 & -\frac{g_2}{\sqrt{2}}v_2 & -{\mu} & 0 & {\rho^e}
& {\rho^{\mu}} & {\rho^{\tau}}\\ \\
0 & 0 & -{\lambda^e}v_2 & {\rho^e} & 2 {u^{ee}_c} & 2 {u^{e{\mu}}_c} &
2 {u^{e{\tau}}_c}\\ \\
0 & 0 & -{\lambda^{\mu}}v_2 & {\rho^{\mu}} & 2{u^{{\mu}e}_c} &
2 {u^{{\mu}{\mu}}_c} & 2 {u^{{\mu}{\tau}}_c}\\ \\
0 & 0 & -{\lambda^{\tau}}v_2 & {\rho^{\tau}} & 2 {u^{{\tau}e}_c} &
2{u^{{\tau}{\mu}}_c} & 2 {u^{{\tau}{\tau}}_c}
\end{array}\right),
\label{neutralino_7x7}
\end{equation}
and
\begin{equation}
m_{3\times7} =
\left(\begin{array}{ccccccc}
-\frac{g_1}{\sqrt{2}}{v'_e} & \frac{g_2}{\sqrt{2}}{v'_e} & 0 & {r^e_c} &
Y_{\nu}^{ee} v_2 & Y_{\nu}^{e{\mu}} v_2 & Y_{\nu}^{e{\tau}} v_2\\ \\
-\frac{g_1}{\sqrt{2}}{v'_{\mu}} & \frac{g_2}{\sqrt{2}}{v'_{\mu}} & 0 &
{r^{\mu}_c} & Y_{\nu}^{{\mu}e} v_2 & Y_{\nu}^{{\mu}{\mu}} v_2 &
Y_{\nu}^{{\mu}{\tau}} v_2\\ \\
-\frac{g_1}{\sqrt{2}}{v'_{\tau}} & \frac{g_2}{\sqrt{2}}{v'_{\tau}} & 0 &
{r^{\tau}_c} & Y_{\nu}^{{\tau}e} v_2 & Y_{\nu}^{{\tau}{\mu}} v_2 &
Y_{\nu}^{{\tau}{\tau}} v_2
\end{array}\right).
\label{neutralino_3x7}
\end{equation}
Note that the top-left $4\times4$ block of the matrix $M_{7\times7}$ is the
usual neutralino mass matrix of the MSSM (see eqn.(\ref{MSSM-neutralino})).
The bottom right $3\times3$ block represents
the Majorana mass matrix for gauge singlet neutrinos, which will be taken as
diagonal (see eqn.(\ref{assumption1})) in the subsequent analysis.
The entries of $M_{7\times7}$ are in
general of the order of the electroweak scale whereas the entries of
$m_{3\times7}$ are much smaller $\sim$ $\cal O$ $(10^{-5})$ GeV.
Hence, the matrix (\ref{neutralino-seesaw}) has a seesaw structure,
which will give
rise to three very light eigenvalues corresponding to three light neutrinos.
The correct neutrino mass scale of $\sim$ $10^{-2}$ eV can easily be obtained
with such a structure of the $10\times10$ neutralino mass matrix. It has
been shown in ref.\cite{c4Ghosh:2008yh} that one can obtain the correct mass-squared
differences and the mixing pattern for the light neutrinos even with the choice
of flavour diagonal neutrino Yukawa couplings in eqn.(\ref{neutralino_3x7}).
Besides, the choice of flavour diagonal neutrino Yukawa couplings
(eqn.(\ref{assumption1})) makes the analysis simpler
with a reduced number of parameters and makes the model more predictive. As
we will show later, it is possible to find out the correct mixing pattern and
the mass hierarchies (both normal and inverted) among the light neutrinos in
such a situation, even at the tree level \cite{c4Ghosh:2008yh}.
In order to obtain the physical neutralino states, one needs to diagonalize
the $10\times10$ matrix $\mathcal{M}_n$. As in the case of MSSM, the symmetric
mass matrix $\mathcal{M}_n$ can be diagonalized with one unitary matrix {\bf{N}}.
The mass eigenstates $\chi^0_i$ are related to flavour eigenstates
$\Psi^0_j$ (eqn.(\ref{neutralino_basis})) as
\begin{equation}
\chi^0_i = \mathbf N_{i1} \widetilde B^0 + \mathbf N_{i2} \widetilde W^0_3 + \mathbf N_{i3} \widetilde H^0_d
+ \mathbf N_{i4} \widetilde H^0_u + \mathbf N_{i,\alpha+4} \nu^c_\alpha + \mathbf N_{i,\alpha+7} \nu_\alpha.
\label{munuSSM-neutralinos-reln}
\end{equation}
where the $10\times10$ unitary matrix {\bf{N}} satisfies
\begin{equation}\label{neutralino_mass_eigenstate_matrixform}
{\mathbf N}^* \mathcal{M}_n {\mathbf N}^{-1} = \mathcal{M}^0_D =
diag(m_{\widetilde \chi^0_i}, m_{\nu_j}),
\end{equation}
with the diagonal neutralino mass matrix denoted as $\mathcal{M}^0_D$.
$i$ and $j$ runs from $1$ to $7$ and $1$ to $3$, respectively. The quantity
$m_{\widetilde \chi^0_i}$ represent neutralino masses. Physical neutrino
masses are being represented by $m_{\nu_j}$.
It is, in general, very difficult to predict the nature of the lightest neutralino
(out of seven $\chi^0_i$) state since that depends on several unknown
parameters. Neutralino mass matrix for $\mu\nu$SSM with complex VEVs
is given in ref.\cite{c4Fidalgo:2009dm}.
\begin{flushleft}
{\it{$\maltese$ Chargino mass matrix}}
\end{flushleft}
Similar augmentation in the charged lepton sector result in a $5\times5$
chargino mass matrix where the charged electroweak gauginos $(-i \widetilde {\lambda}^{\pm}_{2})$
and higgsinos $(\widetilde{H}^+_u,\widetilde{H}^-_d)$ mix with charged leptons through
$R_p$ violating couplings. These mixings are coming from the second term of
eqn.(\ref{munuSSM-superpotential}) and as well as from non-zero left-handed
sneutrino VEVs.
In the weak interaction basis defined by
\begin{equation}
{\Psi^{+T}} = (-i \tilde {\lambda}^{+}_{2}, \widetilde{H}_u^{+}, \ell_{R}^{+}),
~~{\Psi^{-T}} = (-i \tilde {\lambda}_{2}^{-}, \widetilde{H}_d^{-}, \ell_{L}^{-}), \nonumber \\
\label{chargino_basis}
\end{equation}
where $\ell=e,\mu,\tau$.
The charged fermion mass term in the Lagrangian is of the form
\begin{equation}\label{chargino_mass_Lagrangian}
{\mathcal{L}_{charged}^{mass}} = -\frac{1}{2}
\left(\begin{array}{cc}
\Psi^{+^T} & \Psi^{-^T}
\end{array}\right)
\left(\begin{array}{cc}
0_{5\times5} & m_{5\times5}^T \\ \\
m_{5\times5} & 0_{5\times5}
\end{array}\right)
\left(\begin{array}{c}
\Psi^+ \\ \\
\Psi^-
\end{array}\right).
\end{equation}
Here we have included all three generations of charged leptons and assumed
that the charged lepton Yukawa couplings are in the diagonal form. The matrix
$m_{5\times5}$ using eqn.(\ref{Abbrevations}) is given by
\cite{c4Escudero:2008jg,c4Ghosh:2008yh,c4Ghosh:2010zi}
\begin{equation}\label{chargino_mass_matrix}
m_{5\times5} =
\left(\begin{array}{ccccc}
M_2 & {g_2}{v_2} & 0 & 0 & 0 \\ \\
{g_2}{v_1} & {\mu} & -{Y_{e}^{ee}}{v'_e} & -{Y_{e}^{{\mu}{\mu}}}{v'_{\mu}} &
-{Y_{e}^{{\tau}{\tau}}}{v'_{\tau}} \\ \\
{g_2}{v'_e} & -{r^e_c} & {Y_{e}^{ee}}{v_1} & 0 & 0 \\ \\
{g_2}{v'_{\mu}} & -{r^{\mu}_c} & 0 & {Y_{e}^{{\mu}{\mu}}}{v_1} & 0 \\ \\
{g_2}{v'_{\tau}} & -{r^{\tau}_c} & 0 & 0 & {Y_{e}^{{\tau}{\tau}}}{v_1}
\end{array}\right).
\end{equation}
The charged fermion masses are obtained by applying a bi-unitary
transformation like
\begin{equation}\label{chargino_mass_eigenstate_matrixform}
\mathbf U^* m_{5\times5} \mathbf V^{-1} = \mathcal{M}^{\pm}_D,
\end{equation}
where $\mathbf U$ and $\mathbf V$ are two unitary matrices and $\mathcal{M}^{\pm}_D$ is the
diagonal matrix. Relations between the mass $\chi^\pm_i$ and flavour eigenstates
for charginos are same as eqn.(\ref{brpMSSM-charginos-reln}), namely
\begin{eqnarray}
&& \chi^+_i = \mathbf V_{i1} \widetilde W^+ + \mathbf V_{i2} \widetilde H^+_u + \mathbf V_{i,\alpha+2} \ell^+_{\alpha_R}, \nonumber \\
&& \chi^-_i = \mathbf U_{i1} \widetilde W^- + \mathbf U_{i2} \widetilde H^-_d + \mathbf U_{i,\alpha+2} \ell^-_{\alpha_L},
\label{munuMSSM-charginos-reln}
\end{eqnarray}
where $ \widetilde W^\pm \equiv {-i\widetilde {\lambda}}^{\pm}_{2}$.
It is important to note that the off-diagonal elements (except for 12 and 21 elements) of the chargino mass
matrix (eqn. (\ref{chargino_mass_matrix})) either contain $Y^{ij}_\nu$ $({r^{i}_c} = {\sum Y^{ij}_{\nu}
{v^c_{j}}})$ or left-handed sneutrino VEVs $v'_i$, both of which are very small $\sim$ $\cal {O}$ $(10^{-4}$
GeV). This indicates that the physical charged lepton eigenstates will
have very small admixture of charged higgsino and charged gaugino states. So it is safe to assume
(also verified numerically) that these lepton number violating mixing have very little effect on
the mass eigenstates of the charged leptons. Thus, while writing down the
PMNS matrix \cite{c4Pontecorvo:1957cp,c4Pontecorvo:1957qd,c4Maki:1962mu,c4Pontecorvo:1967fh}
(eqn.(\ref{PMNS1})), it is justified to assume that one is working in
the basis where the charged lepton mass matrix is already in the diagonal form \cite{c4Ghosh:2008yh}.
So far all of the neutralinos and charginos are considered in two-component form. Corresponding
four component neutralino, chargino and charge conjugated chargino spinors are respectively defined as
\begin{eqnarray}
& &{\widetilde \chi}^0_i =
\left(\begin{array}{c}
\chi^0_i \\
\overline{\chi^0_i}\\
\end{array}\right),\quad
{\widetilde \chi}_i =
\left(\begin{array}{c}
\chi^+_i \\
\overline{\chi^-_i}\\
\end{array}\right),\quad
{\widetilde \chi}^c_i =
\left(\begin{array}{c}
\chi^-_i \\
\overline{\chi^+_i}\\
\end{array}\right),
\label{neutralino-chargino}
\end{eqnarray}
where $\chi^0_i$ and $\chi^{\pm}_i$ are two component neutral and charged spinors, respectively.
In our analysis the charged leptons are represented by their charged conjugate fields
\cite{c4Nowakowski:1995dx}, which are positively charged.
Unlike the scalar mass squared matrices, eigenvalues of the neutralino or chargino
mass matrix can be either positive or negative.
It is always possible to remove the wrong signs via appropriate rotations.
However, then one should be very
careful about the corresponding Feynman rules. A viable
alternative is to live with the signs of fermion masses ($\eta_i$ for neutralinos and
$\epsilon_i$ for charginos) and incorporate them properly in the respective Feynman rules
\cite{c4Gunion:1984yn}.
For the sake of completeness we also write down the quark mass matrices in $\mu\nu$SSM
in appendix \ref{appenB}.
\section{{\bf N}eutrinos at the tree level}\label{tree-neut}
It has been already emphasized that the $10\times10$ neutralino mass matrix
$\mathcal{M}_n$ possesses a seesaw like structure.
The effective light neutrino mass matrix ${M^{seesaw}_{\nu}}$, arising
via the seesaw mechanism in the presence of explicit lepton number violation,
is in general given by
\begin{equation}
{M^{seesaw}_{\nu}} = -{m_{3\times7}} {M_{7\times7}^{-1}}
{m_{3\times7}^T}.
\label{seesaw_formula}
\end{equation}
With small $\not\!\!{R_p}$, it is possible to carry out a
perturbative diagonalization of the $10\times10$ neutralino mass
matrix (see \cite{c4Schechter:1981cv}), by defining \cite{c4Hirsch:1998kc,
c4Hirsch:2000jt} a matrix $\xi$ as
\begin{equation}
\label{expansion-parameter}
\xi=m_{3\times7}.M^{-1}_{7\times7}.
\end{equation}
If the elements of $\xi$ satisfy $\xi_{ij} \ll 1$, then this can be
used as an expansion parameter to get an approximate analytical
solution for the matrix $\mathbf N$ (see
eqn.(\ref{neutralino_mass_eigenstate_matrixform})). A general expression for the
elements of $\xi$ with simplified assumptions can be written in the form
$\mathcal{A'}a_i + \mathcal{B'}b_i + \mathcal{C'}c_i$, where
\begin{equation}
a_i = Y_{\nu}^{ii} v_2, ~c_i = {v'_i}, ~b_i = (Y_{\nu}^{ii} v_1 + 3 {\lambda}
{v'_i}) = (a_i\cot\beta + 3 \lambda c_i),
\label{specifications}
\end{equation}
with ${i} = {e,\mu,\tau} ~\equiv{1,2,3}$, $\tan\beta = \frac{v_2}{v_1}$ and
$\mathcal{A'},\mathcal{B'},\mathcal{C'}$ are complicated functions of various
parameters of the model \cite{c4Ghosh:2010zi}. The complete expressions for the elements of
$\xi$ \cite{c4Ghosh:2010zi} are given in appendix \ref{appenC}. In deriving
detailed expression for $\xi$'s we neglect the sub-dominant terms $\sim$ $\cal{O}$
($\frac{v'^3}{{\tilde m}^3}$, $\frac{Y_\nu v'^2}{{\tilde m}^2}$,
$\frac{Y_\nu^2 v'}{{\tilde m}}$), where $\tilde m$ is the electroweak (or
supersymmetry breaking) scale.
With the help of eqn.(\ref{expansion-parameter}), eqn.(\ref{seesaw_formula})
reduces to
\begin{equation}
{M^{seesaw}_{\nu}} = -\xi{m_{3\times7}^T}.
\label{seesaw_formula-2}
\end{equation}
Using the favour of eqn.(\ref{assumption1}) in eqn.(\ref{seesaw_formula-2}),
together with the expressions for $\xi^{ij}$ given
in appendix \ref{appenC}, entries for the $3\times3$ matrix ${M^{seesaw}_{\nu}}$
are approximately given as (neglecting terms $\propto$ fourth power in $Y^{ij}_\nu,v'_i$
(separately or in a product) \cite{c4Ghosh:2008yh,c4Fidalgo:2009dm})
\begin{eqnarray}
({M^{seesaw}_{\nu}})_{ij}& \approx& {\frac{v^2_2}{6 \kappa {v^c}}}{Y^{ii}_{\nu}}
{Y^{jj}_{\nu}}(1-3 {\delta{ij}}) \nonumber \\
&-& {\frac{1}{2 M_{eff}}}\left[{v'_i}{v'_j} +{\frac{v_1 {v^c}
(Y^{ii}_{\nu}{v'_j}+Y^{jj}_{\nu}{v'_i})}{\mu}
+{\frac{Y^{ii}_{\nu}Y^{jj}_{\nu}v^2_1 {v^c}^2}{\mu^2}}}\right].\nonumber \\
\label{mnuij-compact2}
\end{eqnarray}
Here we have used
\begin{eqnarray}
& & M_{eff}= M \left[1-{\frac{v^2}{2 M A {\mu}}}\left({\kappa}{v^c}^2
\sin{2\beta} +{\frac{\lambda {v^2}}{2}}\right)\right],\nonumber \\
& &v_2 = v {\rm{sin}{\beta}},~v_1 = v {\rm{cos}}{\beta},~\mu = 3 \lambda v^c,\nonumber \\
& & A = ({\kappa}{v^c}^2 + {\lambda} v_1 v_2),~\frac{1}{M} = \frac{g^2_1}{M_1}
+\frac{g^2_2}{M_2}.
\label{mnuij-compact2-details}
\end{eqnarray}
Before proceeding further it is important to discuss eqn.(\ref{mnuij-compact2}) in
more details \cite{c4Ghosh:2008yh,c4Ghosh:2010zi}.
\vspace*{0.1cm}
\noindent
I.~First consider the limit ${v^c}\rightarrow \infty$ and $v \rightarrow 0$
($\Rrightarrow v_1,v_2 \to 0$).
Immediately eqn. (\ref{mnuij-compact2}) reduces to
\begin{equation}
\label{gauginoseesaw}
({M^{seesaw}_{\nu}})_{ij} \approx -{\frac{{v'_i}{v'_j}}{2 M}}
~~\Rrightarrow m_\nu \sim \frac{(g_1 c_i)^2}{M_1} + \frac{(g_2 c_i)^2}{M_2},
\end{equation}
which is the first part of the second term of eqn.(\ref{mnuij-compact2}). In
this case the elements of the neutrino mass matrix are bilinear in the
left-handed sneutrino VEVs and they appear due to a seesaw effect involving
the gauginos. This is known as the ``gaugino seesaw'' effect and neutrino mass
generation through this effect is a characteristic feature of the bilinear
$R_p$ violating model. This effect is present in this model because we
have seen earlier that the effective bilinear $R_p$ violating terms are
generated in the scalar potential as well as in the superpotential through the
vacuum expectation values of the gauge singlet sneutrinos $(\varepsilon^i=Y^{ij}v^c_j)$.
In gaugino seesaw the role of the {\it{Dirac}} mass terms are played by $g_1 v'_i$ and $g_2 v'_i$,
where $g_1,~g_2$ are the $U(1)$ and the $SU(2)$ gauge couplings respectively
and $v'_i$ ($\equiv c_i$ (eqn.(\ref{specifications}))) stand for the left-handed sneutrino
VEVs. The role of the {\it{Majorana}} masses are played by the gaugino soft
masses $M_1,~M_2$. The gaugino seesaw effect is closely analogous to the
TYPE-I \cite{c4Minkowski:1977sc,c4Yanagida:1979as,c4Glashow:1979nm,c4Mohapatra:1979ia,
c4GellMann:1980vs,c4Schechter:1980gr,c4Schechter:1981cv} + Type-III seesaw mechanism
\cite{c4Foot:1988aq,c4Ma:1998dn} due to simultaneous involvement of a
singlet $(\widetilde B^0)$ and triplet fermion $(\widetilde W^0_3)$
(see section \ref{neut-mass-II}, figure \ref{brpv-seesaw}, diagrams $(a,b)$).
This analogy has been pointed out in ref.\cite{c4Ghosh:2010zi}.
Note that the gaugino seesaw effect can generate mass for only one doublet neutrino,
as shown in eqn.(\ref{gauginoseesaw}).
\vspace*{0.1cm}
\noindent
II.~In the limit $M\rightarrow \infty$, eqn.(\ref{mnuij-compact2}) reduces to
\begin{equation}
\label{ordinaryseesaw}
({M^{seesaw}_{\nu}})_{ij} \approx{\frac{v^2_2}{6 \kappa {v^c}}}{Y^{ii}_{\nu}}
{Y^{jj}_{\nu}}(1-3 {\delta{ij}})\equiv {\frac{a_ia_j}{3 m_{\nu^c}}}(1-3 {\delta{ij}}),
\end{equation}
which corresponds to the ``ordinary seesaw'' effect between the left-handed and
gauge singlet right-handed neutrinos. Remember that the effective Majorana masses for
the gauge singlet neutrinos are given by $m_\nu^c = 2 \kappa v^c$
and the usual Dirac masses are given by $a_i=Y^{ii}_\nu v_2$.
The ordinary seesaw effect can generate, in general, masses for more than one neutrinos.
Thus depending on the magnitudes and the hierarchies of various diagonal
neutrino Yukawa couplings $Y_\nu^{ii}$, one can generate normal or inverted
hierarchy of neutrino masses (combining with the ``gaugino seesaw" effect)
corresponding to atmospheric and solar mass squared differences \cite{c4Ghosh:2008yh}.
It is also
interesting to note that a conventional ordinary seesaw (generated only
through the mixing between left-handed and right-handed neutrinos) in contrast to
eqn.(\ref{ordinaryseesaw}) would give rise to a mass matrix of the form \cite{c4Fidalgo:2009dm}
\begin{equation}
\label{ordinaryseesaw-2}
({M^{seesaw}_{\nu}})_{ij} \approx - {\frac{v^2_2}{2 \kappa {v^c}}}{Y^{{ii}^2}_{\nu}}.
\end{equation}
The off-diagonal contributions as shown in eqn.(\ref{ordinaryseesaw}) are arising from
an effective mixing between the right-handed neutrinos and Higgsinos.
Hence, when right-handed neutrinos are also decoupled $(v^c\rightarrow \infty)$,
the neutrino masses are zero as corresponds to the case
of a seesaw with only Higgsinos \cite{c4Fidalgo:2009dm}.
\subsection{{\bf N}eutrino masses at the tree level}\label{tree-neut-mass}
Eqn.(\ref{mnuij-compact2}) can be re-casted in a compact form
using eqns.(\ref{specifications}) (\ref{mnuij-compact2-details}) as
\begin{equation}
({M^{seesaw}_{\nu}})_{ij} = {\frac{1}{6 \kappa {v^c}}} {a_{i}}
{a_{j}}(1-3\delta_{ij}) + {\frac{2 A {v^c}}{3 \Delta}} {b_{i}}
{b_{j}},
\label{mnuij-compact1}
\end{equation}
or alternatively using eqn.(\ref{specifications}) in a more elucidate form as
\begin{equation}
({M^{seesaw}_{\nu}})_{ij} = f_1 a_i a_j + f_2 c_i c_j + f_3 (a_i c_j + a_j c_i),
\label{mnuij-compact-recasted}
\end{equation}
with
\begin{eqnarray}
f_1 = \frac{1}{6 \kappa v^c} (1-3\delta_{ij}) + \frac{2 A v^c {\rm{cot}}^2\beta}{3 \Delta},
~~f_2 = \frac{2 A \lambda \mu}{\Delta},~~f_3 = \frac{2 A \mu {\rm{cot}}\beta}{3 \Delta}
\label{specifications2}
\end{eqnarray}
and $\Delta = \lambda^2 (v^2_1 + v^2_2)^2 + 4 \lambda \kappa v_1 v_2 {v^c}^2 - 4 \lambda A \mu M$.
It is apparent from eqn.(\ref{mnuij-compact1}) that the second term $(\propto b_ib_j)$ can
contribute to only one neutrino mass, $\propto b^2_i$. However, presence of $(1-3\delta^{ij})$ factor
in the first term assures non-zero masses for other neutrinos. If we concentrate on the normal hierarchical
scheme of light neutrino masses, then with suitable choice of model ingredients it is possible
to generate the atmospheric neutrino mass scale $(\sim$ $\cal O$ $(10^{-11}$ GeV)) from the second term,
whereas relatively small solar scale $(\sim$ $\cal O$ $(10^{-12}$ GeV)) emerges from the first term
of eqn.(\ref{mnuij-compact1}). The imposed order of magnitude difference between the first and
the second term of eqn.(\ref{mnuij-compact1}) through certain choices of model parameters can be used to extract
the eigenvalues of eqn.(\ref{mnuij-compact1}) analytically. Choosing the dominant terms to be $\propto b_ib_j,$
which contribute to only one neutrino mass, it is possible to apply the techniques of degenerate perturbation
theory to extract the effect of the perturbed term $(\propto a_ia_j)$ over the unperturbed one
$(\propto b_ib_j)$ \cite{c4Ghosh:2008yh}. It has to be clarified here that actually in $\mu\nu$SSM
for a novel region of the parameter space $b_i \sim a_i$ \cite{c4Ghosh:2008yh}, however, with a clever
choice of the $\lambda$ and $\kappa$ parameter it is possible to vary the order of magnitude of the
co-efficients in front $(\frac{1}{6\kappa v^c},\frac{2 A v^c}{3 \Delta}$, see eqn.(\ref{mnuij-compact1})).
For the chosen set of parameters (see table \ref{table-tree-parameters}) co-efficients of the $a_ia_j$
term is an order of magnitude smaller compared to that of $b_ib_j$ \cite{c4Ghosh:2008yh}. So the perturbative
approach is well justified. As shown in ref.\cite{c4Ghosh:2008yh} it is possible to extract
simple analytical form for light neutrino masses in this approach. Detailed expressions for
the eigenvectors and eigenvalues of eqn.(\ref{mnuij-compact1})
obtained through perturbative calculations are given in appendix \ref{appenC}. It is interesting to
see from eqn.(\ref{eigenvalues}) that the correction to unperturbed eigenvalues are proportional
to the effect of ordinary seesaw \cite{c4Ghosh:2008yh}.
The numerical values of the solar and atmospheric mass squared differences
$\Delta{m^2_{solar}}$ $(\equiv \Delta{m^2_{21}})$ and $\Delta{m^2_{atm}}$ $(\equiv \Delta{m^2_{31}})$
as obtained from full numerical calculations (Using eqn.(\ref{seesaw_formula}))
and from appropriate analytical formulae (Using eqn.(\ref{eigenvalues}))
have been shown in table \ref{table-tree}\footnote{A typo has been corrected
compared to ref.\cite{c4Ghosh:2008yh}. Also to denote individual neutrino masses, $m_{\nu_i}$
are used instead of $m_i$ (ref.\cite{c4Ghosh:2008yh}).} and the results show good agreement \cite{c4Ghosh:2008yh}.
The numerical calculations have been performed with the help of a code developed by us using
Mathematica \cite{c4math-wolfram}.
In our numerical analysis for the normal hierarchical pattern in
light neutrino masses, we choose ${m_2}|_{max} < 1.0 \times 10^{-11}$ GeV \cite{c4Ghosh:2008yh}.
Results of table \ref{table-tree} are consistent with the three flavour global neutrino data
\cite{c4Schwetz:2008er,c4GonzalezGarcia:2010er} as shown in table \ref{osc-para} in the $3\sigma$ limit.
It is interesting to observe that unlike conventional bilinear $R_p$ violating models, in
$\mu\nu$SSM all three neutrinos are massive itself at the tree level. Consequently, it is
possible to accommodate the three flavour global neutrino data (table \ref{osc-para}) at the tree
level even with the choice of diagonal neutrino Yukawa couplings (see table
\ref{table-tree-parameters}) \cite{c4Ghosh:2008yh}.
\begin{table}[ht]
\centering
\begin{tabular}{c | c || c | c}
\hline \hline
Parameter & Chosen Value & Parameter & Chosen Value \\ \hline \hline
$\lambda$ & $0.06$ & $(A_\lambda \lambda)$ & $-60$ GeV\\
$\kappa$ & $0.65$ & $(A_\kappa \kappa)$ & $650$ GeV\\
$Y^{11}_\nu$ & $4.57\times10^{-7}$ & $(A_\nu Y_\nu)^{11}$ & $1.57\times10^{-4}$ GeV\\
$Y^{22}_\nu$ & $6.37\times10^{-7}$ & $(A_\nu Y_\nu)^{22}$ & $4.70\times10^{-4}$ GeV\\
$Y^{33}_\nu$ & $1.80\times10^{-7}$ & $(A_\nu Y_\nu)^{33}$ & $3.95\times10^{-4}$ GeV\\
$M_1$ & $325$ GeV & $M_2$ & $650$ GeV\\
$m^2_{\widetilde L}$ & $400^2$ GeV$^2$ & $m^2_{\widetilde \nu^c}$ & $300^2$ GeV$^2$\\
\hline \hline
\end{tabular}
\caption{\label{table-tree-parameters}
Parameter choices (consistent with figure \ref{scalar_sector_for_tan_beta_values}) for result
presented in table \ref{table-tree}. Eqn.(\ref{assumption1}) has been used here and we choose
$\tan\beta=10$.}
\end{table}
Both left and right sneutrino VEVs ($v'_i,v^c_i$, respectively) are derived using
the set of parameters given in table \ref{table-tree-parameters}. The relation between the
gaugino soft masses $M_1$ and $M_2$ are assumed to be GUT (grand unified theory) motivated,
so that, at the electroweak scale, we have $M_1 : M_2~=~1:2$.
\begin{table}[ht]
\footnotesize
\centering
\begin{tabular}{c|c|c|c|c|c}
\hline\hline
& \multicolumn{3}{c|}{$m_\nu$ (eV) ($\times 10^3$)}
& $\Delta{m^2_{21}}$(eV$^2$) & $\Delta{m^2_{31}}$(eV$^2$) \\
\cline{2-4}
& $m_{\nu_1}$ & $m_{\nu_2}$ & $m_{\nu_3}$ & $(\times 10^5)$
& $(\times 10^3)$ \\
\hline
\text{Using eqn.(\ref{seesaw_formula})} & 4.169 & 9.970 & 48.23 & 8.203 & 2.307 \\
\hline
\text{Using eqn.(\ref{eigenvalues})} & 4.168 & 9.468 & 47.71 & 7.228 & 2.187 \\
\hline\hline
\end{tabular}
\caption{\label{table-tree}
Absolute values of the neutrino masses and the mass-squared differences for a
sample point of the parameter space \cite{c4Ghosh:2008yh}. Results for full numerical analysis
have been obtained using eqn.(\ref{seesaw_formula}). Approximate analytical expressions of
eqn.(\ref{eigenvalues}) have been used for comparison. Parameter choices are given in
table \ref{table-tree-parameters}.}
\end{table}
\subsection{{\bf N}eutrino mixing at the tree level}\label{tree-neut-mixing}
The expansion parameter $\xi$ (see eqns.(\ref{expansion-parameter-terms})) has been introduced in
eqn.(\ref{expansion-parameter}) to perform perturbative diagonalization of the $10\times10$ neutralino
mass matrix $\mathcal{M}_n$. It is possible to express the neutralino mixing matrix $\mathbf N$
(see eqn.(\ref{neutralino_mass_eigenstate_matrixform})) in leading order in $\xi$ as
\begin{equation}
\mathbf N^* = \left(\begin{array}{cc}
\mathcal{N}^* & 0 \\
0 & U^T
\end{array}\right)
\left(\begin{array}{cc}
1-\frac{1}{2} \xi^{\dagger} \xi & \xi^{\dagger} \\
-\xi & 1-\frac{1}{2} \xi \xi^{\dagger}
\end{array}\right).
\label{neutralino-mixing-matrix}
\end{equation}
The $10\times10$ neutralino mass matrix $\mathcal{M}_n$ can approximately be
block diagonalized to the form {\it{diag($M_{7\times7},{M^{seesaw}_{\nu}}$)}},
by the matrix defined in eqn.(\ref{neutralino-mixing-matrix}). The matrices
$\mathcal{N}$ and $U$, defined in eqn.(\ref{neutralino-mixing-matrix}),
are used to diagonalize ${M_{7\times7}}$ and ${M^{seesaw}_{\nu}}$ in the following
manner (using eqn.(\ref{neutralino_mass_eigenstate_matrixform})),
\begin{eqnarray}
\label{diag-matrix}
& &\mathcal{N}^* M_{7\times7} \mathcal{N}^{\dagger} =
{\rm{diag}} (m_{{\widetilde \chi}^0_i}), \nonumber \\
& &{U}^T {M^{seesaw}_{\nu}} {U} =
{\rm{diag}} (m_{\nu_1},m_{\nu_2},m_{\nu_3}).
\end{eqnarray}
Where $U$ is the non-trivial leptonic mixing matrix, known as PMNS matrix
\cite{c4Pontecorvo:1957cp,c4Pontecorvo:1957qd,c4Maki:1962mu,c4Pontecorvo:1967fh}.
As already stated in section \ref{neut-osc}, a non-trivial neutrino mixing is
a consequence of massive neutrinos. If we adhere to a scenario where $CP$ is preserved,
the PMNS matrix following eqn.(\ref{PMNS1}) can be written as
\begin{equation}
U =
\left(\begin{array}{ccc}
{c_{12}}{c_{13}} & {s_{12}}{c_{13}} & {s_{13}} \\ \\
-{s_{12}}{c_{23}}-{c_{12}}{s_{23}}{s_{13}} & {c_{12}}{c_{23}}
-{s_{12}}{s_{23}}{s_{13}} & {s_{23}}{c_{13}}\\ \\
{s_{12}}{s_{23}}-{c_{12}}{c_{23}}{s_{13}} & -{c_{12}}{s_{23}}
-{s_{12}}{c_{23}}{s_{13}} & {c_{23}}{c_{13}}
\end{array}\right),
\label{PMNS-CPC}
\end{equation}
where $c_{ij} = \cos{\theta_{ij}}$, $s_{ij} = \sin{\theta_{ij}}$.
It is definitely possible to extract the mixing angles from $U$
in a full numerical analysis. However, it is always useful to
do the same with a simplified approximate analytical analysis (if at all possible)
to get an idea about the relative importance of the different parameters.
An analysis of this kind for light neutrino mixing angles using degenerate
perturbation theory has been addressed in ref. \cite{c4Ghosh:2008yh}. We showed that it is
possible to write down the PMNS matrix $U$ as (eqn.(\ref{mneutrino_mixing_numerical}))
\begin{eqnarray}
{U}&=&
\left(\begin{array}{ccc}
{\mathcal{Y}_1} & {\mathcal{Y}_2} & {\mathcal{Y}_3}
\end{array}\right)_{3\times3},
\label{mneutrino_mixing_numerical2}
\end{eqnarray}
where ${\mathcal{Y}_i}$'s are defined in appendix \ref{tree-perturbed}. Using
eqn.(\ref{mneutrino_mixing_numerical2}) it is possible to derive appropriate expressions
for the light neutrino mixing angles $\theta_{13},\theta_{23},\theta_{12}$ as \cite{c4Ghosh:2008yh}
\begin{eqnarray}
\sin^2\theta_{13} = \frac{b^2_e}{b^2_e + b^2_\mu + b^2_\tau}.
\label{reactor_analytical}
\end{eqnarray}
\begin{eqnarray}
\sin^2\theta_{23} = \frac{b^2_\mu}{b^2_\mu + b^2_\tau}.
\label{atmos_analytical}
\end{eqnarray}
\begin{eqnarray}
\sin^2\theta_{12} \approx 1 - {(\alpha^\prime_1
+ \alpha^\prime_2 {\frac{b_e}{b_\tau}})}^2,
\label{solar_analytical}
\end{eqnarray}
where $b_i$'s are given by eqn.(\ref{specifications}). The quantities
$\alpha^\prime_1,\alpha^\prime_2$ are given by eqn.(\ref{alphas}).
It is apparent from eqn.(\ref{reactor_analytical}) that if we want the (13) mixing angle to be
small (which is supported by data, see table \ref{osc-para}) then one must have
$b^2_e \ll (b^2_\mu + b^2_\tau)$. On the other hand, since the (23) mixing angle
$\theta_{23}$ is maximal by nature $(\sim 45^\circ$, see table \ref{osc-para}), it is natural to
expect $b^2_\mu = b^2_\tau$. The formula for solar mixing angle $\theta_{12}$ is a bit complicated.
Nevertheless, in order to have $\theta_{12} \sim 35^\circ$, the square root of the second
term on the right hand side of eqn.(\ref{solar_analytical}) should
be approximately $0.8$. So these approximate analytical formulae clearly help
us to choose suitable corner of parameter space rather than performing a
blind search, which is the power of the analytical approach.
We compare three light neutrino mixing angles as obtained from eqn.(\ref{mneutrino_mixing_numerical2})
to that obtained in full numerical analysis using eqn.(\ref{seesaw_formula}) in
table \ref{table-tree-mixing}. Neutrino masses are taken to be normal hierarchical.
\begin{table}[ht]
\footnotesize
\centering
\begin{tabular}{c|c|c}
\hline\hline
\text{mixing angles in degree} & \text{Using (\ref{seesaw_formula})} &
\text{Using (\ref{mneutrino_mixing_numerical})}\\
\hline
$\theta_{12}$ & 36.438 & 37.287 \\
\hline
$\theta_{13}$ & 9.424 & 6.428 \\
\hline
$\theta_{23}$& 38.217 & 42.675 \\
\hline\hline
\end{tabular}
\caption{\label{table-tree-mixing}
Neutrino mixing angles using eqn.(\ref{seesaw_formula}) and
eqn.(\ref{mneutrino_mixing_numerical}) Parameter choices are given in
table \ref{table-tree-parameters}. These values are consistent with entries of
table \ref{osc-para} in the $3\sigma$ limit \cite{c4Schwetz:2008er}.}
\end{table}
We can see that for this set of chosen parameters (table \ref{table-tree-parameters}), numerical and
approximate analytical results give quite good agreement. Naturally, one would
be interested to check the predictions made in
eqns. (\ref{reactor_analytical}), (\ref{atmos_analytical}),
and (\ref{solar_analytical}) over a wide region in the parameter space and see
the deviations from the full numerical calculations. These are shown in
figures.\ref{neut2313-mixing-tree}, \ref{neut12-mixing-tree} \cite{c4Ghosh:2008yh}.
\begin{figure}[ht]
\vspace{0.5cm}
\centering
\includegraphics[width=10.00cm]{FigsPDF/Sinsq2313nn.pdf}
\caption{Scatter plot of the neutrino mixing angle $\sin^2\theta_{23}$ (left) and $\sin^2\theta_{13}$ (right)
as a function of the ratio $\frac{b^2_\mu}{b^2_\tau}$ and $\frac{b^2_e}{b^2_\mu + b^2_\tau}$. Values of
model parameters are given in table \ref{neut-tree-mix-param}. The lightest neutralino (LN) is either
a bino $(\widetilde B^0)$ or a higgsino $(\widetilde H^0_u,\widetilde H^0_d)$ dominated. Light neutrino mass ordering is normal
hierarchical.}
\label{neut2313-mixing-tree}
\end{figure}
It is apparent from the left diagram of figure \ref{neut2313-mixing-tree} that for $b^2_\mu = b^2_\tau$,
the value of $\sin^2\theta_{23}$ varies in the range $0.41 -- 0.44$, which corresponds to
$\theta_{23}$ between 40$^\circ$ and 42$^\circ$. On the other hand,
eqn.(\ref{atmos_analytical}) tells that for $b^2_\mu = b^2_\tau$, $\sin^2\theta_{23} = 0.5$.
So for a wide region of parameter space result from the
numerical calculation is reasonably close to the prediction from the
approximate analytical formula.
\begin{figure}[ht]
\vspace{0.5cm}
\centering
\includegraphics[width=5.00cm]{FigsPDF/Sinsq12nn.pdf}
\caption{$\sin^2\theta_{12}$ versus $(\alpha_1^{\prime}+\alpha_2^{\prime}\frac{b_e}{b_\tau})^2$ scatter
plot. Parameter choice and mass hierarchy is same as figure \ref{neut2313-mixing-tree}.}
\label{neut12-mixing-tree}
\end{figure}
Also from figure \ref{neut12-mixing-tree} as
$(\alpha_1^{\prime}+\alpha_2^{\prime}\frac{b_e}{b_\tau})^2\rightarrow0.50$,
$\sin^2\theta_{12}$ tends to be maximal, that is $\theta_{12}=45^\circ$,
which is well expected.
\begin{table}[ht]
\centering
\begin{tabular}{c | c || c | c}
\hline \hline
Parameter & Chosen Value & Parameter & Chosen Value \\ \hline \hline
$Y^{11}_\nu$ & $3.55-5.45\times10^{-7}$ & $(A_\nu Y_\nu)^{11}$ & $1.25-1.95\times10^{-4}$ GeV\\
$Y^{22}_\nu$ & $5.55-6.65\times10^{-7}$ & $(A_\nu Y_\nu)^{22}$ & $3.45-4.95\times10^{-4}$ GeV\\
$Y^{33}_\nu$ & $1.45-3.35\times10^{-7}$ & $(A_\nu Y_\nu)^{33}$ & $2.35-4.20\times10^{-4}$ GeV\\
$m^2_{\widetilde L}$ & $400^2$ GeV$^2$ & $m^2_{\widetilde \nu^c}$ & $300^2$ GeV$^2$\\
$\lambda$ & $0.06(0.13)$ & $(A_\lambda \lambda)$ & $-1000\times\lambda$ GeV\\
$\kappa$ & $0.65$ & $(A_\kappa \kappa)$ & $1000\times\kappa$ GeV\\
$M_1$ & $110(325)$ GeV & $M_2$ & $2\times M_1$ GeV\\
\hline \hline
\end{tabular}
\caption{\label{neut-tree-mix-param}
Parameter choices (consistent with figure \ref{scalar_sector_for_tan_beta_values})
for figures \ref{neut2313-mixing-tree}, \ref{neut12-mixing-tree}.
$\lambda=0.06(0.13)$ for a bino(higgsino) dominated lightest neutralino.
Similarly, $M_1=110~(325)$ GeV for a bino (higgsino) dominated lightest neutralino.
Eqn.(\ref{assumption1}) has been used here and we choose $\tan\beta=10$. The set of
chosen parameters are consistent with the constraints of the scalar sector.}
\end{table}
Concerning table \ref{neut-tree-mix-param} it has to be emphasized here that
the allowed regions in the $\lambda -~\kappa$ plane (see figure
\ref{scalar_sector_for_tan_beta_values}) are not very sensitive to the values of
$Y_\nu$ and $A_\nu Y_\nu$ due to their smallness. Hence we choose to vary them randomly
(see table \ref{neut-tree-mix-param}), in order to accommodate the three flavour global neutrino data.
So far we considered eqn.(\ref{mnuij-compact1}) in the limit when with suitable choice of
model parameters the terms $\propto a_ia_j$ can act as perturbation over the second term.
However, the huge parameter space for $\mu\nu$SSM always leaves room for the inverse
situation. In other words there exists suitable corner of parameter space where the
first term of eqn.(\ref{mnuij-compact1}) is the dominant one and then eqn.(\ref{atmos_analytical})
can be expressed as
\begin{eqnarray}
\sin^2\theta_{23} = \frac{a^2_\mu}{a^2_\mu + a^2_\tau}.
\label{atmos_analyticala}
\end{eqnarray}
This is exactly what is shown by figure \ref{neut23a-mixing-tree}.
Note that for $a^2_\mu = a^2_\tau$, the atmospheric mixing angle becomes maximal.
\begin{figure}[ht]
\vspace{0.5cm}
\centering
\includegraphics[width=6.00cm]{FigsPDF/Ntb10nnNCDSSQ23amat-1.pdf}
\caption{Scatter plot of the neutrino mixing angle $\sin^2\theta_{23}$ as
a function of the ratio $\frac{a^2_\mu}{a^2_\tau}$. The lightest neutralino (LN) is
right-handed neutrino $(\nu^c)$ dominated.}
\label{neut23a-mixing-tree}
\end{figure}
In figure \ref{normal_hierarchical_scheme_for_bino-dominated_case}, we have shown
the regions in the various $Y_\nu$ planes satisfying the three flavour global
neutrino data. The values of other parameters are as shown in
table \ref{neut-tree-mix-param} for the case where the lightest neutralino $(\ntrl1)$ is
bino dominated. We can see from these figures that the allowed values of
$Y_\nu$s show a mild hierarchy such that $Y_\nu^{22} > Y_\nu^{11} >
Y_\nu^{33}$ \cite{c4Ghosh:2008yh}.
Similar studies have been performed for the inverted
hierarchical case and the allowed region shows that the magnitudes of the
neutrino Yukawa couplings are larger compared to the case of normal
hierarchical scheme of the neutrino masses with a different hierarchy among
the $Y_\nu$'s themselves ($Y_\nu^{11} > Y_\nu^{22} > Y_\nu^{33}$). In this case
$\sin^2\theta_{12}$ shows an increasing behaviour with the ratio
$b^2_e/b^2_\mu$, similar to the one shown by $\sin^2\theta_{23}$ with
$b^2_\mu/b^2_\tau$ in the normal hierarchical scenario (see figure \ref{neut2313-mixing-tree}).
On the other hand, $\sin^2\theta_{23}$ shows a decreasing behaviour with $b^2_\mu/b^2_\tau$.
In all these cases, the solar and atmospheric mass-squared differences are within
the 3$\sigma$ limits (table \ref{osc-para}).
\begin{figure}[ht]
\vspace{0.5cm}
\centering
\includegraphics[width=13.50cm]{FigsPDF/ynu-ynu-bn_NH.pdf}
\caption{Plots for normal hierarchical scheme of neutrino mass in
$Y^{22}_\nu ~-~Y^{33}_\nu$,~$Y^{11}_\nu ~-~Y^{33}_\nu$ and $Y^{11}_\nu ~-
~Y^{22}_\nu$ plane when the lightest neutralino (LN) is bino dominated.}
\label{normal_hierarchical_scheme_for_bino-dominated_case}
\end{figure}
\section{{\bf N}eutrinos at the loop level}\label{loop-neut}
It is legitimate to ask that what is the motivation for performing loop calculations
in $\mu\nu$SSM when all three neutrinos can acquire masses at the tree level \cite{c4Ghosh:2008yh}?
In fact this is a feature where the $\mu\nu$SSM model is apparently successful over most
of the other models of light neutrino mass generation where loop corrections are unavoidable
in order to account for oscillation data. However, in the regime of
renormalizable quantum field theories, stability of any tree
level analysis must be re-examined in the light of radiative
corrections. Following this prescription, the results of neutrino masses and
mixing will be more robust, once tree level analysis is further improved by
incorporating radiative corrections. The radiative corrections may have
sizable effect on the neutrino data at one-loop level. Thus, although all
three SM neutrinos acquire non-zero masses in the $\mu \nu$SSM even at the
tree level \cite{c4Ghosh:2008yh}, it is interesting to investigate the fate of
those tree level masses and mixing when exposed to one-loop corrections. With
this in view, in the following subsections we perform a systematic study of the neutrino
masses and mixing with all possible one-loop corrections both analytically and
numerically. In the subsequent subsections, while showing the results of one-loop
corrections, we try to explain the deviations (which may or may not be
prominent) from the tree level analysis. The complete set of one-loop diagrams are shown
in figure \ref{one-loop-diagrams}.
Before going into the details, let us
discuss certain relevant issues of one-loop correction and renormalization for
the neutralino-neutrino sector. The most general one-loop
contribution to the unrenormalized neutralino-neutrino two-point function can be expressed as
\begin{equation}
i {\bf \Sigma}^{ij}_{{\widetilde \chi}^0 {\widetilde \chi}^0}(p) = i\{\displaystyle{\not} p \left[P_L
{\Sigma^L_{ij}}(p^2) + P_R {\Sigma^R_{ij}}(p^2)\right] -\left[P_L
{\Pi^L_{ij}}(p^2) + P_R {\Pi^R_{ij}}(p^2)\right]\},
\label{loopgen-unrenorm}
\end{equation}
where $P_L$ and $P_R$ are defined as $\frac{1-\gamma_5}{2}$ and $\frac{1+\gamma_5}{2}$, respectively.
$i,~j~=~1,...,10$ and $p$ is the external momentum. The unrenormalized self-energies $\Sigma^{L,R}$
and $\Pi^{L,R}$ depend on the squared external momentum $(p^2)$. The generic self
energies $\Sigma^{L(R)}_{ij},~\Pi^{L(R)}_{ij}$ of the Majorana neutralinos and neutrinos must
be symmetric in its indices, $i$ and $j$. $\overline{DR}$ scheme \cite{c4Siegel:1979wq,c4Capper:1979ns,
c4Avdeev:1981vf,c4Avdeev:1982xy,c4Jack:1997sr} has been used to regularize one-loop contributions.
In the $\overline{DR}$ scheme\footnote{In $\overline{DR}$ scheme the subtraction
procedure is same as $\overline{MS}$ \cite{c4Passarino:1978jh} scheme and the momentum integrals are also
evaluated with $D$ dimensions. However, the {\it{Dirac algebras}} are done strictly in four dimensions
since only in four dimensions the numbers of fermions and bosons match in the case of
a supersymmetric system.}, the counter-terms cancel only the divergent pieces of the self-energies.
Thus the self energies become finite but depend on the arbitrary scale
of renormalization. To resolve this scale dependency, the tree level masses are promoted to
running masses in which they cancel the explicit scale dependence of the self energies
$\Sigma,\Pi$ \cite{c4Hirsch:2000ef}. The resulting one-loop corrected mass matrix using dimensional
reduction ($\overline{DR}$) scheme is given by
\begin{eqnarray}
\label{one-loop-corrected-mass}
(\mathcal{M}^{\rm{tree + 1-loop}}_{\chi^0})^{ij} &=& {m}_{{\widetilde \chi}^0}(\mu_R)\delta^{ij} +
{\frac{1}{2}}\left(\widetilde \Pi^V_{ij}({{m^2_i}}) + \widetilde
\Pi^V_{ij}({{m^2_j}}) \right.\nonumber\\
&-& {m_{{\widetilde \chi}^0_i}}{\widetilde
\Sigma^V_{ij}}({{m^2_i}})
- \left. {m_{{\widetilde \chi}^0_j}}{\widetilde
\Sigma^V_{ij}}({{m^2_j}})\right), \nonumber\\
\end{eqnarray}
with
\begin{eqnarray}
\label{Sigma-Pi-renomalized}
\widetilde \Sigma^V_{ij} = \frac{1}{2} (\widetilde \Sigma^L_{ij} + \widetilde
\Sigma^R_{ij}),
~~\widetilde \Pi^V_{ij} = \frac{1}{2} (\widetilde \Pi^L_{ij} + \widetilde
\Pi^R_{ij}),
\end{eqnarray}
where the tree level neutralino mass $({m}_{{\widetilde \chi}^0})$ is defined at the
renormalization scale $\mu_R$, set at the electroweak scale. Here, the
word {\it{neutralino mass}} stands for all the {\it{ten}} eigenvalues
of the $10\times10$ neutralino mass matrix. The self-energies $\Sigma,~\Pi$
are also renormalized in the $\overline{DR}$ scheme and denoted
by $\widetilde \Sigma$ and $\widetilde \Pi$ respectively. The detailed
expressions of $\widetilde \Sigma^V_{ij}$ and $\widetilde \Pi^V_{ij}$
depend on corresponding Feynman rules and the Passarino-Veltman functions
\cite{c4'tHooft:1978xw,c4Passarino:1978jh}.
\begin{figure}[ht]
\vspace{0.5cm}
\centering
\includegraphics[width=3.80cm]{FigsPDF/Loopnnh.pdf}
\includegraphics[width=3.80cm]{FigsPDF/Loopnna.pdf}
\includegraphics[width=3.80cm]{FigsPDF/Loopnnz.pdf}
\vspace{1cm}
\includegraphics[width=3.80cm]{FigsPDF/Loopncs.pdf}
\includegraphics[width=3.80cm]{FigsPDF/Loopncw.pdf}
\includegraphics[width=3.80cm]{FigsPDF/Loopnqsq.pdf}
\caption{One-loop diagrams contributing to the neutralino masses. The various
contributions are arising from (clockwise from top left)
(a) neutralino-neutralino-neutral scalar loop,
(b) neutralino-neutralino-neutral pseudoscalar loop,
(c) neutralino-neutralino-$Z^0_{\mu}$ loop,
(d) neutralino-chargino-charged scalar loop,
(e) neutralino-chargino-$W^{\pm}_{\mu}$ loop,
(f) neutralino-quark-squark loop.}
\label{one-loop-diagrams}
\end{figure}
In the next section we will describe our calculational approach.
\section{{\bf A}nalysis of neutrino masses and mixing at one-loop}\label{loop-neut-calc}
In this section we consider the effect of radiative corrections to the light
neutrino masses and mixing. Just for the sake of completeness it is always better to
recapitulate some of the earlier works regarding
one-loop corrections to the neutralino-neutrino sector. The complete set of
radiative corrections to the neutralino mass matrix in the $R_p$ conserving
MSSM was discussed in ref.\cite{c4Pierce:1993gj,c4Pierce:1994ew}, and the leading order
neutrino masses has been derived in ref.\cite{c4Hall:1983id}. One-loop radiative
corrections to the neutrino-neutralino mass matrix in the context of a
$R_P$-violating model were calculated in ref.\cite{c4Hempfling:1995wj} using
't-Hooft-Feynman gauge. In ref.\cite{c4Hirsch:2000ef}, $R_{\xi}$ gauge has been
used to compute the corrections to the neutrino-neutralino mass matrix at
one-loop level in an $R_p$-violating scenario. For our one-loop calculations
we choose to work with 't-Hooft-Feynman gauge, i.e. $\xi=1$.
Neutrino mass generation at the one-loop level in other variants of $R_P$-violating MSSM has
been widely addressed in literature, which are already given in the beginning of
subsection \ref{neut-mass-II}. We note in passing that in a recent reference
\cite{c4Liebler:2011tp} on-shell renormalization
of neutralino and chargino mass matrices in $R_p$ violating models has been addressed,
which also includes the $\mu\nu$SSM.
We begin by outlining the strategy of our analysis. We
start with a general $10\times10$ neutralino matrix, with off-diagonal entries
as well, which has a seesaw structure in the flavour-basis (see
eqn.(\ref{neutralino-seesaw})). Schematically, we can rewrite
eqn.(\ref{neutralino-seesaw}) as,
\begin{equation}
\mathcal{M}_n = \left(\begin{array}{cc}
M_f & m^T_{D_f} \\
m_{D_f} & 0
\end{array}\right),
\label{neutralino-seesaw-schematic}
\end{equation}
where the orders of the block matrices are same as those indicated in
eqn. (\ref{neutralino-seesaw}), and the subscript `$f$'denotes the flavour
basis. Here $M_f$ stands for the $7\times7$ Majorana mass matrix of the heavy
states, while $m_{D_f}$ contains the $3\times7$ Dirac type masses for the left-handed
neutrinos. In the next step, instead of utilizing the seesaw structure
of this matrix to generate the effective light neutrino
mass matrix for the three active light neutrino species, we {\it{diagonalize}} the entire
$10\times10$ matrix $\mathcal{M}_n$. The diagonal $10\times10$ matrix
$\mathcal{M}_D^{0}$ (eqn.(\ref{neutralino_mass_eigenstate_matrixform})) thus contains
tree level neutralino masses, which we symbolically write as \cite{c4Ghosh:2010zi}
\begin{equation}
\mathcal{M}_D^{0} = \left(\begin{array}{cc}
M_m & 0 \\
0 & m_m
\end{array}\right),
\label{neutralino-tree-level-schematic}
\end{equation}
where $M_m~(m_m)$ are the masses of the heavy states (left-handed
neutrinos). Following eqn.(\ref{diag-matrix}) one can write
\begin{eqnarray}
M_m &=& diag (m_{\widetilde \chi^0_1},m_{\widetilde \chi^0_2},m_{\widetilde \chi^0_3},m_{\widetilde \chi^0_4}
,m_{\widetilde \chi^0_5},m_{\widetilde \chi^0_6},m_{\widetilde \chi^0_7}),\nonumber\\
m_m &=& diag (m_{\nu_1},m_{\nu_2},m_{\nu_3}).
\label{mass-mass-p1-p2}
\end{eqnarray}
At this stage we turn on all possible one-loop interactions as shown in
figure \ref{one-loop-diagrams}, so
that the $10\times10$ matrix $\mathcal{M}_D^{0}$ picks up radiatively
generated entries, both diagonal and off-diagonal. The resulting one-loop
corrected Lagrangian for the neutralino mass terms in the $\widetilde \chi^0$ basis,
following eqn.(\ref{weak-basis-Lagrangian-neutralino}), can be written as
\begin{equation}
\label{loop-correcetd-Lagrangian-neutralino-a} {\mathcal{L}^{\prime}}
= -\frac{1}{2}{{\chi^{0^T}}} \left(\mathcal{M}^0_D +
\mathcal{M}^1 \right) {\chi^0} + \text{H.c.},
\end{equation}
where $\mathcal{M}^1$ contains the effect of one-loop corrections. The
$10\times10$ matrix $\mathcal{M}^0_D$ is diagonal, but the matrix
$\mathcal{M}^1$ is a general symmetric matrix with off diagonal
entries.
One can rewrite the above equation, using
eqns.(\ref{munuSSM-neutralinos-reln}) and
(\ref{neutralino_mass_eigenstate_matrixform}), as
\begin{equation}
\label{loop-correcetd-Lagrangian-neutralino-b}
{\mathcal{L}^{\prime}} = -\frac{1}{2}{{\Psi^0}^T} \left(\mathcal{M}_n
+ \mathbf N^T \mathcal{M}^1 \mathbf N\right) {\Psi^0} + \text{H.c.}.
\end{equation}
This is nothing but the {\it{one-loop corrected}} neutralino mass term in the Lagrangian in the
flavour basis. Symbolically \cite{c4Ghosh:2010zi},
\begin{equation}
\label{loop-correcetd-Lagrangian-neutralino-c}
{\mathcal{L}^{\prime}} = -\frac{1}{2}{{\Psi^0}^T}
\mathcal{M}^{\prime} {\Psi^0} + \text{H.c.},
\end{equation}
with the $10\times10$ matrix $\mathcal{M}^{\prime}$ having the form \cite{c4Ghosh:2010zi}
\begin{equation}
\mathcal{M}^{\prime} =
\left(\begin{array}{cc}
M_f + \Delta{M}_f & (m_{D_f} + \Delta{m_{D_f}})^T \\
m_{D_f} + \Delta{m_{D_f}} & \Delta{m}_f
\end{array}\right).
\label{neutralino-tree-plus-one-loop-level-schematic}
\end{equation}
The quantities $\Delta{M}_f$ and $\Delta{m}_f$ stand for one-loop corrections
to the heavy neutralino states and light neutrino states respectively, in the
flavour basis $\Psi^0$. The entity $\Delta{m_{D_f}}$ arises because of the off
diagonal interactions, i.e. between the heavy neutralinos and the light
neutrinos, in the same basis $(\Psi^0)$. Note that all of $\Delta M_f$, $\Delta m_{D_f}$,
$\Delta m_f$ in the $\chi_0$ basis are given by the second term on the right
hand side of eqn.(\ref{one-loop-corrected-mass}). We suitably transform them
into the basis $\Psi^0$ with the help of neutralino mixing matrix
$\mathbf N$. From the order of magnitude approximations\footnote{The
loop corrections are at least suppressed
by a loop factor $\frac{1}{16\pi^2}$ and thus tree level order of magnitude
approximations are still valid.}
the matrix $\mathcal{M}^{\prime}$ once again possesses a
seesaw structure, and one can therefore write down the
one-loop corrected effective light neutrino mass matrix as
\begin{equation} ({M}^{\nu^{\prime}})_{\rm{eff}} \approx
\Delta{m}_f - (m_{D_f} + \Delta{m_{D_f}})(M_f +
\Delta{M}_f)^{-1}((m_{D_f} + \Delta{m_{D_f}})^T).
\label{mass-basis-seesaw-schematic}
\end{equation}
Let us now present an approximate form of
eqn.(\ref{mass-basis-seesaw-schematic}). For simplicity, let us begin by
assuming the quantities present in eqn.(\ref{mass-basis-seesaw-schematic}) to
be c-numbers (not matrices). In addition, assume $M_f \gg \Delta{M}_f$
(justified later), so that eqn.(\ref{mass-basis-seesaw-schematic}) may be
written as,
\begin{equation}
({M}^{\nu^{\prime}})_{\rm{eff}} \approx \Delta{m}_f - \delta \times
{M_f}\left\{\left(\frac{m_{D_f}}{M_f}\right)^2 + 2
\left(\frac{m_{D_f}}{M_f}\right) \left(\frac{\Delta
m_{D_f}}{M_f}\right) + \left(\frac{\Delta m_{D_f}}{M_f}\right)^2
\right\},
\label{mass-basis-seesaw-schematic-2}
\end{equation}
with $\delta = \left(1-\frac{\Delta M_f}{M_f}\right)$. Now, even when $\Delta
m_{D_f}$ $\sim$ $\frac{1}{16 \pi^2}$ $m_{D_f}$ and $\Delta
M_f$ $\sim$ $\frac{1}{16 \pi^2}$ $M_f$,
eqn.(\ref{mass-basis-seesaw-schematic-2}) looks like
\begin{eqnarray}
({M}^{\nu^{\prime}})_{\rm{eff}} &\approx& \Delta{m}_f - {M_f}
\left(1-\frac{1}{16 \pi^2}\right)\left\{\left(\frac{m_{D_f}}{M_f}\right)^2 +
\frac{2}{16 \pi^2} \left(\frac{m_{D_f}}{M_f}\right)^2 \right.\nonumber\\
&+&\left. \frac{1}{256 \pi^4}
\left(\frac{m_{D_f}}{M_f}\right)^2 \right\}.
\label{mass-basis-seesaw-schematic-3}
\end{eqnarray}
Thus, up to a very good approximation one can rewrite
eqn.(\ref{mass-basis-seesaw-schematic-3}) as
\begin{equation}
({M}^{\nu^{\prime}})_{\rm{eff}} \approx
\Delta{m}_f - {M_f} \left(\frac{m_{D_f}}{M_f}\right)^2.
\label{mass-basis-seesaw-schematic-4}
\end{equation}
Reimposing the matrix structure and using eqn.(\ref{seesaw_formula}),
eqn.(\ref{mass-basis-seesaw-schematic-4}) can be modified as,
\begin{equation}
({M}^{\nu^{\prime}})_{\rm{eff}} \approx
\Delta{m}_f + {M^{seesaw}_{\nu}}.
\label{mass-basis-seesaw-schematic-5}
\end{equation}
The eigenvalues of the $3\times3$ one-loop corrected neutrino mass matrix
$({M}^{\nu^{\prime}})_{\rm{eff}}$ thus correspond to one-loop corrected light
neutrino masses. In conclusion, it is legitimate to calculate one-loop
corrections to the $3\times3$ light neutrino mass matrix only
(see eqn.(\ref{mass-basis-seesaw-schematic-5})), and diagonalize
it to get the corresponding one-loop corrected mass eigenvalues \cite{c4Ghosh:2010zi}.
Let us denote the one-loop corrections to the masses of heavy neutralinos and
light neutrinos in the basis $\chi^0$ by $\Delta M$ and $\Delta m$
respectively. The one-loop corrections arising from neutralino-neutrino
interactions is denoted by $\Delta m_D$ in the same basis. The tree level
neutralino mixing matrix $\mathbf N$, in the leading power of expansion matrix $\xi$
(eqn.(\ref{expansion-parameter})), using eqn.(\ref{neutralino-mixing-matrix})
can be written as,
\begin{equation}
\mathbf N =
\left(\begin{array}{cc}
\mathcal{N} & \mathcal{N} \xi^T \\
-U^\dagger \xi^* & U^\dagger
\end{array}\right)=
\left(\begin{array}{cc}
\widetilde N_{7\times 7} & \widetilde N_{7\times 3} \\
\widetilde N_{3\times 7} & \widetilde N_{3\times 3}
\end{array}\right).
\label{neutralino-mixing-matrix-block-form}
\end{equation}
Now from the order of magnitude approximation of $\xi$ (eqn.(\ref{expansion-parameter}))
we get approximately $\xi \sim$ $({m_D^{\nu}}/{M_{\widetilde \chi^0}})$, where $m_D^{\nu}$
represents a generic entry of $m_{3\times7}$ matrix and ${M_{\widetilde \chi^0}}$ that
of $M_{7\times7}$ (see eqn.(\ref{neutralino-seesaw})). So apparently
the entries of the matrices $\widetilde N_{7\times
3},~\widetilde N_{3\times 7}$ suffers a suppression $\sim \cal{O}$
$({m_D^{\nu}}/{M_{\widetilde \chi^0}})$, due to very small
neutrino-neutralino mixing \cite{c4Atre:2009rg}. The quantities $m_D^{\nu}$ $\sim$ $\cal{O}$
($10^{-4}$ GeV) and $M_{\widetilde \chi^0}$ $\sim$ $\cal{O}$ ($10^2$ GeV) represent the Dirac mass
of a left-handed neutrino $(\nu_i)$ and the Majorana mass of a neutralino $(\chi^0_i)$,
respectively. From
eqns.(\ref{loop-correcetd-Lagrangian-neutralino-b}), (\ref{neutralino-mixing-matrix-block-form})
it is easy to figure out the relation between $\Delta m$ and $\Delta m_f$ as,
\begin{equation}
\Delta m_{f} = {\widetilde N_{7\times 3}^T} {\Delta M} {\widetilde
N_{7\times 3}} + {\widetilde N_{7\times 3}^T} {\Delta m_D^T} {\widetilde
N_{3\times 3}} + {\widetilde N_{3\times 3}^T} {\Delta m_D} {\widetilde
N_{7\times 3}} + {\widetilde N_{3\times 3}^T} {\Delta m} {\widetilde
N_{3\times 3}}.
\label{flavour-mass-relation}
\end{equation}
Now as argued earlier, for a Dirac neutrino, the mass is $\lesssim$ $\cal{O}$
$(10^{-4}~\rm{GeV})$, while for a neutralino, the mass is
$\sim$ $\cal{O}$ $(10^{2}~\rm{GeV})$. This means that the
entries of the off-diagonal blocks in
eqn.(\ref{neutralino-mixing-matrix-block-form}) are $\lesssim$ $\cal{O}$
$(10^{-6})$. Therefore, for all practical purpose, one can neglect
the first three terms in comparison to the fourth term on the right
hand side of eqn.(\ref{flavour-mass-relation}). Thus,
\begin{equation}
\Delta m_{f} \approx {\widetilde N_{3\times 3}^T} {\Delta m} {\widetilde N_{3\times 3}}.
\label{flavour-mass-relation-2}
\end{equation}
up to a very good approximation. With this in view, our strategy is to compute
the one-loop corrections in the $\chi^0$ basis first, and then use
eqn.(\ref{flavour-mass-relation-2}) to obtain the corresponding corrections in
the flavour basis. Finally, adding tree level contribution $M^{seesaw}_\nu$ (eqn.(\ref{seesaw_formula}))
to $\Delta m_f$ (eqn.(\ref{flavour-mass-relation-2})), we diagonalize
eqn.(\ref{mass-basis-seesaw-schematic-5}) to obtain the one-loop corrected
neutrino masses. We have performed all calculations in the 't-Hooft-Feynman
gauge. Let us also note in passing that the form of eqn.(\ref{one-loop-corrected-mass})
predicts off-diagonal entries ($i\neq j$). The off-diagonal elements are
responsible for the admixtures between diagonal entries, which become dominant
only when $\left({m}_{{\widetilde \chi}^0_i}-{m}_{{\widetilde \chi}^0_j}\right)~\lesssim(\frac{\alpha}{4 \pi})
\times {\rm{some ~electroweak ~scale ~mass}}$, (using the essence of eqn.(\ref{seesaw-mixing}))
and then, one can choose $p^2 =\overline{m^2} = ({{m^2}_{{\widetilde \chi}^0_i} + {m^2}_{{\widetilde \chi}^0_j}})/2$ for external
momentum\cite{c4Hempfling:1995wj}. Thus, one can conclude that unless the tree level masses are highly
degenerate, the off-diagonal radiative corrections can be neglected for all practical
purposes, when at least one indices $i$ or $j$ refers to a heavy states.
The self-energy corrections contain entries of the neutralino mixing matrix
$\mathbf N$ through the couplings $O^{ff^{\prime}b}$ appearing in Feynman rules
(see, appendix \ref{appenD}) \cite{c4Ghosh:2010zi}. This is because, the
self energies ${\it{\widetilde \Sigma}_{ij}}$ and ${\it{\widetilde \Pi}_{ij}}$ in
general contain products of couplings of the form $O^{ff^{\prime}b}_{i..}O^{ff^{\prime}b}_{j..}$
(see, appendix \ref{appenE}\cite{c4Ghosh:2010zi} for detailed expressions of
$\widetilde \Sigma^V_{ij}$ and $\widetilde \Pi^V_{ij}$). The matrix $\mathbf N$, on the other
hand, contains the expansion parameter $\xi$ in the leading order (see
eqn.(\ref{neutralino-mixing-matrix})). This observation, together with the help
of eqn.(\ref{expansion-parameter-terms}), help us to express the
effective structure of the one-loop corrected neutrino mass matrix as \cite{c4Ghosh:2010zi},
\begin{equation}
[(\mathcal{M}^{\nu^{\prime}})_{eff}]_{ij} = A_1 a_i a_j + A_2 c_i c_j + A_3 (a_i c_j + a_j c_i),
\label{one-loop corrected structure of neutralino mass matrix}
\end{equation}
where $a_i$ and $c_i$ are given by eqn.(\ref{specifications}) and
$A_i$'s are functions of our model parameters and the Passarino-Veltman
functions $(B_0,B_1)$ \cite{c4'tHooft:1978xw,c4Passarino:1978jh,c4Hahn:1998yk} defined in
appendix \ref{appenF}. The form of the loop corrected mass matrix thus obtained is identical
to the tree level one (see, eqn.(\ref{mnuij-compact-recasted}))
with different coefficients $A_1$, $A_2$ and $A_3$ arising due to one-loop
corrections.
Note that the one-loop diagrams in figure \ref{one-loop-diagrams}, contributing to the neutrino mass matrix
are very similar to those obtained in bilinear R-parity violating scenario \cite{c4Hirsch:2000ef,c4Davidson:2000uc,
c4Davidson:2000ne,c4Diaz:2003as,c4Grossman:2003gq,c4Dedes:2006ni}. However, it has been pointed
out in ref.\cite{c4Bartl:2009an}, that there is a new significant contribution coming from the loops containing the
neutral scalar and pseudoscalar with dominant singlet component. This contribution is proportional to the
mass-splitting between the singlet scalar and pseudoscalar states
\cite{c4Hirsch:1997vz,c4Grossman:1997is,c4Dedes:2007ef}.
The corresponding mass splittings for the doublet sneutrinos are much smaller \cite{c4Bartl:2009an}.
In fact the sum of contributions of the singlet scalar $({\widetilde \nu}^c_{n{\mathcal R}})$ and pseudoscalar
states $({\widetilde \nu}^c_{n{\mathcal I}})$ (see diagrams one and two of the top row of figure \ref{one-loop-diagrams})
is $\propto$ $\kappa^2 v^{c^2}$, squared mass difference between
the singlet scalar and pseudoscalar mass eigenstates \cite{c4Bartl:2009an}.
The effect of one-loop correction to light neutrino
masses and mixing has been considered in ref.\cite{c4Bartl:2009an} for one and two generations
of right-handed neutrinos.
To conclude this section we finally concentrate on the one-loop contributions to
light neutrino mixing. The tree level
$3\times3$ orthogonal matrix $U$ diagonalizes the tree level seesaw matrix $M^{seesaw}_\nu$
as shown in eqn.(\ref{diag-matrix}). In a similar fashion the $3\times3$ orthogonal matrix
(in the limit of all phases equal to zero) that
diagonalizes the one-loop corrected neutrino mass matrix
$({M}^{\nu^{\prime}})_{\rm{eff}}$ (eqn.(\ref{mass-basis-seesaw-schematic-5})), can be denoted as
$U'$. Mathematically
\begin{equation}
\label{diag-one-loop-corrected-neutrino-mass}
{{U}^{{\prime}^{T}}} ({M}^{\nu^{\prime}})_{\rm{eff}}
{{U}^{\prime}} ={\rm{diag}} (m'_1,~m'_2,~m'_3),
\end{equation}
with $m'_1$, $m'_2$, $m'_3$ as the three one-loop corrected light neutrino
masses. The matrix $U'$ now can be used (see eqn.(\ref{PMNS-CPC})) to extract
the one loop corrected light neutrino mixing angles, $\theta'_{23},\theta'_{12},
\theta'_{13}$.
In the next section we will discuss the effect of one-loop corrections to the
light neutrino masses and mixing in $\mu\nu$SSM for different light neutrino mass
hierarchy.
\section{{\bf O}ne-loop corrections and mass hierarchies}\label{loop-mass-hierarchy}
Analytical forms for the tree level and the one-loop corrected light neutrino mass matrices
are given by eqn.(\ref{seesaw_formula}) and
eqn.(\ref{one-loop corrected structure of neutralino mass matrix}), respectively.
Note that in both of the equations the first two terms $(\propto a_ia_j,
~\propto~c_ic_j)$ individually can generate only one neutrino mass, $\propto \sum a^2_i$
and $\propto \sum c^2_i$, respectively. These terms are the effect of the {\it{ordinary}}
and the {\it{gaugino}} seesaw, as already discussed in section \ref{tree-neut}.
Together, they can generate two neutrino masses which is sufficient to
satisfy the neutrino oscillation data without the cross term $(a_ic_j+a_jc_i)$.
However, it is the effect of the mixing terms $(a_ic_j+a_jc_i)$ which together
with the first two terms along with different co-efficients for each term
give masses to all three light neutrinos \cite{c4Ghosh:2008yh,c4Ghosh:2010zi}.
In the following three consecutive subsections we will analyze the effect of one-loop radiative corrections
on the light neutrino masses and mixing when the mass orderings are (1) normal, (2) inverted
and (3) quasi-degenerate in nature. The choice of model parameters are given in table
\ref{loop-param} \cite{c4Ghosh:2008yh,c4Ghosh:2010zi}.
\begin{table}[ht]
\centering
\begin{tabular}{c | c || c | c}
\hline \hline
Parameter & Chosen Value & Parameter & Chosen Value \\ \hline \hline
$\lambda$ & $0.10$ & $(A_\lambda \lambda)$ & $-100$ GeV\\
$\kappa$ & $0.45$ & $(A_\kappa \kappa)$ & $450$ GeV\\
$m^2_{\widetilde e^c}$ & $300^2$ GeV$^2$ & $v^c$ & $-895$ to $-565$ GeV\\
$(A_\nu Y_\nu)^{ii}$ & $Y^{ii}_\nu\times1$ TeV & $\tan\beta$ & $10$\\
$M_1$ & $110$ GeV & $M_2$ & $220$ GeV\\
\hline \hline
\end{tabular}
\caption{\label{loop-param}
Choice of parameters for numerical analysis consistent with the EWSB conditions. These
choices are according to the eqn.(\ref{assumption1}). The gaugino soft masses $M_1$ and $M_2$ are
assumed to be GUT (grand unified theory) motivated, so that, at the
electroweak scale, we have $M_1 : M_2~=~1:2$.}
\end{table}
Apart from the right-handed sneutrino VEVs other variables are chosen to be
the left sneutrino VEVs $(v'_i)$ and the flavour diagonal neutrino Yukawa couplings $(Y^{ii}_\nu)$.
These are given in table \ref{loop-param-2} \cite{c4Ghosh:2008yh,c4Ghosh:2010zi}.
\begin{table}[ht]
\centering
\begin{tabular}{c||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$Y^{ii}_\nu \times 10^7$}
& \multicolumn{3}{c}{$v'_i \times 10^5 (\rm{GeV})$ } \\
\cline{2-7}
& $Y^{11}_\nu$ & $Y^{22}_\nu$ & $Y^{33}_\nu$ & $v'_1$
& $v'_2$ & $v'_3$ \\
\hline\hline
\text{Normal hierarchy} & 3.550 & 5.400 & 1.650 & 0.730 & 10.100 & 12.450\\
\text{Inverted hierarchy} & 12.800 & 3.300 & 4.450 & 8.350 & 8.680 & 6.400\\
\text{Quasi-degenerate-I} & 19.60 & 19.94 & 19.99 & 9.75 & 10.60 & 11.83\\
\text{Quasi-degenerate-II} & 18.50 & 18.00 & 18.00 & 9.85 & 10.50 & 10.10\\
\hline\hline
\end{tabular}
\caption{Values of the neutrino Yukawa couplings and the left-handed sneutrino VEVs,
used as sample parameter points for numerical calculations. These are the
values around which the corresponding parameters were varied.
Other parameter choices are given in table \ref{loop-param}.\label{loop-param-2}}
\end{table}
To fit the three flavour global data we consider not only the oscillation
constraints (see table \ref{osc-para}) but also constraints from various
non-oscillation experiments like Tritrium beta decay, neutrinoless double
beta decay and cosmology both for the tree level and the one-loop combined analysis.
\subsection{{\bf N}ormal hierarchy}\label{loop-normal}
In the normal hierarchical pattern of the three light neutrino masses
(individual masses are denoted by $m_i$, $i=1,2,3$),
the atmospheric and the solar mass squared differences, given by
$\Delta m^2_{atm}= m^2_3 - m^2_2$ and $\Delta m^2_{solar}= m^2_2 - m^2_1$,
are largely governed by the higher mass squared in each case, namely,
$m_3^2$ and $m_2^2$, respectively. Before going into the discussion of
the variation of the mass-squared values with the model parameter, some
general remarks are in order. First of all, note that in eqn.(\ref{specifications}),
if we choose $v^\prime_i$ such that $v'_i \gg \frac{Y^{ii}_\nu v_1}{3 \lambda}$,
then $b_i \approx c_i$\cite{c4Fidalgo:2009dm}.
Second, both the tree level and the one-loop corrected light neutrino mass matrix
have similar structure as shown in eqn.(\ref{mnuij-compact-recasted}) and
eqn.(\ref{one-loop corrected structure of neutralino mass matrix}). Due to this
structural similarity we expect both the tree and the one-loop corrected masses and mixing
to show similar type of variations with certain relevant quantities, however
with some modifications, because of the inclusion of the one-loop corrections.
This similarity also indicates that the light neutrino masses and mixing are
entirely controlled by $a_i$ and $c_i$.
\begin{figure}[ht]
\centering
\includegraphics[width=6.00cm]{FigsPDF/masssqesgs1s.pdf}
\includegraphics[width=6.00cm]{FigsPDF/masssqesos1s.pdf}
\vspace{0.2cm}
\includegraphics[width=6.00cm]{FigsPDF/masssqesgs2s.pdf}
\includegraphics[width=6.00cm]{FigsPDF/masssqesos2s.pdf}
\vspace{0.2cm}
\includegraphics[width=6.00cm]{FigsPDF/masssqesgs3s.pdf}
\includegraphics[width=6.00cm]{FigsPDF/masssqesos3s.pdf}
\caption{Neutrino mass squared values ($m^2_i$) versus $\frac{c^4_i}{M^2}$
(left panel) and versus $\frac{a^4_i}{m^2_{\nu^c}}$ (right panel) plots for
the {\it{normal hierarchical}} pattern of light neutrino masses, $i = e,
\mu, \tau$. Parameter choices are shown in tables \ref{loop-param}
and \ref{loop-param-2}.}
\label{numsqNH}
\end{figure}
In this subsection, we show the variation of the neutrino squared
masses ($m^2_i$) and the atmospheric and solar mass squared differences
with the square of the seesaw parameters $\frac{c^2_i}{M}$ and
$\frac{a^2_{i}}{m_{\nu^c}}$ for normal ordering in light neutrino masses.
Results are shown for the tree level
as well as the one-loop corrected neutrino masses. These plots also
demonstrate the importance of one-loop corrections to neutrino masses
compared to the tree level results \cite{c4Ghosh:2010zi}.
Typical mass spectra are shown in figure \ref{numsqNH}. Note that a
particular model parameter has been varied while the others are fixed
at values given in tables \ref{loop-param} and \ref{loop-param-2}. The effective
light neutrino mass matrix given in eqn.(\ref{mnuij-compact1})
suggests that as long as $v'_i \gg \frac{Y^{ii}_\nu v_1}{3 \lambda}$ and
$\kappa \gg \lambda$, the second term on the right hand side of
eqn.(\ref{mnuij-compact1}) dominates over the first term and as a result
the heaviest neutrino mass scale ($m_3$) is controlled mainly by the gaugino
seesaw effect. This is because in this limit $b_i \approx c_i$, and, as discussed
earlier, a neutrino mass matrix with a structure $\sim \frac{c_ic_j}{M}$
can produce only one non-zero neutrino mass. This feature is evident in
figure \ref{numsqNH}, where we see that $m^2_3$ increases as a function of
$c^4_i/M^2$. The other two masses are almost insensitive to $c^2_i/M$. A mild variation
to $m^2_2$ comes from the combined effect of gaugino and ordinary seesaw (see
the $(a_ic_j+a_jc_i)$ terms in eqns.(\ref{mnuij-compact-recasted}),
(\ref{one-loop corrected structure of neutralino mass matrix})).
On the other hand, the two lighter neutrino mass scales ($m^2_2$ and
$m^2_1$) are controlled predominantly by the ordinary seesaw parameters
$a^2_i/{m_{\nu^c}}$. This behaviour is observed in the right panel figures of
figure \ref{numsqNH}. The heaviest neutrino mass scale is not much affected
by the quantities $a^2_i/{m_{\nu^c}}$.
One can also see from these plots that the inclusion of one-loop
corrections, for the chosen values of the soft SUSY breaking parameters,
reduces the values of $m^2_2$ and $m^2_1$, while increasing the
value of $m^2_3$ only mildly. This is because, with such a choice, the
one-loop corrections cause partial cancellation in the generation of $m_1$
and $m_2$. For the heaviest state, it is just the opposite, since the
diagonalization of the tree-level mass matrix already yields
a negative mass eigenvalue, on which the loop correction has an additive effect.
If, with all other parameters fixed, the signs of $\lambda$ and $A_\lambda$ are
reversed (leading to a positive $\mu$ in the place of a negative one), $m_1$,
$m_2$ and $m_3$ are all found to decrease through loop corrections. A flip in the
sign of $\kappa$ and the corresponding soft breaking terms, on the other hand, causes
a rise in all the mass eigenvalues, notably for $m_1$ and $m_2$.
In the light of the discussion above, we now turn to explain the
variation of $\Delta m^2_{atm}$ and $\Delta m^2_{solar}$
with $c^4_i/M^2$ and $a^4_i/{m^2_{\nu^c}}$ shown in figure \ref{gsNH}
and figure \ref{osNH}. For our numerical analysis, in order to
set the scale of the normal hierarchical spectrum, we choose
$m_2|_{max}<0.011~\rm{eV}$. The left panel in
figure \ref{gsNH} shows that $\Delta m^2_{atm}$ increases more
rapidly with $c^4_{\mu,\tau}/M^2$, whereas the variation with
$c^4_e/M^2$ is much slower as expected from figure \ref{numsqNH}.
Similar behaviour is shown for the one-loop corrected
$\Delta m^2_{atm}$. The small increase in the one-loop
corrected result compared to the tree level one is essentially due
to the splitting in $m^2_2$ value as shown earlier. The variation
of $\Delta m^2_{solar}$ with $c^4_i/M^2$
can be explained in a similar manner. Obviously, in this case
the one-loop corrected result is smaller compared to the tree
level one (see, figure \ref{numsqNH}). However, one should note
that $\Delta m^2_{solar}$ falls off with $c^4_{\mu}/M^2$ as
opposed to the variation with respect to the other two
gaugino seesaw parameters. This is due to the fact that $m^2_2$
slightly decreases with $c^4_{\mu}/M^2$ but show a slow increase
with respect to $c^4_{e}/M^2$ and $c^4_{\tau}/M^2$. The dark solid
lines in all these figures show the allowed values of various parameters
where all the neutrino mass and mixing constraints are satisfied.
\begin{figure}[ht]
\centering
\includegraphics[width=6.00cm]{FigsPDF/atmsqvsgs1sNH.pdf}
\includegraphics[width=6.00cm]{FigsPDF/solsqvsgs1sNH.pdf}
\vspace{0.2cm}
\includegraphics[width=6.00cm]{FigsPDF/atmsqvsgs2sNH.pdf}
\includegraphics[width=6.00cm]{FigsPDF/solsqvsgs2sNH.pdf}
\vspace{0.2cm}
\includegraphics[width=6.00cm]{FigsPDF/atmsqvsgs3sNH.pdf}
\includegraphics[width=6.00cm]{FigsPDF/solsqvsgs3sNH.pdf}
\caption{Atmospheric and solar mass squared differences $(\Delta
m^2_{atm},~\Delta m^2_{solar})$ vs $\frac{c^4_i}{M^2}$ plots for the
{\it{normal hierarchical}} pattern of light neutrino masses, $i = e,
\mu, \tau$. The full lines are shown for which only the constraints on
$\Delta m^2_{solar}$ is not within the 3$\sigma$ limit (see table \ref{osc-para}).
The dark coloured portions on these lines are the values of parameters
for which all the neutrino constraints are within the 3$\sigma$ limit. The
red (yellow) coloured lines in the plots correspond to the tree (one-loop corrected)
regions where all the constraints except $\Delta m^2_{solar}$ are within 3$\sigma$
allowed region. Parameter choices are shown in tables \ref{loop-param}
and \ref{loop-param-2}.}
\label{gsNH}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=6.00cm]{FigsPDF/atmsqvsos1sNH.pdf}
\includegraphics[width=6.00cm]{FigsPDF/solsqvsos1sNH.pdf}
\vspace{0.2cm}
\includegraphics[width=6.00cm]{FigsPDF/atmsqvsos2sNH.pdf}
\includegraphics[width=6.00cm]{FigsPDF/solsqvsos2sNH.pdf}
\vspace{0.2cm}
\includegraphics[width=6.00cm]{FigsPDF/atmsqvsos3sNH.pdf}
\includegraphics[width=6.00cm]{FigsPDF/solsqvsos3sNH.pdf}
\caption{Atmospheric and solar mass squared differences $(\Delta
m^2_{atm},~\Delta m^2_{solar})$ vs $a^4_i/m^2_{\nu^c}$ plots for the
{\it{normal hierarchical}} pattern of light neutrino masses with $i
= e, \mu, \tau$. Colour specification is same as described
in the context of figure \ref{gsNH}.
Parameter choices are shown in tables \ref{loop-param}
and \ref{loop-param-2}.}
\label{osNH}
\end{figure}
The variation of $\Delta m^2_{atm}$ and $\Delta m^2_{solar}$ with
$a^4_i/m^2_{\nu^c}$ in figure \ref{osNH} can be understood in a
similar way by looking at the right panel plots of
figure \ref{numsqNH}. $\Delta m^2_{atm}$ shows a very little increase
with $a^4_{e,\mu}/m^2_{\nu^c}$ as expected, whereas the change is more
rapid with $a^4_\tau/m^2_{\nu^c}$ for the range of values considered
along the x-axis. As in the case of figure \ref{gsNH}, the solid dark lines
correspond to the allowed values of parameters where all the neutrino
mass and mixing constraints are satisfied.
For higher values of $a^4_{e,\tau}/m^2_{\nu^c}$, $m^2_2$ increases
very slowly with these parameters (see, figure \ref{numsqNH}) and this is
reflected in the right panel plots of figure \ref{osNH}, where
$\Delta m^2_{solar}$ shows a very slow variation with
$a^4_{e,\tau}/m^2_{\nu^c}$. On the other hand, $m^2_2$ increases
more rapidly with $a^4_\mu/m^2_{\nu^c}$, giving rise to a faster
variation of $\Delta m^2_{solar}$. The plots of figure \ref{osNH} show
that larger values of Yukawa couplings are required in order to satisfy
the global three flavour neutrino data, when one considers one-loop
corrected neutrino mass matrix. However, there are allowed ranges of
the parameters $a^4_i/m^2_{\nu^c}$, where the neutrino data can be
satisfied with both tree and one-loop corrected analysis.
We have also considered the variation of light neutrino mass
squared differences with the effective bilinear $R_P$
violating parameter, $\varepsilon_i = Y^{ij} v^c_j$.
For this particular numerical study we vary both
$Y^{ii}_\nu$ and the right-handed sneutrino VEVs $(v^c_i)$
simultaneously, in the suitable ranges around the
values given in tables \ref{loop-param} and \ref{loop-param-2}.
$\Delta m^2_{atm}$ is found to increase
with $\varepsilon_i$, whereas the solar mass squared
difference decreases with increasing $\varepsilon_i$.
The $3\sigma$ allowed region for the solar and atmospheric mass
squared differences were obtained for the lower values of
$\varepsilon_i$s. In addition, we have noticed that the correlations
of $\Delta m^2_{atm}$ with $\varepsilon_i$ is sharper compared to
the correlations seen in the case of $\Delta m^2_{solar}$.
Next let us discuss the dependence of $\Delta m^2_{atm}$ and
$\Delta m^2_{solar}$ on two specific model parameters, $\lambda$ and
$\kappa$, consistent with EWSB conditions. The loop corrections
shift the allowed ranges of $\kappa$ to lower values with some
amount of overlap with the tree level result. On the other hand,
the allowed ranges of $\lambda$ shrinks towards higher values when
one-loop corrections are included. These results are shown in
figure \ref{L-K-NH}. We note in passing that the mass of the
lightest CP-even scalar decreases with increasing $\lambda$. For example,
$\lambda = 0.15$ can produce a lightest scalar mass of $40 ~\rm{GeV}$,
for suitable choices of other parameters. This happens
because with increasing $\lambda$, the lightest scalar state picks up
more and more right-handed sneutrino admixture. This phenomena as
discussed earlier has serious consequence in the mass of the lightest
Higgs boson in $\mu\nu$SSM (see section \ref{munuSSM-scalar} and also
eqn.(\ref{Higgs-mass-NMSSM-lightest-2})).
\begin{figure}[ht]
\centering
\includegraphics[width=6.00cm]{FigsPDF/atmosvskappaNH.pdf}
\includegraphics[width=6.00cm]{FigsPDF/solarvskappaNH.pdf}
\vspace{0.2cm}
\includegraphics[width=6.00cm]{FigsPDF/atmosvslambdaNH.pdf}
\includegraphics[width=6.00cm]{FigsPDF/solarvslambdaNH.pdf}
\caption{Plots showing the variations of $\Delta m^2_{atm},~\Delta
m^2_{solar}$ with model parameters $\lambda$ and $\kappa$ for
{\it{normal hierarchy}}. Colour specification is same as
described in the context of figure \ref{gsNH}.
Parameter choices are shown in tables \ref{loop-param}
and \ref{loop-param-2}.}
\label{L-K-NH}
\end{figure}
Finally, we will discuss the $\rm{tan}\beta$ dependence of $\Delta
m^2_{atm}$ and $\Delta m^2_{solar}$. These plots are shown in figure
\ref{tanbetaNH}. The quantity $\Delta m^2_{atm}$ decreases with the
increasing values of $\rm{tan}\beta$ and nearly saturates for larger
values of $\rm{tan}\beta$. However, the one-loop corrected result
for $\Delta m^2_{atm}$ is not much different from that at the
tree level for a particular value of $\tan\beta$.
On the other hand, the solar mass squared difference initially
increases with $\tan\beta$ and for higher values of $\tan\beta$
the variation slows down and tends to saturate. The one-loop corrections
result in lower values of $\Delta m^2_{solar}$ for a particular $\tan\beta$.
The darker and bigger points on both the plots of figure \ref{tanbetaNH} are
the allowed values of $\tan\beta$, where all the neutrino experimental data
are satisfied. Note that only a very small range of $\tan\beta$ ($\sim$ 10--14)
is allowed. This is a very important observation of this analysis.
\begin{figure}[ht]
\centering
\includegraphics[width=6.00cm]{FigsPDF/atmsqvstanbNH.pdf}
\includegraphics[width=6.00cm]{FigsPDF/solsqvstanbNH.pdf}
\caption{$\Delta m^2_{atm},~\Delta m^2_{solar}$ vs $\rm{tan}\beta$
plots for the {\it{normal hierarchical}} pattern of light neutrino
masses. The allowed values of $\tan\beta$ are shown by bold points.
Other parameter choices are shown in tables \ref{loop-param}
and \ref{loop-param-2}.}
\label{tanbetaNH}
\end{figure}
Next we will discuss the light neutrino mixing and the effect of
one-loop corrections on the mixing angles. It was shown in
ref.\cite{c4Ghosh:2008yh} that for the normal hierarchical pattern of
neutrino masses, when the parameter $b_i \sim a_i$ (see subsection \ref{tree-neut-mass}),
the neutrino mixing angles $\theta_{23}$ and $\theta_{13}$ can be
written as (with the tree level analysis), (see eqns.(\ref{atmos_analytical}),
(\ref{reactor_analytical}))
\begin{eqnarray}
\sin^2\theta_{23} \approx \frac{b^2_\mu}{b^2_\mu + b^2_\tau},
\end{eqnarray}
and
\begin{eqnarray}
\sin^2\theta_{13} \approx \frac{b^2_e}{b^2_\mu + b^2_\tau}.
\end{eqnarray}
On the other hand, the mixing angle $\theta_{12}$ is a much more
complicated function of the parameters $b_i$ and $a_i$ and we do
not show it here. Now, when $b_i \sim a_i$, we can easily
see from eqn.(\ref{specifications}), that
\begin{eqnarray}
v^\prime_i \sim \frac{Y^{ii}_\nu v_1}{3\lambda}(\tan\beta -1).
\end{eqnarray}
This implies that for $\tan\beta \gg$ 1 (recall that the allowed
range of $\tan\beta$ is $\sim$ 10--14),
\begin{eqnarray}
v^\prime_i \gg \frac{Y^{ii}_\nu v_1}{3\lambda}.
\end{eqnarray}
As we have discussed earlier, for such values of $v^\prime_i$,
the quantities $b_i \approx c_i$. Hence, the mixing angles
$\theta_{23}$ and $\theta_{13}$ can be approximately written as
\begin{eqnarray}
\sin^2\theta_{23} \approx \frac{c^2_\mu}{c^2_\mu + c^2_\tau},
\label{eq-mixing-23}
\end{eqnarray}
and
\begin{eqnarray}
\sin^2\theta_{13} \approx \frac{c^2_e}{c^2_\mu + c^2_\tau}.
\label{eq-mixing-13}
\end{eqnarray}
Naively, one would also expect that $\sin^2\theta_{12}$ should
show some correlation with the quantity $c^2_e/c^2_\mu$. However,
as mentioned earlier, this is a very simple minded expectation since
$\sin^2\theta_{12}$ has a more complicated dependence on the model
parameters (see eqn.(\ref{solar_analytical})).
The variation of all three mixing angles with the corresponding
parameters are shown in figure \ref{mixing-GS-OS}. Note that in order
to generate these plots, we vary only the quantities $c_i$ and
all the other parameters are fixed at the values given in
tables \ref{loop-param} and \ref{loop-param-2}. We have chosen the
range of parameters in such a way that the 3-flavour global neutrino
data are satisfied. The mixing angles have been calculated numerically
by diagonalizing the neutrino mass matrix in eqn.(\ref{mnuij-compact1})
and in eqn.(\ref{one-loop corrected structure of neutralino mass matrix}).
As expected from our approximate analytical expressions, these plots
show very nice correlations of the mixing angles $\theta_{23}$ and
$\theta_{13}$ with the relevant parameters as discussed in
eqns.(\ref{eq-mixing-23}) and (\ref{eq-mixing-13}). For example, note
that when $c_\mu \approx c_\tau$, $\sin^2\theta_{23}$ is predicted to
be $\approx$ 0.5 and that is what we observe in the tree level plot in
figure \ref{mixing-GS-OS}. However, when one-loop corrections are
considered, the value of $\sin^2\theta_{23}$ is predicted to be
somewhat on the lower side of the 3$\sigma$ allowed region. This
can be understood by looking at the left panel plots of
figure \ref{gsNH}, where one can see that the one-loop corrected
results prefer lower values of $c^2_\mu$ and higher values of
$c^2_\tau$. Obviously, this gives smaller $\sin^2\theta_{23}$. On the
other hand, the tree level analysis prefers higher values of $c^2_\mu$
and both lower and higher values of $c^2_\tau$. This gives rise to large
as well as small values of $\sin^2\theta_{23}$.
\begin{figure}[ht]
\centering
\includegraphics[width=7.50cm]{FigsPDF/sinsq23GSNH.pdf}
\vspace{0.2cm}
\includegraphics[width=7.50cm]{FigsPDF/sinsq12GSNH.pdf}
\vspace{0.2cm}
\includegraphics[width=7.50cm]{FigsPDF/sinsq13GSNH.pdf}
\caption{Variation of $\rm{sin}^2\theta_{23}$ with
$\frac{c^2_\mu}{(c^2_\mu + c^2_\tau)}$, $\rm{sin}^2\theta_{12}$ with
$\frac{c^2_e}{c^2_\mu}$, $\rm{sin}^2\theta_{13}$ with
$\frac{c^2_e}{(c^2_\mu + c^2_\tau)}$ for normal hierarchy of light
neutrino masses. Parameter choices are shown in tables \ref{loop-param}
and \ref{loop-param-2}.}
\label{mixing-GS-OS}
\end{figure}
If one looks at the plot of $\sin^2\theta_{13}$ in figure \ref{mixing-GS-OS},
then it is evident that the amount of $\nu_e$ flavour in the heaviest state
$(\nu_3)$ decreases a little bit with the inclusion of one-loop corrections
for a fixed value of the quantity $\frac{c^2_e}{(c^2_\mu + c^2_\tau)}$. Very
small $\sin^2\theta_{13}$ demands $c^2_e \ll c^2_\mu, ~c^2_\tau$. This feature
is also consistent with the plots in figure \ref{gsNH}. The correlation of
$\sin^2\theta_{12}$ with the ratio $c^2_e/c^2_\mu$ is not very sharp as expected
from the discussion given above. However, a large $\theta_{12}$ mixing angle
requires a larger value of this ratio. The effect of one-loop correction is
more pronounced in this case and predicts a smaller value of $\sin^2\theta_{12}$
compared to the tree level result. There is no specific correlation of the mixing
angles with the quantities $a^2_i$ and we do not show them here.
\subsection{{\bf I}nverted hierarchy}\label{loop-inverted}
In this subsection we perform a similar numerical analysis for the
inverted hierarchical scheme of three light neutrino masses.
Recall that for the inverted hierarchical pattern of light neutrino
masses, the absolute values of the mass eigenvalues are such that
$m_2 > m_1\gg m_3$. Thus the solar and the atmospheric mass squared
differences are defined as $\Delta m^2_{atm} = m^2_1 - m^2_3$ and
$\Delta m^2_{solar} = m^2_2 - m^2_1$. In order to generate such a mass
pattern, the choices of neutrino Yukawa couplings $Y_\nu^{ii}$ and the
left-handed sneutrino VEVs $v^\prime_i$ are shown in table \ref{loop-param-2}.
However, these are just sample choices and other choices also exist
as we will see during the course of this discussion. The choices of
other parameters are shown in table \ref{loop-param}. The effect of
one-loop corrections to the mass eigenvalues are such that the absolute
values of masses $m_3$ and $m_1$ become smaller whereas $m_2$ grows in
magnitude. This effect of increasing the absolute value of $m_2$ while
decreasing that of $m_1$ makes it extremely difficult to account for the
present 3$\sigma$ limits on $\Delta m^2_{solar}$.
\begin{figure}[ht]
\centering
\includegraphics[width=6.00cm]{FigsPDF/masssqesgs1sIH.pdf}
\includegraphics[width=6.00cm]{FigsPDF/masssqesos1sIH.pdf}
\vspace{0.2cm}
\includegraphics[width=6.00cm]{FigsPDF/masssqesgs2sIH.pdf}
\includegraphics[width=6.00cm]{FigsPDF/masssqesos2sIH.pdf}
\vspace{0.2cm}
\includegraphics[width=6.00cm]{FigsPDF/masssqesgs3sIH.pdf}
\includegraphics[width=6.00cm]{FigsPDF/masssqesos3sIH.pdf}
\caption{Neutrino mass squared values ($m^2_i$) vs $\frac{c^4_i}{M^2}$
(left panel) and vs $\frac{a^4_i}{m^2_{\nu^c}}$ (right panel) plots for
the {\it{inverted hierarchical}} pattern of light neutrino masses, $i = e,
\mu, \tau$. Parameter choices are shown in tables \ref{loop-param}
and \ref{loop-param-2}.}
\label{numsqIH}
\end{figure}
Typical mass spectra are shown in figure \ref{numsqIH}. Once again
note that a particular model parameter has been varied while the others
are fixed at values given in tables \ref{loop-param} and \ref{loop-param-2}. As it
is evident from these plots, the masses $m_1$ and $m_3$ are controlled
mainly by the parameters $a^2_i/m_\nu^c$, whereas the mass $m_2$ is
controlled by the seesaw parameters $c^2_i/M$ though there is a small
contribution coming from $a^2_i/m_\nu^c$ as well.
Let us now turn our attention to the variation of $|\Delta m^2_{atm}|$
and $\Delta m^2_{solar}$ with $c^4_i/M^2$ and $a^4_i/{m^2_{\nu^c}}$
shown in figure \ref{gsIH} and figure \ref{osIH}. For our numerical analysis,
we have set the scale of $m_3$ as $|m_3|_{max}<0.011~\rm{eV}$. The left panel
in figure \ref{gsIH} shows that $|\Delta m^2_{atm}|$ increases
with $c^4_{\mu,\tau}/M^2$ and decreases with $c^4_e/M^2$. This is
essentially the behaviour shown by $m^2_1$ with the variation of
$c^4_i/M^2$. Similar behaviour is obtained for the one-loop corrected
$\Delta m^2_{atm}$. The decrease in the one-loop corrected result
compared to the tree level one is due to the splitting in $m^2_1$
value as shown in figure \ref{numsqIH}.
\begin{figure}[ht]
\centering
\includegraphics[width=6.00cm]{FigsPDF/atmsqvsgs1sIH.pdf}
\includegraphics[width=6.00cm]{FigsPDF/solsqvsgs1sIH.pdf}
\vspace{0.2cm}
\includegraphics[width=6.00cm]{FigsPDF/atmsqvsgs2sIH.pdf}
\includegraphics[width=6.00cm]{FigsPDF/solsqvsgs2sIH.pdf}
\vspace{0.2cm}
\includegraphics[width=6.00cm]{FigsPDF/atmsqvsgs3sIH.pdf}
\includegraphics[width=6.00cm]{FigsPDF/solsqvsgs3sIH.pdf}
\caption{Atmospheric and solar mass squared differences $(|\Delta
m^2_{atm}|,~\Delta m^2_{solar})$ vs $\frac{c_i^4}{M^2}$ plots for
the {\it{inverted hierarchical}} pattern of light neutrino masses
with $i = e, \mu, \tau$. Colour specification is same as described
in the context of figure \ref{gsNH}. Parameter choices are shown in
tables \ref{loop-param} and \ref{loop-param-2}.}
\label{gsIH}
\end{figure}
The variation of $\Delta m^2_{solar}$ with $c^4_i/M^2$ can be understood
in a similar manner by looking at figure \ref{numsqIH}. As explained earlier,
in the case of $\Delta m^2_{solar}$, the one-loop corrected result is larger
compared to the tree level one. The range of parameters satisfying all the
three flavour global neutrino data are shown by the fewer dark points on
the plots. Note that the increase of $\Delta m^2_{solar}$ at the one-loop
level is such that we do not even see any allowed range of parameters when
looking at the variation with respect to $c^4_{e,\tau}/M^2$. Once again, the
behaviour of $\Delta m^2_{atm}$ and $\Delta m^2_{solar}$ with the change
in the parameters $a^4_i/m^2_{\nu^c}$ (shown in figure \ref{osIH}) can be
explained by looking at the right panel plots of figure \ref{numsqIH}.
\begin{figure}[ht]
\centering
\includegraphics[width=6.00cm]{FigsPDF/atmsqvsos1sIH.pdf}
\includegraphics[width=6.00cm]{FigsPDF/solsqvsos1sIH.pdf}
\vspace{0.2cm}
\includegraphics[width=6.00cm]{FigsPDF/atmsqvsos2sIH.pdf}
\includegraphics[width=6.00cm]{FigsPDF/solsqvsos2sIH.pdf}
\vspace{0.2cm}
\includegraphics[width=6.00cm]{FigsPDF/atmsqvsos3sIH.pdf}
\includegraphics[width=6.00cm]{FigsPDF/solsqvsos3sIH.pdf}
\caption{Atmospheric and solar mass squared differences $(|\Delta
m^2_{atm}|,~\Delta m^2_{solar})$ vs $a^4_i/m^2_{\nu^c}$ plots for
the {\it{inverted hierarchical}} pattern of light neutrino masses
with $i = e, \mu, \tau$. Colour specification is same as described
in the context of figure \ref{gsNH}. Parameter choices are shown in
tables \ref{loop-param} and \ref{loop-param-2}.}
\label{osIH}
\end{figure}
We have also investigated the nature of variation of $|\Delta m^2_{atm}|$
and $\Delta m^2_{solar}$ with $\varepsilon^2_i$, the squared effective
bilinear $R_P$-violating parameters. $|\Delta m^2_{atm}|$ was found to
increase with $\varepsilon^2_i$ (the increase is sharper for $\varepsilon^2_1$),
whereas $\Delta m^2_{solar}$ initially increases very sharply with $\varepsilon^2_i$
(particularly for $\varepsilon^2_1$ and $\varepsilon^2_2$) and then becomes
flat. In the one-loop corrected results we do not find any range of values for
parameters where the neutrino data are satisfied. These plots are not shown here.
The variation of mass squared differences with $\lambda$ and
$\kappa$ have also been analyzed. The variation of
$|\Delta m^2_{atm}|$ and $\Delta m^2_{solar}$ with $\lambda$
and $\kappa$ are found to be opposite to those of normal hierarchical
scenario. The one-loop corrected results do not show any allowed ranges
of $\lambda$ and $\kappa$ (for the chosen values of other parameters)
where the neutrino data can be satisfied.
The $\tan\beta$ dependence of $|\Delta m^2_{atm}|$ and $\Delta m^2_{solar}$
is shown in figure \ref{tanbetaIH}. One can see from these two figures that
$|\Delta m^2_{atm}|$ initially increases and then start decreasing at a
value of $\tan\beta$ around $10$. On the other hand, $\Delta m^2_{solar}$
initially decreases and then start increasing around the same value of
$\tan\beta$. Note that the one-loop corrected result for $|\Delta m^2_{atm}|$
is lower than the corresponding tree level result for $\tan\beta < 10$ whereas
the one-loop corrected result for $\Delta m^2_{solar}$ is lower than the
corresponding tree level result for $\tan\beta > 10$. For the chosen values
of other parameters we see that the one-loop corrected analysis does not
provide any value of $\tan\beta$ where the neutrino data can be satisfied.
\begin{figure}[ht]
\centering
\includegraphics[width=6.00cm]{FigsPDF/atmsqvstanbIH.pdf}
\includegraphics[width=6.00cm]{FigsPDF/solsqvstanbIH.pdf}
\caption{$|\Delta m^2_{atm}|,~\Delta m^2_{solar}$ vs $\rm{tan}\beta$
plots for the {\it{inverted hierarchical}} pattern of light neutrino
masses. Colour specification is same as described
in the context of figure \ref{gsNH}.
Parameter choices are shown in tables \ref{loop-param}
and \ref{loop-param-2}.}
\label{tanbetaIH}
\end{figure}
We conclude the discussion on inverted hierarchy by addressing
the dependence of neutrino mixing angles with the relevant parameters.
In figure \ref{mixing-GS-OS-IH-B} we show the variation of the neutrino
mixing angles with the same set of parameters as chosen for the normal
hierarchical scenario. We notice that for inverted hierarchy
the quantity $\sin^2\theta_{23}$ decreases with increasing
$\frac{c^2_\mu}{(c^2_\mu + c^2_\tau)}$ which is just opposite to that
of the normal hierarchy (see, figure \ref{mixing-GS-OS}). Nevertheless, the
correlation of $\sin^2\theta_{23}$ with $\frac{c^2_\mu}{(c^2_\mu + c^2_\tau)}$
is as sharp as in the case of normal hierarchy. A similar feature is obtained
for the variation with $\frac{a^2_\mu}{(a^2_\mu + a^2_\tau)}$.
On the other hand, the correlations of $\sin^2\theta_{12}$ with
$\frac{c^2_e}{c^2_\mu}$ and $\frac{a^2_e}{a^2_\mu}$ and the
correlations of $\sin^2\theta_{13}$ with
$\frac{c^2_e}{(c^2_\mu + c^2_\tau)}$ and
$\frac{a^2_e}{(a^2_\mu + a^2_\tau)}$ are not very sharp and
we do not show them here. There are allowed values of relevant parameters
where all neutrino data can be satisfied. Remember that, for
the plots with $c_i$s, we varied all the $c_i$s simultaneously, keeping
the values of $a_i$s fixed at the ones determined by the parameters in
table \ref{loop-param-2}. Similarly, for the variation of $a_i$s, the quantities
$c_i$s were kept fixed. The inclusion of one-loop corrections restrict the
allowed values of parameter points significantly compared to the tree level
results.
\begin{figure}[ht]
\centering
\includegraphics[width=6.00cm]{FigsPDF/sinsq23GSbIH.pdf}
\includegraphics[width=6.00cm]{FigsPDF/sinsq23OSaIH.pdf}
\caption{Variation of $\sin^2\theta_{23}$ with
$\frac{c^2_\mu}{(c^2_\mu + c^2_\tau)}$ and
$\frac{a^2_\mu}{(a^2_\mu + a^2_\tau)}$
for inverted hierarchy of light neutrino masses.
Parameter choices are shown in tables \ref{loop-param}
and \ref{loop-param-2}.}
\label{mixing-GS-OS-IH-B}
\end{figure}
\subsection{{\bf Q}uasi-degenerate spectra}
\label{Quasi-degenerate-spectra}
The discussion on the light neutrino mass spectrum remains incomplete
without a note on the so-called ``quasi-degenerate'' scenario. A truly
degenerate scenario of three light neutrino masses is, however, inconsistent
with the oscillation data.
Hence, the quasi-degenerate scenario of light neutrino masses is defined
in such a way that in this case all the three individual neutrino masses are
much larger compared to the atmospheric neutrino mass scale. Mathematically,
one writes $m_1 \approx m_2 \approx m_3 \gg \sqrt{|\Delta m^2_{atm}|}$.
Obviously, the oscillation data suggest that even in such a situation there
must be a mild hierarchy among the degenerate neutrinos. It is important to note that
unlike the normal or inverted hierarchical scheme of light neutrino masses, in
the case of quasi-degenerate neutrinos all three neutrinos must be massive
in order to satisfy oscillation data (see table \ref{osc-para}). In the case of
normal or inverted hierarchical neutrino masses it is possible to accommodate
the three flavour neutrino data even with two massive neutrinos.
In this subsection we have shown that the huge parameter space of $\mu\nu$SSM
always leaves us with enough room to accommodate quasi-degenerate spectrum.
For our numerical analysis, we called a set of light neutrino masses to be
quasi-degenerate if the lightest among them is greater than 0.1 eV. We choose
two sets of sample parameter points which are given below in tabular form
(values of other parameters are same as in table \ref{loop-param}). For these
two sets of neutrino Yukawa couplings ($Y_\nu^{ii}$) and the left-handed
sneutrino VEVs ($v^\prime_i$) we observe the following patterns of light
neutrino masses at the tree level
\noindent (i) Quasi-degenerate-I: $m_3 \gtrsim m_2 \gtrsim m_1 \gg
\sqrt{|\Delta m^2_{atm}|}$
\\
(ii) Quasi-degenerate-II: $m_2 \gtrsim m_1 \gtrsim m_3 \gg \sqrt{|\Delta m^2_{atm}|}.$
\noindent For case (i), we have varied the parameters around the values in
table \ref{loop-param-2} and identified a few extremely fine-tuned points in the
parameter space where either the tree level or the one-loop corrected result
is consistent with the three flavour global neutrino data. Two representative
spectrum as function of $\frac{c^4_e}{M^2}$ and $\frac{a^4_e}{m^2_{\nu^c}}$
are shown in figure \ref{numsqQD}. The mass spectrum for Quasi-degenerate-I case is
analogous to a normal hierarchical scenario whereas that for Quasi-degenerate-II
resembles a inverted spectrum.
\begin{figure}[ht]
\centering
\includegraphics[width=6.00cm]{FigsPDF/masssqesgs1sQD.pdf}
\includegraphics[width=6.00cm]{FigsPDF/masssqesos1sQD.pdf}
\caption{Neutrino mass squared values ($m^2_i$) vs $\frac{c^4_e}{M^2}$
(left panel) and vs $\frac{a^4_e}{m^2_{\nu^c}}$ (right panel) plots for
the {\it{quasi-degenerate}} pattern of light neutrino masses.
Parameter choices are shown in tables \ref{loop-param}
and \ref{loop-param-2}.}
\label{numsqQD}
\end{figure}
As mentioned earlier, one can play with the model parameters and
obtain a spectrum with a different ordering of masses
termed as ``Quasi-degenerate-II" in table \ref{loop-param-2}. However, for
such an ordering of masses, we found that it was rather impossible
to find any region of parameter space where the one-loop corrected result
satisfies all the constraints on neutrino masses and mixing.
Nevertheless, we must emphasize here that it is not a
completely generic conclusion and for other choices of soft SUSY breaking
and other parameters it could be possible to have a spectrum like that
shown in ``Quasi degenerate II" with neutrino constraints satisfied
even at the one-loop level. On the other hand, there exist regions where
neutrino data are satisfied at the tree level with this ordering of masses.
\section{{\bf S}ummary}\label{com-ana-neut}
So in a nutshell in $\mu\nu$SSM it is possible to account for three flavour global
neutrino data itself at the tree level even with the choice of flavour diagonal
neutrino Yukawa couplings. Besides, different hierarchical (normal, inverted, quasi-degenerate)
scheme of light neutrino mass can be accommodated by playing with the hierarchy
in Yukawa couplings. The tree level results of neutrino masses and mixing show appreciable
variation with the inclusion of the one-loop radiative corrections, depending on
the light neutrino mass hierarchy.
\vspace{2cm}
It seems so far that the $\mu\nu$SSM is extremely successful in accommodating massive
neutrinos both with tree level and one-loop combined analysis, consistent with the three
flavour global data (see table \ref{osc-para}). But how to test these neutrino physics
information in a collider experiment, which can give additional checks for the $\mu\nu$SSM
model? Fortunately for us certain ratios of the decay branching ratios of the lightest
neutralino (which is also the LSP for a large region of the parameter space) show
nice correlations with certain light neutrino mixing angle \cite{c4Ghosh:2008yh,c4Bartl:2009an}.
These correlations could act as excellent probes to the $\mu\nu$SSM model in the ongoing era of
the colliders. These issues will be considered in details in the next chapter.
\chapter*{\sffamily{{\bf }}}
\thispagestyle{empty}
\vspace{7cm}
\begin{center}
\sffamily
{\bf{ {{\it To a teacher who is like the pole star to me and many others}}}}\\
\vspace*{2cm}
\begin{flushright}
{\it{ {\Huge {Dr. Ranjan Ray}}}} \\
\vspace*{0.5cm}
{\it{ {{1949 - 2001}}}}
\end{flushright}
\end{center}
\chapter{ \sffamily{{\bf $\mu$}$\nu$SSM: decay of the LSP
}}\label{munuSSM-LSP}
\section{{\bf A} decaying LSP}\label{LSP-dec-1}
We have learned already in section \ref{R-parity} that
the lightest supersymmetric particle (LSP) is absolutely stable so long
as $R_p$ is conserved. Besides, as argued in section
\ref{corre} that the LSP has to be charge and colour neutral \cite{c5Wolfram:1978gp,
c5Dover:1979sn,c5Ellis:1983ew} so long it preserves its stability. Consequently,
only the electrically neutral colourless sparticles remain to be the only possible
choice for the LSP. Interestingly,
when $R_p$ is broken (figure \ref{Rpc-Rpv}, see also section \ref{R-parity}),
any sparticle (the lightest neutralino, chargino \cite{c5Feng:1999fu},
squark, gluino \cite{c5Raby:1997bpa,c5Baer:1998pg,c5Raby:1998xr}, sneutrino
\cite{c5Hagelin:1984wv}, (see also ref.\cite{c5Ellis:1983ew}))
can be the LSP. In a supersymmetric model with broken $R_p$
the LSP will decay into further lighter states namely, into the SM particles.
Apart from the neutrinos rest of these decay products are easily detectable in a collider
experiment and thus can act as a potential probe for the underlying model.
Since $\mu\nu$SSM is an $R_p$-violating supersymmetric model, the LSP
for this model is also unstable and can yield striking signatures at the collider
which we aim to discuss in this chapter.
This remarkable feature
is absent in the conventional $R_p$ conserving supersymmetric models,
where any sparticle decay ends with LSP in the final state and hence
yield large missing energy signatures.
For example if the lightest neutralino $(\ntrl1)$ is the LSP then the following
two and three body decay modes are kinematically possible
\begin{eqnarray}
\ntrl1 &\to& W^{\pm} \ell^{\mp},~Z^0 \nu_k,
~h^0\nu_k,\nonumber\\
&\to&b \bar{b}\nu_k,~\ell^+_i \ell^-_j\nu_k,
~q_i\bar{q_i}\nu_k,~q_i\bar{q}^{\prime}_j\ell^{\mp}_k,~\nu_i\bar{\nu}_j\nu_k.
\label{2-3-body-decays}
\end{eqnarray}
The lightest neutralino $(\ntrl1)$ can be the LSP in a large
region of the parameter space. The three body
decay modes become dominant when mass of the LSP $(m_{\ntrl1})$ is less than that of
the $W$-boson $(m_W)$.
The corresponding Feynman diagrams are given in appendix \ref{appenG}, section \ref{LN-decay}
(figures \ref{LSP-decay2}, \ref{LSP-decay3}).
It is also interesting to note that apart
from these tree level two and three body decays the LSP can also decay into a neutrino
and a photon radiatively \cite{c5Hall:1983id,c5Dawson:1985vr,c5Hempfling:1997je,
c5Mukhopadhyaya:1999gy}.
One more important aspect in the decays of the lightest supersymmetric particle
through $R_p$-violating channel is the appearance of the displaced vertices
\cite{c5Mukhopadhyaya:1998xj,c5Chun:1998ub,c5Choi:1999tq,c5Porod:2000hv,c5DeCampos:2010yu}.
The displaced vertices appear to be macroscopic $(\sim \rm{a~few~mm}$ or larger)
due to the smallness of the associated $R_p$-violating couplings.
A displaced vertex is defined as the distance traversed by a neutral particle
between the primary and the secondary interaction points.
The displaced vertices are extremely useful to remove undesired
backgrounds in case of a collider analysis. The length of the displaced vertices
also vary with the nature of the lightest neutralino or the LSP.
Thus, before proceeding further it is important to discuss about the various LSP
scenario in $\mu\nu$SSM.
We note in passing that in this chapter we concentrate on the two-body decays
only and in the next chapter we will discuss about the three body decays.
\section{{\bf D}ifferent LSP scenarios in $\mu\nu$SSM}\label{lsp-nat}
In the $\mu\nu$SSM the neutralino sector is highly enriched compared to that of the MSSM
due to $R_p$-violating mixing of the MSSM neutralinos with the three generations of left-handed and
right-handed neutrinos. So mathematically in $\mu\nu$SSM with gaugino mass unification
at the GUT scale (that is at the electroweak scale $M_2=2 M_1$), possible LSP natures are described by
\vspace{0.1cm}
\noindent
1.~$\chi^0_1 \approx \mathbf N_{11} \widetilde B^0,~~|\mathbf N_{11}|^2 \sim 1 \Rrightarrow$ bino like $\ntrl1$.
\vspace{0.1cm}
\noindent
2.~$\chi^0_1 \approx \mathbf N_{13} \widetilde H^0_d + \mathbf N_{14} \widetilde H^0_u,
~~|\mathbf N_{13}|^2 + |\mathbf N_{14}|^2 \sim 1 \Rrightarrow$ higgsino like $\ntrl1$.
\vspace{0.1cm}
\noindent
3.~$\chi^0_1 \approx \sum \mathbf N_{i,\alpha+4} \nu^c_\alpha,
~~|\mathbf N_{15}|^2 + |\mathbf N_{16}|^2 + |\mathbf N_{17}|^2 \sim 1
\Rrightarrow$ right-handed neutrino $(\nu^c)$ like $\ntrl1$.
In terms of the model ingredients the LSP nature in $\mu\nu$SSM
depends on the relative dominance of three parameters, (1) the $U(1)$ gaugino
soft mass $M_1$ (see eqn.(\ref{munuSSM-soft})), (2) the higgsino mass parameter or
the $\mu$-term ($=3\lambda v^c$) (see eqns.(\ref{Abbrevations}), (\ref{assumption1}))
and (3) the right-chiral neutrino Majorana mass term, $m_{\nu^c}$ ($=2\kappa v^c$)
(using eqn.(\ref{assumption1}), see eqn.(\ref{specifications-2})) \cite{c5Ghosh:2008yh,
c5Bartl:2009an,c5Bandyopadhyay:2010cu}. Thus we can write
\vspace{0.1cm}
\noindent
I.~$\mu$,~$m_{\nu^c}~>~M_1$ $\Longrightarrow$ LSP bino (gaugino) like.
\vspace{0.1cm}
\noindent
II.~$M_1$,~$m_{\nu^c}~>~\mu$ $\Longrightarrow$ LSP higgsino like.
\vspace{0.1cm}
\noindent
III.~$M_1$,~$\mu~>~m_{\nu^c}$ $\Longrightarrow$ LSP right-handed neutrino
like. Since right-handed neutrinos are singlet under the SM gauge group,
a right-handed neutrino like LSP is often called a ``{\it{singlino}}'' LSP.
It is important to mention that the right sneutrinos $(\widetilde \nu^c)$ are also eligible
candidate for the LSP in $\mu\nu$SSM \cite{c5Ghosh:2008yh,c5Chang:2009dh}. Also as a continuation
of the discussion of the last section, the length of the displaced vertices
can vary from a few mm to a few cm for a bino like LSP to a higgsino like LSP
\cite{c5Ghosh:2008yh,c5Bartl:2009an}.
On the other hand, for a singlino LSP the length of the displaced vertices
can be as large as a few meters \cite{c5Bartl:2009an,c5Bandyopadhyay:2010cu}. None of
these are unexpected since a bino like LSP, being a gaugino, has gauge interactions
and the gauge couplings are $\sim$ $\cal{O}$ $(1)$ couplings whereas a higgsino
like LSP involves smaller Yukawa couplings which is responsible for a smaller decay width
and consequently a larger ($\sim$ a few cm) displaced vertices. A singlino LSP on the other hand
is mostly a gauge singlet fermion by nature and thus couples to other particles via very small
$R_p$-violating couplings, which finally yield a large displaced vertex.
\section{{\bf D}ecays of the lightest neutralino in $\mu\nu$SSM}\label{lsp-decay-calc}
In this section we aim to calculate a few tree level two-body decays of
the lightest neutralino $\ntrl1$ in $\mu\nu$SSM model \cite{c5Ghosh:2008yh}. As stated earlier
we denote the lightest neutralino as $\ntrl1$ when the seven neutralino
masses (see eqn.(\ref{neutralino_mass_eigenstate_matrixform}))
are arranged in the increasing order of magnitude ($\ntrl1$ being the
lightest and $\ntrl7$ being the heaviest).
However, for this chapter from now on, we follow the convention of ref. \cite{c5Ghosh:2008yh}
where the eigenvalues are arranged in reverse order so that $\ntrl7$ denotes
the lightest neutralino. The lightest neutralino considered here is
either the LSP or the next-to LSP (NLSP). The lightest neutralino
mass is set to be more than $m_W$ such that two-body decays dominate.
Two-body and three-body decays of the LSP in $\mu\nu$SSM has been
discussed in a recent ref. \cite{c5Bartl:2009an} with one generation
of right handed neutrino superfield.
Three-body decays of a singlino like lightest neutralino (which is also the
LSP) for $\mu\nu$SSM also has been addressed in ref. \cite{c5Bandyopadhyay:2010cu}.
In this section we mainly concentrate on the two-body decays like
\begin{eqnarray}\label{general_2body-decay}
&&\ntrl7 \longrightarrow W^\pm + \ell^\mp_k\\
&&\ntrl7 \longrightarrow Z + \nu_k,\nonumber,
\end{eqnarray}
where $k=1,2,3\equiv e,\mu,\tau$. The required Feynman rules
are given in appendix {\ref{appenD}}. Let us also remark that the lightest neutralino
can also decay to $h^0 + \nu_k$, if it is kinematically allowed, where $h^0$ is
the MSSM-like lightest Higgs boson (this is true if the amount of admixture
of the MSSM Higgses with the right-handed sneutrinos are very small).
However, for our illustration purposes we
have considered the mass of the lightest neutralino in such a way that this
decay is either kinematically forbidden or very much suppressed (assuming a
lower bound on the mass of $h$ to be 114 GeV). Even if this decay branching
ratio is slightly larger, it is usually smaller than the branching ratios in
the ($\ell^\pm_i + W^\mp$) channel.
Hence, this will not affect
our conclusions regarding the ratios of branching ratios in the charged lepton channel
($\ell_i + W$), to be discussed later. The lightest neutralino decay
${\widetilde \chi}^0_7 \rightarrow \nu + {\widetilde \nu}^c$, where ${\widetilde \nu}^c$
is the scalar partner of the gauge singlet neutrino $\nu^c$, is always very
suppressed.
We will discuss more on this when we consider a $\nu^c$
dominated lightest neutralino in subsection \ref{rhnu-LN}.
Consider the following decay process
\begin{equation}\label{general_decay}
\widetilde{\chi}_i \longrightarrow \widetilde{\chi}_j + V,
\end{equation}
where $\widetilde{\chi}_{i(j)}$ is either a neutralino\footnote{Remember that the neutrinos are
also a part of the extended neutralino matrix (eqn.(\ref{neutralino_basis})).}
or chargino, with mass $m_{i(j)}$ and $V$ is the gauge boson which is either $W^{\pm}$
or $Z$, with mass $m_v$. The masses $m_i$ and $m_j$ are positive.
The decay width for this process in eqn.(\ref{general_decay}) is given
by \cite{c5Gunion:1987yh,c5Franke:1995tf,c5Franke:1995tc}
\begin{equation}\label{general_decay_width_formula}
\Gamma \left(\widetilde{\chi}_i \longrightarrow \widetilde{\chi}_j + V\right) =
\frac{g^2 \mathcal{K}^{1/2}}{32 ~\pi m^3_i m^2_W} \times \left\{\left( G^2_L
+ G^2_R \right)\mathcal{F} - G^*_L G_R
~\mathcal{G} \right \},
\end{equation}
where $\mathcal{F}$, $\mathcal{G}$ are functions of $m_i, m_j, m_v$ and
given by
\begin{eqnarray}\label{general_decay_width_formula_specifications}
& & \mathcal{F}(m_i, m_j, m_v) = \mathcal{K} + 3 ~m^2_v \left( m^2_i + m^2_j
- m^2_v \right), \nonumber \\
& & \mathcal{G}(m_i, m_j, m_v) = 12 ~\epsilon_i \epsilon_j m_i m_j m^2_v,
\end{eqnarray}
with $\epsilon_i(j)$ carrying the actual signs $(\pm 1)$ of the
neutralino masses \cite{c5Gunion:1984yn}. The chargino masses must be positive.
The kinematical factor $\mathcal{K}$ is given by
\begin{equation}\label{kinematical_factor}
\mathcal{K}(m^2_i, m^2_j, m^2_v) = \left( m^2_i + m^2_j - m^2_v \right)^2
- 4 ~m^2_i m^2_j.
\end{equation}
In order to derive eqn.(\ref{general_decay_width_formula}), we have used
the relation $m^2_W = m^2_Z \cos^2\theta_W$ and since $v'_i << v_1, v_2$,
some of the MSSM relations still hold good.
The factors $G_L$, $G_R$ are given here for some possible decay modes
\begin{eqnarray}\label{some_possible_decay_modes}
& &\text{For} \quad \widetilde{\chi}^0_i \longrightarrow \widetilde{\chi}^0_j Z
\nonumber \\
& & G_L = O^{\prime \prime L}_{ji}, \quad G_R = O^{\prime \prime R}_{ji},
\nonumber \\
\quad \
& &\text{For} \quad \widetilde{\chi}^0_i \longrightarrow \widetilde{\chi}^+_j {W^-}
\nonumber \\
& & G_L = O^{L}_{ij}, \quad G_R = O^{R}_{ij},
\end{eqnarray}
where $O^{\prime \prime L(R)}_{ji}$ and $O^{L(R)}_{ij}$ are given by
(using eqns.(\ref{nnz-L-R-couplings}),(\ref{ncw-L-R-couplings}) without
the sign factors $\epsilon_i,\eta_j$)
\begin{eqnarray}\label{O-Op-coups}
O^{{\prime\prime}^L}_{ij} &=& - \frac{1}{2} \mathbf N_{i3} \mathbf N^*_{j3} + \frac{1}{2}
\mathbf N_{i4} \mathbf N^*_{j4} - \frac{1}{2} \mathbf N_{i,k+7} \mathbf N^*_{j,k+7}, \nonumber \\
O^{{\prime\prime}^R}_{ij} &=& - {O^{{\prime\prime}^L}_{ij}}^*, k = 1,2,3, \nonumber \\
O^L_{ij} &=& \mathbf N_{i2} \mathbf V^*_{j1} - \frac{1}{\sqrt{2}} \mathbf N_{i4} \mathbf V^*_{j2},
\nonumber \\
O^R_{ij} &=& \mathbf N^*_{i2} \mathbf U_{j1} + \frac{1}{\sqrt{2}} \mathbf N^*_{i3} \mathbf U_{j2} +
\frac{1}{\sqrt{2}} \mathbf N^*_{i,k+7} \mathbf U_{j,k+2}.
\end{eqnarray}
Now consider the decays shown in eqn.(\ref{general_2body-decay}). At this stage let us
discuss our notation and convention for calculating these decays \cite{c5Ghosh:2008yh}.
The neutralino mass matrix is a 10$\times$10 mass matrix which includes three generations of
the left-handed as well as the gauge-singlet neutrinos (eqns.(\ref{neutralino_7x7}),
(\ref{neutralino_3x7})). If the mass eigenvalues of this
matrix are arranged in the descending order then the three lightest eigenvalues of
this 10$\times$10 neutralino mass matrix would correspond to the three light
neutrinos. Out of the remaining seven heavy eigenvalues, the lightest one
is denoted as the lightest neutralino. Thus, as argued earlier in our notation
$\widetilde{\chi}^0_7$ is the lightest neutralino (LN) and
$\widetilde{\chi}^0_{j+7}, \text{where}$ $j=1,2,3$ correspond to the three light
neutrinos \cite{c5Ghosh:2008yh}. Similarly, for the chargino masses, $\widetilde{\chi}^{\pm}_{l+2}$
($l=1,2,3$) corresponds to the charged leptons $e, \mu, \tau$. Immediately,
with this choice, we can write down different natures of the lightest neutralino as
\vspace{0.1cm}
\noindent
A.~$\chi^0_7 \approx \mathbf N_{71} \widetilde B^0,~~|\mathbf N_{71}|^2 \sim 1 \Rrightarrow$ bino like LN.
\vspace{0.1cm}
\noindent
B.~$\chi^0_7 \approx \mathbf N_{73} \widetilde H^0_d + \mathbf N_{74} \widetilde H^0_u,
~~|\mathbf N_{73}|^2 + |\mathbf N_{74}|^2 \sim 1 \Rrightarrow$ higgsino like LN.
\vspace{0.1cm}
\noindent
C.~$\chi^0_7 \approx \sum \mathbf N_{7,\alpha+4} \nu^c_\alpha,
~~|\mathbf N_{75}|^2 + |\mathbf N_{76}|^2 + |\mathbf N_{77}|^2 \sim 1
\Rrightarrow$ $\nu^c$ like LN.
\vspace{0.1cm}
So for $\widetilde{\chi}^0_{LN} \rightarrow Z + \nu_k$, which is also equivalent
to $\widetilde{\chi}^0_7 \rightarrow Z + \widetilde{\chi}^0_{j+7}$ ($j = 1,~2,~3 $),
one gets from eqn.(\ref{some_possible_decay_modes}) and
eqn.(\ref{O-Op-coups})
\begin{eqnarray}\label{G_L-&-G_R-for-first-of-two-selected-decays}
& & G_L = - \frac{1}{2} \mathbf N_{j+7,3} \mathbf N^*_{73} + \frac{1}{2} \mathbf N_{j+7,4} \mathbf N^*_{74}
- \frac{1}{2} \mathbf N_{j+7,k+7} \mathbf N^*_{7,k+7}, \nonumber \\
& & G_R = -G^*_L,
\end{eqnarray}
where $~j,k = 1,~2,~3$ and this in turn modifies
eqn.(\ref{general_decay_width_formula}) as
\begin{equation}\label{decay-width-for-process-1}
\Gamma \left(\widetilde{\chi}^0_7 \rightarrow Z + ~\widetilde{\chi}^0_{j+7}
\right) = \frac{g^2 \mathcal{K}^{1/2}}{32 ~\pi m^3_{{\widetilde \chi}^0_7}
m^2_W} \times \left\{ 2 ~G^2_L \mathcal{F} + G^{*^2}_L ~\mathcal{G} \right \},
\end{equation}
with $m_i = m_{{\widetilde \chi}^0_7}$, $m_j = m_{\nu_j} \approx 0$
(eqn.(\ref{neutralino_mass_eigenstate_matrixform})) and $m_v = m_Z$.
Let us now consider the other decay which is $\widetilde{\chi}^0_{LN} \rightarrow
W^{\pm} + \ell^{\mp}$ or equivalently $\widetilde{\chi}^0_7 \rightarrow
W^{\pm} + \widetilde{\chi}^{\mp}_j (j=3,4,5)$.
For the process $\widetilde{\chi}^0_7 \rightarrow W^{-} + \widetilde{\chi}^{+}_j$
\begin{eqnarray}\label{decay-width-for-process-2-a}
\Gamma \left(\widetilde{\chi}^0_7 \rightarrow W^-
+ ~\widetilde{\chi}^+_j\right) &=& \frac{g^2 \mathcal{K}^{1/2}}{32 ~\pi
m^3_{{\widetilde \chi}^0_7} m^2_W} \times \left\{ \left(G^2_L + G^2_R \right) \mathcal{F} -G^*_L G_R
~\mathcal{G} \right \},\nonumber \\
G_L &=& \mathbf N_{72} \mathbf V^*_{j1} - \frac{1}{\sqrt{2}} \mathbf N_{74} \mathbf V^*_{j2},\nonumber \\
G_R &=& \mathbf N^*_{72} \mathbf U_{j1} + \frac{1}{\sqrt{2}} \mathbf N^*_{73} \mathbf U_{j2}
+ \frac{1}{\sqrt{2}} \mathbf N^*_{7,k+7} \mathbf U_{j,k+2},\nonumber \\
(k &=& 1,~2,~3),
\end{eqnarray}
where eqn.(\ref{some_possible_decay_modes}) and
eqn.(\ref{O-Op-coups})
have been used. The process $\widetilde{\chi}^0_7 \longrightarrow W^{+}
+ \widetilde{\chi}^{-}_j$ is obtained by charge conjugation of the process
in eqn.(\ref{decay-width-for-process-2-a}).
Note that the neutralino mixing
matrix $\mathbf N$ contains the expansion parameter $\xi$ (eqn.(\ref{expansion-parameter}))
which as shown in appendix \ref{appenC} can be expressed as a function
of the quantities $a_i,b_i,c_i$ (eqn.(\ref{specifications})). On the other
hand as shown in eqns.(\ref{decay-width-for-process-1}), (\ref{decay-width-for-process-2-a})
the decay widths (for $\ntrl7 \to Z+ \nu_j$ and $\ntrl7 \to W^\pm+ \ell^\mp_j$)
contain quadratic power of $\mathbf N$, that is, these decay widths are quadratic in
$\xi$ or even more precisely quadratic in $a_i,b_i,c_i$. This information will
be explored further in the next section.
\section{{\bf L}ight neutrino mixing and the neutralino decay}\label{correlation}
In $\mu\nu$SSM, the light neutrino mixing angles are expressible
in terms of the parameters $a_i,b_i,c_i$ (see eqn.(\ref{specifications})).
These relations were also verified numerically, as shown in figures
\ref{neut2313-mixing-tree}, \ref{neut12-mixing-tree}, \ref{neut23a-mixing-tree}.
Now it has been already argued in the last section that the two-body
decays of the lightest neutralino are also quadratic in $a_i,b_i,c_i$ parameters. Combining
these two pictures we found that in $\mu\nu$SSM the light neutrino mixing angles are
correlated with the lightest neutralino (or LSP) decays, to be more precise with the ratios of the
decay branching ratio $(Br)$ \cite{c5Ghosh:2008yh}.
These correlations are well studied in the context of the $R_p$-violating supersymmetric
model of light neutrino mass generation \cite{c5Mukhopadhyaya:1998xj,c5Chun:1998ub,c5Choi:1999tq,
c5Porod:2000hv,c5DeCampos:2010yu}.
Nevertheless, one should note certain differences in these two cases. In $\mu\nu$SSM
lepton number is broken explicitly in the superpotential by terms which are trilinear
as well as linear in singlet neutrino superfields. In addition to that there are
lepton number conserving terms involving the singlet neutrino superfields with dimensionless neutrino
Yukawa couplings. After the electroweak symmetry breaking these terms can generate the effective bilinear
R-parity violating terms as well as the $\Delta L$ =2 Majorana mass terms for the singlet neutrinos in the
superpotential. In general, there are corresponding soft supersymmetry breaking terms in the scalar potential.
Thus the parameter space of this model is much larger compared to the bilinear $R_p$ violating model.
Hence, in general, one would not expect a very tight correlation between the neutrino mixing angles and the
ratios of decay branching ratios of the LSP. However, under certain simplifying assumptions
\cite{c5Ghosh:2008yh}, one can reduce the number of free parameters and in those cases it is possible that the above
correlations reappear. This issue has been studied in great detail for the two body $\ell^\pm-W^\mp$
final states in ref. \cite{c5Ghosh:2008yh} and for all possible two and three body final states
in ref. \cite{c5Bartl:2009an}. Let us note in passing that such a nice correlation is lost in
the general scenario of bilinear-plus-trilinear
R-parity violation \cite{c5Choi:1999tq}.
Another important difference between $\mu\nu$SSM and the bilinear R-parity violating model in the context of
the decay of the LSP (assumed to be the lightest neutralino in this case) is that in $\mu\nu$SSM the lightest
neutralino can have a significant singlet neutrino ($\nu^c$) contribution. In this case, the correlation between
neutrino mixing angles and decay branching ratios of the LSP is different \cite{c5Ghosh:2008yh,c5Bartl:2009an} compared
to the cases when the dominant component of the LSP is either a bino, or a higgsino or a Wino. This gives us a
possibility of distinguishing between different R-parity violating models through the observation of the
decay branching ratios of the LSP in collider experiments \cite{c5Ghosh:2008yh,c5Bartl:2009an}. In addition, the decay of
the lightest neutralino will show displaced vertices in collider experiments and when the lightest neutralino
is predominantly a singlet neutrino, the decay length can be of the order of several meters for a lightest
neutralino mass in the neighbourhood of 50 GeV \cite{c5Bartl:2009an}. This is very different from the bilinear R-parity
violating model where for a Bino LSP of similar mass the decay length is less than or of the order of a meter
or so \cite{c5Porod:2000hv}.
In references \cite{c5Mukhopadhyaya:1998xj,c5Porod:2000hv,c5Chun:2002rh}
this correlation was studied for a bino like neutralino LSP. However, the correlations
appear for other natures of the lightest supersymmetric particle as well
\cite{c5Hirsch:2002ys,c5Hirsch:2003fe,c5Aristizabal-Sierra:2004cy}.
These inter-relations reflects
the predictive power of a model where the light neutrino mass generation as well as
the lightest neutralino/LSP decays are governed by a common set of small number of parameters.
These correlations are also addressed in a recent review \cite{c5Nath:2010zj}.
So in conclusion, with the help of these nice correlations neutrino mixing angles
can be indirectly measured in colliders by comparing the branching ratios of
the lightest neutralino or the LSP decay modes.
We observe that the correlations between the lightest neutralino decays and neutrino mixing angles
depend on the nature of the lightest neutralino as well as on the
mass hierarchies of the neutrinos, i.e. whether we have a normal
hierarchical pattern of neutrino masses or an inverted one \cite{c5Ghosh:2008yh}.
In this section we look into these possibilities in details with three
different natures of the lightest neutralino. We
consider that the lightest neutralino to be either (1) bino dominated or
(2) higgsino dominated or (3) right-handed neutrino dominated. For each of these
cases we consider both the normal and the inverted hierarchical pattern
of neutrino masses. In the case of a bino or a higgsino like lightest neutralino,
they are also the LSP but for a right-handed neutrino dominated lightest
neutralino it is the NLSP with right handed sneutrino as the LSP \cite{c5Ghosh:2008yh}.
The possibility for a right-handed neutrino or singlino like lightest neutralino
LSP has also been addressed in references \cite{c5Bartl:2009an,c5Bandyopadhyay:2010cu}.
We show that for the different natures of the lightest neutralino, the
ratio of branching ratios of certain decays of the lightest neutralino
correlates with certain neutrino mixing angle. In some cases
the correlation is with the atmospheric angle $(\theta_{23})$ and the reactor angle $(\theta_{13})$
and in other cases the ratio of the branching ratios correlates with the solar mixing
angle $(\theta_{12})$. Nevertheless, there also exists scenarios with no correlations at all. Let us
now study these possibilities case by case \cite{c5Ghosh:2008yh} in three subsequent subsections.
As already mentioned, that the interesting difference between this study and similar studies with $R_p$
violating scenario \cite{c5Mukhopadhyaya:1998xj,c5Chun:1998ub,c5Choi:1999tq,c5Romao:1999up,
c5Porod:2000hv} in the MSSM is the presence of a gauge singlet neutrino dominated lightest
neutralino. We will see later that in this case the results can be very different from
a bino or higgsino dominated lightest neutralino. The lightest neutralino
decays in neutrino mass models with spontaneous R-parity violation have been
studied in ref.\cite{c5Hirsch:2008ur}. Our parameter choices for the next three
subsections are consistent with the constraints of the scalar sector
(section \ref{munuSSM-scalar}).
\subsection{{\bf B}ino dominated lightest neutralino}\label{bino-LN}
According to our choice, at the EW scale the ratio of the $U(1)$ and $SU(2)$
gaugino masses are $M_1:M_2 = 1:2$. If in addition, $M_1<\mu$ and the value of
$\kappa$ is large (so that the effective gauge singlet neutrino mass
$2 \kappa v^c$ is large), the lightest neutralino is essentially bino
dominated and it is the LSP. First we consider the case when the composition
of the lightest neutralino is such that, the bino-component $|N_{71}|^2>$ 0.92
and neutrino masses follow the normal hierarchical pattern. We have observed
that for the bino dominated case, the lightest neutralino
(${\tilde \chi}^0_7$) couplings to $\ell^{\pm}$--$W^{\mp}$ pair
(where $\ell = e, \mu$ or $\tau$) depend on the
quantities $b_i$ along with a factor which is independent of various lepton
generations. Naturally, we would expect that the ratios of various decay
branching ratios such as BR(${\tilde \chi}^0_7 \rightarrow e + W$),
BR(${\tilde \chi}^0_7 \rightarrow \mu + W$), and
BR(${\tilde \chi}^0_7 \rightarrow \tau + W$) show nice correlations with the
quantities $b^2_i/b^2_j$ with $i,j$ being $e, \mu$ or $\tau$. This feature is
evident from figure \ref{nor-bino-bmbebt}. Here we have scanned the parameter
space of the three neutrino Yukawa couplings with random values for a
particular choice of the couplings $\lambda$, $\kappa$ and the associated soft
SUSY breaking trilinear parameters, as well as other MSSM parameters. The
trilinear soft parameters $A_\nu$ corresponding to $Y_\nu$s also vary
randomly in a certain range. In addition we have imposed the condition that
the lightest neutralino (which is the LSP) is bino dominated and neutrino mass
pattern is normal hierarchical.
\begin{figure}[ht]
\centering
\vspace*{0.4cm}
\includegraphics[height=4.00cm]{FigsPDF/nnBDBRmt.pdf}
\includegraphics[height=4.00cm]{FigsPDF/nnBDBRet.pdf}
\includegraphics[height=4.00cm]{FigsPDF/nnBDBRem.pdf}
\caption{Ratio $\frac{Br(\chi^0_7 \longrightarrow \ell_i~W)}{Br(\chi^0_7
\longrightarrow \ell_j~W)}$ versus $\frac{b^2_i}{b^2_j}$ plot for a bino like
lightest neutralino (the LSP) with bino component, $|N_{71}|^2>$~0.92, where
$i,j,k~=~e,\mu,\tau$. Neutrino mass pattern is taken to be normal
hierarchical. Choice of parameters are $M_1=110$ GeV, $\lambda=0.13,
\kappa=0.65, m_{\tilde{\nu}^c}=300~{\rm GeV} ~{\rm and} ~m_{\tilde{L}}=400
~{\rm GeV}$. Mass of the LSP is $106.9$ GeV.
The value of the $\mu$ parameter comes out to be $-228.9$ GeV.}
\label{nor-bino-bmbebt}
\end{figure}
\begin{figure}[ht]
\centering
\vspace*{0.4cm}
\includegraphics[height=4.00cm]{FigsPDF/Tsq23BDnn.pdf}
\includegraphics[height=4.00cm]{FigsPDF/Tsq13BDnn.pdf}
\caption{Ratio $\frac{Br(\chi^0_7 \longrightarrow \mu~W)}{Br(\chi^0_7
\longrightarrow \tau~W)}$ versus $\tan^2\theta_{23}$~(left),
$\frac{Br(\chi^0_7 \longrightarrow e~W)}{\sqrt{Br(\chi^0_7 \longrightarrow
\mu~W)^2+Br(\chi^0_7 \longrightarrow \tau~W)^2}}$ with $\tan^2\theta_{13}$~(right)
plot for a bino dominated lightest neutralino (the LSP) with bino component,
$|N_{71}|^2>$~$0.92$. Neutrino mass pattern is normal hierarchical. Choice of parameters
are same as that of figure \ref{nor-bino-bmbebt}.}
\label{nor-bino-LSP}
\end{figure}
We have checked that the correlations between the ratios of the lightest
neutralino decay branching ratios and $b^2_i/b^2_j$ is more prominent
with increasing bino component of the lightest neutralino. Note that when
$(b_i/b_j)^2\rightarrow 1$ the ratios of branching ratios shown in
figure \ref{nor-bino-bmbebt} also tend to $1$. We have seen earlier that the
neutrino mixing angles $\theta_{23}$ and $\theta_{13}$ also show nice
correlation with the ratios $b^2_\mu/b^2_\tau$ and $b^2_e/b^2_\tau$,
respectively (see figure \ref{neut2313-mixing-tree}). Hence we would expect
that the ratios of the
branching ratios $\frac{{\rm BR}({\tilde \chi}^0_7 \rightarrow \mu W)}
{{\rm BR}({\tilde \chi}^0_7 \rightarrow \tau W)}$ and
$\frac{BR({\tilde \chi}^0_7 \longrightarrow e~W)}
{\sqrt{BR({\tilde \chi}^0_7 \longrightarrow \mu~W)^2
+BR({\tilde \chi}^0_7 \longrightarrow \tau~W)^2}}$ show correlations with
$\tan^2\theta_{23}$ and $\tan^2\theta_{13}$. These correlations are shown in
figure \ref{nor-bino-LSP}. We have seen earlier (see eqn. (\ref{eigenvalues})) that
with the normal hierarchical pattern of the neutrino masses, the atmospheric
mass scale is determined by the quantity $\Omega_b = \sqrt{b^2_e + b^2_\mu
+ b^2_\tau}$. Naturally one would expect that the atmospheric and the reactor
angles are correlated with the $\ell + W$ final states of the lightest
neutralino decays and no correlation is expected for the solar angle. This is
what we have observed numerically. Here we have considered the regions of the
parameter space where the neutrino mass-squared differences and mixing angles
are within the 3$\sigma$ allowed range as shown in table \ref{osc-para}.
Figures \ref{nor-bino-LSP} also
shows the model prediction for the ratios of branching ratios where the
neutrino experimental data are satisfied. For our sample choice of parameters
in figure \ref{nor-bino-LSP}, one would expect that the ratio
$\frac{BR({\tilde \chi}^0_7 \longrightarrow \mu~W)}{BR({\tilde \chi}^0_7
\longrightarrow \tau~W)}$ should be in the range 0.45 to 1.25. Similarly,
the other ratio $\frac{BR({\tilde \chi}^0_7 \longrightarrow e~W)}
{\sqrt{BR({\tilde \chi}^0_7 \longrightarrow \mu~W)^2
+BR({\tilde \chi}^0_7 \longrightarrow \tau~W)^2}}$ is expected in this case
to be less than 0.07. We can also see from figure \ref{nor-bino-LSP} that the
ratio of branching ratios in the ($\mu + W$) and ($\tau + W$) channels becomes
almost equal for the maximal value of the atmospheric mixing angle
($\theta_{23}~=~45^\circ$). On the other hand, we do not observe any
correlation with the solar mixing angle $\theta_{12}$ since it is a
complicated function of $a^2_i$ and $b^2_i$ (see eqn. (\ref{solar_analytical})).
\begin{figure}[ht]
\centering
\vspace*{0.4cm}
\includegraphics[height=4.00cm]{FigsPDF/niBDBRmt.pdf}
\includegraphics[height=4.00cm]{FigsPDF/niBDBRet.pdf}
\includegraphics[height=4.00cm]{FigsPDF/niBDBRem.pdf}
\caption{Ratio $\frac{BR({\tilde \chi}^0_7 \longrightarrow \ell^-_i~W)}
{BR({\tilde \chi}^0_7 \longrightarrow \ell^-_j~W)}$ versus
$\frac{b^2_i}{b^2_j}$ plot for a bino like lightest neutralino (the LSP)
with bino component $|N_{71}|^2>$~0.95, where $i,j,k~=~e,\mu,\tau$. Neutrino
mass pattern is inverted hierarchical. Choice of parameters are
$M_1=105$ GeV, $\lambda=0.15, \kappa=0.65, m_{\tilde{\nu}^c}=300~{\rm GeV}
~{\rm and} ~m_{\tilde{L}}=445~{\rm GeV}$. Mass of the LSP is $103.3$ GeV.
The value of the $\mu$ parameter comes out to be $-263.7$ GeV.}
\label{inv-bino-bmbebt}
\end{figure}
In the case of inverted hierarchical mass pattern of the light neutrinos, the
${\tilde \chi}^0_7$--$\ell_i$--$W$ coupling is still controlled by the
quantities $b^2_i$. Hence the ratios of the branching ratios discussed earlier,
show nice correlations with $b^2_i/b^2_j$ (see figure \ref{inv-bino-bmbebt}).
However, in this case the solar mixing angle shows
some correlation with the ratio $\frac{BR({\tilde \chi}^0_7 \longrightarrow e~W)}
{\sqrt{\sum BR({\tilde \chi}^0_7 \longrightarrow \ell_i~W)^2}}$
with $\ell_i=\mu,\tau$. This is shown in
figure \ref{inv-bino-LSP}. The correlation is not very sharp and some dispersion
occurs due to the fact that the two heavier neutrino masses controlling the
atmospheric mass scale and solar mass-squared difference are not completely
determined by the quantities $b^2_i$ and there is some contribution of the
quantities $a^2_i$, particularly for the second heavy neutrino mass eigenstate.
\begin{figure}[ht]
\centering
\vspace*{0.4cm}
\includegraphics[height=4.00cm]{FigsPDF/Tsq12BDni.pdf}
\includegraphics[height=4.00cm]{FigsPDF/Tsq23BDni.pdf}
\caption{Ratio $\frac{BR({\tilde \chi}^0_7 \longrightarrow e~W)}
{\sqrt{BR({\tilde \chi}^0_7 \longrightarrow \mu~W)^2+BR({\tilde \chi}^0_7
\longrightarrow \tau~W)^2}}$ with $\tan^2\theta_{12}$ (left) plot for a bino
dominated lightest neutralino (LSP) with bino component $|N_{71}|^2>$~0.95.
In the right figure the ratio $\frac{BR({\tilde \chi}^0_7 \longrightarrow
\mu~W)}{BR({\tilde \chi}^0_7 \longrightarrow \tau~W)}$ versus
$\tan^2\theta_{23}$ is plotted. Neutrino mass pattern is assumed to be
inverted hierarchical. Choice of parameters are same as that of
figure \ref{inv-bino-bmbebt}.}
\label{inv-bino-LSP}
\end{figure}
The correlation of the ratio $\frac{BR({\tilde \chi}^0_7 \longrightarrow
\mu~W)} {BR({\tilde \chi}^0_7 \longrightarrow \tau~W)}$ with
$\tan^2\theta_{23}$ shows a different behaviour compared to what we have seen
in the case of normal hierarchical scenario. This is because in the case of
inverted hierarchical mass pattern of the neutrinos, $\tan^2\theta_{23}$
decreases with increasing $b^2_\mu/b^2_\tau$. One can observe from
Figures \ref{nor-bino-LSP} and \ref{inv-bino-LSP} that if the experimental
value of the ratio $\frac{BR({\tilde \chi}^0_7 \longrightarrow e~W)}
{\sqrt{BR({\tilde \chi}^0_7 \longrightarrow \mu~W)^2+BR({\tilde \chi}^0_7
\longrightarrow \tau~W)^2}}$ is $\ll$ 1 then that indicates a normal
hierarchical neutrino mass pattern for a bino-dominated lightest neutralino
LSP whereas a higher value ($\sim$ 1) of this ratio measured in experiments
might indicate that the neutrino mass pattern is inverted hierarchical.
Similarly a measurement of the ratio $\frac{BR({\tilde \chi}^0_7
\longrightarrow \mu~W)}{BR({\tilde \chi}^0_7 \longrightarrow \tau~W)}$ can
also give an indication regarding the particular hierarchy of the neutrino
mass pattern in the case of a bino dominated LSP.
\subsection{{\bf H}iggsino dominated lightest neutralino}\label{higgsino-LN}
When one considers higher values of the $U(1)$ gaugino mass $M_1$, i.e.
$M_1>\mu$ and large value of $\kappa$ (so that the effective gauge singlet
neutrino mass $2 \kappa v^c$ is large), the lightest neutralino is
essentially higgsino dominated and it is the LSP. Naturally one needs to
consider a small value of the coupling $\lambda$ so that the effective $\mu$
parameter ($\mu = 3 \lambda v^c$) is smaller. In order to look at the
lightest neutralino decay branching ratios in this case, we consider a
situation where the higgsino component in ${\tilde \chi}^0_7$ is
$|N_{73}|^2 + |N_{74}|^2 >$ $0.90$. As in the case of a bino dominated LSP,
the generation dependence of the ${\tilde \chi}^0_7$--$\ell_i$--$W$ couplings
comes through the quantities $b^2_i$. However, because of the large value of
the $\tau$ Yukawa coupling, the higgsino--$\tau$ mixing is larger
and as a result the partial decay width of ${\tilde \chi}^0_7$ into ($W + \tau$) is
larger than into ($W + \mu$) and ($W + e$). This feature is shown in
figure \ref{nor-higgsino-bmbebt}, where the ratios of branching ratios are
plotted against the quantities $b^2_i/b^2_j$. The domination of
BR(${\tilde \chi}^0_7 \rightarrow \tau + W$) over the other two is clearly
evident. Nevertheless, all the three ratios of branching ratios show
sharp correlations with the corresponding $b^2_i/b^2_j$. In this figure
the normal hierarchical pattern of the neutrino masses has been considered.
As in the case of a bino LSP, here also the ratios $\frac{BR({\tilde \chi}^0_7
\longrightarrow \mu~W)} {BR({\tilde \chi}^0_7 \longrightarrow \tau~W)}$ and
$\frac{BR({\tilde \chi}^0_7 \longrightarrow e~W)}
{\sqrt{BR({\tilde \chi}^0_7 \longrightarrow \mu~W)^2+BR({\tilde \chi}^0_7
\longrightarrow \tau~W)^2}}$ show nice correlations with neutrino mixing
angles $\theta_{23}$ and $\theta_{13}$, respectively. This is shown in
figure \ref{nor-higgsino-LSP}. However, in this case the predictions for these
two ratios are very different from the bino LSP case. The expected value of
the ratio $\frac{BR({\tilde \chi}^0_7 \longrightarrow \mu~W)}
{BR({\tilde \chi}^0_7 \longrightarrow \tau~W)}$ is approximately between
0.05 and 0.10 in a region where one can accommodate the experimental
neutrino data. Similarly, the predicted value of the ratio
$\frac{BR({\tilde \chi}^0_7 \longrightarrow e~W)}
{\sqrt{BR({\tilde \chi}^0_7 \longrightarrow \mu~W)^2+BR({\tilde \chi}^0_7
\longrightarrow \tau~W)^2}}$ is $\leq$ 0.006. On the other hand, there is
no such correlations with the solar mixing angle $\theta_{12}$.
\vspace{0.5cm}
\begin{figure}[ht]
\centering
\vspace*{0.4cm}
\includegraphics[height=4.00cm]{FigsPDF/nnHDBRmt.pdf}
\includegraphics[height=4.00cm]{FigsPDF/nnHDBRet.pdf}
\includegraphics[height=4.00cm]{FigsPDF/nnHDBRem.pdf}
\caption{Ratio $\frac{BR({\tilde \chi}^0_7 \longrightarrow l_i~W)}
{BR({\tilde \chi}^0_7 \longrightarrow l_j~W)}$ versus $\frac{b^2_i}{b^2_j}$
plot for a higgsino like LSP with higgsino component $(|N_{73}|^2+
|N_{74}|^2)>$~0.95, where $i,j,k~=~e,\mu,\tau$. Neutrino mass pattern is
normal hierarchical. Choice of parameters are $M_1=325$ GeV, $\lambda=0.06,
\kappa=0.65, m_{\tilde{\nu}^c}=300~{\rm GeV} ~\rm{and} ~m_{\tilde{L}}=
400~{\rm GeV}$. Mass of the LSP is $98.6$ GeV.
The value of the $\mu$ parameter comes out to be $-105.9$ GeV.}
\label{nor-higgsino-bmbebt}
\end{figure}
\vspace{0.5cm}
\begin{figure}[ht]
\centering
\vspace*{0.4cm}
\includegraphics[height=4.00cm]{FigsPDF/Tsq23HDnn.pdf}
\includegraphics[height=4.00cm]{FigsPDF/Tsq13HDnn.pdf}
\caption{Ratio $\frac{BR({\tilde \chi}^0_7 \longrightarrow \mu~W)}
{BR({\tilde \chi}^0_7 \longrightarrow \tau~W)}$ versus $\tan^2\theta_{23}$~
(left), $\frac{BR({\tilde \chi}^0_7 \longrightarrow e~W)}
{\sqrt{BR({\tilde \chi}^0_7 \longrightarrow \mu~W)^2+
BR({\tilde \chi}^0_7 \longrightarrow \tau~W)^2}}$ with $\tan^2\theta_{13}$
~(right) plot for a higgsino LSP with higgsino component
$(|N_{73}|^2+|N_{74}|^2)>$~$0.95$. Neutrino mass pattern is normal
hierarchical. Choice of parameters are same as that of
figure \ref{nor-higgsino-bmbebt}.}
\label{nor-higgsino-LSP}
\end{figure}
\vspace{1.3cm}
Similar correlations of the ratios of branching ratios with $b^2_i/b^2_j$
are also obtained for a higgsino dominated LSP in the case where the neutrino
mass pattern is inverted hierarchical. Once again it shows that the
${\tilde \chi}^0_7$ decays to ($\tau + W$) channel is dominant over the
channels ($e + W$) and ($\mu + W$) for any values of $b^2_i/b^2_j$ because
of the larger $\tau$ Yukawa coupling. On the other hand, the correlations with
the neutrino mixing angles show a behaviour similar to that of a bino LSP
with inverted neutrino mass hierarchy though with much smaller values for the
ratios $\frac{BR({\tilde \chi}^0_7 \rightarrow \mu~W)}
{BR({\tilde \chi}^0_7 \rightarrow \tau~W)}$
and $\frac{BR({\tilde \chi}^0_7 \rightarrow e~W)}
{\sqrt{BR({\tilde \chi}^0_7 \rightarrow \mu~W)^2+Br({\tilde \chi}^0_7
\rightarrow \tau~W)^2}}$. These are shown in figure \ref{inv-higgsino-LSP}.
Note that the correlations in this case are not very sharp, especially with
$\tan^2\theta_{12}$. Thus we see that small values of these ratios
(for both normal and inverted hierarchy) are characteristic features of a
higgsino dominated LSP in this model.
\vspace{1.7cm}
\begin{figure}[ht]
\centering
\vspace*{0.4cm}
\includegraphics[height=4.00cm]{FigsPDF/Tsq23HDni.pdf}
\includegraphics[height=4.00cm]{FigsPDF/Tsq12HDni.pdf}
\caption{Ratio $\frac{BR({\tilde \chi}^0_7 \longrightarrow \mu~W)}
{BR({\tilde \chi}^0_7 \longrightarrow \tau~W)}$ versus $\tan^2\theta_{23}$
~(left), $\frac{BR({\tilde \chi}^0_7 \longrightarrow e~W)}
{\sqrt{BR({\tilde \chi}^0_7 \longrightarrow \mu~W)^2+Br({\tilde \chi}^0_7
\longrightarrow \tau~W)^2}}$ with $\tan^2\theta_{12}$~(right) plot
for a higgsino LSP with higgsino component $(|N_{73}|^2+|N_{74}|^2)>$~0.95.
Neutrino mass pattern is inverted hierarchical. Choice of parameters are
$M_1=490$ GeV, $\lambda=0.07, \kappa=0.65, m_{\tilde{\nu}^c}=320 \rm{GeV}
~\rm{and} ~m_{\tilde{L}}=430 \rm{GeV}$. Mass of the LSP is $110.8$ GeV.
The value of the $\mu$ parameter comes out to be $-115.3$ GeV.}
\label{inv-higgsino-LSP}
\end{figure}
\subsection{{\bf R}ight-handed neutrino dominated lightest neutralino}\label{rhnu-LN}
Because of our choice of parameters i.e., a generation independent coupling
$\kappa$ of the gauge singlet neutrinos and a common VEV $v^c$ (see eqn.(\ref{assumption1})),
the three neutralino mass eigenstates which are predominantly gauge singlet
neutrinos are essentially mass degenerate. There is a very small mass
splitting due to mixing. However, unlike the case of a bino or higgsino
dominated lightest neutralino, these $\nu^c$ dominated lightest neutralino
states cannot be considered as the LSP. This is because in this case the
lightest scalar (which is predominantly a gauge singlet sneutrino
${\tilde \nu}^c$) is the lightest supersymmetric particle. This is very
interesting since usually one does not get a ${\tilde \nu}^c$ as an LSP
in a model where the gauge singlet neutrino superfield has a large Majorana
mass term in the superpotential. However, in this case the effective Majorana
mass term is at the EW scale and there is also a contribution from the
trilinear scalar coupling $A_\kappa \kappa$ which keeps the mass of the
singlet scalar sneutrino smaller. It is also very interesting to study the
decay patterns of the lightest neutralino in this case since here one can
probe the gauge singlet neutrino mass scales at the colliders.
Before discussing the decay patterns of the lightest neutralino which is
$\nu^c$ dominated, let us say a few words regarding their production at
the LHC. The direct production of $\nu^c$ (by $\nu^c$ we mean the $\nu^c$
dominated lightest neutralino in this subsection) is negligible because of
the very small mixing with the MSSM neutralinos. Nevertheless, they can be
produced at the end of the cascade decay chains of the squarks and gluinos at
the LHC. For example, if the next-to-next-to-lightest SUSY particle (NNLSP)
is higgsino dominated (this is the state above the three almost degenerate
lightest neutralinos) and it has a non-negligible mixing with $\nu^c$ (remember
that the higgsino--$\nu^c$ mixing occurs mainly because of the
term $\lambda {\hat \nu}^c {\hat H}_d {\hat H}_u$ in the superpotential,
eqn.(\ref{munuSSM-superpotential})), then
the branching ratio of the decay ${\tilde H} \rightarrow Z + \nu^c$ can be
larger than the branching ratios in the $\ell W$ and $\nu Z$ channels.
This way one can produce $\nu^c$ dominated lightest neutralino. Similarly,
a higgsino dominated lighter chargino can also produce gauge singlet neutrinos.
Another way of producing $\nu^c$ is through the decay of an NNLSP ${\tilde \tau}_1$, such as
${\tilde \tau}_1 \rightarrow \tau + \nu^c$.
\vspace{0.5cm}
\begin{figure}[ht]
\centering
\vspace*{0.4cm}
\includegraphics[height=4.00cm]{FigsPDF/nnNCDBRmt_old.pdf}
\includegraphics[height=4.00cm]{FigsPDF/nnNCDBRmt.pdf}
\caption{Ratio $\frac{BR({\tilde \chi}^0_7 \longrightarrow \mu~W)}
{BR({\tilde \chi}^0_7 \longrightarrow \tau~W)}$ versus
$\frac{a^2_\mu}{a^2_\tau}$ (left) and versus $\frac{b^2_\mu}{b^2_\tau}$
(right) plot for a $\nu^c$ like lightest neutralino (${\tilde \chi}^0_7$)
with $\nu^c$ component $(|N_{75}|^2+|N_{76}|^2+|N_{77}|^2)>$~0.99, (left) and
$>$0.97 (right). Neutrino mass pattern is normal hierarchical. Choice of
parameters are for (left) $M_1=405$ GeV, $\lambda=0.29, \kappa=0.07,
(A_\lambda \lambda)= -8.2~{\rm TeV} \times \lambda,(A_\kappa \kappa)= 165
~{\rm GeV} \times \kappa,~m_{\tilde{\nu}^c}=50~{\rm GeV} ~\rm{and}
~m_{\tilde{L}}=825~{\rm GeV}$ and for (right) $M_1=405$ GeV, $\lambda=0.10,
\kappa=0.07, (A_\lambda \lambda)= -2~{\rm TeV} \times \lambda,(A_\kappa \kappa)
= 165~{\rm GeV} \times \kappa, ~m_{\tilde{\nu}^c}=50~{\rm GeV} ~{\rm and}
~m_{\tilde{L}}=825~{\rm GeV}$. Mass of the lightest neutralino is $129.4$ GeV
(left) and $119.8$ GeV (right) respectively.
The values of the $\mu$ parameter are $-803.9$ GeV and $-258.8$ GeV, respectively.}
\label{nor-singlino-bmbt-amat}
\end{figure}
When one considers higher value of the gaugino mass, i.e. $M_1>\mu$ and
a small value of the coupling $\kappa$ (so that the effective Majorana mass of
$\nu^c$ is small, i.e. $m_{\nu^c}=2 \kappa v^c < \mu$), the lightest neutralino is
essentially $\nu^c$ dominated. As we have mentioned earlier, in this case the
LSP is the scalar partner of $\nu^c$, i.e. ${\tilde \nu}^c$. However, the
decay of $\nu^c$ into $\nu + {\tilde \nu}^c$ is suppressed compared to the
decays $\nu^c \rightarrow \ell_i + W$ and $\nu^c \rightarrow \nu_i +Z$ that
we have considered so far. Because of this, in this section we will neglect the
decay $\nu^c \rightarrow \nu + {\tilde \nu}^c$ while discussing the correlation
of the lightest neutralino (${\tilde \chi}^0_7$) decays with the neutrino
mixing angles.
In this case the coupling of the lightest neutralino (${\tilde \chi}^0_7$)
with $\ell_i$--$W$ pair depends on the $\nu^c$ content of ${\tilde \chi}^0_7$.
Note that the $\nu^c$ has a very small mixing with the MSSM neutralino states.
However, in some cases the $\nu^c$ dominated lightest neutralino can have a
non-negligible higgsino component. In such cases the coupling
${\tilde \chi}^0_7$--$\ell_i$--$W$ depends mainly on the quantities $b_i$. On
the other hand, if ${\tilde \chi}^0_7$ is very highly dominated by $\nu^c$,
then the coupling ${\tilde \chi}^0_7$-- $\ell_i$--$W$ has a nice correlation
with the quantities $a_i$. So in order to study the decay correlations of the
$\nu^c$ dominated lightest neutralino, we consider two cases (i) $\nu^c$
component is $>$ $0.99$, and (ii) $\nu^c$ component is $>0.97$ with some
non-negligible higgsino admixture.
The correlations of the decay branching ratio
$\frac{BR({\tilde \chi}^0_7 \longrightarrow \mu~W)}
{BR({\tilde \chi}^0_7 \longrightarrow \tau~W)}$ are shown in
figure \ref{nor-singlino-bmbt-amat} for the cases (i) and (ii) mentioned above.
As we have explained already, this figure demonstrates that in case (i) the
ratio of the branching ratio is dependent on the quantity $a^2_\mu/a^2_\tau$
whereas in case (ii) this ratio is correlated with $b^2_\mu/b^2_\tau$ though
there is some suppression due to large $\tau$ Yukawa coupling.
\vspace{0.5cm}
\begin{figure}[ht]
\centering
\vspace*{0.4cm}
\includegraphics[height=4.00cm]{FigsPDF/nnNCDBRem_old.pdf}
\includegraphics[height=4.00cm]{FigsPDF/nnNCDBRem.pdf}
\caption{Ratio $\frac{BR({\tilde \chi}^0_7 \longrightarrow e~W)}
{BR({\tilde \chi}^0_7 \longrightarrow \mu~W)}$ versus
$\frac{a^2_e}{a^2_\mu}$ (left) and versus $\frac{b^2_e}{b^2_\mu}$
(right) plot for a $\nu^c$ like lightest neutralino (${\tilde \chi}^0_7$)
with $\nu^c$ component $(|N_{75}|^2+|N_{76}|^2+|N_{77}|^2)>$~0.99 (left), and
$>$0.97 (right). Neutrino mass pattern is normal hierarchical. Choice of
parameters are same as that of figure \ref{nor-singlino-bmbt-amat}.}
\label{nor-singlino-bebm-aeam}
\end{figure}
Similar calculations were performed also for the other ratios discussed
earlier. For example, in figure \ref{nor-singlino-bebm-aeam} we have shown the
variations of the ratio $\frac{BR({\tilde \chi}^0_7 \longrightarrow e~W)}
{BR({\tilde \chi}^0_7 \longrightarrow \mu~W)}$ as functions of
$\frac{a^2_e}{a^2_\mu}$ and $\frac{b^2_e}{b^2_\mu}$ for the cases (i) and (ii),
respectively. The variation with $\frac{a^2_e}{a^2_\mu}$ is not sharp and
dispersive in nature whereas the variation with $\frac{b^2_e}{b^2_\mu}$ is
very sharp and shows that in this case the relevant couplings are proportional
to $b_e$ and $b_\mu$, respectively.
\vspace{0.5cm}
\vspace{0.5cm}
\begin{figure}[ht]
\centering
\vspace*{0.4cm}
\includegraphics[height=4.00cm]{FigsPDF/Tsq23NCDnn_old.pdf}
\includegraphics[height=4.00cm]{FigsPDF/Tsq12NCDnn.pdf}
\caption{Ratio $\frac{BR({\tilde \chi}^0_7 \longrightarrow \mu~W)}
{BR({\tilde \chi}^0_7 \longrightarrow \tau~W)}$ versus
$\tan^2\theta_{23}$~(left), $\frac{BR({\tilde \chi}^0_7 \longrightarrow e~W)}
{\sqrt{BR({\tilde \chi}^0_7 \longrightarrow \mu~W)^2
+BR({\tilde \chi}^0_7 \longrightarrow \tau~W)^2}}$ with $\tan^2\theta_{12}$
~(right) plot for a $\nu^c$ dominated lightest neutralino with $\nu^c$ component
$(|N_{75}|^2+|N_{76}|^2+|N_{77}|^2)>$~$0.99$ (left) and $>$ $0.97$ (right).
Neutrino mass pattern is normal hierarchical. Choice of parameters are same
as that of figure \ref{nor-singlino-bmbt-amat}.}
\label{nor-singlino-LN}
\end{figure}
On the other hand, in case (i) only $\tan^2\theta_{23}$ shows a nice
correlation with the ratio $\frac{BR({\tilde \chi}^0_7 \longrightarrow \mu~W)}
{BR({\tilde \chi}^0_7 \longrightarrow \tau~W)}$ (see
figure \ref{nor-singlino-LN}) and $\tan^2\theta_{12}$ or $\tan^2\theta_{13}$
does not show any correlation with the other ratio. The non-linear behaviour
of the ratios of branching ratios in case(i) is due to the fact that the
parameters $Y_\nu$s (which control the $a_i$) appear both in the neutralino
and chargino mass matrices. The charged lepton Yukawa couplings also play a
role in determining the ratios. One can also see that the
prediction for this ratio of branching ratio for case (i), as shown in
figure \ref{nor-singlino-LN}, is in the range $0.5-3.5$, which is larger compared
to the bino dominated or higgsino dominated cases for both normal and
inverted hierarchical pattern of neutrino masses. Also, the nature of this
variation is similar to what we see with the inverted hierarchical pattern of
neutrino masses in the bino or higgsino dominated cases.
In case (ii) none of the neutrino mixing angles show very good correlations
with the ratios of branching ratios that we have been discussing. However, one
can still observe some kind of correlation between $\tan^2\theta_{12}$ and the
ratio $\frac{BR({\tilde \chi}^0_7 \longrightarrow e~W)} {\sqrt{BR({\tilde \chi}^0_7
\longrightarrow \mu~W)^2 +BR({\tilde \chi}^0_7 \longrightarrow \tau~W)^2}}$.
The prediction for this ratio from the neutrino data
is on the smaller side $(\sim 0.07)$.
\vspace{0.5cm}
\begin{figure}[ht]
\centering
\vspace*{0.4cm}
\includegraphics[height=4.00cm]{FigsPDF/niNCDBRmt_old.pdf}
\includegraphics[height=4.00cm]{FigsPDF/niNCDBRmt.pdf}
\caption{Ratio $\frac{BR({\tilde \chi}^0_7 \longrightarrow \mu~W)}
{BR({\tilde \chi}^0_7 \longrightarrow \tau~W)}$ versus
$\frac{a^2_\mu}{a^2_\tau}$ (left) and versus $\frac{b^2_\mu}{b^2_\tau}$
(right) plot for a $\nu^c$ like lightest neutralino (${\tilde \chi}^0_7$)
with $\nu^c$ component $(|N_{75}|^2+|N_{76}|^2+|N_{77}|^2)>$~$0.99$ (left), and
$>~0.97$ (right). Neutrino mass pattern is inverted hierarchical. Choice of
parameters are for (left) $M_1=445$ GeV, $\lambda=0.29, \kappa=0.07,
(A_\lambda \lambda)= -8.2~{\rm TeV} \times \lambda,(A_\kappa \kappa)=
165~{\rm GeV} \times \kappa,~m_{\tilde{\nu}^c}=50~{\rm GeV} ~{\rm and}
~m_{\tilde{L}}=835 \rm{GeV}$ and for (right) $M_1=445$ GeV, $\lambda=0.10,
\kappa=0.07, (A_\lambda \lambda)= -2~{\rm TeV} \times \lambda,(A_\kappa \kappa)
= 165~{\rm GeV} \times \kappa, ~m_{\tilde{\nu}^c}=50~{\rm GeV} ~{\rm and}
~m_{\tilde{L}}=835~{\rm GeV}$. Mass of the lightest neutralino is $129.4$ GeV
(left) and $119.8$ GeV (right) respectively.}
\label{inv-singlino-bmbt-amat}
\end{figure}
\vspace{2.3cm}
With the inverted hierarchical neutrino mass pattern, in case (i)
one observes a sharp correlation of the ratio
$\frac{BR({\tilde \chi}^0_7 \longrightarrow \mu~W)}
{BR({\tilde \chi}^0_7 \longrightarrow \tau~W)}$ with
$\frac{a^2_\mu}{a^2_\tau}$ (see figure \ref{inv-singlino-bmbt-amat}).
The other two ratios $\frac{BR({\tilde \chi}^0_7 \longrightarrow e~W)}
{BR({\tilde \chi}^0_7 \longrightarrow \mu~W)}$ and
$\frac{BR({\tilde \chi}^0_7 \longrightarrow e~W)}
{BR({\tilde \chi}^0_7 \longrightarrow \tau~W)}$ do not show very sharp
correlations with $\frac{a^2_e}{a^2_\mu}$ and $\frac{a^2_e}{a^2_\tau}$,
respectively and we do not plot them here. However, in case (ii) all the
three ratios show nice correlations with the corresponding $b^2_i/b^2_j$. We
have shown this in figure \ref{inv-singlino-bmbt-amat} only for
$b^2_\mu/b^2_\tau$. In this case the variations of the ratios of branching
ratios with neutrino mixing angles are shown in figure \ref{inv-singlino-LN}.
\vspace{0.9cm}
\begin{figure}[ht]
\centering
\vspace*{0.4cm}
\includegraphics[height=4.00cm]{FigsPDF/Tsq23NCDni.pdf}
\includegraphics[height=4.00cm]{FigsPDF/Tsq12NCDni.pdf}
\caption{Ratio $\frac{BR({\tilde \chi}^0_7 \longrightarrow \mu~W)}
{BR({\tilde \chi}^0_7 \longrightarrow \tau~W)}$ versus $\tan^2\theta_{23}$
~(left), $\frac{BR({\tilde \chi}^0_7 \longrightarrow e~W)}
{\sqrt{BR({\tilde \chi}^0_7 \longrightarrow \mu~W)^2
+BR({\tilde \chi}^0_7 \longrightarrow \tau~W)^2}}$ with $\tan^2\theta_{12}$
~(right) plot for a $\nu^c$ dominated lightest neutralino with $\nu^c$
component $(|N_{75}|^2+|N_{76}|^2+|N_{77}|^2)>$~$0.97$. Neutrino mass pattern is
inverted hierarchical. Choice of parameters are same as that of figure
\ref{inv-singlino-bmbt-amat}.}
\label{inv-singlino-LN}
\end{figure}
For the case (i), only $\tan^2\theta_{13}$ shows certain correlation with the
ratio of branching ratio shown in figure \ref{inv-singlino-LN} (right), but we
do not show it here.
Finally, we would like to reemphasize that in all these different cases discussed
above, the lightest neutralino can have a finite decay length which can produce
displaced vertices (also discussed earlier in sections \ref{LSP-dec-1}, \ref{lsp-nat})
in the vertex detectors. Depending on the composition of the
lightest neutralino, one can have different decay lengths. For example, a
bino-dominated lightest neutralino can produce a displaced vertex $\sim$ a few
mm. Similarly, for a higgsino dominated lightest neutralino, decay vertices of
the order of a few cms can be observed \cite{c5Ghosh:2008yh,c5Bartl:2009an}.
On the other hand, if the lightest
neutralino is $\nu^c$ dominated, then the decay lengths can be of the order
of a few meters \cite{c5Ghosh:2008yh,c5Bartl:2009an,c5Bandyopadhyay:2010cu}.
These are very unique predictions of this model which can,
in principle, be tested at the LHC \cite{c5Bandyopadhyay:2010cu}.
The advantage of having large displaced vertices for a singlino like
lightest neutralino makes it easier to kill all of the SM backgrounds
unambiguously. Additionally, it is also difficult to achieve a reasonably large ($\sim$ a
few meter) displaced vertex in the conventional $R_p$-violating model \cite{c5Mukhopadhyaya:1998xj,
c5Choi:1999tq,c5Porod:2000hv}. As a consequence it is rather difficult for
the $R_p$-violating supersymmetric models to mimic
a specific collider signatures of $\mu\nu$SSM, particularly when a
gauge singlet LSP is involved in the process. We will use the favour of large displaced
vertex associated with a singlino like LSP to describe an unconventional
signal of the lightest Higgs boson of $\mu\nu$SSM \cite{c5Bandyopadhyay:2010cu}
in the next chapter.
\chapter{ \sffamily{{\bf $\mu$}$\nu$SSM: Unusual signal of Higgs boson at LHC
}}\label{munuSSM-Higgs}
\section{{\bf H}iggs boson in $\mu\nu$SSM}
In $\mu\nu$SSM $R_p$ is violated through lepton number violation both in the superpotential and
in the soft terms. In this model neutral Higgs bosons of the MSSM mix with
three generations of left and right-handed sneutrinos and thus the neutral
scalar and pseudoscalar squared mass matrices are enhanced $(8\times8)$
over their $2\times2$ MSSM structures \cite{c6Escudero:2008jg,c6Ghosh:2008yh}. In a similar fashion
the charged scalar squared mass matrix is also a $8\times8$ matrix for $\mu\nu$SSM due to
mixing between charged Higgs of the MSSM and charged sleptons \cite{c6Escudero:2008jg,c6Ghosh:2008yh}.
In general the nature of the lightest neutral scalar state can be very different from
that of the MSSM due to the presence of the gauge singlet right-handed
sneutrino component. It has been already shown that $\mu\nu$SSM is capable
of accommodating neutrino data both from tree level \cite{c6Ghosh:2008yh}
and one loop combined analysis \cite{c6Ghosh:2010zi}. With the initiation of
the LHC experiment at CERN it is naturally tempting to see
whether this is capable of producing interesting collider signatures apart from
accommodating the neutrino data.
The issues of Higgs boson discovery have been studied extensively over years
in the literature (see for example \cite{c6Barger:1987nn}).
In this chapter we propose a prodigious signal of Higgs boson in supersymmetry,
having dilepton and four hadronic jets along with large displaced vertices
$(\lower0.5ex\hbox{$\:\buildrel >\over\sim\:$} 3 ~m)$ \cite{c6Bandyopadhyay:2010cu}.
Most of the usual signal of Higgs boson are
impaired by undesired backgrounds and one has to remove them somehow for claiming
a discovery. Often the procedures for background subtraction in turn weaken the
desired signal significantly.
On the other hand, it was well known that the advantage of displaced vertices are
always extremely useful to kill all of the SM backgrounds and also some
of the possible backgrounds arising from the $R_p$ violating MSSM.
Displaced vertices arising from MSSM with $\not\!\!{R_p}$ are usually
much smaller \cite{c6Mukhopadhyaya:1998xj,c6Choi:1999tq,romao-dl,c6Porod:2000hv}
Now in the last chapter we have learned that in $\mu\nu$SSM, with suitable
choice of parameters, a right-handed neutrino like lightest neutralino can be a viable candidate
for the LSP. It was also discussed that since a right-handed neutrino is singlet under the
SM gauge group it can decay only in $R_p$-violating channels through small $R_p$-violating couplings
and consequently the associated displaced vertices can be very large $(\sim {\rm{meter}})$
\cite{c6Bartl:2009an,c6Bandyopadhyay:2010cu}. Indeed these displaced vertices can
kill all of the SM backgrounds as well as backgrounds
arising from MSSM with $\not\!\!{R_p}$ \cite{c6Mukhopadhyaya:1998xj,
c6Choi:1999tq,romao-dl,c6Porod:2000hv}. Furthermore, imprint of
this signal is different from that of the cosmic muons which have definite entry and exit
point in the detector. So this is apparently a clean signal and a discovery, thus is
definite even with small number of signal events.
In the next section we will discuss how to use the favour of
these large displaced vertices associated with a singlino like LSP for proposing
a new kind of signal of Higgs boson \cite{c6Bandyopadhyay:2010cu}.
\section{{\bf T}he Signal}\label{signal}
There are essentially two key features of our analysis, which collectively can lead
to an unusual signal of the Higgs boson in supersymmetry
\vspace{0.1cm}
\noindent
1.~The lightest neutralino LSP $(\ntrl1)$ in the $\mu\nu$SSM with the parameter choice
$M_1,\mu\gg m_{\nu^c}$ (see section \ref{lsp-nat}) can be predominantly composed of
right-handed neutrinos which, as argued earlier will be called a $\nu^c$-like or
a singlino like LSP \cite{c6Bartl:2009an,c6Bandyopadhyay:2010cu}. For the
analysis of ref. \cite{c6Bandyopadhyay:2010cu} we choose
$|\mathbf N_{15}|^2 + |\mathbf N_{16}|^2 + |\mathbf N_{17}|^2 \geqslant 0.70$.
\vspace{0.1cm}
\noindent
2.~A pair of singlino like LSP can couple to a Higgs boson in $\mu\nu$SSM
mainly through couplings like $\nu^cH_uH_d$ (see fig \ref{lam-sing}).
\begin{figure}[ht]
\centering
\vspace*{0.5cm}
\includegraphics[height=2.80cm]{FigsPDF/lam-sing.pdf}
\caption{Feynman diagram for the singlino singlino Higgs couplings. $\beta=2/1$ for $\alpha=d/u$.}
\label{lam-sing}
\end{figure}
The neutralino LSP, $\ntrl1$ in $\mu\nu$SSM can be predominantly
$(\lower0.5ex\hbox{$\:\buildrel >\over\sim\:$} 70\%)$ $\nu^c$-like (also known as a {\it{singlino}} LSP).
$\ntrl1$ being singlet, $\ntrl1 \ntrl1 Z$ or $\ntrl1 q \widetilde q$
couplings \cite{c6Ghosh:2010zi} are vanishingly small, which in turn results in very small
cross-section for direct $\ntrl1$ pair production. On the contrary, the coupling
$\lambda\nu^cH_uH_d$ may produce a large $\ntrl1 \ntrl1 S^0_i$
\cite{c6Ghosh:2010zi} coupling with $\lambda \sim$ $\cal{O}$ $(1)$, where $S^0_i$ are the scalar
states. With the chosen set of parameters (see Table \ref{mass-spectrum})
we obtained $S^0_4 \equiv h^0$, where $h^0$ is the lightest Higgs boson of MSSM.
In addition with heavy squark/gluino masses as indicated in Table \ref{mass-spectrum}
for different benchmark points, production of a singlino LSP through
cascade decays is suppressed.
In the backdrop of such a scenario, production of $h^0$ in gluon fusion channel
followed by the decay process $h^0 \to \ntrl1 \ntrl1$ will be the leading production
channel for the singlino LSP at the LHC. We want to emphasize here that
for the present analysis we choose to work with the tree level mass of the
lightest CP-even Higgs boson $(S^0_4 \equiv h^0)$ of the $\mu\nu$SSM. With loop
corrections the Higgs boson mass can be higher \cite{c6Escudero:2008jg,c6Bartl:2009an}.
For loop corrected Higgs boson mass, the process $h^0\to\ntrl1\ntrl1$ will yield
heavy singlino like LSPs with smaller decay lengths \cite{c6Bartl:2009an}.
However, our general conclusions will not change for a singlino LSP in the
mass range $20-60$ GeV, as long as the decay branching ratio for the
process $h^0\to\ntrl1\ntrl1$ is substantial.
A set of four benchmark points (BP) used for collider studies
compatible with neutrino data \cite{c6Schwetz:2008er}, upto one-loop level
analysis \cite{c6Ghosh:2010zi} are given in Table \ref{mass-spectrum}.
These are sample points and similar spectra can be obtained in a reasonably
large region of the parameter space even after satisfying all the constraints
from neutrino experiments.
\begin{table}[ht]
\centering
\begin{tabular}{ c || c c c c}
\hline \hline
& BP-1 & BP-2 & BP-3 & BP-4 \\ \hline \hline
$\mu$ & 177.0 & 196.68 & 153.43 & 149.12 \\
tan$\beta$ & 10 & 10 & 30 & 30 \\
$m_{h^0}~(\equiv m_{\ns4})$ & 91.21 & 91.63 & 92.74 & 92.83 \\
$m_{\ns1}$ & 48.58 & 49.33 & 47.27 & 49.84 \\
$m_{\nps2}$ & 47.21 & 49.60 & 59.05 & 49.45 \\
$m_{\cs2}$ & 187.11 & 187.10 & 187.21 & 187.21 \\
$m_{\bsq1}$ & 831.35 & 831.33 & 830.67 & 830.72 \\
$m_{\bsq2}$ & 875.03 & 875.05 & 875.72 & 875.67 \\
$m_{\tsq1}$ & 763.41 & 763.63 & 761.99 & 761.98 \\
$m_{\tsq2}$ & 961.38 & 961.21 & 962.46 & 962.48 \\
$m_{\ntrl1}$ & 43.0 & 44.07 & 44.20 & 44.24 \\
$m_{\ntrl2}$ & 55.70 & 57.64 & 61.17 & 60.49 \\
$m_{{\chpm4}}$ & 151.55 & 166.61 & 133.69 & 130.77 \\
\hline \hline
\end{tabular}
\caption{\label{mass-spectrum}
$\mu$-parameter and relevant mass spectrum (GeV) for chosen
benchmark points. $m_{\widetilde{\chi}^{\pm}_{1,2,3}} \equiv ~m_{e,\mu,\tau}$.
Only the relevant masses are shown here. Squark
masses of first two generations are $\sim 800$ GeV, which are not shown here. For our parameter
choices the fourth CP-even scalar state $S^0_4\equiv h^0$ \cite{c6Bandyopadhyay:2010cu}.
The quantities $S^0,P^0,S^\pm,\widetilde \chi^0,\widetilde \chi^\pm$ represent
physical scalar, pseudoscalar, charged scalar, neutralino and chargino states,
respectively. \cite{c6Escudero:2008jg,c6Ghosh:2008yh,c6Ghosh:2010zi}. The heavy quarks
namely, bottom, charm and top masses are computed at the $m_Z$ mass scale or
at the electroweak scale (see ref. \cite{c6Djouadi-Spira} and references therein).}
\end{table}
For the set of specified benchmark points (table \ref{mass-spectrum}), we observe,
the process $h^0\to \ntrl1 \ntrl1$ to
be one of the dominant decay modes of $h^0$ (branching fraction within $35$-$65\%$), while
the process $h^0 \to b \bar{b}$ remains the main competitor.
Different Feynman rules concerning Higgs decays are given in appendix \ref{appenH}. With
a suitable choice of benchmark points (table \ref{mass-spectrum}) two body
decays of $h^0$ into lighter scalar or pseudoscalar states were kept kinematically
disfavoured. Squared matrix elements for the processes $h^0\to b \bar{b}$
and $h^0 \to \ntrl1 \ntrl1$ are also given in appendix \ref{appenH}.
The pair produced singlino like $\ntrl1$ will finally decay into standard
model particles as shown in eqn.(\ref{2-3-body-decays}). For a lightest Higgs
boson mass $m_{h^0}$ as shown in table \ref{mass-spectrum}, mass of a singlino
like $\ntrl1$ $(m_{\ntrl1})$ arising from $h^0$ decay (see figure \ref{lam-sing})
is $< m_W$, and thus three body decays dominate. Out of the five
possible three body final states we choose to work with the specific decay
mode $\ntrl1 \to q_i\bar{q}^{\prime}_j\ell^\pm_k$
to yield a signal $pp\to 2\ell + 4j + X$ in the final state\footnote{The dilepton have
same sign on $50\%$ occurrence since $\ntrl1$ is a Majorana particle.}. This particular
final state is free from neutrinos and thus a reliable invariant mass reconstruction is
very much possible. It has to be emphasized here that as suggested in ref. \cite{c6Barger:1987nn},
a reliable mass reconstruction is possible even for the final states with a single neutrino,
thus apart from the $2\ell + 4j + X$ final state there also exist other equally
interesting final states like $3\ell + 2j + X$ ($\ntrl1 \to q_i\bar{q}^{\prime}_j\ell^\pm_k,
\ntrl1 \to\ell^+_i \ell^-_j\nu_k$), $1\ell + 4j + X$ ($\ntrl1 \to q_i\bar{q}^{\prime}_j\ell^\pm_k
,\ntrl1 \to q_i\bar{q_i}\nu_k$) etc. For the chosen benchmark points,
$Br(\ntrl1 \to q_i\bar{q}^{\prime}_j\ell^\pm_k)$ lies within $8-10\%$.
Squared matrix elements for all possible three body decays of $\ntrl1$
(see eqn.(\ref{2-3-body-decays})) are given in appendix \ref{appenI}.
At this point the importance of a singlino $\ntrl1$ becomes apparent. Since
all the leptons and jets are originating from the decays of a gauge singlet fermion,
the associated displaced vertices are very large $\sim 3-4$ meter, which
simply wash out any possible backgrounds. Detection of these
displaced as well as isolated leptons and hadronic jets can lead to
reliable mass reconstruction for $\ntrl1$ and Higgs boson in the absence of missing
energy in the final state. There is one more merit of this analysis,
i.e., invariant mass reconstruction for a singlino LSP can give
us an estimation of the seesaw scale, since the right-handed neutrinos are
operational in light neutrino mass generation through a TeV scale seesaw
mechanism \cite{c6LopezFogliani:2005yw,c6Ghosh:2008yh} in $\mu\nu$SSM.
It is important to note that in the real experimental ambience, extra jets can arise from
initial state radiation (ISR) and final state radiation (FSR). Likewise semi-leptonic
decays of quarks can accrue extra leptons. Also from the experimental point of view
one cannot have zero missing $p_T$ in the final state. With this set of information
we optimize our chosen signal as
\begin{eqnarray}
(n_{j} \ge 4) + (n_{\ell} \geq 2) + (\not\!\!{p_T} \le 30 ~{\rm{GeV}}),
\label{signal-choice}
\end{eqnarray}
where $n_{j(\ell)}$ represents the number of {\it{jets(leptons}}).
It should also be noted that, similar final states can appear from the decay of
heavier scalar or pseudoscalar states in the model. Obviously, their production cross
section will be smaller compared to $h^0$ and the invariant mass distribution
(some other distributions also) should be different in those cases. So, in a sense
it is possible to discriminate this signal (eqn.(\ref{signal-choice})) from the model
backgrounds. Another possible source of backgrounds can arise from the cosmic muons.
However as discussed earlier, cosmic muons have definite entry and exit points inside a
detector and thus there signatures are different from the proposed signal.
\section{{\bf C}ollider analysis and detection}\label{coll-ana}
{\tt PYTHIA (version 6.4.22) \cite{c6Pythia6.4}} has been used for the purpose of event generation.
The corresponding mass spectrum and decay branching fractions
are fed to {\tt PYTHIA} by using the SLHA interface \cite{c6Skands-SLHA}. Subsequent decays of the
produced particles, hadronization and the collider analysis were performed using {\tt PYTHIA}.
We used {\tt CTEQ5L} parton distribution function (PDF) \cite{c6Lai-CTEQ,c6Pumplin-PDF} for the analysis.
The renormalization/factorization scale $Q$ was chosen to be the parton level center-of-mass energy,
$\sqrt{\hat{s}}$.
We also kept ISR, FSR and multiple interaction on for the analysis. The
production cross-section of $h^0$ via gluon fusion channel for different benchmark points
(table \ref{mass-spectrum}) is shown in table \ref{tabcross}.
\begin{table}[ht]
\centering
\begin{tabular}{ c || c c c c}
\hline \hline
$\sqrt{s}$ & BP-1 & BP-2 & BP-3 & BP-4 \\ \hline \hline
$7$ TeV &6837 &7365 &6932 &6948 \\
$14$ TeV &23150 &25000 &23580 &23560 \\
\hline \hline
\end{tabular}
\caption{\label{tabcross}
Hard scattering cross-section in fb for the process
$gg \to h^0$ for PDF CTEQ5L with $Q = \sqrt{\hat{s}}$.}
\end{table}
We have used {\tt PYCELL}, the toy calorimeter simulation provided in
{\tt PYTHIA}, with the following criteria:
\noindent
I. The calorimeter coverage is $\rm |\eta| < 4.5$ and the segmentation is
given by $\rm\Delta\eta\times\Delta\phi= 0.09 \times 0.09 $ which resembles
a generic LHC detector.
\noindent
II. $\Delta R \equiv \sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}} = 0.5$
has been used in cone algorithm for jet finding.
\noindent
III. $p_{T,min}^{jet} = 10$ GeV.
\noindent
IV. No jet matches with a hard lepton in the event.
In addition, the following set of standard kinematic cuts
were incorporated throughout
\noindent
1. $p_T^{\ell} \geq 5$ GeV and $\rm |\eta| _{\ell} \le 2.5$,
\noindent
2. $|\eta| _{j}\leq 2.5$, $\Delta R_{\ell j} \geq 0.4$,
$\Delta R_{\ell\ell}\geq 0.2,~$
\noindent
where $\Delta R_{\ell j}$ and $\Delta R_{\ell \ell}$ measure the lepton-jet and
lepton-lepton isolation, respectively. Events with isolated leptons, having $p_T\ge 5$
GeV are taken for the final state analysis.
Now depending on the distribution of the transverse decay length it is
possible to study the behaviour of this spectacular signal in different
regions of a generic LHC detector like CMS or ATLAS. For the purpose of
illustration we present a slice like picture of the CMS detector in figure
\ref{CMS-detec} to describe this novel signal in more details.
\begin{figure}[ht]
\centering
\includegraphics[width=11.55cm,keepaspectratio]{FigsPDF/CMS-Slice.pdf}
\caption{Transverse slice from the CMS detector. The maroon square corresponds
to the global muons which travel throughout the detector starting from the interaction
point. The light green square on the other hand corresponds to the stand-alone muons
which leave their imprints only in the muon chamber.}
\label{CMS-detec}
\end{figure}
Let us now analyze this rare signal (see eqn.(\ref{signal-choice})) piece wise
for the CMS detector as shown by figure \ref{CMS-detec}. We choose BP-2 as
the sample benchmark point. To start with we divide the entire detector in
five different regions on the basis of different transverse decay lengths $(L_{T})$
and conduct our analysis. The decay length $(L)$ is given by
\begin{eqnarray}\label{decay-length-defn}
L = c \tau (\beta \gamma),
\end{eqnarray}
where $c$ is the speed of light in vacuum ($=1$ in natural unit system),
$\tau$ is the proper life time and the kinematical factor $\beta\gamma=
\frac{|\vec{p}|}{m}$. Here $|\vec{p}|$ is the magnitude of the three momentum
$=\sqrt{|p_x|^2+|p_y|^2+|p_z|^2}$ and $m$ is the mass of the decaying particle. Now it is in general
difficult to measure the longitudinal component of the momentum $(p_z)$ which
lies along the beam axis, thus we choose to work with the transverse decay length
given by
\begin{eqnarray}\label{decay-length-defn-T}
L_T = c \tau (\beta \gamma)_T,
\end{eqnarray}
where $(\beta\gamma)_T=\frac{\sqrt{|p_x|^2+|p_y|^2}}{m}$.
\vspace{0.2cm}
\noindent
I.~ $L_{T}\leq1~cm$ $\blacktriangleright$ Roughly $10\%$ to $15\%$
of the total number of events appear in this region. These events are close to the
interaction point and may be mimicked by MSSM models with $\not\!\!{R_p}$. Thus we do not consider these
points in our analysis though these are also free from the SM backgrounds.
\vspace{0.2cm}
\noindent
II.~ $1~cm<L_{T}\leq50~cm$ $\blacktriangleright$ There exist reasonable number of events
$(\sim 30\%$ of the total events) with decay length in between {\it{$1$ cm {\rm{and}} $50$ cm}}.
For these events the associated electrons and muons\footnote{$\tau$'s are dropped out
for poor detection efficiency.} will leave charged tracks in the inner silicon tracker
as well as the electrons will deposit their energy at the electronic calorimeter (ECAL).
Associated hadronic jets will also deposit their energy at the hadronic calorimeter (HCAL).
The associated muons are {\it{global}} in nature and leave their marks throughout,
upto the muon chamber starting from few layers on the inner tracker.
It is easy for the conventional triggers to work for this kind of
signal and a reliable mass reconstruction of these displaced hadronic jets and leptons
can lead to a discovery. The number of signal events in this region are shown in table
\ref{tabevents-cut}.
\vspace{0.2cm}
\noindent
III.~ $50~cm<L_{T}\leq3~m$ $\blacktriangleright$ Almost $40\%$ of the total events
appear in this region. The associated electrons and hadronic jets may or may not get
detected in this situation depending on the length of the displaced vertices. However,
the associated muons will leave tracks either in the muon chamber only or in
the muon chamber along with matching tracks in the inner detector also.
The number of signal events in this region are also given
in table \ref{tabevents-cut}.
\vspace{0.2cm}
\noindent
IV.~ $3~m<L_{T}\leq6~m$ $\blacktriangleright$ There exist some number of
events $(\sim 10\%$ of the total number of events) which appear only in the
territory of the muon chamber. In this case the associated electrons get
absorbed in the thick iron yoke of the muon chamber and thus escape detection.
Besides, it is also difficult to identify the hadronic jets as jets in the muon
chamber, rather they appear as noise. The muons are, however leave visible tracks
in the muon chamber {\it{only}} indicating their {\it{stand-alone}} natures.
It is indeed difficult for the conventional triggers to work for this specific signal,
rather this asks for a dedicated special trigger which we believe is a challenging task
for experimentalists. The corresponding number of events in this region for BP-2
are shown in table \ref{tabevents-cut}.
\vspace{0.2cm}
\noindent
V.~ $L_{T}\geq7~m$ $\blacktriangleright$ There also exist a small number of
events $(\sim 4\%)$ where decays occur outside the detector and yield conventional
missing energy signature.
\begin{table}[ht]
\centering
\begin{tabular}{ c || c || c c c}
\hline \hline
& & & No. of events & \\
$\sqrt{s}$ & signal & $L_{T_1}$ & $L_{T_2}$ & $L_{T_3}$\\ \hline\hline
& $\ge 4j + \geq 2\ell + \not\!\!{p_T} \le 30~\rm{GeV}$
&45 &69 &17 \\
7 & $\ge 4j + \geq 2\mu + \not\!\!{p_T} \le 30~\rm{GeV}$
&27 &38 &11 \\
$\rm{TeV}$ & $\ge 4j + \geq 2e + \not\!\!{p_T} \le 30~\rm{GeV}$
&6 &10 &2 \\
& $\ge 4j + 1e +1\mu + \not\!\!{p_T} \le 30~\rm{GeV}$
&12 &21 &4 \\ \hline\hline
& $\ge 4j + \geq 2\ell + \not\!\!{p_T} \le 30~\rm{GeV}$
&234 &373 &98 \\
14 & $\ge 4j + \geq 2\mu + \not\!\!{p_T} \le 30~\rm{GeV}$
&128 &218 &58 \\
$\rm{TeV}$ & $\ge 4j + \geq 2e + \not\!\!{p_T} \le 30~\rm{GeV}$
&37 &45 &16 \\
& $\ge 4j + + 1e +1\mu + \not\!\!{p_T} \le 30~\rm{GeV}$
&69 &113 &24\\
\hline \hline
\end{tabular}
\caption{\label{tabevents-cut}
Number of signal events for $\mathcal{L} =$ $5 ~\rm{fb}^{-1}$
for $\sqrt{s}=7$ and $14~\rm{TeV}$ at different ranges of the decay length for BP-2
with $1~cm<L_{T_1}\le50~cm$, $50~cm<L_{T_2}\le3~m$ and $3~m<L_{T_3}\le6~m$.
$L_{T_i}$s are different transverse decay lengths.}
\end{table}
The number of events for different length of displaced vertices as addressed
earlier are shown in table \ref{tabevents-cut} both for center-of-mass energy
$7$ and $14$ TeV with an integrated luminosity of $5 ~\rm{fb}^{-1}$. Since this
is a background free signal, even with this number of events this spectacular signal can lead to
discovery at 14 TeV run of the LHC with $\mathcal{L} =$ 5 fb$^{-1}$. At 7 TeV the
situation looks less promising and higher luminosity might be required for
discovering such an event. Distribution of the transverse decay length is shown
by figure \ref{Decay-Length}.
\begin{figure}[ht]
\centering
\includegraphics[width=6.05cm,angle=-90]{FigsPDF/DL5fb10l.pdf}
\caption{Transverse decay length distribution of $\ntrl1$ for $\sqrt{s} = 7$
and $14$ TeV with BP-2 for a typical detector size $\sim$ $10~m$ with $\mathcal{L}=$
$5 ~\rm{fb}^{-1}$. Minimum bin size is $10~cm$.
The signal is given by eqn.(\ref{signal-choice}).}
\label{Decay-Length}
\end{figure}
In summary, this signal can give rise to non-standard activities
in the muon chamber with two muons and four hadronic jets. There are, however,
number of events which can leave their imprints not only at the
muon chamber but also in the inner tracker and calorimeters concurrently.
Integrating these two signatures can lead to discovery of an unusual signal
of Higgs boson at the $14$ TeV run of the LHC.
Though with higher luminosity discovery at $\sqrt{s}=7$ TeV is also possible. Indubitably,
development of new triggers and event reconstruction tools are essential.
It is also important to note that the average decay length for a singlino like LSP
is determined by the LSP mass as well as by a set of parameters
$(\lambda,\kappa,v^c,Y^{ii}_\nu,v'_i)$ so
that the constraints on neutrino masses and mixing are satisfied.
Here $v^c$ and $v'_i$ stand for the vacuum expectation values of
the right and left-handed sneutrino fields.
\section{{\bf C}orrelations with neutrino mixing angles}\label{neut-corr}
One of the striking features in $\mu\nu$SSM is that certain ratios of branching fractions of
the LSP decay modes are correlated with the neutrino mixing angles \cite{c6Ghosh:2008yh,c6Bartl:2009an}.
These correlations have been explored in details in chapter \ref{munuSSM-LSP}. A
consequence of the correlation with solar mixing angle $\theta_{12}$
implies $n_{\mu} > n_{e}$ in the final state. Figure \ref{lmlt} shows the lepton
multiplicity distribution for inclusive $\geq 2\ell$ ($\geq 2 \mu~+\geq 2 e+1\mu,1e$)
and exclusive ($\geq 2 \mu$, $\geq 2 e$) for BP-2, without the signal criteria
(eqn.(\ref{signal-choice})). Muon dominance of the higher histograms
(without any isolation cuts) continues to the lower ones even after the application
of $\Delta R_{\ell j},~\Delta R_{\ell \ell}$ cuts. Consequently we observe that the
correlation between $n_e$ and $n_{\mu}$ also appears in the lower
histograms (figure \ref{lmlt}) with a ratio $n_e:n_\mu$ $\sim$ $1:3$.
\begin{center}
\begin{figure}[ht]
\centering
\includegraphics[width=10.55cm]{FigsPDF/lmlt.pdf}
\caption{Lepton multiplicity distribution of signal
for $\sqrt{s} = 7$ and $14$ TeV with $1~\rm{fb}^{-1}$
of integrated luminosity.}
\label{lmlt}
\end{figure}
\end{center}
We present number of events for final state signal (eqn.(\ref{signal-choice}))
in table \ref{tabevents} both for $\sqrt{s}=7$ and $14$ TeV for $\mathcal{L} = 5~ {\rm fb}^{-1}$,
without a cut on the actual $\ntrl1$ decay position (like table \ref{tabevents-cut}). It is important
to note from table \ref{tabevents} that the correlation between $n_e$ and $n_{\mu}$ in the final
state is still well maintained, similar to what was shown in the lower histograms of
figure \ref{lmlt} even with the final state signal topology (eqn.(\ref{signal-choice})).
\begin{table}[ht]
\centering
\begin{tabular}{ c || c || c c c c}
\hline \hline
$\sqrt{s}$ & signal & BP-1 & BP-2 & BP-3 & BP-4 \\ \hline\hline
& $\ge 4j + \geq 2\ell + \not\!\!{p_T} \le 30~\rm{GeV}$
&181 &153 &170 &173 \\
7 & $\ge 4j + \geq 2\mu + \not\!\!{p_T} \le 30~\rm{GeV}$
&100 &85 &97 &100 \\
$\rm{TeV}$ & $\ge 4j + \geq 2e + \not\!\!{p_T} \le 30~\rm{GeV}$
&27 &23 &21 & 23\\
& $\ge 4j + 1e +1\mu + \not\!\!{p_T} \le 30~\rm{GeV}$
&54 &46 &52 &50\\ \hline\hline
& $\ge 4j + \geq 2\ell + \not\!\!{p_T} \le 30~\rm{GeV}$
&1043 &878 & 951& 929 \\
14 & $\ge 4j + \geq 2\mu + \not\!\!{p_T} \le 30~\rm{GeV}$
&580 &463 & 533 & 513\\
$\rm{TeV}$ & $\ge 4j + \geq 2e + \not\!\!{p_T} \le 30~\rm{GeV}$
&160 &139 & 121&129 \\
& $\ge 4j + + 1e +1\mu + \not\!\!{p_T} \le 30~\rm{GeV}$
&306 &279 & 300&290 \\
\hline \hline
\end{tabular}
\caption{\label{tabevents}
Expected number of events of signals for $\mathcal{L} =$ $5 ~\rm{fb}^{-1}$
for $\sqrt{s}=7$ and $14~\rm{TeV}$.}
\end{table}
\section{{\bf I}nvariant mass}\label{inv-mass}
It has been already argued in section \ref{coll-ana} that with a trustworthy
detection of the two isolated and displaced muons and(or) electrons and four associated
hadronic jets a background free signal of this kind can lead to
definite discovery. We have already discussed about the possibility
for invariant mass reconstruction using those leptons and jets, not only
for a singlino like LSP but also for $h^0$. Results of invariant mass reconstruction
for $\ntrl1$ and $h^0$ for BP-2 are shown in figure \ref{invariant-masses}.
We choose $jj\ell$ invariant mass $M(jj\ell)$ for $m_{\ntrl1}$ reconstruction.
Reconstruction of $m_{h^0}$ was achieved through $M(jjjj\ell\ell)$,
invariant mass of $jjjj\ell\ell$ (see eqn.(\ref{signal-choice})).
We take the jets and leptons from the window of
$35\, \rm{GeV}\, \leq M(jj\ell)\leq \, 45\, \rm{GeV}$ to construct
$M(jjjj\ell\ell)$. Even a narrow window like this cannot kill all the
combinatorial backgrounds. As a corollary, effect of combinatorial background
for $m_{\ntrl1}$ reconstruction ($^4C_2$ for $j$ and $^2C_1$ for $\ell$)
also causes long tail for Higgs mass distribution.
\begin{figure}[ht]
\centering
\includegraphics[width=8.55cm]{FigsPDF/invmass.pdf}
\caption{Invariant mass distribution for (a) $\ntrl1$ $(jj\ell )$
and (b) the Higgs boson $(jjjj\ell\ell )$. Plots are shown for
$\sqrt{s} = 7$ and $14$ TeV with $1~\rm{fb}^{-1}$
of integrated luminosity.
Number of events for reconstructing
$m_{\ntrl1}$ for $\sqrt{s} = 7(14)$ TeV
are scaled by a multiplicative factor $4(7)$.}
\label{invariant-masses}
\end{figure}
In conclusion, we have studied an unusual but spectacular signal of
Higgs boson in supersymmetry. This signal can give rise to non-standard activities
in the muon chamber with two muons and four hadronic jets. There are, however,
number of events which can leave their imprints not only at the
muon chamber but also in the inner tracker and calorimeters concurrently.
Integrating these two signatures can lead to discovery of an unusual signal
of Higgs boson at the 14 TeV run of the LHC.
Though with higher luminosity discovery at $\sqrt{s}=$7 TeV is also possible. Indubitably,
development of new triggers and event reconstruction tools are essential. This signal is
generic to a class of models where gauge-singlet neutrinos and $\not\!\!{R_p}$ take part simultaneously
in generating neutrino masses and mixing. Another interesting feature of this study is that
the number of muonic events in the final state is larger than the number of electron events
and the ratio of these two numbers can be predicted from the study of the neutrino mixing angles.
\chapter*{\sffamily{{\bf A}bstract}}
Weak scale supersymmetry has perhaps become the most popular choice for
explaining new physics beyond the standard model. An extension beyond the
standard model was essential to explain issues like gauge-hierarchy
problem or non-vanishing neutrino mass. With the initiation
of the large hadron collider era at CERN,
discovery of weak-scale supersymmetric particles and, of course, Higgs boson
are envisaged. In this thesis we try to discuss certain phenomenological aspects
of an $R_p$-violating non-minimal supersymmetric model, called $\mu\nu$SSM. We show that
$\mu\nu$SSM can provide a solution
to the $\mu$-problem of supersymmetry and can simultaneously accommodate the
existing three flavour global data from neutrino experiments even at the tree
level with the simple choice of flavour diagonal neutrino Yukawa couplings.
We show that it is also possible to achieve different mass hierarchies
for light neutrinos at the tree level itself. In $\mu\nu$SSM, the effect of
$R$-parity violation together with a seesaw mechanism with TeV scale right-handed
neutrinos are instrumental for light neutrino mass generation. We also
analyze the stability of tree level neutrino masses and mixing with
the inclusion of one-loop radiative corrections. In addition, we investigate
the sensitivity of the one-loop corrections to different light neutrino mass orderings.
Decays of the lightest supersymmetric particle were also computed and ratio of certain
decay branching ratios was observed to correlate with certain neutrino mixing
angle. We extend our analysis for different natures of the lightest
supersymmetric particle as well as with various light neutrino mass hierarchies.
We present estimation for the length of
associated displaced vertices for various natures of the lightest
supersymmetric particle which can act as a discriminating feature
at a collider experiment.
We also present an unconventional signal of Higgs boson in supersymmetry
which can lead to a discovery, even at the initial stage
of the large hadron collider running. Besides, we show that a signal of this
kind can also act as a probe to the seesaw scale. Certain other phenomenological
issues have also been addressed.
\chapter{ \sffamily{{\bf
}}\label{appenF}
\section{The ${\mathbf{B_0,~B_1}}$ functions}\label{b0-b1}
The $B_0$ and $B_1$ functions are Passarino-Veltman \cite{App'tHooft:1978xw,AppPassarino:1978jh}
functions defined in the notation of \cite{AppHahn:1998yk} as
\begin{eqnarray}{\label{Passarino-Veltman-Functions}}
\frac{i}{16 {\pi}^2} {B_0(p^2,m^2_{f^\prime_k},m^2_{b_r})} &=&
{(\mu^2)^{4-D}}\int \frac{d^Dq}{(2\pi)^D} \frac{1}{(q^2-m^2_{f^\prime_k})((q+p)^2-m^2_{b_r})},\nonumber \\
\frac{i}{16 {\pi}^2} {B_{\mu}(p^2,m^2_{f^\prime_k},m^2_{b_r})} &=&
{(\mu^2)^{4-D}}\int \frac{d^Dq}{(2\pi)^D} \frac{q_{\mu}}{(q^2-m^2_{f^\prime_k})((q+p)^2-m^2_{b_r})},\nonumber \\
{B_{\mu}(p^2,m^2_{f^\prime_k},m^2_{b_r})} &=& p_{\mu} {B_1(p^2,m^2_{f^\prime_k},m^2_{b_r})}.\nonumber \\
\end{eqnarray}
$D$ is the dimension of the integral. In the $D$ dimension mass dimension $[M]$ for a fermion
is $[M]^{\frac{D-1}{2}}$ and that of a scalar is $[M]^{\frac{D-2}{2}}$. Consequently, the
4-dimensional couplings are scaled by a factor $(\mu^2)^{4-D}$, where $[\mu]=[M]$.
\chapter{ \sffamily{{\bf S}ummary and Conclusion
}}\label{con-sum}
The standard model (SM) of particle physics has already been firmly
established as one of the very successful theories in physics
as revealed by a host of experiments. However, there are issues
where the SM is an apparent failure. Perhaps the severe most problem
of the SM is that the scalar mass is not protected by any symmetry arguments.
Thus the Higgs boson mass (only scalar in the SM) can be as large as the
Planck scale with radiative corrections. It appears that in the SM an unnatural
fine-tuning in the Higgs sector is essential for a Higgs boson mass consistent
with the requirements of theory and experiment.
On the other side, non-vanishing
neutrino masses as have been confirmed by experiments, are impossible
to explain with the SM alone. These shortcomings, as discussed in chapter \ref{SM},
ask for some new physics requirement at and beyond the TeV scale.
As a candidate theory to explain new physics beyond the TeV scale together
with solutions to the drawbacks of the SM, supersymmetry has sought
tremendous attention for the last few decades.
A supersymmetric theory includes new particles
having spin difference of half-integral unit with that of the SM members.
The scalar masses are no longer unprotected and consequently the Higgs boson mass
remains free from quadratic divergences under radiative corrections.
However, missing experimental evidences for sparticles have confirmed
that supersymmetry must be broken in nature so that sparticles remain
heavier compared to their SM partners. It was pointed out in chapter \ref{susy}
that supersymmetry must be broken softly,
so that only logarithmic divergences appear in the
Higgs boson mass which requires sparticle masses around
the TeV scale. This is the prime reason why the discovery of TeV scale superpartners are highly
envisaged at the LHC. The definite mechanism
for supersymmetry breaking remains yet a debatable issue and consequently different
mechanisms exist in literature.
Turning our attention to the neutrino sector it appears that it is possible
to accommodate massive neutrinos in supersymmetric theories either through
$R$-parity violation or using seesaw mechanisms with extra particle content.
It must be emphasized here that in spite of being successful in solving some of the
drawbacks of the SM, supersymmetric theories
are also not free from shortcomings, which in turn result in a wide variant of
models. To mention one, as briefly reviewed in chapter \ref{susy}, the non-minimal supersymmetric
standard model was required to propose a solution to the $\mu$-problem of the minimal version.
Issues of the neutrino masses and mixing remain the prime focus of this thesis.
Requirement of massive neutrinos were essential to explain phenomena like
atmospheric and solar neutrino problem as observed in oscillation experiments.
From experimental constraints, a neutrino mass is expected to be very small.
So it remains to be answered how one can generate tiny neutrino masses consistent with the
oscillation data. Moreover, it also remains to be answered whether the neutrinos
are Dirac or Majorana particles by nature. We review these issues in
chapter \ref{neut} along with different mechanism of light neutrino mass generation
both in supersymmetric and non-supersymmetric theories. The seesaw mechanisms
turn out to be the simple most ways to generate small neutrino masses both in
supersymmetric and non-supersymmetric theories at the cost of enhanced particle
content. But there also exists models of neutrino mass generation through radiative
effects. On the contrary, neutrino mass generation through the violation of $R$-parity
is a pure supersymmetric phenomena without any SM analogue.
Sources of $R$-parity violation can be either spontaneous or explicit.
In the conventional
$R$-parity violating (bilinear and trilinear) models loop corrections are unavoidable
to accommodate neutrino data. Bilinear $R$-parity violating models of neutrino mass generation
have one more striking feature, that is the existence of nice correlations
between the neutrino mixing angles and the lightest supersymmetric particle decay modes.
In addition decays of the lightest supersymmetric particle for these class of models
produce measurable displaced vertices which together with the fore stated correlations
can act as a very promising probe for these models at the colliders.
All of these spectacular features of the $R$-parity
violating models have made them perhaps the most well studied models in the context
of supersymmetry.
Apart from inevitable loop corrections to satisfy three flavour
neutrino data, models with bilinear $R$-parity violation suffer from the naturalness
problem similar to the $\mu$-problem, which is better known as the
$\epsilon$-problem. A new kind of supersymmetric model of neutrino mass generation with
a simultaneous solution to the $\mu$-problem has been introduced in chapter \ref{munuSSM-neut}.
This model is known as the $\mu\nu$SSM which by virtue of an imposed $Z_3$ symmetry
is completely free from naturalness
problem like $\mu$ or $\epsilon$-problem.
$\mu\nu$SSM introduces the gauge singlet right-handed neutrino
superfields (${\hat\nu}^c_i$) to solve
the $\mu$ problem in a way similar to that of NMSSM. These right-handed neutrinos
are also instrumental for light neutrino mass generation in $\mu\nu$SSM.
The terms in the superpotential involving the ${\hat\nu}^c_i$ include the neutrino
Yukawa couplings, the trilinear interaction terms among the singlet neutrino
superfields as well as a term which couples the Higgs superfields to the
$\hat\nu^c_i$. In addition, there are corresponding soft SUSY breaking terms in
the scalar potential. When the scalar components of ${\hat\nu}^c_i$ get VEVs
through the minimization conditions of the scalar potential, an effective
$\mu$ term with an EW scale magnitude is generated.
Explicit $\not\!\!{R_p}$ in $\mu\nu$SSM through lepton number violation
both in the superpotential and in the soft supersymmetry
breaking Lagrangian result in enlarged $(8\times8)$ scalar,
pseudoscalar and charged scalar squared mass matrix. Also
the neutralino and chargino mass matrices are enhanced.
Small Majorana masses of the active neutrinos are generated due
to the mixing with the neutralinos as well as due to the seesaw mechanism
involving the gauge singlet neutrinos. In such a scenario,
we show that it is possible to provide a theory of neutrino masses and mixing
explaining the experimental data, even with a flavour diagonal neutrino Yukawa
coupling matrix, without resort to an arbitrary flavour structure in the
neutrino sector. This essentially happens because of the mixing involved in
the neutralino-neutrino (both doublet and singlet) system mentioned above.
Light neutrino mass generation in $\mu\nu$SSM
is a combined effect of $R$-parity violation and a TeV scale seesaw mechanism using
right handed neutrinos. Alternatively, as shown in chapter \ref{munuSSM-neut}
a combined effect of Type-I and Type-III seesaw mechanisms are instrumental for neutrino
mass generation in the $\mu\nu$SSM.
The TeV scale seesaw mechanism itself is very interesting since it may
provide a direct way to probe the gauge singlet neutrino mass scale at the
LHC and does not need to introduce a very high energy scale in the theory,
as in the case of GUT seesaw.
We present a detailed analytical and numerical work and show
that the three flavour neutrino data can be accommodated in such a scenario.
In addition, we observe that in this model different neutrino mass hierarchies
can also be obtained with correct mixing pattern, at the tree level.
Though all three neutrinos acquire masses at the tree
level, it is always important to judge the stability of tree level analysis
in the exposure of radiative corrections. In chapter \ref{munuSSM-neut} effect of the complete
set of one-loop corrections to the light neutrino masses and mixing are considered. The
tree level and the one-loop corrected neutrino mass matrix are observed to posses similar
structure but with different coefficients arising from the loop corrections.
The effects of one-loop corrections are found
to be capable of altering the tree level analysis
in an appreciable manner depending on the concerned
mass hierarchy. We also explore different regions in the SUSY parameter
space, which can accommodate the three patterns in turn.
In conclusion, $\mu\nu$SSM can accommodate neutrino masses and mixing consistent
with the three flavour global neutrino data for different mass hierarchies at the
tree level itself even with the choice of flavour diagonal neutrino Yukawa
couplings. Inclusion of one-loop radiative corrections to light neutrino masses
and mixing can alter the results of tree level analysis in a significant manner,
depending on the concerned mass orderings.
Correlations between the light neutrino mixing angles with the
ratios of certain decay branching ratios of the lightest supersymmetric
particle (usually the lightest neutralino for a large region of
the parameter space)
in $\mu\nu$SSM have been explored in chapter \ref{munuSSM-LSP}.
These correlations are very similar to the bilinear $\not\!\!{R_p}$ models and
have drawn immense
attention as a test of neutrino mixing at a collider experiment.
However, there exist certain differences between these two scenarios.
In $\mu\nu$SSM
lepton number is broken explicitly in the superpotential by terms which are trilinear
as well as linear in singlet neutrino superfields. In addition to that there are
lepton number conserving terms involving the singlet neutrino superfields with dimensionless neutrino
Yukawa couplings. After the electroweak symmetry breaking these terms can generate the effective bilinear
R-parity violating terms as well as the $\Delta L$ =2 Majorana mass terms for the singlet neutrinos in the
superpotential. In general, there are corresponding soft supersymmetry breaking terms in the scalar potential.
Thus the parameter space of this model is much larger compared to the bilinear R-parity violating model.
Hence, in general, one would not expect a very tight correlation between the neutrino mixing angles and the
ratios of decay branching ratios of the LSP. However, under certain simplifying assumptions
one can reduce the number of free parameters and in those cases it is possible that the above
correlations reappear. As mentioned earlier, we have studied these correlations
in great detail for the two body $\ell^\pm W^\mp$
final states. These nice correlations are lost in the general scenario of bilinear-plus-trilinear
R-parity violation.
Another important difference between $\mu\nu$SSM and the bilinear R-parity violating model in the context of
the decay of the LSP (assumed to be the lightest neutralino in this case) is that in $\mu\nu$SSM the lightest
neutralino can have a significant singlet neutrino ($\nu^c$) contribution. In this case, the correlation between
neutrino mixing angles and decay branching ratios of the LSP is different compared
to the cases when the dominant component of the LSP is either a bino, or a higgsino or a Wino. This gives us a
possibility of distinguishing between different R-parity violating models through the observation of the
decay branching ratios of the LSP in collider experiments. In addition, the decay of
the lightest neutralino will show displaced vertices in collider experiments and when the lightest neutralino
is predominantly a singlet neutrino, the decay length can be of the order of a few meters for a lightest
neutralino mass in the neighbourhood of $50$ GeV. This is very different from the bilinear R-parity
violating model where for a Bino LSP of similar mass the decay length is less than or of the order of a meter
or so.
In a nutshell we have studied the correlations among the ratio of the branching
ratios of the lightest supersymmetric particle decays into $W$-boson and a charged lepton
with different relevant parameters. These correlations are analysed for different natures
of the lightest neutralino which is usually the lightest supersymmetric particle for
a novel region of the parameter space. Besides, effect of different light neutrino mass hierarchies
in the correlation study are
also taken into account. These spectacular and nice correlations together with a measurement of the
displaced vertices can act as an important experimental signature for the $\mu\nu$SSM.
We shift our attention to a different aspect of the $\mu\nu$SSM in chapter \ref{munuSSM-Higgs},
where a new kind of unconventional signal for the Higgs boson in supersymmetry has
been advocated. The basic idea lies in the fact that with suitable choice of model parameters
a right handed neutrino like lightest supersymmetric particle is possible in the $\mu\nu$SSM
and a pair of such gauge singlet fermions can couple to a MSSM like Higgs boson. We show that
with heavy squark and gluino masses,
pair production of the right handed neutrino like lightest supersymmetric particles from the
decay of a MSSM like Higgs boson, produced in the gluon fusion channel at the LHC can
be the dominant source for singlino pair production.
We analyze a specific final state signal with two isolated
and displaced leptons (electron and(or) muon) and four isolated and displaced hadronic jets
arising from the three body decay of a pair of right handed neutrino like
lightest supersymmetric particles.
This particular final state has the advantage of zero missing energy since no neutrinos
appear in the process and thus a reliable Higgs boson mass reconstruction as well as
the same for a right handed neutrino are highly envisaged.
Appearance of reasonably large displaced vertices associated with the gauge singlet nature
of a right handed neutrino are extremely useful to abate any SM backgrounds for this proposed
signal. Besides, presence
of the definite entry and the exit points for the cosmic muons also helps to discriminate this
signal from the cosmic muon signature. Depending on the length of the associated displaced
vertices this rare signal can either leave its imprints in the entire detector, starting
from the tracker to the muon chamber with conventional global muon signature or can leave
visible tracks in the muon chamber only from stand alone muons. The latter case also
requires development of special kind of triggers. Combining the two fore mentioned scenarios
a discovery with this signal criteria is expected with the $14$ TeV run of the LHC.
This unusual signal is also
testable in the $7$ TeV LHC run but requires much higher luminosity
compared to the $14$ TeV scenario. Ratio of the number
of electrons to that of the muons in the final state signal is again observed to show
correlation with the concerned neutrino mixing angle. We present a set of four benchmark points
where the neutrino data are satisfied up to one-loop level. Apart from the Higgs discovery,
a signal of this kind with a faithful mass reconstruction for right handed neutrino like
lightest supersymmetric particle offers a possibility to probe the seesaw scale which is
one more appealing feature of the $\mu\nu$SSM. It must be emphasized here that though
we performed this analysis with tree level Higgs boson mass in $\mu\nu$SSM, but
even for loop corrected Higgs boson mass our general
conclusions will not change for a singlino LSP in the
mass range $20-60$ GeV.
To conclude, $\mu\nu$SSM is a potential candidate for explaining physics beyond the standard
model. Not only this model can accommodate massive neutrinos consistent with the three
flavour global data but at the same time also offers a solution to the $\mu$-problem
of supersymmetry with the use of same set of right handed neutrino superfields. This model
is also phenomenologically very rich and can yield new kind of signatures at collider experiments.
Diverse interesting aspects of the $\mu\nu$SSM have been addressed in this thesis
and more studies are expected to reveal more phenomenological wonders in the near future. There
are a host of areas yet to be explored for this model like the effect of complete one-loop corrections
in the scalar sector, more detailed analysis of new kind of Higgs signal at the colliders, a
comparative study of different lightest supersymmetric particle scenarios in the context
of an accelerator experiment and many more. In a nutshell,
with the LHC running at the corner we expect to explore more wonders of the $\mu\nu$SSM.
\chapter*{\sffamily{ {\bf L}ist of Publications}}
\vspace*{0.02cm}
\begin{center}
{\bf{\sf{\underline{In refereed journals}}}}.
\end{center}
\vspace*{0.02cm}
\begin{itemize}
\item
{\textbf{Radiative
contribution to neutrino masses and mixing in $\mu\nu$SSM.}}\\
{{\em Pradipta Ghosh}},~{\em Paramita Dey},~{\em Biswarup Mukhopadhyaya},~{\em Sourov Roy.}\\
{\bf \hyperref{http://dx.doi.org/10.1007/JHEP05(2010)087}{\textit{Journal
of High Energy Physics} 05 (2010) 087},
\hyperref{http://arxiv.org/abs/arXiv:1002.2705}{arXiv:1002.2705 [hep-ph]}.}
\vspace*{1cm}
\item
{\textbf{Neutrino masses and mixing, lightest neutralino decays
and a solution to the $\mu$ problem in supersymmetry.}}\\
{{\em Pradipta Ghosh}},~{\em Sourov Roy.}\\
{\bf \hyperref{http://dx.doi.org/10.1088/1126-6708/2009/04/069}{\textit{Journal
of High Energy Physics} 04 (2009) 069},
\hyperref{http://arxiv.org/abs/arXiv:0812.0084}{arXiv:0812.0084 [hep-ph]}.}
\end{itemize}
\vspace*{0.02cm}
\begin{center}
{\bf{\sf{\underline{Preprints}}}}.
\end{center}
\vspace*{0.02cm}
\begin{itemize}
\item
{\textbf{An unusual signal of Higgs boson in supersymmetry at the LHC.}}\\
{{\em Priyotosh Bandyopadhyay}},~{{\em Pradipta Ghosh}},~{\em Sourov Roy.}\\
{\bf \hyperref{http://dx.doi.org/10.1103/PhysRevD.84.115022}{\textit{Phys. Rev. D} 84 (2011) 115022},
\hyperref{http://arxiv.org/abs/arXiv:1012.5762}{arXiv:1012.5762 [hep-ph]}\footnote{Now published
as ``Unusual Higgs boson signal in R-parity violating nonminimal
supersymmetric models at the LHC'' in {\bf \hyperref{http://dx.doi.org/10.1103/PhysRevD.84.115022}{\textit{Phys. Rev. D} 84 (2011) 115022}.} }.}
\end{itemize}
\vspace*{0.02cm}
\begin{center}
{\bf{\sf{\underline{In proceedings}}}}.
\end{center}
\vspace*{0.02cm}
\begin{itemize}
\item
{\textbf{Neutrino masses and mixing in $\mu\nu$SSM.}}\\
{\hyperref{http://dx.doi.org/10.1088/1742-6596/259/1/012063}{\textit{2010
J. Phys.: Conf. Ser.} 259 012063},
\hyperref{http://arxiv.org/abs/arXiv:1010.2578}{arXiv:1010.2578 [hep-ph]},}
~{PASCOS 2010}
\item
{\textbf{Neutrino masses and mixing in a supersymmetric model
and characteristic signatures at the LHC.}}\\
{Proceedings of the XVIII DAE-BRNS symposium, Vol.~18, 2008, 140-143.}
\end{itemize}
|
train/arxiv
|
BkiUc804eIXhtfm3O4LC
| 4 | 0.8 |
\section{Introduction}
The Poisson problem with mixed Dirichlet-Neumann boundary conditions deals with conductivity, heat transfer, metallurgical melting, wave phenomena, elasticity and electrostatics in mathematical physics and engineering. The detailed applications can be found in \cite{Dauge}, \cite{Fabrikant}, \cite{Jochmann}, \cite{Lagnese}, \cite{Mazya}, \cite{Rempel}, \cite{Simanca}, \cite{Sneddon}, \cite{Wendland} and the references therein. A common problem of interest found in the literature is the following mixed boundary value (MBVP).
\begin{eqnarray}\label{m1}
\begin{split}
L u&= h~\text{in}~\Omega,\\
u&= f~\text{on}~\Gamma_1,\\
Mu&= g~\text{on}~\Gamma_2,
\end{split}
\end{eqnarray}
where, $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^N$ for $N\geq3$. The boundary of $\Omega$, which will be denoted by $\Gamma$, is the disjoint union of $\Gamma_1$ and $\Gamma_2$ which are subsets of $\Gamma$ such that $\overline{\Gamma}_1\cup\Gamma_2=\Gamma_1\cup\overline{\Gamma}_2=\Gamma$ and $\Gamma_1\cap\Gamma_2=\emptyset$. Further, $L$ is a second order elliptic operator, $M$ is a general first order oblique differential operator on $\Gamma_2$.\\
Lieberman \cite{Lieberman 1,Lieberman 2} considered the problem $\eqref{m1}$ and proved the existence and H\"{o}lder continuity of classical solutions with smooth data. The techniques used in the corresponding Dirichlet problem ($\Gamma_2=\emptyset$) and oblique derivative problem ($\Gamma=\Gamma_2$) of $\eqref{m1}$ are helpful to show the existence of solutions to the mixed boundary value problem. It is worth to mention the work due to Azzam and Kreyszig \cite{Azzam}, as they have provided the regularity result for MBVP in a plane domain with corners, where the Dirichlet data belongs to $C^{2,\alpha}(\Gamma\setminus\{0\})$ and the remaining boundary data is in $C^{1,\alpha}(\Gamma\setminus\{0\})$. The work due to Sykes \cite{Sykes} deals with the boundary regularity of problem $\eqref{m1}$ with Dirichlet and Neumann boundary conditions where $L$ is the Laplacian operator, $h=0$ in $\Omega$,
$g\in L^p(\Gamma_2)$, $f\in W^{1,p}(\Gamma_1)$ for $1<p\leq2$ and the angle between $\Gamma_1$, $\Gamma_2$ should be strictly less than $\pi$ in the interface. Sykes \cite{Sykes} drew motivation from Brown \cite{Brown}. who considered the two boundary data as $f\in H^1(\Gamma_1)$ and $g\in L^2(\Gamma_2)$. \\
Not much of literature is found for MBVP involving a measure data, although Liang and Rodrigues \cite{Liang} considered a problem involving measure data both on the domain and on the boundary $\Gamma_2$. Some work has been done by Gallou\"{e}t \cite{Gallouet}, where the non linearity lies on the boundary with measure supported on the domain $\Omega$ and on the boundary $ \Gamma_2$. The MBVP in \cite{Gallouet} posessess a weak solution $u$ in $ W^{1,q}(\Omega),~\forall~ 1<q<\frac{N}{N-1}$ and the trace of $u$ on $\Gamma$ lies in $W^{1-\frac{1}{q},q}(\Gamma)$, $\forall 1\leq q< \frac{N}{N-1}$.\\
In this article we have considered the following two mixed boundary value problems. The first problem (P1) is
\begin{eqnarray}\label{q1}
\begin{split}
-\Delta u-\lambda^2 u&= h~\text{in}~\Omega,\\
u&= f_1~\text{on}~\Gamma_1,\\
\frac{\partial u}{\partial \hat{n}}&= f_2~ \text{on}~\Gamma_2,\\
\end{split}
\end{eqnarray}
and
\begin{eqnarray}\label{q2}
\begin{split}
-\Delta u-\lambda^2 u&= 0~\text{in}~\mathbb{R}^N\setminus\bar{\Omega},\\
u&= f_1~\text{on}~\Gamma_1,\\
\frac{\partial u}{\partial \hat{n}}&= f_2~ \text{on}~\Gamma_2,\\
\end{split}
\end{eqnarray}
where $u$ satisfies the following conditions at infinity, i.e. $|x|\rightarrow\infty.$\\
For $\lambda=0$
\begin{equation}\label{zero}
u(x)=O(|x|^{2-N}).
\end{equation}
For $\lambda\neq0$ (Sommerfeld's radiation condition)
\begin{equation}\label{nonzero}
\begin{split}
u(x)&=O(|x|^{\frac{1-N}{2}})\\
\frac{\partial u(x)}{\partial |x|}-i\lambda u(x)&=o(|x|^{\frac{1-N}{2}}).
\end{split}
\end{equation}
The second problem (P2) is
\begin{eqnarray}\label{lq1}
\begin{split}
-\Delta u&= \mu~\text{in}~\Omega,\\
u&= f_1~\text{on}~\Gamma_1,\\
\frac{\partial u}{\partial \hat{n}}&= f_2~ \text{on}~\Gamma_2,\\
\end{split}
\end{eqnarray}
and
\begin{eqnarray}\label{lq2}
\begin{split}
-\Delta u&= 0~\text{in}~\mathbb{R}^N\setminus\bar{\Omega},\\
u&= f_1~\text{on}~\Gamma_1,\\
\frac{\partial u}{\partial \hat{n}}&= f_2~ \text{on}~\Gamma_2,
\end{split}
\end{eqnarray}
where $u$ satisfies
\begin{equation}\label{infinity}
|u(x)|+|x||\nabla u(x)|=O(|x|^{2-N}),~\text{as}~|x|\rightarrow\infty.
\end{equation}
Throughout the article $\frac{\partial u}{\partial \hat{n}}$ will represent the normal derivative with respect to the outward unit normal $\hat{n}$ to the boundary, $f_1\in H^{\frac{1}{2}}(\Gamma_1)$, $f_2\in H^{-\frac{1}{2}}(\Gamma_2)$ are boundary data, $h\in \widetilde{H}^{-1}(\Omega)$, $\mu$ will denote a bounded Radon measure, $\lambda\in\mathbb{C}$ with $Im(\lambda)\geq0$ and $\lambda^2$ will be different from the eigenvalues of the Laplacian ($-\Delta$). We will, at some places, refer problem $\eqref{q1},\eqref{lq1}$ as interior problems (IP1), (IP2) respectively and $\eqref{q2},\eqref{lq2}$ as exterior problems (EP1), (EP2) respectively. This work is motivated from the work of Chang \cite{Chang} and Stephan \cite{Stephan} where the authors have used the method of layer potentials to show the uniqueness of solution to the homogeneous mixed bounadry value problem in both interior and exterior domains. Chang \cite{Chang} has shown that for $h=0$ and $\lambda=0$ the solution $u$ belongs to $H^1(\Omega)$ for the interior problem and belongs to $H^1_{loc}(\mathbb{R}^N\setminus\bar{\Omega})$ for the exterior problem. This $u$ also satisfies the following inequality.
$$\norm{u}_{ H^{1/2}(\Gamma_1)}+\norm{\frac{\partial u}{\partial\hat{n}}}_{ H^{-1/2}(\Gamma_2)}\leq C\{\norm{f_1}_{ H^{1/2}(\Gamma_1)}+\norm{f_2}_{ H^{-1/2}(\Gamma_2)}\}$$
where $C$ is independent of $f_1$, $f_2$ and $h$. The novelty of our work is the consideration of two nonhomogenous mixed boundary value problems and a Radon measure $\mu$ as a nonhomogeneous term in (P2), for which the solution space becomes weaker than the Sobolev space $H^1(\Omega)$.
\section{Preliminary definitions and properties of boundary layer potentials}
We will denote several constants by $C$ which can only depend on $\Omega$, $N$ and independent of the indices of the sequences. The value of $C$ can be different from line to line and sometimes, on the same line.\\
For $1\leq p\leq\infty$ and $k$ be a nonnegative integer, the Sobolev space $\{u\in L^p(\Omega):D^\gamma u\in L^p(\Omega),~\text{for}~|\gamma|\leq k\}$ will be denoted by $W^{k,p}(\Omega)$ \cite{Evans} and the norm on vectors in $W^{k,p}(\Omega)$ is defined as
\begin{center}
$\norm{u}_{W^{k,p}(\Omega)}=\sum_{|\gamma|\leq k}\norm{D^\gamma u}_{L^p(\Omega)}$
\end{center}
where $\Omega$ is a domain in $\mathbb{R}^N$. We denote $W_{loc}^{k,p}(\Omega)$ to be the local Sobolev space such that for any compact $K\subset \Omega$, $u\in W^{k,p}(K)$. For $0<\alpha<1$, we define the Sobolev space $W^{\alpha,p}(\Omega)$ as
$$W^{\alpha,p}(\Omega)=\{u\in L^p(\Omega):\norm{u}_{W^{\alpha,p}(\Omega)}^p=\norm{u}_{L^p(\Omega)}^p+\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{n+p\alpha}}dydx<\infty\}.$$
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^N$, $N\geq3$. We now introduce the following Sobolev spaces. For $p=2$, $s\in\mathbb{R}$ and $0\leq\alpha\leq 1$,
\begin{enumerate}
\item$H^s(\mathbb{R}^N)=\{u:\int_{\mathbb{R}^N}(1+|\xi|^2)^{s/2}\widehat{u}(\xi)e^{i2\pi\xi.x}d\xi \in L^2(\mathbb{R}^N)\},~ \widehat{u}$ is the Fourier transform of $u$. This space is a separable Hilbert space.
\item $H^s(\Omega)=\{u|_{\Omega}:u\in H^s(\mathbb{R}^N)\}$
\item $\widetilde{H}^s(\Omega)=\text{Closure of}~C_0^\infty(\Omega)~\text{in}~H^s(\mathbb{R}^N)$. For further details on these Sobolev spaces one may refer to \cite{Hsiao} Chapter 4.
\item $H^\alpha(\Gamma)=\begin{cases}
\{g|_\Gamma: g\in H^{\frac{1}{2}+\alpha}(\Omega)\},&(0<\alpha\leq1)\\
L^2(\Gamma)&(\alpha=0),\end{cases}$
\item $H^{\alpha}(\Gamma_i)= \{g|_{\Gamma_i}:g\in H^{\alpha}(\Gamma)\}$,
\item $\widetilde{H}^{\alpha}(\Gamma_i)= \{g|_{\Gamma_i}:g\in H^{\alpha}(\Gamma),~\text{supp}(g)\subset\overline{\Gamma}_i\},~i=1,2.$
\end{enumerate}
Let $H^{-\alpha}(\Gamma)$ is the dual space of $H^{\alpha}(\Gamma)$, i.e. $H^{-\alpha}(\Gamma)=(H^{\alpha}(\Gamma))^*$. Equivalently $H^{-\alpha}(\Gamma_i)=(\widetilde{H}^{\alpha}(\Gamma_i))^*$ and $\widetilde{H}^{-\alpha}(\Gamma_i)=(H^{\alpha}(\Gamma_i))^*$, for $i=1,2$. \\
We denote $\langle.,.\rangle_\Gamma$ as the duality pairing between $H^\alpha(\Gamma)$ and $H^{-\alpha}(\Gamma)$ given by $\langle f,g\rangle_\Gamma=\int_\Gamma f(z)g(z)ds_z$ for any $f\in H^\alpha(\Gamma)$ and $g\in H^{-\alpha}(\Gamma)$. Similarly, $\langle.,.\rangle_{\Gamma_i}$ is the duality pairing between $H^{\alpha}(\Gamma_i)$ and $\widetilde{H}^{-\alpha}(\Gamma_i)$ (or $H^{-\alpha}(\Gamma_i)$ and $\widetilde{H}^{\alpha}(\Gamma_i)$), $i=1,2$.\\
Since $\widetilde H^{\alpha}(\Gamma_i)$, $i=1,2$, is a reflexive space, the operator $$J:\widetilde H^{\alpha}(\Gamma_i)\rightarrow (\widetilde H^{\alpha}(\Gamma_i))^{**}=(H^{-\alpha}(\Gamma_i))^*$$ is a bijection. Hence, for any $f'\in(H^{-\alpha}(\Gamma_i))^*$ there exists a unique $f\in \widetilde H^{\alpha}(\Gamma_i)$ such that $J(f)=f'$. For $g\in H^{-\alpha}(\Gamma_i)$ we define $\langle\langle.,.\rangle\rangle$ by the duality pairing between $H^{-\alpha}(\Gamma_i)$ and $(H^{-\alpha}(\Gamma_i))^*$ such that
$$\langle\langle f',g\rangle\rangle=\langle g,f\rangle_{\Gamma_i}.$$
\begin{definition}
An open set $\Omega\in \mathbb{R}^N$ is said to be a Lipschitz domain if for each $P\in\partial\Omega$ there exist a rectangular coordinate system, $(x,z)$ such that $x\in \mathbb{R}^{n-1}, z\in \mathbb{R}$, a neighborhood $N(P)=N\subset\mathbb{R}^N$ and a function $\varphi:\mathbb{R}^{n-1}\rightarrow\mathbb{R}$ such that
\begin{enumerate}
\item $|\varphi(x)-\varphi(y)|\leq C|x-y|, ~\forall~x,y\in\mathbb{R}^{n-1}$,
\item $ N\cap\Omega=\{(x,z):z>\varphi(x)\}\cap N$.
\end{enumerate}
\end{definition}
\begin{definition}
The Marcinkiewicz space denoted as ${M}^r(\Omega)$ (or weak $L^r(\Omega)$ space), for every $ 0 < r <\infty$, consists of all measurable functions $g:\Omega\rightarrow\mathbb{R}$ such that
$$m(\{x\in \Omega:|g(x)|>b\})\leq \frac{C}{b^r},~b>0,C<\infty,$$
\end{definition}
\noindent where $m$ is the Lebesgue measure. In fact in the case of bounded domain $\Omega$, for any fixed $\bar{r}>0$ we observe ${M}^r(\Omega)\subset {M}^{\bar{r}}(\Omega)$ for $r\geq \bar{r}$. Furthermore, the embeddings
\begin{equation}\label{marcinq}
L^r(\Omega)\hookrightarrow {M}^r(\Omega)\hookrightarrow L^{r-\epsilon}(\Omega),
\end{equation}
is continuous for every $1<r<\infty$ and $0<\epsilon<r-1$.
\begin{definition}\label{defn}
\textbf{(Fredholm operator)} Let $X~ \text{and}~ Y$ are two Banach spaces and $A$ is a bounded linear operator from $X$ to $Y$. Then $A$ is said to be a Fredholm operator if its kernel (ker($A$)) and cokernel (coker($A$)$~=Y/Range(A)$ are finite dimensional.
\end{definition}
\begin{remark}
\begin{enumerate}
\item The ``Fredholmness" of an operator $A$ ensures that Range$(A)$ is closed.
\item The index of a Fredholm operator $A$ is given by ind$(A)$=dim(ker$(A)$)-dim(coker$(A)$).
\end{enumerate}
\end{remark}
\noindent The following two theorems are borrowed from \cite{Driver} which show the relationship between a Fredholm operator and a compact operator.
\begin{theorem}\label{ft1}
For a bounded linear operator $A:X\rightarrow Y$, the following two statements are equivalent\\
1. $A$ is a Fredholm operator.\\
2. $A$ is an invertible modulo compact operators, i.e. there exist compact operators $C_1,C_2$ and an operator $B$ such that $AB=I+C_1$ and $BA=I+C_2$.
\end{theorem}
\begin{theorem}\label{ft2}
If $A$ is a Fredholm opertor then ind$(A)=0$ iff $A=A_1+A_2$, where $A_1$ is an invertible operator and $A_2$ is a compact operator.
\end{theorem}
\begin{definition}
The space of all finite Radon measures on $\Omega\subset\mathbb{R}^N$, is denoted as $\mathcal{M}(\Omega)$. For $\mu\in \mathcal{M}(\Omega)$ we define $$\norm{\mu}_{\mathcal{M}(\Omega)}=\int_{\Omega}d|\mu|,$$
which is called the `Total variation' norm.
\end{definition}
\noindent We now define the weak solution of the first problem (P1).
\begin{definition}
Let $X$ and $Y$ are two test function spaces defined as $X=\{\varphi\in C^1(\bar{\Omega}):\varphi|_{\Gamma_1}=0\}$ and $Y=\{\zeta\in C_c^1(\mathbb{R}^N\setminus(\Omega)):\zeta|_{\Gamma_1}=0~ \text{and satisfies}~ \eqref{zero}~\text{and}~\eqref{nonzero}\}$.
A function $u\in W^{1,1}(\Omega)$ is a weak solution to the problem $\eqref{q1}$ if it satisfies
\begin{equation}
\int_\Omega \nabla u\cdot\nabla \varphi~ -\int_{\Omega}\lambda^2 u\varphi= \int_\Omega h\varphi +\int_{\Gamma_2}f_2\varphi, ~\forall \varphi\in X.\nonumber
\end{equation}
Similarly a function $u\in W^{1,1}_{loc}(\mathbb{R}^N\setminus\bar{\Omega})$ is said to be a weak solution of $\eqref{q2}$ if
\begin{equation}
\int_{\mathbb{R}^N\setminus\bar{\Omega}} \nabla u\cdot\nabla \zeta~ -\int_{\mathbb{R}^N\setminus\bar{\Omega}}\lambda^2 u\zeta=-\int_{\Gamma_2}f_2\zeta, ~\forall \zeta\in Y.\nonumber
\end{equation}
\begin{remark}\label{remark 1}
Hereafter, a subsequence of a sequence will be denoted by the same notation as that of the sequence. Further a solution will always refer to a weak solution.
\end{remark}
\end{definition}
\noindent We further we denote $\Phi$ as the fundamental solution of
Helmholtz equation for $N\geq3$ which satisfies $-\Delta \Phi-\lambda^2 \Phi=\delta$, where $\delta$ is the Dirac distribution and $\Phi$ is
\begin{eqnarray}
\begin{split}
\Phi(x,y)&\begin{cases}
\frac{1}{(N-2)w_N}\frac{1}{|x-y|^{N-2}},&\text{for}~ \lambda=0\\
\frac{e^{i\lambda|x-y|}}{4\pi|x-y|},&\text{for}~ \lambda\neq 0,~ N=3\nonumber\\ \frac{i}{4}\bigg(\frac{{\lambda}}{2\pi(|x-y|)}\bigg)^{\frac{N-2}{2}}H^{(1)}_{\frac{N-2}{2}}(\lambda|x-y|),&\text{for}~\lambda\neq0,~N\geq3
\end{cases}
\end{split}
\end{eqnarray}
for every $x,y\in\mathbb{R}^N$, $x\neq y$. Here $w_N$ is the measure of the unit sphere in $\mathbb{R}^N$ and $H^{(1)}_m$ denotes the Hankel function of the first kind of order $m$. We next define, boundary layer potentials (single layer and double layer) to solve the homogeneous Helmholtz equation in $\mathbb{R}^N$. Let $g_1\in H^{\alpha}(\Gamma),g_2\in H^{-\alpha}(\Gamma)$ for some $0\leq\alpha\leq1$, then the single layer potential is given by,
\begin{align}\label{o}
v_1(x)&= \mathbb{S}_{\lambda}g_2(x)\nonumber\\
&=\int_\Gamma g_2(y)\Phi(x-y) dy ,~\forall~ x\in \mathbb{R}^N\setminus\Gamma
\end{align}
and the double layer potential is by
\begin{align}\label{p}
v_2(x)&=\mathbb{K}_{\lambda}g_1(x)\nonumber\\
&= \int_\Gamma g_1(y)\frac{\partial}{\partial \hat{n}_y}\Phi(x-y) dy,~\forall~x\in\mathbb{R}^N\setminus\Gamma.
\end{align}
where $ \hat{n}_y$ denotes the unit outward normal to the boundary $\Gamma$. We can see that for $x\in\mathbb{R}^N\setminus\Gamma$ the above two kernels are $C^\infty$ functions on $\Gamma$.\\
If $P\in\Gamma$, then $X(P)$ denotes a cone with vertex at $P$ such that one component is in $\Omega$ which is denoted by $X_i(P)$ and the other is in $\mathbb{R}^N\setminus\bar{\Omega}$ denoted by $X_e(P)$.
\begin{definition}
Let $P\in \Gamma$, then we define
$$S_{\lambda}g_2(P)=\int_\Gamma g_2(y)\Phi(P-y) dy $$ and $$K_{\lambda}g_1(P)=\int_\Gamma g_1(y)\frac{\partial}{\partial \hat{n}_y}\Phi(P-y) dy.$$
\end{definition}
\noindent According to the Lemma 3.8 of \cite{Costabel} the boundary values of the two potentials in $\eqref{o}$ and $\eqref{p}$ are given by
\begin{eqnarray}\label{single1}
\begin{split}
v_1^i(P)&=\lim_{X_i(P),x\rightarrow P}\mathbb{S}_{\lambda}g_2(x)\\&=S_{\lambda}g_2(P),
\end{split}\\
\begin{split}\label{single2}
v_1^e(P)&=\lim_{X_e(P),x\rightarrow P}\mathbb{S}_{\lambda}g_2(x)\\&=S_{\lambda}g_2(P)
\end{split}
\end{eqnarray}
and
\begin{equation}\label{double1}
\begin{split}
v_2^i(P)&=\lim_{X_i(P),x\rightarrow P}\mathbb{K}_{\lambda}g_1(x)\\ &=\left(-\frac{1}{2}I+K_{\lambda}\right)g_1(P),
\end{split}
\end{equation}
\begin{equation}\label{double2}
\begin{split}
v_2^e(P)&=\lim_{X_e(P),x\rightarrow P}\mathbb{K}_{\lambda}g_1(x)\\ &=\left(\frac{1}{2}I+K_{\lambda}\right)g_1(P).
\end{split}
\end{equation}
\noindent In case of inhomogeneous Helmholtz equation $-\Delta u-\lambda^2u=h$ in $\Omega$, where $h\in \widetilde{H}^{-1}(\Omega)$. The Newton potential (or Volume potential) appears in the form, \\
$$\mathbb{N}_\lambda h(x)= \int_{\Omega} \Phi(x-y) h(y) dy, ~x\in \mathbb{R}^N.$$
It is well known that the Newton potential $\mathbb{N}_\lambda:\widetilde{H}^{-1}(\Omega)\rightarrow H^{1}(\Omega)$ is a continuous map by \cite{Mclean,Steinbach}. From \cite{Necas} we know the Dirichlet trace operator, $\gamma_D:H^1(\Omega)\rightarrow H^{\frac{1}{2}}(\Gamma)$ and the Neumann trace operator, $\gamma_N:H^1(\Omega)\rightarrow H^{-\frac{1}{2}}(\Gamma)$, are continuous operators.
The Dirichlet trace operator of $\mathbb{N}_\lambda$ denoted as $\gamma_D\mathbb{N}_\lambda$ is given by
$$\gamma_D\mathbb{N}_\lambda(h(P))=\lim_{x\rightarrow P}\mathbb{N}_\lambda h(x), ~\forall P\in \Gamma.$$ Thus
\begin{align}
\norm{\gamma_D\mathbb{N}_\lambda(h)}_{H^{\frac{1}{2}}(\Gamma)}&\leq C\norm{\mathbb{N}_\lambda(h)}_{H^1(\Omega)}\nonumber\\
&\leq C \norm{h}_{\widetilde{H}^{-1}(\Omega)}.\nonumber
\end{align}
The Neumann trace of $\mathbb{N}_\lambda$ is denoted as $\gamma_N\mathbb{N}_\lambda$ and hence it satisfies
$$\norm{\gamma_N\mathbb{N}_\lambda(h)}_{H^{-\frac{1}{2}}(\Gamma)}\leq C \norm{h}_{\widetilde{H}^{-1}(\Omega)}.$$
Let us fix $\alpha=\frac{1}{2}$. Consider the single layer potential $v_1(x)=\mathbb{S}_\lambda{g}_2(x),~ \text{for} ~g_2\in H^{-\frac{1}{2}}(\Gamma) $. Then $v_1$ solves the Helmholtz equation in $\mathbb{R}^N\setminus\Gamma$. Thus $v_1\in H^1(\Omega)$ for (IP1), $v_1\in H_{loc}^1(\mathbb{R}^N\setminus\bar{\Omega})$ for (EP1) and satisfies $\eqref{zero}$-$\eqref{nonzero}$ at infinity. We now define the ouward normal derivative of $v_1$, i.e. $\frac{\partial v_1}{\partial \hat{n}}$ that belongs to $H^{-\frac{1}{2}}(\Gamma)$. Let us choose $h_1,h_2\in H^{\frac{1}{2}}(\Gamma)$. We will denote $h_1^*,~h_2^*$ to be the extensions of $h_1$, $h_2$ respectively such that
\begin{align}\label{cont}
\norm{h_1^*}_{H^1(\Omega)}\leq C
\norm{h_1}_{H^{\frac{1}{2}}(\Gamma)}, ~
\norm{h_2^*}_{H^1(\mathbb{R}^N\setminus\bar\Omega)}\leq C \norm{h_2}_{H^{\frac{1}{2}}(\Gamma)}
\end{align}
for some constant $C>0$ which does not depend on $h_1$ and $h_2$ by \cite{Jonsson}. Define
\begin{equation}\label{normal}
\begin{split}
\left\langle\frac{\partial v_1}{\partial \hat{n}},h_1\right\rangle_\Gamma&= \int_{\Omega}\nabla v_1\cdot\nabla h_1^*-\int_{\Omega}\lambda^2v_1h_1^*,\\
\left\langle\frac{\partial v_1}{\partial \hat{n}},h_2\right\rangle_\Gamma&= -\int_{\mathbb{R}^N\setminus\bar{\Omega}}\nabla v_1\cdot\nabla h_2^*+\int_{\mathbb{R}^N\setminus\bar{\Omega}}\lambda^2v_1h_2^*,
\end{split}
\end{equation}
We have from Costabel et al. \cite{Costabel} that for every $P\in \Gamma$,
\begin{equation}\label{normal1}
\begin{split}
\frac{\partial v_1^i(P)}{\partial \hat{n}}&=\lim_{X_i(P),x\rightarrow P}\frac{\partial v_1(x)}{\partial \hat{n}}\\&= \bigg(\frac{1}{2}I+K_\lambda^*\bigg)g_2(P)
\end{split}
\end{equation}
and
\begin{equation}\label{normal2}
\begin{split}
\frac{\partial v_1^e(P)}{\partial \hat{n}}&=\lim_{X_e(P),x\rightarrow P}\frac{\partial v_1(x)}{\partial \hat{n}}\\&=\bigg(-\frac{1}{2}I+K_\lambda^*\bigg)g_2(P)
\end{split}
\end{equation}
where $K_{\lambda}^*$ is the adjoint operator of $K_{\lambda}$ defined as
\begin{equation}
K_{\lambda}^*g_1(P)=\int_\Gamma\frac{\partial}{\partial \hat{n}_P}\Phi
(P-y)g_1(y)dy.\nonumber
\end{equation}
Similarly, in case of double layer potential $v_2(x)=\mathbb{K}_\lambda{g_1}(x)$ for ${g_1}\in H^{\frac{1}{2}}(\Gamma)$, we have $\frac{\partial v_2}{\partial\hat{n}}\in H^{-\frac{1}{2}}(\Gamma)$ which satisfies $\eqref{normal}$ and $\eqref{cont}$. Let us define an operator $D_\lambda:H^{\frac{1}{2}}(\Gamma)\rightarrow H^{-\frac{1}{2}}(\Gamma)$ as in \cite{Costabel} such that for every $P\in\Gamma$,
\begin{equation}
D_\lambda{g_1}(P)=\frac{\partial}{\partial \hat{n}_P}\mathbb{K}_\lambda g_1(P)
\end{equation}
and
\begin{align}\label{1double}
\lim\limits_{X_i(P),x\rightarrow P}\frac{\partial}{\partial \hat{n}_x}\mathbb{K}_\lambda g_1(x)&=\lim\limits_{X_e(P),x\rightarrow P}\frac{\partial}{\partial \hat{n}_x}\mathbb{K}_\lambda g_1(x)\nonumber\\&=D_\lambda{g_1}(P).
\end{align}
\begin{lemma}\label{lemma1} The operators
\begin{enumerate}
\item $S_{\lambda}:H^{-\frac{1}{2}}(\Gamma)\rightarrow H^{\frac{1}{2}}(\Gamma)$,
\item $\big(\pm\frac{1}{2}I+{K_{\lambda}}\big):H^{\frac{1}{2}}(\Gamma)\rightarrow H^{\frac{1}{2}}(\Gamma)$,
\item $\big(\pm\frac{1}{2}I+{K^*_{\lambda}}\big):H^{-\frac{1}{2}}(\Gamma)\rightarrow H^{-\frac{1}{2}}(\Gamma)$,
\item $D_{\lambda}:H^{\frac{1}{2}}(\Gamma)\rightarrow H^{-\frac{1}{2}}(\Gamma)$
\end{enumerate}
are continuous by $\cite{Costabel}$.
\end{lemma}
\subsection{Derivation of representation formulae}
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^N$ and $\overline{\Gamma}_1\cup\Gamma_2=\Gamma_1\cup\overline{\Gamma}_2=\Gamma$, $\Gamma_1\cap\Gamma_2=\emptyset$. For $g_1\in \widetilde{H}^{\frac{1}{2}}(\Gamma_i),~ i=1,2$, we denote the zero extension function $\widetilde{g}_1$ of $g_1$ by
\[ \widetilde{g}_1= \begin{cases}
g_1 & in ~\Gamma_i \\
0 & in ~\Gamma\setminus\Gamma_i,~i=1,2.
\end{cases}
\]
Clearly, $\widetilde{g}_1\in H^{\frac{1}{2}}(\Gamma)$. Similarly, for $g_2\in \widetilde{H}^{-\frac{1}{2}}(\Gamma_i),~ i=1,2$, we extend $g_2$ to a function $\widetilde{g}_2\in {H}^{-\frac{1}{2}}(\Gamma)$. We now introduce the following operators.
$$S_{ij}:\widetilde{H}^{-\frac{1}{2}}(\Gamma_i)\rightarrow H^{\frac{1}{2}}(\Gamma_j), ~ S_{ij}g_2=S_\lambda\widetilde{g}_2|_{\Gamma_j} ~\text{for}~ g_2\in \widetilde{H}^{-\frac{1}{2}}(\Gamma_i),$$
$$K_{ij}:\widetilde{H}^{\frac{1}{2}}(\Gamma_i)\rightarrow H^{\frac{1}{2}}(\Gamma_j), ~ K_{ij}g_1=K_\lambda\widetilde{g}_1|_{\Gamma_j} ~\text{for}~ g_1\in \widetilde{H}^{\frac{1}{2}}(\Gamma_i),$$
$$K^*_{ij}:\widetilde{H}^{-\frac{1}{2}}(\Gamma_i)\rightarrow H^{-\frac{1}{2}}(\Gamma_j), ~ K^*_{ij}g_2=K_\lambda^*\widetilde{g}_2|_{\Gamma_j} ~\text{for}~ g_2\in \widetilde{H}^{-\frac{1}{2}}(\Gamma_i),$$
$$D_{ij}:\widetilde{H}^{\frac{1}{2}}(\Gamma_i)\rightarrow H^{-\frac{1}{2}}(\Gamma_j), ~ D_{ij}g_1=D_\lambda\widetilde{g}_1|_{\Gamma_j} ~\text{for}~ g_1\in \widetilde{H}^{\frac{1}{2}}(\Gamma_i).$$
Let $u\in H^1(\Omega)$ be a solution to the Helmholtz equation $-\Delta u-\lambda^2u=h$ in $\Omega$ and $u\in H_{loc}^1(\mathbb{R}^N\setminus\bar{\Omega})$ satisfies $-\Delta u-\lambda^2u=0$ in $\mathbb{R}^N\setminus\bar{\Omega}$ along with $\eqref{zero}-\eqref{nonzero}$. From the Green's second identity we have
$$\int_{\Omega}u\Delta v-v\Delta u =\int_{\Gamma}u\frac{\partial v}{\partial \hat{n}}-v\frac{\partial u}{\partial \hat{n}}~.$$
When we replace $v$ with $\Phi$, the fundamental solution of Helmholtz equation, we obtain the following.
\begin{align}\label{green1}
\int_{\Omega}u(y)\Delta \Phi(x,y)-\Delta u(y) \Phi(x,y)&=\int_{\Gamma}u(y)\frac{\partial \Phi(x,y)}{\partial \hat{n}}-\Phi(x,y)\frac{\partial u(y)}{\partial \hat{n}}\nonumber\\
u(x)&=\int_{\Omega}\Phi(x,y) h(y) -\int_{\Gamma}u(y)\frac{\partial \Phi(x,y)}{\partial \hat{n}}-\Phi
(x,y)\frac{\partial u(y)}{\partial \hat{n}}~.
\end{align}
Let $B_r=\{z\in \mathbb{R}^N: |z|=r\}$ and $D_r=\{x\in\mathbb{R}^N\setminus\bar{\Omega}:|x|<r\}$.
On applying the Green's second identity in the domain $D_r$ we get
\begin{align}\label{green2}
u(x)&=-\int_{D_r}u(y)\Delta \Phi(x,y)-\Delta u(y) \Phi(x,y)\nonumber\\
&=-\int_{B_r}u(y)\frac{\partial \Phi(x,y)}{\partial \hat{n}}-\Phi(x,y)\frac{\partial u(y)}{\partial \hat{n}}+\int_{\Gamma}u(y)\frac{\partial \Phi(x,y)}{\partial \hat{n}}-\Phi(x,y)\frac{\partial u(y)}{\partial\hat{n}}~.
\end{align}
On passing the limit $r\rightarrow \infty$ and by using $\eqref{zero}-\eqref{nonzero}$ we see that $$\int_{B_r}u(y)\frac{\partial \Phi(x,y)}{\partial \hat{n}}-\Phi(x,y)\frac{\partial u(y)}{\partial \hat{n}}\rightarrow 0.$$
Let us denote the Cauchy data as $(\phi,\psi)\in H^{\frac{1}{2}}(\Gamma)\times H^{-\frac{1}{2}}(\Gamma)$, where $u|_\Gamma=\phi$ and ${\frac{\partial u}{\partial \hat{n}}}\big|_\Gamma=\psi$.
On combining $\eqref{green1}$ and $\eqref{green2}$, we can express $u$ as
\begin{equation}\label{represerntation}
\begin{split}
u(x)=\begin{cases}\mathbb{N}_\lambda h(x)-\mathbb{K}_\lambda\phi(x)+\mathbb{S}_\lambda\psi(x),&\text{if}~ x\in\Omega\\
\mathbb{K}_\lambda\phi(x)-\mathbb{S}_\lambda\psi(x),&\text{if}~ x\in\mathbb{R}^N\setminus\bar\Omega.
\end{cases}
\end{split}
\end{equation}
Consider (P1), with the boundary data $u|_{\Gamma_1}=f_1$ and $\frac{\partial u}{\partial\hat{n}}|_{\Gamma_2}=f_2$, where $f_1\in H^{\frac{1}{2}}(\Gamma_1), f_2\in H^{-\frac{1}{2}}(\Gamma_2)$. For simplicity, we restrict ourselves to the interior mixed boundary value problem $\eqref{q1}$. Obviously the corresponding results for the exterior problem $\eqref{q2}$ are obtained by only slight modifications. Furthermore, we say $\mathring{f}_1, \mathring{f}_2$ are the extensions of $f_1$ and $f_2$ respectively which satisfies
\begin{equation}\label{e1}
\norm{\mathring{f}_1}_{H^{\frac{1}{2}}(\Gamma)}\leq C \norm{f_1}_{H^{\frac{1}{2}}(\Gamma_1)}
\end{equation}
and
\begin{equation}\label{e2}
\norm{\mathring{f}_2}_{H^{-\frac{1}{2}}(\Gamma)}\leq C\norm{f_2}_{H^{-\frac{1}{2}}(\Gamma_2)}.
\end{equation}
The above extension is possible since we know $\partial\Gamma_1=\partial\Gamma_2$ and $\Gamma$ is Lipschitz [3].
Let us define $\phi=\mathring{f}_1+\widetilde{g}_1$ and $\psi=\mathring{f}_2+\widetilde{g}_2$, where $\widetilde{g}_1$ and $\widetilde{g}_2$ are arbitrary functions in ${H}^{\frac{1}{2}}(\Gamma)$ and ${H}^{-\frac{1}{2}}(\Gamma)$ respectively. Here $\widetilde{ g}_1$ is the zero extension
of $g_1\in \widetilde{H}^{\frac{1}{2}}(\Gamma_2)$ and $\widetilde{ g}_2$ is the zero extension
of $g_2\in \widetilde{H}^{-\frac{1}{2}}(\Gamma_1)$. The representation $\eqref{represerntation}$ is used to express the solutions of problem $\eqref{q1}$ as
\begin{equation}\label{interior}
u(x)=\mathbb{N}_\lambda h(x)-\mathbb{K}_\lambda(\mathring{f}_1+\widetilde{g}_1)(x)+\mathbb{S}_\lambda(\mathring{f}_2+\widetilde{g}_2)(x).
\end{equation}
On restricting the equation $\eqref{interior}$ to $\Gamma$ we get,
$$\mathring{f}_1+\widetilde{g}_1= \gamma_D\mathbb{N}_\lambda h-\left(-\frac{1}{2}I+K_\lambda\right)(\mathring{f}_1+\widetilde{g}_1)-S_\lambda(\mathring{f}_2+\widetilde{g}_2).$$
On $\Gamma_1$ we have the following,
\begin{align}\label{dirichlet}
f_1&= \gamma_D\mathbb{N}_\lambda h|_{\Gamma_1}-\left(-\frac{1}{2}I+K_\lambda\right)(\mathring{f}_1+\widetilde{g}_1)\Big|_{\Gamma_1}+S_\lambda(\mathring{f}_2+\widetilde{g}_2)\big|_{\Gamma_1}\nonumber\\
&= \gamma_D\mathbb{N}_\lambda h|_{\Gamma_1}-K_{21}g_1-\left(-\frac{1}{2}I+K_\lambda\right)\mathring{f}_1\Big|_{\Gamma_1}+S_{11}g_2+S_\lambda\mathring{f}_2\big|_{\Gamma_1}\nonumber\\
K_{21}g_1-S_{11}g_2 &= -f_1+\gamma_D\mathbb{N}_\lambda h|_{\Gamma_1}-\left(-\frac{1}{2}I+K_\lambda\right)\mathring{f}_1\Big|_{\Gamma_1}+S_\lambda\mathring{f}_2\big|_{\Gamma_1}\nonumber\\
&= F^*({f_1},{f_2},h)~(\text{say}).
\end{align}
Taking the Neumann trace of $\eqref{interior}$ we have $$\mathring{f}_2+\widetilde{g}_2=\gamma_N\mathbb{N}_\lambda h-D_\lambda(\mathring{f}_1+\widetilde{g}_1)+\left(\frac{1}{2}I+K_\lambda^*\right)(\mathring{f}_2+\widetilde{g}_2).$$
Similarly on $\Gamma_2$,
\begin{align}\label{neumann}
f_2&= \gamma_N\mathbb{N}_\lambda h|_{\Gamma_2}-D_\lambda(\mathring{f}_1+\widetilde{g}_1)\big|_{\Gamma_2}+\left(\frac{1}{2}I+K_\lambda^*\right)(\mathring{f}_2+\widetilde{g}_2)\Big|_{\Gamma_2}\nonumber\\
&= \gamma_N\mathbb{N}_\lambda h|_{\Gamma_2}-D_{22}g_1-D\mathring{f}_1\big|_{\Gamma_2}+K^*_{12}g_2+\left(\frac{1}{2}I+K_\lambda^*\right)\mathring{f}_2\Big|_{\Gamma_2}\nonumber\\
D_{22}g_1-K^*_{12}g_2 &=-f_2+\gamma_N\mathbb{N}_\lambda h|_{\Gamma_2}-D_\lambda\mathring{f}_1\big|_{\Gamma_2}+\left(\frac{1}{2}I+K_\lambda^*\right)\mathring{f}_2\Big|_{\Gamma_2}\nonumber \\
&=G^*({f_1},{f_2},h)~(\text{say}).
\end{align}
Clearly $F^*\in H^{\frac{1}{2}}(\Gamma_1)$ and $G^*\in H^{-\frac{1}{2}}(\Gamma_2) $.
Combining equations $\eqref{dirichlet}$ and $\eqref{neumann}$ we get
\[ \left( \begin{array}{cc}
K_{21} & -S_{11} \\
D_{22} & -K^*_{12}
\end{array} \right)
\left( \begin{array}{cc}
g_1 \\
g_2
\end{array} \right)
= \left( \begin{array}{cc}
F^* \\
G^*
\end{array} \right).
\]
We now define a matrix operator $A$ as
\[ A=\left( \begin{array}{cc}
K_{21} & -S_{11} \\
D_{22} & -K^*_{12}
\end{array} \right) \]
where, $A:\widetilde{H}^{\frac{1}{2}}(\Gamma_2)\times \widetilde{H}^{-\frac{1}{2}}(\Gamma_1)\rightarrow {H}^{\frac{1}{2}}(\Gamma_1)\times {H}^{-\frac{1}{2}}(\Gamma_2).$
\subsection{Invertibility of layer potentials.}
For the homogeneous Helmholtz equation with $\lambda=0$, the boundary layer operators $S_{0}:H^{-\frac{1}{2}}(\Gamma)\rightarrow H^{\frac{1}{2}}(\Gamma)$ and $\big(-\frac{1}{2}I+{K_{0}}\big):H^{\frac{1}{2}}(\Gamma)\rightarrow H^{\frac{1}{2}}(\Gamma)$ are bijective operators by \cite{Chang}.
\begin{proposition}\label{inver1} This Proposition is from $\cite{Torres}$ which concludes that for $Im(\lambda)>0$
\begin{enumerate}
\item $S_{\lambda}:L^2(\Gamma)\rightarrow H^{1}(\Gamma)$ is invertible.
\item $\big(\pm\frac{1}{2}I+{K_{\lambda}}\big):L^2(\Gamma)\rightarrow L^2(\Gamma)$ is invertible.
\item $\big(\pm\frac{1}{2}I+{K^*_{\lambda}}\big):L^2(\Gamma)\rightarrow L^2(\Gamma)$ is invertible.
\end{enumerate}
\end{proposition}
\begin{theorem}\label{inver2}
Let $Im(\lambda)>0$. Then $D_\lambda:H^1(\Gamma)\rightarrow L^2(\Gamma)$ is an invertible operator.
\begin{proof}
Let us consider a $g\in L^2(\Gamma)$. From the above Proposition $\ref{inver1}$, $\left(\frac{1}{2}I+K^*_\lambda\right)$ is bijective from $L^2(\Gamma)$ to itself. Hence, there exists a $g'\in L^2(\Gamma)$ such that $(\frac{1}{2}I+K^*_{\lambda})g'=g$.\\
Let $v(x)=\mathbb{S}_\lambda\left(-\frac{1}{2}I+K^*_{\lambda}\right)^{-1}g'(x)$. Then $v$ satisfies the homogenous Helmholtz equation in $\mathbb{R}^N\setminus\bar{\Omega}$. Using the properties $\eqref{single2},\eqref{normal2}$ and the decay conditions at infinity $\eqref{zero}-\eqref{nonzero}$ in the exterior domain we have the following representation for $v$. \begin{equation}\label{r}
v(x)= \mathbb{K}_\lambda f(x)-\mathbb{S}_\lambda g'(x)\nonumber
\end{equation}
where, $f={S}_\lambda(-\frac{1}{2}I+K_\lambda^*)^{-1}g'\in H^1(\Gamma)$. Taking the Neumann trace of $v$ we get
$$ g'=D_\lambda f-\left(-\frac{1}{2}I+K_\lambda^*\right)g'$$
which implies
\begin{equation}\label{q9}
D_\lambda f=\left(\frac{1}{2}I+K_\lambda^*\right)g'.
\end{equation}
Hence, for any $g\in L^2(\Gamma)$, there exists $f\in H^1(\Gamma)$ such that $D_\lambda f=\left(\frac{1}{2}I+K_\lambda^*\right)g'=g.$\\
{\it Claim:} $D_\lambda$ is injective.\\
Suppose there exists $f\in H^1(\Gamma)$ such that $D_\lambda f=0$ on $\Gamma$. We write $v(x)=\mathbb{K}_\lambda f(x)$, for all $x$ in $\mathbb{R}^N\setminus\Gamma$. Hence, $v\in H^1(\Omega)$ is a solution of $-\Delta v-\lambda^2v=0$ in $\Omega$ and $v\in H_{loc}^1(\mathbb{R}^N\setminus\bar{\Omega})$ satisfies $-\Delta v-\lambda^2v=0$ in $\mathbb{R}^N\setminus\bar{\Omega}$ along with $\eqref{zero}-\eqref{nonzero}$. From the equations $\eqref{double1}$-$\eqref{double2}$ we get
\begin{align}
v^i-v^e&=\left(-\frac{1}{2}I+K_\lambda\right)f-\left(\frac{1}{2}I+K_\lambda\right)f\nonumber\\
&=-f\nonumber
\end{align} and from \cite{Torres} we have
\begin{eqnarray}
\begin{split}
D_\lambda f(P)&=\lim_{X_i(P),x\rightarrow P}\frac{\partial}{\partial \hat{n}}v(x)\\
&=\lim_{X_e(P),x\rightarrow P}\frac{\partial}{\partial \hat{n}}v(x).
\end{split}
\end{eqnarray}
Thus,
\begin{align}\label{q}
0&=\langle D_\lambda f,-f\rangle_\Gamma\nonumber\\
&=\Big\langle D_\lambda f,v^i\Big\rangle_\Gamma-\Big\langle D_\lambda f,v^e\Big\rangle_\Gamma\nonumber\\
&=\int_{\mathbb{R}^N}|\nabla v|^2-\int_{\mathbb{R}^N}\lambda^2| v|^2
\end{align}
where the last term in $\eqref{q}$ is due to the fact that $v$ is a solution to the homogeneous Helmholtz equation. As per our assumption $\lambda^2$ is not an eigen value of ($-\Delta$). Hence, using the conditions $\eqref{zero}-\eqref{nonzero}$ we have $v=0$ a.e. in $\mathbb{R}^N$. Since $v$ is continuous across the boundary, we have $-f=v^i-v^e=0$. This implies
$f=0~\text{on}~ \Gamma$. So, $D_{\lambda}$ is injective.
\end{proof}
\end{theorem}
\begin{remark}\label{remark}
The operators $S_\lambda$ and $D_\lambda$ are self-adjoint operators, i.e. $S_\lambda=S_\lambda^*$, $D_\lambda=D_\lambda^*$ (refer Lemma 3.9(a) of $\cite{Costabel})$, where $S_\lambda^*$, $D_\lambda^*$ are the adjoint operators of $S_\lambda$, $D_\lambda$ respectively. Hence, using Proposition $\ref{inver1}$, Theorem $\ref{inver2}$ we obtain $S_\lambda^*:H^{-1}(\Gamma)\rightarrow L^2(\Gamma)$ and $D_\lambda^*:L^2(\Gamma)\rightarrow H^{-1}(\Gamma)$ are invertible operators. Using the properties of real interpolation from Appendix B (Theorem B.2) of $\cite{Mclean}$ on $S_\lambda$, $D_\lambda$ we have
\begin{enumerate}
\item $S_\lambda:H^{-\frac{1}{2}}(\Gamma)\rightarrow H^{\frac{1}{2}}(\Gamma)$,
\item $D_\lambda:H^{\frac{1}{2}}(\Gamma)\rightarrow H^{-\frac{1}{2}}(\Gamma)$
\end{enumerate}
are invertible operators.
\end{remark}
\section{Existence and uniqueness results of (P1).}
\begin{theorem}\label{S}
Let $\Gamma_1\subset\Gamma$, then $S_{11}:\widetilde{H}^{-\frac{1}{2}}(\Gamma_1)\rightarrow {H}^{\frac{1}{2}}(\Gamma_1)$ is a bijective operator.
\begin{proof}
We break the proof into three steps.\\
\textbf{Step 1.} {\it The operator $S_{11}$ is injective.}\\
Assume that there exists $g_2\in\widetilde{H}^{-\frac{1}{2}}(\Gamma_1)$ such that $S_{11}g_2=0$ on $\Gamma_1$. We write $v_1(x)=\mathbb{S}_\lambda\widetilde{g}_2(x)$, $\forall x\in \mathbb{R}^N\setminus\Gamma$, where $\widetilde{g}_2\in {H}^{-\frac{1}{2}}(\Gamma)$ is the zero extension of $g_2$. Hence, from the equations $\eqref{single1}$-$\eqref{single2}$ we have $v_1^i=v_1^e\in {H}^{\frac{1}{2}}(\Gamma)$ and from $\eqref{normal1}$, $\frac{\partial v_1^i}{\partial \hat{n}}-\frac{\partial v_1^e}{\partial \hat{n}}=\widetilde{g}_2.$\\
On replacing $h_1,h_2$ with $v_1^i,v_1^e$ respectively in the equation $\eqref{normal}$ we have
\begin{align}\label{q5}
0&= \langle g_2,S_{11}g_2\rangle_{\Gamma_1}\nonumber\\
&=\langle\widetilde{g}_2,S_\lambda\widetilde{g_2}\rangle_\Gamma\nonumber\\
&=\Big\langle\frac{\partial v_1^i}{\partial \hat{n}},v_1^i\Big\rangle_\Gamma-\Big\langle\frac{\partial v_1^e}{\partial \hat{n}},v_1^e\Big\rangle_\Gamma\nonumber\\
&=\int_{\mathbb{R}^N}|\nabla v_1|^2-\int_{\mathbb{R}^N}\lambda^2 |v_1|^2.
\end{align}
Thus, on using the conditions $\eqref{zero}-\eqref{nonzero}$ we conclude that $v_1=0$ a.e. in $\mathbb{R}^N$. By the continuity of $v_1$ on $\Gamma$, we have $\widetilde{g}_2=\frac{\partial v_1^i}{\partial \hat{n}}-\frac{\partial v_1^e}{\partial \hat{n}}=0$. This implies $g_2=0$ in $\Gamma_1$ and hence $S_{11}$ is injective.\\
\textbf{Step 2.} {\it$S_{11}$ is bounded below.}\\
Suppose there exists a sequence $(g_2^n)\in \widetilde{H}^{-\frac{1}{2}}(\Gamma_1)$ such that $S_{11}g_2^n\rightarrow f$, for some $f\in {H}^{\frac{1}{2}}(\Gamma_1)$. \\
{\it Case 1.} Assume that $(g_2^n)$ is a bounded sequence in $\widetilde{H}^{-\frac{1}{2}}(\Gamma_1)$. Therefore, there exists a subsequence $(g_2^n)$ and $g_2\in \widetilde{H}^{-\frac{1}{2}}(\Gamma_1)$ such that $(g_2^n)$ converges weakly to $g_2$, i.e. $g_2^n\overset{w}\rightharpoonup g_2$ in $\widetilde{H}^{-\frac{1}{2}}(\Gamma_1)$. Let $l\in \widetilde{H}^{-\frac{1}{2}}(\Gamma_1)$. Then we have
\begin{align}
\langle l,f\rangle_{\Gamma_1}&= \langle l, \lim_{n\rightarrow\infty}S_{11}g_2^n\rangle_{\Gamma_1}\nonumber\\
&= \lim_{n\rightarrow\infty}\langle l,S_{11}g_2^n\rangle_{\Gamma_1}\nonumber\\
&=\lim_{n\rightarrow\infty} \langle\langle S^*_{11}l,g_2^n\rangle\rangle\nonumber\\
&=\langle\langle S^*_{11}l, g_2\rangle\rangle\nonumber\\
&=\langle l, S_{11}g_2\rangle_{\Gamma_1}\nonumber.
\end{align}
Since every reflexive space has a unique predual, hence $S_{11}g_2=f$. Therefore, $S_{11}$ has a closed range.\\
{\it Case 2.} Assume that $(g_2^n)$ is an unbounded sequence in $\widetilde{H}^{-\frac{1}{2}}(\Gamma_1)$. Let us denote
$$G^n=\frac{g_2^n}{\norm{g_2^n}_{\widetilde{H}^{-\frac{1}{2}}(\Gamma_1)}}.$$ Hence, $\norm{G^n}_{\widetilde{H}^{-\frac{1}{2}}(\Gamma_1)}=1.$
Therefore, there exists a subsequence $(G^n)$ and $G\in \widetilde{H}^{-\frac{1}{2}}(\Gamma_1)$ such that $G^n\overset{w}\rightharpoonup G$ in $\widetilde{H}^{-\frac{1}{2}}(\Gamma_1)$. Since $S_{11}g_2^n\rightarrow f$ and $\norm{g_2^n}_{\widetilde{H}^{-\frac{1}{2}}(\Gamma_1)}\rightarrow\infty$, we have $S_{11}G^n\rightarrow 0$ in $H^{\frac{1}{2}}(\Gamma_1)$. From Case 1 it easily follows that $S_{11}G=0$, which further implies $G=0$ by the injectivity of $S_{11}$. Using the invertibility of $S_\lambda$ (refer Remark $\ref{remark}$) we obtain
\begin{align}\label{contra}
1&=\norm{G^n}_{\widetilde{H}^{-\frac{1}{2}}(\Gamma_1)}\nonumber\\
&\leq\norm{\widetilde{G^n}}_{{H}^{-\frac{1}{2}}(\Gamma)}\nonumber\\
&\leq C\norm{S_{\lambda}(\widetilde{G^n})}_{{H}^{\frac{1}{2}}(\Gamma)}~(\text{for}~ C>0).
\end{align}
We know that $S_{11}G^n=S_\lambda(\widetilde{G}^n)|_{\Gamma_1}$ and
$S_{12}G^n=S_\lambda(\widetilde{G}^n)|_{\Gamma_2}$.
For $x\neq y$, $\Phi(x-y)$ is a $C^\infty$ function. This implies $S_{12}G^n\rightarrow 0$ in $H^{\frac{1}{2}}(\Gamma_2)$, since $G^n\overset{w}{\rightharpoonup}0$ in $\widetilde{H}^{-\frac{1}{2}}(\Gamma_1)$. Hence, $S_\lambda(\widetilde{G^n})\rightarrow 0$ in $H^{\frac{1}{2}}(\Gamma)$, which is a contradiction to $\eqref{contra}$. Therefore, we conclude that $S_{11}$ has closed range. Thus, $S_{11}$ is bounded below since $S_{11}$ is injective and its range is closed. \\
\textbf{Step 3.} {\it $S_{11}$ has dense range}.\\
Assume that $S_{11}^*g_2=0$ for some $g_2\in \widetilde{H}^{-\frac{1}{2}}(\Gamma_1)$. Hence, for any $l\in \widetilde{H}^{-\frac{1}{2}}(\Gamma_1)$,
\begin{align}
0&=\langle\langle S_{11}^*g_2,l\rangle\rangle\nonumber\\
&=\langle g_2, S_{11}l\rangle_{\Gamma_1} .\nonumber
\end{align}
Choose $l=g_2$. Then by proceeding on similar lines as in step 1 we get $g_2=0$. Since $\text{Kernel}(S_{11}^*)= {\text{Range}(S_{11})^\perp}=\overline{\text{Range}(S_{11})}^\perp$, the injectivity of $S_{11}^*$ implies $S_{11}$ has dense range.\\
Combining the results from the above three steps we conclude that the operator $S_{11}:\widetilde{H}^{-\frac{1}{2}}(\Gamma_1)\rightarrow {H}^{\frac{1}{2}}(\Gamma_1)$ is bijective.
\end{proof}
\end{theorem}
\begin{theorem}\label{D}
Let $\Gamma_2\subset\Gamma$, then the operator $D_{22}:\widetilde{H}^{\frac{1}{2}}(\Gamma_2)\rightarrow {H}^{-\frac{1}{2}}(\Gamma_2)$ is invertible.
\begin{proof}
Similar to the steps in Theorem $\ref{S}$, we will show that $D_{22}$ is injective and bounded below with a dense range.
Assume that there exists $g_1\in \widetilde{H}^{\frac{1}{2}}(\Gamma_2)$ such that $D_{22}g_1=0$ on $\Gamma_2$. We now express $v_2(x)=\mathbb{K}_\lambda\widetilde{g}_1(x), ~\forall x\in\mathbb{R}^N\setminus\Gamma$. From the equations $\eqref{double1}$ and $\eqref{double2}$ we get
\begin{align}
v_2^i-v_2^e&=\left(-\frac{1}{2}I+K_\lambda\right)\widetilde{g}_1-\left(\frac{1}{2}I+K_\lambda\right)\widetilde{g}_1\nonumber\\
&=-\widetilde{g}_1.\nonumber
\end{align}
Thus,
\begin{align}\label{u}
0&=\langle D_{22}g_1,-g_1\rangle_{\Gamma_2}\\
&=\langle D_\lambda\widetilde{g}_1,-\widetilde{g}_1\rangle_\Gamma\nonumber\\
&=\Big\langle D_\lambda\widetilde{g}_1,v_2^i\Big\rangle_\Gamma-\Big\langle D_\lambda\widetilde{g}_1,v_2^e\Big\rangle_\Gamma~~(\text{from the equation}~\eqref{1double})\nonumber\\
&=\int_{\mathbb{R}^N}|\nabla v_2|^2-\int_{\mathbb{R}^N}\lambda^2| v_2|^2.\nonumber
\end{align}
Hence, using the conditions $\eqref{zero}-\eqref{nonzero}$ we have $v_2=0$ a.e. in $\mathbb{R}^N$, since $\lambda^2$ is not an eigen value of ($-\Delta$). By the continuity of $v_2$ in $x\in\mathbb{R}^N\setminus\Gamma_2$ we have $v_2^i-v_2^e=-\widetilde{g}_1=0$. This implies
$g_1=0~\text{in}~ \Gamma_2$. So, $D_{22}$ is injective. \\
On using arguments from Theorem $\ref{S}$, we can show that $D_{22}$ has a closed range and hence it is bounded below. We suppose that $D_{22}^*g_1'=0$ for some $g_1\in\widetilde{H}^{\frac{1}{2}}(\Gamma_2).$ Then for $f\in \widetilde{H}^{\frac{1}{2}}(\Gamma_2)$,
\begin{align}
0&=\langle -D_{22}^*g_1',f\rangle_{\Gamma_2}\nonumber\\
&=\langle\langle- g_1',D_{22}f\rangle\rangle\nonumber\\
&=\langle -g_1, D_{22}f\rangle_{\Gamma_2}.\nonumber
\end{align}
Taking $f=g_1$, then from $\eqref{u}$ we obtain $g_1=0$ in $\Gamma_2$. Hence, $D^*_{22}$ is injective which implies $D_{22}$ has dense range.
Therefore, $D_{22}$ is an invertible operator.
\end{proof}
\end{theorem}
\begin{theorem}\label{A}
The matrix operator $A:\widetilde{H}^{\frac{1}{2}}(\Gamma_2)\times \widetilde{H}^{-\frac{1}{2}}(\Gamma_1)\rightarrow {H}^{\frac{1}{2}}(\Gamma_1)\times{H}^{-\frac{1}{2}}(\Gamma_2)$
is invertible.
\begin{proof}
For any $g_1\in \widetilde{H}^{\frac{1}{2}}(\Gamma_2)$ and $P\in\Gamma_1$, the operator $K_{21}:\widetilde{H}^{\frac{1}{2}}(\Gamma_2)\rightarrow{H}^{\frac{1}{2}}(\Gamma_1)$ is defined as
\begin{align}\label{s}
K_{21}g_1(P)&=K_\lambda\widetilde{g}_{1}(P)\nonumber\\
&=\int_{\Gamma}\frac{\partial}{\partial\hat{n}_y}\Phi(P,y)\widetilde{g}_1(y)dy\nonumber\\
&=\int_{\Gamma_2}\frac{\partial}{\partial\hat{n}_y}\Phi(P,y){g}_1(y)dy.
\end{align}
We can see that the kernel $\frac{\partial\Phi(P,y)}{\partial\hat{n}_y}$ in $\eqref{s}$ is a $C^\infty$ function. Let $(g_1^n)$ be a bounded sequence in $\widetilde{H}^{\frac{1}{2}}(\Gamma_2)$, then there exists a subsequence $(g_1^n)$ and $g_1$ in $\widetilde{H}^{\frac{1}{2}}(\Gamma_2)$ such that $(g_1^n)$ converges weakly to $g_1$. Hence,
\begin{align}
\lim_{n\rightarrow\infty}K_{21}g_1^n(P)&=\lim_{n\rightarrow\infty}\int_{\Gamma_2}\frac{\partial}{\partial\hat{n}_y}\Phi(P,y){g}_1^n(y)dy\nonumber\\
&=\int_{\Gamma_2}\frac{\partial}{\partial\hat{n}_y}\Phi(P,y) {g}_1(y)dy\nonumber\\
&=K_{21}g_1(P).\nonumber
\end{align}
Thus, $K_{21}$ is a compact operator. Similarly we can show that the operator $K^*_{12}$ is also compact.\\
We have
\begin{align}
A&= \left( \begin{array}{cc}
K_{21} & -S_{11} \\
D_{22} & -K^*_{12}
\end{array} \right)\nonumber
\\ &=\left( \begin{array}{cc}
K_{21} & 0 \\
0 & -K_{12}^*
\end{array} \right)
%
+ \left( \begin{array}{cc}
0 & -S_{11} \\
D_{22} & 0
\end{array} \right)\nonumber\\
&=A_1+A_2\nonumber
\end{align}
where $A_1=\left( \begin{array}{cc}
K_{21} & 0 \\
0 & -K_{12}^*
\end{array} \right)$ and $A_2=\left( \begin{array}{cc}
0 & -S_{11} \\
D_{22} & 0
\end{array} \right)$. The matrix $A_2$ is invertible, since $S_{11}$ and $D_{22}$ are invertible operators by Theorem $\ref{S}$ and Theorem $\ref{D}$ respectively. As the operators $D_\lambda$ and $S_\lambda$ are also continuous by \cite{Costabel}, the inverse of $A_2, \text{i.e.}~A_2^{-1}$ is also bounded. We know the operators $K_{21}~\text{and}~K_{12}^*$ are compact operators and hence $A_1$ is also a compact operator.\\
Thus, we can write $A_2^{-1}A=A_2^{-1}A_1+I=C_1+I$ and $AA_2^{-1}=A_1A_2^{-1}+I=C_2+I$ where $C_1,C_2$ are compact operators. Using Theorem $\ref{ft1}$, it is equivalent to say that $A$ is a Fredholm operator. This implies ind$(A)=0$ (by Theorem $\ref{ft2}$). Now to show $A$ is bijective it is sufficient to show $A$ is injective, i.e. dim ker$(A)=0$. \\
{\it Claim:} $A$ is injective.\\
Let us assume that there exist some $g_1\in \widetilde{H}^{\frac{1}{2}}(\Gamma_2)$ and $g_2\in\widetilde{H}^{-\frac{1}{2}}(\Gamma_1)$ such that $A(g_1,g_2)=0.$ Now for $x\in \mathbb{R}^N\setminus\Gamma$, we write
\begin{eqnarray}
v(x)=\begin{cases}
\mathbb{S}_\lambda\widetilde{ g}_2(x)-\mathbb{K}_\lambda\widetilde{ g}_1(x),&\text{if}~x\in \Omega\\
-\mathbb{S}_\lambda\widetilde{ g}_2(x)+\mathbb{K}_\lambda\widetilde{ g}_1(x),&\text{if}~x\in \mathbb{R}^N\setminus\bar\Omega
\end{cases}\nonumber
\end{eqnarray}
Then $v$ satisfies the following problems
\begin{eqnarray}\label{q6}
\begin{split}
-\Delta v-\lambda^2 v&= 0~\text{in}~\Omega,\\
v&= 0~\text{on}~\Gamma_1,\\
\frac{\partial v}{\partial \hat{n}}&= 0~ \text{on}~\Gamma_2,\\
v &\in H^1(\Omega)
\end{split}
\end{eqnarray}
and
\begin{eqnarray}\label{q7}
\begin{split}
-\Delta v-\lambda^2 v&= 0~\text{in}~\mathbb{R}^N\setminus\bar{\Omega},\\
v&= 0~\text{on}~\Gamma_1,\\
\frac{\partial v}{\partial \hat{n}}&= 0~ \text{on}~\Gamma_2,\\
v &\in H^1_{loc}(\mathbb{R}^N\setminus\bar{\Omega}).
\end{split}
\end{eqnarray}
This implies $v|_\Gamma\in {H}^{\frac{1}{2}}(\Gamma)$, $\frac{\partial v }{\partial \hat{n} }\in {H}^{-\frac{1}{2}}(\Gamma)$ and
\begin{align}
0&=\Big\langle\frac{\partial v}{\partial \hat{n}},v\Big\rangle\nonumber\\
&=\int_{\Omega}|\nabla v|^2-\int_{\Omega}\lambda^2|v|^2\nonumber
\end{align}
and
\begin{align}
0&=\Big\langle\frac{\partial v}{\partial \hat{n}},v\Big\rangle\nonumber\\
&=-\int_{\mathbb{R}^N\setminus\bar{\Omega}}|\nabla v|^2+\int_{\mathbb{R}^N\setminus\bar{\Omega}}\lambda^2|v|^2.\nonumber
\end{align}
Thus, $v=0$ a.e. in $\mathbb{R}^N$, since $\lambda^2$ is not an eigenvalue of $(-\Delta)$ and $v$ satisfies the radiation conditions at infinity. On $\Gamma_1$, $v=0$ and hence $v^i-v^e=\pm\widetilde{g}_1=0$ and $\frac{\partial v^i}{\partial \hat{n}}-\frac{\partial v^e}{\partial \hat{n}}=\pm\widetilde{g}_2=0$. Thus, $g_1=0$ and $g_2=0$.
\end{proof}
\end{theorem}
\begin{theorem}\label{main 1}
The mixed boundary value problem $\eqref{q1}$ with $Im(\lambda)>0$ possesses a unique solution $u$ which is represented as
$$u(x)=\mathbb{N}_\lambda h(x)-\mathbb{K}_\lambda(\mathring{f_1}+\widetilde{g_1})(x)+\mathbb{S}_\lambda(\mathring{f_2}+\widetilde{g_2})(x)$$ for unique $g_1\in\widetilde{H}^{\frac{1}{2}}(\Gamma_2)$ and $g_2\in \widetilde{H}^{-\frac{1}{2}}(\Gamma_1)$.
This solution $u$ also satisfies $\eqref{zero}-\eqref{nonzero}$ and
\begin{equation}\label{estimate}
\norm{u}_{H^{\frac{1}{2}}(\Gamma)}+\norm{\frac{\partial u}{\partial \hat{n}}}_{H^{-\frac{1}{2}}(\Gamma)}\leq C\{\norm{f_1}_{H^{\frac{1}{2}}(\Gamma_1)}+\norm{f_2}_{H^{-\frac{1}{2}}(\Gamma_2)}+\norm{h}_{\widetilde{H}^{-1}(\Omega)}\}.
\end{equation}
\begin{proof}
The solvability and uniqueness of problem $\eqref{q1}$ depend on the invertibility of the operator $A$. Due to Theorem $\ref{A}$ we know that $A$ is invertible. Hence, there exists a unique pair $(g_1,g_2)\in\widetilde{H}^{\frac{1}{2}}(\Gamma_2)\times \widetilde{H}^{-\frac{1}{2}}(\Gamma_1)$ such that
$$A(g_1,g_2)=(F^*,G^*),$$
where, $F^*\in H^{\frac{1}{2}}(\Gamma_1)$ and $G^*\in H^{-\frac{1}{2}}(\Gamma_2)$ as defined in equations $\eqref{dirichlet}$ and $\eqref{neumann}$. Then we can represent $$u(x)=\mathbb{N}_\lambda h(x)-\mathbb{K}_\lambda\phi(x)+\mathbb{S}_\lambda\psi(x),$$
where, $\phi=\mathring{f}_1+\widetilde{g}_1$ and $\psi=\mathring{f}_2+\widetilde{g}_2$ are the Cauchy data for the problem $\eqref{q1}$. Since
\begin{align}\label{q8}
\left( \begin{array}{cc}
K_{21} & -S_{11} \\
D_{22} & -K^*_{12}
\end{array} \right)
%
\left( \begin{array}{cc}
g_1 \\
g_2
\end{array} \right)
%
= \left( \begin{array}{cc}
F^* \\
G^*
\end{array} \right),
\end{align}
so we can write
\begin{equation}\label{g1}
g_1=D_{22}^{-1}(K^*_{12}g_2+G^*).
\end{equation}
Substituting the value of $g_1$ in the equation $\eqref{q8}$ we get
\begin{equation}\label{g2}
(S_{11}-K_{21}D_{22}^{-1}K^*_{12})g_2=K_{21}D_{22}^{-1}G^*-F^*.
\end{equation}
We will now show that the operator $H:=S_{11}-K_{21}D_{22}^{-1}K^*_{12}:\widetilde{H}^{-\frac{1}{2}}(\Gamma_1)\rightarrow H^{\frac{1}{2}}(\Gamma_1)$ is invertible. We can then represent $g_1$, $g_2$ as follows.
$$g_1=D_{22}^{-1}G^*+D_{22}^{-1}K_{12}^*H^{-1}(K_{21}D_{22}^{-1}G^*-F^*)$$ and
$$g_2=H^{-1}(K_{21}D_{22}^{-1}G^*-F^*).$$
{\it Claim:} $H=S_{11}-K_{21}D_{22}^{-1}K^*_{12}$ is invertible.\\
We know $K_{21}$ and $K^*_{12}$ are compact operators. So $K_{21}D_{22}^{-1}K^*_{12}$ is also compact. Using Theorem $\ref{ft2}$ we get ind$(H)=0$, since $S_{11}$ is bijective. Thus we only need to show that $H$ is injective.\\
Suppose $H(g_2)=0,$ for some $g_2\in \widetilde{H}^{-\frac{1}{2}}(\Gamma_1)$. We express $w(x)=\mathbb{S}_\lambda\widetilde{g_2}(x)-\mathbb{K}_\lambda \widetilde{D_{22}^{-1}K^*_{12}g_2}(x)$.
We observe that on $\Gamma_1$,
\begin{align}
w& = (S_{11}-K_{21}D_{22}^{-1}K^*_{12})g_2\nonumber\\
&=H(g_2)\nonumber\\
&=0\nonumber
\end{align}
and on $\Gamma_2$,
\begin{align}
\frac{\partial w}{\partial \hat{n}} &= K^*_{12}g_2-D_{22}\{D_{22}^{-1}K^*_{12}\}g_2\nonumber\\
&=0.\nonumber
\end{align}
Therefore, $w$ satisfies $\eqref{q6}$ and $\eqref{q7}$. Following the proof of Theorem $\ref{A}$ we conclude that $g_2=0$. So $H$ is injective hence invertible.\\
Furthermore,
\begin{align}\label{inq1}
\norm {u}_{H^{\frac{1}{2}}(\Gamma)}&= \norm{\mathring{f}_1+\widetilde{g}_1}_{H^{\frac{1}{2}}(\Gamma)}\nonumber\\
& \leq \norm {\mathring{f}_1}_{H^{\frac{1}{2}}(\Gamma)}+\norm{\widetilde{g}_1}_{H^{\frac{1}{2}}(\Gamma)}\nonumber\\
&\leq C\{\norm{f_1}_{H^{\frac{1}{2}}(\Gamma_1)}+\norm{K^*_{12}g_2+G^*}_{H^{-\frac{1}{2}}(\Gamma_2)}\}~(\text{by}~ \eqref{e1}~ \text{and}~ \eqref{g1})\nonumber\\
& \leq C\{\norm{f_1}_{H^{\frac{1}{2}}(\Gamma_1)}+\norm{G^*}_{H^{-\frac{1}{2}}(\Gamma_2)}+\norm{g_2}_{H^{-\frac{1}{2}}(\Gamma_1)}\}\nonumber\\
&\leq C\{\norm{f_1}_{H^{\frac{1}{2}}(\Gamma_1)}+\norm{G^*}_{H^{-\frac{1}{2}}(\Gamma_2)}+\norm{K_{21}D_{22}^{-1}G^*-F^*}_{H^{\frac{1}{2}}(\Gamma_1)}\}~(\text{by}~\eqref{g2})\nonumber\\
&\leq C\{\norm{f_1}_{H^{\frac{1}{2}}(\Gamma_1)}+\norm{f_2}_{H^{-\frac{1}{2}}(\Gamma_2)}+\norm{h}_{\widetilde{H}^{-1}(\Omega)}\}.~(\text{by}~ \eqref{dirichlet}~\text{and}~ \eqref{neumann})
\end{align}
Similarly
\begin{equation}\label{inq2}
\norm{\frac{\partial u}{\partial \hat{n}}}_{H^{-\frac{1}{2}}(\Gamma)}\leq C\{\norm{f_1}_{H^{\frac{1}{2}}(\Gamma_1)}+\norm{f_2}_{H^{-\frac{1}{2}}(\Gamma_2)}+\norm{h}_{\widetilde{H}^{-1}(\Omega)}\}.
\end{equation}
On combining inequalities $\eqref{inq1}$ and $\eqref{inq2}$ we get
$$\norm{u}_{H^{\frac{1}{2}}(\Gamma)}+\norm{\frac{\partial u}{\partial \hat{n}}}_{H^{-\frac{1}{2}}(\Gamma)}\leq C\{\norm{f_1}_{H^{\frac{1}{2}}(\Gamma_1)}+\norm{f_2}_{H^{-\frac{1}{2}}(\Gamma_2)}+\norm{h}_{\widetilde{H}^{-1}(\Omega)}\}.$$
\end{proof}
\end{theorem}
\noindent Proceeding similarly for the exterior problem $\eqref{q2}$ of (P1), the following Theorem can be established.
\begin{theorem}\label{main 2}
For $Im(\lambda)>0$, problem (P1) with given boundary data $f_1\in H^{\frac{1}{2}}(\Gamma_1)$ and $f_2\in H^{-\frac{1}{2}}(\Gamma_2)$ has a unique solution $u$ which is represented as
\begin{eqnarray}
u(x)=\begin{cases}
\mathbb{N}_\lambda h(x)-\mathbb{K}_\lambda(\mathring{f}_1+\widetilde{g}_1)(x)+\mathbb{S}_\lambda(\mathring{f}_2+\widetilde{g}_2)(x),&\text{if}~x\in \Omega\\
\mathbb{K}_\lambda(\mathring{f}_1+\widetilde{g}_1)(x)-\mathbb{S}_\lambda(\mathring{f}_2+\widetilde{g}_2)(x),&\text{if}~x\in \mathbb{R}^N\setminus\bar\Omega
\end{cases}\nonumber
\end{eqnarray}
for unique $g_1\in\widetilde{H}^{\frac{1}{2}}(\Gamma_2)$ and $g_2\in \widetilde{H}^{-\frac{1}{2}}(\Gamma_1)$. The solution $u$ belongs to $H^1(\Omega)$ for (IP1) and belongs to $H^1_{loc}(\mathbb{R}^N\setminus\bar{\Omega})$ for (EP1) satisfying the conditions $\eqref{zero}-\eqref{nonzero}$ at infinity. Furthermore, $u$ satisfies $\eqref{estimate}$.
\end{theorem}
\section{Existence results of (P2).}
Problem (P2) is a mixed boundary value problem of Poisson equation where $\mu\in\mathcal{M}(\bar\Omega)$ is a bounded Radon measure supported on $\Omega$ and the boundary data are $f_1\in H^{\frac{1}{2}}(\Gamma_1)$ and $f_2\in H^{-\frac{1}{2}}(\Gamma_2)$.
\begin{definition}\label{weak solution}
We say a function $u\in W^{1,1}(\Omega)$ is a weak solution to the problem $\eqref{lq1}$ if
\begin{equation}
\int_\Omega \nabla u\cdot\nabla \varphi~ = \int_\Omega \varphi d\mu +\int_{\Gamma_2}f_2\varphi, ~\forall \varphi\in X\nonumber
\end{equation}
where, $X=\{\varphi\in C^1(\bar{\Omega}):\varphi|_{\Gamma_1}=0\}$ is the test function space. Similarly a function $u\in W^{1,1}_{loc}(\mathbb{R}^N\setminus\bar{\Omega})$ is said to be a weak solution of $\eqref{lq2}$ if
\begin{equation}
\int_{\mathbb{R}^N\setminus\bar{\Omega}} \nabla u\cdot\nabla \varphi=-\int_{\Gamma_2}f_2\varphi, ~\forall \varphi\in Z\nonumber
\end{equation}
where, $Z=\{\zeta\in C_c^1(\mathbb{R}^N\setminus\Omega):\zeta|_{\Gamma_1}=0~ \text{and satisfies}~ \eqref{infinity}\}$.
\end{definition}
\noindent We will now approximate $\mu\in\mathcal{M}(\bar\Omega)$ by a smooth sequence $(\mu_n)\subset L^\infty(\Omega)$, in the weak* topology, i.e.
\begin{equation}\label{convergence}
\int_{\Omega} g~ d\mu_n\rightarrow \int_{\Omega} g ~d\mu, ~\forall g\in C(\bar\Omega).
\end{equation}
In order to show the existence of solutions to (P2), we consider the \textquoteleft {\it approximating}\textquoteright ~problems to $\eqref{lq1}$-$\eqref{lq2}$ which are as follows.
\begin{eqnarray}\label{q3}
\begin{split}
-\Delta u_n&= \mu_n~\text{in}~\Omega,\\
u_n&= f_1~\text{on}~\Gamma_1,\\
\frac{\partial u_n}{\partial \hat{n}}&= f_2~ \text{on}~\Gamma_2,\\
\end{split}
\end{eqnarray}
and
\begin{eqnarray}\label{q4}
\begin{split}
-\Delta u_n&= 0~\text{in}~\mathbb{R}^N\setminus\bar{\Omega},\\
u_n&= f_1~\text{on}~\Gamma_1,\\
\frac{\partial u_n}{\partial \hat{n}}&= f_2~ \text{on}~\Gamma_2,\\
\end{split}
\end{eqnarray}
These \textquoteleft {\it approximating}\textquoteright~ problems are special cases of (P1) with $\lambda=0$. The weak formulation to \eqref{q3} is
\begin{equation}\label{formulation}
\int_\Omega \nabla u_n\cdot\nabla \varphi~ = \int_\Omega \varphi \mu_n +\int_{\Gamma_2}f_2\varphi, ~\forall \varphi\in X.
\end{equation}
\begin{theorem}\label{t1}
The problems $\eqref{q3}$ and $\eqref{q4}$ admit a unique solution $u_n$ which is represented as
\begin{eqnarray}\label{t}
u_n(x)=\begin{cases}
\mathbb{N}_0 \mu_n(x)-\mathbb{K}_0(\mathring{f}_1+\widetilde{g}_1)(x)+\mathbb{S}_0(\mathring{f}_2+\widetilde{g}_2)(x),&\text{if}~x\in \Omega\\
\mathbb{K}_0(\mathring{f}_1+\widetilde{g}_1)(x)-\mathbb{S}_0(\mathring{f}_2+\widetilde{g}_2)(x),&\text{if}~x\in \mathbb{R}^N\setminus\bar\Omega
\end{cases}
\end{eqnarray}
for a unique pair $(g_1,g_2)\in\widetilde{H}^{\frac{1}{2}}(\Gamma_2)\times \widetilde{H}^{-\frac{1}{2}}(\Gamma_1)$. The solution $u_n$ belongs to $H^1(\Omega)$ for the problem $\eqref{q3}$ and belongs to $H^1_{loc}(\mathbb{R}^N\setminus\bar{\Omega})$ for the problem $\eqref{q4}$ satisfying the radiation condition $\eqref{infinity}$ at infinity.
\begin{proof}
The invertibility of the operators $S_{11}$ and $D_{22}$ follows from Theorem 3.1 and Theorem 3.2 of \cite{Chang} respectively. Further, following the proof of Theorem $\ref{A}$, Theorem $\ref{main 1}$ one can see that the matrix operator $A$ is invertible and the problems $\eqref{q3}$-$\eqref{q4}$ have a unique solution denoted as $u_n$. The solution $u_n$ can be represented as in $\eqref{t}$ by Theorem $\ref{main 2}$ and satisfies the condition $\eqref{infinity}$ at infinity.
\end{proof}
\end{theorem}
\noindent Now to show that problem in $\eqref{lq1}$, involving measure, possesses a solution $u$ we need to pass the limit $n\rightarrow\infty$ in the weak formulation $\eqref{formulation}$.
\begin{lemma}\label{bdd}
Let us suppose that $u_n$ is a solution of problem $\eqref{q3}$ with $f_1\in H^{\frac{1}{2}}(\Gamma_1)~\text{and}~ f_2\in H^{-\frac{1}{2}}(\Gamma_2)$. Then the sequence $(u_n)$ is bounded in $W^{1,q}(\Omega)~\forall q<\frac{N}{N-1}$.
\begin{proof}
From the continuous embedding $\eqref{marcinq}$ we have
\begin{equation}\label{marcink}
L^{\frac{N}{N-1}}(\Omega)\hookrightarrow {M}^{\frac{N}{N-1}}(\Omega)\hookrightarrow L^{\frac{N}{N-1}-\epsilon}(\Omega).
\end{equation}
If we can show that $(u_n)$ is bounded in ${M}^{\frac{N}{N-1}}(\Omega)$, then this will also imply $(u_n)$ to be bounded in $L^{\frac{N}{N-1}-\epsilon}(\Omega)$. More precisely, we can say $(u_n)$ to be bounded in $L^q(\Omega)$, for every $q<\frac{N}{N-1}$.\\
{\it Claim:} The sequences $(u_n)$ and $(\nabla u_n)$ are bounded in ${M}^{\frac{N}{N-1}}(\Omega)$.\\
Let us fix a constant $a>0$. The truncation function of $u_n$ is defined as
$$T_a(u_n)=\max\{-a,\min\{a,u_n\}\}.$$
Choose $\varphi\in H^1(\Omega)$. Then from Theorem $\ref{t1}$ we have $\frac{\partial u_n}{\partial\hat{n}}\Big|_{\Gamma}=\mathring{f_2}+\widetilde{g_2}$, for a unique ${g_2}\in \widetilde{H}^{-\frac{1}{2}}(\Gamma_1)$. The weak formulation of $\eqref{q3}$ becomes
$$\int_{\Omega}\nabla u_n.\nabla\varphi=\int_{\Omega}\varphi\mu_n+\int_{\Gamma}(\mathring{f_2}+\widetilde{g_2})\varphi$$
Consider $\varphi =T_a(u_n)$, then we get
\begin{align}\label{truncation}
\int_{\Omega}|\nabla T_a(u_n)|^2&\leq
\int_{\Omega}\nabla u_n.\nabla (T_a(u_n))\nonumber\\ &=\int_{\Omega}T_a(u_n)\mu_n+\int_{\Gamma}(\mathring{f_2}+\widetilde{g_2})T_a(u_n)\nonumber\\
&\leq a\int_{\Omega}\mu_n+a\int_{\Gamma}(\mathring{f_2}+\widetilde{g_2}) \nonumber\\
& \leq Ca.
\end{align}
Consider
\begin{align*}
\{|\nabla u_n|\geq k\} & = \{|\nabla u_n|\geq k,u_n< a\} \cup \{|\nabla u_n| \geq k,u_n \geq a\}
\\& \subset \{|\nabla u_n|\geq k,u_n <a\} \cup \{u_n \geq a\} \\&= B_1\cup B_2\subset \Omega
\end{align*}
where $B_1= \{|\nabla u_n|\geq k,u_n <a\}$ and $B_2=\{u_n \geq a\}$. Hence, due to the subadditivity property of the Lebesgue measure \textquoteleft $m$\textquoteright ~we have
\begin{equation}\label{sub}
m(\{|\nabla u_n|\geq k\}) \leq m(B_1) + m(B_2).
\end{equation}
Using the Sobolev inequality, we have
\begin{align}\label{sobolev}
\left(\int_\Omega |T_a(u_n)|^{2^*}\right)^{\frac{2}{2^*}}&\leq \frac{1}{\lambda_1}\int_{\Omega}|\nabla T_a(u_n)|^2 \nonumber\\
&\leq Ca
\end{align}
where $\lambda_1$ is the first eigenvalue of ($-\Delta$). Now we restrict the above inequality $\eqref{sobolev}$ on $B_2$ to get
\begin{align}
& a^2m(\{u_n\geq a\})^{\frac{2}{2^*}}\leq Ca,~(\text{Since}~ T_a(u_n)=a ~\text{in}~ B_2).\nonumber
\end{align}
Thus,
\begin{equation}
m(\{u_n\geq a\})\leq \frac{C}{a^\frac{N}{N-2}},~ \forall a\geq1.\nonumber
\end{equation}
Hence, $(u_n)$ is bounded in ${M}^{\frac{N}{N-2}}(\Omega)$ and also bounded in ${M}^{\frac{N}{N-1}}(\Omega)$. Similarly on restricting $\eqref{truncation}$ on $B_1$, we have
\begin{align}
m(\{|\nabla u_n|\geq k,u_n< a\})&\leq \frac{1}{k^2}\int_\Omega |\nabla T_a(u_n)|^2\nonumber\\
&\leq \frac{Ca}{k^2}, ~\forall a>1.\nonumber
\end{align}
Now the inequality $\eqref{sub}$ becomes
\begin{align}
m( \{|\nabla u_n|\geq k\}) &\leq m(\{B_1\}) + m(B_2)\nonumber\\&\leq \frac{Ca}{k^2} + \frac{C}{a^\frac{N}{N-2}},~ \forall a>1.\nonumber
\end{align}
On choosing $a=k^{\frac{N-2}{N-1}}$ we get
$$ m(\{|\nabla u_n|\geq k\})\leq \frac{C}{k^\frac{N}{N-1}},~ \forall k\geq 1.$$
So $(\nabla u_n)$ is bounded in ${M}^{\frac{N}{N-1}}(\Omega)$. Hence, we conclude that $(u_n)$ is bounded in $W^{1,q}(\Omega)$ for every $q<\frac{N}{N-1}$.
\end{proof}
\end{lemma}
\begin{theorem}\label{main3}
There exists a weak solution $u$ of $\eqref{lq1}$ in $W^{1,q}(\Omega), ~\forall q<\frac{N}{N-1}$.
\begin{proof}
According to Lemma $\ref{bdd}$, $(u_n)$ is bounded in $W^{1,q}(\Omega)$ which is a reflexive space. This implies that there exists a function $u\in W^{1,q}(\Omega)$ such that $u_n$ converges weakly to $u$, i.e. $u_n\overset{w}{\rightharpoonup} u$ in $W^{1,q}(\Omega), ~\forall q<\frac{N}{N-1}$.\\
Thus for $\varphi\in X$,
$$\lim_{n\rightarrow +\infty} \int_{\Omega} \nabla u_n . \nabla\varphi = \int_{\Omega}\nabla u .\nabla\varphi.$$
The sequence $(\mu_n)$ converges to $\mu$ in the weak* topology in the sense given in $\eqref{convergence}$.
On passing the limit $n\rightarrow\infty$ in the weak formulation $\eqref{formulation}$ involving $\mu_n$ we obtain
\begin{equation}
\int_{\Omega}\nabla u.\nabla\varphi=\int_{\Omega}\varphi d\mu+\int_{\Gamma_2}f_2\varphi, ~\forall\varphi\in X.\nonumber
\end{equation}
Hence, a weak solution of $\eqref{lq1}$ in $W^{1,q}(\Omega)$ for every $q<\frac{N}{N-1}$ is guaranteed.
\end{proof}
\end{theorem}
\noindent Now with the consideration of Theorem $\ref{t1}$ and Theorem $\ref{main3}$ we state our main result which is as follows.
\begin{theorem}
There exists a weak solution $u$ of (P2) with $\mu\in\mathcal{M}(\bar\Omega)$, with support is in $\Omega$, as a nonhomogeneous term, $f_1\in H^{\frac{1}{2}}(\Gamma_1)$ and $f_2\in H^{-\frac{1}{2}}(\Gamma_2)$. The solution $u$ belongs to $W^{1,q}(\Omega),~\forall q<\frac{N}{N-1}$ for (IP2) and belongs to $H^1_{loc}(\mathbb{R}^N\setminus\bar{\Omega})$ for (EP2) satisfying equation $\eqref{infinity}$ at infinity.
\end{theorem}
\section*{Acknowledgement}
The author Akasmika Panda thanks the financial assistantship received from the Ministry of Human
Resource Development (M.H.R.D.), Govt. of India. Both the authors also acknowledge the facilities received from the Department of mathematics, National Institute of Technology Rourkela.
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train/arxiv
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\section*{Introduction}
To fix ideas let us consider a hypersurface $f$ inside a polynomial ring $R = \mathbb{F}_p[x_1, \ldots, x_n]$. In order to study the singularities of $f$ one may consider the test ideal filtration $\tau(R, f^\lambda)_{\lambda \geq 0}$ which is a decreasing $\mathbb{Q}$-indexed right-continuous filtration of ideals that is defined in terms of certain $p^{-e}$-linear maps. One has $\tau(R, f^0) = R$ and the smallest $\lambda$ for which $\tau(R, f^\lambda) \neq R$ is called the \emph{$F$-pure threshold}, $\fpt(f)$ for short. It has been known for a long time that if $f \in \mathbb{Q}[x_1,\ldots, x_n]$ and one considers the various reductions $f_p$ of $f$ to positive prime characteristic, then $\fpt(f_p) \xrightarrow{p \to \infty} \lct(f)$ and $\fpt(f_p) \leq \lct(f)$ for almost all $p$ (\cite[Theorem 6.8]{harayoshidagentightclosuremultideals}). Here $\lct(f)$ is the so-called log-canonical threshold which is a similar characteristic zero invariant that is defined using an embedded resolution of singularities. It is moreover conjectured that $\fpt(f_p) = \lct(f)$ for infinitely many $p$ (\cite[Conjecture 3.6]{mustatatakagiwatanabefthresholdsbernsteinsato}).
It has been observed for quasi-homogeneous hypersurfaces that if the log canonical threshold does not coincide with the $F$-pure threshold, then the denominator of the $F$-pure threshold is a $p$th power (see \cite[Theorem 3.5]{hbwzfthresholdshomogeneouspoly} or \cite[Lemma 3.7 (2)]{muellerfptquasihomo}). On the other hand, there are only two known (families of) examples where the $F$-pure threshold does not coincide with the log canonical threshold but the denominator of the $F$-pure threshold is not divisible by $p$ (see \cite[Example 4.5]{mustatatakagiwatanabefthresholdsbernsteinsato} and \cite[Proposition 2.7, Corollary 2.10]{cantonhswbehaviorofsingsatfpt}).
From the point of view of birational geometry $F$-jumping numbers with a denominator divisible by $p$ are special in the sense that the correspondence between certain Cartier linear maps and certain $\mathbb{Q}$-divisors breaks down in this case (e.g.\ \cite{schwedefadjunction}). This correspondence is central to many applications of test ideals in birational geometry.
The purpose of this note is to further illustrate that $F$-pure thresholds whose denominators are divisible by $p$ are special in the following sense. We call a finitely generated $R$-module $M$ endowed with a $p^{-1}$-linear map, i.e.\ an $R$-linear map $\kappa: F_\ast M \to M$ a Cartier module. One can associate to $M$ and $f \in R$ a test module filtration $\tau(M, f^\lambda)_{\lambda \geq 0}$ that has similar properties as in the case $M = R$. In particular, we can form the associated graded $Gr^\lambda M = \bigoplus_{\lambda > 0} \tau(M, f^{\lambda-\varepsilon})/\tau(M, f^\lambda)$. In Section \ref{SectionNilpotenceAssGraded} we will attach a natural Cartier module structure to these summands and show that such a Cartier module is \emph{nilpotent} if and only if the denominator of $\lambda$ is divisible by $p$. Here nilpotent means that some power of the structural map $\kappa$ acts as zero on the module.
This notion of nilpotence is interesting for the following reason. Nilpotent Cartier modules form a Serre subcategory so that we may consider the attached localized category of Cartier crystals. This category is then equivalent (\cite{blickleboecklecartierfiniteness}) to the category of unit $R[F]$-modules of Emerton and Kisin (\cite{emertonkisinrhunitfcrys}) and anti-equivalent to the category of perverse constructible $\mathbb{F}_p$-sheaves on the \'etale site associated to $\Spec R$.
The first main result that we obtain is
\begin{TheoB}[Theorem \ref{TheoNilpotentGr}]
Let $R$ be essentially of finite type over an $F$-finite field, $(M, \kappa)$ be a Cartier module, $f \in R$ and $\lambda$ an $F$-jumping number of the test module filtration of $M$ along $f$. The Cartier structure on $Gr^\lambda M$ defined by $\kappa^e f^{a_e(\lambda)}$ is not nilpotent if and only if $a_e(\lambda) = \lambda(p^e-1)$ and this quantity is an integer. In particular, if the denominator of $\lambda$ is divisible by $p$, then all these Cartier structures are nilpotent.
\end{TheoB}
Finally, we also show, extending and vastly simplifying some results of \cite[Section 4]{blicklestaeblerbernsteinsatocartier}, that test modules admit a simple description, akin to the case of an ideal in a polynomial ring, in many cases. A special case is the following
\begin{TheoB}[cf.\ Theorem \ref{TauFRegularDescription}]
Let $R$ be essentially of finite type over an $F$-finite field, $(M, \kappa)$ an $F$-regular Cartier module and $f \in R$ an $M$-regular element, then one has $\tau(M, f^\lambda) = \kappa^e f^{\lceil \lambda p^e \rceil} M$ for all $e \gg 0$.
\end{TheoB}
There is evidence that $F$-pure thresholds with denominator divisible by $p$ are related to certain arithmetic phenomena like non-ordinarity via the anti-equivalence mentioned above. If $X = V(f) \subseteq \mathbb{P}^{n+1}_k$ defines a smooth quasi-homogeneous Calabi-Yau hypersurface then the $F$-pure threshold of $f$ in $k[x_0, \ldots, x_{n+1}]$ is $1 - \frac{h}{p}$, where $0 \leq h \leq n$ is the order of vanishing of the Hasse-invariant associated to a certain deformation space of $X$ (see \cite{muellerfptquasihomo} for details). In particular, $h \neq 0$ if and only if $X$ is not ordinary. As a special case if $\dim X =1$, then non-ordinariness coincides with the case $H^{1}_{crys}(X)_0 = H^{1}_{\acute{e}t}(X) =0$, where the index $0$ indicates the slope zero part of crystalline cohomology. In this case, the crystalline cohomology is concentrated at slope $\frac{1}{2}$.
Moreover, we will also relate these Cartier structures on $Gr^\lambda M$ to certain Cartier structures obtained from eigenspaces of higher Euler operators that play a crucial role when constructing Bernstein-Sato polynomials in positive characteristic (see \cite{blicklestaeblerbernsteinsatocartier}). Similar results were observed by Bitoun (\cite{bitounbernsteinsatoposchar}) for the case that $M = R$ in the (equivalent) framework of unit $R[F]$-modules. His proof relies on formal properties of $p$-adic expansions. We will show that it is in fact also a formal consequence of the machinery of test modules via Theorem \ref{TheoNilpotentGr}. Our main result in this direction is Theorem \ref{AnotherMainResult}. Since its formulation requires several $\mathcal{D}$-module theoretic we only state a partial result in the introduction and refer the reader to Section \ref{DModApproach} for a detailed exposition.
Let $\mathcal{D}^e_{R[t]}$ be the ring of differential operators of order $\leq p^e -1$. We denote by $\theta_i \in \mathcal{D}^e_{R[t]}$ so-called higher Euler operator $t^i \partial_t^{[i]}$. To the data $(M, \kappa, f, \lambda)$ one associates $\mathcal{D}^e_{R[t]}$-modules (for varying $e$). In order to construct Bernstein-Sato polynomials one looks at certain eigenspaces of the higher Euler operators associated to these $\mathcal{D}^e_{R[t]}$-modules.
\begin{TheoB}[cf.\ Theorem \ref{AnotherMainResult}]
Let $R$ be regular and essentially of finite type over an $F$-finite field, $(M, \kappa)$ an $F$-regular Cartier module and $f$ an $M$-regular element. Then for $e \gg 0$ there is an isomorphism ${F^e}^! Gr^\lambda M \to E_i$, where $E_i$ is a certain eigenspace associated to a certain $\mathcal{D}^e_R[\theta_1, \ldots, \theta_{p^{e-1}}]$-module. This isomorphism induces a transition map ${F^e}^! Gr^\lambda M \to {F^e}^! Gr^\lambda M$. This transition map is induced by a non-nilpotent morphism $C: Gr^\lambda M \to {F^s}^! Gr^\lambda M$ if and only if $\lambda(p^s -1) \in \mathbb{Z}$ and $e,a$ are multiples of $s$. Moreover,
in this case $C$ is the adjoint of $\kappa^s f^{\lambda(p^s -1)}$.
\end{TheoB}
We will obtain a partial generalization of the correspondence between $F$-jumping numbers and zero of Bernstein-Sato polynomials for the case that $(M, \kappa)$ is $F$-regular to arbitrary Cartier modules. The precise statement we prove is
\begin{TheoB}[cf.\ Theorem \ref{LastTheo}]
Let $R$ be an $F$-finite regular ring and $(M, \kappa)$ a Cartier module. Fix a rational number $\lambda$. If for some $e \gg 0$ such that $\lambda(p^e -1) \in \mathbb{Z}$ one has that $\lambda - \frac{\lambda}{p^e}$ is a zero of the Bernstein-Sato polynomial $b_{M,f}^e(s)$, then $Gr_\sigma^\lambda M \neq 0$.
\end{TheoB}
Here $Gr_\sigma M$ is the filtration of $M$ by so-called \emph{non-$F$-pure modules}. We will study their basic properties in Section \ref{Nonfpuremodules} and also explain how they are a generalization of non-$F$-pure ideals studied in \cite{fujinotakagischwedenonlc}. We will also see that some of the pathologies that non-$F$-pure ideals exhibit in comparison to characeristic zero are constrained to the cases where the ``jumps'' have a denominator divisible by $p$.
We start with a short review of test modules in Section \ref{TestmoduleReview}. In Section \ref{SectionNilpotenceAssGraded} we discuss (non)-nilpotence of Cartier structures defined on the associated graded of the test module filtration. Then after recalling the necessary setup we relate these Cartier structures to the construction one uses to obtain Bernstein-Sato polynomials in Section \ref{DModApproach}. In Section \ref{Nonfpuremodules} we introduce the notion of non-$F$-pure module and study its basic properties. In the last section we use these to prove Theorem \ref{LastTheo}.
\subsection*{Acknowledgements} I thank Kevin Tucker and Mircea Musta\c{t}\u{a} for a useful discussion. Part of this paper was conceived while the author was visiting the University of Utah. I thank Karl Schwede for inviting me and for inspiring discussions. The author is supported by grant STA 1478/1-1 of the Deutsche Forschungsgemeinschaft (DFG).
\section{A brief review of test modules}
\label{TestmoduleReview}
In this section we review very briefly the necessary facts on test modules that we need. We refer the reader to \cite{blicklestaeblerfunctorialtestmodules} for a detailed treatment.
Fix an $F$-finite ring $R$. A \emph{Cartier module} is a pair $(M, \kappa)$, where $M$ is an $R$-module and $\kappa: F_\ast^e M \to M$ is an $R$-linear map. In many cases we assume $e =1$. It is however crucial to allow $e \geq 1$ in order to obtain meaningful Cartier structures on certain quotients. If we assume that $\Spec R$ is embeddable into a smooth scheme, then for given $e$ there is a contravariant functor to the category constructible $\mathbb{F}_{p^e}$-sheaves on the \'etale site. If we localize at nilpotent Cartier modules (to be defined below), call the resulting category \emph{Cartier crystals}, then this functor induces an anti-equivalence between Cartier crystals and perverse constructible $\mathbb{F}_{p^e}$-sheaves on the \'etale site (see \cite{schedlmeiercartierpervers} and references therein).
Given a Cartier module $(M, \kappa)$ and $f \in R$ and $\lambda$ a non-negative rational number we can form $R$-linear maps $\kappa^e f^{\lceil \lambda p^e \rceil}: F_\ast^e M \to M$ given by $m \mapsto \kappa^e(f^{\lceil \lambda p^e \rceil} m)$ for varying $e \geq 1$. The collection of these maps with addition induced by the one in $M$ and multiplication by composition form an $\mathbb{N}$-graded subring $\mathcal{C}$ of $\bigoplus_{e \geq 0} \Hom(F_\ast^e M, M)$, where we set $\mathcal{C}_0 = R$. It has both a left and a right $R$-module structure that are related via $r \varphi = \varphi r^{p^e}$ for any homogeneous element $\varphi$ of degree $e$. This ring is a special case of a so-called \emph{Cartier algebra}.
\begin{Def}
Let $R$ be an $F$-finite ring. A \emph{Cartier algebra} is an $\mathbb{N}$-graded ring $\bigoplus_{e \geq 0} \mathcal{C}_e$ with $\mathcal{C}_0 = R$ satisfying the relation $r \varphi = \varphi r^{p^e}$ for any $\varphi \in \mathcal{C}_e$ and $r \in R$.
\end{Def}
As usual $\mathcal{C}_+ = \bigoplus_{e \geq 1} \mathcal{C}_e$. We will write $\mathcal{C}_+^h$ for $(\mathcal{C}_+)^h$.
\begin{Bem}
\label{RoundedCartierAlgebras}
We will mostly use Cartier algebras of the form $\kappa^e f^{\lceil \lambda p^e \rceil}$ in this article. The reader familiar with test ideals may notice that people also often use algebras of the form $\kappa^e f^{\lceil \lambda (p^e -1) \rceil}$. If one computes test modules then both notions yield the same result (cf.\ \cite[Lemma 3.1]{staeblertestmodulnvilftrierung}). However, the categories of crystals are not the same and this will play an important role later on (cf.\ Section \ref{Nonfpuremodules}).
\end{Bem}
A $\mathcal{C}$-module means a left module over $\mathcal{C}$. We will moreover always assume that it is finitely generated as an $R$-module. We call a $\mathcal{C}$-module $M$ nilpotent if $\mathcal{C}_+^a M = 0$ for some (equivalently all) $a \gg 0$. A morphism $\varphi: M \to N$ of $\mathcal{C}$-modules is a \emph{nil-isomorphism} if its kernel and cokernel are nilpotent.
If $M$ is a $\mathcal{C}$-module then the descending chain $\mathcal{C}_+ M \supseteq \mathcal{C}_+^2 M \supseteq \ldots$ stabilizes (see \cite[Proposition 2.13]{blicklep-etestideale}) and we denote its stable member by $\underline{M}$.
\begin{Def}
The test module $\tau(M, f^\lambda)$ is the smallest $\mathcal{C}$-submodule $N$ of $M$ such that the inclusion $H^0_\eta(N)_\eta \subseteq H^0_\eta(M)_\eta$ is a nil-isomorphism for every associated prime $\eta$ of the $R$-module $M$.
\end{Def}
At this point we encourage the reader to take a look at \cite[Sections 1 and 2]{blicklestaeblerfunctorialtestmodules} for further discussion. It is proven in \cite[Theorems 3.4 and 3.6]{blicklestaeblerfunctorialtestmodules} that test modules exist if $R$ is essentially of finite type over an $F$-finite field. Moreover, in this case the test module filtration $\tau(M, f^\lambda)_{\lambda \geq 0}$ is a decreasing right-continuous discrete filtration. Many other formal properties like Brian\c con-Skoda also hold in this more general situation (see \cite[Section 4]{blicklestaeblerfunctorialtestmodules}). We call a number $\lambda$ such that $\tau(M, f^\lambda) \neq \tau(M, f^{\lambda - \varepsilon})$ for all $\varepsilon \leq \lambda$ an \emph{$F$-jumping number}.
Finally, we say that a prime $\eta \in \Spec R$ of a $\mathcal{C}$-module $M$ is an associated prime of $M$ if $H^0_\eta(M)_\eta$ is not nilpotent. These form a subset of the associated primes of the underlying $R$-module.
We call a Cartier module $(M, \kappa)$ \emph{$F$-regular} if $\tau(M, f^0) = M$. The smallest $\lambda > 0$ such that $\tau(M, f^\lambda) \neq \tau(M, f^0)$ is called the \emph{$F$-pure threshold} of $f$ with respect to $M$. More generally, if we have a $\mathcal{C}$-module $N$, where $\mathcal{C}_e = \kappa^e f^{\lceil \lambda p^e \rceil}$, then we say that $N$ is $F$-regular if $\tau(N, f^\lambda) = N$.
If $N$ is a $\mathcal{C}$-module and $\eta_1, \ldots, \eta_n$ its associated primes then we call $c_1, \ldots, c_n$ a \emph{sequence of test elements} if $c_i \notin \eta_i$ and the $\underline{H^0_{\eta_i}(N_{c_i})}$ are $F$-regular. If all associated primes of $M$ are minimal then we only need a single test element and this condition simplifies: If $N$ is a $\mathcal{C}$-module whose associated primes are minimal, then we call $c \in R$ a test element if $c$ is not contained in any minimal prime of $N$ and $\underline{N_c}$ is $F$-regular.
With this notion one has
\begin{Theo}
Let $R$ be essentially of finite type over an $F$-finite field, $(M, \kappa)$ a Cartier module and $f \in R$. Then there exists a sequence of test elements $c_1, \ldots, c_n$ for the $\mathcal{C}$-module $M$, where $\mathcal{C}_e = \kappa^e f^{\lceil \lambda p^e \rceil}$, and one has
\[\tau(M, f^\lambda) = \sum_{e \geq e_0} \sum_{i=1}^n \mathcal{C}_{e} c_i^{a_i} \underline{H^0_{\eta_i}(M)} \]
for any $e_0 \geq 0$ and any $a_i \geq 1$.
Moreover, if $M$ only has minimal associated primes, then this simplifies to
\[\tau(M, f^\lambda) = \sum_{e \geq e_0} \mathcal{C}_e c^a \underline{M}.\]
\end{Theo}
\begin{proof}
See \cite[Theorem 3.4, Theorem 3.6]{blicklestaeblerfunctorialtestmodules} for the general case and \cite[Theorem 3.11]{blicklep-etestideale} for the special case
\end{proof}
It is mostly this presentation that we will be used in this article. Also note that we will prove shortly that, if $(M, \kappa)$ is $F$-regular, then one has in fact $\tau(M, f^\lambda) = \kappa^e f^{\lceil \lambda p^e \rceil} M$ (Theorem \ref{TauFRegularDescription} below).
\section{Nilpotence of the associated graded}
\label{SectionNilpotenceAssGraded}
Throughout this section $R$ is a ring essentially of finite type over an $F$-finite field. This assumption is imposed to ensure existence of test modules (see \cite[Theorem 3.6]{blicklestaeblerfunctorialtestmodules}; in our setup this automatically implies existence of a sequence of test elements -- cf.\ \cite[Remark 3.7]{blicklestaeblerfunctorialtestmodules}) and discreteness of the filtration. Granting these notions our arguments work for arbitrary $F$-finite rings.
Fix a Cartier module $(M, \kappa)$ and $f \in R$.
In \cite[Proposition 4.5]{staeblertestmodulnvilftrierung} the author defined a Cartier structure on the associated graded of the test module filtration. Namely, if $Gr^\lambda(M) = \tau(M,f^{\lambda-\varepsilon})/\tau(M, f^\lambda)$, then $\kappa f^{\lceil \lambda(p-1)\rceil}$ operates on this quotient. In fact, more generally $\kappa^e f^{\lceil \lambda(p^e-1)\rceil}$ operates on $Gr^\lambda(M)$ and one easily checks if $Gr^\lambda M$ is nilpotent with respect to this Cartier structure, then also with respect to the one above.
While this definition may seem ad hoc we point out that if $i: \Spec R/(f) \to \Spec R$ denotes the natural inclusion and $f$ is a non-zero-divisor on $R$ then for any Cartier module $(M, \kappa)$ on $\Spec R$ the induced Cartier structure on $R^1i^! M$ is given by $\kappa f^{p-1}$ (cf.\ \cite[Example 3.3.12]{blickleboecklecartiercrystals}). By Brian\c con-Skoda one easily sees that the support of $Gr^\lambda(M)$ is contained in $\Spec R/(f)$. We will in fact see shortly that these Cartier structures are very natural.
The next lemma was already proven in \cite[Proposition 3.2]{staeblertestmodulnvilftrierung} for the case that $M$ has only minimal primes. We give a simplified proof here.
\begin{Le}
\label{CartierTauDivisionByP}
Let $(M, \kappa)$ be a Cartier module and $f \in R$. Then for all $\lambda \geq 0$ we have $\kappa(\tau(M, f^\lambda)) = \tau(M, f^{\frac{\lambda}{p}})$.
\end{Le}
\begin{proof}
By virtue of \cite[Theorem 3.4]{blicklestaeblerfunctorialtestmodules} we have \[\tau(M, f^\lambda) = \sum_{i=1}^n \sum_{e \geq e_0} \kappa^e f^{\lceil \lambda p^e\rceil} c_i \underline{H^0_{\eta_i}(M)}\] for any $e_0 \geq 0$, where the $\eta_i$ are the associated primes of $M$ and the $c_i$ form a sequence of test elements in the sense of \cite[Definition 3.1]{blicklestaeblerfunctorialtestmodules}. We thus have \begin{align*} \kappa (\tau(M,f^\lambda)) &= \sum_{i=1}^n \sum_{e \geq e_0} \kappa^{e+1} f^{\lceil \frac{\lambda}{p} p^{e+1}\rceil} c_i \underline{H^0_{\eta_i}(M)} \\&= \sum_{i=1}^n \sum_{e \geq e_0 +1} \kappa^{e} f^{\lceil \frac{\lambda}{p} p^{e}\rceil} c_i \underline{H^0_{\eta_i}(M)} = \tau(M, f^{\frac{\lambda}{p}}). \end{align*}
\end{proof}
\begin{Le}
\label{Remarkable}
Let $(M, \kappa)$ be a Cartier module and $f \in R$. For any integer $a_e(\lambda)$ the map given by $\kappa^e f^{a_e(\lambda)}$ induces a Cartier structure on $Gr^{\lambda} M$ if and only if $a_e(\lambda) \geq \lceil \lambda (p^e -1)\rceil$.
\end{Le}
\begin{proof}
Using Lemma \ref{CartierTauDivisionByP} and Brian\c con-Skoda (\cite[Proposition 4.1]{blicklestaeblerfunctorialtestmodules}) one has \[\kappa^e f^{a_e(\lambda)} \tau(M, f^\lambda) = \tau(M, f^{\frac{\lambda - a_e(\lambda)}{p^e}}).\] For this to induce a Cartier structure on the quotient we must have $\frac{\lambda - a_e(\lambda)}{p^e} \geq \lambda$. Equivalently, $a_e(\lambda) \geq \lambda(p^e -1)$ and since $a_e(\lambda)$ is an integer this is equivalent to $a_e(\lambda) \geq \lceil \lambda (p^e -1)\rceil$. One similarly checks that in this case $\kappa^e f^{a_e(\lambda)} \tau(M, f^{\lambda - \varepsilon}) \subseteq \tau(M, f^{\lambda - \varepsilon})$ using that the test module filtration is decreasing.
\end{proof}
The first main result of this section is
\begin{Theo}
\label{TheoNilpotentGr}
Let $(M, \kappa)$ be a Cartier module, $f \in R$ and $\lambda$ an $F$-jumping number of the test module filtration of $M$ along $f$. The Cartier structure on $Gr^\lambda M$ defined by $\kappa^e f^{a_e(\lambda)}$ is not nilpotent if and only if $a_e(\lambda) = \lambda(p^e-1)$ and this quantity is an integer. In particular, if the denominator of $\lambda$ is divisible by $p$, then all these Cartier structures are nilpotent.
\end{Theo}
\begin{proof}
We may write $\lceil \lambda(p^e-1) \rceil = \lambda(p^e -1) + \delta$ with $0 \leq \delta < 1$ and $\delta = 0$ if and only if $\lambda(p^e -1 )$ is an integer. By Brian\c con-Skoda (\cite[Proposition 4.1]{blicklestaeblerfunctorialtestmodules}) we have \[\kappa^e f^{\lambda(p^e -1) + \delta} \tau(M, f^{\lambda -\varepsilon}) = \kappa^e \tau(M, f^{\lambda- \varepsilon + \lambda(p^e -1) + \delta}) = \kappa^e \tau(M,f^{\lambda p^e + \delta - \varepsilon}).\]
Now we use Lemma \ref{CartierTauDivisionByP} and obtain \[\kappa^e \tau(M, f^{\lambda p^e + \delta -\varepsilon}) = \tau(M, f^{\lambda + \frac{\delta - \varepsilon}{p^e}}).\] Note in fact, that we may take any $\varepsilon'$ such that $\varepsilon> \varepsilon' >0 $ and still have $\tau(M,f^{\lambda -\varepsilon'}) = \tau(M, f^{\lambda-\varepsilon})$. Hence, if $\delta >0$ this actually forces $\varepsilon < \delta$. But then $\tau(M,f^{\lambda + \frac{\delta - \varepsilon}{p^e}}) \subseteq \tau(M, f^\lambda)$ which shows that the Cartier structure is nilpotent. The same argument also shows nilpotence for any $e_a(\lambda) > \lceil \lambda(p^e -1)\rceil$.
\end{proof}
\begin{Bem}
Note that with the notation of the proof of Theorem \ref{TheoNilpotentGr} if $\mu < \lambda$ is the previous $F$-jumping number then necessarily $\lambda- \mu \leq \delta$. In particular, if $\lambda$ is the $F$-pure threshold, then it actually follows that if $\delta \neq 0$ then $\delta \geq \lambda$ since we can take any $0 < \varepsilon < \lambda$ without changing $\tau(M, f^{\lambda-\varepsilon})$. Still assuming that $\lambda$ is the $F$-pure threshold and writing $\delta = \lceil \lambda(p^e -1) \rceil - \lambda(p^e -1) \geq \lambda$ we obtain equivalently that $\lceil \lambda(p^e -1) \rceil \geq \lambda p^e$. From this one easily deduces that $\lambda \notin (\frac{a}{p^e}, \frac{a}{p^e -1})$ for any integer $0 \leq a \leq p^e -1$. This is the analogue of \cite[Proposition 4.3 (ii)]{blicklemustatasmithdiscretenesshypersurfaces} (which is proved using similar techniques) for modules. Also note that the result of \cite{blicklemustatasmithdiscretenesshypersurfaces} requires $R$ to be regular and $F$-finite while the above argument works for any $F$-finite ring
\end{Bem}
\begin{Bsp}
Consider the cusp $f = x^2 + y^3 \in \mathbb{F}_p[x,y] = R$. If $\lambda$ denotes the $F$-pure threshold then $\tau(R, f^\lambda) = (x,y)$ and for $p > 3$ one has (see \cite[Example 3.4]{mustatabernsteinsatopolynomialspositivechar}) \[\lambda =
\begin{cases}
\frac{5}{6} & p \equiv 1 \mod 3,\\
\frac{5}{6} - \frac{1}{6p} & p \equiv 2 \mod 3.
\end{cases}\] Moreover, $\lambda = \frac{1}{2}$ for $p = 2$ and $\lambda = \frac{2}{3}$ if $p =3$. So if $p \equiv 1 \mod 3$ we can take $e =1$ and obtain $\frac{5}{6} (p -1) \in \mathbb{Z}$. Since the quotient $Gr^\lambda$ is just $\mathbb{F}_p$ and we know that the obtained Cartier structure is not nilpotent it has to be $\kappa = \id$.
Of course, for any $p$ one has $Gr^\lambda = \mathbb{F}_p$ (as an $R$-module quotient) so that it admits the non-nilpotent Cartier structure $\kappa = \id$.
\end{Bsp}
We end this section by proving a simple description of $\tau(M, f^\lambda)$ in the case where $(M, \kappa)$ is $F$-regular. This is very useful for computations. We will use it in the next section to extend the relation of $F$-jumping numbers and zeros of Bernstein-Sato polynomials (\cite{blicklestaeblerbernsteinsatocartier}) to $F$-regular Cartier modules.
\begin{Theo}
\label{TauFRegularDescription}
Let $(M, \kappa)$ be a Cartier module and $f$ an $M$-regular element. If $(M, \kappa)$ is $F$-regular, then $\tau(M, f^\lambda) = \kappa^e f^{\lceil \lambda p^e \rceil} M$ for all $e \gg 0$.
\end{Theo}
\begin{proof}
By $F$-regularity we have $M = \tau(M, f^0)$. Hence, by Brian\c con-Skoda (\cite[Proposition 4.1]{blicklestaeblerfunctorialtestmodules}) and Lemma \ref{CartierTauDivisionByP} we get \[\kappa^e f^{\lceil \lambda p^e \rceil} M = \kappa^e f^{\lceil \lambda p^e \rceil} \tau(M, f^0) = \kappa^e \tau(M, f^{\lceil \lambda p^e\rceil}) = \tau(M, f^{\lceil \lambda p^e \rceil p^{-e}}).\]
Since $\lambda \leq \frac{\lceil \lambda p^e\rceil}{p^e} \leq \lambda + \frac{1}{p^e}$ we conclude by right-continuity that $\tau(M, f^{\lceil \lambda p^e \rceil p^{-e}}) = \tau(M, f^{\lceil \lambda p^e \rceil})$ for all $e$ sufficiently large.
\end{proof}
\begin{Ko}
\label{TauStuff}
Let $(M, \kappa)$ be a Cartier module and $f$ an $M$-regular element. Then $\tau(M, f^\lambda) = \tau(\tau(M, f^0), f^\lambda)$. In particular, we have $\tau(M, f^\lambda) = \kappa^e f^{\lceil \lambda p^e \rceil} \tau(M, f^0)$.
\end{Ko}
\begin{proof}
Clearly, $\tau(M, f^0) \subseteq M$ so that $\tau(\tau(M, f^0), f^\lambda) \subseteq \tau(M, f^\lambda)$ by \cite[Proposition 1.15]{blicklestaeblerfunctorialtestmodules}.
For the other inclusion, by definition $\tau(M, f^0)$ is the smallest submodule of $M$ for which the inclusions $H^0_\eta(\tau(M, f^0))_\eta \subseteq H^0_\eta(M)_\eta$ are nil-isomorphisms with respect to $\kappa$ for all associated primes $\eta$ of $M$. Being a nil-isomorphism here just means that some power of $\kappa$ annihilates the cokernel. But then a fortiori some power of $\mathcal{C}_+ = \bigoplus_{e \geq 1} \kappa f^{\lceil \lambda p^e \rceil}$ acts as zero on this cokernel. By definition of $\tau(M, f^\lambda)$ this shows $\tau(M, f^\lambda) \subseteq \tau(M, f^0)$. Now we may apply $\tau(-, f^\lambda)$ to this inclusion to obtain $\tau(M, f^\lambda) \subseteq \tau(\tau(M, f^0), f^\lambda)$, where we again use \cite[Proposition 1.15]{blicklestaeblerfunctorialtestmodules} and the fact that $\tau$ (for a fixed Cartier algebra) is idempotent.
The final claim is an immediate application of Theorem \ref{TauFRegularDescription}.
\end{proof}
\begin{Bem}
The assumption on the $F$-regularity cannot be omitted, that is, if $M$ is not $F$-regular, then $\tau(M, f^\lambda) \neq \kappa^e f^{\lceil \lambda p^e\rceil} M$ in general. Consider for example $R = \mathbb{F}_p[x,y]$ and the Cartier module $M =\mathbb{F}_p[x,y]\cdot y^{-1} \subseteq j_\ast \mathbb{F}_p[x,y,y^{-1}]$, where $j: D(y) \to \Spec R$, with Cartier structure induced by localization.
Then $M$ is not $F$-regular since $\mathbb{F}_p[x,y]$ is a proper submodule that generically agrees with $M$. It is easy to see that $M$ is $F$-pure. Moreover, $y$ is a test element for $(M, \kappa)$ and therefore $\tau(M, x^\lambda) = x^{\lfloor \lambda \rfloor} R$ while $\kappa^e x^{\lceil \lambda p^e \rceil} M$ is not even contained in $R$ since $y^{-1}$ is a fixed point for the Cartier operation.
Similarly, if $M$ is not $F$-pure and has only minimal associated primes then $\tau(M, f^\lambda) \neq \kappa^e f^{\lceil \lambda p^e\rceil} \kappa^a c M$ in general. For example, consider the Cartier module $M = k[x] \cdot x^{-n} \subseteq k[x]$ with $n \geq 2$ and take $\lambda = 0$ and $f = x$. Clearly, $c = x$ is a test element and $\underline{M} = \kappa^a k[x] \cdot x^{-n} = k[x] \cdot x^{-1}$. We see that $\tau(M, x^0) = k[x]$ by the theorem. However, $\kappa^a x M = k[x] \cdot x^{-1}$ for all $a \gg 0$ so that (since $\lambda =0$) $\kappa^e x^{0} k[x] \cdot x^{-1} = k[x] \cdot x^{-1}$.
\end{Bem}
\section{Cartier structures on the associated graded induced by differential operators}
\label{DModApproach}
Throughout this section we assume that $R$ is a regular ring essentially of finite type over an $F$-finite field. Regularity of $R$ is critical since we need that $F_\ast^e R$ is a flat $R$-module to ensure that ${F^e}^!$ is exact.
The goal of this section is to show that the Cartier structures defined on the associated graded of $\tau(M, f^\lambda)$ at the beginning of Section \ref{SectionNilpotenceAssGraded} correspond to the Cartier structures obtained on quotients of eigenspaces of certain $\mathcal{D}^e_R[\theta_1, \theta_p, \ldots, \theta_{p-1}]$-modules associated to $(M, f^\lambda)$ for varying $e$. In particular, we will recover and generalize results of Bitoun (\cite{bitounbernsteinsatoposchar}).
We start by recalling the necessary $\mathcal{D}$-module theoretic notions (see \cite{mustatabernsteinsatopolynomialspositivechar} and \cite{blicklestaeblerbernsteinsatocartier} for more elaborate discussions). For any ring $R$ containing $\mathbb{F}_p$ the ring of $\mathbb{F}_p$-linear differential operators $\mathcal{D}_R \subseteq \End_{\mathbb{F}_p}(R)$ in the sense of Grothendieck (\cite[Definition 16.8.1]{EGAIV-4}) admits the so-called \emph{$p$-filtration} $\mathcal{D}_R = \bigcup_{e \geq 0} \mathcal{D}^e_R$, where $\mathcal{D}^e_R \cong \End_R(F^e_\ast R)$ with $\colim_{e\geq 0} \mathcal{D}^e_R = \mathcal{D}_R$ (see \cite{chasepfiltration}). In particular, we see that for any $R$-module $M$ there is a natural right action of $\mathcal{D}^e_R$ on ${F^e}^! M = \Hom_R(F_\ast^e R, M)$.
\begin{Konv}
Unless otherwise specified modules over rings of differential operators will always be \emph{right} modules.
\end{Konv}
\begin{Bsp}
If $R = k[t_1, \ldots, t_n]$ is a polynomial ring over a perfect field $k$, then $\mathcal{D}_R^e = k[\partial_{t_i}^{[1]}, \ldots, \partial_{t_i}^{[p^e -1]} \, \vert \, i =1, \ldots, n]$, where the $\partial_{t_i}^{[j]}$ are \emph{divided power operators} that act as follows \[\partial_{t_i}^{[j]} \bullet (t_1^{a_1} \cdots t_n^{a_n}) = \binom{a_i}{j} t_1^{a_1} \cdots t_{i-1}^{a_{i-1}} t_i^{a_i - j} t_{i+1}^{a_{i+1}} \cdots t_n^{a_n}.\] Moreover, one has $[\partial_{t_i}^{[a_i]}, \partial_{t_j}^{[a_j]}] = 0$ for $i \neq j$.
As an aside we also mention that since $k$ is perfect $\mathcal{D}_R$ coincides with $k$-linear differential operators.
\end{Bsp}
Assume from now on that $R$ is a regular essentially of finite type over an $F$-finite field and $f \in R$. We consider the graph embedding $\gamma: \Spec R \to \Spec R[t], t \mapsto f$. Given a Cartier module $M$ on $R$ we obtain a Cartier structure on $\gamma_\ast M$ via the natural isomorphism $\gamma_\ast F_\ast \cong F_\ast \gamma_\ast$ which we will from now on suppress from notation. Note that $\mathcal{D}^e_{R[t]}$ contains the differential operator $\theta_{a} = t^{a} \partial_{t}^{[a]}$ for any $a < p^{e}$. In fact, $\mathcal{D}^e_{R[t]} = \mathcal{D}^e_R[t, \partial_t^{a} \, \vert \, a < p^e]$. We call the $\theta_a$ the \emph{(higher divided power) Euler operators}.
\begin{Le}
Let $(M, \kappa)$ be a Cartier module on $R$ and $\gamma: \Spec R \to \Spec R[t], t \mapsto f$ the graph embedding. Then the adjoint of the Cartier structure ${F^{e}}^!\gamma_\ast M \to {F^{e+a}}^! \gamma_\ast M$ is given by \begin{align*}\Hom_{R[t]}(F_\ast^e R[t], \gamma_\ast M) &\longrightarrow \Hom_{R[t]}(F_\ast^{a+e} R[t], \gamma_\ast M) \\ \varphi &\longmapsto \kappa^a \circ F_\ast^e \varphi. \end{align*}
\end{Le}
\begin{proof}
First of all, note that duality of finite morphisms yields a map \[\gamma_\ast M \longrightarrow {F^a}^! \gamma_\ast M,\quad m \longmapsto [s \mapsto \kappa^e(\gamma^\#(s)m)],\] where $\gamma^{\#}$ denotes the induced map $R[t] \to R, t \mapsto f$, corresponding to $F_\ast^a \gamma_\ast M \to \gamma_\ast M$. Applying ${F^e}^!$ (which is just a hom-functor) we get a map \[F^{e} \gamma_\ast M \longrightarrow {F^{e}}^! {F^a}^! \gamma_\ast M, \quad\varphi \longmapsto [r \mapsto [s \mapsto \kappa^a(\gamma^\#(r)(\varphi(s)))]].\] Now by tensor-hom adjunction we have ${F^e}^! {F^a}^! \cong {F^{e+a}}^!$ which sends a map as above to the map $[s \mapsto \kappa^a(\varphi(\gamma^\#(s)))]$.
\end{proof}
Fix a Cartier module $M$. Recall that ${F^e}^! \gamma_\ast M$ admits a $\mathcal{D}^e_{R[t]}$-module structure and hence in particular a $\mathcal{D}^e_R[\theta_1, \theta_p, \ldots, \theta_{p^{e-1}}]$-module structure. As such it admits a decomposition into generalized eigenspaces ${F^e}^! \gamma_\ast M = \bigoplus_{i \in \mathbb{F}_p^e} E_{i}$, where $\theta_{p^j}$ acts on $E_i$ by multiplication with $i_j$ (see \cite[Lemma 3.1]{blicklestaeblerbernsteinsatocartier} for details). As follows from Lucas' Theorem (see \cite[\S XXI]{lucasbinom}) the projection ${F^e}^! \gamma_\ast M \to E_i$ is induced by $\pi: F_\ast^e R[t] \to F^e_\ast R[t], R t^m \mapsto 0$ unless $m = i_1 +i_2p + \ldots + i_e p^{e-1}$ in which case $r t^m \mapsto rt^m$.
Moreover, this eigenspace decomposition is preserved by morphisms. If we consider a $\mathcal{D}^e_R[\theta_1, \theta_p, \ldots, \theta_{p^{e-1}}]$-module as a $\mathcal{D}^{e-1}_R[\theta_1, \theta_p, \ldots, \theta_{p^{e-2}}]$-module, then these eigenspace decompositions are compatible with each other in the sense that if $M = \bigoplus_{i \in \mathbb{F}_p^{e-1}} E_i$ is an eigenspace decomposition, then the $E_i = \bigoplus_{j \in \mathbb{F}_p} E_{i_1, \ldots, i_{e-1}, j}$ are the eigenspaces as $\mathcal{D}^{e-1}_R[\theta_1, \theta_p, \ldots, \theta_{p^{e-2}}]$-module.
Note that one has a natural map $\gamma_\ast {F^e}^! M \to {F^e}^! \gamma_\ast M, \varphi \mapsto \varphi(\gamma^{\#})$. By abuse of notation we will write $\gamma_\ast C^e(M)$ for the image of $C^e(M) = \{ [r \mapsto \kappa^e(rm)] \, \vert \, m \in M\}$ under this natural map.
A key technical result (\cite[Corollary 5.3]{blicklestaeblerbernsteinsatocartier}) yields that for all $e \gg 0$ the $(i_1, \ldots, i_e)$-eigenspace of the quotient \[N_e := (\gamma_\ast C^e(M))\mathcal{D}^e_R[\theta_1, \theta_p, \ldots, \theta_{p^{e-1}}]/ (\gamma_\ast C^e(fM))\mathcal{D}^e_R[\theta_1, \theta_p, \ldots, \theta_{p^{e-1}}]\] is ismorphic as a $\mathcal{D}^e_R$-module to \[ P_e := C^e(f^{i_1 + i_2p + \ldots + i_ep^{e-1}} M) \cdot \mathcal{D}^e_R/C^e(f^{1 + i_1 + i_2 p + \ldots + i_e p^{e-1}}M) \cdot \mathcal{D}^e_R.\]
This isomorphism is given as follows. Given $\varphi \in N_e$ that is contained in the $(i_1,\ldots,i_e)$-eigenspace one has $\varphi(rt^m) = 0$ for all $m < p^e$ if $m \neq i_1 + i_2p + \ldots +i_ep^{e-1}$. The image of $\varphi$ under this isomorphism is the map $r \mapsto \varphi(rf^{i_1 + i_2 p + \ldots +i_e p^{e-1}})$.
With this notation and under the additional assumption that $M$ is \emph{$F$-regular} it follows that $P_e$ and hence the $(i_1, \ldots, i_e)$-eigenspace of $N_e$ above is naturally isomorphic as a $\mathcal{D}^e_R$-module to \[{F^e}^! Gr^{i} M := {F^e}^! (\tau(M, f^{(i_1 + i_2p + \ldots + i_ep^{e-1})/p^e})/\tau(M, f^{(1+i_1 + i_2p + \ldots +i_e p^{e-1})/p^e}))\] for all $e \gg 0$. For the case that $M$ has only minimal associated primes, see \cite[Corollary 4.8]{blicklestaeblerbernsteinsatocartier} and note that ${F^e}^!$ is exact since $R$ is regular. In the general $F$-regular case, we can use Theorem \ref{TauFRegularDescription} above instead. In fact, for $n \in \mathbb{N}$ we have $C^e(f^n M) = \{[r \mapsto \kappa^e(rf^n m)] \, \vert\, m \in M\} \subseteq \Hom(F_\ast^e R, M)$ while ${F^e}^! \tau(M) \subseteq {F^e}^! M = \Hom(F_\ast^e R, M)$ and the natural inclusion $C^e(f^m M) \to {F^e}^! M$ induces the isomorphism.
We stress that \cite[Corollary 4.8]{blicklestaeblerbernsteinsatocartier} is the only test module theoretic input of \cite{blicklestaeblerbernsteinsatocartier}. Thus, by virtue of Theorem \ref{TauFRegularDescription} the result \cite[Theorem 5.4]{blicklestaeblerbernsteinsatocartier} immediately generalizes to the more general framework of $F$-regular Cartier modules in the sense of \cite{blicklestaeblerfunctorialtestmodules}. In fact, Theorem \ref{TauFRegularDescription} is more precise, since it guarantees an equality $\kappa^e f^m M = \tau(M, f^{\frac{m}{p^e}})$ for any $m \in \mathbb{Z}$ and $e \geq 1$, thereby removing an inaccuracy in the proof of \cite[Theorem 5.4]{blicklestaeblerbernsteinsatocartier} -- the application of \cite[Corollary 4.8]{blicklestaeblerbernsteinsatocartier} is not correct since one may have to enlarge the numerator.
We will prove a partial generalization of this result to arbitrary Cartier modules in the last section.
With these preliminaries we can prove the
\begin{Prop}
\label{EigenspaceRestrictionCartier}
Let $M$ be a Cartier module on $R$ and $f \in R$ a non-zero-divisor on $M$. Then the transition map $\gamma_\ast {F^{e}}^! M \to {F^{e+a}}^! \gamma_\ast M$ induces a map $N_{e} \to N_{e+a}$. Furthermore, if $E_i$ is the $(i_1, \ldots, i_e)$-eigenspace of $N_e$ and $E_{i,j}$ the $(i_1, \ldots, i_e, j_{e+1}, \ldots, j_{e+a})$- eigenspace of $N_{e+a}$ then we get an induced map
\[\alpha: \begin{xy} \xymatrix@1{E_i \ar[r]^i & N_e \ar[r]& N_{e+a} \ar[r]^{\pi} & E_{i,j},}\end{xy}\] where $\pi$ is the projection onto the eigenspace and $i$ is the natural inclusion.
If, in addition, $M$ is $F$-regular, then via the isomorphisms $E_i \to {F^e}^! Gr^i M$, $E_{i,j} \to {F^{e+a}}^! Gr^{i,j} M$ and adjunction we get an induced Cartier structure $F_\ast^{a} {F^e}^! Gr^i M \to {F^e}^! Gr^i M$ which is given by $\varphi \mapsto \kappa^a f^{j_{e+1} + j_{e+2} p + \ldots + j_{e+a}p^{e+a-1}} \circ \varphi$.
\end{Prop}
\begin{proof}
Recall that the isomorphism $E_i \to {F^e}^! Gr^i$ is given by \[\varphi \longmapsto [r \mapsto \varphi(r f^{i_1 +i_2p + \ldots i_ep^{e-1}})].\] Similarly, the image of $\alpha(\varphi)$ in ${F^{e+a}}^! Gr^{i,j} M$ is given by \[r \longmapsto \pi \kappa^a \varphi(rf^{i_1 + i_2 p + \ldots i_e p^{e-1}} r).\] We claim that this map coincides with the map
\[\psi_j: r \longmapsto \kappa^a \varphi(r f^{i_1 + i_2 p + \ldots i_e p^{e-1} + j_{e+1} p^e + \ldots j_{e+a}p^{e+a-1}}).\]
Note that $\{ rt^l \, \vert \, r \in R$ with $0 \leq l \leq p^{e+a} -1$\} is a set of generators for $F_\ast^{e+a} R[t]$.
Since $N_{e+a}$ is the direct sum of its eigenspaces the claim comes down to verifying that the equality \[\sum_{j \in \mathbb{F}_p^a} \psi_j = \kappa^a \varphi\] holds in $N_{e+a}$, where we lift the $\psi_j$ to $N_{e+a}$ by taking the natural section of the projection onto the eigenspace.
Hence, $\psi_j$ evaluated at \[t^{i_1 + i_2 p + \ldots + i_ep^{e-1} + l_{e+1} p^e + \ldots l_{e+a} p^{e+a -1}}\] coincides with \[\kappa^a\varphi(f^{i_1 + i_2 p + \ldots + i_ep^{e-1} + l_{e+1} p^e + \ldots l_{e+a} p^{e+a -1}})\] if $j = l$ and is zero otherwise. In particular, we see that the claimed identity holds.
Now since we may view $\varphi$ as a map $\varphi: F_\ast^e R \to M$ we have that \[\psi_j(r) = \kappa^a f^{j_{e+1} + j_{e+2} p + \ldots j_{e+a} p^{a-1}} \varphi(f^{i_1 + i_2 p + \ldots + i_e p^{e-1}}r )\] which shows the claim.
\end{proof}
\begin{Bem}
We note that taking the projection onto the corresponding eigenspace in Proposition \ref{EigenspaceRestrictionCartier} is necessary (see \cite[Example 6.15]{mustatabernsteinsatopolynomialspositivechar}).
\end{Bem}
Next, recall that any $\lambda \in [0,1)$ admits a unique non-terminating $p$-adic expansion $\lambda = \sum_{i \geq 1} \frac{c_i(\lambda)}{p^i}$ with $0 \leq c_i(\lambda) \leq p-1$. Moreover, one obtains for any $e \geq 1$ that $\sum_{i=1}^e \frac{c_i(\lambda)}{p^i} = \frac{\lceil \lambda p^e\rceil -1}{p^e}$.
Hence, if $\lambda \in [0,1)$, then we can identify for all $e \gg 0$ ${F^e}^! Gr^{\frac{\lceil \lambda p^e \rceil}{p^e}} M$ as a $\mathcal{D}^e_R$-module with the $(c_1(\lambda), \ldots, c_e(\lambda))$-eigenspace of $N_e$. Moreover, since the test module filtration is discrete we may assume, by choosing a larger $e$ if necessary, that \[\tau(M, f^{\frac{\lceil \lambda p^e \rceil}{p^e}})= \tau(M, f^\lambda) \text{ and } \tau(M, f^{\frac{\lceil \lambda p^e \rceil -1}{p^e}}) = \tau(M, f^{\lambda - \varepsilon}).\]
In other words $Gr^{\frac{\lceil \lambda p^e \rceil}{p^e}} M = Gr^\lambda M$.
\begin{Le}
\label{SufficientCartier}
For $e \gg 0$ the morphism \[\begin{xy}\xymatrix@1{{F^e}^! Gr^{\lambda} M \ar[r]^-\cong& E_i \ar[r]^\alpha& E_{i,j} \ar[r]^-\cong & {F^{e+a}}^! Gr^{\lambda} M} \end{xy},\] where $\alpha$ is the map in Proposition \ref{EigenspaceRestrictionCartier}, is of the form ${F^e}^! C$ for a morphism $C: Gr^\lambda M \to {F^a}^! Gr^\lambda M$ if $\lambda (p^s -1)$ is an integer for some $s \in \mathbb{Z}$ and both $e,a$ are multiples of $s$.
\end{Le}
\begin{proof}
$\lambda(p^s -1)$ is an integer if and only if the denominator of $\lambda$ is not divisible by $p$. This is equivalent to the fact that the $p$-adic expansion of $\lambda$ is strictly periodic with period length dividing $s$. In this case, one has $\lambda(p^{s}-1) = p^{s} \sum_{i=1}^{s} \frac{c_i(\lambda)}{p^i}$ (see e.g.\ \cite[Lemma 2.6]{hbwzfthresholdshomogeneouspoly}). Write $e = e' s$ and $a = a' s$ Then by Proposition \ref{EigenspaceRestrictionCartier} and the previous observation the Cartier structure ${F^e}^! Gr^\lambda M \to {F^{e+a}}^! Gr^\lambda M$ is given by the adjoint of \[\kappa^a f^{\lceil \lambda (p^a -1) \rceil} = (\kappa^s f^{\lceil \lambda (p^s -1) \rceil})^{a'}\] and we may set \[C = [m \longmapsto [r \mapsto (\kappa^s f^{\lceil \lambda (p^s-1) \rceil})^{e'}(rm)]].\]
\end{proof}
\begin{Bem}
Note that the Cartier structure on ${F^e}^! Gr^\lambda M$ is only giving us information on $Gr^\lambda M$ if we have a map $C$ as in Lemma \ref{SufficientCartier}. Quite generally, in this case, $C$ is a nil-isomorphism (see e.g.\ \cite[Lemma 2.2]{staeblerunitftestmoduln}) so that as crystals $Gr^\lambda M$ and ${F^e}^! Gr^\lambda M$ coincide. Otherwise ${F^e}^! Gr^\lambda M$ does, at least a priori, not encode more information than any other faithful functor.
\end{Bem}
We come to the main result of this section:
\begin{Theo}
\label{AnotherMainResult}
Let $R$ be an $F$-finite regular ring, $(M, \kappa)$ an $F$-regular Cartier module and $f$ an $M$-regular element. Assume that $\lambda \in \mathbb{Q}$ is an $F$-jumping number of the test module filtration of $M$ along $f$. Then for $e \gg 0$ we consider the map $C: {F^e}^! Gr^\lambda M \to {F^{e+a}}^! Gr^\lambda M$ obtained by the isomorphism of Lemma \ref{SufficientCartier}.
If $\lambda (p^s -1) \in \mathbb{Z}$, then $C$ is induced by a morphism $Gr^\lambda \to {F^s}^! Gr^\lambda M$ if and only if $s \mid e$ and $s \mid a$. In this case, the morphism is given by the adjoint of $\kappa^s f^{\lambda(p^s -1)}$ and is not nilpotent.
If $\lambda(p^s -1) \notin \mathbb{Z}$, then there is a morphism $Gr^\lambda M \to {F^s}^! Gr^\lambda M$ inducing $C$ if and only if $s \mid e$, $s = a$ and $\lceil \lambda (p^s - 1) \rceil$ is of the form $c_1 p^{s-1} + c_2 p^{s-2} + \ldots + c_s$, where $\frac{c_1}{p} + \frac{c_2}{p^2} + \ldots + \frac{c_s}{p^s}$ is the truncated $p$-adic expansion of $\lambda$. In this case, the morphism is nilpotent and given by the adjoint of $\kappa^a f^{\lceil \lambda(p^a -1) \rceil }$.
\end{Theo}
\begin{proof}
If $\kappa: F_\ast^e M \to M$ is a Cartier structure then its adjoint is given by $C: M \to {F^e}^! M, m \longmapsto [r \mapsto \kappa(rm)]$. In what follows we will not distinguish between $C$ and $\kappa$.
The Cartier structure $C$ is of the form $\kappa^a f^{b(a)}$ for some $b(a) \in \mathbb{N}$. In particular, the Cartier structure $Gr^\lambda M \to {F^s}^! Gr^\lambda M$ is necessarily also of the form $\kappa^s f^?$. But then, using Lemma \ref{Remarkable}, it is not nilpotent if and only if $? = {\lambda(p^s -1)}$ with $\lambda(p^s -1) \in \mathbb{Z}$. Clearly, if some power of $\kappa^s f^{\lambda(p^s -1)}$ induces $C$ then we must have $s \mid e$ and $s \mid a$. The converse is the assertion of Lemma \ref{SufficientCartier}.
For the addendum note that it is clearly necessary that $s \mid e$ and $s \mid a$. If we write the $p$-adic expansion of $\lambda$ as $\lambda = \sum_{i \geq 1} c_i p^{-i}$, then the existence of such a morphism $\kappa^s f^{u_s(\lambda)}: Gr^\lambda M \to {F^s}^! Gr^\lambda M$ is equivalent to a commutative diagram of the form
\[\begin{xy} \xymatrixcolsep{7pc}\xymatrix{Gr^\lambda M \ar@/^3pc/[rr]^{\kappa^{e+a} f^{u_s(\lambda) \frac{p^{e+a} -1}{p^s-1}}} \ar[r]^{\kappa^e f^{u_s(\lambda) \frac{p^{e}-1}{p^s-1}}}& {F^e}^! Gr^\lambda M \ar[r]^{\kappa^a f^{c_a + c_{a-1}p + \ldots + c_1 p^{a-1}}}& {F^{e+a}}^! Gr^\lambda M.}\end{xy}\]
By Lemma \ref{Remarkable} we may write $u_s(\lambda) = \lceil \lambda( p^s -1)\rceil + m$ for some non-negative integer $m$. Using this, the diagram is commutative if and only if we have the following equality
\[(\lceil \lambda (p^s -1) \rceil + m) \frac{p^e -1}{p^s -1} p^a + c_a + c_{a-1} p + \ldots + c_1 p^{a-1} = (\lceil \lambda (p^s -1) \rceil +m) \frac{p^{e+a} -1}{p^s -1}.\]
This simplifies to
\[c_a + c_{a-1} p + \ldots + c_1 p^{a-1} = (\lceil \lambda(p^s -1) \rceil +m)\frac{p^a -1}{p^s -1}.\] Note that $\lceil \lambda(p^s -1) \rceil = c_1 p^{s-1} + c_2 p^{s-2} + \ldots + c_s + \varepsilon$, where $\varepsilon \in \{0,1\}$ depending on whether the fractional part is negative or positive. Using this formula and writing $a =s \tilde{a}$ we may further expand the equation to
\begin{equation*}\begin{split} c_a + c_{a-1} p + \ldots + c_1 p^{a-1} ={} &c_1 p^{a-1} + c_2 p^{a-2} + \ldots + c_s p^{a-s} + (m + \varepsilon )p^{a-s} \\
& +c_1 p^{a-1-s} + c_2 p^{a-2-s} + \ldots + c_s p^{a-2s} + (m + \varepsilon) p^{a-2s}\\
& \,\,\,\vdots\\
& +c_1 p^{s-1} + c_2 p^{s-2} + \ldots + c_s + (m + \varepsilon)\end{split}\end{equation*} where we have $\tilde{a}$ rows. The summands in the first row of the right hand side except the last one all occur on the left hand side. Subtracting this from both sides we observe that the left hand side is $< p^{a-s}$. Hence, if $m + \varepsilon$ is positive, then equality cannot hold. Thus, we must have $m = \varepsilon = 0$.
If $m = \varepsilon = 0$, then the last summand of each row vanishes. If moreover $a =s$, then equality holds. If $s < a$, then there must exist $0 \leq i \leq s$ with $c_i \neq c_{rs + i}$ for some $1 \leq r < \tilde{a}$ . Hence, equality cannot hold
\end{proof}
\begin{Bsp}
The second case of the theorem occurs for instance for any $F$-jumping number of the form $\frac{ap}{p^2 -1}$ with $a < p$ (and necessarily $p > 2$).
If the denominator of $\lambda$ is divisible by $p$ then the second case of the theorem can only occur if $\lambda = \frac{a}{p^nb}$ with $p \nmid b$ and $a > b$. Indeed, assume that $a \leq b$. Denote the coefficients of the $p$-adic expansion of $\lambda$ by $c_i$ and the coefficients of the expansion of $\frac{a}{b}$ by $d_i$.
Then $c_1 = \ldots = c_n =0$ and $c_{n+i} = d_i$ for $i \geq 1$. In particular, the fractional part of $\lambda(p^s-1)$ is then non-negative since it is of the form \[\sum_{i \geq 1} \frac{c_i}{p^{i-s}} - \sum_{i \geq 1}\frac{c_i}{p^i}.\] Since $\lambda(p^s -1)$ is not an integer by assumption its fractional part is strictly positive so that the second case of the theorem cannot occur.
\end{Bsp}
\begin{Que}
Are there $F$-pure thresholds which actually satisfy the second case of Theorem \ref{AnotherMainResult}?
\end{Que}
A possible strategy may be to consider quasi-homogeneous polynomials $f \in k[x_1, \ldots, x_n]$ whose Jacobian ideal coincides, up to radical, with the irrelevant ideal. For then a likely candidate for the $F$-pure threshold is the log canonical threshold \[\frac{\sum \deg(x_i)}{\deg f}.\] If $\fpt(f) \neq \lct(f)$ then the denominator of the $F$-threshold has to be a $p$th power provided that $p \geq (n-1)\deg f - \deg f - n +1$. So one would need $f$ with $\fpt(f) = \lct(f)$ and $f$ having degree $pb$ (since $\lct(f) = \frac{\sum_{i=1}^n \deg x_i}{\deg f}$).
Finally, we show that these results continue to hold if we work with the image of the natural map $\gamma_\ast M \to \colim_{e \geq 0} {F^e}^! \gamma_\ast M$ instead of $\gamma_\ast M \to {F^e}^! \gamma_\ast M$ for varying $e$. This is the setting of \cite{mustatabernsteinsatopolynomialspositivechar} (for $M = R$). Indeed, note that by \cite[Proposition 3.5]{blicklestaeblerbernsteinsatocartier} one has $\colim_{e \geq 0} {F^e}^! M = \gamma_+ \colim_{e \geq 0} {F^e}^! M$ as right $\mathcal{D}_{R[t]}$-modules. If $M = \omega_R$ then $\colim_{e \geq 0} {F^e}^! \omega_R \cong \omega_R$ and the equivalence of right and left $\mathcal{D}_{R[t]}$-modules yields $\gamma_+ R = R[t]_{t-f}/R[t]$ which is what is considered in \cite{mustatabernsteinsatopolynomialspositivechar}.
Before we proceed, we need to recall one more notion from the theory of Cartier crystals. If $(M, \kappa)$ is a Cartier module then we can consider the union of all nilpotent submodules $M_{\text{nil}}$. This is again a nilpotent Cartier module and we define $\overline{M} = M/M_{\text{nil}}$. Note, in particular, that the natural projection $M \to \overline{M}$ is a nil-isomorphism.
We note the following
\begin{Le}
\label{NilpotentsPushforward}
Let $i: \Spec R/I \to \Spec R$ be a closed immersion and $(M, \kappa)$ a Cartier module. Then $i_\ast \overline{M} = \overline{i_\ast M}$.
\end{Le}
\begin{Theo}
\label{Herrgottssack}
Assume that $(M, \kappa)$ is an $F$-regular Cartier module satisfying $\overline{M} = M$. Write $\gamma_+ \mathcal{M} = \colim_{e \geq 0} {F^e}^! \gamma_\ast M$ and denote the natural maps ${F^e}^! \gamma_\ast M \to \gamma_+ \mathcal{M}$ by $\varphi_e$. Then for any $e$ the quotients \[\varphi_0(\gamma_\ast C^e(M)) \mathcal{D}^e_R[\theta_1, \theta_p, \ldots, \theta_{p^{e-1}}]/\varphi_0(\gamma_\ast C^e(fM)) \mathcal{D}^e_R[\theta_1, \theta_p, \ldots, \theta_{p^{e-1}}] \] and \[\gamma_\ast C^e(M) \mathcal{D}^e_R[\theta_1, \theta_p, \ldots, \theta_{p^{e-1}}]/\gamma_\ast C^e(fM) \mathcal{D}^e_R[\theta_1, \theta_p, \ldots, \theta_{p^{e-1}}]\] are naturally isomorphic as $\mathcal{D}^e_R[\theta_1, \theta_p, \ldots, \theta_{p^e-1}]$-modules. In particular, we obtain $\mathcal{D}^e_R$-isomorphisms of generalized eigenspaces.
\end{Theo}
\begin{proof}
First of all, note that the natural map $\varphi_e: {F^e}^! \gamma_\ast M \to \gamma_+ \mathcal{M}$ is $\mathcal{D}^e_{R[t]}$-linear.
Next, observe that by Lemma \ref{NilpotentsPushforward} we have $\overline{\gamma_\ast M} = \gamma_\ast M$. It follows that the map $\gamma_\ast M\to {F^e}^! \gamma_\ast M$ is injective for all $e$ (cf. \cite[Lemma 6.20]{staeblertestmodulnvilftrierung}). Hence, the natural map $\varphi_0$ is also injective. Note that $\varphi_0$ factors as $\gamma_\ast M \xrightarrow{\gamma_\ast C^e} {F^e}^! \gamma_\ast M \xrightarrow{\varphi_e} \gamma_+ \mathcal{M}$ for any $e \geq 1$. Since $\varphi_e$ is $\mathcal{D}^e_{R[t]}$-linear we have \[\varphi_e(\gamma_\ast C^e(M) \mathcal{D}^e_R[\theta_1, \theta_p, \ldots, \theta_{p^{e-1}}]) = \varphi_0(\gamma_\ast C^e(M)) \mathcal{D}^e_R[\theta_1, \theta_p, \ldots, \theta_{p^{e-1}}],\] and a similar assertion holds for $\gamma_\ast C^e(fM)$.
We conclude that the quotients \[\varphi_0(\gamma_\ast C^e(M)) \mathcal{D}^e_R[\theta_1, \theta_p, \ldots, \theta_{p^{e-1}}]/\varphi_0(\gamma_\ast C^e(fM)) \mathcal{D}^e_R[\theta_1, \theta_p, \ldots, \theta_{p^{e-1}}] \] and \[\gamma_\ast C^e(M) \mathcal{D}^e_R[\theta_1, \theta_p, \ldots, \theta_{p^{e-1}}]/\gamma_\ast C^e(fM) \mathcal{D}^e_R[\theta_1, \theta_p, \ldots, \theta_{p^{e-1}}]\]are naturally isomorphic as $\mathcal{D}^e_R[\theta_1, \theta_p, \ldots, \theta_{p^e-1}]$-modules since $\varphi_0$ is injective. In particular, we obtain $\mathcal{D}^e_R$ isomorphisms of generalized eigenspaces.
\end{proof}
\begin{Prop}
Let $(M, \kappa_M)$, $(N, \kappa_N)$ be Cartier modules and $f \in R$ $M$-regular and $N$-regular. If $\varphi: M \to N$ is a nil-isomorphism then we get an induced nil-isomorphism $Gr^\lambda M \to Gr^\lambda N$ with respect to any of the Cartier structures $\kappa_?^e f^{a_e(\lambda)}$, where $? = M$ or $N$ and $a_e(\lambda) \geq \lceil \lambda(p^e -1)\rceil$.
\end{Prop}
\begin{proof}
By \cite[Theorem 2.8]{staeblerunitftestmoduln} the restriction of $\varphi$ induces a surjective map $\tau(\varphi): \tau(M, f^\lambda) \to \tau(N, f^\lambda)$. By \cite[Lemma 2.1, proof of Theorem 2.8]{staeblerunitftestmoduln} we may assume that both $M, N$ are $F$-pure. In particular, $\varphi$ is surjective. Now the claim follows from \cite[Theorem 5.8]{staeblertestmodulnvilftrierung}.
\end{proof}
In particular, given an $F$-regular Cartier module $M$ we obtain nil-isomorphisms $Gr^\lambda M \to Gr^\lambda \overline{M}$. Since for $\overline{M}$ working on the $e$th level or in the colimit induces natural isomorphisms of the generalized eigenspaces of the quotients by Theorem \ref{Herrgottssack} we see that we obtain the same nilpotence results if we construct the quotients by working with the colimit.
\section{Non-$F$-pure modules}
\label{Nonfpuremodules}
In this section we study a generalization of the \emph{non-$F$-pure ideal} or \emph{$\phi$-fixed ideal} to modules. These non-$F$-pure ideals were first introduced in \cite{fujinotakagischwedenonlc} and are further developed and studied in \cite{schwedecanlinearsystem} and \cite{hsiaoschwedezhangtoriccartier}. They are the characteristic $p$ analog of the so-called \emph{non-lc ideal} and have applications to birational geometry.
The importance of these for us is that we will prove a connection between zeros of Bernstein-Sato polynomials and certain non-$F$-pure-modules. These non-$F$-pure modules will form a decreasing and discrete filtration of the ambient Cartier module. However, there will be no continuity properties even if $M$ is $F$-regular.
The actual connection with Bernstein-Sato polynomials will be discussed in the next section. Here we just develop the basic theory of non-$F$-pure modules.
Let us however briefly motivate why it is interesting to consider non-$F$-pure modules in this context.
In the situation of the previous section, the generalized eigenspaces always admit a description of the form \[C^e (f^m M) \cdot \mathcal{D}^e_R/ C^e(f^{m+1} M)\cdot \mathcal{D}^e_R \cong {F^e}^!(\kappa^e f^m M/\kappa^e f^{m+1} M).\] A first naive guess (based on \cite[Lemma 4.3]{blicklestaeblerbernsteinsatocartier} which asserts that the statement is true if $M$ is $F$-regular) may be that $\kappa^e f^m M$ (for $e \gg 0$) should correspond to $\mathcal{D}_+^e M = \underline{M}$, where $\mathcal{D}_+ = \bigoplus_{e \geq 1} \kappa^e f^{\lceil \lambda p^e \rceil} R$ with $m + 1= \lceil \lambda p^e \rceil$.
In practice, we replace the algebra $\mathcal{D}$ with the algebra $\mathcal{C} = \bigoplus_{e \geq 0} \kappa^e f^{\lceil \lambda (p^e -1) \rceil} R$. Both algebras, $\mathcal{C}$ and $\mathcal{D}$ yield the same notion of test module (see \cite[Lemma 3.1]{staeblertestmodulnvilftrierung}).
Unless specified otherwise $R$ in this section is an arbitrary $F$-finite ring.
\begin{Def}
Fix an $F$-finite ring $R$. Given a Cartier module $(M, \kappa)$ and an $M$-regular element $f$ we denote by $\mathcal{C}$ the Cartier algebra generated by the $\kappa^e f^{\lceil \lambda (p^e -1)\rceil}$. We define the \emph{non-$F$-pure submodule} $\sigma(M, f^\lambda)$ as $\mathcal{C}_+^h M$ for all $h \gg 0$.
\end{Def}
Note that by \cite[Proposition 2.13]{blicklep-etestideale} the descending chain $\mathcal{C}_+^h M \supseteq \mathcal{C}_+^{h+1} M \supseteq \ldots$ stabilizes for \emph{arbitrary} Cartier algebras in any $F$-finite ring.
\begin{Bem}
\begin{enumerate}[(a)]
\item{The reason for working with the algebra $\mathcal{C}$ rather than $\mathcal{D}$ is that Lemma \ref{NonFpureIdealSimpleDescr} does not hold in this context (cf.\ Example \ref{DLemmadontwork} below).}
\item{This is not the definition that is used in \cite{fujinotakagischwedenonlc}. We will see in Corollary \ref{DefsCoincide} below that these definitions agree.}
\end{enumerate}
\end{Bem}
\begin{Le}
\label{NonFpureIdealSimpleDescr}
Let $R$ be an $F$-finite ring and let $(M, \kappa)$ be a Cartier module. Given an $M$-regular element $f$ and a rational number $\lambda$ we may consider the Cartier algebra $\mathcal{C}_+ = \bigoplus_{e \geq 0} \kappa^e f^{\lceil \lambda (p^e -1)\rceil} R$ acting on $M$. For all $a \gg 0$ such that $\lambda (p^a -1) \in \mathbb{Z}$ we have $\sigma(M,f^\lambda) = \kappa^a f^{\lambda (p^a -1)} M$.
\end{Le}
\begin{proof}
Note that by definition $\sigma(M, f^\lambda) = \underline{M} = \mathcal{C}_+^h M$ for all $h \gg 0$. We first deal with the inclusion from right to left. Fix a natural number $h$ such that $\mathcal{C}_+^h M = \mathcal{C}_+^{h+1} M$. Take any $e$ such that $\lambda (p^{e} -1) \in \mathbb{Z}$ and set $a = eh$. Then \[ \underbrace{\kappa^e f^{\lambda(p^e -1)} \cdots \kappa^e f^{\lambda (p^e -1)}}_{h \text{ times}} M = \kappa^a f^{\lambda (p^a-1)} M\] which shows the desired inclusion. Also note that, say, taking $e$ minimal and replacing $h$ with $h+1$ shows that $e(h+1)$ also works. So that the inclusion holds for all $a \gg 0$ such that $\lambda(p^a-1) \in \mathbb{Z}$.
Given $a$ such that $\lambda(p^a-1) \in \mathbb{Z}$ we show that for any $h \geq a$ one has $\mathcal{C}_+^h M \subseteq \kappa^a f^{\lambda (p^a -1)} M$. It suffices to show that \[ \kappa^{e_1} f^{\lceil \lambda(p^{e_1} -1)\rceil} \cdots \kappa^{e_h} f^{\lceil \lambda (p^{e_h} -1) \rceil} M \subseteq \kappa^a f^{\lambda (p^a-1)} M\] for all $e_1, \ldots, e_h \geq 1$.
Write $e = e_1 + \ldots + e_h$ and $e = la + r$ (note that $l \geq 1$). Then we have \begin{multline*} \kappa^{e_1} f^{\lceil \lambda(p^{e_1} -1)\rceil} \cdots \kappa^{e_h} f^{\lceil \lambda (p^{e_h} -1) \rceil} M \subseteq \kappa^e f^{\lceil\lambda(p^e -1)\rceil} M\\= \kappa^{la} f^{\lambda(p^{la} -1)} \kappa^r f^{\lambda (p^r - 1) + \delta} M \subseteq \kappa^{la} f^{\lambda(p^{la} -1)} M = \kappa^a f^{\lambda(p^a-1)}M,\end{multline*} where $\delta = \lceil \lambda(p^{e} -1) \rceil - \lambda(p^e -1)$.
\end{proof}
\begin{Bsp}
\label{Running1}
In this example we point out that the assumption that $\lambda(p^a -1)$ is an integer in Lemma \ref{NonFpureIdealSimpleDescr} cannot be omitted. Let $k$ be an $F$-finite field of characteristic $2$. Consider the Cartier module $M = k[x] \frac{1}{x}$ with Cartier structure induced from the usual one on $k[x]$ (i.e., $x^{p-1} \mapsto 1, x^i \mapsto 0$ for $0 \leq i \leq p-2$). Explicitly, $\kappa(\frac{g}{x}) =\frac{1}{x} \kappa(g x^{p-1})$.
$M$ is $F$-pure but not $F$-regular (a test element is given by $x$). We filter $M$ along $f = x^2$ and take $\lambda = \frac{1}{2}$. Then for any $e \geq 1$ we have
\[ \kappa^e f^{\lceil \lambda(p^e -1)\rceil} M = \kappa^e f^{2^{e-1}} \frac{1}{x} k[x] = \kappa^e x^{2^e -1} k[x] = k[x].\]
On the other hand, for any $e \geq 2$ we have $\mathcal{C}_+^e M \subseteq \mathcal{C}_+^2 M$. Elements of $\mathcal{C}_+^2 M$ in turn are sums of elements of \[ \kappa^{a} f^{\lceil \lambda(p^a -1) \rceil} \kappa^b f^{\lceil \lambda(p^b -1)\rceil} m = \kappa^{a+b} f^{2^{a +b -1}} f^{2^{b -1}} m \in \kappa^{a+b} x^{2^{a+b} -1} x^{2^b} k[x] \subseteq (x),\] where $a, b \geq 1$. We conclude that $\mathcal{C}_+^e M \subseteq (x)$ for $e \geq 2$.
\end{Bsp}
As remarked earlier, Lemma \ref{NonFpureIdealSimpleDescr} does not hold if we use the Cartier algebra $\mathcal{D} = \bigoplus_{e \geq 1} \kappa^e f^{\lceil \lambda p^e\rceil}R \oplus R$:
\begin{Bsp}
\label{DLemmadontwork}
Consider $M = k[x]\frac{1}{x}$ endowed with its usual Cartier structure. Take $f = x$ then if $\mathcal{D} = \bigoplus_{e \geq 1} \kappa^e f^{\lceil \lambda p^e\rceil} R \oplus R$ and $\lambda = 1$ one has $\mathcal{C}_+^h M \subsetneq \kappa^a f^{\lceil \lambda p^a \rceil} M$ for all $a, h \gg 0$. Indeed, for any $a \geq 1$ the right hand side coincides with $\kappa^a x^{p^a -1} k[x] = k[x]$. For the left hand side note that by \cite[Lemma 4.1]{blicklestaeblerbernsteinsatocartier} one has $\mathcal{D}_+ M = \kappa^a f^{\lceil \lambda p^a \rceil} M$ for all $a \gg 0$. Hence, for $h \gg 0 $ we have \[\mathcal{D}_+^h M = \mathcal{D}_+^{h+2} M = \mathcal{D}_+^h \kappa^a x^{p^a} \kappa^a x^{p^a} k[x]\frac{1}{x} = \mathcal{D}_+^h \kappa^{a} x^{p^{a}} k[x] = \mathcal{D}_+^h (x).\] Since $x$ is a test element for $M$ one observes that $\tau(M, x^1) = (x)$. In particular, $(x)$ is $F$-pure with respect to the Cartier algebra $\mathcal{D}$. Hence, $\mathcal{D}_+^h (x) = (x)$.
\end{Bsp}
The following example further illustrates that the filtration obtained via $\mathcal{D}$ also has all the shortcomings of the one obtained via $\mathcal{C}$.
\begin{Bsp}
We work in the situation of Example \ref{Running1} but use $\mathcal{D}_+$. Then $\mathcal{D}_+^h M \neq \kappa^a f^{\lceil \lambda p^a \rceil} M$ for all $h, a \gg 0$. One has $\kappa^a f^{ \lambda p^a} M = \kappa^a x^{p^a -1} k[x] = k[x]$. If we had an equality above then in particular $\mathcal{D}_+ k[x] = \mathcal{D}_+ \kappa^a f^{ \lambda p^a} M = \kappa^a f^{\lambda p^a }M = k[x]$. But for any $e \geq 1$ we have $\kappa^e f^{\lambda p^e} k[x] = \kappa^e x^{p^e} k[x] = (x)$.
Furthermore, using this filtration there will be no continuity properties. For instance, if $(M, \kappa)$ is any $F$-pure non-$F$-regular Cartier module whose associated primes are all minimal and $f$ a test element then for $\lambda = 0$ one obtains $\mathcal{D}_+^h M = M$ while for any $0 < \lambda \ll 1$ one obtains $\mathcal{D}_+^h M = \tau(M, f^0)$ using \cite[Lemma 4.1]{blicklestaeblerbernsteinsatocartier} and Corollary \ref{TauStuff}. So the filtration is not right-continuous. On the other hand, if $(M, \kappa)$ is $F$-regular then the filtration coincides with the test module filtration and is thus right-continuous but not left-continuous.
\end{Bsp}
\begin{Prop}
\label{DiscreteDecreasing}
Let $R$ be $F$-finite and $(M, \kappa)$ a Cartier module. Fix an $M$-regular element $f \in R$. Then the filtration $\sigma(M, f^\lambda)$ is discrete and decreasing.
\end{Prop}
\begin{proof}
In order to show that the filtration is decreasing simply observe that for any $e$ one has $\prod_{i=1}^e \kappa^{e_i} f^{\lceil (\lambda +\varepsilon)(p^e -1) \rceil} M \subseteq \prod_{i=1}^e \kappa^{e_i} f^{\lceil \lambda(p^e-1)\rceil}M$. Discreteness now follows by the same argument as in \cite[Theorem 4.18]{blicklep-etestideale}
\end{proof}
In \cite{fujinotakagischwedenonlc} the authors introduce some other candidates for the non-$F$-pure ideal and ask whether they all coincide. In what follows we write $\mathcal{C}$ for the Cartier algebra generated in degree $e$ by $\kappa^e f^{\lceil \lambda (p^e -1)\rceil}$. The following variants are considered:
\begin{enumerate}[(a)]
\item{For a fixed $n \gg 0$ define $\sigma_n(M, f^\lambda) = (\mathcal{C}_{\geq n})^h M$ for $h \gg 0$.}
\item{Suppose that $\lambda(p^e -1) \in \mathbb{Z}$ for some sufficiently large and sufficiently divisible $e$. Write $\mathcal{D}$ for the algebra that is given by $\bigoplus_{a \geq 0} \kappa^{ea} f^{\lambda (p^{ea} -1)}$. Then set $\sigma'(M, f^\lambda) = \mathcal{D}_+^h M$ for $h \gg 0$.}
\end{enumerate}
Once again note that by \cite[Proposition 2.13]{blicklep-etestideale} all these notions are well-defined for any $F$-finite ring $R$.
We point out that if one studies non-$F$-pure ideals in the context of non-principal ideals (i.e. if $f$ is replaced by an ideal $\mathfrak{a} \subseteq R$), then one works with the integral closures of the $\mathfrak{a}^{\lceil \lambda(p^e -1) \rceil}$ making things more subtle (if $R$ is normal then any principal ideal generated by a non-zero-divisor is integrally closed -- see e.g.\ \cite[Proposition 10.2.3]{brunsherzog}).
As an application of Lemma \ref{NonFpureIdealSimpleDescr} we show that these notions all coincide in our situation of a principal ideal. First we record an elementary
\begin{Le}
\label{DecrChain}
Let $R$ be $F$-finite and $(M, \kappa)$ a Cartier module. Given a rational number $\lambda$ such that $\lambda (p^a -1) \in \mathbb{Z}$ we have that the $\kappa^{ea} f^{\lambda(p^{ea} -1)} M$ form a decreasing chain for varying $e$.
\end{Le}
\begin{proof}
One has $\kappa^{2a} f^{\lambda (p^{2a} -1)} M = \kappa^a f^{\lambda (p^a -1)} \kappa^a f^{\lambda(p^a -1)} M \subseteq \kappa^a f^{\lambda(p^a -1)} M$.
\end{proof}
\begin{Ko}
\label{DefsCoincide}
Let $R$ be essentially of finite type over an $F$-finite field. If the denominator of $\lambda$ is not divisible by $p$, then one has \[\sigma(M, f^\lambda) = \sigma_n(M, f^\lambda) = \sigma'(M, f^\lambda)\] for $n \gg 0$. In general, one still has $\sigma(M, f^{\lambda}) = \sigma_n(M, f^{\lambda})$ for all $n \gg 0$.
\end{Ko}
\begin{proof}
Note that one clearly has containments $\sigma'(M, f^\lambda) \subseteq \sigma_n(M, f^\lambda) \subseteq \sigma(M, f^\lambda)$ whenever these objects are defined.
Let us first consider the case $\lambda (p^a -1) \in \mathbb{Z}$ for some $a$. It suffices to show that $\sigma(M, f^\lambda) \subseteq \sigma'(M, f^\lambda)$. By Lemma \ref{NonFpureIdealSimpleDescr} we have $\sigma(M, f^\lambda) = \kappa^b f^{\lambda (p^b -1)} M$ for infinitely many $b$. In particular, if $\sigma'(M, f^\lambda) = \mathcal{D}_+^h M$ for some fixed $h \gg 0$ and $\mathcal{D}_+ = \bigoplus_{a \geq 1} \kappa^{ea} f^{\lambda{p^{ea} -1} }R$ then we may arrange for $b = eh$. Now use Lemma \ref{DecrChain}.
Let us now consider the case of an arbitrary $\lambda$. Fix $h \gg 0$ such that $(\mathcal{C}_{\geq n})^h M = \sigma_n(M, f^\lambda)$ and $\mathcal{C}_+^h M = \sigma(M, f^\lambda)$. An argument just as in the second part of the proof of Lemma \ref{NonFpureIdealSimpleDescr} shows that for any $a \geq n$ one has $\mathcal{C}_+^a M \subseteq \mathcal{C}_{\geq n} M$. We conclude that $\mathcal{C}_+^h M = \mathcal{C}_+^{ah} M \subseteq \mathcal{C}_{\geq n}^h M$.
\end{proof}
As is well-known to experts one cannot expect any continuity properties for the filtration $\sigma(M, f^\lambda)$ (not even in the case $M = R$). For completeness we give an example:
\begin{Bsp}
Consider the case of Example \ref{Running1} (i.e. $M = k[x] \frac{1}{x}, f = x^2$) but this time for a field $k$ of characteristic $\neq 2$. We claim that $\mathcal{C}_+^e M = M$ for any $\lambda \leq \frac{1}{2}$ but $\mathcal{C}_+^e M = (x)$ for any $\lambda > \frac{1}{2}$. Since $2$ and $p$ are coprime we can use Lemma \ref{NonFpureIdealSimpleDescr} and obtain
\[ \mathcal{C}_+^e M = \kappa^e x^{p^e -1} \frac{1}{x} k[x] = k[x] \] for $\lambda = \frac{1}{2}$ and hence for all $\lambda \leq \frac{1}{2}$ since the filtration is decreasing. If $\lambda > \frac{1}{2}$ then $\lambda (p^e -1) > \frac{1}{2}(p^e -1)$. Choosing $e$ in such a way that the left-hand side is an integer we get that $\lambda (p^e -1) \geq \frac{1}{2}(p^e -1) +1$. Using Lemma \ref{NonFpureIdealSimpleDescr} once more we compute
\[ \kappa^e f^{\lambda(p^e -1)} M \supseteq \kappa^e x^{p^e +1} M = (x).\]
Now assume that $\chara k = 2$. We claim that the filtration is not left-continuous at $\lambda = \frac{1}{2}$ in this case. In Example \ref{Running1} we saw that $\sigma(M, f^\lambda) \subseteq (x)$. We will show that for all $\varepsilon > 0$ one has $\sigma(M, f^{\lambda - \varepsilon}) = R$. We may assume that $(\lambda - \varepsilon)(p^a -1) \in \mathbb{Z}$ for some $a \gg 0$. Then using Lemma \ref{NonFpureIdealSimpleDescr} we obtain \[\sigma(M, f^{\lambda - \varepsilon}) = \kappa^a f^{(\lambda - \varepsilon)(p^a -1)} M = \kappa^a x^{2^a - 2^{a+1}\varepsilon - 1 + 2\varepsilon} \frac{1}{x} k[x]. \] Since the exponent is non-zero and $\leq 2^a -1$ we get $\sigma(M, f^{\lambda -\varepsilon}) = R$.
Finally, note that the Cartier module $(M, \kappa)$ is isomorphic to the Cartier module $(k[x],\kappa x^{p-1})$ via the multiplication by $x$ map $k[x]\frac{1}{x} \to k[x]$.
\end{Bsp}
\begin{Prop}
\label{PLeftContinuity}
Assume that $R$ is an $F$-finite ring and $(M, \kappa)$ a Cartier module. If the denominator of $\lambda$ is divisible by $p$ then $\sigma(M, f^\lambda) = \sigma(M, f^{\lambda+\varepsilon})$ for all $0 < \varepsilon \ll 1$.
\end{Prop}
\begin{proof}
Since the filtration is decreasing by Proposition \ref{DiscreteDecreasing} we only need to show the inclusion from left to right.
Next note that $\mathcal{C}_+^h \subseteq \mathcal{C}_{\geq h}$. By definition we have $\sigma(M, f^\lambda) = \mathcal{C}_+^h M = \mathcal{C}_+^{h+ g} M \subseteq \mathcal{C}_+^h \mathcal{C}_{\geq g}M$ for all $h \gg 0$ and all $g$. Hence, any element of $\mathcal{C}_+^{h} \mathcal{C}_{\geq g} M$ may be written as \[ \prod_{i=1}^{h} \kappa^{e_i} f^{\lceil \lambda(p^{e_i} -1)\rceil} \kappa^{e_0}f^{\lceil \lambda(p^{e_0} -1)\rceil}m\] with $e_0 \geq g$.
Fix $h$ such that $\mathcal{C}_+^h M = \sigma(M, f^\lambda)$.
If we write $\lceil \lambda (p^{e_i} -1) \rceil = \lambda(p^{e_i} -1) + \delta_i$ with $0 < \delta_i < 1$ for $0 \leq i \leq h$ and $e = \sum_{i=0}^h e_i$, then we obtain that any element of $\mathcal{C}_+^{h} \mathcal{C}_{\geq g}$ is of the form
\[ \kappa^e f^{\lambda(p^e -1) + \sum_{i=0}^h \delta_i p^{\sum_{j=i+1}^h e_j}}.\]
Now we set \[ \varepsilon' = \lim_{e_h \to \infty} \frac{\sum_{i=0}^h \delta_i p^{\sum_{j = i+1}^h e_j}}{p^e -1}.\] Note that this sequence converges to $\varepsilon'$ from above and that $\varepsilon' > 0$. Thus by perturbing a little bit we may choose $\varepsilon > \varepsilon'$ such that the denominator of $\lambda + \varepsilon$ is not divisible by $p$ and such that \[\kappa^e f^{\lambda(p^e -1) + \sum_{i=0}^h \delta_i p^{\sum_{j=i+1}^h e_j}} m \in \kappa^e f^{(\lambda + \varepsilon)(p^e -1)} M = \sigma(M, f^{\lambda + \varepsilon})\] holds for all $e_h$ sufficiently large and divisible (using Lemma \ref{NonFpureIdealSimpleDescr}).
\end{proof}
It is observed in \cite[Remark 14.11]{fujinotakagischwedenonlc} that, contrary to the situation in characteristic zero, one does not have $\sigma(R, f^\lambda) = \tau(R, f^{\lambda - \varepsilon})$ if $R$ is ($F$-)regular.
However, we have the
\begin{Prop}
Let $R$ be essentially of finite type over an $F$-finite field, $(M, \kappa)$ an $F$-regular Cartier module and $f$ an $M$-regular element. If the denominator of $\lambda$ is not divisible by $p$, then $\sigma(M, f^{\lambda}) = \sigma(M, f^{\lambda - \varepsilon}) = \tau(M, f^{\lambda -\varepsilon})$.
\end{Prop}
\begin{proof}
By our assumption on $\lambda$ we find $e \gg 0$ such that $\sigma(M, f^\lambda) = \kappa^e f^{\lambda (p^e -1)} M$. Moreover, by discreteness we may assume that $\lambda - \varepsilon$ also has denominator not divisible by $p$ and that the same $e$ computes $\sigma(M, f^{\lambda - \varepsilon}$). By $F$-regularity we have $M = \tau(M, f^0)$. Using Brian\c con-Skoda (\cite[Proposition 4.1]{blicklestaeblerfunctorialtestmodules}) and Lemma \ref{CartierTauDivisionByP} we then have $\kappa^e f^{\lambda(p^e-1)} M = \tau(M, f^{\lambda - \frac{1}{p^e}})$ and $\kappa^e f^{(\lambda - \varepsilon)(p^e -1)} M = \tau(M, f^{\lambda - \varepsilon - \frac{\lambda -\varepsilon}{p^e}})$ and by discreteness of $\tau$ these coincide.
\end{proof}
\begin{Que}
Is the filtration $\sigma(M, f^\lambda)$ left-continuous if the $F$-jumping numbers of $\tau(M, f^{\lambda})$ all have a denominator that is not divisible by $p$?
\end{Que}
\begin{Prop}
Let $R$ be essentially of finite type over an $F$-finite field, $(M, \kappa)$ an $F$-regular Cartier module and $f$ an $M$-regular element.. Then for all $\lambda \geq 0$ one has $\sigma(M, f^{\lambda + \varepsilon}) = \tau(M, f^\lambda)$ for all $0 < \varepsilon \ll 1$.
\end{Prop}
\begin{proof}
Since $M$ is $F$-regular we have $\tau(M, f^0) = M$. As the filtration $\sigma$ is discrete and decreasing there exists $0 < \varepsilon \ll 1$ such that $\sigma$ is constant in $(\lambda, \lambda + \varepsilon)$. In particular, we may assume that the denominator of $\lambda + \varepsilon$ is not divisible by $p$. Hence, we can apply Lemma \ref{NonFpureIdealSimpleDescr} and obtain
\[ \sigma(M, f^{\lambda + \varepsilon}) = \kappa^e f^{(\lambda + \varepsilon)(p^e -1)} M = \tau(M, f^{\lambda + \varepsilon - \frac{\lambda + \varepsilon}{p^e}}) = \tau(M, f^{\lambda}),\] where we use Brian\c con-Skoda (\cite[Proposition 4.1]{blicklestaeblerfunctorialtestmodules}) and Lemma \ref{CartierTauDivisionByP} for the second equality and right-continuity of $\tau$ (and the fact that we may choose a larger $e$) for the last equality.
\end{proof}
This correspondence breaks down however in the non $F$-regular case.
\begin{Bsp}
Take $M = k[x,y] \frac{1}{x}$ with the usual Cartier structure, $k$ any $F$-finite field of positive characteristic. We filter along $f =y$. Then $\tau(M, f^0) = k[x,y]$ and, in particular, $\tau(M, f^1) = (y)$. For $\sigma$ we obtain $\sigma(M, f^\lambda) = k[x,y] \frac{1}{x}$ for $\lambda \leq 1$ and $\sigma(M, f^{1 + \varepsilon}) = (y) \frac{1}{x}$.
\end{Bsp}
\section{Bernstein-Sato polynomials attached to Cartier modules and the filtration $\sigma$}
\begin{Def}
Given an $F$-finite ring $R$, a Cartier module $(M, \kappa)$ and an $M$-regular element $f$ we define $Gr_\sigma^\lambda M = \sigma(M, f^\lambda)/\sigma(M, f^{\lambda+\varepsilon})$ for any $0< \varepsilon \ll 1$.
\end{Def}
\begin{Le}
\label{NonNilpotentInfo}
Let $R$ be an $F$-finite ring, $(M, \kappa)$ a Cartier module and $f$ an $M$-regular element. Then $\kappa^a f^{\lceil \lambda (p^a -1)\rceil}$ defines a Cartier structure on $Gr^\lambda_\sigma M$. If $Gr^\lambda_\sigma M \neq 0$, then this Cartier structure is non-nilpotent if and only if $\lambda (p^a -1) \in \mathbb{Z}$.
\end{Le}
\begin{proof}
We have $\sigma(M, f^\lambda) = \mathcal{C}_+^{h} M = \mathcal{C}_+^{h+1} M \supseteq \kappa^a f^{\lceil \lambda (p^a -1)\rceil} \mathcal{C}_+^h M$. We may assume that $(\lambda + \varepsilon)(p^e -1) \in \mathbb{Z}$ for some $e \gg 0$ and $\sigma(M, f^{\lambda + \varepsilon}) = \kappa^e f^{(\lambda + \varepsilon)(p^e -1)} M$. Write $\lceil \lambda(p^a -1) \rceil = \lambda(p^a -1) + \delta$. Then we compute
\[ \kappa^a f^{\lceil \lambda (p^a -1)\rceil} \sigma(M, f^{\lambda+\varepsilon}) = \kappa^{a+e} f^{\lambda(p^{e+a} -1) + \varepsilon(p^e -1) +\delta p^e} M.\]
Replacing $\varepsilon$ by $2 p^a \varepsilon'$ and possibly choosing a larger $e$ so that \[\kappa^e f^{(\lambda + \varepsilon')(p^e -1)} M = \sigma(M, f^{\lambda+\varepsilon'}) = \sigma(M,f^{\lambda + \varepsilon})\] we obtain that $\varepsilon' 2 p^a (p^e -1) \geq \varepsilon' (p^{e+a} -1)$. Hence, we get the desired containment \[\kappa^a f^{\lceil \lambda (p^a -1) \rceil} \sigma(M,f^{\lambda +\varepsilon}) \subseteq \sigma(M, f^{\lambda + \varepsilon}).\]
For the supplement, if $\lambda(p^a -1) \in \mathbb{Z}$, then $\sigma(M, f^\lambda) = \kappa^e f^{\lambda(p^e -1)} M$ for all $e \gg 0$ such that $\lambda(p^e -1) \in \mathbb{Z}$. So $\kappa^a f^{\lambda(p^a-1)} \sigma(M, f^\lambda) = \sigma(M, f^\lambda)$. Conversely, we may write $\lceil \lambda(p^a -1) \rceil = \lambda(p^a -1 ) + \delta = (\lambda + \delta')(p^a -1)$. By construction the denominator of $\lambda + \delta'$ is not divisible by $p$ so that we obtain for suitable $e \gg 0$
\begin{multline*}(\kappa^a f^{\lceil \lambda (p^a - 1)\rceil})^e \sigma(M, f^\lambda) = \kappa^{ea} f^{(\lambda + \delta')(p^{ea} -1)} \sigma(M, f^\lambda) \\ \subseteq \kappa^{ea} f^{(\lambda + \delta')(p^{ea} -1)} M = \sigma(M, f^{\lambda + \delta'}) \end{multline*}
where we use Lemma \ref{NonFpureIdealSimpleDescr} for the last equality. As the filtration is decreasing this shows nilpotence.
\end{proof}
Assume that $R$ is regular and $F$-finite.
Let us now recall from \cite{blicklestaeblerbernsteinsatocartier} the notion of Bernstein-Sato polynomial attached to a Cartier module $(M, \kappa)$ and an $M$-regular element $f$.
As in Section \ref{DModApproach} we denote by $\gamma: \Spec R \to \Spec R[t]$ the graph embedding $t \mapsto f$ and may consider, for any $e \geq 1$, the right $\mathcal{D}^e_R[\theta_1, \ldots, \theta_{p^{e -1}}]$-module
\[N_e := (\gamma_\ast C^e(M))\mathcal{D}^e_R[\theta_1, \theta_p, \ldots, \theta_{p^{e-1}}]/ (\gamma_\ast C^e(fM))\mathcal{D}^e_R[\theta_1, \theta_p, \ldots, \theta_{p^{e-1}}].\] Now let $\Gamma^e_f \subseteq \mathbb{F}_p^e$ be the set of those $i = (i_1, \ldots, i_e)$ for which the corresponding $i$-eigenspace with respect to the action of the $\theta_1, \theta_p, \ldots, \theta_{p^{e-1}}$ is non-trivial. Then by lifting $\mathbb{F}_p$ to $\{0, \ldots, p-1\} \subseteq \mathbb{Z}$ we define the $e$th \emph{Bernstein-Sato polynomial} $b^e_{M,f}(s)$ as \[\prod_{i \in \Gamma^e_f} (s - (\frac{i_e}{p} + \ldots + \frac{i_1}{p^{e-1}})).\]
\begin{Theo}
\label{LastTheo}
Let $R$ be an $F$-finite regular ring and $(M, \kappa)$ a Cartier module. Fix a rational number $\lambda$. If $\lambda(p^e -1)$ is a zero of the Bernstein-Sato polynomial for $e \gg 0$ such that $\lambda(p^e -1) \in \mathbb{Z}$, then $Gr_\sigma^\lambda$ is non-trivial.
\end{Theo}
\begin{proof}
First of all, we may assume that $(M, \kappa)$ is $F$-pure (use \cite[Lemma 2.1]{staeblerunitftestmoduln}).
By construction the zeros of the Bernstein-Sato polynomial are in $[0,1]$. Fix $0 < \varepsilon \ll 1$ such that $\lambda + \varepsilon \in \mathbb{Z}_{(p)}$ and take $e \gg 0$ such that (using Lemma \ref{NonFpureIdealSimpleDescr}) $\sigma(M, f^\lambda) = \kappa^e f^{\lambda(p^e-1)} M$ and $\sigma(M, f^{\lambda + \varepsilon}) = \kappa^e f^{(\lambda + \varepsilon)(p^e -1)} M$. Note that one has $\kappa^e f^{\lambda (p^e -1)} f M \supseteq \kappa^e f^{\lambda (p^e-1}) f^{\varepsilon(p^e -1)} M$ by possibly choosing a larger $e$.
Now $Gr_\sigma^\lambda M$ is non-trivial if and only if $\kappa^e f^{\lambda(p^e-1)} M \neq \kappa^e f^{(\lambda + \varepsilon)(p^e -1)} M$.
Since $R$ is regular the functor ${F^e}^!(-) = \Hom(F_\ast^e R, -)$ is fully faithful (i.e. Frobenius is finite flat -- \cite{kunzpositive}) so that this is further equivalent to \[{F^e}^! (\kappa^e f^{\lambda(p^e -1)} M) \neq {F^e}^! (\kappa^e f^{(\lambda + \varepsilon)(p^e -1)} M).\]
Using \cite[Lemma 4.6]{blicklestaeblerbernsteinsatocartier}, one obtains \[{F^e}^!(\kappa^e f^{\lambda(p^e -1)} M) = \mathcal{C}^e(f^{\lambda(p^e-1)} M) \mathcal{D}^e_R\] and similarly for ${F^e}^!(\kappa^e f^{(\lambda + \varepsilon)(p^e -1)} M)$. Hence, $Gr_\sigma^\lambda M \neq 0$ if and only if \[\mathcal{C}^e(f^{\lambda(p^e-1)} M) \mathcal{D}^e_R \neq \mathcal{C}^e(f^{(\lambda + \varepsilon)(p^e-1)} M) \mathcal{D}^e_R \]
Let us write $\lambda (p^e -1) = i_1 + \ldots + i_e p^{e-1}$, with $i_j \in \{0, \ldots, p-1\}$, and assume that $\lambda(p^a -1)$ is a zero of the Bernstein-Sato polynomial. Then, by (left)exactnes of ${F^e}^!$, we conclude that
\[\mathcal{C}^e (f^{i_1 + \ldots + i_e p^{e-1}} M) \mathcal{D}^e_R \neq \mathcal{C}^e (f^{1+ i_1 + \ldots + i_e p^{e-1}} M) \mathcal{D}^e_R \supseteq \mathcal{C}^e (f^{(\lambda+\varepsilon)(p^e -1)} M) \mathcal{D}^e_R.\]
In particular, $Gr_\sigma^\lambda M$ is non-zero as claimed.
\end{proof}
\begin{Bsp}
The following example illustrates that the other implication of Theorem \ref{LastTheo} is not true. Take $M = k[x]\frac{1}{x}$, $f = x$ and $\lambda =1$. Then $\kappa^ax^{p^a -1} x M = \kappa^a x^{p^a-1} M = k[x]$ while $\kappa^a x^{p^a - 1} x^{\varepsilon(p^a -1)} M = (x)$ as soon as $\varepsilon(p^a - 1) \geq 2$.
\end{Bsp}
\begin{Bem}
\begin{enumerate}[(a)]
\item{If $(M, \kappa)$ is $F$-regular then all non-nilpotent information is recovered. Indeed, for suitable $e \gg 0$ one has $\sigma(M, f^{\lambda+\varepsilon}) = \kappa^e f^{(\lambda + \varepsilon)(p^e -1)} M = \tau(M, f^{\lambda + \varepsilon - \frac{\lambda + \varepsilon}{p^e}})$, and $\kappa^e f^{\lambda(p^e -1)} f M = \tau(M, f^{\lambda + \frac{1 - \lambda}{p^e}})$ using Theorem \ref{TauFRegularDescription}. By right-continuity of $\tau$ these two quantities coincide for $0 < \varepsilon \ll 1$ and $e \gg 0$ suitable.}
\item{Note that in characteristic zero one has that if $\lambda$ is a jumping number of the multiplier ideal filtration then it is a zero of the Bernstein-Sato polynomial. Here the situation is the other way around.}
\end{enumerate}
\end{Bem}
|
train/arxiv
|
BkiUbg85qsFAfmeJo12V
| 5 | 1 |
\section{Introduction}
Because the neutral ($a=3,8$) members of the
$SU(3)_F$ octet of vector currents,
$J^a_\mu = \bar{q}\gamma_\mu {\frac{\lambda^a}{2}}q$
(with $\lambda^a$ the usual Gell-Mann matrices),
couple to fermions in the Standard Model, it is possible
to use experimental data on the spectral functions
associated with correlators involving these currents
to determine certain quantities of phenomenological
interest. For example, defining
the scalar correlators, $\Pi^{ab}(q^2)$, by means of
\begin{equation}
i\int\, d^4x\, \exp (iqx)\, \langle 0\vert T\left(
J^a_\mu (x) J^b_\nu (0)\right) \vert 0\rangle \equiv
\left( q_\mu q_\nu -q^2 g_{\mu\nu}\right) \Pi^{ab}(q^2 ),
\label{scalarcorrelator}
\end{equation}
and the corresponding spectral functions, $\rho^{ab}(q^2)$, as usual,
by $\rho^{ab}(q^2)={\frac{1}{\pi}}{\rm Im}\, \Pi^{ab}(q^2)$,
one finds that (1) integrating the difference
$\rho^{33}(q^2)-\rho^{88}(q^2)$
with the weight
function occuring naturally (due to kinematics) in the finite energy sum rule
(FESR) treatment of hadronic $\tau$ decays\cite{taurefs}
produces a sum rule from which one can, in principle,
determine the running strange quark mass, $m_s(\mu )$\cite{narisonms}, and
(2) integrating the same difference $\rho^{33}(q^2)-\rho^{88}(q^2)$
with weight function $w(s)=1/s$ produces a sum rule from which one
can extract one
of the $6^{th}$ order low-energy constants (LEC's), $Q$, appearing
in the $6^{th}$ order version of the effective chiral Lagrangian\cite{gk96}.
(See Ref.~\cite{gl85} for a discussion of chiral perturbation theory
(ChPT) and the method of effective chiral Lagrangians in general,
Ref.~\cite{fsq6} for the form of the ${\cal O}(q^6)$ terms in
the effective Lagrangian
in the most general case,
and Ref.~\cite{gk95} for both a discussion of the subset of these terms
surviving when one restricts one's attention
to vacuum correlators, and a definition of $Q$.)
Of course, $J^3_\mu$ and $J^8_\mu$ do not couple separately in the
Standard Model, but only in the combination
\begin{equation}
J^{EM}_\mu =J^3_\mu +{\frac{1}{\sqrt{3}}}J^8_\mu
\label{EMcurrent}
\end{equation}
which gives the light quark ($u,d,s$) part of the electromagnetic (EM) current.
Thus, what is measured in $e^+e^-\rightarrow hadrons$, is
not the desired quantities,
$\rho^{33}$ and $\rho^{88}$, separately, but the combination
\begin{equation}
\rho^{EM}(q^2)=\rho^{33}(q^2)+{\frac{2}{\sqrt{3}}}\rho^{38}(q^2)
+{\frac{1}{3}}\rho^{88}(q^2)\ .
\label{EMrho}
\end{equation}
In the isospin symmetry limit, $\rho^{38}$ would vanish and, since
one could then classify the final hadronic states according to their
G-parity, it would be straightforward to separate
the isovector ($33$) and isoscalar ($88$) components of
the EM spectral function.
In the presence of isospin breaking, however,
this process is no longer so straightforward. The most obvious
experimental signature of the presence of isospin-breaking in
$e^+e^-\rightarrow hadrons$ is the interference shoulder in the
$e^+e^-\rightarrow \pi^+\pi^-$ cross-section in the $\rho$-$\omega$
region\cite{barkovetc}.
The $e^+e^-\rightarrow \omega\rightarrow \pi^+\pi^-$ contribution
to $\rho^{EM}$ is clearly, to leading order in isospin-breaking,
to be associated with $\rho^{38}$ and hence is usually removed
explicitly in analyzing the data. This removal is accomplished by
(1) fitting the parameters of
a model for the total $e^+e^-\rightarrow \pi^+\pi^-$
amplitude, consisting of $\rho$, $\omega$ and possible
background contributions, to the experimental data, (2) removing the
$\omega$ contribution once the fit has been performed and (3) squaring
the modulus of the
remaining $\rho$ contribution and identifying
this result with the $\rho$ contribution to $\rho^{33}$\cite{hocker}.
While this procedure
{\it does} remove one source of isospin breaking contamination
from the nominal
$\rho^{33}$ so extracted, it is easy to see that other such contaminations
still remain. Indeed, once one allows
isospin-breaking, the physical $\rho$ and $\omega$ are admixtures of pure
isovector and isoscalar states, the size of the admixture of
the ``wrong'' isospin
component being governed by the scale of isospin breaking.
As a consequence,
the intermediate $\rho$ contribution to $\rho^{38}$, for example,
does not vanish. In fact, if one denotes the pure isovector
$\rho$ state by $\rho^{(0)}$ and the pure isoscalar $\omega$ state
by $\omega^{(0)}$, one expects $\rho$ contributions to $\rho^{38}$
from two sources: (1) that due to $\rho^{(0)}$-$\omega^{(0)}$ mixing
(a one particle reducible contribution, with coupling of the
isovector current to the $\rho^{(0)}$ component and the isoscalar
current to the $\omega^{(0)}$ component of the $\rho$),
and (2) that due to the ``direct'' (one particle
irreducible, $1PI$) coupling of the $\rho^{(0)}$ component to the
isoscalar current (such a coupling being unavoidable in any hadronic
effective Lagrangian based on QCD).
Thus, removing the contribution due to the intermediate
state $\omega$ from the $e^+e^-\rightarrow \pi^+\pi^-$ cross-section,
while removing part of the $\rho^{38}$ contribution, does not remove
it all. One is then left with,
{\it not} the desired quantity, $\rho^{33}$, but rather with a combination of
$\rho^{33}$ and the residual part of $\rho^{38}$ associated with
the intermediate $\rho$ state (plus possible additional
such contaminations from elsewhere in the spectrum).
Similar isospin-breaking flavor $38$
contributions exist for $e^+e^-\rightarrow \omega\rightarrow 3\pi$,
complicating the extraction of the isoscalar spectral function.
Corrections for such isospin breaking effects, {\it which are unavoidable so
long as no process exists in which only one of the two neutral
flavor currents couples}, are thus necessary if one
wishes to perform phenomenological analyses of the type mentioned
above. Such corrections would also be important in performing
precision tests of CVC, which involve a comparison of
$\rho^{33}$ and the charged
isovector spectral function $\rho^{(+)}$, measured in hadronic
$\tau$ decays (see, for example, Ref.~\cite{ALEPH}).
It is easy to see that, to be able to make these corrections (at least
in the region below $s\sim 2$ GeV$^2$, where the EM spectral function
is, experimentally, resonance dominated), it is sufficient to determine
the isospin-breaking vector meson decay constants.
Let us first
clarify notation. We define the flavor $3$ and $8$ vector meson
decay constants via
\begin{equation}
\langle 0\vert J^a_\mu \vert V(k)\rangle = m_V F^a_V\, \epsilon_\mu (k)
\end{equation}
where $V=\rho ,\omega ,\phi ,\cdots$,
$\epsilon_\mu (k)$ is the vector meson polarization vector,
and $a=3,8$. $F^{(3)}_\rho$, $F^{(8)}_\omega$ and $F^{(8)}_\phi$ are non-zero in the limit of
isospin symmetry; $F^{(8)}_\rho$, $F^{(3)}_\omega$ and $F^{(3)}_\phi$ zero in
the absence of isospin breaking. The experimentally determined
EM decay constants, $F^{EM}_V$, are then given by
\begin{equation}
F^{EM}_V=F^3_V+{\frac{1}{\sqrt{3}}}F^8_V\ .
\label{femphysical}
\end{equation}
Thus, for example, the broad $\rho$ contribution to $\rho^{EM}$,
usually taken to be associated purely with $\rho^{33}$, consists not only of
a flavor $33$ contribution proportional to $\left[ F^3_\rho\right]^2$,
but also of a flavor $38$ contribution proportional to
${\frac{1}{\sqrt{3}}}F^3_\rho F^8_\rho$. The $\omega$
contribution to $\rho^{EM}$, similarly,
contains both a flavor $88$ part proportional
to $\left[ F^8_\omega\right]^2$ and a flavor $38$ part proportional
to ${\frac{1}{\sqrt{3}}}F^3_\omega F^8_\omega$. The flavor $38$
parts, in both cases, are present only due to isospin breaking, and
have to be removed from the experimental $\rho$ and $\omega$
contributions to $\rho^{EM}$ in order to obtain the corresponding
$\rho$ contribution to $\rho^{33}$ and $\omega$ contribution to
$\rho^{88}$.
It is important to stress
at this point that the conventional ``few-percent'' rule-of-thumb
for estimating the size of isospin-breaking effects, which
might lead one to expect such effects to be numerically negligible,
is inapplicable
in the cases involving $\rho^{33}(q^2)-\rho^{88}(q^2)$
discussed above.
This
is true for a number of reasons. First, because the
difference of spectral functions is itself
flavor-breaking,
the {\it relative} importance of isospin breaking is enhanced by
a factor $\sim 3$, characteristic of the inverse of the
scale of flavor-breaking.
Second, the
effect of $\rho$-$\omega$ mixing naturally produces corrections
for the $\rho$ contribution to $\rho^{33}$ and
$\omega$ contribution to $\rho^{88}$ which are opposite in sign;
the effects therefore add
when the difference is taken. Finally,
there is a natural numerical enhancement which makes the
size of the correction needed to remove
the $\rho^{38}$ part of the $\omega$ contribution to $\rho^{EM}$,
and hence isolate $\rho^{88}$, larger than naively expected\cite{krmspec}.
The latter two points are discussed in somewhat more detail in Section II
below.
In what follows, we evaluate the isospin-breaking vector meson
decay constants by performing a QCD sum rule analysis of the
isospin-breaking vector current correlator $\Pi^{38}$. The vector
meson spectral contributions are, in this case, proportional to
$F^3_V F^8_V$, so that a determination of this product, in combination
with the experimental determination of $F^{EM}_V$, given in terms
of $F^3_V$ and $F^8_V$ above, allows a separate determination of
$F^3_V$ and $F^8_V$. The rest of the paper is organized as follows.
In Section II we discuss qualitative expectations for the pattern
of isospin-breaking corrections
based on the
structure of the leading (chiral) order terms in the vector meson
effective chiral Lagrangian, as well as semi-quantitative
expectations for their probable
scale which can, using this perspective, be obtained from
experimental data. In Section III, we discuss
briefly the form of QCD sum rules employed (a version of FESR), and
the advantages of this approach. In Section IV, we discuss the
input used for the hadronic and OPE sides of the sum rules employed, and
present our results.
Some advantages of the approach, in particular {\it viz a vis} the handling
of the $D=6$ terms in the OPE of the $38$ correlator, will also
be discussed here. Finally, in Section V we summarize, and make some
brief comments on the phenomenological significance of our results.
\section{Chiral Constraints and the Scale of Isospin-Breaking Corrections}
ChPT provides both an underlying conceptual
framework and systematic procedure\cite{ccwz}
for writing down the most general effective Lagrangian relevant to
a given set of hadronic states which fully incorporates the symmetries
of QCD and implements the broken symmetries (such as chiral symmetry)
with the same pattern of symmetry breaking as occurs in QCD.
Although the resulting effective Lagrangian, ${\cal L}_{eff}$,
is non-renormalizable, and hence contains a infinite number of
low-energy constants (LEC's) (coefficients of those terms allowed
by the symmetry arguments that go into constructing ${\cal L}_{eff}$),
it is possible to formulate the theory
in such a way that
only a finite number of such terms appear to a given order in the
chiral, or low-energy expansion. (For so-called ``heavy'' fields,
those whose masses are non-vanishing in the chiral limit, this
requires a re-formulation in terms of velocity-dependent
fields\cite{heavybaryon,heavyvector}.)
The leading order terms in this
expansion (in which light quark masses, $m_q$, $q=u,d,s$, count
as ${\cal O}(q^2)$, with $q$ representing some soft external momentum),
incorporate the leading constraints associated either with chiral
symmetry, or the symmetry pattern of its breaking.
The heavy field implementation of ChPT given in Ref.~\cite{heavyvector}
for the vector mesons and their interactions with the members of the
pseudo-Goldstone boson pseudoscalar octet
provides some useful information about the pattern of isospin-breaking
mixing in the vector meson sector. One must bear in mind that, in
general, mixing in field theory is more complicated than in Hamiltonian
quantum mechanics. This is because, in an effective field theory,
one in general has (and cannot, in fact, avoid) momentum-dependent
mixing terms. This means that there are off-diagonal elements of the
wave-function renormalization matrix and, as a consequence, the
relation between the original, unmixed fields and the final, renormalized
and diagonalized fields, involves the product of a symmetric matrix and
a rotation (and hence is not itself a rotation). (See, for example, the
discussion of the effects of $\pi^0$-$\eta$ mixing in Ref.~\cite{gl85};
an expanded version is also given as part of the discussion of
the treatment of the mixed-isospin axial current correlator in
Ref.~\cite{kma3a8}.)
From the heavy vector meson
field formulation, however, one sees that the leading (in chiral order)
term in ${\cal L}_{eff}$ generating isospin-breaking mixing involves
one power of the quark mass matrix and no derivatives\cite{heavyvector},
and hence
produces no off-diagonal contributions to the wave-function renormalization
matrix. The {\it leading order} mixing effect thus results in a
physical $\rho$ and $\omega$ basis which is related by a rotation
to the original pure isospin $\rho^{(0)}$, $\omega^{(0)}$ basis, just
as in the quantum mechanical case. The ``wrong'' isospin
$\omega^{(0)}$ admixture in the physical $\rho$ state is thus equal
in magnitude, but opposite in sign,
to that of the $\rho^{(0)}$ admixture in the physical $\omega$ state,
at this order in the chiral expansion.
While higher (chiral) order corrections exist, this pattern should
remain approximately valid, even at higher order.
Let us now consider the vector meson decay constants. Since the
chiral limit is also $SU(3)_F$ symmetric, the leading order
term generating the vector meson decay constants is necessarily
$SU(3)_F$-symmetric.
When one now considers
the effects of flavor- and isospin-symmetry breaking (recalling
that both are generated by the quark mass matrix, and hence both
are produced by the same set of terms in the effective Lagrangian),
there are two potential sources of flavor and isospin breaking in the vector
meson decay constants. The first is that associated with higher
order terms, involving at least one power of
the quark mass matrix, coupling the external photon field to
the vector meson nonet, the second that induced by the leading
quark-mass-dependent term, responsible for mixing, discussed above.
The leading order mixing effect simply reproduces the
standard leading order $SU(3)_F$ mixing analysis\cite{heavyvector},
leading to near ideal mixing in the vector meson sector.
As is well-known, the combination of ideal mixing and neglect of
flavor-breaking in the EM couplings of the unmixed states leads
to the prediction that the vector meson EM decay constants, measured
experimentally in $V\rightarrow e^+e^-$,\cite{pdg98}
should be in the proportions $F^{(0)}_\rho :F^{(0)}_\omega :F^{(0)}_\phi
= 3:1:-\sqrt{2}$, where the superscript $(0)$ indicates that the
couplings refer to the ideally mixed, but isospin pure
vector meson states. That this prediction
is borne out by experiment represents empirical evidence that,
despite the potential $SU(3)_F$-breaking photon coupling contributions
being of the same formal order as effects induced by mixing,
the former are numerically suppressed relative to the latter.
Since flavor breaking and isospin breaking are generated by the same
terms in the effective Lagrangian, this implies that isospin
breaking in the vector meson decay constants should also be dominated
by mixing effects.
If we take this point of view then, up to sub-leading corrections,
we find, for the physical $\rho$ and $\omega$ decay constants, now
including isospin breaking and taking into account the
relation $F^{(0)}_\rho \simeq 3F^{(0)}_\omega$,
\begin{eqnarray}
F^{EM}_\rho&=&F^{(0)}_\rho -\epsilon F^{(0)}_\omega
\simeq F^{(0)}_\rho\left( 1-\frac{\epsilon}{3}\right) \nonumber \\
F^{EM}_\omega&=&F^{(0)}_\omega +\epsilon F^{(0)}_\rho
\simeq F^{(0)}_\omega\left( 1+3\epsilon \right) \, ,
\end{eqnarray}
where $\epsilon$ is the leading order mixing angle, defined via
\begin{equation}
\rho = \rho^{(0)} -\epsilon\, \omega^{(0)} \, ,\qquad
\omega =\omega^{(0)} +\epsilon\, \rho^{(0)}\ .
\end{equation}
We note two relevant features of these results: (1) because of the
dominance by mixing, the corrections required to convert the
pure isovector $F^{(0)}_\rho$ coupling to the experimental
$F^{EM}_\rho$ coupling is opposite in sign to that required
to convert the pure isoscalar $ F^{(0)}_\omega$ coupling
to the $F^{EM}_\omega$ and, (2) because of the pattern of ideal
mixing and the numerical suppression of the isoscalar current
relative to the isovector current in $J^{EM}_\mu$, the magnitude
of the correction is a factor of $9$ larger in the $\omega$
than in the $\rho$ case. This does not mean that, somehow,
isospin breaking has become huge in the $\omega$ case, but
rather reflects the expected natural suppression of the $\omega$ coupling
relative to that of the $\rho$, which is confirmed by experiment.
In general, but particularly in light of the large numerical enhancement
just discussed, it
is crucial to find a means of estimating the isospin-breaking
contributions to the vector meson EM decay constants. In view of
the discussion above, although there exist additional
effects beyond those associated with mixing at
leading order, one can get a rough idea of the
size expected for the isospin-breaking
vector meson decay constants by considering experimental information
on $\rho$-$\omega$ mixing, and ignoring all non-mixing effects.
Although crude, this estimate provides an additional qualitative
constraint for our later sum rule analysis.
In order to obtain the parameter $\epsilon$ describing $\rho$-$\omega$
mixing at leading order, it is sufficient to determine the
off-diagonal element, $\Pi_{\rho\omega}$, of the vector meson
self-energy matrix. In the past, values for $\Pi_{\rho\omega}$
in the range $\sim -4000$ MeV$^2$ have been quoted, based on
simplified analyses of $e^+e^-\rightarrow \pi^+\pi^-$ data in the
interference region which effectively assume that the one-particle
irreducible $\omega^{(0)}\pi^+\pi^-$ vertex is zero, even in the presence
of isospin breaking. Since effective operators which
generate such a coupling exist in the vector meson effective
Lagrangian, however, this assumption is unphysical (in the sense of being
incompatible with QCD). Once one
includes contributions to the $\omega\rightarrow\pi\pi$ amplitude
generated both by $\rho$-$\omega$ mixing {\it and} the 1PI vertex
(whose strength we will denote by $g_{\omega\pi\pi}^{(0)}$),
the analysis of the experimental data is somewhat more complicated but,
in principle, allows a separate determination of both $\Pi_{\rho\omega}$
and the isospin-breaking ratio of couplings of the isospin pure states
$G=g_{\omega\pi\pi}^{(0)}/g^{(0)}_{\rho\pi\pi}$\cite{mow,otw}.
An important feature of the analysis framework developed in Ref.~\cite{mow}
is that, from it, one understands that the smallness of previously quoted
errors for $\Pi_{\rho\omega}$ is an artifact of the unphysical
assumption $G=0$, and does not survive the more general treatment.
It is worth outlining the reason why this is the case since, in doing
so, it will become clear that it is a difficult task to improve the
experimental situation sufficiently to really pin down the mixing
contribution.
The contribution of the {\it physical} (i.e., mixed-isospin) $\omega$
to the amplitude for $e^+e^-\rightarrow \pi^+\pi^-$ is obtained
experimentally by determining the timelike pion form factor, $F_\pi (q^2)$,
in the interference region and fitting it to a form
\begin{equation}
F_\pi (q^2)\propto \left[ {\frac{1}{q^2-m_\rho^2}}+{\frac{A^{i\phi}}
{q^2-m_\omega^2}}\right] \, +\, {\rm background}
\label{fpi}
\end{equation}
where $m_V^2$ are the complex pole positions,
$m_V^2=\hat{m}_V^2-{\rm i}\hat{m}_V\Gamma_V$, and the fit parameter,
$\phi$, is known as the ``Orsay phase''. The $\omega$ contribution
in Eq.~(\ref{fpi}) is generated by the coupling of the physical
$\omega$ to $\pi^+\pi^-$ which, as discussed above, has two sources:
1PR ($\rho^{(0)}$-$\omega^{(0)}$ mixing), and 1PI (associated with the
$\omega^{(0)}\pi\pi$ vertex). The physical coupling is given, in
terms of these contributions, by
\begin{equation}
g_{\omega\pi\pi}=g^{(0)}_{\omega\pi\pi}+\epsilon\, g^{(0)}_{\rho\pi\pi}\ ,
\label{gomega}
\end{equation}
where, as usual, the superscript $(0)$ indicates couplings of the unmixed
isospin pure states. In the (physically plausible) approximation
that one assumes saturation of the imaginary part of $\Pi_{\rho\omega}$
by $\pi\pi$ intermediate states, one finds
\begin{equation}
{\rm Im}\, \Pi_{\rho\omega}({m}_\rho^2)=-G\hat{m}_\rho\Gamma_\rho
\label{impi}
\end{equation}
and hence, in the narrow $\rho$-$\omega$ interference region,
\begin{equation}
\Pi_{\rho\omega}\simeq\tilde{\Pi}_{\rho\omega}-{\rm i}G
\hat{m}_\rho\Gamma_\rho
\label{fullpi}
\end{equation}
where $\tilde{\Pi}_{\rho\omega}$ is now real.
The mixing angle $\epsilon$ is then given by\cite{mow}
\begin{eqnarray}
\epsilon&=&{\frac{\Pi_{\rho\omega}(m_\rho^2)}{m_\omega^2-m_\rho^2}}
\nonumber \\
&=&-{\rm i}z\tilde{T}-zG\ ,
\label{epsilon}
\end{eqnarray}
where
\begin{equation}
z\equiv {\frac{{\rm i}\hat{m}_\rho\Gamma_\rho}{m_\omega^2-m_\rho^2}},
\qquad\qquad \tilde{T}\equiv {\frac{\tilde{\Pi}_{\rho\omega}(m_\rho^2)}
{\hat{m}_\rho\Gamma_\rho}}\ .
\label{zttilde}
\end{equation}
One then finds, upon substitution of Eq.~(\ref{epsilon})
into Eq.~(\ref{gomega}), that
\begin{equation}
g_{\omega\pi\pi}=\left[ G(1-z)+{\frac{\tilde{\Pi}_{\rho\omega}(m_\rho^2)}
{\hat{m}_\rho\Gamma_\rho}}\right] \, g^{(0)}_{\rho\pi\pi}\ .
\label{renard}
\end{equation}
In many places in the literature, one finds the approximation
$m_\omega^2-m_\rho^2\simeq {\rm i}m_\rho\Gamma_\rho$ employed.
The deviation of the quantity $z$
from $1$ represents the error made in employing this
approximation. Since ${\rm Re}\, z\simeq 1$ and ${\rm Im}\, z$
is small ($\sim .2$ to $.3$), one might think it rather safe to set
$z=1$ in the above analysis (this approximation was, in fact,
made uniformly in analyses previous to the discussion of Ref.~\cite{mow}).
If it were true that this approximation were reliable, then the
effect of $G$ in Eq.~(\ref{renard}) would cancel exactly\cite{renardarg},
and the experimental data would determine the real part of
$\Pi_{\rho\omega}$ in the interference region with the usually
quoted errors ($\tilde{\Pi}_{\rho\omega}(m_\rho^2)=-3844\pm 271$
MeV$^2$; see Ref.~\cite{otw,twold,cb87} and earlier references cited therein).
Unfortunately, it turns out that the approximation is both
misleading and unreliable. The reason is that, although
$z$ is approximately real and near $1$, $(1-z)$ is dominantly
imaginary. Since the denominator of the second term in
Eq.~(\ref{renard}) is also dominantly imaginary, the two terms
add nearly constructively. Were the phases of these terms to be
actually identical, of course, it would be impossible to use
experimental information to separate them, no matter how
precise that data. Fortunately,
there is a small phase difference which, at least in principle,
means that a determination, with sufficient accuracy,
of both the magnitude, $A$, and
phase, $\phi$, of the $\omega$ contribution
to $F_\pi$,
would allow separate determination of $G$ and $\tilde{\Pi}_{\rho\omega}$.
From this, one would be able to reconstruct $\Pi_{\rho\omega}$ and hence
determine $\epsilon$. There is, however, a practical difficulty,
which limits the accuracy obtainable using present experimental
data. This is a result of the fact that the
phase difference of the two terms is not large, so one is
rather close to being in the
``identical phase'' situation, for which a separate extraction of the
two terms would be impossible.
If one takes the updated numerical
analyses of Ref.~\cite{otw}, for example, one sees that values
of $\tilde{\Pi}_{\rho\omega}$ between $-4000$ and $-8000$ MeV$^2$
are allowed by present data (with a central value $\sim -6800$ MeV$^2$),
and that, while the central extraction for $G$ is moderately
large $\sim .1$, $G=0$ is only $2{\frac{1}{2}}\, \sigma$ distant.
A significant improvement in this situation would require a significant
reduction of the errors in the determination of the Orsay phase.
Unfortunately, the prospects for seeing
such an improvement in the near future are remote, at present.
Although, with present experimental accuracy, the errors in the
determination of $\epsilon$ are disappointingly large, we
can, nonetheless, as explained above, use the range of values obtained in
Refs.~\cite{mow,otw} to help set a rough scale for the
expected size of those corrections required to go from $F^{EM}_\rho$
to $F^3_\rho$, and $F^{EM}_\omega$ to $F^8_\omega$. Using the
central values for the four fits given in Table 1 of Ref.~\cite{otw},
one finds
that $F^3_\rho$ is {\it less than} $F^{EM}_\rho$ by between $0.3$ and $3.8\%$
(the former corresponding to fixing $G=0$ by hand, the latter to
the MOW and A solutions contained in the Table of Ref.~\cite{otw}) and
$F^8_\omega$ {\it greater than} $F^{EM}_\omega$ by between
$2.6$ and $24.6\%$. We will see that the solutions obtained below
via the sum rule analysis satisfy these rather loose constraints,
and, in fact, provide an alternate determination
which is to be favored since it both has considerably smaller
errors and is free of the
uncertainties associated with effectively truncating at leading
chiral order (which goes into the estimate just discussed).
\section{QCD Sum Rules and the Choice of the FESR Method}
As is well-known, the properties of unitarity and analyticity
lead to the existence of various (appropriately subtracted) dispersion
relations for typical hadronic correlators, $\Pi (q^2)$.
By the term ``dispersion relations'' we mean here
both those relations
based on the Cauchy representation theorem, and those based on Cauchy's
theorem (the vanishing of the integral of an analytic function
over a closed contour entirely in the region
of analyticity). The generic term ``QCD sum rules''
describes those versions of these relations obtained by making
kinematic restrictions which allow one to
take advantage of the asymptotic freedom of QCD.
In the first case, one simply writes
the representation theorem for
$\Pi (q^2)$ with $q^2$ large and spacelike;
in the second, a choice
of the integration contour is made so
that at least part of the contour
lies in the region of large spacelike momenta.
In either case, the point of the kinematic restriction is to
take advantage of the fact that, in the region of large
spacelike momenta,
computational techniques based on
the operator product expansion
(OPE)/perturbative QCD (pQCD)
are applicable.
The generic form of the sum rules generated using the
representation theorem is then, up to possible subtractions,
\begin{equation}
\Pi (q^2)=
\int^\infty_{s_{th}}\, ds {\frac{\rho (q^2)}{s-q^2}}\ ,
\label{normsr}
\end{equation}
where the $s_{th}$ is the lowest physical
threshold in the channel in
question and the spectral function, $\rho$, is defined as usual by
$\rho ={\frac{1}{\pi}}{\rm Im}\, \Pi$.
The LHS is to be computed using the OPE/pQCD, while
the RHS is given in terms of measured spectral data and/or some spectral
ans\"atz (involving a limited number of unknown
parameters) for the unmeasured part of the spectral function.
As first pointed out by Shifman, Vainshtein and Zakharov (SVZ)\cite{svz},
the utility of such relations is greatly improved by Borel transformation.
The effect of the Borel transform is (1) to replace the
weight $1/\left( s-q^2\right)$ in the hadronic spectral
integral on the RHS of Eq.~(\ref{normsr}) with
$exp\left( -q^2/M^2\right)$, where $M$, the Borel mass, is a
parameter of the transformation, (2) to kill all subtraction terms
and (3) to create a factorial suppression of the contributions
of higher dimensional operators on the OPE side of the equation
($c/\left( Q^2\right)^n\rightarrow c/(n-1)! M^{2n}$).
Ideally, on the hadronic side, one would like to
choose $M$ as small as possible, in order to suppress the contributions
from the large-$s$ part of the integral, for which
the spectral function will typically be complicated, and difficult
to model. In contrast, to improve the convergence of the OPE
when truncated at operators of relatively low dimension, one
wants $M$ as large as possible. To be usable, SVZ sum rules thus require
a compromise:
one must hope to find a ``stability window'' in $M$, i.e., a region of
$M$ values
for which both contributions from the complicated part of the
spectral distribution (on the hadronic side) and from
the highest dimension operators (on the OPE side) are not too large.
Typically, because of the compromise nature of the choice of
stability window, this means that neither the contributions
from the large-$s$ part of the spectrum, nor that from the highest
dimension operator retained on the OPE side, are
negligible\cite{svz,review,book,skeptics}.
For the case of sum rules based on Cauchy's theorem, a common choice
of integration contour
(and the one we will employ in the analysis below) is that
shown in Fig.~1. The radius, $s_0$, of the circular
part of the contour is to be taken large enough that the OPE,
to the order available, is reliable in the spacelike region of the circle.
The resulting
sum rule is then generically of the form
\begin{equation}
{\frac{-1}{2\pi {\rm i}}}\,
\oint_C\, dq^2\, w(q^2) \Pi (q^2)\ =\ \int_{s_{th}}^{s_0}\,
dq^2\, w(q^2) \rho (q^2)\ ,
\label{fesrbasic}
\end{equation}
where $w(q^2)$ is any function of $q^2$ analytic in the region of
integration, and
$C$ denotes the circular part of the contour, traversed counterclockwise
(from above to below the cut). The OPE is then to be
used on $C$ (see also the discussion
to follow), while spectral data and/or a spectral ans\"atz is
to be employed in the ``hadronic'' integral on the RHS. Such sum rules
are called, generically, finite energy sum rules (FESR's), and
have usually been employed with integer-power weights
($w(s)=s^k$, with $k=0,1,2,\cdots$)\cite{fesrrefs,bpr}, though
the standard theoretical treatment of hadronic $\tau$ decays,
which is of the FESR type, involves a more complicated weight,
as determined by kinematics\cite{taurefs}.
It is useful, for the discussion which follows,
to recall at this point the distinction
between ``local'' and ``semi-local''
duality.
For a typical hadronic correlator, $\Pi (q^2)$, as above,
the OPE is expected to be reliable,
not only for $q^2$ large and spacelike, but also for those
$q^2$ on any circle of sufficiently large
radius in the complex $q^2$-plane, apart possibly from
some
region whose size is hadronic in scale about the timelike real
axis (where the
effects of confinement are expected to be important)\cite{pqw}.
The term ``local duality'', in this context, refers to the
postulate (underlying, for example,
all statements giving the number of subtractions
required to make a given dispersion relation
convergent in QCD) that, once one is in a region $q^2\sim s_0$ for
which
the separation between adjacent
resonances is small compared to the typical resonance width,
the region of validity of the OPE on the
circle of radius $s_0$ extends all the way down to the
real timelike axis. This is equivalent to saying that
the hadronic spectral function can then be calculated
using the OPE. This is what allows, for example,
the hadroproduction ratio, $R$,
in $e^+e^-$ scattering to be computed
theoretically in the asymptotic region using perturbative methods.
Similarly, ``semi-local duality'' refers
to the idea that, at somewhat lower (``intermediate'') scales,
where local duality is no longer valid,
nonetheless, averaged over some range of (timelike)
momenta, the mean values given by using either the actual
hadronic spectral function or
the OPE version thereof should be the same. It is important
to understand that, empirically, the condition that
resonance spacing be {\it much smaller} than typical resonance widths
is crucial to the validity of local duality. Indeed, one can
test local duality through the application of various FESR's
in the case of the isovector vector channel, for which
the hadronic spectral function is very accurately measured in
hadronic $\tau$ decays\cite{ALEPH}. One finds that, even at
scales as large as $m_\tau^2\simeq 3.2$ GeV$^2$, and even though
the experimental spectral function appears rather featureless
in this region, nonetheless, local duality is rather poorly
satisfied\cite{kmfesr}. (Note that in this
channel, at these scales, the separation
of resonances is comparable to their widths.)
In light of the above discussion, we distinguish
three (timelike) kinematic regimes: ``low'' (the region
of narrow, well-isolated resonances), ``intermediate'' (the
region of the validity of semi-local duality) and ``high''
(the region of the validity of local duality, for which
the hadronic spectral function can be reliably obtained
using pQCD/OPE methods). The distinction is important because
of the ways, described above, in which sum rules are practically implemented.
Typically, one attempts to use known information, contained in the
OPE (i.e., $a(Q^2)=\alpha_s(Q^2)/\pi$,
and the values of various
vacuum condensates), to place constraints on the parameters
of the (hopefully, physically-motivated)
ans\"atz for the corresponding unknown (or not fully-known)
spectral function. This attempt will, of course, be successful only
if (1) one is able to
work at scales for which the OPE, truncated at operators
of relatively low dimension, and at a given perturbative order for
the Wilson coefficients of these operators, is well-converged,
and (2) one has a {\it qualitative} form for the hadronic
spectral ans\"atz which is both physically plausible and does
not involve a large number of unknown parameters. The latter
condition can, realistically, be satisfied only if the
scale up to which one needs the spectral ans\"atz is not too far into
the intermediate region. Once the higher resonances start to
overlap, and various multiparticle background processes start
to become important, the general form of the spectral function
will become increasingly difficult to anticipate in advance
(until, that is, one is at sufficiently large scales that
local duality finally becomes valid).
In the SVZ version of QCD sum rules,
as explained above, it is only rarely possible to choose
the stability window for the analysis in
such a way as to avoid contributions
from the region where either one does not know the
qualitative form of the spectral function, or where, even if one
does, to implement it fully would involve
the use of more free parameters than could
be determined reliably, given the limited amount of information
available in the truncated OPE\cite{skeptics}.
Conventionally, this problem is dealt with
by employing a spectral ans\"atz in which (1) the low-$s$
region is assumed to be dominated by
one or two low-lying resonance contributions
and (2) the intermediate- and high-$s$ region
is approximated using the local duality version of the spectral function,
which one assumes to start at some ``continuum threshold'',
$s_0$. It is
well-known that this form of ``continuum ans\"atz''
represents a rather crude approximation,
and hence can create significant uncertainties in the analysis
if the contributions from the continuum part of the spectral
integral are large for $M$ values in the stability window. Typically,
one attempts to minimize this problem by avoiding $M$ values
for which this is the case, but the cost
of doing so is poorer convergence on the OPE side of the
sum rule and hence increased uncertainties associated
with neglect of higher dimension contributions.
A similar problem exists for the integer-power weighted version
of FESR's. One advantage of the FESR approach is that,
in contrast to the SVZ method, the choice of scale appearing
on the OPE side of the sum rule ($s_0$)
is not constrained by the requirement of working in a stability window.
One is, of course, still forced to choose $s_0$ not
too far into the intermediate
region, since otherwise the spectral ans\"atz would be
too complicated to allow a reliable analysis.
The possibility of working at such intermediate scales
in the FESR approach can,
nonetheless,
represent a practical advantage
over the SVZ approach in certain cases. This is true of those channels
for which the stability window of the SVZ analysis lies at
relatively low $M$ ({\it e.g.}, $M\sim 1$ GeV$^2$, as found
for many applications
in the literature). In such cases, the
larger scale of the FESR analysis leads to an improvement of both the
convergence by operator dimension and
convergence by perturbative order of the Wilson coefficients
on the OPE side of the sum rule.
Unfortunately, this advantage is usually more than offset
by an increase in the difficulties associated with the use of the
local duality approximation in the intermediate region. The
reason is obvious:
with integer-power weights, the region near the timelike real axis
where the circular part of the contour joins the cut, does not
have the exponential suppression present for the ``continuum''
contributions in the SVZ approach.
The problems that result can be quantified
in the case of the isovector vector channel, where the hadronic
spectral function is known experimentally.
As shown in Ref.~\cite{kmfesr},
the errors in integer-power weighted FESR's, even at
scales as high as $m_\tau^2$,
can be very large, despite the fact that the OPE at this scale is both
dominated by the leading ($D=0$) perturbative term, and
rather rapidly converging. (A more detailed discussion of why it is that
one cannot judge the degree of
validity of the local duality approximation simply by
looking at the degree of convergence of the OPE in a given region
is given in Ref.~\cite{kmfesr}.)
These problems, encountered in employing
the conventional, integer-power-weighted version of
FESR's, are, however, not intrinsic to the FESR approach. Indeed,
there is at least one example of a non-integer-power-weighted FESR that
is very well-satisfied: that giving the hadronic $\tau$
decay widths in terms of an integral over the circle of radius
$s_0=m_\tau^2$ of the product of the OPE for the isovector vector
current correlator and the weight function
$w_\tau (s)=\left( 1-s/m_\tau^2\right)^2 \left( 1+2s/m_\tau^2\right)$
(where the dominant input parameter in the OPE representation is
$a(m_\tau^2)$, which can be taken as obtained by running the
value measured at the $Z$ mass down to the $\tau$ scale).
The reason for the success of this sum rule is well-known: the
juncture of the cut and circular portions of the FESR contour
corresponds to the edge of hadronic phase space and hence,
because of kinematics, the weight function $w_\tau (s)$ has a
(double) zero at $s=m_\tau^2$. This zero suppresses contributions
from the portion of the circular part of the contour near the
real timelike axis for which the OPE representation of the correlator
is not reliable (at intermediate scales like $m_\tau^2$)\cite{taurefs}.
This immediately suggests that the appropriate way to implement
FESR's in other
channels is, not with integer-power weights, but rather with
weights having a zero at $s=s_0$. In Ref.~\cite{kmfesr} it
was shown that, in the isovector vector channel, where one can
check the procedure explicitly, weight functions of either the form
\begin{equation}
w_s(s)=\left( 1-{\frac{s}{s_0}}\right) \left( 1+A{\frac{s}{s_0}}\right) \ ,
\label{singlepinch}
\end{equation}
having a single zero at $s=s_0$, or
\begin{equation}
w_d(s)=\left( 1-{\frac{s}{s_0}}\right)^2 \left( 1+A{\frac{s}{s_0}}\right) \ ,
\label{doublepinch}
\end{equation}
having a double zero at $s=s_0$, produce extremely well-satisfied
FESR's for a wide range of values of $s_0$ and
the continuous parameter, $A$. In addition,
using {\it only} the OPE representation, for a range of $A$ and $s_0$,
and fitting the parameters of a sum-of-resonances ans\"atz
to this representation, results in a very good reconstruction
of the hadronic spectral function, including a determination
of the $\rho$ decay constant accurate to within a few $\%$\cite{kmfesr}.
In order
to have a compact terminology for use in describing the weight
families in Eqs. (\ref{singlepinch}) and (\ref{doublepinch}),
we will refer to $w_s(s)$ and $w_d(s)$ as single-pinch
and double-pinch weights, respectively.
The freedom to vary $A$ plays a role analogous to that of the
variation of $M$ within the stability window in an SVZ-style analysis,
in that it strengthens the sum rule constraints on the parameters
of the spectral ans\"atz by working with a range of different
weight profiles. An additional advantage, at least if one wishes to
determine not just the parameters of the lowest
resonance in the channel but also those associated with higher
resonances, is that the weight function
can actually be arranged to be larger in the second resonance
region than in the first.
In what follows, in light of its success in
the isovector vector channel, we will employ the FESR framework, using both
the single- and double-pinch weight families, to investigate
the isospin-breaking vector current correlator,
$\Pi^{38}$, defined above. As usual in the FESR framework,
we will work at scales as high as possible, compatible with
the constraint of having a tractable and physically sensible
spectral ans\"atz for $s<s_0$.
Since little is known about the vector meson resonance spectrum
beyond the second excited resonance region, and since including even
the second excited resonance region would lead to a spectral
ans\"atz with more parameters than are generally tractable for the present
analysis, we are forced
to work at scales no higher than $\sim 2.8$ GeV$^2$.
Since the separation of the
first and second excited vector meson resonance regions
is comparable to the resonance widths
(the $\rho^\prime$ and $\omega^\prime$ lie at $1419$ and $1452$
MeV, the $\rho^{\prime\prime}$ and
$\omega^{\prime\prime}$ at
$1723$ MeV and $1649$ MeV, respectively\cite{pdg98}),
it is clear that, at these scales, we are not yet in the
region of the validity of local duality. This makes the
use of the single- and double-pinch families crucial to
the reliability
of the FESR analysis.
In order to maintain as good convergence
as possible on the OPE side of the two sum rule families, while at the
same time allowing enough variation in $s_0$ to get a
good determination of the parameters of the spectral ans\"atz,
we also restrict our attention to scales, $s_0$, greater
than $2$ GeV$^2$.
\section{Details of the Analysis}
Since the general framework to be employed in the analysis has been outlined
in the previous section, it remains only to discuss
the input required on the hadronic and OPE sides
of the various sum rules.
We begin with the hadronic side.
We take, as our ans\"atz for the hadronic spectral function, a sum
of resonance contributions. For the scales used in the
analysis, the resonances present in the region of
the hadronic spectral integral are
the $\rho$, $\omega$, $\phi$, $\rho^\prime$ and $\omega^\prime$.
(Although the tails of the $\rho^{\prime\prime}$ and
$\omega^{\prime\prime}$ intrude slightly into the hadronic
integration region for $s_0$ near $2.8$ GeV$^2$,
their contributions are strongly suppressed
by the zeros in the weight functions. We have checked that
including an effective, combined
$\rho^{\prime\prime}$-$\omega^{\prime\prime}$ contribution in
the spectral ans\"atz has negligible effect on the extracted
$\rho$, $\omega$ and $\phi$ spectral strength parameters.)
We thus include contributions, written in terms of
Breit-Wigner resonance forms,
for all these resonances. Because the separation of the
$\rho^\prime$ and $\omega^\prime$ is much smaller than either
of their widths, and also to reduce the number of free parameters
in the spectral ans\"atz, we have combined the latter two contributions.
The strong overlap of the two resonances would,
in any case, prevent one from being able to sensibly extract separate
$\rho^\prime$ and $\omega^\prime$ strength parameters by means of
any sum rule analysis of $\Pi^{38}$.
The spectral ans\"atz
then has the form
\begin{equation}
\rho^{38} (q^2)=
{\frac{1}{4\sqrt{3}}}\left[ f_\rho \hat{\delta} \left( q^2-m_\rho^2\right)
-f_\omega \hat{\delta} \left( q^2 -m_\omega^2\right) +
f_\phi\hat{\delta} \left( q^2 -m_\phi^2\right)
+f_{\rho^\prime\omega^\prime}\delta \left( q^2-
\bar{m}_{\rho^\prime\omega^\prime}^2\right)\right]\ ,
\label{specansatz}
\end{equation}
where
\begin{equation}
\hat{\delta}\left( q^2-m^2\right)\equiv {\frac{1}{\pi}}
{\frac{m\Gamma}{\left( q^2-m^2\right)^2+m^2\Gamma^2}}
\label{wideresonanceapprox}
\end{equation}
(with $\Gamma$ the width of the resonance in question).
This expression reduces
to $\delta \left( q^2-m^2\right)$ in the narrow width approximation (NWA).
The minus sign in front of $f_\omega$ and the factor of
$1/4\sqrt{3}$ are
conventional; inclusion of the former ensures
that $f_\omega$ and $f_\rho$ become equal
in the limit that the spectral contributions in the $\rho$-$\omega$ region
are generated entirely by leading order $\rho$-$\omega$ mixing.
For the combined $\rho^\prime$-$\omega^\prime$ contribution we
have taken average values for the effective mass and width.
$f_\rho$, $f_\omega$, $f_\phi$ and $f_{\rho^\prime\omega^\prime}$
are free parameters, to be determined from the matching of
hadronic and OPE sides of the single- and double-pinch sum rules,
for a range of $s_0$, $A$ values.
A few comments are in order concerning the form
of the ans\"atz above, and the physical meaning of the
parameters to be extracted from the analysis which follows.
The first concerns the
need for the inclusion of a $\phi$ contribution.
Note that the correlator $\Pi^{38}$ is very closely related to that,
$\Pi^{\rho\omega}$, obtained by dropping the strange part of
the hypercharge current from $\Pi^{38}$ (the OPE's for
the two correlators are, in fact, identical to three-loop order).
The latter correlator has been
studied in a number of earlier SVZ-style analyses\cite{svzro,hhmk,kmro,ijl}.
In the earliest of these, the NWA
was employed for all resonances, and no $\phi$ contribution was included in
the spectral ans\"atz\cite{svzro,hhmk}. As pointed out
in Ref.~\cite{kmro}, however, the existence of significant cancellations
between the NWA $\rho$ and $\omega$ contributions (which would be
exact in the limit of mixing dominance and equality of $\rho$
and $\omega$ masses) means that a $\phi$ contribution,
even if significantly smaller than the {\it individual} $\rho$
and $\omega$ contributions, could nonetheless be important.
Performing the sum rule analysis with a $\phi$ contribution included
shows that this is, indeed the case\cite{kmro}.
Including the $\phi$ contribution
in the spectral ans\"atz also cured an unphysical feature of
the solutions obtained earlier, which did not include it\cite{kmro}.
The analysis of Ref.~\cite{kmro},
however, still employed the NWA
for all resonances.
The second point concerns the
need to incorporate the $\rho$ width into the analysis.
Because, again, of the high degree of
cancellation between the NWA $\rho$ and $\omega$ contributions,
it was pointed out that
the precise degree of this cancellation might well be
sensitive to whether or not the difference between
the $\rho$ and $\omega$ widths
was retained in the spectral ans\"atz\cite{ijl}.
The subsequent analysis of Ref.~\cite{ijl} showed that this is, indeed,
the case: the spectral parameters,
$f_V$, decrease by factors $\sim 6$ when one employs the
physical widths in place of the NWA.
The third point concerns the interpretation of the higher resonance
strength parameters, $f_\phi$ and $f_{\rho^\prime\omega^\prime}$.
It is, of course, very natural to take the spectral function
to be resonance dominated.
Moreover, the near-threshold region of the spectral function has been
computed to two loops in ChPT\cite{km2loop},
and one can see from this result that the corresponding low-$s$ background
contribution to the relevant spectral integrals is
tiny compared to that from the $\rho$-$\omega$ region. The
case of the $\phi$, however, is less clear, since background
contributions above the $\rho$-$\omega$ region are
not amenable to as reliable estimates, and hence might not
be similarly negligible. In the ans\"atz as written,
such physical contributions, if present, could only be mocked up
(approximately) by additional effective contributions to the
$\phi$ and $\rho^\prime$-$\omega^\prime$ strengths. Thus, one
must use some caution in interpreting, for example, the
extracted $f_\phi$
in terms of the physical resonance parameters $F^3_\phi$ and $F^8_\phi$ --
some portion of $f_\phi$ could actually correspond
to an averaged version of background contributions in the region
between $\rho$-$\omega$ and $\rho^\prime$-$\omega^\prime$.
The quality of the agreement between the hadronic and OPE sides
of our sum rules is, however, {\it post facto} evidence in
favor of resonance dominance, and hence also in
favor of the possibility of interpreting
$f_\phi$ in terms of $F^3_\phi$ and $F^8_\phi$.
Let us turn, then, to the input on the OPE side of the sum rules. We
will discuss the contributions, in turn, by operator dimension.
Since the correlator in question is isospin-breaking, the only dimension
$D=0$ contribution to $\Pi^{38}$ is electromagnetic
(we adhere, here, to common usage,
according to which the leading mass-dependent perturbative terms
are labelled $D=2$).
We retain
only the leading order (2-loop) graph in this case.
The $D=2$ contributions are dominated by the strong interaction
terms proportional to $(m_d-m_u)^2$.
To 3-loop order, the results for these terms follow from the 3-loop
expressions for the correlator involving a flavor-non-diagonal
current and its conjugate\cite{ck93}, since
the perturbative contributions involving
two quark loops and a purely gluonic intermediate state
(present for flavor diagonal
currents but not for flavor-non-diagonal currents)
do not enter until 4-loop order. The resulting expressions
are given in the Appendix. To evaluate them, we require the
running masses, $m(Q^2)$, and running strong coupling,
$\alpha_s(Q^2)$. These can be obtained once the values are determined
at any fixed scale, $\mu_0$. Since the 4-loop $\gamma$\cite{gamma4} and
$\beta$\cite{beta4}
functions for QCD are now known, we have employed these
when running the masses and coupling (explicitly, we
solve the RG equations exactly, using the truncated 4-loop
$\gamma$ and $\beta$ functions as input).
As input for the running coupling, we take $\mu_0 =m_\tau$ and
use the latest (1998) value for $\alpha_s(m_\tau^2)$
obtained by the ALEPH Collaboration
in their analysis of non-strange hadronic $\tau$ decays\cite{ALEPH98}.
(The analysis of the strange
decays employed previous theoretical results for the $D=2$ terms,
proportional to $(m_s-m_u)^2$, which turn out to be in
error\cite{kmmsprob,chetyrkin,pp}; the value obtained in the global
ALEPH analysis must, therefore, be excluded.)
The situation with the light quark mass
difference $\delta m\equiv (m_d-m_u)$ is somewhat more complicated.
We first write
\begin{equation}
\delta m=\left({\frac{m_d-m_u}{m_d+m_u}}\right) (m_d+m_u)
\equiv r(m_d+m_u)\ .
\label{delm}
\end{equation}
The isospin-breaking mass ratio, $r$, is known, from a number
of ChPT analyses, to be $r=0.288\pm 0.037$\cite{leutwylermq}, which
would allow one to determine $\delta m$ if $m_d+m_u$ were
known.
The most recent determination of $m_d+m_u$
is that based on an integer-power-weighted FESR analysis of the
isovector pseudoscalar channel\cite{bpr,prades97}.
In this analysis, the pion pole contribution to the spectral function
is known experimentally but the
continuum contribution is not. The authors of Refs.~\cite{bpr,prades97},
therefore, constructed an ans\"atz for the unmeasured
continuum contribution. It turns out
that the continuum portion of the resulting model
spectral function provides
roughly $3/4$ of the contribution to the extracted
value of $(m_d+m_u)^2$.
Unfortunately, it has recently been shown, using
the FESR framework discussed above, that this continuum ans\"atz
is unphysical\cite{kmfesr}, so one cannot employ the values of
Refs.~\cite{bpr,prades97}.
If $m_s$ were known (at some scale),
then one could straightforwardly determine $m_d+m_u$ (at that same
scale) using the known
(scale-independent) ratio of masses, $r_s=2m_s/(m_d+m_u)=24.4\pm 1.5$,
obtained by Leutwyler\cite{leutwylermq} using ChPT.
Unfortunately, the situation is also somewhat
complicated for $m_s$. A number of recent analyses produce values
of $m_s (1\ {\rm GeV}^2)$ (in the $\overline{MS}$ scheme)
ranging from $\sim 110$ MeV to $\sim 210$ MeV, often with rather
large errors\cite{jm,cps,cfnp,narison3388,km3388,kmmsprob,chetyrkin,jnew}.
Because the analyses based either on flavor breaking in hadronic $\tau$
decays\cite{kmmsprob,chetyrkin} or Narison's
$\tau$-decay-like sum rule for
$\Pi^{33}-\Pi^{88}$\cite{narison3388,km3388}
have rather large errors resulting
from experimental uncertainties which are unlikely to be significantly
improved in the near future,
the most favorable approach would appear to be that
based on various sum rule treatments of the strange scalar
channel, where the dominant $K\pi$ part of the spectral function
is in principle determined, via the Omnes representation of the timelike
scalar $K\pi$ form factor, in terms of experimental $K\pi$ phase
shifts and $K_{e3}$ data\cite{jm}. The most recent analyses of
this channel\cite{cfnp,jnew} employ the SVZ framework, and
produce values
$m_s(1\ {\rm GeV}^2)=125-160$ MeV\cite{cfnp}, and
$160\pm 30$ MeV\cite{jnew}. (The same low-$s$ part
of the spectral function is
used in both analyses; the only difference between the two lies
in the treatment
of the ``continuum''. The results of Ref.\cite{jnew},
in addition, show no stability window
for $m_s$). Preliminary
work using the FESR framework discussed above indicates
that residual errors associated with the use of the local duality
approximation in the continuum region
remain, for this channel, when one uses the SVZ approach.
(See, e.g., the results of Ref.~\cite{kmfesr}. From these one can see
(1) that using the central values for the parameters describing
the fit to the $K\pi$ phases from Refs.~\cite{jm,cfnp}, together with
the central value from the $m_s$ range from Ref.~\cite{cfnp},
one obtains rather poorly satisfied families of FESR's, and (2) that using
the spectral function of Refs.~\cite{jm,cfnp,jnew},
again with central values for the fit parameters, the FESR analysis, in fact,
produces $m_s$ values larger by $\sim 20$ MeV than those obtained
in the analysis
of Ref.~\cite{cfnp}). Although work on the extraction
of $m_s$ is still in progress\cite{mgb},
we conclude already from the preliminary results noted above, that
$m_s(1\ {\rm GeV}^2)\sim 165$ MeV, probably with errors
$\sim \pm 20$ MeV or less. The ChPT ratio then produces
$(m_d+m_u)(1\ {\rm GeV}^2)\simeq 13.5$ MeV, with errors $\sim\pm 2$ MeV.
For any value in this range it turns out that the $D=2$ contributions
are at the $\sim 15\%$ (or less) level of the $D=4$ contributions, and
the resulting errors lead, therefore, to very small ($\%$ level)
uncertainties in the final results. Since these uncertainties
are much smaller
than those generated by the uncertainty in the isospin-breaking mass
ratio $r$, we have employed the central value
$(m_d+m_u)(1\ {\rm GeV}^2)=13.5$ MeV, and retained only the uncertainty
in $r$, in the analysis which follows.
The $D=4$ contributions are much more straightforward.
Although in principle both those $D=4$ terms
proportional to the isospin-breaking mass difference,
$\delta m$,
and those proportional to the isospin-breaking condensate difference,
$<\bar{d}d>-<\bar{u}u>$, appear in the OPE of $\Pi^{38}$,
the latter are numerically tiny compared to the former.
The dominant $D=4$ contribution can then be written in terms
of $r$ and the combination $(m_d+m_u)<\bar{q}q>$, which we
can take from
the GMO relation
\begin{equation}
(m_d+m_u)<\bar{q}q>=-m_\pi^2f_\pi^2\ .
\label{gmo}
\end{equation}
The dominant uncertainties for the $D=4$ terms thus result from
those on $r$.
The phenomenological situation is not so favorable in the
case of the $D=6$ condensates. Usually, in the absence
of pre-existing determinations of the relevant condensates, one
makes estimates based on the vacuum saturation approximation (VSA).
It is well known that, in situations where it has been possible to
perform phenomenological checks by extracting the total
$D=6$ contribution from data, the VSA has proven to
significantly underestimate
these contributions\cite{vacsat}. Usually one simply replaces
the factor $\alpha_s<\bar{q}q>^2$, which is produced by the VSA,
by an effective scale-independent factor, written
$\rho^\prime\alpha_s<\bar{q}q>^2$. The parameter, $\rho^\prime$, then
represents the deviation from the VSA. Ideally, it should either
be possible to determine $\rho^\prime$ from data, or the $D=6$
contributions should be small, for the sum rule in question.
In our case, neither of these conditions holds. In particular,
because we are forced to work at scales as low as $2$ GeV$^2$
in order to constrain the spectral parameters, the $D=6$
contributions can, for $s_0\sim 2$ GeV$^2$, and
certain values of $A$ employed in our analysis, approach $\sim 40\%$
of the leading $D=4$ term. Fortunately, it turns out, as
we will see explicitly below, that by working with both the
single- and double-pinch weight families, we can actually
obtain a rather good determination of the $D=6$ contribution
to the correlator (albeit it as
a function of $r$) by insisting on the consistency
of the results obtained from the two different sum rule
families.
In the Appendix, it is shown that the VSA leads
to an expression for the $D=6$ contribution to $\Pi^{38}$
proportional to
\begin{equation}
\alpha_s \left(\langle \bar{d}d\rangle^2
-\langle \bar{u}u\rangle^2\right)\ =\
\gamma \left( \alpha_s \langle \bar{q}q\rangle^2\right)\ ,
\label{vacsat}
\end{equation}
where $\langle \bar{q}q\rangle$ is the average of the $u$ and $d$
condensates, and
\begin{equation}
\gamma\equiv {\frac{\langle \bar{d}d\rangle}{\langle \bar{u}u\rangle}}-1\ ,
\label{gamma}
\end{equation}
describes isospin-breaking in the light quark condensates. In order
to compare the deviation from the VSA in the isospin-breaking channel
with that in the analogous isospin-conserving isovector vector ($ab=33$)
channel, we write the re-scaled version of the RHS of Eq.(\ref{vacsat})
in the form
\begin{equation}
\rho^\prime\alpha_s \left(\langle \bar{d}d\rangle^2
-\langle \bar{u}u\rangle^2\right)\ =\
\rho_{red}\gamma \left(\rho \alpha_s \langle \bar{q}q\rangle^2\right)\ ,
\label{defnrhored}
\end{equation}
where $\rho$ is the parameter describing the deviation from the VSA in
the $33$ channel, and one has, phenomenologically,\cite{narisonrho}
\begin{equation}
\rho \alpha_s \langle \bar{q}q\rangle^2 = (5.8\pm 0.9)\times 10^{-4}
\ {\rm GeV}^6\ .
\label{value33rho}
\end{equation}
With this definition, $\rho_{red}$ reduces to $1$
in the limit that the deviation from the VSA is the same in the $33$ and
$38$ channels.
The consistency procedure for fixing the
$D=6$ contribution to $\Pi^{38}$, together with the phenomenological
input of Eq.~(\ref{value33rho}), of course,
determines only the
product $\rho_{red}\gamma$.
In
presenting our results for $\rho_{red}$ below, we have taken
$\gamma \simeq -0.008$, which represents an average of
the previous determinations listed in Ref.\cite{hhmk}, bar one.
(We omit the value based on an analysis of
baryon splittings because it implies
(via the 1-loop ChPT relations between flavor-breaking and
isospin-breaking in the light quark condensate\cite{gl85})
$\langle \bar{s}s\rangle / \langle \bar{u}u\rangle >1$, which
appears unphysical).
We will discuss the determination of $\rho_{red}$ in more detail below
when we present the results of the analysis.
The last point in need of discussion concerns the way in which we handle
the integrals
on the OPE side of the various FESR's.
Two options exist in the literature. The first, sometimes
called the ``fixed order expansion'', involves first
expanding $\alpha_s(Q^2)$ and the
mass factors, generically $m(Q^2)$, in terms of
$\alpha_s(s_0)$ and $m(s_0)$. The coefficients of the
perturbative expansions in powers of $\alpha_s(s_0)$
are then polynomials in $\ell n(s/s_0)$\cite{kniehl},
and the desired contour integrals can thus be
written in terms of
elementary integrals involving logarithms and powers of $s$.
The integrated OPE expressions which result
involve $m(s_0)^2$, multiplied by a power series in $\alpha_s(s_0)$.
There is, of course, in this expression, the usual residual
dependence on the choice of scale $s_0$ for the expansions
discussed above, which results from truncating the full
perturbative series at fixed order. The second alternative,
often referred to as ``contour improvement'',
involves numerically integrating the factors
$\left[ m(Q^2)\right]^k \left[\alpha_s(Q^2)\right]^j s^\ell$
around the circular contour in the $s=-Q^2$ plane\cite{pledib}.
It is known that this has the effect of simultaneously
improving the convergence of the perturbative series and
reducing the residual scale dependence\cite{pledib,bpr,prades97}.
As a result, we have evaluated all the integrals on the OPE
sides of our sum rules using this approach.
Let us now turn to the results, which are presented in the Table.
As explained above, the dominant uncertainty is due to that
in the ChPT determination of $r$. We have, therefore,
tabulated the results for the range of values corresponding
to the errors on $r$ quoted by Leutwyler\cite{leutwylermq}.
All results are based on matching the hadronic and OPE sides
of the two sum rule families for $s_0$ in the range $2.0$ to $2.8$
GeV$^2$, and with $A$ in the range $2$ to $5$ for the
single-pinch case and $3$ to $6$ in the double-pinch case.
The choice of range of $A$ in each case
has been made so as to keep the convergence of
the perturbative series for the $D=2$ term under control.
It is worth mentioning that the quality of the match between the OPE
and hadronic sides which results after the fitting of the spectral parameters
is significantly better for Leutwyler's central value of $r$.
The value of the $D=6$ VSA-violating parameter, $\rho_{red}$,
given in the Table, is determined by
requiring that the valueds of $f_\rho$ obtained using
the single- and double-pinch weight families are the same.
The sensitivity of
$f_\rho$ to variations in $\rho_{red}$
(true also of the other $f_V$), as well
as the difference in the $\rho_{red}$ dependence of $f_\rho$
for the single- and double-pinch analyses,
is shown in Figure 2.
The fact that,
once $\rho_{red}$ has been determined by the requirement
of the consistency of the two output $f_\rho$ values, all the rest of the
spectral parameters, determined using either the single-
or double-pinch weights, also become consistent
is strong evidence in favor of the reliability of the analysis.
Note that (1) the possibility of determining the correction
to the VSA for the $D=6$ operators, and (2) the inclusion of both
the $D=2$ terms and the ${\cal O}(\alpha_s ,\alpha_s^2)$
contributions to the Wilson
coefficient of the $D=4$ term, are features not present in
previous analyses of the analogous isospin-breaking $\Pi^{\rho\omega}$
correlator.
Although the value of $\rho_{red}$, determined as just described,
depends somewhat on $r$, this dependence is not
strong, and we obtain $\rho_{red} =1.15\pm 0.15\pm 0.2$. The first
error corresponds to that in Eq.~(\ref{value33rho}),
the second to that on $r$. We see that the violation
of the VSA is very similar in both the $33$ and $38$ channels.
The importance, in
reducing the errors on the determinations of the
spectral parameters, $f_V$, of being able to determine $\rho_{red}$
is also evident from Fig.~2.
Having determined the $D=6$ contributions by self-consistency,
the errors on the extracted values of $f_V$ are determined
solely by those on $r$, and are $\sim 10-15\%$, completely
correlated with $r$.
Having extracted the parameters $f_V$, it is straightforward
to determine the isospin violating decay constants. One finds
\begin{eqnarray}
F^8_\rho &=&2.4\pm 0.3\ {\rm MeV} \nonumber \\
F^3_\omega &=& -3.4\pm 0.4\ {\rm MeV}\nonumber \\
F^3_\phi &=& 0.33\pm 0.02\ {\rm MeV}\ ,
\label{decayconst}
\end{eqnarray}
where the errors reflect those on the input isospin-breaking mass
ratio, $r$.
\section{Summary and Discussion of Phenomenological Consequences}
A number of useful general
observations follow from the analysis above. First, we
have found that the violation of the
VSA for the $D=6$ condensates is very similar
in the isospin-breaking ($38$) and isospin-conserving
($33$) vector current channels. Second, we have seen that,
although the analysis
can be
performed successfully for any $r$ in the range given
by Leutwyler, the central value of that range
is preferred, in the sense of
giving the best match between OPE and hadronic sides of both the
single- and double-pinch families of
sum rules. Finally, we have demonstrated that
the FESR
method, particularly when implemented using both the single- and
double-pinch weight families, is very effective, allowing
a determination of the $\rho^{38}$ spectral parameters, $f_V$,
with rather small errors.
These errors (between $10$ and $15\%$ for $f_\rho$,
$f_\omega$ and $f_\phi$) are a factor
of $3$ smaller than those obtained in the earlier analysis
of Ref.~\cite{krmspec} based on results of an SVZ
analysis of $\Pi^{\rho\omega}$\cite{ijl}.
In order to obtain this level of reduction,
the ability to self-consistently determine $\rho_{red}$
was crucial.
Let us now turn to the phenomenological consequences of our
results. First note that the corrections required
to convert the measured contribution of the vector meson, $V$,
to the EM spectral function, $\rho^{EM}$,
into the corresponding contribution to either $\rho^{33}$ (for $V=\rho$)
or $\rho^{88}$ (for $V=\omega ,\ \phi$) are given by the ratios
\begin{eqnarray}
\left[ \frac{F^{3}_\rho}{F^{EM}_\rho}\right]^2&=& 0.982\pm 0.0021
\nonumber \\
\left[ \frac{F^{8}_\omega}{\sqrt{3}\, F^{EM}_\omega}\right]^2&=&
1.154\pm 0.017 \nonumber \\
\left[ \frac{F^{8}_\phi}{\sqrt{3}\, F^{EM}_\phi}\right]^2&=&
1.009\pm 0.001\ ,
\label{corrections}
\end{eqnarray}
where the numerical values follow from those in Eq.~(\ref{decayconst}).
The size of the deviations of the $\rho$ and $\omega$
corrections from $1$ are reduced by
$\sim 15-20\%$ from those obtained in the earlier
analysis \cite{krmspec}; that for the $\phi$ is increased, but remains
small. In all cases the errors have been reduced by
a factor of $3$ or more.
Note that the first of these corrections is the one relevant to precision
tests of CVC. Note also that, as claimed above, the corrections
given in Eqs.~(\ref{corrections}),
for both the $\rho$ and $\omega$, lie in
the corresponding ranges produced by the estimate of Section II.
With the results given in Eq.~(\ref{corrections}), it is now
possible to correct the EM data used as input to the
inverse moment chiral sum rule for the $6^{th}$ order
LEC, $Q$. The sum rule is given by\cite{gk95,gkinv}
\begin{eqnarray}
&&\int_{4m_\pi^2}^\infty ds \ \frac{(\rho^{33}-\rho^{88})(s)}{s}
=
{\frac{16 (m_K^2-m_\pi^2)}{3F^2}} Q (\mu^2)
+ {\frac{1}{48 \pi^2}} \log \left( {\frac{m_K^2}{m_\pi^2}}\right) \nonumber \\
&&\qquad\qquad
+\left(\frac{L_9^r(\mu^2)+L_{10}^r(\mu^2)}
{2 \pi^2 F^2}\right) \left[ m_\pi^2 \log \left({\frac{m_\pi^2}{\mu^2}}\right)
- m_K^2\log\left({\frac{m_K^2}{\mu^2}}\right)\right]\ ,
\label{invchmom}
\end{eqnarray}
where $\mu$ is the renormalization scale of the effective chiral
theory and $L_k^r$ are the usual renormalized $4^{th}$ order LEC's
of Gasser and Leutwyler\cite{gl85}.
This sum rule was evaluated in Ref.~\cite{gkinv} using as input
EM data for the isoscalar spectral function and
both EM and $\tau$ decay data
for the isovector spectral function.
The corrections above, required for the EM data, were not
considered in this analysis. It is not clear, from our reading of
the discussion of Ref.~\cite{gkinv}, exactly what
the relative weightings of
$\tau$ decay and EM data in the determination of
the $\rho^0$ contribution to the LHS above actually were.
Since, however, $\tau$ decay data is considerably
more precise than electroproduction data, we have assumed in what
follows that the determination is dominated by
$\tau$ decay data.
To the extent that this is true,
we need only make corrections to the (nominally) isoscalar
$\omega$ and $\phi$ contributions. The result of this exercise
is a shift of $Q(m_\rho^2)$ from $(3.7\pm 2.0)\times 10^{-5}$
to
\begin{equation}
Q(m_\rho^2)=(2.4\pm 2.0)\times 10^{-5}\ .
\label{Q3388value}
\end{equation}
(For reference, making, instead, the somewhat perverse
assumption that the determination of the $\rho^0$
contribution was dominated by
EM data, one would find $Q(m_\rho^2)=(2.0\pm 2.0)\times 10^{-5}$.
The full correction is dominated by that to the $\omega$ contribution.
The reason this correction is so much larger than the others
has been discussed above.)
One should bear in mind, in
interpreting these results, that (1) there are, in principle,
additional corrections to be made to the nominal isovector
and isoscalar contributions at higher $s$, and (2) the $\bar{K}K2\pi$
contributions were taken to be purely isoscalar in the analysis
of Ref.~\cite{gkinv}. Because the separations within the higher
isovector and isoscalar resonance pairs,
$\rho^\prime$-$\omega^\prime$ and
$\rho^{\prime\prime}$-$\omega^{\prime\prime}$, are much smaller than
the resonance widths, it is not possible to use sum rule methods to
extract the individual isospin-breaking decay constants of
these resonances. As such, we are unable to estimate the size
of the former corrections. Were the $\bar{K}K2\pi$ states to
have a significant isovector component, the effect would be
to raise $Q(m_\rho^2)$.
An alternate method for determining $Q$ is
based on the observation that $Q$
occurs not only in the inverse chiral moment sum rule above,
but also in the 2-loop ChPT expression for
$\Pi^{38}(0)$\cite{km2loop}.
It is, thus, possible to make an
independent estimate by using the fitted spectral
ans\"atz for $\Pi^{38}$ to compute $\Pi^{38}(0)$, assuming
negligible contribution from the portion of the spectrum
above $2.8$ GeV$^2$. One obtains, from this exercise,
\begin{equation}
Q(m_\rho^2)=(3.3\pm 0.4)\times 10^{-5}\ .
\label{Q38value}
\end{equation}
Since there is no positivity constraint on $\rho^{38}(s)$
one does not know in which direction this result would be changed by
corrections due to the small higher-$s$ part of the spectral
integral. The results of a study
of the effect of including two combined spectral contributions,
one for the $\rho^{\prime}$-$\omega^{\prime}$ and one for the
$\rho^{\prime\prime}$-$\omega^{\prime\prime}$, however, shows
negligible change in $Q(m_\rho^2)$,
suggesting that
such corrections are unlikely to be numerically significant.
Since the two independent determinations of
$Q$ are completely consistent, within errors,
the conclusions that $Q(m_\rho^2)\simeq 3\times 10^{-5}$
is considerably strengthened.
The last phenomenological application of our results concerns
the effect on Narison's $\tau$-decay-like sum rule for $m_s$.
Since a detailed discussion of the way in which one implements
the isospin-breaking corrections is given in Ref.~\cite{km3388},
we report here only the results of employing the improved
determinations of the correction factors determined above.
In doing so we will also take the opportunity
to update the input parameters to
the analysis of Ref.~\cite{km3388}, employing the newer (1998)
ALEPH value of $\alpha_s(m_\tau^2)$\cite{ALEPH98}.
One finds, for example, using
$\tau$ decay data for the isovector input, that the average over
the values of $m_s(1\ {\rm GeV}^2)$ extracted using the set of
scales $s_0=1.4$, $1.5$ and $1.6$ GeV$^2$ in the sum rule analysis
is shifted from $138$ MeV to
$146$ MeV ($147$ MeV if one retains the 1997 ALEPH
value of $\alpha_s$ as input).
Unfortunately, the errors on this value associated
with uncertainties in the experimental input are still very large,
$\sim \pm 50$ MeV at least, and this uncertainty cannot be appreciably
reduced without a significant improvement in the accuracy
of the determination of the experimental
$\omega\rightarrow e^+e^-$ and $\phi\rightarrow e^+e^-$ widths.
As such, although the central value is brought into better
agreement with that discussed above, little more can learned
from the Narison sum rule, at the present time.
\acknowledgements
KM would like to acknowledge the ongoing support of the Natural
Sciences and Engineering Research Council of Canada, and to thank
Terry Goldman, Andreas H\"ocker,
Heath O'Connell, Derek Leinweber, Tony Thomas, and Tony Williams
for useful discussions.
|
train/arxiv
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